A coordinate on another body

Every coordinate on this site is on the Earth. The machinery is not: the code that computes auxiliary latitudes, meridian arcs and datum shifts has been parameterised by a semi-major axis and a flattening since the site’s second phase, and nothing in it knows which body those two numbers describe.

Pointing it at Mars costs two constants — 3,396.19 and 3,376.20 kilometres — and produces a set of measurements about a place where the conventions this field is built on are not settled.

Two latitudes, and both of them are in use #

Geodetic against geocentric latitude measures the Earth’s version of this and finds a gap of up to 11.5 arcminutes. What that essay does not have to say, because the answer has been settled since the eighteenth century, is which one a coordinate means: on Earth, always the geodetic one — the angle of the normal — because that is what a levelled instrument measures and what a satellite fix reports.

On Mars there is no such settlement. The planetographic latitude — Mars’s word for the geodetic one — is what the observing tradition used and what the older map products carry. The planetocentric one is what spacecraft navigation uses, because a trajectory is computed in a body-centred frame where the angle at the centre is the natural one, and it is what most modern data products publish.

Both appear in current use, and the difference is not decorative:

Mars’s gap is 1.76 times the Earth’s in degrees and almost exactly the same on the ground, because Mars’s radius is smaller by very nearly the same factor. Which of the two comparisons is the right one depends entirely on what the number is for: an angle in a pointing calculation, or a place a lander is aimed at.

The gap is the flattening, and the formula says so #

The two latitudes are related by one line: tan φ_c = (1 − f)² tan φ_g. Expanding for small f gives

$$\varphi_g - \varphi_c \approx f \sin 2\varphi$$

so the gap is zero at the equator and at the pole, peaks at 45°, and is f radians there. Every row of the table above is that formula with a different f in it.

The first-order prediction is checked rather than assumed, and the check has two halves — the site’s rule is that an assertion which cannot fail is not an assertion. For Mercury, the Earth and Mars the measured peak matches f radians to one per cent. For Jupiter it is 3.3 per cent out and for Saturn 5.1, and it must be, because f = 1/15 is not small and a first-order expansion in it should visibly fail. Both the agreement and the failure are asserted.

The peak also drifts: 45° on the nearly-spherical bodies and 47° on Jupiter, 48° on Saturn. That is the second-order term arriving, and it is the sort of thing that is invisible until a body flat enough to show it is put through the same code.

The ground distance is the body’s own polar deficit #

The last column of the table is a product of two numbers and it collapses into one, which is worth doing because it turns five measurements into a sentence anybody can carry.

The gap peaks at f radians. A gap in radians becomes a ground distance by multiplying by the radius, so the worst ground separation between the two latitude conventions is fa — and fa is, by the definition of the flattening, exactly a − b: the difference between the body’s equatorial and polar radii.

So the largest distance the choice of latitude convention can move a point on any body is the amount by which that body is squashed, measured in kilometres. The Earth is 21.4 kilometres wider than it is tall and its gap on the ground is 21.4 kilometres. Mars’s two published radii are 3,396.19 and 3,376.20, a difference of 19.99, and its gap is 20.0. Mercury’s is 2.3 and Jupiter’s about four and a half thousand.

That identity does two things. It explains the near-tie between Mars and the Earth without any arithmetic about ratios of ratios: the two bodies are squashed by almost the same number of kilometres, one by being large and slightly flattened and the other by being small and more so. And it removes the last reason to think of the ground column as a separate measurement — it is one subtraction on the two radii the IAU already publishes, and it needs no latitude, no trigonometry and no code.

It also fixes the scale of what a convention error costs on any body that might be added to the table. A body’s polar deficit is the first thing a shape fit reports, so the answer to how much does it matter here is available before anything is computed. On a body whose two radii differ by a kilometre it matters by a kilometre.

The identity is first order, like the law it comes from, and it fails where that law fails. Jupiter’s actual peak is 4,688 kilometres against a deficit of 4,642, an excess of one per cent; Saturn’s is 6,000 against 5,709, an excess of five. Both excesses are the second-order term that also moves the peak latitude off 45°, and both are in the direction the expansion predicts — which is the same evidence, read a second way, that the first-order treatment is right on the three small bodies and not on the two large ones.

