Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Stellar Atmospheres: Radiative Equilibrium Radiative Equilibrium Energy conservation 1 Stellar Atmospheres: Radiative Equilibrium Radiative Equilibrium Assumption: Energy conservation, i.e., no nuclear energy sources Counter-example: radioactive decay of Ni56 Co56 Fe56 in supernova atmospheres Energy transfer predominantly by radiation Other possibilities: Convection e.g., H convection zone in outer solar layer Heat conduction e.g., solar corona or interior of white dwarfs Radiative equilibrium means, that we have at each location: Radiation energy absorbed / sec = Radiation energy emitted / sec integrated over all frequencies and angles 2 Stellar Atmospheres: Radiative Equilibrium Radiative Equilibrium Absorption per cm2 and second: d dv (v) I 4 Emission per cm2 and second: v 0 d dv (v) 4 0 Assumption: isotropic opacities and emissivities Integration over d then yields dv (v) J 0 v 0 0 dv (v) (v) J v S v dv 0 Constraint equation in addition to the radiative transfer equation; fixes temperature stratification T(r) 3 Stellar Atmospheres: Radiative Equilibrium Conservation of flux Alternative formulation of energy equation In plane-parallel geometry: 0-th moment of transfer equation dH v J v Sv dt Integration over frequency, exchange integration and differentiation: d H v dv J v S v dv 0 dt 0 0 H H v dv const 0 4 Teff 4 4 dK v 0 H v dv 4 Teff 0 d dv because of radiative equilibrium for all depths. Alternatively written: 0 d ( fv J v ) 4 dv Teff d 4 (1st moment of transfer equation) (definiton of Eddington factor) 4 Stellar Atmospheres: Radiative Equilibrium Which formulation is good or better? I Radiative equilibrium: local, integral form of energy equation II Conservation of flux: non-local (gradient), differential form of radiative equilibrium I / II numerically better behaviour in small / large depths Very useful is a linear combination of both formulations: d ( f v J v ) A J v S v dv B dv H 0 0 0 d A,B are coefficients, providing a smooth transition between formulations I and II. 5 Stellar Atmospheres: Radiative Equilibrium Flux conservation in spherically symmetric geometry 0-th moment of transfer equation: 1 2 r H v S v J v 2 r r r 2 H v dv r 2 S v J v dv 0 r 0 0 1 r H v dv const L 2 16 0 2 because L 16 2 R 2 H 6 Stellar Atmospheres: Radiative Equilibrium Another alternative, if T de-couples from radiation field Thermal balance of electrons Q H QC 0 QffH 4 ne N j ff, j (v, T ) J v dv j 0 3 hv kT 2 hv C Qff 4 ne N j ff, j (v, T ) J v 2 e dv c j 0 vlk Q 4 nl bf, lk (v) J v 1 dv v l ,k 0 H bf vlk Q 4 nk bf, lk (v) J v 1 v l ,k 0 C bf QcH ne nm qlm (T )hvlm 2hv 3 hv kT dv J v 2 e c l ,m QcC ne nl qlm (T )hvlm l ,m 7 Stellar Atmospheres: Radiative Equilibrium The gray atmosphere Simple but insightful problem to solve the transfer equation together with the constraint equation for radiative equilibrium Gray atmosphere: Moments of transfer equation dH v dK v J v Sv II H v with dt d d Integration over frequency I I dH J S d II Radiative equilibrium I J S dK H d (J v Sv )dv ( J v Sv )dv J S 0 and because of conservation of flux II dH 0 d d 2K dK 0 K c c from II follows c H , c2 see below 1 2 1 2 d d 8 Stellar