What Is the Gravitational Quadrupole Moment of a Homogeneous Ellipsoid?

[Osman123]

Homework Statement

The surface of a homogeneous body of mass m is the ellipsoid
(x/a)^2 + (y/b)^2 + (z/c)^2 = 1 with a,b,c>0.

What is the gravitational quadrupole moment dyadic of this body?

Homework Equations

This is my first confusion: what is the right equation?

In my Clssical Mechanics lect notes, it says

Q = ∫$\rho$(r)(3r.r-r^2.1) dr

where 1 is the unit dyadic. But there is no hint on how to apply this formula!

The Attempt at a Solution

Since I am concurrently doing EM, I remember using Legendre polynomials to solve both electric/magnetic quadrupole expansion-type question. So the question is, is it a must to use that weird integral in 2. above or can the answer be obtained using Legendre polynomials and somehow expanded into a 3 * 3 matrix?

Thank you very much in advance!

P.S. also I usually solve my doubts by browsing online, but on gravitational quadrupole moment I can't find any links at all that show how to do the integration. What should I search under? GR? Or something else. Thanks once again.

 

[Thaakisfox]

You don't need GR or anything the like. All you need is to calculate that integral over the volume of the ellipsoid. As the ellipsoid is homogenous the mass density is just a constant.
So calculate that integral over an ellipsoid which is an exercise of vector analysis. I would suggest switching to ellipsoidal coordinates. It's pretty ugly...