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 .
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 , 2013 , 2015a ) 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 } 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 2009 ) .
However , comparing the values of the flattenings obtained for such range of \gamma with those obtained from MESSENGER ’ s measurements ( Perry et al .
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 .