A longitude that drifts with the rotation rate

Every essay in this ladder so far has been about the shape of the body being mapped — whether it is a sphere at all, and what the same projection does on a different one: whether it is a sphere, an ellipsoid of revolution, a triaxial ellipsoid or none of those; where the cut goes; what a conformal map of it looks like. Shape is what makes a planetary map hard, and it is not what makes a planetary coordinate uncertain.

A latitude on another body is a measurement of the body, and a coordinate without its system is not a location in any case. A longitude is worse than that. It is a measurement of a landmark, carried forward by a rotation rate:

$$W(d) = W_0 + \dot W\,d,$$

where $d$ is the number of days since the epoch. So a longitude is a statement about a date as much as about a place, and every error in $\dot W$ is a longitude error that grows without bound as the date recedes from the epoch.

What the working group actually publishes #

The International Astronomical Union’s working group on cartographic coordinates publishes, for each body, three numbers that fix its orientation in space and a fourth that fixes where longitude zero is: the right ascension and declination of the north pole, the prime meridian angle $W_0$ at the epoch, and the rate $\dot W$.

The prime meridian is defined by a landmark. Mars’s is the crater Airy-0; Mercury’s is Hun Kal, a small crater chosen in 1970 to sit at exactly 20° west; Vesta’s is Claudia; the Moon’s is the mean direction of the Earth. Jupiter has no surface at all, so its System III longitude is defined by the rotation of its magnetic field, which is the only thing on the planet that is plausibly rigid.

That construction is a datum in this field’s own sense — a set of marked points and a date — with one crucial difference. An Earth datum’s marks are all on the body and stay there; a planetary prime meridian is one mark plus a clock, and the clock is the part that is uncertain.

The rate is quoted to a stated number of places, and that is the error #

The right error scale here is not a formal uncertainty from somebody’s fit, because none is published beside the rate. It is the number of decimals actually printed.

Mars’s rate is 350.891982443297 degrees a day, to twelve decimals. Vesta’s is 1617.3329428, to seven. Phobos’s is 1128.8445850, to seven. Mercury’s is 6.1385108, to seven. A number written to seven decimals is a claim that the eighth is unknown, and half a unit in the last place is what a user of the table is entitled to assume.

That gives a drift that is trivially computable and startlingly uneven.

The angle does not know how fast the body turns #

Here is the first thing that is not obvious and that the table of constants does not make visible.

Six of these seven rates are published to seven decimals. Every one of those six therefore drifts by exactly the same angle after the same elapsed time — $9.13 \times 10^{-4}$ degrees after fifty years — because the drift is the rate error times the time and the rate error is a fixed number of degrees per day. Vesta turns 1,641 times a year and Venus turns 1.5 times a year, and their prime meridians drift by identical amounts.

What separates them on the ground is the radius, and only the radius. Jupiter, at 71,492 km, reaches 1,139 metres. Venus reaches 96. Vesta, at 286 km, reaches 4.6. Phobos, at 13 km, reaches 21 centimetres.

The exposure is the radius times the published precision, and nothing else. That is a statement about the table rather than about the solar system: the working group quotes every rate to a similar number of places, and the bodies differ in size by four orders of magnitude, so the resulting positional uncertainty differs by four orders of magnitude too, with no relation to how well any of them is known.

And the relative precision runs the other way #

This is the inversion worth carrying out of the essay. A fixed number of decimals is not a fixed precision: it is an absolute precision, and the relative precision it amounts to depends entirely on the size of the number. So the fastest rotator in the table is specified a thousand times more tightly, in relative terms, than the slowest — and the two are equally exposed, because a longitude cares about the absolute error.

Mars is the exception and the reason is the obvious one. Its rate is published to twelve decimals because a great deal of mapping depends on it, and the result is a drift of half a millimetre a century against Jupiter’s two kilometres. Precision in a table is a decision about how much somebody cared, and the decision is visible in the digit count.

