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Home , Spectroscopy, Stellar classification

6. Stellar spectra

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

1 Occupation numbers: LTE case

Absorption coefficient: κν = niσν à calculation of occupation numbers needed

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

2 Excitation in LTE

Boltzmann excitation equation

nij: number density of 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

2 gij = 2i for = (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 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 à j+1 initial state: n1j & statistical weight g1j final state: n1j+1 + free electron & statistical weight g1j+1 gEl

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

3 gEl: volume in phase space normalized to smallest possible volume (h ) for electron:

2 orientations 4 Ionization: Saha’s formula

using Boltzmann formula

Sum over all final states: integrate over all phase space

- ionization falls with n (recombinations) Saha 1920 e - ionization grows with T 5 Ionization: Saha’s formula

Generalize for arbitrary levels (not just ground state):

using Boltzmann’s equation

also 6 Ionization: Saha’s formula

Using electron pressure P instead of n (P = n kT) e e e e

which can be written as:

2 with Pe in dyne/cm and Ej in eV

7 Ionization: Saha’s formula

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

+ + + + T (K) n(H ) / n(H) n(H) /[n(H) + n(H )] n(H ) /[n(H) + n(H )] 1.2 1 4,000 2.46E-10 1.000 0.246E-9 0.8 6,000 3.50E-4 1.000 0.350E-3 0.6 0.4 8,000 5.15E-1 0.660 0.340 H I H II 0.2 10,000 4.66E+1 0.021 0.979 0 H fractional ionization 12,000 1.02E+3 0.000978 0.999 4000 6000 8000 14,000 9.82E+3 0.000102 1.000 100001200014000160001800020000 Temperature 16,000 5.61E+4 0.178E-4 1.000

8 On the partition function

Partition function for neutral H

E E = 13.6 eV i0 · ion g = 2i2 i0 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

2 idea: orbit radius r = a0 i divergent (i is main ) à there must be a max i corresponding

to the finite spatial extent of atom rmax introduces a pressure imax dependence of U 9 An example: pure hydrogen 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 ):

α(T)

10 * The LTE occupation number ni

3/2 From Saha’s equation: n1j+1 ne g1j+1 2¼mkT Ej = 2 e¡ kT n g h2 1j 1j µ ¶

+ Boltzmann: in LTE we can express the bound level

occupation numbers as a function of T, ne and the ground-state occupation number 2 3/2 gij 1 h Eij of the next higher ionization stage. kT ni¤ := nij = n1j+1 ne e g1j+1 2 2¼mkT * µ ¶ Note ni is the occupation number used to calculate bf-stimulated and bf-

11 and temperature: application of Saha – and Boltzmann formulae

temperature (spectral type) & pressure ( 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 …

13 The pioneers of stellar spectroscopy… at work…

14

15 Annie Jump Cannon

16 Stellar classification

Approximate Surface Type Main Characteristics Temperature Singly ionized lines either in emission or absorption. Strong O > 25,000 K 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 11,000 - 25,000 B II, Fe III.

Hydrogen lines at maximum strength for A0 , decreasing thereafter. Neutral metals stronger. Fe II prominent A0-A5. H and He lines weaken with increasing 7,500 - 11,000 A luminosity. O I, Si II, Mg II, Ca II, Ti II, Mn I, Fe I-II.

Metallic lines become noticeable. G-band starts at F2. H lines decrease. CN 4200 6,000 - 7,500 F increases with luminosity. Ca II, Cr I-II, Fe I-II, Sr II. Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. once- G 5,000 - 6,000 ionized ) grow in strength. CN 4200 increases with luminosity.

Metallic lines dominate, H weak. Weak blue continuum. CN 4200, Sr II 4077 increase K 3,500 - 5,000 with luminosity. Ca I-II.

Molecular bands of TiO noticeable. CN 4200, Sr II 4077 increase with M < 3,500 luminosity. Neutral metals. 17 online Gray at NED nedwww.ipac.caltech.edu/level5/Gray/frames.html

Stellar classification

18 Stellar classification

ionization (I.P. 25 eV)

The definition of the break between the O-type stars The spectral type can be judged easily by the ratio of the and the B-type stars is the absence of lines of ionized strengths of lines of He I to He II; He I tends to increase in helium (He II) in the spectra of B-type stars. The lines strength with decreasing temperature while He II decreases of He I pass through a maximum at approximately B2, in strength. The ratio He I 4471 to He II 4542 shows this trend and then decrease in strength towards later (cooler) clearly. 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 19 O- spectral types

20 SiIV SiIII

B0Ia B-star spectral types

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

22 Stellar classification

Balmer lines indicate stellar luminosity à atmospheric density à line broadening

Spectral class ion and ionization potential

He II C III N III O III Si IV O 24.6 24.4 29.6 35.1 33.5 Are the ionization levels for different elements HII C II N II O II Si III Fe III B observed in a given spectral type consistent with a 13.6 11.3 14.4 13.6 16.3 16.2 “single” temperature? Mg II Ca II Ti II Cr II Si II Fe II A-M 7.5 6.1 6.8 6.8 8.1 7.9 23 Cecilia Payne-Gaposchkin

24 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 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.

25 Cecilia Payne-Gaposchkin

26 Stellar continua: opacity sources

27 Stellar continua: the Balmer jump

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)

ionization changes are reflected in changes in the continuum 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) =

28 Stellar continua: the Balmer jump

from and Boltzmann’s equation

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

if b-f transitions dominate continuous opacity à the Balmer discontinuity increases with decreasing T à measure T from Balmer jump 29 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: N / N(H0) 1 10-8 N / N(H0) 6 10-10 (Paschen continuum) 2 ' £ 3 ' £

- 0 0 - N(H )/ N(H ): > N3 / N(H ): 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 à higher in dwarfs than supergiants/

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

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