Abstract
This paper deals with the application of the creep tide theory (Ferraz-Mello, Celest Mech Dyn Astron 116:109, 2013a) to the rotation of close-in satellites, Mercury, close-in exoplanets, and their host stars. The solutions show different behaviors with two extreme cases: close-in giant gaseous planets with fast relaxation (low viscosity) and satellites and Earth-like planets with slow relaxation (high viscosity). The rotation of close-in gaseous planets follows the classical Darwinian pattern: it is tidally driven toward a stationary solution that is synchronized with the orbital motion when the orbit is circular, but if the orbit is elliptical, it has a frequency larger than the orbital mean motion. The rotation of rocky bodies, however, may be driven to several attractors whose frequencies are \(1/2,1,3/2,2,5/2,\ldots \) times the mean motion. The number of attractors increases with the viscosity of the body and with the orbital eccentricity. The final stationary state depends on the initial conditions. The classical example is Mercury, whose rotational period is 2/3 of the orbital period (3/2 attractor). The planet behaves as a molten body with a relaxation that allowed it to cross the 2/1 attractor without being trapped but not to escape being trapped in the 3/2 one. In that case, the relaxation is estimated to lie in the interval \(4.6< \gamma < 27 \times 10^{-9}\,{\mathrm{s}}^{-1}\) (equivalent to a quality factor roughly constrained to the interval \(5<Q<50\)). The stars have a relaxation similar to the hot Jupiters, and their rotation is also driven to the only stationary solution existing in these cases. However, solar-type stars may lose angular momentum due to stellar wind, braking the rotation and displacing the attractor toward larger periods. Old, active host stars with big close-in companions generally have rotational periods larger than the orbital periods of the companions. The paper also includes a study of energy dissipation and the evolution of orbital eccentricity.
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Notes
Supersynchronous means that the angular rotation velocity is larger than the mean motion, i.e., the rotation period is smaller than the orbital period. It is often referred to in the literature as a pseudosynchronous solution.
The tidal torques derived in theories based on energy dissipation due to tides instead of lags (e.g., Hut 1981; Eggleton et al. 1998) are the same as in Darwinian theories with lags proportional to frequencies. They also predict pseudosynchronous stationary rotation with \(\varOmega \simeq n(1+6e^2)\).
The Cayley functions introduced here correspond to degree 3 in a / r – since \(\epsilon _\rho \propto (a/r)^3\). More general definitions, corresponding to higher powers, will be introduced in Sect. 7. These functions are equivalent to the Hansen coefficients preferred by other authors, and the equivalence is given by \(E^{(n)}_{q,p}=X^{-n,q}_{2-p}\) (Correia et al. 2014).
This definition of \(\nu \) is obviously different from that adopted in pure Keplerian approaches (as in Paper I), where the possible precessions of the node and pericenter were not considered, i.e., \({\dot{\varpi }}=0\). (NB: Here, \(n={\dot{\ell }}\) is the anomalistic mean motion.)
We do not use the words resonance and capture because the dynamics of this approach is not pendulumlike. Rather, we have attractors and basins of attraction.
In fact, to obtain the maps, a first-order integrator would be enough. \(y'\) is too small, and we are allowed to assume that y is constant (that is, \(\sigma _k\) is constant) on the right-hand side and just integrate over one cycle of the periodic terms, that is,
$$\begin{aligned} \varDelta \left( \frac{\nu }{n}\right) =-\frac{3 \pi M \overline{\epsilon }_\rho }{(M+m)}\sum _{k\in \mathbb {Z}} E_{2,k}^2 \sin 2{\overline{\sigma }}_k . \end{aligned}$$The results are almost the same as those shown. This equation is, in fact, just a translation of Eq. (37) to the used adimensional variables.
To obtain the dissipation in the other body, the equations are the same, but with the meanings of \({\mathsf {M}}\) and \({\mathsf {m}}\) interchanged.
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Acknowledgments
This study was funded by the National Research Council, CNPq, Grant 306146/2010-0. The improvement of the text by Springer’s language editor is acknowledged.
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The results of this paper were presented as Paper DDA 202.01 at the 45th Annual Meeting of the Division on Dynamical Astronomy of the American Astronomical Society, 2014, Philadelphia, USA.
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Ferraz-Mello, S. Tidal synchronization of close-in satellites and exoplanets: II. Spin dynamics and extension to Mercury and exoplanet host stars. Celest Mech Dyn Astr 122, 359–389 (2015). https://doi.org/10.1007/s10569-015-9624-5
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DOI: https://doi.org/10.1007/s10569-015-9624-5