Curvature of the Mind

Thoughts from a Recreational Physicist

Gravitational slingshots

March 1, 2012 by Andy Brummer Leave a Comment

In my last few posts, I’ve been trying to characterize different potentials through the shapes of their orbits (gravitational, harmonic, lorentz). In the middle of it, I came across a post on bad astronomy about gravitational slingshots. I figured it would be a perfect opportunity to use these images to show the effect in a different way.

To start, I’m sending a beam of particles directly towards the planet in question, just like my previous demos. This time I slowly increase the speed of the planet and the orbit shapes change accordingly. It was pretty hard to see what was going on with the first few renderings, so I started by increasing the intensity and color of the orbit with the speed of the particle. Things were starting to look better, but they didn’t really start to illuminate the dynamics until I subtracted off the initial velocity of the test particles and only showed particles that were moving faster than their initial speed.

This one shows the default case of a stationary target. The particles accelerate as they get close and then slow down to their initial speed as they move a way.

moving target

This is the first shot of a moving target. You can see there are now a few particles that are being shot back at a higher speed, extending past the first line of acceleration from the last image.

There are a few more features visible now as the speed picks up. Since these are lorentz potentials, you can see a more pronounced central core of paths that pass right by the target and are only slightly deflected. The same thing happens with particles further out, but there is a sweet spot that generates two beams. The faster the target moves, the less deflection is seen. With a slow target, the final trajectory is almost a 180, but the angle decreases more and more as the target speed increases, though the final boost in speed increases as well.

I wanted to finish up with one image that captured the gravitational result. There are many similarities, but some significant differences. The infinity at dead center means that there isn’t a max deflection angle like there is in the lorentz case. As you you approach dead center, there will always be a set of bound states. This leads to simpler images, and really those trajectories aren’t all that interesting as far as the slingshot goes. Hyperbolic orbits are the only ones that get launched somewhere.

Anyway I hope these images illustrate some of the features of this process. I’ve learned a lot putting them together, both about these processes, and how to illustrate them in a way that makes the physics visible to the naked eye.

Filed Under: Images, Physics Tagged With: diffeq, generative art, gravity, physics

The best surprises come from unexpected places

February 15, 2011 by Andy Brummer Leave a Comment

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This time I'm back with some more physics visualizations with a flat 2d canvas. I'm skipping over some demos of basic physics to get at some orbital mechanics animations that I found surprising. I've derived and calculated solutions for two objects gravitationally bound to each other from my freshman physics classes back in college. Then I did it again with more sophisticated mathematics, and again when I did quantum mechanics of the atom. As I think about it, I was writing BASIC programs back in high school to simulate the 3 body problem. All of those equations and simulations had some pretty severe limits. For one they only involved one or two bodies in motion. What you are looking at here are roughly 185 test particles orbiting one massive object like planets in circular orbits around the sun. While each of these test particles start close to each other, their mutual interactions are ignored. If they weren't there would be much more complicated dynamics going on. What caught me off guard was how fast the inner bodies were moving in relation to the outer planets. Whenever I had pictured the slow ponderous motion of the planets, I had pictured them moving more or less like a uniform disk. Whoa, was I wrong. The inner bodies are just whipping around at a frenetic pace, while the outer ones just plod along at a snail's pace. In fact there is a rather conspicuous divergence in the speed of motion as the distance between the particles decreases. I plan to have something more to say about that in the future. You might be wondering why I picked 185 test bodies? In this case it comes from looking at orbits in the range of 5 to 375. Which corresponds to Mercury (.4AU) to Neptune (30AU) a ratio of 1 to 75. If our solar system was build from evenly spaced bodies in circular orbits, this is what it would look like. So when I set this up, I never expected the slow graceful curve of the spiral slowly winding around the center. As I play with it, it seems so obvious, but that's why I find this stuff so fascinating. I've calculated and simulated these same orbits for well over 20 years now, and they can still surprise and awe me with just the slightest change of perspective.

Filed Under: Experiments, Physics Tagged With: canvas, gravity, html5, javascript, physics, simulation

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