RIP Kepler, how shall we call your orbit? Does this cyclic flip-flop process have a name?

Question

So long, and thanks for all the Fish Planets

The captions in the NASA Ames Research Center video What Will Happen to NASA’s Kepler Spacecraft? read as follows:

• NASA’s Kepler space telescope found thousands of planets outside our solar system. With its mission completed, the telescope will remain 94 million miles away in an orbit trailing Earth.

• Kepler is traveling in a somewhat larger, slower orbit than Earth. Over time it will trail farther and farther behind.

• By 2060, Kepler will fall so far behind that Earth will almost catch up to it.

• As Earth approaches, its gravity will tug at the telescope and send Kepler into a closer, faster orbit around the Sun.

• In its closer, faster orbit, Kepler will slowly speed ahead of our planet.

• In 2117, the telescope will almost catch up to Earth from behind. Earth’s gravity will tug on the telescope as before, but this time pull it back into the wider, slower orbit.

• For the foreseeable future the pattern repeats as Kepler is tugged inside and outside Earth’s orbit. The telescope never comes closer than a million miles to Earth — more than four times the distance from Earth to the Moon.

This suggest that roughly every 108 years Kepler's orbit will complete one full cycle, spending the first half, or one synodic period (also here) in a heliocentric orbit higher and slower than Earth's, then another synodic period in a faster, lower one.

Question: RIP Kepler, how shall we call your orbit? Does this cyclic flip-flop process have a name?

animated GIF:

Answer 1

Kepler's is a horseshoe orbit:

A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body (such as Earth). The orbital period of the smaller body is very nearly the same as for the larger body, and its path appears to have a horseshoe shape as viewed from the larger object in a rotating reference frame.

The loop is not closed but will drift forward or backward slightly each time, so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'.

Answer 2

As Russell Borogove already noted, the NASA video in your question describes a classical horseshoe orbit. Whether that's a correct description of Kepler's actual orbit is another matter, which I will address below.

However, in any case, the animation shown starting at about 0:27 in the video does not illustrate a typical "horseshoe bounce" interaction. As noted in the comments, the animation shows the Earth and the spacecraft entering the frame side by side, with the same true anomaly, which is impossible for a horseshoe orbit.

In fact, it's clear that the animation is physically inaccurate in other ways too, like in its depiction of the Moon's orbit. It's hard to say whether the animation actually shows any physically realistic orbital interaction at all, or whether it's just an animated "artist's sketch" with the Earth and the Kepler spacecraft moving "on rails" along arbitrary non-physical paths.

A (probably) more accurate animation (created by Tony Dunn / Orbitsimulator.com) of Kepler's orbit can be found as an illustration to this seti.org article:

As this animation shows, the orbit indeed features some horseshoe-like behavior, but it's also clear that it's not a "pure" horseshoe orbit. In particular, the eccentricity of Kepler's orbit causes it to trace a looping corkscrew path when viewed from a reference frame co-rotating with the Earth, instead of a clean horseshoe.

Also, while the animation shown above does feature a nice textbook "bounce" during the ~2060 encounter, the next encounter around the year 2110 instead shows the spacecraft passing the Earth and interacting with it, but continuing in a < 1y period orbit. Presumably, this is also due to the eccentricity (and possibly the inclination) or Kepler's orbit complicating the situation and causing the dynamics of the encounter to depend on the relative phases of Earth and Kepler in their orbits.

In any case, this is contrary to the NASA video, which claims that there should be another "bounce" around 2117, and that this process should continue "for the foreseeable future". At this point, I cannot definitely say which of these contradictory predictions is correct, although personally, on the balance of the evidence, I'd be more inclined to trust the Orbitsimulation.com animation, if only because it seems to make fewer simplifications and "artistic liberties". Of course, I cannot even completely rule out the possibility that both are wrong.