Lamé's problem of gravitational viscoelasticity: the isochemical, incompressible planet

Abstract

SUMMARY We consider a spherical, isochemical, incompressible, non-rotating fluid planet and study infinitesimal, quasi-static, gravitational-viscoelastic perturbations, induced by surface loads, of a hydrostatic initial state. The analytic solution to the incremental field equations and interface conditions governing the problem is derived using a formulation in terms of the isopotential incremental pressure measuring the increment of the hydrostatic initial pressure with respect to a particular level surface of the gravitational potential. This admits the decoupling of the incremental equilibrium equation from the incremental potential equation. As result, two mutually independent (4 X 4) and (2 X 2) first-order ordinary differential systems in terms of the mechanical and gravitational quantities, respectively, are obtained, whose integration is algebraically easier than that of the conventional (6 X 6) differential system. In support of various types of application, we provide transfer functions, impulse-response functions and Green’s functions for the full range of incremental field quantities of interest in studies of planetary deformations. The functional forms in the different solution domains involve explicit expressions for the Legendre degrees n = 0, n = 1 and n 2 2, apply to any location in the interior or exterior of the planet and are valid for any type of generalized Maxwell viscoelasticity and for arbitrary surface loads.