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i was reading a book on astrodynamics and was trying to understand the mathematical treatment of the Earth gravitational potential . i kinda understand the main idea , after reaching the following equation of the potential in terms of the Legendre polynomials :

##\alpha = r_{Q}/r##

i am really interested in the geometrical interpretation of this equation which is known as MacCullagh’s method , the 1st , 2nd and the third expansion terms was some how understandable in which they some how determine the moment of inertia and different mass distribution contribution to the non sphereicity , but i am curious abut the 4th term :

it some how seems like we start the series with the regular spherical potential and each successive term held some kind a contribution to the non sphericity so every term should have a physical meaning , right! , what this term mean is it also some kind of a fancy combination of moment of inertial around some point or something ?!