A ray from the centre hits the surface twice

Eleven rungs of this anchor map bodies that are not the Earth. A body that is not an ellipsoid; a body with three unequal axes; a body with no sea level; a body whose longitude drifts with its own rotation rate.

Every one of them rests on an assumption none of them states. The surface is star-shaped about the centre — a ray from the origin meets it exactly once — which is what makes a radius a function of direction, what makes a longitude and a latitude a coordinate, and what makes the standard small-body shape model, a table of radii on a direction grid, a representation of anything at all.

The body #

Two balls of radius 1 with centres at ±1.5, joined by a cylindrical neck of radius 0.35, with the origin at the middle of the neck.

Three numbers, stated rather than fitted, and crude on purpose: what is being demonstrated is a property of the radial parameterisation rather than of any real nucleus, and a reader can recompute every figure below from those three. The origin is inside the neck and outside both lobes, which is the case a contact binary is actually in — 67P/Churyumov–Gerasimenko’s centre of mass is in its neck.

A ray leaving the origin at a shallow angle to the long axis leaves the neck through its side, travels through empty space, and enters a lobe. Three crossings, where a star-shaped body has one: out through the side of the neck, across the gap, in through the near face of the lobe and out through its far side — with the middle pair separated by a stretch of nothing at all.

What was computed, and how #

The body is a predicate: given a point, is it inside? The crossings along a ray are found by marching the predicate outward and refining every change by bisection, so nothing depends on a formula for the surface — which matters, because the surface of a union has no single formula and that is the point.

The directions are sampled uniformly on the sphere — uniform in longitude and in the sine of latitude, which is the same closed form every area on this site rests on — so the share reported below is a solid angle rather than a count over a grid that is denser at the poles.

On the stated body, 10.9 per cent of the sky has more than one crossing, and the worst direction has three.

The control #

An ellipsoid, run through the identical machinery: 1,600 directions, and not one of them crosses more than once.

That is the check the rung needs. A ray-marching routine with an off-by-one in it, or a bisection that reported a spurious crossing at the surface, would show multiple hits on a convex body too — and a measurement whose whole content is a count of crossings has no other way to be wrong.

The neck decides how much of the sky is lost #

A thinner waist is a larger hole in the coordinate system, and the two lobes do not move at all. That is the whole geometry in one column: the failure is not about the lobes being far apart, it is about the neck being thin enough that a ray can get out of it before it reaches one.

The band has a closed form, and a ceiling #

The neck ladder is a monotone column and it hides a bound. The three-crossing directions form a band about the long axis, and both of its edges can be written down from the body’s three numbers.

A ray leaves the origin at an angle θ to the axis. It stops being able to reach a lobe when the quadratic for the ray’s intersection with a unit ball centred at 1.5 has no real root, which is at cos θ = √5/3, or θ = 41.81°. It starts escaping through the side of the neck when its exit from the cylinder of radius a happens before the cylinder joins the lobe, at tan θ = a / (1.5 − √(1 − a²)).

Three crossings need both, so the band runs between those two angles, and its solid-angle share of the sphere is the difference of the two cosines:

Five rows agreeing to within half a percentage point, from a ray-marching measurement and from two arctangents, which is the check the table needed and did not have.

The outer edge of the band never moves. It is 41.81° at every neck radius, because it is decided by the angular size of a lobe seen from the origin and the lobes do not move. Only the inner edge travels, sliding towards zero as the waist thins.

Which gives the column a ceiling the measurements only approach. As the neck goes to nothing the band becomes the whole cone of directions that reach a lobe at all, and the share tends to 1 − cos 41.81° = 25.46 per cent. The thinnest neck measured, at 0.1, is already at 23.97 — within a point and a half of a limit no thinner neck can pass.

So the failure does not grow without bound as the body becomes more extreme. It saturates at a quarter of the sky, and what fixes that quarter is not the neck at all but how much of the sky the two lobes occupy. A body with larger lobes or a shorter separation would lose more of its sky; one with the same neck and more distant lobes would lose less. The neck decides how quickly the ceiling is approached and the lobes decide where the ceiling is.

What a shape model actually describes #

A small body’s shape is published as a table of radii on a direction grid — that is the format, and every mission product uses it — the same kind of format decision a projection defined by a table has an interpolation in it prices for a map. Handed a body with three crossings in a direction, such a table has to store one number, and it stores the outermost.

So the body the model describes is the star-shaped hull about the origin: everything the real body has, plus everything between an inner crossing and the outer one along the same ray. That is the neck filled in, out to where the lobes would be.

One and two thirds per cent of the volume, at the stated neck, invented by the file format. Not by an error in anybody’s processing, not by a limit on the data — by the decision to represent a surface as a radius per direction, which is a decision made before any measurement.

And it is worse than a volume error suggests, because the invented volume is all in one place. A density computed from the model’s volume is wrong by that fraction; a moment of inertia computed from it is wrong by more, because the neck is near the origin where the moment arm is short and the shape of the error is concentrated exactly where the body’s own structure is most interesting.

Where the coordinate goes #

The consequence for mapping is sharper than the consequence for volume.

A planetocentric latitude and longitude name a direction, and the surface point they refer to is where the ray in that direction meets the surface. Where there are three such points, the coordinate names three places, and the convention that picks the outermost is a convention rather than a fact.

So on eleven per cent of this body:

• A coordinate does not name a place. Two surface features can have the same latitude and longitude, which is precisely the failure no map of the whole sphere is one to one establishes for a projection, arriving one level earlier — in the coordinate system rather than in the map of it.
• A map of the body cannot show all of it. Anything on the inner surface of the neck is behind something, and a projection of a radial coordinate has nowhere to put it.
• And a distance is not a distance. The great-circle arc between two directions is a distance on the sphere of directions and has no relation to the distance across the body’s own surface, which may have to go round a lobe.

