Math · chapter 6 of 13
Two big, one small
Let the third body be a pebble too light to pull back. Now the problem has a fence you can draw, five places where a pebble can sit still, and a number that says which side of the fence you are on.
Most useful three-body arrangements have one body that does not matter to the other two. A spacecraft between the Earth and the Sun. A pebble near Jupiter. Set its weight to zero — it still gets pulled, it does not pull — and the problem changes character. The two heavy bodies now run a plain two-body orbit, which is solved, and the pebble moves in a field that repeats.
Take the next step and spin the paper. Put the two heavy bodies on a turntable that rotates with them, so they sit still on the page. In that turning frame the pebble feels gravity from both plus the outward throw of the spin, and all three of those can be rolled into one hill-and-valley surface.
On that surface there is a quantity the pebble cannot change: Jacobi's constant. Carl Gustav Jacob Jacobi found it in 1836 and it is the only conserved quantity this case has left. Rearranged, it says the pebble's speed is fixed by where it is — so at the places where the arithmetic would need the speed to be negative, the pebble cannot go. Draw that boundary and you have drawn a fence. On the inside, the pebble is trapped with one body; loosen the fence and a gate opens between them.
Five places on the surface are flat — the pull and the throw cancel exactly. L1, L2, L3 sit on the line through the two bodies and are saddles: a pebble there slides off, slowly, which is why the James Webb telescope at L2 has to nudge itself back every three weeks. L4 and L5 sit at the corners of equilateral triangles with the two bodies, and they are hilltops that nevertheless hold, provided the lighter of the two heavy bodies is under about 4% of the total. Jupiter clears that by a mile, and its L4 and L5 have been collecting asteroids for four billion years — over ten thousand catalogued.
The same surface gives you the Hill sphere: how close a moon has to be for a planet to keep it rather than the Sun taking it. For the Earth that radius is about 1.5 million km, and the Moon sits at a quarter of it.
Reading it: Jacobi's constant: twice the hill-and-valley surface at the pebble's position, minus its speed squared. It does not change, ever.
μ is the small body's share of the two weights. The first term is the spin's outward throw, the other two are the two pulls.
Reading it: Speed squared cannot be negative, so the pebble is shut out of everywhere the surface sits below its own constant.
The boundary where it equals zero is the zero-velocity curve. Drawn in the demo, and it is a fence with gates that open in a fixed order as C comes down.
Reading it: The triangle points hold onto things only if the smaller heavy body is under about four percent of the pair.
Sun and Jupiter: 0.00095. Earth and Moon: 0.0121 — both under, both hold. Pluto and Charon: 0.104, over the line.
Reading it: A planet's grip reaches out about its distance from the Sun times the cube root of its share of weight.
Earth: 1.5 million km. The Moon at 0.384 million km is comfortably inside, which is the whole reason we have one.
This one runs in the browser, with JavaScript on. The words above it stand on their own.
Sources
- Wikipedia, Jacobi integral
- Wikipedia, Lagrange point
- Wikipedia, Hill sphere
- NASA, Webb orbit at L2
- IAU Minor Planet Center, Jupiter Trojans
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