Diss ETH No. 19005

The influence of geometry on the FUV and X-ray driven chemistry in star formation

A dissertation submitted to ETH Zurich for the degree of Doctor of Sciences presented by Simon Bruderer Dipl. Phys. ETH born August 20, 1979 citizen of Herisau (AR) accepted on the recommendation of Prof. Dr. Arnold O. Benz, examiner Prof. Dr. Michael R. Meyer and Prof. Dr. Ewine F. van Dishoeck co-examiners 2010

An electronic version of this thesis is available at http://e-collection.ethbib.ethz.ch.

F¨ ur Mama, Papa, Mirjam und Laura

Front Cover: A schematic illustration of the proposal of direct FUV irradition of outflow walls discussed in this thesis meets near-infrared observations of the high-mass star forming region AFGL 2591.

Abstract

This thesis is devoted to the study of the chemistry in envelopes of young stars. Stars form in molecular clouds, composed of huge amounts of molecular gas and small dust grains. Gravitation contracts the cloud until protostars are born in the densest region of the cloud. Low-mass stars with masses similar to our sun slowly reach the main-sequence, while they disperse the cloud that gave birth to them by outflows, stellar winds and radiation. In this young phase, a planetary system surrounding the protostar may form. High-mass stars, with masses larger than about eight solar masses, reach the main-sequence while still deeply embedded in their natal cloud. Their luminosity may exceed 10000 times the solar luminosity. Due to their high surface temperature, most radiation is emitted at ultraviolet wavelengths. Also, strong X-ray radiation has been observed towards both low-mass and high-mass protostars. As the cloud collapses, angular momentum must be carried away to prevent the protostar from being disrupted by rotational forces. The observed fast outflows of molecular gas together with magnetic fields are believed to carry away the angular momentum. Outflows can etch large cavities into the envelope. If the shape of the cavity allows protostellar radiation to escape and to directly irradiate the envelope at the edge of the cavity a peculiar region is formed. These so called outflow-walls are characterized by a high temperature and strong far ultraviolet (FUV) radiation that can significantly alter the chemical composition. For example a large amount of simple molecules like light hydrides (OH, OH+ , CH, CH+ , SH, SH+ , NH, NH+ ) is found in the outflow-walls. The Herschel Space Observatory allows for the first time to study the infrared line radiation of light hydrides at sufficient angular and spectral resolution. In this work, we study the proposal of directly irradiated outflow walls by observations and numerical models. We concentrate on the study of the high-mass star forming region AFGL 2591, situated in the constellation of Cygnus at a distance of approximately 1 kpc (about 3000 light years). In the introduction of the work (Chapter 1), we give a short review of star formation and the involved physical processes. In particular, we discuss the formation of low-mass and high-mass stars in the interstellar medium, the feedback of the protostar to its environment and the physical structure of a envelope surrounding a young star. The chemical reaction network, the formation of molecular lines and heating by continuum radiation is summarized. We discuss the foundations for numerical modelling of the processes. In Chapter 2, we introduce a novel approach to model chemical abundances. We adopt a previously published computer code (Doty et al. 2002, 2004; St¨auber et al. 2004, 2005) to calculate chemical abundances in gas with strong UV irradiation. The code is used to calculate a grid of

vi chemical abundances, consisting of about 100000 models. Abundances can then be quickly interpolated from this look-up table. This method is orders of magnitude faster than previous models and allows to study envelopes with a realistic multi-dimensional geometry rather than spherical symmetry, assumed in most previous models. The accuracy of the approach is verified and found to be very good with a mean deviation of 35 % to fully calculated models, very small in comparison with the uncertainties of the modeling. We find that cosmic-ray and X-ray driven chemistry is very similar, except for regions with very strong X-ray irradiation. This has the observational consequence that molecular tracers of X-rays are hard to distinguish from cosmic-ray ionization tracers. A two dimensional axisymmetric model of the high-mass star forming region AFGL 2591 is introduced in Chapter 3, to study the proposal of directly irradiated outflow walls. Based on a density structure, constrained from observations (van der Tak et al. 1999), the far ultraviolet irradiation of the envelope is calculated using a Monte Carlo radiative transfer code that considers the scattering and attenuation by dust grains. The temperature of the gas is next calculated, self-consistently with the chemical abundances. The chemical abundances enter through the cooling rates, for example of the atomic fine structure line of ionized carbon (C+ at 158 µm). We obtain the chemical abundances with the method introduced in the previous chapter. Using the new model, we study the molecular emission of ionized carbon monoxide (CO+ ). Synthetic maps of the emission are calculated and compared to observations. We find that the small layer of the outflow walls entirely dominates the molecular flux. The model explains both the detection with the JCMT single-dish telescope (St¨auber et al. 2007) and the non-detection in new data obtained with the Submillimeter Array interferometer. Chapter 4 presents new observations of AFGL 2591 carried out using the Submillimeter Array. In combination with observations by Benz et al. (2007), we find emission of CS and HCN along the outflow region. The regions with molecular emission are spatially coincident, which are interpreted by a chemical model as dense and hot regions irradiated with FUV radiation. This is in agreement with the proposal of directly irradiated outflow walls. A new method and computer code to model molecular line radiation is introduced in Chapter 5. Based on the escape probability approximation, it is much faster than other radiative transfer methods. The method is implemented to model line radiation of 1D spherical symmetric, 2D axisymmetric or 3D regions. We discuss a method to accelerate convergence, verify the code by different benchmark problems and study the accuracy of the method. We find good agreement with exact methods. In Chapter 6, we extend the axisymmetric model of AFGL 2591 from Chapter 3, using a Monte Carlo radiative transfer method to self-consistently calculate the heating by dust continuum radiation. A grid of models with different shapes of the outflow cavity and protostellar properties is calculated. We find a relatively small dependence of the FUV irradiated and heated amount of gas depending on the cavity shape, as long as the geometry allows the FUV radiation to escape from the innermost region. However, the size of the warm region with temperatures above 100 K but without FUV irradiation depends strongly on the shape of the cavity. This region is important for molecules like for example water. The model is used together with the code for molecular line radiation from Chapter 5, to predict line fluxes of light hydrides that will be observed with the Herschel Space Observatory. We find that the emission of outflow wall enhanced species should be easily detectable. Conclusions and an outlook on future work end the thesis (Chapter 7). Zurich, March 2010

Simon Bruderer

Zusammenfassung

Diese Dissertation widmet sich dem Studium der Chemie in H¨ ullen von jungen Sternen. Sterne entstehen in Molek¨ ulwolken, welche aus riesigen Mengen von molekularem Gas und kleinen Staubk¨ornern bestehen. Die Wolken werden durch die Gravitation kontrahiert, bis Protosterne in ihren dichtesten Regionen entstehen. Massearme Sterne (mit Masse ¨ahnlich unserer Sonne) erreichen langsam die Hauptreihe, w¨ahrend ihre Ausfl¨ usse, Sternwind und Strahlung, die H¨ ulle, aus der sie geboren worden sind, aufl¨osen. In dieser jungen Phase des Sternenlebens kann auch ein Planetensystem um den Stern gebildet werden. Massereiche Sterne (mit Massen gr¨osser als ungef¨ahr der achtfache Sonnenmasse) erreichen die Hauptreihe bereits, wenn sie noch tief in ihrer H¨ ulle eingebettet sind. Die Leuchtkraft solcher Sterne kann st¨arker als die 10000 fache Sonnenleuchtkraft sein. Wegen ihrer hohen Oberfl¨achentemperatur wird die meiste Strahlung im Ultravioletten emittiert. Bei massearmen und massereichen Protosternen wird starke R¨ontgenstrahlung beobachtet. Wenn die H¨ ulle wegen der Gravitationskraft kollabiert, muss Drehmoment abgef¨ uhrt werden, um den Stern vor dem Zerreissen durch Zentrifugalkr¨afte zu bewahren. Dies geschieht durch schnelle molekulare Ausfl¨ usse in Kombination mit Magnetfeldern. Die molekularen Ausfl¨ usse k¨onnen grosse Hohlr¨aume in die H¨ ulle des Protosterns fressen. Sofern die Form dieser Hohlr¨aume es zul¨asst, kann Strahlung vom Protostern entweichen und die H¨ ulle am Rande der Hohlr¨aume direkt bestrahlen, wodurch eine Region mit speziellen physikalischen Bedingungen entsteht. Diese sogenannten Ausflussw¨ande sind durch eine hohe Temperatur und starke Bestrahlung im fernen Ultraviolett charakterisiert. Dadurch wird die chemische Zusammensetzung stark ver¨andert. Zum Beispiel entsteht eine grosse Menge leichter Hydride (OH, OH+ , CH, CH+ , SH, SH+ , NH, NH+ ). Mit dem Weltraum-Teleskop “Herschel” k¨onnen solche Hydride zum ersten Mal mit einer guten r¨aumlichen und spektralen Aufl¨osung beobachtet werden. In dieser Arbeit studieren wir das Szenario direkt bestrahlter Ausflussw¨ande mit Beobachtungen und numerischen Modellen. Wir konzentrieren uns auf das massereiche Sternentstehungsgebiet AFGL 2591, das sich im Sternbild des Schwan in einer Entfernung von etwa 1 kpc (3000 Lichtjahre) befindet. In der Einleitung zu dieser Arbeit (Kapitel 1) wird eine kurze Zusammenfassung der Sternentstehung und der beteiligten physikalischen Effekte gegeben. Wir besprechen die Entstehung von massearmen und massereichen Sternen in der Interstellaren Materie, die R¨ uckkopplung des jungen Sterns auf seine Umgebung und die physikalische Struktur von H¨ ullen um Protosterne. Eine Zusammenfassung u ¨ ber chemische Reaktionen, molekulare Strahlungsprozesse sowie die Heizung durch Kontinuumstrahlung des Staubs wird ebenso gegeben. Die numerische Modellierung der besprochenen Prozesse wird kurz erl¨autert.

viii In Kapitel 2 wird eine neuartige Methode zur Modellierung von chemischen H¨aufigkeiten eingef¨ uhrt. Wir passen ein zuvor publiziertes Computerprogramm (Doty et al. 2002, 2004; St¨auber et al. 2004, 2005) an, um chemische H¨aufigkeiten in Regionen mit starker Bestrahlung im fernen Ultraviolett zu berechnen. Mit dem Programm wird eine Tabelle von chemischen H¨aufigkeiten angelegt, welche aus etwa 100000 Eintr¨agen besteht. Die H¨aufigkeiten k¨onnen dann aus dieser Tabelle interpoliert werden. Die vorgestellte Methode ist wesentlich schneller als fr¨ uhere Rechnungen. Dies erlaubt H¨ ullen mit realistischer Geometrie zu berechnen und nicht nur Modelle in sph¨arischer Symmetrie wie in fr¨ uheren Arbeiten. Wir verifizieren die Genauigkeit der Interpolationsmethode und finden mittlere Abweichungen von 35 %, was sehr klein ist im Vergleich zu den Unsicherheiten der Modellrechnung. Wir finden sehr kleine Unterschiede in der Chemie von Regionen mit R¨ontgenbestrahlung im Vergleich mit Regionen wo kosmische Strahlen (Protonen, α-Teilchen) wirken. Dies bedeutet, dass diese Regionen durch Beobachtung schwer unterscheidbar sind. In Kapitel 3 wird ein zweidimensionales Modell des massereichen Sternentstehungsgebiets AFGL 2591 konstruiert, damit das Szenario von direkt bestrahlten Ausflussw¨anden studiert werden kann. Wir nehmen Achsensymmetrie an. Das Modell basiert auf einer Dichtestruktur, die durch Beobachtungen gewonnen wurde (van der Tak et al. 1999). In einem ersten Schritt wird die UltraviolettStrahlung in der H¨ ulle mit eine Monte-Carlo Strahlungstransport-Programm berechnet, unter Ber¨ ucksichtigung von Streuung und Absorption durch Staub. Die Gas-Temperatur wird dann selbstkonsistent mit den chemischen H¨aufigkeiten berechnet. Dies ist n¨otig, da die K¨ uhlungsraten von den H¨aufgkeiten abh¨angen. Ein wichtiges Beispiel f¨ ur einen K¨ uhlungsprozess ist die atomare FeinstrukturLinie von ionisiertem Kohlenstoff, die bei 158 µm strahlt. Mit dem neuen Modell studieren wir die molekulare Strahlung von ionisiertem Kohlenstoff-Monoxid (CO+ ). Karten (Bilder) der molekularen Emission werden simuliert und mit Beobachtungen verglichen. Wir finden, dass alle Strahlung des Molek¨ uls aus der d¨ unnen Schicht der Ausflussw¨anden kommt. Unser Modell erkl¨art die von St¨auber et al. (2007) mit dem JCMT Teleskop beobachtete Emission des Molek¨ uls. Es kann aber auch erkl¨aren, wieso in neu durchgef¨ uhrten Beobachtungen mit dem Submillimeter Array Interferometer das Molek¨ ul nicht detektiert wurde. Kapitel 4 stellt neue Beobachtungen von AFGL 2591 vor, welche mit dem Submillimeter Array durchgef¨ uhrt wurden. Die Beobachtungen wurden mit jenen von Benz et al. (2007) kombiniert. Wir finden Strahlung von CS und HCN entlang der Ausfluss-Region. Die Strahlung der beiden Molek¨ ule ist r¨aumlich korreliert. Mit einer Modellrechnung k¨onnen wir diese Korrelation als Region mit heissem und dichtem Gas unter starker UV-Bestrahlung deuten. Diese Beobachtung stimmt mit dem Vorschlag der direkt bestrahlten Ausflussw¨anden u ¨ berein. Kapitel 5 bespricht eine neue Methode zur Berechnung von molekularer Linenemission und ihre Implementierung als Computerprogramm. Die Methode geht von der “Escape-Probability” Approximation aus und ist wesentlich schneller als andere Rechenmethoden. Die Methode wird implementiert, um sph¨arisch symmetrische, achsensymmetrische oder beliebige dreidimensionale Probleme zu l¨osen. Wir besprechen eine Technik, um die Konvergenz der Methode zu beschleunigen. Der ¨ Code wird mit verschiedenen Test-Problemen verifiziert. Wir finden eine gute Ubereinstimmung der Resultate mit jenen einer Rechnung ohne Approximationen. In Kapitel 6 wird das achsensymmetrische Modell aus Kapitel 3 mit einem Strahlungstransport Code erweitert, um die Heizung durch die Staubkontinuum-Strahlung selbstkonsistent zu berechnen. Eine Anzahl Modelle mit verschiedenen Formen der Ausfluss-Kavit¨at und verschiedenen Eigenschaften des jungen Protosterns wird berechnet. Wir finden eine relativ kleine Abh¨angigkeit der warmen Masse mit starker UV-Bestrahlung von der Form, solange die Form das Entweichen der Strahlung aus dem innersten Teil zul¨asst. Die Gr¨osse der warmen Region ohne starke UV-Bestrahlung h¨angt hingegen stark von der Form ab. Diese Region ist wichtig, da eine grosse Menge Wasser in ihr entstehen kann. Die Modelle werden zusammen mit dem Strahlungstransport-Code aus Kapitel 5

ix benutzt, um Linienfl¨ usse von leichten Hydriden zu berechnen, welche mit dem Weltraum-Teleskop “Herschel” beobachtet werden. Wir finden, dass die in den Ausflussw¨anden verst¨arkten Spezies leicht detektierbar sein sollten. Die Arbeit schliesst mit einer Zusammenfassung und einem Ausblick auf m¨ogliche zuk¨ unftige Arbeiten (Kapitel 7). Z¨ urich, im M¨arz 2010

Simon Bruderer

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Contents

Abstract

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Zusammenfassung

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1 Introduction 1.1 The formation of stars . . . . . . . . . . . . . . . . . . . 1.1.1 The formation of stars in the interstellar medium 1.1.2 The collapse of a core to a star . . . . . . . . . . 1.1.3 High-mass star formation . . . . . . . . . . . . . . 1.2 The envelope of a forming high-mass star . . . . . . . . . 1.2.1 Physical structure . . . . . . . . . . . . . . . . . . 1.2.2 Chemical structure . . . . . . . . . . . . . . . . . 1.2.3 Molecular line radiative transfer . . . . . . . . . . 1.2.4 Modeling envelopes of YSOs . . . . . . . . . . . . 1.3 Outline: Directly irradiated outflow walls . . . . . . . . .

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2 A grid of chemical models, Method and Benchmarks 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A grid of chemical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Temperature and density dependence . . . . . . . . . . . . . . . . . 2.2.2 FUV driven chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 X-ray driven chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Total ionization rate versus X-rays . . . . . . . . . . . . . . . . . . 2.2.5 Calculation the X-ray ionization rate ζH2 (FX , N(H), TX ) . . . . . . . 2.2.6 The X-ray ionization rate ζH2 (FX , N(H), TX ) for a thermal spectrum 2.2.7 Multidimensional interpolation . . . . . . . . . . . . . . . . . . . . 2.3 Chemical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Sulphur bearing species . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A spherical model of AFGL 2591 . . . . . . . . . . . . . . . . . . .

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CONTENTS 2.4.2 FUV driven chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Utilizing the grid of chemical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Multidimensional chemical modeling, Irradiated outflow 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Chemistry of CO+ . . . . . . . . . . . . . . . . . . . . . . 3.3 An axisymmetric model of AFGL 2591 . . . . . . . . . . . 3.3.1 Density structure . . . . . . . . . . . . . . . . . . . 3.3.2 FUV and X-ray radiative transfer . . . . . . . . . . 3.3.3 Temperature calculation . . . . . . . . . . . . . . . 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Directly irradiated outflow walls . . . . . . . . . . . 3.4.2 CO+ abundance . . . . . . . . . . . . . . . . . . . . 3.4.3 Comparison with JCMT observations . . . . . . . . 3.4.4 Comparison with SMA observations . . . . . . . . . 3.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . 4 Evidence of warm and dense material 4.1 Introduction . . . . . . . . . . . . . . 4.2 Observations and data reduction . . . 4.3 Results . . . . . . . . . . . . . . . . . 4.4 Interpretation . . . . . . . . . . . . . 4.5 A scenario . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . .

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walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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the outflow of a high-mass YSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 A computer program for fast multidimensional sion 5.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.2 Method . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Radiative Transfer in molecular lines . . 5.2.2 The escape probability approach . . . . . 5.3 Implementation . . . . . . . . . . . . . . . . . . 5.3.1 Limitations and possible extensions . . . 5.4 Applications . . . . . . . . . . . . . . . . . . . . 5.4.1 A two-level problem (Problem 1) . . . . 5.4.2 A collapsing cloud in HCO+ (Problem 2) 5.4.3 A grid of water models (Problem 3) . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . .

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modelling of molecular line emis. . . . . . . . . . .

6 The influence of geometry on the abundance and 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Dust Radiative Transfer . . . . . . . . . . 6.2.2 Molecular Excitation Analysis . . . . . . . 6.2.3 A grid of density structures/cavity shapes 6.3 Physical structure . . . . . . . . . . . . . . . . . . 6.3.1 Total mass . . . . . . . . . . . . . . . . . . 6.3.2 Dust temperature above 100 K . . . . . .

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excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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of diatomic hydrides127 . . . . . . . . . . . . . 128 . . . . . . . . . . . . . 130 . . . . . . . . . . . . . 130 . . . . . . . . . . . . . 131 . . . . . . . . . . . . . 133 . . . . . . . . . . . . . 134 . . . . . . . . . . . . . 136 . . . . . . . . . . . . . 138

CONTENTS

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6.6 6.7 6.8

6.3.3 Gas temperature above 100 K . . . . . . . . . . 6.3.4 X-ray and FUV irradiation . . . . . . . . . . . . Chemical Abundances . . . . . . . . . . . . . . . . . . 6.4.1 Standard and spherically symmetric model . . . 6.4.2 Comparing different geometries . . . . . . . . . 6.4.3 X-ray driven chemistry . . . . . . . . . . . . . . Molecular excitation and line fluxes . . . . . . . . . . . 6.5.1 Excitation: Level population . . . . . . . . . . . 6.5.2 Line fluxes . . . . . . . . . . . . . . . . . . . . . Observing diatomic hydrides with Herschel and ALMA Limitations of the model . . . . . . . . . . . . . . . . . Conclusions and outlook . . . . . . . . . . . . . . . . .

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7 Summary and outlook

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A Implementation of the UMIST 06 network

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B A Monte Carlo code for FUV radiative transfer 169 B.1 The FUV luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 B.2 An analytical expression for LFUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 C Thermal balance calculation C.1 Heating rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Cooling rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Benchmark test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D Dust radiative transfer calculation 177 D.1 Benchmark tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 D.2 Local radiation field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Acknowledgements

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List of publications

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Curriculum Vitæ

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xiv

CONTENTS

Chapter 1

Introduction

“Hier ist wahrhaftig ein Loch im Himmel!” (“Truly, there is a hole in the sky”) William Herschel (1738-1822)

Where do planets, the sun and stars come from? Ever since, mankind has asked this question and found different answers. Creation myths explain the creation of heaven and earth with memorable stories. Every culture found its way to explain the origin of the Earth and the firmament. For example the Indian Brahmanda Purana states that the universe hatches from a golden egg. In the Norse mythology, where the “world of men” is created from the giant Ymir. His blood becomes the seas and lakes, his flesh the Earth, his bones the mountains and his teeth the rocks. The first book of the Hebrew Bible, the book Bereishit (literally “in the beginning”), explains the creation of heaven and earth. However, the book starts with the second letter of the hebrew alphabet, the i (beth), rather than the ℵ (aleph) and we are left with the question of the very beginning. After God has created heaven and earth, he separates light from darkness on the first day, separates the waters and creates the sky on the second day and set lights in the sky to separate days and years, creating the sun, the moon, and the stars on the fourth day. Science, and in particular astronomy and astrophysics, explains the world with other methods. Ren´e Descartes (1596-1650) divorced Theology and Philosophy, claiming that the world can be understood by rational means. Not much later, astronomy and physics got married when Isaac Newton (1643-1727) realized that the same physical laws that describe dynamics on Earth also explain the motion of planets and the moon. This cleared the ground for the first physical theories about star and planet formation1 : Emanuel Swedenborg’s (1688-1772) “Nebula Hypothesis” has been further developed by Immanuel Kant (1724-1804) in his “Allgemeine Naturgeschichte und Theorie des Himmels” (Kant 1755). He argues that the solar system origins from a primordial nebula (“Urnebel”) which gradually collapses due to gravity. Independently of Kant’s work, Pierre-Simon Laplace (17491

Historical overviews are given in e.g. Woolfson 1993, Hamel 2002, Schulz 2005 or Nussbaumer 2005

2

1. Introduction

1827) proposed in his “Exposition du syst`eme du monde” (1796) a similar theory: A slowly rotating spherical cloud cools and collapses under the gravitational force. As a result of angular momentum conservation, the nebula collapses to a flat disk, in which the star and planets form. By the time Kant’s and Laplace’s work was published, William Herschel has observed and cataloged several hundred emission nebulae using his ever growing telescopes. Doubts about the Kant-Laplace hypothesis came up with the knowledge that the sun contains less than 1 % of the angular momentum of the solar system, while it consists of 99.9 % of the mass. Because of this angular momentum problem, the theory was abandoned at the beginning of the 20th century. In order to solve the angular momentum problem, many other models about the formation of planets were suggested. In the model by Georges Louis Marie Leclerc de Buffon (1707-1788), a comet collides with the sun and knocks out material that forms planets. In the same vein, Chamberlin (1901) and Moulton (1905) proposed a model starting with a very active (young) sun periodically erupting material which forms planets. In the tidal model of Jeans (1917), a massive star passing by pulls material out of the sun which then forms planets. Otto Schmidt (Schmidt 1944) proposed an “accretion theory” where the sun in its present form passes through a dense interstellar cloud and emerges surrounded by an envelope. McCrea (1960) and Woolfson (1964) state that planets are first formed and then captured by the sun. While these theories contain remarkable elements like the high activity in young stars, which was later indeed measured, they all have flaws. For example they fail to explain the formation of the slowly rotating sun. To make further progress in the field, more observations and knowledge of the physical processes involved in star formation were needed. Much observational progress was achieved since William Herschel spotted “holes in the sky”. Spectroscopic studies by Hartmann (1904) revealed a stationary and very narrow absorption line of calcium in the spectroscopic binary δ Orionis, which he correctly attributed to the interstellar medium. Barnard (1919) presented observations of 182 “dark markings of the sky”. A proof for interstellar extinction was delivered by Trumpler (1930) and ruled out Herschel’s explanation of “holes” being regions devoid of stars. Between 1937 and 1940, the first molecules in the interstellar medium were detected with CH (Swings 1937), CN (McKellar 1940) and CH+ (Douglas & Herzberg 1941). Around the same time, Herschel’s “holes in the sky” were identified to be sites of star formation (Bok 1948). Visible light has been the only source of information for astronomy for several millennia. Since Karl Jansky has discovered extraterrestrial radio radiation in the early 1930s, radio astronomy became an important part of astronomy. Sensitive receivers and antennas for wavelengths of a few 10 cm were developed for the use in RADAR devices during World War II. They allowed Ewen & Purcell (1951) and Muller & Oort (1951) the detection of the 21 cm hyperfine transition of neutral hydrogen, earlier predicted by van de Hulst in 1945. Radio astronomy provided in the following years much insights in the composition of the interstellar medium. Many of the 230 molecules2 detected so far have been identified upon their radio emission. The first molecule identified by radio observations was hydroxyl (OH) in 1963 (Weinreb 1963). Pure rotational lines of the abundant carbon monoxide (CO) were first detected by Wilson et al. (1970): A new important tool to study the structure and dynamics of the interstellar medium was established. The most abundant interstellar molecule, molecular hydrogen (H2 ), was detected in the same year (Carruthers 1970) by observations of ultraviolet radiation from a rocket. The study of the molecular universe started and astrochemistry emerged as a new branch of astronomy. Cold star forming regions often emit only in infrared wavelengths and infrared astronomy is essential for their exploration. The foundations for infrared astronomy have already been laid by William Herschel in 1800. He was initially an amateur astronomer and full time musician, and contributed to astronomy by the discovery of Uranus, several moons of Saturn and catalogs of nebulae. In 1800, he 2

As of March 8th 2010 (Including isotopomers, from astrochemistry.net)

1.1. The formation of stars

3

found a new type of radiation, which he called “calorific rays”. He carried out an experiment with a thermometer put in the spectra of the solar light. Moving the thermometer through the spectra from the blue side to the red side, he noticed that the temperature did not stop rising, even when the thermometer was moved away from the red side to the new invisible radiation. The new radiation however was not accessible to astronomy for a long time. Besides the difficult detector technology, much of the infrared radiation is absorbed by the Earth’s atmosphere and cannot be studied from the ground. Important contributions to our knowledge about star formation have thus been made by satellites: The Infrared Astronomical Satellite (IRAS) carried out a full sky survey at wavelengths3 of 12, 25, 60 and 100 µm in 1983. IRAS detected 500000 infrared sources and discovered disks around evolved stars. Between 1995 and 1998, the Infrared Space Observatory (ISO) performed the first spectroscopic observations in the mid and far-infrared. ISO studied the composition of interstellar grains and ices and found infrared dark clouds, only seen in absorption against the bright diffuse mid-infrared emission of the Galactic plane (e.g. van Dishoeck 2004). Spectroscopy of submillimeter wavelengths has been carried out 1998-2005 by the Submillimeter Wave Astronomy Satellite (SWAS) and ODIN 2001-present. Both satellites focus on the ground state rotational line of water at 557 GHz. Water is an important constituent of the interstellar and circumstellar medium for the chemical evolution and physical conditions. The Spitzer Space Telescope observes since 2003 the infrared sky, mostly at near infrared wavelengths and detected e.g. water in protoplanetary disks (Carr & Najita 2008). The atmosphere is however not completely opaque to submillimeter radiation and observations and at certain frequency range away from lines of atmospheric water (e.g. between 200-300 GHz or around 350 GHz) can done from the ground. On top of Mauna Kea (Hawaii), the James Clerk Maxwell Telescope (JCMT) and the Caltech Submillimeter Observatory (CSO) single-dish telescope operate since 1987 and 1988, respectively. Since water vapor is responsible for the attenuation of the infrared radiation, the new Atacama Pathfinder Experiment (APEX) telescope profits from the extremely dry conditions in the Atacama desert. The exploration of new wavelengths brought much new knowledge, like for example the discovery of massive molecular outflows, which can carry away an important part of the angular momentum of the accreted material (e.g. Kwan & Scoville 1976 or Zuckerman et al. 1976). The first direct observation of a protoplanetary disk using the Hubble Space Telescope in 1994 (O’dell & Wen 1994) was followed by the first detection of a planet orbiting another star (Mayor & Queloz 1995). Much progress has also been made in the theoretical understanding of star formation, for example with the studies of Shu (1977) on hydrodynamics of a collapsing protostellar envelope. Our new scenario of the star and planet formation still resembles the ideas of Laplace – however, today we have the chance to observe, analyze, model. . . and hopefully understand the physics and chemistry of star formation. Our knowledge is increasing rapidly in this exciting time with the Herschel Space Telescope ready to observe, the Atacama Large Millimeter/submillimeter Array (ALMA) having the “first light” within a year and the James Webb Space Telescope being launched in a few years.

1.1 1.1.1

The formation of stars The formation of stars in the interstellar medium

In our current understanding, stars form in the coldest and densest regions of the interstellar medium (e.g. Stahler & Palla 2005; Tielens 2005). The interstellar medium (ISM) can be roughly characterized by the state of hydrogen, which is either (i) neutral atomic (H, HI-regions); (ii) ionized atomic 3

1 µm is 10−6 m, an overview of different wavelengths is given in Figure 1.5

4

1. Introduction

Table 1.1: Phases of the galactic interstellar medium (Tielens 2005) Phase Ionized (HII)

Hot intercloud (HIM) HII regions Warm ionized (WIM) Neutral (HI) Warm neutral (WNM) Cold neutral (CNM) Molecular (H2 ) Molecular clouds

ntot [cm−3 ] 3 × 10−3 1 − 105 0.1 0.5 50 > 200

T M [K] [109 M⊙ ] 106 ∼ 0.1 4 10 0.05 8000 1.0 8000 2.8 80 2.2 10 1.3

Volume filling factor [%] ∼ 50 25 30 1 0.05

(H+ , HII-regions) or (iii) molecular (H2 -regions). The regions are nearly pure with thin transition layers in between. Besides hydrogen, other ingredients of the ISM are heavier elements and small dust grains. Heavier elements like helium, carbon or oxygen have abundances similar to the solar system. Dust grains, mixed with the gas, are responsible for the extinction of background light (e.g. Draine 2003a). They are composed of silicates and carbonaceous material and incorporate most of the Si, Mg and Fe in the ISM. Grains account for about 1% of the mass of the ISM and have typical sizes of the order of 0.1 µm. Their size distribution can be fitted with a power-law dn/da ∝ a−3.5 and a minimum and maximum size of a− = 50 ˚ A and a+ = 2500 ˚ A, respectively (Mathis et al. 1977). The ISM is composed of different phases (e.g. Cox 2005), summarized in Table 1.1 for our Galaxy. An enormous range of temperatures and densities is covered by the ISM! Hot intercloud material (HIM) reaches a million Kelvin and is heated and ionized through supernova explosions and shocks, driven by stellar winds of early type stars. This tenuous “coronal gas” cools to the warm and ionized material (WIM) by X-ray emission and adiabatic expansion. UV photons from hot O and B stars ionize and heat the WIM and HII regions. While HII regions are directly associated with nearby massive O and B stars, the WIM is penetrated by strayed photons which found a way to escape the HII regions. After a short living O or B star disappears, the ionized gas in his neighborhood recombines to neutral gas. Much of the matter in the ISM is in the form of warm neutral (WNM) or cold neutral material (CNM). The phases correspond to the two possible thermally stable conditions of equilibrium between heating and cooling. Heating by FUV radiation on small dust grains and polycyclic aromatic hydrocarbons (PAHs) is balanced by cooling from [CII] emission (CNM) or Lyα and [OI] emission (WNM). The phases may coexist in pressure equilibrium. The CNM is bound in discrete clumps (HI clouds) and forms the basis for the densest and coldest phase of the interstellar medium, molecular clouds. Molecules form in the ISM, if the interstellar radiation field is sufficiently shielded to prevent photodissociation (e.g. Snow & McCall 2006). The main constituent of molecular clouds is H2 . Second is carbon monoxide CO, with an abundances of order 10−4 relative to H2 . The formation of H2 takes place on dust grains, as the direct radiative association of two hydrogen atoms is too inefficient. Carbon monoxide is formed primarily by the reaction of C+ + OH → CO+ + H, followed by the quick reaction of CO+ with H2 to HCO+ , which then recombines with electrons to CO. Both H2 and CO are photodissociated by absorption of photons only in discrete lines. Thus, molecules at the edge of the cloud absorb all photons in these lines and shield molecules inside. Due to this self-shielding, the transition from H to H2 is very sharp and at a low visual extinction. A visual extinction of AV = 1 corresponds to a hydrogen column density of about 1.86 × 1021 cm−2 (Bohlin et al. 1978). If the visual extinction AV to the center of a cloud is below ∼ 0.3, most of the hydrogen is in atomic form. Diffuse clouds with 0.3 < AV < 1 have a considerable part (> 10%) of hydrogen

1.1. The formation of stars

5

Table 1.2: Typical physical conditions in molecular clouds (Snow & McCall 2006; Bergin & Tafalla 2007) Cloud Type

ntot [cm−3 ] Giant molecular cloud 100 Cold dark cloud 500 Clumps 103 − 104 Core 104 − 105

R T M [pc] [K] [M⊙ ] 50 15 105 2-15 10 103 − 104 0.3-3 10-20 50-500 0.03-0.2 8-12 0.5-5

in molecular form. In these clouds, simple molecules like CH, CH+ or OH, which are resistant to photodissociation, can be detected by absorption in visual or UV wavelengths. Translucent clouds (1 < AV < 2) shield their interior sufficiently so that the transition between ionized (C+ ) and neutral (C) and molecular carbon (CO) takes place. In dark clouds with AV > 5, almost all carbon and hydrogen is in molecular from. Due to efficient molecular cooling, temperatures drop to 10 K and the mass of such clouds is sufficient to be self-gravitating. Most star formation takes place in giant molecular clouds (GMCs), which give birth to both high-mass (M∗ & 8M⊙ ) and low-mass (M∗ . 1M⊙ ) stars (e.g. Bergin & Tafalla 2007). The Orion molecular cloud is an example of a nearby GMC. Due to the heating by the forming O and B stars, these GMCs are usually warmer than the cold dark clouds, where only low-mass star form. An example of a cold dark cloud complex is the Taurus-Auriga region. Star forming clouds have a selfsimilar and fractal spatial structure. A possible characterization of the hierarchical substructures is given in Table 1.2. In this table, the cloud (∼ 10 pc) is composed of clumps (∼ 1 pc), coherent regions in the position-velocity space. Within these clumps, cores (∼ 0.1 pc) are defined as gravitationally bound, single-peaked regions where individual stars or stellar systems form. How do cores emerge from clouds? Clouds are quasi-static and close to the equilibrium during their lifetime. To maintain this situation, support over gravitational collapse is necessary. The virial theorem states (e.g. McKee & Ostriker 2007; Stahler & Palla 2005), 1¨ I = 2T +2U +M +W 2

(1.1)

with the moment of inertia I of the cloud, and energy contributions by macroscopic motion (T ≥ 0), thermal motion (U > 0), magnetic field (M ≤ 0) and the gravitational potential (W ≤ 0). Thus, different ways to prevent collapse may exist. Observationally, contribution of both magnetic fields and macroscopic (turbulent) motion may compete with gravitation. In the first scenario, magnetic fields bind to the (partially ionized) gas and prevent the collapse by flux freezing (e.g. Benz 2002). The neutral component of the gas may however slowly diffuse through the ions and the cloud eventually collapses. This process is called ambipolar diffusion. In the scenario of gravoturbulent star formation (e.g. Mac Low et al. 1998), supersonic turbulence leads to strong density fluctuations, followed by gravitational collapse. The turbulences however quickly decay, as shocks convert mechanical energy into heat. Young stars embedded in the cloud may replace the energy. The controversy between this “fast mode” of star formation, with the cloud never reaching equilibrium, and the “slow mode”, with the cloud in equilibrium, is still ongoing and partially refers to the difficult age determination of the clouds (Bergin & Tafalla 2007). In triggered star formation finally, shocked material from a nearby supernova explosion may compress the cloud and initiate the gravitational collapse. Similarly, collisions on galactic scales may induce starbursts as the cloud gets compressed by tidal

6

1. Introduction

forces. Thus, the question on the fragmentation of clouds to star forming cores is far from being clear.

Figure 1.1: Star formation in the interstellar medium. Adopted from Shaw 2006. The picture of star formation in the interstellar medium is summarized in Figure 1.1. A star forming giant molecular cloud is embedded in the cold and neutral interstellar medium, separated by a thin layer where the transition of HI to H2 takes place. Turbulences in the molecular cloud lead to a fragmentation of the cloud to clumps and cores. Stars then form in the small dense cores. The newly formed stars interact in various ways with the cloud giving birth to them: Young high-mass stars ionize their environment and create HII-regions. Forming stars drive massive molecular outflows into the cloud and drive shocks. These processes eventually disperse the molecular cloud and prevent further star formation. More evolved, but still young stars are found in the neighborhood of the molecular cloud.

1.1.2

The collapse of a core to a star

The cores collapse gravitationally to individual stars or clusters of stars. The evolution of a solar-like (low-mass) star with M∗ . 1M⊙ from a cloud core to a main sequence star with a planetary system in summarized in Figure 1.2 (Shu et al. 1987; McKee & Ostriker 2007). a.) Prestellar core A cloud core of a fraction of a parsec in size contracts quasi-statically. While the ambipolar diffusion takes place on timescales of 106 yrs, a centrally concentrated core forms. The density profile of such a prestellar core approximately follows the power law ρ ∝ r −2 in the outer part of the core and but is less steep in the inner part ρ ∝ r −1 (r < 2500 − 5000 AU) (Ward-Thompson et al. 1994; Bergin & Tafalla 2007). b.) Young Accreting Protostar (Class 0) The process of star formation starts when turbulence and magnetic fields can no longer support the core against gravitational collapse. As the information about the central collapse may only propagate with sound speed, the core collapses

1.1. The formation of stars

7

from inside out. A deeply embedded protostar forms and builds up its mass by accretion from the infalling envelope and an accretion disk. The density of the envelope within 10000 AU can be described by a power law ρ ∝ r −α , with α between 1 and 2 (e.g. Jørgensen et al. 2002). Bi-polar molecular outflows are driven by high-velocity jets and wide-angle winds that originate close to the protostar. Many details of the jet-acceleration are unclear, but the mechanism is likely associated to the interaction between the accretion disk and the stellar magnetic field. The jets and outflows are essential to carry away the excess angular momentum and allow further accretion. This phase lasts for about 105 yrs (Evans et al. 2009) c.) Evolved Accreting Protostar (Class 1) The main accretion of the protostar finishes. The outflows are slower and wider. They etch cavities into the envelope and then disperse the envelope gradually. Since the onset of the gravitational collapse, about 105 yrs have elapsed. d.) Classical T Tauri (Class 2) The protostar is no longer embedded in the envelope and becomes visible in optical wavelengths. An optically thick, massive protoplanetary disk (0.01 − 0.1M⊙ ) with radius of order 100 AU orbits the protostar. Broad optical and UV emission lines and in particular Hα emission emerge from shocks, associated with winds and infall. In the disk, planets probably form during this phase of evolution. e.) Weak Line T Tauri (Class 3) At an age of 106 − 107 yrs, most of the disk is removed, likely by protostellar X-ray or EUV radiation. The protostar has fully developed and a well defined position in the Hertzsprung-Russel diagram, slightly above the main sequence. f.) Main-Sequence Star The post-T Tauri star reaches the main sequence on the Hayashi track (Hayashi 1961). After 107 yrs, a ZAMS (zero age main sequence) star is born. Observations of the spectral energy distribution (SED) support this picture of low-mass star formation and have led to the classification scheme (e.g. Lada 1987; Andre et al. 1993). The spectral energy distribution varies with the evolution due to the increasing temperature of the protostar and the decreasing extinction by the envelope (inset in Figure 1.2). The spectral slope between 2.2 µm and 24 µm, d log(λFλ ) αIR = , (1.2) dλ with the flux Fλ depending on the wavelength, is used to distinguish between the different classes: Class 0 No emission at 2.2 µm detected: The source emits only light that has been reprocessed by the envelope to far infrared wavelengths. Class 1 αIR > 0.3: The bulk of the emission is still at far infrared wavelengths, some direct contribution by the protostar can however already be seen. Class 2 −2 < αIR < −0.3: In addition to the pre-main sequence radiation, an excess in UV wavelengths due to accretion hot-spots and a strong contribution by the disk govern the SED. Class 3 αIR < −2: The spectra is almost a pure black body with only little contribution by the disk. This classification scheme has however been doubted in recent time, as for example the geometry of the source may affect the SED strongly (e.g. Masunaga & Inutsuka 2000). The distinction between class 0 and class 1 sources for example may only reflect a higher density of the surrounding cloud rather than another evolutionary state (Jayawardhana et al. 2001).

8

1. Introduction

Figure 1.2: The evolutionary picture of low-mass star formation after Shu et al. (1987), adopted from Jonkheid (2006). The spectral energy distribution (SED) for each step in evolution is given as inset. The protostellar contribution is given by a red dashed line and the total SED by a green line.

1.1.3

High-mass star formation

High-mass stars (M∗ & 8M⊙ ) are rare and short-living. For a Salpeter-Initial Mass Function (dn/dlog(M) ∝ M −1.35 ; Salpeter 1955), only about 2% of the newly born stars are high-mass stars. The lifetime of a star on the main sequence is approximately tMS = 1010 (M∗ /M⊙ )/(L∗ /L⊙ ) yrs, with the stellar (bolometric) luminosity L∗ . Thus, while our sun has an approximate lifetime of 1010 yrs, a B3 star (M∗ = 8M⊙ , L∗ = 2000L⊙ ) may only live 4 × 107 yrs. Nevertheless, O and B stars play an important role in the ISM due to their interaction with the environment: Their supernova explosions are the main source of heavy elements. Their winds, massive outflows and expanding HII-regions are important for the mixing of the ISM, which profoundly affects star formation (e.g. Zinnecker & Yorke 2007). Their UV radiation dominates the interstellar radiation field and the heating of the ISM. High-mass star formation is not well understood for several reasons. Compared to low-mass young stellar objects (YSOs), high-mass protostars are usually at larger distances and more obscured by the envelope in the earliest phase of formation. Also, they are rare and evolve more quickly, thus less objects are available for studies. To distinguish high-mass and low-mass YSOs, the accretion timescale compared to the contraction timescale of the protostar is used (McKee & Ostriker 2007; Cesaroni 2005). The accretion timescale is defined by tacc = M∗ /M˙ acc , with the mass accretion rate of the protostar M˙ acc . The contraction

1.1. The formation of stars

9

timescale is given by the Kelvin-Helmholtz timescale, tKH = GM∗2 /R∗ L∗ . This timescale states how long the energy from gravitational contraction can maintain the luminosity of the star (tKH = W/L∗ ). Low-mass star formation is defined by tacc < tKH , whereas high-mass star formation by tacc > tKH . This definition roughly states that low-mass stars have completed their accretion phase before hydrogen burning starts, while high-mass stars still accrete material when they reach the main-sequence. The evolutionary sequence of high-mass stars is less clear than for low-mass stars, but likely as follows (e.g. Beuther et al. 2007; Zinnecker & Yorke 2007): 1. Formation of high-mass starless cores (HMSCs) in infrared dark clouds (IRDCs, Egan et al. 1998), probably induced by gravoturbulent cloud fragmentation. Only a few true objects of this class have been detected so far. 2. Gravitational collapse of cores to “protostellar embryos”. 3. Further accretion of the objects as they evolve towards the main-sequence. High-mass protostellar objects (HMPOs) are formed. 4. Young massive stars disrupt the natal cloud by their winds, outflows and UV radiation. The sequence of compression, collapse, accretion and disruption can occur simultaneously within the molecular cloud. The formation of the HMPOs remains controversial and different scenarios exist. In the “monolithic collapse” scenario (McKee & Tan 2003), high-mass stars form like low-mass stars by accretion of ambient gas with a sufficiently large accretion rate. In the competitive accretion scenario (Bonnell & Bate 2002), massive stars form via coalescence of intermediate mass protostars with M∗ < 8M⊙ . Historically, stellar mergers in very dense systems have also been suggested to form massive stars, but are now believed to be important only for the formation of the most massive stars. In favor for the scenario of a monolithic collapse as a scaled-up version of low-mass star formation are the detection of disks (e.g. Cesaroni et al. 2006) and powerful bi-polar outflows (e.g. Beuther et al. 2002b). The lack of high-mass starless cores (HMSCs) and the reproduction of the observed IMF on the other hand is in favor for the competitive accretion and coalescence scenario. Due to the feedback of the high-mass star to its environment, massive star formation is in any case not just a scaled-up version of low-mass star formation. The high luminosity of the forming high-mass star leads to radiation pressure and stops further accretion. The Eddington luminosity Led =

4πcGM∗ , κDust

(1.3)

with the (effective) dust opacity per unit mass κDust , would lead to an upper limit of about 13M⊙ for the mass of the star. Several ways to overcome the problem of radiation pressure have been proposed (e.g. McKee & Ostriker 2007). For example the effects of a geometry where the protostellar radiation can escape through cavities etched out by the outflow (Krumholz et al. 2005). The radiation emitted by a forming high-mass star can soon be approximated by a black body, as the star quickly reaches the main sequence. The radiation field is then characterized by the bolometric luminosity Lbol and the effective surface temperature T∗ . A forming O or B star with T∗ > 104 K, emits most radiation in the UV with photon energy above 1 eV. Colder low-mass stars with temperature T∗ < 6000 K produce less UV photons in their photosphere, but may still emit FUV radiation produced by hot accretion shocks. X-ray emission (≈ 0.1 − 100 keV) is observed towards both young low-mass and high-mass stars (e.g. G¨ udel et al. 2007; Feigelson et al. 2007). The high temperatures of 107 - 108 K necessary for X-ray emission are attributed usually to magnetic activity, for example from the reconnection of field lines between the disk and protostar (e.g. Feigelson &

10

1. Introduction

Montmerle 1999). Shocks of the strong winds by high-mass stars may also produce X-rays (e.g. Parkin et al. 2009). The radiative feedback of the forming O or B star to its environment has led to the following observationally driven picture of evolution (e.g. Churchwell 2002): As the young star lights up, it heats up the cloud and forms a Hot Molecular Core (HMC) with a rich molecular inventory. The HMCs may probably coexist with small < 0.01 pc and dense > 106 cm−3 hypercompact HII regions (HCHIIs). They evolve to larger (< 0.1 pc) and less dense (104 cm−3 ) ultracompact HII regions (UCHIIs) which last of order 105 yrs and finally form extended HII regions. Molecular outflows, likely driven by collimated jets, are another way of feedback of the forming massive star to its environment (e.g. Arce et al. 2007). Many observational studies (e.g. Shepherd & Churchwell 1996; Beuther et al. 2002b; Kim & Kurtz 2006; Qiu et al. 2007) on molecular outflows of massive YSOs have been carried out, since they are spatially resolved even at the large distance to most high-mass star forming regions. They find that the collimation of massive outflows is roughly independent of mass (Kim & Kurtz 2006) and the outflow mass proportional to the mass of the core over a wide range of masses (Beuther et al. 2002b). This led to the suggestion that high-mass star formation is indeed very similar to low-mass star formation. However, as disks and collimated outflows have not been detected yet towards the most massive young stars with M∗ & 25M⊙ (Cesaroni et al. 2007) the question about the similarity between low-mass and high-mass star formation remains open. The effect of outflows and jets on the environment of the forming high-mass star is manifold: Outflows blow large scale polar cavities free (e.g. Preibisch et al. 2003). On impact to quiescent material, shocks heat up gas to post-shock temperatures of (Stahler & Palla 2005) T ≈ 3 × 10

5



vshock 100 km s−1

2

K,

(1.4)

with the shock velocity vshock . For the high velocities observed in outflows (10 − 100 km s−1 ) and jets (up to 500 km s−1 ), post-shock temperatures are sufficient to produce UV and soft-X ray emission (e.g. Neufeld & Dalgarno 1989). This radiation can further affect the environment. Shocks of jets are also manifested in the so-called Herbig Haro (HH) objects (e.g. Reipurth & Bally 2001). They can be observed in atomic lines like [HI], [OI] or [SII] and show chains of knots along the jet and the bow-shock of the jet. We conclude that forming high-mass stars interact in various ways with their environment and are important “engines” in the ISM. Much of their formation is however unclear and in particular the earliest, deeply embedded phase remains unclear. In this work, we discuss the feedback of a forming high-mass star on the molecular envelope surrounding it. In particular, we will concentrate on the effect of the high-energy radiation on the chemical composition of the gas.

1.2 1.2.1

The envelope of a forming high-mass star Physical structure

The physical structure of a YSO envelope is given by the density, temperature and velocity depending on position. Observationally, the structure is studied by a combination of dust continuum radiation and molecular line radiation. For example, van der Tak et al. (1999) studied the high-mass star forming region AFGL 25914 by a combination of images of the dust continuum in submillimeter wavelength (450 µm, 850 µm), the spectral energy distribution (SED) in a wide range of wavelengths 4

Air Force Geophysics Laboratory Survey at 4.2 µm, 11.0 µm, 19.8 µm, and 27.4 µm (Price & Walker 1976)

1.2. The envelope of a forming high-mass star

11

(∼ 2.2 − 6000 µm) and molecular line observations of carbon monoxide (CO) and carbon sulfide (CS). They assume the envelope to be spherically symmetric and constrain the structure by fitting with dust and line radiative transfer calculations. A large sample of envelopes of both low-mass and high-mass YSOs has been studied in this way (e.g. van der Tak et al. 2000; Jørgensen et al. 2002; Beuther et al. 2002a; Mueller et al. 2002; Doty et al. 2005). As an example, the physical structure of AFGL 2591 obtained by van der Tak et al. (1999) and Doty et al. (2002) is given in Figure 1.3. Note that the velocity profile is a rough estimation of an inside-out collapse. In the outer part of the envelope, the hydrogen density ntot = 2n(H2 ) + n(H) is about 105 cm−3 and rises to above 107 cm−3 in the inner part. Also, the temperature spans the range of below 30 K to above 400 K. 500 400 Temperature [K]

Density ntot [cm−3 ]

107

106

b.) Temperature

0

300 200 100

105

0 103 104 Distance to the Protostar [AU]

c.) Velocity

Gas Temperature Dust Temperature Velocity [km s−1 ]

a.) Density

103 104 Distance to the Protostar [AU]

−1 −2 −3 −4

103 104 Distance to the Protostar [AU]

Figure 1.3: Physical structure of the high-mass star forming region AFGL 2591 in a spherically symmetric model. The density, dust and gas temperature are obtained from van der Tak et al. (1999) and Doty et al. (2002). The velocity profile assumes free fall. Theoretically, the physical structure of the infalling envelope is determined by magneto-hydrodynamical equations. Neglecting the magnetic field is not a bad assumption, since only those regions collapse where ambipolar diffusion has “cleared” the magnetic field. The set of hydrodynamical equations for a self-gravitating sphere then consists of the continuity equation and the momentum equation, ∂ρ 1 ∂(r 2 ρv) + 2 = 0 ∂r r ∂r ∂v 1 ∂P GM(r) ∂v +v = − − , ∂t ∂r ρ ∂r r

(1.5) (1.6)

with the density ρ, the mass M(r) within a radius r, and the pressure P and the velocity v. To close this system, an equation of state (e.g. P = ρcs , with the sound velocity cs ) is necessary. A popular self-similar analytical solution of this system, assuming an isothermal sphere as initial condition, has been presented by Shu (1977). The solution predicts that a core in hydrostatic equilibrium has a density profile ρ ∝ r −2 . When collapse starts, for example by an external increase of pressure, the core collapses from inside out with an expansion wave at sound velocity. In the collapsing part of the cloud, the density profile follows ρ ∝ r −3/2 . An important concept in this context is the BonnorEbert mass (Ebert 1955; Bonnor 1956), which gives the maximum mass that can be stabilized over gravitational collapse by pressure. More realistic solutions of the non-isothermal cloud are given by e.g. McLaughlin & Pudritz (1997) and yield similar density profiles. If the cloud is supported by turbulence, the predicted density profile follows ρ ∝ r −1 (Lizano & Shu 1989; McLaughlin & Pudritz

12

1. Introduction

1996). Observational studies mentioned above indeed yield density profiles ρ ∝ r −α , with α between 1 and 2. Dust continuum radiation in the form of images or spectral energy distribution (SED) is compared to theoretically predicted density profiles by a dust radiative transfer calculation (Figure 1.4). The protostellar radiation with energy below the hydrogen ionization energy of 13.6 eV is mostly absorbed by dust grains (Figure 1.5). The dust grains are heated up by absorption of radiation and emit the same amount of energy, but at infrared and far-infrared wavelengths assuming equilibrium. Stellar radiation is thus “reprocessed” to longer wavelength by dust and the SED of a YSO envelope peaks in the far-infrared. At temperatures above ≈ 1500 K, dust evaporates. Infrared radiation by dust may heat dust grains at other positions of the envelope. Thus, for the calculation of the dust temperature and dust continuum radiation from an assumed density profile, it is necessary to solve the equation for the balance between absorption and emission at every position of the envelope, Z Z Z
stellar dust κλ Iλ + Iλ dλdΩ = 4π κλ Bλ (Tdust )dλ , (1.7)

with the dust opacity per unit mass κλ , the intensity of the direct stellar and ambient dust radiation, Iλstellar and Iλdust , respectively, and the black body function for the temperature of the dust grains, Bλ (Tdust ). The left hand side of the equation requires the knowledge of the intensity of dust radiation at different wavelengths and in different directions. The intensity is obtained from the solution of a differential equation along rays, the radiative transfer equation dIλ = κλ (Bλ (Tdust ) − Iλ ) , ds

(1.8)

with the path element ds. This problem can for example be solved by a Monte Carlo approach (Bjorkman & Wood 2001, Chapter 6), where photons are traced from the protostar until they leave the envelope. The dust opacity can be approximated by κλ ∝ λ−β (Figure 1.5). Assuming that the dust is optically thin (i.e. only direct stellar photons are important for the dust heating), the dust temperature can be approximated by (e.g. Jørgensen et al. 2006), q/2

Tdust (r) ∝ Lbol r −q

(1.9)

with the bolometric luminosity Lbol , distance to the protostar r, and q = 2/(β + 4). Thermal equilibrium between different components (gas, dust and electrons) is not necessarily reached in the tenuous environment of a YSOs envelope. Detailed calculations (Doty & Neufeld 1997) however show that the temperature exchange between dust and gas is sufficient in a large part of YSOs envelopes and the temperature thus similar. In the outermost part of the envelopes however, density is too low and due to more efficient line cooling (by CO), the gas temperature can drop below the dust temperature. In regions with very strong direct irradiation by the protostar, as for example in outflow walls (Chapters 3,4 and 6), the gas temperature may rise significantly over the dust temperature and reach several 100 to 1000 K, while the dust temperature is still of order a few 100 K. In these regions, the gas temperature has to be calculated explicitly. The gas temperature can be obtained from the balance of heating and cooling rates X X ∂ǫ = Γtot − Λtot = Γi − Λi = 0 , ∂t i i

(1.10)

with the inner energy of the gas ǫ and various heating and cooling rates Γi and Λi , respectively. When cooling and heating timescales are sufficiently short, equilibrium can be assumed. Main heating mechanisms are the photoelectric effect on small dust grains or PAHs, where a fast electron

1.2. The envelope of a forming high-mass star

13

Figure 1.4: Schematic illustration of radiative transfer in the dust continuum. The protostar emits photons in UV/optical wavelengths. The photons scatter on dust and are absorbed by dust. They heat up dust and dust emits IR photons that can further heat dust at other positions. hit out by a FUV photon heats the gas or H2 pumping, where a FUV photon pumps H2 and the subsequent collisional deexcitation releases energy to the gas. Important cooling mechanisms are atomic fine structure emission [CII] at 158 µm, [OI] at 63 µm, H2 rovibrational emission, and, deeper in the cloud, also molecular emission in rotational lines and dust-gas temperature exchange. For the calculation of the cooling rates, the abundance of the coolants has to be known and the gas temperature thus depends on the molecular composition of the gas (Chapter 3). Molecular line emission can also be used as tracer for the density and temperature. Since the molecular structure between different molecules varies grossly, different lines can be used to probe different parts of the envelope. For example, the pure rotational transitions J = 1 → 0 at 115.271 GHz of carbon monoxide (CO) probes gas with very low density and temperature, while higher density and temperature regions can be probed for example with the J = 7 → 6 transition of carbon sulfide at 342.883 GHz. Note however that the molecular line emission also depends on the abundance of the molecule. Only molecules with well known abundances are suitable as probe of the physical structure. Unlike dust emission, molecular lines also yield information on the velocity structure by the shape of the molecular line. Molecular lines however only give velocity information integrated along the line of sight. Thus, to constrain velocity fields, a theoretically predicted velocity field is fitted to observed line shapes (e.g. Brinch 2008). The process of calculating molecular line fluxes by line radiative transfer is summarized in Section 1.2.3.

1.2.2

Chemical structure

The huge range of physical conditions within a YSOs envelope is reflected in strong spatial variations of the chemical composition. This is a result of the strong dependence of chemical reaction on physical parameters like the temperature or the UV flux. This section only gives a short summary on the chemistry of YSOs envelopes. More comprehensive reviews, in particular of the chemistry in envelopes of high-mass YSOs, are given in van Dishoeck & Blake (1998), van der Tak (2005), van der Tak (2008) and van Dishoeck (2009).

14

1. Introduction FUV

Gas Absorption

10−20 10−21 10−22 10−23

Lyman edge (13.6 eV)

10

σ ∝ λ2.5

−19

Compton Scattering

Cross section (cm2 /H-atom)

10−18

V

Near IR

Far IR

Radio

Dust Absorption Dust Extinction Gas Extinction

σ ∝ λ−2

EUV

Silicate absorption

10−17

soft X-ray

Dust working function (6 eV)

hard X-ray

10−24 10−25 10−26 10−5

10 keV 10−4

1 keV 10−3

1 eV

100 eV 10−2

10−1

100

101

102

103

Wavelength [µm]

Figure 1.5: Extinction cross section for dust and gas absorption per hydrogen atom. Adopted from Montmerle (2001). The outer part of the envelope with temperatures ≪ 100 K and densities above 104 cm−3 shows only little chemical activity, similar to dark clouds or prestellar cores. Many molecules are frozen out onto dust grains, if the temperature is low enough to prevent thermal evaporation and the density is high enough (& 104 cm−3 ) for sufficiently many collisions between dust and molecules (Jørgensen et al. 2005b). While H2 O and CO2 have a large binding energy and evaporate at temperatures around 100 K, CO is more volatile and already evaporates at ≈ 18 K. Chemical reactions take place in the gas phase but also on grain surfaces. On the grain surface, sufficiently light species such as atomic hydrogen, oxygen, carbon or nitrogen can migrate by tunneling or thermal hopping (e.g. Tielens 2005) and react with species frozen out onto the grain surface. Such grain-surface reactions with hydrogen (hydrogenation, e.g. Gibb & Little 2000) lead for example from C to CH4 or from O to H2 O. Also more “complex”5 molecules like formaldehyde (H2 CO) or methanol (CH3 OH) can be produced on grain surfaces (e.g. Bisschop et al. 2007). After formation on the grain surface, various mechanism can release molecules into the gas phase, like for example thermal evaporation or cosmic ray induced evaporation. Reactions in the gas-phase also take place in this cold environment. At such temperatures, many important reactions are between ionized and neutral species (e.g. Herbst & Klemperer 1973), since they mostly do not require additional thermal energy to break up chemical bonds before new bonds are created. Such an activation energy can be considerable for reactions between neutrals and they are thus mostly very slow at low temperature. However, there are also neutral-neutral reactions proceeding quickly at low temperatures, like e.g. CN + C2 H2 (Woon & Herbst 1997). An important driver of chemistry in this part of the envelope is ionization by cosmic rays. In inner regions, the temperature rises to above 100 K and higher densities are reached. In these hot-cores, a rich chemistry takes place. Temperature is now high enough for H2 O, CO2 , H2 S and 5

In astrochemistry, molecules consisting of more than six atoms are considered being “complex” (Herbst & van Dishoeck 2009)

1.2. The envelope of a forming high-mass star

15

many others, to evaporate from ice mantles to the gas phase. Temperature is also high enough to overcome the activation energy of neutral-neutral reactions. For example at temperatures above 200-300 K, water is built up quickly a pfrom OH (e.g. Graff & Dalgarno 1987). The radial size of 4 hot-core region is approximately 15 Lbol /L⊙ AU and for a high-mass protostar with Lbol = 10 L⊙ of order 1500 AU (e.g. Bisschop et al. 2007). Typical species detected towards hot-cores are complex organics, like e.g. formaldehyde (H2 CO), acetonitrile (CH3 CN) and many others (e.g. Herbst & van Dishoeck 2009). A driving force of chemistry in hot-cores may be X-rays (e.g. Krolik & Kallman 1983; Lorenzani & Palla 2001; St¨auber et al. 2005). Having a much lower extinction cross section than UV photons (Figure 1.5), they may penetrate further into the envelope (St¨auber et al. 2004). Shocked gas is another region with a distinct chemistry. In shocks, kinetic energy is converted to a high temperature (e.g. Draine & McKee 1993; Flower et al. 2003; Flower 2007). Depending on the shock velocity vs and the magnetic field, a magnetic precursor can “warn” the upstream gas about the incoming shock wave and the hydrodynamical variables remain continuous. In a Cshock (Continuous) with velocity vs . 50 km −1 , a gas temperature of a few 1000 K is reached and molecules which require a high temperature for formation are built up (e.g. OH, CH+ , SH, H2 O, SO, SO2 ). Also volatile molecules (e.g. SiO or CH3 OH) are released from grains by “sputtering”: The ion-neutral drift in magnetized regions causes collisions between grains and H2 molecules which erode the grains (e.g. Flower et al. 1996). In faster shocks (vs & 50 km s−1 ), gas is heated up to 105 K, a large amount of UV radiation is produced and molecules are dissociated. In a J-shocks (Jump), hydrodynamical variables undergo a sudden change in a short distance. Observationally, molecular emission from shocks often shows broad and asymmetric lines (e.g. Wakelam et al. 2005a; Jørgensen et al. 2004a). Photodissociation regions (PDRs) are irradiated by a strong far UV field (e.g. Hollenbach & Tielens 1999). The far UV (FUV) radiation field is defined by the energy range between the hydrogen ionization energy (13.6 eV) and the average dust work function (6 eV), as radiation in the EUV with energy above 13.6 eV is quickly absorbed and the photoelectric effect on dust grains is an important heating mechanism. The strength of the FUV radiation field is usually expressed in units of the interstellar radiation field (ISRF; Habing 1968 or Draine 1978) with approximately 1G0 ≡ 1 ISRF ≡ 1.6 × 10−3 erg cm−2 s−1 . The conversion factor from the radiation field defined by Draine (1978) to Habing (1968) is 1.7. In Figure 1.6, abundances and cooling rates of important PDR species are given together with the gas and dust temperature. The plots show a cut through a prototypical PDR with the FUV radiation entering from the left side. The depth is given by the attenuating dust column density, expressed in visual extinction. A density of ∼ 2 × 105 cm−3 and a FUV irradiation of G0 = 105 ISRF is assumed. At AV < 2, the gas temperature reaches more than 1000 K. Carbon is fully ionized and H2 dissociated. Due to H2 self-shielding, the transition from H to H2 is very sharp and takes place at AV < 2. The main coolants in this top layer of the PDR are atomic fine structure lines. For low-density PDRs, the carbon [CII] 158 µm line dominates, while for this high-density PDR, the oxygen [OI] 63 µm line cools more. Deeper in the cloud, at 2 < AV < 4, C+ decreases and CO takes over, with a thin layer of relatively high neutral carbon. Due to the lower abundance and the lower temperature, atomic fine structure cooling becomes unimportant and the PDR is mainly cooled by molecular emission. Even deeper, at AV > 4, sulfur S is not ionized anymore. An important parameter for the chemistry is the ionization fraction xe = n(e)/ntot decreasing from ∼ 10−4 at AV < 2 to below ∼ 10−7 at AV = 10. The electron fraction for example governs how quickly ionized species recombine. The layered structure of the PDR is a result of the increasing dust opacity for shorter wavelengths, thus photons with higher energy are already absorbed in the top layer of the PDR. As a result of the S ionization energy (1197 ˚ A∼ 10.36 eV) compared to the carbon ionization energy (1101 ˚ A∼ 11.26 eV), S+ is found deeper in the PDR compared to C+ . PDRs are physically complicated regions, due to various interactions between the chemical network,

16

1. Introduction

heating and cooling rates. Various numerical models have been constructed for the study of such regions (e.g. R¨ollig et al. 2007 for a comparison of different models).

