SST self-gravitating disk collapse

Описание: The program is called Space Simulation Toolkit, you can buy it or try the demo here: https://sstgame.com
Also check out SST discord server:   / discord  

00:00 - some text at the start
00:17 - first version
05:16 - second version

Caution: some maths (and physics) ahead

What's really cool about this simulation is how at the start all the particles are orbiting the center of mass in unison, as if the whole thing was solid. This is due to the uniformity of this disk.

Suppose there's a particle at a distance R from the center. Then all the attraction from particles outside of the range R would be cancelled out*. So we can estimate the attraction to that particle, knowing the mass of that "sub-disk" with radius R. The density here is the same everywhere, so its mass M only depends on its area, or a square of the radius. So M ~ R^2

The grav. force equation is almost like in real world, but, since this is a 2d simulation, the force decreases proportionally to the distance (and not to the distance^2). So it's kinda like this: F = G * m * M / R. If we don't account for constants (mass of one particle IS a constant, and it equals 1 anyway), we get this: F ~ M / R. Or, since we know what M is, F ~ R.

This might already look familiar, because many pendulums work on principals that look like ax ~ x. But we can go further.

Since the particle is moving roughly in a circular motion, its acceleration a should obey the formula for centripetal acc. So, A = V^2 / R, where V is the speed of the particle, and A is its acceleration.

2nd law of Newton: F = m*A, or F ~ A, so F ~ V^2 / R

But we already know that F ~ R, so R ~ V^2 / R,
which means: R^2 ~ V^2, or V ~ R.

Now, since the speed is proportional to the distance from from the center, this means all the particles should be orbiting with the same time period, which is exactly what we observe at the start.

Then, everything collapses and nothing is distributed evenly any longer.
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*(I don't remember a proof, so take this as granted, but there's a similar thing with the Hollow Earth: according to known physics, there should be a zero-g environment everywhere inside it, because all the mass is outside this zone, evenly distributed on a surface of a sphere)