Stealing speed from a planet, which does not notice

A spacecraft flies past a planet, is pulled in, swings round and leaves — and departs the solar system faster than it arrived, having burned nothing. Voyager 2 gained about 18 km/s from four such encounters, which is more than any rocket then existing could have supplied.

The manoeuvre looks like something for nothing, and the resolution is a change of viewpoint. In the planet’s frame nothing is gained at all: the spacecraft leaves at exactly the speed it arrived. It is only when the planet’s own motion is added back that the gain appears, and the gain comes out of the planet’s orbit.

The frame that makes it obvious #

Work in the frame moving with the planet. In that frame the spacecraft approaches from far away, follows a hyperbola round the planet, and recedes to far away again.

Energy is conserved in that frame — the planet is stationary in it and its gravitational field is unchanging — so the spacecraft’s speed at a great distance afterwards equals its speed at a great distance before. What has changed is the direction, by the turn angle of the hyperbola.

Now transform back to the Sun’s frame by adding the planet’s orbital velocity to both vectors. The incoming and outgoing velocities were equal in length but different in direction, so adding the same vector to both gives two results of different lengths. The rotation has become a change of speed.

That is the whole mechanism. There is no exotic physics; there is a rotation in one frame and a vector addition in another, and the manoeuvre exists because velocity is frame-dependent while a rotation is not.

The figure’s circle makes it visible: the incoming and outgoing tips lie on a circle of radius $v_\infty$ centred on the planet’s velocity vector. Every possible outcome of the flyby is somewhere on that circle, and the turn angle picks the point.

What the planet pays #

Conservation of momentum is not violated, and following it through gives a satisfying number.

The spacecraft gains momentum, so the planet loses exactly as much. Jupiter’s mass is about $10^{24}$ times a Voyager’s, so the velocity change it suffers is smaller by that factor — of order $10^{-25}$ m/s. Over the age of the solar system, at one such encounter per century, the accumulated effect would be far below any conceivable measurement.

It is worth writing that number out, because the ratio is the point. A Voyager masses 722 kg and Jupiter $1.9\times10^{27}$ kg. A 10 km/s gain for the spacecraft therefore costs Jupiter $3.8\times10^{-24}$ m/s — a displacement, integrated over the remaining lifetime of the Sun, of about half a nanometre. Nothing that could be called an instrument will ever register it, and the conservation law is nonetheless exactly satisfied.

The transaction is nonetheless real, and at larger scales it matters. The same mechanism, applied to planetesimals rather than spacecraft, is thought to have moved the outer planets substantially during the solar system’s early history: Jupiter scattered small bodies inward and migrated outward in response, and Neptune migrated outward by several astronomical units. Gravity assists rearranged the solar system before anybody used one.

How much is available #

The gain depends on the turn angle and on the geometry of the approach.

The turn angle grows as the pass gets closer and as the approach speed gets lower. In the limit of a grazing pass with a slow approach, the turn approaches 180° — a complete reversal in the planet’s frame — and the heliocentric speed change reaches $2v_{\text{planet}}$.

That limit is the hard ceiling: no single flyby can change a spacecraft’s heliocentric speed by more than twice the planet’s orbital speed. Jupiter moves at 13.1 km/s, so a Jupiter flyby is worth at most 26.2 km/s, and in practice much less because a grazing pass at low approach speed is not compatible with actually going anywhere.

Direction decides the sign. Passing behind the planet — trailing it — the spacecraft is pulled forward and gains. Passing in front, it is pulled backward and loses. Both are used: outbound missions pass behind, and missions heading sunward pass in front to shed the Earth’s orbital motion.

That last sentence is the manoeuvre’s central frustration and it is worth stating as an inequality. The turn angle satisfies $\sin(\delta/2) = 1/(1 + r_p v_\infty^2/\mu)$, so it falls as the approach speed rises. A spacecraft crawling into Jupiter’s neighbourhood can be turned through 160°; one arriving at 10 km/s relative to the planet is turned through perhaps 40°. The assist is largest exactly when the spacecraft has the least energy to work with, and it shrinks as the trajectory gets more ambitious.

