Radiative stellar structure

In a radiative stellar region, photons carry luminosity down the temperature gradient. Dimensional scaling of the radiative-diffusion equation gives $L\sim R{T}_c^4/({\kappa }_c{\rho }_c)$.

A stellar core in which radiative diffusion in a star carries most of the energy, rather than convection. Its formation in a pre-main-sequence star breaks the fully convective assumption behind an idealized Hayashi track.

The radiative temperature gradient needed to carry a spherical stellar luminosity follows by dividing radiative diffusion in a star by the hydrostatic pressure support equation. It is a logarithmic pressure gradient, not $dT/dr$ itself. Comparing it with the adiabatic temperature gradient by the Schwarzschild criterion determines whether purely radiative transport is locally unstable at uniform composition.

Opacity is an extinction cross-section per unit mass. It controls the photon mean free path and therefore the radiative temperature gradient in an optically thick star.

The negative hydrogen ion supplies important continuum opacity in cool hydrogen-rich stellar atmospheres. A useful local approximate law is $\kappa ={\kappa }_0{\rho }^{1/2}{T}^9$; it is not valid over arbitrary temperatures and compositions. In an ideal gas power-law opacity radiative envelope matched to ${\nabla }_{\mathrm{phot}}=1/8$, it gives ${\nabla }_{\mathrm{rad}}=-1/3+(11/24)(T/T_{\mathrm{eff}}{)}^{9/2}$, rising rapidly toward convective instability.

Let $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the Stefan–Boltzmann law. The gas sound speed used here satisfies $c_s^2=\mathcal{R}T$, and $\Omega$ is the local orbital frequency. For the local opacity approximation $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, a gas-supported optically thick alpha disk has $\rho \sim \Sigma /H$ and $\mathcal{R}T\sim {\Omega }^2{H}^2$. Integrated viscous heating is $F_{+}\sim \alpha \Sigma {\Omega }^3{H}^2$. Radiative diffusion gives $F_{-}\sim (\sigma /{\kappa }_0)T^{-6}{\Sigma }^{-4/3}{H}^{1/3}\sim (\sigma /{\kappa }_0){\mathcal{R}}^6{\Omega }^{-12}{\Sigma }^{-4/3}{H}^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}{\Sigma }^{-7/41}$ and $c_s\propto r^{6/41}{\Sigma }^{-7/41}$ in Keplerian rotation. Radial powers therefore require a surface-density profile; they are not determined by opacity alone. The formula is local to the stated opacity regime and small disk aspect ratio.

The Kramers opacity law has $\kappa ={\kappa }_0\rho T^{-7/2}$ for bound-free and free-free absorption under its idealized ionized-gas assumptions.

Electron-scattering opacity is approximately independent of density and temperature in a fully ionized nonrelativistic plasma of fixed composition. For hydrogen mass fraction $X$, ${\kappa }_{\mathrm{es}}\simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$.

Free-free opacity is absorption caused when a photon is absorbed during a Coulomb encounter between an electron and an ion. In the Kramers approximation its Rosseland mean scales approximately as ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$.

In an optically thick stellar region, radiative diffusion carries luminosity according to

In the optically thick radiative diffusion in a star approximation, the heat flux is $\mathbf{F}=-\chi \nabla T$. The conductivity depends on opacity, mass density and temperature. It relates a material temperature gradient to energy transport and must not be confused with the emergent effective temperature.