Tidal evolution of the Keplerian elements

Abstract

We address the expressions for the rates of the Keplerian orbital elements within a two-body problem perturbed by the tides in both partners. Formulae for these rates have appeared in the literature in various forms, at times with errors. We reconsider, from scratch, the derivation of these rates and arrive at the Lagrange-type equations which, in some details, differ from the corresponding equations obtained previously by Kaula (1964).We also write down detailed expressions for da/dt , de/dt and di/dt , to order e 4 . They differ from Kaula's expressions which contain a redundant factor of M/(M + M ′ ) , with M and M ′ being the masses of the primary and the secondary. As Kaula was interested in the Earth-Moon system, this redundant factor was close to unity and was unimportant in his developments. This factor, however, must be removed when Kaula's theory is applied to a binary composed of partners of comparable masses.We have found that, while it is legitimate to simply sum the primary's and secondary's inputs in da/dt or de/dt , this is not the case for di/dt . So our expression for di/dt differs from that of Kaula in two regards. First, the contribution due to the dissipation in the secondary averages out when the apsidal precession is uniform. Second, we have obtained an additional term which emerges owing to the conservation of the angular momentum: a change in the inclination of the orbit causes a change of the primary's plane of equator.Recall that these expressions are general. In our case, the two-body problem perturbed by tides, H is independent of Θ and Θ ′ . So the partial derivatives of the Hamiltonian over these two quantities in equations (35) and (36) become zero. 9

Cassini states of a rigid body with a liquid core

“…In this paper, we analyse the Cassini states of a hollow rigid body filled with a perfect fluid using the Poincare-Hough model (Poincaré, 1910;Hough, 1895) assuming a low ellipticity of the cavity. The study is performed in a non-canonical Hamiltonian formalism exploiting as much as possible the properties of rotations and of spin operators (Boué andLaskar, 2006, 2009;Boué et al, 2016;Boué, 2017;Boué et al, 2017;Boué and Efroimsky, 2019). For convenience, the Hamiltonian and the equations of motion are given in terms of matrices as in Ruiz, 2015, 2017), but the coordinate system in which these matrices are written is arbitrary and only chosen at the end of the calculation.…”

The dynamical evolution of close-in binary systems formed by a super-Earth and its host star

“…To describe the dynamical evolution, including the orbital and rotational motions, of a binary system constituted of a rocky planet and its host star, we make use of the equations of motion derived by Boué & Efroimsky (2019), which are valid for the case when the tidal dissipation within both bodies is considered, including the disturbing potential due to the permanent-triaxial shape of the planet. We consider a system of two bodies, the host star and a rocky planet.…”

“…As both orbits are defined with respect to the corresponding equatorial planes of each body, then the aforementioned reference systems are related by a rotation from one system to an inertial reference system and then to the system attached to the other body. However, this rotations only affect the angles that give the orientation of the orbits in the space, that is, ω, i and Ω, while a, e, and M are the same in both systems (Boué & Efroimsky 2019).…”

“…The rotational state of the star and the planet are completely determined by the corresponding Euler angles, that is, the precession, ψ k , nutation, ε k , and proper rotation angles, θ k -the latter being reckoned from a fixed direction in an inertial reference system -as well as their time rates ψk , εk and θk , respectively. As we see later in this paper, under the gyroscopic approximation, it is enough to follow the time evolution of ε k and θ k (Boué & Efroimsky 2019).…”

“…The orbital dynamics is described by a set of Lagrange-type planetary equations which give, on one hand, the time evolution of the major semiaxis, the eccentricity, and the mean anomaly along with, on the other hand, the time evolution of the inclination, the argument of the pericenter and the longitude of the ascending node of the common orbit as seen from each body. The respective equations were derived by Boué & Efroimsky (2019), which correspond to Eqs. ( 116)-( 121) of the cited work and we reproduce them here:…”