An aim bought with the last day of the approach

A flyby of a moon multiplies its own aiming error: the turn a moon gives a spacecraft depends steeply on how close the craft passes, so a few kilometres of error in the aim become tens of metres a second of error in the velocity afterwards, and hundreds of kilometres of miss at the next encounter. A tour of a giant planet’s moons that brakes on the moons before braking with the engine depends on aiming every flyby to within a few kilometres. The large missions that have flown such tours did it with large antennas, tracking from the Earth, and navigation teams on the ground.

The essay that worked out the sensitivity ended with a question about small spacecraft: craft of a few hundred kilograms with small antennas, for which ground-based navigation delivers an aim of tens of kilometres rather than a few. A camera on board, locating the moon against the stars and computing its own corrections, could in principle do better. This essay asks how much better, and what it costs — and finds that the answer is a trade between time and propellant with a floor under it set by the moon itself.

A camera that sees better as it gets closer #

Navigation from the ground measures a spacecraft’s distance and speed by the radio signal and its direction against distant quasars, and from those it predicts where the craft will pass the moon. The prediction degrades with the size of the craft’s antenna — a weaker signal gives noisier measurements — and with the uncertainty in the moon’s own position, which is known to a few kilometres for the large moons of Jupiter and Saturn. For a small craft, thirty kilometres is a reasonable delivery error from the ground.

A camera measures something different: the direction from the craft to the moon, against the background stars. An angle is not a position — a camera cannot see how far — but on a flyby approach the range is known well enough from the approach itself, and the quantity that matters, the aim point in the plane through the moon perpendicular to the approach, is the angle times the range. A camera that locates the moon’s centre to twenty microradians — a small fraction of a pixel, achievable by fitting the image of a disc — knows the aim to twenty microradians times the distance. A day out, at six kilometres a second, the distance is half a million kilometres and the aim is known to ten kilometres. Six hours out it is known to two and a half.

The camera’s knowledge improves in proportion to the time remaining. That is the whole attraction of navigating on board: the last day of the approach is the day in which the target is seen best, and it is the day the ground, with its hour-long light time and its analysis cycle, can hardly use.

The same improvement makes corrections expensive #

Knowing the aim is not the same as achieving it. A correction made with time-to-go $t$ moves the aim point by the velocity change times $t$, so removing an error $\sigma$ costs $\sigma/t$. The ground’s thirty-kilometre error, removed a day out, costs a third of a metre a second; removed six hours out, a metre and a half; removed an hour out, eight. Waiting for better knowledge makes the correction cheaper to aim and dearer to execute, in exactly inverse proportion.

The hero figure draws both. The error left at the flyby falls as the correction is made later, because the craft knows more, until it meets a floor. The cost rises as the correction is made later, because the same error has to be removed in less time. With a metre a second to spend on the final correction — which for a small craft on a tour of dozens of flybys is a generous allowance — the best plan is to correct about nine hours out and accept an aim of about four kilometres. A day out would cost a third as much and deliver eleven kilometres; an hour out would cost eight times as much and deliver nothing better than the floor.

The floor is the part of the picture that a simple account misses, and it decides where the trade stops.

A moon too close to find its centre #

Far from a moon, the camera sees a small bright disc and finds its centre by fitting a model of a lit sphere; the error is a fraction of a pixel, and the twenty microradians assumed here is typical of a good narrow-angle camera on a large disc. Close to a moon, the disc fills the field. Its centre is then off the image altogether, and has to be inferred from the curve of the lit limb, compared with a model of the moon’s shape. Real moons are not perfect spheres — Europa has ridges, Titan has a thick hazy atmosphere whose apparent edge depends on the wavelength, Enceladus is slightly flattened — and fitting a limb gives the centre to a fraction of the radius rather than a fraction of a pixel. A few tenths of a per cent is a good result.

That puts a floor under the knowledge. For a Europa-sized moon it is about three kilometres, reached about six hours out; for Titan, about five, reached eleven hours out; for a small moon like Enceladus, half a kilometre, reached only in the last quarter-hour. After the floor the camera brings no further improvement in the aim, and every later correction is paid for without being bought. The floor can be beaten, but only by a different measurement: imaging surface features against a map made on an earlier pass, which is how landers navigate in their final minutes and how the closest flybys of a well-mapped moon are refined. For a first encounter there is no map.

