Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

The planet pays, and it shows

A gravity assist takes energy from a planet and gives it to a spacecraft, and the planet's loss is exactly the spacecraft's gain. For a two-tonne probe past Jupiter that loss is unmeasurable. Do it with a hundred Earth masses of icy debris and the same bookkeeping moves Neptune outward by several astronomical units.

Assumes Gravity assist, Tisserand parameter and Orbital transfer.

16 min read 8 figures Two bodies, and no moreScales that defeat intuition

A spacecraft can steal speed from a planet, which does not notice. The first rung of this anchor established the mechanism and used exactly that phrase, and the parenthesis was doing real work: the energy has to come from somewhere, and the somewhere is the planet’s orbit.

This rung takes the parenthesis seriously. The planet does not notice because the exchange is divided by a mass ratio of ten to the twenty-two, not because the exchange does not happen. Do the same encounter with something whose total mass is comparable to the planet’s own, and the planet moves.

A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 1 A flyby drawn as a velocity triangle. The incoming and outgoing speeds relative to the planet are equal and the turn angle is set by how close the approach was; what changes in the heliocentric frame is the vector sum, and it can be as large as twice the planet’s own speed. The planet’s velocity changes by the same momentum divided by its own mass, in the opposite direction, and the two changes are exactly antisymmetric because nothing else is present to take a share.

The exchange, written as momentum

In the planet’s frame the encounter is a hyperbola: the small body comes in at some speed, is turned through an angle, and leaves at the same speed. Its momentum has changed by the vector difference of two equal-magnitude vectors, which is at most twice the incoming momentum — an unbound conic, with a definite asymptotic direction at each end.

The planet’s momentum has changed by exactly the negative of that. There is no third body and no dissipation, so momentum conservation is exact and instantaneous.

Converting to velocities divides by the masses. A two-tonne spacecraft gaining ten kilometres a second past Jupiter changes Jupiter’s velocity by ten kilometres a second times two tonnes divided by two times ten to the twenty-seven kilograms — about ten to the minus twenty-three metres a second. Over the age of the solar system, at one spacecraft a decade, that is a displacement smaller than a proton.

A gravity assist with a 90° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 2 The same triangle at a right-angle turn, which is the geometrically simplest case and the one worth having the numbers for. At ninety degrees the change in velocity relative to the planet is 2\sqrt2 times the approach speed, and its direction is at forty-five degrees to both asymptotes — so the projection onto the planet’s own motion, which is what the heliocentric gain is, depends entirely on how the encounter is oriented. The identity that the planet’s recoil is exactly the negative of this vector holds whatever the orientation; only the useful component of it is a matter of aiming.
A gravity assist with a 35° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 3 And a shallow passage, which is what most encounters in a real tour are. A 35° turn takes about a quarter of what a deep one does, and it is used not to gain speed but to change direction — to set up the next encounter, or to move the spacecraft out of the ecliptic. The planet pays in both cases and by the same accounting: whatever the spacecraft gains, the planet’s orbit loses, in the ratio of their masses.

What the small body cannot change

The exchange is constrained, and the constraint is the subject of the number that survives the encounter. One combination of the small body’s orbital elements is unchanged by the flyby, so the set of orbits reachable from a given starting orbit is one curve rather than a region. Two things follow. A single planet cannot deliver a spacecraft anywhere: repeated flybys of one body move it along one curve, and getting off requires either a manoeuvre or another planet. And the same conservation, read from the planet’s side, means that a planet gains angular momentum when the small body loses it and vice versa — the bookkeeping that makes the same planet worth visiting three times — so the direction the planet migrates depends on where the small bodies end up.

A gravity assist with a 130° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 4 The same encounter turned harder. The velocity gained is 2v∞sin⁡(θ/2)2v_\infty\sin(\theta/2) projected onto the planet’s motion, so a 130° turn extracts nearly twice what a 68° one does from the same approach — and the turn angle is set by how deep the passage is, bounded below by the planet’s own radius and its atmosphere. Every gravity assist is a negotiation between how much can be taken and how close the spacecraft is willing to go.

A hundred Earth masses of debris

Now do the arithmetic that this rung exists for.

The early solar system contained a disc of icy planetesimals outside Neptune, of total mass estimated at tens of Earth masses. Each of those bodies that encounters Neptune is scattered — inward, outward, or out of the system — and each scattering exchanges angular momentum with the planet in exactly the way the hero figure draws.

