Tidal evolution of the Keplerian elements

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 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 (Rev Geophys 2:661–684, 1964). We also write down detailed expressions for \({\mathrm{d}}a{/}{\mathrm{d}}t\), \({\mathrm{d}}e{/}{\mathrm{d}}t\) and \({\mathrm{d}}i{/}{\mathrm{d}}t\), to order \(e^4\). They differ from Kaula’s expressions which contain a redundant factor of \(M{/}(M+M^{\,\prime })\), with M and \(M^{\,\prime }\) 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 \({\mathrm{d}}a{/}{\mathrm{d}}t\) or \({\mathrm{d}}e{/}{\mathrm{d}}t\), this is not the case for \({\mathrm{d}}i{/}{\mathrm{d}}t\). So our expression for \({\mathrm{d}}i{/}{\mathrm{d}}t\) 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.