Uniform-density planetary pressure profile
= Uniform-density planetary pressure profile
{title2=$P(r)=P_c(1-r^2/R^2),\quad P_c=3g_s^2/(8\pi G)$}
For a spherical <incompressible planetary interior>, $M(r)=4\pi\rho r^3/3$ gives $dP/dr=-4\pi G\rho^2r/3$. 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.