A Spectral Method to Study the Tides of Laterally Heterogenous Bodies

The tidal response of a planet or moon depends on its internal structure. To obtain it, it is common to assume the body is spherically symmetric. This allows to efficiently obtain the tidal deformation of a body by projecting the viscoelastic-gravitational equations into a basis of spherical harmonics, eliminating longitude and latitude from the equations and leaving the radius as the only independent variable. However, real planets and moons are not spherically symmetric, their internal properties (e.g., density, shear modulus, viscosity) do not only vary with radius but also with latitude and longitude. Finite element methods, which rely on directly solving the 3D governing equations, can be used to study the tides of such bodies. However, these models are notably more complicated to run and more computationally expensive than spectral methods based on spherical harmonics.

We present a spectral method to compute the tides of non-spherically symmetric bodies. Using the properties of tensor spherical harmonics we transform the 3D system of equation into a set of coupled 1D equations for each spherical harmonic degree and order. The method offers insight into how lateral heterogeneities couple internal modes of different degrees and orders and allows to efficiently compute the tidal response of bodies in which internal properties change radially and laterally. We obtain the tidal response of bodies with different types of long-wavelength lateral rheology variations and discuss how geodetic observations can be used to constrain such variations.

Figure: Gravitational perturbation arising from a degree 2 order 0 tidal forcing for different geographical variations of the shear modulus, indicated between brackets. The (0,0) case corresponds to the response of a spherically symmetric body. For all cases the peak-to-peak variations of the shear modulus is 10%.