What an aim is worth, flyby by flyby #

The figures express the trade in kilometres of aim and metres a second of correction, and the tour converts one into the other again on the far side of the moon. A flyby three hundred kilometres above Ganymede at six kilometres a second turns the craft’s velocity strongly, and each kilometre of error in the aim becomes about a third of a metre a second of error in the outgoing velocity, which has to be corrected a few days later at about that cost. An aim of four kilometres therefore costs about a metre and a third after the flyby, on top of the metre spent achieving it before; an aim of eleven kilometres, bought for a third of a metre a second a day out, costs three and a half afterwards.

So there is a best place on each menu that is not at either end. Spending a little more before the flyby to aim better saves more than it costs after, until the aim reaches the floor, beyond which spending more before saves nothing after. For a Ganymede flyby, whose larger disc puts the limb floor at about five kilometres, the total — before plus after — is smallest when the final correction is made about half a day out and the aim is five or six kilometres: a total of about two and a half metres a second per flyby, against the hundred and ninety metres a second that a braking flyby of Ganymede saves on arrival. The aim a small craft can achieve is affordable many times over; the question is only how it compares with the tour’s tolerance for drift between encounters.

A small moon reverses the emphasis. Enceladus is a tenth of Europa’s size, its limb floor is half a kilometre, and it is reached only in the last quarter-hour; a flyby of it is limited by the propellant and the delay rather than by the moon’s shape. Its gravity is also weak, so a flyby of it turns the craft very little and multiplies aim errors very little. The moons whose flybys most need a precise aim are the large ones, whose turns are strong, and those are exactly the moons whose size puts a floor under how precisely a camera can find them on a first approach. The geometry that makes a moon useful for braking is the geometry that limits how well it can be aimed at.

What a better camera buys #

Each curve in the figure is a camera’s menu: every point is a choice of when to make the final correction, and the curve says what that choice costs and delivers. Between the ground’s thirty kilometres and the limb floor, the aim improves in proportion to the propellant, and a better camera shifts the curve down by its own factor: with a metre a second, a ten-microradian camera reaches the floor, a thirty-microradian one six kilometres, a hundred-microradian one twenty-one. The camera’s angular accuracy and the propellant are interchangeable within that range — each halving of the angle is worth a doubling of the propellant.

Beyond the floor nothing is interchangeable. A better camera reaches the floor sooner and more cheaply but does not lower it; more propellant buys nothing once there. For a large moon the floor is the limit of optical navigation on a first encounter, and the four kilometres the hero figure gives for a metre a second is close to it. That is the sense in which a small craft’s aim is set by geometry rather than by its antenna: once the craft navigates itself, the moon’s own shape decides how close it can aim.

Faster encounters are harder #

The speed of the encounter enters twice. A faster craft is further from the moon at a given time-to-go, so the camera’s knowledge is worse at any given moment; and the correction has to remove the same error in the same time, so it costs the same. To stay within a fixed budget, the correction has to be made at the same time-to-go, from further away. The aim degrades in proportion to the speed once it has left the floor: twice as fast, twice the error.

That lands hardest on the first encounter of a tour. A spacecraft arriving at a giant planet meets its first moon at fourteen kilometres a second or more, on the approach that the tour’s braking flybys are designed to slow; the later encounters, after the speed has been reduced, are at four to six. The flyby whose aim matters most — because it begins the tour and sets every encounter after it — is the one the camera can aim worst, and it is also the one where the flyby’s own sensitivity to aim is smallest. The two dependences on speed run in opposite directions: slow flybys turn more for a given miss and are easier to aim; fast flybys turn less and are harder.

The hours between the image and the burn #

The camera’s advantage depends on how quickly its images become corrections. A correction computed from an image taken some hours earlier uses the knowledge of that earlier moment, when the moon was further away. If the delay is short compared with the time at which the floor is reached, it hardly matters: with an hour from image to burn, the error still reaches the floor. If it is comparable, the floor is reached later or not at all: with twelve hours, the correction never sees the moon from close enough, and the aim stops at five kilometres however late the burn is made.

