Polytropic mass-radius relation ($R\propto K^{n/(3-n)}{M}^{(1-n)/(3-n)}$)

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.

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 $Y_n=-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$. Hydrostatic scaling gives $GM/R=(n+1)(Y_n/{\xi }_1)\mathcal{R}{T}_c$. 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.

For a finite regular polytrope with fixed index, the scale ${\alpha }^2\propto K{\rho }_c^{(1-n)/n}$ implies $M^{n-1}{R}^{3-n}\propto K^n$. Thus, for $n\ne 3$, $R\propto K^{n/(3-n)}{M}^{(1-n)/(3-n)}$. For $n=3/2$, $R\propto K{M}^{-1/3}$. Fixed composition makes the cold electron degeneracy pressure coefficient approximately fixed, but a convective ideal gas has $K$ controlled by entropy. A stellar sequence with changing entropy therefore need not follow the fixed-$K$ negative slope.