Stability of the distant satellites of the 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 10⁹ 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 _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: (1) many prograde and retrograde satellites can survive out to radii 0.5r _H and 0.7r _H, respectively, while some coplanar retrograde satellites of Jupiter and Neptune can survive out to r _H; (2) stable orbits do not exist within the Hill sphere at high ecliptic inclinations when the semimajor axis is large enough that the solar tide is the dominant non-Keplerian perturbation; (3) there is a gap between r _H and 2r _H in which no stable orbits exist; (4) at distances ≳2r _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 10r _H; (5) 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 ≳2r _H could survive for the lifetime of the solar system around Uranus, Neptune, and, perhaps, Jupiter.