Spin orbit resonance cascade via core shell model. Application to Mercury and Ganymede

Abstract

We discuss a model describing the spin orbit resonance cascade. We assume that the primary has a two-layer (core-shell) structure: it is composed by a thin solid crust and an inner and heavier solid core that are interacting due to the presence of a fluid interface. We assume two sources of dissipation: a viscous one, depending on the relative angular velocity between core and crust and a tidal one, smaller than the first, due to the viscoelastic structure of the core. We show how these two sources of dissipation are needful for the capture in spin-orbit resonance. The crust and the core fall in resonance with different time scales if the viscous coupling between them is big enough. Finally, the tidal dissipation of the viscoelastic core, decreasing the eccentricity, brings the system out of the resonance in a third very long time scale. This mechanism of entry and exit from resonance ends in the $1:1$ stable state.

Figures (4)

• Figure 1: Geometry of the system. The triaxialities of core and crust have been exaggeratedly increased to make them graphically appreciable.
• Figure 2: Graphical representation of the condition that ensure the capture in resonance: the energy dissipation |$\Delta E_{eff}$| between two maxima of the potential is bigger than the corresponding potential variation |$\Delta V_{eff}$|. The scale on the $y$-axis has been voluntarily increased.
• Figure 3: Numerical solution for the system with initial conditions $(\gamma; v_{\gamma}; \eta; v_{\eta}) = (0.1; 1000\,\frac{1}{y}; 0.1; 50\,\frac{1}{y})$, i.e. when $v_{\eta}$ does not satisfy condition \ref{['eq.condition_existence']}. In the picture on the left we have plotted $v_{\gamma}$ vs time for $100$ time steps. As we can see, after about $30$ time steps $v_{\gamma}$ reaches the value of $v_{\eta}$ and oscillates around it. In the picture on the right we have plotted $v_{\eta}$ vs time for $100000$ time steps. As we can see, $v_{\eta}$ remain almost constant: indeed he decreases very slowly on a very long time scale.
• Figure 4: Numerical solution for the system with initial conditions $(\gamma; v_{\gamma}; \eta; v_{\eta}) = (0.1; 1000\,\frac{1}{y}; 0.1; 5\,\frac{1}{y})$, i.e. when $v_{\eta}$ satisfies condition \ref{['eq.condition_existence']}. In the picture on the left we have plotted $v_{\gamma}$ vs time for $200$ time steps. As we can see, after about $50$ time steps $v_{\gamma}$ vanishes and oscillates around zero. In the picture on the right we have plotted $v_{\eta}$ vs time for $10^7$ time steps. As we can see, $v_{\eta}$ vanishes but on a much longer time scale (on the order of $10^6$ iterations).