 Uniform-density planetary pressure profile

For a spherical incompressible planetary interior, $M(r)=4\pi \rho r^3/3$ gives $dP/dr=-4\pi G{\rho }^{2}r/3$. With $P(R)=0$, $P(r)=2\pi G{\rho }^2(R^2-r^2)/3$. The surface gravity $g_s=4\pi G\rho R/3$ then yields $P_c=3g_s^2/(8\pi G)$. Add an imposed surface pressure to the entire profile if needed.