Astronomy & Celestial Mechanics
the sky, computed
Concept

Node regression — where it appears

The steady drift of an orbit's ascending node caused by the primary's equatorial bulge. Its rate goes as the cosine of the inclination, so it reads one way as a measurement of the oblateness and the other as the design equation for a sun-synchronous orbit.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

orbits · Perturbations 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.

orbits · Perturbations

Named alongside it

The objects these essays reach for when they reach for this one.

Osculating elementsArgument of latitudeDisturbing functionGauss's variational equationsGreat inequalityLow-thrust transferOberth effectOblatenessOrbital elementsOrbital energyPerturbationPerturbing acceleration

All concepts