= Polytropic mass-radius relation
Eliminating central density from the Lane-Emden scalings gives
$$
R\propto K^{n/(3-n)}M^{(1-n)/(3-n)}
$$
for $n\ne3$. In particular, $R\propto KM^{-1/3}$ for $n=3/2$.
With $Y_n=-\xi_1^2\theta'(\xi_1)$, the exact rearranged relation is
$$
M=A_nK^{n/(n-1)}R^{(3-n)/(1-n)},
$$
where
$$
A_n=4\pi Y_n
\left(\frac{n+1}{4\pi G}\right)^{n/(n-1)}
\xi_1^{(3-n)/(n-1)}.
$$
For $n=3$, the radius dependence disappears and the mass is proportional to $K^{3/2}$.
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