# Rotation and figure evolution in the creep tide theory: a new approach and application to Mercury

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## Abstract

This paper deals with the rotation and figure evolution of a planet near the 3/2 spin–orbit resonance and the exploration of a new formulation of the creep tide theory (Folonier et al. in Celest Mech Dyn Astron 130:78, 2018). This new formulation is composed by a system of differential equations for the figure and the rotation of the body simultaneously (which is the same system of equations used in Folonier et al. 2018), different from the original one (Ferraz-Mello in Celest Mech Dyn Astron 116:109–140, 2013; Celest Mech Dyn Astron 122:359–389, 2015a. arXiv: 1505.05384) in which rotation and figure were considered separately. The time evolution of the figure of the body is studied for both the 3/2 and 2/1 spin–orbit resonances. Moreover, we provide a method to determine the relaxation factor \(\gamma \) of non-rigid homogeneous bodies whose endpoint of rotational evolution from tidal interactions is the 3/2 spin–orbit resonance, provided that (i) an initially faster rotation is assumed and (ii) no permanent components of the flattenings of the body existed at the time of the capture in the 3/2 spin–orbit resonance. The method is applied to Mercury, since it is currently trapped in a 3/2 spin–orbit resonance with its orbital motion and we obtain \(4.8 \times 10^{-8}\) s\(^{-1} \le \gamma \le 4.8 \times 10^{-9}\) \(\mathrm{s}^{-1}\). The equatorial prolateness and polar oblateness coefficients obtained for Mercury’s figure with such range of values of \(\gamma \) are the same as the ones given by the Darwin–Kaula model (Matsuyama and Nimmo in J Geophys Res 114, E01010, 2009). However, comparing the values of the flattenings obtained for such range of \(\gamma \) with those obtained from MESSENGER’s measurements (Perry et al. in Geophys. Res. Lett. 42, 6951–6958, 2015), we see that the current values for Mercury’s equatorial prolateness and polar oblateness are 2–3 orders of magnitude larger than the values given by the tidal theories.

## Keywords

Planetary tide Mercury Flattenings Spin-orbit## Notes

### Acknowledgements

We thank the two referees for the fruitful discussions about tides and equilibrium figures, which led to an improvement of this paper. This investigation is funded by the National Research Council, CNPq, Grant 302742/2015-8 and by FAPESP, Grants 2016/20189-9 and 2017/25224-0. This investigation is part of the thematic project FAPESP 2016/13750-6.

### Compliance with ethical standards

### Conflict of interest

The authors declare no competing interests.

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