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LetThe enclosed mass isThis is continuous at the core boundary. Hydrostatic equilibrium gives $dP/dr=-Gm(r)\rho (r)/r^2$, with $P(R_p)=0$. DefineSince $F^{\prime }(r)=Gm(r)(A-br)/r^2$, the mantle pressure isInside the uniform core, the shell theorem gives $m(r)=4\pi {\rho }_c{r}^3/3$, so

The four stellar structure equations aretogether with the perfect-gas equation of stateHere $a_{\mathrm{rad}}$ is the radiation constant, $ϵ$ is the specific stellar energy-generation rate, and $\kappa$ is the opacity.

Put $x=r/R$. Integrating the prescribed density gives the enclosed massSince $m_R=M$,Integrating hydrostatic equilibrium inward from $P(R)=0$ givesThe ideal-gas law then givesIn particular,