# Roughly how many kinds of closed or periodic orbits are there in the circular restricted three-body problem?

Question: Roughly how many kinds of closed or periodic orbits are there in the circular restricted three-body problem? Has anyone made an attempt to enumerate and classify/categorize them?

Notes: If there is a distinction between closed and periodic required to answer the question, you can choose either one. It is possible that an answer can be found in references contained in this answer.

The Circular, Restricted Three Body Problem or CR3BP or CRTBP is the mathematical problem where the orbit of a third object of negligible mass responds to the gravity field of two massive objects in circular orbit around a their center of mass.

Commonly people think of the two bodies as the Sun and Earth, or the Earth and the Moon in the approximation that there are no other bodies and these two are in circular orbits, and the third body is a spacecraft or small asteroid of negligible mass. Early work by the famous mathemagician Joseph-Louis Lagrange discussed the five equilibrium points or Lagrange Points where the acceleration on the third body is zero in the rotating frame.

The CR3BP is a purely mathematical problem and is helpful as a starting point for thinking about orbits of small bodies in realistic scenarios where the effect of two bodies dominate, such as a sun-planet or planet-moon system.

Most orbits of the third body in the CR3BP are chaotic and non-repeating (non-periodic or non-closed). However there are certain cases where the orbit are. After a certain period of time, the body will pass through its exact starting point with the same exact velocity vector and go on to repeat the same path. This is where the depth and complexity of the problem really begins.

See this answer to What sort of orbital elements are used to describe halo orbits? and have a gander at Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem and then answers to What sort of orbital elements are used to describe halo orbits?

• Does the ratio of mass of the two first objects influence the possible orbits? Orbits that may be calculated only by numerical simulation and not by analytical solution are very hard to count and their periodicity relies only on the numeric calculation. Are there newer analytical solutions not from Lagrange?
– Uwe
Jun 5 '18 at 13:39
• @Uwe purely numerical, and yes indeed some might exist for some ratios but not others. It's quite a beautiful mess!
– uhoh
Jun 5 '18 at 13:46
• It should be noted that every orbit in a pure three body problem is bound unless it results in a collision ('escape velocity' doesn't really exist for a system of 3 bodies because gravity has infinite range). And consequently, all orbits (even hyperbolic or parabolic orbits) must be either periodic or chaotic. But the distinction between the two is completely arbitrary: formally speaking, periodic orbits in a three body problem are chaotic. So the short answer to this question is that no orbit is periodic or closed in any three body problem. Aug 13 '20 at 22:59
• @WilliamMiller please consider posting that as an answer! I'd like to see "no orbit is periodic or closed in any three body problem" supported, and an answer post will provide the room to do so.
– uhoh
Aug 13 '20 at 23:55