Title:

Finitetime stability criteria for sunperturbed planetary satellites

The inability of the c2H stability criterion in the general threebody problem to guarantee Hill (or hierarchical) stability in the case of a planetary satellite perturbed by the Sun when the planet's orbital eccentricity is nonzero, leads to the search for a finitetime stability criterion applicable to such cases. Instead of looking for a stability criterion that guarantees stability for all time, a search is made for one which is valid for a finite length of time and which provides an estimate of that finite time. The finitetime stability method involves applying a series of increasingly less pessimistic stability criteria that are each valid for finite lengths of time. The successive levels of the finitetime stability method are based on the natural periodic cycles found in the planetsatelliteSun system. At each level the stability criteria method takes the most pessimistic viewpoint. Choosing the eccentricity to be the best indicator of an approaching unstable situation, it assumes that the worst possible change in the satellite's eccentricity over the specified cycle is added on to the satellite's eccentricity every period of that cycle. When the eccentricity accumulates to some arbitrarily chosen upper limit, the system is taken to be approaching an unstable situation. The time required for the eccentricity to reach this upper limit is then a measurable minimum lifetime for the satellite system. The finitetime stability method is developed for both a circular and elliptical coplanar restricted threebody model of planetsatelliteSun systems. Applications of the method to the main satellites of Jupiter, Saturn and Uranus produce minimum durations ranging from 1 x 10e6 to 1x10e9 years for Jupiter's satellites, 7x10e5 to 1 x 10e10 years for Saturn's satellites and 1 x 10e9 to 9x10e11 years for Uranus's satellites. Extension of the finitetime stability method to include the elliptic coplanar restricted threebody model for planetary satellites, produces minimum durations that are similar to the equivalent results for the circular case, only slightly smaller. The failure of the c2H criterion in the general threebody problem to guarantee the stability of any of the satellites found in the solar system when the eccentricity of the planet is included in the problem, suggests that the c2H criterion is far too stringent a test for most of the real cases of interest in the solar system. Application of the finitetime stability method failed to provide useful results for several highly Sunperturbed outer satellites, such as the Earth's Moon, Jupiter's outer satellites and Saturn's Phoebe. In the case of the Earth's Moon, a search is made for a larger natural period which will produce more reasonable results. A description is given of certain historically known cycles associated with highnumber near commensurabilies among the synodic, anomalistic and nodical lunar months and the anomalistic year. In particular, the properties of the Saros cycle are studied. The Saros is a period of 6,585. 32 days or approximately 18 years and 10 or 11 days, depending on the number of leap years in the interval. The Saros has been known since Babylonian times as the time that elapses between successive repetitions of a particular sequence or family of solar and lunar eclipses. It is a cycle formed by highnumber commensurabilities between the synodic, anomalistic and nodical lunar months. Using eclipse records and the JPL ephemeris, any dynamical configuration of the EarthMoonSun system (within the framework of the main lunar problem) is shown to repeat itself closely after one Saros period. The role played by mirror configurations in reversing solar perturbations on the lunar orbit is examined and it is shown that the EarthMoonSun system moves in a nearly periodic orbit of period equivalent to the Saros. The Saros cycle is therefore the logical base period to use in the application of the finitetime stability method to the lunar problem. The Saros cycle is also the natural averaging period of time by which solar perturbations can be most effectively removed in any search into the long term evolution of the lunar orbit. The Saros cycle, with its ability to reverse solar perturbations, may have relevance to the stability of any system which contains a saroslike cycle. Since tidal evolution affects the periods which form the commensurabilites within the Saros cycle, the Moon's orbit may not have had in the past and possibly will not have in the future a sarostype cycle to cancel solar perturbations. Also studied therefore is the probability that a dynamical threebody system will contain a saroslike cycle (ie that a set of commensurabilities between the three periods can be found to within a given accuracy and with integer multiples of less than a given upper limit). Unfortunately, the Moon has only about a 25% chance of finding a saroslike cycle within its orbital dynamics. A possible sequence of saros cycles that the Moon's orbit might evolve through is calculated, if only tidal friction is considered.
