{title2=$P_c\ge GM^2/(8\pi R^4)$}

The <hydrostatic pressure support equation> in enclosed-mass coordinate is $dP/dm=-Gm/(4\pi r^4)$. Since $r(m)\le R$, integration from surface pressure zero to the centre gives
$$
P_c=\int_0^M\frac{Gm}{4\pi r(m)^4}\,dm\ge\frac{GM^2}{8\pi R^4}.
$$
If <mass density> decreases outward, the mean density within $r$ is at least the global mean, so $r(m)\le R(m/M)^{1/3}$. This improves the bound to
$$
P_c\ge\frac{3GM^2}{8\pi R^4},
$$
with equality for uniform <mass density>. Centrally concentrated planets need larger central <pressure> to support their inner mass.


 Hydrostatic lower bound on planetary central pressure

ID: hydrostatic-lower-bound-on-planetary-central-pressure