A flyby that multiplies its own aiming error

A gravity assist costs no propellant, and that is usually where its account ends. The planet or moon turns the spacecraft’s velocity relative to it through an angle that depends on how close the spacecraft passes, and in the frame of the planet’s orbit that turn is a change of speed — a gain on one side of the moon, a loss on the other. A moon of Jupiter, passed at a few hundred kilometres, can brake an arriving spacecraft by a couple of hundred metres per second before its capture burn, and a sequence of moon flybys after capture can reshape its orbit round the planet dozens of times over with no more fuel than the corrections between them.

The corrections are the part the account leaves out, and they are not incidental. The turn a flyby produces depends on exactly where the spacecraft passes, and a spacecraft never passes exactly where it was aimed. The flyby turns the aiming error along with the velocity, and it turns it into a velocity error. What that error costs, and how quickly it grows if nobody corrects it, is the price of the free turn — and for the small spacecraft now being designed to reach the outer planets without large engines or large antennas, it is the price that decides whether the free turn can be used at all.

How the turn depends on the aim #

A spacecraft approaching a moon on a hyperbola is aimed at a point in the B-plane: the plane through the moon’s centre perpendicular to the incoming relative velocity. Aiming at a plane rather than at the moon itself is how every flyby is targeted, because the point where the incoming asymptote pierces that plane fixes the hyperbola completely. Its distance from the moon’s centre, the impact parameter $b$, sets how far the flyby turns the velocity. With $a_h = \mu/v_\infty^2$ — the hyperbola’s semi-major axis, a length fixed by the moon’s mass $\mu$ and the encounter speed $v_\infty$ — the turn angle δ satisfies

$$\tan\frac{\delta}{2} = \frac{a_h}{b}.$$

An error in $b$ changes δ; an error across the aim point, perpendicular to the direction towards the moon’s centre, does not change δ but rotates the plane in which the turn happens. Either way the outgoing velocity is moved. Along the aim, the change in angle per kilometre is $2a_h/(b^2 + a_h^2)$; across it, the plane rotates by $1/b$ per kilometre, and the outgoing velocity moves by $2v_\infty\sin(\delta/2)$ times that. When the flyby is far from grazing, so that $a_h \ll b$, both reduce to the same number,

$$\frac{\Delta v}{\Delta b} \approx \frac{2 v_\infty a_h}{b^2},$$

and a miss of a given size costs the same velocity whichever way it goes.

At a typical tour speed of six kilometres a second, a flyby three hundred kilometres above Ganymede turns the velocity by about ten degrees, and each kilometre of aim error becomes a third of a metre per second of velocity error. Io is more sensitive at low altitudes, because it is dense and its surface is close to its centre; Europa and Callisto, less massive or larger, are less. All four fall off at altitudes of a thousand kilometres or more, where the flyby turns less and its turn changes more gently with the aim. The sensitivity is highest precisely where the flyby is most useful, at low altitude, where the turn is largest.

A circle in, a circle out #

The equality of the along and across sensitivities is a first-order result, and it can be checked directly against the exact geometry.

The simulated flybys, each computed from the exact hyperbola, map a round cloud of aim errors into a round cloud of velocity errors, with the radius the linear rule predicts to within a few per cent. That has a practical meaning for navigation. The uncertainty of a spacecraft’s approach is usually an elongated ellipse in the B-plane — the direction along the line of sight from the Earth is measured better, by the Doppler shift, than the directions across it — and the flyby transfers that ellipse, scaled by one number, into the velocity uncertainty on departure. The shape of the navigation error survives the flyby; only its units change.

The encounter speed at which aiming is hardest #

The sensitivity depends on the encounter speed in a way that at first looks contradictory and turns out to be the same trade that governs every flyby. A slow encounter is turned through a large angle, but a large change in the angle of a slow velocity is a small change in velocity. A fast encounter moves quickly, but the moon hardly turns it, and its small turn hardly depends on the aim. The product has a maximum, at a little over three kilometres a second for Ganymede, and falls on either side. A fast arrival, at fourteen kilometres a second, is half as sensitive to aiming errors as a tour flyby at six — and gains, as the braking argument showed, a correspondingly smaller change of speed. Ease of aiming and usefulness come together, and a flyby that is easy to aim is one that does little.

An aim error a week later #

A velocity error does not stay a velocity error. It becomes a position error at the next encounter.