Atmospheres: Radiative Equilibrium The gray atmosphere Relations (I) und (II) represent two equations for three quantities S,J,K with pre-chosen H (resp. Teff) Closure equation: Eddington approximation K 1 3J S J 3K 3H 3c2 III Source function is linear in Temperature stratification? In LTE: S ( ) B(T ( )) insert into III : with H 4 T 4 T 3H 3c2 4 Teff we get: 4 4 3 T ( ) Teff4 3c2 4 IV c2 is now determined from boundary condition ( =0) 9 Stellar Atmospheres: Radiative Equilibrium Gray atmosphere: Outer boundary condition Emergent flux: 1 H (0) S ( ) E2 ( )d with S from III 20 1 3H 3c2 E2 ( )d 20 3 H E2 ( )d c2 E2 ( )d 2 0 0 with t l En (t )dt 0 H ( 0) from (IV): l! 1 and E2 (t ) e t tE1 (t ) ln 2 1 3 1 1 2 H c c H 2 2 2 3 2 3 3 2 2 T 4 Teff4 , S 3H (from III) 4 3 3 10 Stellar Atmospheres: Radiative Equilibrium Avoiding Eddington approximation Ansatz: J ( ) 3H ( q ( )) generaliza tion of III q ( ) Hopf function J ( ) 3 4 Teff ( q ( )) 4 Insert into Schwarzschild equation: J ( ) S J integral equation for J 1 q( ) q( ) E1 d (*) integral equation for q, see below 20 Approximate solution for J by iteration (“Lambda iteration“) J (1) 3H ( 2 3) i.e., start with Eddington approximation 2 1 1 J ( 2 ) J (1) 3H ( 2 3) 3H E 2 ( ) E3 ( ) 3 3 2 (was result for linear S) 11 Stellar Atmospheres: Radiative Equilibrium At the surface 0 , E2 (0) 1 , E3 (0) 1 2 2 1 1 J ( 2 ) 3H 3H 0.583 3 3 4 exact: q(0)=0.577…. At inner boundary , E2 () 0 , E3 () 0 2 J (2) 3H 3 Basic problem of Lambda Iteration: Good in outer layers, but does not work at large optical depths, because exponential integral function approaches zero exponentially. Exact solution of (*) for Hopf function, e.g., by Laplace transformation (Kourganoff, Basic Methods in Transfer Problems) Analytical approximation (Unsöld, Sternatmosphären, p. 138) q ( ) 0.6940 0.1167 e 1.972 12 Stellar Atmospheres: Radiative Equilibrium Gray atmosphere: Interpretation of results Temperature gradient d 4 dT 3 4 T 4T 3 Teff d d 4 The higher the effective temperature, the steeper the dT ~ Teff4 temperature gradient. d dT dT The larger the opacity, the steeper the (geometric) temperature dt d gradient. Flux of gray atmosphere LTE: Sv Bv (T ( )) 1 1 H v ( ) Bv (T ( )) E2 (t )dt Bv (T ( )) E2 ( t )dt 2 20 with hv kTeff , T Teff 3 4 ( q( )) 1/ 4 H d H v dv and H 4 p ( ) hv kT p ( ) Teff4 H v dv E2 (t ) E2 ( t ) 4 kTeff 4 k 4 3 H ( ) / H Hv 3 2 dt dt H d Teff4 h hc exp( p ( )) 1 exp( p ( )) 1 0 1 2 hv3 4 k 3 k 3 2 c 2 h h3 v 3 13 Stellar Atmospheres: Radiative Equilibrium Gray atmosphere: Interpretation of results Limb darkening of total radiation I( 0, ) S( ) B(T( )) 4 4 3 2 T ( ) Teff 4 3 I(0, ) 2 / 3 2 3 (1 cos ) I(0,1) 1 2 / 3 5 2 i.e., intensity at limb of stellar disk smaller than at center by 40%, good agreement with solar observations Empirical determination of temperature stratification measure I ( 0, ) S ( ) S ( ) B(T ( )) T Observations at different wavelengths yield different Tstructures, hence, the opacity must be a function of wavelength 14 Stellar Atmospheres: Radiative Equilibrium The Rosseland opacity Gray approximation (=const) very coarse, ist there a good mean value ? What choice to make for a mean value? gray transfer equation 