The table that is a fit

Ask where Mars will be on a given evening and something will answer to a fraction of an arcsecond. It is easy to assume that the answer came out of Kepler’s laws with a few corrections bolted on, and that the corrections are what the last three centuries were for.

That is not what happens. The answer comes out of a file — a set of coefficients called an ephemeris, DE440 or INPOP or EPM depending on who made it — and the file is the output of a fit. Somebody integrated the equations of motion of the whole solar system numerically, compared the result against every observation anybody has recorded since about 1913, adjusted the starting positions and velocities and a few dozen physical constants until the disagreement was as small as it could be made, and then wrote the resulting trajectory down as a polynomial approximation.

Nothing in that description is a formula for where Mars is. It is an estimation problem, and the interesting properties of the answer are the properties estimation problems have.

The integration is the easy part #

The dynamical model is not in doubt and has not been for a long time. Every planet, the Moon as a separate body, several hundred of the largest asteroids, the post-Newtonian corrections to the point-mass interaction, the Earth’s and the Moon’s figures and tides, and the Moon’s own rotation — all of it goes into a set of second-order differential equations and gets integrated.

The integrator matters, but not in the way one might expect. Over the two centuries an ephemeris spans, a modern scheme run at a modern step size makes an error far below anything observable; what it must not do is drift, because a numerical energy leak turns into an along-track error that grows without bound. The starting conditions are what is unknown. Six numbers per body, plus the masses, plus the astronomical unit or its modern replacement, plus a set of constants describing the Earth’s and the Moon’s interiors — a few hundred parameters in total, none of which can be measured except through their effect on where things are seen to be.

What the observations actually are #

The fit is against a pile of heterogeneous data, and the heterogeneity is the point.

The oldest usable material is meridian-circle astrometry: a telescope constrained to swing only in the north–south plane, and an observer timing the moment a planet crosses the wire. That gives two angles. It is good to somewhere between a third of an arcsecond and a second, and the error is dominated not by the observer but by the star catalogue the position is measured against.

From 1964 there is radar ranging to Venus, Mercury and Mars — a distance rather than an angle, from the round-trip time of a pulse, good to a kilometre or so at first and to a hundred metres by the end.

From 1976 there is spacecraft ranging: a transponder on an orbiter or a lander returns a signal, and the round-trip time gives the distance to metres and later to less than one. A lander is better still, because it also carries the planet’s rotation.

And from the 1990s there is very-long-baseline interferometry of a spacecraft against a quasar, which supplies the two angles the ranging cannot, at a few nanoradians.

Each of those requires an observation model as elaborate as the dynamical one. A radar echo has to be corrected for the solar corona it passed through, the relativistic delay in the Sun’s field, the station’s position on a rotating and deforming Earth, the planet’s topography at the reflection point, and the difference between the time kept at the station and the time the equations are integrated in. Any of those, got wrong, appears in the solution as a change in a planet’s orbit.

Why the answer is anisotropic #

The most surprising property of a modern ephemeris is how unequal it is. Mars’s heliocentric position is known to a few metres. Neptune’s is uncertain by thousands of kilometres. The two numbers appear in the same file, produced by the same integration, in the same units.

The reason is not the integration and not the distance. It is that a fit is only as good as its data, and the data is wildly unequal.

Mars has been ranged continuously for half a century by half a dozen orbiters. Every one of those measurements is a distance along the Earth–Mars line to better than a metre, and there are hundreds of thousands of them. Neptune has one flyby, in 1989, giving a handful of range and angle points at one epoch, plus optical astrometry going back to its discovery in 1846 — which sounds impressive until one notices that its period is 165 years, so the whole optical record covers slightly more than one revolution.

The range measurement’s other property is that it is one-dimensional. A transponder gives the distance along the line of sight and says nothing whatever about the two directions across it. That is why a planet with an orbiter is known radially some ten to twenty times better than it is known altogether, and why VLBI — which measures exactly the two coordinates ranging cannot — is worth its considerable trouble.

The masses nobody can weigh #

The largest systematic error in the inner solar system is not an instrument. It is the asteroid belt.

Several hundred main-belt asteroids are massive enough to perturb Mars’s orbit at a level the ranging can see. Their masses are almost entirely unknown: only a few dozen have been visited, or have satellites, or have produced a measurable deflection of another asteroid during a close approach. The rest are known only as a brightness and a rough size.

