= Mass-averaged temperature maximum in a partially degenerate polytrope
{title2=$\overline T_{M,\max}=\frac{3\mu G^2M^{4/3}}{49\mathcal RA}$}
For a homologous <stellar polytrope> of index $3/2$, take $P=K\rho^{5/3}+\mathcal R\rho T/\mu$ and zero surface pressure. The <stellar virial theorem>, with $\Omega=-6GM^2/(7R)$, gives
$$
\overline T_M=\frac{2\mu GM}{7\mathcal RR}-\frac{\mu A M^{2/3}}{3\mathcal RR^2},\qquad A=3K\int_0^1\left(\frac{3\rho}{4\pi\bar\rho}\right)^{2/3}dq.
$$
The normalized density profile fixes $A$ independently of mass and radius at fixed composition. Writing $\overline T_M=\alpha/R-\delta/R^2$ shows that its maximum is $\alpha^2/(4\delta)=3\mu G^2M^{4/3}/(49\mathcal RA)$, at $R=7A M^{-1/3}/(3G)$. Below this radius contraction increases the fraction of support provided by <electron degeneracy pressure> and lowers the thermal temperature. A fixed normalized profile also fixes the ratio of central to mass-averaged temperature, giving a <hydrogen-burning minimum mass>.
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