A moon that two bodies could not have caught

The region a planet may keep a moon in was established first, and the same radius then became a unit of dynamical distance. Both take for granted that something is inside the sphere. Getting it there is a separate problem, and it has a clean impossibility result at the centre of it.

A small body approaching a planet on a heliocentric orbit is, to an excellent approximation, a test particle in the circular restricted three-body problem. That problem conserves one quantity — the Jacobi constant — and the conservation is exact rather than approximate. It is what decides whether the neck between the Sun’s domain and the planet’s is open or shut, and a body that arrives through an open neck departs through the same open neck.

Why the neck is the whole argument #

The rotating frame’s effective potential has three collinear stationary points, and the one between the Sun and the planet is the relevant one. Above a critical value of the Jacobi constant the region a body may occupy has two separate components — a lobe around the Sun and a lobe around the planet — with no path between them. Below it the two lobes touch, then merge, and passage becomes possible.

That gives the statement its force. Permanent capture is not “the body ends up bound”; it is “the neck through which the body entered has closed behind it”. And closing the neck means raising the Jacobi constant above the critical value, which is a change in a conserved quantity.

There is a weaker thing that does happen without any dissipation, and it is worth separating out because it is often mistaken for capture. A body can enter the lobe with the neck open, orbit the planet several times, and leave — sometimes after decades or centuries. This is temporary capture, it is common, and it is what a comet does when it is briefly a satellite of Jupiter. Two dozen such episodes have been observed or reconstructed. Every one of them ended.

Which trajectories transit, and why that is not a matter of aim #

The passage through an open neck is more structured than a hole in a wall.

The consequence for capture is that the set of orbits which enter is measure-zero-adjacent rather than generic: it is a tube, with a definite cross-section, and a body has to be inside it. That tube is the reason capture rates are computed numerically rather than estimated from a cross-section, and it is the same structure a low-energy transfer between planets exploits deliberately.

It also explains an otherwise puzzling feature of temporary captures. They are not brief grazing episodes; they last for many orbits, because a body that enters through the tube arrives near the periodic orbit at the neck and lingers there before falling either inward or back out. The residence time is set by how closely the initial condition approached the manifold, and it has no upper bound short of the manifold itself.

The three ways out of the impossibility #

Since gravity between three bodies will not do it, something else must. There are exactly three candidates and all three have operated somewhere.

A gas drag. While the planet was forming it was surrounded by a circumplanetary disc, and a body passing through gas loses energy continuously. In the rotating frame this raises the Jacobi constant smoothly, and a body that entered the lobe with the neck barely open leaves with it shut. The mechanism is efficient and it has a fatal date stamp: it stops working when the gas disperses, a few million years after the planet forms.

A collision. Two bodies that happen to be inside the lobe at the same moment can collide or pass close enough to exchange energy, and one of them is left bound. The rate is proportional to the square of the number density of small bodies in the region, so it favours epochs when the planetesimal population was large and it produces satellites in groups rather than singly — since the collision fragments share an orbit.

A third body. A binary approaching the planet is torn apart by the tidal field: one component is ejected, carrying away energy, and the other is left bound. This works at any epoch, needs no gas, and it produces single satellites on a wide range of orbits.

How much has to change, and what that is worth #

The impossibility is absolute and the quantity involved is small, which is worth putting numbers to because the two facts together are what make the three mechanisms plausible rather than exotic.

A body drifting in from a heliocentric orbit near the planet’s arrives with a Jacobi constant just below the critical value — if it were much below, the neck would be wide and the body would sail through without ever being near the planet, and if it were above, the neck would be shut and the body could not approach at all. The interesting arrivals are the ones within a fraction of a per cent of the threshold.

For a planetary mass ratio the gap between the value at which the neck opens and the value at which a body is safely inside the lobe is of order the square of the Hill radius in units of the separation, which at Jupiter is a part in two hundred of the total. Converting that into a velocity change at the point of closest approach, a body passing at half a Hill radius needs to lose of order ten to a hundred metres a second — out of an orbital speed around the planet of a kilometre a second at that distance.

