Sun and Planets
gravity m=1 x=0 y=0 m=0.001 x=1 y=0 vx=0 vy=1 m=0.001 x=2.2 y=0 vx=0 vy=sqrt(1/2.2)
Open in the app The dials and keys named below are the app's.
The setup
A star of mass 1 sits at the origin with no velocity given, so it starts at rest. Two planets of mass 0.001 start out along the x axis, at 1 and 2.2, each moving straight across its own radius. At a thousandth of the star's mass they barely feel each other, which is why this looks like two clean circles rather than a three-body problem.
Why those speeds
A circular orbit needs the pull to be exactly the turning force: M/r² = v²/r, so v = √(M/r) = √(1/r). That is the vy=1 on the inner planet and the vy=sqrt(1/2.2) ≈ 0.674 on the outer, written as an expression, which the spec accepts. Divide the circumference by the speed and the period is 2π r^1.5. Inside that is 2π ≈ 6.28 seconds, outside 2π · 2.2^1.5 ≈ 20.5, a ratio of 3.26. Period squared against radius cubed is Kepler's third law, published in 1619, and it falls out of the same one-line sum.
What the star does
Watch the middle. Giving the star no velocity leaves the system with a total momentum of 0.001 · 1 + 0.001 · 0.674 ≈ 0.00167. So the whole picture slides gently upward, a tenth of a unit in a minute, with the star's own small wobble on top. Nothing is wrong: momentum is conserved, and it was not zero to begin with.
Try
- Stop the drift: add vx=0 vy=-0.001674 to the star's line, and the middle of the plot stays put.
- Add a third planet on a line of its own, m=0.001 x=4 y=0 vx=0 vy=sqrt(1/4). It circles out at 4 with a period near 2π · 8 ≈ 50 seconds.
- Slow the outer planet, vy=0.5. That is too slow for a circle, so it falls into an ellipse that dives in to 0.83 and swings back out.
- Top in the deck for the orrery view, both orbits seen flat.