Astronomy & Celestial Mechanics
the sky, computed
Concept

Clock bias — where it appears

The unknown offset of a navigation receiver's clock from the time kept by the satellites it measures. It lengthens every range equally, so it is solved for as a fourth unknown alongside the three coordinates of position.

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

Also named here as coverage, dilution of precision, geometry matrix, satellite navigation — the same set of essays touches all of them, so they are one junction rather than several.

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.

CoverageDilution of precisionElevation maskGeometry matrixLeast-squaresSatellite navigationFrozen orbitGround tracksWalker constellation

All concepts