 Solution

For a regular spherical star, mass conservation gives $m^{\prime }=4\pi r^2\rho$. Multiplying the hydrostatic equilibrium equation by $r^2/\rho$ and differentiating therefore yieldsAt the center, finite density implies $m(r)=O(r^3)$, so a regular pressure satisfies $P(0)=P_c<\infty$ and $P^{\prime }(0)=0$. At a free outer boundary in vacuum, $P(R)=0$; a surrounding medium would instead prescribe its external pressure.

Differentiate the auxiliary function, using both equations:where enclosed mass is nonzero. Since $m^2/r^4\to 0$ at the regular center, $F(0)=P_c$, whereas $F(R)=G{M}^2/(8\pi R^4)$. Integrating the strictly negative derivative givesWith nonzero external pressure, the corresponding bound is $P_c>P(R)+G{M}^2/(8\pi R^4)$.