If the scatterings were symmetric the exchanges would cancel. They are not symmetric, and the asymmetry is where the migration comes from. A body scattered inward is passed to Uranus, then Saturn, then Jupiter, and is eventually ejected from the solar system entirely by Jupiter — the transfers between them are free in the sense the second rung of this anchor’s neighbour describes — which is massive enough to eject rather than merely pass along. A body scattered outward returns to Neptune and is scattered again.

So the inward-scattered bodies are removed from the system and the outward-scattered ones are not. Neptune, Uranus and Saturn each lose angular momentum to bodies they scatter inward and regain it from bodies they scatter outward, and the net for the three outer planets is a gain: they migrate outward. Jupiter, which does the ejecting, loses and migrates inward.

A gravity assist with a 160° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 5 The deep encounter, at a hundred and sixty degrees, which is nearly a reversal. This is what Jupiter does to a planetesimal it ejects: the body arrives, is turned almost back along its own path, and departs with a heliocentric speed close to the sum of its approach speed and twice the planet’s — which for Jupiter is enough to exceed the solar system’s escape speed at that distance. The turn angle is bounded by how close the passage can be, and for a body with no surface to hit and no atmosphere to enter the bound is the planet’s own radius.

The numbers work out at several astronomical units for Neptune and a few tenths for Jupiter, which is a very unequal migration and follows directly from the mass ordering.

The record it left

The migration is not a hypothesis with no evidence. It left a signature in the Kuiper belt that is as close to a fossil as dynamics produces.

As Neptune moved outward, its mean-motion resonances moved with it, sweeping through the disc. A resonance sweeping outward through a population of small bodies is a resonance approaching them convergently, and convergent approach captures.

The result should be a population of objects trapped in Neptune’s resonances, at eccentricities pumped up by the continued migration after capture, with a distribution that records how far the planet moved. That population exists: Pluto is its largest member, and the objects sharing its three-to-two resonance are called plutinos — a lock of the same kind that makes a chain of planets evidence of how they arrived. The eccentricities are the quantitative part. A body captured into a resonance and then dragged outward has its eccentricity pumped by an amount that depends on how far it travelled, and the observed eccentricity distribution of the plutinos requires Neptune to have moved outward by about seven astronomical units.

Why the direction is not obvious

The result that Jupiter moves inward while the other three move outward looks arbitrary until the accounting is done carefully, and it is worth doing because it is the part of the story most often stated without justification.

Consider Neptune alone, with a disc of planetesimals both inside and outside its orbit. A body scattered outward gains angular momentum, so Neptune loses; a body scattered inward loses, so Neptune gains. If every body that is scattered comes back for another encounter, the exchanges are reversible and nothing accumulates.

A gravity assist with a 20° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 6 The gentle encounter, at twenty degrees, which is the overwhelming majority of them. Most planetesimals pass far from the planet and are barely deflected, and the exchange is correspondingly small — but there are enormously more of them than there are deep encounters, and the integrated effect of many shallow passes is comparable to that of a few close ones. That is why the migration is computed as an integral over impact parameters rather than as a count of close approaches, and why a disc’s total mass rather than its densest part sets the answer.

What breaks the reversibility is a sink. Bodies scattered inward eventually reach Jupiter, and Jupiter’s escape speed at its own orbit exceeds the solar system’s escape speed there — uniquely among the planets — so a body handed to Jupiter is likely to be thrown out of the system rather than returned. It never comes back, and Neptune keeps the angular momentum it gained by sending it.

A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 7 The same encounter against a planet moving at the Earth’s orbital speed rather than Jupiter’s. The triangle’s proportions change entirely: the planet’s velocity now dominates the approach speed, so the heliocentric gain is a larger multiple of v∞v_\infty and the geometry is less forgiving. That ratio — the approach speed against the planet’s own — is what decides whether a planet can eject a body, and it is why Jupiter is the sink. A planet whose orbital speed is high relative to the local escape speed cannot throw anything out, however hard it turns it.

Bodies scattered outward have no such sink. They return, and the exchange is undone.

So the asymmetry is not in the scattering but in what happens afterwards, and it depends on the existence of one planet massive enough to eject. A solar system without a Jupiter would have had a very different migration history, and possibly none at all.

