= Homologous polytropes at fixed central temperature
{title2=$\frac{GM}{R}=(n+1)\frac{Y_n}{\xi_1}\mathcal RT_c$}
For finite-radius regular <polytropes> of one fixed index $n$ and composition, let $\xi_1$ be the first zero of the <Lane-Emden function> and $Y_n=-\xi_1^2\theta'(\xi_1)$. Hydrostatic scaling gives $GM/R=(n+1)(Y_n/\xi_1)\mathcal RT_c$. Fixed central <temperature> therefore implies $M\propto R$. This is not the fixed-entropy-constant sequence; if the index or composition changes, their prefactors cannot be discarded.
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