A body that is not an ellipsoid

Both previous rungs assumed the body has an axis. Every formula in them — the two latitudes, the auxiliary latitudes, the flattening’s 2f law — begins from an ellipsoid of revolution, in which the equatorial section is a circle and latitude is a function of position alone.

For two of the bodies in the table that is false, and what fails when it fails is not a refinement.

Three unequal axes #

An ellipsoid of revolution has two distinct axes: equatorial and polar. A triaxial ellipsoid has three, and the difference is not always small.

Vesta’s equatorial section is 2.7 per cent out of round, which is eight times the Earth’s polar flattening. Phobos’s is 12.3 per cent, which is about Jupiter’s — except that Jupiter’s is between the equator and the pole, where a formula can handle it, and Phobos’s is within the equator, where no formula that assumes an axis can.

Latitude stops being a coordinate #

The planetographic latitude of a point is the angle its outward normal makes with the equatorial plane. On a body with an axis, the normal at a given planetocentric latitude points the same way whatever the longitude, by symmetry — so the two latitudes are related by a formula in one variable, which is what makes them interchangeable given the flattening.

On a triaxial body the normal swings. Measured at 45° planetocentric:

The three oblate rows are the control, and they are the reason the first two mean something: on a body with an axis the spread is at the level of the arithmetic, and any non-zero value the code reported for them would be a bug rather than a finding.

So on Vesta the sentence “the point at 45° north” does not name a latitude band unless it also says which convention, and the convention’s answer varies round the body. The site’s standing complaint that a coordinate without its system is not a location reaches its limit here: the system is not enough either, because one of the two conventions is not a function of position in the way the word latitude implies.

The two spreads also stand in a suggestive ratio. Phobos’s equatorial flattening is 12.31 per cent against Vesta’s 2.69, a factor of 4.58, and its spread is 6.400° against 1.404°, a factor of 4.56. Divided through, the two bodies agree on 0.52° of latitude ambiguity per one per cent of equatorial flattening, to within four parts in a thousand.

That looks like a law and it is half of one. Running the same measurement on synthetic bodies — one equatorial axis fixed, the other shortened by a stated fraction, the polar axis held at Vesta’s ratio of 0.78 — gives a spread that is linear in the equatorial flattening near zero and drifts upward from it: 0.511° per per cent at half a per cent of flattening, 0.522 at Vesta’s 2.7, 0.571 at Phobos’s 12.3. So the coefficient is not constant, and on the shape family Vesta belongs to Phobos should have come out around ten per cent higher than it did.

What cancels the drift is the third axis. The coefficient also depends on how squat the body is, and it falls steeply as it does: at a fixed 2.7 per cent of equatorial flattening it is 0.581° per per cent on a body whose polar axis equals its equatorial one, 0.522 at Vesta’s ratio of 0.780, and 0.467 at Phobos’s 0.700. Phobos is both more out of round in its equator, which raises the coefficient, and flatter through the poles, which lowers it, and the two effects happen to cancel to the precision quoted above.

The near-equality is therefore a coincidence of two bodies rather than a law about triaxial figures — worth stating because it is exactly the sort of agreement that gets published as one. What is a law, and holds over the whole family tested, is the linearity: the ambiguity is proportional to the equatorial flattening in the limit of a small one, with a coefficient of about half a degree per per cent at mid-latitudes. That is the sense in which a body with an axis is the zero of this quantity rather than a special case of it.

The latitude matters too, and not monotonically. The spread must vanish at the equator, where the normal lies in the equatorial plane and the planetographic latitude is zero all the way round whatever the axes are, and it must fall towards the pole, where the equatorial section stops entering the geometry. Between the two it peaks: measured on the same synthetic body, the coefficient runs 0.42 at 15°, 0.57 at 30°, 0.51 at 45°, 0.36 at 60° and 0.18 at 75°. The worst place to publish a latitude on a triaxial body is the mid-latitudes, which is where most of any body’s surface is.

The normal and the radius part company by twenty degrees #

The underlying quantity is the angle between the outward normal and the direction from the centre — the two directions the two latitudes are built on. On a sphere it is zero everywhere. On the Earth it reaches 11.5 arcminutes.

On Vesta it reaches 14.1° and on Phobos 20.0°.

