Astronomy & Celestial Mechanics
the sky, computed
Concept

Least-squares fit — where it appears

The choice of model parameters that minimises the sum of squared differences between the data and the model, each weighted by its uncertainty. It is the standard way of estimating orbits and other parameters from many imperfect measurements.

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

Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft.

The table that is a fit

A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics.

orbits · Ephemerides The heaviest unseen planet Saturn's orbit allows, by distance. The largest mass, in Earth masses, a body at each distance could have without leaving more than 20 metres rms in Saturn's range residuals over a 13-year arc once Saturn's orbit has been refitted, from a single-planet calculation. The three lines are the most favourable direction for detection, the median over the sky, and the least favourable. Every line rises as the cube of the distance, because the tidal field falls as the mass over the distance cubed: the ephemeris measures that ratio and nothing else. At 500 AU the bound is 2.0 Earth masses in the best direction, 5 in the median and 11 in the worst. The shaded box is the range of masses and distances proposed for a planet beyond Neptune: in this calculation some of its directions are excluded and some are not. The real analyses fit every planet and every data type together and reach tighter limits; what they share with this one is the cube and the dependence on direction.

A distant planet felt only as a stretch

An undiscovered planet far beyond Neptune pulls the Sun and Saturn almost equally, and an ephemeris, which measures only where Saturn is relative to the Sun, cannot see a pull that moves both together. What it can see is the difference — a tidal stretch falling as the mass over the cube of the distance — and most even of that is absorbed by refitting Saturn's orbit. What is left decides which planets the solar system can be hiding.

orbits · Ephemerides The atmosphere's delay, by how high the spacecraft stands. The one-way delay of a radio signal through the troposphere, in metres of range, against the elevation of the line of sight, for an atmosphere whose refractivity falls exponentially with a hydrostatic scale height of 8 km and a wet one of 2 km. At the zenith it is 2.42 m. Integrated along a straight ray through a curved atmosphere (solid), it is 5.0 m at 30°, 13.4 m at 10° and 24.6 m at 5°. The flat-atmosphere rule, zenith delay over the sine of the elevation (dashed), agrees above about twenty degrees and then runs away, giving 46.2 m at 3° where the curved atmosphere gives 35.9: a ray at a low angle leaves the atmosphere through the top of a shell, not through the side of a slab. Ray bending, ignored here, adds centimetres below about five degrees.

Two metres overhead and thirty at the horizon

A radio signal crossing the lower atmosphere is delayed by about two and a half metres of range straight up and by more than ten times that near the horizon. Tracking a spacecraft means converting the first number into the second, and the conversion — the mapping function — is where almost all of the troposphere's error lives: under a millimetre above thirty degrees, centimetres to decimetres in the last ten. And the last ten degrees are exactly the observations the geometry cannot do without.

spaceflight · Radiometric navigation

Named alongside it

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

DegeneracyEphemerisA nuisance parameterA post-fit residualSpacecraft rangingAsteroid massBarycentric dynamical time (TDB)Elevation cutoffFit intervalKuiper beltLunar laser rangingMapping function

All concepts