# What is a "synodic frame"? Can one be defined for an elliptical orbit?

A description of a "synodic frame" was one of the main ingredients in this excellent answer to a question about an animation of a funky-looking orbit that several different and distinctly interesting aspects to it.

As part of the background for this question I plotted historical orbit data for SOHO in a rotating frame centered on a point in space that is associated with the Sun-Earth L1 libration point. I say associated with because in real-world orbits, and not just circular-restricted ones, L1 is more of a "Gestalt" (or "area") than a specific, mathematically well-defined point.

If you look at the lengths to which I had to go to explain the frame of this plot, you can see that I could have used a better term, if one exists.

So here I'd like to ask both What is a "synodic frame"? so we could have a great answer and useful definition to link back to in the future. But I'd like to also ask Can one be defined for an elliptical orbit?

Here's the plot from the question where the math is described. It is the orbit of SOHO shown in a home-grown coordinate system. Note that the "Gestalt", faux-L1, or "instantaneous-L1" red point on the left is fixed in this frame, and the position of the Earth as a function of time, the blue elongated sausage on the right, defines only the axis.

• A comment, since I know just enough on this subject to be dangerous (and dangerously wrong). Yes, it's possible. Properly defining a synodic frame in the case of elliptical orbits a key aspect of the elliptical restricted three body problem, or ER3BP for short. There are various approaches. One is to scale distance and time non-linearly so as to make the orbit circular. This results in rather peculiar fictitious forces that can be treated as perturbations in the case of small eccentricity. Commented May 20, 2017 at 8:58
• @DavidHammen this is an interesting comment. When reading published literature last year, I chose not to download any of the ER3BP papers I came across for no rational reason except that I thought I should "skip that" for now out of concern I'd get confused. Now perhaps it's time to give those a second look. I hadn't thought about it, but now it's obvious, that's exactly what I should read. The primary question here is still "What is a “synodic frame”?" If you or anyone are interested in simply answering that and mentioning existence of ER3BP, that would certainly be a sufficient answer here.
– uhoh
Commented May 20, 2017 at 10:00