Does the distance to L2 vary?

Discussion of Lagrange point L2 and the JWST seem to be dropping out of the news cycle, so I thought I should ask this question while the topic is still warm.

The distance of L2 for the sun/earth two body system depends on the distance between the sun and the earth (and the masses of these two bodies). Typical popular accounts of this L2 point place it at approximately 1.5 million kilometers from earth.

However, because the earth’s orbit around the sun is elliptical (not circular), it occurs to me that the distance from the earth to this L2 would be closer to the earth at perihelion, and further at aphelion.

Is that correct? If so, what is the range of distances of L2 from the earth, over the course of an earth year?

Does the distance to L2 vary?

Yes, it does. The Earth is in a more or less elliptical orbit about the Sun. I wrote "more or less" because the Moon is a factor, and also because Venus, Jupiter, and the rest of the solar system are also factors.

The circular restricted three body problem (CR3BP) is based on a rotating frame that rotates at a uniform rate so as to keep the locations of the two primary bodies fixed. The third body in the CR3BP is assumed to have a very small mass, so very small that it does not perturb the circular orbits of the primary bodies. (That's the meaning of the word "restricted" in CR3BP.) The CR3BP has five Lagrange points at which the net force is zero. This net force includes the fictional centrifugal force.

It is possible to extend the C3RBP to elliptical orbits. This is the elliptical restricted three body problem (ER3BP). To keep the locations of the two primary bodies fixed, it is necessary to introduce a frame that rotates at a non-uniform rate (the true anomaly rate) and whose length scale pulsates. The math in this rotating / pulsating frame is beyond ugly. That said the five CR3BP Lagrange points have well-defined analogues in the ER3BP.

JPL Horizons has data for 8 Lagrange points:

ID# Name Designation IAU/aliases/other
31 SEMB-L1 Lagrange
32 SEMB-L2 Lagrange
34 SEMB-L4 Lagrange
35 SEMB-L5 Lagrange
3011 EM-L1 Lagrange
3012 EM-L2 Lagrange
3014 EM-L4 Lagrange
3015 EM-L5 Lagrange

"EM" is for the Earth - Moon Lagrange points, "SEMB" is for the Sun - (Earth-Moon barycentre) points. I guess they think L3 is too boring to compute. ;)

"L2" is often called the Sun-Earth L2 point. But it makes more sense to use the Sun - (Earth-Moon barycentre) Lagrange points.

Here's a 2 year plot, with a 1 day time step, created using my range script from this answer. It shows the distance from the SEMB L2 to the Earth body centre, and to the Earth-Moon barycentre. You can see the influence of the Moon on the former.

As you can see, the minimum distance is near the start of the year, and perihelion of the EMB is around the 5th of January.

The Lagrange points are only defined for the circular restricted three body problem (CR3BP) and nowhere else, and so they are only fuzzy, vague concepts when discussed in the context of real world where orbits are elliptical and there are several gravitational bodies present.

There are certainly equations for the distance to L1 and L2 but these only have meaning for circular orbits, you can't put a variable distance into them and then watch the L1 and L2 points "breathe in and out" over the year. So to the question

Does the distance to L2 vary?

The correct answer is "No, they are only mathematically defined when the orbit is circular".

There are still halo-like orbits around L1/L2-like regions into which we can put spacecraft, and we can call those "halo orbits". That's because Earth's orbit is pretty close to circular so the real world orbit is similar to a proper halo orbit from the CR3BP. That's what JWST and several other spacecraft take advantage of.

• The elliptical restricted three body problem (ER3BP) uses a time-varying rotation rate and length scale that makes the two massive bodies remain at fixed positions in the rotating-pulsating frame. The math is rather ugly, but analogs of each of the five Lagrange points for the CR3BP do exist in the ER3BP. Commented Feb 17, 2022 at 14:13
• One of the key tools that is being used for the trajectory design and analysis of NASA's Gateway project is Copernicus. Copernicus can calculate the location of the Earth-Moon Lagrange points as perturbed not only by the elliptical nature of the Moon's orbit but also as perturbed by solar gravity (and other third bodies). Unfortunately, Copernicus is categorized as a U.S. Government Purpose Release, which means ordinary people cannot obtain it. Commented Feb 17, 2022 at 14:29
• @DavidHammen I think that should be elevated to the status of a new and separate answer.
– uhoh
Commented Feb 17, 2022 at 15:19
• You might want to retract this incorrect answer. To be honest, I don't understand the upvotes. Commented Feb 18, 2022 at 9:38
• @DavidHammen I disagree that it's incorrect. I'm betting that if we asked Lagrange or Euler they'd both be on my side. ;-)
– uhoh
Commented Feb 18, 2022 at 10:46

Yes, the sun-L2 and earth-L2 distances must vary over time. The simplest argument is:

• The earth moves in an ellipse
• L2 is defined as the point beyond earth, on a line between sun and earth, where earth and sun gravity balance correctly to create the combined gravitational force that's exactly right for an earth-synchronous orbit
• If earth-L2 stayed the same, then the suns contribution to gravity at L2 would vary. It doesn't seem very likely that it would vary as required for a synchronous movement at L2, and in fact if you crunch the numbers, it would not.
• therefore earth-L2 doesn't stay the same.