Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S26 / 1 / c / Solution

The radiation pressure at a photospheric temperature near $5800$ K is only about $0.3$ Pa. For an illustrative photospheric gas density $3\times 10^{-4}\mathrm{kg}{\mathrm{m}}^{-3}$ and mean molecular weight near $1.3$, the ideal gas pressure is about $10^4$ Pa. Thus gas pressure dominates those outer layers by many orders of magnitude. Both the temperature and density are needed: low temperature alone is not a complete comparison.

For homologous stars, hydrostatic and ideal-gas estimates give $T\sim \mu m_{H}GM/(k_{B}R)$ and $\rho \sim M/R^3$. Thereforewith a structural coefficient independent of mass only under the homology assumption. As mass rises, radiation becomes increasingly important. A rough solar interior normalization of $10^{-3}$-$10^{-2}$ suggests equality after a mass increase by tens, while realistic radiation-dominated structure and composition shift the scale; stars of order $10^2{M}_{\odot }$ are a useful estimate for strong radiation support. The very small photospheric ratio must not be used as a universal normalization for their whole interiors.

Radiation also transfers outward momentum. Comparing radiative acceleration $\kappa L/(4\pi r^{2}c)$ with gravity gives the Eddington luminosityNear this scale, radiatively driven winds and envelope instabilities can remove matter or impede further accretion. Opacity, composition, rotation, mass loss and formation history affect the limiting stellar mass. The Eddington condition is a luminosity-to-mass constraint, not a unique composition-independent maximum stellar mass.