The growth is simple: the velocity error multiplied by the time. A kilometre of aim error at Ganymede becomes about two hundred kilometres a week later, and a tour of flybys a week or two apart would, left alone, turn a kilometre of error at one moon into a miss of hundreds of kilometres at the next, and that into a trajectory that never meets the third moon at all. The real growth is not quite linear — the spacecraft is in orbit round Jupiter between flybys, and its orbit stretches some errors and compresses others over a revolution — but the order of magnitude is right, and it makes the essential point. A multi-flyby tour is an unstable sequence, in which each flyby amplifies the errors of the one before, by factors of a hundred or more.

That is not a defect of flyby tours but a property of any trajectory that uses close encounters. A millimetre a second at launch becomes seventy-six kilometres at Mars for the same reason: the arrival is sensitive to the departure, and sensitivity compounds through every encounter. The remedy is also the same. The error is measured, and corrected, while it is still small — typically a few days after each flyby, once enough tracking has accumulated to see the new trajectory, and again a few days before the next, to trim the aim. A tour of thirty flybys is a tour of sixty or more correction manoeuvres, each of them small.

What the corrections cost #

The correction after a flyby costs about the velocity error the flyby created — the aim error times the sensitivity, a third of a metre per second per kilometre at Ganymede. With a kilometre of error, a correction costs a fraction of a metre per second, and a sequence of a dozen flybys costs a few metres per second in total: a small fraction of what the flybys provide, and a small fraction of what a spacecraft’s propellant budget reserves for such things. That is the regime the large tours have operated in. The flagship missions to Jupiter and Saturn delivered their flybys to within a few kilometres, using optical navigation — photographs of the target moon against the background stars, taken in the days before each flyby — to supplement the radio tracking, and their statistical correction budgets were a few metres per second per flyby at most.

The scaling is linear, and at some aim error the corrections eat the saving. For twelve Ganymede flybys, the crossover is at about forty kilometres per axis: beyond that, the propellant spent correcting the tour exceeds what the first braking flyby saved. A spacecraft navigated by radio tracking alone, far from the Earth, with a small antenna and infrequent tracking passes, can have errors in the tens of kilometres at Jupiter’s distance, and is therefore closer to the crossover than it looks. For such a spacecraft, a moon tour is not free; it is a trade of propellant for navigation, and whether the trade is favourable depends on how well the craft can see where it is going.

Seeing the target #

The difference between a few kilometres and a few tens of kilometres is largely the difference between being tracked from the Earth and seeing the moon. Radio tracking measures the spacecraft’s range and velocity along the line of sight from the Earth with great precision, but its position across the line of sight — the two directions that matter most in the B-plane — comes only from the slow rotation of the Earth and the geometry of the tracking stations, or from interferometry against a quasar, and at Jupiter’s distance an angular error of a few nanoradians is a few kilometres. The moon’s own position is known from its ephemeris, which has improved with every mission but is itself uncertain by kilometres. The aim is the difference between two positions, each uncertain.

Optical navigation measures that difference directly. A camera photographing the moon against the stars sees the direction to the moon in the spacecraft’s own frame, and in the last days before the flyby the moon grows from a point to a disc whose limb can be located to a fraction of a pixel. The aim point’s error shrinks as the spacecraft approaches, and a correction made two or three days before the flyby, informed by those images, delivers it to a kilometre or two. A small spacecraft with a small camera can do this too, which is why the answer to whether small craft can use moon tours is not purely a question of antennas.

An expiry date for every tour #

The growth from one flyby to the next can be read as a rate. A factor of two hundred in a week is a factor of e roughly every day and a third, and a tour of flybys a week apart behaves, between corrections, like a system with a Lyapunov time of about a day. That is the same kind of statement as the expiry date on a prediction of a chaotic orbit: beyond a few e-foldings, knowing the present state to any practical precision says nothing about where the spacecraft will be. For the planets of the Solar System the expiry date is tens of millions of years; for a spacecraft in a moon tour it is a couple of weeks.

The difference from natural chaos is that the spacecraft can act. A chaotic asteroid’s future is lost; a spacecraft’s is repaired every few days by a burn of a fraction of a metre per second, and the sensitivity that makes the uncorrected tour unpredictable is what makes the corrected one cheap to steer. A small change in the aim at one moon produces a large change at the next, so a small correction buys a large redirection. Tour designers exploit this deliberately: a trajectory that would need a large burn to change the sequence of moons it visits can be retargeted, by a correction of a few metres per second placed before a flyby, into a different sequence altogether. The amplifier works in both directions.

