Generalized Maxwell Love numbers

TL;DR

This paper derives analytical formulas for Love numbers of a homogeneous sphere with generalized Maxwell rheology, revealing stability and providing explicit solutions for complex shear moduli and inversion methods.

Contribution

It introduces new analytical forms of complex shear modulus for generalized Maxwell bodies and explores their application to calculating Love numbers for homogeneous spheres.

Findings

Sphere is asymptotically stable for all parameters

Analytical Love numbers are available for systems with fewer than five Maxwell bodies

Laplace inversion methods can be applied without numerical discretization

Abstract

By elementary methods, I study the Love numbers of a homogeneous, incompressible, self-gravitating sphere characterized by a generalized Maxwell rheology, whose mechanical analogue is represented by a finite or infinite system of classical Maxwell elements disposed in parallel. Analytical, previously unknown forms of the complex shear modulus for the generalized Maxwell body are found by algebraic manipulation, and studied in the particular case of systems of springs and dashpots whose strength follows a power-law distribution. We show that the sphere is asymptotically stable for any choice of the mechanical parameters that define the generalized Maxwell body and analytical forms of the Love numbers are always available for generalized bodies composed by less than five classical Maxwell bodies. For the homogeneous sphere, real Laplace inversion methods based on the Post-Widder formula can be applied without performing a numerical discretization of the n-th derivative, which can be computed in a "closed-form" with the aid of the Faa di Bruno formula.