Analytic Solutions II: Rutten & Rosseland Mean Opacity

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

2. Rosseland Mean Extinction: In order to recast the diffusion approximation into the familiar expression for the total flux F as a function of the geometrical radial temperature gradient dT / dz, we use the Rosseland Mean Extinction coefficient: Where kR(z) = aR(z) / r(z). It averages the extinction similarly to the formula for combining parallel resistors: 1/R = 1/R1 + 1/R2 + … The extinction represents resistance to the photon flux which favours the more transparent spectral windows. The Planck function temperature sensitivity, dBn /dT enters as a weighting function for the same reason. It produces larger flux from a given spatial temperature gradient at frequencies where it is large.

3. The total energy flow is now given by: Where u is the total energy density defined before u = (4s / c) T4. This diffusion equation is 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 . In the solar interior l is only a few millimetres, making the optical depth from the surface tn ~ 1011, so the diffusion approximations are very accurate.

4. The Eddington Approximation Using the equations for J and K at large depth we get the First Eddington Approximation, often called the Eddington approximation Validity: It is exact for isotropic radiation. It is also exact, at any depth tn, when In(t,m) can be expanded in odd powers of m, with all even coefficients ai = 0 in This implies the Eddington approximation may hold for tn < 1, in contrast to the approximations at large depth and the 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. The major assumption in the gray atmosphere is that the opacity is independent of frequency: dkn / dn = 0. The ERT simplifies considerably since we can integrate over frequency. where We 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) A useful quantity relation is the frequency integrated Planck function:

7. Using the frequency integrated terms, the moment equations for the ERT now become: Radiative equilibrium implies the flux is constant throughout the atmosphere, so dH / dt = 0, since H = F / 4. This then gives S(t) = J(t) and K(t) = H(t + q), since H(t) = H (constant) and q is a constant of integration. Using the Eddington approximation that J(t) = 3K(t), we get:

8. Using the second Eddington approximation, J(t = 0) = 2H, we get q = 2/3, which gives the Milne-Eddington approximation for a gray atmosphere: since F = 4H and S = J. Here F = (s/p)Teff4. The exact solution is usually written as: where q(t) is the Hopf function which varies slowly with t.

9. Temperature Structure: Assuming LTE gives S(t) = B(t) = (s/p)T4 and using the Milne-Eddington approx above, we get In the above we get Teff = T(t = 2/3) as it should. Limb Darkening: Using the Eddington_Barbier surface relation I(0,m) = S(t = m) gives I(0,m) = F(2 + 3m), so the centre-to-limb variation for a gray star is: This gives I(0,0) / I(0,1) = 0.4, which is in excellent agreement Solar limb darkening.