Astronomy & Celestial Mechanics
the sky, computed
Concept

Corrected target iteration — where it appears

A way of repairing a trajectory designed in a simplified field: fly it in the real one, measure the miss, and re-aim at the target minus the miss. It converges when moving the aim moves the arrival by less than the aim moved.

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

A two-body arc flown in the real field. How far from its target a spacecraft arrives when it flies the velocity Lambert's two-body problem gives, through the Earth's real field with its equatorial bulge, against the angle of the transfer, from a 400 km orbit to a point on a 420 km orbit, on a logarithmic scale. In a pure two-body field every solution arrives within a metre of the target; exactly 180° is left out, where the two points and the Earth's centre are in line and the transfer's plane is not determined. In the real field the miss grows with the transfer: at 51.6° inclination, 4.2 km after 60° and 43.0 km after 300°. An equatorial orbit misses most, because the bulge's extra pull is strongest over the equator and acts on it all the way round; an inclined orbit feels it only part of the time but also has its plane turned.

An answer from two bodies, flown through a flattened Earth

Lambert's problem gives the exact velocity that carries a spacecraft between two points in a given time — exactly, for a spherical Earth. The real Earth bulges at the equator, by a thousandth of its central pull, and a spacecraft flying the exact two-body answer between two points in low orbit arrives tens of kilometres from where it was sent. The repair is to keep the two-body solver and lie to it about the target, and the lie converges by a factor of a hundred each time it is told.

orbits · Lambert's problem An iteration that slows toward the straight-through transfer. The contraction ratio of the corrected-target iteration — how much each pass shrinks the miss — against the transfer angle, for a transfer between orbits of 51.6° inclination at 400 and 420 km, with the Earth's bulge at its real strength and at 10 and 30 times it, on a logarithmic scale. The iteration converges where the ratio is below 1 (dashed). At the real strength it shrinks the miss 1857-fold per pass at 150°, 83-fold at 178° and 8-fold at 181°: always convergent, ever slower. At 10 times the strength it fails between 181° and 184°; at 30 times it fails between 182.5° and 192°. The slowing is geometric: near 180° the transfer's plane is set by a sideways shift of the aim point, so correcting the bulge's sideways miss tilts the whole arc.

A correction that tilts the arc it meant to aim

A transfer designed with Lambert's two-body solution and flown through the Earth's flattened field misses its target, and the standard repair is to aim at the target minus the miss and solve again. That converges fast — until the transfer angle approaches 180°. There the two end points and the Earth's centre are nearly in line, the arc's plane is decided by a sideways shift of the aim, and correcting the bulge's sideways miss rotates the whole arc. The iteration slows by a factor of a hundred, and with a larger perturbation it runs away. Correcting only within the plane fixes it, at the cost of a plane change paid at arrival.

orbits · Lambert's problem

Named alongside it

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

Differential correctionJ2Lambert's problemOblatenessPerturbationsNodal regressionPlane changeRendezvousTransfer planeThe two-body problem

All concepts