An arrival the Sun hands over

A spacecraft arriving at a planet on a hyperbola must burn to stay. In two-body mechanics there is no way round it: an orbit that comes in from infinity goes back out to infinity, and the only question is where along the hyperbola to brake so that the burn is smallest. The planet cannot capture anything by itself, because its gravity is conservative — whatever energy the spacecraft gains falling in, it loses climbing out.

But a planet is not alone. It orbits the Sun, and the Sun’s gravity, in the planet’s rotating frame, is a tide that stretches the region round the planet along the line to the Sun. There are two points on that line, the Lagrange points L1 and L2, where the planet’s pull, the Sun’s and the centrifugal force of the rotating frame balance. Between them lies a region the planet controls, and the boundary of that region has necks at the two points — narrow passages through which a body with just the right energy can pass from the Sun’s domain into the planet’s, and back.

A spacecraft that arrives through a neck, with almost no speed relative to the planet, finds itself looping round the planet without having burned anything. It is captured, for a while. This essay is about that capture: how it happens, how long it lasts, why it is so sensitive to the arrival, and what it saves.

A problem with one conserved number #

The simplest setting is Hill’s problem: a planet on a circular orbit round the Sun, a massless spacecraft, and the planet’s rotating frame, with distances scaled so that the planet’s Hill radius — the distance of L1 and L2 — is $3^{-1/3}$, about 0.69, and time scaled so the planet’s year is $2\pi$. The equations of motion are

$$\ddot x - 2\dot y - 3x = -\frac{x}{r^3}, \qquad \ddot y + 2\dot x = -\frac{y}{r^3},$$

with $x$ pointing away from the Sun and $y$ along the planet’s motion. The $3x$ is the Sun’s tide and the centrifugal force together; the terms in the velocities are the Coriolis force; the right-hand side is the planet.

These equations have one conserved quantity, the Jacobi constant $C = 3x^2 + 2/r - \dot x^2 - \dot y^2$, and its value decides everything about where a body can go. For a body far from the planet on a circular heliocentric orbit offset by $b$ from the planet’s, it is $\tfrac34 b^2$. At the Lagrange points the critical value is $3^{4/3} \approx 4.33$. A body whose Jacobi constant is above that cannot cross between the Sun’s region and the planet’s at all; the curve where its velocity would have to be zero closes the necks. Below it, the necks are open, and their width grows as the constant falls.

So an arriving body whose heliocentric orbit is offset from the planet’s by less than about 2.4 Hill units has the energy to get in. Whether it does, and what it does once there, is decided by the dynamics, not by the energy.

Why it comes in, and why it leaves #

The body in the figure was never moving fast relative to the planet. Its heliocentric orbit was nearly the planet’s own, a little further from the Sun and therefore a little slower, so in the planet’s frame it drifted backward along the orbit, slowly, until the planet’s gravity began to bend its path. That is the opposite of a hyperbolic arrival: a spacecraft on a Hohmann transfer to Mars meets the planet at two and a half kilometres a second, crosses the Hill region in a few days and is gone unless it brakes. The drifting body meets it at a few metres a second and lingers.

Lingering is not the same as staying. Once inside, the body moves under the planet’s pull and the Sun’s tide together, and its orbit round the planet is not an ellipse. Each loop is perturbed by the tide, which is strongest along the Sun–planet line and pushes the orbit’s apocentre outward when it points that way. After a few loops the body finds itself near a neck again with the right velocity to pass through, and it leaves — into a heliocentric orbit near the one it came from, on either side of the planet’s.

The Jacobi constant guarantees that it can leave. It entered through an open neck and the neck is still open; nothing in the conservative dynamics can close it behind the body. A permanent capture needs the constant to rise above the critical value, and only a non-conservative push — a burn, a drag, a collision — can do that. Ballistic capture is temporary by construction.

Which arrivals are kept #

The outcome depends on where the body came from, and the dependence is the defining feature of the problem.

The jumps are the signature of chaos in the restricted three-body problem. Two arrivals that differ slightly approach the planet on nearly the same path, but the tide amplifies the difference on every loop, and after two or three loops the paths have nothing to do with each other. A smooth band of arrivals that each loop once is interrupted by narrow windows in which they loop three, four or five times, and the windows contain narrower windows of their own.

For Mars, a ten-thousandth of a Hill unit is about 160 kilometres. A spacecraft’s heliocentric orbit is not known to 160 kilometres at the start of its cruise, and a mission designed to arrive in one of these windows has to be targeted precisely and corrected on the way in. That is not unusual — interplanetary arrivals are routinely targeted to kilometres — but it means a ballistic capture is designed, not stumbled into.

