If you want to use the Kohn-Sham method to study the many body system of gravitationally interacting planets and the Sun, the Hamiltonian would look similar, but with gravitational interactions instead of Coulomb interactions (as these would be negligible because of the large distances between the objects).
– The kinetic energy term would here describe the kinetic energy of the planets orbiting the Sun.
– The Hartree-energy term would here be replaced by a term to describe the gravitational potential energy felt by one planet and generated by all the other planets.
– The external potential is here the gravitational potential energy exhibited by the Sun.
– There would also be an exchange-correlation energy term.