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

TL;DR

This paper demonstrates that the spherical harmonic expansion of gravitational potential may fail to converge below the Brillouin sphere, even for potentials arbitrarily close to a given planet, highlighting limitations in the expansion's convergence.

Contribution

It shows that for any planet, there exist nearby planets with gravitational potentials whose spherical harmonic expansions do not extend below the Brillouin sphere, revealing non-convergence issues.

Findings

Spherical harmonic expansions can fail below the Brillouin sphere.

Non-convergence occurs even for potentials close to a given planet.

The result holds in an appropriate $C^0$-sense.

Abstract

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