The twins make the practical point. A spacecraft cannot plan to be held for five revolutions and do its insertion on the fifth, because a navigation error far smaller than any spacecraft can avoid at the start of its cruise would turn five revolutions into one, or into an impact. What it can plan is to arrive in the capture region with the right energy and direction, and to take its burn early — while it is already bound and the burn is small — rather than relying on the capture lasting.

How long captures last #

The scans use arrivals from circular heliocentric orbits. Real arrivals have small eccentricities as well, which widen the range of conditions that enter the Hill region and change which ones are kept.

These are short captures — a revolution or two, which near the Hill radius of Mars is a year or two. That is long by the standard of a hyperbolic flyby, which crosses the Hill region in days, and short by the standard of a mission. It is long enough for what it is used for: to give a spacecraft time, at a large distance from the planet where it is moving slowly, to make a small burn that lowers its orbit inside the region where the tide cannot pull it back out.

The planar Hill problem understates the captures somewhat. In three dimensions, with the planet’s eccentric orbit and with the arrival direction free, the set of capture trajectories is larger and includes longer ones; the missions that have used ballistic capture have found trajectories that loop for months before the first manoeuvre. The shape of the survival curve — a steep fall with a thin tail — is the same.

What the capture saves #

The saving comes at the insertion burn, and it is easiest to see by comparing two ways of entering the same final orbit.

The saving has a simple source. A spacecraft on a hyperbola at closest approach is moving at the local escape speed plus whatever its excess adds, and must remove all of that excess and then some. A spacecraft already captured is moving at slightly less than the local escape speed, and must remove only the difference between that and the circular speed. The excess on a Hohmann arrival at Mars is two and a half kilometres a second, which at closest approach is worth about three-quarters of a kilometre a second of extra speed, by the leverage the planet’s own gravity gives a burn deep in its well.

The saving is not free. A spacecraft does not arrive with zero excess by accident: its heliocentric orbit has to be shaped so that it reaches the planet’s neighbourhood almost co-moving with the planet, which usually means a longer transfer, arriving ahead of or behind the planet and letting it catch up, with more energy spent at departure from the Earth. Studies of ballistic capture at Mars find that part of the arrival saving is returned at departure and part is kept, depending on the transfer, and that the transfer takes months longer.

What makes the trade attractive is where each cost falls. Propellant burned at Mars has already been carried across interplanetary space and paid for, many times over, in the launch; propellant burned at departure has not. A kilogram saved at arrival is worth several at departure. And the capture removes a single critical event: a spacecraft on a hyperbola that fails to burn on time is gone, while one already captured has months to try again.

Where it has been flown #

The first spacecraft to use ballistic capture was Hiten, a small Japanese probe launched in 1990 to study the Moon. Its propulsion was inadequate for a conventional lunar transfer after its first mission phase, and a trajectory was found that went first far out towards the Sun–Earth Lagrange points, let the Sun’s gravity reshape its orbit, and then fell back towards the Moon slowly enough to be captured weakly, arriving in 1991 with a fraction of the propellant a direct transfer would have needed. The method used the tubes of trajectories that lead through the necks of the three-body problem before those tubes had been named.

Low-energy lunar transfers of the same kind have been flown since by several missions to the Moon, taking three or four months instead of three or four days, and the European–Japanese mission to Mercury was designed to arrive by a weak capture, approaching the planet slowly enough to be held by it while its insertion burns were made. No mission has yet used ballistic capture at Mars, where it has been proposed as a way of arriving with a smaller spacecraft or with a stage that does not have to work at a single precise moment.

Why the Moon was first #

The Moon is the easiest place to try this, and the reasons are scales. Its Hill radius in the Earth’s gravity is about sixty thousand kilometres, a sixth of its distance from the Earth, so the region it controls is large compared with the precision of a spacecraft’s navigation. Its orbital period is a month, so a capture that lasts a revolution or two near its Hill radius lasts weeks rather than years. And there is a fourth body to help: the Sun, whose tide on the Earth–Moon system can reshape a spacecraft’s orbit far beyond the Moon, out near the Sun–Earth Lagrange points a million and a half kilometres away, and send it back towards the Moon slowly.

That is the route the low-energy lunar transfers take. A spacecraft leaves the Earth on an orbit reaching out towards the Sun–Earth L1 or L2, lingers there where the Earth’s and the Sun’s gravity nearly balance, and is turned by the Sun’s tide onto a path that falls back towards the Earth–Moon system with almost exactly the Moon’s orbital energy. It arrives at the Moon’s neck slowly, and is captured weakly. The transfer takes three or four months instead of three or four days, and saves a few hundred metres a second of braking at the Moon, which for a small spacecraft is a large fraction of what it can carry.

