Radiation Transport

Part 4: Radiative Diffusion — Energy Flow as a Leaking Fog

The random walk tells us why transport is slow; stellar structure needs a law for how much luminosity a temperature gradient can carry. A thermal radiation field has energy density $u_\text{rad} = aT^4$ (with $a = 7.56 \times 10^{-15}~\text{erg cm}^{-3}\text{K}^{-4}$). If the interior is hotter than the exterior, $u_\text{rad}$ decreases outward, and that gradient drives a net outward flux:

Luminosity is flux times area, $L(r) = 4\pi r^2 F_\text{rad}$:

Reading the Math: the diffusion luminosity

The flux law carries a derivative, $d(aT^4)/dr$, so we run the signature move again — now on the temperature profile — and read off how a star’s luminosity scales with its mass and radius.

① Approximate the derivative. Temperature falls from $T_c$ at the center to essentially nothing at the surface over a distance $\sim R$:

$\frac{d(aT^4)}{dr} \;\approx\; \frac{aT_\text{surf}^4 - aT_c^4}{R - 0} \;\approx\; -\frac{aT_c^4}{R}.$

The coefficient out front, $c/(3\kappa\rho)$, is the radiative diffusion coefficient $D = c\ell/3$ — Part 2’s tiny mean free path $\ell$ in one symbol. A small $\ell$ makes $D$ small, which is the random walk of Part 3.

② Extract the scaling. Substitute into $F = -(c/3\kappa\rho)\,d(aT^4)/dr$, form the luminosity $L = 4\pi R^2 F$, and replace $\rho \sim M/R^3$:

$L \;\sim\; 4\pi R^2 \cdot \frac{c}{3\kappa\rho}\cdot\frac{aT_c^4}{R} \;\sim\; \frac{4\pi a c}{3\kappa}\,\frac{T_c^4 R^4}{M}.$

Read the exponents — but note $T_c$ is not free. Reading 5 substitutes the hydrostatic core temperature $T_c \sim \mu G M m_p/(k_B R)$; the four factors of $R$ cancel and one power of $M$ survives, leaving the headline mass–luminosity law $L \propto M^3$. Hydrostatic equilibrium (Reading 2) built the first half of that result; this diffusion law is the second.

③ Name the assumption. The audit gains a transport row:

Same deal as Reading 2: right exponents, approximate coefficient. The scaling $L \propto M^3$ is robust; the prefactor needs the true opacity and structure.

At fixed luminosity, high opacity makes radiative transport less efficient, so the star needs a steeper temperature gradient to carry the same energy. If the required gradient becomes too steep, radiation is no longer the preferred mechanism and convection can take over.