Astronomy & Celestial Mechanics
the sky, computed
Concept

Perigee — where it appears

The point in an orbit round the Earth that is closest to it, and the moment of passing it. The Moon's perigee distance varies around 363,000 kilometres, and the direction of the perigee turns once in about 8.85 years under the Sun's pull. For other central bodies the general word is pericentre.

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

What a departure loses to taking time. The gravity loss of a departure burn — the velocity a rocket spends beyond what an instantaneous impulse would need — against its initial thrust-to-weight ratio, on logarithmic axes, for departures from a 300-km orbit to hyperbolic excess speeds of 0.5, 3, 6 km/s, an exhaust speed of 4.41 km/s and thrust held along the velocity. The impulsive costs are 3.21, 3.60, 4.74 km/s. At a thrust-to-weight of 1.2 the losses are 4, 5, 11 m/s; at 0.1, 298, 396, 708; at 0.03, 1485, 1785, 2546. For a stage of high thrust the loss falls as about the inverse square of the thrust (slope -1.98 at the high end), because the burn arc shrinks in proportion and the loss grows as the square of the arc.

A burn that cannot happen at one point

Every transfer cost in orbital mechanics is computed as though the engine fired for an instant. A real departure burn runs for minutes or hours, and the vehicle travels a long arc while it does — climbing, slowing, and delivering thrust ever further from the point where it would have done the most good. The extra velocity that costs is the gravity loss, and it grows as the square of the arc: negligible for a strong stage, ruinous for a weak one, and recoverable by splitting the burn at the price of time.

spaceflight · Orbital transfer The first-order perigee rate is right for a slower Sun, and half wrong for ours. The rate at which an orbit's perigee turns, in turns per orbit, against the ratio m of the Sun's motion to the orbit's, from separate integrations of a satellite with the Moon's eccentricity under a solar tide on a circular path. The first-order rate (3/4)m² gives 0.0042 at the Moon's m of 0.0748, a perigee period of 18 years. The integrations give 0.0085, 8.9 years; the Moon's perigee takes 8.85. At m = 0.01 the integration gives 8.3e-5 against a first-order 7.5e-5. So the disagreement is not a matter of neglected forces: it depends on how large m is.

A perigee that moved twice as fast as Newton

The Sun's tide on the Moon should turn the line of its perigee once in eighteen years. It takes less than nine. For sixty years that was the outstanding failure of the law of gravitation, and in 1747 Clairaut told the Paris Academy that the inverse-square law would have to be amended. Two years later he withdrew the remark. The law was right and the calculation was unfinished: the series for the perigee has a small parameter that is not small, and its second term is seventy per cent of its first.

orbits · Perturbations

Named alongside it

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

Apsidal precessionΔvEvectionFinite-burnGravity lossHyperbolic excess speedInverse square lawLunar theoryOberth effectPerturbation theorySpecific impulseThrust-to-weight

All concepts