from now): then, approaches close to the Hill sphere of Venus are possible (0.0075 AU, nearly 19,500 yr from now).

Model 2
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Our previous calculations clearly indicate that, in the case of 2002 VE$_{68}$, encounters with the Earth as close as 0.002 AU are possible. This is 0.8 times the distance between the Earth and the Moon. Almost certainly, the dynamical role of our natural satellite on the outcome of these close encounters cannot be neglected. In order to further investigate this claim, we will repeat the calculations including the perturbations by the eight major planets and the Moon on 2002 VE$_{68}$ for the nominal orbit in Table \[elements\]. Now, the Earth-Moon system is resolved as two separate bodies and assumed to be made of point mass objects; the only force acting between them is Newtonian gravitation, i.e. tidal dissipation is neglected.

Figure \[mld2\] confirms that 2002 VE$_{68}$ has been a quasi-satellite of Venus for some time although in these calculations it entered the quasi-satellite phase about 7,000 yr ago not 11,000 yr ago like in Model 1. In the time interval -7,000 to +1,000 yr from now the evolution of the mean longitude difference is quite similar to 