Amplitude estimation of the Moons spheroidal excitation induced by a gravitational wave

Gravitational waves from astrophysical sources interact with elastic bodies and their interaction can be presented in terms of the normal mode excitations. In this regards, the GWs are described as forces driving these oscillations. In the community, two theories emerged on how the GW metric perturbation couples with the elastic body. One theory relates to Dysons paper (Dyson, 1969), where the GW force is coupled with the elastic body through the gradient of the shear modulus. The second one is related to Webers paper (Weber, 1960), where the GW is coupled with the elastic body through a Newtonian tidal forcing. Here, we present analytical displacement solutions to both of these theories by using the Green tensor formalism, where the induced displacement is calculated as a double integral of the convolution between the impulse response of the elastic body and the GW force term. Eventually, we examine what are the key ingredients to obtain the GW response of the Moon. It has been shown that Moon is extremely seismically quiet with an upper limit on the background noise that is lower than the Earth one by at least 3 orders of magnitude. This is one of the main driving reasons why the Moon is considered as a unique environment for a gravitational astronomy. Therefore, to conduct our study, we introduce several approximations: firstly, GWs are monochromatic waves defined by a scalar metric value, a polarization tensor and a propagating vector; secondly, we consider non-rotating anelastic Moon; thirdly, the derivation is obtained in the Moons reference system. We study, what are the main differences of the induced responses from the two theories, and what are the levels of the excitation amplitudes for a published lunar model. Next, we also scrutinise how the estimated excitation amplitude depends on the regolith structure by altering the initial lunar model and using different regolith models. We discuss what are the prospect of detecting these signals with future GW detectors build on the Moon.Dyson, F. J. (1969). Seismic response of the earth to a gravitational wave in the 1-Hz band. Astrophysical Journal, vol. 156, p. 529, 156, 529.Weber, J. (1960). Detection and generation of gravitational waves. Physical Review, 117(1), 306.