Forward and adjoint calculations of gravitational potential in heterogeneous, aspherical planets

TL;DR

This paper presents a computational package for accurately calculating gravitational potentials and their sensitivities in aspherical, heterogeneous planets, demonstrated on Phobos, with detailed methods and error analysis.

Contribution

It introduces a novel spectral method for gravitational potential calculations in complex planetary models, including adjoint sensitivity kernels and error quantification.

Findings

Successfully computed gravitational potential of Phobos with heterogeneous models

Derived and validated sensitivity kernels for potential functionals

Quantified errors in perturbation expansions

Abstract

We have developed a computational package for the calculation of numerically exact internal and external gravitational potential, its functional derivatives and sensitivity kernels, in an aspherical, heterogeneous planet. We detail our implementation, utilizing a transformation of the Poisson equation into a reference domain, as well as a pseudospectral/spectral element discretisation. The use of the forward solver within the package is demonstrated by calculating the gravitational potential of Phobos with homogeneous and heterogeneous density models. Equations for the first-order perturbation expansion of potential in the referential formulation are found, and the magnitude of the error is quantified based on the exact method. The adjoint Poisson equation is derived, and from it the sensitivity kernels for objective functionals of the potential, which are calculated for Phobos. The expression for perturbations to an objective functional is reduced to a single body and surface integral. Finally, a relation is obtained between the sensitivity kernels which must be satisfied when using a computational domain different to the physical domain.