Circular Orbits | Planetary Motion | NBody Simulation

I took a few hours yesterday to start on a simulation for a job application. Part of the simulation involves trying to figure out how to create stable nbody orbits, at least stable for some amount of time to look interesting.

Turns out that the mathematics beyond constructing a circular orbit of just two bodies is a bit far-fetch’d. We can see in this link that solving for the velocity of each planet is very straightforward. Let me take a screenshot of the final equations here:

To me this makes a lot of sense and matches my own derivation, albeit with a slightly different final form (just due to some algebra differences). I plug in the equation into my simulation and hit run, however there’s just not enough velocity to keep an orbit. See for yourself (the simulation is slowed down a lot compared to later gifs):

Each white circle is a body (like a planet) and the red circle is the system’s barycenter.

So I tinkered a bit and found out the velocity is too dim by a factor of sqrt( 2 ). My best guess is that the equations I’m dealing with have some kind of approximation involved where one of the bodies is supposed to be infinitely large. I don’t quite have the time to look into this detail (since I have much more exciting things to finish in the simulation), so I just plugged in my sqrt( 2 ) factor and got this result:

Here’s the function I wrote down to solve for orbit velocity in the above gif:

void Orbit( Body* a, Body* b )
{
	float r = Length( a->xf.p - b->xf.p );
	float d = r * (a->m + b->m);

	a->v.Set( 0, -sqrtf( (G * b->m * b->m) / d ) * sqrtf( 2.0f ) );
	b->v.Set( 0, sqrtf( (G * a->m * a->m) / d ) * sqrtf( 2.0f ) );
}

I’m able to run the simulation for thousands of years without any destabilization.

If any readers have any clue as to why the velocity is off by a factor of rad 2 please let me know, I’d be very interested in learning what the hiccup was.

Next I’ll try to write some code for iteratively finding systems of more than 2 bodies that have a stable configuration for some duration of time. I plan to use a genetic algorithm, or maybe something based off of perturbation theory, though I think a genetic algorithm will yield better results. Currently the difficulty lies in measuring the system stability. For now the best metrics I’ve come up with is measuring movement of the barycenter and bounding the entire simulation inside of a large sphere. If the barycenter moves, or if bodies are too far from the barycenter I can deem such systems as unstable and try other configurations.

One particularly interesting aspect of the barycenter is if I create a group of stationary bodies and let them just attract each other through gravity, the barycenter seems to always be dead stationary! The moment I manually initialize a planet with some velocity the barycenter moves, but if they all start at rest the barycenter is 100% stable.

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