calculate the gravitational attraction?

Question

in a strictly Newtonian system:

if GMM/d^2 gives the force of attraction between two sperical planetary masses (of homogenous density) the size and mass of the sun Ms, ether

prove,using integration or whatever, that the formula gives the force of attraction EXACTLY EVEN WHEN the two planets' surfaces are less than 1 inch apart
(i.e. regarding the masses as large spherical gas clouds almost touching each other)
OR
calculate the % difference in terms of their masses between the actual force of attraction and the force of attraction as approximated by GMM/d^2
when the two planets' surfaces are less than 1 inch apart.

if your answer contains the double integral of:

[density(r1)]. [density(r2)]. [dV1]. [ dV2] ... / (r1-r2)^2

explain how this can be integrated.

Answer 1

Hi. The tidal forces cause both objects to become non-spherical. They would be asymmetrical ellipsoids, similar to an egg. I have no idea how to calculate the problem but I CAN say that if they are rotating then there is no answer. Tidal friction would slow the bodies quickly.

Answer 2

The exterior gravitational field of any spherical distribution is identical to the case where all the mass is concentrated in the center.

This is Newton's Second Theorem of Gravitation. Look it up. You can do it by hand or it's immediately derivable from Gauss's Theorem.