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

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

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

• Just out of curiosity: do you have a rank-list with points community members gain answering your quizz-like questions? Commented Dec 7, 2020 at 6:29
• @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.
– uhoh
Commented Dec 7, 2020 at 6:33
• The narrator says "towards the Sun", "towards the Earth" and "away into the outer solar system". Seems about right. Commented Dec 7, 2020 at 7:30
• Worst animation ever! So many errors I wouldn't know where to start correcting them. Commented Dec 7, 2020 at 7:32

The pseudo-potential/effective potential for the circular restricted 3-body problem is the sum of the gravitational and the centrifugal potential, cf. Fig. 1.

$$\uparrow$$ Fig. 1: The 2D orbital plane with equipotential lines and the 5 Lagrange points. (From Wikipedia.)

The main shortcoming of the linked video is that it fails to mention the role of the Coriolis force. Due to the Coriolis force, a test mass $$m$$ (without propulsion) will drift along (instead of perpendicular to) equipotential lines! ($$\leftarrow$$ This fact alone ensures that the pseudo-potential is still physically relevant, cf. OP's title question.) The drift is often superimposed with a circular motion of angular velocity $$-2\Omega$$, cf. Fig. 2.

$$\uparrow$$ Fig. 2: A possible horseshoe orbit along equipotential lines. $$\star$$ is the Sun, while $$\bullet$$ is Jupiter.

For a derivation, explicit formulas, and more information, see e.g. part III of my Phys.SE answer here and my Space.SE answer here.

TL;DR: The Coriolis force explains the stability of Lagrange points $$L_4$$ & $$L_5$$, cf. e.g. this Phys.SE post.

• Thanks for your answer! It's great when a new user digs in and writes a well-sourced answer to an old question. I'm used to the standard derivation of the circular restricted three-body problem in the rotating frame but it never really sunk in that it's simply the same math we use for the Coriolis force (though it really should have) :-)
– uhoh
Commented Aug 30, 2021 at 23:45
• Here is a YouTube video by Scott Manley from Oct 2021 that gets the physics right: youtube.com/watch?v=7PHvDj4TDfM Commented Dec 30, 2021 at 18:36
• I'm still waiting for someone to plot the full pseudopotential surface which includes the Coriolis term and therefore constantly changes shape as the velocity changes, instead of sticking with a plot of the zero-velocity potential and just talking for four minutes about Coriolis while showing the (boring, unresponsive) gravitational potential surface.
– uhoh
Commented Dec 30, 2021 at 21:03

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!

These are as much comments as answers, but they do provide some different perspectives.
[This has been edited based on helpful pointers from @uhoh. The earlier comment about the total energy of a test particle being conserved was not correct.]

1. Chladni patterns (sand on a vibrating plate).
In the presence of two large masses in circular orbit, the conserved quantity (akin to energy) is the Jacobi Integral. The Jacobi integral, EJ is the sum of the gravitational potential energy plus the centrifugal potential energy plus the kinetic energy (as seen in the rotating frame). The plots of quasi-potential, EP correspond to the sum of the two potential terms and omit the kinetic energy. Since EJ for a small test particle remains constant (conserved), the kinetic energy (~speed-squared) is the difference between the fixed value of EJ and the quasi-potential surface, EP. So, given a value for EJ for the particle, its speed can be found as sqrt(EJ-EP). Unfortunately this is speed not velocity, so it doesn't help establish the trajectory, but it does tell us something about the probability of finding the particle in a particular place. The probability of finding the particle in an elementary volume of space is inversely proportional to its speed (among other things).

This is the basis through which distinct patterns arise in sand on a vibrating plate, as first detailed by Ernst Chladni. The grains of sand incidentally acquire lateral velocities proportional to the magnitude of the vibrations at a particular point on the plate. These velocities are smallest at nodes in the vibrational modes. The probability of finding grains is greatest where their lateral velocities are lowest - i.e. they spend most of their time at the vibrational nodes (image below).

In general, the motion of our test particle will be chaotic, even with only two large masses present. Particles will have their maximum quasi-potential energy and lowest kinetic energy in the rotating frame when they are close to the L4, L5 points. Lowest kinetic energy means lowest velocity. This means that the particle has highest probability of being found close to L4, L5. This is especially true of particles whose value of EJ is exactly equal to the maximum quasi-energy, EP (which is reaches its maximum at L4, L5). Their speed will slow to zero as they approach the Lagrange points and thus they are almost certain to be found there.

2. Simple Harmonic Motion
For situations where the panetary mass is much smaller than the central mass, the L4, L5 points still exist, but the forces from the planetary mass will be very small. In this case, we can consider the test particle to be in a simple Keplarian orbit around the central mass. Assume the orbit of the planetary mass is circular - as will be the L4,L5 orbits. Assume our test particle is in the same orbit as L4 or L5, but slightly perturbed. We will observe it from a rotating frame attached to L4 or L5.

If the perturbation is made to the plane of the orbit (still circular), then the test particle will appear to oscillate up and down (perpendicular to the plane of the planetary orbit) in simple harmonic motion with the same period as the orbital period.

If the perturbation is made to the eccentricity (still in plane), then the test particle will again appear to move with simple harmonic motion around the Lagrange point. In this case, because of the Coriolis effect, the orbit will be an ellipse with axes in the ratio 2:1 (if the perturbation is larger, the ellipse becomes 'kidney bean shaped').

So, viewed from a rotating frame attached to the Lagrange point, very small perturbations from an L4 or L5 orbit will result in a particle making small elliptical orbits around L4 or L5 with the same period. However, these are not Keplerian orbits. They are orbits with simple harmonic motion such as where the central force is proportional to distance (rather than inverse-square) and the potential is quadratic.

From this perspective, there is no equivalent point mass at the L4, L5 points. Instead, locally, they can be considered to be the center of a sphere of constant frictionless mass density (dark matter!) within which these simple harmonic orbits can persist. The required mass density is the mass of the large central object divided by the volume of a sphere that the planetary orbit would enclose, M/[(4/3)pi.R3]. This ensures that the period of these simple harmonic motions will be the same as the orbital period of the Lagrange points

• I really like this insight and the statistical analogy to particles on a Chladni plate! There has been some fairly recent work done in the stability of chaotic three-body motion based on statistical analysis as well, I'll have to go looking to find it. In the mean time, while the kinetic + potential energy sum is a conserved quantity in two-body orbits (double it and it's called $C_3$ or characteristic energy) that doesn't apply to motion of a third massless body in a two massive body system...
– uhoh
Commented Sep 1, 2021 at 22:26
• (i.e. the CR3BP). Instead, the quantity that is actually conserved (in the synodic or rotating frame) is apparently $C_J$, the Jacobi integral. The zero velocity surface (shown in the question) is that expression evaluated at zero velocity. As soon as a particle starts moving, the surface changes shape!
– uhoh
Commented Sep 1, 2021 at 22:34
• @uhoh You are quite right, of course. Voyager would never have left the Solar system if its energy were conserved! Commented Sep 2, 2021 at 3:48
• @uhoh I hadn't seen the Jacobi integral before, but it does seem to be simply the zero velocity surface plus the velocity-squared (as measured in the rotating frame). Commented Sep 2, 2021 at 3:51
• @uhoh there's still something I can't quite reconcile between simple harmonic motion around the Lagrange point vs. its speed being the sqrt(Ej-Ep). I'll try and post it as another question. Commented Sep 10, 2021 at 2:01