Figure 1.6: A photodominated region PDR. Adopted from Tielens & Hollenbach (1985). An X-ray dominated region (XDR) is heated and ionized by X-ray photons. XDRs have traditionally been studied in the context of active galactic nuclei (AGNs; e.g. Maloney et al. 1996; Meijerink & Spaans 2005). For the same incident flux, larger temperatures are reached in an XDR compared to a PDR, having consequences on the molecular composition and the emitted radiation (Meijerink et al. 2007). Young protostars emit a small fraction of their bolometric luminosity in X-rays. For low-mass protostars LX /Lbol ∼ 10−4 (G¨ udel et al. 2007), while for high-mass YSO it only amounts −7 to LX /Lbol ∼ 10 (e.g. Berghoefer et al. 1997). For a B star with Lbol = 104 L⊙ , LX ∼ 1030 erg s−1 . For this luminosity, the high-density gas (106 cm−3 ) of a YSO envelope can only be heated to 100 K by X-ray irradiation at distances smaller than 10 AU (Meijerink et al. 2007). Except in the innermost regions, with for example a disk, the influence of X-rays on the envelope of a high-mass YSO is restricted to providing an additional source of ionization. The chemical composition in such X-ray irradiated gas is very similar to gas with high cosmic ray irradiation (Chapter 2).

Chemical models Chemical models reproduce the chemical network by a numerical calculation. For a parcel of gas, not coupled to other parcels by flows, the chemical evolution can be modeled by the rate equations (e.g. Tielens 2005), X dni X kjl nj nl , (1.11) = kj nj + dt j jl with the abundance of a species i, ni (cm−3 ), and the rate coefficients for reactions involving one reaction partner kj (s−1 ) or two reaction partners kjl (cm3 s−1 ). Only one reaction parter is involved

1.2. The envelope of a forming high-mass star

17

in photodissociation reactions. In the tenuous gas of YSO envelopes, three-body reactions can be neglected. This system of coupled, non-linear and stiff, ordinary differential equations (ODEs) can be solved numerically for the temporal evolution of all species in the reaction network. For example the DVODE algorithm (Brown et al. 1989) has proven to yield stable and accurate solutions for a wide range of parameters (e.g. Nejad 2005). Once a solution is obtained, the accuracy can be checked by the conservation of elements. In situations with short timescales (e.g. strong FUV irradiation), steady state conditions can be assumed. The system of rate equations is then reduced to system of non-linear equations. That system, complemented with the conservation of elements, can be solved by a Newton-Raphson algorithm. To close the above system of equations, either initial conditions for time-dependent solutions or the elemental composition for steady state solutions are required. As a simple example, we discuss the abundance of HCO+ , a tracer for ionization (e.g. Jørgensen et al. 2004b; van der Tak & van Dishoeck 2000). The formyl cation (HCO+ ) is mainly formed by the reaction of CO with H+ 3, + CO + H+ (1.12) 3 → HCO + H2 .

+ Cosmic rays or X-rays ionized H2 and H+ 2 then quickly react with H2 to H3 . The main destruction mechanism of HCO+ is recombination with electrons,

HCO+ + e− → CO + H ,

(1.13)

but reactions with H2 O and negatively charged dust grains can also destroy the molecule. Taking into account only the two reactions given above, the rate equation reads dnHCO+ = kform nCO nH+3 − kdestr nHCO+ ne− , dt

(1.14)

with the abundance ni of a species i = {HCO+ , CO, . . .} and the rate coefficients kform and kdestr for formation and destruction, respectively. The timescale for formation can be obtained assuming (i) the abundance of HCO+ is much lower than the equilibrium abundance; (ii) H+ 3 is mainly destroyed + + + + by reaction with CO to form HCO , thus n˙ HCO = −n˙ H3 = kform nCO nH3 , and (iii) the abundance of CO is constant. We then obtain for the timescale τ  −1 dnHCO+ 1 τ = nHCO+ . (1.15) ≈ dt kform nCO This is an important result, as many reactions involve a partner with much higher abundance, like for example CO or H2 . The reaction rate coefficients kj and kjl determine how fast a reaction is. Important types of chemical reactions in astrochemical reaction networks are summarized in Table 1.3. The rate coefficient given in the table is only a rough indication, as the coefficients depend on the reactants and the physical conditions. The coefficients of the first group of reactions (types 1-8) can be expressed by the modified Arrhenius equation (e.g. Shaw 2006),   Ea β k(T ) = αT exp − cm3 s−1 , (1.16) kB T with the temperature T , an activation energy Ea and a pre-exponential factor αT β , usually a weak function of temperature. The temperature dependence differs between the reaction types. For example ion-molecule reactions (type 1) have no activation barrier and proceed at the temperature independent Langevin rate r α ∼ 10−9cm3 s−1 , (1.17) k = 2πq µ

18

1. Introduction

with the electric charge q, the polarizability α and the reduced mass µ of the colliding species. The temperature dependence of neutral-neutral reactions (type 2) is varying, as the van der Waals interaction potential depends on the reactants. Some reactions can only be accurately expressed by the modified Arrhenius equation in a limited temperature range and require the use of individual expressions to cover the whole temperature range. Photoionization and photodissociation rates by protostellar FUV or X-ray radiation (types 9 and 10) are obtained from the integral Z k=

Fλ σλ dλ ,

(1.18)

with the local flux of the radiation field Fλ depending on the wavelength λ, and the photoionization or photodissociation cross section σλ . Since the cross section at FUV wavelengths is not known for all species, and to facilitate the calculation, the rate coefficients are often approximated by the fit equation k = G0 Cexp(−γAV ), with the FUV field G0 in units of the interstellar radiation field, the absorption towards the FUV source AV , and constants C and γ (e.g. van Dishoeck 1988). The constants depend on the incident radiation field. Direct photoionization and photodissociation dominate the FUV driven chemistry, but the X-ray driven chemistry is governed by ionizations through fast photoelectrons from direct X-ray ionizations (type 11; e.g. Maloney et al. 1996; Yan 1997; St¨auber et al. 2005). Cosmic ray induced reactions (types 12 and 13) are given by rates proportional to the cosmic ray ionization rate ζc.r. ∼ 5.6 × 10−17 s−1 (e.g. Doty et al. 2004), k = α ζc.r. .

(1.19)

Cosmic ray ionizations create a cascade of fast electrons, that can excite H2 in the Lyman-Werner bands. On radiative decay of the excited state, internal FUV radiation is generated that can ionize molecules (e.g. Gredel et al. 1989). The same mechanism is also triggered by photoelectrons from X-ray ionizations. Grain surface reactions are another class of important reactions (types 14-16). These reactions are difficult to implement into astrochemical models, since only a discrete amount of reaction sites are available on the grain surfaces. This discrete nature of the problem cannot be represented in the continuous approach using rates. While the approach using “modified rates” (e.g. Caselli et al. 1998; Garrod 2008; Garrod et al. 2009) solves part of this problem, many models still only implement the important formation of H2 on dust grains (Hollenbach & Salpeter 1971). A considerable fraction of the dust grains are negatively charged, due to electrons sticking to the grain surface (e.g. Umebayashi & Nakano 1980). Ionized molecules can thus recombine with negatively charged grains (type 15) (e.g. Aikawa et al. 1999). Photoelectric heating ionizes dust grains (type 16). Like the photoionization by FUV radiation (type 9), these rates also scale with the FUV field. Freeze-out and evaporation can either be incorporated by rates (e.g. Visser et al. 2009), or by different initial conditions for warm and cold regions. Different databases of reaction rates for astrochemical applications are available. The two most popular are the UMIST database for astrochemistry6 (Millar et al. 1997a; Woodall et al. 2007) and the OSU-network7 , maintained by E. Herbst. Both networks contain of order 500 species connected by about 5000 reactions. A bulk part of the reactions are ion-neutral and neutral-neutral reactions. Both networks provide a limited amount of grain surface reactions and no X-ray induced reactions. Reactions involving isotopologues (e.g. deuterium or 13 C) are also not included in these reaction networks. 6 7

www.udfa.net www.physics.ohio-state.edu/˜eric/research.html

1.2. The envelope of a forming high-mass star

19

Table 1.3: Important reaction types.

1 2 3 4 5 6 7 8 9 10 11 12 13

Type ion-neutral neutral-neutral charge exchange dissociative recombination radiative recombination associative detachment radiative association ion neutralisation FUV photodissociation FUV photoionization X-ray photoionization/dissociation X-ray induced electron impact cosmic-ray ionization cosmic-ray dissociation cosmic-ray induced photoprocess

14 grain surface reaction 15 grain surface recombination 16 grain photoionization a

γFUV : FUV photons. b : at 500 K. γX : X-ray − ephoto :fast photoelectrons. hν: radiation.

Examplea C+ + OH → CO+ + H OH + H2 → H2 O + H O + H+ → O+ + H CO+ + e− → C + O HCO+ + e− → CO + H H + H− → H2 + e− C + H → CH + hν H− + H+ 2 → H2 + H CO + γFUV → C + O C + γFUV → C+ CO + γX → C+ + O+ + 2e− photo + − CH + e− → CH + 2e photo − H2 + c.r. → H+ 2 + e H2 + c.r. → H+ + H− CO + γFUV → C + O C + γFUV → C+ g:H + H → H2 g− + HCO+ → g + H + CO g + γFUV → g+

Rate coefficient 10−9 cm3 s−1 10−13 cm3 s−1 (b ) 5 × 10−9 cm3 s−1 (b ) 10−7 cm3 s−1 10−7 cm3 s−1 10−9 cm3 s−1 10−19...−12 cm3 s−1 3 × 10−7 cm3 s−1 10−9 s−1 10−9 s−1 10−9 cm3 s−1 10−9 cm3 s−1 10−17 s−1 10−17 s−1 10−14 cm3 s−1 10−14 cm3 s−1 10−9 cm3 s−1 10−7 cm3 s−1 10−9 s−1

photons. “c.r.”: cosmic rays, “g”: dust grains,

Uncertainties in the reaction rates unfortunately are a problem, as many rates are difficult to measure or to calculate. The latest versions of the OSU and UMIST database give estimated uncertainties of the rate coefficients. In the UMIST06 database (Woodall et al. 2007), for example, the accuracy of about 30 % of the rates is estimated to be better than 25 %, while 60 % are only known to a factor of two. As the uncertainties in the rates propagate in a non-linear way to derived abundances, Monte Carlo calculations have to be carried out to estimate uncertainties in abundances (e.g. Wakelam et al. 2006; Vasyunin et al. 2008). Other systematic uncertainties stem from reactions not included in the network and uncertainties in the initial conditions. Results obtained from chemical models are considered to be in good agreement with observations, if they agree to within a factor of about 3 (Doty et al. 2004).

1.2.3

Molecular line radiative transfer

To compare abundances derived from chemical models with observations, the molecular emission has to be modeled. Molecules can be excited electronically, vibrationally and rotationally. Electronic excitation requires an energy of order 1 eV and corresponds to temperatures of order 10000 K (e.g. Burton et al. 1992). Electronically excited molecules decay by emission of optical or UV photons. Vibrational and rotational excitation correspond to a much lower energy of order 0.1 eV (1000 K) and 10−3 eV (10 K). Thus, vibrational and rotational lines, emitted in infrared and submillimeter wavelengths, probe the cold environment of YSOs envelops. Many of the lines observable with the

20

1. Introduction

heterodyne instrument HIFI onboard the Herschel Space Observatory (de Graauw et al. 2009) are pure rotational lines and we will concentrate on them in this section. The energy levels of a linear molecule, assumed to be a rigid rotor, is given by E(J) =

~ J(J + 1) ≡ BJ(J + 1) , 2I

(1.20)

with the moment of inertia I, the rotational constant B and the rotational quantum number J. For a more accurate level energy, corrections due to centrifugal distortion have to be taken into account. Molecules breaking the linear structure (e.g. H2 O or NH3 ) require considering additional rotation axes. Depending on the molecular structure, coupling between the angular momentum of the molecular rotation, the electronic spin and the nuclear spin occur and lead to a splitting of the level into a “fine structure” (electronic spin) or “hyperfine structure” (nuclear spin). In particular cases (e.g. CH, SH, OH), the orbital motion of the electrons can couple to the molecular rotation leading to so called Λ-type doubling. The level population of many molecules in an YSOs envelope is not in the local thermal equilibrium (LTE). Thus, the level population has to be obtained by considering different excitation and deexcitation mechanism: (i) collisions with abundant species (H2 , H, He and electrons) that do not lead to a destruction of the molecule; (ii) spontaneous decay of an excited level by emission of a line photon and (iii) radiative pumping and photon induced deexcitation. The radiation field for pumping and deexcitation comprises the 2.7 K cosmic microwave background radiation, dust continuum emission and radiation by the same molecule or other molecules. Very reactive molecules like CH+ or CO+ are more likely destroyed than excited in collisions (St¨auber & Bruderer 2009 for CO+ and Chapter 6 for CH+ ). Their level population is governed by the initial level population. The following paragraphs describe the processes involved in molecular excitation and line radiation. A schematic illustration of line radiative transfer is given in Figure 1.7. The solution of the rate equation (e.g. van der Tak et al. 2007) N

N

X dni X = nj Pji − ni Pij , dt j6=i j6=i

(1.21)

yields the population ni of the level, i = 1, . . . , N. For most problems, steady state conditions can be assumed and thus n˙ i = 0. Rate coefficients Pij for the transition between level i and j are given by  Aij + Bij hJij i + Cij (Ei > Ej ) Pij = (1.22) Bij hJij i + Cij (Ei < Ej ) ,

with Ei , the energy of level i. Collisional and radiative excitation and deexcitation enter by the collision rates Cij and the Einstein coefficients for spontaneous and induced emission Aij and Bij , respectively. The average intensity of the radiation is given by hJij i. Collision rates depend on the kinetic temperature Tkin and the density of the collision partner nCol . They are tabulated in the form of collision rate coefficients Kij (Tkin ) = Cij /nCol . The detailed balance relation gi Cij = gj Cjiexp (−hν/kTkin ), with the statistical weights gi , gj , relating the upward and downward collision rates. It is sufficient to know the Einstein Aij coefficients, since gl Blu = gu Bul and Bul = Aul c2 /(2hνij3 ), with the transition frequency νul . Levels are only substantially populated if excitation can keep balancePwith spontaneous decay. This defines the important concept of the critical density ncrit = Aul / i Kui giving the density necessary for collisions to compensate spontaneous decay. Molecules with high transition frequency 3 or a large permanent dipole have larger critical densities, since Aul ∝ µ2 νul . In Table 1.4, an overview of Einstein-A coefficient, critical density and level energy is given for selected lines of CO, HCO+

1.2. The envelope of a forming high-mass star

21

and CH+ . The table shows a wide range of upper level energy and critical density. Different lines can thus be used to probe different parts of the envelope. For example the critical density of the HCO+ J = 4 → 3 transition is not reached in the outer part of the envelope and this line can be used as tracer of the inner region of the envelope. The CH+ lines on the other hand probe regions with very high densities and high temperatures. However, they are not accessible to ground based telescopes, since the atmosphere is not transparent at the line frequencies. Table 1.4: Transitions frequency νul , energy of the upper level Eu , Einstein-A coefficient Aul and critical density ncrit of selected molecular lines (after Tielens 2005 and Chapter 6). The critical density is given for collisions with H2 . Molecule CO

HCO+

CH+

Transition Ju → Jl 1→0 2→1 3→2 7→6 1→0 3→2 4→3 1→0 2→1 3→2

νul [GHz] 115.3 230.8 346.0 806.5 89.2 267.6 356.7 835.1 1669.2 2501.3

Eu [K] 5.5 16.5 33.2 155.0 4.3 26.0 43.0 40.0 120.0 240.2

Aul [s−1 ] 7 × 10−8 7 × 10−7 3 × 10−6 3 × 10−5 3 × 10−5 1 × 10−3 3 × 10−3 6 × 10−3 3 × 10−2 2 × 10−1

ncrit [cm−3 ] 1 × 103 7 × 103 2 × 104 2 × 105 2 × 105 4 × 106 1 × 107 5 × 107 3 × 108 7 × 108

The line radiation can be obtained from the radiative transfer equation, once the level population is known. The radiative transfer equation reads (e.g. Rybicki & Lightman 1979) dIν = −αν Iν + jν , (1.23) ds with the absorption and emission coefficients αν and jν , respectively. Note that at infrared and submillimeter wavelengths, scattering can be neglected. The absorption and emission coefficients are given by hνul φν nu Aul + κν ρDust (1.24) 4π hνul φν (nu Bul − nl Blu ) + κν ρDust Bν (TDust ) , (1.25) jν = 4π where φν denotes the normalized line profile function. The line shape depends on the velocity of the gas, microturbulence, thermal and natural linewidth. The dust contribution to the emission and absorption is controlled by the opacity per unit mass κν , the Planck function for the dust temperature TDust by Bν (TDust ) and the mass density of dust ρDust . Note that in the above equations, we have neglected that quantities like the level population, dust mass density or dust temperature can vary along the line of sight. The radiation that emerges from the envelope is obtained from the solution of the radiative transfer equation along lines of sight from the observer to the background. We define the optical depth by Z αν =

s

αν (s′ )ds′ ,

τν =

0

(1.26)

22

1. Introduction

along the line of sight from the observer (s′ = 0) to the background (s′ = s). The intensity is then given by the formal solution of the radiative transfer equation, Z τν ′ −τν S(τν′ )e−(τν −τν ) dτν′ , (1.27) Iν (τν ) = Iν (0)e + 0

with the source function S ≡ jν /αν . In radio astronomy, the intensity is often expressed by the radiation temperature Tν defined in the Rayleigh-Jeans limit by Tν =

c2 I . 2 ν 2kB νul

(1.28)

In order to simulate the molecular line measured by a telescope, a synthetic map is calculated and then convolved with the beam (point spread function) of the telescope.

Figure 1.7: Schematic illustration of molecular line radiative transfer in a YSOs envelope. In regions with low density, most molecules are in the ground state (J = 0). In regions with higher temperature and density, molecular levels with higher J are populated. The level population depends on collisional and radiative excitation. Radiation stems from molecular line radiation, microwave background radiation or dust radiation. To obtain the radiation that emerges from the envelope, the radiative transfer equation is solved along lines of sight.

Radiative transfer models To calculate molecular line emission, the envelope is first divided into cells. The rate equation (Equation 1.21) is then solved in each cell. This requires the local radiation field, Z Z 1 φν Iν (Ω)dΩdν , (1.29) hJij i = 4π with the line profile function φν and the intensity for different directions and frequencies Iν (Ω). The intensity Iν (Ω) is obtained from the radiative transfer equation (Equation 1.23), where the level

1.2. The envelope of a forming high-mass star

23

population enters. Thus, all cells are coupled by the molecular line radiation, making the problem of line radiative transfer computationally tremendously costly. The straight-forward approach of iteratively solving the rate equations, the radiative transfer equation and then calculating the local radiation field (Lambda-Iteration method) lacks convergence, because information is spread very slowly in regions with high optical depth. A huge improvement is the so called Accelerated Lambda Iteration technique (ALI; e.g. Rybicki & Hummer 1991), where the local and remote radiation field are treated differently. For the calculation of the integrals in hJij i, the Monte Carlo method can be used (e.g. Hogerheijde & van der Tak 2000). A much faster, but approximative method is the popular escape probability approach. It is based on the observation that a photon either escapes the envelope or pumps a molecule at another place in the envelope (e.g. Sobolev 1960; de Jong et al. 1975). The radiation field can then be expressed by hJij i ≈ Bνij (Tex ) (1 − β(τij )) , (1.30)

with the Planck function Bνij (Tex ) for the excitation temperature8 Tex , obtained from the local level population. The probability β(τij ) for a photon to escape the cloud only depends on the optical depth τij . Thus, the radiation field only depends on the local level population and the optical depth. This greatly simplifies the calculation. The escape probability approach is often used for simple geometries like a slab or sphere, but can be extended to arbitrary geometries (Chapter 5). Various molecular constants are required to carry out a radiative transfer calculation. The level energy and Einstein-coefficient are given accurately for many important molecules and lines in the JPL catalog9 (Pickett et al. 1998) and the Cologne Database for Molecular Spectroscopy CDMS10 (M¨ uller et al. 2001). Many collision rates, however, are still unknown or affected by a large uncertainty. A collection of collision rates is given in the Leiden Atomic and Molecular Database LAMDA database11 (Sch¨oier et al. 2005).

1.2.4

Modeling envelopes of YSOs

The goal of the models constructed in the following chapters is to explain observed molecular line fluxes in order to constrain properties that are not directly accessible to observations. Examples are the age of the envelope or the FUV and X-ray radiation of the protostar. Due to the high extinction towards the protostar (Figure 1.5), X-rays and FUV radiation may not be observed directly. Molecules that are enhanced e.g. by FUV radiation can act as indirect tracers. Several approaches to model the molecular line emission from a YSOs envelope are reasonable and adequate. The simplest approach assumes the envelope to be a slab with homogeneous physical conditions and the molecules in the local thermal equilibrium. This approach yields the excitation temperature and column density of the molecule.R The column density is defined as the number of s molecules integrated along the line of sight, N = 0 n(s)ds, with the abundance n(s) depending on the position along the line of sight. The approach has the advantage of being very simple and does not depend on many molecular constants that might be affected by uncertainties. The method can be applied to large spectral surveys and many sources (e.g. Bisschop et al. 2007). However, this model only yields limited information on the physical and chemical structure and composition of the envelope. In a more sophisticated model, first a dust radiative transfer calculation is carried out (Section 1.2.1) to derive the density and temperature structure. Molecular line emission is modeled by a detailed radiative transfer calculation (Section 1.2.3) assuming a fixed abundances relative to 8

Defined by ni gj = nj gi exp(−hν/kB Tex ) spec.jpl.nasa.gov 10 www.cdms.de 11 www.strw.leidenuniv.nl/˜moldata 9

24

1. Introduction

the H2 density. Such models have been successfully applied by e.g. Jørgensen et al. (2004b, 2005b) to study differences in the chemical compositions of low-mass envelopes or to measure freeze-out of molecules. An advantage of the approach is that it gives an impression of the physical and chemical structure, while there are not many free parameters. The method however cannot explain the molecular abundance in terms of the physical conditions or the age of the envelope. Ab initio models calculate both the physical and chemical structure of the envelope based on a few parameters, like the age, the envelope mass or the spectrum of the protostar. In turn, they can be used to constrain such properties. Using this type of models Doty et al. (2002) and Doty et al. (2004) have determined the chemical age, cosmic ray ionization rate and freeze-out temperatures of high-mass and low-mass YSOs. In Rodgers & Charnley (2003), evidence is found that the core must be stabilized in some way in order to explain species with long chemical timescales. Evidence for protostellar X-rays is found by St¨auber et al. (2005). This type of models however has many free parameters. Uncertainties may enter by numerous molecular constants. The ultimate ab initio model would couple a radiative magneto-hydrodynamical simulation with chemical network. However, would we be able to understand the result of such a simulation and – more important – would we be able to relate the results of such a model to the observations? In this work, we use an ab initio approach, however do not consider any hydrodynamics. This is only justified for situations, where the chemical timescales are much shorter than the dynamical timescales. Our approach calculates the dust and gas temperature separately, the later selfconsistently with the chemical evolution. An approximative method is then used to calculate the molecular excitation and line emission. The flowchart of the calculation is summarized in Figure 1.8: 1. First, the density profile of the envelope and the spectrum of the protostar are assumed or determined from observations. 2. A dust radiative transfer calculation is carried out in order to determine the dust temperature. 3. The local radiation field is calculated by a radiative transfer calculation. The calculation takes into account the scattering and attenuation by dust grains. The radiation field is later used to e.g. calculate the photodissociation rates or the heating rates by FUV radiation. 4. Chemical abundances and the gas temperature are calculated self-consistently. For this purpose, the equilibrium between cooling and heating rates and the chemical abundances are calculated iteratively until convergence. The cooling rates require the calculation of the atomic excitation by a radiative transfer calculation. 5. Chemical abundances and the physical structure are used as input for a line radiative transfer calculation to predict molecular line fluxes. Models including this degree of complexity have so far mostly assumed spherical symmetry. Many observations however indicate that envelopes are far from being spherically symmetric, and in particular the chemically active outflow region cannot be modeled in spherical symmetry. In this work, we introduce a two-dimensional axisymmetric models that can, if necessary, be relatively easily extended to a three dimensional model. To make the calculation in two- or three-dimensional geometries feasible, we introduce a number of novel approaches. For example chemical abundances are obtained by interpolation from a grid of precalculated abundances depending on physical parameters like the density, temperature or FUV irradiation. This approach makes the calculation several orders of magnitude faster. Molecular line emission is modeled by a fast method based on the escape probability approach.

1.3. Outline: Directly irradiated outflow walls

25

Figure 1.8: Flowchart of the modeling process used in this work.

1.3

Outline: Directly irradiated outflow walls

In this thesis, we study the high-mass protostar AFGL 2591 (Figure 1.9 and 1.10) both theoretically and observationally. We construct a two-dimensional model using earlier work on that source (e.g. van der Tak et al. 1999; Doty et al. 2002; St¨auber et al. 2004, 2005). Based on observational evidence (St¨auber et al. 2007 and Chapter 4), we study the proposal of directly irradiated outflow walls: The strong protostellar FUV radiation escapes through a cavity, etched out by outflows. Theoretical models predict the shape of such a cavity to be convex (Cant´o et al. 2008) and to allow direct irradiation of the protostar to the walls. In this scenario, the outflow walls are heated and photoionized, similar to photo-dominated regions (PDRs). Typical PDR tracers such as C+ and also hydrides (e.g. CH+ ) are built up in the outflow walls. The predicted hydrides will be observable for the first time using the Herschel Space Observatory. In addition to the outflow walls, also a chemically active hot-core region (T > 100 K) exists. This region is irradiated and ionized by protostellar X-rays, and ionization tracers such as HCO+ are enhanced. Chapter 2 presents the grid of chemical models, an interpolation method to quickly read out chemical abundances from a precalculated grid of chemical models. Differences between cosmic ray induced and X-ray induced chemistry are studied. Chapter 3 introduces the two-dimensional model of AFGL 2591. The proposal of a directly irradiated outflow wall is presented and the calculation of the gas temperature and the FUV field using a Monte Carlo radiative transfer code is discussed. The model is used to explain the detection of CO+ using the JCMT (St¨auber et al. 2007). The non-detection of CO+ in new data, obtained with the Submillimeter Array (SMA), is also explained by the model. New SMA observations of CS and HCN lines in AFGL 2591 are presented in Chapter 4. We find evidence for warm and FUV irradiated material along the outflow of the source. In Chapter 5, an escape probability code for the calculation of molecular line emission is introduced. We verify the code with different benchmark applications and study the accuracy of the approximations used in the method.

26

1. Introduction

The two-dimensional model is extended with a self-consistent calculation of the dust temperature in Chapter 6. We combine the model with the escape probability code from the previous chapter. In this chapter, different shapes of the outflow cavity are studied. Use the model to predict the line fluxes of hydrides that can be observed for the first time using the Herschel Space Observatory.

1.3. Outline: Directly irradiated outflow walls

27

Figure 1.9: Schematic illustration of the physical and chemical structure in AFGL 2591. The line of sight towards the observer is given as suggested by van der Tak et al. (1999).

Figure 1.10: AFGL 2591 in the near infrared (J, H and K band at 1.2, 1.6 and 2.2 µm; C. Aspin using NIRI on Gemini North). Note that the outflow-direction is to the right of the image.

28

1. Introduction

Chapter 2

A grid of chemical models, Method and Benchmarks

Abstract: Upcoming facilities such as the Herschel Space Observatory or the Atacama Large Millimeter Array will deliver a wealth of molecular line observations of young stellar objects (YSOs). Based on line fluxes, chemical abundances can then be estimated by radiative transfer calculations. To derive physical properties from abundances, the chemical network needs to be modeled and fitted to the observations. This modeling process is however computationally exceedingly demanding, particularly if in addition to density and temperature, far-UV (FUV) irradiation, X-rays, and multidimensional geometry have to be considered. We develop a fast tool, suitable for various applications of chemical modeling in YSOs. A grid of the chemical composition of the gas having a density, temperature, FUV irradiation and X-ray flux is pre-calculated as a function of time. A specific interpolation approach is developed to reduce the database to a feasible size. Published models of AFGL 2591 are used to verify the accuracy of the method. A second benchmark test is carried out for FUV sensitive molecules. The novel method for chemical modeling is more than 250,000 times faster than direct modeling and agrees within a mean factor of 1.35. The tool is distributed for public use. Main applications are (1) fitting physical parameters to observed molecular line fluxes and (2) deriving chemical abundances for two- and three-dimensional models. They will be presented in two future publications of this series. In the course of devloping the method, the chemical evolution is explored: we find that X-ray chemistry in envelopes of YSOs can be reproduced by means of an enhanced cosmic-ray ionization rate with deviations less than 25%, having the observational consequence that molecular tracers for Xrays are hard to distinguish from cosmic-ray ionization tracers. We provide the detailed prescription to implement this total ionization rate approach in any chemical model. We further find that the

30

2. A grid of chemical models, Method and Benchmarks

abundance of CH+ in low-density gas with high ionization can be enhanced by the recombination of doubly ionized carbon (C++ ) and suggest a new value for the initial abundance of the main sulphur carrier in the hot core.

Simon Bruderer, Steven D. Doty & Arnold O. Benz The Astrophysical Journal Supplement Series 183, 179 (2009)

2.1

Introduction

There is an interesting but little explored phase in star formation when the cloud core collapses, but the protostar is still deeply embedded. In this phase, the temperature of the interior envelope exceeds 100 K, outflows are observed, protoplanetary disks form, and the protostar begins to radiate in UV and X-rays. These physical processes can best be observed in low temperature lines, some atomic but most molecular. A wealth of new line data of young stellar objects (YSOs) will be available soon by upcoming facilities like the Herschel Space Observatory or the Atacama Large Millimeter Array (ALMA; e.g. van Dishoeck & Jørgensen 2008 for a recent review). Two steps are necessary to constrain physical parameters by molecular line observations: (1) radiative transfer calculations are applied to estimate the abundance of the observed molecule, and (2) modeling the chemical network relates the abundance to physical parameters such as age, density, temperature, far-UV (FUV), and X-ray irradiation by the protostar. Although the present knowledge of these networks is still limited by unknown reactions on grain surfaces, in some cases it is possible to derive chemical abundances from a set of initial physical parameters. The abundances form the bases for radiative transfer calculations. The resulting line fluxes can then be compared to the observed fluxes. Finally, the input physical parameters for the chemical modeling are changed until the derived line fluxes fit the observed ones, for instance producing a minimum in a χ2 test. Chemical modeling simulates a network of molecular species reacting with each other (e.g. Doty et al. 2002). Contrary to an atomic gas, molecular abundances are not conserved, but depend on local physical parameters. In chemical simulations, the abundances of different species are related to the change of one species in a set of non-linear coupled differential equations. Solving these equations is a time-consuming task for a chemical network consisting of several hundred species connected by thousands of reactions. While early astrochemical models of envelopes of YSOs assumed a fixed density and temperature (e.g. Leung et al. 1984), more recent models considered space-dependent chemistry (e.g. Caselli et al. 1993 or Millar et al. 1997b). So far, chemical models have only been fitted to observations under the assumption of spherically symmetric, one-dimensional physical parameters (Doty et al. 2002; St¨auber et al. 2005). This is a questionable assumption as YSOs have inherently two-dimensional or three-dimensional geometries with outflow cavities, disks etc. For such geometries the number of cells for which chemical modeling has to be performed increases from about 100 to 104 and more. Fitting requires repetition of many similar calculations. It becomes inefficient and beyond present resources for two- or three-dimensional geometries and for a large sample of sources. In a dynamical situation, the “chemical age” (evolution time since the start from an initial composition) may be different from the age of the YSO. The chemistry in a gravitationally collapsing envelope has been studied by Rodgers & Charnley (2003) and Lee et al. (2004). Doty et al. (2006) have constructed a model including infall as well as the evolution of the central stellar object. They conclude that the chemical evolution of an ice-evaporated hot core appears to be younger than the time since the formation of the YSO by a factor of 4 to 10. “Pseudo time-dependent” models keep the physical structure constant with time, while the chemical abundances evolve. Such models

2.1. Introduction

31

nevertheless are reasonable approximations if the chemical composition evolves rapidly compared to the change of the relevant physical parameters, like for example in regions with strong FUV irradiation. Undoubtedly, the ionization of molecular hydrogen, H2 , followed by fast ion-molecule reactions involves extremely rapid processes. Ionized molecules are a driving force of the chemistry in the envelope of YSOs (Dalgarno 2006). Cosmic rays account for ionizations mainly in the outer part of the envelope. In the inner part, high-energy radiation from the YSO may be the dominant source of ionization. Accretion phenomena and shocks may account for FUV radiation in low-mass stars, while hot high-mass stars emit copious FUV photons mostly from their hot photosphere. Direct photoionization through FUV photons leads to a particular chemistry as observed in photo dissociation regions (PDRs). FUV radiation is attenuated by dust (e.g. Montmerle 2001). Unless a low-density region allows FUV radiation to escape from the innermost region of the envelope, its influence on the chemistry is restricted to a small volume behind the irradiated surface (St¨auber et al. 2004). The observations of St¨auber et al. (2007) revealed a much larger amount of several FUV sensitive molecules than predicted by their spherically symmetric models. These authors suggested outflows acting as paths for FUV photons to escape and penetrate the high-density gas in the border region between the outflow and the envelope. A multidimensional geometry is needed to model these “outflow-walls”. In addition to FUV, young stars are known to be strong emitters of X-rays with luminosities up to 1032 erg s−1 in the 1 - 100 keV band (e.g. Preibisch & Feigelson 2005). Magnetic activity is believed to be the origin of the X-rays, but the exact mechanism remains unclear. X-ray photons have a smaller absorption cross-section than FUV (∝ λ3 ) and penetrate deeper into dense material than FUV. X-ray ionization thus decreases mostly by geometric dilution, ∝ r −2 with distance r to the source. Various studies on the influence of X-ray irradiation on the chemistry of molecular clouds have been carried out (Krolik & Kallman 1983, Maloney et al. 1996, Lepp & Dalgarno 1996, Yan 1997, St¨auber et al. 2005). For low-mass YSOs, the presence of X-rays has implications on the disc and thus planet formation (e.g. Ilgner & Nelson 2008; Meijerink et al. 2008). In an early stage of star formation, FUV and X-ray radiation are absorbed by the large opacity of the envelope and are therefore not directly observable. The point of evolution when FUV and X-ray activity sets in is unknown. The goal of this chapter is to provide a fast and simple method for chemical modeling. A large set of pre-calculated abundances for different ages and physical conditions (density, temperature, etc.) is used to quickly interpolate the chemical abundances for individual conditions in a model with physical conditions that change with position. The main problem is to reduce the high-dimensional space of physical parameter to a size that fits current hardware resources, but keep the accuracy of the interpolation at an acceptable level. Based on this new method for chemical modeling, observed data can be quickly fitted to physical parameters such as the chemical age or the X-ray luminosity. Also detailed two-dimensional or three-dimensional models may be constructed quickly allowing to study the influence of the geometry on observable parameters. This chapter is organized as follows: in the first section, we describe the grid approach. We discuss the relevant parameters for the chemical composition and the similarity between X-ray and cosmic-ray-induced chemistry. In Section 2.3, the chemical model (network and reaction rates) used for this chapter is briefly described. Two benchmark tests on realistic applications are carried out and discussed in the following section. Section 2.5 describes the application of the method. In the following chapter of this thesis, we will present a detailed multidimensional chemical model of a high-mass star-forming region applied to the enigmatic CO+ molecule and a parameter study of two-dimensional effects on the interpretation of line observations as expected to be observed with Herschel and ALMA.

32

2.2

2. A grid of chemical models, Method and Benchmarks

A grid of chemical models

Chemical models solve for the molecular abundances using the rate equations X dn(i) X ′ ki,j,k · n(j) · n(k) + S(i) = ki,j · n(j) + dt j j,k

(2.1)

which relate the temporal evolution of a species, labeled by i, to the number density n(j), n(k) [cm−3 ] ′ of some species j and k. The constants of proportionality ki,j [s−1 ] and ki,j,k [cm3 s−1 ] depend on physical properties such as temperature, cosmic-ray ionization rate, or FUV flux. Some applications require an additional source term S(i) to account, e.g., for molecules evaporating from dust grains or spatial flows. Many of the reactions relevant in the interstellar environment at low temperature are difficult or impossible to measure in terrestrial conditions. In standard networks for astrochemical applications, a majority of reaction rates are thus only known to within a factor of 2 (Woodall et al. 2007). Furthermore, uncertainties in the rate coefficients may grow due to the non-linear nature of the rate equations (Eq. 2.1). Studies on the quantitative uncertainty of the chemical abundances have been carried out only recently by Wakelam et al. (2005b, 2006) and Vasyunin et al. (2008). Considering their results, it seems reasonable to require the interpolated abundances to agree within a factor of 2 with the fully calculated values. The basis of our method is to interpolate abundances from a grid of physical parameters relevant for the chemical composition. These physical parameters form a multidimensional space, where the dimensions correspond to temperature, density, cosmic-ray ionization rate and several parameters for the impact of high-energetic radiation (X-rays and FUV irradiation). For each grid point and all molecules in the chemical network, the temporal evolution is stored in a database. Therefore, time enters as an additional dimension. In order to explore a wide range in parameter space, we assume a rectangular grid: for every grid point along one axis, all possible combinations of grid points along all other axes are calculated. The total number of models is obtained by multiplying the number of grid points along each dimension. Thus, it is crucial to keep the number of dimensions and grid points as low as possible, while it must still be sufficient to guarantee the required interpolation accuracy. As an example for problems to be met by the interpolation, we show the water fractional abundance with respect to H2 in Fig. 2.1a: a parcel of gas and dust at a gas density between 104 cm−3 and 108 cm−3 irradiated by X-rays has been modeled in a similar way as St¨auber et al. (2007). For this figure, the fractional abundances are given at the time (the so called chemical age) of 5 × 104 yr according to the age of the high-mass protostellar object AFGL 2591 (St¨auber et al. 2005). The presentation of Fig. 2.1 with logarithmic scales in both dimensions suggests the use of linear interpolation of the fractional abundance in log-log space. The straight (red/grey) lines in the figure show interpolated fractional abundances. The temperatures marked by arrows at the top of the figure act as interpolation points. Two temperatures close to 100 K have been used, since water is assumed to evaporate at this temperature from dust grains. For an accurate interpolation in the temperature range between ≈ 200 and 400 K, a couple of interpolation points is required. The reaction OH + H2 → H2 O + H with high activation energy becomes important in this temperature range, leading to little net variation in the water abundance at T > 400 K, while at lower temperature H2 O is destroyed by X-rays (St¨auber et al. 2006).

2.2. A grid of chemical models a.)

b.)

2 · 10−4 1 · 10−4

n(H)=104 cm−3

5 · 10−5

n(H)=106 cm−3 8

n(H)=10 cm

Fractional Abundance n(X)/n(H2 )

Fractional Abundance n(H2 O)/n(H2 )

5 · 10−4

33

−3

2 · 10−5 1 · 10−5 −6

5 · 10

2 · 10−6 −6

10−4

10−6

H2 O HCO+

10−8

H2 CO 10−10

H+ 3

1 · 10

50

100

200

500

104

Gas Temperature [K] 5 · 10−4

−5

10

10−6

S

10−8

CS SO2

H2 S

10−10

Fractional Abundance n(H2 O)/n(H2 )

Fractional Abundance n(X)/n(H2 )

10−4

10−9

106

107

108

109

Total Density n(H) [cm−3 ]

c.)

10−7

105

d.)

2 · 10−4 1 · 10−4 5 · 10−5 2 · 10−5

5 × 103 yr

1 · 10−5

2.1 × 104 yr

5 · 10−6

2.5 × 104 yr 5 × 104 yr

2 · 10−6 1 · 10−6

−11

10

10

100

1000

Gas Temperature [K]

10

100

1000

Gas Temperature [K]

Figure 2.1: Fractional abundance of different species depending on physical properties (temperature, density). The arrows indicate the positions of the interpolation points. The red lines show the results of the interpolation in log-log space. a.) Water in a region with X-ray irradiation versus temperature for three different densities. b.) Density dependence of four selected species related to the ionization fraction. c.) Sulphur-bearing species in the temperature range of the grid. d.) X-ray irradiated water at different chemical ages versus temperature.

2.2.1

Temperature and density dependence

The dependence of abundance on parameters can be much simpler than the temperature sensitivity of water: the fractional abundances of H3 , HCO+ , H2 O , and H2 CO depending on the total gas density are shown in Fig. 2.1b. For that plot, a temperature slightly above the water evaporation temperature of 100 K has been assumed. No X-ray or FUV irradiation is considered, but cosmic

34

2. A grid of chemical models, Method and Benchmarks

−16 rays ionize H2 and account for the production of H+ 3 . A cosmic-ray ionization rate ζc.r. = 4 × 10 −1 s , found by Doty et al. (2004) in IRAS 16293-2422 is used for this plot. Higher values than the “standard” value of a few times 10−17 s−1 have also been suggested by van der Tak & van Dishoeck (2000). This shows the necessity to include the cosmic-ray ionization rate as a dimension for the interpolation approach. The rate of ionizations by cosmic-ray particles and thus the density of H+ 3 is + independent of the H2 density. Therefore, the fractional abundance of H3 is approximately inversely proportional to the density. The same effect is found for HCO+ , produced by the reaction H+ 3 + CO → HCO+ + H2 . In contrast, water and formaldehyde are not directly related to the ionization rate and their fractional abundances do not vary more than an order of magnitude in the explored density range. As can be seen in Fig. 2.1a and 2.1b, the fractional abundance is a much stronger function of temperature than of density. This is due to the exponential dependence of some reaction rates on temperature. Reactions between neutrals that do not involve radicals or atoms often have considerable activation barriers. Their reactions rate is proportional to the Boltzmann factor exp(−Ea /kb T ) in the usual Arrhenius expression, where Ea denotes the activation energy, kB is the Boltzmann constant and T the kinetic temperature of the gas. Selecting grid points in temperature is most important for the accuracy of the interpolation. In addition to water, sulphur-bearing species also pose problems: many reactions have a large activation energy and their fractional abundance is thus expected to be especially dependent on temperature. Indeed, Fig. 2.1c shows a strong dependence of H2 S, CS, SO2 and atomic sulphur on temperature. Again, we overplot interpolated abundances onto the fully calculated results. For temperatures below 100 K, only a small number of grid points is needed, but the hot core regime of T > 100 K requires a much larger amount in order to keep deviations small. How can we select the grid points along the temperature axis of the grid? Two methods have been tested: (1) an unbiased method and (2) hand-placed points. For method (1), the grid points are chosen based on the number of reactions with activation temperature in a certain temperature range. The activation temperature of a reaction is obtained by scaling to the activation energy of the very important reaction OH + H2 → H2 O + H which proceeds at a gas temperature above ≈ 250 K (Fig. 2.1a and St¨auber et al. 2006). The grid points are then placed to properly sample temperature ranges with higher activation temperatures. For method (2), the grid points are placed by hand based on parameter studies like Fig. 2.1a. The selection of species given by St¨auber et al. (2005) and other important molecules are considered at different chemical ages. The interpolation quality of the two methods has been tested using the technique presented in Section 2.4.1. We find a considerably better interpolation accuracy using the grid with hand-placed grid points for the following reason: the unbiased approach does not take into account the relative importance of different reactions: e.g. the reaction between OH and H2 , despite its substantial influence on the chemical network since the key molecule water is involved. As Fig. 2.1a shows for this reaction, it is not sufficient to put two grid points near 250 K in order to obtain a good interpolation quality. The temporal evolution of the water abundance reveals another difficulty in the selection of the grid points (Fig. 2.1d): for chemical ages between a few times 103 and 5 × 104 yr, the fractional abundance cannot be interpolated well using the same set of grid points: the gradient of the water abundance for 100 < T < 250 K and T > 250 K changes with the chemical evolution. Thus, the base point for the connection between the low and high water abundance shifts from 100 K to 250 K. A similar effect is also observed in Fig. 2.1a, where different total densities lead to different timescales of the water destruction. While the fractional abundance of water varies by approximately two orders of magnitude, species connected to its network may be affected even stronger. For example the fractional abundance of H2 S having its main destroyer (HCO+ ) in common with H2 O increases

2.2. A grid of chemical models

35

by many orders of magnitude between 250 K and 600 K (Fig. 2.1c). We finally decided to implement 23 hand picked points at 8, 12, 17, 40, 59.9, 60.1, 99.9, 100.1, 120, 180, 230, 250, 300, 400, 600, 800, 1000, 1400, 1800, 2200, 2600, 3000, and 3400 K for our work. This range thus covers conditions from cold dark clouds to hot PDR-like regions, heated by FUV photons. The selection of the grid points for the density axis is less critical due to the smooth abundance profile along this dimension. We implement one grid point per order of magnitude in density. The density is taken to be within 104 - 109 cm−3 , sufficient to model envelopes of low-mass and high-mass YSOs (Jørgensen et al. 2002, Maret et al. 2002, van der Tak et al. 2000). Table 2.1: Parameters for the grid of chemical models, their physical range in the envelope of YSOs, and the required number of models to achieve a good interpolation accuracy (≈ a factor of 2 for most species/physical conditions). The parameters are defined and described in the text. Parameter Range Straightforward Implementation: Density n 104 − 109 cm−3 Temperature T 8 − 3400 K X-ray flux FX 10−6 − 10 erg cm−2 s−1 a Plasma temperature TX 107 − 3 × 108 K X-ray Attenuationa N(Htot ) 1021 − 1024 cm−2 Cosmic-ray ionization rate ζcr 10−17 − 10−15 s−1 FUV flux G0 0 − 107 ISRF FUV attenuationa τ 0.1 - 10 AV Improved Implementation: Density n Temperature T Total ionization rate ζtot (ζcr , FX ,TX ,N(Htot )) α(G0 ,τ ) β(G0 ,τ )a

104 − 109 cm−3 8 − 3400 K 10−17 − 10−12 s−1 arbitrary units arbitrary units

Grid points 6 23 14 4 6 5 9 10 1.7 × 107 6 23 11 15 5 1.1 × 105

a

The models for G0 =0 or FX =0 do not depend on the attenuating column density and plasma temperature. The same simplification is possible for high values of α(G0 ,τ ).

2.2.2

FUV driven chemistry

Reaction rates for photodissociation or ionization of molecules by FUV radiation can be written as (e.g. van Dishoeck et al. 2006) Z Emax k= J(E)σ(E)dE [s−1 ] . (2.2) Emin

The mean intensity of the FUV radiation field J(E) [cm−2 s−1 erg−1 ] times the cross-section for photoabsorption and ionization σ(E) [cm2 ] is integrated between the average dust work function (Emin ≈ 6 eV) and the ionization energy of hydrogen (Emax = 13.6 eV). For these photon energies, photodissociation proceeds through line absorption and absorption of the continuum by dust. Line

36

2. A grid of chemical models, Method and Benchmarks

absorption occurs by other species or by the considered species itself (the so-called self-shielding). In principle, the exact shape of the spectra is thus required to calculate the chemical rates. As an approximation, van Dishoeck (1988) used the spectral shape of the interstellar radiation field (ISRF; e.g. Habing 1968 or Draine 1978) and fitted rate coefficients to the equation [s−1 ] ,

k = G0 · C · exp (−γ · τ )

(2.3)

where G0 is a scaling factor to the interstellar radiation field and τ = Av /1.086, AV being the extinction at visual wavelengths due to dust. It is calculated from the total hydrogen column density by the conversion factor N(Htot ) = 2N(H2 ) + N(H) ≈ 1.87 × 1021 · Av cm−2 (Bohlin et al. 1978 for a dust reddening of Rv ≈ 3.1). The values of the constants C and γ in Eq. (2.3) depend on the spectrum, used for the fitting. For hot FUV sources with T > 20 000 K, it is safe to use fits to the interstellar radiation field, dominated by O and B stars. G0 is then scaled with respect to the average interstellar flux of 1.6 × 10−3 erg cm−2 s−1 at photon energies between 6 and 13.6 eV. For colder sources such as accretion hot spots, the intensity of the spectra decreases at lower wavelengths, and C and γ should be adjusted (see van Dishoeck et al. 2006). The value of γ depends furthermore on the adopted dust model as well as on the range of Av used for the fitting.

10 -1

6

10−10 1e

1e-16

5

10−11 14

1e

1e-

4

6 1e

-1

2

10−13

14 1e-

1e12

-1 0

10−14

α

1e

2 1e -1

1e-

14

0.5 1e16

10−12

2

-1

1e

1e

4 -1 1e

6 -1

1

1e

τ [Av ]

-1

-1

1e

2

12

1e-

n(CO+ )/n(H2 )

-16

10−15

β

0.2

10−16 0.1 100

102

104

106

G0 [ISRF] Figure 2.2: Fractional abundance of CO+ for different FUV fluxes (G0 ) and FUV attenuation (τ ). The solid contour lines and grey scale give the fractional abundance of the molecule. The gas density was chosen to be 106 cm−3 , while the temperature is fixed at 550 K. The curvilinear coordinate system (α, β) given in dotted line is used for the interpolation in the (G0 , τ ) - plane. The grid points are given by grey/red dots.

2.2. A grid of chemical models

37

St¨auber et al. (2004) studied the influence of FUV radiation on the chemistry in envelopes of YSOs and found CO+ to be a good tracer in high temperature regions. Observations of that molecule by St¨auber et al. (2007) revealed a surprisingly high abundance which cannot be explained in terms of their spherical models. In chapter 3, we will thus present a detailed chemical model in twodimensional using the interpolation approach introduced in this chapter. The dependence of the fractional abundance of CO+ on the radiation field G0 and the attenuation τ is presented in Fig. 2.2. The fractional abundance peaks at radiation strengths of 103 to 104 times the ISRF for low extinction (τ ≈ 0.1). Self-shielding can lead to a high H2 abundance at extinctions τ < 0.1. We restrict the range of our chemical grid to values τ > 0.1, since for the planned applications, the uncertainty in the geometry can easily account for errors of τ on the order of 0.1. At high extinction, with τ > 20, the influence of the FUV radiation is negligible even for field strengths of 107 times the ISRF. In absence of any attenuation, this FUV field corresponds to an early B star with a temperature of 30 000 K and a luminosity of 2 × 104 L⊙ at a distance of about 1 000 AU. An implementation of the chemical grid using τ and G0 as interpolation axes would lack sufficient interpolation accuracy. Another approach was therefore developed: contour plots of FUV sensitive molecules (e.g. C+ , C, CO, and hydrocarbons Cx Hy ) similar to Fig. 2.2 are calculated for different temperature regimes. The contour lines are then used to fit a curvilinear coordinate system as indicated by (red/grey) dots in Fig. 2.2. A unique function relates the physical units of τ and G0 to arbitrary units of a coordinate system denoted by α and β. In this way, the interpolation quality can be greatly improved while the number of grid points is kept constant (Table 2.1). One grid point is placed at very high extinction and no FUV irradiation to represent a model without any influence of FUV irradiation.

2.2.3

X-ray driven chemistry

Rates for the direct photoionization by X-ray photons can be evaluated in a similar way as for FUV radiation using Eq. (2.2). The local intensity of the X-ray emission of a thermal plasma is approximated by   LX N E J(E, r) ≈ · exp (−σphoto (E) · N(Htot )) , (2.4) · · exp − 4πr 2 E kTX

where E is the photon energy, r the distance to the X-ray source, TX the temperature of the X-ray emitting plasma and N(Htot ) the attenuating column density. The normalization factor N is evaluR ated from the luminosity of the X-ray source LX [erg s−1 ] using LX = 4πr 2 Junatt. (E, r)EdE, where Junatt. denotes Eq. (2.4) without attenuation (N(Htot ) ≡ 0). We thus obtain for the normalization   −1   1 Emax Emin N = − exp − · exp − . (2.5) kTX kTX kTX

The photoionization cross-section σphoto is obtained by summing up the contributions of all species including heavy elements in the solid phase (St¨auber et al. 2005). For a photon energy above 10 keV, inelastic compton scattering (described by σcompton ) governs the total X-ray cross-section (σtot = σphoto + σcompton ). The influence of elastic compton scattering and line emission in the X-ray spectra to the chemical composition can be neglected (St¨auber et al. 2005). Unlike FUV radiation, secondary processes are more important for the chemistry than direct photoionization (e.g. Maloney et al. 1996, St¨auber et al. 2005): fast photoelectrons and Auger electrons can ionize other species very efficiently. The rate of ionization per hydrogen molecule due to the impact of secondary electrons can be written as Z Emax E dE , (2.6) ζ H2 = J(E, r)σtotal (E) W (E)x(H2 ) Emin

38

2. A grid of chemical models, Method and Benchmarks

where x(H2 ) ≈ 0.5 denotes the fractional abundance of H2 with respect to the total hydrogen density. The mean energy per ion pair is given by W (E). Like cosmic rays, secondary electrons can excite H2 , hydrogen, and helium atoms. The electronically excited states decay back to the ground state by emitting FUV photons. Mostly photons of the Lyman-Werner bands of H2 contribute to this internally generated FUV field. This field can also photodissociate and photoionize other species, and hence influence the chemistry. In a straightforward implementation, three different parameters are required to parameterize the impact of X-rays on the chemistry: (1) the X-ray flux, as defined by FX ≡ LX /4πr 2 , with the X-ray luminosity LX and the distance to the X-ray source r. (2) The attenuating column density N(Htot ) and (3) the temperature TX of the X-ray emitting plasma. The relevant range for each grid dimension is given in Table 2.1. It is chosen to cover the range of high-mass and low-mass (class 0/1) sources as modeled by St¨auber et al. (2005, 2006). The required number of 14 grid points in the dimension of the X-ray flux can be explained referring to the example of the very strong dependence of the water fractional abundance on this parameter. In Fig. 2.5, the H2 O fractional abundance for a gas temperature of 120 K is shown: at an X-ray flux of 10−4 erg s−1 cm−2 , the abundance drops by about two orders of magnitude. As discussed by St¨auber et al. (2006), this drop depends on X-ray irradiation and the chemical age. Thus, many grid points along the dimension of FX are required to sufficiently sample this drop. Together with five points along the dimension for the cosmic-ray ionization rate ζcr [s−1 ], we would obtain 1.7 × 107 grid points for a straightforward implementation (Table 2.1). This large number can be reduced by a factor of 100 using the following approximation. Instead of the X-ray model described in St¨auber et al. (2005), we make use of the fact that direct photoionization processes due to X-rays can be neglected for the chemical abundances: instead of the X-ray model, the H2 ionization rate ζH2 is calculated by Eq. (2.6) and the parameter for the cosmic-ray ionization rate is increased by this rate. A virtually enhanced cosmic-ray ionization rate thus acts as a proxy for the total ionization rate: 0 ζtot ≡ ζcr = ζcr + ζH2 (FX , N(H), Tx ) (2.7)

0 The “standard” cosmic-ray ionization rate ζcr ≈ 5.6 × 10−17 s−1 and the ionization rate of H2 due to the impact of secondary electrons ζH2 (Fx , N(H), TX ) form together the total ionization rate, which enters the chemical model as effectively increased cosmic-ray ionization rate. In this vein, Doty et al. (2004) used an increased cosmic-ray ionization rate as a proxy for an additional, internal source of ionization. Our approach treats the photodissociation reactions due to internally produced FUV photons exactly in the same way as the X-ray model by St¨auber et al. (2005): rates by Gredel et al. (1989) for cosmic-ray-induced FUV reactions are implemented therein with the same total ionization rate. This approach provides a simple recipe to include X-ray-induced reactions with an arbitrary X-ray spectrum into chemical models: the average energy per ion pair W (E) and X-ray cross-sections σphoto and σcompton are given in Sect. 2.2.5. Using Eq. (2.6) the total ionization rate ζH2 (FX , N(H), TX ) can be calculated. Pre-calculated ionization rates for a thermal X-ray spectrum (Eq. 2.4) are given in Sect. 2.2.6.

2.2.4

Total ionization rate versus X-rays

The approach to use a total ionization rate instead of the previous X-ray model is tested with a spherical model of AFGL 2591 as given in St¨auber et al. (2005). In this model, temperature and density vary with distance to the central source. The density structure is given in the form of a power law in radius (van der Tak et al. 1999) and the temperature structure is taken from the detailed thermal balance calculations of Doty et al. (2002). The temperature and density distributions are

2.2. A grid of chemical models

39

given in Fig. 2.3. The model is devided into 30 shells of approximately constant column density in radial direction. In this way, more shells are placed in the warm, chemically active part of the envelope. To resolve the water evaporation properly, assumed to be at 100 K, an additional shell is set to the corresponding position. Fewer positions are needed in the outer, colder, and thus chemically less active region. Calculations with more shells show insignificant differences in observable quantities derived from the models. More points may be necessary for other temperature/density profiles as presented in Fig. 2.3. When the interpolation method for chemical abundances is used to construct such a spherical model, the number of points can be easily increased due to the gain in speed using that approach. For the implementation, we suggest to bisect the points until the derived observable quantities converge.

500 450

2 · 107

400

1 · 107

Density Temperature

300

2 · 106

250

1 · 106

200

5 · 105

150

2 · 105

100

Density [cm−3 ]

Temperature [K]

350

5 · 106

1 · 105

50

5 · 104

0 1016

1017

Radial Position [cm] Figure 2.3: Density and temperature in the spherical model of AFGL 2591. Vertical lines on top of the figure indicate the position of the shells used for the calculation. Figure 2.4 shows the radial dependence of the abundances for the main molecules CO, H2 O, CO2 as well as for those 10 molecules predicted to be the best X-ray tracers by St¨auber et al. (2005) + + + (HCN, HNC, H2 S, CS, CN, SO, HCO+ , HCS+ , H+ 3 and N2 H ). The ionized hydrides CH , SH , and + NH , discussed later in this section, are given in addition. The X-ray models are indicated by black lines. Results from the model with a total ionization rate ζ given by Eq. (2.7) are presented by thick (grey/red) lines. A model without protostellar X-ray radiation and models with X-ray luminosity LX = 1030 , 1031 , and 1032 erg s−1 are shown. Protostellar FUV radiation is included in the same way as St¨auber et al. (2005) assuming a flux of G0 = 10 at 200 AU. The agreement between the model including X-rays and with a total ionization rate is very good for the shown set of species, with deviations less than 25% in the fractional abundance.

40

2. A grid of chemical models, Method and Benchmarks

1 · 10−3

10−3

10−4

−4

5 · 10−4

2 · 10−4

1 · 10−4

10−6

10−7

10−6

10−7

10−8

10

10−8 1017

10−9 1016

10−5

1017

1016

10−6

10−6

1017

10−6

H2 S

10−6

CS

CN

10−7

10−7 10−7

Fractional Abundance n(X)/n(H2 )

HCN −6

10−5

1016

HNC

−5

10

1017

10−5

CO2

10

10−7 1016

10−5

H2 O

CO

10−7

10−8

10−8

10−8

10−9

10−9

10−10

10−8

10−9 10−10

10−9

10−11

10−11 1016

1017

10−10 1016

10−4

1017

10−6

1017

1016

10−9

HCO 10−7

1017

10−7

+

SO 10−5

10−12 1016

HCS

H3 +

+ 10−8

10−10

−8

10−9

10 10−6

10−11 10−9

10−7

10−10 LX = 1030 (ζ)

10−12

10−10

10−11

LX = 1031 (ζ) LX = 1032 (ζ)

10−8

10−11 1016

1017

10−8

10−13 1016

1017

1016

10−10

1017

10−9

−12

1017

10−13

SH+

CH+ 10

1016

10−8

N2 H+

10−9

10−12

NH+ 10−14

10−10 10−10

10−14

10−11 No X-rays

10−12

LX = 1030 (X-ray)

10−15 10−11

10−16

LX = 1032 (X-ray)

10−16

10−12

LX = 1031 (X-ray)

10−13

10−18

10−14

10−13 1016

1017

1016

1017

10−17 1016

1017

1016

1017

Radial Position [cm]

Figure 2.4: Abundances in a spherical AFGL 2591 model calculated by the X-ray model of St¨auber et al. (“X-ray”, black lines) compared to a model with a total ionization rate given by Eq. (2.7) (“ζ”, grey/red lines). Four different models are shown for X-ray luminosities of 0, 1030 , 1031 , and 1032 erg s−1 . This very good agreement between X-ray models and models with a total ionization rate has observational consequences: in order to use molecular lines as tracers for protostellar X-ray radiation, the effects of cosmic-ray ionization and X-ray-induced ionizations have to be disentangled by spatial or excitation information on the abundance. This can be obtained from high-J lines with a high critical density (e.g. observed by the upcoming HIFI spectrometer onboard the Herschel Space Observatory) or by comparing visibility amplitudes from high angular resolution interferometer observations (e.g.

