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Rotation and figure evolution in the creep tide theory: a new approach and application to Mercury. (English) Zbl 1451.70043

Summary: 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 [H. A. Folonier et al., ibid. 130, No. 12, Paper No. 78, 23 p. (2018; Zbl 1445.70007)]. 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 [loc. cit.], different from the original one [the third author, ibid. 122, No. 4, 359–389 (2015; Zbl 1322.70023)] 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 \(4.8 \times 10^{-8} s^{-1} \leq \gamma \leq 4.8 \times 10^{-9}\text{s}^{-1}\). 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 [I. Matsuyama and F. Nimmo, “Gravity and tectonic patterns of Mercury: effect of tidal deformation, spin-orbit resonance, nonzero eccentricity, despinning, and reorientation”, J. Geophys. Res. 114, No. E1, Article ID E01010 (2009; doi:10.1029/2008JE003252)]. However, comparing the values of the flattenings obtained for such range of \(\gamma\) with those obtained from MESSENGER’s measurements [M. E. Perry et al., “The low-degree shape of Mercury”, Geophys. Res. Lett. 42, No. 17, 6951–6958 (2015; doi:10.1002/2015GL065101)], 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.

MSC:

70M20 Orbital mechanics
70K28 Parametric resonances for nonlinear problems in mechanics
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