1. Poissons equation, gravitational field.

A spherical planet of radius (3/2)R is composed of two solid regions of different, but constant, mass distributions. The core is located at r<=R and has a uniform density p0 and total mass M. The outer region has mass distribution p1 and total mass (19/2)/M

Obtain a general expression for the gravitational field g(r), using poissons equation, at a general point r from the planets centre.

Also determine the gravitational field experienced by a satelite orbiting the planet at a general distance r>(3/2)R, express the answer in terms of the known mass of the planet core.

Any help greatly appreciated.

2. The potential $\displaystyle g$ satisfies $\displaystyle \Delta g=0$, and so $\displaystyle g=g(r)=c(r)/r^2, \ 0<r\leq (3/2)R$ for a radial function $\displaystyle c(r)$. Easily, $\displaystyle c=GM(r)$, $\displaystyle G$ being the constant of universal attraction and $\displaystyle M(r)$ being the total mass up to the radius $\displaystyle r$. Now, graph $\displaystyle g$ for $\displaystyle r$, considering the cases $\displaystyle r\leq R$ and $\displaystyle r>R$.

And for the last part, use the fact that $\displaystyle M=(4/3)\pi p_0R^3$.