2.2. A grid of chemical models

41

Benz et al. 2007). What are the limits of this approach? The X-ray flux FX = LX /4πr 2 at a distance of 20 AU from a source with an X-ray luminosity of 1032 erg s−1 is about 90 erg s−1 cm−2 . Meijerink & Spaans (2005) used similar field strengths when modeling X-ray dominated regions (XDRs) in galactic nuclei. Thus, we explored the validity of the ionization rate approximation up to X-ray fluxes of 100 erg s−1 cm−2 . All molecules in the chemical network have been tested for significant deviations between the two approaches in the range from 0 to 100 erg s−1 cm−2 . Excellent agreement as in the example of water (Fig. 2.5g) was found for most species. For our application in envelopes of YSOs, the ionization rate approach thus can be safely used. A small set of species revealed discrepancies between the two approaches under certain conditions. They are presented in Fig. 2.5. There are different reasons for deviation. They are discussed in the following paragraphs. The negatively charged species H− , OH− , and CN− significantly deviate since the X-ray approach does not include a path for the formation of H− by the ionization of H2 (Fig. 2.5a). The UMIST database lists the reaction H2 + c.r. → H+ + H− for this process, resulting in a much higher abundance of this species. Subsequently, the abundances of OH− and CN− also increase. The X-ray approach only includes H− formation by radiative association (H + e− → H− + γ). Thus, our total ionization rate approach to is an improvement. At low X-ray fluxes the abundance of O+ is higher in the X-ray model than in the total ionization rate approach (Fig. 2.5b). In the X-ray approach, direct photoionization CO + γX → C+ + O+ + 2e− can compete with the relatively inefficient formation of O+ through the reaction of He+ with CO2 , OCS, or SO. The fractional abundance of O+ thus increases due to the presence X-ray photons. However, it remains below a few times 10−12 insignificant for our applications (Sect. 4.1). At higher X-ray fluxes, O+ is formed in the charge exchange reaction of H+ with O, and the two approaches agree well. For the hydride ions SH+ and CH+ at low density and high X-ray flux, the X-ray approach predicts higher abundances than the ionization rate approach. This discrepancy can be explained by doubly ionized atoms added by St¨auber et al. (2005) to the chemical network, but not included in the UMIST database. Thus, doubly ionized atoms lack in the total ionization rate approach. At + high densities, doubly ionized atoms are not important because X + H+ 3 → XH + H2 dominates XH+ formation. Abel et al. (2008) found the branching ratio of S++ + H2 to be important for the efficiency of SH+ production through doubly ionized atoms. St¨auber et al. (2005) only included X++ −13 + H2 → X+ + H+ cm3 s−1 ). However, 2 with rate coefficients taken from Yan (1997) (k ≤ 10 in the later version of their code used here, They implemented the formation of hydride ions as the main product with a rate coefficient of 10−9 cm3 s−1 (Yan 1997), similar to the value of Abel et al. (2008) for SH+ , but about an order of magnitude higher for CH+ . Doubly ionized atoms become important for low densities and high irradiation. For a given Xray flux, the number of ionization processes per volume is constant, independent of the total gas density. The electron density at a high X-ray flux of 102 erg cm−2 s−1 varies only by a factor of about 2 between a density of 104 and 108 cm−3 . The amount of O+ is large, accounting for a short destruction time-scale of H2 in the reaction H2 + O+ → OH+ + H. H2 is however the precursor of + H+ 3 which is subsequently also reduced (Fig. 2.5h). In this regime, the main production of SH and + ++ + + CH is through X + H2 → XH + H . NH+ enhancement in the X-ray approach is due to another effect. The charge exchange reactions of oxygen and nitrogen with H2 have rate coefficients of 3 × 10−11 cm3 s−1 and 10−9 cm3 s−1 , much faster than for sulphur and carbon. The chemical network of St¨auber et al. (2005) follows Yan (1997) and thus assumes the formation of OH+ and NH+ through doubly ionized species to be negligible. The slightly enhanced abundance of NH+ in the X-ray approach compared to the total ionization

42

2. A grid of chemical models, Method and Benchmarks

rate approximation is explained by the larger amount of N+ at low density, since its production by the reaction of N++ + H → H+ + N+ is faster than the reaction between N++ and H2 . In conclusion, the total ionization approach should not be used for low densities where the X-ray fluxes exceed 10 erg s−1 X-ray flux. For SH+ and CH+ , the X-ray flux limit is lower (Fig. 2.5). We note that the X-ray approach is also incomplete in this regime as, e.g., the important vibrationally excited H2 with a high energy deposition of X-rays per density HX /n > 10−25 erg cm3 s−1 (Yan 1997) is not included. Nevertheless, the total ionization rate approach is well applicable for models of YSO envelopes, since density models by Jørgensen et al. (2002) or van der Tak et al. (2000) predict a gas density larger than 106 cm−3 close to the protostar where X-ray irradiation is significant (i.e., exceeding than the cosmic-ray effects). Indeed, the derived abundances of SH+ , CH+ , and NH+ in Fig. 2.4, show no difference between the two approaches. For our application, the total ionization approach is not only more elegant, but adequate.

2.2.5

Calculation the X-ray ionization rate ζH2 (FX , N (H), TX)

For the calculation of the X-ray ionization rate ζH2 (FX , N(H), TX ) using Eq. (2.6), the cross-sections for photo- and comptonionization (σphoto (E) and σcompton (E)) and the mean energy per ion pair W (E) are required. We use cross-sections calculated using the X-ray model of St¨auber et al. (2005) to which we refer for the source of the atomic and molecular constants. The elemental composition given in Table 2.4 is assumed. Since the photoionization cross-section of a molecule can be approximated by adding cross-sections of the contained atoms, the total photoionization cross-section does not depend on the abundance of the molecular species as long as the elemental composition does not change. The cross-sections are shown in Fig. 2.6 for a photon energy between 100 and 105 eV. The photoionization cross-section, given by the solid line, is approximated by a power law, defined piecewise for an energy range between Emin and Emax . Table 2.2 gives cross-sections at the boundaries of each energy interval. The cross-section at an energy E ∈ [Emin , Emax ] can then be calculated using σphoto (E) ≈ 10log10 (σmin )·(1−α)+log10 (σmax )·α ,

(2.8)

with α = (log(E) − log(Emin ))/(log(Emax ) − log(Emin )). The deviation of the fit to the calculated cross-section is less than 5% in the given energy range. Interpolation intervals are indicated by ticks at the bottom of the figure. The cross-section for inelastic compton scattering is dominated by H2 and H. This process does not contribute to the total cross-section at low energy. We therefore use the fit from St¨auber et al. (2005) to the XCOM database (NIST) for the energy above 1 keV. With x = log10 (E [eV]), the cross-section reads

σcompton =

 2.869674 × 10−23 − 2.6364914 × 10−23 · x    2   +7.931175 × 10−24 · x2 − 7.74014 × 10−25 · x3 [cm ]   −2.374894 × 10−24 + 1.423853 × 10−24 · x    −1.70095 × 10−25 · x2 [cm2 ]

for E ≤ 10 keV (2.9) for E > 10 keV

Finally, the mean energy per ion pair W (E) can be approximated by (Dalgarno et al. 1999)   23.65 − (log10 (E[eV]) − 2) × 2.7 [eV] for 0.1 keV < E ≤ 1 keV W (E) = (2.10)  20.95 [eV] for E > 1 keV .

2.2. A grid of chemical models

43

10−8

H− / 500 K

O+ / 500 K

10−9

10−6

10−10 10−9 10−11 10−12

10−12

10−13 10−15 10−14 10−6

10−4

10−2

100

102

10−6

10−4

10−2

100

102

10−2

100

102

10−2

100

102

10−2

100

102

10−8 10−8

SH+ / 500 K

CH+ / 80 K 10−9

Fractional Abundance n(X)/n(H2 )

10−10 10−10 10−11 10−12

10−12

10−13 10−14

10−14 10−6

10−4

10−2

100

102

10−8

10−6

10−4

10−8

CH+ / 500 K

NH+ / 500 K

10−9

10−9

10−10

10−10

−11

−11

10

10

10−12

10−12

10−13

10−13

10−14

n(H)=108 cm−3 (ζ) n(H)=106 cm−3 (ζ) n(H)=104 cm−3 (ζ)

10−14 10−6

10−4

10−2

100

102

10−6

10−4

10−3

H2 O / 120 K

10−6

10−4

H+ 3 / 500 K

10−5 10−8 10−6 10−7

n(H)=108 cm−3 (FX ) 6

n(H)=10 cm

10−8

−3

10−10

(FX )

n(H)=104 cm−3 (FX ) 10−9

10−12 10−6

10−4

10−2

100

102

10−6

10−4

X-ray Flux FX [erg s−1 cm−2 ]

Figure 2.5: Abundance versus X-ray luminosity of species with significant deviations between the X-ray model (black lines) compared to the results of the model with enhanced cosmic-ray ionization rate (grey/red lines). The temperature is indicated on the top. Models for different total hydrogen densities (104 , 106 , and 108 cm−3 ) are given to point out the largest effects. Water exhibiting little deviation is shown as an example for comparison.

44

2. A grid of chemical models, Method and Benchmarks

Cross Section [cm2 ]

10

σphoto σcompton σtot

−20

10−22

10−24

10−26

102

103

104

105

Energy [eV] Figure 2.6: Photoionization and compton cross-section versus photon energy. The ticks at the bottom of the figure indicate the supporting points for the interpolation of σ given in Table 2.2.

Table 2.2: Fitting parameters for the X-ray cross-section. a(b) means a × 10b . Emin [eV] 100 291 404.7 538 724 857 870 1311 1846 2477 3203 4043 5996 7124 8348 28900

Emax [eV] 291 404.7 538 724 857 870 1311 1846 2477 3203 4043 5996 7124 8348 28900 100000

σmin [cm2 ] 6.02(−20) 3.03(−21) 1.22(−21) 7.97(−22) 3.99(−22) 2.59(−22) 2.89(−22) 1.06(−22) 4.61(−23) 2.23(−23) 1.11(−23) 5.89(−24) 1.91(−24) 2.24(−24) 1.50(−24) 4.24(−26)

σmax [cm2 ] 2.71(−21) 1.15(−21) 5.29(−22) 3.59(−22) 2.54(−22) 2.48(−22) 9.75(−23) 4.13(−23) 2.03(−23) 1.09(−23) 5.76(−24) 1.89(−24) 1.15(−24) 1.45(−24) 4.24(−26) 1.04(−27)

2.2. A grid of chemical models

2.2.6

45

The X-ray ionization rate ζH2 (FX , N (H), TX) for a thermal spectrum

For many applications, it is sufficient to adopt a thermal X-ray spectrum (Eq. 2.4). In this case, the ionization rate only depends on the geometrically diluted flux FX , the attenuating column density N(H) and the plasma temperature TX . Figure 2.7 shows ζH2 (FX , N(H), TX ) depending on the attenuating column density. The “standard” cosmic-ray ionization rate of 5.6 × 10−17 s−1 is given by a dotted line. The top panel gives the ionization rate for three different plasma temperatures at a fixed flux of 10−2 erg s−1 cm−2 . A hotter spectrum results in a larger number of photons at high energy which can penetrate further into the envelope. Most photons of a cold spectrum (≈ 106 K) however are quickly absorbed. For the bottom panel, the plasma temperature has been fixed to 7 × 107 K. Since the X-ray intensity depends linearly on the flux FX , the ionization rates scale in the same way. The difference in the ionization rate, when the integral in Eq. (2.6) is evaluated between 1 and 100 keV (solid line) and 0.1 and 100 keV (dashed line) is small at column densities larger than 1021 cm−2 , except for a small difference due to the normalization N (Equations 2.4 and 2.5). Up to column densities of a few times 1021 cm−2 , photons with an energy below 1 keV can however contribute significantly to the ionization rate. In Table 2.3, the ionization rate for an X-ray flux of 1 erg s−1 cm−2 is given. The columns correspond to different plasma temperatures and the rows to different attenuating column densities. These ionization rates can be scaled linearly to an arbitrary value of the X-ray flux. For a point-like X-ray source, the flux at a distance r from the source is obtained from the X-ray luminosity by FX = LX /4πr 2 . The impact of X-rays on the chemical composition of a YSO envelope is less depending on the geometry compared to FUV radiation (Chapters 3 and 6) due to the smaller absorption cross-section of X-rays. For the density profile adopted in the spherical model of AFGL 2591, the ionization rate ζH2 (FX , N(H), TX ) decreases approximately as the geometrical dilution (∝ r −2 , St¨auber et al. 2005). For other density profiles, attenuation by photoabsorption may significantly decrease the ionization rate. For the implementation of the total ionization rate approach into chemical models, we provide a fit-equation for ζH2 , which is accurate to within ≈ 10 % for plasma temperatures 107 < TX < 3 × 108 K and column densities N(Htot ) between 1022 and 1024 cm−2 ,

ζH2 = 3.56 × 10

−2

·



LX 30 10 erg s−1

2   100 AU 2 · · 10a+bx+cx s−1 , r

(2.11)

with the X-ray luminosity LX [erg s−1 ], the distance to the X-ray source r [AU] and the attenuating column density x = log10 (N(Htot ) cm− 2). The cofficients a, b and c change with plasma temperature and can be obtained from a = −17957.273149 + 6071.404798y − 685.922103y 2 + 25.952759y 3 b = 1718.671224 − 583.303675y + 66.175787y 2 − 2.516039y 3 c = −41.338087 + 14.079062y − 1.604290y 2 + 0.061291y 3 ,

(2.12) (2.13) (2.14)

with y = log10 (TX ), the plasma temperature TX in K. Note, that the high accuracy of the coefficients is necessary, because the numbers enter in the exponent (Eq. 2.11).

46

2. A grid of chemical models, Method and Benchmarks 10−12

ζ [s−1 ]

10

3×106 K 3×107 K 3×108 K

−13

10−14 10−15 10−16 10−11

1 - 100 keV 0.1 - 100 keV

10−12

ζ [s−1 ]

10−13

10

10−14 10−15

0.1

10−16

Cosmic rays

10−17

10−3

10−5

10−18 1021

1022

1023

1024

1025

N(Htot ) [cm−2 ]

Figure 2.7: Ionization rates assuming a thermal X-ray spectrum. The X-axis gives the attenuating column density. Top: Ionization rate for different plasma temperatures for an X-ray flux of 10−2 erg s−1 cm−2 . Bottom: Ionization rate for different X-ray fluxes (10−5 , 10−3, 0.1, and 10 erg s−1 cm−2 ). Solid line: Eq. (2.6) is integrated from 1 - 100 keV. Dashed line: Eq. (2.6) is integrated from 0.1 100 keV.

Table 2.3: The ionization rate ζH2 in s−1 at FX = 1 erg s−1 cm−2 . a(b) means a × 10b . The columns give values for different plasma temperatures [K] and the rows corresponds to column densities between 1020 and 1025 cm−2 . N(Htot ) 1(20) cm−2 1(21) cm−2 1(22) cm−2 1(23) cm−2 1(24) cm−2 1(25) cm−2

3(6) K 4.8(-11) 4.2(-12) 6.9(-14) 6.9(-17) 6.8(-22) 1.8(-30)

1(7) K 2.4(-11) 4.5(-12) 3.7(-13) 9.6(-15) 4.5(-17) 8.6(-22)

3(7) K 1.0(-11) 2.5(-12) 4.0(-13) 3.9(-14) 2.0(-15) 1.7(-17)

1(8) K 3.4(-12) 9.8(-13) 2.2(-13) 4.5(-14) 9.4(-15) 1.4(-15)

3(8) K 1.2(-12) 3.7(-13) 1.0(-13) 3.3(-14) 1.4(-14) 6.4(-15)

2.3. Chemical model

2.2.7

47

Multidimensional interpolation

The relevant physical parameters for the molecular/atomic fractional abundance evolution are the temperature T , the density n, two parameters for the FUV flux, and the total ionization rate. The implemented grid thus consists of five dimensions and a total of about 1.1 × 105 grid points in the improved implementation. The range and number of points for each dimension are given in Table 2.1. For the interpolation of the abundance at a specific physical condition ~λ ≡ (λ1 , λ2 , λ3 , λ4 , λ5 ), a multidimensional interpolation in logarithmic space is used: the neighboring interpolation points of ~λ are found in each dimension d = {1, 2, . . . ,
5}, such that λda < λd < λdb . Then, the abundances at the interpolation points, x λ1i , λ2j , λ3k , λ4l , λ5m are read out of the grid for all combinations of i, j, k, l, m = {a, b}. The interpolated abundance x is obtained from X

5 (2.15) log10 (x) = αi1 αj2 αk3 αl4 αm · log10 x λ1i , λ2j , λ3k , λ4l , λ5m . i,j,k,l,m={a,b}

The weights αid along each dimension are defined by αad = 1 − β and αbd = β. The position within the hypercube, normalized to the interval between 0 and 1 is given by     β = log10 (λd ) − log10 (λda ) / log10 (λdb ) − log10 (λda ) .

(2.16)

The most time-consuming step of the interpolation is the calculation of the total ionization rate by the integral of Eq. (2.6) and the inversion of the coordinate transformation (α, β) → (G0 , τ ). Both functions are given in tabulated form in the interpolation routine. This allows to obtain approximately 10 000 fractional abundances per second on a standard personal computer. Using the full chemical model, the calculation of the same number of abundance evolutions would require about 80 hours of CPU time.

2.3

Chemical model

In this chapter, we start from the chemical model introduced by Doty et al. (2002, 2004) and St¨auber et al. (2004, 2005). Thus, we describe only changes to their model in this section: the chemical network has been updated from the UMIST 97 (Millar et al. 1997a) to the UMIST 06 database for astrochemistry (Woodall et al. 2007). We implement the standard UMIST 06 database without the dipole-enhanced rates. For technical reasons, fluorine bearing species have not been included. Appendix A gives details of the implementation of the new chemical network. The rates for cosmicray and X-ray-induced FUV reactions implemented in St¨auber et al. (2005) were a factor of 2 too high due to an error in the previously used database (comment by S. Doty in Woodall et al. 2007) and are now at their correct values. Charge exchange on PAH or small grains can be important for the ionization balance (Wakelam & Herbst 2008, Maloney et al. 1996). FUV irradiation enhances the number of positively charged grains. The rates for this process were accidentally divided by the total density in the implementation of St¨auber et al. (2005) and are now at the correct value. The influence on the chemical abundance in their applications is however small. Grain surface reactions are not taken into account except for the formation of H2 , where the rate given in Draine & Bertoldi (1996) is used. We have extended the chemical model with self-shielding of CO and H2 using the shielding factors given in Lee et al. (1996) and Draine & Bertoldi (1996), respectively. They depend on the column densities N(H2 ) and N(CO) between the FUV source and the modeled parcel of gas. Since we do not

48

2. A grid of chemical models, Method and Benchmarks

include any “non-local” parameter as dimension in the chemical grid, an approximation is needed: prior to the calculation of a model, the optical depth dependence of the H2 abundance at the given physical conditions (density, temperature, and FUV irradiation) is calculated for a simple steadystate model considering only the H2 photodissociation and formation on dust with a fixed density and temperature. The column density can be read out of this toy model and shows good agreement with the examples given in Draine & Bertoldi (1996). As a rough approximation of the CO column density, we assume a fixed ratio of N(CO)/N(H2 ) = 2 × 10−4. In the papers of Doty et al./St¨auber et al., the system of stiff ordinary differential equations (Eq. 2.1) is solved using the DDRIV3 algorithm1 . This solver is numerically unstable for chemical networks involving reactions with short timescales, e.g. evaporation from dust or photodissociation at high values of G0 . The DVODE solver2 proved to be more robust and much faster in a large range of physical parameters. Similar characteristics in comparing the two algorithms were found by Nejad (2005). The equations for the conservation of elements revealed a better accuracy of the DVODE solver. A single run of the chemical model on a standard personal computer takes about 30 s, and the calculation of the whole grid thus about 900 hours of CPU time. Distribution on several CPUs is easily possible and allows to build various chemical grids, e.g., for changed chemical networks or different initial conditions in relatively short time.

2.3.1

Initial conditions

In order to solve the first-order differential rate equations (Eqs. 2.1), initial conditions for the abundances have to be assumed. If the chemical evolution were traced starting from diffuse cloud conditions, a purely atomic composition would have to be assumed and the physical conditions such as FUV irradiation or density would have to be changed during the evolution (e.g. Lintott & Rawlings 2006). Here, we follow the approach of Doty/St¨auber et al., who started at dark cloud conditions with a molecular composition (Table 2.5). In this way, uncertainties of the physical evolution during this first phase and/or reactions on dust grains less affect the chemical composition. The evaporation of species frozen out on ice is not taken explicitly into account with evaporation-type reactions but approximated with different sets of initial abundances as in the models of Doty/St¨auber et al. Evaporation and freeze-out reactions are implemented in the model. However, they slow down the calculation significantly due to short timescales at high temperature and are therefore not activated. Table 2.4: Total elemental composition. a(b) means a × 10b . Element H He C N O S P Si Cl 1 2

http://www.netlib.org/slatec http://www.netlib.org/ode

ntot (X)/ntot 1.0 8.5(-2) 3.5(-4) 1.0(-4) 5.4(-4) 2.0(-5) 1.0(-8) 3.5(-5) 8.3(-8)

Element Fe Ne Na Mg Al Ar Ca Cr Ni

ntot (X)/ntot 3.2(-5) 1.4(-4) 2.1(-6) 4.0(-5) 3.1(-6) 3.8(-6) 2.2(-6) 4.9(-7) 1.8(-6)

2.3. Chemical model

49

Table 2.5: Initial conditions in the gas phase relative to the total hydrogen density (ntot = 2n(H2 ) + n(H)). If no other reference is given, we follow Doty et al. (2002, 2004) and St¨auber et al. (2004, 2005). All species in the network without specification are initially set to an absolute abundance of 10−8 cm−3 (effectively zero). a(b) means a × 10b . Species H2 CO CO2

nGas (X)/ntot 0.5 1.8(-4) 1.5(-5) H2 O 7.5(-5) O 4(-5) H2 S 2(-5) S 9.1(-8) N2 3.5(-5) CH4 5(-8) C2 H4 4(-8) C2 H6 5(-9) H2 CO 6(-8) CH3 OH 5(-7) He 8.5(-2) He+ 2(-10) H 5(-8) H+ 3(-10) + H3 2(-9) HCO+ 3(-9) H3 O+ 5(-10) − Grain 1.9(-8) e− 7.5(-9)

Remark

T T T T T T T T T T

< 100 ≥ 100 < 100 ≥ 100 < 100 ≥ 100 < 100 ≥ 100 < 100 ≥ 100

T T T T

< 60 ≥ 60 < 60 ≥ 60

K K K K K K K K K K

K K K K

a a

a a b b,c a

a d a d e e e e e e e,f e

Remarks: a assumed to be frozen-out onto dust grains at cold temperatures or to be not abundant in hot regions, b see Sect. 2.3.2, c Aikawa et al. (2008), d Evaporation temperature from Doty et al. (2004), e St¨auber et al. (2004, 2005), f Negatively charged grains (Maloney et al. 1996)

The total elemental abundances given in Table 2.4 are used for the calculation of the photoionization cross-section (Eq. 2.6), where heavy elements locked into dust grains are also taken into account. The values are taken from Yan (1997), except for helium, where the values assumed by St¨auber et al. (2005) have been adopted.

50

2.3.2

2. A grid of chemical models, Method and Benchmarks

Sulphur bearing species

Various sulphur-bearing molecules are predicted by St¨auber et al. (2005) to be good tracers of X-ray radiation. Despite the large uncertainty in many of the reaction rates involving sulphur-bearing species (Wakelam et al. 2004), these molecules might be important for applications of the grid. To better constrain the initial molecular conditions, we use the multitude of observed sulphur-bearing species found toward AFGL 2591 by van der Tak et al. (2003), van der Tak et al. (1999), and St¨auber et al. (2007). Spherical models of AFGL 2591 (compare Sect. 2.2.4) are calculated for different sets of initial abundances, main sulphur carriers, and chemical ages. The abundances are used as input for a full non-LTE radiative transfer calculation using the RATRAN code (Hogerheijde & van der Tak 2000) to model line fluxes. A χ2 test is carried out to compare the modeled line fluxes with the observations. For this chapter we adopt S (T < 100 K) and H2 S (T > 100K) to be the main initial sulphur carriers. The abundance for the cold part follows Aikawa et al. (2008) to be 9.1 × 10−8 relative to the total hydrogen density. For the hot core, the abundance of SO is chosen to reproduce the jump in the abundance between the cold and hot part as observed by Benz et al. (2007). The adopted sulphur abundance is about the solar abundance (compare Snow & Witt 1996 and Asplund et al. 2005) indicating no or only minor sulpur depletion on dust grains in the hot core. Goicoechea et al. (2006) modeled a PDR with a relatively low FUV irradiation of χ = 60 ISRF (Draine) and reported a sulphur abundance of about a factor of 4 lower than the value adopted in this chapter to reproduce their observations. It may be caused by the massive impact of FUV irradiation in the vicinity of AFGL 2591, raising the temperature severely and resulting in an even higher sulphur abundance in the gas phase.

2.4

Benchmarks

In this section, the accuracy of the interpolation method is verified. For two realistic problems – possible applications of the chemical grid – the interpolated abundances are compared to abundances calculated using the chemical model. The results are considered satisfactory if they agree within a factor of 2 of each other for the reasons noted in Sect. 2.2.

2.4.1

A spherical model of AFGL 2591

A first test of the chemical grid is performed using the spherical model of AFGL 2591 as introduced in Sect. 2.2.4. Figure 2.8 shows the radial dependence of the abundances for the main molecules CO, H2 O, CO2 as well as for those 10 molecules predicted to be the best X-ray tracers by St¨auber + + et al. (2005) (HCN, HNC, H2 S, CS, CN, SO, HCO+ , HCS+ , H+ 3 , and N2 H ). CO which will be addressed in chapter 3 and the two molecules, C6 H and HCNH+ , having the largest deviation are shown in addition. Solid lines give the results of the chemical model, while interpolated abundances are shown by dashed lines. Models with no protostellar X-ray radiation and an X-ray luminosity of LX = 1032 erg s−1 (St¨auber et al. 2005) are given in thin and thick lines, respectively. We assume a chemical age of 5 × 104 yr. The shaded region indicates a range of a factor of 2 compared to the fully calculated model and marks the goal of our interpolation approach. Indeed, most molecules comply with the aimed accuracy for the models with and without X-rays. To provide an unbiased check for a larger set of molecules, a statistical approach is used. The goal of chemical modeling is to obtain characteristics which can be compared with observations. In the following, we check the column density of all molecules in the chemical R model for agreement between the grid and full calculation. The radial column density, Nradial = n(r)dr, with the radial distance r, gives a measure for molecules observed in absorption. For molecules observed in emission, the

2.4. Benchmarks beam-averaged column density is more appropriate. It is defined by RR n(z, p)G(p) 2πp dp dz R Nbeam = , G(p) 2πp dp

51

(2.17)

with z being the coordinate along the line of sight and p the impact parameter perpendicular to z. G(p) is a beam response function (e.g. a gaussian). This approach using column densities does not incorporate the molecular excitation. Thus, it cannot be used to compare models directly with observations. Computationally much more demanding radiative transfer calculations would have to be carried out instead. For our benchmark purposes however, column densities are sufficient. Four different spherical models of AFGL 2591 without protostellar X-rays and with X-ray luminosity of LX = 1030 , 1031 , and 1032 erg s−1 are compared. Three different chemical ages of 3 × 103 , 5×104 , and 3×105 yr are considered to trace possible deviations due to the temporal evolution of the fractional abundances. For these 12 models, the radial (Nradial ) and beam-averaged (Nbeam ) column density are obtained for all molecules in the chemical network. We assume a 14” beam corresponding approximately to a JCMT or Herschel beam and adopt a distance of 1 kpc to the source. To compare grid results to fully calculated models, the factorial deviation Y ≡ max (Ngrid /Nfull, Nfull /Ngrid) is introduced. The fraction of molecules having a factorial deviation larger than a certain value is shown in Fig. 2.15. Only molecules with a column density larger than 1011 cm−2 are considered for the following reason: assuming optically thin radiation, the line strengths corresponding to this column density for a molecule at maximal excitation, a typical Einstein-A coefficient of 10−3 s−1 , and a frequency of 350 GHz is about 40 mK km s−1 , a lower limit for observations with current and upcoming facilities. Compared with the H2 column density of order 1023 cm−2 , it corresponds to a fractional abundance of about 10−12 which marks approximately the possible limit of a detection within an observation time of a few hours. Table 2.6 lists the molecules with the largest deviations. The largest disagreement is found in the radial column density of OH− with a factorial deviation up to a factor of 9.6. This deviation is explained by the different paths for H− formation in the two models (Sect. 2.2.4). It is thus an improvement of the total ionization approach. A problematic disagreement larger than a factor of 5 is found however for HCNH+ (Y = 5.4) and C6 H (Y = 5.2). Figure 2.8 shows the radial dependence of the fractional abundances for these two species at the chemical age and X-ray luminosity where the largest deviation occurs. How can such large deviations be explained? In the case of C6 H at low X-ray fluxes, it is surprisingly an effect of the ionization rate. Figure 2.9 presents the fractional abundance of C6 H versus total ionization rate obtained from the chemical model and interpolated from the chemical grid. Three different chemical ages are given. For 5 × 104 yr, we find a deviation of about a factor of 5 due to the interpolation over a sharp peak in the fractional abundance (bold vertical bar), which is not resolved by the grid. This peak shifts to higher ionization rates with time. A proper sampling would thus require a large number of grid points to cover all chemical ages. At ionization rates below this peak, C6 H is formed by the reaction C + C5 H2 → C6 H. C stems from the X-rayor cosmic-ray-induced dissociation of CO. At an older chemical age (or higher ionization rate), the “late-phase” molecule SO2 is photodissociated to SO and O. Reaction of C6 H + O then decreases the fractional abundance of C6 H. At very high ionization rate, the bulk of C6 H is produced by the electron recombination of C7 H+ .

52

2. A grid of chemical models, Method and Benchmarks

1 · 10−3

10−3

10−4

10

5 · 10−4

10−12

H2 O

CO −4

CO2 10

CO+ 10−13

−5

10−14 10−5

10−6 10−15

2 · 10−4

1 · 10−4

10−6

10−7

10−7 1016

1017

10−8 1016

10−5

1017

10−17 1016

10−5

1017

1016

10−5

HCN 10−6

10−16

HNC

1017

10−6

H2 S

10−6

10−6

CS 10−7

Fractional Abundance n(X)/n(H2 )

10−7 10−7

10−7

10−8

10−8

10−8

10−8

10−9 10−9 10−10 10−9

10−9 1016

1017

10−11 1016

10−6

1017

10−4

1017

1016

10−6

CN

10−7

10−10 1016

10−9

HCO+

SO 10−7

10−5

1017

HCS+ 10−10

10−8 10−8 10−9

10−6

10−11 10−9

10−10 10−7

10−12

10−8 1016

1017

10−11 1016

10−7

10−12

10−10

10−11

1017

10−13 1016

1017

1016

10−8

H3 + 10−8

10−7

N2 H +

10−9

C6 H

10−10

HCNH+

10−8

10−10

Age: 3 × 105 yr LX = 1030 (Full)

10−12

10−9

LX = 1030 (Grid)

10−9 −11

10

Age: 5 × 104 yr

10−10

10−14

10−12

LX = 1032 (Full) LX = 1032 (Grid)

10−10

No X-rays (Full) No X-rays (Grid)

−11

10

1017

LX = 1032 (Full)

10−13

10−16

10−11

LX = 1032 (Grid) 10−12

10−14 1016

1017

10−18 1016

1017

10−12 1016

1017

1016

1017

Radial Position [cm]

Figure 2.8: Abundances of major species, X-ray/UV tracers and problem cases in a spherical model of AFGL 2591. The plots show a comparison between results calculated by the chemical model (solid line) and interpolated from the grid of chemical models (dashed line). The grey region shows a deviation of a factor of 2 from the chemical model. Note the different vertical scales of individual panels. The thick lines correspond to a model with a stellar X-ray luminosity of 1032 erg s−1 and the thin line to a model without X-rays. All abundances are given for a chemical age of 5 × 104 yr, except for HCNH+ (3 × 105 yr).

2.4. Benchmarks

53

Table 2.6: List of molecules with the largest factorial deviation of the grid interpolation from the fully calculated model found in a benchmark test with a spherical model of AFGL 2591. The column densities of both approaches are given along with the factorial deviation Y . Species

Nfull Ngrid Y LX [cm−2 ] [cm−2 ] [erg s−1 ] Radial column densities Nradial OH− 1.9(10) 1.8(11) 9.6 1032 − OH 1.1(11) 8.9(11) 7.9 1031 C6 H 3.2(12) 6.2(11) 5.2 0 + HCNH 6.4(13) 1.2(13) 5.1 1030 C7 H 2.1(12) 4.2(11) 5.0 0 HCOOCH3 5.7(11) 1.1(11) 4.9 0 H− 1.3(11) 5.9(11) 4.5 1032 HS+ 8.5(11) 2.0(11) 4.3 1031 2 H− 1.4(11) 6.0(11) 4.3 1032 − H 1.5(11) 6.2(11) 4.2 1032 CH2 CHCN 2.2(11) 5.2(10) 4.2 1031 + HSO2 4.9(12) 1.2(12) 4.0 0 + HS2 8.5(11) 2.1(11) 4.1 1031 Beam-averaged column densities Nbeam CH3 OH 8.3(12) 2.7(12) 3.0 0 CH3 OCH3 5.7(11) 2.0(11) 2.8 0 H2 S 2.2(15) 1.0(15) 2.2 0 HC7 N 3.3(11) 1.6(11) 2.1 0 HSO+ 3.1(11) 1.5(11) 2.1 0 2 S2 1.4(12) 2.8(12) 2.0 0

Age [yr] 3(3) 3(3) 5(4) 3(5) 5(4) 5(4) 5(4) 3(5) 3(5) 3(3) 5(4) 3(5) 5(4) 5(4) 5(4) 5(4) 5(4) 3(5) 3(5)

Most contribution to the radial column density of HCNH+ is from the innermost part of the model. In this region, HCNH+ is mainly formed by the reaction of HCN with H3 O+ . Both molecules are enhanced by X-rays and FUV irradiation at temperatures above 250 K, where water is not destroyed due to X-ray irradiation (St¨auber et al. 2006). An X-ray luminosity higher than 1030 erg s−1 is sufficient to dominate the effect of FUV enhancement. At lower X-ray luminosities however, the FUV flux dominates the HCNH+ abundance. Due to the particular shape of the contour lines of its fractional abundance depending on G0 and τ , this molecule is difficult to interpolate. Figure 2.10 shows the fractional abundance of HCNH+ in the (G0 , τ )-plane along with the coordinate system used for the gridding and interpolation (Sect. 2.2.2). Physical conditions of the innermost region of the AFGL 2591 model and a chemical age of 3 × 105 yr were chosen for this figure. In general, the agreement between the grid results and the fully calculated abundances is excellent. The mean deviation of hYbeam i = 1.12 for the beam-averaged column densities and hYradial i = 1.28 for the radial column densities indicate agreement for the majority of the species. A total number of 1661 (Nradial ) and 1563 (Nbeam ) column densities are considered for the means, respectively. The slightly higher mean deviation of the radial column density is explained by the larger weight of the chemically active hot core region. The mean deviation does not vary by more than 0.04 with chemical age.

54

2. A grid of chemical models, Method and Benchmarks

10−10 10

−11

3 × 103 yr

10

−12

3 × 105 yr

5 × 104 yr

Cosmic rays

Fractional Abundance n(C6 H)/n(H2 )

10−9

10−13 10−14 10−15 10−16 10−17

10−16

10−15

10−14

10−13

10−12

Ionization Rate ζ [s−1 ]

Figure 2.9: Illustration of the C6 H problem: fractional abundance of C6 H versus ionization rate calculated by the full chemical model (black lines) and from the grid of chemical models (grey/red lines). Three different chemical ages are given. The grid points are indicated at the top of the figure. The cosmic-ray ionization rate of 5.6 × 10−17 s−1 is given by the dotted line. 10−6 1e-08

10

10−8

5 1

08

-

0

-1

-1

1e

10

1

4

-1

1e

10−10

14

1e-

1e12

1e

1e -

τ [Av ]

2

1e

2

0.5

1e-1

10−14

2

0 1e-1

0.2

10−12

n(HCNH+ )/n(H2 )

2 1e-1

0 e-1

0.1 0.1

1

10

100

1000

G0 [ISRF]

Figure 2.10: Illustration of the HCNH+ problem: fractional abundance of HCNH+ (grey scale and contours) at a chemical age of 3 × 105 yr and physical conditions corresponding to the innermost part of the AFGL 2591 model with LX = 1030 erg s−1 . The dotted grey/red lines indicate the coordinate system used for the interpolation and the solid points correspond to grid points.

2.4. Benchmarks

2.4.2

55

FUV driven chemistry

In the spherically symmetric models of AFGL 2591, protostellar FUV radiation cannot penetrate further into the envelope than a few hundred AUs (St¨auber et al. 2004). This is, however, not the case when an outflow cavity allows FUV photons to escape and irradiate a larger volume of gas. This two-dimensional situation will be addressed in the next chapters of this thesis. Here we carry out the following benchmark test. A (plane parallel) region with a fixed temperature of 80, 100, and 550 K and a density of 106 cm−3 is considered. The incident radiation field is assumed to be 1, 10, 102 , 104 , and 106 times the ISRF and no geometrical dilution (i.e., G0 ∝ r −2 ) is taken into account. The dust column density, however, attenuates the FUV radiation. It is given by the optical depth τ using the conversion factor introduced in Sect. 2.2.2. Figure 2.11 shows the fractional abundance versus the optical depth τ . The main molecules CO, H2 O, CO2 , several PDR related species, and the molecules showing the largest deviations are selected for Fig. 2.11. For most species only the results for a temperature of 550 K are shown, since FUV irradiation can considerably heat the gas through the photoelectric effect on dust grains. For reasons of clarity not all fractional abundances are given in this plot. A value reflecting observable quantities is needed for the statistics comparing interpolated to fully calculated abundances (Fig. 2.15). We use the column density of a molecule i between τ = 0.1 and τ = 15 calculated by Z 15 21 Ni = 1.87 × 10 xi (τ ) dτ [cm−2 ] , (2.18) 0.1

where xi (τ ) is the fractional abundance relative to the total gas density of the species i depending on the optical depth τ . All molecules in the chemical network are considered at five different values of G0 and three temperatures. Only column densities larger than 1011 cm−2 are selected following the argument given in Sect. 2.4.1. A total of 2096 column densities are taken into account. The mean deviation for this benchmark is found to be hYFUV i = 1.35, and thus higher than the values in the previous section, but well within the goal of a factor of 2. Table 2.7 lists species, for which large factorial deviations in the column density between the grid and the fully calculated model were found. HSO+ shows the largest deviation amounting to a factor of Y = 5.6 for a high FUV flux of G0 = 106 times the ISRF at a temperature of 100 K. The deviations can be explained by the sharp peak in fractional abundance at τ ≈ 11. It is a result of the formation + + + through SO + H+ 3 → HSO + H2 or SO + HCO → HSO + CO. The peak in the abundance is also found in the fractional abundance of SO due to the competition of two FUV related reactions: at τ < 11, SO is formed by the photodissociation of SO2 and destroyed by the reaction with OH leading to SO2 . At larger τ however, SO dissociation becomes more important and its abundance thus decreases. This peak is not resolved by the current implementation of the grid as can easily be seen in a plot of the fractional abundance depending on G0 and τ (Fig. 2.12). While large deviations of HSO+ occur at high τ , the O+ 2 fractional abundance disagrees at low optical depth (Fig. 2.13). What is the reason for these peaks in fractional abundance? Abundances in FUV irradiated regions are controlled by rate coefficients of the form k ∝ exp (−γ · τ ), where the product of the optical depth times a coefficient enters (see Eq. 2.3). Photodissociation rates thus drop exponentially with τ . They become unimportant for the chemical abundance in a narrow interval of τ . It should be noted, however, that γ is also affected by uncertainties at a level of at least 10% due to the dust model used for fitting the rate and the range of τ over which the rates have been fitted (van Dishoeck et al. 2006). To reveal areas in the (G0 , τ ) plane where deviations occur, the mean factorial deviation for all three temperatures and all species is shown in Fig. 2.14. It shows that large areas agree well with hY i < 2, while other regions show a larger disagreement, however still mostly within a factor of 4.

56

2. A grid of chemical models, Method and Benchmarks

Fractional abundance n(X)/n(H2 )

A narrow region in the (G0 , τ ) plane is found to have mean deviations larger than a factor of 10. It coincidences with the values of G0 and τ where a large deviation of HSO+ has been found in a previous paragraph. As the statistics in Fig. 2.15 and the low mean deviation of the column density show, this local deviation is, however, sufficiently averaged out when observable quantities are derived from the grid.

10−4 10−3

C+ / 550 K

Fractional abundance n(X)/n(H2 )

10−3

CO / 550 K

10−6

−7

10

G0 =1 10

−9

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10−9

10−8

G0 =104 G0 =106

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10−12 5

7.5

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12.5

15

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5

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10−5

−4

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OH / 550 K

−4

5

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10

12.5

15

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10

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15

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10

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15

CO2 / 550 K

10−6

10−6 10−7

10−8

10−8 10−8 10−9 10−10 10−6 10

10−10

10−10 2.5

Fractional abundance n(X)/n(H2 )

C / 550 K

10−6

10−6

2.5

5

7.5

10

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15

2.5 10−6

CN / 550 K

5

7.5

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HNC / 550 K

2.5 10−4

5

HCN / 550 K

−8 −8

10

10−6 −10

10

10

−10

−12

10

10−12

10−14

10−14 2.5

Fractional abundance n(X)/n(H2 )

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5

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10−8

10−10 2.5

5

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5

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CO+ / 550 K

O+ 2 / 550 K

HSO+ / 100 K −10

−10

10

10

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10−12 10

−12

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10−16

10−16 2.5

5

7.5 τ

10

12.5

15

2.5

5

7.5 τ

10

12.5

15

2.5

5

Figure 2.11: Fractional abundances of FUV sensitive species between τ = 0 and τ = 15. For this presentation, the temperature is fixed at the given value and the density is assumed to be 106 cm−3 . Results for an incident FUV field of 1, 102 , 104 , and 106 ISRF are shown. The thick line corresponds to the result calculated by the chemical model while the thin line gives the results from the grid interpolation. The grey region indicates the range of a factor of two relative to the calculated model specified for agreement of the interpolation.

2.4. Benchmarks

57

0 1e-11e-12

10

4

1e-10

10 1e- e-12 1 14 1e-

5

10

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16

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2 1 14

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0.5

1e-

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0.2

10−16 10−17

0.1 10

0

10

2

10

4

10

6

G0 [ISRF]

Figure 2.12: Fractional abundance of HSO+ (black solid line) for different FUV fluxes (G0 ) and FUV attenuation (τ ). The total density was chosen to be 106 cm−3 , while the temperature is fixed at 100 K. The grid points of the curvilinear coordinate system are shown in (grey/red) dots and black dotted line. The value of G0 (106 ) where large deviations were found is indicated by a vertical red/grey dotted line.

10−10

10 1e-14

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-14

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4

1 e-

1

2 1e

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-1

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-1

1e

2

1e-1 0

1e-1

2

0.5 10−14

0.2 10−15 0.1 100

102

104

106

G0 [ISRF]

Figure 2.13: Fractional abundance of O+ 2 (black solid line) for different FUV fluxes (G0 ) and FUV attenuation (τ ). The total density was chosen to be 106 cm−3 , while the temperature is fixed at 550 K. The grid points of the curvilinear coordinate system are shown in (grey/red) dots and black dotted line. The value of G0 (102 ) where large deviations were found is indicated by a vertical red/grey dotted line.

58

2. A grid of chemical models, Method and Benchmarks

10 5

hYi 10 -

τ [Av ]

2 4 - 10 1 2-4 0.5

0-2

0.2 0.1 100

102

104

106

G0 [ISRF] Figure 2.14: Mean factorial deviation Y between the interpolated and fully calculated abundances of all species in the chemical network (grey scale). G0 is the FUV field strength in units of the ISRF and τ the optical depth to the FUV source. The curvilinear coordinate system (α, β) given in dotted line is used for the interpolation in the (G0 , τ ) - plane. The grid points are given by grey/red dots.

Table 2.7: List of molecules with the largest factorial deviation of the grid interpolation from the fully calculated model found in a benchmark test with a FUV irradiated zone. The column densities of both approaches are given along with the factorial deviation Y. Species HSO+ O+ 2 CH3 OCH3 CH3 OH HC3 N C2 N

Nfull [cm−2 ] 3.0(12) 1.3(12) 2.5(12) 5.3(12) 2.9(14) 1.0(14)

Ngrid [cm−2 ] 5.3(11) 2.6(11) 6.1(11) 1.3(12) 7.1(13) 2.5(13)

Y 5.6 4.8 4.1 4.0 4.0 4.0

G0 [ISRF] 106 102 106 106 106 10

T [K] 101 550 101 80 101 101

2.5. Utilizing the grid of chemical models

59

Fraction of Species

1

0.95

0.9

0.85

AFGL 2591 model (Nbeam ) AFGL 2591 model (Nradial ) UV region

0.8 2

4

6

8

10

Factorial Deviation of Column Density NFull / NGrid

Figure 2.15: Statistics on the accuracy of the interpolation method for chemical abundances: the fraction of species within a factorial deviation given in the X-axis is shown. Both benchmarks to the spherical AFGL 2591 model (Sect. 2.4.1) and the FUV model (Sect. 2.4.2) are given. For the AFGL 2591 model, the comparisons of radial and beam-averaged column densities are shown.

2.5

Utilizing the grid of chemical models

The grid of chemical models is accessible through Abundance Modelling of young stellar Objects Under protostellar Radiation (AMOUR) available for public use at http://www.astro.phys.ethz.ch/chemgrid.html The page provides an online form to retrieve interpolated fractional abundances from the grid of chemical models (Fig. 2.16). This form also calculates the X-ray ionization rate ζH2 for a particular X-ray flux FX , X-ray attenuation N(Htot ), and plasma temperature TX . For large modeling tasks or to include the interpolation method in other codes, the page also offers a FORTRAN code and the necessary molecular data tables in binary and ASCII format for download. Documentation is available at the website which describes the input/output parameters, the technical implementation, the format of the data tables, and shows example input/output data. There exist a number of restrictions for the currently provided implementation which must be kept in mind when applying the method: the chemical composition in the molecular data table has been calculated for a specific chemical network (the UMIST 2006 database,Woodall et al. 2007) using one set of initial conditions (Table 2.5). The current implementation does not include grain surface reactions except for H2 formation on dust. The medium is assumed to be static and physical conditions (e.g., temperature or density) have been assumed to be constant over time. These are however only restrictions of the currently distributed molecular data tables. Further tables for other chemical networks, initial conditions, time-dependent parameters, etc. can be obtained from the authors on a collaborative basis.

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2. A grid of chemical models, Method and Benchmarks

2.6

Conclusions and outlook

Starting from the chemical model of Doty et al. (2002, 2004) and St¨auber et al. (2004, 2005), we have introduced a new method to simulate the chemical evolution of YSO envelopes based on interpolation in a pre-calculated grid. The chemical model has been revised and the relevant physical parameters for the chemical composition of the gas are discussed. Benchmark tests have been carried out to verify the accuracy of the interpolation method. We conclude the following: 1. Accurate chemical modeling of a multidimensional envelope of YSOs is possible using a fast, grid-based interpolation method (Sect. 2.4). Comparison of observable quantities (beamaveraged line fluxes) yields a mean factorial deviation between fully calculated and interpolated values of 1.35. This is more accurate than the uncertainties introduced by observations and chemical rate coefficients. Our method is more than five orders of magnitude faster than the full calculation (Sect. 2.2.7). 2. The X-ray models by St¨auber et al. (2005) can be reproduced using an enhanced cosmic-ray ionization rate as a proxy for X-ray irradiation (Sect. 2.2.4). In the relevant physical range for the application in YSO envelopes, the agreement in the fractional abundance is within 25%. The implementation of the this approach is described in detail in Sect. 2.2.5 and allows to include the effect of X-ray irradiation to chemical models in a simple way. 3. Ionization by X-rays and cosmic rays cannot be easily distinguished by molecular tracers. Spatial information on the abundance is thus needed to disentangle protostellar X-ray and cosmic-ray ionization (Sect. 2.2.4) 4. For the formation of CH+ in low density gas (104 cm−3 ) with high X-ray irradiation (> 1 erg −1 cm−2 ) recombination of doubly ionized carbon with H2 (C++ + H2 → CH+ + H+ ) is important. 5. Increasing the initial abundance in the hot core region (T > 100 K) of the main sulphur carrier improves the agreement between models and observations in a high-mass star-forming region (Sect. 2.3.2). 6. Exploring the parameter space, we find regions with high gradients in molecular abundances. This is where small changes in the physical parameters yield large variations in abundance. As an example, interpolation of fractional abundances of an FUV irradiated region is difficult because photodissociation rates depend exponentially on the optical depth (Sect. 2.4.2). We have compensated for this by adopting a curvilinear coordinate system and a high number of grid points (Sect. 2.2.2). The current grid method is limited by the assumption of the initial composition and fixed chemical rates. For different assumptions, the database needs to be recalculated. Using this fast chemical interpolation method, the radiative transfer calculation becomes the bottleneck in computing time to interpret data. Recently introduced escape probability methods, such as the exact method by Elitzur & Asensio Ramos (2006) or an approximative multidimensional code by Poelman & Spaans (2005) can speed up this step of the modeling. This chapter shows the possibility of interpolation for chemical modeling. In future publications, we will demonstrate major applications in multidimensional chemical modeling of YSOs and fast data-fitting to interpret observations. The gain in speed allows to carry out parameter studies on the influence of the geometry on the interpretation of observations. Furthermore, it will be possible to apply detailed chemical models to a large set of sources and draw conclusions on physical properties based on statistics.

2.6. Conclusions and outlook

61

Acknowledgments We thank Ewine van Dishoeck, Pascal St¨auber, Susanne Wampfler and an anonymus referee for useful discussions. Michiel Hogerheijde and Floris van der Tak are acknowledged for use of their RATRAN code. The submillimeter work at ETH is supported by the Swiss National Science Foundation grant 200020-113556 (SB and AOB). This work was partially supported under grants from The Research Corporation and the NASA grant NNX08AH28G (SDD).

Figure 2.16: Web interface for the grid of chemical models AMOUR.

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2. A grid of chemical models, Method and Benchmarks

Chapter 3

Multidimensional chemical modeling, Irradiated outflow walls

Abstract: Observations of the high-mass star-forming region AFGL 2591 reveal a large abundance of CO+ , a molecule known to be enhanced by far UV (FUV) and X-ray irradiation. In chemical models assuming a spherically symmetric envelope, the volume of gas irradiated by protostellar FUV radiation is very small due to the high extinction by dust. The abundance of CO+ is thus underpredicted by orders of magnitude. In a more realistic model, FUV photons can escape through an outflow region and irradiate gas at the border to the envelope. Thus, we introduce the first two-dimensional axisymmetric chemical model of the envelope of a high-mass star-forming region to explain the CO+ observations as a prototypical FUV tracer. The model assumes an axisymmetric power law density structure with a cavity due to the outflow. The local FUV flux is calculated by a Monte Carlo radiative transfer code taking scattering on dust into account. A grid of precalculated chemical abundances, introduced in chapter 2, is used to quickly interpolate chemical abundances. This approach allows us to calculate the temperature structure of the FUV-heated outflow walls self-consistently with the chemistry.

64

3. Multidimensional chemical modeling, Irradiated outflow walls

Synthetic maps of the line flux are calculated using a raytracer code. Single-dish and interferometric observations are simulated and the model results are compared to published and new JCMT and SubMillimeter Array (SMA) observations. The two-dimensional model of AFGL 2591 is able to reproduce the JCMT single-dish observations and also explains the non-detection by the SMA. We conclude that the observed CO+ line flux and its narrow width can be interpreted by emission from the warm and dense outflow walls irradiated by protostellar FUV radiation.

Simon Bruderer, Arnold O. Benz, Steven D. Doty, Ewine F. van Dishoeck & Tyler L. Bourke The Astrophysical Journal 700, 872 (2009)

3.1

Introduction

Forming high-mass O and B stars with a surface temperature larger than ≈ 104 K emit most of their radiation in far UV (FUV) wavelengths. After a not yet well defined point of the evolution, also Xrays are emitted. When still deeply embedded in their natal cloud, most of this high-energy radiation is absorbed by the large column density of gas and dust toward the protostar. It is thus not available to direct observation but does influence the composition of the molecular envelope through heating and photoionization. An essential ingredient of star formation is bipolar outflows transporting the excess angular momentum outwards. When the fast, low-density gas of molecular outflows or jets expands into the surrounding molecular cloud, large cavities may result (e.g., Preibisch et al. 2003). Along these outflow cavities, FUV radiation may escape and irradiate the high density material at the border of the cavity. A surprisingly large amount of CO+ has been detected in the high-mass star-forming region AFGL 2591 by St¨auber et al. (2007). Their radiative transfer calculations indicate fractional abundances of order 10−10 , much higher than 10−15 to 10−14 predicted by dark cloud models. CO+ has previously not been detected toward envelopes of young stellar objects (YSOs), but has been seen in photodominated regions (PDRs; e.g., Jansen et al. 1995 or Fuente et al. 2003). Strong FUV radiation in these regions heats the gas and drives a peculiar chemistry which enhances the abundance of CO+ . The detection of CO+ in envelopes of YSOs is thus strong evidence for the feedback of the protostar on the envelope by high-energy irradiation. In this chapter, we will study CO+ as a prototypical species enhanced by FUV irradiation in high-mass YSOs. Chemical models solve for the evolution of the abundances of molecules and atoms. They simulate a network of species reacting with each other. The network consists of different types of chemical reactions. In dark clouds, for example, all FUV radiation is shielded by a large column density of dust and the chemistry is dominated by cosmic-ray ionization and reactions between neutral and ionized species (e.g., Doty et al. 2002). In the innermost part of a YSO envelope, X-rays may dominate the ionization (St¨auber et al. 2005). They influence the chemistry mainly by secondary fast photoelectrons hitting molecules and atoms. X-ray-induced chemistry is very similar to cosmicray-induced chemistry (chapter 2). In regions with a strong FUV irradiation, direct photoionization of species with ionization potential < 13.6 eV drive the chemistry. The influence of FUV radiation on the envelope of YSOs has been studied by St¨auber et al. (2004). So far, a spherically symmetric structure of the envelope has been assumed for chemical models of envelopes of high-mass star-forming regions (e.g., Viti & Williams 1999, Doty et al. 2002, St¨auber et al. 2004, 2005). While the abundances derived using this spherical model agree for many species with observations of specific sources, they fail to explain the amount of CO+ . Due to the attenuation

3.1. Introduction

65

by dust, FUV radiation cannot escape the innermost few hundred AU and the volume of gas where CO+ may be formed is too small in spherical models to reproduce the observations. Another possible formation mechanism of CO+ is X-ray irradiation. Unlike FUV photons, X-rays are mainly attenuated by geometrical dilution (∝ r −2 ) due to their smaller absorption cross-section proportional to λ3 . X-ray radiation can thus penetrate much deeper into the envelope on scales of a few thousand AU. Protostellar X-ray luminosities of up to 1032 erg s−1 in the 1-100 keV band are observed (Preibisch & Feigelson 2005). For this luminosity, however, the X-ray flux is too low to enhance CO+ sufficiently to match with observations. The interaction zone between outflow and infall has been proposed before as sites of anomalous chemistry and molecular excitation. Spaans et al. (1995) have explained the strong CO(6-5) emission in narrow lines observed toward many low-mass YSOs by heating of the cavity wall through protostellar FUV photons. Compact emission of HCO+ associated with the outflow wall has been detected by Hogerheijde et al. (1998) in the Class 0 object L 1527. A possible explanation is mixing of partially ionized gas from the outflow with infalling material leading to a thin layer of dense and ionized gas, where species with a high dipole moment (e.g., HCO+ or HCN) can be excited efficiently by collisions with electrons (Hogerheijde 1998). Detailed three-dimensional radiative transfer studies by Rawlings et al. (2004) on the other hand prefer a scenario with an enhanced abundance of HCO+ due to shock liberation and photoprocessing of molecular material stored in ice mantles. Observational evidence for such an interaction layer in a high-mass star-forming region are found by Codella et al. (2006) by studying the line profiles of different molecular species. Recently, van Kempen et al. (2009b) find evidence for UV photons escaping through the outflow cones and impacting the walls for the low-mass protostar HH 46. In this chapter, we will introduce a chemical model that implements a non-spherical density structure taking an outflow cavity into account. A proper treatment of the FUV irradiated outflow walls requires two-dimensional chemical models with high spatial resolution. This is however computationally excessively expensive, especially if the temperature structure is calculated self-consistently with the chemical abundances. In chapter 2, we have introduced a new method for fast chemical modeling. A precalculated grid of chemical abundances depending on different physical parameters such as the density, temperature or FUV flux is calculated and the abundances can then be obtained by interpolation in this database. Using different benchmark tests, we found very good agreement between fully calculated chemical abundances and interpolated values, while the interpolation approach is more than 5 orders of magnitudes faster. This speed-up allows us to quickly construct detailed chemical models with a complex geometry. We construct a detailed two-dimensional chemical model of the high-mass star-forming region AFGL 2591. The goal is to explain the high abundance of CO+ measured by the JCMT singledish telescope (St¨auber et al. 2007) as well as the non-detection by the Submillimeter Array (SMA) interferometer (Benz et al. 2007 and new data, reported in Sect. 3.4.4). In this chapter, we assume a fixed geometry and consider only CO+ . The chapter is organized as follows: in section 3.2, we briefly discuss the chemistry of CO+ . The next section introduces the two-dimensional model of AFGL 2591: we describe the modeling process, the radiative transfer of the FUV radiation, and the calculation of the temperature structure. In section 3.4, we present results of the chemical model. Modeled fluxes and synthetic maps are compared to JCMT and SMA observations.

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3. Multidimensional chemical modeling, Irradiated outflow walls

3.2

Chemistry of CO+

The fractional abundance of CO+ depending on the temperature is given in Fig. 3.1 for different gas densities, X-ray fluxes [erg s−1 cm−2 ] and FUV irradiation in units of the interstellar radiation field (ISRF). A parcel of gas has been modeled for this figure using the chemical model described in chapter 2. The chemical model solves for the evolution of the abundances of a network of chemical species starting from initial abundances. We assume a chemical age of 5 × 104 yr as suggested by St¨auber et al. (2005). Chemical timescales are very short in regions with strong FUV or X-ray irradiation, where a significant amount of CO+ is produced. Equilibrium conditions are thus reached very quickly and the chemical age does not influence the resulting abundances. Photodissociation and ionization rates depend on the attenuating column density between the FUV/X-ray emission and the modeled parcel of gas. We assume an optical depth of τ = 1 (FUV) and a hydrogen column density of 1020 cm−2 (X-rays) to derive the irradiating spectra. 10−9

CO+ / UV

CO+ / UV

n(H2 )=104 cm−3

CO+ / UV

n(H2 )=106 cm−3

n(H2 )=108 cm−3

n(CO+ )/n(H2 )

10−10 10−11

G0 =0 ISRF G0 =10 ISRF G0 =103 ISRF G0 =105 ISRF G0 =107 ISRF

10−12 10−13 10−14 10−15 10−16 1000 500 10−9

200 100

50

20

10

1000 500

200 100

50

20

10

CO+ / X-rays n(H2 )=106 cm−3

CO+ / X-rays n(H2 )=104 cm−3

1000 500

50

20

10

CO+ / X-rays n(H2 )=108 cm−3

10−10 n(CO+ )/n(H2 )

200 100

FX =0 erg s−1 cm−2 FX =10−3 erg s−1 cm−2 FX =0.1 erg s−1 cm−2 FX =10 erg s−1 cm−2

10−11 10−12 10−13 10−14 10−15 10−16 1000 500

200 100 50 Temperature [K]

20

10

1000 500

200 100 50 Temperature [K]

20

10

1000 500

200 100 50 Temperature [K]

20

10

Figure 3.1: Parameter study of CO+ : The fractional abundance is shown for a temperature range between 10 and 1500 K and a hydrogen density of 104 , 106 and 108 cm−3 . For comparison, observed abundances are of order 10−10 . Top: The abundance is given for different FUV fluxes between 0.1 and 107 ISRF at τ = 1 in units of AV . Bottom: The abundance is given for different X-ray fluxes between 10−10 and 10 erg cm−2 s−1 for an attenuating hydrogen column density of 1020 cm−2 .

In gas with strong FUV irradiation, CO+ is efficiently produced by the reaction C+ + OH → CO + H (St¨auber et al. 2007, Fuente et al. 2006). Direct photoionization of carbon atoms accounts for C+ , while OH is produced by the photodissociation of water. At temperatures below 100 K, +

3.3. An axisymmetric model of AFGL 2591

67

H2 O is frozen-out and the amount of OH is reduced. Between 100 K and about 250 K, the reaction O + H2 → OH + H does not proceed due to an activation barrier and OH is destroyed through photodissociation. The amount of OH increases with higher temperature and thus CO+ is produced. At very high FUV fields, CO+ is photodissociated and its abundance again decreases. In X-ray irradiated gas, ionization of CO through secondary electrons accounts for the production of CO+ . The abundance of CO+ is thus constant with density and the fractional abundance nearly inversely proportional to the total density. In low-density gas with high X-ray irradiation, the production of CO+ through C+ + OH can also be important as Figure 3.1 shows at a density of 104 cm−3 and an X-ray flux of 10 erg cm−2 s−1 . In this regime, the fractional abundance of CO+ increases with temperature. CO+ is destroyed by H2 leading to HOC+ and HCO+ with equal branching ratios. In regions with very high FUV irradiation, H2 is photodissociated to atomic hydrogen and CO+ is destroyed by the reaction CO+ + H → CO + H+ . To produce the observed fractional abundances of order 10−10 (St¨auber et al. 2007) by X-rays, a flux higher than 10 erg cm−2 s−1 is needed for a density of 106 cm−3 . For an X-ray luminosity of 1032 erg s−1 as found by St¨auber et al. (2005) for AFGL 2591 and in the extreme case of no attenuation, this flux can only be achieved in the innermost 60 AU. In a spherical model of AFGL 2591, X-rays may enhance the abundance of CO+ up to a few times 10−13 , and St¨auber et al. (2005) have thus suggested the molecule as a tracer for X-rays. X-rays are however not able to enhance the abundance to 10−10 on larger distances to the source. The FUV flux of a young B star on the other hand is still sufficient at a distance of almost 10 000 AU to produce a fractional abundance exceeding 10−10 if no dust attenuates the FUV field. We thus consider CO+ to be a tracer for FUV irradiation rather than for X-rays. A short note concerning the combined influence of X-rays and FUV radiation: in the vicinity of a strong FUV source, CO is dissociated and X-rays cannot produce CO+ efficiently. For example, the calculation of the parameter study including X-rays at a density of 104 cm−3 has been repeated in the presence of an FUV field of 104 ISRF. For this condition, the fractional abundance does not exceed 10−14 and we thus expect the fractional abundance in a low density and X-ray/FUV irradiated outflow region to be very low.

3.3

An axisymmetric model of AFGL 2591

Figure 3.2 summarizes the process for an axisymmetric model of AFGL 2591. We first assume a density structure in Sect. 3.3.1. This density distribution is used in the following section to calculate the local FUV and X-ray field at every position of the model. The high-energy flux together with the density structure then enters the thermal balance calculation, which solves for the temperature structure. Since cooling and heating depend upon the composition of the gas, the temperature must thus be calculated self-consistently with the chemical abundances which are read out of the grid of chemical models presented in chapter 2. The abundance of CO+ is then obtained using the same approach. Synthetic maps are calculated using a raytracer and convolved to the appropriate telescope beam for comparison with single-dish observations. To compare model results to interferometric observations, the synthetic maps are converted into visibility amplitudes and reduced in the same way as interferometric observations.

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3. Multidimensional chemical modeling, Irradiated outflow walls

Figure 3.2: Flowchart of the modeling process for the axisymmetric model of AFGL 2591.

3.3.1

Density structure

One goal of this chapter is to study the difference in the modeled CO+ line strength using a spherically symmetric one-dimensional and an axisymmetric two-dimensional model. For the two-dimensional model we thus assume a density distribution following the power law of the spherically symmetric one-dimensional model, except for an outflow-region (Fig. 3.3a). We implement n(H2 ) = n0 (r0 /r) with n0 = 5.8 × 104 cm−3 and r0 = 2.7 × 104 AU, as proposed by van der Tak et al. (1999). For the separation between the outflow region and the envelope, we use the function z=



1 10 000 tan2 (α/2)



× r2 ,

(3.1)

where z and r denote coordinates along the outflow axis and perpendicular to the outflow (in units of AU). The full opening angle α at z = 10 000 AU is assumed to be ≈ 62◦ , approximately in agreement with the high resolution mid-IR observations by Preibisch et al. (2003). The full opening angle of the outflow cone at z = 30 000 AU is about 40 degrees. The choice of a parabolic outflow cavity is supported by theoretical predictions (Cant´o et al. 2008) and observations (Velusamy & Langer 1998). It is also compatible with the mid-IR observations of AFGL 2591. Vibrationally excited CO at a velocity of −200 km s−1 , emitted in outflow gas has been observed by van der Tak et al. (1999). Assuming an infall velocity in the envelope of order 10 km s−1 , pressure 2 2 equilibrium (ρin vin = ρout vout ) requires a density ratio of 2.5 × 10−3 between the two regions. In the outflow region, the density of the power law distribution is thus assumed to be reduced by this ratio. As in the spherical models of Doty et al. and St¨auber et al., we do not model the innermost ≈250 AU. We make use of the symmetry and only model positive values of r and z. In this quadrant, the model consists of 300 cells along the outflow (z-axis) and 450 cells parallel to the midplane (r-axis) thus a total of 135 000 cells. The resolution is thus about 100 AU × 67 AU.

