Forbidden Regions and where to Find Them

Jacobi and Lagrange

So far we have explored the Jacobi Integral and the Lagrange points. Now, we are ready to combine the two into one of the most useful concepts in the Circular restricted 3-Body Problem (CR3BP), forbidden regions. You’ve seen them in my visualizations before, but now we’ll finally go over what they are.

Note: this is an ongoing series on the CR3BP. If you want to get ahead on your own these are some good books on the material and astrodynamics in general (Book 1, Book 2, Book 3, Book 4).

Recap of Jacobi

The Jacobi constant, is an energy like integral that relates position to velocity.

$J = 2\Omega - v^2 = (x^2+y^2)+\frac{2(1-\mu)}{d}+\frac{2\mu}{r}-v^2$

If we have the spacecrafts position, and velocity, we know it’s Jacobi constant. Let’s say we have Jacobi constant at t=0

$J_0=2\Omega_0 - v_0^2$

And we want to see the “farthest” out that the spacecraft can go. At this “farthest” point, much like with potential and kinetic energy, we should have traded all our velocity for position and now we’ll have zero velocity. Let’s call the Jacobi constant at this point the zero velocity constant (ZVC)

$J_{ZVC} = 2\Omega_{ZVC} + 0 = (x_{ZVC}^2+y_{ZVC}^2)+\frac{2(1-\mu)}{d_{ZVC}}+\frac{2\mu}{r_{ZVC}}$

Any object will have the same Jacobi constant until there is a non-gravitational force acting on it or more energy is added to the system, so we know these two Jacobi values are the same

$J_0= J_{ZVC} = (x_{ZVC}^2+y_{ZVC}^2)+\frac{2(1-\mu)}{d_{ZVC}}+\frac{2\mu}{r_{ZVC}}$

We now have one equation, and three unknowns, X, Y, and Z. These X,Y, and Z will form a closed hyper curve that the spacecraft cannot enter, this is the forbidden region. The hyper curve, is sometimes also called the zero velocity curve, or zero velocity surface because any object with the same Jacobi constant that’s on the curve will have zero velocity. For the rest of this post I’ll be abbreviating zero velocity curve as ZVC.

Let’s do a clarifying example

Example:

I’ve picked an initial state for our spacecraft

$\boldsymbol{X} = [.1,.491,0,-.7,.7,0] = [x,y,z,\dot{x},\dot{y},\dot{z}]$

Let’s calculate it’s Jacobi value in the Earth Moon system (μ = 0.0122 ).

J = 3.2178

Plotting the ZVC for this Jacobi value gives us

Note I’ve also plotted a zoomed in area of around the moon and two of the Lagrange points that we are going to be inspecting later in this post.

Now let’s see how well that bounds our trajectory

We now see that the ZVC acts, as we before discovered, a forbidden region bounding where our spacecraft can, and can’t go.

Lagrange Ties In

We know the Jacobi value is an energy like quantity. Increasing the velocity, increases the energy, but it decreases the Jacobi value. Through this connection we know that as we increase the energy of the system the Jacobi value goes down. Let’s start decreasing the Jacobi constant (increasing the energy) and see how hour ZVC evolves

That’s interesting. As we decreased the Jacobi constant, both cavities started growing, but I’ve marked off two really important parts of the diagram.

The two parts that I’ve marked were two of the collinear Lagrange points (link). With L1, the one I circled, both cavities seem to be approaching it. I paused the simulation when the ZVC had the same Jacobi constant as the L1 Lagrange point. Any particle with a Jacobi value larger than this will be stuck in either one of the two cavities or outside the system. It doesn’t have enough energy to cross any of these forbidden regions. The second thing we should note is at the Lagrange point on the other side of the moon. Around L2 we see a small divot beginning to form on the outside.

L2

Let’s now watch as we decrease the Jacobi constant some more

By decreasing the Jacobi value (increasing the energy) we’ve opened up an earth moon gateway. Before, the forbidden region had locked us into trajectories around earth, but now our spacecraft now has enough energy to be able to travel to the moon. At L2, the bulge has now become more pronounced, but has not yet opened. I paused the simulation now when the Jacobi value of the ZVC is equal to the Jacobi value of L2. This means that we cannot leave or enter the earth moon system just yet. We’ll need to decrease the Jacobi constant just a bit more.

L3

Continuing on with decreasing the Jacobi constant

We’ve now opened up our second gateway at L2. Our spacecraft now has enough energy to leave the earth moon system. Notice, that we haven’t gotten rid of the forbidden region in it’s entirety. In future post’s we’ll use this to design some pretty un-intuitive orbits.

L4 & L5

We now have these teardrop shaped forbidden regions around L4 and L5. Can we make the forbidden region disappear completely?

It turns out that we can make the ZVC disappear completely. At least in the X-Y plane. Because the Jacobi constant is a 3-dimensioanl quantity, there will always exist some zero velocity surface in the X-Z axis.

Gotta go Fast

Earlier int this post we had an example that demonstrated how good the ZVC was at bounding the spacecrafts trajectory. Now let’s pump in some energy by raising the velocity of the spacecraft and seeing what new regions our spacecraft can go to. Note: In this GIF I only increase the velocity 2%

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If you can’t wait for next weeks post and want some more Gereshes I suggest

An Introduction to Error

Asteroid Wars – Part 1

My Undergraduate EDC

Forbidden Regions and where to Find Them – The 3-Body Problem