Term Paper

Introduction to Spectroscopy Matthias Kobelt 21.11.2014

Table of contents

Introduction Historical background Spectroscopy Physics of atoms and molecules Stellar surfaces and atmospheres Radiative transfer List of figures List of references

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Introduction - Rainbow

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One of the most famous phenomena of the light spectrum is the rainbow.

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The first correct explanation of the rainbow was from Dietrich von Freiberg in 1304.

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Ren´ e Descartes and Isaac Newton explained it completely in terms of geometrical optics.

[1]

[2]

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Historical background

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The first telescopes had too much chromatic aberration.

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Newton separated in 1666 sun light with prisms in its colors and recombined it.

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Thomas Melville discovered the emission spectra of flames in the early eighteenth century.

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William Wollaston found in 1802 the dark lines, but he thought that they are the natural boundaries between the different colors.

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Joseph von Fraunhofer saw 1814 almost 600 lines in the spectrum of the sun. This was possible not only because he used a slit but also because of his better prisms.

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In 1823 Fraunhofer was able to measure wavelengths and so he did. He labelled the nine most prominent lines, still known today as Fraunhofer lines.

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Historical background

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Alexandre Becquerel took the first picture of the solar spectrum in 1842.

[3] I

A decade later Jean Foucault discovered, that the spectrum of a sodium flame contains partly the lines from the solar spectrum.

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In 1859 Kirchhoff and Bunsen formulated Kirchhoff’s law: λ (T ) = constant kλ (T )

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With this knowledge the discovery of elements became really fast.

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Historical background

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In 1862 Andres ˚ Angstrom found hydrogen in the Sun.

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In 1863 Norman Lockyer found Helium lines in the solar spectrum. The next 30 years it was not found on earth.

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In 1864 William Huggins identified hydrogen, iron, sodium and calcium in stars.

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Huggins developed the procedure of comparing a spectrum with the spectrum of an artificial light source.

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Angelo Secchi classified in 1863 stars according to the appearance of their spectra in four simple classes.

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The star classification was soon extended. They were classified by the complexity of their spectra. A, B, C...

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The system used today categorise by temperature but with the same categories, so it becomes complex: Oh Be A Fine Girl/Guy Kiss Me

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Historical background

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In 1885 Johann Balmer found a law, to predict the wavelengths of the hydrogen lines:   1 1 ν=R − 2 2 n1 n2

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Joseph Thomson discovered the electron in 1897.

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Albert Einstein explained the photo effect in 1905.

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In 1913 Niels Bohr presented a model of the Atoms. So energy difference between electron orbits correspond to lines in the spectrum.

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Karl Jansky discovered radio waves from space in 1933.

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Microwaves, ultraviolet x- and gamma ray only can be measured without atmosphere, so is done since 1960.

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Spectroscopy - Different types

There are several types of spectroscopy, however only three of them are usable in astrophysics currently. I

Atomic and molecular absorption and emission spectroscopy (mostly)

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Fluorescence spectroscopy

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Spectroscopy of solids

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Spectroscopy - A spectrograph

[4]

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Spectroscopy - Example for spectral atlas

[5]

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Physics of atoms and molecules - Motivation

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The structure of atoms and molecules determines their spectra.

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This can be calculated with quantum mechanics.

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But the Bohr-Sommerfeld model is enough for many purposes in astro spectroscopy.

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Physics of atoms and molecules - Bohr model

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In QM electrons have the de Broglie wavelength λ = h/p

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In a quantum-mechanical view only the orbits are allowed whose length are integer multiples of the de Broglie wavelength, due to the momentum of the orbit.

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Since the electron is captured by the nucleus due to the coulomb force the possible radii are given by: r=

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0 n2 h2 πZe2 m

The energy for an electron in orbit “n” is given by: E=−

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[6]

Z 2 e4 m 820 n2 h2

Energy differences only depends on −2 n−2 1 − n2 , so we got Balmers law.

[7]

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Physics of atoms and molecules - Bohr Sommerfeld model I

Sommerfeld introduced three optimisations: 1 Use of the reduced mass instead of the electron mass. 2 Elliptical orbits are allowed → second degree of freedom: l 3 Allows relativistic effects, so the electron can precess. So energy dependence of l.

E=−

   α2 Z 2 1 3 Z 2 e4 µ 1 + − 820 n2 h2 n l+1 4n

5Å

n=1, n=2, n=2, n=3, n=3, n=3,

4Å 3Å

l=0 l=0 l=1 l=0 l=1 l=2

2Å 1Å

6Å

4Å

2Å

1Å

2Å

4Å

6Å

8Å

10 Å

2Å 3Å 4Å 5Å

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Stellar surfaces and atmospheres - The sun

Spectroscopy useful to study Stars → need to know their structure

1 Core 2 Radiative zone 3 Convective zone 4 Photosphere 5 Chromosphere 6 Corona 7 Sunspot 8 Granulation 9 Solar prominence

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Stellar surfaces and atmospheres - Optical depth

The optical depth describes the opacity of a medium Z r2 τν = κν ρdr with r2 > r1 r1