3.3. An axisymmetric model of AFGL 2591 3 · 104

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a.)

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0 0

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100 r [AU]

2 · 104

3 · 104

Averaged τ [AU V ]

3 · 104

69

0

0 0

1 · 104

r [AU]

2 · 104

3 · 104

Figure 3.3: Physical structure of the (axisymmetric) two-dimensional model of AFGL 2591: a.) density structure (Sect. 3.3.1). The border between the outflow and infall-region is indicated by the dotted line. b.) Attenuated FUV flux calculated by the FUV radiative transfer code (Sect. 3.3.2). The gray/red arrow indicates the width of the outflow wall until G0 is attenuated to 1 ISRF. c.) Temperature structure (Sect. 3.3.3) d.) Averaged value of hτ i in units of AUV (Sect. 3.3.2).

3.3.2

FUV and X-ray radiative transfer

High-energy radiation with photon energy in the range of 6 - 13.6 eV (FUV radiation) and 0.1 100 keV (X-rays) penetrates molecular gas, ionizes molecules and heats the gas. The local FUV and X-ray field at every position are thus required to calculate the temperature structure and the chemical abundances. For the local X-ray intensity, we take into account the distance r and the total hydrogen column density N(Htot ) [cm−2 ] between the X-ray source and the local position. For an X-ray luminosity LX [erg s−1 ], the local intensity is thus proportional to LX /4πr 2 × exp (−σphoto (E) · N(Htot )), where σphoto (E) [cm2 ] is the photoionization cross-section at a photon energy E [eV]. We follow St¨auber et al. (2005) and assume an X-ray emission by a thermal plasma at a temperature of 7 × 107 K and a protostellar X-ray luminosity of 1032 erg s−1 . In their spherically symmetric models, the attenuating column density in the innermost 250 AU is a free parameter found to be about 3 × 1022 cm−2 . For the two-dimensional model, the same value is adopted in the midplane (z = 0). Since density in the outflow is a factor of 400 lower, the column density between source and outflow walls can be of order

70

3. Multidimensional chemical modeling, Irradiated outflow walls

1020 cm−2 , and photons with energies between 0.1 and 1 keV are not absorbed. Unlike St¨auber et al. (2005), we thus take into account these photons in the two-dimensional model. The central FUV source is assumed to emit a blackbody spectrum with a temperature of 3 × 104 K and a luminosity of 2 × 104 L⊙ (St¨auber et al. 2004). At a distance of 200 AU, the FUV flux in absence of any attenuation corresponds to 2.3×108 times the ISRF. Unlike elastic compton scattering for X-rays (St¨auber et al. 2005), scattering of FUV photons on dust grains can be important due to the similar wavelength of FUV radiation and the dust size. A Monte Carlo simulation in the vein of van Zadelhoff et al. (2003) calculates the local FUV strength (Appendix B). Photon packages starting at the central source are traced through the density model, while scattering and attenuation by dust is taken into account. For simplicity, we do not calculate the FUV spectra explicitly, but deal with the scaling factor G0 in units of the ISRF. To read out chemical abundances from the grid of chemical models, the visual extinction τ [AV ] is needed. We obtain the local extinction from   i hIatt ii , (3.2) hτ i = − log i hIunatt ii i i where Iatt is the local flux for a photon package i. The unattenuated flux Iunatt is only geometrically diluted. Brackets denote averaging over all photon packages passing through a cell. We implement the cross-section on dust given in Draine (2003b) for average Milky Way dust with RV =3.1 and C/H=56 ppm in polycyclic aromatic hydrocarbons (PAHs). This choice however has a minor effect on the model results, as photionization no longer dominates in regions, where a higher value of RV ≈ 5.5 is expected. Since the adopted approach corresponds to the calculation at one FUV wavelength, we use the dust properties at a photon energy of 9.8 eV, in the middle of the 6 - 13.6 eV FUV band. For the conversion N(Htot )/AV = 1.87 × 1021 cm−2 (Bohlin et al. 1978) and an extinction cross-section of 1.29 × 10−21 cm2 per H-atom at 9.8 eV, we obtain a conversion factor between the FUV and visual extinction of AUV /AV = 2.4. For the density profile of Sect. 3.3.1, the calculated FUV flux hIatt i and extinction hτUV i are given in Fig. 3.3b and Fig. 3.3d. The attenuated flux shows the region where FUV radiation influences the chemistry, as this value enters the heating rates and the photodissociation rates of the form k = G0 · C · exp (−γ · τv ), with C and γ fitting constants (e.g., van Dishoeck 1988) and G0 the FUV flux in units of the ISRF. Note that the unattenuated flux hIunatt i has to be used together with the attenuation τv to read out the abundances from the chemical grid.

3.3.3

Temperature calculation

Traditional PDR models (e.g., Tielens & Hollenbach 1985) find the photoelectric effect of FUV photons on small dust grains and PAHs to be the dominant heating process in regions with strong FUV irradiation. In the dense outflow walls of the two-dimensional model, this process can easily heat the gas to temperatures above 250 K. For temperatures between 250 K and several hundred K, the fractional abundance of CO+ is a strong function of temperature (Sect. 3.2). It is thus important to carry out a detailed calculation of the temperature profile in the outflow wall. Input parameters are the gas density n [cm−3 ], the FUV field given by G0 [ISRF] and τ [AV ] and the energy deposition per density by X-ray photons HX [erg s−1 ]. To obtain the gas temperature Tgas , the thermal balance has to be solved. We assume steady state conditions. Thus, the equilibrium between the total heating (Γtot ) and cooling (Λtot ) rate yields the temperature. Different physical processes contribute to the individual rates Γi and Λi , where i denotes processes. The inner energy per parcel of gas ǫ [erg cm−3 ] is given by

3.3. An axisymmetric model of AFGL 2591

71

δǫ = Γtot − Λtot δt X = Γi (n, G0 , τ, HX , xH2 , . . . , N(CO), . . . , Tgas , Tdust ) i

−

X i

Λi (n, G0 , τ, HX , xH2 , . . . , N(CO), . . . , Tgas , Tdust ) ≡ 0 .

Important examples for heating processes are H2 collisional de-excitation following pumping through FUV photons and the photoelectric effect. The cooling rate consists of gas-grain collisional cooling and atomic/molecular line emission (e.g., the [CII] line at 158 µm). An overview of the implemented cooling and heating processes is given in Appendix C. Heating and cooling rates depend on the composition of the gas. For example the C+ atomic line cooling depends on the fractional abundance xC+ of ionized carbon. To obtain a self-consistent solution, the composition of the gas is read out from the grid of chemical models. Because the temperature also enters as a parameter for the interpolation of the fractional abundances, the equation for the thermal balance has to be solved iteratively. Molecular or atomic line emission contributes to the cooling only if the photons can escape. A radiative transfer calculation is thus required to obtain proper cooling rates. For the atomic fine structure lines, we implement an escape probability code. As a simplification, the radiative transfer is not carried out in full two-dimensions, but from the modeled point to the closest point of the outflow wall (Fig. 3.4). The probability β for a photon to escape from the outflow wall to the cavity is thus obtained from the molecular/atomic column density (e.g., N(CO) for CO) along this line. z outflow cavity outflow wall

N(H2 ), N(CO), . . . →β

inflow region r

Figure 3.4: Schematic overview on the two-dimensional model. In the outflow wall, the simplified radiative transfer for heating and cooling is calculated between between the local position and the closest point of the outflow region. The column density (e.g., N(CO) or N(H2 )) along this line is used to calculate the escape probability β of a photon.

A fully self-consistent temperature model requires a two-dimensional radiative transfer calculation of the dust continuum radiation to obtain the dust temperature Tdust . As a simplification, we use the dust temperature of the spherically symmetric model of Doty et al. (2002) for the region where the FUV radiative transfer calculation predicts an optical depth larger than 10. In regions with τV < 10, the dust temperature is obtained from the analytical expression given by Hollenbach et al. (1991). Using this approximation, the temperature structure remains consistent with the spherically symmetric model for temperatures below 100 K. In the particular case of CO+ , the relevant temperature range is T > 250 K, and the interpolation to the spherically symmetric temperature profile is thus unimportant.

72

3.4 3.4.1

3. Multidimensional chemical modeling, Irradiated outflow walls

Results Directly irradiated outflow walls

The adopted separation between envelope and outflow (z ∝ r 2 ) yields a “flared” geometry that allows direct irradiation of the dense outflow walls by the central FUV source. This results in a much larger FUV irradiated volume compared to a “linear” separation (z ∝ r) where FUV photons can only penetrate the outflow walls if they are scattered on dust in the outflow (Fig. 3.5). Scattering is however relatively inefficient. Direct irradition

z

z

outflow

Only scattered radiation

outflow Scattering on dust

outflow wall

FU

V

rad

iat

ion

outflow wall

envelope

envelope r

r

Figure 3.5: Schematic view of the axisymmetric model with conical and concave outflow cavity.

What is the size scale of the outflow walls? We define the width of the outflow wall by the region with G0 > 1 along a cut of constant z. For example at z = 15 000 AU, this corresponds to a thickness of about 1 130 AU. The outflow cavity thus allows long-range streaming of FUV radiation to large distances from the central source. Photon packages do not necessarily follow cuts of constant z. The straight ray path from the FUV source to the far end of the outflow wall at z = 15 000 AU propagates for 9 000 AU in the outflow wall. Scattering thus extends the outflow wall. For comparison, the Monte Carlo code was re-run with scattering switched off. This run yielded a width of the outflow wall of only about 470 AU at z = 15 000, about 2.5 times smaller than the width calculated including scattering effects. This demonstrates the importance of scattering for the strengths of the local FUV field. The width of the outflow wall with G0 > 1, 10 or 100 ISRF depending on z is given in Fig. 3.6 along with the error due to the finite resolution of the model. An FUV-enhanced layer along the outflow with a width of a few hundred AU is thus produced by the protostellar FUV radiation. While the width defined by G0 = 1 increases with z, the layer with G0 > 100 has an approximately constant width of 280 ± 30 AU. This is a consequence of the adopted geometry: for the G0 > 100 layer, scattering is less important than for the G0 = 1 region. Unscattered radiation at the surface layer of the outflow wall travels a longer way in the high-density region for higher values of z due to the larger impact angle. This effect compensates with the flux at the surface of the outflow wall ∝ 1/d2, where d is the distance from the FUV source to the surface of the outflow wall, and the attenuation decreasing with 1/d for the implemented density profile.

3.4. Results

73

1500 Width of Outflow Wall [AU]

T = 100 K 1250

G0 = 1 ISRF

1000 750

G0 = 10 ISRF

500 250

G0 = 100 ISRF 0 0

5 · 103

1 · 104

1.5 · 104

z-Axis [AU]

2 · 104

2.5 · 104

Figure 3.6: Width of the outflow wall depending on the distance along the outflow axis. The figure shows the width defined by the region with temperature > 100 K (line) and with FUV flux larger than 1, 10 or 100 times the ISRF (error bars).

Only a small fraction of the envelope in a thin layer on the outflow wall is heated by FUV photons to temperatures above 100 K. The temperature along cuts of fixed values of z is shown in Fig. 3.7. For z < 5000 AU, the temperature at the surface exceeds 1000 K, consistent with the surface temperatures obtained from PDR models for similar conditions (e.g., Kaufman et al. 1999). The width of the region with T > 100 K in the outflow wall is approximately constant with z (Fig. 3.6) as the G0 = 100 layer. In the region with T ≈ 100 K, the heating is dominated by the photoelectric effect ΓP E ∝ n · G0 , with n being the density. The main coolant is atomic oxygen which is approximately thermalized and the cooling rate is thus ΛO ∝ n. As found in the calculation, the gas temperature is thus approximately independent of density and thus follows G0 . An overview of the mass and volume of different regions of the axisymmetric two-dimensional model compared to the spherical one-dimensional model is given in Table 3.1. Indeed, the volume heated to T > 100 K is more than two orders of magnitude larger in the two-dimensional model, while the mass of the same region is only ≈ 8 times larger due to the adopted density gradient. Similarly, the volume irradiated by G0 > 1 ISRF is almost 4 orders of magnitude larger in the twodimensional model, while the mass of the gas is about 2 orders of magnitude larger. We note that the size of the region with G0 > 1 ISRF in the spherical one-dimensional model heavily depends on FUV attenuating column density in the innermost 250 AU. For Table 3.1, the extreme case of no attenuation between the protostar and a radius of 250 AU was assumed. Physical parameters such as the cavity shape or the density ratio between infalling envelope and outflowing gas enter the modeling. A complete study is beyond the scope of this study and will be presented in chapter 6. Here we discuss some general trends. A large surface directly irradiated by protostellar FUV radiation is required for a significant volume and mass of the dense and FUV-heated material in the outflow wall. By changing the outflow angle α by ±20◦ or the separation between

74

3. Multidimensional chemical modeling, Irradiated outflow walls

outflow region and envelope to z ∝ r α with α = 1.5 − 2.5, the volume and mass with T > 100 K vary only by a factor of about 2. On the other hand, a higher density in the outflow region affects the model much more due to the higher extinction of FUV radiation. Increasing the density ratio nout /nin from 2.5 × 10−3 to 10−2 reduces the mass of the T > 100 K region by a factor of 4, while the volume decreases by about an order of magnitude. These results for the relative insensitivity with respect to angle and shape, and relative sensitivity to density, underline one of the key results of this chapter – namely the importance of direct irradiation on the chemical composition of outflow cavity walls. Table 3.1: Volume and mass of different regions of the axisymmetric model compared to the spherically symmetric model. Region

Mass [M⊙ ] Axisymmetric two-dimensional model: Outflow 0.014 Envelope 44.7 Envelope (T > 100 K) 0.53 Envelope (G0 > 1) 1.65 Envelope (G0 > 1 and T < 100 K) 1.1 Spherical one-dimensional model: Envelope 50.3 Envelope (T > 100 K) 0.07 Envelope (G0 > 1) 0.01

Volume [AU3 ] 1.1 × 1013 1.0 × 1014 8.2 × 1011 3.0 × 1012 2.2 × 1012 1.1 × 1014 6.5 × 109 4.8 × 108

The calculation of the FUV flux and gas temperature in the axisymmetric two-dimensional model suggests a region with temperature T < 100 K but FUV irradiation G0 > 1 of potentially FUV irradiated ices. While a detailed discussion of these ices is beyond the scope of this chapter, it is interesting to note two things. First, such a region exists due to the direct irradiation allowed by the concave outflow walls. Second, the potential mass in such a region is 1.1M⊙ . Together, these facts suggest that further exploration of irradiated ices is warranted.

3.4.2

CO+ abundance

Given the physical parameters for an axisymmetric two-dimensional model of AFGL 2591 defined previously, the grid of chemical models (chapter 2) can now be used to quickly read out the abundances of CO+ . Since no outflow features have been observed in lines of CO+ by St¨auber et al. (2007), we assume there is no CO+ in the outflow cavity. Also, the discussion in Sect. 3.2 indicates a very low fractional abundance of CO+ in the outflow. The fractional abundance of CO+ is given in Fig. 3.7 along with the temperature and the density for cuts parallel to the midplane of the model. In a thin layer at the outflow wall, CO+ is enhanced. This layer matches with the FUV-heated gas at temperatures above 250 K. The CO+ layer in the midplane (z = 0) is considerably larger due to the scattering of FUV radiation from both outflows along the positive and negative z-axis. At the surface of the outflow wall, the fractional abundance of CO+ exceeds 10−10 for values of z up to a few thousand AU. At z = 25 000, the fractional abundance of CO+ is still a few times 10−11 .

3.4. Results

75

10−9

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Figure 3.7: CO+ in the two-dimensional axisymmetric model of AFGL 2591. Slices at different values of z are shown. The x-axis corresponds to the distance along the cut to the outflow region. The plots give the fractional abundance x(CO+ ) = n(CO+ )/n(H2 ), the temperature, and the density in solid, dashed and dotted lines, respectively. The figure in the top left shows a spherical model of the same source. The gray rectangle indicates the innermost 250 AU where no modeling has been performed. Since the chemical timescales of CO+ in regions with abundance larger than a few times 10−11 are shorter than 100 yr, the evolutionary age of the source does only affect the modeling results if the FUV production of the protostar has not set in yet or the outflow cavity is not large enough yet. While direct evidence for protostellar FUV radiation of the source cannot be given, the mid-IR observations of the source by Preibisch et al. (2003) suggest an outflow cavity with size of at least 10 000 AU along the outflow axis. For the measured flow velocity of a few hundred km s−1 (van der Tak et al. 1999), the expansion to 10 000 AU would require < 500 yr much less than the inferred age

76

3. Multidimensional chemical modeling, Irradiated outflow walls

since YSO formation estimated to be a few times 105 yr (Doty et al. 2002, 2006). For comparison, results of the spherical one-dimensional model are given in the top left of Fig. 3.7. In this model, FUV radiation of the central source cannot escape the innermost part and the abundance of CO+ is thus very small at fractional abundances below 10−13 . Outflow walls in the two-dimensional model increase the surface irradiated by FUV and thus enhance the total amount of CO+ . Mixing between warm and ionized outflow material with the envelope can also induce a particular chemistry in a thin layer along the outflow. Charge-exchange reactions in this mixing layer enhance the abundance of CO+ . The width of the mixing layer is however small due to the high electron abundance leading to short recombination timescales. While a detailed study (Taylor & Raga 1995, Lim et al. 1999b or Nguyen et al. 2000) is beyond the scope of this chapter, a simple model predicts the amount of CO+ produced in this mixing layer to be enhanced, however insufficiently to explain observations.

CO+ production in a mixing layer As a toy-model for the mixing layer between the warm and ionized outflow and the envelope, we assume the outflow material to be in the form of electrons (e− ) and ionized hydrogen (H+ ). The chemical evolution of a parcel of gas consisting of a mixture of outflow and inflow material is modeled similarly to Sect. 3.2. Free parameters are the temperature and the initial abundances of e− and H+ . The peak fractional abundance of the temporal evolution of CO+ is given in Fig. 3.8 together with the temporal evolution of the fractional abundance for selected temperatures and initial ionizations. 100

a.) Peak fractional abundance of CO+

b.) Temporal evolution of CO+ A

10−2

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A T=885 K, xe =1.4(-1)

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1(-9)

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−12

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B n(H2 )=1(4) cm−3 B n(H2 )=1(3) cm−3 10−11

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10

n(CO+ )/n(H2 )

Ionization Fraction

−10

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Figure 3.8: a.) Peak fractional abundance of the temporal evolution of CO+ for different gas temperatures and initial ionization fractions is displayed by marked isocontours and grayscale. The density was chosen to be 106 cm−3 . b.) Temporal evolution of the fractional abundance at points A, B, C, D (black lines). For a temperature of 82 K and an ionization of 10−3 (point B), densities of 103 cm−3 , 104 cm−3 , and 105 cm−3 are given in red lines. For temperatures above 60 K, formaldehyde (H2 CO) is evaporated from ice mantles. The reaction H2 CO + H+ → CO+ + H2 + H forms CO+ and it is destroyed by the reaction with H2 . For very high initial ionization fractions (> 10−2 ), the dissociative recombination CO+ + e− → C + O with rate √ coefficient k ∝ 1/ T becomes important and the peak fractional abundance is weakly temperature dependent. Due to the short recombination timescale of e− with H+ and grains, the CO+ abundance peaks at young chemical ages and then decreases quickly.

3.4. Results

77

The column density along the mixing layer is N(CO+ ) ≈ L · n · xCO+ , with the width of the mixing layer L, the gas density n and the peak fractional abundance xCO+ . We set L = vA · t, with the Alfv´en velocity vA ≤ 10 km s−1 and the time t, defined by the point of evolution where the abundance of CO+ has dropped by 1 order of magnitude compared to the peak. The peak fractional abundance xCO+ and the product t · n are approximately independent of density as Fig. 3.8b shows. On the other hand, t · xCO+ is roughly independent of temperature and peaks at an electron fraction of about 10−3, where t ≈ 5 × 105 s and xCO+ ≈ 2 × 10−10 for a density of 106 cm−3 . We obtain for the upper limit of the column density due to mixing N(CO+ ) ≈ vA · t · n · xCO+ ≈ 108 cm−2 . In the FUV model of Sect. 3.3, the column density of CO+ along constant z is of order 1010 cm−2 . We conclude that mixing alone cannot reproduce the observed amount of CO+ .

3.4.3

Comparison with JCMT observations

Two rotational lines of CO+ with ∆v ≈ 4 km s−1 have been detected by St¨auber et al. (2007) at the center position of AFGL 2591 using the JCMT. We follow their arguments and assume the lines to be detected. The observed line fluxes are given in Table 3.2 along with molecular data of CO+ taken from the JPL database (Pickett et al. 1998). The Einstein-A coefficients are scaled to a dipole moment of 2.77 D (Rosmus & Werner 1982). Table 3.2: Molecular data of CO+ and integrated line flux at the center position of AFGL 2591 observed by the JCMT (St¨auber et al. 2007). Transition 3 25 3 27

-

2 32 2 52

Frequency [GHz] 353.741 354.014

Aul −3 −1

[10 s ] 1.58 1.7

Eup gu [K] 33.94 6 33.99 8

Line Flux [K km s−1 ] 0.5 0.27

Synthetic spectra are calculated to compare the abundance modeling results to these observations. The excitation of CO+ is needed to model the line flux. However, the mechanism is unclear since CO+ is more likely to be destroyed than excited in collision with H2 . However, electrons and atomic hydrogen are abundant at the surface of PDRs and can excite CO+ as discussed in Andersson et al. (2008). Possibly, CO+ is formed in an excited state. In this chapter, we follow St¨auber et al. (2007) and assume a fixed excitation temperature Tex . The effects of collisional excitation by electrons and atomic hydrogen and excited formation are being studied by St¨auber & Bruderer (2009). For the transitions discussed in this section and assuming an excited formation, they find a flux increase by a factor of only ≈ 3 compared to the peak local thermal equilibrium (LTE) flux at an excitation temperature of ≈ 30 K. To constrain rotational temperatures, transitions for different upper levels are necessary. However, only tentative detections of CO+ in YSOs for Jup ≥ 14 are reported (Ceccarelli et al. 1998). We will thus follow Black (1998) and assume Tex =30 K, slightly higher than St¨auber et al. (2007) (20 K) or Fuente et al. (2006) (10 K). For an upper energy level Eu , the level population is proportional to e−Eu /kTex /Q(Tex ), with Q(T ) being the partition function. Using Tex = 30 K, the population of the J = 3 levels is approximately maximized. The emission of optically thin lines scales with the population of the upper level. Compared to Tex = 30 K, the line flux is thus about about 20% weaker for Tex = 20 K and a factor of 3.5 for Tex = 10 K or Tex = 300 K. An excitation temperature of a few hundred Kelvin is expected for collisional excitation by electrons and atomic hydrogen (Andersson et al. 2008). We conclude that a different excitation temperature, within the

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3. Multidimensional chemical modeling, Irradiated outflow walls

expected range or assuming an excited formation of CO+ , changes the modeled line flux less than an order of magnitude. Synthetic line fluxes are obtained from the solution of the radiative transfer equation. The two lines considered in this chapter have a very low optical depth with τ < 0.08 for Tex =30 K at an assumed line width and molecular column density of 4 km s−1 and 9 × 1012 cm−2 , respectively. This column density corresponds to an emitting region with a diameter of 60 000 AU at a density of 105 cm−3 with a CO+ fractional abundance of 10−10 . For our application, we can thus neglect self-absorption and the integrated flux is obtained from the velocity integrated radiative transfer equation Z Z hc3 gu Aul e−Eu /kTex n(s) ds , (3.3) Tmb dv = 8πk ν 2 Q(Tex ) LOS

with ν being the line frequency, gu the statistical weight of the upper level, Aul the Einstein-A coefficient [s−1 ] and n(s) the CO+ density [cm−3 ]. The integral is taken along the line of sight (LOS). To have comparable results to the older literature, we assume a distance of 1 kpc to AFGL 2591, following Benz et al. (2007) and van der Tak et al. (1999). A larger distance of 1.7 kpc as suggested by Schneider et al. (2006) would however not change the main conclusions of this chapter. The line flux of the spherical one-dimensional model calculated using this method agrees within a few percent with results from SKY of the RATRAN radiative transfer code (Hogerheijde & van der Tak 2000). Maps of the integrated line flux from the two-dimensional model at different inclination angles are shown in Fig. 3.9. The angular resolution of the images is 0.05′′ . We give only results for the 3 27 - 2 52 transition at 354.014 GHz in this section. The modeled flux of the other fine structure line can be obtained by scaling with 0.7, the ratio of the Einstein-A coefficients and the statistical weights. The modeled maps are convolved with a 14′′ FWHM Gaussian to account for the angular resolution of the JCMT. The effect of different inclination angles is clearly visible in unconvolved maps, but the convolved maps are almost independent of inclination angle.

0◦

10◦

20◦

10

6 4 1

R

∆δ [”]

2 0 -2 -4

0.1

-6 -8 0.01 8

◦

◦

Integrated Intensity

8

79

Tmb dv [K km s−1 ]

3.4. Results

◦

30

40

50

60◦

70◦

80◦

90◦

30◦ (14” Beam)

90◦ (14” Beam)

6 4 ∆δ [”]

2 0 -2 -4 -6 -8

8 6 4 ∆δ [”]

2 0 -2 -4 -6 -8

8 6 4 ∆δ [”]

2 0 -2 -4 -6 -8 -8

-6

-4

-2

0 ∆α [”]

2

4

6

8

-8

-6

-4

-2

0 ∆α [”]

2

4

6

8

-8

-6

-4

-2

0

2

4

6

8

∆α [”]

Figure 3.9: Synthetic maps of the velocity integrated CO+ line at 354.014 GHz (3 27 − 2 52 ) for different inclination angles of the source. The inclination angle is shown with the gray/red arrow onto a symbolic density map in the bottom left corner of each map. Two plots in the lower right corner give maps convolved with a 14′′ JCMT beam.

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The predicted line fluxes for the JCMT at the center position are given in Table 3.3. They only vary between 0.16 and 0.14 K km s−1 for an inclination of 0◦ and 90◦ , respectively. This reflects a slightly larger fraction of the outflow walls in the telescope beam for low inclination angles. The difference is however small since most emission is from regions close to the center. Indeed, about between 15% and 30% of the modeled JCMT flux stems from within a radius of 1′′ from the center, corresponding to 1 000 AU at the adopted distance. An inclination angle of 30◦ was suggested by van der Tak et al. (1999) and is thus displayed in this and the following section. The line flux of the CO+ line at 354.014 GHz as obtained from the spherically symmetric onedimensional model is with 1.1 × 10−4 K km s−1 about 3 orders of magnitude weaker than observed. On the other hand, the results of the axisymmetric two-dimensional model agree to within a factor of 2 which we consider as a good agreement given all uncertainties entering the modeling. The 3 25 2 23 line at 353.741 GHz however disagrees by about a factor of 5, as the observed line ratio of 1.85 (=[353.741 GHz] / [354.014 GHz]) is larger than the ratio of 0.7 as predicted by the models. Possible explanations are a different excitation mechanism of the fine structure levels or line blending of the 353.741 GHz line, e.g. by the 33 SO2 (JKp Ko = 194,16 → 193,17 ) line at 353.741 GHz (St¨auber et al. 2007). To explain the line ratio by a different excitation mechanism, a grossly different excitation temperature would however be required, as discussed earlier in this section. Table 3.3: Modeled and observed maximum velocity integrated fluxes of the CO+ 3 27 − 2 25 line at 354.014 GHz. (a extended configuration, b compact configuration, c Not detected: 3σ upper limit,d Detected) Inclination SMA (Ext.a ) SMA (Both.b ) ◦ −1 −1 [ ] [Jy Beam km s ] [Jy Beam−1 km s−1 ] Axisymmetric two-dimensional model: 0 0.05 0.23 10 0.06 0.25 20 0.05 0.26 30 0.06 0.30 40 0.06 0.38 50 0.11 0.44 60 0.16 0.47 70 0.17 0.45 80 0.16 0.44 90 0.16 0.43 Spherical one-dimensional model: < 0.01 < 0.01 Observed: < 2.3c < 0.91c

JCMT [K km s−1 ] 0.16 0.16 0.16 0.15 0.15 0.15 0.14 0.14 0.14 0.14 1.1 · 10−4 0.27d

3.4. Results

3.4.4

81

Comparison with SMA observations

Interferometric observations of the two CO+ lines in AFGL 2591 have been carried out using the Submillimeter Array (SMA)1 . Benz et al. (2007) have used the extended configuration of the array with projected baselines covering the range between 38 and 214 kλ (32.2 m - 181.2 m). New observations have been carried out in the compact configuration on 2006 April 14. These data cover a projected baseline range of 10.6 - 82 kλ (9 m - 69.4 m). The frequency setting is the same as for the extended array observation, covering the range between 342.6-344.6 GHz in the LSB and 352.6-354.6 GHz in the USB. Data of this observation is discussed in chapter 4. The rms noise per velocity bin of 0.5 km s−1 is 0.46 Jy Beam−1 (extended array), 0.21 Jy Beam−1 (compact array) and 0.18 Jy Beam−1 when data of both arrays are combined using natural weighting. The better weather conditions and shorter baselines result in a higher sensitivity of the compact array √ observations. The detection limit for the integrated line can be estimated from 1σ = 1.2 δV · δv Trms , where Trms is the rms noise per frequency bin, δV ≈ 4 km s−1 is the expected linewidth, and δv = 0.5 km s−1 is the velocity resolution (Maret et al. 2004). The factor 1.2 accounts for the uncertainty of the absolute calibration of about 20%. For a 3σ detection, an integrated flux of 2.3 Jy km s−1 (extended array) and 0.91 Jy Beam−1 km s−1 (compact and extended arrays combined) is thus necessary. The CO+ line at 354.014 GHz has not been detected at a 3σ level in either configurations of the SMA. The 353.741 GHz line is not detected in the extended configuration observations. In the data of the combined array however, a line is found at approximately 353.741 GHz. Is the CO+ line blended by 33 SO2 (JKp Ko = 194,16 → 193,17 ) at the same frequency? Several other lines of 33 SO2 are within the observed frequency setting, but are not detected despite their presumably higher intensity due to a larger Einstein-A coefficient, lower critical density, and upper level energy. CO+ is thus more likely blended by an unidentified line. The blended line has a velocity integrated line flux of 4.9 Jy Beam−1 km s−1 corresponding to a 25σ detection. For the predicted line ratio ([353.741 GHz] / [354.014 GHz]) of 0.7 and the ratio of 1.85 as observed by the JCMT, the CO+ line at 354.014 GHz should have been detected. We thus conclude CO+ is not detected by the SMA observations. Is this non-detection consistent with the modeled line flux of the two-dimensional model? To answer that question, synthetic maps (Fig. 3.9) are converted to visibility amplitudes for the (u,v)coverage of the SMA observation using the MIRIAD2 task uvmodel. The simulated visibilities are then reduced in the same way as SMA observations: they are inverted and the clean algorithm is applied on the resulting maps. Simulated velocity integrated maps for inclination angles of 30◦ and 90◦ are presented in Fig. 3.10. The maximum flux of the simulated maps for the extended array is 0.17 Jy Beam−1 km s−1 at an inclination of 70◦ . For the (u,v)-coverage of the combined array, the peak flux is 0.47 Jy Beam−1 km s−1 at an inclination angle of 60◦ . These integrated fluxes correspond to less than 1σ and the non-detection is thus consistent with the models. As the simulated SMA maps in Fig. 3.10 show, this non-detection is explained by the limited sensitivity of our observations rather than by over-resolving effects of the interferometer.

1

The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. 2 http://sma-www.cfa.harvard.edu/miriadWWW/manuals/ manuals.html

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Figure 3.10: Simulated SMA observations [Jy Beam−1 km s−1 ] (gray/red contour lines) overlaid on modeled velocity integrated maps [K km s−1 ] (black/white map). Negative fluxes are shown by dashed contour lines. The half-power beam is given in the bottom left corner. For comparison, a 3σ detection with the SMA requires a flux of 2.3 (only extended array) / 0.91 (compact+extended array) Jy Beam−1 km s−1 .

3.5. Conclusions and outlook

3.5

83

Conclusions and outlook

We have used the grid of chemical models introduced in chapter 2 to construct a detailed twodimensional model of the high-mass star-forming region AFGL 2591. The spherically symmetric one-dimensional models by Doty et al. (2002) and St¨auber et al. (2004, 2005) have been extended with a low-density outflow region allowing protostellar FUV radiation to escape from the innermost region. In the two-dimensional model, FUV irradiates a larger surface and thus a larger volume compared to spherically symmetric models. The model is used to simulate the line fluxes of CO+ as a prototypical FUV tracer. The main conclusions of this chapter are: 1. The existence of concave outflow cavity walls allows the efficient, long-range streaming of FUV radiation to large distances from the central source (Sect. 3.4.1). 2. A thin FUV-enhanced layer in the outflow is produced, having a thickness of a few hundreds of AU, and G0 > 100 ISRF (Sect. 3.4.1). 3. A large mass of FUV irradiated material – more than a solar mass – can reside in the outflow walls. This combined with the large extent can lead to a situation where the molecular emission from the outflow walls dominates the total line flux in single-dish observations (Sections 3.4.1 and 3.4.3) 4. Detailed two-dimensional modeling of CO+ including these outflow walls is consistent with single-dish (Sect. 3.4.3) and interferometric observations (Sect. 3.4.4), while the flux in a onedimensional model is orders of magnitude too low. This indicates the need for multidimensional chemical models for the interpretation of FUV sensitive molecules. 5. The strong gradients in density and FUV flux require an accurate calculation of the gas temperature structure and inclusion of scattering of FUV radiation in the outflow walls, which can extend the width of the FUV enhanced region significantly (Sect. 3.4.1). 6. A region of high FUV (G0 > 1 ISRF) and low temperature (T < 100 K) leads to the strong possibility of ice mantle processing by FUV photons in the outflow walls (Sect. 3.4.1). 7. The abundance of CO+ is also enhanced in a mixing layer between the ionized and warm outflow and the envelope. Mixing alone can however not explain the observed amount of CO+ (Sect. 3.4.2). This chapter shows the application of the grid of chemical models (chapter 2) for the construction of a multidimensional chemical model of a YSO envelope. This interpolation method simplifies the construction of two or three dimensional chemical models considerably and the gain in speed allows the self-consistent calculation of the temperature structure with the chemical abundances (Sect. 3.3.3). This chapter shows that such detailed modeling including the influence of high-energy irradiation and the geometrical shape of the envelope can be necessary even to explain spatially unresolved single-dish observations. Some hydrides (e.g., SH+ or NH+ ) will be observable for the first time using the upcoming HIFI spectrometer onboard Herschel. They are expected to be chemical tracers of FUV/X-ray radiation (St¨auber et al. 2004, 2005, 2007) and further study will be carried out to investigate the influence of the geometry on these species for the interpretation of the Herschel data. Direct imaging of the outflow region will be available with the Atacama Large Millimeter/submillimeter Array (ALMA). These high spatial resolution observations will deliver additional constraints on the geometry.

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Acknowledgments We thank Kaspar Arzner, Pascal St¨auber, and Jes Jørgensen for useful discussions, and the anonymous referee for his/her valuable comments. Michiel Hogerheijde and Floris van der Tak are acknowledged for use of their RATRAN code. This chapter was partially supported under grants from The Research Corporation and the NASA grant NNX08AH28G (SDD). Astrochemistry in Leiden is supported by the Netherlands Research School for Astronomy (NOVA) and by a Spinoza grant from the Netherlands Organization for Scientific Research (NWO). The submillimeter work at ETH is supported by the Swiss National Science Foundation grant 200020-113556.

Chapter 4

Evidence of warm and dense material along the outflow of a high-mass YSO

The spherical cow is dead, long live the spherical cow! Steven Doty

Abstract: Outflow cavities in envelopes of young stellar objects (YSOs) have been predicted to allow far-UV (FUV) photons to escape far from the central source, with significant observable effects, especially if the protostar is a forming high-mass star suspected of emitting a copious amount of FUV radiation. Indirect evidence of this picture has been provided by models and unresolved single-dish observations, but direct high-resolution data are necessary for confirmation. Previous chemical modeling has suggested that CS and HCN are good probes of the local FUV field, so make good target species. In this chapter, we directly probe the physical conditions of the material in the outflow walls to test this prediction. Interferometric observations of the CS(7-6) and HCN(4-3) rotational lines in the high-mass star-forming region AFGL 2591 are carried out in the compact and extended configuration of the SubMillimeter Array (SMA). The velocity structure was analyzed, and integrated maps compared to K-band near-IR observations. A chemical model predicts abundances of CS and HCN for a gas under protostellar X-ray and FUV irradiation, and was used in conjunction with the data to distinguish between physical scenarios. We find CS and HCN emission in spatial coincidence in extended sources displaced up to 7′′ from the position of the young star. Their line widths are small, excluding major shocks. Chemical

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model calculations predict an enhanced abundance of the two molecules in warm, dense, and FUV irradiated gas. Hot dust observed between the molecular emission and the outflow accounts for the necessary attenuation to prevent photodissociation of the molecules. The SMA data suggest that the outflow walls are heated and chemically altered by the FUV emission of the central high-mass object, providing the best direct evidence yet of large-scale direct irradiation of outflow walls.

Simon Bruderer, Arnold O. Benz, Steven D. Doty & Tyler L. Bourke Astronomy and Astrophysics Letters 503, 13 (2009)

4.1

Introduction

Powerful outflows and jets driven by young embedded stars can etch large cavities in their natal cloud (e.g. Velusamy & Langer 1998). In addition, the forming star releases energy as radiation. Young O or B stars emit much of their radiation in far-UV (FUV, 6 - 13.6 eV) wavelengths. These highenergy photons can heat and photoionize molecular gas and induce a peculiar chemistry, observed in photon-dominated regions (PDRs, e.g. Hollenbach & Tielens 1997). Due to the large extinction by dust, only the innermost few hundred AU of the envelope are influenced by FUV radiation. Cavities evacuated by the outflow thus allow the FUV radiation to reach longer distances. They may irradiate an increased volume and mass of the envelope along the border of the cavity in this way. Indirect evidence for such a dense and warm outflow wall is found in chapter 3 where single-dish observations of the FUV enhanced molecule CO+ are explained using a detailed 2D chemical model including an outflow cavity. Low-mass YSO interferometric observations by Hogerheijde et al. (1998) reveal warm material along the outflow. Using HIFI/Herschel, previously inaccessible frequency bands can be observed with good angular and spectral resolution. This allows the first study of the chemistry of hydrides (e.g. CH+ or SH+ ), predicted to be excellent tracers of warm and FUV irradiated gas (St¨auber et al. 2004). The WISH Herschel guaranteed time key-program1 will observe hydrides, providing indirect insight on the proposed FUV irradiated outflow walls. A direct confirmation of this physical picture, however, requires spatially resolved, high-resolution observations of FUV tracing molecules. In this chapter, we present new submillimeter interferometric observations of molecules tracing FUV radiation. In combination with near-IR observations and chemical models, we provide the first direct evidence of dense, warm, and FUV irradiated material along the outflow of a high-mass star forming region.

4.2

Observations and data reduction

The SubMillimeter Array (SMA)2 has observed AFGL 2591 in the extended configuration with projected baselines between 32.2 m - 181.2 m (Benz et al. 2007). New observations were carried out in the compact configuration with projected baselines between 9 m - 69.4 m. At the observed frequency and a distance of 1 kpc to the source (van der Tak et al. 1999), the combined array is thus sensitive to spatial structures in a wide range of length scales between about 600 and 12000 AU. 1

http://www.strw.leidenuniv.nl/WISH/ The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. 2

4.3. Results

87

The frequency setting was the same for both observations (342.6-344.6 GHz in the LSB and 352.6-354.6 GHz in the USB). It covers the J = 4 → 3 transition of HCN at 354.506 GHz and the J = 7 → 6 transition of CS at 342.884 GHz. We focus here on secondary sources besides the continuum peak and thus concentrate on the two molecules that show such features. Data was calibrated in the same way as in Benz et al. (2007). The primary frequency resolution corresponds to a velocity resolution of 0.35 km s−1 . For the weighting of the visibility amplitudes, Brigg’s robustness parameter was chosen to be 0. The FWHM size of the reconstructed beam is 1.24′′ × 0.99′′ at a position angle of 43.4 degrees.

4.3

Results

Velocity-integrated and cleaned maps of the SMA observations are presented in Fig. 4.1. The molecular emissions are overlayed on K-band (2.2 µm) observations. A high-resolution bispectrum speckle interferometry image by Preibisch et al. (2003) covers the inner part of the SMA maps at a resolution of 0.17′′ , while an observation obtained in the commissioning phase of NIRI at Gemini North shows the whole SMA field of view at a resolution of ≈ 0.4′′ . The fitted outflow direction and opening angle (Preibisch et al. 2003) are indicated by green lines. Line spectra at selected positions (A, A1 , A2 , and B) are given in Fig. 4.2. The spectra show the mean brightness temperature averaged in an area equivalent to the synthesized beam. The K-band image is saturated at the peak position (A). Results of a Gaussian fit, line fluxes, and coordinates of the spectra are given in Table 4.1. The maps from the combined compact and extended configuration of the SMA show the emission peak at the same position (A) as those from the extended configuration (Benz et al. 2007). The increased sensitivity of the combined array by a factor of ≈ 3 together with a better (u,v)-coverage allows weak emission to be detected on larger scales. A striking feature of this extended emission is the good spatial correlation of the CS(7-6) and HCN(4-3) emission. Also, most secondary sources are concentrated along the border of the western outflow. This outflow is known to point towards us at an inclination angle similar to the opening angle (van der Tak et al. 1999). The line of sight from the observer to the protostar is thus approximately following the border of the outflow. The alignment of the other outflow (in the eastern direction pointing away from us) is unclear since it has not been detected here in molecular emission and in the K-band. Assuming the same axis for both outflows, a slight extension along the northeastern wall of the outflow is found in the HCN(4-3) map. About perpendicular to the outflow axis, position A1 shows molecular emission at projected distances of up to 3400 AU in HCN and 4100 AU in CS. Along the border of the western outflow, the HCN and CS emission at position A2 is about at the same angle relative to the RA-axis as the secondary source found by Benz et al. (2007), but detected here out to a distance four times farther (≈ 4200 AU). A detailed map of position B is given as an inset in Fig. 4.1. It shows a spatial displacement of the molecular emission compared to the K-band image by Preibisch et al. (2003) towards the exterior of the projected outflow.

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Figure 4.1: Velocity-integrated emission of HCN(4-3) (blue contours) and CS(7-6) (red contours) observed by the SMA in AFGL 2591. Contours show isophotes at 5σ, 7σ, etc. Negative values are indicated by dashed lines. The half-power beam of the SMA is given in a corner of the maps. Gray scale shows the K-band at 2.2 µm. The green asterisks give positions discussed in the text, Fig. 4.2 and Table 4.1.

4.3. Results

89

Are the emitting regions associated with the outflow or with the envelope? The velocity VLSR at positions A, A1 , A2 , and B is obtained from a Gaussian fit and the first moment of the spectra (Table 4.1). Both values are consistent, due to the Gaussian shape of the spectra, except for HCN at positions A and A1 . At the continuum peak (A), HCN is skewed to red velocities as found by Benz et al. (2007). Our observations do however not show a self-absorption feature at the systemic velocity of the envelope (Vsys = −5.5 km s−1 ). The missing self-absorption may stem from to the relative contribution of the innermost optically thick region to the beam being greater for the smaller beam of the extended array. The HCN lines are generally broader than the CS lines, as observed by the JCMT, namely ∆V = 3.32 km s−1 (CS) and ∆V = 4.6 km s−1 (HCN) (van der Tak et al. 1999). The ratio between the linewidth of CS and HCN is similar to the JCMT value, except for position A2 , but the HCN spectra at that position tentatively shows a self-absorption signature at Vsys . The linewidths are comparable to the JCMT observations at positions A1 and A2 but are broader at positions A and B. The broader line at the continuum peak (A) is possibly a result of the larger infall velocity on scales of the SMA beam (free-fall velocity Vff = 7.5 km s−1 at 500 AU, Benz et al. 2007) compared to the JCMT beam (Vff = 2.6 km s−1 at 7000 AU). The near-IR observations show a clumpy border of the outflow cavity (Preibisch et al. 2003), and the velocity dispersion between several clumps contributing to the emission may cause the broader line at position B. The velocities at A2 and B are consistent with the systemic velocity, while the center position and A1 are slightly red-shifted with V −Vsys < 2 km s−1 . As Vsys was obtained from JCMT observations, it may possibly not be applicable for the innermost 1000 AU. Typical shock indicators as line asymmetry, shifted velocity compared to Vsys , large linewidths, and location near the apex of the bow-shock are missing in positions A, A1 , and B. Our observations can, however, not exclude shocks because the observed linewidth may be smaller than the shock velocity if the shock propagates at an angle to the line of sight. We conclude that the extended emission is associated with the accreting envelope. The spectrally integrated line fluxes F of the two lines are correlated, with flux ratios (FHCN /FCS ) between 1.57 and 1.77. The total flux of the SMA observations compared to the JCMT is 177 to 435 Jy km s−1 (41 %) for CS and 285 to 873 Jy km s−1 (33 %) for HCN. Is the missing flux of the interferometer consistent with the single-dish observation despite the good (u,v)-coverage of the combined array? A flux density of 1.8 Jy beam−1 km s−1 (2.6σ-detection) for CS and 4.1 Jy beam−1 km s−1 (3.7σ-detection) for HCN would amount by distributing the missing flux equally over the area of the JCMT beam, where no line at a 5σ level was detected by the SMA. The missing flux is below the SMA threshold of 0.7 (CS) and 1.1 (HCN) Jy beam−1 km s−1 (for 1σ) even if no flux was resolved out by the interferometer. The SMA flux within the hot-core region of 1000 AU (Doty et al. 2002, about 1′′ ) from the continuum peak is 78 (CS) and 143 (HCN) Jy km s−1 . The contribution of the extended emission detected by the SMA compared to the JCMT flux is thus about 23 % for CS and 16 % for HCN.

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Figure 4.2: Spectra of HCN(4-3) and CS(7-6) in AFGL 2591 at different positions (Fig. 4.1). A Gaussian fitted to the spectra is given with dashed line. Bars at the bottom of each spectrum indicate the first moment and the velocity of the fitted Gaussian with the range of the error. A vertical line at -5.5 km s−1 indicates the systemic velocity of the envelope obtained from JCMT observations.

c

b

a

HCN(4-3) CS(7-6) HCN(4-3) CS(7-6) HCN(4-3) CS(7-6) HCN(4-3) CS(7-6)

Line

Coordinates α / δ (J2000) 20h 29m 24s. 87 / +40◦ 11’ 19.5” 20h 29m 24s. 90 / +40◦ 11’ 22.0” 20h 29m 24s. 59 / +40◦ 11’ 20.4” 20h 29m 24s. 48 / +40◦ 11’ 22.4”

Gaussian fit R R
1st moment T V dV / T dV integrated spectra

B

A2

A1

A

Position

a b a a VLSR VLSR ∆VFWHM TPeak Fc FHCN /FCS (km s−1 ) (km s−1 ) (km s−1 ) (K) (Jy beam−1 km s−1 ) -4.21±0.04 -4.7±0.1 8.4±0.1 67.7±0.7 67.8±1.1 1.77 -5.14±0.03 -5.15±0.07 5.50±0.07 65.2±0.7 38.3±0.7 -3.6±0.1 -4.5±0.2 3.3±0.4 15.4±1.2 7.8±1.1 1.59 -4.77±0.09 -4.5±0.1 2.8±0.2 16.0±1.0 4.9±0.7 -5.4±0.2 -5.3±0.5 5.8±0.4 11.7±0.8 9.1±1.1 1.57 -5.31±0.08 -5.2±0.1 2.8±0.2 17.2±1.0 5.8±0.7 -3.8±0.4 -4.3±1.1 8.8±0.9 8.1±0.7 7.8±1.1 1.63 -5.0±0.3 -4.6±0.7 6.1±0.6 7.9±0.7 4.8±0.7

Table 4.1: Observed properties of the CS(7-6) and HCN(4-3) lines. The velocity VLSR , the linewidth ∆V , the line peak T , the velocity integrate flux F and the flux ratio FHCN /FCS are given for different positions (Fig. 4.2). The conversion factor K/(Jy beam−1 ) is 7.88 (HCN) and 8.64 (CS).

4.3. Results 91

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4. Evidence of warm and dense material along the outflow of a high-mass YSO

4.4

Interpretation

The spatial coincidence of the CS and HCN emission in all secondary sources is not trivial. For interpretation, we use the chemical model by Doty et al. (2002) and St¨auber et al. (2004, 2005) (see chapter 2 for details). The models calculate the temporal evolution of chemical abundances in a parcel of gas irradiated by protostellar FUV radiation. Figure 4.3 presents the modeled abundances of CS and HCN relative to the H2 density as a function of temperature for different densities and FUV irradiation. The abundances are given for a chemical age of 5 × 104 yrs as suggested by St¨auber et al. (2005). We note however that the main conclusions from this section are not sensitive to the chemical age due to very short chemical time scales for strong FUV irradiation. The FUV luminosity of AFGL 2591 is estimated3 to be 4×1037 erg s−1 assuming Lbol = 2×104 L⊙ and Teff = 3 × 104 K (van der Tak et al. 1999). At a distance of 1000 AU, this yields G0 = 9 × 106 ISRF (interstellar radiation fields, 1.6 × 10−3 erg s−1 cm−2 ) in the absence of any attenuation. The equivalent hydrogen column density for an optical depth of 1 is only about 8 × 1020 cm−2 , so that FUV heating and photodissociation is a “surface”-effect acting only on the edge of a high-density region. The model abundance of CS (Fig. 4.3) shows a distinct step at 100 K due to the evaporation of sulphur from dust grains assumed at this temperature. The initial relative abundance of sulphur in the models is 2 × 10−5 for T ≥ 100 K (in H2 S) and 9.1 × 10−8 for T < 100 K (in S). FUV irradiation destroys CS, except for T > 250 K at high density, when the reaction C + S2 → CS + S competes with photodissociation. Below 250 K, H2 + OH → H2 O + H does not proceed and the photodissociation of water and OH liberates much atomic oxygen (Charnley 1997). Subsequently, O reacts with S2 to SO and sulphur is not available for the formation of CS. The formation of HCN by the reaction H2 + CN → HCN + H is very temperature dependent because of an activation energy of 820 K. Under FUV irradiation, CN is enhanced for temperatures above 250 K by the reaction C + NS → S + CN. At lower temperature, atomic oxygen destroys NS resulting in a lower HCN abundance. Most important for interpreting the SMA data indicating simultaneous enhancements of HCN and CS, the chemical models require a dense, warm gas with temperatures above 250 K and FUV irradiation. Under these conditions, the abundance of SO is reduced to a level consistent with the nondetection (0.8 Jy beam−1 km s−1 for 1σ) of extended emission of the SO(87 -77 ) line in our observations. In photon-dominated regions (PDRs), FUV photons efficiently heat the gas through the photoelectric effect on dust grains. The minimal distance of position B to the central source is 7000 AU. In the absence of any attenuation, the FUV field would have decreased to 2 × 105 ISRF or lower. An FUV flux of ≈ 3 × 102 (n/106 cm−3 ) ISRF is required at a gas density of n ≥ 106 cm−3 to produce a PDR surface temperature of ≈ 1000 K (Kaufman et al. 1999). For example, at a density of 107 cm−3 , above the critical density of the observed CS transition, an FUV flux of about 3 × 103 ISRF would heat the gas to 1000 K, while CS and HCN are not photodissociated. The attenuation from 2 × 105 to 3 × 103 ISRF requires an optical depth of τ = 4.2.

3

Appendix B.1 gives a discussion of the sensitivity of LFUV on Teff .

4.4. Interpretation

93

n(CS)/n(H2)

10−5

CS / n(H2)=104 cm−3

CS / n(H2)=107 cm−3

10−6 10−7

T high n(H2) high

10−8 10−9 10−10 10−11

n(HCN)/n(H2)

10−4

1000 500

200 100

50

20

HCN / n(H2)=104 cm−3

10−5

G0=0 ISRF G0=102 ISRF G0=103 ISRF G0=105 ISRF

10−6 10−7

10 1000 500

200 100

50

20

10

HCN / n(H2)=107 cm−3 T high n(H2) high

10−8 10−9 10−10

1000 500

200 100

50

temperature (K)

20

10 1000 500

200 100

50

20

10

temperature (K)

Figure 4.3: Modeled abundance of HCN (bottom panels) and CS (top panels) relative to H2 as a function of temperature. The models assume a far-UV (FUV) radiation of 0 (black), 102 (red), 103 (green), and 105 ISRF (blue line). Gas densities of 104 and 107 cm−3 are shown. The density gives the initial number of hydrogen molecules per cm3 in the chemical model. Another mechanism for simultaneously increasing the abundance of HCN and CS would be ionization by protostellar X-ray radiation (St¨auber et al. 2005). The cross-section of X-rays is much lower than for FUV radiation, and they can thus influence the chemistry within a few 1000 AU of the source. A temperature > 250 K, required to enhance CS and HCN by X-rays, can however not be reached by X-ray heating at distances of a few 1000 AU (even assuming an unusually high X-ray luminosity of ≈ 1032 erg s−1 , St¨auber et al. 2005). FUV-heated gas with additional X-ray irradiation cannot be ruled out, as X-rays would further enhance the abundance of HCN and CS. However, X-rays are not necessary in this context, as the FUV present in heating the gas also explains the CS and HCN observations. As discussed before, minor shocks cannot be excluded by the observations and could also contribute to the FUV field (van Kempen et al. 2009b).

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4. Evidence of warm and dense material along the outflow of a high-mass YSO

4.5

A scenario

The observed morphology and chemical models suggest the following scenario. Protostellar FUV radiation escapes trough a low-density region of the outflow, observed in the near-IR. The radiation penetrates the high-density material at the border of the outflow, where it heats the gas and induces a PDR-like chemistry in the accreting envelope. For observed distances to the protostar, the FUV field has to be attenuated at the surface of the outflow wall to prevent molecules from being photodissociated. Indeed, position B in Fig. 4.1 shows molecular emission outside the projected outflow. A thin layer of dense dust situated between the molecular emission and the outflow is seen in the K-band and may act as an “FUV shield” for the molecules. For positions A1 and A2 , the geometry is less evident. The emission at A1 within the projected outflow region can stem either from the front or back side of the outflow wall. For the small angle between the line of sight and the outflow as proposed by van der Tak et al. (1999), region A2 can either be associated to the western or eastern outflow walls.

4.6

Conclusions

Submillimeter interferometric observations of the high-mass star-forming region AFGL 2591 in CS(76) and HCN(4-3) were carried out. The high angular resolution reveals a spatial coincidence of extended CS and HCN emission, which is interpreted by the help of a chemical model as warm, dense, and FUV-irradiated gas. In combination with K-band near-IR images, these results provide the first direct evidence of FUV irradiated outflow walls. We conclude: • Protostellar FUV radiation alters the chemical composition in envelopes of a high-mass YSO on large scales up to several 1000 AU, if a low-density outflow cavity allows radiation to escape. • The contribution of the FUV-enhanced HCN and CS emission on distances > 1000 AU to the observed single-dish flux is about 25 %. This chapter shows the power of direct imaging the chemistry as a probe of physical conditions. The high angular resolution of prospective ALMA observations are ideal for studying the structure and geometry of outflow walls. The contribution of the outflow walls can dominate the total flux even in unresolved single-dish observations (e.g. for CO+ , chapter 3) and good knowledge of the geometry will be required for the interpretation of upcoming Herschel/HIFI observations.

Acknowledgments We thank Cecilia Ceccarelli, Michael Meyer, Michael Rissi, and Pascal St¨auber for useful discussions. The chapter was supported by the Swiss National Science Foundation grant 200020-113556 (SB), a grant from The Research Corporation (SDD) and the NASA grant NNX08AH28G (SDD). This chapter makes use of observations obtained at the Gemini Observatory (acquired through the Gemini Science Archive), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership.

Chapter 5

A computer program for fast multidimensional modelling of molecular line emission

Abstract: New facilities such as the Herschel Space Observatory or ALMA will deliver a wealth of molecular line observations with unprecedented angular resolution and in new frequency bands. Molecular emission can be used as tracer for physical and chemical conditions in the emitting material. Since the level populations of molecules are often not in the local thermal equilibrium, radiative transfer calculations have to be carried out for the analysis. Chemical models of star forming regions suggest that the geometry (e.g. outflow regions, disks etc.) plays a crucial role and two or even three dimensional models need to be constructed. Multidimensional radiative transfer is computationally too expensive to calculated numerous models in order to fit observations with synthetic lines. Here, we develop an efficient approximative method to model model molecular line emission from 1D, 2D or 3D physical/chemical models. The method is based on the well-known escape probability formalism, however calculates the excitation in different cells and takes the lineshape explicitly into account. To improve the convergence and thus accelerate the calculation, a technique similar to the accelerated lambda iteration (ALI) approach is used. In several benchmark tests and realistic applications, we test the method against exact radiative transfer solutions. These applications cover the range from simple two-level molecules in a homogeneous cloud to water in the envelope of a young stellar objecte. We find that the line fluxes derived from the new method agrees mostly to within 30 % despite the much simpler structure of the calcula-

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5. A computer program for fast multidimensional modelling of molecular line emission

tion, which accelerates the calculation often by a factor of 100 or more. Thus, the method presented here allows to model molecular line emission of complex multidimensional physical/chemical models quickly. The gain in speed allows to interpret the large amount of new observations in terms of more complex and complete models. Simon Bruderer et al., in preparation

5.1

Introduction

Molecular lines and atomic fine structure emission contains a lot of information on the physics and chemistry of the interstellar medium as for example envelopes of young stellar objects (YSOs) on which we focus in this chapter. The combination of line fluxes, line ratios and line shapes are determined by the temperature and density of collision partners (e.g. H2 or electrons), abundance of a species and the velocity field. Lines thus probe various parameters. For example, Jørgensen et al. (2004b) studied abundances of molecules in envelopes of embedded YSOs by their rotational lines. Brinch et al. (2007) constrained the shape, velocity structure, inclination and molecular abundances of the class I object L1489 using a two-dimensional radiative transfer model. These and many other studies fit models to observations, keeping abundances constant. Chemical models (e.g. Bruderer et al. 2009b) however predict strong gradients in abundances, for example some molecules are strongly enhanced in the warm and FUV irradiated region along the outflow of a YSO. Among such species are simple hydrides (e.g OH or CH+ , Chapter 6) which can be observed with the Herschel Space Observatory for the first time with high angular and spectral resolution. To model the hydride emission from the warm region along outflows, two-dimensional models are required. Such more complete models can however only be applied on larger samples, if the line radiation modelling process is accelerated. Different methods for analyzing molecular line emission exist. The simplest method assumes that the excitation temperature in different lines is the same, the emission stems from the same region of the source, and the optical depth is low. The excitation temperature and column density of a species can then be estimated from a so called “rotation diagram” or “Boltzmann plot” (e.g. Blake et al. 1987). That method however does not explain the obtained excitation temperature in terms of the physical conditions in the emitting region. A more elaborated approach calculates the level populations of a molecule by solving the rate equations which incorporate the effects of collisional or radiative excitation and deexcitation. In the “escape probability” approach (e.g. de Jong et al. 1975 or van der Tak et al. 2007), the simplification of a single homogeneous emitting zone is still kept and the ambient radiation field is estimated using an analytical expression. Due to the low number of free parameters and fast calculation, the method allows to quickly constrain column densities, kinetic temperatures or volume densities using a χ2 fit (e.g. Jansen et al. 1995). Also, the escape probability approach can be used to study the excitation of a molecule in a wide range of the parameter space or under special condition such as excited formation of the molecule (St¨auber & Bruderer 2009). Radiative transfer models account for the physical structure of the emitting medium, for example an infalling envelope of a YSO. In an YSO envelope, temperature and density are of order 20 K and 104 cm−3 in outer regions and increases to several 100 K and above 107 cm−3 close to the forming star. The molecular excitation thus varies with position, and different lines of the same species probe different regions. Radiative transfer models thus calculate the population depending on the position and account for the density and temperature stratification and the velocity field. Mostly, a spherical geometry is assumed with physical parameters depending on the distance to the protostar only. Level populations at different positions are connected through line emission by radiative excitation and

5.2. Method

97

deexcitation. The solution of this non-linear problem can be very time-consuming. Widely used methods are the Monte Carlo approach (e.g. Bernes 1979 or Hogerheijde & van der Tak 2000) and the Accelerated Lambda Iteration (ALI) method (e.g. Rybicki & Hummer 1991). A review of the different methods for radiative transfer in molecular lines is given in van Zadelhoff et al. (2002). Multidimensional radiative transfer models are in principle possible (e.g. the 2D version of Ratran; Hogerheijde & van der Tak 2000 or Rawlings et al. 2004). They are however extremely timeconsuming and lack of convergence in geometries with a huge dynamical range in temperature, density and thus excitation. This prevents the application of such more sophisticated models on larger samples of sources or the creation a grid of many different geometrical and chemical models. Recent developments of radiative transfer on unstructured grids can accelerate the calculation up to a certain level (e.g. LIME; Brinch et al., in preparation). For example, a model of CO in a spherically symmetric envelope implemented in 3D with 303 cells requires of order one hour of calculation time but may take much longer for “difficult” molecules such as water (Brinch 2008). The complex mathematical structure of the problem requires the calculation of the interaction of many cell in different lines and frequency channels. It will thus remain time-consuming to obtain exact solutions of the problem and approximative, but much faster, methods have various applications. They can for example be used to set limits for fitting or finding initial guesses for exact methods or to explore the possible parameter space by calculating large grids of models. In this chapter, we present a new approximative method to model molecular line emission from complex multidimensional models. It is based on the escape probability approximation and calculates the excitation in different, coupled cells. Such multizone-escape probability methods have also been presented by Elitzur & Asensio Ramos (2006), Apruzese et al. (1980), Giuliani et al. (2005) or Poelman & Spaans (2005, 2006). The method presented here is similar to the approach by Poelman et al. in some ways, however considers the lineshape explicitly and implements methods to accelerate the convergence. This chapter is organized as follows: In the first section, we introduce the problem of molecular excitation. Details and limitations of the implementation are given in Section 5.3. In Section 5.4, we study different benchmark tests and applications of the method and discuss its accuracy.

5.2

Method

In this section, we briefly introduce the formalism for molecular line radiative transfer and excitation. For a more detailed discussion of line radiative transfer, we refer to Elitzur (1992) or Rybicki & Lightman (1979).

