6. Stellar spectra. excitation and ionization, Saha s equation stellar spectral classification Balmer jump, H -

Transcription

1 6. Stellar spectra excitation and ionization, Saha s equation stellar spectral classification Balmer jump, H - 1

2 Occupation numbers: LTE case Absorption coefficient: = n i 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 2

3 Excitation in LTE Boltzmann excitation equation n ij : number density of atoms in excited level i of ionization stage j (ground level: i=1 neutral: j=0) g ij : statistical weight of level i = number of degenerate states E ij excitation energy relative to ground state g ij = 2i 2 for hydrogen = (2S+1) (2L+1) in L-S coupling log n ij =log g ij E ij (ev) 5040 n 1j g 1j T The fraction relative to the total number of atoms of in ionization stage j is n ij n j = g ij U j (T) e E ij/kt U j (T )= X i g ij e E ij/kt U j (T) is called the partition function 3

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 j+1 initial state: final state: n 1j & statistical weight g 1j n 1j+1 + free electron & statistical weight g 1j+1 g El n 1j+1 (v) dx dy dz dp x dp y dp z R R n1j+1 (v)dv d 3 p = n 1j+1 number of ions in groundstate with free electron with velocity in (v,v+dv) in phase space g El : volume in phase space normalized to smallest possible volume (h 3 ) for electron: g El =2 dx dy dz dp x dp y dp z h 3 2 spin orientations R dx dy dz = R dv = V =1/ne dp x dp y dp z =4πp 2 dp =4πm 3 v 2 dv 4

5 Ionization: Saha s formula n 1j+1 (v) dv dp x dp y dp z = g 1j+1 g El e (E 1 j + 2 mv 2 )/kt n 1j g 1j using Boltzmann formula n 1j+1 (v) dv dp x dp y dp z = g 1j+1 2dV dp x dp y dp z e [E j+ 1 n 1j g 1j h 3 2m (p 2 x +p2 y +p2 z )]/kt Sum over all final states: integrate over all phase space n 1j+1 = n 1j g 1j+1 g 1j 2 V h 3 e E j/kt Z Z Z e 1 2mkT (p2 x +p2 y +p2 z ) dp x dp y dp z Z e x2 dx = π (2πmkT ) 3/ 2 Saha 1920 n 1j+1 n e n 1j =2 g 1j+1 g 1j µ 2π mkt h 2 3/ 2 e E j kt - ionization falls with n e (recombinations) - ionization grows with T 5

6 Ionization: Saha s formula Generalize for arbitrary levels (not just ground state): n j = X n ij n j+1 = i=1 X i=1 n ij+1 n ij+1 = n 1j+1 g ij+1 g 1j+1 e E ij+1/kt using Boltzmann s equation n j+1 = n 1j+1 g 1j+1 X g ij+1 e E ij+1/kt i=1 U j+1 (T) : partition function also n j = n 1j g 1j U j (T) n j+1 n e n j =2 U j+1(t) U j (T) µ 2πmkT h 2 3/2 e E j kt 6

7 Ionization: Saha s formula Using electron pressure P e instead of n e (P e = n e kt) n j+1 n j P e =2 U j+1(t) U j (T) µ 2πm h 2 3/ 2 (kt) 5/2 e E j kt which can be written as: n log j+1 U 10 = log 10 P e +log j+1 (T ) 10 n j U j (T ) +2.5log 10 T 5040 T E j with P e in dyne/cm 2 and E j in ev 7

8 Ionization: Saha s formula Example: H at P e = 10 dyne/cm 2 (~ solar pressure at T = T eff ) T (K) n(h + ) / n(h) n(h) /[n(h) + n(h + )] n(h + ) /[n(h) + n(h + )] 4, E E-9 6, E E-3 8, E , E , E , E , E E H fractional ionization H I H II Temperature

9 On the partition function Partition function for neutral H atom E i0 E ion = 13.6 ev U 0 (T)= X g i0 e E i0/kt i=1 g i0 = 2i 2 U 0 (T ) 2 e E ion /kt infinite number of levels partition function diverges! reason: Hydrogen atom level structure X i 2 calculated as if it were alone in the universe not realistic cut-off needed i=1 divergent idea: orbit radius r = a 0 i 2 (i is main quantum number) there must be a max i corresponding to the finite spatial extent of atom r max 4π 3 r3 max = 1 N = 4π 3 (r 0i 2 max) 3 i max introduces a pressure dependence of U 9

