Where a planet is and where it is seen

A planetary ephemeris is a table of positions, and a position is not an observable. Nobody has ever measured where Mars is. What is measured is a direction on the sky at a particular instant, or a round-trip radio delay, or a Doppler shift — and every one of those is separated from the position by machinery larger than the precision being sought.

The gap is not small and it is not a refinement. Mars near opposition is seen about twenty-five arcseconds from where it is, which is fifty thousand times the residuals of a modern fit. An ephemeris that got the corrections wrong and the dynamics right would be useless; an ephemeris that got the corrections right and the dynamics slightly wrong would be almost as good as the real thing. Most of the work of fitting one is in the translation rather than in the mechanics.

The equation that contains its own answer #

The light-time condition is one line and the line is circular:

$$\tau = \frac{\left|\mathbf{r}(t - \tau) - \mathbf{R}(t)\right|}{c}.$$

The observer is at $\mathbf{R}$ at the instant of observation $t$; the target was at $\mathbf{r}(t-\tau)$ when it emitted. To evaluate the right-hand side the answer is needed. The standard resolution is the obvious one: guess $\tau = 0$, evaluate, use the result as the next guess, and repeat.

What makes it converge, and converge fast, is that changing $\tau$ by $\delta$ changes the distance by at most $v\delta$, so the error is multiplied by $v/c$ on every pass. For Mars, $v/c$ is $8 \times 10^{-5}$. The initial error is the full distance the planet travels while its light is in transit — some eighteen thousand kilometres — and after three passes it is under a millimetre. Ephemeris codes therefore do not test for convergence; they run a fixed number of iterations, because the number is known in advance and the test would cost more than the extra pass.

The correction is not optional at any level of precision anybody has ever worked at. Rømer measured the light-time itself in 1676 by watching Io’s eclipses run late as the Earth receded, and the twenty-two minutes he reported across the diameter of the Earth’s orbit is exactly this quantity. What has changed since is only that it is now applied rather than discovered.

There is a subtlety in the equation that is easy to miss and expensive to get wrong. The light-time is not the distance divided by $c$ evaluated at any single instant — it is the distance between two events at different times, one on the target’s worldline and one on the observer’s. For a target whose distance is changing quickly the difference between “the distance now” and “the distance at emission” is itself of order $v\tau$, which is the same size as the effect being computed. Codes that compute a light-time from a current range and then apply it as a delay are correct to first order and wrong at the metre level, which is exactly the level modern ranging works at.

A second subtlety concerns which event the light left. For a radar bounce the signal leaves the station at one time, reflects at another and returns at a third, so there are two light-time equations and they have to be solved in sequence rather than doubled. The up-leg and the down-leg differ by the target’s motion during the round trip — for Mars near opposition, about two thousand kilometres — and a code that assumes the two legs are equal has introduced an error a thousand times the measurement precision.

Two things called aberration, and only one of them is the observer’s #

The second correction is a direction rather than a delay, and there are two of them that get confused because both are a velocity divided by the speed of light.

The observer’s own motion tilts every arriving ray by $v_\oplus/c$, which is 20.4955 arcseconds and is the same for every object in the sky — the annual aberration Bradley found while looking for something else. It is a property of the telescope’s velocity, not of the target, and applying it is a rotation of the entire sky.

The target’s own motion during the light-time is a different statement. The body moved, so the direction the light came from is the direction of where the body was, and the offset in angle is the displacement divided by the distance. The displacement is $v_{\rm targ}\tau$ and the distance is $c\tau$, so the angle is $v_{\rm targ}/c$ — and the light-time cancels out completely.

This is why an ephemeris carries three different kinds of place and three different names for them. The geometric place is where the body is, in the chosen frame, at the given instant — the thing the integration produces and the thing nobody observes. The astrometric place is the direction after the light-time has been applied but before aberration: it is what a catalogue position of a star is, and it is the right thing to compare a planet against a star with. The apparent place has aberration and light deflection applied as well, and it is where a telescope must actually point.

Getting the wrong one is a classic and expensive mistake, because all three agree to within an arcminute and the residuals being chased are microarcseconds.