5.2.1

Radiative Transfer in molecular lines

To calculate molecular line fluxes, the density ni (~r) (cm−3 ) of the energy levels i = 1, . . . , N at all positions ~r of the model have to be known. In practice, the source model is discretized into cells with constant physical conditions (temperature, density etc). The position ~r can then be expressed by an index for the cell, e.g. nki for the density of level i in cell k. The level populations for each cell k is obtained by solving the rate equations, N

N

X X dnki Pijk = 0 , = nkj Pjik − nki dt j6=i j6=i

(5.1)

where we assumed the level populations to be constant with time. Additional terms may be necessary to account for formation in an excited state and destruction (e.g. St¨auber & Bruderer 2009). This

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5. A computer program for fast multidimensional modelling of molecular line emission

effect is however only important for very reactive species (e.g. CO+ or CH+ ). The rate coefficients Pijk (s−1 ) for the line transition between level i and j are given by  Aij + Bij hJijk i + Cijk (Ei > Ej ) k (5.2) Pij = Bij hJijk i + Cijk (Ei < Ej ) , with Ei , the energy of level i. Here, the collisional and radiative excitation enter by the collision rate coefficients Cijk and the Einstein coefficients for spontaneous and induced emission (Aij and Bij ) and the ambient radiation field hJijk i. The collision rates are tabulated (e.g. Sch¨oier et al. k 2005) in the form of temperature dependent rates Kij (Tkin ) = Cijk /nkCol (cm3 s−1 ), with the kinetic k k temperature Tkin (K) and the density of the collision partner nkCol (cm−3 ). The rates Kij (Tkin ) are the Maxwellian average of the collision cross section between the molecule and a collision partner. Knowledge of either the excitation or deexcitation rate is sufficient, as the detailed balance relation
k k k gj Cij = Cji gi exp −hν/kTkin yields the other. The detailed balance relations ensures that the level populations are thermalized at high density (Tex = Tkin ) independent of the collision rates. Depending on the modeled region, collisions with H2 , atomic hydrogen and electrons have to be taken into account. The ambient radiation field is the specific intensity Iν integrated over the line profile and averaged over all directions. It is given by (for the sake of clarity, we omit the label for the cell k in the following), Z Z 1 φij (ν)Iν (Ω)dνdΩ , (5.3) hJij i ≡ 4π with the normalized line profile function    1 ν  2 √ exp − ν − νij − ~v · ~n ij /∆νD , (5.4) φij (ν) = ∆νD π c

where ∆νD is the Doppler-width of the line, νij is the frequency of the transition i → j, ~v the velocity of the gas at the local position and ~n the unity vector pointing to the direction of light propagation. The Doppler-width accounts for the natural line width, the thermal broadening and broadening due to microturbulence. Usually, microturbulence contributes most, except in very cold environments such as dark cloud cores. The intensity Iν (Ω) used for the calculation of the ambient radiation field (Eq. 5.3) is obtained from solutions of the radiative transfer equation in different directions. For applications at wavelengths longer than mid-infrared, to which we will restrict in this chapter, scattering can be neglected. The radiative transfer equation for the line radiation, including an overlapping dust continuum and assuming complete redistribution of frequencies then reads (e.g. Takahashi et al. 1983) dIν = − (αL,ν + κD,ν ρD ) Iν + αL,ν SL + κD,ν Bν (TD ) , ds

(5.5)

with the line absorption coefficient αL,ν , the dust opacity κD,ν , the dust mass density ρD and the line source function SL = Bν (Tex ). Here, Bν (T ) is the Planck function for the excitation temperature1 Tex or the dust temperature TD . The line absorption coefficient for the transition i → j is given by αL,ν =

hνij φij (ν) (nj Bji − ni Bij ) 4π

(5.6)

and defines the optical depth by dτν ≡ (αL,ν + κD,ν ρD )ds. The line and dust optical depth are defined similarly by dτL,ν ≡ αL,ν ds and dτD,ν ≡ κD,ν ρD ds. 1

   g n Tex ≡ hνij / k log gij nji

5.2. Method

99

A self-consistent solution of the radiative transfer problem needs to fullfill the rate equations (Eq. 5.1) with the ambient radiation field (Eq. 5.3), where the level populations of the same and other cells enter through the solution of the radiative transfer equation (Eq. 5.5). This non-linear and coupled problem is thus mostly solved iteratively by starting with assumed level populations in the cells. The ambient radiation field hJij i is then computed and new level populations are obtained from the rate equations. This steps are repeated until convergence. This “Lambda-Iteration” approach however often converges very slowly or not at all, if high optical depths are reached. An approach to overcome these problems is the so-called Accelerated Lambda Iteration (ALI) method which treats radiation from local and remote cells differently (e.g. Rybicki & Hummer 1991 or Hogerheijde & van der Tak 2000).

5.2.2

The escape probability approach

The most time-consuming step in the iteration scheme described before is the solution of the radiative transfer equation along different rays and frequencies to obtain the ambient radiation field. The formal solution of the radiative transfer equation (Eq. 5.5) is Z τ ′ BG −τν Iν (Ω) = Iν e + Sν e−τν dτν′ , (5.7) 0

with the backgroud field IνBG (e.g. the CMB) and the source function Sν comprising dust- and line-contribution, αL,ν SL + κD,ν Bν (TD ) . (5.8) Sν = αL,ν + κD,ν ρD The integral in Eq. 5.7 is evaluated between the local position and the outer edge of the modeled region. In the “classical” escape probability approach, the level populations are assumed to be constant in the modeled region, hence the line-source function is constant. Neglecting dust contribution (κD,ν = 0), the ambient radiation field can then be obtained from hJij i = (1 − β(τ ))Bν (Tex,ij ) + I BG β(τ ) ,

(5.9)

with the escape probability 1 β(τ ) ≡ 4πV

Z Z Z Z

′

φij (ν)e−τν dτν′ dνdΩdV .

(5.10)

This integral is obtained from the ambient radiation field (Eq. 5.3) together with the formal solution (Eq. 5.7) averaged over the volume V of the modeled region. Analytical functions for β(τ ) exist and the solution of the radiative transfer problem is substantial simplified by this approach. Functions for β(τ ) for different geometries (slab, sphere etc.), line shapes (Voigt, Gaussian) or velocity structures are given in the literature (e.g. Kastner 1993). In a non-homogeneous medium, where the level populations cannot be assumed to be constant, different approaches to apply the escape probability method exist. Elitzur & Asensio Ramos (2006), Apruzese et al. (1980) or Giuliani et al. (2005) split up Eq. 5.7 into contributions from different cells. They the ambient radiation field in terms of the level populations in other cells, P express ′ k kk ′ kk ′ hJij i = k′ K (τij )ni with coupling constants K kk . Using this expression, the rate equations yield a large system of non-linear equations for the level populations in each cell, allowing a direct solution of the radiative transfer problem. In the context of stellar atmospheres, this elegant approach has been proposed long ago (e.g Avrett 1971). However, an efficient calculation of the coupling constants

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5. A computer program for fast multidimensional modelling of molecular line emission

is only possible for slab geometry (Elitzur & Asensio Ramos 2006) or spherically symmetric problems (Apruzese et al. 1980, Giuliani et al. 2005 and Yun et al. 2009). The approximative method by Poelman & Spaans (2005, 2006), on the other hand can be implemented easily in complex geometries. They assume the source function to be constant along the ray for the solution of Eq. 5.7 and calculate hJijk i from Eq. 5.9. The escape probability β(τ ) is estimated from the optical depth at the line center, R edge τ = cell k dτν=line center . The integral is calculated along a ray from the local position (cell k) to the edge of the modeled region taking the varying opacity in different cells into account. In this chapter, we implement the assumption of a constant source function according to Poelman et al., we however calculate the escape probability in different frequency (or velocity) channels. We thus account for the exact line shape and thus the effects of the velocity structure and/or high optical depth, where photons may only escape in the line-wings. We obtain the ambient radiation field from (Takahashi et al. 1983) hJij i = (1 − ǫij ) Bνij (Tex,ij ) + (ǫij − ηij ) Bνij (TD ) + ηij Bνij (TCMB ) ,

(5.11)

with ǫij ηij

Z Z τD,ν + τL,ν e−(τL,ν +τD,ν ) 1 φ(ν) = dνdΩ and 4π τL,ν + τD,ν Z Z 1 φ(ν)e−(τL,ν +τD,ν ) dνdΩ , = 4π

(5.12)

where τL,ν and τD,ν are the line and dust optical depth along rays from the local cell to the edge of the modeled region, Z edge Z edge τL,ν = αL,ν ds and τL,ν = κD,ν ρD ds . (5.13) cell k

cell k

A simple physical interpretation of this approximative radiation field (Eq. 5.11) exists: ǫij is the probability, that a photon escapes line absorption and does not contribute to the local radiation field. The term (1 −ǫij )Bνij (Tex,ij ) in Eq.5.11 is thus the contribution by line photons. ηij is the probability that a photon escapes both dust and line absorption. Conversely, it gives the probability that a photon from an external radiation field contributes to the local radiation field (term ηij Bνij (TCMB )). The contribution by dust photons is proportional to ǫij − ηij , the amount of dust absorption. To derive Eq. 5.11 and 5.12 from the formal solution of the radiative transfer equation (Eq. 5.7) with the total source function (Eq. 5.8), the following approximations are made: (i) Along a ray, the line and dust source functions SL and Bνij (TD ) are assumed to be constant. (ii) The fraction between line and dust source function contributing to the total source are assumed to be constant along a ray, thus τL,ν αL,ν τD,ν κD,ν ρD ≈ and ≈ τL,ν + τD,ν αL,ν + κD,ν ρD τL,ν + τD,ν αL,ν + κD,ν ρD

(5.14)

at each position along a ray. The approximation used here couples different cells of the model by the optical depth in other cells only, while the excitation is assumed to be the same as at the local position. This considerably simplifies the calculation of the ambient radiation field. Instead of the time consuming solution of the radiation transfer equation by the numerical integration of Eq. 5.7 only opacities have to be summed up along a ray. The implemented approximation is good, if the region which contributes significantly to the local intensity is either small (e.g. large velocity gradient) or homogeneous. Due to the non-linear nature of the problem, it is generally not possible to estimate how much these

5.3. Implementation

101

approximations affect the level populations without calculating the full, exact solution. In Section 5.4 we will carry out different benchmark problems to test the applicability of the escape probability approach.

5.3

Implementation

The method presented before has been implemented in a Fortran 77 code. In this section, we give an overview of the code and refer to a separate manual for details on the input/output parameters and for instructions on how to run the code. The code consists of two programs, SPEC which is first run to solve for the level populations. The raytracer TRAC then calculates synthetic maps and convolves them with the appropriate telescope beam for comparison with observations. Three different versions of the programs are provided, for spherically symmetric problems (hereafter 1D-version), for axi-symmetric problems (2D-version) and on a Cartesian coordinate system in three dimension (3D-version). The adopted coordinate systems are shown in Fig. 5.1. For the 2D-version, symmetry on the xy-plane is assumed. In the 1D-version, the cells consist of different shells with user-defined radii. For the 2D and 3D-version, equally spaced cells in one direction (dr/dz in 2D and dx/dy/dz in 3D) are used. This assumption simplifies the time-consuming calculation of the line/dust opacities (Eq. 5.13) and the raytracing with TRAC considerably. It may however pose problems, if a wide dynamical range of scales has to be covered. To avoid an excessively high number of cells, the code allows to “group” cells: The excitation is only calculated in one representative cell of the group an then copied to all other cells of the group. The grouping of cells with similar physical/chemical conditions has to be provided by the user.

Figure 5.1: Adopted geometry and coordinate system (red) in the 1D, 2D and 3D-version of the code. The directions for averaging Eq. 5.12 are given in green (for the sake of clarity, some diagonal directions in the 3D-version are omitted).

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5. A computer program for fast multidimensional modelling of molecular line emission

A flowchart of the program is given in Fig. 5.2: 1. SPEC reads in the source model providing information on geometry, kinetic temperature, velocity structure, density of the collision partners and gas/dust ratio. The code can deal with an arbitrary number of collision partners. Molecular data in the LAMDA2 format (Sch¨oier et al. 2005) and dust opacities in the format wavelength (µm) / opacity (cm2 g−1 ) (e.g. Ossenkopf & Henning 1994) are read in. The cells are then initialized with thermalized level populations. 2. Level populations are updated in a global iteration over the cells in the model until convergence. Levels are considered converged if subsequent level populations have factorial deviations < 10−4 . Only levels with a fractional population > 10−6 are checked. A single cell is updated in a local iteration: Opacities of different directions/frequencies are calculated (Eq. 5.13), ǫkij and ηijk are obtained and the rate equations are solved with hJij i from Eq. 5.11. The local iteration is repeated until convergence or a maximum number of local iterations is reached. The maximum number of local iterations is K times the number of the global iteration (i.e. in the first global iteration, the local iteration is repeatet at most K times). 3. The level populations are written out. 4. TRAC solves the radiative transfer equation (Eq. 5.5) along different rays/frequencies using the level populations obtained by SPEC. The resulting spectral map is convolved with the appropriate telescope beam for comparison with observations. The local iteration approach accelerates convergence, as it prevents the code to perform many global iterations for a few very optically thick cells, which usually require a high number of iterations to converge. Increasing the maximum number of local iterations by K (≈ 5 − 20) for each global iteration is experimentally found to be efficient. This can be explained by the “information” that needs to be spread in lines with moderate optical depth during the first few global iterations, before cells with high optical depth should converge for a fast global convergence. Another advantage of the local iteration is, that the code needs to recalculate opacities only locally in the inner loop. In practice, this is currently only implemented in the 3D-version, as the rays in the 1D/2D-version may intersect the same cell again after they have left it. The extrapolation method after Ng (1974) has been tested, but was found to hamper convergence in some cases and is not used here. The idea of the local-iteration acceleration scheme is somehow similar to the ALI-approach in exact radiative transfer, which also treats the optically thin part different as the optically thick part in order to allow “information” to propagate quickly in less optically thick regions. To obtain ǫij and ηij from Eq. 5.12, the dust and line opacity for different rays and frequencies have to be calculated. This is the most time-consuming step of the method. The number of frequencies used here is of order 20 per spectral line. As a further approximation, we only take into accout very few angular directions. In the 1D and 2D-version, 2 directions and 4 directions, respectively, are implemented (green arrows in Fig. 5.1). The directions follow a mean photon escape angle to the radial direction of about 60◦ (Apruzese 1981) and the z-axis (for the 2D version). For the 3D-version, the cardinal direction and diagonals are calculated. Thus 26 directions in total are considered. The optical depth within one cell can still be meaningful and we thus average ǫij and ηij over the local R1 opacity using hf i = 0 f (τoutside + x · τlocal )dx, with the opacity in the same cell (τlocal ) and from the border of the local cell to the edge of the model (τoutside ). 2

http://www.strw.leidenuniv.nl/˜moldata/

5.3. Implementation

SPEC

103

1. Initialisation Read model, dust opacity and molecular data Initialise level population in LTE, iglobal = 0

2. Iteration over all cells iglobal = iglobal + 1; ilocal = 0

global iteration

local iteration

ilocal = ilocal + 1 Update local level population

Calculate τL,ν , τD,ν for different ν and Ω (Eq, 15) Calculate ǫij , ηij and hJij i (Eq. 11, 12) Solve rate equations (Eq. 1) → new level population

no

Level population converged in local iteration or ilocal ≥ K × iglobal ? yes

no

Level population converged in all cells compared to previous global iteration ? yes 3. Write out level population

TRAC

4. Calculate synthetic maps using Eq. 5 Convolve map with telescope beam Figure 5.2: Flowchart of the code.

The raytracer TRAC produces synthetic spectral maps, similar to SKY in the Ratran code (Hogerheijde & van der Tak 2000). Our code however implements an adaptive refinement in order to resolve small but intense features of the map such as walls along outflows (Bruderer et al. 2009b). The 1Dversion makes use of the spherical symmetry and calculates the map only along a radial ray from the center outwards. The raytracer starts with an equally spaced grid. If the velocity integrated line intensity of neighboring pixels differs by more than a predefined fraction, the cell is refined. In the 2D and 3D-version the refinement is performed on a quadtree, dividing each pixel in 4 new pixel, when refinement is necessary. This step is repeated until convergence or the maximum recursion depth is reached. Because standard image formats (e.g. the FITS format) support only equally spaced pixels, the most commonly used task of convolving the maps and subtracting the background are already performed by the code. We note that the 1D/2D version of the raytracer may also be useful in combination with Ratran to resolve fine details that are not covered by their raytracer running on a fixed grid (or with non-adaptive refinement in some parts of the image).

104

5.3.1

5. A computer program for fast multidimensional modelling of molecular line emission

Limitations and possible extensions

The program presented here calculates the excitation of only one molecule at the same time. Line overlap between different molecules is thus not accounted for. Also, overlap of hyperfine components of the same molecules (e.g. OH or HCN) is not considered. Extensions of the current code could incorporate line-overlap using the approach presented in Lockett & Elitzur (1989), Cesaroni & Walmsley (1991) or Doel et al. (1990). In special physical conditions (for example low-density with high background radiation field), level populations of some molecules (e.g. water or OH) may be inverted resulting in negative opacities (τL,ν < 0). Hence the intensity along a ray is amplified. A proper modelling of such “masers” requires e.g. a fine sampling of directions for the calculation of hJij i (Elitzur 1992), for which the current implementation is not suited. If too many lines have saturated masers (about τ < −1), non-masing lines may also be affected. The code thus warns the user about lines with line opacities smaller than -1.

5.4

Applications

Compared to exact ALI or Monte Carlo radiative transfer calculations, the method implemented in SPEC is computationally often faster by more than a factor of 100. For example, the calculation of a water model discussed in Section 5.4.3 can take several hours/days for the calculation using Ratran, while SPEC calculates a model within a few minutes. The method discussed here may thus accelerate the line modelling process enormously. However, as approximations are used in the method, it is important to find the limitations of the approach. In this section, we thus calculate several benchmark problems and realistic applications in order to study the properties of the method implemented in SPEC. We will focus on the accuracy of the method rather than on the improvement in speed, since the calculation time depends on various parameters like the convergence criterion, the type of machine and others. Since there are no analytical solutions for most problems/applications discussed in this section, we check the accuracy of the method implemented in SPEC by comparing to an exact radiative transfer code. We use the Monte Carlo radiative transfer code Ratran (Hogerheijde & van der Tak 2000). This code has been extensively tested against other codes (e.g. van Zadelhoff et al. 2002). Like SPEC and TRAC, Ratran consists of two different codes for the calculation of the level population (AMC) and the raytracing to obtain maps (SKY). The Monte Carlo code has been run until a signal-to-noise of 50 is reached and the level population should thus be exact to a level of about 2 %.

5.4.1

A two-level problem (Problem 1)

As a first benchmark test, we use the 1D, 2D and 3D version of the code and calculate the level population of a fictive 2-level molecule in a homogeneous sphere. The molecular data is taken in agreement with the 101 − 110 rotational groundstate line of water, motivated by the challenging water radiative transfer calculations. The velocity structure is assumed to be static or a strong gradient with velocity proportional to the radial distance (V (r) = α × r). No external radiation field (CMB or dust radiation) is taken into account. Parameters of the problem are summarized in Table 5.1. The advantage of this benchmark problem, suggested by D. Neufeld (see van der Tak et al. 2005 and the webpage http://www.sron.rug.nl/˜vdtak/H2O/), is that an analytical solution exists, while the numerical calculation is difficult. The difficulties stem from high optical depths, while the level population is not thermalized, because the density of the collision partner is much lower than the critical density. For very high molecular abundances, the radiation field helps thermalizing the level

5.4. Applications

105

population and the problem is again simple to calculate, as it is in the optically thin case of very low abundances. Table 5.1: Problem 1: Parameters Parameters Physical Parameters Radius Density Temperature Molecular abundance Line width Velocity field (V (r) = α × r) Molecular Data Einstein-A coefficient Frequency Statistical weight Collision ratea Critical Density a

of the two-level benchmark problem

0.001-0.1 pc ncol = 104 cm−3 Tkin = 40 K xmol = nmol /ncol = 10−10 − 1 ∆V = 1 km s−1 α = 0 or 100 km s−1 pc−1 Aul = 3.458 × 10−3 s−1 νul = 556.936 GHz gu = gl = 9 Cul = 2.18 × 10−10 cm3 s−1 ncrit = 1.6 × 107 cm−3

Clu is obtained from Clu = Cul gu /gl exp (−hν/kTkin)

The problem has been calculated with logarithmically spaced 200 shells in the 1D version of SPEC and Ratran. For the 2D and 3D version of SPEC, a 150 x 150 and 200 x 200 x 200 grid, respectively, has been used. For the calculation of the 3D version, different cells are grouped as described before and only a few hundred cells are calculated. The 3D version has also been tested without this grouping, but at a lower spatial resolution, and the results are very similar. Analytical Solution This section develops an analytical expression for the luminosity L (erg s−1 ) of the line emission from the homogeneous sphere. The analytical expression is used in the next section for comparison with the numerical calculation. The luminosity from the numerical calculation is obtained by calculating spectral maps, either with TRAC or SKY. The maps are convolved with a large beam of radius Rbeam = 10 pc (FWHM) covering the whole cloud. The line luminosity is calculated using Z 3 2kνul L = 4π × TA dv , (5.15) c3 R with the convolved and velocity integrated line flux TA dv (K km s−1 ). For the analytical expression, we use Z L = Aul hνul βnu dV , (5.16) 2 πRbeam



and thus an expression for the product of the upper level density nu (cm−3 ) times the escape probability β has to be obtained in the following. The rate equation for the statistical equilibrium (Eq. 5.1) relates the escape probability and the density of the upper and lower level nu , nl (cm−3 ) (nu +nl = nmol ). Using that the net absorption, corrected for stimulated emission are the photons that do not escape, (nl Blu − nu Bul )hJul i = nu (1 − β)Aul ,

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5. A computer program for fast multidimensional modelling of molecular line emission

the rate equation reads nl ncol Clu = nu ncol Cul + nu βAul , thus

Clu /Cul nu = , crit nl 1 + βn ncol

(5.17)

using the definition of the critical density (ncrit ≡ Aul /Cul ). In the static situation (α = 0), consider that the critical density is much higher than the density of the collision partner (ncrit /ncol = 1600) and the level population can only be thermalized (nu /nl ≈ −1 Clu /Cul ) if the optical depth is high and thus β small. For line opacities τul larger than a few, β ≈ τul and τul of order a few 100 is require for thermalization. This is only reached for xmol = nmol /ncol > 10−6 . For lower xmol , it is save to assume nl ≈ nmol and β > ncol /ncrit . Thus Clu /Cul β −1 Clu nmol ncol nu Clu ncol = −1 ncrit ≈ , and βnu ≈ . nl β + ncol βAul Aul

(5.18)

The product βnu is thus constant throughout the cloud. Using Eq. 5.16, we obtain for the line luminosity L = hνul nmol ncol Clu V ,

(5.19)

with the volume V of the cloud. This expression states that collisional deexcitation can be neglected and every collisional excitation results in photons that escape the cloud despite the considerable optical depth. The expression is exact, as long as the approximation nl ≈ nmol is fulfilled, but it does not make any statement about the level population that may depend on the position. For a large velocity gradient (α > 0) at small radii, the Sobolev or Large Velocity Gradient (LVG) approximation (e.g. Sobolev 1960) applies, β = β(τul ) and τul are then given by 1 − e−τul and τul   gu c3 Aul nl − nu . = 3 8πνul α gl

β(τul ) =

(5.20)

τul

(5.21)

These equations form together with Eq. 5.17 a non-linear equation for nu and β which can be solved numerically. It is important to emphasize that this solution is only exact in the limit of small radii, but is a good approximation except in a thin shell on the surface with size of the order ∆V /α. Thus if the luminosity is dominated by this region, the approximation should not be used anymore. For the parameters adopted here, the LVG solution should not be used for abundance above about 10−6 .

Results and discussion The spectra for the static model and a molecular abundance xmol = 10−7 is presented in Figure 5.3a as an example. The figure gives the beam convolved lines, obtained from Ratran and the 1D version of SPEC. Comparing both lines, we find good agreement in the line wings (|v| > 0.75 km s−1 ), while the line center (|v| < 0.75 km s−1 ) is predicted to be stronger by the exact solution (Ratran).

5.4. Applications

107

2.5 · 10−4

a.) Spectra Escape Probability

SPEC (1D)

Ratran (SKY)

10−1 100

τ

10−2

10

10−4

1 · 10−4

10−5

1 5 · 10

Escape Probability

10−3

1.5 · 10−4

τ

Intensity [K]

2 · 10−4

Ratran

−5

10−6 0.1 −3

−2

−1

0

1

2

3

Velocity [km s−1 ]

5 · 10

−5

2 · 10

−5

10−12

100 50

10

5 · 10−6

5

2 · 10−6

2

10−13

1 · 10

−6

1

5 · 10−7 2 · 10−7 1 · 10−7

10−14

nu Ratran

Icenter

nu SPEC (1D)

τwing

SL = Bν (Tex )

Iwing

0.5 0.2

τcenter

10−15

Intensity (erg s−1 cm−2 Hz−1 sr−1 )

20

1 · 10−5

τ

Upper Level Population nu (cm−3 )

b.) Line Formation

0.1 0

1 · 10

17

2 · 10

Radius (cm)

17

3 · 10

17

Figure 5.3: Problem 1: a.) Line emission for a molecular abundance xmol = 10−7 obtained with Ratran and SPEC. The absorption along a ray through the center of the cloud and the escape probability for a photon from the center of the cloud are given in dashed. b.) Level population obtained from Ratran and SPEC. The source function, the cumulative line opacity and the intensity along a radial ray are given for the Ratran results. The line shape can be understood by the opacity which peaks at the line center and is smaller in the line wings. This is explained by Figure 5.3b which shows the level population obtained from Ratran and SPEC depending on the radial position. The source function, the opacity and the intensity are given for the Ratran results. The opacity for the line center and the wings are summed up from the the edge of the cloud to the current position. The intensity is obtained from the formal solution of the radiative transfer equation along a ray from the edge to the current position, Z r I= S(r)e−τ (r) dτ (r) , (5.22) Edge

108

5. A computer program for fast multidimensional modelling of molecular line emission

with the source function S(r) depending on the radial position and τ (r) from the edge to the current position. Figure 5.3b thus shows that the intensity in the line center already reaches its value after about 1/50 of the cloud and does not increase further in the inner region. The line center is thus only sensitive to the source function, and thus the level population, in the outer part of the cloud. The line wings on the other hand are sensitive also to the inner part of the cloud. Differences between Ratran and SPEC in the intensity at the line center are thus a result of the upper level population predicted too low by SPEC in the outer part of the cloud. Why is the excitation at the edge of the cloud underestimated by SPEC? To calculate the local intensity, SPEC assumes the excitation to be constant along a path. However, Eq. 5.18 states that the upper level population depends on the escape probability which is higher at the edge of the cloud and nu ∝ 1/β thus decreased. Neglecting the decrease of the upper level thus leads to a local intensity which is too high in the inner region since the method presumes the excitation to be equally high in the whole cloud. Conversely, the local intensity and thus the excitation is underpredicted at the edge. Note that this is only true if nl ≈ nmol and the absorption coefficient does not vary much with nu . If this is not fulfilled anymore, the method is more likely to underestimate the excitation as τ ∝ (nl gu /gl − nu ) is decreased with nu increasing. Thus, β is reduced due to stimulated emission. Level populations dropping towards the edge of the model is common, not only because of less effective line trapping as discussed here. For example, in models of YSO envelopes with power-law density profiles, collisional excitation becomes less effective to the edge of the cloud. The inaccuracy in the calculated line flux by this effect depends on the problem and is large if the sparsely populated region still contributes significantly to the total flux. The static model with different molecular abundances is discussed in Figure 5.4a and 5.4c and the corresponding beam convolved spectra are given in Figure 5.5. In Figure 5.4a, the line luminosities obtained from Ratran and the 1D, 2D and 3D version of SPEC and their deviation to Ratran and the analytical solution are shown. The analytical solution can be applied up to abundances of about 10−7 . In this range, the results from SPEC agree to within better than 10 % to the analytical solution. The comparison between SPEC and Ratran agrees within 33 % for all abundances except for 10−6 , where deviations amount to 67 %. For this abundance, the solution obtained from Ratran exceeds the analytical solution by about 80 %. As the level population are thermalized for higher abundances and collisional deexcitation becomes important, the luminosity is however expected to drop, as seen in the solution of the LVG approximation (Figure 5.4b. Thus it remains unclear if the deviation is real or a bias of the method employed in Ratran. Note that the agreement of different versions of SPEC is very similar. This is also seen in the similar line shape of different version in Figure 5.5. The good agreement at low or high abundances is not surprising as these are the computationally simple situations where the radiative pumping is unimportant (optically thin) or the level population is thermalized by the radiation field (optically very thick). Level populations of the static model (Figure 5.4c show a thermalized population (xu = nu /nmol ≈ 0.34, from nu /nl = Clu /Cul ) for abundances of about xmol = 10−5 , while for xmol below ≈ 10−10 , the optically thin limit of xu = nu /nmol = Clu nCol /Aul ≈ 3.2 × 10−4 is reached. For abundances in between, the effect discussed in the previous paragraph is seen: The level population is underestimated in the outer part up to abundances of 10−7 , while it is overestimated in the inner part. For an abundance of 10−6 , the excitation is underestimated by SPEC. Black and grey dots on the level population of Ratran show the point, where τ = 1 is reached in the line center (black) and at a velocity of 1 km s−1 (grey). For abundances below 10−8 , both line center and line wings are optically thin, while they are optically thick for this abundance (line center) and higher abundances (both line center and wings). The region which is traced by the center and wings shifts outwards for higher abundances. The good agreement of the luminosities found for xmol = 10−8 despite the level populations differing by 30 % is thus a result of the excitation being similar at the τ = 1 surface.

5.4. Applications

109

b.) Luminosity (Expanding)

1032 1030 10

28

Analytical

SPEC (1D)

Ratran

SPEC (2D)

1 channel

SPEC (3D)

Line Luminosity (erg s−1 )

Line Luminosity (erg s−1 )

a.) Luminosity (Static) 1032

LVG Approximation

1030

Ratran SPEC (1D)

10

SPEC (2D)

28

SPEC (3D)

to LVG Approximation

to Analytical 50 30 %

0 to Ratran

50

Deviation (%)

Deviation (%)

50

0 to Ratran

50

0

0 10−9

10−6

100

10−3

10−9

100

c.) Level Population (Static) xmol = 10−5 xmol = 10−6

10−1 Ratran

−7

SPEC (1D)

10−2

xmol = 10

−8

xmol = 10 10−3

10

xmol = 10−9 xmol = 10−10

−4

0

1 · 1017

2 · 1017

Radius (cm)

100

10−3

Molecular Abundance (nmol ) Upper Level Population xu = nu /nmol

Upper Level Population xu = nu /nmol

Molecular Abundance (nmol )

10−6

3 · 1017

100

d.) Level Population (Expanding) xmol = 10−4

10−1

xmol = 10−5 −6

xmol = 10 10−2

Ratran

LVG Approximation

SPEC (1D)

10−3

10

xmol = 10−7

xmol = 10−8 xmol = 10−9

−4

0

1 · 1017

2 · 1017

Radius (cm)

3 · 1017

Figure 5.4: Problem 1: Comparison of line luminosities, obtained from Ratran and SPEC with analytical solutions: a.) and b.) Luminosity in the static/expanding cloud. Below the luminosities, a comparison to the analytical solution/LVG approximation and to Ratran is given. c.) and d.) Level population in the static/expanding cloud. The grey/black points in c.) give the position of the τ = 1 surface.

110

5. A computer program for fast multidimensional modelling of molecular line emission

2

10

SPEC (1D)

SPEC (3D)

SPEC (2D)

Ratran

A xmol = 10−10 B xmol = 10−9 C xmol = 10−8 D xmol = 10−7 E xmol = 10−6 (x 10) F xmol = 10−5 (x 100) G xmol = 10−4 (x 1000)

Intensity [K]

100

G F

10−2

E 10−4

D 10−6

C B A

10−8 −3

−2

−1

0

1

2

3

Velocity [km s−1 ]

Figure 5.5: Problem 1: Line profiles for the static model with different abundances. The lines are beam convolved and have an offset of a factor of 10, 100 and 1000 for abundances of 10−6 , 10−5 and 10−4 . The expanding model is presented in Figure 5.4b and 5.4d. Comparing the luminosities, we again find agreement within 30 % to the analytical solution (LVG approximation) and an agreement within 50 % to Ratran. The differences peak for abundances, where the level population is not in the optically thin regime (xmol < 10−8) and not thermalized (xmol > 10−3 ). While the deviations of the 1D and 3D versions of SPEC are similar, those of the 2D version are about 15 % larger. This is due to the choice of the directions for the calculation of the escape probability which is adequate for applications e.g. in disks or outflow walls, but less favorable for the problem calculated here. The level populations presented in Figure 5.4d are fairly constant, with smaller decreases towards the edge compared to the static situation. Since the LVG approximation also calculates level populations, they can also be compared to the ones obtained from the numerical calculation. We find good agreement between the LVG approximation and the results obtained from SPEC in the inner region, where the LVG approximation is applicable. The raytracer TRAC is also tested in this section, as the numerical calculation of the luminosity by Eq. 5.15 requires the beam convolved and velocity integrated line luminosity. The good agreement with the analytical solutions at low or very high optical depth indicates that the implementation is accurate. We have further tested the raytracer by comparing the results from the 1D version of TRAC with SKY from Ratran by raytracing the population of AMC with both raytracers. We find agreement within better than 1 % for all models. The method implemented in SPEC can deal with both dynamical and static models. This is a result of the escape probability being calculated from opacities in different velocity (frequency) channels (Eq. 5.12). If only the line center is accounted for and the escape probability is obtained from β(τ ) = (1 − e−τ )/τ , the escape probability is grossly underestimated as Figure 5.3 shows. The excitation is thus overestimated and the resulting luminosities are far too high as the 1 channel calculation in Figure 5.4 shows. Expressions for β(τ ) to correct for the line shape exist for specific

5.4. Applications

111

situations like the static case or a linear velocity field (e.g. Takahashi et al. 1983). These expression however cannot deal with arbitrary velocity fields in combination with optically thick lines as the approach implemented in SPEC does. We conclude that the results obtained with the new method agree mostly within 30 % to the exact solution in the line luminosities, while differences in the level populations can be much larger, especially at the edge of the cloud where the source function changes significantly and the approximations made in SPEC are not fulfilled anymore. Thus, depending on the problem, different parts of the modeled region are important for the accuracy of the calculated spectra. For the simple benchmark problem calculated here, the approximation of a constant source function made in SPEC are fulfilled in large parts of the envelope and further benchmark problems should thus include a density and temperature stratification.

5.4.2

A collapsing cloud in HCO+ (Problem 2)

A more realistic problem suggested by van Zadelhoff et al. (2002) is calculated as a second test. The problem consists of a collapsing cloud with a temperature, density and velocity gradient. Also, the intrinsic line width is a function of the position. The physical conditions are motivated by the envelope of a class 0 low-mass object in the inside-out collapse phase. The pure rotational emission of HCO+ is calculated and thus the population of several levels connected by several radiative transitions is studied. We assume a constant molecular abundance. This problem is challenging because the critical density of HCO+ lines is larger than the H2 density, while high optical depths are reached. Thus, radiative pumping is important and the level population is neither in the local thermal equilibrium nor in the optically thin case. In addition, the infall in combination with the temperature gradient leads to the formation of complicated line shapes which will be discussed in Section 5.4.2. 1 · 106

0.2

5 · 105

18

0

14

12

Density (cm3 )

16

Temperature Density Line Width Infall Velocity

5

1 · 10

5 · 104

−0.2

2 · 104

−0.4

4

1 · 10

5 · 103

Velocity (km s−1 )

Temperature (K)

2 · 105

−0.6

2 · 103 10

1 · 103

16

2 · 10

16

5 · 10

1 · 10

17

Radius (cm)

17

2 · 10

−0.8

17

5 · 10

Figure 5.6: Problem 2: Physical model for the benchmark problem of HCO+ in a collapsing cloud.

112

5. A computer program for fast multidimensional modelling of molecular line emission

A comparison between different radiative transfer methods using this model has been carried out by van Zadelhoff et al. (2002). Unlike SPEC, all codes participating in this comparison study are “exact codes” of Monte Carlo, ALI or local linearization type. Differences between the models are calculated using the mean of all solutions, since no analytical solution exists for this problem. Like their work, we will restrict the discussion on the J = 1 and J = 4 energy level. Comparing the population of the J = 1 and J = 4 levels, they find differences between the codes of order 1 % (J = 1) and 5 % (J = 4) for an HCO+ abundance of 10−9 relative to H2 . For the higher abundance of 10−8 , differences increase to 2 % (J = 1) and 15 % (J = 4). This is likely, since the optical depth reached for the lower abundance are of order τ = 1 − 10, while they increase to τ = 10 − 100 for the higher abundance making the problem computationally more difficult. Also, the J = 4 level is more difficult to calculate due to the higher critical density. We will thus study here, how much the approximative solution obtained from SPEC deviates from the exact solution. The benchmark problem is calculated using the 1D, 2D and 3D version of SPEC. A grid with individually spaced 50 shells is used in 1D, while the 2D and 3D version use grids of 150 x 150 and 150 x 150 x 150 cells. For the 2D and 3D calculation, the cells are grouped to only 150 cells. The 2D calculation has been repeated calculating all 150 x 150 cells with essentially the same result. For example, the level population in the midplane differs by less than 1 %. We use molecular data of HCO+ from the LAMDA database (Sch¨oier et al. 2005) taking into account 30 transitions between 31 energy levels. The molecular abundance is set to 10−9 and 10−8 as in van Zadelhoff et al. (2002). In addition, we calculate for an abundance of 10−7 . For comparison, we use the results from Ratran. To obtain line intensities, raytracing is performed with TRAC, assuming a distance of 250 pc. The maps are convolved to a 29′′ IRAM 30 m beam (J = 1 → 0) or a 14′′ JCMT beam (J = 4 → 3). The accuracy of the raytracer TRAC is also tested for this problem. Using the level population obtained from Ratran, both raytracer TRAC and SKY are run. The maps from SKY are further processed using the Miriad package (Qi 2005). We again find very good agreement, though the line wings show deviations of a few percent. This is explained by the adaptive mesh technique implemented in TRAC which resolves the inner part of the envelope more precisely. Results The normalized level populations of the J = 1 and J = 4 levels for the different abundances are shown in Figure 5.7. Presented are the level populations obtained from Ratran and the 1D, 2D and 3D version of SPEC. Also the level population in the LTE and under optically thin conditions are given. For the optically thin population, Eq. 5.1 is solved assuming hJij i = 0. The optically thin case thus neglects pumping of the molecule by line radiation or CMB photons. The J = 1 level is out of LTE for the abundance of 10−9 and 10−8 , while an abundance of 10−7 is sufficient to thermalized the population in the inner region (< 10−17 cm). The agreement between SPEC and Ratran in the level population of the J = 1 level is typically better than 5 % for all abundances. Larger deviations are only found at the edge of the model with abundances of 10−8 and 10−7 . Deviations between the 1D, 2D and 3D version of SPEC are similar in the outer part of the envelope, while they differ in the inner part with a velocity gradient. This is likely due to the lower spatial resolution of the 2D and 3D version in combination with the different angular sampling for the calculation of the escape probability. The J = 4 level population shows a stronger dependence on the abundance and distance to the source compared to the J = 1 level. For an abundance of 10−9 , the population is close to the optically thin case, while for an abundance of 10−7 it is thermalized in the inner part (< 1017 cm) but still drops to the optically thin case at the edge of the cloud. The agreement between SPEC and Ratran for the optically thin case of an abundance of 10−9 is typically within 10 %, except for the 3D calculation in the inner region, where it increases to 40 %. For higher abundances (10−8 and

5.4. Applications

113

10−7 ), deviations are generally larger, typically of order 20 to 30 %. They amount to up to 60 % at a radius of 8 × 1016 cm for an abundance of 10−8 and at the outer edge of the model with an abundance of 10−7 . Not surprisingly, we find largest deviations between the exact and approximative solution in regions, where the transition between the thermalized inner region and the optically thin population at the edge takes place. In this region, the source function changes considerably and the approximations used in SPEC are not well fulfilled anymore. Deviations between the 1D, 2D and 3D version are mainly found in inner regions, like in the J = 1 level. Compared to different exact methods for radiative transfer (van Zadelhoff et al. 2002), SPEC clearly deviates more. This is expected, as the level population and thus the source function changes significantly within the cloud, and the approximations made in SPEC are not well fulfilled. An example thereof is the homogeneous model discussed in Section 5.4.1 (Problem 1).

15 10 5 0 5 · 1016 1 · 1017 2 · 1017 Radius (cm)

Normalized Level Population

10−4

Deviation (%)

10−5 60 40 20 0 2 · 1016

5 · 1016 1 · 1017 2 · 1017 Radius (cm)

5 · 1017

Normalized Level Population

SPEC (3D)

0.3 15 10 5 0

10−1

10−3

SPEC (2D)

0.4

2 · 1016

d.) J = 4 (Abundance 10−9 )

SPEC (1D)

0.5

5 · 1017

10−2

Ratran LTE Thin

0.6

0.7

5 · 1016 1 · 1017 2 · 1017 Radius (cm)

10−4

10−5 60 40 20 0 2 · 1016

5 · 1016 1 · 1017 2 · 1017 Radius (cm)

5 · 1017

SPEC (1D) SPEC (2D) SPEC (3D)

0.5 0.4 0.3 15 10 5 0

10−1

10−3

Ratran LTE Thin

2 · 1016

e.) J = 4 (Abundance 10−8 )

10−2

c.) J = 1 (Abundance 10−7 )

0.6

5 · 1017

Normalized Level Population

0.3

b.) J = 1 (Abundance 10−8 )

Deviation (%)

SPEC (3D)

0.4

10−1 Normalized Level Population

SPEC (2D)

0.5

2 · 1016

Deviation (%)

SPEC (1D)

Normalized Level Population

Ratran LTE Thin

0.6

0.7

Deviation (%)

a.) J = 1 (Abundance 10−9 )

Deviation (%)

Deviation (%)

Normalized Level Population

0.7

5 · 1016 1 · 1017 2 · 1017 Radius (cm)

5 · 1017

f.) J = 4 (Abundance 10−7 )

10−2

10−3

10−4

10−5 60 40 20 0 2 · 1016

5 · 1016 1 · 1017 2 · 1017 Radius (cm)

5 · 1017

Figure 5.7: Problem 2: Normalized level population for the J = 1 and J = 4 levels obtained from Ratran and the 1D, 2D and 3D version of SPEC. The population in the LTE and the optically thin case are also given. How do inaccuracies in the level population affect the derived line shape and flux? In Figure 5.8, the beam convolved spectra of the J = 1 → 0 and the J = 4 → 3 lines are given for the level populations discussed before. We find that the LTE or optically thin calculation is clearly inappropriate for this problem, except for the J = 4 → 3 line at an abundance of 10−9 . Note that the J = 1 → 0 lines in the optically thin case are seen in absorption against the cosmic microwave background. Deviations between Ratran and SPEC are different from line to line and different parts of a line (center, wing) also show different degrees of deviation. This is a result of the complex line formation which is studied in the next section. In the J = 1 → 0 line for abundances of 10−8 and

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5. A computer program for fast multidimensional modelling of molecular line emission

10−7 , we find the line center to be underpredicted, while the line wings are overpredicted. In the J = 4 → 3 for an abundance of 10−8 , the blue line wing is overpredicted by the 2D and 3D version but underpredicted by the 1D version, while all three versions overpredict the red line wing. For an abundance of 10−7 , the blue wing is underpredcited by the 1D version. a.) J=1→0 (Abundance 10−9 )

6

b.) J=1→0 (Abundance 10−8 )

8

c.) J=1→0 (Abundance 10−7 )

Ratran (Sky)

3

Ratran

7

5

SPEC (1D)

6

SPEC (2D)

4

LTE Thin

5 T (K)

2

T (K)

T (K)

SPEC (3D)

3

3

2

1

4

2 1

0

1

0 −1

−0.5

0

0.5

0 −1

1

V (km s−1 )

d.) J=4→3 (Abundance 10−9 )

−0.5

0

0.5

−1

1

V (km s−1 )

e.) J=4→3 (Abundance 10−8 )

6

−0.5

0

0.5

1

0.5

1

V (km s−1 )

f.) J=4→3 (Abundance 10−7 )

Ratran (Sky)

0.3

Ratran

5

2

SPEC (1D) SPEC (2D)

4

Thin

T (K)

LTE

T (K)

T (K)

SPEC (3D)

0.2

1

3 2

0.1

1 0

0 −1

−0.5

0 V (km s−1 )

0.5

1

0 −1

−0.5

0 V (km s−1 )

0.5

1

−1

−0.5

0 V (km s−1 )

Figure 5.8: Problem 2: Line shape of the J = 1 → 0 and the J = 4 → 3 lines obtained from Ratran and the 1D, 2D and 3D version of SPEC. The spectra obtained from the population in the LTE and the optically thin case are also given. For comparison, a range of ±30 % to the Ratran model is indicated by a grey region. The insets on the right upper corner show a larger y-scale of the spectra. Quantitatively, the agreement in the line intensity is mostly within 30 %, except for the line center of the J = 1 → 0 line for abundances of 10−8 and 10−7 . Also, the line wings of the J = 4 → 3 line for an abundance of 10−8 deviate more then 30 %. The velocity integrated intensity is compared in Table 5.2. It shows an agreement of the SPEC results compared to Ratran mostly within 30 %, with the best agreement found in the J = 4 → 3 line for an abundance of 10−9 . The largest deviation with a disagreement of 35.3 % is found in the J = 4 → 3 line for an abundance of 10−8 (3D version of SPEC). These line also shows the largest differences between the 1D, 2D and 3D version. The largest deviation is found in a line with a considerable optical depth, which is however not sufficient to thermalize the level population. To set deviations of order 30 % in a context, consider the following: A typical telescope, operating at submillimeter wavelengths, has a calibration uncertainty of up to 30 %. Also, the uncertainty in the collision rate coefficients is large and often reaches 50 % if not a factor of 2 (Sch¨oier et al. 2005). If the abundances that enters the calculation are derived from chemical models (e.g. Bruderer et al. 2009b), an uncertainty of a factor of a few in this parameter is realistic. We will thus refer deviations

5.4. Applications

115

smaller than 30 % as “good agreement”, however keep in mind that this definition is rather arbitrary and better precision is desirable for some applications. Table 5.2: Problem 2: Factorial deviation of the velocity integrated intensity between the exact result (Ratran) and SPEC. The deviations for lines assuming local thermal equilibrium or an optically thin situation is also given. Line xHCO+ LTE thin SPEC 1D 2D 3D −9 J = 1 → 0 10 207.8 -130.9 17.6 25.9 20.6 J = 1 → 0 10−8 188.2 -121.5 5.0 11.7 4.8 −7 J = 1 → 0 10 143.5 -117.4 -8.0 2.1 -9.1 J = 4 → 3 10−9 2390.6 -20.2 4.0 20.6 12.6 −8 J = 4 → 3 10 429.3 -39.0 -11.5 13.5 35.3 −7 J = 4 → 3 10 153.0 -62.7 -14.7 -1.2 2.4 Line formation How do inaccuracies in the level population obtained with SPEC (Fig. 5.7) propagate to deviations in the calculated line (Fig. 5.8)? To answer this question, the formation of the beam convolved lines is discussed. Two different effects have to be taken into account simultaneously. First, the spectra are formed along the line of sight on rays parallel to the radius but with different impact radii (Figure 5.9). Second, the spectra at different impact radii are convolved with the telescope beam. To study both effects, Figures 5.10 and 5.11 show maps with spectra depending on the impact radius (middle panels) and the contribution function for a line of sight with impact radius corresponding to 0.5 FWHM of the telescope beam (top panels). For simplicity, the figure only gives the result obtained from the level population calculated with Ratran. Note that differences in the level population discussed in Figure 5.7 do not directly translate into differences in the lines since also the lower level (ground state and J = 3 level, respectively) are involved in line formation.

p Line of Sight Impact Radius l r

Figure 5.9: Problem 2: Geometry used for the discussion of the line formation.

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5. A computer program for fast multidimensional modelling of molecular line emission

The contribution function CBFν (l) allows to study how much a path element contributes to the solution of the radiative transfer equation Eq. 5.7. It is defined by CBFν (l) ≡ S(l)e−τν (l)

dτν , dl

(5.23)

with the source function S(l) at the position l on the path, the cumulative opacity from the edge of the cloud to the current position τ (l) and the opacity per unit length dτ /dl (Pavlyuchenkov et al. 2008). For Figures 5.10 and 5.11, we normalize the contribution R ∞ function to its peak value. The convolution consists of calculating the integral Iν = 0 G(p)Iν (p)pdr, with the (normalized) beam response function G(p) and the intensity Iν (p) depending on the impact radius p. The weighting G(p)p peaks at an impact radius of 0.42 times the FWHM of the telescope beam. In Figures 5.10 and 5.11, we use an impact radius corresponding to half of the telescope beam for the display of the contribution function. For the adopted JCMT 14′′ and IRAM 29′′ beam, the weighting G(p)p is given in the middle panel for an abundance of 10−7 . The formation of the line along the ray with impact radius 0.5 FHWM is discussed in the top panel of Figures 5.10 and 5.11. The maps (middle panels in Figures 5.10 and 5.11) show that the emission of the J = 1 → 0 lines are considerably extended, while the J = 4 → 3 lines are closer to the center. The emission in the line wings (|v| > 0.5 km s−1 ) is more concentrated to the center in both lines, as the infall in the inner region “shifts” the gas out of the line center with high opacity and radiation of the inner region (1017 cm) can escape. The beam convolved spectra and the spectra for an impact radius of 0.5 FWHM (lower panels in Figures 5.10 and 5.11) agreement well except in the line wings. Thus, despite the lower weight G(p)p in the center, the strong emission in the line wings from the central region dominates the line wings of the beam convolved spectra. In the J = 1 → 0 line, the emission in the line center (|v| < 0.25 km s−1 ) is dominated by outer regions (> 1017 cm). The τ = 1 surface and the main contribution moves outward with higher abundances. For an abundance of 10−7, all emission in the line center is from beyond about 3 × 1017 cm. The line center of the J = 4 → 3 line is formed further in, even for the highest abundance. This is due to the excitation of the J = 3 and J = 4 levels which are only populated in the inner region. Thus both emission and absorption in the outer region are low. The line wings (|v| > 0.25 km s−1 ) of the J = 4 → 3 line emerge from the innermost, infalling region (< 1017 cm). The blue wing solely stems from the region just behind the center of the cloud moving towards the observer. The blue wing thus has less absorbing material in the foreground and is stronger than the red wing. The main contribution to the wings of the J = 1 → 0 line is also from inside 1017 cm, however material from outside contributes as well and thus the line is not skewed to the blue. The relative contribution of inner and outer part shifts to the inner part for higher abundances. How do inaccuracies in the level population obtained with SPEC propagate to deviations in the calculated line? While quantitative statements are difficult to draw due to the combination of different effects entering the line formation, we can say the following: Deviations in the wings of the J = 1 → 0 line stem from different regions, depending on the abundance. For an abundance of 10−9 , the wings arise from regions outside 1017 cm, where the level population indeed is too high. For higher abundances, the wings emerge from regions inside 1017 cm, and thus also correspond to regions with deviations in the level population. The line center of the J = 1 → 0 line emerges from the outermost part of the envelope for abundances above 10−9 and deviations thus are explained by the effect discussed in Problem 1. For the J = 4 → 3 line at an abundance of 10−8 , deviations in the wings are explained by the region inside 5 × 1016 cm (blue wing) and between 5 × 1016 − 1017 cm (red wing). For the abundance of 10−7 , the blue wing obtained using the 1D version of SPEC is too low as it emerges from a small region between 3 × 1016 cm and 5 × 1016 cm.

5.4. Applications

J =1→0 Abundance 10−9 CBF

Abundance 10−7

0.01, 0.1 0.1Peak

−2 · 1017

Infall

Line of Sight

0 Infall 17

τ = 1, 3, 10

4 · 1017

Map

2 · 1017

0.1, 1 3, 10 K

1 · 1017

0.5 FWHM

0 6

G(p) p

2 · 10

Abundance 10−8

Impact Radius

Impact Radius p (cm)

Distance along Path (cm)

−4 · 1017

Line

100

p=0.5 FWHM Convolved

4

10

τ

2

1

0

0.1 −1

−0.5

0 Velocity

0.5

1 −1

−0.5

0 Velocity

0.5

1 −1

−0.5

0

0.5

τ

Intensity (K)

117

1

Velocity

Figure 5.10: Problem 2: Formation of the J = 1 → 0 lines for different abundances. Top panels: Contribution function normalized to the peak. Contours give 0.3, 0.1 and 0.01 of the peak contribution function (black) and τ = 1, 3, 10 (red). Middle panels: Map of spectra for different impact radii. In the plots for the abundance of 10−7 , also the beam is shown. Contours give an intensity of 0.1, 1, 3 and 10 K. Bottom panels: Line spectra together with line opacity τ at an impact radius 0.5 FWHM (solid) and convolved with the beam (dashed).

Conclusion We conclude that the approximative method implemented in SPEC allows the calculation of models with a complex physical structure, including density, velocity and temperature gradients. The level population shows considerable deviations of up to 60 % compared to the exact solution, especially in regions where the level population makes the transition from the optically thin to the thermalized region. Most of the radiation however does not emerge from the “transition layer” between the two regions in this benchmark problem and the resulting lines agree mostly to within 30 %, which we consider as a good agreement given all other uncertainties which enter in such a calculation. This

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5. A computer program for fast multidimensional modelling of molecular line emission

problem raises the question whether the approximative method implemented in SPEC works well in a larger range of parameters, e.g. with different density slopes or chemical abundances which depend on the position.

−2 · 1017

Abundance 10−8

Abundance 10−7

0.01, 0.1 0.1Peak Infall

2 · 1017

τ = 1, 3, 10

Map 0.1, 1 3, 10 K

1 · 1017 0.5 FWHM 0 6

G(p) p

4 · 1017

Infall

Impact Radius

2 · 1017

Line of sight

0

Line

100

p=0.5 FWHM Convolved

4

10

τ

2

τ

Intensity (K)

Impact Radius p (cm)

Distance along Path (cm)

−4 · 1017

J =4→3 Abundance 10−9 CBF

1

0

0.1 −1

−0.5

0 Velocity

0.5

1 −1

−0.5

0 Velocity

0.5

1 −1

−0.5

0

0.5

1

Velocity

Figure 5.11: Problem 2: Formation of the J = 4 → 3 lines for different abundances. Top panels: Contribution function normalized to the peak. Contours give 0.3, 0.1 and 0.01 of the peak contribution function (black) and τ = 1, 3, 10 (red). Middle panels: Map of spectra for different impact radii. In the plots for the abundance of 10−7 , also the beam is shown. Contours give an intensity of 0.1, 1, 3 and 10 K. Bottom panels: Line spectra together with line opacity τ at an impact radius 0.5 FWHM (solid) and convolved with the beam (dashed).

5.4. Applications

5.4.3

119

A grid of water models (Problem 3)

The rotational lines of water span a wide range in upper level energy and critical density. They are thus good probes of the physical conditions. Due to the complicated level structure and high abundance, radiative transfer calculations of water lines are however very time-consuming. In addition to very high optical depths reached in the ground state lines, water also has maser activity in some lines and may be pumped by dust radiation. In this third problem, we will study the water excitation calculated by SPEC and compare it with the exact result. In preparation for the upcoming Herschel Space Observatory, van Kempen et al. (2008) have calculated a grid of water radiative transfer models using Ratran. The grid consists of 108 models which depend on five different parameters (Figure 5.12). The parameters are the bolometric luminosity of the protostar Lbol (2, 5 and 7 L⊙ ), the exponent of the power-law density profile α (1.5 or 2), the density at a distance of 1000 AU to the protostar nH2 (r0 ) (4 × 105 , 106 and 5 × 106 cm−3 ) and the chemical abundance of water in the inner region x0 (10−4 and 10−6 relative to H2 ) and the outer region xD (10−6 , 10−7 and 10−8 ). The region with abundance x0 and xD are defined by the temperature of 100 K, where water evaporates from dust grains. A velocity profile after Shu (1977) is implemented. 109

1 · 10−4

250

2 · 10−5

2 · 10

−6

1 · 10−6 5 · 10−7

10 Density (cm3 )

5 · 10

−6

8

Temperature Density Infall Velocity Abundance

107

106

nH2 (r) ∝ r−α

nH2 (r0 )

2 · 10−7 1 · 10−7

xD

200

150

100

−1

−2

Velocity (km s−1 )

1 · 10

0

Temperature (K)

Abundance oH2 O/H2

5 · 10−5

−5

300

x0

−3

50 −4

105 1015

1016

Radius (cm)

Figure 5.12: Problem 3: Example physical model for the water radiative transfer models (Lbol = 2L⊙ , α = 1.5, nH2 (r0 ) = 4 × 105 cm−3 , x0 = 10−4 and xD = 10−6 ) For the calculation of the radiative transfer models, para and ortho water can be treated separately. An ortho to para ratio of water of 3:1 is used. For simplicity, we will only discuss ortho water in this section. Also, we will only carry out the calculation using the 1D version of SPEC. The same molecular and dust properties as van Kempen et al. (2008) are used. We also use the same grid for the calculation which consists of 200 shells. Note that the calculation with Ratran has been defined as convergened by a signal-to-noise ratio of 5 and the level population can be affected with an uncertainty of order 20 %. To compare the results of SPEC and Ratran, we calculate beam convolved spectra of the water lines that can be observed using the Herschel Space Observatory. Molecular data and the size of the Herschel beam is summarized in Table 5.3. The raytracing and beam convolution is performed using TRAC.

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5. A computer program for fast multidimensional modelling of molecular line emission

Table 5.3: Problem 3: Molecular data and Herschel-beam sizes of ortho-H2O. Transition Frequency νul Eup Beam (GHz) (K) (”) 110 − 101 556.936 61.0 39 212 − 101 1669.905 114.4 13 221 − 110 2773.977 194.1 8 221 − 212 1661.008 194.1 13 303 − 212 1716.770 196.8 12 312 − 221 1153.127 249.4 19 312 − 303 1097.365 249.4 21 321 − 312 1162.912 305.3 19 423 − 303 2640.474 323.5 8 Convergence The calculation of one model of the grid using SPEC took about 5-10 minutes and the whole grid of water models can be calculated in a few hours on a standard personal computer. The calculation of one of the models using Ratran on the other hand takes as long as a day. This corresponds to an acceleration by a factor of more than 100. In this section, we discuss the convergence of one of the water models to test the improvement by the local iteration approach (Section 5.3). The convergence of the model with Lbol = 25L⊙ , α = 1.5, nH2 (r0 ) = 4 × 105 cm−3 , x0 = 10−4 and xD = 10−7 is studied in the following. Figure 5.13 gives the maximum factorial change in the level population of all cells per global iteration. A model including the local iteration approach and a model without are calculated. The model including the local iteration shows a peak change in the first few iterations. This is due to levels and cells that are far out of LTE conditions, used as initial level populations. The changes then decrease quickly and are below 10−3 after the 13th iteration. Convergence is then slower but after the 26th iteration, all relative changes are below 10−4 and the model is considered converged. The model without local iteration does not show such a peak and smaller changes until the 7th iteration. However, the convergence then is very slow and the changes not below 0.1 until the 40th iteration. The convergence then is even slower and convergence is not reached even after 10000 global iterations. For one step of the model with local iteration, the local radiation field and the rate equations are solved several times. Depending on the convergence of a single cell, a maximum of 7000 solutions per cell is calculated until global convergence of the model is reached. The model without the local iteration acceleration however still shows changes of order 10−2. Thus, the local iteration approach is necessary to reach convergence at all. This is similar to the convergence behaviour in exact radiative transfer. Note that the approach of locally updating the level population in a cell until convergence is also employed in the Ratran code (Hogerheijde & van der Tak 2000).

5.4. Applications

121

With local iteration Without local iteration

Maximum Factorial Change

104

102

100

10−2

Converged

10−4 0

5

10

15

20

25

30

35

40

Global Iteration Figure 5.13: Problem 3: Convergence history of a water model discussed in Section 5.4.3. The maximum change of the levels between subsequent global iterations is shown for a model with and without the local iteration acceleration. Results An example of spectra obtained from SPEC and Ratran is given in Figure 5.14. This figure shows the water-lines given in Table 5.3, observable by Herschel. The model with Lbol = 2L⊙ , α = 1.5, nH2 (r0 ) = 4×105 cm−3 , x0 = 10−4 and xD = 10−6 is used. For many lines, we find a similar behaviour as discussed before, with SPEC lines too low in the line center and too high in the line wings. To quantify the accuracy of the solution, we use the factorial deviation of the beam convolved and continuum subtracted line. The factorial deviation between the exact and approximative solution is within about 1.4 (40 %). How well is the agreement of the water lines obtained using the exact method and the approximative method implemented in SPEC? A statistic of the factorial deviation of all water lines, derived from the Ratran and SPEC level populations is given in Figure 5.15. The Figure shows the fraction of water lines with deviation smaller than a certain value. A total of 108 models with 9 lines each, thus 972 water lines have been taken into account. We find the 69.4 % of the lines have deviations less than 1.3 (about 30 %), which we consider as a good agreement (Section 5.4.2). We define disagreement by more than a factor of 1.75 as a bad agreement and find 6.9 % of the lines to disagree by more than this factor. The largest disagreement is by a factor of 5.7. These outliers will be discussed in the next section. The average deviation is a factor of 1.28. We conclude that the overall agreement of the water lines obtained by SPEC compared to an exact radiative transfer is generally good, even if such a complicated problem as water excitation is calculated.

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5. A computer program for fast multidimensional modelling of molecular line emission

0.8

a.) 110 − 101 (556 GHz)

1

Factorial Deviation: 1.01

b.) 212 − 101 (1669 GHz)

0.5

Factorial Deviation: 1.31

0.8

c.) 221 − 110 (2773 GHz)

Factorial Deviation: 1.09

0.4

0.6

AMC SPEC

T (K)

0.4

0.3

AMC SPEC

T (K)

T (K)

0.6

AMC SPEC

0.4

0.2

0.2

0.1

0.2

0

0 −5 −4 −3 −2 −1

0

1

2

3

4

V (km s−1 )

0.4

0 −5 −4 −3 −2 −1

5

0

1

2

3

4

−5 −4 −3 −2 −1

5

V (km s−1 )

d.) 221 − 212 (1661 GHz)

0.5

Factorial Deviation: 1.37

0

1

2

3

4

5

V (km s−1 )

e.) 303 − 212 (1716 GHz)

0.175

Factorial Deviation: 1.25

f.) 312 − 221 (1153 GHz)

Factorial Deviation: 1.10

0.15

0.4 0.3

0.125

AMC SPEC

T (K)

AMC SPEC

0.2

T (K)

T (K)

0.3

AMC SPEC

0.1

0.2 0.075 0.1 0.1

0

0 −5 −4 −3 −2 −1

0

1

2

3

4

0

1

2

3

4

−5 −4 −3 −2 −1

5

V (km s−1 )

g.) 312 − 303 (1097 GHz)

0.25

Factorial Deviation: 1.11

0.12

0.025 −5 −4 −3 −2 −1

5

V (km s−1 )

0.14

0.05

0

1

2

3

4

5

V (km s−1 )

h.) 321 − 312 (1162 GHz)

0.4

Factorial Deviation: 1.34

i.) 423 − 303 (2640 GHz)

Factorial Deviation: 1.14

0.2 0.3

0.1

AMC SPEC

T (K)

0.08

T (K)

T (K)

0.15

AMC SPEC

AMC SPEC

0.2

0.1 0.06 0.1 0.05

0.04

0.02

0 −5 −4 −3 −2 −1

0

1

V (km s−1 )

2

3

4

5

0 −5 −4 −3 −2 −1

0

1

V (km s−1 )

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

V (km s−1 )

Figure 5.14: Problem 3: Water lines obtained from SPEC and Ratran for the model with Lbol = 2L⊙ , α = 1.5, nH2 (r0 ) = 4 × 105 cm−3 , x0 = 10−4 and xD = 10−6. The lines are convolved to the Herschel beam. The grey shaded area gives a deviation of 30 % to the Ratran results.