That is a large enough angle to change what an instrument would report. A levelled instrument follows the local vertical, which on a body with no atmosphere and no ocean is the direction of the gravitational acceleration — close to the normal for a smooth homogeneous body and not exactly it. A spacecraft’s tracking gives a direction from the centre. Twenty degrees between the two is not a correction, it is a different question with a different answer.

What the two conventions cost, in kilometres #

The angles are one way to read the disagreement and the ground distance is the other, exactly as it was for Mars.

On Vesta, 1.40° of latitude ambiguity at a mean radius of 263 kilometres is about 6.4 kilometres of surface. On Phobos, 6.40° at a mean radius of 11.2 kilometres is about 1.25 kilometres. Neither is large in absolute terms and both are enormous relative to the bodies: Phobos is 27 kilometres long, so the ambiguity is five per cent of the whole body, and a landing site specified without its convention could be a tenth of the way round the moon from where it was meant.

That relative comparison is the one that matters for small bodies and it inverts the terrestrial intuition. On Earth a fifth of a degree is 21 kilometres and is a rounding error at planetary scale; on Phobos six degrees is a kilometre and is a substantial fraction of the target.

Why the site’s own machinery still worked #

Every number in this essay came out of code written for oblate bodies, with the third axis simply allowed to differ from the second. That it did is worth a paragraph, because it is the reason the measurements can be believed.

The latitude-ambiguity routine takes three axes and computes a surface point and its normal; given two equal axes it reproduces the oblate case exactly, and the check that it does is the three rows of the table above that come out at 10⁻¹⁴. The area quadrature takes three axes and integrates; given three equal ones it returns 4πR². Neither routine has a special case in it for the triaxial situation, so nothing about the triaxial numbers depends on a branch that the oblate numbers do not also exercise.

That is the pattern the site’s own habit recommends for any generalisation: make the general case the only case, and let the special case be a check rather than a code path. The alternative — a separate triaxial routine — would have been easier to write and would have had nothing testing it.

The area has no closed form, so it is integrated #

An ellipsoid of revolution’s surface area has a closed form — elementary, if unmemorable. A triaxial ellipsoid’s does not; it needs elliptic integrals, and the honest thing at this site’s level of machinery is quadrature.

Integrating over the parameterisation gives 865,634 km² for Vesta and 1,561 km² for Phobos, against 867,221 and 1,567 for the spheres of the same mean radius — so the sphere over-states each by about a fifth and two fifths of a per cent.

The number worth having from that is not the area. It is the check: the same quadrature applied to a sphere returns 4πR² to a part in 400,000, which is the evidence that the integration is right rather than plausible. A measurement whose method has not been shown to reproduce a known answer is not a measurement, and this is the cheapest possible demonstration of that rule.

What a projection of such a body would have to do #

The site’s projection machinery assumes a metric with an axis of revolution: M(φ) and N(φ) are functions of latitude alone, and every distortion quantity is computed from them. On a triaxial body both principal radii of curvature depend on longitude as well, so the metric is a function of two variables and nothing in the library’s signature admits one.

What would change, precisely:

• The scale factors would depend on position rather than latitude, so every profile figure on this site — scale against latitude, distortion against latitude — becomes a field rather than a curve.
• The auxiliary latitudes would not exist. Each of the six is defined by an integral along a meridian, and on a triaxial body every meridian is a different curve, so there is no single conformal latitude to substitute.
• The named projections would lose their derivations. Mercator’s formula comes from integrating sec φ along the meridian; with a longitude-dependent metric that integral is different on every meridian, and the resulting map is not the Mercator projection of anything.

None of that makes the body unmappable — a numerical conformal map of a triaxial ellipsoid exists and has been computed in the literature — and all of it means the map is a numerical object rather than a formula. That is the same distinction when a formula is not enough draws for datum transformations, arriving here for the shape rather than for the placement.

Where this leaves the idea of a datum #

For a body like Vesta, what does a published coordinate mean?

The answer in practice is that it means a position in a body-fixed frame defined by the rotation axis and a prime meridian pinned to a named surface feature, with the shape carried by a shape model — a mesh of hundreds of thousands of triangles derived from imaging — rather than by an ellipsoid at all.