The tours #

The technique turned the outer solar system from unreachable into routine.

Mariner 10, 1974, used Venus to reach Mercury — the first flight use, on a suggestion by a graduate student, Michael Minovitch, who had worked the mathematics out on an IBM 7090 in 1961 and had considerable difficulty getting anyone to believe it.

Voyager 2, launched 1977, used Jupiter, Saturn, Uranus and Neptune in sequence. That alignment recurs every 175 years. A direct Hohmann transfer to Neptune would have taken about 30 years and arrived with no fuel for anything; Voyager 2 arrived in 12 and visited three other planets on the way.

Cassini used Venus twice, Earth once and Jupiter once to reach Saturn — a seven-year, 3.5-billion-kilometre route for a destination 1.2 billion kilometres away. Flying further to arrive sooner is the standard shape of an assisted trajectory.

Parker Solar Probe uses Venus seven times in the other direction, shedding orbital energy at each pass to work its way inward. Reaching the Sun is harder than leaving the solar system, and the flybys are how the difference is paid.

What it costs instead #

Nothing is free, and what an assist spends is time and opportunity.

Time. Assisted trajectories are longer in distance and usually in duration than the direct transfer, when a direct transfer is possible at all. Cassini’s seven years to Saturn against a direct six is a modest example; the Parker sequence takes seven years to reach its final orbit.

Windows. Synodic periods — which are just the harmonic law applied to two orbits at once — govern when each encounter is possible, and a multi-flyby route requires several alignments to coincide. That is what makes an opportunity like Voyager’s rare rather than merely inconvenient, and it is why missions slip by years when a launch window is missed.

Precision. Each flyby amplifies whatever error the spacecraft arrives with. A small navigational error at Jupiter becomes a large one at Saturn, so assisted missions require frequent small corrections and very accurate tracking. Voyager’s aim point at Neptune, twelve years after launch, had to be met to within about 100 km.

Risk. Every flyby is an event that cannot be repeated. A failure during the encounter loses the entire remaining mission.

What was actually measured, and the millimetres left over #

Everything above is theory that has been flown, and flown trajectories are tracked well enough to check it. The check is a radio one: the spacecraft’s carrier signal is returned coherently, the Doppler shift gives the line-of-sight velocity, and after a long integration the residual is measured in fractions of a millimetre per second. That is the accuracy against which a gravity assist’s predicted velocity change is compared.

Almost every flyby matches. Six did not.

Between 1990 and 2005, six Earth flybys — Galileo twice, NEAR, Cassini, Rosetta and MESSENGER — showed unexplained velocity changes at the closest approach. Galileo’s first, in December 1990, came out 3.92 mm/s faster than predicted. NEAR’s in 1998 was 13.46 mm/s fast. Cassini’s in 1999 was 2 mm/s slow. The numbers are minute against an orbital speed of tens of kilometres per second — parts in $10^{10}$ — and they were far outside the tracking error, which was well under a tenth of a millimetre per second.

The residuals became known as the flyby anomaly, and for about fifteen years there was no accepted explanation. An empirical formula was found that fitted all six, involving the declinations of the incoming and outgoing trajectories, which is the kind of result that is either a deep clue or a coincidence with six data points. Later analyses attributed most of it to mismodelled effects in the tracking itself — Earth’s radiation pressure, the thermal recoil of the spacecraft, the reference frame used for the atmospheric delay — and later flybys, tracked with better models, have shown nothing. The anomaly is now generally regarded as closed, without any single moment at which it was solved.

What the episode demonstrates is the standard the measurement operates at. A manoeuvre whose whole content is a vector addition is confirmed to a few parts in $10^{10}$, and the discrepancies that survive at that level are millimetres per second of thermal radiation leaving one side of a spacecraft slightly warmer than the other.