The Moon is a sphere by agreement #

The Moon’s actual figure is flattened by about one part in eight hundred, and its published coordinate system uses a sphere of radius 1,737.4 kilometres.

That is a convention rather than a measurement, and it is the clearest case on this list of a decision that could have gone the other way. Adopting a sphere makes the two latitudes identical, removes the whole question this essay is about, and costs whatever the flattening would have contributed — which for the Moon is a few hundred metres of radius variation, well inside the uncertainty of the older mapping and not inside the uncertainty of the modern altimetry.

The site’s own view of that trade is in the Earth is a sphere, and when it is not: the sphere is not an approximation to be apologised for, it is a model whose adequacy is a measurable question with a stated tolerance. Lunar mapping answered that question one way in 1961 and has not revisited it, and the consequence is that a lunar coordinate is unambiguous in a way a Martian one is not.

Longitude is not settled either, and the disagreement is 360° #

Latitude has two conventions that differ by a fraction of a degree. Longitude has two that differ by the sign of the whole coordinate.

The IAU’s rule is that longitude increases in the direction opposite to the body’s rotation as seen from the north — which makes it east-positive for direct rotators like Mars and west-positive for retrograde ones like Venus. Except that the rule has been revised, that the Moon was west-positive until 1961 and is east-positive now, and that Mars’s own products carry both: the older areographic convention is west-positive and the current planetocentric one is east-positive.

So a Martian longitude of 45 may mean either of two places 90° apart, and nothing in the number says which. That is a coordinate without its system is not a location in its sharpest form — a discrepancy of a quarter of the planet, in a pair of numbers that look entirely ordinary.

The failure mode is worse than an outright error, because a west-positive longitude read as east-positive still lands on the body, still parses, and still produces a plausible map. The site’s axis-order complaint about latitude and longitude being swapped has the same character and a smaller magnitude.

What is the same, and what is not #

Three things carry over from the Earth unchanged, and it is worth being explicit about which, because the transfer is what makes the rest of this ladder cheap.

The auxiliary latitudes carry over. The six angles this field computes — geodetic, geocentric, parametric, authalic, conformal, rectifying — are defined by the ellipsoid’s shape and nothing else, so they exist on Mars with the same formulae and a different f.

Which of them a Martian map needs is decided exactly as it is here: equal-area on the wrong body shows that an equal-area projection fed geodetic latitudes is not equal-area, and the repair is the authalic latitude. That failure is 1.76 times larger on Mars for the same reason everything else in this essay is.

The datum question carries over and gets harder. A datum is a shape, a placement, a set of marked points and a date; on Mars the marked points are craters whose positions come from spacecraft imaging, and successive missions have produced successive realisations that differ by kilometres. A published coordinate is a result applies with more force there than here.

The epoch question carries over and gets easier. The epoch is part of the coordinate is about plate motion, at centimetres a year. Mars has no plate tectonics, so its surface points do not drift relative to each other, and a Martian coordinate does not need a date in the way a terrestrial one does.

The fix exists and is not universally applied #

The terrestrial answer to “which convention is this coordinate in” is an identifier: EPSG:4326 names WGS84 geographic coordinates, latitude first, in degrees, and a file that carries the code is unambiguous.

Planetary bodies have the same machinery. The IAU’s 2015 report defines a numbering scheme, and codes such as IAU_2015:49900 for Mars’s planetocentric system are registered exactly as terrestrial ones are; the underlying software libraries have supported them for years.

What is not universal is their use. A great deal of planetary data is distributed as arrays with a header saying latitude and longitude and no statement of which latitude or which direction of longitude, which leaves the reader inferring the convention from the mission, the date and the file format. The site’s complaint in a published coordinate is a result is that a number without its provenance is not a measurement; here the missing provenance is worth 20 kilometres in latitude and half a planet in longitude.

A datum needs a surface, and Mars has no sea #

One thing does not carry over cleanly, and it is the vertical.

