1

6. Stellar spectra

excitation and ionization, Saha’s equation

stellar spectral classification

Balmer jump, H-

2

Occupation numbers: LTE case

Absorption coefficient: κν = niσν$ à 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

3

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

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

The fraction relative to the total number of atoms of in ionization stage j is

Uj (T) is called the partition function

4

Ionization in LTE: Saha’s formula

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

Consider ionization process j à j+1

initial state: n1j & statistical weight g1j

final state: n1j+1 + free electron & statistical weight g1j+1 gEl

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:

2 spin orientations

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Ionization: Saha’s formula

Sum over all final states: integrate over all phase space

Saha 1920 -  ionization falls with ne (recombinations)

-  ionization grows with T

using Boltzmann formula

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Ionization: Saha’s formula

Generalize for arbitrary levels (not just ground state):

using Boltzmann’s equation

also

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Ionization: Saha’s formula

Using electron pressure Pe instead of ne (Pe = nekT)

which can be written as:

with Pe in dyne/cm2 and Ej in eV

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Ionization: Saha’s formula

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

T (K) 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

00.20.40.60.8

11.2

400060008000

10000

12000

14000

16000

18000

20000

Temperature

H f

ract

iona

l ioniza

tion

H I H II

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On the partition function

Partition function for neutral H atom

Ei0 · Eion = 13.6 eV

gi0 = 2i2

divergent idea: orbit radius r = a0 i2

(i is main quantum number)

à there must be a max i corresponding

to the finite spatial extent of atom rmax

imax introduces a pressure dependence of U

infinite number of levels à partition function diverges! reason: Hydrogen atom level structure calculated as if it were alone in the universe not realistic à cut-off needed

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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):

α(T)

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The LTE occupation number ni*

From Saha’s equation:

+ Boltzmann: 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 bf-spontaneous emission

n¤i := nij = n1j+1 negijg1j+1

1

2

µh2

2¼mkT

¶3/2eEijkT

n1j+1 nen1j

= 2g1j+1g1j

µ2¼mkT

h2

¶3/2e¡

EjkT

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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

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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 Approximate Surface Temperature Main Characteristics

O > 25,000 K 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.

B 11,000 - 25,000

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.

A 7,500 - 11,000

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.

F 6,000 - 7,500 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.

G 5,000 - 6,000 Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. once-ionized 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.

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Stellar classification

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

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Stellar classification

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

ionization (I.P. 25 eV)

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O-star spectral types

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B-star spectral types SiIV SiIII

SiII

B0Ia

B0.5Ia

B1Ia

B1.5Ia

B2Ia

B3Ia

B5Ia

B8Ia

B9Ia

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Stellar classification

(I.P. 6 eV)

H & K strongest

T high enough for single ionization, but not further

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Stellar classification

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

Balmer lines indicate stellar luminosity

Gravity à atmospheric density

à line broadening

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Cecilia Payne-Gaposchkin

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Stellar classification

Saha’s equation à 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

ionization changes are reflected in changes in the continuum flux

for λ > λ(Balmer limit) no n=2 b-f transitions possible à drop in absorption

à atmosphere is more transparent

à observed flux comes from deeper hotter layers

à higher flux BALMER JUMP

Balmer discontinuity (when H^- absorption is negligible T > 9000 K) =

From the ground we can measure part of Balmer (λ < 3646 A) and Bracket (λ > 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)

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Stellar continua: the Balmer jump

from

and Boltzmann’s equation

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

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

0.033 at T = 10,000 K

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Stellar continua: H-

apply Saha’s equation to H- (H- is the ‘atom’ and H0 is the ‘ion’)

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

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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 à higher in dwarfs than supergiants

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