= Contraction-limited central temperature
{title2=$T_{\max}\propto\mu_iM^{4/3}$}
A fixed <stellar homology> profile with $P=\rho\mathcal RT/\mu_i+K_{\rm nr}\rho^{5/3}$ gives
$$
T_c=\frac{\mu_i}{\mathcal R}\left(AGM/R-BK_{\rm nr}M^{2/3}/R^2\right),
$$
where $A,B>0$. As a function of $1/R$ this is a concave quadratic; its maximum is $A^2G^2\mu_iM^{4/3}/(4BK_{\rm nr}\mathcal R)$, at $R=2BK_{\rm nr}M^{-1/3}/(AG)$. At the maximum, the central <electron degeneracy pressure> and thermal pressure are equal. If this temperature is insufficient for sustained <hydrogen burning>, the object becomes a <brown dwarf>. Only the positive-temperature segment represents a physical sequence.
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