Astronomy & Celestial Mechanics
the sky, computed
Series

Ephemerides — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · orbits
  2. Three passes, and the fourth is below the noise. How far the position is still wrong after each pass of the light-time iteration, for five targets, on a logarithmic scale of kilometres. The first pass uses the body's position now and is wrong by exactly the distance the body travels while its light is in transit — 6254 kilometres for Mars near opposition, which is about twenty-five arcseconds and larger than any residual in an ephemeris fit. Each further pass multiplies the error by the body's speed divided by the speed of light, so the lines are straight and their slopes are that ratio and nothing else. Three passes take every one of these below a metre, which is why the loop in an ephemeris code has a fixed trip count rather than a convergence test. The same argument run backwards is why the correction cannot be applied to the observation instead: the direction light arrived from is what the observation is, and the position it corresponds to is not known until the body's orbit is.

    Where a planet is and where it is seen

    An ephemeris is fitted to observations, and no observation is of a position. It is of a direction light arrived from, at a time that is not the time the light left, bent by a Sun that is nowhere near the line of sight. Three corrections stand between the two, and all three are larger than the residuals.

    part 2 · orbits
  3. 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.

    part 3 · orbits
  4. The Moon pushed towards the Sun, once a month. The change in the Earth–Moon distance, in millimetres, if the Earth's gravitational self-energy fell towards the Sun differently from the rest of its mass by a fraction η = 0.001 — the term that would be present at the synodic period, the cycle of lunar phases, in an integration of the Moon's orbit in the Sun's full field. The amplitude is 11.4 mm, which is 11.4 metres times η: the Moon's orbit would be stretched towards the Sun at new moon and squeezed at full, or the reverse, depending on the sign. The shaded band is ±2 mm, roughly the precision of a modern lunar laser ranging normal point. A violation a thousandth of the size of the effect it tests would be twice that precision.

    A Moon that would lean towards the Sun

    Part of the Earth's mass is its own gravitational binding energy — a few parts in ten billion. If that part fell towards the Sun even slightly differently from the rest, the Earth and the Moon would fall at different rates and the Moon's orbit would be pushed sunward once a month by about eleven metres for every unit of the difference. Fifty years of laser ranging to the Moon have looked for that push at the millimetre level, against a term the Sun's ordinary gravity puts at the same frequency a hundred kilometres high.

    part 4 · orbits

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