Radiation Transport

Section 7 of 7

Reference and Synthesis

Reference Tables

Radiation Transport at a Glance

Quantity	Formula	Sun Value
Opacity (electron scattering)	$\kappa_\text{es} \approx 0.34~\text{cm}^2/\text{g}$	$0.34~\text{cm}^2/\text{g}$
Mean free path (core)	$\ell = 1/(\kappa\rho)$	$\sim 0.02~\text{cm}$
Number of scatterings	$N \sim (R/\ell)^2$	$\sim 10^{25}$
Optical depth	$\tau \sim \kappa\rho R \sim R/\ell$	$\sim 3.5 \times 10^{12}$
Photon diffusion time	$t_\text{diff} \sim R^2/(\ell c)$	$\sim 2.6 \times 10^5~\text{yr}$
Radiation pressure (core)	$P_\text{rad} = aT^4/3$	$\sim 1.3 \times 10^{14}~\text{dyn cm}^{-2}$
Pressure ratio	$P_\text{rad}/P_\text{gas}$	$\sim 4 \times 10^{-4}$
Eddington luminosity	$L_\text{Edd} = 4\pi GMc/\kappa$	$\sim 1.5 \times 10^{38}~\text{erg s}^{-1}$

Symbol Legend

Symbol	Meaning	CGS Units
$\kappa$	Opacity (cross-section per unit mass)	$\text{cm}^2/\text{g}$
$\rho$	Mass density	$\text{g}/\text{cm}^3$
$\ell$	Mean free path	cm
$\tau$	Optical depth	dimensionless
$N$	Number of scatterings	dimensionless
$F_\text{rad}$	Radiative flux	$\text{erg cm}^{-2}\text{s}^{-1}$
$a$	Radiation constant	$7.56 \times 10^{-15}~\text{erg cm}^{-3}\text{K}^{-4}$
$P_\text{rad}$	Radiation pressure	$\text{dyn}/\text{cm}^2$
$\mu$	Mean molecular weight	dimensionless

Summary: Energy’s Tortuous Journey

Stars are opaque — the photon mean free path in the solar core is $\sim 0.2~\text{mm}$ , about a hundred billion times smaller than the solar radius.
The random walk explains the diffusion time: a one-zone estimate gives $t_\text{diff} \sim 2.6 \times 10^5~\text{yr}$ , and realistic models stay in the same $10^5$ -year range.
Radiative diffusion transports energy down the temperature gradient; the random walk explains why it is slow, the diffusion equation how much luminosity a gradient can carry.
Convection takes over when radiative diffusion becomes too inefficient — energy is then carried by rising and sinking fluid.
Radiation pressure ( $P_\text{rad} = aT^4/3$ ) is negligible in the Sun but grows as $T^4$ ; in very luminous stars it competes with gravity and sets the Eddington luminosity.

Where we stand · Reading 4 of 5

The through-line of Module 3: what holds a star up, and what makes it shine?

✓ Settled. The fusion energy of Reading 3 escapes by radiative diffusion — a photon random walk that takes $\sim 10^5\,\mathrm{yr}$ to cross the Sun, throttled by opacity. Where the required gradient would exceed the adiabatic (Schwarzschild) value, convection takes over. Running Reading the Math on the diffusion law gives a luminosity scaling $L \sim a c\,T_c^4 R^4/(\kappa M)$ — half of the main-sequence mass–luminosity law.

? Still open. That scaling still hides $T_c$ . Substitute what sets it and the whole thing collapses to a single power of mass — but which power, and why so steep? What is the mass–luminosity law, and what reads off the main sequence from one number?

→ Next. The synthesis. Reading 5 combines hydrostatic equilibrium, energy generation, and this transport law into $L \propto M^3$ — and reads the entire main sequence as a one-parameter family in mass.

Quick retrieval: photons move at $c$ — why does energy still take $\sim 10^5\,\mathrm{yr}$ to cross the Sun?

Glossary

Convection: Energy transport by the bulk motion of fluid, driven by buoyancy: hot material rises, cool material sinks, and energy is carried by moving matter rather than diffusing radiation. It takes over where the radiative temperature gradient would be too steep to be stable.
Eddington luminosity: The luminosity at which radiation’s outward force balances gravity, $L_\text{Edd} = 4\pi GMc/\kappa$ (about $3.8 \times 10^4\,L_\odot$ for the Sun). A structural ceiling, not a hard cutoff — stars near it drive winds and instabilities.
Mean free path: The average distance a photon travels between interactions, $\ell = 1/(\kappa\rho)$ — about $0.02~\text{cm}$ in the solar core. When $\ell \ll R$ , radiation cannot stream out; it diffuses.
Opacity: A measure of how strongly matter absorbs or scatters radiation per unit mass, $\kappa$ (units $\text{cm}^2\,\text{g}^{-1}$ ). Larger opacity means a shorter photon mean free path and more resistance to radiative energy flow.
Optical depth: The dimensionless thickness of a medium in mean free paths, $\tau \sim \kappa\rho R \sim R/\ell$ . $\tau \ll 1$ is optically thin (photons stream freely); $\tau \gg 1$ is optically thick (photons interact many times).
Radiation pressure: The pressure exerted by a photon field through its momentum, $P_\text{rad} = aT^4/3$ for an isotropic thermal field. It grows as $T^4$ — negligible in the Sun, but dominant in very massive, hot stars.
Radiative diffusion: Energy transport by photons random-walking down a temperature gradient through an optically thick medium. The diffusive flux is $F_\text{rad} = -(c/3\kappa\rho)\,d(aT^4)/dr$ — throttled by opacity, driven by the gradient.
Random walk: A path of many steps each taken in an independent random direction. The net displacement grows only as $\sqrt{N}\,\ell$ (not $N\ell$ ), which is why crossing a distance $R$ needs $N \sim (R/\ell)^2$ steps and makes radiative transport slow.