for the next 150 yr in a coordinate system that rotates with Venus. It appears to follow a precessing kidney-shaped retrograde path when viewed from Venus over the course of a Venusian year. This result is equivalent to that in Fig. 1 in Mikkola et al. (2004). In Fig. \[mld\] we plot the mean longitude difference of 2002 VE$_{68}$ and Venus: its value currently librates around 0$^{\circ}$, i.e. the object oscillates around the mean longitude of Venus. This figure is equivalent to Fig. 2 in Mikkola et al. (2004). In both Figs. \[qs\] and \[mld\], data from Model 3 are plotted (see details below). We confirm that the minor planet 2002 VE$_{68}$ is a quasi-satellite of Venus, it has remained in its present orbit for thousands of years and will continue trapped in this resonant state for another 500 years.

Model 1
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Here we show the results from our Model 1 that includes the perturbations by the eight major planets on 2002 VE$_{68}$ for the nominal orbit in Table \[elements\]. The mean longitude difference of 2002 VE$_{68}$ and Venus is displayed in Fig. \[mld1\]. As pointed out above, the relative mean longitude currently librates around 0$^{\circ}$. 2002 VE$_{68}$ has 