So they are carried as free parameters. The fit solves for the masses of the three hundred or so largest bodies along with everything else, and the numbers that come out are, for many of them, the only mass estimates that exist. This has a consequence worth stating plainly. When a parameter is fitted alongside the thing one actually wants, the two become correlated: an error in one can be partly absorbed by an error in the other. Two ephemerides can therefore agree about Mars’s position to a metre while disagreeing about the mass of a particular asteroid by a factor of two, and neither of them is wrong. The mass is not a measurement of the asteroid; it is the value that made the Mars residuals smallest, given every other choice the solution made.

The honest way to read such a number is to ask what would happen if the asteroid’s true mass were different. For Mars’s position the answer is almost nothing, because another asteroid would take up the slack. For the asteroid, the answer is everything.

A table is only a table where its data is #

Every ephemeris is issued with a fit interval — the span of epochs the observations cover — and it is usually possible to evaluate it well outside that span. Doing so is not using a better ephemeris. It is extrapolating one.

This is why ancient observations are treated the way they are. A Babylonian record of a lunar eclipse is not something to be checked against a modern ephemeris; it is an observation, and it goes into the fit. The famous result that the Earth’s rotation has been slowing — that the accumulated clock error since 700 BC is about four hours — comes from exactly that: eclipse records placed where the modern extrapolation says they should not be, with the discrepancy attributed to the length of the day rather than to the orbits.

What the frame is measured against #

There is one more thing being solved for, and it has no analogue in a two-body problem.

The integration produces positions in a coordinate frame defined by the dynamics — the barycentre of the solar system, with axes fixed by the equations. Observations are made in a frame defined by objects: the quasars of the International Celestial Reference Frame, which are far enough away to have no measurable motion.

Tying the two together is its own estimation problem, and the tie is currently good to about a milliarcsecond. Everything downstream inherits it. A star’s position, its parallax and its proper motion are all measured in the quasar frame; a spacecraft’s trajectory is computed in the dynamical one; and the manoeuvre that puts the second onto a target measured in the first carries the error of the link between them.

Why there is more than one #

If an ephemeris were a computation, there would be one of them. There are three, maintained independently on three continents, and the differences between them are the best available estimate of how good any of them is.

Each group makes its own decisions, and none of the decisions is forced by the data. Which observations to include and which to discard as unreliable; how to weight a nineteenth-century meridian-circle transit against a spacecraft range; how many asteroids to model individually and how to represent the rest; which relativistic terms to carry; how to model the Sun’s oblateness and the solar plasma the radar signals cross.

Those choices are defensible in every case and different between groups, and the resulting tables disagree. For the inner planets, where spacecraft ranging dominates, the agreement is at the level of tens of metres — remarkable, and consistent with the formal errors. For the outer planets it is worse, and for Uranus and Neptune the differences reach hundreds of kilometres, which is many times either group’s stated uncertainty.

That discrepancy is the useful number. A formal uncertainty comes out of a covariance matrix and describes only the noise the fit was told about; it cannot see a systematic in the observation model or a wrong choice about asteroid masses. Three independent analyses of overlapping data are a crude but genuine sample of those, and where they diverge by more than they claim, the claim is what is wrong.

The practice that follows is standard in the field and worth stating: a mission that needs a planet’s position to a stated accuracy checks it against two ephemerides rather than one, and treats the difference as the error.

The time the table is a function of #

An ephemeris is a table against time, and the time is not the one on a wristwatch.

General relativity makes clock rates depend on gravitational potential and on motion, so a clock on the Earth’s surface does not keep the same time as an imaginary clock at rest at the solar system’s barycentre. The difference is not a constant offset: it varies over the year as the Earth moves through the Sun’s potential on its eccentric orbit, with an amplitude of about 1.66 milliseconds and a period of a year.

The equations of motion an ephemeris integrates are written in barycentric coordinates, so their time argument is barycentric dynamical time, and an observation timestamped by a terrestrial clock has to be converted before it can be compared with the table. Getting that wrong introduces an annual error of a millisecond and a half, which for a spacecraft moving at tens of kilometres a second is tens of metres, and for a pulsar timed to microseconds is catastrophic.

The conversion itself is a series with several hundred terms, computed from the same ephemeris — so the timescale and the positions are solved together, and using a table with a time argument it was not built for is a subtle way of getting a wrong answer that looks reasonable.

The astronomical unit went the other way at the same time. It used to be a fitted parameter — the length that made the Earth’s orbit come out right — and since 2012 it has been a defined constant, exactly 149,597,870,700 metres. What is fitted instead is the Sun’s gravitational parameter, which is what the dynamics actually depends on and what the observations actually constrain. The quantity that was measured for two centuries is now a conversion factor, and the thing it used to be a proxy for is measured directly.