The repair, and what it costs #

There is a standard answer and it is not a better coordinate system. It is to give up the radial parameterisation entirely and represent the surface as a mesh — a list of vertices and triangles, with no distinguished origin and no radius function.

That is what every modern small-body shape model is, and it removes the problem completely: a mesh represents any surface, star-shaped or not, exactly.

What it costs is everything built on the coordinate. A mesh has no latitude and no longitude, so there is no graticule, no rectangular data product, no way to index a raster, and no way to say where something is except by naming a facet. The mission products that carry a radius table alongside the mesh carry it because the table is what everything downstream can read.

The trade is between representing the body and being able to index it, and it is forced in the same way every trade in this collection is: a coordinate system is a homeomorphism onto a piece of the plane or the sphere, and a surface that is not homeomorphic to a sphere in the right way does not admit one.

What survives, which is more than nothing #

It would be a misreading to conclude that nothing can be mapped on such a body, and the division is worth drawing because it is the practical guidance.

The lobes are fine. Each lobe on its own is star-shaped about its own centre, so a coordinate system centred in a lobe maps that lobe faithfully — and mission products for bilobed nuclei do exactly this, with a separate frame per lobe and a stated relation between them. What that gives up is a single coordinate for the whole body, which is the thing an index needs.

The outer surface is fine. Everything visible from far away in a given direction is at the outermost crossing, so a radius table is a correct model of what a distant observer sees — which is what a lightcurve or a radar return measures, and is why radial models fitted to those data are not wrong about the data.

It is the neck that has no coordinate, and the neck is where the interesting geology is: the surface there is the youngest, the most active, and the part every mission to such a body was sent to look at.

That ordering is uncomfortable and it is the honest summary. The representation is adequate everywhere except the place it was chosen to study.

Where the model stops #

The body is three numbers. Real contact binaries have irregular lobes, a neck that is not a cylinder, and a centre of mass that is not on the axis. The direction of the effect is unchanged and every number would be.

The march has a resolution. A feature thinner than the marching step is missed, which is stated rather than hidden — the neck’s own thickness is what decides the answer, so a coarse march would report a body that is star-shaped when it is not. The steps used are two thousand over the body’s reach, which resolves the thinnest neck measured by a factor of forty.

And the origin is a choice. A body may be star-shaped about some point without being star-shaped about its centre of mass, and moving the origin into a lobe would make more of the sky single-valued and put the coordinate system’s own centre somewhere with no physical meaning. Nothing here searches for the best origin, and whether one exists for a given body is a real question with a real answer.

The same failure, one level up #

There is a version of this that has nothing to do with contact binaries, and it is worth naming because it is much commoner.

Any body with an overhang has it. A cliff that overhangs, a boulder resting on a slope, an arch: each is a place where a ray from the centre crosses the surface three times, over a solid angle far too small to show in a table like the one above but not too small to matter to whatever is being mapped there. The Earth has them too, and a national height model — one height per horizontal position — has exactly this failure at every overhanging cliff in the country.

So the assumption being examined is not exotic. It is the assumption that a surface is a graph — one value of one coordinate for each value of the others — and it is inside every raster elevation model, every radius table and every bathymetric grid. A height that is not a length prices what a height means; this prices whether there is one.

The difference is only in scale. On the Earth the failure is a few square kilometres of cliff face; on a contact binary it is eleven per cent of the sky.

A last number, because it is the one a mission planner would want. The share of the sky with no radius is 10.9 per cent at the stated neck, and the share of the surface area that is unreachable is smaller — the inner faces of the neck are a small part of a large body. The two are different quantities and the first is the one that matters for a coordinate system, because a coordinate has to name a direction before it can name a place.

The generalisation #

A coordinate system is a claim about topology, and the claim is usually inherited rather than checked.

The Earth’s surface is a topological sphere and star-shaped about its centre to a very good approximation, so latitude and longitude work, and every convention built on them works. Carrying that machinery to another body carries the assumption with it, silently, because the assumption was never written down — it is inside the word radius.

This collection meets the same shape wherever a construction is transplanted. A coordinate on another body is about the conventions that do not transfer; a body that is not an ellipsoid is about the reference surface that does not; a map of a body with three axes is about the formulae that need rewriting.

What is different here is that the thing that fails is not a formula or a convention but the existence of the coordinate. There is no better latitude, no correction term, no more careful definition. Eleven per cent of the directions have three answers and the question was malformed.

Who found it, and when #

Radial shape models for small bodies date from the first radar and lightcurve inversions in the 1980s, and the format — a radius on a latitude–longitude grid — is still in use. The move to polyhedral models came with the first close flybys, and the reason given is resolution rather than topology.

The topological reason is well known to anybody who has tried to fit a radial model to a bilobed nucleus, and appears in the shape-modelling literature as a practical remark rather than as a measurement. What the numbers above supply is the size of it: eleven per cent of the sky at a neck a third of a lobe wide, and a volume error of one and two thirds per cent from the format alone.

Where the ladder goes next #

Twelve rungs map bodies with the wrong shape, the wrong field, the wrong rotation and now the wrong topology. What none of them reaches is a body with no stable shape at all: a rubble pile reconfigures, a comet loses metres of surface per perihelion, and a coordinate frame tied to surface features on such a body is a frame whose definition erodes. That is the epoch is part of the coordinate with the ground itself as the thing that expires.