0-th moment 1st moment dI (S I ) dz dH (S J ) 0 dz dK H dz non-gray dI v (v)( S v I v ) dz dH v (v)( S v J v ) dz dK v (v) H v dz For each of these 3 equations one can find a mean , with which the equations for the gray case are equal to the frequency-integrated non-gray equations. Because we demand flux conservation, the 1st moment equation is decisive for our choice: Rosseland mean of opacity 15 Stellar Atmospheres: Radiative Equilibrium The Rosseland opacity 1 dK v 1 dK dv (v) dz R dz 0 H v dv const 0 1 dK v dv (v) dz 1 0 with Eddington approximat ion K 1 / 3 J and LTE J B : dK R dz 1 dBv dv (v) dz dBv dBv dT 1 dB d 4 4 3 dT 0 with and T T dB R dz dT dz dz dz dz dz 1 dBv dv (v) dT 1 0 4 3 R T Definition of Rosseland mean of opacity 16 Stellar Atmospheres: Radiative Equilibrium The Rosseland opacity The Rosseland mean 1 R is a weighted mean of opacity 1 with weight function dBv (v ) dT Particularly, strong weight is given to those frequencies, where the radiation flux is large. The corresponding optical depth is called Rosseland depth z Ross ( z ) R ( z )dz 0 For Ross 1 the gray approximation with R is very good, i.e. 3 T 4 ( Ross ) Teff4 ( Ross q( Ross )) 4 17 Stellar Atmospheres: Radiative Equilibrium Convection Compute model atmosphere assuming • Radiative equilibrium (Sect. VI) temperature stratification • Hydrostatic equilibrium pressure stratification Is this structure stable against convection, i.e. small perturbations? • Thought experiment Displace a blob of gas by r upwards, fast enough that no heat exchange with surrounding occurs (i.e., adiabatic), but slow enough that pressure balance with surrounding is retained (i.e. << sound velocity) 18 Stellar Atmospheres: Radiative Equilibrium Inside of blob outside T Tad Tad (r r ) T Trad Trad (r r ) ad ad (r r ) rad rad (r r ) r T (r ), (r ) T (r ), (r ) ad (r r ) rad (r r ) further buoyancy, unstable ad (r r ) rad (r r ) gas blob falls back, stable d ad d rad unstable dr dr stable k with ideal gas equation p= T and pressure balance adTad = radTrad AmH i.e. dTad dr dTrad dr unstable stable Stratification becomes unstable, if temperature gradient dTad dr 19 rises above critical value. Stellar Atmospheres: Radiative Equilibrium Alternative notation Pressure as independent depth variable: AmH p hydrostatic equation: dp g eff dr g eff dr k T kT dr dp AmH g eff p (ideal gas) AmH AmH dT dT T d (ln T ) g eff g eff dr k dp p k d (ln p ) d (ln Tad ) d (ln Trad ) unstable d (ln p ) d (ln p ) stable Schwarzschild criterion Abbreviated notation d (ln Tad ) d (ln Trad ) ; rad d (ln p ) d (ln p ) ad rad stable ad 20 Stellar Atmospheres: Radiative Equilibrium The adiabatic gradient dQ 0 (no heat exchange) dQ dE pdV (1st law of thermodynamics) dE cV dT internal energy cV dT pdV 0 (*) Internal energy of a one-atomic gas excluding effects of ionisation and excitation 3 3 E NkT cV Nk 2 2 But if energy can be absorbed by ionization: 3 cV Nk 2 Specific heat at constant pressure cp Q T p const cp cV Nk dE dV p dT dT cV p p const d ( NkT p) Nk cV p dT p 21 Stellar Atmospheres: Radiative Equilibrium The adiabatic gradient Ideal gas: pV NkT Vdp pdV NkdT cp cV dT dT Vdp pdV cp cV (**) from(*) with (**) cV Vdp pdV pdV 0 cp cV /pV cp cV