This page is an old scribble; I have a newer attack on the same question that's marginally less incomplete.
Classical theory can cleanly describe the gravitational field of a spherically symmetric shell of matter: it yields a constant gravitational potential (hence no gravitational field) in its interior and appears, to its surroundings, as though all its mass were concentrated at its centre. From this one can build up the gravitational field of a spherically symmetric ball, whose density depends only on radius from its centre, by treating it as a composite of such shells.
I propose, here, to consider the analogous case for a ball
, such as
my home planet, which is not quite strictly spherical - because it rotates, and
the consequent centrifugal
effect causes it to bulge at its equator - on
the theory that denser material sinks below lighter, so that we can suppose that
the density of the body, at each point, depends only on the effective
gravitational potential at that point; and the dependency is simply that the
density is a (non-strictly) monotonically decreasing function of potential.
Note that the potential is negative, so density increases as potential gets
bigger, but decreases as it gets nearer to zero.
Treating the body as rigid enough that we can ignore Coriolis effects, we thus take it to be rotating with some fixed angular velocity about some fixed axis. Take that axis as our z axis, so chosen that the angular velocity, w, is positive. Employ a cylindrical radial co-ordinate, R, which measures distance from the axis; and an angular longitude co-ordinate m about this axis. The system is naturally symmetric under rotations about its spin axis, hence under changes in the origin of m.
Then the effective gravitational potential alluded to above is simply the (negative) gravitational potential, φ, a non-rotating frame would report plus a w.w.r.r/2 term, equal to half the square of the velocity (angular velocity times cylindrical radius) at each point of our rigid body. My hypothesis is then that the density of matter in the ball is simply given as ρ = F(w.w.r.r/2 + φ) for some monotonically decreasing ({positive reals}: F :{reals}), with ρ tacitly zero where F is undefined.
The gravitational potential will then satisfy the integral equation:
in which the square of distance(q,p) will be the sum of squares of z(q)-z(p), r(q).sin(m(q)-m(p)) and r(p)-r(q).cos(m(q)-m(p)); the sum of squares of the last two is just r(q).r(q) +r(p).r(p) -2.r(p).r(q).cos(m(q)-m(p)).
Written by Eddy.