with Fig. \[distance2\] for Model 2 although close encounters with Venus after the end of the quasi-satellite phase tend to be more distant; however, encounters as close as 0.002 AU are observed about 18,000 yr into the future. As in the previous case, encounters with Mercury as close as 0.0025 AU are possible. In contrast, the closest encounters with the Earth are now at 0.006 AU but still well within its Hill sphere.

Discussion
==========

The asteroid 2002 VE$_{68}$ follows a rather eccentric orbit ($e\approx0.4$). In general, minor body trajectories crossing the paths of one or more planets are rapidly destabilized by scatterings resulting from close planetary approaches. The lifetime of the orbits of such objects can be relatively short. But this is only true if the orbital inclination is small, 2002 VE$_{68}$ moves in a relatively highly inclined orbit ($i \approx 9^{\circ}$). In the Solar System and for a minor body moving in an inclined orbit, close encounters with major planets are only possible in the vicinity of the nodes. The distance between the Sun and the nodes is given by $r = a (1 - e^2) / (1 \pm e \cos \omega)$, where the “+” sign is for 