2. Basic assumptions for stellar atmospheres

Transcription

1 . Basic assumptions for stellar atmospheres 1. geometry, stationarity. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1

2 1. Geometry Stars as gaseous spheres spherical symmetry Exceptions: rapidly rotating stars Be stars v rot = km/s (Sun v rot = km/s) For stellar photospheres typically: r/r << 1 Sun: R o = 700,000 km photosphere r = 300 km r/r = 4 x 10-4 cromosphere r = 3000 km r/r = 4 x 10-3 corona r/r ~ 3

3 As long as r/r << 1 : plane-parallel symmetry consider a light ray through the atmosphere: r/r << 1 r/r ~ 1 path of light lines of constant temperature and density Plane-parallel symmetry very small curvature () Solar photo/cromosphere, 3 dwarfs, (giants) Spherical symmetry significant curvature ( ) Solar corona, supergiants, expanding envelopes (OBA stars), novae, SN

4 Homogeneity & stationarity We assume the atmosphere to be homogeneous Counter-examples: sunspots, granulation, non-radial pulsations clumps and shocks in hot star winds magnetic Ap stars and stationary Most spectra are time-independent: t = 0 (if we don t look carefully enough ) Exceptions: explosive phenomena (SN), stellar pulsations, magnetic stars, mass transfer in close binaries 4

5 . Conservation of momentum & mass consider a mass element dm in a spherically symmetric atmosphere The acceleration of the mass element results from the sum of all forces acting on dm, according to Newton s nd law: 5 d dm v(r,t) df F dt i i

6 a. Hyostatic equilibrium d dm v(r,t) df F dt i i Assuming a hyostatic stratification: v(r,t)=0 0 df i i Mr dm Gravitational forces: dfgrav G g( r) dm r dp Gas Pressure forces: df p, gas AP( r ) A P( r) A Radiation forces: df g ( r) dm absorbed photon momentum rad rad dt 6

7 In equilibrium: dp 0 dfi g( r) dm A g i raddm and substituting dm = A : dp gr ( ) A A grada0 dp ( r)[ g( r) g ] rad Hyostatic equilibrium in spherical symmetry 7

8 Approximation for g(r) The mass within the atmosphere M(r) M(R) << M(R)=M * M(R) = M(r) = M * g(r) = G M * /r Example: take a geometrically thin photosphere r M phot Rr R R R 3 3 R 4 3 r R [1 3 1] 4R r 3 R [( ) ] [ (1 ) ]

9 Example: the sun R = 7 x cm, r = 3 x 10 7 cm, ρ ~ m H N = 1.7 x 10-4 x g/cm 3 : M phot = 3 x 10 1 g M phot /M = 10-1 Moreover in plane-parallel symmetry: r/r << 1 g(r) = const g = G M * /R 9 Main sequence star log g = 4 (cgs: cm/s ) supergiant white dwarf 8 Sun 4.44 Earth 3.0

10 Barometric formula Assume: plane-parallel symmetry g rad << g ideal gas: P kt RT N kt m g H Note: and: 10 1 d 1 dt T dp ( rgr ) ( ) k: Boltzmann constant = 1.38 x erg K -1 m H : mass of H atom = 1.66 x 10-4 g : mean molecular weight per free particle = <m> / m H R = k N A = x 10 7 erg/mole/k

11 Barometric formula dp ( rgr ) ( ) k d dt [ T ] m H g kt 1 d 1 dt [ ] g m T negligible by assumption H 1 d gmh kt and: H kt gm μ H pressure scale height H(Sun) = 150 km 11 solution: ( ) ( ) r r e 0 rr H 0

12 dp ( rgr ) ( ) ( ) ( ) r r e 0 rr H 0 Approximate solution: Barometric formula exponential decline of density scale length H ~ T/g explains why atmospheres are so thin 1

13 b. radial flow Let us now consider the case in which there are deviations from hyostatic equilibrium and v(r,t) 0. Newton s law: d v( rr, t t) v( r, t) v(r,t) lim dt t 0 t d dm v(r,t) dfi dt i Taylor expansion v v vr ( rt, t) vrt (, ) r t r t and r vt v v v t t d lim v v v(r,t) r t v dt t 0 t t r In general: 13

14 For the assumption of stationarity /t = 0 d dt v(r,t) v v r 14 d v dm v(r,t) dm v df dt r i i dp dm A g( r) dm A g rad dm dp dv [ gr ( ) grad v ] new term

15 equation of continuity, mass conservation d dt. M M 4 r v v. M 4 r.. dv M M d v v d 3 4 r 4 r r dp dv From: [ gr ( ) grad v ] and from the equation of state: dp kt d v soun d mh d v v r dv v d Assuming again that the 15 temperature gradient is small

