Justification of the two-bulge method in the theory of bodily tides

TL;DR

This paper proves that the two-bulge method in bodily tides modeling is mathematically equivalent to the complex amplitude method, arising naturally from Fourier expansion in linear tidal theory with arbitrary rheology.

Contribution

It demonstrates that the two-bulge approach is not an approximation but directly derived from Fourier expansion in a general linear tidal framework.

Findings

Two-bulge method is equivalent to complex amplitude method.

The two-bulge approach naturally follows from Fourier expansion.

Two-bulge method can be more practical in certain calculations.

Abstract

Mathematical modeling of bodily tides can be carried out in various ways. Most straightforward is the method of complex amplitudes, which is often used in the planetary science. Another method, employed both in planetary science and astrophysics, is based on decomposition of each harmonic of the tide into two bulges oriented orthogonally to one another. We prove that the two methods are equivalent. Specifically, we demonstrate that the two-bulge method is not a separate approximation, but ensues directly from the Fourier expansion of a linear tidal theory equipped with an arbitrary rheological model involving a departure from elasticity. To this end, we use the most general mathematical formalism applicable to linear bodily tides. To express the tidal amendment to the potential of the perturbed primary, we act on the tide-raising potential of the perturbing secondary with a convolution operator. This enables us to interconnect a complex Fourier component of the tidally generated potential of the perturbed primary with the appropriate complex Fourier component of the tide-raising potential of the secondary. Then we demonstrate how this interrelation entails the two-bulge description. While less economical mathematically, the two-bulge approach has a good illustrative power, and may be employed on a par with a more concise method of complex amplitudes. At the same time, there exist situations where the two-bulge method becomes more practical for technical calculations.