Astronomy & Celestial Mechanics
the sky, computed
Concept

Navigation — where it appears

The determination of a spacecraft's position and velocity from tracking data, and the planning of corrections that bring it to its target. Deep-space navigation uses the Doppler shift and range of radio signals and, near a target, optical images.

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

An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.

A position measured from a frequency

A spacecraft is unresolvable and unreachable, and everything known about where it is comes from two scalars — a round-trip light time and a Doppler shift. Neither is an angle. The orbit solution returns two angles anyway, because the antenna is bolted to a rotating planet.

spaceflight · Radiometric navigation How far a small error at departure moves the arrival at Mars. The distance by which a craft on the Hohmann transfer from the Earth's orbit to Mars's misses its aim point at the nominal arrival time, against an error in its departure speed, on logarithmic axes, integrated numerically for the 259-day flight. An error along the direction of motion (solid) moves the arrival by 76 km for every millimetre a second; one across it (dashed) by 23 km. The relation is linear over the whole range drawn: a metre a second at departure — the precision of a good launch — is 76 thousand kilometres at Mars, several times the planet's own radius and a thousand times what a landing needs. The ratio is the corresponding element of the state transition matrix, and it is the number every navigation plan is built around.

A millimetre a second at launch, seventy-six kilometres at Mars

A craft on its way to Mars arrives seventy-six kilometres from where it was aimed for every millimetre a second its departure speed was wrong. No launch is that good, so no trajectory is flown as designed; it is flown as corrected, by Newton's method run on the matrix that says how the arrival depends on the departure — which doubles its correct digits at every step and makes a two-body sketch into a real trajectory in two iterations.

spaceflight · Patched conics Bits bought with carrier, until the carrier cannot hold. For the Voyager-like link at 165 AU, the thermal Doppler noise of the carrier loop over a 60 s count, in millimetres a second, against the data rate the remaining power carries, as the modulation index rises. The noise stays under the 0.03 mm/s the solar plasma and the troposphere already impose all the way to the index of 81.8 degrees, where the carrier falls to the 15 dB-Hz a loop needs to stay locked and the noise is 0.012 mm/s. Past that line the Doppler is not noisier; it is gone. The limit on the data's share is the loop's lock, not the precision.

One signal that carries the bits and the Doppler

A spacecraft is navigated by the same radio signal that brings its data home, and the transmitter's watts have to be divided between the two jobs by a single number, the modulation index. Near the Earth the division hardly matters. Out past the planets, where the inverse square has taken almost everything, the carrier needs a fixed minimum to stay locked, and every watt above it is data — so the most distant spacecraft is flown at the edge of losing the signal it is navigated by.

spaceflight · Radiometric navigation

Named alongside it

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

Coherent transponderDeep-space networkLight timeTwo-way DopplerThe B-planeDeclinationDelta-DORDifferential correctionDiurnal signatureDual-frequencyHohmann transferNewton's method

All concepts