Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S02 / 2

Let $r$ be radial position, $\mathcal{F}=L/(4\pi r^2)$ the outward radiative energy flux, and $P_r=a{T}^4/3$ the radiation pressure. A thin layer absorbs a fraction $\kappa \rho dr$ of an incident directed beam. Since a photon carries momentum $E/c$, the outward force per unit volume deposited by the radiation is $\kappa \rho \mathcal{F}/c$. Balance it against the radiation pressure gradient:Substituting $d{P}_r/dr=(4a/3)T^{3}dT/dr$ yields the radiative diffusion equationThe sign expresses outward energy flux down the temperature gradient. The approximation assumes optically thick material and near local thermodynamic equilibrium, with an appropriate mean opacity.

The printed attenuation law describes the loss from an incident beam; it is not generally the radial derivative of the net luminosity, since absorption is accompanied by emission. For a concrete counterexample, in a steady envelope without local energy generation the net luminosity is constant even though $\kappa \rho >0$ and $dT/dr<0$. Applying $dL/dr=-\kappa \rho L$ to that net luminosity would contradict energy conservation. The local momentum deposition argument supplies the physically correct derivation.

Write $m(r)$ for enclosed mass. The stellar hydrostatic equation is $dP/dr=-Gm\rho /r^2$. If radiation pressure dominates, compare it directly with the radiative diffusion result for $d{P}_r/dr$:This is the limiting Eddington luminosity scaling, assuming fixed composition and electron-scattering opacity.

If gas pressure dominates, use stellar homology: characteristic density $\rho \sim M/R_{\ast }^3$ and pressure $P\sim G{M}^2/R_{\ast }^4$. The ideal gas law then givesEstimating $|dT/dr|\sim T/R_{\ast }$ in radiative diffusion givesThus at fixed $\mu$ and constant electron-scattering opacity, $\boxed{L\propto M^3}$. These are stellar homology estimates for radiative stars, not exact relations valid for arbitrary composition and internal transport.

The same gas-pressure stellar homology estimates give $T\propto \mu M/R_{\ast }$ and $\rho \propto M/R_{\ast }^3$. With Kramers' opacity law $\kappa \propto \rho T^{-7/2}$, radiative diffusion givesConsequently the claimed exponent five additionally requires the approximate low-mass main-sequence mass-radius relation $R_{\ast }\propto M$ and fixed composition:The supplied opacity and pressure assumptions alone do not establish that mass-radius relation. For example, a homologous fixed-radius family obeys those scaling assumptions but gives $L\propto M^{11/2}$, not $M^5$. More generally $R_{\ast }\propto M^q$ gives $L\propto M^{(11-q)/2}$. Very low-mass stars can be convective, where this radiative model ceases to apply.