Mercury’s frequency-dependent k2 Love number from MORE: constraints on mantle relaxation timescales at tidal forcing frequencies

Introduction

Mercury will soon be visited by the BepiColombo mission, whose scientific objectives span a wide range of topics in physics and planetary science. Investigating its interior and origin is crucial for understanding the formation and evolution of the Solar System. To this aim, the Mercury Orbiter Radio science Experiment (MORE) will enable the determination of the static gravity field and Mercury’s potential Love number k₂^[1], which quantifies the response of a planetary body’s gravity potential to tidal forcing.

In many cases, a static tide approximation is adopted, leading to the determination of a single k₂ value. For Mercury, previous geodetic studies have generally adopted this approximation, that effectively neglects any frequency dependence and phase lag in the tidal response. However, a more realistic description of planetary bodies would instead require viscoelastic rheological models, in which the material response depends on the frequency of the forcing^[4]. Since tidal forcing can be expressed as a Fourier sum of different tidal modes, each associated with a distinct frequency^[2], in this description the Love number emerges as a frequency-dependent quantity, k₂(ω_2mpq)^[3], acting on each tidal component according to its forcing frequency. 

For many Solar System bodies, especially those on low-eccentricity orbits and in a 1:1 spin-orbit resonance, the contribution of secondary modes is negligible compared with that of the mode associated with the orbital frequency. However, Mercury’s peculiar orbital and rotational dynamics characterized by its large eccentricity (e=0.2056) and 3:2 spin-orbit resonance, may provide a unique opportunity to measure the frequency-dependent dynamical Love number  k₂(ω_2mpq). Here, we explore the scientific return and the feasibility of measuring the dynamical Love number k₂ using future measurements from BepiColombo mission. The internal structure of Mercury is indeed widely debated^[5,6,8], and we suggest that the recovery of Mercury's frequency-dependent Love number would offer an opportunity to better understand its mantle relaxation timescales, over the range of periods characteristic of the tidal forcing.

Methods and Results

We first assess the expected accuracy of the Love number estimation by simulating range and Doppler measurements in Ka-band over the two-year extended orbital phase and processing them through a least-squares estimation to recover the spacecraft trajectory, gravity field, dynamical Love numbers, and rotational parameters. Once the accuracies are estimated, we generate a set of toy models and, using ALMA3^[7], we compute the Love number over the range of frequencies that compose the tidal forcing signal. These models span the range of interior architectures proposed in the current literature for Mercury, accounting for different polar moments of inertia, CMB radius, mantle configurations, and lithospheric rigiditis. They aim at providing a framework to assess whether the recovered dynamical Love number could discriminate between different interior scenarios.
Finally, our work shows that measurement of the dynamical Love number can significantly improve our understanding of Mercury’s mantle relaxation timescales, providing new and essential constraints on its internal structure. 

Acknowledments

A.C. and G. M. acknowledge support from the Italian Space Agency (2022-16-HH.1-2024)

Bibliography

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