Astronomy & Celestial Mechanics
the sky, computed
Concept

Least-squares — where it appears

The fitting method that chooses model parameters to minimise the sum of squared residuals between data and model. With independent Gaussian errors it gives the most probable parameters, and its covariance matrix says how the errors combine.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc.

Five directions and no distance among them

An image of a moving point records an angle and throws the range away, so an orbit has to be assembled out of angles alone. How many angles are needed is not a detail of the method — it is the whole of what a determination is.

orbits · Orbit determination Four satellites, and which four. Two sets of four navigation satellites seen on the sky from below, zenith at the centre and the horizon at the rim, with circles at 30° and 60° elevation. Left: one overhead and three low and spread around the horizon. Right: four bunched in one part of the sky. Each set fixes a position, and the error it leaves is the range error multiplied by the dilution of precision — 2.0 for the spread set, 62.0 for the bunched one. The same four range measurements, made with the same precision, give positions 31 times more uncertain when the satellites are bunched.

Four satellites in view, and which four

A satellite navigation receiver needs four satellites to fix its position, and a constellation is designed to keep at least four above every place. Four is the count, not the answer. The same four ranges, measured equally well, give a position thirty times worse when the satellites are bunched in one part of the sky than when they are spread around it, and the constellations carry their extra satellites to pay for the moments when the geometry, not the count, fails.

spaceflight · Ground tracks 6 satellites over the Moon's south pole. The horizontal (solid) and vertical (dashed) dilution of precision for a receiver near the Moon's south pole over 48 hours, with 6 satellites in elliptical frozen orbits — semi-major axis 6541.4 km, eccentricity 0.6, inclination 56.2°, apolune over the south pole, each in its own plane — and a 5° horizon mask, logarithmic. Gaps are times with fewer than four satellites in view, when no fix is possible: 28 per cent of the time. When there is a fix, the horizontal geometry is good — median HDOP 1.7 — and the vertical is bad, median VDOP 6: the satellites near apolune all stand at much the same elevation, and a receiver cannot tell whether every range is long because it is lower or because its clock is late.

A height the map already knows

A navigation service for the Moon's south pole needs only a handful of satellites, if they are put in elliptical orbits that hang for hours over the pole at the high point of each lap. But satellites hanging over a pole all stand at much the same height in the sky, on a cone, and four ranges from a cone cannot tell whether the receiver is lower or its clock is late. On the surface the receiver knows its height already, from a map of the terrain — and that one number, used as a measurement, does more for the service than two more satellites.

spaceflight · Ground tracks

Named alongside it

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

Clock biasCoverageDilution of precisionElevation maskGeometry matrixSatellite navigationAstrometryEccentricityEpochError propagationFrozen orbitGround tracks

All concepts