Tidal Response of a Laterally Heterogeneous Moon and its Potential Application to GRAIL Observations

The Moon displays a number of hemispherical asymmetries near its surface, including mare basalts, crustal thickness variation and topography. However, there remain important unresolved questions about the origin of these global asymmetries, especially their potential relationship to deep interior dynamic processes. The recently completed GRAIL mission aims to investigate the interior structure of the Moon by measuring the lunar gravity field to unprecedented precision. We propose to use GRAIL's measurements to constrain the thermochemical structure of the lunar mantle, by solving for tidal variations in the lunar gravity field and searching for specific tidal terms that result from lateral heterogeneities in elastic moduli. Based on perturbation theory, we have developed a semi-analytic method of calculating the elastic tidal response of the lunar mantle with small lateral heterogeneities in shear modulus. In our specific case, a small lateral heterogeneity of spherical harmonic (1, 1) is introduced into the shear wave speed Vs throughout the mantle, and a harmonic degree-2 tidal force is applied. We calculate the tidal response up to second order in the perturbation. The zero-th order response represents that of a homogeneous Moon and is of the same harmonic as the tidal force. The coupling of the zero-th order response with (1, 1) structure excites a 1st order response at spherical harmonic degree 3, while further coupling of the degree-3 response with (1, 1) structure excites a 2nd order response at degrees 2 and 4. Of all the secondary modes generated, the degree-3 and degree-2 modes have the largest amplitude. For an overall lateral heterogeneity of δVs /Vs ~5%, the degree-3 response is ~2% of the zero-th order signal (i.e., degree-2), while the 2nd order coupling causes the degree-2 response itself to change by ~0.3%. The larger the lateral heterogeneity in Vs is, the larger the 1st and 2nd order responses will be, and the 2nd order response grows more rapidly with increasing heterogeneity and may become even larger than the 1st order response. We suggest that GRAIL observations may have the accuracy to resolve meaningful Vs variations. Our perturbation analysis also provides a true 3-D benchmark for 3-D finite element solutions of the tidal response computed using CitcomSVE as reported in Zhong et al. [2012]. Our calculations show excellent agreement between CitcomSVE and the perturbation analysis not only for the primary response at degree-2 but also for responses at secondary modes that may be 4 or 5 orders of magnitude smaller.