The conserved quantity that limits every tour #

Momentum bookkeeping says the planet pays. There is a second and much more restrictive bookkeeping, and it constrains what any sequence of flybys can possibly achieve.

In the restricted three-body problem — Sun, planet on a circular orbit, and a spacecraft of negligible mass — there is a quantity conserved along the spacecraft’s whole trajectory, including through every encounter with the planet. In terms of the spacecraft’s heliocentric orbit it is the Tisserand parameter,

$$T = \frac{a_p}{a} + 2\sqrt{\frac{a}{a_p}(1-e^2)}\,\cos i,$$

with $a_p$ the planet’s orbital radius. A flyby can change the semi-major axis, the eccentricity and the inclination as much as the geometry allows, but it cannot change this combination of them.

The consequence is sharp. A spacecraft using Jupiter alone is confined to a two-dimensional surface in the space of orbits, and any orbit off that surface is unreachable no matter how many Jupiter flybys are performed or how they are arranged. Getting off the surface requires a burn, or a different planet — which is precisely why multi-planet tours exist. Each new planet supplies its own conserved quantity and its own surface, and a trajectory hops between them.

The parameter was invented for a completely different purpose. Félix Tisserand introduced it in the 1890s to identify comets: a comet perturbed by Jupiter into an apparently new orbit could be recognised as the same object, because the value of $T$ was unchanged. It is still used that way — a body with $T$ between 2 and 3 is a Jupiter-family comet almost by definition — and the same invariant that identifies a comet across a century of perturbation tells a mission designer which trajectories are off the table.

The two frames, side by side #

The manoeuvre is impossible to picture because it requires holding two descriptions at once, and each of them is simple on its own. In the planet’s frame nothing is gained: the spacecraft arrives and departs at the same $v_\infty$, having been turned. That is a two-body hyperbola with an exact solution.

In the Sun’s frame the speed changes, so the energy changes, so the semi-major axis changes — the spacecraft leaves on a different orbit from the one it arrived on. Both descriptions are complete and neither can be drawn on the other’s axes, which is the whole reason the manoeuvre reads as a trick.

Every flyby that adds speed has a mirror image that removes it, and the removal is what makes arriving anywhere possible.

A spacecraft reaching an outer planet on a minimum-energy transfer arrives with several kilometres per second of excess speed relative to the planet, and shedding that to enter orbit is expensive — Cassini’s Saturn orbit insertion was a 96-minute burn costing 626 m/s, and it was the largest single manoeuvre of the mission. Passing in front of a moon on the way in subtracts velocity by the same triangle that adds it when passing behind, and the saving is propellant that does not have to be carried from Earth.

Cassini used Titan for this repeatedly. Over thirteen years it flew more than a hundred Titan flybys, each one reshaping the orbit — changing inclination, raising or lowering periapsis, rotating the orbital plane — for essentially no propellant. The mission’s entire tour of the Saturnian system was designed as a sequence of Titan encounters, with small burns used only to aim the next one.

That is the technique’s mature form: not a single boost on the way out, but a long series of encounters used as a free control surface.

Coming back for another one #

The ceiling of twice the planet’s orbital speed applies to a single encounter. A sequence of encounters with the same planet is not bounded that way, and arranging one is a standard technique.

The requirement is that the spacecraft return. After a flyby its heliocentric orbit has a new period, and if that period is a simple ratio of the planet’s — three of the spacecraft’s to two of the planet’s, say — then both bodies arrive back at the same point together and another encounter happens.

Such a resonant return costs nothing to set up beyond the aiming, since the flyby that produced the new orbit can be aimed to give whatever period is wanted within its available range. So a mission can chain encounters with one planet indefinitely, each one turning the velocity a little further.