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τν big → optical thick

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τν small → optical thin

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τν = 1 defines the radius of the sun, for every wavelength

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Radiative transfer - Intensity loss and gain

Knowledge about physical model for the radiation transport in the atmosphere ⇒ get information out of measured spectra

How does the radiation intensity change while passing a layer of material? I

Absorption

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Scattering

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Emission

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Radiative transfer - Intensity loss and gain

Absorption depends on I

Incoming intensity

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Radiation frequency

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Material density

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Material temperature

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Present elements dIν cos(θ) = −Iν κν ρdr

κ contains also the scattering coefficient. The emission can be characterised by the coefficient ν , so the total intensity change is: dIν cos(θ) = (−Iν κν ρ + ν ) dr

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Radiative transfer - Intensity loss and gain

With dτ = −κν ρdr and dIν cos(θ) = (−Iν κν ρ + ν ) dr we get:   ν dIν cos(θ) = Iν − dτ = (Iν − Sν ) dτ κν ρ Here the quotient of emission and absorption is Sν , the source function, which is in general an arbitrary complicated function.

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Radiative transfer - Local thermodynamic equilibrium

Need to know source function ⇒ find easy model

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Temperature gradient → no thermodynamic equilibrium!

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Consider small volume with not too low ρ → LTE

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So the source function can be described by the Kirchhoff-Planck function: Sν = Bν (T ) =

1 2hν 3 hν c2 e kT −1

This means that the sun can be seen as a black body with a temperature around 6000K. So the continuum spectrum is explained With other formulas in LTE the wings of most of the spectral lines and the entire profile of weak lines can be explained.

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Radiative transfer - Formation of spectral lines

Question: Why are there only absorption and no emission lines? I

A continuous spectrum comes from the inner sun

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Most of the lines form in the photosphere, rarely also in chromosphere and corona

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If a photon with a certain wavelength is absorbed by an atom/ion → two possibilities: 1 Emission of several photons with other wavelength 2 Re-emission with same wavelength

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These photons are emitted equally in all directions

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If 1 → intensity of the original wavelength reduced If 2 → two possibilities:

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a Absorbed again, so start from beginning b Photo-effect → wavelength disappears I

Just some photons with this certain wavelength can leave the sun → absorption line at this wavelength

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No emission lines because the photons emitted at possibility 1 can be absorbed in the same way

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Radiative transfer - Line shapes

In reality the absorption lines are not perfectly sharp. They are broadened by several effects, the main ones are: I

Natural broadening

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Doppler broadening

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Collision broadening

Additionally there are some other effects, which are less powerful.

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The end

2500

idealer Schwarzer Körper (Temperatur 5900 K) extraterrestrische Sonnenstrahlung (Luftmasse AM0) terrestrische Sonnenstrahlung (Luftmasse AM1,5)

2000

1500

1000

500

UV 250

sichtbarer Bereich IR 500

750

1000

1250

1500

Wellenlänge / nm

1750

2000

2250

[11]

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List of figures

[1] A rainbow 9.11.14: https://en.wikipedia.org/wiki/Rainbow#mediaviewer/File: WhereRainbowRises.jpg [2] Rainbow model 9.11.14: https://de.wikipedia.org/wiki/Regenbogen#mediaviewer/File: Rainbow1.svg [3] Solar spectrum 20.11.14: http://www.bu.edu/astronomy/files/2009/09/spectrum_merged.jpg [4] A simple spectrograph 20.11.14: http: //www.ipf.uni-stuttgart.de/lehre/online-skript/optik/spektrograph.gif [5] Solar spectral atlas 20.11.14: http://bass2000.obspm.fr/solar_spect.php?WL=6530&DW=70&sel_ resol=0.01&Find.x=15&Find.y=21 [6] Bohr model 15.11.14: https://de.wikipedia.org/wiki/Bohrsches_Atommodell#mediaviewer/ File:Bohr-atom-PAR.svg [7] Standing electron wave on orbit Kitchin - Optical Astronomical Spectroscopy from 1995 23/ 25

List of figures

[8] Bohr-sommerfeld model 14.11.14: https://de.wikipedia.org/wiki/Bohr-sommerfeldsches_Atommodell# mediaviewer/File:Bohr-sommerfeld_Atommodell_%28Elektronenbahnen%29.svg [9] The sun 18.11.14: https://de.wikipedia.org/wiki/Sternaufbau#mediaviewer/File: Sun_diagram.svg [10] Radiative transfer Prialnik - An Introduction to the Theory of Stellar Structure and Evolution - Second edition from 2010 [11] The spectrum of the sun 20.11.14: https://de.wikipedia.org/wiki/Sonnenstrahlung#mediaviewer/File: Sonne_Strahlungsintensitaet.svg

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List of references

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Kitchin - Optical Astronomical Spectroscopy from 1995

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Prialnik - An Introduction to the Theory of Stellar Structure and Evolution Second edition from 2010

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Stix - The Sun - Second edition - Corrected second printing from 2004

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Novotny - Introduction to stellar atmospheres and interiors from 1973

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