This is why the loop has to be closed on board. Navigation through the ground has a delay of the light time to the planet and back — one and a half hours for Jupiter, more than two for Saturn — plus the time to downlink the images, process them, design the correction, check it and uplink it, which for a careful team is most of a day. Such a loop can use images from a day or two out, which give the aim to ten or twenty kilometres, and cannot use the last day of the approach, in which the target is seen best. An onboard loop, computing its corrections minutes after its images, can use all of it. The first spacecraft to navigate themselves this way — a technology demonstrator that flew past a comet in 2001, the impactor that hit another comet in 2005 and the spacecraft that hit an asteroid’s moon in 2022 — all used the last hours of their approaches because no loop through the Earth could have.

Two corrections spend less than one #

The cost of a late correction is set by how much error it has to remove, and nothing requires the final correction to remove all of the ground’s thirty kilometres. A coarse correction a day out, when removing error is cheap, can bring the aim from thirty kilometres to ten or so; the final correction then has a third of the error to remove, and costs a third as much. Optimising the time of the first correction for each choice of the second, the pair costs about a third of the single correction for the same final aim.

The principle is the one every navigated trajectory uses, from Mars transfers to moon tours: a sequence of corrections, each made as soon as the knowledge justifies it and sized to the error that knowledge reveals, spends less than one correction made late. What onboard navigation changes is the time scale. A ground-navigated tour plans its corrections days apart; an onboard one can plan them hours apart in the last day, and the economy of the sequence carries over.

The camera and the ground together #

Nothing in the trade requires the camera to work alone, and in practice it does not. Radio tracking from the Earth measures the spacecraft’s distance and line-of-sight speed with a precision no camera can approach, and it measures the craft’s position relative to the planet and the moon system as a whole; the camera measures the direction to the moon with a precision no radio measurement can approach in the last day. A navigation filter combining the two gives the trajectory from the radio data and the aim point from the images, each where it is best.

What onboard navigation changes is who closes the last part of the loop. The ground can deliver its best estimate a day or two before the flyby; the craft then refines the aim from its own images and makes the final correction itself. The large missions did the image processing on the ground, with a day’s delay, and still achieved kilometre-level aims at Titan and the Galilean moons because their radio tracking was excellent; a small craft with weaker radio tracking needs the camera to do more, and needs it later, which is exactly the part the ground’s delay cannot reach.

What the trade assumes #

The approach is a straight line at constant speed. In the last hours the moon’s gravity bends the approach, and the mapping from a velocity change to a change in the aim is no longer simply the time-to-go; it remains linear and well known, and the corrections are computed with it, but the figure’s simple cost overstates the leverage of a correction made within the moon’s sphere of influence.

The camera’s accuracy is a single angle. In practice it depends on the moon’s apparent size, its phase — a crescent’s centre is harder to find than a full disc’s — and the brightness of the background stars that fix the camera’s pointing. The twenty microradians assumed is plausible for a large moon seen at a good phase from a distance of a day or so, and optimistic for a crescent.

The moon’s position is assumed known. The camera measures the direction to the moon, not the moon’s position in space, and an error in the moon’s ephemeris shifts the aim the camera achieves relative to the moon not at all — the camera aims at the moon it sees — but shifts the flyby relative to the trajectory the next encounter was designed for. For the Galilean and Saturnian moons, whose positions are known to a few kilometres, that error is comparable to the floor.

And the execution error is two per cent of each correction. Small thrusters have a minimum impulse below which they cannot fire accurately, and a correction of a few centimetres a second may be all error; for the coarse correction that matters little, and for a late final correction on a small craft it can add a floor of its own.

Still open: navigating a whole tour without the ground #

A single flyby can be navigated on board to within the limb floor for about a metre a second. A tour is dozens of flybys, each setting up the next, and the errors of one become the starting errors of the next. What has not been demonstrated is a craft navigating a whole tour itself: choosing, at each encounter, how much of its propellant to spend on aim against how much to keep for the encounters still to come, with the ground monitoring rather than steering.

The trade drawn here is the unit of that problem. Each flyby offers a menu of aim against cost; the tour’s design decides how much error each flyby can tolerate, given how strongly it multiplies into the next; and the craft has to choose a point on each menu with only its own images to go on. A small spacecraft that does that has replaced its large antenna with a camera and an algorithm — and the tours of the giant planets’ moons, which have so far needed the largest missions to fly, become something a craft the size of a desk could attempt.