Gravity game set and some thoughts on probability and chaos.

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jotabe
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Gravity game set and some thoughts on probability and chaos.

Postby jotabe » Mon Mar 22, 2010 6:44 am

First, check this out ^_^
http://phet.colorado.edu/sims/my-solar- ... em_en.html

It's a small gravity simulation, that allows a good range of masses and start velocities. Go ahead, play around, try stuff and have fun, i'll wait.

Done? Well, maybe you should play a bit more, but that's ok. If you have tried out the 2-body problem, that's pretty boring. Depending on the mass difference you can have sun-earth-like orbits, or 2 intertwinning elipses.
Fun starts with 3 bodies. If you keep large mass differences, you can have sun-earth-moon-like systems, or even sun-earth-venus/mars, depending on the initial velocities. Still, these systems are pretty stable... unless you manage to get the "planet" orbits overlapping. If you aren't careful, the gravitatory pull of a heavy planet might send the lighter planet out of the system (of course, this is just because of the limited time frame: it would simply move to a much wider elliptical orbit.
But try 4 bodies, or 3 bodies of comparable masses. Now that's where it's at. It's very difficult to get stable orbits... and even then, stable is just an overstatement: getting configurations where no planets crash or go flying off. Sometimes it might seem that it works for a while, but 2 planets get too close to each other and problems start.

Gravity, according to Newton's Gravitational pull, it's linear. So, for 2 bodies the system is very easy to calculate. For more than 2 bodies, it ceases to be linear. The trajectory of one of the planets depends on the position of all the other bodies. And these positions change along time... so going to be an awful amount of feedback. Only thing we can do is to calculate this system numerically, using what we call a linear approximation. To do this, we stablish a time-step, a window of time after which we recalculate the system, linearly again. Since every non-linear system behaves linearly when the time-step is infinitely close to zero, the smaller the time-step is, the more accurate our calculation.

And this is how we wade through chaos.

Just three pics to view how chaos is actually chaotic, predictable only with probabilities.

This is the initial state:
Image

Now, if we make fast, inaccurate calculations, we obtain this:
Image
The large time-step makes that the trajectories have longer straight-line segments, so when bodies are close to each other, the probability of impact increases. Approximation leaves out important data, so the system quickly "crashes" into a simple, linear 2-body system.

If we go for accuracy, instead, there are no crashes:
Image
We can't talk about orbits proper, here. Energy can be transferred from one planet to the others, and viceversa. Being fair, we can only talk of probabilities of a planet being at a certain orbit. Probabilities of where to find planets, which depend on their initial position, and their mass. We could talk about planetary orbitals, instead of planet orbits.
Sounds familiar? This is what happens for atomic electrons. The difference is that electrons have probabilities because of their quantum nature: a single electron wouldn't have a definite orbit either.

Why don't we observe this in our solar system: the key is mass difference (the Sun mass is incomparably larger than the planetary masses) and orbit separation. Under this condition, the gravitation problem for each planet becomes again a 2-body problem, and stable orbits return.
This doesn't always happen however: we have found many hot-jupiters in very small orbits around small stars. This contradicts our current theories on planetary system formation, so it's been hypotesized that these large planets have started at much farther orbits, but eventually expelled their siblings out of the system, imparting them part of their energy, and hence going to lower orbits.

I hope this is easily readable and understandable ^_^ any question, or advice on how to write it better, please don't be shy :D
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