Astronomy & Celestial Mechanics
the sky, computed
Concept

Keyhole — where it appears

A small region in the plane of an asteroid's close approach to a planet through which it must pass to return and collide years later. Moving an asteroid out of a keyhole needs only kilometres of shift, not an Earth radius.

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

After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging.

An error that is nearly all in one direction

A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume.

orbits · Orbit determination How hard a hovering spacecraft pulls an asteroid. The velocity change a spacecraft hovering beside a 200-m asteroid gives it in a year by gravity alone, in millimetres per second, against the hovering distance from the asteroid's centre in asteroid radii, for spacecraft of 5 t, 20 t, 50 t. The pull depends on the spacecraft's mass and the distance, not on the asteroid's mass. A 20-tonne tractor at 1.5 radii gives 1.9 mm/s a year; at 3 radii, a quarter of that. Closer is better, and the limit on how close is set by the exhaust: inside the shaded region the engines would have to point so steeply outward to keep their 20° plumes off the surface that holding station costs more than the pull is worth.

A tow with no rope

A spacecraft parked beside an asteroid pulls it by gravity alone, with a force set by the spacecraft's mass and its distance and nothing else. Twenty tonnes hovering a hundred and fifty metres from the centre of a two-hundred-metre asteroid moves it by two millimetres a second in a year. That is the gentlest engine there is, and its limit is not the pull but the exhaust: the spacecraft must thrust away from the rock to stay put without blasting it, and the angle that takes decides how close it can hover.

spaceflight · Low-thrust transfer

Named alongside it

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

Along-track driftAlong-track errorArc lengthClone orbitsCondition numberCovarianceGravitational focusingGravity tractorImpact probabilityKinetic impactorThe line of variationsLow-thrust transfer

All concepts