If Geographos is a tidally distorted object, it had to encounter a planet at some time in the past. Not just any encounter will do, however. Tidal forces drop off as the inverse cube of the distance between the bodies, such that distant encounters far outside the planet's Roche limit cause negligible damage to the rubble pile. High velocity trajectories past a planet leave little time for tidal forces to modify the rubble pile's shape. Thus, we need to estimate the probability that Geographos has made a close slow encounter with Earth or Venus.
The orbits of ECOs evolve chaotically. Many of them have orbits which allow them to encounter multiple planets and the terrestrial planet region is crisscrossed with secular and mean-motion resonances ([Froeschlé et al., 1995,Michel et al., 1997]). For these reasons, it is impossible to accurately track the orbital motion of any ECO more than a few hundred years into the past or future. The only way, therefore, to assess the likelihood that Geographos had a planetary encounter in the past is to numerically integrate its orbit with that of many clones, in the hope that broad evolution patterns can be readily characterized. To this end, following the procedure of Michel et al., (1996), we used a Bulirsch-Stoer variable step-size integration code, optimized for dealing accurately with close encounters, to track the evolution of 8 Geographos-like test clones. We integrated the nominal orbit with a=1.246 AU, e=0.335, ; the other clones were defined by slightly changing their orbital parameters one at a time. All of the planets were included except Pluto. Orbital parameters were provided by the JPL's Horizons On-line Ephemeris System v2.60 (Giorgini et al. 1998). Each clone was followed for 4 Myr.
In general, we determined the orbital evolution of the clones to be controlled by two mechanisms: close encounters with Earth and overlapping secular resonances and involving the mean precession frequencies of the nodal longitudes of Earth and Mars's orbits (Michel and Froeschlé 1997, Michel 1997). We found that 5 of the 8 clones (62.5%) had their inclinations increased by these resonances. This trend opens the possibility that these mechanisms could have affected Geographos's orbit in the past and consequently that its inclination has been pumped up from a lower value. Similarly, 6 of the 8 clones (75%) had their orbital eccentricities increased by the and secular resonances with Venus and Jupiter. Lower eccentricities and inclinations in the past imply that close approaches near Earth were even more likely to occur, and to happen at the low velocities conducive for tidal disruption, in agreement with integrations by other groups ([Froeschlé et al., 1995]). Thus, these integrations moderately increase our confidence that Geographos has been stretched by tides in the past.