# This orbit looks wrong near a Lagrange point. Is it?

On a completely unrelated forum, i came across the following graphic:

The orbit seems wrong to me, especially the first curve. From the initial trajectory, I would expect the orbit to have been in a clockwise direction, or it to have been a slingshot instead of a capture. What a wouldn't have expected is what appears to be an attraction by L1. As I understand it, a Lagrange point is like a 'hill' in spacetime (as opposed to a gravity well), and so the apparent effect should be an acceleration away from the point.

Is the orbit shown in the graphic wrong, or is my understanding of orbital mechanics lacking, having only been influenced by KSP?

If the trajectory is incorrect, what should it look like?

• This is the trajectory of J002E3. Correctness-wise this should be real. – Mys_721tx May 18 '17 at 12:43
• @Mys_721tx Worthy of an answer--the only expansion needed would be a side by side of the original and modified animations. – called2voyage May 18 '17 at 13:04
• @called2voyage yes. I would hope for some explanation of what I might be getting wrong as well though, – Baldrickk May 18 '17 at 13:05
• @Baldrickk Yes, that would be ideal. I just wanted to point out that Mys_721tx doesn't have to have those details to make an answer. Comments are temporary. – called2voyage May 18 '17 at 13:09
• To my knowledge, L1 is not a hill, but a saddle. Can some elaborate on that? Does that (partially) help to understand the trajectory? – Franky May 19 '17 at 5:42

Is the orbit shown in the graphic wrong, or is my understanding of orbital mechanics lacking, having only been influenced by KSP?

It's not an either-or question.

The graphic is "wrong" from the perspective of an Earth-centered inertial frame. That graphic instead uses a synodic frame, a frame that rotates with the Earth's orbit about the Sun. You can tell that this is the case by the fact that the L1 point along with the Earth are fixed. In the roughly one year interval shown in that animated graphic, the L1 point would have made a full revolution about the Earth from the perspective of an ECI frame. The rotation of the synodic frame is largely what's responsible for the apparent spirograph behavior of J002E3 in the graphic. (Other than label changes, the graphic is correct, by the way.) The orbit would look considerably more mundane from the perspective of an ECI frame.

This is where your KSP-understanding of orbital mechanics steers you wrong. While the orbit wouldn't have those neat petals from the perspective of an ECI frame, it would still look markedly non-Keplerian. The object's ECI velocity would make the object appear to be on a hyperbolic trajectory for much of each orbit. Yet it mysteriously turns around and orbits, six times, before escaping. KSP cannot display this behavior because it uses the patched conic approximation.

So why use this weird rotating frame?

This is the circular restricted three body problem, with the added twist of a fourth body, the Moon. I'll ignore the Moon. The circular restricted three body problem asks about the behavior of a body of negligible mass (e.g., J002E3) under the gravitational influence of two larger bodies orbiting circularly about one another. The term "restricted" means the third body's mass is so small that it essentially doesn't perturb the orbits of the two larger bodies.

The synodic frame turns out to be be very useful in analyzing the three body problem. While the third body's energy and angular momentum are not conserved quantities in this frame, a new quantity, the Jacobi integral, is conserved, in this frame.

Objects with little energy ($C_j>4$), depicted in the upper left sub-image, are restricted to orbiting the smaller mass (e.g., the Earth) or the larger mass (e.g., the Sun). There's a forbidden region surrounding the smaller that objects with a large Jacobi integral cannot enter. Objects in low Earth orbit are stuck there, and objects will sufficiently low energy outside the forbidden zone can't hit us unless perturbed by something else.

Something interesting happens with a Jacobi integral of about 3.9: A keyhole opens up around the L1 point. Objects with sufficient energy can enter the vicinity of the smaller mass (e.g., the Earth). This is exactly what happened to J002E3. It changed from orbiting the Sun to temporarily orbiting the Earth by passing through that keyhole.

Another keyhole, this one about the L2 point, opens up with an even lower Jacobi integral. You can see this in the graphic that shows J002E3's orbit. In late October 2002, J002E3 came close to the L2 point. It didn't have quite enough energy and it didn't come quite close enough, but if it had, it could have escaped six months prior to when it did. Instead, J002E3 had to wait another six months when it's strange orbit brought it close enough to the L1 point that it could finally escape.

It's very difficult to see these behaviors from the perspective of an ECI frame. It's quite easy to see them from the perspective of a synodic frame, once you know what to look for.

As noted by Mys_721tx in comments, this is the trajectory of the object J002E3 -- believed to be the S-IVB upper stage launched with the Apollo 12 mission -- and the animation is accurate.

As David Hammen explains in his (better) answer, the animation is showing a rotating frame of reference centered on the Earth. Because of the rotation, and the fact that J002E3 is initially in a solar orbit, the initial part of the trajectory is misleading. In the solar frame of reference, Earth is moving at high speed toward the top of the image, and at the start, J002E3 is moving faster than it and so is going to pass above/in front of Earth. As it passes L1, though, Earth's gravity begins to work more strongly on it, pulling it backwards/sideways to its trajectory, which slows it down, so it falls behind Earth.

KSP experience doesn't help you with this because it uses patched-conic approximation, where only one body at a time influences the trajectory of a spacecraft. You will notice in KSP that your trajectory changes direction abruptly when a new sphere of influence is entered; this is because of the change in reference frame, not the gravity of the body you're approaching.

• So my problem is that I was assuming a solar reference frame with the 'camera' tracking the Earth, instead of Earth's frame of reference? That makes a lot of sense. I guess if it was the former, the orbit trail should really be trailing off 'behind' the planet and forming a wavy line or corkscrew, instead of loops as it orbits. – Baldrickk May 18 '17 at 14:39
• Yep. Here it's extra-weird, because the moon's track around the earth is a fairly-regular wavy/loopy line in the solar frame, and J002E3 is being bounced around by the moon's gravity as well. – Russell Borogove May 18 '17 at 16:01
• This is not so much an Earth-centered frame as it is a synodic frame, with the view shifted toward the Earth. Notice how both the L1 point and the Earth are not moving. This is a frame that rotates once per year. – David Hammen May 18 '17 at 18:49
• Good point -- I'm afraid I'm not formally educated in this. – Russell Borogove May 18 '17 at 20:43

The original .gif that the graphic from the question was based on can be found on the wikipedia page for J002E3:

As noted by the other two answers, this is the actual path taken by J002E3.

For those who play KSP, there are different options for how the patched conics are displayed. While it cannot accurately model the path, it can show the behaviour shown in the graphic near L1, which is the cause of the confusion leading to this question.