Regular central expansion of the TOV equations ($m=\frac{4\pi }{3}{\rho }_c{r}^3+O(r^5)$)

For a smooth centre with density ${\rho }_c$ and pressure $p_c$, the Tolman–Oppenheimer–Volkoff equation gives $m(r)=4\pi {\rho }_c{r}^3/3+O(r^5)$ and $p(r)=p_c-2\pi ({\rho }_c+p_c)({\rho }_c/3+p_c)r^2+O(r^4)$. A fixed barotropic stellar equation of state leaves one physical central parameter; the additive lapse constant fixes time normalization.