Braking on a moon before braking with the engine

A spacecraft flying past a planet cannot change its speed relative to the planet. It arrives on a hyperbola and leaves on one, at the same speed, turned through an angle set by how close it came. What it gains or loses is measured in some other frame, where the planet itself is moving: add the planet’s velocity back to the turned relative velocity, and the spacecraft can come out faster or slower than it went in. That is how probes have been flung outward through the Solar System for fifty years, and the same exchange works in the other direction. Turn the relative velocity the other way, and the spacecraft comes out slower.

At a giant planet, being slower is exactly what a spacecraft wants. It arrives from interplanetary space on a hyperbola with several kilometres per second of excess speed, and it has to burn off enough of it to be captured — an orbit-insertion burn of several hundred to over a thousand metres per second, made at the closest point to the planet, often the largest single manoeuvre of the mission. Any energy removed before that burn is propellant not carried across the Solar System. Jupiter has four large moons on circular orbits, moving at ten to seventeen kilometres per second, and an arriving spacecraft falls past their orbits on its way in.

A small turn and a large lever #

The turn is small. A spacecraft falling toward Jupiter crosses Ganymede’s orbit at about sixteen kilometres per second and meets Ganymede, itself moving at eleven, at a relative speed of nearly fourteen. Ganymede’s gravity at 300 kilometres’ altitude turns a fourteen-kilometre-per-second trajectory through about two degrees. That is a feeble deflection by the standards of a planetary flyby, where turns of tens of degrees are routine.

What makes it worth having is the lever. The change in the spacecraft’s velocity from a turn through a small angle δ is about δ times the relative speed — two degrees of fourteen kilometres per second is about half a kilometre per second — and the change in its orbital energy in Jupiter’s frame is the moon’s velocity times that change, if the turn is aimed to point against the moon’s motion. A fast moon and a fast encounter multiply a small turn into a large exchange of energy. For the Ganymede flyby the spacecraft leaves the encounter about 320 metres per second slower in Jupiter’s frame than it arrived.

The energy removed is Ganymede’s gain. The moon’s orbit is enlarged by an amount a planet pays and never notices: a two-tonne probe changing the orbital energy of a moon of 1.5 × 10²³ kilograms moves it by less than the width of an atom.

Speed lost at the moon, saved at periapsis #

The 320 metres per second lost at Ganymede’s orbit turn into only 191 saved at the insertion burn, and the difference is not a loss. It is the Oberth effect running in reverse. The flyby removes a fixed amount of orbital energy. The insertion burn is made at periapsis, five Jupiter radii from the centre, where the spacecraft is moving much faster than at Ganymede’s distance, and a given energy corresponds to a smaller change of speed where the speed is higher. Energy, not speed, is what is conserved between the two points, so the saving in the burn is the energy removed divided by the speed at periapsis.

That conversion is also why the insertion burn is made at periapsis at all: an engine firing where the spacecraft is fastest removes the most energy per metre per second of propellant. The moon removes energy where the spacecraft is slower, and so less efficiently in these terms, but it does so without propellant. The two combine: the moon takes what it can on the way in, and the engine takes the rest at the point where each metre per second is worth most.

Which moon brakes best #

The four Galilean moons differ in the three things that matter.

Mass sets how sharply a moon can turn the spacecraft; Ganymede is the most massive, Io second, Callisto third and Europa, at about half of Io’s mass, a distant fourth. The relative speed sets how much turning a given mass achieves — the turn angle falls as the square of the encounter speed — and here the geometry favours the inner moons more than might be expected. Deep in Jupiter’s well the spacecraft is moving fast, but so is the moon, and their velocities are more nearly aligned, so Io is met more slowly than Ganymede. And the moon’s own orbital speed multiplies the whole exchange: Io moves at seventeen kilometres per second, Callisto at eight.

Put together, Io wins. It was used for exactly this purpose once: the Galileo spacecraft, arriving in December 1995, flew within a thousand kilometres of Io a few hours before its insertion burn, and the flyby reduced the burn substantially. No mission since has done it again, and the reason is not orbital mechanics.

