DDA 2015 – Forced libration of tidally synchronized planets and moons

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)


Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.


  • Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth
    • spectrum of harmonics
    • dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$
  • Geometry offorcedlibrations
    • longest axis of planet tries to align with line of centers (but can’t)
    • triaxial torque (Eckhardt 1981) $\ddot{\theta}\approx 6 n^2 \dfrac{B-A}{C^3}\left[e\, – \dfrac{31}{16}e^2 + O\left(e^5\right)\right] \sin M$
  • Problem: Io
    • striking difference with moon
    • dominant term now $\dfrac{\dot{\theta}}{n}\, – 1 \propto +\sin M$
  • Tidal torques
    • torque = triaxial + tidal
    • tidal = secular + periodic
      • secular can dominate in the “linear” regime
        • plot: $\ddot{\theta}$ vs. $\dfrac{\dot{\theta}}{n}\, – 1$
      • It’s not the amplitude that’s important, but the gradient, which provides a subtle but powerful feedback.
      • tiny interval of pert freqs where secular torque can be important
  • Problem: doesn’t work for Moon
    • because Moon is not sitting on that linear portion of $\dfrac{\dot{\theta}}{n}\, – 1$
    • $\therefore$ need further harmonic terms

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