5.4. Applications

123

1 0.9 0.8

Bad Agreement

Fraction of Lines

0.7 0.6

Good Agreement

0.5 0.4 0.3 0.2 0.1 0 1

1.2

1.4

1.6

1.8

2

Factorial Deviation Figure 5.15: Problem 3: Fraction of water lines derived from the grid of water models with factorial deviation less than a certain value. To study the molecular lines deviating most, Table 5.4 gives average factorial deviations, maximum factorial deviations and number of lines with disagreement by more than a factor of 1.75 for different classes of models. The classes consists of the different physical conditions and also the different lines. We find that considerable differences in the agreement between the lines exist (2-10 in Tab. 5.4). For example, the 556 GHz (110 - 101 ) groundstate line has the largest average factorial deviation and many cases with disagreement larger than a factor of 1.75. On the other hand, the 2640 GHz (423 - 303 ) line does not deviate much having a maximum factorial deviation of 1.57. Clearly, lines with high critical density and upper level energy agree better. However, also the size of the beam is important, as the example of the 1162 GHz (321 - 312 ) line shows. Thus, lines which emit mostly from the inner (more thermalized) region agree better. However, as we take dust radiation and attenuation into account, the inner part of the envelope can at the shorter wavelengths be shielded by the dust attenuation and the lines emerge from further out, despite the lower excitation. The effect of different physical conditions and chemical abundance on the accuracy is more difficult to study, as different parameters affect each other. For example, the largest deviation in all lines is found in the 1669 GHz (212 - 101 ) line of a model with Lbol = 2L⊙ , α = 1.5, nH2 (r0 ) = 106 cm−3 , x0 = 10−4 and xD = 10−6 , where total line and beam integrated deviation amounts to a factor of 5.68 (Figure 5.16). This is because water in the outer part of the model can still considerably contribute to the total line flux and those molecules are pumped by molecules further in. This however requires the special condition of density and luminosity, as models with larger density would shield the inner part to the outer part in these wavelength due to dust absorption. Models with increased luminosity are larger and the relative contribution of the more accuratly determined inner part is larger. Indeed for an abundance combination of x0 = 10−4 and xD = 10−6 , only this particular model shows a deviation larger than a factor of 1.75.

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5. A computer program for fast multidimensional modelling of molecular line emission

Beam (13”)

a.) Lbol = 2L⊙ , α = 1.5, nH2 = 1(6), x0 = 1(−4), xD = 1(−6)

10−1

2.5

Line: 221 - 212 (1661 GHz) Deviation: 5.68 Ratran SPEC

1.5 10−2 1

212 221 212 221

10−3

10−4 1014

(Ratran) (Ratran) (SPEC) (SPEC)

0.5

0 1015

1016

1017

−4

Radius (cm)

−2

0

2

4

Velocity (km s−1 )

b.) Lbol = 7L⊙ , α = 2, nH2 = 4(5), x0 = 1(−6), xD = 1(−7)

Beam (39”)

10−1

1

Line: 110 - 101 (556 GHz) Deviation: 4.29 Ratran SPEC

0.6

10−2 0.4

101 110 101 110

10−3

10−4 1014

(Ratran) (Ratran) (SPEC) (SPEC)

0.2

0 1015

1016

1017

−4

−2

0

Velocity (km s

Radius (cm)

2 −1

4

)

c.) Lbol = 2L⊙ , α = 2, nH2 = 4(5), x0 = 1(−4), xD = 1(−8)

0.25

100

Line: 110 - 101 (556 GHz) Deviation: 4.01 Ratran SPEC

Beam (39”)

10−1

−2

10

101 110 101 110

10−3

10−4 1014

(Ratran) (Ratran) (SPEC) (SPEC)

0.2

0.15

0.1

0.05

0 1015

1016

1017

−4

Radius (cm)

−2

0

2

4

Velocity (km s−1 )

d.) Lbol = 2L⊙ , α = 2, nH2 = 4(5), x0 = 1(−4), xD = 1(−8) 100

1

Line: 212 - 101 (1669 GHz) Deviation: 3.24 Ratran SPEC

0.8

0.6

10−2 0.4

10−4 1014

101 212 101 212

(Ratran) (Ratran) (SPEC) (SPEC)

Intensity (K)

Beam (13”)

10−1

10−3

Intensity (K)

Normalize Population

0.8

Intensity (K)

Normalize Population

100

Normalize Population

2

Intensity (K)

Normalize Population

100

0.2

0 1015

1016

Radius (cm)

1017

−4

−2

0

2

4

Velocity (km s−1 )

Figure 5.16: Problem 3: Water lines with largest deviation between the exact solution and SPEC. The population of the upper and lower level together with the spectra of the line is given. The green vertical line indicates the Herschel beam.

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125

Table 5.4: Problem 3: Agreement between Ratran and SPEC for the grid of water models. Modells #a Avg. Max. Badb 1 All 972 1.28 5.68 67 2 556 GHz Line 108 1.64 4.29 35 3 1669 GHz Line 108 1.32 3.24 11 4 2773 GHz Line 108 1.12 1.63 0 5 1661 GHz Line 108 1.36 5.68 6 108 1.29 2.91 5 6 1716 GHz Line 7 1153 GHz Line 108 1.14 1.76 1 8 1097 GHz Line 108 1.17 2.43 2 9 1162 GHz Line 108 1.30 1.98 7 10 2640 GHz Line 108 1.17 1.57 0 11 α = 1.5 486 1.25 5.68 24 12 α = 2.0 486 1.31 4.29 43 13 nH2 (r0 ) = 4(5) cm−3 324 1.35 4.29 35 14 nH2 (r0 ) = 1(6) cm−3 324 1.29 5.68 23 −3 15 nH2 (r0 ) = 5(6) cm 324 1.19 3.07 9 16 Lbol = 2L⊙ 324 1.28 5.68 30 17 Lbol = 7L⊙ 324 1.30 4.29 20 18 Lbol = 25L⊙ 324 1.26 2.76 17 19 x0 = 1(−4), xD = 1(−6) 162 1.35 5.68 18 20 x0 = 1(−4), xD = 1(−7) 162 1.26 2.06 9 21 x0 = 1(−4), xD = 1(−8) 162 1.26 4.01 10 22 x0 = 1(−6), xD = 1(−6) 162 1.29 3.00 12 23 x0 = 1(−6), xD = 1(−7) 162 1.28 4.29 8 24 x0 = 1(−6), xD = 1(−8) 162 1.23 2.52 10 a b

Total number of lines. Number of lines with factorial deviation larger than 1.75

While individual combinations of physical/chemical conditions determine the accuracy of individual lines, we can still give some trends. From Table 5.4, we conclude the following: (i) A steeper density gradient (α = 2) results in larger deviations as it leads to lower densities and thus less thermalized levels in the outer part (Lines 11 & 12 in Tab. 5.4). (ii) Higher densities result in better agreement, as the level population is more thermalized by collisions (Lines 13-15 in Tab. 5.4). (iii) The bolometric luminosity leaves the agreement relatively unchanged, with some better agreement in the lines with generally large disagreement for higher luminosities. This is due to the models being larger for higher luminosities and thus the relative contribution of the inner, more termalized population is larger. (iv) Chemical abundances (19-24 in Tab. 5.4) affect the accuracy only little with the exception of the combination x0 = 10−4 and xD = 10−6 , as discussed before. We conclude that while clearly some extreme physical conditions may simplify the calculation considerably and result in good accuracy, it is difficult to predict upon the physical conditions whether the agreement between SPEC and the full radiative transfer is good. Which model disagree most? For the four lines where the largest deviations have been found, we give in Figure 5.16 the beam convolved lines and the level population of the upper and lower level. The four models have in common that SPEC underestimates the level population in the outer parter, where the level population may still be considerably and contributes significantly to the observed

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5. A computer program for fast multidimensional modelling of molecular line emission

flux. For these models, the outer part is not shielded by dust. Note that due to the too steep decrease of the higher level populations, the 101 groundstate is overpredicted and thus the line opacity of the 556 GHz (110 - 101 ) and 1669 GHz (212 - 101 ) lines are too large. We conclude that the new method is able to deal with complex models of the water radiation from protostellar envelopes including the effects of dust pumping. However, some outliers with deviations of up to a factor of a few exist. These are mainly lines with low frequency and low upper level energy, which have very high optical depth and, in addition, the large Herschel beam picks up more of the outer part of the envelope. In this part, the new method usually yields less accurate level populations. However, the agreement depends on the detailed physical conditions.

5.5

Conclusions

In this chapter, we have developed a fast approximative method to calculate the molecular excitation and line radiation. It can be applied on 1D, 2D or 3D models. The method is based on the well-known escape probability formalism but can be applied on any complex physical structures. It still keeps the computational advantages of the standard escape probability approach. Despite the method is much simpler to implement and faster than ALI or Monte-Carlo codes, it generally shows a good agreement. In order to test the applicability of the approximation made in the method, different problems are calculated and the results compared to exact solutions. The main conclusions of this chapter are: 1. As simple multizone escape probability method can used to model line fluxes of complex physical structures and molecules (e.g. water) with deviations in the calculated line flux typically of order less than 30 % (Section 5.4.1, 5.4.2 and 5.4.3). 2. Deviations exceeding the 30 % limit, between the exact and approximative method are found, if the line radiation emerges from region where the level population is in transition between the thermalized and optically thin case (Section 5.4.2). 3. The approach of locally repeating the calculation until convergence considerably accelerates the convergence (Section 5.3 and 5.4.3). 4. Taking into account the exact lineshape for the calculation of the escape probability and e.g. allowing photons to escape through linewings improves the accuracy considerably (Section 5.4.1). The method presented here can be applied in many different astrophysical enviroments from AGB envelopes to protoplanetary discs. The fast calculation allows to study many different models with a complex geometry and will thus be a valuable tool to study upcoming high-resolution data such as e.g. from the ALMA. In combination with the grid of chemical models by Bruderer et al. (2009c), the method allows to quickly calculate molecular line fluxes based on a detailed physical/chemical model (e.g. Chapter 6).

Acknowledgments We would like to thank Steven Doty, Pascal St¨auber, Arnold Benz and Susanne Wampfler for useful discussions and Tim van Kempen for providing his grid of water radiative transfer models in electronic form. The work was supported by the Swiss National Science Foundation grant 200020-113556.

Chapter 6

The influence of geometry on the abundance and excitation of diatomic hydrides

Abstract: The Herschel Space Observatory opens the sky for observations in the far infrared at high spectral and spatial resolution. A particular class of molecules will be directly observable; light diatomic hydrides and their ions (CH, OH, SH, NH, CH+ , OH+ , SH+ , NH+ ). These simple constituents are important both for the chemical evolution of the region and as tracers of high-energy radiation. If outflows of a forming star erode cavities in the envelope, protostellar far UV (FUV; 6 < Eγ < 13.6 eV) radiation may escape through such low-density regions. Depending on the shape of the cavity, the FUV radiation then irradiates the quiescent envelope in the walls along the outflow. The chemical composition in these outflow walls is altered by photoreactions and heating via FUV photons in a manner similar to photo dominated regions (PDRs). In this chapter, we study the effect of cavity shapes, outflow density, and of a disk with the two-dimensional chemical model of a high-mass young stellar object introduced in Chapter 3. The model has been extended with a self-consistent calculation of the dust temperature and a multi-zone escape probability method for the calculation of the molecular excitation and the prediction of line fluxes. We find that the shape of the cavity is particularly important in the innermost part of the envelope, where the dust temperatures are high enough (& 100 K) for water ice to evaporate. If the cavity shape allows FUV radiation to penetrate this hot-core region, the abundance of FUV de-

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stroyed species (e.g. water) is decreased. On larger scales, the shape of the cavity is less important for the chemistry in the outflow wall. In particular, diatomic hydrides and their ions CH+ , OH+ and NH+ are enhanced by many orders of magnitude in the outflow walls due to the combination of high gas temperatures and rapid photodissociation of more saturated species. The enhancement of these diatomic hydrides is sufficient for a detection using the HIFI and PACS instruments onboard Herschel. The effect of X-ray ionization on the chemistry is found to be small, due to the much larger luminosity in FUV bands compared to X-rays.

Simon Bruderer, Arnold O. Benz, Pascal St¨auber & Steven D. Doty Accepted by the Astrophysical Journal

6.1

Introduction

Observations of young stellar objects (YSOs) envelopes reveal a rich variety of morphologies (e.g. Arce & Sargent 2006, Jørgensen et al. 2007). Many of the observed features are associated with outflows: For example, strong high-J CO emission along a cavity, etched out by the outflow, is found in the class I object HH 46 (van Kempen et al. 2009b). They explain the CO narrow emission with a warm and FUV heated region, with FUV photons from either the bow shock, internal jet working surfaces or the accretion disk boundary layer. In this scenario, FUV radiation may travel freely in the low-density cavity without being absorbed or scattered. Similarly, Bruderer et al. (2009a) have observed molecular tracers for warm, dense and FUV irradiated gas along the outflow of the highmass young stellar object AFGL 2591. Since the line shape does not show clear signs of shocks, they suggest the FUV radiation to be of protostellar origin. Indeed, the hot photosphere of a luminous young O or B star emits mostly at FUV wavelengths and a cavity that allows protostellar radiation to escape has also been observed in that source (Preibisch et al. 2003). This scenario of irradiated outflow walls along a cavity, swept out by an outflow, has also been used to explain other observations. For example, Jørgensen (2004) found emission of FUV enhanced CN along an outflow of a low-mass class 0 source. Van Kempen et al. (2009c) explain the water maser emission at 183 GHz in Ser SMM 1 with the increased water abundance in the warm, FUV or shock heated outflow walls. Outflow walls can also be sites of anomalous molecular excitation. For example Hogerheijde et al. (1998, 1999) observe strong emission of HCN and HCO+ along the outflow of L 1527 and Ser SMM 1. These molecules have a large dipole moment and are efficiently excited by collisions with electrons, mixed from the ionized outflow into the outflow wall. Rawlings et al. (2004) explain the strong HCO+ emission by shock liberation of molecules from ice mantles followed by photoprocessing. Spaans et al. (1995) model the strong 13 CO J = 6 → 5 emission in narrow lines observed toward several low-mass sources (e.g. TMR-1) by FUV heated outflow walls. Bruderer et al. (2009b) (Chapter 3) explain the observed high abundance of the FUV enhanced molecule CO+ in AFGL 2591 by a concave cavity that allows direct irradiation onto the outflow walls. Diatomic hydrides and their ions (XH, XH+ ) are corner-stones of the chemical network. For example water at high temperature (& 250 K) is mainly formed by the reaction of OH + H2 . Due to their chemical properties, discussed in this work, they trace dense, warm and FUV or X-ray irradiated gas like in outflow walls. Such tracers are valuable to constrain the FUV or X-ray radiation of embedded sources. This is of particular interest for the innermost part of low-mass YSOs with protoplanetary disks being ionized and heated by FUV radiation and X-rays (e.g. Bergin 2009). Here, we discuss the use of light diatomic hydrides with the common elements (X=O, C, N, S) as tracers of protstellar high-energy radiation. The chemistry of other diatomic hydrides with heavier

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129

(e.g. X=Fe) and/or less abundant elements (e.g. X=F) is not discussed here, as their chemistry is poorly known and their use as tracers currently limited. However, with the ongoing search for new diatomic hydrides using the new Herschel Space Observatory (e.g.Cernicharo et al. 2010), this sitation may change. For example Neufeld et al. (2010) report the detection of HF using the HIFI instrument onboard Herschel. Due to the dissociation energy of HF being larger than of H2 , HF can be formed in an exothermic reaction of F with H2 and HF may become the dominant reservoir of fluorine (Neufeld & Wolfire 2009). The silicon bearing SiH and SiH+ may be interesting in the context of shocked regions where “sputtering” from dust grains (e.g. Flower et al. 1996) can increase the gas phase abundance of silicon. The Herschel Space Observatory and Band 10 receivers of the upcoming Atacama Large Millimeter and submillimeter Array (ALMA) allow the study of important FUV and X-ray tracers, like diatomic hydrides, at high sensitivity and with good spatial and spectral resolution for the first time. This raises the questions: What is the expected abundance and line flux of FUV enhanced species in the scenario of directly irradiated outflow walls? From which region of the envelope does the molecular radiation emerge and how are the molecules excited? How dependent are observable quantities (line fluxes or line ratios) on characteristics that cannot be constrained directly from observations (e.g. geometry in the innermost part of the envelope and chemical age)? To attack such questions, detailed numerical models of the chemistry of YSOs envelopes can be used. Models of the chemistry of YSO envelopes have so far mostly assumed a slab or spherical symmetry and a static physical structure (e.g. Viti & Williams 1999, Doty et al. 2002). In situations with chemical timescales longer than dynamical timescales, the chemistry needs to be coupled with the dynamical evolution (e.g. Doty et al. 2006). A particular example for this situation is the slow sulfur chemistry (Wakelam et al. 2004). Other models focus on the effect of protostellar FUV and X-ray radiation on the chemistry of a spherical envelope (St¨auber et al. 2004, 2005). Due to the high extinction of FUV radiation by dust, they find the chemistry to be modified only in the innermost few hundred AU. X-rays on the other hand, have a much lower absorption cross section. They may thus penetrate further into the envelope and dominate the ionization in the entire hot core. To study the effects of an outflow cavity that allows FUV radiation to escape and irradiate a larger part of the envelope, two-dimensional axisymmetric models are necessary. Non-spherical chemical models of YSO envelopes are presented by Brinch et al. (2008), Visser et al. (2009) and in Chapter 3. These models assume axisymmetry. The first two are applied on low-mass sources and include the effects of infall to study the chemical evolution from the cloud to the disk. The model in Chapter 3 is applied on a high-mass star forming region. It calculates the gas temperature self-consistently with the chemical abundances similar to models of photo dominated regions (PDRs; e.g. Kaufman et al. 1999, Meijerink & Spaans 2005), but it does not account for the dynamical evolution. The model presented in Chapter 3 considers FUV (6 < Eγ < 13.6 eV) and X-ray (Eγ > 100 eV) induced chemistry. In this chapter, we further develop the model introduced in Chapter 3. It is based on the grid of chemical models introduced in Chapter 2 (Bruderer et al. 2009c). We extend the model with two important tools: (i) with a dust radiative transfer method for a self-consistent calculation of the dust temperature and (ii) a multi-zone escape probability method to study the molecular excitation and radiation. We apply the model to AFGL 2591 to answer the questions raised above. In particular we predict line fluxes of two selected diatomic hydrides (SH and CH+ ), having typical properties of diatomic hydrides with high temperatures needed for formation and high critical densities. These two molecules will be observed in AFGL 2591 by the Water In Star-forming regions with Herschel (WISH1 ; van Dishoeck et al. 2010, in prep.) guaranteed time key program. This chapter is organized as follows. We first describe the details of the model in section 6.2. 1

www.strw.leidenuniv.nl/WISH

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

The physical structure of different models is discussed in section 6.3. The chemical abundances obtained from these physical structures are presented in section 6.4, and compared to the results of one-dimensional models. The molecular excitation and line radiation is then studied in section 6.5, and the scientific prospectives of hydride observations with Herschel are discussed in 6.6. Limitations of the model used in this chapter are given in Section 6.7.

6.2

Model

In this section, we briefly describe the calculation flow of the models used here. Details on the new parts of the model, the dust radiative transfer and the molecular excitation calculation, are given in Section 6.2.1 and 6.2.2, respectively. The modeling process starts with assuming a density structure and the spectrum of the central source. A dust radiative transfer calculation then solves for the dust temperature. The local FUV and X-ray radiation is calculated at every point of the model as described in Section 6.2.1. In the next step, the gas temperature is obtained by equating the heating and cooling rates (Chapter 3). Since these rates depend on the chemical abundances (e.g. of the atomic coolants C+ and O) the calculation of the gas temperature has to be performed simultaneously with the chemistry. This time-consuming step is facilitated by the grid of chemical models introduced in Chapter 2. The grid consists of a database of precalculated abundances depending on physical parameters (density, temperature, FUV flux, X-ray flux, cosmic-ray ionization rate and chemical age; see Chapter 2 for a definition of these parameters) from which abundances can be interpolated quickly. The interpolation approach is also used to obtain the abundances of all other species (e.g. diatomic hydrides). The molecular excitation is calculated using the escape probability approach described in Section 6.2.2 and yields the modeled line radiation, thus the observable quantities.

6.2.1

Dust Radiative Transfer

The dust temperature in steady state condition is obtained by equating the total dust emission Γemit (erg s−1 g−1 ) with the total absorption Γabs (erg s−1 g−1 ). Dust may absorb either stellar photons or ambient emission by dust. Assuming a single dust temperature TDust , we can write the energy balance at one position of the envelope as Z Z Z Γabs = κλ Iλ (Ω)dλdΩ = 4π κλ Bλ (Tdust )dλ = Γemit , (6.1) with κλ (cm2 g−1 ) the dust absorption opacity per mass, Iλ (Ω) (erg s−1 cm−2 µm−1 sr−1 ) the intensity depending on the direction and wavelength and Bλ (TDust ) the Planck function. To solve this coupled problem, we implement the Monte Carlo approach presented by Bjorkman & Wood (2001). The results of the new code have been verified with the benchmark tests suggested by Ivezic et al. (1997) and Stamatellos & Whitworth (2003) and also by comparing the results with the DUSTY code (Ivezic & Elitzur 1997; Nenkova et al. 1999). For the models of AFGL 2591, we adopt the dust opacities by Ossenkopf & Henning (1994) (column 5). This dust opacities have been found to explain the observation of AFGL 2591 well and yield a “standard” dust to molecule mass ratio of about 100 (van der Tak et al. 1999). For the calculation of the FUV intensity, the dust opacities are extended to shorter wavelengths (λ < 1µm) with the absorption and scattering properties by Draine (2003a), available on his web-page (similar to Chapter 3). An example of calculated dust temperature is given in Figure 6.1 for the model used in Chapter 3. The figure also shows the calculation domain which consists of 5 nested grids.

6.2. Model

131

6000

z (AU)

150

4000

100

2000

50

0

Dust Temperature Tdust (K)

200

0 0

2000

4000

6000

r (AU) Figure 6.1: Inner region of the calcalultion grid and the dust temperature of an axisymmetric model of AFGL 2591. The dust radiative transfer code also calculates the local mean intensity of scattered and attenuated stellar photons with a Monte Carlo method similar to van Zadelhoff et al. (2003). The analytical expression for the photodissociation and ionization rates (e.g. van Dishoeck 1988) implemented in Chapter 2 require the calculation of the unattenuated FUV flux G0 [ISRF] and the visual extinction AV . We estimate these quantities by the mean intensities at both edges of the FUV band (6 and 13.6 eV) and at 1 micron. We have tested the approach by comparing photoionization and dissociation rates obtained from the fit and calculated by the FUV cross sections of van Dishoeck et al. (2006)2. We find agreement within a factor of two in the chemically relevant range of attenuation (AV < 10). This is reasonable provided that we do not implement the same dust properties and spectral shape of the emitting spectra (see Section 6.7 for a discussion of the spectral shape). The FUV calculation is also tested against the results in Chapter 3. We find good agreement, for example the mass and volume of material irradiated with an attenuated flux larger than 1 ISRF agree to within 30 %.

6.2.2

Molecular Excitation Analysis

For the calculation of the level population of SH and CH+ , a multi-zone escape probability method is used. This section briefly discusses the method. Further details of the code are given in Chapter 5. The excitation of a molecule or atom (i.e. the population of different energy levels) is controlled by collisional and radiative processes. The species can be excited or deexcited by collisions with H2 , H and electrons or by radiation either from the dust continuum or molecular emission from the 2

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

same or other molecules. The population ni (cm−3 ) of level i at one position is described by the rate equations (e.g. van der Tak et al. 2007) N

N

X dni X = nj Pji − ni Pij + Fi − ni Di ≡ 0 , dt j6=i j6=i

(6.2)

where N is the number of levels considered, Fi (cm−3 s−1 ) and Di (s−1 ) are the chemical formation and destruction rates and Pij (s−1 ) is given by  Aij + Bij hJij i + Cij (Ei > Ej ) Pij = (6.3) Bij hJij i + Cij (Ei < Ej ) . Here, Aij (s−1 ) and Bij (erg−1 cm2 Hz) are the Einstein coefficients for spontaneous and induced emission, respectively. The collisional excitation and deexcitation rates Cij (s−1 ) are obtained from the sum of ncol Kij (Tkin ) for different collision partners, with the density of the collision partners ncol and the collision rate coefficients Kij (Tkin ) (cm3 s−1 ) at a kinetic temperature Tkin . The level population is assumed to be constant with time and thus dni /dt = 0 in Eq. 6.2. The level distribution at formation may govern the level population if a molecule is destroyed in collisions rather than excited, like CH+ in collisions with H, H2 or electrons. We take the effects of excited formation and destruction into account by the terms Fi and ni Di in Eq. 6.2. Due to the short chemical time-scales, we assume the abundance to be constant. At formation, we assume the level population to follow a Boltzmann distribution with temperature Tform (St¨auber & Bruderer 2009; van der Tak et al. 2007). The destruction is assumed to be independent of levels. Hence, X X F = Fi = F gi e−Ei /kTform /Q(Tform ) i

=

X i

i

ni Di = D

X

ni = Dn(CH+ ) ,

(6.4)

i

with the partition function Q(T ), the statistical weight gi of a level with energy Ei . For the calculation of the destruction rate D, we take reactions with H, H2 and electrons into account. Thus D = k1 n(H) + k2 n(H2 ) + k3 n(e− ), with the reaction rate coefficients k1 , k2 and k3 . In this approach, the formation temperature Tform remains a free parameter. Since the level populations at different positions of a model are coupled by the radiation field, the solution of a radiative transfer problems can be very time-consuming, especially in multi-dimensional geometries or for molecules with high optical depth like water. We thus use the escape probability approach including continuum emission and absorption presented by Takahashi et al. (1983) and approximate the ambient radiation by hJij i ≈ (1 − ǫij )B(Tex,ij , νij ) + (ǫij − ηij )B(Tdust , νij ) +ηij B(TCMB , νij ) ,

(6.5)

with the line frequency νij and the Planck functions B(T, ν) for the excitation temperature Tex , the temperature of the dust Tdust and the cosmic microwave background TCMB . The escape probabilities ǫij and ηij give the probability for a photon to escape line absorption (ǫij ) or to escape both line and dust absorption (ηij ). They are calculated using Eq. 3.11 - 3.13 given in Takahashi et al. (1983). The necessary optical depth of dust and lines is obtained from the summation along different rays for different frequencies. This approach couples different cells and the problem is solved iteratively, similar to one-zone escape probability codes (e.g. RADEX, van der Tak et al. 2007). To improve convergence, we use

6.2. Model

133

an ALI-like acceleration mechanism (e.g. Rybicki & Hummer 1991 or Hogerheijde & van der Tak 2000). To test the code, we have run several benchmark problems, e.g. van der Tak et al. (2005), with results typically within 30 % to the exact solution. A similar method has also been successfully used by Poelman & Spaans (2005). The velocity field and the intrinsic line width enter the excitation calculation by the ambient molecular radiation field. These parameters are however only important for optically thick lines and the line shape. Optically thin lines are only little affected. The predicted velocity field by Doty et al. (2006) (after McLaughlin & Pudritz 1997) for young chemical ages of less than a few times 104 yrs is approximately static outside a few 1000 AU and we thus assume a static situation. The line width mainly due to microturbulence is assumed to be 1.6 km s−1 (Doppler parameter, e.g. St¨auber et al. 2007) except in the outflow wall, where a larger line width of 4.2 km s−1 is assumed as indicated by interferometric observations (Bruderer et al. 2009b).

6.2.3

A grid of density structures/cavity shapes

A spherical model of AFGL 2591 has been constructed by van der Tak et al. (1999, 2000) based on maps of the dust continuum, the spectral energy distribution (SED) and molecular line emission. In their fitting, they find good agreement on larger scales. However, continuum emission at shorter wavelengths (warm dust) requires a shallower density profile in the innermost 10′′ − 20′′ . They conclude that an inner region with roughly constant temperature may be required to reproduce all observations. Jørgensen et al. (2005a) encountered similar problems of an excess emission in short wavelengths when fitting the continuum emission of IRAS 16293-2422. They concluded that a large spherical cavity can explain this excess. However, Crimier et al. (2010) can explain both SED and maps by a higher bolometric luminosity. In AFGL 2591, new high resolution observations at 24.5 µm by de Wit et al. (2009) show extended emission at scales of up to 10′′ in the direction of the outflow. This emission stems from the outflow walls (Bruderer et al. 2009a), supporting the scenario of a cavity. For this chapter, we adopt the density structure of the spherical model by van der Tak et al. (1999). This choice allows to study the differences of the two-dimensional chemical model including an outflow cavity with the spherically symmetric models by Doty et al. (2002) and St¨auber et al. (2004, 2005). The cavity shape is assumed to follow a power-law with index b, z=

z0b−1

1 rb , b tan(α/2)

(6.6)

where z and r are coordinates along the outflow axis and perpendicular to the outflow, respectively. The full opening angle α of the outflow is given at a distance z0 . As in Chapter 3, we use z0 = 10000 AU, approximately the scale on which Preibisch et al. 2003 have probed the opening angle of the outflow cavity. Cavity shapes for different choices of α and b are given in Figure 6.2. The solid line indicates the model used in Chapter 3. Figure 6.2 also shows the line of sight toward us crossing the outflow wall at a shallow angle, as suggested by van der Tak et al. (1999). A further parameter of the model is the density in the cavity, which we assume to be reduced by a constant factor γ = nout /nin compared to the spherical density model at the same radius. This is motivated by the pressure equilibirum along the outflow wall (Chapter 3). Note that for γ = 2.5 × 10−3 , used in most of the models, the results are independent of that assumption as the opacity in FUV wavelengths along the outflow is low. A grid of different models is discussed in the following sections. Starting from the model used in Chapter 3 (referred as model Standard in the following) we explore the parameter range by varying the outflow opening angle α, the power-law index b, cavity density γ and the bolometric luminosity

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

3 · 104

nout nin

Line of sight

Outflow

γ=

z [AU]

2 · 104

Envelope z0

1 · 104

z ∝ rb α = 62◦ , b = 2 α = 62◦ , b = 1.5 α = 62◦ , b = 2.5 α = 40◦ , 80◦ , b = 2

α/2

0 0

1 · 104

r [AU]

2 · 104

3 · 104

Figure 6.2: Separation between envelope and outflow using the power-law description (Equation 6.6). The line of sight suggested by van der Tak et al. (1999) is given by an arrow. of the protostar Lbol (Table 6.1). As in Chapter 3, we assume the central source to emit a black body spectrum with a temperature of 3 × 104 K following van der Tak et al. (1999) (see Section 6.7). Thus, the FUV field has photons that can dissociate CO and H2 . A compact structure with a systematic velocity gradient has been detected in AFGL 2591 by van der Tak et al. (2006) using interferometric observations. They propose a circumstellar disk to be the source of this radiation and constrain a mass of ≈ 0.8 M⊙ within a radius of 400 AU to the protostar. While a detailed modeling of the disk is beyond the scope of this study, model Disk implements a toy-model of a disk embedded in the envelope of the model Standard. The toy-model consists of a massless (in relation to the protostar) Keplerian disk (e.g. Fischer et al. 1996) with a surface density Σ ∝ r −0.75 , a scale height r/z = 0.15 and a mass of 0.8 M⊙ within a radius of 400 AU to the protostar. The density distribution of the disk and the inner envelope is given in Figure 6.6.

6.3

Physical structure

Different physical conditions such as gas and dust temperature, density, FUV and X-ray irradiation govern the chemical composition and molecular excitation. As a first step to quantify these physical properties of the different envelopes models, we study the amount of material with distinct conditions (e.g. a gas temperature above 100 K). Certain conditions like warm temperatures, a high density and a FUV field stronger than 1 ISRF (interstellar radiation field) can be necessary for the formation and excitation of a molecule. Combinations of these regions (e.g. high density and strong FUV field)

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135

Table 6.1: Parameters of the adopted models. Model α b γ = nout /nin Lbol ◦ [ ] [L⊙ ] Standard 62 2 2.5(-3) 2(4) Diska 62 2 2.5(-3) 2(4) Opening angle α20 20 2 2.5(-3) 2(4) 40 2 2.5(-3) 2(4) α40 α80 80 2 2.5(-3) 2(4) α100 100 2 2.5(-3) 2(4) Cavity density γ0 62 2 0 2(4) γ0.01 62 2 1(-2) 2(4) γ0.1 62 2 0.1 2(4) Cavity shape b1.5 62 1.5 2.5(-3) 2(4) b2.5 62 2.5 2.5(-3) 2(4) Source luminosity L2e3 62 2 2.5(-3) 2(3) L1e4 62 2 2.5(-3) 1(4) 62 2 2.5(-3) 4(4) L4e4 a(b) = a × 10b a Parameters like in the model Standard, but with a disk of mass 0.8 M⊙ in the innermost 400 AU.

can be relevant for the formation and excitation of molecules and thus the predicted line strength that can be compared with observations. While the definition of such physical regions is arbitrary, it gives a simple overview on differences between models and thus the influence of the geometry. In Table 6.2, we show the absolute mass (in M⊙ ) and mass relative to Model Standard (M/Mstandard ) for the following regions: • Tdust , Tgas > 100 K. Several molecules only form at high temperature, like for example the diatomic hydrides discussed in this chapter. We report on the region with dust and gas temperature above 100 K, which is assumed to be the water evaporation temperature (Chapter 2). • FUV. FUV radiation alters the chemical composition by photoprocesses (ionization and dissociation) and also through heating of the gas. The FUV region is defined by the FUV flux larger than 1 ISRF. We recall, however, that the structure of the outflow walls is more complicated and some species only exist in different layers, for example the C+ /C/CO layer structure also seen in PDRs. • X-rays. X-rays affect the chemistry mostly by secondary electrons created by fast photoelectrons ionizing H2 (Maloney et al. 1996; St¨auber et al. 2005). Assuming an X-ray luminosity of 1032 erg s−1 in the 0.1 - 100 keV band as suggested by St¨auber et al. (2005), heating by X-rays can be neglected (Chapter 3 and St¨auber et al. 2005). While the influence of X-rays is very similar to that of a high cosmic-ray ionization rate (Chapter 2), there are differences in FUV irradiated regions (e.g. St¨auber et al. 2006 for a discussion of the water abundance in FUV and

136

6. The influence of geometry on the abundance and excitation of diatomic hydrides X-ray irradiated regions). The X-ray region is defined by the H2 ionization rate larger than the “standard” cosmic-ray ionization rate of ζc.r. = 5.6 × 10−17 s−1 by a factor of about five (ζ > 3 × 10−16 s−1 ). This anticipates the uncertainty in the observational determination of the cosmic-ray ionization rate (van der Tak & van Dishoeck 2000, Doty et al. 2002, Indriolo et al. 2007).

• T > 100 K & No FUV. An important region of the envelope is the part which has a temperature above 100 K but no FUV irradiation. Hot core species which can be photodissociated may exist only in that region. The region is defined by the temperature T = max(Tdust , Tgas ) (see Section 6.4) and the FUV region, defined above. The spherical model is also given in Table 6.2. For comparison with earlier modeling, we follow St¨auber et al. (2004, 2005) and assume a visual extinction of τV = 0 and G0 = 10 ISRF at the inner edge of the modeled region at 200 AU and use the gas and dust temperature by Doty et al. (2002). For the model Spherical, a plot of the radial dependence of density and temperature is given in Figure 3.3. For this model, the radial size of the regions discussed here are: 1900 AU (Tdust > 100 K ), 1100 AU (Tgas > 100 K and T > 100 K (No FUV)), 224 AU (FUV ), 2300 AU (X-rays and only X-rays). This shows the small size of the FUV irradiated and heated region in a spherical model compared to the models including and outflow wall. The extent of the different regions of model Standard is indicated in Figure 6.3 and the inset of Figure 6.7. The X-ray luminosity of AFGL 2591 cannot be measured directly do to the high column density towards the protostar, but upper limits of 1.6 × 1031 erg s−1 are given in Carkner et al. (1998). However, they assume an attenuating column density to the source of only NH = 1022 cm−2 , much smaller than the radial column density of about NH = 3 × 1023 cm−2 of the density profile used here. Note that the column density derived from the density profile can be affected by uncertainties e.g. in the adopted dust opacity (van der Tak et al. 1999). For the plasma temperature of 7×107 K assumed here (Chapter 3), the X-ray luminosity may be a factor of about 10 higher assuming an attenuating column density of NH = 3 × 1023 cm−2 (e.g. Section 2.2.5). We thus consider the adopted value of the X-ray luminosity of 1032 erg s−1 as an upper limit.

6.3.1

Total mass

The total mass of the envelopes (Mtot ) of the various models is very similar with maximum differences of 25 % compared to model Standard (Table 6.2). The situation is different if only the inner part of the envelope is considered. We give the mass within a radius of 4000 and 20000 AU to the protostar (M4000 and M20000 ) following the Herschel beam at the shortest/longest wavelengths of PACS/HIFI (60-625 µm) assuming a distance of 1 kpc to the source (van der Tak et al. 1999). Clearly, there is foreground material within the Herschel beam. In this chapter, however, we concentrate on molecules with lines having high critical densities. They thus trace only the inner regions due to the density gradient. However, dust or line radiation from inner regions may also pump molecules further out (e.g. van Kempen et al. 2008). A detailed discussion which regions are traced by molecular emission requires the calculation of the excitation as presented in Section 6.5. Models with a large opening angle (α100 ) or the toy-disk have masses M4000 which are 3 times smaller and 10 times larger, respectively, compared to the model Standard. More than half of the envelope mass is outside a radius of 20000 AU and differences between individual models are less evident in M20000 and not seen in Mtot . We conclude that different cavity shapes affect the total mass only in the inner part of the envelope, with the largest changes found for models with a large/small outflow opening angle α or a disk embedded in the envelope.

Tdust > 100 K 0.20 6.7(-2) 3.0 1 15.6 2.3 1.5 0.72 0.49 1.1 0.91 0.71 1.2 0.91 6.9(-2) 0.41 2.3

M20000 33.5 18.7 1.8 1 1.2 1.2 1.1 0.87 0.68 1 1 1 1.01 0.99 1 1 1

a(b) = a × 10b . a Defined by max(M/Mstandard , Mstandard/M ) > 3. b Only in the X-ray region and not in the FUV region. c Region with T = max(Tgas , Tdust ) > 100 K outside the FUV region.

Model Mtot M4000 Absolute Values (M⊙ ) Spherical 49.4 0.94 Standard 44.3 0.45 Relative to Model Standard Spherical 1.1 2.1 Standard 1 1 Disk 1.1 9.1 Opening angle α20 1.1 1.8 α40 1.1 1.5 α80 0.91 0.64 α100 0.75 0.36 Cavity density γ0 1 1 γ0.01 1 1 γ0.1 1 1 Cavity shape b1.5 0.99 1.3 b2.5 1 0.76 Source luminosity L2e3 1 1 L1e4 1 1 L4e4 1 1 0.37 0.76 1.3

0.90 1.1

1.2 0.46 1.2(-2)

0.36 0.60 1.4 1.8

9.6(-2) 1 2.3

6.8(-2) 0.71

Tgas > 100 K

0.66 0.89 1.1

0.95 1.1

1.1 0.73 3.8(-3)

0.25 0.60 1.4 1.8

3.0(-4) 1 1.2

5.7(-4) 1.9

FUV

1 1 1

0.76 1.1

1.0 0.93 0.65

0.70 0.87 0.90 0.73

0.42 1 1.3

0.29 0.70

X-rays

1.2 1.0 0.96

0.85 0.90

1.0 1.0 1.4

1.3 1.2 0.47 9.4(-2)

0.92 1 0.82

0.29 0.32

only X-raysb

0 8.9(-2) 5.0

4.3 8.9(-2)

0.71 1.8 5.5

17.0 6.7 0 0

8.5 1 83.7

6.7(-2) 7.9(-3)

T > 100 K (No FUV)c

Table 6.2: Mass of different regions in the 2D models and the spherical model. The upper part of the table gives absolute values of the mass (in M⊙ ), while the lower part of the table gives factorial deviations a relative to the model Standard. Factorial deviations larger than a factor of three are highlighted. The total mass of the envelope and the mass within a radius of 4000 AU and 20000 AU to the protostar are given by Mtot , M4000 and M20000 , respectively. “1” means that the value is by definition equal to the model Standard. The spatial extent of the regions is discussed in Section 6.3.

6.3. Physical structure 137

138

6.3.2

6. The influence of geometry on the abundance and excitation of diatomic hydrides

Dust temperature above 100 K

The mass with dust temperature above 100 K approximately scales with M4000 for models with the disk and different cavity shape (Models α20-α100, b1.5, b2.5 ). This is explained by the dust temperature depending only on the distance to the protostar and not on the density in optically thin, spherically symmetric models. For this case, the radius for a given temperature varies as L0.5 bol (e.g. −1 Jørgensen et al. 2006). For the density structure ∝ r , the mass with Tdust > 100 K should scale ∝ Lbol . Models L2e3, L1e4 and L4e4 depend slightly stronger on Lbol due to the cavity removing more of the dense material in the inner part compared to larger distances to the protostar. Changing the density in the cavity (Models γ0, γ0.01 and γ0.1 ) affects the mass with Tdust > 100 K by the attenuation of direct stellar radiation in FUV wavelengths that heats the dust in the outflow walls. We conclude that the amount of material with dust temperature above 100 K is within a factor of two to the model Standard for models with a different cavity shape, but larger for the models including a Disk and with a lower bolometric luminosity (L2e3 ).

6.3.3

Gas temperature above 100 K

Gas with Tgas > 100 K is more than an order of magnitude less present in the spherical model and model γ0.1 compared to the model Standard. This underlines the impact of the empty cavity to extensively enhance warm gas by escaping FUV radiation. Model γ0.1 has even less warm gas than the spherical model, due to a more efficient cooling through the low-density outflow region. In all other models, the deviation of the mass with Tgas > 100 K compared to model Standard is smaller than a factor of three. In particular, the models with a different bolometric luminosity (L2e3, L1e4 and L4e4 ) scale less than with Lbol due to the PDR surface temperature scaling with less than G0 for densities below about 106 cm−3 (e.g. Kaufman et al. 1999, Meijerink et al. 2007). We conclude that the mass with gas temperatures above 100 K changes most if the geometrical situation is similar to the spherically symmetric model. These are models with a small outflow opening angle or a high density of absorbing material in the outflow.

6.3.4

X-ray and FUV irradiation

Regions with high-energy irradiation (FUV and/or X-rays) are given in Figure 6.3 for models Standard, disk and α20. They show an FUV region along the outflow cavity. The X-ray region is less concentrated to the surface and also includes the envelope to distances of up to a few 1000 AU. A region with only X-ray influence follows the FUV irradiated surface layer in the vicinity of the protostar (distance < 5000 AU). Some surface concentration is also seen in X-rays due to photons with energy between 0.1 and 1 keV which are attenuated by a relatively low column density (of order 1022 cm−2 ). Harder X-rays with Eγ > 1 keV require a column density of order 1024 cm−2 to be attenuated (Chapter 2). A disk can provide such a high column density and shield the envelope even for hard X-rays (Figure 6.3). The mass of the X-ray/FUV irradiated gas is the same for most models within a factor of two relative to model Standard. Exceptions are the FUV irradiated mass in the spherical model and model γ0.1 being orders of magnitude smaller. Smaller opening angles (Model α20 ) yield a larger incident angle of the FUV radiation and thus a thinner outflow wall. This is reflected in the FUV irradiated mass and also the mass with Tgas > 100 K. For large opening angles (α100 ) the FUV radiation penetrates to the midplane of the flat structure and the region with only X-ray irradiation is much smaller. In the model including a disk, the mass suffering X-ray irradiation is not much different despite the shielding seen in Figure 6.3. It is caused by the larger density in the innermost part at the surface of the disk.

6.3. Physical structure

139

a.) Standard (inner region) X-rays X-rays and FUV FUV

1 · 104 z [AU]

b.) Disk 1 · 104

5 · 103

5 · 103 Not irradiated

Not irradiated

0

0 0

3 · 104

5 · 103

1 · 104

0

5 · 103

1 · 104

d.) α = 20◦

c.) Standard (whole model)

z [AU]

1 · 104 2 · 104 5 · 103

1 · 104

Not irradiated Not irradiated

0

0 0

1 · 104 2 · 104 r [AU]

3 · 104

0

5 · 103 r [AU]

1 · 104

Figure 6.3: FUV and X-ray irradiated regions in the model Standard and the models with a disk and α = 20◦ . FUV and X-ray irradiated regions as defined in Section 6.3 are given in green (only X-rays), orange (only FUV) and red (both X-rays and FUV). The largest variations between different models are found for the mass of the region with T > 100 K but no FUV irradiation. This region is shown in yellow in the Figures 6.7 and 6.6 for models Standard, b1.5, b2.5, α20 and disk. A thin layer of FUV heated gas is followed by an infrared heated region. If the region close to the protostar is geometrically thin (b2.5, α80 and α100 ) FUV radiation can penetrate to the midplane and no material with T > 100 K but no FUV irradiation exists. Geometrically thin/thick means that the distance in vertical direction between the midplane and the outflow is small/large. In models with a geometrically thick inner region (b1.5, α80 and α100 ) or more mass in the innermost region (disk ) the amount of IR heated gas with T > 100 can be even larger than in the spherically symmetric model. The strong sensitivity of this mass on the bolometric luminosity is the result of the dust mass with temperature above 100 K depending more on the bolometric luminosity than the gas temperature (Section 6.3.2 and 6.3.3).

140

6. The influence of geometry on the abundance and excitation of diatomic hydrides

We conclude that the mass with X-ray irradiation is not considerably affected by the geometry, while the mass with FUV irradiation shows differences for those models that are similar to the spherical situation (small opening angle or high density of absorbing material in the outflow). The region with temperature above 100 K but no FUV irradiation depends considerably on the geometry and the bolometric luminosity of the source. Models with a geometrically thin inner part of the envelope (for example a large outflow opening angle) have considerably less material with temperatures above 100 K but no FUV irradiation.

6.4

Chemical Abundances

Chemical abundances, their evolution, differences between physical models and the impact of X-rays are studied in this section. We first study fractional abundances n(X, r ′ )/ntot (r ′ ), with n(X, r ′ ) and ntot (r ′ ), the density of the species and the total hydrogen density (ntot = 2n(H2 ) + n(H)) depending on the position. The fractional abundance gives an overview of the spatial variation of the molecule. Next, the volume averaged fractional abundance is discussed. The volume averaged abundance of a species X within a radius r is defined by Z Z ′ hXir = n(X, r )dV ntot (r ′ )dV . (6.7) |r ′ | 100 K. The evaporated water is quickly photodissociated and its abundance is not increased. The higher amount of sulphur in the gas phase leads to an enhancement of SH and SH+ which can been seen. Also affected are C+ and CH, which have an increased abundance in colder regions due to reaction partners being less abundant. C+ is destroyed in reactions with atomic sulphur, while CH is destroyed with atomic oxygen. Among the species discussed here (SH, SH+ , CH and CH+ ), only SH reaches abundances in this region that are higher compared to the outflow walls. Thus, SH emission may be dominated by this region with sulphur evaporated to the gas phase (Section 6.4.2 and 6.5.2). We next consider HCO+ which is predicted to be an ionization tracer (e.g. van der Tak & van Dishoeck 2000 or Savage & Ziurys 2004). In the top layer of the outflow wall, it is formed by the reaction of H2 with HOC+ and CO+ . The huge enhancement of CO+ by four orders of magnitude in a thin layer along the outflow has been discussed in Chapter 3. Going deeper into the outflow-wall, HCO+ is destroyed by electron recombination and then enhanced by the reaction of CO with H+ 3. + The H3 ion is a key species for the X-ray driven chemistry (Maloney et al. 1996 or St¨auber et al. 2005) and formed by cosmic-ray or X-ray ionization of H2 followed by a quick reaction with H2 . The + high abundance of H+ 3 at the surface of the outflow wall is maintained by the reaction NH + H2 →

144

6. The influence of geometry on the abundance and excitation of diatomic hydrides

N + H+ 3 . Deeper in the outflow wall, the high electron fraction leads to quick electron recombination of H+ . 3 An important difference between X-ray and FUV driven chemistry is the electron fraction. For fluxes of 1 erg s−1 cm−2 (G0 ≈ 1000 ISRF) the electron fraction at a total density of 106 cm−3 is approximately x(e− ) = n(e− )/ntot ∼ 10−4 for FUV but only 10−6 for X-rays. Since the luminosity in the FUV band is 4 orders of magnitude higher than the X-ray luminosity, the large electron fraction is mainly produced by FUV radiation. Since this electron fraction leads to the destruction of H+ 3 , the influence of X-rays is dwarfed by FUV radiation. In addition, many molecules predicted to be X-ray tracers by St¨auber et al. (2005), are much more enhanced by FUV compared to X-rays. Particular examples are CH+ and OH+ , which both have fractional abundances below 10−12 in the spherical model but increase to more than 10−8 in the outflow walls. We conclude that the effect of X-rays compared to FUV on the chemistry is small in a geometry which allows the much stronger FUV radiation to escape from the innermost part. Volume averaged abundances How is the total amount of a molecule/atom altered by the enhancement or absence in the outflow wall? For example a thin layer of strongly enhanced CO+ is found in Chapter 3 to enhance the total amount by three orders of magnitude in agreement with observations. We note that CO+ is related to the diatomic hydrides studied here through formation by C+ + OH → CO+ + H. FUV destroyed species on the other hand, should not decrease their total abundance significantly since only a relatively small fraction of the envelope mass is cut out by a cavity (Section 6.3). In Table 6.3, we give volume averaged abundances within a radius of 4000 and 20000 AU for the model Standard and compare it to the spherical model. In the following we consider a deviation larger than a factor of three to be significant, since similar models (Doty et al. 2004) qualify a factorial deviation of this factor between a model and observation as a good agreement. Outflow wall enhanced species indeed have a significantly enhanced volume averaged abundance compared to the spherically symmetric model. Most enhanced are the ionized molecules C+ , CH+ , OH+ , NH+ , H2 O+ and the electron fraction with an enhancement larger than 3 orders of magnitude and up to almost 7 orders of magnitude. On the other hand, it is found that the FUV destruction of H2 O, CO2 and SH reduces the volume averaged abundance less, by at most a factor of 100. Both enhancement and reduction are seen more in the abundance averaged over 4000 AU rather than 20000 AU. The width of the region with FUV enhancement is important too. For example the volume averaged abundance of SH+ is only increased by a factor of about two despite the enhancement by about 2 orders of magnitude in a very thin layer on the surface of the outflow wall. Similarly, the abundance of NH is only increased in the inner part of the envelope due to the strong temperature dependence of the abundance and the volume averaged abundance is only increased by a factor of two. This strong temperature dependence and consequently the concentration to the inner part also explains the largest difference of enhancement of NH+ between the volume averages of 4000 AU and 20000 AU. The temporal evolution is studied in Table 6.3 by the volume averaged abundances within 20000 AU for a chemical age of 1011 s (3 × 103 yrs) and 1013 s (3 × 105 yrs), and compared to the chemical age of 5 × 104 yrs found by St¨auber et al. (2005). The abundances averaged over 4000 AU are not shown since they are less time dependent. The outflow wall enhanced species have very short chemical time-scale for their formation and thus do not show any temporal evolution. We find that CH+ in the enhanced region only requires of order 10 yrs to reach the final fractional abundance to within a factor of two. The time-scales of OH+ , NH+ and H2 O+ are similarly short. Assuming an upper limit of 10 km s−1 for the Alfv´en and sound velocity, the irradiated gas would move up to a distance of about 25 AU within the chemical timescale which justifies our approach to ignore

6.4. Chemical Abundances

145

Table 6.3: Volume averaged abundances hXir of selected species for the model Standard (“2D”) and the spherically symmetric model (“1D”). Abundances averaged over a radius of 4000 and 20000 AU from the protostar and assuming different chemical ages and protostellar X-ray luminosities are given. The ratio of averaged abundances between the model Standard and the spherically symmetric model is given on the right. Model X-rays [erg s−1 ] Chem. age [yrs] Averaged [AU] H2 e− H+ 3 CH CH+ OH OH+ SH SH+ NH NH+ C+ C CO CO2 H2 O H2 O+ H3 O+ HCO+ O

2D 1(32) 5(4) 20000 5(-1) 8(-6) 3(-10) 6(-10) 4(-10) 2(-8) 1(-10) 3(-11) 1(-11) 1(-9) 1(-12) 7(-6) 5(-7) 2(-4) 5(-9) 2(-7) 5(-11) 4(-10) 2(-9) 5(-5)

2D 1(32) 5(4) 4000 5(-1) 5(-5) 3(-10) 2(-9) 4(-9) 1(-7) 3(-9) 2(-11) 2(-10) 2(-9) 3(-11) 3(-5) 2(-6) 1(-4) 4(-9) 9(-8) 5(-10) 1(-9) 1(-9) 8(-5)

2D 5(4) 4000 5(-1) 5(-5) 5(-11) 2(-9) 4(-9) 1(-7) 3(-9) 2(-11) 2(-10) 9(-10) 3(-11) 3(-5) 7(-7) 1(-4) 5(-9) 6(-8) 5(-10) 9(-10) 4(-10) 8(-5)

2D 1(32) 3(3) 20000 5(-1) 6(-6) 3(-10) 9(-10) 4(-10) 2(-8) 1(-10) 2(-10) 1(-11) 1(-9) 1(-12) 4(-6) 1(-7) 2(-4) 1(-9) 5(-8) 5(-11) 3(-10) 2(-9) 5(-5)

2D 1(32) 3(5) 20000 5(-1) 8(-6) 3(-10) 6(-10) 4(-10) 2(-8) 1(-10) 2(-11) 1(-11) 1(-9) 1(-12) 7(-6) 6(-7) 2(-4) 1(-8) 3(-7) 5(-11) 4(-10) 2(-9) 4(-5)

1D 1(32) 5(4) 20000 5(-1) 1(-8) 3(-10) 5(-11) 4(-16) 9(-9) 2(-14) 2(-10) 9(-12) 5(-10) 1(-15) 4(-10) 4(-8) 2(-4) 3(-8) 8(-7) 4(-14) 1(-9) 4(-9) 4(-5)

1D 1(32) 5(4) 4000 5(-1) 2(-8) 4(-10) 8(-11) 6(-16) 1(-8) 3(-14) 1(-9) 9(-11) 7(-10) 2(-15) 7(-10) 3(-8) 2(-4) 7(-8) 6(-7) 7(-14) 2(-9) 8(-9) 3(-5)

1D -a 5(4) 4000 5(-1) 6(-9) 6(-11) 2(-11) 6(-17) 3(-9) 4(-15) 3(-9) 2(-12) 6(-11) 3(-16) 1(-10) 1(-8) 2(-4) 9(-7) 5(-6) 1(-14) 6(-10) 2(-9) 3(-5)

Ratio 2D/1D 1(32) 1(32) 5(4) 5(4) 5(4) 20000 4000 4000 1.0 0.9 0.9 8(2) 3(3) 8(3) 1.0 0.7 0.8 11 30 1(2) 1(6) 8(6) 7(7) 1.9 9.5 43 6(3) 9(4) 7(5) 0.1 0.01 6(-3) 1.1 2.1 1(2) 2.4 2.3 15 8(2) 2(4) 1(5) 2(4) 4(4) 2(5) 11 70 51 1.0 0.8 0.8 0.2 0.06 5(-3) 0.3 0.1 0.01 1(3) 7(3) 5(4) 0.4 0.6 1.5 0.5 0.2 0.2 1.3 2.5 2.4

a(b) = a × 10b . a FUV also switched off.

the dynamical evolution of the gas. This is further justified by the fact that diatomic hydrides produced in the outflow walls diffusing into the cloud are quickly destroyed by reactions with other species. To test this, a chemical model presented in Chapter 2 has been run for conditions of the cold and not irradiated gas but with initially increased abundances of the diatomic hydrides. Since the C+ enhanced region is thicker, contributions from colder and less irradiated regions with longer timescales are important and the volume averaged abundance shows some temporal evolution (within a factor of two).

6.4.2

Comparing different geometries

The influence of different physical structures (Section 6.3) on chemical abundances is studied in this section. Tables 6.4 and 6.5 give the volume averaged abundances of molecules studied in the previous section. A chemical age of 5 × 104 yrs and an X-ray luminosity of LX = 1032 erg s−1 are used. The abundance is averaged over 4000 AU (all models) and 20000 AU (selected models). To obtain the total amount of a molecule, the mass of the region over which is averaged (M4000 and M20000 ) is given relative to the model Standard. The molecules which were found in Section 6.4.1 to be significantly enhanced/destroyed or unchanged are marked with “+”/“-” or “0”, respectively. For the outflow wall enhanced species CH+ , OH+ , SH+ , NH+ , C+ , H2 O+ and electrons, the

146

6. The influence of geometry on the abundance and excitation of diatomic hydrides

Table 6.4: Ratios of the volume averaged abundances (Eq. 6.7) to the model Standard. A chemical age of 5 × 104 yrs and a protostellar X-ray luminosity of 1032 erg s−1 are assumed. The mass over which is averaged (M4000 and M20000 ) is given relative to model Standard. Model Within 4000 AU Spherical Standard Disk α20 α40 α80 α100 γ0 γ0.01 γ0.1 b1.5 b2.5 L2e3 L1e4 L4e4 Within 20000 AU Spherical Standard Disk α20 γ0.1 L2e3

M4000 2.1 1 9.1 1.8 1.5 0.64 0.36 1 1 1 1.3 0.76 1 1 1 M20000 1.8 1 1.2 1.2 1 1

H2 0a 1.1 1 1.1 1.0 1.0 1.0 0.9 1.0 1.0 1.1 1.0 1.0 1.0 1.0 1.0

e− + 4(-4) 1 1(-1) 0.2 0.5 1.5 2.2 1.0 0.8 4(-2) 0.6 1.4 0.5 0.8 1.3

H+ 3 0 1.4 1 3(-2) 1.1 1.2 0.7 0.4 1.0 1.0 1.6 1.0 1.0 1.2 1.1 0.9

CH (+) 3(-2) 1 0.2 0.2 0.4 2.9 8.0 0.9 0.9 9(-2) 0.6 1.7 1.7 1.3 0.9

CH+ + 1(-7) 1 7(-2) 0.2 0.5 1.8 2.9 1.0 0.8 9(-3) 0.6 1.6 0.5 0.9 1.2

OH (+) 0.1 1 0.7 0.9 0.5 1.9 3.8 0.8 0.8 0.2 0.8 1.4 1.1 1.1 0.8

OH+ + 1(-5) 1 6(-2) 0.3 0.6 1.2 1.5 0.9 0.5 1(-2) 0.6 1.4 0.1 0.8 1.0

SH 1(2) 1 2.5 1(2) 4.0 0.4 0.1 0.9 1.4 4.0 3(1) 0.6 0.8 0.7 2.7

SH+ + 0.5 1 8(-2) 0.3 0.6 1.6 2.3 0.8 0.9 0.3 0.7 1.4 0.6 1.0 0.9

NH 0 0.4 1 5(-2) 1.1 1.1 1.0 1.2 0.9 1.0 1.8 1.4 1.0 1.2 1.3 0.7

1.0 1 1.0 1.0 1.0 1.0

1(-3) 1 0.8 0.2 1(-2) 0.5

1.0 1 0.8 1.0 1.1 1.0

9(-2) 1 0.9 0.4 0.3 1.2

9(-7) 1 0.7 0.2 2(-3) 0.2

0.5 1 1.6 0.8 0.7 0.8

2(-4) 1 0.5 0.3 8(-3) 8(-2)

6.8 1 1.0 5.7 1.2 1.3

0.9 1 0.6 0.6 0.4 0.5

0.4 1 0.8 0.7 1.1 1.0

a(b) = a × 10b . a Ratio to the spherical model within 4000 AU (Table 6.3; + more than 103 , - less than 0.1, (+) more than 3, 0 within 0.1 to 3).

volume averaged abundance is most changed in the models Disk, α20 and γ0.1. While for α20 and γ0.1 averaging over 4000 and 20000 AU leads to a significantly lower abundance, the differences in the Disk model are only seen for the average over 4000 AU. However, when scaling with the higher mass of the Disk model within 4000 AU, the total amount of these charged species remains approximately constant. This reflects that the fractional abundance of outflow wall enhanced species is approximately inversely proportional to the density (Chapter 2). In addition, the higher density yields a lower gas temperature and thus a decreased amount of molecules which require a high temperature for formation (e.g. CH+ ). Quantitatively, we find that the volume averaged abundances of outflow wall enhanced species behave similarly as the FUV irradiated mass, discussed in Section 6.3. The FUV destroyed molecules SH, H2 O and CO2 have a similar dependence of the volume averaged abundance. Like the region without FUV irradiation but temperature above 100 K (Section 6.3), they are more dependent on the opening angle (α20 -α100 ) and the shape (b1.5 and b2.5 ) than the FUV enhanced molecules. Quantitatively, however, the dependence of the volume averaged abundance on the physical models differs more between these three species than for the outflow wall enhanced species. This is due to different effects entering the abundances of SH, H2 O and CO2 , like evaporation combined with a different depth dependence of the photodissociation rates. For example, the larger dependence of CO2 on the opening angle for models α80 and α100 is a result of the concentration of the molecule towards the center combined with the CO2 photodissociation rate

6.4. Chemical Abundances

147

Table 6.5: Ratios of the volume averaged abundances (Eq. 6.7) to the model Standard. A chemical age of 5 × 104 yrs and a protostellar X-ray luminosity of 1032 erg s−1 are assumed. The mass over which is averaged (M4000 and M20000 ) is given relative to model Standard. Within 4000 AU Spherical Standard Disk α20 α40 α80 α100 γ0 γ0.01 γ0.1 b1.5 b2.5 L2e3 L1e4 L4e4 Within 20000 AU Spherical Standard Disk α20 γ0.1 L2e3

M4000 2.1 1 9.1 1.8 1.5 0.64 0.36 1 1 1 1.3 0.76 1 1 1 M20000 1.8 1 1.2 1.2 1 1

NH+ +a 6(-5) 1 8(-2) 0.3 0.7 1.2 1.6 0.7 0.6 0.1 0.6 1.3 0.2 0.9 0.8

C+ + 3(-5) 1 0.1 0.2 0.5 1.6 2.4 1.0 1.0 3(-2) 0.6 1.4 0.7 1.0 1.1

C (+) 2(-2) 1 0.2 0.4 0.8 1.1 1.3 1.0 1.0 0.9 0.7 1.2 0.9 0.9 1.2

CO 0 1.3 1 1.3 1.3 1.2 0.8 0.5 1.0 1.0 1.3 1.1 0.9 1.1 1.0 1.0

CO2 2(1) 1 3(2) 4.7 3.0 4(-2) 2(-4) 1.2 1.3 3.2 2.5 0.2 0.7 1.0 1.4

H2 O 6.8 1 7(1) 4.7 2.5 0.1 0.2 1.2 1.1 2.5 2.3 0.3 1.6 1.3 1.0

H2 O+ + 1(-4) 1 8(-2) 0.2 0.4 1.6 2.4 0.9 0.9 2(-2) 0.6 1.6 0.6 1.0 1.2

H3 O+ 0 1.6 1 9(-2) 0.4 0.6 1.7 2.8 1.0 0.9 0.4 0.6 1.4 1.0 0.8 1.1

HCO+ 0 5.9 1 6(-2) 1.5 1.5 0.5 0.5 1.0 1.1 2.0 1.3 0.7 1.1 1.1 0.9

O 0 0.4 1 0.5 0.6 0.7 1.3 1.8 1.0 1.0 0.5 0.8 1.2 0.8 0.9 1.1

1(-3) 1 0.5 0.4 0.1 0.2

7(-5) 1 0.8 0.2 4(-3) 0.5

9(-2) 1 0.9 0.4 0.1 1.1

1.0 1 1.0 1.0 1.0 1.0

6.3 1 5(1) 3.3 1.4 1.6

3.5 1 6.1 1.8 1.3 1.4

1(-3) 1 0.7 0.2 7(-3) 0.2

2.4 1 0.7 1.7 0.9 0.9

2.2 1 0.7 1.8 1.2 1.0

0.8 1 1.0 0.8 0.8 0.9

a(b) = a × 10b . a Ratio to the spherical model within 4000 AU (Table 6.3; + more than 103 , - less than 0.1, (+) more than 3, 0 within 0.1 to 3).

dropping faster with extinction compared to H2 O and SH. On the other hand, SH is more enhanced in the models α20 and b1.5 due to the factor of 10 larger increase in abundance in the hot-core region. Quantitatively, photodissociated and hot-core enhanced species thus have a more complex dependence on the geometry than FUV enhanced species. Shocks, which are not accounted for here, may release H2 O and CO2 to the gas phase. In a fast J-shock with vS = 80 km s−1 and a density of 105 cm−3 , the width of dust grains with temperature above 100 K is of order 700 AU (Hollenbach & McKee 1989). Thus along the outflow walls, H2 O and CO2 molecules released by shocks from the grain surface are likely photodissociated (Figure 6.4 and 6.5). A detailed modeling of shocks along the outflow walls including the strong protostellar FUV irradiation is warranted. + + Of the molecules deemed insensitive to geometry in Section 6.4.1 (H2 , H+ 3 , NH, CO, H3 O , HCO + + + and O), the presence of a disk (Figure 6.6) show significant differences for H3 , NH, H3 O , HCO . This is again due to the combination of a lower gas temperature due to the higher density in the disk and subsequently faster recombination rate. The total amount of these molecules thus remains approximately the same as in the model Standard. Figure 6.7 shows the abundance of the FUV enhanced CH+ and the FUV destroyed SH for models Standard, b2.5, b1.5 and α20. In the midplane, the models b1.5 and α20 have a higher abundance of SH by two orders of magnitude, while the outflow wall with enhanced CH+ abundance is much thinner. The region without FUV irradiation but temperature above 100 K is shown in the inset

148

6. The influence of geometry on the abundance and excitation of diatomic hydrides

of the upper panel. This region coincides with the region having an SH fractional abundance larger than 10−10 , shown as inset in the lower panel. The region with fractional abundance larger than 10−8 shows a layer structure with the molecule being more photodissociated closer to the outflow wall. At larger distances (z = 2000 and z = 10000), the differences between the models are smaller, as found in the previous section for the volume averaged abundances. The abundances of SH, CH+ and H2 O in the Disk model are given in Figure 6.6. The layer with high temperature and enhanced CH+ abundance (disk atmosphere) is thinner than in the outflow wall models in Figure 6.7 because of the higher density in the disk surface. Due to the larger density dependence of the SH abundance compared to H2 O and CO2 , SH does not reach a fractional abundance of 10−8 as in the models 1.5 and α20. The water abundance on the other hand is even higher in the FUV shielded midplane of the Disk model compared to the outflow wall models. This is also seen in Tables 6.4 and 6.5, where the volume averaged abundances of CO2 and H2 O are much more enhanced compared to SH. Total Density ntot [cm−3 ] 2000

FUV, T > 100 K FUV, T < 100 K T > 100 K T < 100 K

z [AU]

4000 1000

3000

0

0

10−4

108 10

0

1000

1000

2000 z=1000

z=200

0

z=0

1000

2000 3000 r [AU]

4000

−7

10

5000

10

10−11 10−12

10−15

105

H2 O

0

1000 Distance to Outflow [AU]

SH −9

10

10−10 n(SH)/ntot

n(H2 O)/ntot

10−10

10−14

10−6 10−7 10−8 10−9

10−11 10−12 10−13

10−10 10

10−9

10−13

106

10−5

−11

z=0 AU z=200 AU z=1000 AU z=0 AU (X-rays) z=200 AU (X-rays) z=1000 AU (X-rays)

−8

109

7

2000

CH+

1010

n(CH+ )/ntot

5000

10−14 0

1000 Distance to Outflow [AU]

0

1000 Distance to Outflow [AU]

Figure 6.6: Density structure and regions with T > 100 K and/or FUV irradiation of the model Disk. Abundances of CH+ , SH and H2 O are given along cuts of constant z (in the midplane and at z = 200, 1000 AU). The model with X-rays (dashed line) assumes an X-ray luminosity of 1032 erg s−1 .

