CNO homology with density-dependent inverse-cubic opacity ($L\propto {\mu }^7{M}^5,R\propto {\mu }^{3/8}{M}^{5/8}$)

For gas-pressure-supported stellar homology with specific nuclear heating $ϵ={ϵ}_0\rho T^{13}$ and opacity $\kappa ={\kappa }_0\rho T^{-3}$, the dimensionless radiative coefficient is proportional to $L{\mu }^{-7}{M}^{-5}$. The nuclear coefficient is proportional to ${\mu }^{13}{M}^{15}/(L{R}^{16})$. Holding both and the dimensionless profiles fixed gives the displayed scalings. The Stefan–Boltzmann law then gives $T_e\propto {\mu }^{25/16}{M}^{15/16}$ and $L\propto {\mu }^{-4/3}{T}_e^{16/3}$. The coefficients, pressure regime and composition assumptions must be held as specified; these relations are not universal laws for massive stars.