Jupiter keeps only the visitors already nearly bound

Three bodies from other planetary systems have been seen passing through the inner solar system, each on an orbit with no period, each leaving as fast as it came. Counting them turned out to be mostly a measurement of the survey that found them: how far a body of a given size can be seen, how many weeks it spends close enough, how strongly the Sun’s gravity gathers slow visitors towards it. That census ended on a possibility none of the three raised. A visitor slow enough, passing close enough to a giant planet, might lose enough energy to stay.

The obstacle is the one every capture meets. Two bodies cannot capture each other: a hyperbolic orbit about one body stays hyperbolic unless something else takes energy away. For a visitor and the Sun, the something else is a planet, and the planet that can take the most is Jupiter. The question is how much it can take, and from whom.

A flyby turns velocity and does not shorten it #

Seen from Jupiter, a passing body follows a hyperbola of its own. It approaches at some relative speed $U$, swings around the planet, and leaves at the same speed $U$ in a different direction. That is the whole of what a flyby does in the planet’s frame: it rotates the relative velocity through an angle $\delta$, larger for closer and slower passes,

$$\sin\frac{\delta}{2} = \frac{1}{1 + r_p U^2/\mu_J},$$

with $r_p$ the closest approach and $\mu_J$ Jupiter’s gravitational parameter. In the Sun’s frame the same rotation changes the body’s speed, because the body’s heliocentric velocity is Jupiter’s velocity plus the relative one. That is how a spacecraft steals speed from a planet, and run the other way it is how a visitor could give speed up.

The diagram has the Sun’s frame at its origin. Jupiter’s velocity is an arrow of 13.06 kilometres a second. The dashed circle is the escape speed from the Sun at Jupiter’s distance, 18.47 kilometres a second; any body whose velocity ends inside it is bound to the Sun. A visitor arriving with three kilometres a second to spare — its speed far from the Sun — reaches Jupiter’s distance at $\sqrt{3^2 + 18.47^2} = 18.71$ kilometres a second, just outside the circle. Its velocity relative to Jupiter is the arrow from Jupiter’s velocity to its own, here 30.7 kilometres a second because the visitor is moving largely against the planet.

A flyby keeps the length of that relative arrow and swings it about Jupiter’s velocity, so every possible outcome lies on a circle of radius 30.7 kilometres a second centred on the tip of Jupiter’s arrow. Not all of that circle is reachable: a pass that grazes the cloud tops turns the relative velocity by 81°, and nothing can turn it further. The reachable arc dips just inside the escape circle, and the slowest outcome is 17.7 kilometres a second — bound, by less than one kilometre a second.

The diagram contains the whole argument. The arc is long because the relative speed is large, but it crosses the escape circle at a shallow angle, so only a small part of it is inside. The visitor’s speed around the Sun can be changed by the flyby, but the amount is limited by how much the arc reaches inside the circle, which is a few kilometres a second at best. A visitor that arrives with less than that to spare can be kept. One that arrives with more cannot be kept by any flyby at all.

How the chance falls with speed #

The diagram shows one geometry. The fraction of visitors captured has to be averaged over all of them: every direction of arrival, and every offset at which a visitor can pass the planet.

The calculation draws arrival directions uniformly on the sphere. That is not an assumption about where visitors come from so much as a consequence of how orbits work: a population whose velocities are spread evenly in direction far from the Sun is spread evenly in direction at every point near it, because a theorem of Liouville’s guarantees that the density of orbits in position and velocity is carried unchanged along them. Offsets from Jupiter are drawn uniformly across a disc of 0.35 astronomical units, about the region in which Jupiter’s gravity dominates the Sun’s, and each encounter is weighted by the rate at which visitors cross that disc, proportional to their speed relative to the planet.

For visitors arriving with half a kilometre a second to spare, 48 per cent of the passes within that disc leave them bound. Such a visitor reaches Jupiter’s distance barely faster than escape, and almost any deflection that slows it relative to the Sun keeps it; roughly half of all deflections do. At one kilometre a second the fraction is 41 per cent, at three it is seven, at eight it is about one in a thousand, and at eighteen it has fallen to a few in a hundred thousand. Not one of the encounters sampled at 26.3 kilometres a second, ʻOumuamua’s speed, captured anything.

The fall is steep because two things worsen together. A faster visitor needs more speed removed, so it needs its relative velocity turned through a larger angle; and a faster visitor meets Jupiter at a higher relative speed, for which any given offset gives a smaller turn. The visitor must pass closer, and the area of offsets close enough shrinks as the square of the required closeness.