6.4. Chemical Abundances Standard: CH+ 2000

FUV, T > 100 K FUV, T < 100 K T > 100 K T < 100 K

z [AU]

n(CH+ )/ntot

10−8 10−9 10−10

Standard: SH 10−6

1000

10−8

−11

10

0

10−12

0

10−13

1000

2000

0

b = 2.5: CH+ 2000

10−10

1000

−11

10

0

10−12

0

−13

1000

2000

10−14 0

10−11

0

1000 2000 3000 Distance to Outflow [AU] 2000 z

10

0

−12

10

1000

10−8

1000

0

−13

1000

2000

0

10−9

r

0

−10

10

1000 2000

10−11 10−12

r [AU]

10

10−14

10−13 0

10−14

1000 2000 3000 Distance to Outflow [AU]

α = 20: CH+ 2000

0

2000 z

10

10−10 −11

10

0

10−12 −13

0

1000

2000

r [AU]

10

1000

10−8

1000

10−14

n(SH)/ntot

z [AU]

10−7

−9

1000 2000 3000 Distance to Outflow [AU]

α = 20: SH 10−6

10−8 n(CH+ )/ntot

1000 2000

b = 1.5: SH

−11

10

r

0

10

10−6

n(SH)/ntot

z [AU]

n(CH+ )/ntot

10−10

−15

0

−10

10−7

10

10−7

10−9

10−14

1000 2000 3000 Distance to Outflow [AU]

b = 1.5: CH+ 2000

−9

10

10−8

10−13

10−8

−15

2000 z

X < 10−10 10−8 < X < 10−10 1000 X > 10−8 (X = n(SH)/ntot )

10−12

r [AU]

10

1000 2000 3000 Distance to Outflow [AU]

b = 2.5: SH 10−6

n(SH)/ntot

z [AU]

n(CH+ )/ntot

10

10−7

0

10−7

−9

10

10−11

10−14

1000 2000 3000 Distance to Outflow [AU]

10−8

−15

10−10

10−13

10

10−7

10−9

10−12

r [AU]

−14

10−15

z=0 AU z=2000 AU z=10000 AU z=0 AU (X-rays) z=2000 AU (X-rays) z=10000 AU (X-rays)

10−7 n(SH)/ntot

10−7

149

0

10−9

r

0

−10

10

1000 2000

10−11 10−12 10−13

0

1000 2000 3000 Distance to Outflow [AU]

10−14

0

1000 2000 3000 Distance to Outflow [AU]

Figure 6.7: Abundances of CH+ and SH for the models Standard, b2.5, b1.5 and α20. The abundances are given for cuts of constant z assuming no protostellar X-ray radiation (solid line) and an X-ray luminosity of 1032 erg s−1 (dashed line). As an inset to the plots for CH+ , the regions with FUV irradiation and/or temperature > 100 K are given. The inset to the plots for SH show the abundance of SH divided into regions with fractional abundances > 10−8 , < 10−10 and in between.

150

6.4.3

6. The influence of geometry on the abundance and excitation of diatomic hydrides

X-ray driven chemistry

X-rays affect only the innermost region due to geometrical dilution ∝ r −2 (Section 6.3). In Table 6.3 we thus give the volume averaged abundance within 4000 AU. In the two-dimensional model (Standard ), the largest differences between the model for an X-ray luminosity of 1032 erg s−1 and no X-rays are found for H+ 3 with an X-ray enhancement of a factor of 6. The abundances of NH, C + and HCO are enhanced by a factor of about three. Compared to the spherically symmetric model, C and HCO+ are enhanced on a similar level, while NH is much less enhanced. Destruction of CO2 and water by X-rays has been found in the spherical models of St¨auber et al. (2005) and St¨auber et al. (2006), respectively. Since the abundances of H2 O and CO2 are already decreased by FUV radiation, this effect is not seen in the two-dimensional model. The extra enhancement of diatomic hydrides due to X-rays over FUV processing is much smaller in the two-dimensional models compared to the spherical model. As discussed in Section 6.4.1, the enhancement in the outflow walls by FUV is much larger than by X-rays. Only the volume averaged abundance NH, which is X-ray sensitive but not much enhanced by FUV radiation shows some dependence on X-ray irradiation (Figure 6.4). However note that NH has high energy levels and thus is more excited in the outflow walls so that the effect of X-rays on the line flux is probably very small. However, a detailed calculation of the line flux should be carried out once collision rates of NH have become available. The X-ray enhancement/destruction in different models is studied in Table 6.6 by the ratio of the volume averaged abundance including protostellar X-ray emission (LX = 1032 erg s−1 ) and no X-rays + within 4000 AU. The X-ray enhanced species H+ 3 , NH and HCO are similarly enhanced for the 2D models except for models α80, α100 and Disk where the enhancement in smaller. Models α80 and α100 have less mass with only X-rays but no FUV irradiation (Section 6.3) and the molecules are photodissociated in the region with X-ray irradiation. In the Disk model, however, these molecules are less FUV dissociated due to the higher density and even enhanced in the warm disk atmosphere. This enhanced layer dominates over the X-ray enhanced region and thus the X-ray enhancement is smaller. X-ray destruction of H2 O and CO2 (see St¨auber et al. 2005, 2006) is remarkable in models which have a larger region with temperature above 100 K but no FUV irradiation (Models Spherical, α20, α40, γ0.1 and L4e4 ). Models with a smaller or similar amount of H2 O or CO2 behave similarly to the model Standard, discussed before. We conclude that the influence of X-rays on the chemical composition is relatively small in this scenario of FUV irradiated outflow walls. Molecules predicted to be X-ray tracers (St¨auber et al. 2005) and in particular diatomic hydrides are much more enhanced by FUV radiation compared to X-rays. Exceptions are the water and CO2 destruction by X-rays which is seen in some of the models that are similar to spherical geometry (e.g. small outflow opening angle). Also, HCO+ and H+ 3 are considerably enhanced for some models. The differences between the models and the uncertainties of the chemical model are however as large as the extra enhancement by X-rays. For example the total amount of HCO+ (obtained from Tables 6.4, 6.5 and 6.6) within 4000 AU is larger for the model α40 without X-rays compared to the model α80 with X-rays. Thus, it is difficult to establish X-ray tracers for these models with very strong FUV irradiation. In envelopes of low-mass YSOs, however, FUV radiation could be less dominant compared to X-rays. Several effects control the relative importance of FUV and X-rays. The colder photosphere of low-mass YSOs emits less FUV. Bow shocks, internal jet shocks and accretion disk boundary layer can however still produce a considerable FUV field (van Kempen et al. 2009a,c). On the other hand, most commonly observed low-mass YSOs are closer. Since the X-ray luminosity of low-mass YSOs is of similar strengths compared to high-mass YSOs (e.g. Nakajima et al. 2003) the X-ray irradiated gas may fill the beam and thus the influence of X-rays on observable molecular lines becomes stronger.

6.5. Molecular excitation and line fluxes

151

Table 6.6: Enhancement or destruction by X-rays for selected species. The ratio of the volume averaged abundance within 4000 AU between a model with LX = 1032 erg s−1 and no protostellar 32 X =10 X =0 X-ray emission is shown (hXiL4000 /hXiL4000 ). Model M4000 H+ NH CO2 H2 O HCO+ 3 Spherical 2.1 6.6 12 0.08 0.1 4.0 Standard 1 5.6 1.8 1.0 1.6 3.5 Disk 9.1 1.8 1.1 0.7 0.7 1.1 α20 1.8 5.9 2.8 0.01 0.1 4.1 α40 1.5 6.0 2.2 0.03 0.4 4.4 α80 0.64 4.2 1.2 1.5 1.3 1.7 α100 0.36 2.6 1.0 1.9 1.0 1.0 5.9 2.0 0.9 1.5 3.7 γ0 1 γ0.01 1 5.5 1.9 0.5 1.5 3.7 γ0.1 1 6.6 1.9 0.05 1.0 5.2 b1.5 1.3 5.3 1.6 0.03 0.4 4.0 b2.5 0.76 5.4 1.6 1.4 2.2 2.6 L2e3 1 6.7 2.1 2.1 1.6 3.5 L1e4 1 5.6 1.6 1.3 1.6 3.6 L4e4 1 5.3 2.0 0.05 1.3 3.3 A detailed discussion is warranted. Very high resolution interferometric observations of FUV and X-ray tracer with ALMA will also help to disentangle the effects of FUV and X-ray irradiation. In particular the X-ray tracer NH, discussed before, has a line at 946 GHz within ALMA band 10. High-J HCO+ lines (e.g. the J = 10 → 9 line at 892 GHz) which probe the innermost part of the envelope are also within ALMA band 10.

6.5

Molecular excitation and line fluxes

Observable quantities of the models presented here are molecular or atomic line fluxes. In this section, we study different excitation effects of diatomic hydrides, like collisional or radiative pumping. We use a multi-zone escape probability method described in Section 6.2.2. Herschel observations are then simulated from the excitation calculation. For simplicity, we restrict the discussion to two prototypical diatomic hydrides, one found to be destroyed (SH) and enhanced (CH+ ) in the outflow wall. Line fluxes of other species will be presented later. Molecular data for the line frequency and Einstein coefficients are obtained from the CDMS (M¨ uller et al. 2001) for CH+ and the JPL database (Pickett et al. 1998) for SH. The molecular collision rates of many diatomic hydrides are poorly known and extrapolation is thus necessary. For SH-H2 , we adopt the OH-H2 rates by Offer & van Dishoeck (1992), scaled for the reduced mass. Since collision rates for ortho and para H2 exist, we use a thermal ortho to para ratio3 . For CH+ -H2 and CH+ -H, we scale the CH+ -He rates by Hammami et al. (2008) and Hammami et al. (2009). As CH+ is abundant in regions with electron fraction of order 10−4 , we also consider excitation by electron impact (Lim et al. 1999a). A general discussion on the uncertainty of collision rates is given in Sch¨oier et al. (2005). Transitions observable by Herschel (HIFI/PACS) are summarized in Tables 6.7 and 6.8. They give the transition wavelength or frequency (νul , λul ), the Einstein-A coefficient Aul , the energy of the 3

oH2 /pH2 = min(3, 9 × exp(−170.6/Tgas))

152

6. The influence of geometry on the abundance and excitation of diatomic hydrides

upper level Eul . Transitions of CH+ with Jup = 2 − 6 can be observed by PACS, while the J = 2 → 1 and J = 1 → 0 lines at 835.079 GHz and 1669.170 GHz, respectively, are accessible to HIFI. The SH lines are in HIFI bands, except for the lines at 1383 GHz, which are given for completeness. For SH, only the strongest hyperfine components are used and overlapping components have been P combined. The critical density (ncrit = Aul / l Kul ) gives the necessary density of collision partners for substantial population of the upper level. Table 6.7: Molecular data of SH. The critical density thermal ortho to para ratio of H2 . νul [J,F/Parity] [GHz] 2 Π1/2 3/2, 1− → 1/2, 1+ 866.947 2 Π1/2 3/2, 2+ → 1/2, 1− 875.267 2 Π3/2 5/2, 3+ → 3/2, 2− 1382.910 2 Π3/2 5/2, 3− → 3/2, 2+ 1383.241 2 Π1/2 5/2, 2+ → 3/2, 1− 1447.012 2 Π1/2 5/2, 3− → 3/2, 2+ 1455.100 2 Π3/2 7/2, 4− → 5/2, 3+ 1935.206 2 Π3/2 7/2, 4+ → 5/2, 3− 1935.847

Table 6.8: Molecular data of CH+ . The λul Aul [J] [µm] [s−1 ] 1 → 0 359.0 6.4(-3) 2 → 1 179.6 6.1(-2) 3 → 2 119.9 2.2(-1) 4 → 3 90.0 5.4(-1) 5 → 4 72.1 1.1(0) 6 → 5 60.2 1.9(0)

is given for a temperature of 100 K and a Aul [s−1 ] 1.9(-3) 1.5(-3) 9.4(-3) 9.4(-3) 1.6(-3) 1.6(-3) 3.5(-2) 3.5(-2)

Eul ncrit (H2 ) [K] [cm−3 ] 571 4.3(6) 572 4.7(6) 66 3.8(7) 66 4.7(7) 641 2.6(7) 642 3.0(7) 159 1.2(8) 159 1.4(8)

critical density is given for a temperature of 500 K. Eul ncrit (H2 ) ncrit (H) ncrit (e− ) [K] [cm−3 ] [cm−3 ] [cm−3 ] 40 5.3(7) 3.8(7) 3.0(4) 120 2.5(8) 1.9(8) 1.3(5) 240 7.1(8) 5.2(8) 3.8(5) 400 1.6(9) 1.2(9) 7.8(5) 600 3.2(9) 2.3(9) 1.7(6) 838 5.8(9) 4.2(9) 3.0(6)

We use wavelength (µm) for CH+ as most lines are observable with PACS and frequency (GHz) for SH (HIFI). Levels of CH+ are labeled by the rotational quantum number J. The molecular structure of SH is similar to hydroxyl (OH) with a 2 Π electronic ground-state, Λ-type doubling and hyperfine splitting. Levels are thus denoted by 2 ΠΩ , J, F + /−, with the total angular momentum Ω, the rotational angular momentum J, the nuclear spin F and the parity (+/-). The CH+ and SH molecules show typical properties of diatomic hydrides, with high critical densities (> 106 cm−3 for H2 collisions) and large energy separation between subsequent levels. This is due to the large rotation constant combined with a large dipole moment. Line frequencies thus mostly lie above the atmospheric windows. Excitation mechanism other than collisional excitation are likely. For example, pumping by dust continuum radiation or formation in an excited state have been suggested by Black (1998) for CH+ .

6.5.1

Excitation: Level population

The calculated level populations of SH and CH+ at different positions of model Standard are given in Figure 6.8. For SH, the position in the midplane with peak abundance and at the edge of the cloud

6.5. Molecular excitation and line fluxes

153

are presented. For CH+ , we show positions with peak abundances along cuts through the midplane and at z = 2000 AU and z = 10000 AU (Section 6.4.1). In order to study the importance of different excitation mechanism, we give the normalized level populations depending on the upper level energy for the following models: • All. Collisional and radiative excitation considered. • No Dust. Dust radiation switched off by setting the dust opacity to zero. This model allows to study the importance of dust pumping. • No Radiation. Ambient radiation field switched off (hJij i = 0). This model together with the previous allows to study the importance of pumping by line emission. • No Collisions. Density of the collision partners set to zero. This situation corresponds to pumping purely by dust radiation. • LTE. Level population in the local thermal equilibrium (LTE) with the gas temperature. This model is given for comparison as the assumption of LTE is often made due to missing collision rates. • Tform = 100 K, Tform = 3000 K (Only CH+ ). Collisional excitation, radiative excitation and excited formation considered with formation temperature Tform (Section 6.2.2). We adopt a Tform of order of the kinetic temperature (3000 K) or much lower (100 K). For SH, the population of the hyperfine quadruplet is combined. We note that the level population can only be translated directly to emissivity per volume in the case of optically thin radiation. Figure 6.9 gives different components (line emission, dust emission, cosmic microwave background and stellar photons) of the ambient radiation field hJij i at positions A (SH) and C (CH+ ). The CH+ level population is far out of LTE. It peaks in the J = 1 state in position C and the ground state in positions D and E. For all positions, the population rapidly decreases to higher J levels and from position C to E for the same level. Collisional excitation (model No Collisions) governs the excitation with increasing differences to model All for higher J. A model without electron excitation but collisions with H and H2 has level populations within 20 % to the model All. Electron excitation is thus unimportant relative to other effects. Models No Dust and No Radiation differ less relative to model All in CH+ compared to SH, thus radiation is less important. Unlike SH, the models No Dust and No Radiation differ, due to the contribution of line photons to the ambient radiation field. At position C, line photons dominate the radiation field up to the J = 5 → 4 transition (Figure 6.9 right). Optical depths of up to τ ≈ 1 (line and dust) for vertical cuts through the outflow wall are reached in the lower J transition. For levels connected by these transitions, we find the largest differences between models No Dust and No Radiation. We conclude that CH+ is mainly collisionally excited with small contributions from pumping by line trapping and dust radiation. The importance of collisions for the excitation of CH+ despite the density of the collision partners below the critical density can be understood by the high kinetic temperature. Radiative transitions connect only levels with ∆J = 1 due to selection rules. Rates for radiative pumping are proportional to the lower level population of a pumping transition and thus decrease quickly to higher J. Collisional excitation is not restricted to ∆J = 1 and can pump over several levels, especially for high kinetic temperatures, when the upward rates are similar or even higher than the downward rates (Clu gl = Cul gu exp(−∆E/kTkin )). At position C in the model Standard, the radiative excitation rate decreases from about 10−2 (J = 1 → 2) to 10−5 cm−3 s−1 (J = 9 → 10), for Blu hJul i and Bul hJul i of order 0.1 s−1 and a CH+ density of 0.4 cm−3 . The collisional excitation rates from the most populated levels (J ≤ 3) to a

6. The influence of geometry on the abundance and excitation of diatomic hydrides

A

5 · 105

2000

2 · 105

Position D Position C Position B Position A

0

1 · 104 2 · 104 r [AU]

0

1 · 105

3 · 104

10

Population (CH+ , Position C)

100 Level Energy [K]

0K

10−2 10−3 10−4 10−5 10

100 1000 Level Energy [K] All LTE No Dust

No Radiation No Collisions Tform = 100 K

10−4

10−6 10

10

J=5 J=6 J=7 J=8 J=9 J=10

−1

J=4

10

0K

10−2 10−3 10−4 10−5 10−6

100 Level Energy [K]

1000

Population (CH+ , Position E)

10

100 1000 Level Energy [K]

Normalized Level Population

10

10

0

J=3

−1

0K −2

Population (CH+ , Position D) J=2

10

100

1000

0

Normalized Level Population

Normalized Level Population

= 1/2 = 3/2 = 5/2

10−6

5 · 104

0

10−6

2 Π1/2 , J 2 Π1/2 , J 2 Π1/2 , J

0K

10−4

Normalized Level Population

1 · 104

C

10

Π3/2 , J = 7/2

0 Position E 0

Population (SH, Position B)

2

1 · 106

Envelope

Π3/2 , J = 5/2

2 · 10

−2

J=1

z [AU]

2 · 106

2000

4

2

5 · 106

Outflow

Population (SH, Position A) 100 Π3/2 , J = 3/2

1 · 107

2

Total Density ntot [cm−3 ]

Groundstate

3 · 104

Normalized Level Population

154

10

10−1

0K

10−2 10−3 10−4 10−5 10−6

10

100 1000 Level Energy [K]

Tform = 3000 K Thermal population (129 K)

Figure 6.8: Normalized level population of SH (top panels) and CH+ (bottom panels) at different positions. The positions A-E are given in the density plot in the top/left panel. A zoom-in of the innermost region is given as inset. The level energy of the ground state has been shifted from 0 K to 15 K for the representation in the logarithmic plot. higher level is found to be about 5 × 10−3 s−1 cm−3 into J = 4 and slightly less for higher J, for rate coefficients Kul of order a few times 10−10 cm3 s−1 and a density of 107 cm−3 . Thus, for J = 4, radiative and collisional rates are similar, but collisions dominate for higher J. This is in agreement with the findings in the previous paragraphs. Excited formation affects the level population relatively little with considerable differences only for higher J levels at lower densities (position E) and a high formation temperature (Tform = 3000 K). At position C, the total formation rate of CH+ is 6 × 10−3 s−1 cm−3 , larger than the collisional excitation, but the effect on the level population is small due to the distribution over different levels following the Boltzmann statistic. Due to this distribution, which supports the thermalization to the formation temperature, the influence of excited formation is mostly seen in higher J levels for Tform = 3000 K, where collisional excitation competes less with molecules formed in an excited state. Observation of the J = 2 → 1 up to J = 6 → 5 of CH+ using the Infrared Space Observatory (ISO) have been reported by Cernicharo et al. (1997) towards the planetary nebula NGC 7027. They can be explained by a single excitation temperature of 150 ± 20 K for a kinetic temperature of about 300-500 K and densities of a few times 107 cm−3 . It is interesting to note that, at position C of our model, the level population up to J = 6 can be well fitted by a single Boltzmann distribution with a temperature of 129 K. Note that Cernicharo et al. (1997) have used N2 H+ collision rates and a slightly higher density. Our calculation thus suggests that the observed line ratio can be explained

6.5. Molecular excitation and line fluxes Radiation Field (SH, Position A)

10−12 −13

10

10−14 10−15 10−16 −17

10

10−18 10−19

10−9 Intensity [erg s−1 cm−2 Hz−1 ]

Intensity [erg s−1 cm−2 Hz−1 ]

10−11

155

800

1200 1600 Frequency [GHz]

2000

Radiation Field (CH+ , Position C)

10−10 10−11 10−12 10−13 10−14 10−15

Total Line Cmb Dust Stellar

200 100 50 Wavelength [µm]

Figure 6.9: Radiation field of SH and CH+ at positions A and C, respectively. The positions are given in Figure 6.8. even with a much higher kinetic temperature as is required for the formation. A more detailed model of their source is however required to draw further conclusions. The SH level population is out of LTE like CH+ . The molecule is however destroyed by FUV radiation and thus abundant in regions with lower temperatures compared to CH+ . The level population is thus concentrated in the ground state for both positions A and B. Levels of the 2 Π1/2 ladder with level energy above 500 K are sparsely populated with relative population below 10−4 . The agreement of the model No Collisions with the model All in the 2 Π3/2 states indicates that these level are pumped by dust radiation. The 2 Π1/2 states on the other hand are connected by lines with lower frequency and thus weaker dust radiation field. These levels are pumped by both collisions and dust radiation.

6.5.2

Line fluxes

Calculated line fluxes of SH and CH+ for different models are given in Table 6.9. Synthetic frequency integrated maps and spectra at selected positions for the J = 1 → 0 and J = 6 → 5 lines of the model Standard are shown in Figure 6.10. Maps and spectra are obtained from the level population discussed in the previous section. Synthetic maps are calculated by integrating the radiative transfer equation. To simulate observations, the maps are convolved with the appropriate Herschel beam, centered on the source and assumed to be a Gaussian. For the raytracing, we assume the same physical parameters as in Chapter 3. These are a distance of 1 kpc, an inclination of 30◦ and a position angle of the outflow direction of 96◦ . The predicted CH+ fluxes are remarkably strong, even for higher J lines. The integrated intensities correspond to fluxes of order a few times 10−19 W cm−2 and are easily detectable with PACS or HIFI. Note that they are similar to those reported towards NGC 7027. Strong lines for higher J transitions despite the low level population found in the previous section are due to the smaller beam 2 at shorter wavelengths and the emissivity (in K km s−1 ) per excited molecule ∝ Aul /νul , increasing by a factor of 10 from the J = 1 → 0 to the J = 6 → 5 transition. Using a Boltzmann diagram, corrected for line absorption, the model Standard fits well for a column density of 7 × 1012 cm−2 and an excitation temperature of about 148 K (Slab at LTE ). Different radiative transfer models of CH+ derived from the chemical model Standard have similar fluxes in the J = 1 → 0 line, with deviation smaller than about 30 %. An exception is the model

6. The influence of geometry on the abundance and excitation of diatomic hydrides 156

Model CH+ Wavelength [µm] Herschel-Beam [”] Standarda Slab at LTEc Standard (2 kpc) Standard (LTE) Standard (Tform = 100 K) Standard (Tform = 3000 K) Standard (0◦ ) Standard (90◦ ) Standard (25.8”) Standard (Noturb) Disk α20 α100 SHe Frequency [GHz] Herschel-Beam [”] Standarda Standard (2 kpc) Standard (LTE) Standard (0◦ ) Standard (90◦ ) Disk α20 α100

Line Flux [K km s−1 ]b 1→0 359.0 25.8” 12.3 9.8 4.3 12.0 14.9 15.8 14.8 16.1 12.3 9.4 8.2 4.9 28.8 Π1/2(3/2→1/2) 2

1382.910 15.6” -2.1(-3)d -9.7(-4) 3.2(-1) 5.4(-3) -6.1(-2) -9.9(-4) -1.5(-1) 1.7(-3)

[K km s−1 ] 2→1 179.6 12.9” 23.9 22.5 8.9 113.1 26.4 30.8 29.5 23.6 7.5 19.0 11.7 10.1 40.7 Π3/2(5/2→3/2) 2

1447.012 14.9” 4.6(-5) 2.1(-6) 8.8(-4) 1.2(-4) 5.7(-6) 2.4(-5) 1.8(-4) 9.8(-5)

[K km s−1 ] 3→2 119.9 8.6” 20.2 22.4 8.0 333.0 21.8 29.4 26.1 17.3 2.9 19.3 7.3 10.7 29.8 Π1/2(5/2→3/2)

2

1455.100 14.8” 5.0(-5) 2.4(-6) 8.5(-4) 1.3(-4) 6.6(-6) 2.6(-5) 2.0(-4) 1.1(-4)

[K km s−1 ] 4→3 90.0 6.5” 10.5 13.6 4.2 612.3 11.0 18.1 14.1 8.0 0.9 11.2 2.6 7.2 14.6 Π1/2(5/2→3/2)

2

1935.206 11.1” 5.8(-4) 8.7(-5) 9.1(-2) 9.2(-4) -7.9(-4) 2.8(-3) 3.8(-2) 2.8(-4)

[K km s−1 ] 5→4 72.1 5.2” 4.6 5.3 1.9 871.9 4.7 9.7 6.3 3.2 0.3 5.0 0.8 3.5 6.7 Π3/2(7/2→5/2)

Table 6.9: Predicted molecular line fluxes of SH and CH+ .

2

866.947 24.9” 3.8(-5) 3.2(-6) 8.2(-5) 8.4(-5) 9.9(-6) 1.8(-5) 1.6(-4) 8.2(-5)

a(b) = a × 10b . a If no other indication is given, the inclination is 30◦ and the distance 1 kpc (see Section 6.2.2) b The conversion factor from K km s−1 to W cm−2 is 2.7 × 10−14 /(λ[µm])3 . c Fit to model Standard for a CH+ column density of 7 × 1012 cm−2 and Tex = 148 K. d Negative fluxes mean line in absorption. See Table 6.7 for the exact label of the transition including fine structure. e

2

[K km s−1 ] 6→5 60.2 4.3” 2.1 1.6 0.9 1064.9 2.1 5.4 3.0 1.3 0.1 2.2 0.3 1.6 3.3 Π3/2(7/2→5/2)

1935.847 11.1” 4.4(-4) 6.1(-5) 9.0(-2) 7.3(-4) -7.6(-4) 2.3(-3) 3.0(-2) 2.1(-4)

6.5. Molecular excitation and line fluxes

157

Standard (2 kpc) which assumes a distance of 2 kpc instead of 1 kpc to the source, as suggested by Schneider et al. (2006). For higher J transitions, fluxes however differ significantly between the models. Assuming the level population in the LTE (Standard (LTE)) yields fluxes more than two orders of magnitude larger compared to the model Standard. Excited formation increases the line fluxes of all lines by less than 20 % for Tform = 100 K. Assuming Tform = 3000 K, a larger increase is found for higher J lines and amounts to a factor of 3 for the J = 6 → 5 line. Inclination affects the line flux by the combination of line and dust opacity, together with the beam containing different parts of the model. For example, the dust opacity reaches about 1.2 for the J = 6 → 5 line, while it is below 0.07 for the J = 1 → 0 line. Thus, the flux of the J = 6 → 5 line in the model Standard (90◦ ) (edge on) is about a factor of two smaller compared to the model Standard (0◦ ) with a line of sight parallel to the outflow. Compared to the model Standard, the inclination changes the flux mostly by less than 40 %. We note that for other sources with larger total column density, the dust attenuation may be important even for longer wavelengths. Increasing the distance to 2 kpc finally, decreases the fluxes by a factor of about 2.5. The frequency integrated maps and spectra of model Standard given in Figure 6.10 show that the concentration of the emission to the innermost region is more pronounced for the J = 6 → 5 line compared to the J = 1 → 0. This is also reflected in a model (Standard (25.9”)), convolved to a 25.8” beam. The flux ratios of that model to model Standard decreases almost as fast as the beam dilution factor, indicating that most of the emission is within the beam. The profile of the J = 1 → 0 line shows considerable self-absorption in the region where the line of sight follows the outflow wall due to the opening angle being similar to the inclination (positions A and B). The optical depth at the line center at these positions are 7 (A) and 19 (B). Further out, the optical depth is about 3 (C), which is about twice the value found in the Section 6.5.1 for a ray crossing the outflow wall once. The optical depth is even small for distances larger than position C. The J = 6 → 5 line has much smaller line optical depths below 0.2 (position A) compared to the J = 1 → 0 line. Beam convolved spectra of both lines however, do not show a self-absorption dip, suggesting that most of the emission comes from regions with moderate optical depth. The considerable line optical depths for lines of sight along the outflow walls raises the question, how much the modeled line fluxes depend on the assumed intrinsic line shape and velocity structure. HIFI can observe the J = 1 → 0 and J = 2 → 1 lines of CH+ spectrally resolved but with a beam corresponding to more than 10000 AU at the assumed distance of 1 kpc. This is insufficient to directly probe the velocity structure along the outflow wall. It can however be used to constrain the average line width along the outflow wall. Spatially resolved interferometric observations reveal broader lines in the outflow wall without much velocity shift to the systemic velocity (Bruderer et al. 2009a). This has been implemented in model Standard (Section 6.2.2). A model without increased microturbulence in the outflow walls (Standard (Noturb)) has been calculated and shows total line fluxes within 20 % to model Standard. We conclude that the while the modeled line shape may depend strongly on the assumed intrinsic line and velocity structure, the line fluxes are pretty independent. Different chemical models (α20, α100 and Disk ) affect the line fluxes of CH+ by a combination of the chemical abundance, the temperature and density structure and also the beam containing different regions of the model. For example the total amount of CH+ in the models α20, α100 and Disk within 4000 AU relative to the model Standard are 0.64, 0.36 and 1.04, respectively (Tables 6.4 and 6.5). The line fluxes relative to model Standard are thus similar to the abundance ratios for models α20 and Disk in low-J lines. For α100, the line is however much stronger due to the larger mass with high temperature (Table 6.2). For higher J lines, the situation is different. The lines of the model Disk get weaker with J relative to model Standard, while those of model α20 get stronger. This is due to the fact, that the abundance of CH+ decreases in model Disk in regions with high

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

0

10 5

B C

−10 −10

−5

0 5 ∆α [”]

2 1

10

A 0

10

Outflow

5

B

−5 −5

0 ∆α [”]

Spectra, CH+ , J = 1 → 0

10

Spectra, CH+ , J = 6 → 5

20

A

8

A

Intensity [K]

25

15

B

10 5

2 1

5

6 4 Convolved x 10 2

C 0

20

Tmb dv [K km s−1 ]

20

Outflow

−5

Intensity [K]

50

50

R

A

5

Integrated Intensity

5

100

Integrated flux, CH+ , J = 6 → 5 (60 µm, 4976 GHz) 100

∆δ [”]

Beam FWHM

R

∆δ [”]

200 10

Integrated Intensity

Integrated flux, CH+ , J = 1 → 0 (359 µm, 835 GHz)

Tmb dv [K km s−1 ]

density, while in the model α20 more material with high density is heated by FUV. We conclude that the low-J lines of CH+ are better tracers for the chemical abundance than higher J lines, as the higher J lines depend more on the temperature structure. The difference between the models is of order of about 3, similar to the uncertainty of the chemical model. Details of the geometry can thus not be obtained solely by observations of diatomic hydrides.

B

Convolved x 10

−12 −9 −6 −3 0 3 6 Velocity [km s−1 ]

9

12

0

−12 −9 −6 −3 0 3 6 Velocity [km s−1 ]

9

12

Figure 6.10: Synthetic velocity integrated maps (top panels) and spectra (bottom panels) of the CH+ J = 1 → 0 (left panels) and J = 6 → 0 transitions (right panels) for the model Standard. The appropriate Herschel beam (FWHM) is given by a circle. Contour lines give integrated intensities of 10 and 100 K km s−1 . The continuum is subtracted from the spectra and the red (gray) spectrum is convolved to the Herschel beam and multiplied by 10. The spectra at positions A, B and C are given in a pencil beam.

6.6. Observing diatomic hydrides with Herschel and ALMA

159

The SH lines are very weak, with most line fluxes clearly below the detection limit. This is remarkable since the volume averaged abundance of SH (Table 6.3) is only about two orders of magnitude smaller than for CH+ . The weak fluxes can be explained by the poor excitation in regions with high abundance of SH, due to the large energy separation of the levels (Section 6.5.1). Lines of the 2 Π1/2 ladder with upper level energies of several 100 K have line fluxes below 10−3 K km s−1 . Lines between 2 Π3/2 states can reach line fluxes of a few times 10−2 K km s−1 owing to their level energies similar to the dust or gas temperature in regions where SH is abundant. These lines can also be seen in absorption if a sufficient column density of cold dust lies between the SH emitting region and the observer. Unfortunately, this feature is most prominent in the 2 Π3/2(5/2→3/2) line at 1383 GHz which cannot be observed by PACS or HIFI. The abundance of SH within 4000 AU of the models α20, α100 and Disk relative to model Standard are 180, 0.04 and 23 (Tables 6.4 and 6.5). This is only reflected in lines of the 2 Π3/2 levels which are excited in larger parts of the envelope compared to the 2 Π1/2 levels. Because of the larger amount of warm material, lines obtained from model α100 can be stronger than from the model Standard despite the lower abundance. We conclude that the excitation in the warm region of the FUV outflow walls is essential for observable lines of diatomic hydrides. Thus, observations or nondetections do not necessarily reflect column densities, due to high critical densities combined with large energy separations between subsequent energy levels.

6.6

Observing diatomic hydrides with Herschel and ALMA

The previous sections have shown that light diatomic hydrides (CH, NH, SH, OH) and their ions belong to a particular class of molecules that are expected to probe the outflow walls of YSO envelopes. This is due to three factors relating to formation and excitation of the diatomic hydrides: i.) Owing to considerable activation barriers for the formation of many of the diatomic hydrides, a high temperature of several 100 K is needed for their formation. ii.) The chemical network leading to the formation is initiated by photoionization and large FUV fields are required for the formation of diatomic hydrides. iii.) Due to high critical densities and a large energy separation between the molecular levels, a high density and temperature is needed for excitation of the diatomic hydrides. Protostellar FUV radiation can escape through a low density cavity, etched out by the outflow, and irradiate the outflow walls separating the outflow and envelope. Our work predicts the abundance and excitation of diatomic hydrides in this thin outflow walls to govern the total line flux and result in fluxes that can be easily detected. Diatomic hydrides thus are predicted to be valuable tracers for the chemistry and physics of the outflow walls, which are in many ways similar to the upper atmosphere of protoplanetary disks. The Herschel Space Observatory provides a unique opportunity to observe diatomic hydrides, as their lines are above atmospheric windows. Beyond the relatively large sample of hydride lines in the HIFI spectral range, which is targeted by the WISH Herschel guaranteed time key program in this source, PACS observations of higher J lines may yield important constraints on the hottest regions. A particular example are high-J lines of CH+ which are predicted to be detectable by PACS. For this particular source, the relatively large beam of Herschel does not allow to spatially resolve the outflow walls and our model do also not indicate that the predicted line fluxes can be used easily as tracer for the exact shape of the cavity. In other sources with a good balance between distance and affected outflow area (e.g Serpens-FIRS 1), the good spatial resolution of the PACS camera may allow to spatially resolve different transitions of CH+ . The upcoming Atacama Large Millimeter and submillimeter Array (ALMA) can observe diatomic hydrides at very high angular resolution using band 10 receivers (787-950 GHz). For example the J = 1 → 0 line of the 13 CH+ isotope is shifted sufficiently to be inside an atmospheric window

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

(Falgarone et al. 2005). Also OH+ at 909 GHz (Wyrowski et al. 2010) has been detected from ground. Menten et al. (2010) has detected SH+ at 683 GHz (ALMA band 9) and thus the SH+ line at 893 GHz might be observable with ALMA band 10. Furthermore, lines of NH (946 GHz) and SH (866 and 875 GHz) are observable with ALMA band 10 receivers. Herschel observations can thus be complemented in the near future with high-resolution interferometric data to constrain parameters like the geometry in the innermost region.

6.7

Limitations of the model

The scenario of directly irradiated outflow walls by radiation of the protostar is not the only possible to alter the chemical composition in the outflow walls. For example mixing between the warm and ionized outflow with the envelope has been suggested by Taylor & Raga (1995) (see also Raga et al. 2002, 2007). The observed abundance of CO+ , however, cannot be explained by mixing solely (Chapter 3). Major dissociative shocks may occur at the boundary layer between outflow and inflow and produce FUV radiation that irradiates the quiescent envelope. To replace the FUV field of direct protostellar radiation with shocks, a rather high shock velocity is required. For example, in the model Standard at z = 2000 AU, vs ≈ 75 km s−1 is required (Neufeld & Dalgarno 1989). Such conditions are clearly met for the bow shock or internal jet working surfaces. Along the outflow walls, however, the velocity perpendicular to the quiescent envelope is likely smaller. In the simple model of the Couette flow solution (e.g. Lizano & Giovanardi 1995) neglecting turbulence, the mean velocity in the entrainment/mixing is along the outflow cavity and depends linearly on the distance to the cavity edge. The width of this layer and the substructure of the velocity including turbulences are however not easily constrained. Observationally, no clear signs of shocks in the outflow walls have been found in interferometric observations of AFGL 2591 by Bruderer et al. (2009a). We conclude that further study of the combined effects of mixing and shocks is warranted, especially in the context of low-mass star formation with less or cooler protostellar FUV radiation. Several simplifications have been made to make the calculation feasible. As discussed above, the physical structure does not account for shocks and mixing in the outflow layer or a clumpy surface of the outflow wall. The chemical models used here do not consider non-thermal desorption (e.g. photodesorption). The photodesorption yields are not well known for many species. However, the change in abundance at the 100 K evaporation temperature of H2 O, CO2 and H2 S is small in the presence of a strong FUV field such as in the outflow wall. Thus the photodesorption would not affect the results of the outflow wall enhanced species considerably. The drop of the water abundance in the region with FUV irradiation but temperatures below 100 K would be mitigated by photodesorption. FUV dissociation and ionization processes are treated here in the rate approach which does not allow to specify the protostellar radiation field explicitly. The surface temperature of the protostar is however not well constrained by observations and would affect the chemistry if the temperature were not hot enough to dissociate CO (Spaans et al. 1995). The models do not account for vibrationally excited H2 (e.g. Sternberg & Dalgarno 1995) which may help to overcome the activation barrier and further increase the abundance of e.g. CH+ or OH. Grain surface reactions, except for H2 formation and charge exchange with grains, are not considered in the models. For species with longer chemical timescales (chemical clocks, e.g. many sulfur bearing species, Wakelam et al. 2004) the effects of infall must be taken into account.

6.8. Conclusions and outlook

6.8

161

Conclusions and outlook

We have used a detailed chemical model of the high-mass star-forming region AFGL 2591 to study the effect of different shapes of a concave low-density cavity on the chemistry. The cavity, etched out by the outflow, allows FUV radiation to escape the innermost region and irradiate the walls between outflow and envelope. We study especially those molecules that are newly observable with the HIFI/PACS instrument onboard Herschel and in particular light diatomic hydrides (CH, NH, SH, OH). The model used in this chapter is based on the two-dimensional axisymmetric chemical model introduced in Bruderer et al. (2009b), extended by a self-consistent calculation of the dust temperature and an escape probability method to calculate the molecular excitation. The model considers protostellar FUV and X-ray irradiation and makes use of the grid of chemical models presented in Bruderer et al. (2009c). The main conclusions of the chapter are: 1. Protostellar FUV radiation can enhance the abundance of light diatomic hydrides and their ions by photoionization/dissociation processes and heating in the outflow-walls considerably. The overall amount of CH+ , OH+ and NH+ in the envelope is orders of magnitude larger than in spherically symmetric geometry (Section 6.4). Similarly enhanced are C+ and H2 O+ . These species thus appear to be clear tracers of extended FUV irradiation and thus the geometric structure of the inner regions of YSOs. 2. Depending whether the vertical height of the envelope from the midplane to the outflow (geometrically thin or thick), the size and mass of the infrared heated hot-core region with temperature above 100 K, but no FUV irradiation, can change substantially (Section 6.3). Geometrically thin inner regions allows FUV radiation to penetrate to the midplane and destroy the bulk of important species such as CO2 and H2 O. The shape of the cavity is thus also important for species which are destroyed by FUV radiation (Section 6.3.4 and 6.4.2). 3. Provided that FUV radiation can escape the innermost region, the effects of different geometries (opening angle of the outflow, cavity shape, density in the cavity) on the total amount of species enhanced in the outflow-wall (CH, CH+ , OH+ , NH+ , C+ and H2 O+ ) are relatively small. Deviations are typically less than a factor of three (Section 6.4.2). 4. In outflow walls, the influence of protostellar X-ray irradiation on the chemistry and in particular on the diatomic hydrides is relatively small with respect to the much larger enhancement by FUV radiation (Section 6.4.3). 5. The large separation between the energy levels of diatomic hydrides requires a hot and dense gas for excitation. Thus, the line fluxes of CH+ , predicted to be enhanced in the hot outflow wall, reach fluxes of several K km s−1 even in the J = 6 → 5 line. On the other hand, SH, is predicted to be destroyed in the outflow wall, but to be abundant in the envelope. Thus, despite the volume averaged abundance of SH being only two orders of magnitude lower than for CH+ , it is not sufficiently excited in the cool envelope to be detectable (Section 6.5.1 and 6.5.2). 6. Excited formation of CH+ does not considerably affect observable line fluxes. Exceptions are high-J lines, if a large formation temperature is assumed and the ambient density is low (Section 6.5.1 and 6.5.2). This suggests that diatomic hydrides are important tracers of warm and FUV irradiated material. Diatomic hydrides are chemically closely related to important species like water. Studying diatomic hydrides will provide important insight in the chemistry of YSO envelopes. Not only the chemistry

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6. The influence of geometry on the abundance and excitation of diatomic hydrides

of the diatomic hydrides is interesting, as they may ultimately also be used a tracers for physical conditions (like the FUV radiation). Herschel will for the first time observe diatomic hydrides with a good sensitivity and angular resolution and making these tracers available. ALMA and in particular band 10 will allow to continue the study of diatomic hydrides at very high angular resolution. Since diatomic hydrides have not been studied thoroughly in the past, much molecular data and in particular collisional excitation rates are missing. More studies on these processes will be necessary in order to analyse Herschel observations. In Chapters 2, 3 and 6, we have introduced a fast method for chemical modeling of envelopes of young stellar objects and studying the molecular excitation. The numerical speed up allows to construct more detailed models including multidimensional geometries. The method will also facilitate the study of larger samples of observations as they will soon be available by new facilities with high sensitivity and larger spectral coverage as for example the upcoming Atacama Large Millimeter/submillimeter Array. Ultimately, extensions of the methods presented here may also be useful to study the chemical/physical evolution of protoplanetary disks.

Acknowledgments We thank Ewine van Dishoeck, Susanne Wampfler, Cecilia Ceccarelli, Michiel Hogerheijde and our referee, Tim van Kempen, for useful discussions. The publicly available RATRAN code (Hogerheijde & van der Tak 2000) has simplified the development of the escape probability code considerably. Submillimeter astronomy at ETH Zurich is supported by the Swiss National Science Foundation grant 200020-121676 (SB, AOB and PS). SDD is supported by the Research Corporation and the NASA grant NNX08AH28G.

Chapter 7

Summary and outlook

Wir stehen selbst entt¨auscht und sehn betroffen Den Vorhang zu und alle Fragen offen. Bertolt Brecht (1898 - 1956)

In this work, we have studied the influence of geometry on the chemical composition of a YSO envelope. In particular, we have investigated a geometry where protostellar FUV radiation can escape through a cavity, etched out by the outflow. If the shape of the cavity allows direct irradiation of the envelope by protostellar radiation, a thin layer of FUV ionized and heated gas is created (Chapter 3 and 6). In these outflow-walls, “traditional” PDR tracers (for example C+ ), but also simple molecules like CO+ or hydrides (OH, OH+ , CH+ , and others), are enhanced by orders of magnitude. The enhancement is predicted to be sufficently strong to dominate the emission from the envelope, even for unresolved single-dish observations. The discussion of this scenario is important, as for example the radiation that escapes through the cavity can reduce the radiation pressure and thus allow further accretion (Krumholz et al. 2005). The shape of the outflow cavity is found to have a relatively small influence on species enhanced in the outflow-wall, as long as the shape allows the FUV radiation to escape and irradiate the envelope at larger distance. Differences between models with different shapes, for example opening angles of the outflow, are within an order of magnitude. This is compared to the enhancement by directly irradiated outflow-walls over spherically symmetric models of up to 4 orders of magnitude. Conversely, some FUV destroyed species are more geometry dependent. In the innermost ∼ 1500 AU, heating by FIR photons can yield a region with temperatures above 100 K but without strong FUV irradiation. In that region, H2 O and CO2 can be highly abundant. FUV radiation can however penetrate this region and destroy these molecules, if the envelope is geometrically thin, i.e. the distance from the midplane to the outflow in vertical direction is small, like for example in a large outflow opening angle. The influence of X-rays is found to be small in this scenario compared to spherically symmetric models. In spherically symmetric models, X-rays are predicted to be important, due to their smaller cross section compared to FUV radiation that allows X-rays to penetrate deeper into the envelope.

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7. Summary and outlook

The large importance of FUV compared to X-rays in the outflow-wall scenario is due to the fact that a high-mass YSO emits most of its radiation in UV wavelengths LFUV ∼ Lbol . The X-ray luminosity is much smaller LX ∼ 10−4 Lbol and X-rays are thus less effective in heating the gas. FUV radiation, that escapes through the outflow cavity, may heat gas even at several 10000 AU to above 100 K, while X-rays can heat the gas to that temperature only in the innermost few AU. Observational evidence for the scenario of directly irradiated outflow-walls has been found in the high-mass star forming region AFGL 2591. In that source, high-resolution near IR observations indicate the existence of a cavity. We have carried out new observations using the Submillimeter Array (SMA) interferometer and find spatially resolved emission of outflow-wall enhanced species at the projected edge of the outflow cavity (Chapter 4). Also, unresolved observation of the outflow-wall enhanced CO+ can only be explained with this scenario (Chapter 3).

Diagnostic Value The strength and shape of a molecular line is controlled by the combination of different physical and chemical conditions like the abundance, temperature or density. Promising tracers for physical or chemical conditions are those molecules that only form and are excited under very peculiar conditions. We find for example hydrides (e.g. CH+ , OH+ ; Chapter 6) to have a very temperature dependent formation mechanism and they are only excited in warm, FUV heated, regions. These hydrides thus have a high diagnostic value to determine the amount of warm gas with strong FUV irradiation and are thus a good probe for the scenario of directly irradiated outflow-walls. Upcoming data from the Herschel Space Observatory will deliver an important contribution to prove the scenario proposed here. They can thus be used as tracer for the geometry – however, given the uncertainties in the modeling, it seems currently not possible to quantify details of the geometry purely from spatially unresolved observations. Spatially resolved observations of outflow-wall species, observable from the ground (e.g. CO+ or 13 CH+ ), will be carried out by the Atacama Large Millimeter/submillimeter Array (ALMA).

Outlook and possible extension of the work The work presented here can be extended in various directions. Clearly further observational tests of the models presented here are needed. One step in this direction are the Herschel observations carried out at this moment by the WISH guaranteed time key program1 . In the first part of this thesis, the observations of the radiation diagnostics subprogram of WISH have been prepared. The subprogram will observe different hydrides and hydride ions in both high-mass and low-mass star forming regions. In the foreseeable future, the Herschel data can be combined with ALMA and the James Webb Space Telescope. The high resolution of these observatories will allow to study the chemical stratification through the outflow-walls. Different extensions and additional applications of the models presented here are evident. For example: • The models presented here can also be applied to low-mass sources. We expect that for these sources with less FUV radiation in comparison with X-ray radiation, the relative influence of X-rays is larger and may result in observable effects of X-rays. • Simple species like CO or C+ with a well known chemical network seem to be good diagnostics of physical conditions (e.g. warm gas). For example high-J lines of CO can be used to probe a wide range of temperatures and thus disentangle effects of FUV or shock heating. 1

http://www.strw.leidenuniv.nl/WISH/

165 • Several extensions of the model can be useful to explain observations. For example the influence of excited H2 on the formation of the hydrides can be added and could result in an extended layer of hydrides behind the hot outflow walls. Gas-grain reactions like photodesorption or hydrogenation could also alter the chemical composition in this intermediate layer between outflow-wall and envelope. More accurate FUV dissociation and ionization rates, obtained from the cross section and the FUV spectra can be useful, especially if the FUV source is colder and produces only few photons with energy above 11 eV, which is necessary to dissociate CO. If the envelope is irradiated by such a colder radiation field with temperature . 10000 K, also the amount of hydrides like e.g. CH+ would be reduced. • The physical model could be extended to include the protoplanetary disk in a more selfconsistent way as Chapter 6. This would include the solution of the hydrostatic equations e.g. along vertical cuts through the disk. Detailed models of protoplanetary disks will obviously be interesting and timely, as they can be spatially resolved soon. The high resolution of upcoming facilities will also resolve the separation between the outflow cavity and the envelope. This separation is clumpy and governed by turbulence as observations in the near-infrared show. Thus, three-dimensional models may ultimately by necessary to model such observations at high resolution. • Coupling of the hydrodynamical evolution to the chemical model will be requird for predictions of species with longer time-scales. However, such models will have many free parameters that are required to be determined by observations. Thus, semianalytical models in the vein of Visser (2009) will be an adequate first step in this direction, before full hydrodynamical simuations in combination with chemical networks are carried out. Examples of species with long time-scales are sulfur bearing molecules (e.g. SO, SO2 ). Once the uncertainties in their chemical network are reduced, they may serve as chemical clocks.

166

7. Summary and outlook

Appendix A

Implementation of the UMIST 06 network

This appendix gives details on the implementation of the chemical network, used in this thesis. The UMIST 06 database for astrochemistry (Woodall et al. 2007) lists an interval [Tmin , Tmax ] for the recommended temperature range for each reaction rate. For some reactions, different rates are given for distinct temperature ranges. The paper lists several rates, for which no extrapolation to low temperatures of ≈ 10 K should be done. We switch off those reactions for temperatures below Tmin . All other reaction rates are extrapolated to lower temperatures with a few exceptions: some reactions of the collider type (“CL” in Woodall et al. 2007) have a negative activation energy. To stabilize the chemical network, we keep their rates constant at temperatures below 500 K or 1000 K, or switch the reaction entirely off below 500 K (cf. Table A.1). Other reactions involve significant extrapolation, i.e., Tmin > 300 K, and the rates do not decrease from 100 K to 10 K. Certain reactions are thus switched off outside their temperature range as given in the database. The reaction OH + CN → OCN + H is not switched off, since the OSU database1 (Smith et al. 2004) lists this reaction with the same rate description. The reaction O + H2 CO → CO + OH + H was introduced in UMIST 99 and has severe consequences for the abundance of formaldehyde: it is mainly destroyed by this reaction and the abundance drops by about two orders of magnitude at steady-state conditions. The line flux, modeled with RATRAN (Hogerheijde & van der Tak 2000) of a spherical model of AFGL 2591, is short by about two orders of magnitude compared to observations by van der Tak et al. (1999). Thus we switch this reaction off below the recommended temperature range of 1750 K - 2575 K. Furthermore, reactions with a rate more than an order of magnitude faster at 10 K than at 100 K are kept constant below their minimum temperature, k(T < Tmin ) = k(Tmin ).

1

www.physics.ohio-state.edu/˜eric/research.html

168

Implementation of the UMIST 06 network

Table A.1: Reactions not extrapolated to lower temperature in the implemented network (cf. text). The rates at 10 and 100 K are obtained using the UMIST 06 database. The number of the reaction in the UMIST database is given in the first column. No.

Remark Reaction

158 239 242 243 291 301 338 420 430 445 457 491 515 4084 4552 4554 4555 4558 4563 4568 4571 4573 4574 4575 4579 4581 4582 4583

f d d d d d d d e d f f d f c f f f f d b a a a a f a a

CH + CH3 OH → CH3 + H2 CO CH2 + CH2 → C2 H3 + H CH2 + O → CO + H2 CH2 + O → HCO + H CH3 + OH → H2 CO + H2 CH3 + O2 → HCO + H2 O O + H2 CO → CO + OH + H NH2 + NO → N2 + OH + H OH + CN → OCN + H OH + HOCH → CO2 + H2 + H NH3 + CN → HCN + NH2 CN + O2 → OCN + O HCO + HCO → CO + CO + H2 H + OH → H2 O + γ CO + M → O + C + M H + CH3 → CH4 + M H + O → OH + M H + OH → H2 O + M H + O2 → O 2 H + M H2 + N → NH2 + M C + C → C2 + M C + O → CO + M C+ + O → CO+ + M C + O+ → CO+ + M O + O → O2 + M O + SO → SO2 + M OH + OH → H2 O2 + M O2 H + O2 H → H2 O2 + O2 + M

k(10K) [cm3 s−1 ] 1.8(-7) 3.3(-11) 8.0(-11) 5.0(-11) 1.7(-12) 1.7(-12) 1.0(-10) 1.5(-12) 7.0(-11) 1.8(-11) 2.0(-9) 1.6(-9) 3.6(-11) 3.3(-14) < 1.0(-30) 2.8(-26) 1.3(-30) 1.3(-28) 2.2(-28) 1.0(-26) 1.9(197) 4.9(67) 4.9(69) 4.9(69) 3.2(-9) 4.8(-28) 2.2(-2) 2.8(10)

k(100K) [cm3 s−1 ] 2.1(-9) 3.3(-11) 8.0(-11) 5.0(-11) 1.7(-12) 1.7(-12) 1.0(-10) 1.5(-12) 7.0(-11) 1.8(-11) 1.0(-10) 3.2(-11) 3.6(-11) 6.6(-16) < 1.0(-30) 4.5(-28) 1.3(-31) 1.4(-30) 1.2(-31) 1.0(-26) 1.7(-9) 9.6(-19) 9.6(-17) 9.6(-17) 9.4(-32) 6.9(-30) 1.9(-28) 3.6(-29)

Remarks: a Kept constant below 500 K, k(T < 500K) = k(500K). b Kept constant below 1000 K, k(T < 1000K) = k(1000K). c Switched off below 500 K, k(T < 500K) = 0. d Switched off outside recommended temperature range, k(T 6∈ [Tmin , Tmax ]) = 0. e Also listed in the OSU database, thus kept at recommended rate. f Kept constant below Tmin , k(T < Tmin ) = k(Tmin ).

Appendix B

A Monte Carlo code for FUV radiative transfer

In this appendix, we discuss the implementation of the radiative transfer calculation of the FUV radiation. Unlike the work of van Zadelhoff et al. (2003), our calculation is implemented on a three dimensional Cartesian grid and thus already able p to solve three-dimensional problems. To improve convergence, points with constant z and r = x2 + y 2 are averaged for the axi-symmetric model in this work. To save memory, only cells with positive values of x, y, and z are stored, but photon packages can of course still travel from one quadrant to another. The flow of the calculation is as follows. 1. A photon package is started from the FUV source at an arbitrary direction with a luminosity of Lpackage = Lsource /(8 · Nphoton ) [erg s−1 ], where Lsource is the luminosity of the FUV source in the 6 - 13.6 eV band. Nphoton is the number of photon packages in the simulation run. The factor 8 takes into account that only cells with positive values of x, y, and z are stored. 2. As the photon package travels along a straight line, Lpackage is attenuated by exp (−τloc ) with τloc = σext n(x, y, z) ∆s, the local FUV optical depth. The optical depth depends on σext , the extinction cross-section, n(x, y, z) the local density and ∆s the step size of the code. ThePtotal extinction, that a photon package suffers from the source to the current position τtot = τloc is stored. To avoid discretization errors, especially close to the source, we use the analytical expression for the local density. 3. If the photon package passes the border of a cell, the local intensity is updated. The attenuated intensity is obtained from Iatt (x, y, z) =

X

Lpackage ×

|~nphoton · ~ncell | , A

where the sum is taken over all photon packages passing the cell. The surface of the cell wall which the photon package just passed is given by A [cm2 ] and the directions of the photon package and the normal of the cell separation is given by ~nphoton and ~ncell , respectively.

170

A Monte Carlo code for FUV radiative transfer

4. The unattenuated intensity Iunatt is calculated using the same expression as Iatt , except Lpackage is replaced by Lpackage × exp (+τtot ). 5. If a random number [0,1] is larger than exp (−τscat ), with τscat = σext n(x, y, z) ∆s, the scatteringoptical depth, a new direction of the photon package is chosen following the implemented phase function φ(∆). For our work, we use the tabulated function of Draine (2003b) for an average Milky Way dust with RV = 3.1 and C/H=56 ppm in PAHs. 6. For each photon package, steps 2-5 are repeated, until Lpackage is smaller than a certain threshold or the package left the simulated area. 7. The code propagates 105 photon packages and then averages for the attenuated intensity Iatt in the axi-symmetric model. Three subsequent solutions are used to calculate the signal-to-noise ratio (S/N), defined by   i i Iatt − hIatt i −1 . S/N ≡ max i i=1,3 hIatt i

Only cells with an averaged intensity larger than 10−4 ISRF are considered. Convergence is reached, if the S/N exceeds 50 in all cells. This means less than 2% difference to the average. The application in this work requires 9 × 106 photon packages to achieve this accuracy.

As a benchmark test, the code is re-run for the density distribution in Sect. 3.3.1, but with scattering switched off. In this way, an analytical solution can be compared to the result of the code. For the analytical solution, only the distance r and column density N(H) [cm−2 ] to the FUV source are needed to derive the local FUV flux [erg s−1 cm−2 ] F loc =

LUV × exp (−τUV ) 4πr 2

with the FUV luminosity LUV = 4.2 × 1037 erg s−1 and the optical depth at FUV wavelengths τUV = 2.4(N(H)[cm−2 ])/1.87 × 1021 . The conversion factor of 1 ISRF ≡ 1.6 × 10−3 erg s−1 cm−2 yields the flux G0 in units [ISRF]. The agreement between the code and the analytical calculation is tested for points with G0 > 10−2 . A very good agreement is found, with deviations less than 50%.

B.1

The FUV luminosity

As an input quantity for the FUV Monte Carlo code, the FUV luminosity LFUV is required. Assuming the protostar to emit a blackbody spectrum, this quantity depends on the bolometric luminosity Lbol and the effective temperature Teff . While Lbol of the embedded protostar can be determined relatively well from photometry in the IR and is assumed to be given in the following, only rough estimations of Teff are available, since photons are absorbed or redistributed to longer wavelengths by the high dust and gas column density toward the source. The Stefan-Boltzmann law requires 4 Lbol = 4πR2 σTeff , hence R = R⊙



T⊙ Teff

2 s

Lbol L⊙

(B.1)

with the source radius R, the Stefan-Boltzmann constant σ, and the solar temperature, radius, and luminosity (T⊙ , R⊙ , and L⊙ ). The FUV band is limited by the Lyα edge (13.6 eV, λmin = 912 ˚ A)

The FUV luminosity

171

at short wavelengths and the average dust working function at long wavelengths (6 eV, λmax = 2067 ˚). For temperatures between 2.4 × 104 K and 5.6 × 104 K, the peak of the blackbody intensity A Bλ (Teff ) is within the FUV band (Wien’s displacement law, λmax [˚ A] = 5.1 × 107 /(T [K])). The FUV luminosity LFUV is obtained from integrating Bλ (Teff ) between λmin and λmax by Z λmin 2 πBλ (Teff ) dλ (B.2) LFUV = 4πR λmax Z xb 2 4 T⊙ x3 60σ R⊙ × dx , (B.3) = Lbol × 3 x π L⊙ xa e − 1 with xa = hc/kTeff λmin and xb = hc/kTeff λmax . For xa = 0 and xb → ∞, the integral in Eq. B.3 is π 4 /15, and Stefan-Boltzmanns law is recovered. In Fig. B.1, the FUV luminosity depending on Teff is given for Lbol = L⊙ . At a temperature of 2.7 × 104 K, where LFUV peaks, the ratio LFUV /Lbol is 0.55. Considering temperatures below this peak, the FUV luminosity is within a factor of 3 for Teff > 1.2 × 104 K and a factor of 10 for Teff > 9 × 103 K. We note that this is valid independently of Lbol , since LFUV ∝ Lbol .

