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6. Stellar spectra excitation and ionization, Saha’s equation stellar spectral classification Balmer jump, H- 1 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 2 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 4 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 5 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 6 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 7 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 8 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 9 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) 10 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… 13 The pioneers of stellar spectroscopy… at work… 14 Annie Jump Cannon 15 Annie Jump Cannon 16 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. 17 online Gray atlas at NED nedwww.ipac.caltech.edu/level5/Gray/frames.html Stellar classification 18 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 19 O-star spectral types 20 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 22 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 23 Cecilia Payne-Gaposchkin 24 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. 25 Cecilia Payne-Gaposchkin 26 Stellar continua: opacity sources 27 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) 28 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 29 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 31