Astronomy & Celestial Mechanics
the sky, computed
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Orbital elements — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it.

    Six numbers that fix an orbit for all time, and the sixth is the awkward one

    Five of the orbital elements describe a curve that never changes. The sixth says where on it the body is, and it is the only one that has to keep being measured.

    part 1 · orbits
  2. Three rotations, applied in order. An orbit of eccentricity 0.45 carried into space by the three orientation angles, one panel per rotation: the argument of periapsis ω = 40°, then the inclination i = 42°, then the longitude of the ascending node Ω = 55°. The last panel applies the same three angles in the reverse order and arrives somewhere else, because rotations about different axes do not commute.

    Three rotations that put an orbit in space, and they do not commute

    An orbit's orientation takes three angles. Give the same three angles in a different order and the orbit ends up somewhere else — which is why the convention is part of the data.

    part 2 · orbits
  3. The angle that has nowhere to be measured from. An eccentricity vector carried round a circle of radius 0.034 centred at 0.031 — which is what a secular perturbation does to one, a forced eccentricity with a free one turning about it. Below, the two components e cos ϖ and e sin ϖ, which are smooth, bounded and perfectly ordinary throughout. Above, the longitude of pericentre read off them, which is not: as the eccentricity passes its minimum of 0.0030 the pericentre sweeps through most of a circle, at up to 4080° per unit time against the 12° the free vector itself turns in the same interval. Nothing has happened to the orbit. The pericentre is a place on the orbit, and a nearly circular orbit does not have one — so ω, and Ω with it at zero inclination, are angles measured from a feature that is not there. The equinoctial elements are the pair drawn below, and a propagator written in them steps through this instant without noticing it.

    The elements that stop existing

    An orbit needs six numbers, and three of the usual six are angles measured from features a perfectly ordinary orbit may not have. At zero eccentricity there is no pericentre to measure from, and the arithmetic knows it.

    part 3 · orbits

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