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: Δr/R << 1 Sun: R o = 700,000 km photosphere Δr = 300 km Δr/R = 4 x 10-4 cromosphere Δr = 3000 km Δr/R = 4 x 10-3 corona Δr/R ~ 3

3 As long as Δr/R << 1 : plane-parallel symmetry consider a light ray through the atmosphere: Δr/R << 1 Δr/R ~ 1 path of light lines of constant temperature and density Plane-parallel symmetry very small curvature (α β) Solar photo/cromosphere, 3 dwarfs, (giants) Spherical symmetry significant curvature (α β) 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 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 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 = A P( 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 ρ : dp gr ( ) ρa A + gradρa= 0 dp = ρ( 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 8 4π 4π Δr ΔM phot ρ R+Δr R = R + R 3 3 R 4π 3 Δr ρ R [ ] = ρ4πr Δr 3 R [( ) ] ρ [ (1 ) ]

9 Example: the sun R = 7 x cm, Δr = 3 x 10 7 cm, ρ ~ m H N = 1.7 x 10-4 x g/cm 3 : ΔM phot = 3 x 10 1 g ΔM phot /M = 10-1 Moreover in plane-parallel symmetry: Δr/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 = ρkt ρ μ = RT μ = N kt m g H and: 1 dρ 1 dt >> ρ T k: Boltzmann constant = 1.38 x erg K -1 m H : mass of H atom = 1.66 x 10-4 g μ: mean molecular weight per free particle = <m> / m H R = k N A = x 10 7 erg/mole/k 10 dp = ρ( rgr ) ( )

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

12 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( r+δ r, t +Δt) v( r, t) v(r,t) = lim dt Δ t 0 Δt d dm v(r,t) = dfi dt i Taylor expansion v v vr ( + Δ rt, +Δ t) = vrt (, ) + Δ r+ Δt r t and Δ r = vδt v v v Δ t + Δt d = lim v v v(r,t) r t = + v Δ dt Δ t 0 t t r 1 In general: d r r r r r α( r,t) = α(r,t) + v(r,t)grad α(r,t) dt t

13 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r d v dm v(r,t) = dm v = df dt r i i dp dm = Aρ g( r) dm A + g rad dm 13 dp dv = ρ[ gr ( ) grad + v ]

14 equation of continuity, mass conservation d dt. M = M = 4π r ρ v dp dv From: = ρ[ gr ( ) grad + v ] and from the equation of state: dp kt dρ v = = soun d mh μ dρ.. v =. M 4π r ρ dv M M dρ v v dρ = = 3 4π r ρ 4π r ρ r ρ 14 dv v d ρ ρv = ρ v r

15 dρ v ( vsound v ) = ρ[ 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 ] dρ rad v sound v = ρg 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 15 v esc = 100 to 1000 km/s for main sequence and supergiant stars (v sound /v esc ) << 1

16 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 π r F r = const = luminosity L 4 ( ) 16 Plane-parallel: r ~ R ~ const F(r) ~ const

17 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 hν ν ν = ν( ) ν = ν (erg cm s Hz sr ) hν / 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 17 radiation follows the Planck function qualitatively

18 properties of Planck function a. B ν (T 1 ) > B ν (T ) (monotonic) for every T 1 > T (no function crossing) b. ν max /T = const Wien s displacement law c. Stefan-Boltzmann law: (total flux) F = π Bν ( T) dν = σt 0 σ = ergcm s 1 K 4 4 λ max = ÅforB T λ λ max = ÅforB T ν hν/kt >> 1: Wien B ( T) ν 3 hν c e - hν / kt 18 hν/kt << 1: Rayleigh-Jeans Rybicki & Lightman, 79 Bν ( T) hν kt c

19 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 ) = 4π v e dv π kt 19 most probable speed: v = ( kt m) ( π ) average speed: v = 8kT m r.m.s. speed: v = 3 rms p ( kt m) 1/ 1/ 1/

20 . Energy level distribution Boltzmann s equation for the population density of excited states in TE upper level: u n n ( Eu El) u u kt l = g e g l g u, g l : statistical weights = J+1: number of degenerate states) = n for Hyogen 0 lower level: l n n En n n kt tot = i g e u u = ge i Ei kt partition function

21 3. Kirchhoff s s law For a thermal emitter in TE εν emission coefficient ν ( ) κ = absorption coefficient = B T ν as a consequence of Boltzmann s equation for excitation and Maxwell s velocity law. 1

22 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

23 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 ν (r) B ν [T(r)] I ν is a superposition of Planck functions originating at different depths in the atmosphere radiation transfer 3

24 Local Thermodynamic Equilibrium LTE remains valid as long as the energy distribution of the local volume dv is not influenced by the photons absorbed (energy absorbed is small compared with the energy density of the gas, which is mostly set by the rate of electron collisions) LTE fails when the rate of photon absorption >> rate of electron collisions : I B ( T ) T α 1 ν ν α : nt e 0.5 LTE bad assumption at low n e, high T (hot stars, cromospheres, coronae, supergiants) 4