How many decimals a body actually needs #

The exposure is a product of three things — the rate’s last printed decimal, the elapsed time and the radius — so it inverts into the number a table ought to carry: how many decimals a rate needs to hold a stated ground tolerance over a stated span.

Half a unit in the seventh decimal is 5 × 10⁻⁸ degrees a day, which over a year is 1.83 × 10⁻⁵ degrees. Multiplied by each body’s radius that is a drift rate on the ground:

Read down that table and the working group’s own decision about Mars stops looking like carefulness and starts looking like arithmetic. At seven decimals a Martian longitude drifts a metre a year, which crosses a lander’s landing ellipse inside a mission’s planning horizon; at twelve it drifts half a millimetre a century. Mars is quoted to twelve decimals because seven is not enough for Mars and twelve is generous, and there is nothing between them anybody needed.

The general rule is one logarithm. To hold a tolerance t over T days on a body of radius R, the rate needs about log₁₀(RT/t) decimals, with R and t in the same units and the angle in degrees carrying the usual factor. Jupiter holding a hundred metres for a century needs nine; Mars holding a metre for a century needs ten. Both are more than the seven that six of the seven bodies are published to, and neither is a demanding requirement on the underlying measurement — it is a requirement on how many of its digits get printed.

That is the useful reframing of the whole rung. The drift computed here is not a statement about how well anybody knows these rotation rates. It is a statement about the width of a column in a table, and the column is the interface.

The shape ladder and the clock, side by side #

It is worth putting the two kinds of planetary coordinate error next to each other, because this ladder has spent ten rungs on one of them.

The latitude problem is a convention problem. Two definitions exist, they differ by a computable amount that depends on the flattening, and the fix is a sentence in a metadata field. Once the sentence is there the error is gone, permanently, for every date.

The longitude problem is an extrapolation problem. One definition exists, everybody uses it, and it produces a different answer every year because it contains a rate that is not exact. No sentence fixes it; only a re-observation does, and a re-observation moves the archive.

That is why the two belong in the same anchor and why the second one had to wait for the first ten rungs. A reader who has not seen how carefully the shape is treated has no scale against which to notice that the clock is treated less carefully.

The error is a rigid turn #

Every other error in this field is a field: a datum shift moves different places by different amounts, the seven parameters each do something different to different parts of the world, and the pattern is what makes them fittable. A rotation-rate error has no pattern at all.

That has a consequence for detection. A datum error can be found from within the body’s own coordinate system, by comparing two observations of the same marks; a rotation-rate error cannot, because it moves everything together. Finding it needs an observation tied to something outside the body — a star, a spacecraft’s radio link, an occultation — which is exactly how these rates are determined and re-determined.

Revising the rate moves every longitude but one #

This is the part that catches archives. When a rate is revised, the working group does not simply publish a new number: it publishes a new $W_0$ as well, chosen so that the landmark is in the right place at the epoch the new observations were made. Every longitude at every other date changes, and the change is proportional to the interval from that epoch.

So a coordinate published in 1990 and a coordinate published in 2020 for the same feature can differ, on the same body, with the same instrument, because the elements were revised in between and the two are extrapolations from opposite sides of a pin. Nothing about the feature moved.

The epoch is part of the coordinate makes the corresponding argument for the Earth, where the cause is plate motion and the size is centimetres a year. The planetary case has the same shape and a different mechanism: on Earth the ground moves relative to the frame, and here the frame moves relative to the ground.

Which of the two is worse #

The comparison is worth making because it is not obvious which way it goes.

Terrestrial plate motion is a few centimetres a year, differential across the surface, and modelled to millimetres by velocity fields that are themselves fitted and published. A planetary rotation-rate error is uniform, unmodelled, and — for a body whose rate is quoted to seven decimals and whose radius is large — tens of metres a decade.

So for Jupiter the rotation term is two orders of magnitude larger than anything plate motion does on Earth, and for Phobos it is two orders smaller. The quantity is not “planetary coordinates are worse”; it is that the shape of the uncertainty is different, and a user importing terrestrial intuitions imports the wrong shape.

