# 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?

• 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, 2020 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, 2020 at 23:55
• @DialFrost Zhang (2022) is quite a broad and deep review of other people's work, but unlike Doedel et al. (2011) (also here) it doesn't mean to provide an enumeration or list of distinct orbital solutions. Instead it's more of an enumeration or list of approaches. Beautiful work for sure though!
– uhoh
May 8 at 2:28
• @DialFrost "...then don't you have your answer already?" That I posted this question after linking to it suggests that I am not sure the 2011 paper is the final word - is a complete, exhaustive enumeration. In fact, I don't think it even claims to be exhaustive itself, and I'm no expert on this topic.
– uhoh
May 8 at 2:51
• Another 2006 paper that is possibly relevant, at the moment no other finds google.com/url?sa=t&source=web&rct=j&url=https://… May 8 at 6:07

I am 100% not an expert in this topic, but after a while of researching and googling, I've decided to post a really short explanation answer that can expanded on.

The CR3BP describes the dynamics of a body with negligible mass under the gravitational influence of two bodies of nonzero mass, called the primaries, where the primaries move in circular orbits around their barycenter. Let $$(x,y,z)$$ denote the position of the negligible mass body in a rotating barycentric coordinate system, where the $$x$$-axis points from the larger primary, of mass $$m_1$$, to the smaller primary of mass $$m_2$$. The $$z$$-axis is orthogonal to the orbital plane and the $$y$$-axis completes the orthogonal coordinate system.

One major parameter that you need to know is $$\mu$$, where it represents the ratio of the mass of the smaller primary to the larger primary: $$\mu={m_2}/{(m_1+m_2)}$$. The units are chosen so that the distance between the primaries and the sum of the masses of the primaries are equal to 1. There are quite a few equations needed for this problem, for example the equations of motion for the larger and smaller primaries located coordinates $$(\mu,0,0)$$ and $$(1-\mu,0,0)$$ respectively:

$$x^{''}=2y^{'}+x-(1-\mu)(x+\mu)r_1^{-3}-\mu(x-1+\mu)r_2^{-3}$$

$$y^{''}=-2x^{'}+y-(1-\mu)yr_1^{-3}-{\mu}yr_2^{-3}$$

$$z^{''}=-(1-\mu)zr_1^{-3}-{\mu}zr_2^{-3}$$

where $$'$$ denotes a derivative with respect to time, and where:

$$r_1=\sqrt{(x+\mu)^2+y^2+z^2}$$

$$r_2=\sqrt{(x-1+\mu)^2+y^2+z^2}$$

One other thing that you need to know is that these 3 equations have 1 integral of motion, namely energy or Jacobi constant:

$$E=\frac{x^{r_2}+y^{r_2}+z^{r_2}}{2}-U(x,y,z)-\frac{\mu(1-\mu)}{2}$$

where,

$$U=\frac{1}{2}(x^2+y^2)+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}$$

Now, the above 3 equations of motion have 5 equilibria in the orbit plane of the primaries (where $$z=0$$), called lagrange or libration points: If those 3 equations where to rewritten as a first order system (the first derivative with respect to time) in $$R^6$$, then its Jacobian evaluated at the libration points L1, L2 and L3 has 2 pairs of purely imaginary eigenvalues, which gives rise to well-known families of periodic orbits, namely the Lyapunov orbits, L1, L2, and L3, and the Vertical orbits, V1, V2 and V3. This is just one of the conditions to achieve many of the families: Based on this table, there is 16 types of periodic orbit families. I'm not sure if there are more families, but if anyone finds anything you can create an answer or comment here :).

(note: the 1st line is the libration points, they are not orbits but points of equilibrium, the last line has branch points, which also aren't orbits, therefore 18 - 1 - 1 = 16)

Periodic solutions for Earth-moon system (with reference to the table 1): Here is some history and numbers of the families of periodic solutions based on wikipedia:

• In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.

• In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem. This was followed in 2018 by an additional 1223 new solutions for a zero-angular-momentum system of unequal masses.

• In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem. The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".

Sources used:

One way of categorising CR3BP orbits is by pseudo-potential topology, that is, by the conserved quantity in the co-rotating frame of reference, as well as location.

This is a practical, spaceflight-orientent view

The problem, of course, being that most of these aren't orbits in the mathematical sense (which, I get, is what the question is really about). They are just "close enough" that trajectory planners can pretend they are orbits. # 1. Pseudo-potential limited

## 1a. Limited by primary's gravitational field.

Objects that can't cross from the inner region over the L1 neck. These are more or less elliptical orbits, with the secondary's influence modelled as a perturbation

## 1b. Limited by secondary's gravitational field.

Objects similarly trapped around the secondary, not able to cross nether the L1 nor L2 necks.

## 1c. L1 neck crossers

Can be found in the well of both the parent and the secondary, but limited to that.

## 1d. Limited by the centripetal field.

Can similarly not access neither of the inner regions, due to not being able to cross the L2 neck.

## 1e. Quasi-moons

Can cross both the L1 and L2 necks, but with such a low margin that an extended number of revolutions are taken inside either the primary or secondary well before escape.

# 2. High pseudo-potential

## 2a. L4 and L5 Lagrange points.

Orbits in kidney shapes around L4 and L5 on the pseudo-potential peak.

## 2c. Horseshoe orbits

A tadpole low enough to switch lobes over L3, orbiting in a "C" shape in the co-rotating frame.

# 3. Co-linear orbits

## 3b. Lissajous orbits

Quasi orbits around the same points

# 4. Repeat flyby orbits

## 4a. Same point flybys

With arbitrarily small corrections, every flyby of the secondary can be made to encounter the secondary again after some integer multiple of revolutions. This is useful for navigation in a moon system, like Cassini used Titan.

## 4b. Different point flybys

Encountering the secondary at varying locations, like a NRHO or a flyback lunar cycler. Degenerates into a captured moon as the encounters get sufficiently frequent.

• Cool! So by this enumeration method "Roughly how many...?"
– uhoh
May 9 at 7:43
• It's arbitrary enough that I can't think of something better than "four broad categories" May 9 at 8:09