ABSTRACT
In spherical polar coordinates (r, θ, φ), where r is the distance to the origin, the equation is:
∂2
∂r2 + 2
r
∂
∂r + 1
r2
[
1 sin2 θ
∂2
∂φ2 + 1
sin θ ∂
∂θ
(
sin θ ∂
∂θ
)]
= 4πGρ. (7.3)
The formal solution to this equation (in Cartesian coordinates) is:
(x, y, z) = − ∫
Gρ(x ′, y′, z′) |x − x′| dx
′dy′dz′ (7.4)
where x = (x, y, z) is the point at which the gravity is being calculated, and τ is a finite volume containing all the mass contributing to the potential (Margenau and Murphy, 1956, Section 7.17). Implicit in this solution is the boundary condition that vanish at infinity at least as fast as 1/|x|. Thus, it is usual to set the potential equal to zero at infinity. With the standard definition that the gravitational force per unit mass g = −∇, it is easily shown that the potential at the surface of a sphere of radius R and mass M is
= −GM R
(7.5)
representing the work done by the forces of gravity on a particle of unit mass as it moves, in an otherwise empty universe, from an infinite distance to the surface of the sphere.
