Generalized Runcorn's theorem and crustal magnetism

TL;DR

This paper generalizes Runcorn's theorem to higher dimensions, explores magnetic field bounds for realistic shapes, and analyzes magnetization in cooling spherical bodies, advancing understanding of planetary magnetism.

Contribution

It extends Runcorn's theorem to arbitrary dimensions, investigates magnetic bounds for non-spherical shapes, and models magnetization in cooling spherical bodies.

Findings

Orthogonality of harmonic function gradients explains Runcorn's result.

Derived bounds on external magnetic fields for realistic geometries.

Analyzed magnetization in cooling spherical astrophysical bodies.

Abstract

During the era of NASA's Apollo missions, Keith S. Runcorn proposed an explanation of discrepancy between the Moon's negligible global magnetic field and magnetized samples of lunar regolith, based on identical vanishing of external magnetic field of a spherical shell, magnetized by an internal source which is no longer present. We revisit and generalize the Runcorn's result, showing that it is a consequence of a (weighted) orthogonality of gradients of harmonic functions on a spherical shell in arbitrary number of dimensions. Furthermore, we explore bounds on external magnetic field in the case when the idealized spherical shell is replaced with a more realistic geometric shape and when the thermoremanent magnetization susceptibility deviates from the spherical symmetry. Finally, we analyse a model of thermoremanent magnetization acquired by crustal inward cooling of a spherical astrophysical body and put some general bounds on the associated magnetic field.