Question

Consider a planet entirely constituted by water,

How the pressure P(h) varies in function of the deepness h?

Answer 1

consider a planet entirely made up of water
then let radius of planet = R
mass of planet = M

now at depth h
pressure = P(h)
now
dP = rho(h)*g*dh
here rho(h) is density of water as function of h

now
let bulk moduluis of water be K
then
K = rho(h)dP/d(rho)

hence
K = [rho(h)]^2*g*dh/d(rho)
(rho^(-2))d*(rho)*K = g*dh
integrating
K(1/rhos - 1/rhoc) = g(R - 0)
hence
K(1/rhos - 1/rho) = g(R - h)
1/rho = 1/rhos - g(R - h)/K
rho = K*rhos/(K - g*rhos*(R - h))

hence
dP = K*rhos*g*dh/(K - g*rhos*(R - h))
where rhos is density of water at the surface
integrating
Ps - P = K*rhos*g*ln([K - g*rhos*R]/[K - g*rhos*(R - h)])/g*rhos
P(h) = Ps - K*ln[[K - g*rhos*R]/[k - r*rhos(R - h)]]