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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.

For the constant-density model, $m(r)=4\pi \rho r^3/3=M{r}^3/R^3$. This is the incompressible Interior Schwarzschild metric, replacing the earlier finite-slope barotropic stellar equation of state. The Tolman–Oppenheimer–Volkoff equation separates asOn the left, partial fractions giveOn the right the antiderivative is $\frac{3}{4\rho }\log (1-8\pi \rho r^2/3)$. Applying the surface condition $p(R)=0$, and writing $s(r)=\sqrt{1-2M{r}^2/R^3}$, $s_R=\sqrt{1-2M/R}$, givesSolving for the pressure recovers the Interior Schwarzschild solution:In particular the central pressure of a constant-density relativistic star isThe denominator must be positive for a regular positive-pressure centre, requiring $M/R<4/9$. As that limiting compactness is approached, the central pressure diverges, in agreement with Buchdahl's theorem.

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.