Dynamics of tidal synchronization and orbit circularization of celestial bodies

Abstract

We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p : q synchronous rotation.

A minimal dynamical model for tidal synchronization and orbit circularization

“…In that model the dynamics of the secondary is three-dimensional and the point masses of the secondary are unequal, but because the orbital elements do not change, the loss of stability is not the result of a temporal evolution as in our case. The figure also shows the centre-ofmass energy E c ; our numerical experience is that the change of E c is exponential only in a 1:1 resonance [7]. In other resonant states the temporal change in E c is approximately linear (for an explanation see Eq.…”

“…In dissipative cases (γ > 0) there is only one attractor, the 1:1 resonance, and a circular orbit [7]. The approach towards the attractor typically occurs through a series of resonances.…”

“…Fig. 7 The shift of function α(t) with eccentricity. The line styles mark the eccentricity values as in Fig.…”

“…In previous work we introduced the model [7]; here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring.…”

“…The model is well-defined and easy to treat numerically in its original form [7]. Nevertheless, we carry out a large distance Taylor expansion (l ≪ r) over the exact equations of motion.…”