Ten metres a second is a small amount of dissipation. A few weeks in a tenuous circumplanetary disc supplies it; so does an encounter with a body a hundredth of its own mass; so does being half of a binary that gets separated, where the energy is exchanged between the two components exactly as it is in a gravity assist and the departing half carries away far more than is needed. The threshold is low enough that the question is never whether a mechanism can supply the energy but whether the geometry brings the body to the right place at the right time.

That also explains the observed eccentricities. A body captured by the minimum possible dissipation is left on an orbit barely inside the lobe, with an apocentre near the stability limit and an eccentricity of a few tenths — which is what the irregular satellites have. A mechanism that removed much more energy would leave close, circular satellites, and those exist too and are called regular.

The capture that was watched, and the one that was not #

Two objects are worth naming, because between them they bracket what the argument can and cannot say.

The first is a comet that was a satellite of Jupiter for about sixty years. Backward integration of its orbit puts its capture in the late 1920s, and the capture was a straightforward temporary one: the comet entered through the neck with its Jacobi constant unchanged and would have left again. It did not leave, because on one of its passages it came inside the distance at which Jupiter’s tidal field exceeds its own self-gravity, broke into twenty-one pieces, and struck the planet two years later.

That is a fourth outcome, and it is not capture. The comet’s Jacobi constant never rose; its residence simply ended on the planet rather than through the neck. The episode is the clearest demonstration available that temporary capture happens routinely and that nothing about it is permanent — and the fact that the best-documented case ended in a collision rather than an escape says something about how deep into the lobe a temporarily captured body can wander.

The second is Neptune’s largest moon. It is retrograde, it is a fifth of one per cent of Neptune’s mass, and it is far too large to have formed in orbit going the wrong way — a satellite that accreted from a circumplanetary disc inherits the planet’s rotation, and this one does not. It is a captured body, and its composition matches the objects beyond Neptune rather than anything else.

Its present orbit is nearly circular at 0.003 of Neptune’s Hill radius, which no capture mechanism produces. Capture leaves a body near the edge of the lobe on an eccentric orbit; what brought this one in and circularised it is tidal dissipation inside the moon itself, acting over a few hundred million years, which also destroyed whatever regular satellite system Neptune had by scattering it. The evidence for that is Neptune’s remaining inner moons, which are anomalous in exactly the way a disrupted and re-accreted system should be.

So the one capture nobody disputes is also the one whose capture orbit is entirely erased. The mechanism most favoured for it — a binary arriving and being separated — is favoured on grounds of efficiency rather than of evidence, and the orbit that would have distinguished it circularised away.

The evidence is a population, not an orbit #

None of the three leaves a mark on the orbit it produces. A satellite’s semi-major axis, eccentricity and inclination today are the result of four and a half billion years of subsequent evolution, and the capture episode is not recoverable from them.

What is recoverable is the shape of the distribution, and that is where the argument is actually made.

The clustering is the strongest single piece of evidence. Jupiter’s retrograde irregulars fall into three groups sharing a semi-major axis, an eccentricity and an inclination to within a few per cent, and the obvious reading — that each group is the debris of one larger body — is supported by their colours, which match within a group and differ between groups. A capture episode that produced one body per event cannot make a group. A collision inside the Hill sphere can, and so can a single capture followed by a later break-up.

The second piece is the mass ratio between the systems. Jupiter, Saturn, Uranus and Neptune have irregular satellite populations of comparable total mass despite differing by a factor of twenty in planet mass, which is what a capture mechanism operating on a common reservoir of small bodies would give and not what a gas-drag mechanism scaling with the planet’s own disc would.

The third is a date. If capture happened during the gas phase, the satellites should be as old as the planets; if it happened during a later rearrangement of the outer solar system, they should be contemporary with the other debris populations. The crater record on the satellites themselves would settle it, and one of them has been visited.

What was actually measured #

For a satellite: a semi-major axis, an eccentricity and an inclination, obtained from an astrometric arc and fitted like any other orbit, with the complication that these orbits are perturbed strongly enough by the Sun that a Keplerian element quoted without an epoch is not a number.

