How do tadpole, horseshoe co-orbital states arise and how are they stable?

Question

A recent paper in Nature "Planetary science: Reckless orbiting in the Solar System" (Morais & Namouni, 2017) presents the following series of four co-orbital states:

While I understand the development of the quasi-satellite and trisectrix states, I am unable to inuit the development of the tadpole and horseshoe co-orbital states.

From explanation provided in the article:

The tadpole and horseshoe shapes arise because the planet's gravitational attraction alters the body's orbital path — the body goes through a cycle of catching up with the planet and falling behind, seeming to change direction from the perspective of the planet.

How, when approaching the planet, does the body "fall behind" instead of continuing to accelerate toward the planet?

The only means that comes to mind is if the planet orbits faster than the body, but then I imagine it would

1. No longer be considered co-orbital (because it would have a different orbital period), and
2. Would eventually be overtaken by the planet.

How, then, do these first two co-orbital states work?

Citation: Nature 543, 635–636 (30 March 2017) doi:10.1038/543635a

Answer 1

If the body is in front of the planet (relative to the planet's orbital motion) and a little further from the sun, it will orbit the sun slightly slower than the planet.

As it is slower, the planet will slowly catch it up. (It takes many "years" for the planet to get close to the body.)

As the planet catches up with the body, the gravitational effect of the planet will tend to pull back on the body, this causes the body to lose energy and begin to fall slighly towards the sun.

The slight fall towards the sun cause the body to move to a lower orbit (relative to the sun) and so speed up. It appears paradoxical, but it the fact that the gravitational drag of the planet on the body actually causes it to speed up.

The body speeds up and so begins to move away from the planet.

When the body is makes an angle of more than 60 degrees from the planet, the gravitational pull of the planet now has the opposite effect. It combines with the pull of the sun to cause the body to accelerate, and (apparently paradoxically) this acceleration causes the body to move to a more distant orbit, and slow down.

There are two things key to understanding this. First remember that the diagrams you show are drawn with a fixed planet. I.e. the "camera" is rotating at the same rate as the planet takes to orbit the sun, so the planet appears fixed. If you don't rotates the camera, you would see that the body actually has a nearly elliptical orbit, relative to the sun. Secondly you need to grasp the idea that, in orbit, drag doesn't slow you down. Drag causes you to fall towards the sun and speed up.

How did bodies get into such orbits? Basically by luck. They happened to have orbits that were close to Earth's orbit, and by a lucky effect of the gravity of the other planets they were dislodged from their regular orbit of the Sun to a tadpole orbit. Once there, they can remain in their orbit for tens fo thousands of years. Even so, only one object is known to exist in a tadpole orbit relative to Earth.

Answer 2

How, when approaching the planet, does the body "fall behind" instead of continuing to accelerate toward the planet?

This is fundamentally the gravity assist problem. In the 2 body system, the smaller object falls towards the Earth, accelerates, misses, then flies away from the earth giving back the velocity in flying away that it added flying towards. The net velocity change relative to the Earth is close to zero during the pass.

In the 3 body system, relative to the sun, that all changes. The asteroid effectively adds or removes velocity relative to the sun. If it slows down, it falls closer to the sun, if it speeds up, it moves away.

The horseshoe orbit is a relatively perfect dance where it does this back and forth, crossing the Earth's orbital path for dozens or hundreds, maybe even thousands or oribts, looking like a horseshoe from Earth's frame of reference. From the sun's frame of reference, the asteroid can't make up it's mind if it wants to be closer than the Earth or further than the Earth.