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Three-body spacecraft orbits1 are regularly discussed here and from time to time someone will include a pseudo-potential plot from Wikipedia in their explanation.

The discussion often goes south when it tries to reconcile that L4 and L5 sit at potential maxima and yet an object placed there is stable; if you give it a small perturbation or "kick" it will stick around in the general area. It's this kind of stability that allows planets (especially Jupiter) to collect and retain Trojan asteroids associated with their Sun-planet L4 and L5.

Today I saw the "It's Just Astronomical!" video The Parking Spaces of Space: The Lagrange Points and the explanations provided seem unphysical and wrong.

It shows a pseudo-potential surface and some balls rolling on it, and explains how objects will drift towards the Sun, drift towards the Earth, or drift away into the outer solar system!

If "into" were scratched out and changed it to "towards" and it were made clear that this is only the initial motion, this might be salvageable.

But the animations show the balls following paths going well down into those three pseudo-potential "wells" in ways that simply can't happen in reality.

Question:

  1. Why shouldn't we illustrate spacecraft trajectories on top of these static pseudo-potential surfaces?
  2. What will really happen to these three objects?
  3. What will happen to the pseudo-potential surface as soon as they do start moving?

"bonus points" for any sighting of a proper animation that includes a dynamic pseudo-potential plot and an object moving across or on top of it.


This graph shows us the effective potential energy and it tells us how objects will move.

An object who’s orbit starts here will slowly drift towards the Sun.

An object starting here will slowly drift towards the Earth.

And an object starting here will slowly drift away into the outer solar system.

Screen shot from "The Parking Spaces of Space: The Lagrange Points" video Screen shot from "The Parking Spaces of Space: The Lagrange Points" video

Screen shot from "The Parking Spaces of Space: The Lagrange Points" video Screen shot from "The Parking Spaces of Space: The Lagrange Points" video

Screen shots from linked video including closed-caption text; click for full size.


1halo, Lissajous, near-rectilinear halo, etc.

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  • $\begingroup$ Just out of curiosity: do you have a rank-list with points community members gain answering your quizz-like questions? $\endgroup$ – CallMeTom Dec 7 '20 at 6:29
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    $\begingroup$ @CallMeTom no and not interested in that stuff I ask a lot of questions and the idea is to build a collection of good answers to on-topic questions through a collaborative effort. It's all about the answers. $\endgroup$ – uhoh Dec 7 '20 at 6:33
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    $\begingroup$ The narrator says "towards the Sun", "towards the Earth" and "away into the outer solar system". Seems about right. $\endgroup$ – Everyday Astronaut Dec 7 '20 at 7:30
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    $\begingroup$ Worst animation ever! So many errors I wouldn't know where to start correcting them. $\endgroup$ – Everyday Astronaut Dec 7 '20 at 7:32
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What the pseudo-potential surface is: A plot of a 3D function.
What the pseudo-potential surface is not: A hill to make a marble roll around on.

All confusion around this thing comes from making the assumption that it is a regular 3D surface, which will behave just as you expect it to if you were to drop a marble on it.

You can make the same nonsensical assumption on a regular 2D graph. Take the Voyager 2 velocity graph, for instance. You could imagine a marble rolling around on top of it too, drawing all sorts of wildly incorrect physical interpretations from it like "getting stuck in the gravity assist valleys", or "gaining more speed as it rolls downhill from the Sun".

People are used to 2D plots not working like that. What makes it plausible that the pseudo-potential surface should act like that at first glance is that it has some similar properties: "Standing still" requires being on a "flat" spot. Rolling "uphill" will cause you do slow down, going "downhill" will increase your velocity.

Why shouldn't we illustrate spacecraft trajectories on top of these static pseudo-potential surfaces?

We shouldn't? I think it is an excellent to plot repeated flyby trajectories and 3-body orbits. Just make it clear that it's a rotating frame of reference. Rotating frames are unintuitive, but also useful. Perhaps producing some educational videos of rotating frames would be better than making (incorrect) videos about a subject that requires knowledge of rotating frames?

What will really happen to these three objects?

They would move around (or, the two main ones are static), just not like you would expect them to if this was a literal physical well. An object would move according to [set of differential equations] instead of [familiar set of differential equations].

What will happen to the pseudo-potential surface as soon as they do start moving?

Nothing, as this is an idealised circularly restricted three body problem. Perhaps what you are looking for is the actual potential energy plot of a two body system? It's a trumpet-shaped well for the main object, with a smaller trumpet shaped well orbiting it ("moving") for the secondary object. And contrary to the pseudo-potential surface, you can roll marbles around in it!

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  • $\begingroup$ Thanks for your post! I'll dig in and give it some thought over the next day or two. $\endgroup$ – uhoh Dec 8 '20 at 11:55
  • $\begingroup$ er... next week or two... $\endgroup$ – uhoh Dec 23 '20 at 20:19

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