Astronomy & Celestial Mechanics
the sky, computed
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Perturbations — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The osculating semi-major axis of a perturbed orbit. The semi-major axis a test particle would have if the perturber vanished, computed from its position and velocity at every step of an integration over 26 orbits of the perturber. It is not constant: a short-period ripple rides on a slow trend, and only the trend accumulates.

    Elements that do not stay constant

    Six numbers fix an orbit for all time, and the phrase is only true in a universe containing two bodies. Add a third and the six start moving — some of them wandering and returning, one or two of them drifting in one direction forever, and the difference between those two behaviours is the whole of celestial mechanics after Newton.

    part 1 · orbits
  2. The same thrust is worth 2.08 times more at perigee, and out of plane it is worth nothing at all. The rate of change of the semi-major axis under a unit acceleration in each of the three directions, against position around an orbit of eccentricity 0.35, from Gauss's variational equations. The along-track curve carries the factor p/r = 1 + e cos f and therefore peaks at perigee, where the same impulse is worth 2.08 times what it is worth at apogee — the whole of the Oberth effect, arriving as a term in a differential equation rather than as an argument about kinetic energy. The radial curve is antisymmetric about apoapsis and integrates to exactly zero over a revolution: pushing outwards for half an orbit and being pushed back for the other half changes the energy by nothing, which is checked here by quadrature and comes out at -9.0e-18. And the out-of-plane response is identically zero at every point of the orbit, because W is perpendicular to the velocity and does no work. An orbital plane can be rotated without touching the energy, and that is why a plane change is so expensive: none of what is spent goes anywhere useful.

    Which direction moves which element

    Resolve a small force into three components and Gauss's equations say exactly what each one does. The out-of-plane component can rotate an orbit and can never change its energy; the along-track component owns the semi-major axis outright and is worth more at perigee than at apogee by a factor that is pure geometry.

    part 2 · orbits
  3. The two halves of what is left over. The disturbing potential of a perturber on a test particle, against the difference in longitude between them, at a semi-major axis ratio of 0.62. The upper curve is the direct term — the perturber's own attraction, which peaks at conjunction where the separation is smallest and falls to 1/(1+α) half a turn later. The lower one is the indirect term, which exists only because the coordinates are centred on a primary that is itself being accelerated, and which is a pure cosine of the longitude difference. The indirect term is the larger of the two over 3 per cent of the circle, and it averages to exactly zero while the direct term averages to something positive. Everything that happens slowly in a planetary system comes from that asymmetry: the part that survives averaging is not the part that dominates the instantaneous force.

    The series that is subtracted

    The two-body problem is solved, so nobody solves it twice. Every planetary theory since Newton begins by taking that solution away and asking what is left — and what is left is an infinite series whose terms are stacked in a hierarchy that makes the first half-dozen of them enough.

    part 3 · orbits

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