Choosing a flyby for its aim as well as its turn #

The sensitivity figures suggest one design freedom that is easy to overlook. A flyby’s altitude is usually chosen for the turn it gives: as low as is safe, for the largest change of velocity. But the sensitivity falls with altitude too, and not in proportion. Raising a Ganymede flyby from three hundred kilometres to a thousand cuts the velocity error per kilometre of aim by about a third, from 0.32 to 0.22 metres per second, while reducing the turn by only about a fifth, from ten degrees to eight, because the turn falls off roughly as the inverse of the impact parameter and the sensitivity as its inverse square. A tour of slightly higher flybys needs a few more of them for the same total change, and each needs a smaller correction; for a spacecraft whose navigation is poor, that trade can favour altitudes well above the minimum.

The same logic chooses between moons. Io is the most effective braking moon and the most sensitive to aim; it also sits deep in Jupiter’s radiation belts. Ganymede and Callisto are less effective and less sensitive, and far safer. The large tours flown so far have used Ganymede and Callisto for most of their shaping, and the reason usually given is radiation; the aim is a second reason pointing the same way.

The same arithmetic at a planet #

None of this is peculiar to moons. A flyby of the Earth or Venus at a few hundred kilometres has an impact parameter of order ten thousand kilometres and an encounter speed of several kilometres a second, and its sensitivity comes out at a few tenths of a metre per second per kilometre as well — the same order as at Ganymede, because the ratio of the encounter speed to the impact parameter is similar. What is different is time. Interplanetary flybys are separated by months, so an error has time to be measured precisely and corrected with small burns long before it matters; the flybys of a moon tour are separated by days, and the corrections must be fitted into the gaps. The sensitivity is a property of each encounter; the difficulty of a tour is the sensitivity multiplied by its pace.

What the calculation leaves out #

The sensitivity drawn is the flyby’s own, in isolation. A real tour has the spacecraft in orbit round Jupiter between flybys, and the orbit transforms the post-flyby velocity error before the next encounter, stretching errors along the orbit and folding them across it over a revolution; the growth figure’s linear rule is a guide to the order of magnitude, not a navigation plan. The correction cost is taken as equal to the velocity error, which is right for a correction made soon after the flyby; one made later, when the error has become a position error, can be cheaper or dearer depending on the geometry, and tours are designed so that the corrections fall where they are cheapest.

Everything about the moon is idealised as a point mass. The largest moons have gravity fields that are not quite central, and at a few hundred kilometres above Ganymede the departure from a point mass is enough to move the outgoing trajectory by kilometres, which is why the moons’ fields are measured, in part, by the very flybys that use them. And the analysis treats the aim error as random and Gaussian. The largest real errors are often systematic — a moon’s ephemeris wrong in a consistent direction, an unmodelled force on the spacecraft from its own radiated heat — and those produce errors that do not average away between flybys but accumulate, which is what the corrections are really for.

The trade, restated #

A gravity assist looks like something for nothing: a large change of velocity with no propellant, bought with a few months of waiting. The aim-sensitivity argument says what it is bought with instead. Every flyby is an amplifier of navigation error, with a gain of order a third of a metre per second per kilometre at the moons of Jupiter, and a gain of hundreds between one flyby and the next. The free turn is paid for in knowledge of where the spacecraft is — in tracking, in photographs, in the ephemerides of the moons — and in the small manoeuvres that cash that knowledge in. For a large spacecraft with a large antenna and a good camera the price is negligible. For a small one it is the main cost of the mission design.

Still open: how few kilometres a small spacecraft can aim to #

The designs now under study for small craft to the outer planets — a few hundred kilograms, with small antennas, travelling fast on direct trajectories — would use moon flybys both to brake before capture and to shape their orbits afterwards. Whether they can depends on the aim they can achieve at fourteen kilometres a second on arrival and six in the tour. A camera and an autonomous navigation system that locates the moon against the stars on board, without waiting for images to be sent to the Earth and analysed, could bring the aim to within a few kilometres regardless of the antenna; demonstrations of on-board optical navigation on asteroid missions suggest it is feasible, but none has yet flown a tour of close flybys of a giant planet’s moons. The next such attempt, and how many metres per second its corrections cost, will say whether the flybys are free for small spacecraft too.