 Low-density polytropic mass divergence

For $p=K{\rho }^{1+1/n}$, the Lane-Emden mass formula gives $M\propto {\rho }_c^{(3-n)/(2n)}$ at fixed $K$, while $M/R\propto {\rho }_c^{1/n}$. For $3<n<5$, the Lane-Emden equation has a finite-radius solution, yet $M\to \infty$ as ${\rho }_c\to 0$. The relativistic Tolman–Oppenheimer–Volkoff equation approaches this limit since $p_c/{\rho }_c\to 0$. Monotone pressure and density alone therefore do not impose a finite stellar maximum mass.