6. Stellar spectra

excitation and ionization, Saha’s equation stellar spectral classification Balmer jump, H-

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Occupation numbers: LTE case Absorption coefficient:  = ni  calculation of occupation numbers needed

LTE each volume element in thermodynamic equilibrium at temperature T(r) hypothesis: electron-ion collisions adjust equilibrium difficulty: interaction with non-local photons LTE is valid if effect of photons is small or radiation field is described by Planck function at T(r) otherwise: non-LTE

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Excitation in LTE

Boltzmann excitation equation nij: number density of atoms in excited level i of ionization stage j (ground level: i=1 neutral: j=0) gij: statistical weight of level i = number of degenerate states Eij excitation energy relative to ground state

gij = 2i2 for hydrogen

log

= (2S+1) (2L+1) in L-S coupling

nij g 5040 = log ij − E ij (eV ) n1j g1j T

The fraction relative to the total number of atoms of in ionization stage j is nij gij = e−Eij /kT nj Uj (T )

Uj (T ) =

X i

gij e−Eij /kT

Uj (T) is called the partition function 3

Ionization in LTE: Saha’s formula

Generalize Boltzmann equation for ratio of two contiguous ionic species j and j+1

Consider ionization process j  j+1 initial state:

n1j & statistical weight g1j

final state:

n1j+1 + free electron & statistical weight g1j+1 gEl RR n1j+1 (v) dx dy dz dp x dpy dpz n1j+1(v )dV d3p = n1j+1

number of ions in groundstate with free electron with velocity in (v,v+dv) in phase space

gEl: volume in phase space normalized to smallest possible volume (h3) for electron: gEl = 2

dx dy dz dpx dpy dpz h3

R

dx dy dz =

R

dV = ∆V = 1/ne

dpx dpy dpz = 4πp2 dp = 4πm3v2 dv 2 spin orientations

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Ionization: Saha’s formula

n1j +1(v) 1 2 g dV dpx dpy dpz = 1j+1 gEl e −(Ej + 2 mv )/kT n1j g1j

using Boltzmann formula

n1j +1 (v) 1 g1j +1 dp x dpy dpz − [Ej+ 2m (p 2x+p2y +p2z )]/k T dV dpx dpy dpz = 2 dV e n1j g1j h3

Sum over all final states: integrate over all phase space n1j+1

∆V g = n1j 1j +1 2 3 e− Ej/k T g1j h

Z∞ Z∞ Z∞

e

1 − 2 mkT (p2x +p2y +p2z )

dpx dpy dpz

−∞ −∞ −∞

Z∞

2

e−x dx =

√ π

−∞

(2πmkT )3/ 2

Saha 1920

g n1j +1 ne = 2 1j+1 n1j g1j

µ

2π mkT h2

¶3/ 2

E

e

− kTj

- ionization falls with ne (recombinations) - ionization grows with T

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Ionization: Saha’s formula

Generalize for arbitrary levels (not just ground state): nj =

∞ X

nij

nj+1 =

i=1

nij+1 = n1j +1

nj +1

∞ X

nij+1

i=1

gij+1 − Eij+1 /k T e g1j+1

using Boltzmann’s equation

∞ n1j +1 X = gij+1 e− Eij+1 /k T g1j +1 i=1

Uj +1 (T ) :

also

nj =

n1j Uj (T ) g1j

partition function

U (T ) nj +1 ne = 2 j +1 nj Uj (T )

µ

2πmkT h2

¶3/2

E

− kTj

e

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Ionization: Saha’s formula

Using electron pressure Pe instead of ne (Pe = nekT) U (T ) nj +1 Pe = 2 j+1 nj Uj (T )

µ

2πm h2

¶ 3/ 2

Ej

(kT )5/2e − kT

which can be written as: log10

nj +1 U (T ) 5040 = −0.1761 − log10 P e + log10 j +1 + 2.5 log10 T − Ej nj Uj (T ) T

with Pe in dyne/cm2 and Ej in eV

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Ionization: Saha’s formula

n(H+) / n(H)

n(H) /[n(H) + n(H+)]

n(H+) /[n(H) + n(H+)]

4,000

2.46E-10

1.000

0.246E-9

6,000

3.50E-4

1.000

0.350E-3

8,000

5.15E-1

0.660

0.340

10,000

4.66E+1

0.021

0.979

12,000

1.02E+3

0.000978

0.999

14,000

9.82E+3

0.000102

1.000

16,000

5.61E+4

0.178E-4

1.000

1.2 1 0.8 0.6 0.4 0.2 0

HI

H II

40 0 0 60 0 0 80 0 10 0 00 12 0 0 0 14 0 00 16 0 00 18 0 0 0 20 0 0 00

T (K)

