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?
This is my first confusion: what is the right equation?
In my Clssical Mechanics lect notes, it says
Q = ∫[itex]\rho[/itex](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 cant find any links at all that show how to do the integration. What should I search under? GR? Or something else. Thanks once again.