The vocabulary is worth memorising because it is a record of the operations rather than an arbitrary set of names. Each word marks a stage of the pipeline, and knowing which stage a published number belongs to is the difference between a comparison and a category error. A catalogue star’s position is astrometric by construction — the catalogue removed aberration when it was compiled — so comparing a planet’s apparent place against a star’s catalogue place is comparing two things twenty arcseconds apart. This is the sort of mistake that survives review, because the numbers look right and the discrepancy looks like a systematic in the data.

The related quantity that has to be applied to the direction, between the light-time and the aberration, is the gravitational deflection of the incoming ray. For a target on the far side of the solar system it is the same effect as the Shapiro delay in the next section, seen as a bend rather than a delay: 1.75 arcseconds at the solar limb, falling as the inverse of the impact parameter, and still four milliarcseconds at ninety degrees from the Sun. Unlike the delay, this one does fall as a power, so it can be neglected in a way the delay cannot — but not at the microarcsecond level, where the deflection by Jupiter has to be included for a source passing within a degree of it.

A delay that is not a distance #

The third correction is the one with no Newtonian counterpart. A radio signal passing near the Sun arrives later than the straight-line distance divided by $c$ would predict, by

$$\Delta t = \frac{4GM_\odot}{c^3}\ln\!\left(\frac{r_1 + r_2 + R}{r_1 + r_2 - R}\right)$$

for the round trip. It is a logarithm of the impact parameter, which means it falls off extremely slowly.

The slowness of the fall-off is the practically important part. A term that went as an inverse square would be negligible away from conjunction and could be handled as a special case; a logarithm is a few tens of microseconds everywhere. Every range measurement in the solar system carries it, and it has to be modelled with the same care as the orbit.

What was actually measured #

The Shapiro delay is the correction in this essay that has been measured rather than merely applied, and the measurement is one of the sharpest tests of general relativity there is.

The coefficient of the logarithm is proportional to $(1 + \gamma)/2$, where $\gamma$ is the parameter measuring how much space curvature a unit mass produces — one in general relativity, zero in Newtonian gravity with light bending only through the equivalence principle. Shapiro proposed the test in 1964 and the first radar bounces off Mercury and Venus gave it to a few per cent by 1971. The Viking landers, ranging from the surface of Mars with a transponder rather than a passive reflector, took it to a part in a thousand in 1979.

The best measurement is Cassini’s, in June 2002, made during a solar conjunction on the way to Saturn. Cassini used a multi-frequency link — X-band up, X and Ka down — precisely so that the solar plasma’s contribution, which scales as the inverse square of the frequency, could be measured and removed rather than modelled. The result was $\gamma - 1 = (2.1 \pm 2.3)\times 10^{-5}$: consistent with general relativity, and the tightest constraint on that parameter from any experiment.

Note what makes the measurement possible. The delay is huge near conjunction and the plasma noise is huge there too, and the two are separated by their different frequency dependence rather than by their size. This is the same trick as dispersion in a pulsar’s arrival times, used for the opposite purpose: there the frequency dependence is the signal and here it is the contaminant.

One historical detail makes the Cassini result sharper than it looks. The plasma removal is not a small correction that happened to be handled well: at a solar elongation of a fraction of a degree the plasma delay at X-band is larger than the relativistic delay, and it varies on timescales of minutes as blobs of corona cross the line of sight. A single-frequency experiment at that elongation measures the corona. What the multi-frequency link does is measure the corona too, at a second frequency, and take the difference — the same manoeuvre as separating dust from distance by observing in two colours, and for the same reason: a contaminant with a known frequency dependence is not a contaminant, it is a second observable.

The other two corrections are not tested in the same sense, because they are kinematic rather than dynamical: the light-time equation follows from the constancy of $c$, and aberration follows from the addition of velocities. What is tested, continuously, is that the whole assembly is self-consistent. An ephemeris fits optical positions from the nineteenth century, radar ranges from the 1960s, and spacecraft ranges from the 1970s onwards, and those data types are sensitive to the corrections in completely different combinations. A systematic error in the light-time model would show up as a data-type-dependent residual, and it does not.

It is worth recording what happens when the corrections are simply omitted, because the failure is instructive rather than catastrophic. An ephemeris fitted to optical observations with no light-time correction does not blow up; it converges, to an orbit whose mean longitude is offset by about the light-time times the mean motion and whose other elements absorb the rest. The fit is internally consistent and predicts future positions almost as well as the correct one, for as long as the observing geometry stays similar. It fails the moment a new data type arrives with a different geometry — which is exactly what happened when radar ranging joined optical astrometry in the 1960s, and is why that decade produced a step change in the solar system’s known scale rather than a gradual improvement.

