Gravity Disc
f(r) = 0.05/r²
Open in the app The dials and keys named below are the app's.
The setup
0.05/r² is Newton's law of gravitation with every mass set to 1 and G = 0.05. The value is positive at every distance, so every pair pulls together and nothing ever pushes. Forty of those unit masses start scattered over a disc, set turning as one at Swirl 0.3, so a mass at distance d from the centre starts out moving at 0.3·d. Nothing ever leaves the floor plane: this is a flat problem.
The force
If all forty masses sat at the centre, a body three units out would feel 0.05 · 40 / 3² ≈ 0.22. A circular orbit there would want a turn rate of √(0.22/3) ≈ 0.27, so the disc opens turning a little faster than a circle.
But the pull is nothing like that smooth. A body's nearest neighbour starts about half a unit away and pulls with 0.05/0.5² = 0.2, as hard as the whole cloud together.
Why it comes apart
With forty point masses the graininess wins. Two bodies fall together, whip round each other and come out as a tight pair moving faster than either arrived. The rest of the swarm pays for it and sinks a little deeper. Repeat this and the cloud sheds its outer members: the mean distance from the centre grows from 2 at the start to 4.7 after twenty seconds, to 8 after forty and to 11 after a minute, by when most of the swarm has crossed the edge of the plot. This is evaporation, the reason real star clusters lose stars one by one.
History
Erik Holmberg ran the first N-body experiment at Lund Observatory in 1941, without a computer. He stood light bulbs where the stars were and read the light with a photocell, because light falls off as 1/r² just as gravity does. Between steps he moved each bulb by hand, and so watched two galaxies pass.
Try
- Top: from above the swarm reads as the plane figure it is.
- Set Swirl to 0: with nothing holding the disc up it falls to a knot in about five seconds, and the pile-up throws most of it straight back out.
- Set Swirl to 0.6: twice the turn rate is far more than this much mass can hold, and the disc flies apart from the first second.
- Change the formula to 0.2/r²: four times the pull, and the disc has fallen in by two seconds.
- Set Bodies to 8: few enough to follow one at a time, and a slingshot is easy to catch.