cV dp dV dV cp cV 0 p V V cV dp dV cp 0 p V cV cp cV d (ln V ) d (ln p ) definition: : cp cV d (ln V ) 1 d (ln p ) 22 Stellar Atmospheres: Radiative Equilibrium The adiabatic gradient needed: d (ln T ) d (ln p ) ad T pV / Nk ln T ln p ln V ln( Nk ) d (ln T ) d (ln V ) 1 d (ln p ) d (ln p ) d (ln T ) 1 1 1 d (ln p ) 1 ad rad 1 stable Schwarzschild criterion 23 Stellar Atmospheres: Radiative Equilibrium The adiabatic gradient • 1-atomic gas cV 3 2 Nk 5 3 cp cV Nk 5 2 Nk ad 2 5 0.4 • with ionization 1 ad 0 convection starts effect • Most important example: Hydrogen (Unsöld p.228) ad 2 x x 2 5 2 EIon kT 5 x x 2 5 2 EIon kT 2 2 f (T ) f (T ) f (T ) with ionization degree x 2N N 2N 24 Stellar Atmospheres: Radiative Equilibrium The adiabatic gradient ad 2 x x 2 5 2 EIon kT 5 x x 2 5 2 EIon kT 2 2 x f (T ) f (T ) f (T ) 2N 2 N N 25 Stellar Atmospheres: Radiative Equilibrium Example: Grey approximation 2 d (ln T ) d ln 3 1 d 4d 4 2 3 T 4 ( ) 3 Teff4 2 4 3 4 ln T ln 3 Teff4 ln 2 4 3 hydrostatic equation: dp g d Ansatz: Ap b ( here a mass absorption coefficient) dp g 1 b 1 g g 1 integrate p d A b 1 A Ap b 1 (b 1) d (ln p ) 1 dp 1 g g 1 d p d p Ap b Ap b 1 (b 1) d ln T d (b 1) rad d ln p d 4 2 3 rad becomes large, if opacity strongly increases with depth (i.e. exponent b large). pb The absolute value of is not essential but the change of with depth (gradient) rad large (> ad ): convection starts, -Effekt 26 Stellar Atmospheres: Radiative Equilibrium Hydrogen convection zone in the Sun -effect and -effect act together Going from the surface into the interior: At T~6000K ionization of hydrogen begins ad decreases and increases, because a) more and more electrons are available to form H and b) the excitation of H is responsible for increased bound-free opacity In the Sun: outer layers of atmosphere radiative Video inner layers of atmosphere convective In F stars: large parts of atmosphere convective In O,B stars: Hydrogen completely ionized, atmosphere radiative; He I and He II ionization zones, but energy transport by convection inefficient 27 Stellar Atmospheres: Radiative Equilibrium Transport of energy by convection Consistent hydrodynamical simulations very costly; Ad hoc theory: mixing length theory (Vitense 1953) Model: gas blobs rise and fall along distance l (mixing length). After moving by distance l they dissolve and the surrounding gas absorbs their energy. l H (r ) H = pressure scale height mixing length parameter =0.5 2 Gas blobs move without friction, only accelerated by buoyancy; detailed presentation in Mihalas‘ textbook (p. 187-190) 28 Stellar Atmospheres: Radiative Equilibrium Transport of energy by convection Again, for details see Mihalas (p. 187-190) For a given temperature structure compute Fconv ( r ) flux conservation including convective flux Frad (r ) 4 Teff Fconv (r ) iterate new temperature stratification T ( r ) with ad rad 29 Stellar Atmospheres: Radiative Equilibrium Summary: Radiative Equilibrium 30 Stellar Atmospheres: Radiative Equilibrium Radiative Equilibrium: d ( f v J v ) A J v S v dv B dv H 0 0 0 d Schwarzschildt Criterion: d (ln Tad ) d (ln Trad ) unstable d (ln p) d (ln p) stable Temperature of a gray Atmosphere 3 4 2 T Teff 4 3 4 31