16 d v ( vsound v ) [ g( r) grad ] r Hyodynamic equation of motion mvesc ( r) mm with escape velocity : = G mrg vesc ( r) gr r re-write g ( v ) ( )[1 4 ] d rad v sound v g r gr ( ) vesc ( r) When v << v sound : practically hyostatic solution (v = 0) for density stratification. This is reached well below the sonic point (where v = v sound ). example: v sound = 6 km/s for solar photosphere, 0 km/s for O stars 16 v esc = 100 to 1000 km/s for main sequence and supergiant stars (v sound /v esc ) << 1

17 3. Conservation of energy Stellar interior: production of energy via nuclear reactions Stellar atmosphere: negligible production of energy the energy flux is conserved at any given radius Fr ( ) energy area time In spherical symmetry: r F(r) = const F(r) ~ 1/r r F r const luminosity L 4 ( ) 17 Plane-parallel: r ~ R ~ const F(r) ~ const

18 4. Concepts of Thermodynamics Radiation field Consider a closed cavity in thermodynamic equilibrium TE (photons and particles in equilibrium at some temperature T). The specific intensity emitted (through a small hole) is (energy per area, per unit time, per unit frequency, per unit solid angle) (Planck function): 3 h ( ) (erg cm s Hz sr ) h / kt I d B T d d c e 1 black body radiation: universal function dependent only on T not on chemical composition, direction, place, etc. Note: stars don t radiate as blackbodies, since they are not closed systems!!! But their 18 radiation follows the Planck function qualitatively

19 properties of Planck function a. B (T 1 ) > B (T ) (monotonic) for every T 1 > T (no function crossing) b max /T = const c. Stefan-Boltzmann law: (total flux) Wien s displacement law F B ( T) d T ergcm s 1 K 4 4 λ m a x = λ m a x = T T Å Å f o r B λ f o r B ν h/kt >> 1: Wien B ( T) 3 h c e - h / kt 19 h/kt << 1: Rayleigh-Jeans Rybicki & Lightman, 79 B ( T) h kt c

20 Concepts of Thermodynamics Gas particles 1. Velocity distribution In complete equilibrium this is given by Maxwell distribution (needed, e.g., for collisional rates): 3/ mv m kt f ( vdv ) 4 v e dv kt 0 most probable speed: v kt m average speed: v 8kT m r.m.s. speed: v 3kT m rms p 1/ 1/ 1/

21 . Energy level distribution Boltzmann s equation for the population density of excited states in TE upper level: u n n ( EuEl) u u kt l g e g l g u, g l : statistical weights = J+1: number of degenerate states) = n for Hyogen 1 lower level: l n n En n n kt tot i g e u u ge i Ei kt partition function

22 3. Kirchhoff s s law For a thermal emitter in TE emission coefficient ( ) absorption coefficient B T as a consequence of Boltzmann s equation for excitation and Maxwell s velocity law.

23 Local Thermodynamic Equilibrium A star as a whole (or a stellar atmosphere) is far from being in thermodynamic equilibrium: energy is transported from the center to the surface, iven by a temperature gradient. But for sufficiently small volume elements dv, we can assume TE to hold at a certain temperature T(r) Local Thermodynamic Equilibrium (LTE) good approx: stellar interior (high density, small distance travelled by photons, nearlyisotropic, thermalized radiation field) bad approx: gaseous nebulae (low density, non-local radiation field, optically thin) stellar atmospheres: optically thick but moderate density 3

24 Local Thermodynamic Equilibrium 1. Energy distribution of the gas determined by local temperature T(r) appearing in Maxwell s/ Boltzmann s equations, and Kirchhoff s law. Energy distribution of the photons photons are carriers of non-local energy through the atmosphere I (r) B [T(r)] I ν is a superposition of Planck functions originating at different depths in the atmosphere radiation transfer 4

25 Spring 013 LTE LTE vs NLTE each volume element separately in thermodynamic equilibrium at temperature T(r) 1. f(v) dv = Maxwellian with T = T(r). Saha: (n p n e )/n 1 ~ T 3/ exp(-h 1 /kt) 3. Boltzmann: n i / n 1 = g i / g 1 exp(-h 1i /kt) However: volume elements not closed systems, interactions by photons LTE non-valid if absorption of photons disrupts equilibrium 5

26 Spring 013 LTE vs NLTE NLTE if rate of photon absorptions >> rate of electron collisions I (T) ~ T, >1» n e T 1/ LTE valid: non-valid: low temperatures & high densities high temperatures & low densities 6

27 Spring 013 LTE vs NLTE in hot stars Kuitzki