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Fundamental Stellar Parameters Radiative Transfer Stellar Fundamental Stellar Parameters Radiative Transfer Stellar Atmospheres Hydrostatic and Radiative Equilibrium Grey Approximation and Mean Opacity Convection Classical Stellar Atmospheres Synthetic Spectra Solar Abundances Equations of Stellar Structure Nuclear Reactions in Stellar Interiors Stellar Atmospheres Introduction Synthetic stellar spectra used for the determination of chemical element abun- dances were, until recently, based on one-dimensional model stellar atmospheres. At any optical (or geometric) depth, gravity is exactly balanced by pressure • (gas, electron and radiation) – hydrostatic equilibrium A constant temperature is maintained at every depth; radiative energy • received from below is equal to the energy radiated to layers above although frequency redistribution will have occurred – radiative equilibrium. Atmosphere assumed to be static and convection treated with crude mixing • length theory, using an empirically determined mixing length. Opacities needed are dependent on abundances sought. • Model stellar atmosphere may be obtained empirically, in the case of the Sun, by direct measurement of limb-darkening; for other cases where only Fν is ob- servable, an iterative calculation is needed with some realistic estimate used as a starting approximation. Hydrostatic Equilibrium { I P (r + dr) Consider a cylindrical element of mass ∆M and surface area dS dS between radii r and r + dr r + dr in a star. r Mass of gas in the star at smaller radii = m = m(r). O P (r) ∆M g 1 Hydrostatic Equilibrium { II Radial forces acting on the element Gravity (inward) • G m ∆M Fg = − r2 Pressure Difference (outward – net force due to difference in pressure • between upper and lower faces) F = P (r) dS P (r + dr) dS p − dP = P (r) dS P (r) + dr dS − dr × dP = dr dS − dr × If ρ is gas density in element ∆M = ρ dr dS. Inward and outward forces must balance in hydrostatic equilibrium: G m ρ dr dS dP 0 = dr dS − r2 − dr dP G m = ρ (First Equation of Stellar Structure) dr − r2 = g ρ (in stellar atmosphere) − Hydrostatic Equilibrium { II Stellar Evolution Stellar Atmospheres Hydrostatic and Radiative Equilibrium Hydrostatic Equilibrium { II 2016-03-02 Radial forces acting on the element Gravity (inward) • G m ∆M Fg = For hydrostatic equilibrium to prevail,− ther2 pressure gradient must exactly balance the forcePressure of Differencegravity (outwardat every – net point force from due to the difference stellar in pressure centre to the • surface. A slabbetween of area upperdS and,lower thickness faces) dr, density ρ and mass ∆M at F = P (r) dS P (r + dr) dS distance r from the stellarp centre has− a total mass m(r) interior to it. dP Calculating the force of gravitation= P (r) dS (FPg()r) and + settingdr dS it equal to the − dr × pressure gradient force (Fp) across the slab, leads to the Equation of dP = dr dS Hydrostatic Equilibrium. The− gravitationaldr × force per unit mass in the stellar atmosphereIf ρ is gas density is simply in elementg, ∆ theM = accelerationρ dr dS. due to gravity. Inward and outward forces must balance in hydrostatic equilibrium: G m ρ dr dS dP 0 = dr dS − r2 − dr dP G m = ρ (First Equation of Stellar Structure) dr − r2 = g ρ (in stellar atmosphere) − Radiative Equilibrium { I Radiative equilibrium requires the flux at any depth to be constant. ∞ ∞ F (r) = Iν cos θ dΩ dν = 4π Hν dν Z0 I4π Z0 2 2 ∞ 4πr F (r) = 4πr . 