Astronomy & Celestial Mechanics
the sky, computed
Series

Conic sections — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

    Every orbit one force allows, and the number that picks between them

    Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

    part 1 · orbits
  2. A vector that does not change, drawn five times. The Laplace–Runge–Lenz vector A = v × L − GM r̂, constructed at five points of one orbit at e = 0.45 under a force ∝ r^−2, each drawn from the body rather than from the focus so that its length and direction can be compared point by point. Under the inverse square every one of them is the same vector: identical to machine precision in length and in direction. Its length is 0.4500, which is the eccentricity, and it points at pericentre — so the orbit's orientation is a conserved quantity and not a constant of integration, which is why the ellipse does not turn.

    The third thing that is conserved

    Energy fixes an orbit's size and angular momentum fixes its shape. Neither fixes which way it points — so a curve that closes needs a third conserved quantity, and only two force laws in the universe supply one.

    part 2 · orbits

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