https://insu.hal.science/insu-03718976Correia, Alexandre C. M.Alexandre C. M.CorreiaIMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides - INSU - CNRS - Institut national des sciences de l'Univers - Observatoire de Paris - PSL - Université Paris sciences et lettres - Université de Lille - SU - Sorbonne Université - CNRS - Centre National de la Recherche ScientifiqueValente, Ema F. S.Ema F. S.ValenteTidal evolution for any rheological model using a vectorial approach expressed in Hansen coefficientsHAL CCSD2022Extended body dynamicsDissipative forcesStellar systemsPlanetary systemsNatural satellitesRotation of celestial bodies[SDU] Sciences of the Universe [physics]POTHIER, Nathalie2022-07-10 11:23:102023-04-20 16:14:052022-07-10 11:23:10enJournal articles10.1007/s10569-022-10079-31We revisit the two-body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of motion are derived from the tidal potential developed in Fourier series expressed in terms of Keplerian elliptical elements, so that the variation of dissipation with amplitude and frequency can be examined. However, this method introduces multiple index summations and some orbital elements depend on the chosen frame, which is prone to confusion and errors. Here, we develop the quadrupole tidal potential solely in a series of Hansen coefficients, which are widely used in celestial mechanics and depend just on the eccentricity. We derive the secular equations of motion in a vectorial formalism, which is frame independent and valid for any rheological model. We provide expressions for a single average over the mean anomaly and for an additional average over the argument of the pericentre. These equations are suitable to model the long-term evolution of a large variety of systems and configurations, from planet-satellite to stellar binaries. We also compute the tidal energy released inside the body for an arbitrary configuration of the system.