Astronomy & Celestial Mechanics
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Effective potential — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The effective potential, for three angular momenta. The radial motion of an orbiting body is one-dimensional motion in an effective potential: the attraction −GM/r plus the centrifugal term L²/2r² that the angular momentum contributes. The barrier at small radius is what stops a body with any angular momentum at all from reaching the centre, and the bottom of each well is the circular orbit.

    The wall that angular momentum builds

    A body falling toward a star almost never arrives. Sideways motion, which looks like a detail of the initial conditions, turns the attraction into a well with a wall around the middle of it — and the wall is why it costs more fuel to hit the Sun than to leave the solar system.

    part 1 · orbits
  2. The two force laws whose orbits close. The apsidal angle — the angle swept from periapsis to the next apoapsis — against the exponent of the force law, for F ∝ r^p. The dashed curve is the near-circular limit π/√(3+p), which has a closed form; the solid curve is the same angle for an orbit of eccentricity 0.4, computed by quadrature of ∫(L/r²)dr/√(2(E−U)) between its two turning points, with U the effective potential. An orbit closes when the apsidal angle is a rational multiple of π, and an orbit closes at every eccentricity only where the two curves meet: p = −2 at exactly 180° and p = +1 at exactly 90°, which is Bertrand's theorem. The quadrature returns 180.0000° and 90.0000° at those two exponents and departs from the near-circular curve by 1.4° at p = 0. The angle diverges as p approaches −3, where the circular orbit stops being stable and there is no well left to oscillate in.

    Only two force laws let an orbit come back

    That a planet returns to the same point of its own path after one lap is not a fact about orbits. It is a fact about the exponent in the force, and out of the whole continuum of attractions only two — the inverse square, and a spring — bring every bound orbit back to where it started.

    part 2 · orbits
  3. The same well after the primary has lost 45 per cent of its mass. Two effective potentials for one body: the solid curve before the primary loses mass and the faint one after, both at the same angular momentum, because a central force of any strength exerts no torque. The well shallows and its floor moves out from r = 1.00 to 1.82. The body's own level moves with it — from E = -0.420 to -0.127 — and the two horizontal lines are drawn where the radial action is conserved, which puts the turning points at 0.71–1.67 before and 1.30–3.03 after. The ratio between them is 2.3333 in both, so the orbit is the same shape at a larger size: everything about the body's path has scaled and its eccentricity of 0.4 has not moved. That is what a slow change leaves behind, and it is not what a sudden one leaves.

    The well moves, and the body keeps its share of it

    When the Sun becomes a white dwarf it will throw away half its mass, and every planet's orbit will swell by the same factor. Their eccentricities will not change at all — provided the loss is slow, and the only meaning "slow" has here is slow compared with one orbital period.

    part 3 · orbits

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