As the mass of the sun is way larger than those of the planets, the Born-Oppenheimer approximation would hold pretty well when modeling the sun as a nucleus and the planets as electrons, each with the corresponding mass ofcourse. Using the Kohn-Sham equations, we can split this many body problem into single body problems, which are way easier to solve. Ofcourse the electromagnetic potentials should be replaced by gravitational potentials. This will change the Kohn-Sham equations, but i am not sure exactly how, since i don’t know how they are derived. Anyways, I guess you can apply the same procedure to split it into single body problems.