H fractional ionization

Example: H at Pe = 10 dyne/cm2 (~ solar pressure at T = Teff)

Temperature

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On the partition function

Partition function for neutral H atom

U0 (T ) =

∞ X

gi0 e− Ei0 /k T

i=1

Ei0 ≤ Eion = 13.6 eV gi0 = 2i2 U0 (T ) ≥ 2 e −Eion /kT

infinite number of levels  partition function diverges! reason: Hydrogen atom level structure ∞ X calculated as if it were alone in the universe i2 not realistic  cut-off needed i=1

divergent

idea: orbit radius r = a0 i2 (i is main quantum number)  there must be a max i corresponding to the finite spatial extent of atom rmax

4π 3 1 4π rmax = (r0 i2max)3 = 3 N 3

imax

introduces a pressure dependence of U

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An example: pure hydrogen atmosphere in LTE Temperature T and total particle density N given: calculate ne, np, ni

From Saha’s equation and ne = np (only for pure H plasma):

(T)

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The LTE occupation number ni* g1j+1 n1j+1 ne =2 n1j g1j

From Saha’s equation:

n1j

g 1 = n1j +1 ne 1j g1j +1 2

µ

+ Boltzmann:

gij 1 n∗i := nij = n1j+1 ne g1j+1 2

h2 2πmkT

¶3/2

e

h2 2πmkT

2πmkT h2

¶3/2

E

− kTj

e

Ej kT

nij gij −E1 i /kT = e n1j g1j

µ

µ

¶3/2

e

Eij kT

in LTE we can express the bound level occupation numbers as a function of T, ne and the ground-state occupation number of the next higher ionization stage. Note ni* is the occupation number used to calculate bf-stimulated and bfspontaneous emission 11

Stellar classification and temperature: application of Saha – and Boltzmann formulae

temperature (spectral type) & pressure (luminosity class) variations + chemical abundance changes.

Qualitative plot of strength of observed Line features as a function of spectral type 12

Stellar classification and temperature: application of Saha – and Boltzmann formulae

The pioneers of stellar spectroscopy…

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The pioneers of stellar spectroscopy… at work…

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Annie Jump Cannon

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Annie Jump Cannon

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Stellar classification Type O

B

Approximate Surface Temperature > 25,000 K

11,000 - 25,000

Main Characteristics Singly ionized helium lines either in emission or absorption. Strong ultraviolet continuum. He I 4471/He II 4541 increases with type. H and He lines weaken with increasing luminosity. H weak, He I, He II, C III, N III, O III, Si IV. Neutral helium lines in absorption (max at B2). H lines increase with type. Ca II K starts at B8. H and He lines weaken with increasing luminosity. C II, N II, O II, Si II-III-IV, Mg II, Fe III. Hydrogen lines at maximum strength for A0 stars, decreasing thereafter. Neutral metals stronger. Fe II prominent A0-A5. H and He lines weaken with increasing luminosity. O I, Si II, Mg II, Ca II, Ti II, Mn I, Fe I-II.

A

7,500 - 11,000

F

6,000 - 7,500

G

5,000 - 6,000

Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. onceionized calcium) grow in strength. CN 4200 increases with luminosity.

K

3,500 - 5,000

Metallic lines dominate, H weak. Weak blue continuum. CN 4200, Sr II 4077 increase with luminosity. Ca I-II.

M

< 3,500

Molecular bands of titanium oxide TiO noticeable. CN 4200, Sr II 4077 increase with luminosity. Neutral metals.

Metallic lines become noticeable. G-band starts at F2. H lines decrease. CN 4200 increases with luminosity. Ca II, Cr I-II, Fe I-II, Sr II.

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online Gray atlas at NED nedwww.ipac.caltech.edu/level5/Gray/frames.html

Stellar classification

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Stellar classification ionization (I.P. 25 eV)

The spectral type can be judged easily by the ratio of the strengths of lines of He I to He II; He I tends to increase in strength with decreasing temperature while He II decreases in strength. The ratio He I 4471 to He II 4542 shows this trend clearly.