The order matters, and so does the frame #

Applying three corrections requires deciding what order to apply them in, and the order is not arbitrary. Light-time comes first, because it decides which position is being talked about. Gravitational deflection is applied to that direction next. Aberration comes last, because it is a transformation of the observer’s frame and everything before it is expressed in the frame the ephemeris lives in.

That frame is worth naming. A modern ephemeris is expressed in barycentric coordinates with barycentric coordinate time as its independent variable — a time coordinate that runs at a different rate from a clock on Earth by about a part in $10^8$, for reasons the collection meets in the six kinds of second. Every position in the table is at a coordinate time, every observation is at a proper time, and the conversion between them is itself part of the model.

This is why an ephemeris is distributed as a set of Chebyshev coefficients together with a piece of software, rather than as a table. The table alone would be ambiguous. What has to be shipped is the definition of what the numbers mean, and most of that definition is the content of this essay.

Where the picture stops #

The picture stops in three places, and the third is the one that has started to matter.

The observer’s position has to be known too. Everything above treats $\mathbf{R}(t)$ as given, and it is not: a station on the Earth’s surface has to be located to centimetres in a frame that is rotating, precessing, wobbling and deforming under load. For the most precise tracking the observer’s own model is the larger error source.

A body is not a point. What is observed optically is a photocentre — the brightness-weighted centre of a partly illuminated disc — and it is displaced from the centre of mass by up to a fraction of the body’s radius depending on phase. For Mercury and Venus that is tens of milliarcseconds, and it is a systematic that varies with elongation in exactly the way an orbital error would.

And two of the three corrections are frame-dependent in a way the third is not. The Shapiro delay quoted above is the value in one particular coordinate system, and a different but equally valid coordinate choice moves the split between “the light took longer” and “the path was longer”. What is invariant is the total round-trip proper time, which is what is measured. The decomposition is a bookkeeping convention, and reporting it as though it were a physical effect that could be separately confirmed is a category error — though the coefficient is physical, which is what makes Cassini’s number meaningful.

Why this is a rung on the ephemeris ladder rather than a footnote #

An ephemeris is a fit, and what a fit does is compare a model with observations. The comparison happens in the space of the observations, never in the space of the model. So the machinery that converts a position into an observed quantity is not a post-processing convenience: it is half the model, and the half that is easier to get wrong, because it is the half without a conservation law to check it against.

There is a second reason the translation deserves a rung of its own. The corrections are not independent of the quantities being fitted, and some of them are nearly degenerate with those quantities. A light-time error looks like an error in the semi-major axis, because both delay the observed phase of the orbit. A systematic error in the aberration constant looks like a rotation of the reference frame. A mis-modelled Shapiro coefficient looks like a range bias that depends on elongation, which is to say like an eccentricity error. In each case the fit will absorb the mistake into a parameter and report a small residual, and nothing in the residuals will say that anything is wrong.

That is why the corrections are validated by consistency across data types rather than by looking at the fit quality. Optical positions carry aberration and no range; radar carries range and Shapiro delay and no aberration; spacecraft Doppler carries a time derivative of everything. A model error in any one correction produces disagreement between the data types, and the disagreement is what is monitored.

The point generalises past the solar system. The same three corrections appear, with different sizes, in pulsar timing, in very long baseline interferometry, and in the reduction of every space astrometry mission. In each of them the translation from state to observable is a longer piece of code than the dynamics, and in each of them a mistake in it looks exactly like a discovery.

A concluding observation, and it is the reason this material is worth a reader’s time rather than a code comment. Every one of these three corrections was, when it was first written down, a discovery about the universe: the finite speed of light, the addition of velocities, the curvature of space near a mass. All three are now boilerplate in a subroutine that runs before the interesting part of the calculation begins. That is what a successful physical theory looks like from the inside — not a result that is celebrated but a term that is applied without comment, and whose omission would be caught immediately by a residual nobody is looking at.

Where the ladder goes next #

The obvious next rung is what happens when the three corrections are not separable from the parameters being fitted — when the light-time and the semi-major axis trade against each other in a way no single observation can distinguish. The rung after that is the timing model of a pulsar, where the same corrections appear with a clock so good that the Earth’s own orbit has to be solved for as part of the fit.