The moon that sits in the belts #

Io orbits inside Jupiter’s radiation belts, in a torus of sulphur and oxygen ions shed by its own volcanoes, where the particle flux is thousands of times that at the Earth’s belts. Trapped particles that nothing removes accumulate there to intensities that degrade electronics in hours. Galileo’s close Io passage on arrival used a substantial fraction of the radiation dose its electronics were designed for, and the spacecraft suffered radiation-induced faults during later Io encounters.

The missions to Jupiter since have taken Ganymede instead. It orbits in a region where the radiation is far lower, and although it saves less, it saves enough to be worth the geometry. The European mission to Jupiter’s icy moons is planned to fly past Ganymede shortly before its insertion burn, for exactly this reason. The choice is a trade between a hundred metres per second of propellant and a fraction of the spacecraft’s lifetime radiation budget, and the orbital mechanics only supply one side of it.

The advantage Io would bring is easier to see against the burn itself.

A hundred kilograms of propellant, for a spacecraft carrying a tonne or more of it, is the margin mission designers argue over — worth having, and not worth a spacecraft. Galileo’s designers took it, planning an Io pass on the day of arrival when a single encounter’s radiation dose, at the start of the mission, was a price the spacecraft could afford. The later missions, planned from the start around years of operation in Jupiter’s radiation, give the margin up.

The moon has to be there #

A flyby needs the moon to be at the crossing point when the spacecraft is, and that fixes the arrival to within hours. The Galilean moons orbit Jupiter in between two and seventeen days, and an interplanetary trajectory arriving at Jupiter is determined years in advance by its launch date and its route. So the arrival time is adjusted in the final months of cruise with small trajectory corrections, shifting the moment the spacecraft crosses the moon’s orbit until the moon is there.

The geometry constrains the approach as well. To remove energy, the flyby has to turn the spacecraft’s relative velocity backward against the moon’s motion, which means passing ahead of the moon — between the moon and the direction it is heading — so that the moon’s gravity pulls the spacecraft back. Aiming a flyby is done in a plane perpendicular to the approach, and a miss of a few kilometres in that plane changes the turn, and so the energy removed, by a measurable amount; the arrival hours before the insertion burn leave little time to correct it. Galileo’s Io flyby and the insertion burn that followed were executed within a single day by the spacecraft’s own stored sequence, with the light travel time to Jupiter, over forty minutes, ruling out any intervention from the ground.

Fast arrivals gain less #

The saving depends on how fast the spacecraft arrives, and it depends in the wrong direction for the missions that need it most.

A faster spacecraft meets the moon at a higher relative speed and is turned through a smaller angle, so the energy removed falls. At the same time the insertion burn it faces rises. The fraction the moon can take is therefore largest for slow arrivals, which already have cheap captures, and smallest for fast ones, which do not. A mission that saves years by flying a faster transfer to Jupiter pays for it at arrival, and the moon cannot pay much of the difference.

That is one reason missions to the outer planets have tended to fly slow, gravity-assisted routes rather than fast direct ones: the arrival speed a route leaves the spacecraft with decides the insertion burn, and slow arrivals are both cheaper to capture and easier to brake with a moon.

A fixed amount off a variable bill #

The moon removes a fixed amount of energy, whatever orbit the spacecraft is captured into, and the insertion burn depends on that orbit.

That is why missions capture into long, loose orbits first. A 200-day orbit costs about half as much to enter as a 30-day one, and the moon takes nearly a quarter of that. The orbit is then tightened over the following months not by the engine but by more flybys: each pass of a moon on the long first orbit can be aimed to remove more energy, shortening the period step by step. A tour of Jupiter’s moons, like the one that shaped every orbit of the Galileo mission and Cassini’s thirteen years among Saturn’s, is this same braking repeated dozens of times, with the Tisserand invariant of each moon confining what each pass can do.

The capture flyby is the first step of that tour, taken before there is an orbit at all. Without it the first orbit would have to be longer, or the insertion burn bigger; with it, the tour begins already braked.

What was measured #

Galileo’s arrival day is the one case where the braking has been done and measured. The spacecraft’s velocity was tracked by the Doppler shift of its radio signal throughout, and the Io flyby’s effect appeared as a change in that shift consistent with a turn of the predicted size; the insertion burn that followed, about fifty minutes long, then completed the capture into a first orbit of about seven months. Io’s gravity field, measured on the same pass from the deflection, gave its mass and its moment of inertia to within a few per cent — the flyby that braked the spacecraft was also a measurement of whether Io’s heavy material had sunk into a core. The same Doppler record that confirmed the brake weighed the brake.

