Introduction NOTE: This is a very graphics-heavy post and may take some time to load. Lagrange points are those points in the three-body system where the tug of gravity from the two primary bodies cancel out and any spacecraft stays in the same relative position with the two primary bodies. If you are in the rotating…
Intro Last year we had a 2 part series on how to land on the moon using a suicide burn. There, the condition was that we could only turn on the engine once, and it had to be fully throttled. But most landers can control their thrust, so in this post we’re going to go…
Space Debris From dead satellites to flecks of paint, there are over 500,000 pieces of space debris currently being tracked. Space debris, and the problems they can cause is a hot topic with everything from a major Hollywood movie to TED talks being given about them. In this post I’m going to go over a less…
Everything Orbits About Something In our solar system, Moons orbit around planets, which themselves orbit around the sun. Well, actual they orbit about the system barycenter, the combined center of mass between the two bodies. But can we get an object to orbit around a region without a barycenter? Yes! (Kinda) There are actually a…
Lissajous curves From orbits around Lagrange Points, to double pendulums, we often run into a family of loopy, beautiful, curves. These are called Lissajous curves, and describe complex harmonic motion. In layman terms, Lissajous curves appear when an object’s motion’s have two independent frequencies. Today, we’ll explore another system that produces Lissajous curves, a double…
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.…
Balancing a Marble Imagine you had a marble and you wanted to put it at the point between the Earth and the Moon where the force of gravity was equal from both. The two forces would cancel out, and the marble would remain in that location forever. If the sun and the earth were not…
Recap In the last two posts on what is the 3-body problem and the dynamics of the three body problem I’ve shown you the following GIF with a spacecrafts trajectory in the earth moon system. The trajectory is bouncing around what I’ve labeled a ‘forbidden region’. In this post we’ll go over the number behind…
Chaos I’ve recreated one of my favorite mathematical demonstrations below. Three double pendulums, all starting with near identical initial conditions, all rapidly diverging. I love this demonstration because it’s such a great example of chaotic dynamics. What is Chaos? Chaotic dynamics, in a nutshell, means that a system is extremely sensitive to initial conditions. That…
Recap In the last post we introduced the 3-body problem along with some constraints that turned it into the circular restricted 3-Body Problem (CR3BP). In the beginning of that post, I had a GIF that traced out a trajectory in the Earth-Moon system. In this post, we will derive the equations of motion for the Circular…