What was actually measured

Three things, and they are of very different kinds.

The spacecraft case is measured directly and to absurd precision. Doppler tracking of a probe during a flyby measures its velocity change to millimetres per second, which is a part in ten million of the change itself, and the agreement with the predicted assist is complete. The planet’s recoil in the spacecraft case has never been measured and never will be. It is smaller than the noise in every ephemeris by twenty orders of magnitude.

The migration is measured indirectly, from the orbital distribution of the trans-Neptunian objects. The resonant populations, their eccentricities, and the sharp outer edge of the classical belt at the two-to-one resonance are all consistent with an outward migration of about seven astronomical units, and they are not consistent with none.

The bodies that were paid and did not leave

The accounting above has a sink — Jupiter ejects, and what is ejected does not come back — and it treats ejection as a single outcome. It is not. A body given slightly less than escape speed does not leave, and where it ends up is a second record of the same transaction.

Escape from the solar system at Jupiter’s distance takes 18.5 kilometres a second, and a body given 18.4 is on an orbit with a semi-major axis of tens of thousands of astronomical units. Its period is millions of years, its aphelion is a substantial fraction of the way to the nearest star, and at that distance the Sun’s grip is weak enough that passing stars and the Galaxy’s own tide can alter its orbit — raising its perihelion out of the planetary region, so that it never encounters Jupiter again and is stranded.

That is the Oort cloud, and it is the population of bodies the giant planets nearly ejected. Its members carry, in their energies, the record of how much was given to them: an orbit at 10410^4 astronomical units is one that received 99.99 per cent of the energy needed to leave.

The arithmetic of the sink therefore has three outcomes rather than two, and the proportions matter for the migration budget. Perhaps a per cent of the scattered mass ends up stranded in the cloud; the rest is genuinely ejected, and a small fraction is scattered inward to be destroyed or accreted. Since only the departures count towards the planets’ angular-momentum gain, and a stranded body is a departure for every practical purpose, the distinction changes the migration estimate hardly at all — but it is the reason the solar system has a comet reservoir at all, and the reason that reservoir’s mass is an independent constraint on how much material the disc held.

What left, and what it took with it

The total is worth writing down, because it is the one part of the story where the solar system’s books nearly balance.

Angular momentum is conserved in the whole system. The outer planets gained; Jupiter lost; and the difference left with the ejected material. Estimates of the primordial disc put tens of Earth masses beyond Neptune, and the great majority of it is now somewhere between the Oort cloud and interstellar space.

That is a large amount of material to have removed, and it has a consequence nobody set out to predict. A disc massive enough to move Neptune seven astronomical units is massive enough to have destabilised the giant planets themselves, and the current family of models has them beginning in a compact chain of mean-motion resonances which the disc eventually breaks — after which the planets scatter off one another for a few million years before settling.

Those models very often eject a planet. A fifth giant, of roughly Neptune’s mass, makes the surviving four end up in the right places considerably more often than a four-planet start does, and it is thrown out of the system in the process. The same bookkeeping that moves Neptune outward by an astronomical unit per few Earth masses of debris will, given a planet-sized participant, remove the participant — and an ejected ice giant is exactly the kind of object that, encountering another star’s disc a few billion years later, arrives as an unbound visitor with a small ∣a∣|a| and a large speed at infinity.

The observational test of that suggestion is not available in this system and may be available in the population. If ejected ice giants are a common product of planetary systems settling, they are a component of the interstellar object population, and the distribution of speeds and sizes of the visitors passing through here is the only sample of them anybody will get. Two of the three known objects are small; whether anything planet-sized is out there is a question about a survey’s depth rather than about dynamics.

What can be said now is that the three known visitors span a factor of two in speed and were found by surveys with quite different sensitivities, so the distribution they sample is not yet the distribution that exists. A fourth object would improve the estimate of the number density by more than a fourth measurement of any of the first three.

Until then the ejected-planet hypothesis is supported by simulations that need it and by nothing observed, which is a fair description of most of what is believed about the first hundred million years here.

The same trick, applied deliberately

There is one place where the planet’s recoil is not negligible and is used on purpose, and it belongs here because it is the same calculation with the mass ratio reversed.