The definition of the break between the O-type stars and the B-type stars is the absence of lines of ionized helium (He II) in the spectra of B-type stars. The lines of He I pass through a maximum at approximately B2, and then decrease in strength towards later (cooler) types. A useful ratio to judge the spectral type is the ratio of He I 4471/Mg II 4481.

A DIGITAL SPECTRAL CLASSIFICATION ATLAS R. O. Gray http://nedwww.ipac.caltech.edu/level5/Gray/Gray_contents.html

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O-star spectral types

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SiIII

SiIV

B-star spectral types

B0Ia B0.5Ia B1Ia B1.5Ia

SiII

B2Ia B3Ia B5Ia B8Ia B9Ia 21

Stellar classification (I.P. 6 eV)

H & K strongest T high enough for single ionization, but not further

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Stellar classification Balmer lines indicate stellar luminosity Gravity  atmospheric density  line broadening

Are the ionization levels for different elements observed in a given spectral type consistent with a “single” temperature?

Spectral class

ion and ionization potential

O

He II C III N III O III Si IV 24.6 24.4 29.6 35.1 33.5

B

HII C II N II O II Si III Fe III 13.6 11.3 14.4 13.6 16.3 16.2

A-M

Mg II Ca II Ti II Cr II Si II Fe II 7.5 6.1 6.8 6.8 8.1 7.9

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Cecilia Payne-Gaposchkin

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Stellar classification Saha’s equation  stellar classification (C. Payne’s thesis, Harvard 1925) The strengths of selected lines along the spectral sequence.

variations of observed line strengths with spectral type in the Harvard sequence.

Saha-Boltzmann predictions of the fractional concentration Nr,s/N of the lower level of the lines indicated in the upper panel against temperature T (given in units of 1000 K along the top). The pressure was taken constant at Pe = Ne k T = 131 dyne cm-2. The T-axis is adjusted to the abscissa of the upper diagram in order to obtain a correspondence between the observed and computed peaks.

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Cecilia Payne-Gaposchkin

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Stellar continua: opacity sources

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Stellar continua: the Balmer jump From the ground we can measure part of Balmer ( < 3646 A) and Bracket ( > 8207 A) continua, and complete Paschen continuum (3647 – 8206 A) Provide information on T, P. Spectrophotometric measurements in UV (hot stars), visible and IR (cool stars) ionization changes are reflected in changes in the continuum flux for  > (Balmer limit) no n=2 b-f transitions possible  drop in absorption  atmosphere is more transparent  observed flux comes from deeper hotter layers  higher flux BALMER JUMP Balmer discontinuity (when H^- absorption is negligible T > 9000 K) =

κbf (> 3650) κbf (n = 3) + ... κbf (n = 3) = bf ' bf κbf (< 3650) κ (n = 2) + κbf (n = 3) + ... κ (n = 2)

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Stellar continua: the Balmer jump from

κbf = σbf n Nn

and Boltzmann’s equation

κbf ∼

σnbf ∼

1 n5ν 3

Nn ∼ Ntot

gn −En /kT e = n2 Ntot e −En /k T Un

1 − En /k T e n3

κbf (> 3650) 8 − (E3 −E2) /kT ' e κbf (< 3650) 27

e.g. 0.0037 at T = 5,000 K 0.033 at T = 10,000 K

if b-f transitions dominate continuous opacity  the Balmer discontinuity increases with decreasing T  measure T from Balmer jump

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Stellar continua: H-

apply Saha’s equation to H- (H- is the ‘atom’ and H0 is the ‘ion’) N(H 0) ne = 4 × 2.411 × 1021 T 3/2 e −8753/T − N(H ) log10

N(H 0) 5040 = 0.1248 − log EI P + 2.5 log T − e 10 10 N(H − ) T

under solar conditions: N(H-) / N(H0) ' 4 × 10-8 at the same time: N2 / N(H0) ' 1 × 10-8

N3 / N(H0) ' 6 × 10-10 (Paschen continuum)

N(H-)/ N(H0): > N3 / N(H0): b-f from H- more important than H b-f in the visible 30

Stellar continua

at solar T H- b-f dominates from Balmer limit up to H- threshold (16500 A). H0 b-f dominates in the visible for T > 7,500 K.

Balmer jump smaller than in the case of pure H0 absorption: instead of increasing at low T, decreases as H- absorption increases Max of Balmer jump: ∼ 10,000 K (A0 type) H- opacity ∝ ne  higher in dwarfs than supergiants

Balmer jump sensitive to both T and Pe in A-F stars

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