Influence of Energy Dissipation on Spin Pole Precession Trajectories for Synchronous Rotators

We present an analysis of the influence of tidal energy dissipation on the precessional motion of the spin pole of a synchronously rotating body, whose orbit pole is also precessing. In the absence of dissipation, this problem has been frequently analyzed in a Hamiltonian context. The Cassini states, or fixed points of the motion of the spin pole, correspond to tangent intersections of the Hamiltonian function with the unit sphere. When dissipation is added to the dynamics, the locations of the fixed points change. We present a new geometrical characterization of the nullclines of the dissipative flow, or motion of the spin pole, and use it to find the locations of the dissipative fixed points. Using the Poincaré−Hopf index theorem, we show that in the case where there are four fixed points, one is a saddle, and depending on the sign of the dissipation rate parameter, the other three comprise either two sources and one sink or one source and two sinks. We also examine the domains of attraction of the sinks and present an analysis of their stability to initial conditions and model parameters.