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8. Analytic Solutions II

8. Analytic Solutions II. Rutten: Rosseland Mean Opacity & Diffusion Approx Gray atmosphere Limb darkening. Rosseland Mean Extinction Coefficient a R : Recast diffusion approximation so F is function of d T / d z , use:. Where k R ( z ) = a R ( z ) / r ( z ).

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8. Analytic Solutions II

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  1. 8. Analytic Solutions II • Rutten: • Rosseland Mean Opacity & Diffusion Approx • Gray atmosphere • Limb darkening

  2. Rosseland Mean Extinction CoefficientaR: Recast diffusion approximation so F is function of dT / dz, use: Where kR(z) = aR(z) / r(z). Averages extinction similar to parallel resistors: 1/R = 1/R1 + 1/R2 + … Extinction = resistance to photon flux which favours transparent spectral windows. dBn /dT enters used for same reason: produces larger flux from a given spatial temperature gradient at frequencies where it is large.

  3. Total energy flow: u = (4s / c) T4 Also called the radiation conduction equation. It says a negative temperature gradient is required to let net radiative flux diffuse outwards through a star by thermal absorptions and re-emissions with a mean free photon path l = 1/rkR . Solar interior l ~ few mm => optical depth from surface tn ~ 1011, so diffusion approximations very accurate.

  4. The Eddington Approximation Using equations for J and K at large depth Validity: Exact for isotropic radiation. Exact for any tn when even coefficients ai = 0 in: Eddington approximation may hold for tn < 1, in contrast to approximations at large depth and diffusion equation which requires LTE and therefore only holds for tn > 1.

  5. Why Do Gray Atmosphere • Opacity independent of frequency • Only true gray opacity is electron scattering • Demonstrates radiative equilibrium • Can relate to more general/realistic situations • Can get exact solution: test approximate numerical techniques • Can set exam questions…

  6. Major assumption: opacity independent frequency: dkn / dn = 0. ERT simplifies considerably since we can integrate over frequency. where Now assume: 1. Radiative equilibrium: Total radiative flux is constant throughout atmosphere: dF / dt = 0 2. Atmosphere is in LTE: Hence Sn(T) = Bn(T) and S(T) = B(T) We’ll use Stefan-Boltzmann:

  7. ERT moment equations (L6) now become: Radiative equilibrium => constant flux, so dH / dt = 0 (H = F / 4) This gives S(t) = J(t) and K(t) = H(t + q) , since H(t) = H (constant) and q = integration constant. Eddington approx, J(t) = 3K(t), gives:

  8. Second Eddington approx, J(t = 0) = 2H, gives q = 2/3 Milne-Eddington gray atmosphere approx: since F = 4H and S = J. Here F = (s/p)Teff4. Exact solution usually written: q(t) = Hopf function which varies slowly with t.

  9. Temperature Structure: LTE gives S(t) = B(t) = (s/p)T4 and using approx above, gives This gives Teff = T(t = 2/3) Limb Darkening: Eddington_Barbier, I(0,m) = S(t = m), gives I(0,m) = F(2 + 3m), so centre-to-limb variation for a gray star is: Gives I(0,0) / I(0,1) = 0.4, agreeing with Solar limb darkening.

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