A spacecraft that has to change a small body’s orbit — an asteroid on a collision course, say — can do it by flying alongside and letting its own gravity pull. The exchange is the same antisymmetric one, and the magnitude is the spacecraft’s mass times the interaction time divided by the square of the separation. A one-tonne craft hovering a hundred metres from a small asteroid for a year changes its velocity by a fraction of a millimetre a second, which over decades is a displacement of thousands of kilometres.

That is a real proposal rather than an illustration, and what makes it work is precisely the fact this essay is about: the exchange is exact, instantaneous and symmetric, so there is no efficiency to be lost. Every gram-metre-per-second the spacecraft gains, the asteroid loses.

The contrast with the planetary case is only in the ratio. Against Jupiter a spacecraft is a twenty-two-decade rounding error; against a hundred-metre asteroid it is a percent-level perturber, and the same equation describes both.

A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.
Fig. 8 The same turn at nearly twice the approach speed, which is the other lever in the triangle. Doubling v∞v_\infty doubles the momentum exchanged at fixed turn angle — so a fast arrival gains more, and it is also harder to turn, since the deflection depends on the approach speed through sin⁡(δ/2)=1/(1+rpv∞2/μ)\sin(\delta/2) = 1/(1 + r_p v_\infty^2/\mu). The two effects run in opposite directions and their product has a maximum: there is an optimal approach speed for extracting the most from a given planet, and it is of order the planet’s own escape speed at the closest permissible passage.

One further point about the migration measurement deserves emphasis, because it is the reverse of the usual situation. Almost every dynamical history in this collection is reconstructed from a present-day distribution and is therefore only as good as the assumption that nothing else could have produced it. Here the alternative hypotheses are unusually constrained: a resonant population with the observed eccentricities requires a sweeping resonance, a sweeping resonance requires a moving planet, and the amount of movement is fixed by the eccentricities rather than fitted to them.

What is not fixed is the timescale. A migration of seven astronomical units over ten million years and one over five hundred million years leave nearly identical resonant populations, because capture is adiabatic in both cases. Distinguishing them needs a different observable — the fraction of objects captured, which depends on the rate — and that fraction is degenerate with how much mass the disc had. The distance is measured; the speed is not, and the two candidate speeds correspond to physically different histories — a slow depletion of the disc over hundreds of millions of years, or a brief instability that rearranged the outer solar system in a few. Both leave seven astronomical units of migration behind them and only one leaves a scattered disc as well.

It is worth noticing what kind of statement the whole essay has been making, because it is unusually clean for a subject built on reconstructions. Momentum conservation in an isolated encounter is not a model, an approximation or a fit; it is an identity, and every quantitative claim above is that identity divided by a mass. What varies between the spacecraft case and the planetesimal case is nothing but the denominator. That is why the two ends of the essay can be treated with the same arithmetic despite differing by twenty-two orders of magnitude in the mass involved and by nine in the number of encounters. The uncertainty enters only when the population of small bodies has to be described — how much mass it held, how it was distributed in semi-major axis, and how many of its members Jupiter could reach. Those are the free parameters of every migration model, and they are constrained by the outcome rather than measured beforehand, which is the ordinary situation in this subject and worth naming rather than glossing.

Where the ladder goes

The next rung is the instability that the migration is now usually embedded in: a model in which the outer planets begin in a compact resonant chain, the chain is broken by the planetesimal disc, and the resulting brief phase of close encounters produces the migration in a few million years rather than a few hundred.

The other direction is what the same bookkeeping does inside a gas disc rather than a planetesimal one, where the exchange partner is a fluid rather than a population of point masses and the torque is calculated at resonances rather than summed over encounters.

Where this goes next

A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun. Stealing speed from a planet, which does not notice A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity. Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between. The same planet, three times A flyby cannot change the encounter speed, only its direction, so a tour has to be designed in the space of what is conserved. One pass moves a spacecraft along a single curve and no further than the planet can bend it — and reaching a distant target means walking that curve, returning to the same planet again and again. The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed. The number that survives the encounter A comet that passes Jupiter comes away with every orbital element changed. One combination of them is not changed, and it is enough to recognise the comet afterwards, to sort the comet families, and to say where a spacecraft can and cannot go.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Angular momentumClose encounterGravity assistHyperbolic orbitKuiper beltMomentum conservationOrbital energyPlanet migrationPlanetesimalResonance captureTisserand parameter