What limits the chain is that each successive encounter has a different approach geometry, and the useful ones run out. The invariant described above still holds — every orbit in the chain has the same Tisserand parameter with respect to that planet — so the chain moves along a curve rather than filling a region, and it reaches the end of the accessible curve eventually.

There is a refinement that extends the chain considerably and is worth naming because it exploits the ceiling rather than fighting it. A small burn applied far from the planet, at the right point of the orbit, changes the period slightly; the changed period alters where the spacecraft meets the planet next time; and the altered geometry gives a different and larger turn. The propellant buys a change in phasing rather than in speed, and the leverage is an order of magnitude better than spending the same propellant on the velocity directly.

That technique is what makes a long tour affordable, and it is the reason a mission’s propellant budget is dominated by small deterministic manoeuvres between encounters rather than by anything at the encounters themselves.

Where the model stops #

Two bodies at a time. The analysis treats the flyby as a two-body hyperbola inside the planet’s sphere of influence and a two-body ellipse outside it, stitched at the boundary. The patched-conic approximation has no rigorous justification and is accurate enough that the corrections are small burns rather than redesigns.

Impulsive encounter. The flyby is treated as instantaneous. It is not — a Jupiter encounter lasts days — and the Sun’s pull during it is a real perturbation.

A point-mass planet. Close passes feel the planet’s oblateness and its tidal gradient, and a pass through an atmosphere feels drag, which is aerogravity assist and has never been flown.

No thrust. Combining a burn with the flyby, at the closest point where the Oberth effect is strongest, beats either alone — a powered flyby, used by several missions.

The figures share a limitation that is worth being explicit about, because it is the same one that makes the manoeuvre seem impossible. Each of them shows a velocity diagram, not a trajectory: the arrows are speeds, the circle is a locus of possible outcomes, and no part of the picture is a path through space. The trajectory itself is a hyperbola in one frame and a piece of an ellipse in another, and the whole difficulty of understanding gravity assists is that the two pictures cannot be drawn on the same axes.

There is one turn the geometry allows and the physics almost never does, and drawing it makes the limit concrete.

The bound is worth stating as a formula because it explains the architecture of every tour ever flown. The turn angle satisfies $\sin(\delta/2) = 1/e$, and the eccentricity of the flyby hyperbola is set by how close the periapsis comes: a fast approach at a large distance gives a nearly straight line, a slow approach skimming the cloud tops gives the largest turn available. So the deliverable change is largest for slow encounters with massive, compact planets, and it is exactly zero for a spacecraft moving much faster than the planet’s own escape speed at the closest approach it can survive.

That is why Jupiter is used for nearly everything and Mars for very little. Jupiter’s escape speed at the cloud tops is 60 km/s and a typical approach speed is 6; Mars’s is 5 and a typical approach is 3. The available turn at Jupiter is most of a reversal and at Mars it is a few tens of degrees, and no amount of navigation changes the ratio.

It also explains why a tour’s later flybys deliver less than its earlier ones. Each assist raises the spacecraft’s speed relative to the Sun, and for a subsequent encounter with the same planet that means a higher approach speed, a lower eccentricity for the same periapsis, and a smaller turn. The sequence is self-limiting, and the limit is reached from below by exactly the process that is trying to escape it.

The ladder from here #

Later rungs: the hyperbolic flyby geometry, and the turn angle as a function of impact parameter. The velocity triangle worked through algebraically. Powered flybys and the Oberth effect combined. The Voyager grand tour and its alignment. Multi-flyby trajectory design and porkchop plots. Planetary migration by planetesimal scattering. The Tisserand parameter, which is conserved across flybys and constrains what any sequence can achieve. Aerogravity assist. And the reverse manoeuvre, used to slow down, which is how a spacecraft is delivered into orbit around a planet it arrives at far too fast.

Minovitch’s 1961 calculations were treated as an academic curiosity by a laboratory that was at the time struggling to reach Venus. Thirteen years later the same laboratory flew the first assist, and no mission to the outer solar system has been designed without one since.