Compressible viscoelastodynamics of a spherical body at long timescales and its isostatic equilibrium

The problem of compressibility in modelling of viscoelastic deformations of planetary bodies is still a topic under discussion. Studies facing this topic discuss the error when considering a stratification of layers with constant material parameters. But homogeneous compressible layers imply that the initial state is not stable. So, any perturbationmethod applied to this type of state results in an ill-posed problem, evident in a denumerable infinite set of modes in the spectral representation of the solution. Based on the analytic solution of Cambiotti and Sabadini, we discuss any violation from the stable Adams-Williamson condition to result in unphysical behaviour where we concentrate here on the consequences for the horizontal displacement and deformation within the mantle due to surface loading. This focus motivates to revisit the Longman paradox, which discusses the boundary conditions for a compressible fluid core.