Lagrange Points of the Circular Restricted Three-Body Problem
Two bodies orbiting each other, a star and a planet, a planet and its moon, drag a whole landscape of gravity around with them. Ride in the rotating frame where the two stay fixed and add the outward centrifugal pull, and five special points appear where everything balances and a third small body can sit still relative to the big two. Three of them, L1, L2 and L3, lie on the line through the two masses; they are saddles, balanced but unstable, a marble on a ridge that rolls off at the slightest push (which is why spacecraft at L1 and L2, like the James Webb telescope, must nudge themselves to stay). The other two, L4 and L5, sit at the tips of equilateral triangles and are genuinely stable when one mass is much heavier than the other, trapping swarms of Trojan asteroids. The scene maps the effective potential with its zero-velocity contours and the five points; drop a test body and watch it librate or escape. The diagnostic is the potential along the line of the two masses, where the three collinear points sit at its crests.
WHAT TO TRY
- Release the test body near L4 at a small mass ratio: it loops in a stable tadpole orbit, never leaving (a Trojan).
- Push the mass ratio above 0.0385: L4 and L5 turn red, and the body spirals away, the stability is lost.
- Release near L1 or L3 and watch it slide off the saddle and fall toward a primary, those points are never stable.
- Drag the test body anywhere and let go; the zero-velocity contour through it bounds where it can go.
- Switch to the inertial frame: the two masses orbit the barycentre, the five points sweep round with them, and a Trojan near L4 traces its real looping path through space.