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 [itex]J_n[/itex] I really don't know how to approach the problem. Can somebody help me out?

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

a) Find a general expression for coefficients [itex]J_n[/itex].

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

 

[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