Introduction to the 3-Body Problem

A Whole New Ballgame

In the past, this blog has covered the 2-Body problem which is solvable (Kinda) and the general n-body problem which isn’t. We have a lot of insight into how the 2-body dynamics work, but unfortunately, they break down as soon as we get away from any major body. If you want to orbit earth you’ll be fine, but if you want to go to the moon you’re going to have problems. The n-body problem is a much better match for real dynamics, but we have almost no analytical insights into it.
Is there some compromise that would give us a better understanding of space dynamics while still being able to be analyzed? What about if we added one more body to the general 2-Body problem to make it the 3-body problem? That would still be too hard for us to solve analytically, we’d need a few more constraints to analyze it. This post will be the first in a series looking at the circular restricted 3-Body problem (CR3BP) which, while still not solvable, is able to give us a lot of insight into multibody dynamics. Unfortunately, we’ll have to leave most of our techniques from the 2-body problem behind as the CR3BP is a whole new ballgame.

Trajectory in the Earth-Moon System

General 3-Body Problem

The rules of the classical 3-Body problem are simple

We have three bodies
They are all point sources and we know their masses
They only interact gravitationally

Given complete state information (instantaneous position, velocity, time) can we predict their motion into the future?
We can’t solve the general 3-body problem analytically. Henry Poincare proved this in 1887. However, if we impose some smart constraints on it, we can get a lot of insight into the full 3-body problem while at the same time making our calculations much easier. The most common set of constraints is labeled the Circular Restricted 3-Body Problem (CR3BP)

CR3BP

Let’s start with the second word in the name, restricted. In this context, it means that one body is much smaller than the other two. For notational sake, the smaller body will always be the third one.
$m_1,m_2>>m_3$
This means we can ignore the gravitational tug from the smaller body on the larger ones. We’ll go over how realistic these restrictions are in the next section, but for now imagine the Earth, the Moon, and a communications satellite. Does the satellite really affect the earth and the moon’s trajectories in a meaningful way? No.
The second restriction is from the first word in the name, circular. Ignoring the third, smaller mass, let’s take the center of mass of the first two bodies, also called the barycenter. The two massive bodies will orbit around the barycenter in circular orbits.

How Good are Those Constraints

Whenever we place constraints on a model, it often makes it easier to analyze, but it takes the model farther away from reality. We need to always balance these two factors. If the general form of the problem is unsolvable, it doesn’t help us at all. On the other hand, if we’ve added so many constraints that the model no longer reflects reality, then it also doesn’t help us. How well do the constraints of the CR3BP match reality?
Let’s start with the circularity constraints. We measure the circularity of an orbit by measuring its eccentricity. If its eccentricity is 0, it’s a perfect circle, and if it’s 1 it’s a parabola. Let’s see what the eccentricity of a few plants/major bodies are

Earth (around Sun) – 0.0549
Moon (around Earth) – 0.0162
Charon (around Pluto) – 0.00005
Mars (around Sun) – 0.0933
Europa (around Jupiter) -0.009

Aside from Mars, these all are extremely small and restricting the planets to a circular orbit in the CR3BP is a relatively realistic assumption.
Now, let’s probe the other assumption, that we can ignore the pull of the third object on the first two. Let’s look at the first problem the CR3BP was applied to, predicting the motion of the moon. For this our three bodies are the Sun, the Earth, and the Moon.
The gravitational force of one body on a second body can be represented as the following
$F_{12}= \frac{GM_1M_2}{R_{12}^2}$
We know that the moon is closer to the earth so let’s see the ratio of the force of the Moon on the Earth to the force of the Sun on the Earth
$\frac{M_{moon}}{M_{sun}}(\frac{r_{earth,moon}}{r_{sun,earth}})^2\approx.005$
This means that the effect of the moon on the earth is .005 times the effect of the sun on the earth which again means that for the Earth-Moon system the CR3BP is a good approximation.
Note: If you want to explore the system dynamics a little more, figure out whether the Sun or the Earth has more of an effect on the motion of the moon.
We’ve now gone and seen that the CR3BP is a good approximation for the Earth-Moon system. You can go through this same procedure for a satellite, and you’ll once again find that the CR3BP is a good approximation for that case too.

Euler – Right Round (feat. Jacobi)

The final topic that I want to introduce is the rotating reference frame. Imagine we’re looking at the Earth-Moon system. We know that both Earth and the moon rotate about the barycenter in a circle. Their rotational rates are related to their masses and positions. We’ll go over this more in the next post, but imagine a line from the center of the Earth to the center of the Moon and made that one axis of our reference frame. If we let the reference frame rotate around at the same speed as the moon we can find structure that was hidden in the inertial frame. I’ll let the following video tracing out the path of a comet demonstrate. On the left, we have the inertial frame and on the right a frame that rotates with the Sun-Jupiter system.

Note: This rotating reference frame is the same one I used to plot the GIF at the begging.

The Path Forward

Unlike the two-body problem, the CR3BP also gives us a jumping off point into dealing with higher order n-body problem. This post is merely an introduction to the CR3BP and much like the 2-body problem, I’ll be creating a whole series for it. The early parts will cover topics like the dynamics of the problem, the Jacobi integral, forbidden regions, and Lagrange points. The latter parts, not yet planned out yet will deal with more advanced aspects of 3-body dynamics, like it’s chaotic dynamics.

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Announcement: Journal Club

As a new grad student, I’ve been wanting to get better at reading scientific literature. The more you do something the better you get at it, so I set a goal of reading a paper a day and writing up a small informal blurb about each one. Collecting each week’s blurbs might make for an interesting second weekly post on Gereshes, but I’ve decided to try it on the newsletter first. I’ll still only be sending out the newsletters on a weekly basis, so no extra emails, but each one will have a section that says Journal Club. This is just an experiment, and I may axe it at any time, but I’ll probably continue using the newsletter to try out new ideas.

Introduction to the 3-Body Problem