10 An example: pure hydrogen atmosphere in LTE Temperature T and total particle density N given: calculate n e, n p, n i From Saha s equation and n e = n p (only for pure H plasma): (T) 10

11 The LTE occupation number n i * From Saha s equation: n 1j+1 n e n 1j =2 g 1j+1 g 1j µ 2πmkT h 2 3/2 e E j kt n 1j = n 1j+1 n e g 1j 1 g 1j+1 2 µ h 2 2πmkT 3/2 e E j kt + Boltzmann: n i := n ij = n 1j+1 n e g ij 1 g 1j+1 2 n ij n 1j = g ij g 1j µ h 2 2πmkT e E 1i/kT 3/2 e E ij kt in LTE we can express the bound level occupation numbers as a function of T, n e and the ground-state occupation number of the next higher ionization stage. Note n i * is the occupation number used to calculate bf-stimulated and bfspontaneous emission 11

12 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

13 Stellar classification and temperature: application of Saha and Boltzmann formulae The pioneers of stellar spectroscopy 13

14 The pioneers of stellar spectroscopy at work 14

15 Annie Jump Cannon 15

16 Annie Jump Cannon 16

17 Stellar classification Type O Approximate Surface Temperature > 25,000 K B 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. 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 G 5,000-6,000 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. 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. 17

18 online Gray atlas at NED nedwww.ipac.caltech.edu/level5/gray/frames.html Stellar classification 18

19 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 A DIGITAL SPECTRAL CLASSIFICATION ATLAS R. O. Gray 19

20 O-star spectral types 20

21 B0Ia SiIV SiIII B-star spectral types B0.5Ia B1Ia B1.5Ia SiII B2Ia B3Ia B5Ia B8Ia B9Ia 21

22 Stellar classification (I.P. 6 ev) H & K strongest T high enough for single ionization, but not further 22

23 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 O B A-M ion and ionization potential He II C III N III O III Si IV HII C II N II O II Si III Fe III Mg II Ca II Ti II Cr II Si II Fe II

24 Cecilia Payne-Gaposchkin 24

25 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 N r,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 P e = N e 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

26 Cecilia Payne-Gaposchkin 26

27 Stellar continua: opacity sources 27

28 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 ( 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) = κ bf (> 3650) κ bf (< 3650) = κ bf (n =3)+... κ bf (n =2)+κ bf (n =3)+... ' κbf (n =3) κ bf (n =2) 28

29 Stellar continua: the Balmer jump from κ bf = σ bf n N n σ bf n 1 n 5 ν 3 and Boltzmann s equation N n N tot g n U n e E n/kt = n 2 N tot e E n/kt κ bf 1 n 3 e E n/kt κ bf (> 3650) κ bf (< 3650) ' 8 27 e (E 3 E 2 ) /kt e.g at T = 5,000 K at T = 10,000 K if b-f transitions dominate continuous opacity the Balmer discontinuity increases with decreasing T measure T from Balmer jump 29

30 Stellar continua: H - apply Saha s equation to H - (H - is the atom and H 0 is the ion ) N(H 0 ) n e N(H ) = T 3/2 e 8753/T N(H 0 ) log 10 N(H ) = log 10 P e +2.5log 10 T 5040 T under solar conditions: N(H - ) / N(H 0 ) ' E I at the same time: N 2 / N(H 0 ) ' N 3 / N(H 0 ) ' (Paschen continuum) N(H - )/ N(H 0 ): > N 3 / N(H 0 ): b-f from H - more important than H b-f in the visible 30

31 Stellar continua at solar T H - b-f dominates from Balmer limit up to H - threshold (16500 A). H 0 b-f dominates in the visible for T > 7,500 K. Balmer jump smaller than in the case of pure H 0 absorption: instead of increasing at low T, decreases as H - absorption increases Max of Balmer jump: 10,000 K (A0 type) H - opacity n e higher in dwarfs than supergiants Balmer jump sensitive to both T and P e in A-F stars 31