For the population: a survey completeness, which is the hard part. Irregular satellites are faint — the smallest known are a kilometre across, at magnitude 26 — and a census that is incomplete in a size-dependent way cannot support an argument about how many there are. The deepest surveys reach about a kilometre at Jupiter and a few kilometres at Neptune, so the four populations are compared at different limits and the comparison requires extrapolating a size distribution.

For the composition: broadband colours, and for a handful of objects a low-resolution spectrum. The colours are the evidence that groups share a parent, and they are also the evidence that the irregulars resemble outer-solar-system bodies rather than main-belt ones — which is a statement about where the reservoir was.

And for one object, a great deal more. Phoebe was photographed at close range, and what was found was a heavily cratered, dense, water-ice-rich body with a surface unlike any inner-solar-system object. Its density of about 1.6 grams per cubic centimetre is too high for a pure ice body and too low for rock, and it is the single best piece of evidence that the irregular satellites came from the same reservoir as the objects beyond Neptune.

What the argument cannot settle #

The Jacobi constant is not exactly conserved for a real satellite. The restricted problem assumes the planet is on a circular orbit and the satellite is massless. Jupiter’s eccentricity of 0.048 breaks the first assumption, and an almost conserved quantity drifts — in exactly the way the Tisserand parameter does for a comet. The drift is slow, and over the age of the solar system it is not nothing. Whether it is ever enough to capture a body on its own is a live question and the answer appears to be no, but it is a numerical answer rather than a theorem.

The three mechanisms are not exclusive. A body slowed by gas and then struck is captured by both, and the observed population is presumably a mixture. Nothing in the data separates a mixture from any one mechanism at the level of an individual object.

And the reservoir is a model. Every capture calculation needs to know how many bodies of what size were on what orbits at the relevant epoch, and none of that is observed. It is taken from a model of the early solar system, and the capture efficiencies that come out are efficiencies relative to that model.

The figures show a planar problem. The restricted three-body problem drawn here is two-dimensional, and the satellites are not. An inclined trajectory has the same Jacobi constant and a different geometry, and the neck it must pass through is a surface rather than a curve. The impossibility result is unaffected — the conservation law does not care about dimension — but the tube’s cross-section, and therefore the capture rate, does.

The generalisation #

What makes this argument work is that the obstacle is a conservation law rather than a difficulty.

A great many statements in dynamics have the form “this is hard”, and they are arguments about rates and cross-sections that a sufficiently favourable case can defeat. This one has the form “this is impossible”, and the difference is that no amount of luck helps: a two-body encounter inside a third body’s field cannot produce a bound orbit, at any impact parameter, at any speed, for any masses, ever.

The value of an impossibility is that it converts an observation into a requirement. Every irregular satellite in the solar system is a statement that something removed energy from a passing body, and the argument does not depend on knowing how often capture happens or how efficient any mechanism is. It is the same move as reading an ocean’s presence off a moon’s response to a tide: the presence follows from a law, and only the details need a model.

The second half is less comfortable. An impossibility narrows the candidates and does not choose among them, and here it leaves three — each of which works, each of which operates in a different epoch, and none of which leaves a signature on a single orbit. A clean theorem can produce a question that only a population can answer, and the population in this case is forty objects, half of which were discovered in the last twenty-five years.

Still open: whether a tilted orbit is caught the same way #

What comes next is the case the figures here cannot draw. Everything in this essay is planar, and the irregular satellites are anything but: their inclinations run across the whole range, and the transit tube’s cross-section, the residence time and the capture rate all depend on the inclination of the approach. Whether the observed retrograde preference is a capture preference or a survival one is the question, and the two produce the same distribution today.

Beside it sits what happens to a satellite once it is in. A capture leaves an orbit that is not the orbit observed, and the difference is four and a half billion years of the Sun’s secular perturbation — which does not act uniformly across inclination, and which is why the census above has a hole in the middle of it.