= Radiative homology with fifth-power hydrogen burning and inverse-cubic opacity
{title2=$L\propto\mu^7M^5/Z,\quad R\propto X^{1/4}Z^{1/8}\mu^{-1/4}M^{1/4}$}
For gas-pressure-supported <stellar homology> with <opacity> $\kappa=\kappa_0Z\rho T^{-3}$ and <stellar energy-generation rate> $\epsilon=\epsilon_0X^2\rho T^5$, <radiative diffusion in a star> gives $L\propto RT_c^7/(Z\rho_c^2)\propto\mu^7M^5/Z$. Integrating the heating rate instead gives $L\propto X^2\mu^5M^7/R^8$. Equality in <stellar thermal equilibrium> yields the displayed <radius> law and $T_c\propto\mu^{5/4}M^{3/4}X^{-1/4}Z^{-1/8}$. This fixes the <dimensionless> profiles and coefficients, and neglects <radiation pressure>; it is a model comparison rather than a universal law for all <main sequence> stars. At fixed <mass> and <metal mass fraction>, homogeneous <hydrogen burning> gives the <homogeneous fuel-depletion luminosity feedback> through $L\propto\mu^7$.
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