What the calculation leaves out #

The flybys here are planar and happen where the inbound hyperbola crosses each moon’s circular orbit, with the turn aimed to remove as much energy as possible. Real arrival trajectories are not in the moons’ plane, and the moon has to be at the right place at the right time, which fixes the arrival date to within hours and constrains the approach direction. The best-aimed flyby may not be the one the timing allows. The insertion burn is computed at the same periapsis with and without the flyby, but the flyby also changes the periapsis slightly, which a real trajectory corrects with the aim point; and the burn is treated as instantaneous, while a real insertion burn lasts most of an hour and loses some efficiency to being spread out.

The periapsis of five Jupiter radii is a typical choice, and a lower one makes the insertion burn cheaper through the Oberth effect while taking the spacecraft deeper into the radiation. Mission designers balance exactly the trade the moons do: every choice that saves propellant near Jupiter costs radiation dose.

An energy ledger #

It helps to put the exchange in the units that are actually conserved. Per kilogram of spacecraft, capture into a 200-day orbit means removing about 22 megajoules of orbital energy: sixteen to cancel the arriving hyperbola’s excess and six more to reach the bound orbit. The Ganymede flyby removes about five megajoules per kilogram, a little under a quarter of the total, which is why it saves a little under a quarter of the burn. The flyby removes that energy at Ganymede’s distance, where the spacecraft moves at sixteen kilometres per second and a change of 320 metres per second is worth five megajoules; the engine then removes the rest at periapsis, where the spacecraft moves faster still and every metre per second of burn is worth more energy than anywhere else on the trajectory.

That ledger also shows why a flyby can capture a spacecraft outright in a favourable case. If the arrival speed were low enough, the energy a moon removes could exceed the spacecraft’s hyperbolic excess energy altogether, and the spacecraft would leave the encounter bound to Jupiter without any burn. For a 5.6-kilometre-per-second arrival the excess energy is about sixteen megajoules per kilogram, three times what one Ganymede flyby removes, so capture by a single moon is out of reach. At arrival speeds below about three kilometres per second, reachable only by long, slow routes through the inner Solar System, one or two moon flybys could in principle do the whole job. The trajectories that exploit this are among the most delicate in mission design, because a spacecraft captured by a moon’s pull alone is barely bound and its first orbit is long and sensitive to every perturbation.

The same move at other planets #

The principle is general. At Saturn, Titan is massive enough to brake an arriving spacecraft, though Cassini, arriving in 2004, captured with its engine alone and began its long series of Titan flybys only afterwards. Titan is nearly as massive as Ganymede, but it orbits Saturn at under six kilometres per second, about half Ganymede’s speed, so the lever the moon’s own motion supplies is shorter and a Titan flyby removes less energy for the same turn. Titan offers something else instead: a thick atmosphere, through which a spacecraft could fly deliberately and lose energy to drag rather than to gravity, and proposals for Saturn orbiters have considered braking that way. At Neptune, Triton would be the brake for any future orbiter, and its retrograde orbit adds a twist: a spacecraft arriving in the direction of Neptune’s rotation meets Triton head-on, at a high relative speed that its gravity can turn only a little. At the Earth, the Moon has been used both ways: spacecraft have swung past it to raise their orbits and to leave the Earth, and a spacecraft returning from deep space can use a lunar flyby to shed enough energy to be captured into a long Earth orbit.

In every case the exchange is the same one — a moving body’s velocity added to a relative velocity turned by its gravity — and the question is only which direction the turn is aimed. The gravity assist that made the outer planets reachable is also the brake that makes their moons orbitable.

Still open: how much a moon can do for a small, fast spacecraft #

The designs now being studied for small spacecraft to the outer planets — craft of a few hundred kilograms, launched on direct fast trajectories without years of inner-planet flybys — face the combination the figures describe as worst: fast arrivals, expensive insertion burns and small savings from moons. For them the question is whether a sequence of moon flybys, beginning before capture and continuing after, can replace most of the insertion burn altogether, trading propellant for months of careful navigation among the moons. The answer depends on how precisely a small spacecraft can be navigated to a flyby a few hundred kilometres above a moon, arriving at fourteen kilometres per second, which is a question about tracking and aim as much as about gravity.