The sun would act like a nucleus and the planets would act like electrons, but the Coulomb interaction would become negligible in comparison to the gravitational interaction. In the KS equations, the Hamiltonian would still contain the same terms, where the terms would be:
– kinetic energy of the i-th quasi particle
– the Hartree energy would be the energy of that quasi-particle resulting from the gravitational interaction with all the other particles
– the external potential would be the gravitational field of the Sun
– the exchange correlation potential
However, I’m not sure what the exchange correlation would mean in this case.