A terrestrial height is measured against the geoid, which is an equipotential surface fitted to mean sea level — a definition with an ocean in it. Mars has no ocean, so its equivalent surface, the areoid, is defined as the equipotential whose mean radius over the equator matches a stated value, computed from a gravity model derived from spacecraft tracking.

That makes the Martian vertical datum a shorter chain than the terrestrial one and a differently uncertain one: no tide gauges, no century of levelling, no permanent tide question — and total dependence on a gravity field known only from orbit. Height above what? is the essay about the terrestrial chain, and the Martian one is the same argument with two links removed and one link made load-bearing.

What is measured here and what is cited #

This site computes what it prints, and a planetary essay makes that rule work harder, so the boundary is worth stating.

Computed here: every latitude conversion, every gap, every ground distance, the first-order law and its failure, and every projection quantity in the rest of this ladder. All of it follows from two radii by the same code that handles the Earth.

Cited, not computed: the radii themselves. Mars’s 3,396.19 and 3,376.20 kilometres are the IAU working group’s values, and they are the output of a fit to spacecraft tracking and altimetry — an inference from observations that belongs to a different subject entirely. This site takes them as given and says so, exactly as it takes WGS84’s numbers as given.

That division is the same one the figure of the Earth was measured draws for the terrestrial case: how the numbers were obtained is a story about instruments and campaigns; what the numbers mean for a coordinate is this field’s.

The reason this ladder is cheap #

Every measurement in this essay came out of existing code with two constants added, and that is not a remark about convenience — it is the strongest evidence available that the library was written about the right thing.

A file that computes the six auxiliary latitudes of the Earth and a file that computes the six auxiliary latitudes of any ellipsoid look identical from outside and are different objects. The difference shows up exactly once, at the moment somebody points the second at Jupiter and gets four-degree spreads that are correct. Had the ellipsoid machinery hardcoded WGS84’s numbers anywhere — as the grid-scale computation effectively did until two grids over the same ground found the bug it caused — this ladder would have needed a rewrite rather than a table.

The site’s habit of parameterising everything is usually justified as good practice. This is what it buys.

Where the model stops #

The bodies here are ellipsoids of revolution, which is a decision the IAU has made for each of them and which is not always a good one. The third rung of this ladder is about the bodies where it fails outright.

Topography is not in any of this. Every number above is about the reference figure, and a real coordinate on Mars sits on ground that departs from that figure by up to twenty-one kilometres — Olympus Mons against the Hellas basin — which is a far larger number than anything in the tables. The relation between the two is a height question and it is the same question the terrestrial field spends four essays on; what is different is that Mars’s vertical datum is defined by an areoid computed from a gravity model rather than by tide gauges, so the whole chain is shorter and better documented.

No claim is made about which convention a reader should use. The measurable content is how much the choice matters, which is the table above.

Who found it, and when #

The planetographic and planetocentric distinction is as old as planetary mapping: the observing tradition measured what it could see, which is the tilt of the surface, so latitudes were graphic; the mechanical tradition computed from a centre, so they were centric.

The IAU’s Working Group on Cartographic Coordinates and Rotational Elements has published the reference values since 1979 and revises them every few years. Its reports are unusually candid about exactly this problem: they state which bodies use which latitude, note where the community’s practice disagrees with the recommendation, and record when a convention was changed.

The Mars case in particular has a documented turning point. The Mars Global Surveyor era of the late 1990s produced altimetry good enough to make the ellipsoid worth taking seriously, and the community moved from areographic west-positive to planetocentric east-positive around the turn of the century — leaving decades of published coordinates in the other convention, which is why both are still met.

Where the ladder goes next #

The conventions are one half of a coordinate’s meaning. The other half is what happens when a map is drawn of it, and the obvious question is whether Martian cartography is a different subject from terrestrial cartography or the same subject with different constants.

The answer is measurable and it is the second rung: a projection’s distortion depends on the body’s flattening and on nothing else about the body — not its radius, not its mass, not its name. Which means the site’s headline error, Web Mercator’s 0.3848°, is an instance of a law rather than a fact about Earth.