That is the same physics expressed in the number that survives the encounter. The Tisserand parameter combines a body’s semi-major axis, eccentricity and inclination into a quantity a flyby of Jupiter leaves almost unchanged, and it is a measure of the relative speed $U$. A visitor’s Tisserand parameter is fixed by its speed and direction of arrival; a captured orbit with the same value must exist for capture to be possible at all. For slow visitors such orbits exist in abundance. For fast ones, the Tisserand value is one no bound orbit has, and the flyby can only exchange one unbound orbit for another.

The same arithmetic that brakes a spacecraft #

The limit is familiar from the other direction. A spacecraft arriving at Jupiter faces the visitor’s problem with the frames swapped: it approaches the planet too fast to stay in orbit about it, and the cheapest way to stop is to burn its engine deep in the planet’s gravity well. Flybys of the planet itself cannot help — a hyperbola about Jupiter stays a hyperbola about Jupiter — but flybys of its moons can, and missions to the outer planets brake on a moon before braking with the engine, shaving a few hundred metres a second off the burn by turning their velocity relative to a moon.

The visitor uses Jupiter as the spacecraft uses a moon. Both are three-body manoeuvres, both remove energy relative to the larger body by turning velocity relative to the smaller one, and both are limited in the same way: the turn cannot exceed what a grazing pass provides, and the speed removed cannot exceed a fraction of the smaller body’s own orbital speed. A moon moving at about ten kilometres a second around Jupiter can take a few hundred metres a second from a spacecraft; Jupiter moving at thirteen around the Sun can take a few kilometres a second from a visitor. The difference is only of scale.

Capture is ejection run backwards #

Every orbit a flyby can produce, run backwards in time, is an orbit a flyby could have started from. A visitor captured from a hyperbolic orbit onto a bound one with an aphelion of several hundred astronomical units is, reversed, a bound body with that aphelion thrown out of the solar system by one pass of Jupiter. The second happens constantly. Most long-period comets that come in from the Oort cloud for the first time and cross Jupiter’s orbit leave on an orbit Jupiter has changed, and a large fraction of them leave on hyperbolic orbits, never to return. The solar system is a steady exporter of comets into interstellar space by exactly the encounters this essay runs in reverse.

That symmetry gives a check. The rate at which Jupiter captures interstellar bodies onto a given set of bound orbits equals the rate at which it ejects bodies from those orbits onto the corresponding hyperbolic ones, multiplied by the ratio of the densities of the two populations in that part of orbital space. Jupiter ejects many comets because the solar system’s own comets are numerous on Jupiter-crossing orbits; it captures few visitors because visitors slow enough to occupy the reversed orbits are a few in a thousand of a population that is itself sparse. The calculation in the figures is the same balance, computed forwards.

Captures come from the slow tail #

A capture rate needs the speeds of real visitors, and those follow the speeds of the stars whose planetary systems ejected them. A body thrown out of a planetary system leaves its star at a few kilometres a second, small against the random motions of stars in the Galaxy, so it inherits its parent star’s velocity, and its speed relative to the Sun is the star’s.

The visitors in the figure move like the young stars of the Galaxy’s thin disc, the population whose motion ʻOumuamua’s arrival speed and direction most resembled. Their speeds relative to the Sun peak in the twenties of kilometres a second, and only 0.4 per cent arrive with less than five kilometres a second to spare. Those 0.4 per cent supply 95 per cent of the captures. The captures peak at two to three kilometres a second, where the visitors are still rare but the chance of capture has not yet collapsed; the very slowest visitors, below one kilometre a second, are captured readily and are so rare that they contribute less.

The total is a capture rate of $3.5 \times 10^{-3}$ a year for every interstellar object per cubic astronomical unit of interstellar space. A captured visitor is therefore a sample not of interstellar objects in general but of the slowest few in a thousand: bodies whose parent stars moved with the Sun to within a few kilometres a second. Stars that close to the Sun’s motion are the Sun’s recent neighbours — siblings from the same birth cluster, or stars that formed nearby at about the same time. A captured interstellar object, if one could be identified, would be more likely than a passing one to have come from a star the Sun once knew.

The orbits a capture leaves #

A capture is barely a capture. The visitor arrives just above escape and leaves just below it, so the orbit it is left on is enormous: its aphelion, the farthest point from the Sun, lies hundreds of astronomical units away, and its perihelion lies inside Jupiter’s orbit, because it was at Jupiter’s distance when it was caught.

