Dynamical Stability of the Laplace Resonance

Abstract

We analyse the stability of the de Sitter equilibria in multi-resonant planetary systems. The de Sitter equilibrium is the dynamical state of the Laplace resonance in which all resonant arguments are librating. The sequence of equilibria exists all along the possible states balancing resonance offsets and forced eccentricities. Possible additional new-de Sitter equilibria may exist when at least one of the forced eccentricities is large (the paradigmatic case is Gliese-876). In the present work, these families of equilibria are traced up to crossing exact commensurability, where approximate first-order solutions diverge. Explicit exact location of the equilibria are determined allowing us to verify the Lyapunov stability of the standard de Sitter equilibrium and of the stable branches of the additional ones.

Figures (7)

• Figure 1: De Sitter-Sinclair equilibria for the Galilean system: continuous lines correspond to solution \ref{['XE1E']}; dashed lines to the approximate solutions (\ref{['YE1']}-\ref{['XE1']}). The verse of the proximity parameter $\omega$ is reversed. The vertical thick lines give the actual observed value ($\omega = -0.00325$) and the instability threshold ($\omega = -0.00078$) predicted by applying \ref{['LUNST']}.
• Figure 2: De Sitter-Sinclair equilibria for the Galilean system: comparison between the analytical solutions (\ref{['XE1E']}, continuous lines) and the exponential fit (\ref{['XEXP1']}, dashed lines). The color code is the same as in Fig.\ref{['fig:Gali1']}
• Figure 3: De Sitter-Sinclair equilibria for the Galilean system: comparison between the analytical solutions (\ref{['XE1E']}), dashed lines, and exact solutions obtained by a root-finding method of the complete model including the 2nd-order terms in eccentricity (continuous lines). The color code is the same as in Fig.\ref{['fig:Gali1']}.
• Figure 4: Idealised pictures of the Galilean system (top panel) compared with GJ-876 (lower panel). Each snapshot corresponds to a revolution period of Io or planet-c (red dots); in the upper plots Europa (green dots) and Ganymede (blue dots) can be in conjunction with Io one at a time; in the lower plots, planet-b (green dots) and planet-e (blue dots) can together be in conjunction with planet-c.
• Figure 5: New deSitter for the GJ-876 system: $X_1$ solutions with $\lambda = 0$. The vertical thick lines give the actual observed value ($\omega = 0.1159$) and the bifurcation threshold ($\omega = 0.0909$).