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A regular central expansion of the TOV equations requires $m(0)=0$. Once ${\rho }_c$ is chosen, the barotropic stellar equation of state fixes $p_c=p({\rho }_c)$. Smoothness at the centre forces the radial first derivatives of $p$ and $\rho$ to vanish. Expanding the Tolman–Oppenheimer–Volkoff equation givesSince $dp/d\rho >0$, the equation of state is locally invertible, and these coefficients determine the corresponding energy density expansion as well. Integrating the regular solution outward gives the surface at the first zero of the pressure, when such a finite surface exists. The ordinary differential equation uniqueness theorem then fixes the interior and its exterior mass from ${\rho }_c$.

The additive constant ${\Phi }_c$ is a time-coordinate normalization, not a second physical parameter. Matching to the Schwarzschild metric with time normalized at infinity fixes it. Therefore the regular stellar solutions, modulo time rescaling, have one physical parameter: ${\rho }_c$. This statement uses a suitably smooth fixed equation of state and refers to central densities for which the solution actually forms a regular star; not every central energy density need give a finite-radius configuration.