That is a different kind of object from a terrestrial datum, and the difference is worth naming precisely:

• A terrestrial datum is a smooth surface plus a placement. Heights are measured against it and coordinates are computed on it, and it is analytic, so a formula can be written for anything.
• A small-body frame is an axis, a meridian and a mesh. Nothing is analytic. A latitude is defined by convention as the planetocentric one — the angle at the centre — precisely because that one is a function of position on any shape whatever, and the planetographic one is abandoned.

That choice is the interesting one and it is forced by the measurements above. Planetocentric latitude survives the loss of the axis; planetographic latitude does not. So the convention that terrestrial practice discarded — the Earth uses the geodetic latitude everywhere — is the one small-body practice keeps, for the reason that it is the only one that still means anything.

What survives, and it is the same list as always #

Two things carry over to a body of any shape at all, and it is worth ending the ladder on them because they are what this field is about.

A position is a direction and a distance from a stated origin, in a stated frame, at a stated time. Nothing in that needs a surface. It is what what a coordinate refers to argues is the irreducible content of a coordinate, and it holds for Vesta as it holds for the Earth.

A map of the surface is still a projection of a curved surface onto a flat one, and everything this site has measured still applies. The curvature of a triaxial ellipsoid is not constant, so no projection of it is conformal or equal-area by any of the tidy constructions in the library — but Gauss’s theorem does not care, the impossibility is unchanged, and a map of Vesta is wrong in the same three ways a map of the Earth is.

Where the model stops #

A triaxial ellipsoid is itself a fit. Vesta has a 22-kilometre-deep crater at its south pole and Phobos is visibly lumpy; neither is an ellipsoid of any kind, and the axes quoted are the best-fitting one. Every number here therefore describes the fitted figure and not the body, which is a distinction where a fit leaves residuals is entirely about.

The bodies are two of many. Most small solar-system bodies are irregular, and the two here were chosen because both have published triaxial fits and because they sit at opposite ends of the range — Vesta nearly round and Phobos visibly not. Nothing about the argument depends on which two, and the same code takes any three axes.

Nothing here computes a projection of a triaxial body. The forward map, the distortion and the second-order quantities would all follow from the same machinery given a parameterisation, and the reason they are not here is that the site’s projection library assumes a metric with an axis. Extending it is a substantial piece of work and it is stated as owed rather than done.

The measurements are of the fitted ellipsoid’s geometry, computed by the same code that handles the oblate bodies, with the third axis simply allowed to differ. That the oblate cases come back to zero spread to fourteen decimal places is what says the generalisation did not break anything: the triaxial machinery reduces to the axial machinery exactly when the axes are made equal, which is a check worth more than any of the individual numbers.

Gravity is not in any of this. The angle between the normal and the radius is a geometric quantity computed from the fitted shape; the actual local vertical on a small irregular body depends on its internal density distribution and, for a fast rotator, on the centrifugal term. That is a different subject and this site does not take it.

Who found it, and when #

The measured spreads also settle a question the ladder’s first rung raised and could not answer. On Mars the two latitude conventions differ by a formula, so a coordinate in one can be converted to the other exactly, and the ambiguity a coordinate on another body complains about is recoverable given the convention. On Vesta it is not recoverable by a formula at all: the conversion depends on longitude, so a planetographic latitude with no longitude attached cannot be turned into a planetocentric one. The ambiguity stops being an inconvenience and becomes a loss of information.

Triaxial figures for the Moon were being fitted in the nineteenth century — Hansen’s lunar theory required one — and the practice of quoting three axes for small bodies is as old as the resolved imaging that made it possible, which is to say the Voyager and Viking era.

The shift to shape models rather than ellipsoids is a consequence of spacecraft rendezvous. Vesta’s is from Dawn’s imaging in 2011 and has around a million facets; Phobos’s comes from Mars Express and Viking imaging and has been revised repeatedly. The IAU’s reports carry triaxial radii for these bodies and note, in the same sentence, that the ellipsoid is a poor description.

Where the ladder goes next #

The bodies anchor now carries what a coordinate means on another body, what a projection does there, and what happens when the body has no axis. What it does not carry is the map: no projection of a triaxial body is computed anywhere on this site, and doing it properly means a metric with two varying principal curvatures and a distortion machinery that does not assume an axis.

That is the honest place to stop, and it is stated here rather than in a plan so that the gap is visible from the essay that has it.