6. Stellar spectra excitation and ionization, 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 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 2 gij = 2i 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 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 ions 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 spin 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 atom 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 quantum number) 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 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) 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- spontaneous 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 Approximate Surface Type Main Characteristics Temperature Singly ionized helium lines either in emission or absorption. Strong ultraviolet 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 stars, 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 calcium) 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 titanium oxide TiO noticeable. CN 4200, Sr II 4077 increase with M < 3,500 luminosity. Neutral metals. 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 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-star 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 Gravity 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 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 (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 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) = 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).