 Polytropic mass-radius relation

For a regular finite-radius stellar polytrope of fixed polytropic index $0<n<5$, zero surface pressure and $P=K{\rho }^{1+1/n}$, the Lane-Emden variables for a stellar polytrope give $R\propto K^{1/2}{\rho }_c^{(1-n)/(2n)}$ and $M\propto K^{3/2}{\rho }_c^{(3-n)/(2n)}$. Eliminating the central mass density yieldsFor $n=3/2$, this becomes $R\propto K{M}^{-1/3}$; changing $K$ changes the sequence, as explained by the entropy dependence of a polytropic mass-radius relation.

With first zero ${\xi }_1$ and $Y_n=-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$, an alternative rearrangement isThe exception here differs from that in the first formula. At $n=1$, $R=\sqrt{\pi K/(2G)}$ is independent of central mass density, so radius cannot be inverted to determine mass. At $n=3$, $M\propto K^{3/2}$ is independent of central mass density, so mass cannot determine radius. These are structural degeneracies, not divergences of the physical stars.