Somehow related question that was puzzling me since forever - how do you calculate cumulative gravity (including expected speed of light limit on the spread of gravitational waves) at a given point around a planet, i.e. not around a single point with a mass, but an irregular large body?
Didn't completely understand what you mean with the gravitational waves and speed of light but for calculating gravitational force in a point you basically need to have a function describing the form of the object(and the density if the object doesn't have uniform density) and then do a (weighted)integral regarding the force excerted by each infinitesimal point in the object.
I meant your satellite and planet are moving, hence gravity is a bit behind your actual location due to a speed limit on gravitational waves propagation, i.e. each part of the planet is contributing gravitational force at different times from the past. Something like when you make a photo of a nebula roughly 700ly away and the top of nebula is 698ly away from you whereas the bottom is 705ly - what you get in the picture are pixels from different times, not a single time snapshot of the nebula.
For a body that's nearly spherical like the Earth you do a multipole expansion. Most of the gravity comes from a monopole (just a point mass at the center of gravity). Then you calculate (or measure) the contribution to the gravitational field from a quadrupole moment, and then an octupole moment and so on.
