Neptune and Pluto orbits

Here you see the orbits of Neptune and Pluto. Pluto is sometimes closer to the Sun than Neptune but its inclination and relative position in orbit keep it from coming too close to Neptune.

If Pluto's closest approaches to Neptune begin to get too close from a westerly direction then Neptune's gravity will accelerate it into a higher orbit with a longer period so the close approaches begin to move further away. The opposite happens if the approaches get too close from a westerly direction. The net is Pluto "librates" around a period that is exactly 2/3 that of Neptune.

Neptune and Pluto strobe

This is a "strobe" view of the planets. The picture is painted every 30089.5 days which is half Neptune's orbital period. So you see Netune pop up twice, once on each side of its orbit.

Pluto pops up three times due to the 3:2 resonance. But you can see that both objects leave short trails as they librates around their average positions. 

 

Neptune and Pluto have orbits that are in 2:3 resonance. That means Pluto goes round the Sun twice for every three times Neptune goes round. Pluto's orbit is quite inclined and eccentric so sometimes it is closer to the Sun than Neptune.

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The simulation includes Solar System planets (system barycentres) from Mars to Pluto. Yoshida 6th order integration, variable step time.