The two changes are of a piece, and they say something about what the subject now regards as fundamental. A length that everything was measured in became a convention; a timescale that nobody could observe directly became the independent variable; and the quantities left over as measurements are the ones the equations of motion actually contain.

That is worth carrying to any table of numbers with a long history behind it: the quantities that survive as measurements are the ones the underlying equations are written in, and the ones that get promoted to definitions are usually the ones an earlier generation could measure best.

What this means for reading one #

Three habits follow, and they are the reason this rung exists.

A quoted position has a covariance, and it is not isotropic. The uncertainty of a planet’s position has a radial part, an along-track part and a cross-track part, and they differ by orders of magnitude. Quoting a single number for “the accuracy of the ephemeris” throws away the only part of the answer that is decision-relevant.

A derived constant from an ephemeris is a fitted parameter, not a measurement. The masses of the asteroids, the mass of the Sun in terms of the astronomical unit, the Earth’s tidal parameters and the Moon’s moments of inertia all come out of this one solution, correlated with each other and with every planet in it.

A residual is not an error. The quantity a fit minimises is the disagreement between the observation and the model, and it contains both. A pattern in the residuals of one data type may be a defect in the orbit, or in the observation model, or in the instrument — and deciding which is not a statistical question. The clearest example on record is the corona: Mercury’s radar residuals showed a systematic that was eventually traced to the solar plasma the signal had crossed, not to Mercury, and correcting the plasma model changed the planet’s fitted orbit.

And the accuracy is a property of a spacecraft programme. The uncertainty in Saturn’s position fell by three orders of magnitude between 2004 and 2017 not because anybody had a better theory but because Cassini was there and its range was measured every few days. When it ended, the uncertainty began growing again — linearly at first, and then faster.

One practical matter is worth setting out, because it is the part of an ephemeris a user actually touches and it is not the integration. The solution is not distributed as a trajectory or as a set of elements. It is distributed as Chebyshev coefficients: the interval is cut into granules of a few days to a month, and within each granule every coordinate of every body is represented by a polynomial of order ten to fifteen, fitted to the integrated trajectory to well inside its own uncertainty. Evaluating a position is then a few dozen multiplications rather than an integration, which is what makes an ephemeris usable inside a spacecraft’s navigation filter and inside a planetarium at the same time. The representation is exact enough to be invisible — the fitting error is centimetres where the solution’s own error is metres — and it carries a consequence: the file’s size, and therefore which bodies and which span a given release contains, is a distribution decision rather than a scientific one, which is why the same solution ships in half a dozen incompatible subsets.

What the file does not contain is the covariance. The fit produces one, of order a thousand parameters square, and it is not distributed with the coefficients — so the uncertainty quoted for a planet’s position is almost always reconstructed from published summaries rather than propagated from the matrix that produced it. A user who needs a rigorous error on a derived quantity has to obtain the covariance separately, and for most releases it is not published at all. That is the practical form of the third habit below, and it is the reason the differences between the three independent solutions do the work that a covariance would.

A last property of the representation is worth knowing because it bites in practice. The granules are fitted independently, so the polynomial in one does not match the polynomial in the next to machine precision at the join: the position is continuous to well inside the solution’s own error and the velocity has a discontinuity of order a millimetre a second at every granule boundary. For a planetarium that is nothing. For a numerical integration that uses the ephemeris to supply perturbing accelerations it is a source of noise at the boundaries, and integrators that step across them without noticing accumulate an error that looks like a physical effect. The remedy is standard and unglamorous — evaluate the perturbing bodies from a single granule wherever possible, or smooth across the join — and it is the kind of detail that separates a table from the trajectory it represents.

Where this ladder goes next #

This rung has established what the object is: a fit, whose accuracy is a property of its observations and whose parameters are correlated.

The rung above takes the covariance seriously. If the position of a planet is a random variable, so is the answer to every question asked of it — a close-approach distance, an occultation time, a launch window — and propagating the uncertainty through those calculations is a different skill from computing the nominal answer.

Beside it lies the lunar ephemeris, which is a harder problem than the planetary one and better solved: lunar laser ranging measures the Earth–Moon distance to a millimetre, the Moon’s own rotation is solved for alongside its orbit, and the result is the most stringent test of general relativity available in the solar system.

And below it, as the habit this rung is really about: a table of positions is the output of an inference, and an inference has error bars that depend on where it is evaluated. Reading a number out of an ephemeris without asking which observations paid for it is the same mistake as reading a mass out of a catalogue without asking whether it was measured or derived.