Radiation Transport

Section 6 of 7

Radiation Pressure

Part 6: Radiation Pressure

Photons carry not only energy but momentum: $p = E/c$ . So when radiation is absorbed or scattered, it exerts a force on matter. In an isotropic thermal radiation field, the radiation pressure is:

This is why radiation transport is also a force problem.

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.

Worked Example 3Radiation Pressure vs. Gas Pressure in the Sun

Problem

Compare $P_\text{rad} = aT^4/3$ with the ideal-gas pressure $P_\text{gas} \approx \rho k_B T/(\mu m_H)$ at the solar core ( $T_c \approx 1.5 \times 10^7~\text{K}$ , $\rho_c \approx 150~\text{g cm}^{-3}$ , mean molecular weight $\mu \approx 0.61$ ).

StepRadiation pressure

$P_\text{rad} = \frac{aT_c^4}{3} \approx 1.3 \times 10^{14}~\text{dyn cm}^{-2}$

StepGas pressure

$P_\text{gas} \approx \frac{(150)(1.38 \times 10^{-16})(1.5 \times 10^7)}{(0.61)(1.67 \times 10^{-24})} \approx 3.0 \times 10^{17}~\text{dyn cm}^{-2}$

Dimensional check

Both reduce to $\text{erg cm}^{-3} = \text{dyn cm}^{-2}$ (a pressure) ✓.

Result

$P_\text{rad}/P_\text{gas} \approx 4 \times 10^{-4}$ — radiation pressure is tiny in the Sun. But $P_\text{rad} \propto T^4$ while $P_\text{gas} \propto T$ (at fixed $\rho$ ), so radiation pressure grows far faster with temperature — which is why it dominates in very massive stars.

Deriving the Eddington luminosity

How luminous can a star become before radiation pushes outward as strongly as gravity pulls inward? Assume spherical symmetry, steady state, and electron-scattering opacity. The outward radiative acceleration on a shell is $g_\text{rad} = \kappa F/c = \kappa L/(4\pi r^2 c)$ ; the inward gravitational acceleration is $g = GM/r^2$ . Setting them equal (the $r^2$ cancels) gives the Eddington luminosity:

A force-balance ceiling: below $L_\text{Edd}$ gravity wins; near it, radiation reshapes the envelope and drives strong winds; above it, hydrostatic balance becomes very hard to maintain.

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.

Schematic showing a star on the left and a gas parcel in the envelope on the right. An inward arrow labeled gravity points from the parcel toward the star, and an outward arrow labeled radiative acceleration points away from the star. Text boxes show g equals GM over r squared, g_rad equals kappa L over four pi r squared c, and the Eddington limit condition g_rad equals g. — Figure 7The Eddington luminosity comes from a force balance on a gas parcel. Gravity pulls inward with g = GM/r^2, while radiation pushes outward with g_rad = kappa L / (4 pi r^2 c). Setting them equal gives the luminosity at which radiation pressure competes directly with gravity.ASTR 201 (generated)

Worked Example 4The Sun's Eddington Luminosity

Problem

Compute $L_\text{Edd}$ for the Sun with $\kappa \approx 0.34~\text{cm}^2\text{g}^{-1}$ , $M_\odot = 2.0 \times 10^{33}~\text{g}$ .

StepPlug into 4πGMc/κ

$L_\text{Edd} = \frac{4\pi (6.674 \times 10^{-8})(2.0 \times 10^{33})(3.0 \times 10^{10})}{0.34} \approx 1.5 \times 10^{38}~\text{erg s}^{-1}$

Dimensional check

$\left[\dfrac{GMc}{\kappa}\right] = \dfrac{(\text{cm}^3\text{g}^{-1}\text{s}^{-2})(\text{g})(\text{cm s}^{-1})}{\text{cm}^2\text{g}^{-1}} = \text{g cm}^2\text{s}^{-3} = \text{erg s}^{-1}$ ✓.

Result

$L_\text{Edd} \approx 1.5 \times 10^{38}~\text{erg s}^{-1} \approx 3.8 \times 10^4\,L_\odot$ . The Sun is far below this, so radiation pressure is dynamically negligible for solar structure — but massive stars come much closer because their luminosities rise faster than linearly with mass.

Problem

$L_\text{Edd} \propto M/\kappa$ . (1) At fixed opacity, if stellar mass doubles, what happens to $L_\text{Edd}$ ? (2) Why do the most massive stars come closest to this limit even though $L_\text{Edd}$ rises with mass?