The spherical symmetry of massive bodies

[rob60]

Homework Statement

Consider the study of the motion of a two bodies system interacting with only gravitational forces.
If the two bodies (or even one of them) has not spherical symmetry, how will you proceed? Indeed the Earth and the moon does not have spherical symmetry mass distributions but normally we consider them as..
thank you

Homework Equations

The Attempt at a Solution

 

[kuruman]

If one of the masses is spherical, you can easily find its gravitational field. Then you place the non-spherical mass in that field and do a numerical integration to find the net force on it. If the first mass is not spherical, then you first need to do a numerical integration to find its gravitational field.

This "brute force" method will work with any mass and requires no approximations or assumptions. It basically calculates the net force between the two masses by adding over all possible pairs dm₁ and dm₂.

 

[rob60]

Thank you, I've understand.
It is true that if the body is small enough it can be considered set in a field of parallel forces (with obvious and simple results) but this is an approximation that works fine, i think, for artificial satellites and for the bodies on a 'human scale'.
In astronomical calculations on the motions of planets and satellites, however, I wonder how it is take in account the 'non-symmetry' of bodies.
Do you know any reference?