 Solution

Assume homologous, gas-pressure-supported main sequence stars with nearly constant composition and mean molecular weight $\mu$, radiative energy transport and approximately constant opacity $\kappa$. Hydrostatic and mass scalings giveThe radiative diffusion equation gives the luminosity scalingTherefore, under these explicit assumptions,The radius cancels. Radiation-pressure dominance in the most massive stars changes the relation towards the Eddington luminosity scaling $L\propto M$, so the cubic law is an idealized gas-pressure homology result rather than an exact universal massive-star law.