Astronomy & Celestial Mechanics
the sky, computed
Concept

Perturbation — where it appears

The departure of a real orbit from the two-body one it would follow if nothing else were pulling on it. The word covers everything from another planet's pull to atmospheric drag to the pressure of sunlight, and what unites them is that each is small enough to be treated as a correction.

Named by 3 essays across 3 fields — 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 A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

exoplanets · Transit-timing The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

An answer obtained along a path that was not taken

Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all.

gravitation · Impulse approximation

Named alongside it

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

Adiabatic invariantChoppingClose encounterCoulomb logarithmCross-sectionDisturbing functionDynamical frictionEphemerisGreat inequalityHyperbolic orbitImpulseImpulse approximation

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