# Do horseshoe orbits have anything to do with Lagrange points? Do words fail us here?

I said

(2010 SO16 is associated with Lagrange point L3 but wanders so far behind and ahead of it that the orbit is called "horseshoe"...

Not really. L3 is unstable. Horseshoe orbiters are in effect "alternating trojans" that switch between L4 and L5, with L3 as a transit point.

All of this breaks down in real solar systems with elliptical orbits and many perturbing bodies, but let's constrain ourselves to CR3BP rules

• two bodies have substantial mass (Sun, Earth) and 2010 SO16's mass can be ignored.
• Sun and Earth have circular orbits around a common center of mass
• all motion is in one plane, it's a 2D problem.

Questions:

1. are there closed, periodic 2D planar orbits in the CR3BP that are good models for horseshoe orbits?
2. can we say that horseshoe orbits "associated" with any of the Lagrange points at all, or does this kind of language fail us when applied to horseshoe orbits?
3. is either of us right? or both? or neither?

note: I'm not looking for opinions or "ways of looking at it". If there is a solid, supportable way to answer, hopefully with a little scholarly, authoritative sourcing, that will be great. But for the purposes of this question just qualitative insights or another way to look at it is's won't be so helpful in this case. Thanks!

• Or could you say that "horseshoe" orbits are extreme halo orbits around L3? Mar 17, 2019 at 15:10

@Diane’s answer to the question Ordering of the Lagrange points describes how different so-orbital situations are connected with one another. The curves drawn there represent "zero-velocity" curves in the co-rotating frame. These are not the true orbits; but they serve as bounds to the actual orbits. They may also approximate orbits that remain close to the reference planet's orbit and thus have low orbital velocities relative to the corotational frame.

At a low energy relative to the co-orbital frame the zero-velocity curve consists of three branches, one "inner" branch orbiting the Sun, another "moon" branch orbiting the planet, and the "outer" branch orbiting both bodies. When we increase the energy, which corresponds to decreasing the Jacobi constant JC, the curves collide, merge and split again to give the various co-orbital configurations. In order if increasing co-orbital frame energy:

1. The inner and moon branches collide at L1, both in the zero-velocity approximation and in the exact orbits (the Lagrange points are, of course, true zero-velocity points). The branches then merge to give a quasi-satellite configuration.

2. The quasi-satellite curve next meets the outer branch as L2, and another merger takes place. This is the horseshoe orbit configuration.

3. The inner and outer loops of the horseshoe collide at L3 and the horseshoe splits into two trojan-type orbits, one surrounding each of the remaining Lagrange points L4 and L5.

• Thanks for your answer, I'll take some time and give it a thorough read. In the first paragraph, I'm uncomfortable with zero-velocity equi-pseudo-potential curves being 'close approximations of orbits'. Is this just space lore, or is there an authoritative source that can be found to back this up?
– uhoh
Mar 19, 2019 at 1:34
• As long as you are staying close to the reference planet's orbit, which by definition is stationary in the frame, then relative orbital velocities in the true orbits are slow (unless you are also close to the planet itself, which introduces its own gravitational acceleration). That's when ZVC's get "good". Mar 19, 2019 at 2:09
• Is there then an authoritative source you can cite for this, or a calculation you can do or link to that shows a periodic orbit following a zero velocity curve over one complete period (not just a little section of it)? I don't believe this is true, but I will be very happy to find out otherwise!
– uhoh
Mar 19, 2019 at 2:18
– uhoh
Mar 19, 2019 at 2:24
• yes indeed Section 4.1.1 (top of page 64) found here says: The general shape of a simple horseshoe orbit resembles that of a ZVC corresponding to a value of Jacobi constant between that of CL2 and CL3.
– uhoh
Mar 22, 2019 at 1:37

There are good descriptions of horseshoe orbits here: https://engineering.purdue.edu/people/kathleen.howell.1/Publications/masters/2011_Howsman.pdf