Time analysis for temporary gravitational capture: Satellites of Uranus

Abstract

We study the problem of gravitational capture in the framework of the Sun-Uranus-particle system. Part of the space of initial conditions is systematically explored, and the duration of temporary gravitational capture is measured. The location and size of different capture-time regions are given in terms of diagrams of initial semimajor axis versus eccentricity. The other initial orbital elements - inclination (i), longitude of the node (Ω), argument of pericenter (ω), and time of pericenter passage (τ) - are first taken to be zero. Then we investigate the cases with ω = 90°, 180°, and 270°. We also present a sample of results for Ω = 90°, considering the cases i = 60°, 120°, 150°, and 180°. Special attention is given to the influence of the initial orbital inclination, taking orbits initially in opposition at pericenter. In this case, the initial inclination is varied from 0° to 180° in steps of 10°. The success of the final stage of the capture problem, which involves the transformation of temporary captures into permanent ones, is highly dependent on the initial conditions associated with the longest capture times. The largest regions of the initial-conditions space with the longest capture times occur at inclinations of 60°-70° and 160°. The regions of possible stability as a function of initial inclination are also delimited. These regions include not only a known set of retrograde orbits, but also a new sort of prograde orbit with inclinations greater than zero.