What this does to a mosaic #

The practical consequence is in image products rather than in coordinates.

A mosaic is assembled by projecting many images into a common frame, and each image’s position in that frame comes from where the spacecraft was and which way it was pointing — both known well — and from where the body’s prime meridian was at the moment of exposure, which is $W_0 + \dot W d$. Two images taken years apart are therefore registered with rate errors that differ by the elapsed time between them.

What that produces is not a blur. It is a seam: two strips of surface, each internally sharp, offset from each other along the direction of rotation by an amount proportional to their separation in time. On a body where the rate is good the seam is sub-pixel and invisible; on one where it is not, it is a feature that looks geological.

The remedy is the obvious one and is a great deal of work. Every image is re-projected with a single consistent set of elements, and where the elements have been revised the whole mosaic is rebuilt rather than patched. That is why planetary image archives version their control networks and why a coordinate quoted from one version cannot be compared with a coordinate from another — the same discipline a published coordinate is a result argues for on Earth, with a different reason behind it.

The landmark is the definition and it is a crater #

There is one more asymmetry with the Earth and it is uncomfortable.

A terrestrial datum’s marks are brass bolts and stone posts, and each has a plaque and a maintenance record. A planetary prime meridian is a small crater identified in an image, and the coordinate is the position of a feature whose centre is a judgement about pixels. Hun Kal is about 1.5 km across on a body 2,440 km in radius; putting its centre wrong by a tenth of its diameter is a longitude error of 0.0035°, which is 150 metres at Mercury’s equator.

That error does not grow with time — it is a constant offset, absorbed into $W_0$ — so it is the smaller problem of the two and it is the one nobody can reduce without re-imaging the crater.

A drift that is measurable is a drift that is bounded #

There is a reassuring half to this and it should be said.

The rates in the table are not guesses. They come from decades of tracking, occultation timings and spacecraft radio science, and the reason Mars’s carries twelve decimals is that it has been measured that well. The exposure computed here is what the published precision implies, which is an upper bound on the real uncertainty and often a generous one — a working group that prints seven decimals may know the eighth perfectly well and have declined to print it.

So the honest reading of the figures is not “planetary longitudes are wrong by a kilometre”. It is that a user who has only the table has no way to know they are not, and that the table is the interface almost everybody uses. The difference between an uncertainty and a published uncertainty is the whole subject of the seven parameters have their own uncertainty, one field over, and it has the same answer: the number a user can defend is the one they were given.

Where the model stops #

The linear model $W = W_0 + \dot W d$ is not the whole of what the working group publishes. Several bodies carry periodic terms — the Moon’s has thirteen — and a few carry a quadratic in $d$ for a secular acceleration. Those are physics rather than uncertainty, and they are omitted here because they do not change the argument: an error in a periodic term is bounded, and an error in the linear rate is not.

What is also omitted is the pole. The right ascension and declination of the rotation axis have their own published precision and their own drift, and an error there is a tilt rather than a turn — it moves latitudes as well as longitudes, and it is not rigid. That is a different measurement and it is a rung this ladder has not written.

Who found it, and when #

The convention is old and the caution is not. The IAU working group’s reports have carried a standing note since the 1980s that longitudes computed from superseded elements should be recomputed rather than reused, which is the correct advice and is followed unevenly, because a longitude in a table looks like a property of a place rather than of a date — the same misreading two parameter sets, one transformation records for terrestrial datums.

The sharpest statement of the problem came out of Mars mapping in the 1990s, when several early spacecraft datasets turned out to be internally consistent and mutually offset by a few kilometres — the signature of a rate revision applied to some products and not others, on a body where a kilometre is a lander’s landing ellipse.

Where the ladder goes next #

The bodies anchor has spent eleven rungs on what a coordinate on another body refers to: the shape it is measured against, the cut the map was made along, and now the clock it was carried forward by. What it has not asked is what happens when two bodies’ coordinate systems have to be related to each other — which is a datum transformation with no common points, because no mark is on both.