Forward and Adjoint Calculations of Gravitational Potential in Heterogeneous, Aspherical Planets

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 spherical reference domain, as well as a pseudospectral/spectral element discretization. The mapping from the aspherical physical planet to the spherical reference domain enables the use of a radially varying spherical harmonic expansion, with the expansion coefficients represented with a radial spectral element mesh. The method is accurate to numerical precision, whilst avoiding the computational costs associated with large‐scale numerical meshes. 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 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.