4π Hν dν = constant = L Z0 For a plane-parallel atmosphere r R and ' ∞ 4π Hν dν = constant = σ Teff Z0 Integrating radiative transfer equation in plane-parallel geometry dIν µ = (κν + σν)(Iν Sν) dx − − over all angles, gives: 1 1 1 dIν 1 µ dµ = (κν + σν)(Iν Sν) dµ 2 1 dx −2 1 − Z− Z− d 1 1 µ Iν dµ = (κν + σν)(Jν Sν) dx 2 1 − − Z− dHν = (κν + σν)(Jν Sν) dx − − Radiative Equilibrium { II Integrating over frequency: d ∞ ∞ Hν dν = (κν + σν)(Jν Sν) dν dx − − Z0 Z0 The integral on the left-hand side is constant and therefore: ∞ (κ + σ )(J S ) dν = 0 ν ν ν − ν Z0 Substituting κν σν Sν = Bν + Jν κν + σν κν + σν gives ∞ [(κ + σ )J κ B σ J ] dν = 0 ν ν ν − ν ν − ν ν Z0 ∞ κ [J B (T )] dν = 0 ν ν − ν Z0 At every depth, the temperature T has to assume a value which satisfies the above equation. Radiative Equilibrium { II Stellar Evolution Stellar Atmospheres Hydrostatic and Radiative Equilibrium Radiative Equilibrium { II 2016-03-02 Integrating over frequency: d ∞ ∞ No energy is generated inHν adν static= stellar(κν + atmosphere,σν)(Jν Sν) dν the process of dx − − radiation transfer simplyZ0 redistributesZ0 radiation in frequency. The monochromaticThe integral on flux the integrated left-hand side over is constant all frequencies and therefore: is therefore invariant ∞ throughout a stellar atmosphere(κ + andσ )(J is theS ) stellardν = 0 luminosity once further ν ν ν − ν integrated over the areaZ of0 the star. Accordingly, at each depth in a Substituting stellar atmosphere, there mustκ beν a specificσ temperatureν which ensures Sν = Bν + Jν (given the monochromatic opacityκν + σν at eachκν frequency)+ σν that the monochromaticgives flux integrated over all frequencies received from below is ∞ the same as that radiated[(κν + toσν the)Jν layerκν B above.ν σν Jν] dν = 0 − − Z0 ∞ κ [J B (T )] dν = 0 ν ν − ν Z0 At every depth, the temperature T has to assume a value which satisfies the above equation. Grey Approximation Approximate radiative equilibrium integral using a frequency-averaged absorption coef- ficient κ¯ placed in front of the integral: ∞ κ¯ [J B (T )] dν = 0 ν − ν Z0 With 4 ∞ ∞ ∞ ∞ σT J = Jν dν H = Hν dν K = Kν dν B = Bν dν = π Z0 Z0 Z0 Z0 radiative equilibrium then implies J = B 4 4πH = σTeff . In LTE and radiative equilibrium S = B = J and with a mean optical depth dτ¯ =κ ¯ dx 1 ∞ J(τ¯) = Λτ¯ [J(t)] = J(t) E1( t τ¯ ) dt 2 | − | Z0 with an exact solution J(τ¯) = B(¯τ) = (σ/π)T (¯τ)4 = constant [τ¯ + q(¯τ)] × and boundary condition 4 4 T (τ¯) = 0.75 Teff [τ¯ + q(τ¯)] where q(τ¯) 2/3 is Hopf’s function. ∼ Grey Approximation Stellar Evolution Stellar Atmospheres Grey Approximation and Mean Opacity Grey Approximation 2016-03-02 Approximate radiative equilibrium integral using a frequency-averaged absorption coef- ficient κ¯ placed in front of the integral: In the Eddington Approximation∞ dKν =dτν = Hν which may be integrated κ¯ [J B (T )] dν = 0 ν − ν over τν to give K = H(τ + c)Z0 in the grey approximation where all With quantities are independent of frequency and H is independent4 of depth. ∞ ∞ ∞ ∞ σT J = Jν dν H = Hν dν K = Kν dν B = Bν dν = Here c is a constant. Since the Eddington Approximation alsoπ gives Z0 Z0 Z0 Z0 Jν = 3 Kν radiativeit follows equilibrium that thenJ implies= 3H(τ + c) and so 4 4 J = B T (¯τ) = 0:75Teff [¯τ + c]. 