The plane-parallel

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Radiative transfer solved: the grey atmosphere, the diffusion limit and the curve of growth

Problem

The plane-parallel, time-independent transfer equation is

\mu\,\dfrac{\partial I_\nu}{\partial \tau_\nu} = I_\nu - S_\nu

, and the angular moments of the specific intensity are defined as

J_\nu = \tfrac{1}{2}\int_{-1}^{1} I_\nu\,\mathrm{d}\mu

H_\nu = \tfrac{1}{2}\int_{-1}^{1} I_\nu\mu\,\mathrm{d}\mu

K_\nu = \tfrac{1}{2}\int_{-1}^{1} I_\nu\mu^2\,\mathrm{d}\mu

. ((a)) Take the zeroth and first angular moments of the transfer equation to derive

\mathrm{d}H_\nu/\mathrm{d}\tau_\nu = J_\nu - S_\nu

and

\mathrm{d}K_\nu/\mathrm{d}\tau_\nu = H_\nu

, and show that for a field that is isotropic in angle (

I_\nu

independent of

\mu

K_\nu = J_\nu/3

, so that

P_{\rm rad} = u/3

with

u = aT^4

for a blackbody field. ((b)) The core of a

10\,M_\odot

main-sequence star has

T_c = 3.0\times10^{7}\,\mathrm{K}

and a total central pressure of

5.0\times10^{15}\,\mathrm{Pa}

. Treating the deep interior radiation field as isotropic and thermalised, compute the central radiation pressure and the fraction of the total central pressure it supplies.

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