Non-convergence of the spherical harmonic expansion of gravitational potential below the Brillouin sphere: The continuous case

For a singleton planet, P, with gravitational potential, V, we show that for each ɛ > 0, there exists a planet P′ with gravitational potential V′, with (P′, V′) “ɛ-close” to (P, V) (in an appropriate C0-sense), for which the spherical harmonic expansion of V′ does not extend more than a distance ɛ below the Brillouin sphere of P′.