4 4πH = σTeff . In LTE and radiative equilibrium S = B = J and with a mean optical depth dτ¯ =κ ¯ dx 1 ∞ J(τ¯) = Λτ¯ [J(t)] = J(t) E1( t τ¯ ) dt Since J = S = B in the grey approximation,2 | the− | Schwarzschild-Milne Z0 equation forwith the an exact Eddington solution Flux gives forτ ¯ = 0 J(τ¯) = B(¯τ) = (σ/π)T (¯τ)4 = constant [τ¯ + q(¯τ)] × 1 and1 boundary condition 1 1 1 4 4 H = J(t) E2(t) dt T=(τ¯) =3 0H.75 Teff [τ¯ + qt(τ¯ E)] 2(t)dt + c E2(t)dt 2 where0 q(τ¯) 2/3 is Hopf’s function.2 0 0 Z ∼ Z Z Grey Approximation Stellar Evolution Stellar Atmospheres Grey Approximation and Mean Opacity Grey Approximation 2016-03-02 Approximate radiative equilibrium integral using a frequency-averaged absorption coef- ficient κ¯ placed in front of the integral: The standard integral ∞ κ¯ [J B (T )] dν = 0 ν − ν Z0 With 1 s s! 4 ∞ ∞ t En dt =∞ ∞ σT J = Jν dν H = Hν dν K = Ksν +dνn B = Bν dν = 0 π Z0 ZZ0 Z0 Z0 radiative equilibrium then implies gives J = B 1 1 1 1 4 t En dt = 4πHand= σTeff . En dt = 0 3 0 2 In LTE andZ radiative equilibrium S = B = J and withZ a mean optical depth dτ¯ =κ ¯ dx Therefore c = 2=3. 1 ∞ J(τ¯) = Λτ¯ [J(t)] = J(t) E1( t τ¯ ) dt 2 | − | Z0 with an exact solution J(τ¯) = B(¯τ) = (σ/π)T (¯τ)4 = constant [τ¯ + q(¯τ)] × and boundary condition 4 4 T (τ¯) = 0.75 Teff [τ¯ + q(τ¯)] where q(τ¯) 2/3 is Hopf’s function. ∼ Rosseland Mean Opacity - I The second moment of the transfer equation in planar geometry dIν µ = Iν Sν dτν − is +1 +1 +1 d 2 µ Iν dµ = µ Iν dµ Sν µ dµ dτν 1 1 − 1 Z− Z− Z− dKν = Hν dτν dKν = κνHν dx − In the grey approximation ∞ 1 dKν 1 dK dν = = H κ dx κ¯ dx − Z0 ν ∞ 1 dK ν dν 1 κ dx = Z0 ν κ¯ dK dx Rosseland Mean Opacity - I Stellar Evolution Stellar Atmospheres Grey Approximation and Mean Opacity Rosseland Mean Opacity - I 2016-03-02 The second moment of the transfer equation in planar geometry dIν The result dKν =dτν = Hν follows becauseµ = I theν S secondν integral on the dτν − right-hand sideis of the line above is clearly zero. +1 +1 +1 d 2 µ Iν dµ = µ Iν dµ Sν µ dµ dτν 1 1 − 1 Z− Z− Z− dKν = Hν dτν dKν = κνHν dx − In the grey approximation ∞ 1 dKν 1 dK dν = = H κ dx κ¯ dx − Z0 ν ∞ 1 dK ν dν 1 κ dx = Z0 ν κ¯ dK dx Rosseland Mean Opacity - II As τ , K (1/3) J , J B and → ∞ ν → ν ν → ν dKν 1 dBν 1 dBν dT = dx → 3 dx 3 dT dx Substitution in above mean opacity expression gives definition for Rosseland Mean Opacity ∞ 1 dBν(T ) dν 1 0 κν dT = Z κ¯Ross ∞ dBν(T ) dν dT Z0 which has the properties: large weight for low-opacity (more transparent to radiation) regions, • at large τ the T structure is accurately given by T 4 = 0.75 T 4 [τ + q(τ )], • eff Ross Ross κ¯ used in stellar interiors and • Ross for stellar atmospheres using κ¯Ross allows an approximate temperature stratifica- • tion to be obtained and used as a starting point for further iterations.
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