Stability of Distant Satellites of Giant Planets in the Solar System

We conduct a systematic survey of the regions in which distant satellites can orbit stably around the four giant planets in the solar system, using orbital integrations of up to $10^9$ yr. In contrast to previous investigations, we use a grid of initial conditions on a surface of section to explore phase space uniformly inside and outside the planet's Hill sphere (radius $r_{\rm H}$; satellites outside the Hill sphere sometimes are also known as quasi-satellites). Our confirmations and extensions of old results and new findings include the following: (i) many prograde and retrograde satellites can survive out to radii $\sim 0.5r_{\rm H}$ and $\sim 0.7r_{\rm H}$, respectively, while some coplanar retrograde satellites of Jupiter and Neptune can survive out to $\sim r_{\rm H}$; (ii) stable orbits do not exist within the Hill sphere at high ecliptic inclinations when the semi-major axis is large enough that the solar tide is the dominant non-Keplerian perturbation; (iii) there is a gap between $\sim r_{\rm H}$ and $2r_{\rm H}$ in which no stable orbits exist; (iv) at distances $\gtrsim 2r_{\rm H}$ stable satellite orbits exist around Jupiter, Uranus and Neptune (but not Saturn). For Uranus and Neptune, in particular, stable orbits are found at distances as large as $\sim 10r_{\rm H}$; (v) the differences in the stable zones beyond the Hill sphere arise mainly from differences in the planet/Sun mass ratio and perturbations from other planets; in particular, the absence of stable satellites around Saturn is mainly due to perturbations from Jupiter. It is therefore likely that satellites at distances $\gtrsim 2r_{\rm H}$ could survive for the lifetime of the solar system around Uranus, Neptune, and perhaps Jupiter.