Geophysics: Dynamic form factor and Equipotential surfaces

[NaturalSymphony]

I've got the following problem which I need help with. I'm used to calculating coefficients when the problem is about ellipsoids and first order approximations. But when it comes to spheres and coefficients $J_n$ I really don't know how to approach the problem. Can somebody help me out?

Consider a spherical planet of radius $a$. This planet has two spherical holes inside of radius $a/4$, both situated on the $X_3$ axis (vertical) at a distance of $3a/4$ from the center of the planet. If the planet were homogeneous, it's mass would me $M$.

a) Find a general expression for coefficients $J_n$.

b) Draw approximately the equipotential surface that goes through the planet's pole, when we consider only a first order approximation, and we don't consider the rotation.

data: $G=6.67\times 10^{-11} \ m/kg \ s^2$; $M=5.973\times 10^{24} \ kg$

 

[Asim Riaz]

; a=6.371\times 10^6 \ mThe coefficients J_n are coefficients used to express the gravitational potential of a rotating body in terms of spherical harmonics. In general, for a body of mass M and radius a, the coefficient J_n can be expressed as:J_n = -GM/a^nFor this particular problem, we can calculate the coefficients J_0 and J_2 as follows:J_0 = -GM/a^0 = -GMJ_2 = -GM/a^2 = -GM/6.371^2 x 10^12 = -1.86 x 10^-7 GM