 Solution

Regularity at the center requires $m(0)=0$ and a finite central density ${\rho }_c$. The equation of state fixes $p_c=p({\rho }_c)$, and the regular central series beginsFor each admissible ${\rho }_c$, ordinary-differential-equation uniqueness determines $m,p,\rho$ outward until the first zero of $p$, which defines $R$. The remaining additive constant in $\Phi$ is fixed by matching to the exterior Schwarzschild time coordinate. Thus smooth stars form a one-parameter family labelled uniquely by ${\rho }_c$.