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

Abstract

For a singleton planet $P$ with gravitational potential $V$, we show that for each $\epsilon >0$ there exists a planet $P^{\prime }$ with gravitational potential $V^{\prime }$, with $(P^{\prime },V^{\prime })$ "$\epsilon$-close" to $(P,V)$ (in an appropriate $C^0$-sense) for which the spherical harmonic expansion of $V^{\prime }$ does not extend more than a distance $\epsilon$ below the Brillouin sphere of $P^{\prime }$.

arXiv comment8 pages, 1 figure