At Mars the same idea is harder. There is no fourth body to reshape the approach, the Hill region is a million kilometres across but Mars itself is a small target at its centre, and a revolution near the Hill radius takes a year, during which the spacecraft is exposed to everything that can go wrong in a long mission. The dynamics are identical in their scaled form; the practicalities are not.

Nature’s captures #

The irregular moons of the giant planets — small bodies on distant, inclined, often retrograde orbits — look like captures. Their orbits are unlike anything a moon formed in a disc round its planet would have, and they occupy the outer part of each planet’s Hill region, out to about half its radius, exactly where a body arriving through a neck would loop. The problem is the one this essay has described: a ballistic capture is temporary, and a body captured by two bodies alone cannot stay captured. Something has to remove energy while the body is inside.

The candidates are gas drag in an extended envelope round a young giant planet, collisions with other bodies already in orbit, a close encounter between two bodies arriving together that leaves one bound, and the planet’s own growth, which deepens its gravity under a temporarily captured body faster than the body can leave. Each converts a temporary capture into a permanent one, and each leaves a different signature in the moons’ orbits. The irregular moons are therefore a population of ballistic captures made permanent by something other than a rocket, and the statistics of their orbits are a record of which mechanism did it.

The capture and the spiral #

The case where ballistic capture matters most is the one the essay on arriving slowly ended on: a spacecraft with an electric engine, too weak to brake from a hyperbola in the few hours of a close approach. Such a spacecraft must arrive with almost no excess anyway, because it cannot remove a large excess quickly, and that is exactly the condition for ballistic capture. The weak capture holds it at the edge of the Hill region for long enough that the engine, thrusting for weeks, can lower its orbit out of the tide’s reach and begin the spiral inward.

For an electric spacecraft the capture is not an optimisation but the only way in: the alternative is a slow spiral that begins only after an excess has been removed that the engine cannot remove in time. The capture turns an arrival the engine cannot make into one it can.

Where the model stops #

A planar Hill problem. The figures use Hill’s approximation — a small planet, a distant Sun, a circular planetary orbit — in two dimensions. Mars’s orbital eccentricity changes its Hill radius by ten per cent over a year, and the third dimension adds inclination to the arrival; both change which trajectories are captured and for how long, and the real design calculations use the full ephemeris.

The planet’s moons and shape ignored. Close to a planet, its flattening and its moons perturb an orbit; at the distances where ballistic capture happens they are small, but the trajectory after the first burn has to include them.

Revolutions counted crudely. The number of revolutions is the winding angle accumulated while inside the Hill radius, divided by a full turn. Captures that dip in and out of that radius are counted in pieces, and a different boundary would give somewhat different numbers.

The departure cost not computed. The figure of insertion burns compares the arrival alone. The heliocentric transfer that delivers a spacecraft into a capture is longer and costs more at the Earth, and that cost depends on the launch date and the transfer; the net saving is a mission-design result, not a property of the dynamics.

The surprising connection #

The necks at L1 and L2 that admit a slow spacecraft into a planet’s Hill region are the same necks through which a star in a close binary loses mass to its companion. In both cases the boundary is the critical surface of the restricted three-body problem, and in both cases what passes through it is material whose energy is just enough — gas at the star’s surface, a spacecraft at the end of a long, careful transfer. The geometry that lets a white dwarf feed on its companion is the geometry that lets a small probe arrive at a planet without a heavy braking stage.

And the chaos that makes the capture unpredictable is the same that makes the three-body problem unsolvable in closed form. The arrivals that loop five times and the arrivals that loop once are separated by a fraction of a per cent in their starting conditions, which is why no formula gives the capture time. A capture is found by integrating many arrivals and choosing one, and then flown by correcting towards it.

Still open: capture at Mars with no burn at all #

Every ballistic capture so far has ended with a burn, because the capture itself is temporary. A body captured into a long-lived quasi-stable orbit round a planet without any non-conservative force is possible in principle in the full problem — some irregular moons of the giant planets are thought to have been captured temporarily and then made permanent by gas drag or collisions in the early solar system — but nothing equivalent is available to a spacecraft at Mars except its own engine. Whether a trajectory exists that is captured for long enough, and lowered far enough by the Sun’s tide alone, for an electric engine of realistic thrust to finish the job before the capture ends, is the question on which a propellant-light Mars orbiter turns.