B5

LFUV (erg s−1 )

1033

B2 B0 O5

B9 A0 A5

32

10

F0 G0

1031

K0 1030

K5

1029 5 · 103

1 · 104

2 · 104

temperature (K)

5 · 104

Figure B.1: Luminosity in the FUV band depending on Teff for Lbol = L⊙ . The spectral classification is indicated by red circles. How does this temperature dependence for exmple affect the results of the models in Sect. 4.4? In the absence of any attenuation, LFUV > 6 × 1035 erg s−1 is required to provide the necessary FUV field of 3 ×103 ISRF at position B for heating. Assuming the bolometric luminosity to be correct, the temperature needs to be higher than 6800 K. For a temperature of 1.5 × 104 K instead of 3 × 104 K, the FUV luminosity decreases by a factor of 2 and the required column density for attenuation (Sect. 4.4) reduces to τ = 3.5. We conclude that the modeling results are not affected by Teff as long as the temperature exceeds about 104 K.

172

B.2

A Monte Carlo code for FUV radiative transfer

An analytical expression for LFUV

For a convenient calculation of LFUV , an analytical expression is provided, accurate to within 10% for Teff < 5 × 104 K. From x = hc/kT λ (Eq. B.3) and the peak position xmax = 2.821 of x3 /(ex − 1), Wien’s displacement law follows. Above x ≈ xmax , the integrand may be approximated by x3 e−x (Wien’s law) which can be analytically integrated. Figure B.2 shows the functions together with the border of the integration xa = 1.578 × 105 /Teff [K] and xb = 6.962 × 104 /Teff [K]. We obtain LFUV ≈ 5.93 × 1032 × Lbol [L⊙ ] × ITeff

(erg s−1 )

(B.4)

x

with ITeff = (x3 + 3x2 + 6x + 6) e−x |xba . 3

x

x /(e − 1)

2

3 −x

x e

5 · 104

xb (Teff ) xa (Teff )

integrand

1.5

2 · 104 Z

1

xb

xa

x3 dx ex − 1

1 · 104

0.5 xa

0 0

temperature Teff (K)

30000K

5 · 103

xb 5

10

15

x Figure B.2: The integrand in Eq. B.3 (black) and Wien’s law (green). The border of the integration (xa , xb ) for the FUV band at different temperatures can be read off from the red and blue lines. As an example, the temperature of 3 × 104 K is shown.

Appendix C

Thermal balance calculation

In this appendix, references for the heating and cooling rates used in Sect. 3.3.3 are given and the temperature balance calculation is benchmarked with PDR models.

C.1

Heating rates

Photoelectric heating: The heating rate of photoelectrons from small dust grains and PAHs is ΓP E = 10−24 ǫ n G0 erg cm−3 s−1 , with the density n, the attenuated FUV field G0 and an efficiency ǫ. The efficiency depends on the ionization fraction of the grains, since the work function increases for ionized dust grains. Bakes & Tielens (1994) give an √ analytical fit for ǫ as a function of the ratio γ between ionization and recombination rate (γ = G0 T /ne , with the electron density ne and the temperature T ). H2 heating: Different heating processes involving molecular hydrogen are considered: (1) collisional de-excitation of vibrationally pumped H2 by FUV photons (FUV-pumping). (2) Formation heating: H2 forming on dust releases part of the binding energy to the gas. (3) Photodissociation of H2 heats the gas. The rates for process (1) and (3) depend on the local FUV field. Since line absorption is responsible for the pumping and dissociation of the molecule, we reduce the local FUV field by the self-shielding factor β(τ ) (Draine & Bertoldi 1996). Rates for (1) and (2) are implemented by the analytical expression of R¨ollig et al. (2006). The rate of process (3) is taken from Meijerink & Spaans (2005). C ionization: Photoionization of atomic carbon, C + γUV → C+ + e− , releases an energy of 1.06 eV to the gas and thus contributes to the heating. We follow Tielens (2005) for the implementation. X-ray heating: Fast photoelectrons produced by X-ray photons lose part of their energy through Coulomb interaction with thermal electrons. The heating rate is given by ΓX = η n HX , where HX [erg s−1 ] is the energy deposition of X-rays in the gas (e.g., St¨auber et al. 2005) per density. The efficiency factor η depends on the H/H2 ratio and the ionization fraction. An analytical fit for η to

174

Thermal balance calculation

detailed calculations is given in Dalgarno et al. (1999). Cosmic ray heating: For low degrees of ionization x < 10−4 , about 9 eV per primary ionization through a cosmic ray particle is used to heat the gas. The heating rate is thus Γcr = 1.5 × 10−11 ζH2 n erg cm−3 s−1 .

C.2

Cooling rates

Atomic fine structure lines: Forbidden fine structure lines of O, C, and C+ are important coolants at the surface of the FUV irradiated zone. We consider the [OI] 63 µm, [OI] 146 µm, [CI] 369 µm, [CI] 609 µm, and [CII] 158 µm lines. The cooling rate is obtained from Λline = Aul hνul β(τul ) nu (X) , with the Einstein-A coefficient Aul [s−1 ] of the transition u → l, νul [s−1 ] the line frequency, τul the optical depth, β(τul ) the escape probability, and nu (X) [cm−3 ], the density of the species X in the excited level u. An escape probability formalism (e.g., Tielens 2005) is used to calculate the level population iteratively. Collisional excitation by H, H2 , and electrons is taken into account - the abundance of the collision partners is read out of the chemical grid. We use the same collision rate coefficients as Meijerink (2006). The optical depth τ is calculated using the molecular column density to the outflow cone (Sect. 3.3.3). H2 line cooling: Vibrational lines of H2 can contribute to the cooling of the gas. Due to the large gap between the ground state and the first excited state, corresponding to about 6 000 K, we use the two-level approximation given in R¨ollig et al. (2006) as a simplification. Again, self-shielding is taken into account for the radiative pumping by FUV photons. Gas-grain cooling/heating: The difference in temperature between Tgas and the Tdust leads to a transfer of heat. This can be cooling close to the FUV irradiated zone, where Tdust < Tgas , or heating deeper in the envelope. The rates are proportional to Tdust − Tgas . We implement the results of Hollenbach & McKee (1989) for a minimal grain size of 10 ˚ A. Cooling by CO and H2 O: Molecules can contribute to the gas cooling by rotational lines at low temperature and rovibrational lines at higher temperature. We include line cooling by CO and H2 O using the fitted rates of Neufeld & Kaufman (1993) (for T > 100 K) and Neufeld et al. (1995) (for T 6 100 K). The rates depend on the molecular column density, the temperature, and the density of the collision partner. Their work considered excitation by H2 . Excitation by electrons and atomic hydrogen is taken into account by the approximation given in Yan (1997) and Meijerink & Spaans (2005). The molecular column density is obtained in the same way as for the cooling through atomic fine structure lines. Cooling by isotopes (13 CO, C18 O, H18 2 O) can be important due to the smaller optical depth. Isotope ratios by Wilson & Rood (1994) are used to scale the molecular density and the column density. Recombination: Recombination of electrons with grains and PAHs can cool the gas. We implement the fit-results by Bakes & Tielens (1994) for our calculation. This rate also depends on the ratio γ between ionization and recombination rate, due to increased Coulomb interaction in gas with a high ionization fraction.

Benchmark test

175

Lyα emission: Atomic hydrogen excited by electron collisions emit Lyα photons and thus contribute to the cooling. This process is only efficient at temperatures higher than a few thousand Kelvin. We use the cooling rate given in Sternberg & Dalgarno (1989).

C.3

Benchmark test

We compare our calculation of Tgas to PDR codes, in a similar situation as model V4 in the PDR comparison study by R¨ollig et al. (2007). The test consists of a plane parallel slab at a density of 105.5 cm−3 and an FUV irradiation of χ = G0 /1.71 = 105 . We assume an X-ray flux of about 0.04 erg s−1 cm−2 consistent to the AFGL 2591 model at z = 1 000 AU assuming an X-ray luminosity of LX = 1032 erg s−1 . Differences in the resulting gas temperature, when X-rays are switched off, are however negligible.

1 · 104

a.) Gas and Dust Temperature

5 · 103

±σ

Temperature [K]

2 · 103 1 · 103 5 · 102 2 · 102 1 · 102 5 · 101

Tgas - This work Tdust - This work

2 · 101 1 · 101

Tgas - PDR Comparison study, Average of different PDR models 0.1

0.2

0.5

b.) Fractional Abundances

Fractional abundance n(X)/nTot

100

1

2

5

10

20

Attenuation τ [Av ]

H H2

10−2

e−

CO

O

10−4

C

+

10−6

C 10−8

H2 O

10−10 0.1

0.2

0.5

1

2

5

10

20

Attenuation τ [Av ]

Figure C.1: Benchmark test of the heating/cooling calculation. a.) Result of the thermal balance calculation described in Appendix C. The gray shaded region gives the range of the PDR comparison study by R¨ollig et al. (2007) (one standard deviation, mean=average=dotted line). b.) Fractional abundances of species relevant to the thermal balance calculation. Figure C.1a shows the dust and gas temperature obtained with the thermal balance calculation. As expected, the gas temperature is much higher than the dust temperature for low optical depth,

176

Thermal balance calculation

while both temperatures agree well at high optical depth and density (Doty & Neufeld 1997). The mean of the gas temperature obtained with different PDR codes is given by a dotted line, along with one standard deviation given by the gray shaded region. The results of our code agrees very well with the results of the PDR codes, while the calculation only takes a few seconds.

10−15

a.) Heating Rates Photoelectric (1) (1)

Heating Rate [erg cm−3 s−1 ]

10−16

Total

(3)

H2 pumping (2) H2 formation (3) H2 dissociation (4)

−17

10

(4)

C ionization (5)

−18

(5)

X-ray (6)

−19

(6)

10

Cosmic rays (7)

10

(2)

Gas-grain (8) (8)

10−20 10−21 (7)

10−22 0.1 10−15

0.5

2

Total

(1)

10

1

(4) (3)

−17

10

5

10

20

Attenuation τ [Av ]

b.) Cooling Rates

−16

Cooling Rate [erg cm−3 s−1 ]

0.2

(2)

H2 (1)

CII 158 µm (6)

Recomb. (2)

Molecules (7)

OI 63 µm (3)

CI 369 µm (8)

Gas-grain (4)

CI 609 µm (9)

OI 146 µm (5)

(5)

−18

10

(6)

10−19

(7)

10−20 10−21

(9)

(8)

−22

10

0.1

0.2

0.5

1

2

5

10

20

Attenuation τ [Av ]

Figure C.2: Benchmark test of the heating/cooling calculation. a.) Heating rates, the line on top gives the total rate. b.) Cooling rates, the line on top gives the total rate. Figure C.1b gives the abundances of molecules involved in heating or cooling processes. As Meijerink & Spaans (2005), we use a “one-line” approximation for the H2 and CO self-shielding and are thus not able to reproduce the exact position of the H/H2 transition. Note the very low water abundance due to the destruction of water by X-rays (St¨auber et al. 2006). In Fig. C.2a, the rates of each heating process are given along with the total heating rate. At low optical depth, photoelectric heating and H2 -pumping are the main heating sources, while dust-gas coupling heats the gas at a high optical depth (τ > 6 in the example). X-ray heating does not contribute significantly to the heating rate, but the destruction of the important coolant H2 O by X-rays slightly enhances the temperature. Fig. C.2b shows rates of different cooling processes. At the edge of the calculated region, H2 rovibrational lines govern the cooling. [OI] 63 µm and molecular cooling is important at higher optical depth. Due to the low abundance of water, the main molecular coolant is CO.

Appendix D

Dust radiative transfer calculation

Detailes and benchmark tests of the dust radiative transfer calculation used in Chapter 6 are discussed in this appendix. The goal of the calculation is to obtain the dust temperature Tdust at every position of the 1D, 2D or 3D model. The temperature of the dust is calculated in the equilibrium between energy absorbed (Γabs ) and emitted (Γemit ) by the dust, Z Z Z Γabs = κλ Iλ dλdΩ = 4π κλ Bλ (Tdust )dλ = Γemit , (D.1) with the dust opacity per unit mass κλ , the incident radiation field Iλ and the Planck function for the dust temperature Bλ (Tdust ). To obtain the incident radiation field, the radiative transfer equation including scattering has to be solved for different wavelengths and directions. In this work, we use a Monte Carlo technique proposed by Lucy (1999) and Bjorkman & Wood (2001). The protostar with a luminosity Lbol provides the energy entering the system. We divide the luminosity into N photon packages that are traced through the envelope. The envelope is devided into different cells. For each photon package we carry out the following calculation: 1. A photon package of energy δL = Lbol /N is emitted and an random direction is chosen. We assign a wavelength λ which is randomly chosen from the spectrum of the protostar. To obtain a random number x of a given probability distribution P (x), we use Rx P (x′ )dx′ 0 R ξ= ∞ ∈ [0, 1] , (D.2) P (x′ )dx′ 0 where ξ is a random number between 0 and 1. In this work, we assume the spectrum of the protostar to be a black body function.

2. Using a random number ξ ∈ [0, 1], we find number τ = − ln(1 − ξ) for the optical depth that the photon package travels until the next scattering or absorption event. We then solve the equation Z s

τ=

(κλ + σλ ) ρ(s)ds ,

0

(D.3)

178

Dust radiative transfer calculation along the direction of the photon package for the distance s to the next scattering or absorption event (e.g. Figure 1.4). Here, κλ and σλ are the extinction and absorption cross-sections per mass, respectively, and ρ(s) is the dust mass density. A further random number ξ ∈ [0, 1] and the albedo aλ = σλ /(κλ + σλ ) are then used to determine if a photon is absorbed or scattered. If ξ < aλ , the event is scattering and otherwise, it is treated as absorption.

3. Scattering The new photon direction is selected using Eq. D.2 with the Henyey-Greenstein function 1 1 − g2 Pcos(θ) d(cos(θ)) = d(cos(θ)) , (D.4) 2 (1 − 2g cos(θ) + g 2 )3/2 with the anisotropy factor g, that depends on the dust properties and the wavelength (e.g. Draine 2003a). The photon package is reemitted at the same wavelenght but in a different direction.

Absorption The number of absorbed photons Ni is counted in each cell and increased in an absorption event. Using the dust mass of the cell M, we can estimate the energy absorbed in a cell and thus calculate the dust temperature, Z Ni δL new κλ Bλ (TDust )= . (D.5) 4πM new This yields the increase of the temperature in the cell ∆T with TDust → TDust +∆T = TDust . In order to maintain the radiative equilibrium, we must reemit the photon package at a new frequency following the emissivity of the dust, jλ = κλ Bλ (TDust + ∆T ). Using the expression κλ Bλ (TDust + ∆T ) and Eq. D.2, we can derive a new wavelength λ. However, the correct dust temperature is not known before all photon packages have been emitted and we thus need to account for the initially too low temperature and subsequently wrong distribution function for the wavelength. One can show (Baes et al. 2005) that the correct probability density function which accounts for the previously too low temperature is κν Bν (TDust + ∆T )dν κν Bν (TDust + ∆T )dν − Ni R . (D.6) Pν dν = (Ni + 1) R κν Bν (TDust )dν κν Bν (TDust )dν

We use this function to derive a new wavelength of the photon package assign a new direction. The photon packages are emitted isotropically.

4. Step 2 and 3 are repeated until the photon package has left the envelope. Statistics on the photon packages leaving the modeled region yields images of the dust radiation and the spectral energy distribution (SED). The algorithm is implemented in the Fortran code “Staubi”. To speed up the calculation, Equations D.4, D.5 and D.6 are precalculated in a look-up table.

D.1

Benchmark tests

In order to test the method and its implementation, we carry out two benchmark tests (e.g. Stamatellos & Whitworth 2003). The first test consists of a spherical envelope with a density distribution ∝ r −2 and an optical depth of 1000 at 1 µm. This envelope is irradiated by an external isotropic blackbody emission. By the law of thermodynamics, the envelope must adopt the temperature of the blackbody also, the radiation emitted by the envelope must agree with the external radiation field.

Benchmark tests

179

In Figure D.1, we show the temperature distribution and the SED calculated with Staubi, in perfect agreement with the expectations from thermodynamics. 1000

a.) Temperature

104

T = 50 K T = 250 K T = 750 K

103 Normalized Flux (Fλ /F )

Temperature (K)

500

b.) SED and Bλ (T )/(σT 4 /π)

200 100 50

101

100

T = 50 K T = 250 K T = 750 K

20

102

10−1

10 2 · 10

15

5 · 10

15

1 · 10

16

r (cm)

2 · 10

16

1

16

5 · 10

10

100

1000

Wavelength λ (µm)

Figure D.1: Benchmark test I: Radiative Transfer results of a sphere irradiate by an external isotropic blackbody emission. The density structure is ∝ r −2 and has a radial optical depth of 1000 at 1 µm. In a second benchmark test, we follow Ivezic et al. (1997) and again consider a spherical envelope with a constant density or a density distribution ∝ r −2 . The dust optical depth at 1 µm along the radius of the envelope is either 1, 10 or 100. A simple function is assumed for the dust absorptionand scattering-cross section, with qabs,λ = κλ /κ1µm and qscat,λ = σλ /σ1µm and qabs,λ = qscat,λ = 1

(D.7)

for λ < 1 µm and 1 λ2 1 = λ4

qabs,λ =

(D.8)

qscat,λ

(D.9)

for λ ≥ 1 µm. We assume isotropic scattering. Since this benchmark problem has no analytical solution, we compare the results of Staubi with those from the 1D code Dusty (Ivezic & Elitzur 1997). In Figure D.2, the results of Staubi and Dusty are given. They show a good agreement in both temperature profile and the spectral energy distribution, except for some noise in the temperature distribution of models with low optical depth. This results from the low number of absorptions due to the low optical depth and thus the low statistics to calculate the dust temperature. Regions with low optical depth are an inherent problem of the method used here, but an extension can be used to overcome the problem (e.g. Dullemond & Dominik 2004). Also, the method used here is not suitable for regions with very high optical depth (e.g. in the midplane of a protoplanetary disk) as

180

Dust radiative transfer calculation

photon packages may scatter for a long time befor leaving the region of high optical depth. This code can however be modified to treat such regions by a diffusion equation (e.g. Pinte et al. 2009 and references therein).

1000

a.) Temperature (ρ = ρ0 )

1000 500

Temperature (K)

Temperature (K)

500

b.) Temperature (ρ = ρ0 · y −2 )

200 100

τ1 µm = 1 (Dusty) τ1 µm = 10 (Dusty)

50

τ1 µm = 100 (Dusty)

200 100 50

τ1 µm = 1 (Staubi) τ1 µm = 10 (Staubi)

20

20

τ1 µm = 100 (Staubi) 10

10 1

10

100

1000

1

10

y = r/rDust

100

1000

y = r/rDust

c.) SED (ρ = ρ0 )

100

d.) SED (ρ = ρ0 · y −2 )

10−1 Normalized Flux (λFλ /F )

10−1 Normalized Flux (λFλ /F )

100

10−2

10−3

10−2

10−3

10−4

10−4

10−5

10−5 0.1

1

10 Wavelength λ (µm)

100

1000

0.1

1

10

100

1000

Wavelength λ (µm)

Figure D.2: Benchmark test II: Results of a comparision between the existing code Dusty and the new radiative transfer code (Staubi). a.) and c.) Temperature profile and Spectral Energy Distribution (SED) for three different radial opacities at 1 µm assuming a constant density sphere. b.) and d.) The same figure as a.) and b.) but for a sphere with a density ∝ r −2 .

Local radiation field

D.2

181

Local radiation field

For the calculation of the photoionization rates, the local FUV radiation field is required. The radiation field can be calculated using the Monte Carlo approach described in Appendix B, used in Chapter 3. However, for the calculation of axisymmetric models, the following algorithm, similar to van Zadelhoff et al. (2003) has proven to be faster and more flexible. As for the dust temperature calculation, we assign N photon packages, each a fraction of the protostellar luminosity, Lbol πBλ (T∗ ) L∗,λ = (D.10) Lλ (0) = N σT∗4 N with the Stefan-Boltzmann constant σ, the surface temperature of the protostar T∗ and the bolometric luminosity Lbol of the protostar. The equation follows immediately from the Stefan-Boltzmann law. The calculation is carried out for different wavelength bins, unlike the dust temperature calculation. The photon packages are traced through the envelope in a similar way as described in the previous section. We account for scattering, however the absorption is here not a discrete event but takes place at each step of length ds, the energy assigned to the photon package decreases as Lλ (s + ∆s) = Lλ (s)e−∆τλ ,

(D.11)

with the opacity ∆τλ per step. To calculate the mean intensity, the sum Jλ =

1 − e−∆τλ,i 1 X Lλ,i ∆si , 4πV i ∆τλ,i

(D.12)

is followed in each cell. Here, V is the volume of the cell, and the sum is taken over all photon packages passing through the cell. The sum requires the step-lenght ∆si , the current energy Lλ,i and the optical depth per step ∆τλ,i of each package. We have compared results of this method with the code used in Chapter 3 and found good agreement (within 30 %), for example in the mass and volume of the region with the attenuated FUV flux G0 > 1 of the model Standard (Chapter 3). The chemical models, used in this work, approximate the photodissociation and photoionization rates k by the fit equation k = CG0 exp(−γτV ), with fitting constants C and γ. We thus have to estimate G0 and τV in order to obtain abundances from the grid of chemical models (Chapter 1). For the models presented in Chapter 6, we calculate the mean intensity at the border of the FUV band, at 6 eV (2070 ˚ A) and at 13.6 eV (911 ˚ A). An addition wavelength of 10000 ˚ A(1 µm) is found to be necessary to determine G0 accurately. We use   ref κV J911 J2070 τV = − ln (D.13) ref κ911 − κ2070 J2070 J911 J10000 κ10000 /κV τV e , (D.14) G0 = ref J10000 with the opacity κλ at wavelength λ and a reference radiation field Jλref (e.g. the Drain ISRF; Draine 1978). We have tested the method by comparison of the photoionization and dissociation rates derived with this approach and the fully calculated rates, obtained using the cross-sections (Section 6.2.1).

182

Bibliography

Bibliography

Abel, N. P., Federman, S. R., & Stancil, P. C. 2008, ApJ, 675, L81 Aikawa, Y., Herbst, E., & Dzegilenko, F. N. 1999, ApJ, 527, 262 Aikawa, Y., Wakelam, V., Garrod, R. T., & Herbst, E. 2008, ApJ, 674, 984 Andersson, S., Barinovs, G ¸ ., & Nyman, G. 2008, ApJ, 678, 1042 Andre, P., Ward-Thompson, D., & Barsony, M. 1993, ApJ, 406, 122 Apruzese, J. P. 1981, Journal of Quantitative Spectroscopy and Radiative Transfer, 25, 419 Apruzese, J. P., Davis, J., Duston, D., & Whitney, K. G. 1980, Journal of Quantitative Spectroscopy and Radiative Transfer, 23, 479 Arce, H. G. & Sargent, A. I. 2006, ApJ, 646, 1070 Arce, H. G., Shepherd, D., Gueth, F., Lee, C., Bachiller, R., Rosen, A., & Beuther, H. 2007, Protostars and Planets V (University of Arizona Space Science Series), 245 Asplund, M., Grevesse, N., & Sauval, A. J. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 336, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ed. T. G. Barnes, III & F. N. Bash, 25 Avrett, E. H. 1971, Journal of Quantitative Spectroscopy and Radiative Transfer, 11, 511 Baes, M., Stamatellos, D., Davies, J. I., Whitworth, A. P., Sabatini, S., Roberts, S., Linder, S. M., & Evans, R. 2005, New Astronomy, 10, 523 Bakes, E. L. O. & Tielens, A. G. G. M. 1994, ApJ, 427, 822 Barnard, E. E. 1919, ApJ, 49, 1 Benz, A., ed. 2002, Plasma Astrophysics (Kluwer) Benz, A. O., St¨auber, P., Bourke, T. L., van der Tak, F. F. S., van Dishoeck, E. F., & Jørgensen, J. K. 2007, A&A, 475, 549

184

Bibliography

Berghoefer, T. W., Schmitt, J. H. M. M., Danner, R., & Cassinelli, J. P. 1997, A&A, 322, 167 Bergin, E. A. 2009, in Physical Processes in Circumstellar Disks Around Young Stars, ed. P. Garcia, (University of Chicago Press: Chicago) Bergin, E. A. & Tafalla, M. 2007, ARA&A, 45, 339 Bernes, C. 1979, A&A, 73, 67 Beuther, H., Churchwell, E. B., McKee, C. F., & Tan, J. C. 2007, Protostars and Planets V (University of Arizona Space Science Series), 165 Beuther, H., Schilke, P., Menten, K. M., Motte, F., Sridharan, T. K., & Wyrowski, F. 2002a, ApJ, 566, 945 Beuther, H., Schilke, P., Sridharan, T. K., Menten, K. M., Walmsley, C. M., & Wyrowski, F. 2002b, A&A, 383, 892 Bisschop, S. E., Jørgensen, J. K., van Dishoeck, E. F., & de Wachter, E. B. M. 2007, A&A, 465, 913 Bjorkman, J. E. & Wood, K. 2001, ApJ, 554, 615 Black, J. H. 1998, in Chemistry and Physics of Molecules and Grains in Space. Faraday Discussions No. 109, 257 Blake, G. A., Sutton, E. C., Masson, C. R., & Phillips, T. G. 1987, ApJ, 315, 621 Bohlin, R. C., Savage, B. D., & Drake, J. F. 1978, ApJ, 224, 132 Bok, B. J. 1948, Harvard Observatory Monographs, 7, 53 Bonnell, I. A. & Bate, M. R. 2002, MNRAS, 336, 659 Bonnor, W. B. 1956, MNRAS, 116, 351 Brinch, C. 2008, PhD thesis, Leiden Observatory Brinch, C., Crapsi, A., Hogerheijde, M. R., & Jørgensen, J. K. 2007, A&A, 461, 1037 Brinch, C., van Weeren, R. J., & Hogerheijde, M. R. 2008, A&A, 489, 617 Brown, P. N., Byrne, G. D., & Hindmarsh, A. C. 1989, SIAM J. Sci. Stat. Comput., 10, 1038 Bruderer, S., Benz, A. O., Bourke, T. L., & Doty, S. D. 2009a, A&A, 503, L13 Bruderer, S., Benz, A. O., Doty, S. D., van Dishoeck, E. F., & Bourke, T. L. 2009b, ApJ, 700, 872 Bruderer, S., Doty, S. D., & Benz, A. O. 2009c, ApJS, 183, 179 Burton, W. B., Elmegreen, B. G., & Genzel, R., eds. 1992, The galactic interstellar medium Cant´o, J., Raga, A. C., & Williams, D. A. 2008, Revista Mexicana de Astronomia y Astrofisica, 44, 293 Carkner, L., Kozak, J. A., & Feigelson, E. D. 1998, AJ, 116, 1933

Bibliography

185

Carr, J. S. & Najita, J. R. 2008, Science, 319, 1504 Carruthers, G. R. 1970, ApJ, 161, L81 Caselli, P., Hasegawa, T. I., & Herbst, E. 1993, ApJ, 408, 548 —. 1998, ApJ, 495, 309 Ceccarelli, C., Caux, E., Wolfire, M., Rudolph, A., Nisini, B., Saraceno, P., & White, G. J. 1998, A&A, 331, L17 Cernicharo, J., Decin, L., Barlow, M. J., Agundez, M., Royer, P., Vandenbussche, B., Wesson, R., Polehampton, E. T., De Beck, E., Blommaert, J. A. D. L., Daniel, F., De Meester, W., Exter, K. M., Feuchtgruber, H., Gear, W. K., Goicoechea, J. R., Gomez, H. L., Groenewegen, M. A. T., Hargrave, P. C., Huygen, R., Imhof, P., Ivison, R. J., Jean, C., Kerschbaum, F., Leeks, S. J., Lim, T. L., Matsuura, M., Olofsson, G., Posch, T., Regibo, S., Savini, G., Sibthorpe, B., Swinyard, B. M., Vandenbussche, B., & Waelkens, C. 2010, ArXiv e-prints Cernicharo, J., Liu, X., Gonzalez-Alfonso, E., Cox, P., Barlow, M. J., Lim, T., & Swinyard, B. M. 1997, ApJ, 483, L65 Cesaroni, R. 2005, Ap&SS, 295, 5 Cesaroni, R., Galli, D., Lodato, G., Walmsley, C. M., & Zhang, Q. 2007, Protostars and Planets V (University of Arizona Space Science Series), 197 Cesaroni, R., Galli, D., Lodato, G., Walmsley, M., & Zhang, Q. 2006, Nature, 444, 703 Cesaroni, R. & Walmsley, C. M. 1991, A&A, 241, 537 Chamberlin, T. C. 1901, ApJ, 14, 17 Charnley, S. B. 1997, ApJ, 481, 396 Churchwell, E. 2002, ARA&A, 40, 27 Codella, C., Viti, S., Williams, D. A., & Bachiller, R. 2006, ApJ, 644, L41 Cox, D. P. 2005, ARA&A, 43, 337 Crimier, N., Ceccarelli, C., Maret, S., Bottinelli, S., Caux, E., Kahane, C., Lis, D. C., & Olofsson, J. 2010, ArXiv e-prints Dalgarno, A. 2006, Proceedings of the National Academy of Science, 103, 12269 Dalgarno, A., Yan, M., & Liu, W. 1999, ApJS, 125, 237 de Graauw, T., Whyborn, N., Caux, E., Phillips, T., Stutzki, J., Tielens, A., G¨ usten, R., Helmich, F., Luinge, W., Martin-Pintado, J., Pearson, J., Planesas, P., Roelfsema, P., Saraceno, P., Schieder, R., Wildeman, K., & Wafelbakker, K. 2009, in EAS Publications Series, Vol. 34, EAS Publications Series, ed. L. Pagani & M. Gerin, 3–20 de Jong, T., Dalgarno, A., & Chu, S.-I. 1975, ApJ, 199, 69 de Wit, W. J., Hoare, M. G., Fujiyoshi, T., Oudmaijer, R. D., Honda, M., Kataza, H., Miyata, T., Okamoto, Y. K., Onaka, T., Sako, S., & Yamashita, T. 2009, A&A, 494, 157

186

Bibliography

Doel, R. C., Gray, M. D., & Field, D. 1990, MNRAS, 244, 504 Doty, S. D., Everett, S. E., Shirley, Y. L., Evans, N. J., & Palotti, M. L. 2005, MNRAS, 359, 228 Doty, S. D. & Neufeld, D. A. 1997, ApJ, 489, 122 Doty, S. D., Sch¨oier, F. L., & van Dishoeck, E. F. 2004, A&A, 418, 1021 Doty, S. D., van Dishoeck, E. F., & Tan, J. C. 2006, A&A, 454, L5 Doty, S. D., van Dishoeck, E. F., van der Tak, F. F. S., & Boonman, A. M. S. 2002, A&A, 389, 446 Douglas, A. E. & Herzberg, G. 1941, ApJ, 94, 381 Draine, B. T. 1978, ApJS, 36, 595 —. 2003a, ARA&A, 41, 241 —. 2003b, ApJ, 598, 1017 Draine, B. T. & Bertoldi, F. 1996, ApJ, 468, 269 Draine, B. T. & McKee, C. F. 1993, ARA&A, 31, 373 Dullemond, C. P. & Dominik, C. 2004, A&A, 421, 1075 Ebert, R. 1955, Zeitschrift fur Astrophysik, 37, 217 Egan, M. P., Shipman, R. F., Price, S. D., Carey, S. J., Clark, F. O., & Cohen, M. 1998, ApJ, 494, L199 Elitzur, M., ed. 1992, Astrophysics and Space Science Library, Vol. 170, Astronomical masers Elitzur, M. & Asensio Ramos, A. 2006, MNRAS, 365, 779 Evans, N. J., Dunham, M. M., Jørgensen, J. K., Enoch, M. L., Mer´ın, B., van Dishoeck, E. F., Alcal´a, J. M., Myers, P. C., Stapelfeldt, K. R., Huard, T. L., Allen, L. E., Harvey, P. M., van Kempen, T., Blake, G. A., Koerner, D. W., Mundy, L. G., Padgett, D. L., & Sargent, A. I. 2009, ApJS, 181, 321 Ewen, H. I. & Purcell, E. M. 1951, Nature, 168, 356 Falgarone, E., Phillips, T. G., & Pearson, J. C. 2005, ApJ, 634, L149 Feigelson, E., Townsley, L., G¨ udel, M., & Stassun, K. 2007, Protostars and Planets V (University of Arizona Space Science Series), 313 Feigelson, E. D. & Montmerle, T. 1999, ARA&A, 37, 363 Fischer, O., Henning, T., & Yorke, H. W. 1996, A&A, 308, 863 Flower, D. 2007, Molecular collisions in the interstellar medium (Cambridge University Press) Flower, D. R., Le Bourlot, J., Pineau des Forˆets, G., & Cabrit, S. 2003, Ap&SS, 287, 183 Flower, D. R., Pineau des Forets, G., Field, D., & May, P. W. 1996, MNRAS, 280, 447

Bibliography

187

Fuente, A., Garc´ıa-Burillo, S., Gerin, M., Rizzo, J. R., Usero, A., Teyssier, D., Roueff, E., & Le Bourlot, J. 2006, ApJ, 641, L105 Fuente, A., Rodrıguez-Franco, A., Garcıa-Burillo, S., Martın-Pintado, J., & Black, J. H. 2003, A&A, 406, 899 Garrod, R. T. 2008, A&A, 491, 239 Garrod, R. T., Vasyunin, A. I., Semenov, D. A., Wiebe, D. S., & Henning, T. 2009, ApJ, 700, L43 Gibb, A. G. & Little, L. T. 2000, MNRAS, 313, 663 Giuliani, J. P., Petrov, G. M., Apruzese, J. P., & Davis, J. 2005, Plasma Sources Sci. Technol., 14, 236 Goicoechea, J. R., Pety, J., Gerin, M., Teyssier, D., Roueff, E., Hily-Blant, P., & Baek, S. 2006, A&A, 456, 565 Graff, M. M. & Dalgarno, A. 1987, ApJ, 317, 432 Gredel, R., Lepp, S., Dalgarno, A., & Herbst, E. 1989, ApJ, 347, 289 G¨ udel, M., Briggs, K. R., Arzner, K., Audard, M., Bouvier, J., Feigelson, E. D., Franciosini, E., Glauser, A., Grosso, N., Micela, G., Monin, J.-L., Montmerle, T., Padgett, D. L., Palla, F., Pillitteri, I., Rebull, L., Scelsi, L., Silva, B., Skinner, S. L., Stelzer, B., & Telleschi, A. 2007, A&A, 468, 353 Habing, H. J. 1968, Bull. Astron. Inst. Netherlands, 19, 421 Hamel, J. 2002, Geschichte der Astronomie (Franckh-Kosmos Verlags-GmbH & Co.) Hammami, K., Owono, L. O., Jaidane, N., & Lakhdar, Z. B. 2008, Journal of Molecular Structure: THEOCHEM, 853, 18 Hammami, K., Owono Owono, L. C., & St¨auber, P. 2009, A&A, 507, 1083 Hartmann, J. 1904, ApJ, 19, 268 Hasegawa, T. I. & Herbst, E. 1993, MNRAS, 261, 83 Hayashi, C. 1961, PASJ, 13, 450 Herbst, E. & Klemperer, W. 1973, ApJ, 185, 505 Herbst, E. & van Dishoeck, E. F. 2009, ARA&A, 47, 427 Hogerheijde, M. 1998, PhD thesis, Leiden Observatory Hogerheijde, M. R. & van der Tak, F. F. S. 2000, A&A, 362, 697 Hogerheijde, M. R., van Dishoeck, E. F., Blake, G. A., & van Langevelde, H. J. 1998, ApJ, 502, 315 Hogerheijde, M. R., van Dishoeck, E. F., Salverda, J. M., & Blake, G. A. 1999, ApJ, 513, 350 Hollenbach, D. & McKee, C. F. 1989, ApJ, 342, 306

188

Bibliography

Hollenbach, D. & Salpeter, E. E. 1971, ApJ, 163, 155 Hollenbach, D. J., Takahashi, T., & Tielens, A. G. G. M. 1991, ApJ, 377, 192 Hollenbach, D. J. & Tielens, A. G. G. M. 1997, ARA&A, 35, 179 —. 1999, Reviews of Modern Physics, 71, 173 Ilgner, M. & Nelson, R. P. 2008, A&A, 483, 815 Indriolo, N., Geballe, T. R., Oka, T., & McCall, B. J. 2007, ApJ, 671, 1736 Ivezic, Z. & Elitzur, M. 1997, MNRAS, 287, 799 Ivezic, Z., Groenewegen, M. A. T., Men’shchikov, A., & Szczerba, R. 1997, MNRAS, 291, 121 Jansen, D. J., Spaans, M., Hogerheijde, M. R., & van Dishoeck, E. F. 1995, A&A, 303, 541 Jayawardhana, R., Hartmann, L., & Calvet, N. 2001, ApJ, 548, 310 Jeans, J. H. 1917, MNRAS, 77, 186 Jonkheid, B. J. 2006, PhD thesis, Leiden Observatory Jørgensen, J. K. 2004, A&A, 424, 589 Jørgensen, J. K., Bourke, T. L., Myers, P. C., Di Francesco, J., van Dishoeck, E. F., Lee, C.-F., Ohashi, N., Sch¨oier, F. L., Takakuwa, S., Wilner, D. J., & Zhang, Q. 2007, ApJ, 659, 479 Jørgensen, J. K., Hogerheijde, M. R., Blake, G. A., van Dishoeck, E. F., Mundy, L. G., & Sch¨oier, F. L. 2004a, A&A, 415, 1021 Jørgensen, J. K., Johnstone, D., van Dishoeck, E. F., & Doty, S. D. 2006, A&A, 449, 609 Jørgensen, J. K., Lahuis, F., Sch¨oier, F. L., van Dishoeck, E. F., Blake, G. A., Boogert, A. C. A., Dullemond, C. P., Evans, II, N. J., Kessler-Silacci, J. E., & Pontoppidan, K. M. 2005a, ApJ, 631, L77 Jørgensen, J. K., Sch¨oier, F. L., & van Dishoeck, E. F. 2002, A&A, 389, 908 —. 2004b, A&A, 416, 603 —. 2005b, A&A, 435, 177 Kant, I. 1755, Allgemeine Naturgeschichte und Theorie des Himmels (Zeitz, Bei W. Webel, 1798. Neue aufl.) Kastner, S. O. 1993, Space Science Reviews, 65, 317 Kaufman, M. J., Wolfire, M. G., Hollenbach, D. J., & Luhman, M. L. 1999, ApJ, 527, 795 Kim, K. & Kurtz, S. E. 2006, ApJ, 643, 978 Krolik, J. H. & Kallman, T. R. 1983, ApJ, 267, 610 Krumholz, M. R., McKee, C. F., & Klein, R. I. 2005, ApJ, 618, L33

Bibliography

189

Kwan, J. & Scoville, N. 1976, ApJ, 210, L39 Lada, C. J. 1987, in IAU Symposium, Vol. 115, Star Forming Regions, ed. M. Peimbert & J. Jugaku, 1–17 Lee, H.-H., Herbst, E., Pineau des Forets, G., Roueff, E., & Le Bourlot, J. 1996, A&A, 311, 690 Lee, J.-E., Bergin, E. A., & Evans, II, N. J. 2004, ApJ, 617, 360 Lepp, S. & Dalgarno, A. 1996, A&A, 306, L21 Leung, C. M., Herbst, E., & Huebner, W. F. 1984, ApJS, 56, 231 Lim, A. J., Rabad´an, I., & Tennyson, J. 1999a, MNRAS, 306, 473 Lim, A. J., Rawlings, J. M. C., & Williams, D. A. 1999b, MNRAS, 308, 1126 Lintott, C. J. & Rawlings, J. M. C. 2006, A&A, 448, 425 Lizano, S. & Giovanardi, C. 1995, ApJ, 447, 742 Lizano, S. & Shu, F. H. 1989, ApJ, 342, 834 Lockett, P. & Elitzur, M. 1989, ApJ, 344, 525 Lorenzani, A. & Palla, F. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 243, From Darkness to Light: Origin and Evolution of Young Stellar Clusters, ed. T. Montmerle & P. Andr´e, 745 Lucy, L. B. 1999, A&A, 344, 282 Mac Low, M., Klessen, R. S., Burkert, A., & Smith, M. D. 1998, Physical Review Letters, 80, 2754 Maloney, P. R., Hollenbach, D. J., & Tielens, A. G. G. M. 1996, ApJ, 466, 561 Maret, S., Ceccarelli, C., Caux, E., Tielens, A. G. G. M., & Castets, A. 2002, A&A, 395, 573 Maret, S., Ceccarelli, C., Caux, E., Tielens, A. G. G. M., Jørgensen, J. K., van Dishoeck, E., Bacmann, A., Castets, A., Lefloch, B., Loinard, L., Parise, B., & Sch¨oier, F. L. 2004, A&A, 416, 577 Masunaga, H. & Inutsuka, S. 2000, ApJ, 531, 350 Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425 Mayor, M. & Queloz, D. 1995, Nature, 378, 355 McCrea, W. H. 1960, Royal Society of London Proceedings Series A, 256, 245 McKee, C. F. & Ostriker, E. C. 2007, ARA&A, 45, 565 McKee, C. F. & Tan, J. C. 2003, ApJ, 585, 850 McKellar, A. 1940, PASP, 52, 187 McLaughlin, D. E. & Pudritz, R. E. 1996, ApJ, 469, 194

190

Bibliography

—. 1997, ApJ, 476, 750 Meijerink, R. 2006, PhD thesis, Leiden Observatory Meijerink, R., Glassgold, A. E., & Najita, J. R. 2008, ApJ, 676, 518 Meijerink, R. & Spaans, M. 2005, A&A, 436, 397 Meijerink, R., Spaans, M., & Israel, F. P. 2007, A&A, 461, 793 Menten, K., Wyrowski, F., Belloche, A., et al. 2010, A&A, in press Millar, T. J., Farquhar, P. R. A., & Willacy, K. 1997a, A&AS, 121, 139 Millar, T. J., MacDonald, G. H., & Gibb, A. G. 1997b, A&A, 325, 1163 Montmerle, T. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 243, From Darkness to Light: Origin and Evolution of Young Stellar Clusters, ed. T. Montmerle & P. Andr´e, 731 Moulton, F. R. 1905, ApJ, 22, 165 Mueller, K. E., Shirley, Y. L., Evans, II, N. J., & Jacobson, H. R. 2002, ApJS, 143, 469 Muller, C. A. & Oort, J. H. 1951, Nature, 168, 357 M¨ uller, H. S. P., Thorwirth, S., Roth, D. A., & Winnewisser, G. 2001, A&A, 370, L49 Nakajima, H., Imanishi, K., Takagi, S., Koyama, K., & Tsujimoto, M. 2003, PASJ, 55, 635 Nejad, L. A. M. 2005, Ap&SS, 299, 1 Nenkova, M., Ivezic, Z., & Elitzur, M. 1999, LPI Contributions, 969, 20 Neufeld, D. A. & Dalgarno, A. 1989, ApJ, 340, 869 Neufeld, D. A. & Kaufman, M. J. 1993, ApJ, 418, 263 Neufeld, D. A., Lepp, S., & Melnick, G. J. 1995, ApJS, 100, 132 Neufeld, D. A., Sonnentrucker, P., Phillips, T. G., Lis, D. C., De Luca, M., Goicoechea, J. R., Black, J. H., Gerin, M., Bell, T., Boulanger, F., Cernicharo, J., Coutens, A., Dartois, E., Kazmierczak, M., Encrenaz, P., Falgarone, E., Geballe, T. R., Giesen, T., Godard, B., Goldsmith, P. F., Gry, C., Gupta, H., Hennebelle, P., Herbst, E., Hily-Blant, P., Joblin, C., Kolos, R., Krelowski, J., Martin-Pintado, J., Menten, K. M., Monje, R., Mookerjea, B., Pearson, J., Perault, M., Persson, C., Plume, R., Salez, M., Schlemmer, S., Schmidt, M., Stutzki, J., Teyssier, D., Vastel, C., Yu, S., Cais, P., Caux, E., Liseau, R., Morris, P., & Planesas, P. 2010, ArXiv e-prints Neufeld, D. A. & Wolfire, M. G. 2009, ApJ, 706, 1594 Ng, K.-C. 1974, J. Chem. Phys., 61, 2680 Nguyen, T. K., Hartquist, T. W., & Williams, D. A. 2000, Ap&SS, 271, 171 Nussbaumer, H. 2005, Das Weltbild der Astronomie (vdf Hochschulverlag)

Bibliography

191

O’dell, C. R. & Wen, Z. 1994, ApJ, 436, 194 Offer, A. R. & van Dishoeck, E. F. 1992, MNRAS, 257, 377 Ossenkopf, V. & Henning, T. 1994, A&A, 291, 943 Parkin, E. R., Pittard, J. M., Hoare, M. G., Wright, N. J., & Drake, J. J. 2009, MNRAS, 400, 629 Pavlyuchenkov, Y., Wiebe, D., Shustov, B., Henning, T., Launhardt, R., & Semenov, D. 2008, ApJ, 689, 335 Pickett, H. M., Poynter, I. R. L., Cohen, E. A., Delitsky, M. L., Pearson, J. C., & Muller, H. S. P. 1998, Journal of Quantitative Spectroscopy and Radiative Transfer, 60, 883 Pinte, C., Harries, T. J., Min, M., Watson, A. M., Dullemond, C. P., Woitke, P., M´enard, F., & Dur´an-Rojas, M. C. 2009, A&A, 498, 967 Poelman, D. R. & Spaans, M. 2005, A&A, 440, 559 —. 2006, A&A, 453, 615 Preibisch, T., Balega, Y. Y., Schertl, D., & Weigelt, G. 2003, A&A, 412, 735 Preibisch, T. & Feigelson, E. D. 2005, ApJS, 160, 390 Price, S. D. & Walker, R. G. 1976, The AFGL four color infrared sky survey: Catalog of observations at 4.2, 11.0, 19.8, and 27.4 micrometer., ed. Price, S. D. & Walker, R. G. Qi, C. 2005, The MIR Cookbook, Harvard-Smithonian Center for Astrophysics Qiu, K., Zhang, Q., Beuther, H., & Yang, J. 2007, ApJ, 654, 361 Raga, A. C., de Colle, F., Kajdiˇc, P., Esquivel, A., & Cant´o, J. 2007, A&A, 465, 879 Raga, A. C., Noriega-Crespo, A., & Vel´azquez, P. F. 2002, ApJ, 576, L149 Rawlings, J. M. C., Redman, M. P., Keto, E., & Williams, D. A. 2004, MNRAS, 351, 1054 Reipurth, B. & Bally, J. 2001, ARA&A, 39, 403 Rodgers, S. D. & Charnley, S. B. 2003, ApJ, 585, 355 R¨ollig, M., Abel, N. P., Bell, T., Bensch, F., Black, J., Ferland, G. J., Jonkheid, B., Kamp, I., Kaufman, M. J., Le Bourlot, J., Le Petit, F., Meijerink, R., Morata, O., Ossenkopf, V., Roueff, E., Shaw, G., Spaans, M., Sternberg, A., Stutzki, J., Thi, W.-F., van Dishoeck, E. F., van Hoof, P. A. M., Viti, S., & Wolfire, M. G. 2007, A&A, 467, 187 R¨ollig, M., Ossenkopf, V., Jeyakumar, S., Stutzki, J., & Sternberg, A. 2006, A&A, 451, 917 Rosmus, P. & Werner, H.-J. 1982, Molecular Physics, 47, 661 Rybicki, G. B. & Hummer, D. G. 1991, A&A, 245, 171 Rybicki, G. B. & Lightman, A. P. 1979, Radiative processes in astrophysics (New York, WileyInterscience)

192

Bibliography

Salpeter, E. E. 1955, ApJ, 121, 161 Savage, C. & Ziurys, L. M. 2004, ApJ, 616, 966 Schmidt, O. Y. 1944, Dokl. Akad. Nauk. USSR, 6, 45 Schneider, N., Bontemps, S., Simon, R., Jakob, H., Motte, F., Miller, M., Kramer, C., & Stutzki, J. 2006, A&A, 458, 855 Sch¨oier, F. L., van der Tak, F. F. S., van Dishoeck, E. F., & Black, J. H. 2005, A&A, 432, 369 Schulz, N. S. 2005, From Dust To Stars Studies of the Formation and Early Evolution of Stars (Springer) Shaw, A. M. 2006, Astrochemistry: from astronomy to astrobiology (Wiley) Shepherd, D. S. & Churchwell, E. 1996, ApJ, 457, 267 Shu, F. H. 1977, ApJ, 214, 488 Shu, F. H., Adams, F. C., & Lizano, S. 1987, ARA&A, 25, 23 Smith, I. W. M., Herbst, E., & Chang, Q. 2004, MNRAS, 350, 323 Snow, T. P. & McCall, B. J. 2006, ARA&A, 44, 367 Snow, T. P. & Witt, A. N. 1996, ApJ, 468, L65 Sobolev, V. V. 1960, Moving envelopes of stars (Harvard University Press) Spaans, M., Hogerheijde, M. R., Mundy, L. G., & van Dishoeck, E. F. 1995, ApJ, 455, L167 Stahler, S. W. & Palla, F. 2005, The Formation of Stars (Wiley-VCH) Stamatellos, D. & Whitworth, A. P. 2003, A&A, 407, 941 St¨auber, P., Benz, A. O., Jørgensen, J. K., van Dishoeck, E. F., Doty, S. D., & van der Tak, F. F. S. 2007, A&A, 466, 977 St¨auber, P. & Bruderer, S. 2009, A&A, 505, 195 St¨auber, P., Doty, S. D., van Dishoeck, E. F., & Benz, A. O. 2005, A&A, 440, 949 St¨auber, P., Doty, S. D., van Dishoeck, E. F., Jørgensen, J. K., & Benz, A. O. 2004, A&A, 425, 577 St¨auber, P., Jørgensen, J. K., van Dishoeck, E. F., Doty, S. D., & Benz, A. O. 2006, A&A, 453, 555 Sternberg, A. & Dalgarno, A. 1989, ApJ, 338, 197 —. 1995, ApJS, 99, 565 Swings, P. 1937, MNRAS, 97, 212 Takahashi, T., Silk, J., & Hollenbach, D. J. 1983, ApJ, 275, 145 Taylor, S. D. & Raga, A. C. 1995, A&A, 296, 823

Bibliography

193

Tielens, A. G. G. M. 2005, The Physics and Chemistry of the Interstellar Medium (Cambridge University Press) Tielens, A. G. G. M. & Hollenbach, D. 1985, ApJ, 291, 722 Trumpler, R. J. 1930, PASP, 42, 214 Umebayashi, T. & Nakano, T. 1980, PASJ, 32, 405 van der Tak, F. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 387, Astronomical Society of the Pacific Conference Series, ed. H. Beuther, H. Linz, & T. Henning, 101 van der Tak, F., Neufeld, D., Yates, J., Hogerheijde, M., Bergin, E., Sch¨oier, F., & Doty, S. 2005, in ESA Special Publication, Vol. 577, ESA Special Publication, ed. A. Wilson, 431–432 van der Tak, F. F. S. 2005, in IAU Symposium, Vol. 227, Massive Star Birth: A Crossroads of Astrophysics, ed. R. Cesaroni, M. Felli, E. Churchwell, & M. Walmsley, 70–79 van der Tak, F. F. S., Black, J. H., Sch¨oier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, A&A, 468, 627 van der Tak, F. F. S., Boonman, A. M. S., Braakman, R., & van Dishoeck, E. F. 2003, A&A, 412, 133 van der Tak, F. F. S. & van Dishoeck, E. F. 2000, A&A, 358, L79 van der Tak, F. F. S., van Dishoeck, E. F., Evans, II, N. J., Bakker, E. J., & Blake, G. A. 1999, ApJ, 522, 991 van der Tak, F. F. S., van Dishoeck, E. F., Evans, II, N. J., & Blake, G. A. 2000, ApJ, 537, 283 van der Tak, F. F. S., Walmsley, C. M., Herpin, F., & Ceccarelli, C. 2006, A&A, 447, 1011 van Dishoeck, E. F. 1988, in Rate Coefficients in Astrochemistry. Editors, T.J. Millar, D.A. Williams, Kluwer Academic Publishers, Dordrecht, ed. T. J. Millar & D. A. Williams, 49 van Dishoeck, E. F. 2004, ARA&A, 42, 119 —. 2009, Astrochemistry of Dense Protostellar and Protoplanetary Environments, ed. Thronson, H. A., Stiavelli, M., & Tielens, A., 187 van Dishoeck, E. F. & Blake, G. A. 1998, ARA&A, 36, 317 van Dishoeck, E. F., Jonkheid, B., & van Hemert, M. C. 2006, in Chemical Evolution of the Universe. Faraday Discussions No. 133, 231 van Dishoeck, E. F. & Jørgensen, J. K. 2008, Ap&SS, 313, 15 van Kempen, T. A., Doty, S. D., van Dishoeck, E. F., Hogerheijde, M. R., & Jørgensen, J. K. 2008, A&A, 487, 975 van Kempen, T. A., van Dishoeck, E. F., G¨ usten, R., Kristensen, L. E., Schilke, P., Hogerheijde, M. R., Boland, W., Menten, K. M., & Wyrowski, F. 2009a, A&A, 507, 1425 van Kempen, T. A., van Dishoeck, E. F., G¨ usten, R., Kristensen, L. E., Schilke, P., Hogerheijde, M. R., Boland, W., Nefs, B., Menten, K. M., Baryshev, A., & Wyrowski, F. 2009b, A&A, 501, 633

194

Bibliography

van Kempen, T. A., Wilner, D., & Gurwell, M. 2009c, ApJ, 706, L22 van Zadelhoff, G.-J., Aikawa, Y., Hogerheijde, M. R., & van Dishoeck, E. F. 2003, A&A, 397, 789 van Zadelhoff, G.-J., Dullemond, C. P., van der Tak, F. F. S., Yates, J. A., Doty, S. D., Ossenkopf, V., Hogerheijde, M. R., Juvela, M., Wiesemeyer, H., & Sch¨oier, F. L. 2002, A&A, 395, 373 Vasyunin, A. I., Semenov, D., Henning, T., Wakelam, V., Herbst, E., & Sobolev, A. M. 2008, ApJ, 672, 629 Velusamy, T. & Langer, W. D. 1998, Nature, 392, 685 Visser, R. 2009, PhD thesis, Leiden Observatory Visser, R., van Dishoeck, E. F., Doty, S. D., & Dullemond, C. P. 2009, A&A, 495, 881 Viti, S. & Williams, D. A. 1999, MNRAS, 305, 755 Wakelam, V., Caselli, P., Ceccarelli, C., Herbst, E., & Castets, A. 2004, A&A, 422, 159 Wakelam, V., Ceccarelli, C., Castets, A., Lefloch, B., Loinard, L., Faure, A., Schneider, N., & Benayoun, J.-J. 2005a, A&A, 437, 149 Wakelam, V. & Herbst, E. 2008, ApJ, 680, 371 Wakelam, V., Herbst, E., & Selsis, F. 2006, A&A, 451, 551 Wakelam, V., Selsis, F., Herbst, E., & Caselli, P. 2005b, A&A, 444, 883 Ward-Thompson, D., Scott, P. F., Hills, R. E., & Andre, P. 1994, MNRAS, 268, 276 Weinreb, S. 1963, Nature, 200, 829 Wilson, R. W., Jefferts, K. B., & Penzias, A. A. 1970, ApJ, 161, L43 Wilson, T. L. & Rood, R. 1994, ARA&A, 32, 191 Woodall, J., Ag´ undez, M., Markwick-Kemper, A. J., & Millar, T. J. 2007, A&A, 466, 1197 Woolfson, M. M. 1964, Royal Society of London Proceedings Series A, 282, 485 —. 1993, QJRAS, 34, 1 Woon, D. E. & Herbst, E. 1997, ApJ, 477, 204 Wyrowski, F., Menten, K. M., Guesten, R., & Belloche, A. 2010, A&A, accepted Yan, M. 1997, PhD thesis, Harvard University Yun, Y. J., Park, Y., & Lee, S. H. 2009, A&A, 507, 1785 Zinnecker, H. & Yorke, H. W. 2007, ARA&A, 45, 481 Zuckerman, B., Kuiper, T. B. H., & Rodriguez Kuiper, E. N. 1976, ApJ, 209, L137

Acknowledgements

After about 4 years, 10000 lines of FORTRAN77 code, as many lines of Python code, a few papers accepted, one paper rejected, several observing proposals rejected, a few won, four offices in two buildings, 2 3/4 keyboards hacked to dead, done the same to 3 computers, about 6 visits in Leiden, 5 room-mates annoyed with my “loud handling” of the keyboard, about 1000 times eating in the canteen, 3 1/2 years of waiting for Herschel data, many nights of working until 4 a.m., once missing the last bus down from H¨onggerberg,. . . , this thesis is done. Is it done? No, it is not – I did not model any low-mass sources, did not model many of the flowers in the weeds of lines currently observed by Herschel, have only carried out few comparison of the models with observations, have neglected various chemical and physical processes involved in star formation. . . Is it really done? Yes it is – I had the opportunity to work in a very interesting field, could develop skills in various areas from numerical analysis to presentations, could train my willpower to continue when it was not easy, learn about the physics and chemistry in star forming regions and I also met many people during my time as a PhD student. I would like to thank Prof. Arnold Benz for giving me the possibility to make all that valuable experience. Thany you Arnold, for your patience, confidence, encouragement and ongoing interest in my work. Many of the ideas collected in this thesis come from Steven Doty and Ewine van Dishoeck – Thank you for letting me share your thoughts, and for the warm welcome in Leiden and Granville, for your encouragement and many interesting discussions! I appreciated very much the collaboration with my colleagues and “combatants” at the Institute of Astronomy: Nadine Afram, Kaspar Arzner, Andrea Banzatti, Marina Battaglia, Kevin Briggs, Esther B¨ unzli, Bartosz Dabrowski, Carolin Dedes, Dominique Fluri, Laure Fouchet, Vincent Geers, Daniel Gisler, Adrian Glauser, Paolo Grigis, Manuel G¨ udel, Ren´e Holzreuter, Andreas James, Franco Joos, Lucia Kleint, Owen Matthews, Hansueli Meyer, Michael Meyer, Christian Monstein, Sascha Quanz, Christian Sennhauser, Alexander Shapiro, Pascal St¨auber, Jan Stenflo, Rolf Walder, Susanne Wampfler and Richard Wenzel. I thank Peter Steiner for his computer support and Marianne Chiesi and Barbara Codoni for their help in administrative issues. Most of this work has been done with a computer. A computer does not help much without good software – I appreciate the contribution of many volunteers to open source programs or operating systems like Linux, Python, or the excellent PYX package that has been used to create most of the figures in this thesis.

196

Acknowledgements

I thank Arnold Benz, Susanne Wampfler, Christoph Bruderer and Mirjam Bruderer for proof reading the manuscript. What would my life at ETH have been without friends? Thanks Martin Bernet, Mirko Birbaumer, Andrea Boller, Christian Knobel, Michael Rissi for being here, for many recreative discussions at the bistro and for many night-time working sessions. . . I also thank many other friends I met during my time in Z¨ urich! I am most thankful for my family, Mama, Papa and Mirjam and Laura for their support and their patience in any way one could wish.

List of publications

1. Chemical Modelling of Young Stellar Objects, I. Method and Benchmarks S. Bruderer, S. D. Doty & A. O. Benz The Astrophysical Journal Supplement Series 183, 179 (2009) 2. Multidimensional Chemical Modelling, II. Irradiated outflow walls S. Bruderer, A. O. Benz, S. D. Doty, E. F. van Dishock & T. L. Bourke The Astrophysical Journal 700, 872 (2009) 3. Excitation and abundance study of CO+ in the interstellar medium P. St¨auber & S. Bruderer Astronomy and Astrophysics 505, 195 (2009) 4. Evidence for warm and dense material along outflows of YSOs S. Bruderer, A. O. Benz, T. L. Bourke & S. D. Doty Astronomy and Astrophysics Letters 503, 13 (2009) 5. Herschel-PACS spectroscopy of the low-mass protostar HH 46 T. A. van Kempen, L. E. Kristensen, G. J. Herczeg, R. Visser, E. F. van Dishoeck, S. Wampfler, S. Bruderer, A. O. Benz, S. D. Doty, C. Brinch, M. R. Hogerheijde, J. K. Jørgensen, and the instrument team, and the WISH guaranteed time key program team. Astronomy and Astrophysics 518, L121 (2010) 6. Multidimensional Chemical Modeling of Young Stellar Objects, III. The Influence of Geometry on the Abundance and Excitation of Hydrides S. Bruderer, A. O. Benz, P. St¨auber & S. D. Doty Accepted by the Astrophysical Journal 7. Detection of interstellar oxidaniumyl: abundant H2 O+ towards the star-forming regions DR21, Sgr B2, and NGC6334 V. Ossenkopf, H. S. P. M¨ uller, D.C. Lis, P. Schilke, T. A. Bell, S. Bruderer, E. Bergin, C. Ceccarelli, C. Comito, J. Stutzki et al. Astronomy and Astrophysics 518, L111 (2010)

198

List of publications

8. Herschel-HIFI detections of hydrides towards AFGL 2591. Envelope emission versus tenuous cloud absorption S. Bruderer, A. O. Benz, E. F. van Dishoeck, M. Melchior, S. D. Doty, F. van der Tak, P. St¨auber, S. F. Wampfler, C. Dedes, U.A. Yildiz, L. Pagani, T. Giannini, and the HIFI instrument team and the WISH guaranteed time key program team. Accepted by Astronomy and Astrophysics 9. Hydrides in young stellar objects: Radiation tracers in a protostar-disk-outflow system A. O. Benz, S. Bruderer, E. F. van Dishoeck, P. St¨auber, S. F. Wampfler, M. Melchior, C. Dedes, F. Wyrowski, S. D. Doty, F. van der Tak, and the HIFI instrument team and the WISH guaranteed time key program team. Accepted by Astronomy and Astrophysics 10. Herschel observations of the hydroxyl radical (OH) in young stellar objects S.F. Wampfler, G.J. Herczeg, S. Bruderer, A. O. Benz, E. F. van Dishoeck, L. E. Kristensen, R. Visser, S. D. Doty, M. Melchior, T. A. van Kempen, U. A. Yildiz, C. Dedes, J. R. Goicoechea, A. Baudry, and the HIFI instrument team and the WISH guaranteed time key program team. Accepted by Astronomy and Astrophysics 11. Water in low-mass star-forming regions with Herschel, HIFI spectroscopy of NGC 1333 L. E. Kristensen, R. Visser, E.F. van Dishoeck, U. A. Yildiz, S. D. Doty, G. J. Herczeg, F.-C. Liu, B. Parise, J. K. Jrgensen, T. A. van Kempen, C. Brinch, S. F. Wampfler, S. Bruderer, A. O. Benz, M. R. Hogerheijde, E. Deul, and the HIFI instrument team and the WISH guaranteed time key program team. Accepted by Astronomy and Astrophysics 12. The origin of the [CII] emission in the S140 PDRs - new insights from HIFI C. Dedes, M. R. R¨ollig, B. Mookerjea, Y. Okada, V. Ossenkopf, S. Bruderer, A.O. Benz, M. Melchior, C. Kramer, M. Gerin, R. G¨ usten, and the HIFI instrument team and the WADI guaranteed time key program team. Accepted by Astronomy and Astrophysics

Curriculum Vitæ

Name

Simon Bruderer

Date of birth

August 20, 1979

Place of birth

Wattwil, Switzerland

1986 - 1995

Primary and secondary school in Gossau (St. Gallen)

1995 - 1999

Kantonsschule am Burggraben, St. Gallen

1999

Matura Typus C

1999 - 2000

Database development at St. Galler Kantonalbank Military service

2000 - 2006

Studies of Physics at ETH Z¨ urich

2006

Dipl. Phys. ETH Diploma Thesis: Calculation and observation of X-ray-sensitive molecules in envelopes of young stellar objects

2006 - 2010

Research assistant at the Radio Astronomy and Plasma Physics Group, Institute of Astronomy, ETH Z¨ urich

2010

Doctoral Thesis The influence of geometry on the FUV and X-ray driven chemistry in star formation Supervisor: Prof. Dr. Arnold O. Benz

Member of the Swiss Society of Astrophysics and Astronomy (SGAA)