Astronomy & Celestial Mechanics
the sky, computed
Concept

A nuisance parameter — where it appears

A quantity a model must include to fit the data but that is not itself the target of the measurement, such as an instrument offset or an unknown mass. Fitting it alongside the wanted parameters can absorb part of the signal being sought.

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

Also named here as spacecraft ranging — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

EphemerisLeast-squares fitA post-fit residualSpacecraft rangingAsteroid massBarycentric dynamical time (TDB)DegeneracyFit intervalKuiper beltLunar laser rangingObservation modelOrbital elements

All concepts