The inclinations are the signature. Visitors arrive from every direction, and a flyby turns their velocity relative to Jupiter by a limited angle, so the captured orbits keep much of the spread they arrived with. Seventy-one per cent of them are inclined by more than thirty degrees to Jupiter’s orbit, and 37 per cent go round the Sun in the opposite direction to the planets. The distribution leans prograde — capture works best when the visitor’s velocity can be turned to line up with Jupiter’s own, which favours orbits already running roughly with it — but the lean is modest.

Nothing formed in the solar system looks like that. The asteroids, the Kuiper belt and the comets that come from it were all made in a flat disc turning one way, and their orbits are overwhelmingly prograde and near the plane. Retrograde small bodies exist — a few long-period comets, which come from the Oort cloud’s spherical swarm, and a handful of asteroids on retrograde orbits among the centaurs and near Jupiter — and several have been proposed as captured visitors. One asteroid sharing Jupiter’s orbit backwards was argued in 2018 to have been captured from interstellar space when the planets were young; the argument rests on its orbit being stable for longer than the solar system has existed, which requires an origin before the planets settled, and it has been contested on the grounds that the solar system’s own scattered population can produce such orbits too. A large captured population would show itself not in any one orbit but in the statistics — an excess of high inclinations and retrograde motion among bodies crossing Jupiter’s orbit with aphelia of hundreds of astronomical units.

How many there should be #

A captured orbit does not last. It crosses Jupiter’s orbit, so at every return it can meet Jupiter again; and it is so weakly bound that a modest kick in either direction changes its fate. The next close encounter may throw it out again, as the reverse of the capture, or send it inwards onto a shorter, more tightly bound orbit, or into the Sun.

The census of visitors put the density of ʻOumuamua-sized bodies at about 0.2 per cubic astronomical unit, uncertain by an order of magnitude. At that density Jupiter captures one every fifteen hundred years. The number present at any time is the capture rate multiplied by how long a captured orbit survives, and the survival time is itself a random walk. Each return past Jupiter changes the orbit’s binding energy by an amount comparable to the spread that marks a comet as a first-time visitor, and a weakly bound orbit that random-walks in energy reaches zero, and is lost, after some tens of returns. With periods of thousands of years for aphelia of hundreds of astronomical units, that is ten thousand to a few hundred thousand years.

The expected number of captured hundred-metre visitors in the solar system is therefore of order ten to a hundred. They would be at large distances most of the time, faint, and crossing the inner solar system for a few months every few thousand years, which is why none has been recognised; any survey that found one would see an object on a Jupiter-crossing orbit with an aphelion of hundreds of astronomical units and, a third of the time, a retrograde inclination — and would have no way of distinguishing it from a long-period comet from the Oort cloud except by its composition.

What the calculation leaves out #

The calculation uses patched conics: a straight hyperbola about Jupiter inside its sphere of influence and a Keplerian orbit about the Sun outside it, with Jupiter on a circular orbit. Real captures happen partly in the region where both bodies’ gravity matters, through the slow passages near Jupiter’s Lagrange points that let a body be held temporarily before it is released or kept, and those passages can capture visitors the patched-conic picture would let go. The other planets are absent; Saturn captures a smaller fraction because its orbital speed and mass are smaller, and the Sun alone, with a nearby star passing, can capture into the Oort cloud on much longer timescales by an entirely different mechanism. The survival time is estimated rather than integrated, and the density of interstellar objects is uncertain by an order of magnitude, which moves the final number by the same factor.

What does not depend on those simplifications is the shape of the result: a single flyby of Jupiter cannot remove more than a few kilometres a second from a visitor’s speed, so only the slowest visitors are kept, and they keep the spread of directions they arrived with.

Still open: whether the solar system was born holding them #

The present capture rate is small because the visitors are fast. That was not always so. The Sun formed in a cluster of stars moving together within a few kilometres a second, each ejecting comets and planetesimals as its planets formed, and the interstellar objects passing through the young solar system came mostly from those siblings — at exactly the low speeds this calculation says Jupiter captures efficiently. Simulations of that early cluster suggest that a substantial fraction of the Oort cloud, and perhaps some of the bodies on distant, inclined orbits beyond Neptune, may have been captured from the Sun’s birth siblings rather than formed around the Sun. The test is chemical: a comet formed around another star would carry that star’s ratios of isotopes, and an orbit no longer remembers where a body came from, but its composition does. The solar system may hold a population of captured visitors far larger than the present rate implies, kept since the cluster dispersed, and the way to find them is spectroscopy of the long-period comets rather than more orbital dynamics.