# Do Lagrangian points apply to eccentric orbits or binary systems?

The assumptions about the Lagrangian points being stable (...in the traditional meaning of the word: not moving around; they are unstable in the mathematical sense, except for arguably L4, L5) that I've commonly encountered are:

• a pure 2-body system, with any satellites that sit at these points being of negligible mass.

• the central body is vastly more massive than the orbiting one

• the orbiting one moves in a circular orbit.

I wonder though, do they remain reasonably stable - allow for a satellite to sit there with only minimum of station-keeping, in cases of:

1. the planet's orbit being eccentric, not a perfect circle
2. The two massive bodies being of comparable mass, orbiting a common barycenter (though both in circular orbits). Specifically, how would they morph/move?

bonus: what about a ternary system, where a distant planet orbits something like a contact-binary star? Does the idea of Lagrangian points assume uniformly round celestial bodies?

• For 2), there's this funny looking constant. L4 and L5 are stable if m1/m2 is greater than (25 +3*sqrt(69))/2, approximately 24.96. As for how they move, L4/L5 and and the two bodies always forms a equilateral triangle – SE - stop firing the good guys Sep 10 '18 at 10:02
• apparently @DavidHammen knows just enough about things like this to be dangerous. – uhoh Sep 10 '18 at 11:17
• Somewhere there is also an answer or a comment by @DavidHammen about the stability of a point midway between two stars, perhaps in Astronomy SE, and there might be something there that relates to Lagrange points. dunno... – uhoh Sep 10 '18 at 11:42
• @uhoh: I'm quite sure for circular orbits, similar masses still have L1, L2, L3 of old, but as masses change, L1 shift towards the body that gets lighter, and L2 of one body becomes L1 of the other. Can't quite picture how L4 and L5 would behave. – SF. Sep 10 '18 at 11:46
• As am I. There are a lot of questions in your question! I'll add another answer to my answer. – uhoh Sep 10 '18 at 11:48

1. the planet's orbit being eccentric, not a perfect circle

Apparently @DavidHammen knows just enough about things like this to be dangerous. I can't speak to the topic mathematically, but in the figure the little blue cigar is the Earth oscillating towards and away from the Sun as it moves in its slightly elliptical orbit, and the cool looking orbit is in this case SOHO, who's been in a halo orbit for more than 20 years, faithfully executing it's station keeping maneuvers towards or away from the Sun, except when it didn't (see Roberts 2002 linked there).

The near-rectilinar halo orbit of the proposed, future stargate err... ,wormhole, err... gateway will be a (near-rectilinear) halo orbit (1, 2 and answers therein) around the Earth-Moon L1 and/or L2, and the eccentricity of that is much larger than that of Earth's heliocentric orbit ($\epsilon$=0.055 vs 0.017).

I always get my science fictions mixed up. (Space Force!, Rocket Racing!, etc.)

Also, Chang'e-4's radio link to the Earth Queqiao relay satellite will be a the Earth-Moon L2.

So for significantly elliptical systems, yes. For very significantly elliptical systems, someone else will have to answer about the ER3BP, and of course, someone has already asked Are there any natural circular orbits? to begin with.

1. The two massive bodies being of comparable mass, orbiting a common barycenter (though both in circular orbits). Specifically, how would they morph/move?

In this answer I show just how to calculate the positions of L1 and L2 for two masses. I'll try to make a plot of how the two points move as the ratio of the two masses changes, that will take a half-day, need to get a stable 10-20 before looking at the stable points.

GIF: SOHO orbit, data from Horizons, plot from here below left: Top-down view, eleven years. right: View from the side, one year. (Sun to the left) James Webb Telescope prototype orbit, data from Horizons, plot from here