 Solution

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.