The same planet, three times

The rung below established what a single flyby does: in the planet’s frame the spacecraft’s speed is unchanged and only its direction turns, and adding the planet’s own velocity back converts that turn into a change in heliocentric speed that cost no propellant.

It also established the limitation, though not what to do about it. Because $v_\infty$ is conserved, a flyby is not a manoeuvre that can be repeated until the desired orbit is reached. It is a rotation, and a rotation of a fixed-length vector has a bounded set of outcomes.

Designing a mission to the outer solar system means finding a sequence of such rotations, and the sequence has to be designed in coordinates that respect what cannot change.

What is conserved, and what that means #

In the circular restricted three-body problem the Jacobi constant is exactly conserved, and for a spacecraft far from the planet it reduces to the Tisserand parameter

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

That combination is unchanged by any number of flybys of that planet. It is not approximately unchanged; within the assumptions of the model it is a constant of the motion.

Rewriting it in terms of the encounter speed makes the constraint transparent. With the planet on a circular orbit of unit radius and unit speed, and $\tilde{v} = v_\infty/v_p$,

$$T = 3 - \tilde{v}^2,$$

so a contour of constant $T$ is a contour of constant $v_\infty$ and vice versa. Every orbit a spacecraft can be put on by flybys of one planet lies on one curve, labelled by the speed at which it first arrived.

The pump angle #

The convenient coordinate along the contour is the pump angle $\alpha$: the angle between the encounter velocity $\mathbf{v}_\infty$ and the planet’s own velocity.

Everything follows from adding the two vectors. The heliocentric speed after the encounter is

$$v^2 = v_p^2 + v_\infty^2 + 2v_p v_\infty\cos\alpha,$$

so with $a_p = 1$,

$$\frac{a}{a_p} = \frac{1}{1 - \tilde{v}^2 - 2\tilde{v}\cos\alpha}.$$

Small $\alpha$ means $\mathbf{v}_\infty$ points along the planet’s motion, which is the fastest heliocentric orbit available and the largest semi-major axis. Large $\alpha$ means it points backwards, giving the slowest and smallest.

A flyby changes $\alpha$ by rotating $\mathbf{v}_\infty$, and the largest available rotation is set by how close the spacecraft may pass:

$$\sin\frac{\delta}{2} = \frac{\mu}{\mu + r_p v_\infty^2}.$$

That last point deserves emphasis because it is counterintuitive and it shapes every mission. A large $v_\infty$ puts a spacecraft on a contour that reaches far out, which is what an outer-planet mission wants. It also makes each flyby nearly useless, because $\mu/(\mu + r_pv_\infty^2)$ collapses. Arriving fast means having a good contour and no ability to move along it, and arriving slowly means the opposite.

Resonant returns #

There is a further constraint that turns a continuous curve into a set of stopping places.

To use the same planet again, the spacecraft has to come back to it — which means its heliocentric period must be commensurate with the planet’s. If it makes $p$ revolutions while the planet makes $q$, the two meet again after $q$ planet-years, and

$$\frac{a}{a_p} = \left(\frac{q}{p}\right)^{2/3}.$$

Those are the resonant returns, and they are the only points on the contour from which a second free encounter is available. A tour is therefore a sequence of hops between resonances, each hop no larger than the turn angle allows. Galileo’s route to Jupiter is the standard example. Launched from the Shuttle with a stage too small to reach Jupiter directly, it flew past Venus once and the Earth twice — the second Earth flyby coming after a two-year 1:2 resonant orbit — and only then had the heliocentric energy for the cruise. The sequence is called VEEGA, and it took three years to reach the point a direct launch would have reached at once.

Cassini did the same thing with two Venus flybys, one Earth and one Jupiter: VVEJGA, six and a half years to Saturn.

The arithmetic is worth stating in the currency that decides such things. A direct Hohmann-like transfer to Saturn needs a departure energy far beyond what any launch vehicle of the period could give a five-tonne spacecraft; the VVEJGA sequence needed a departure that reached only Venus. The difference between those two departure energies is several kilometres per second, and by the rocket equation several kilometres per second is a factor of several in launch mass. The tour was not an optimisation; it was the difference between the mission existing and not existing.

A worked walk #

The figure at the top is worth reading as a sequence rather than as a curve.

A spacecraft arrives at the Earth at 10.4 km/s with a pump angle of 152°, which puts it on a heliocentric orbit of semi-major axis 0.66 astronomical units — inside the Earth’s, which is where a Venus flyby leaves it.

The first Earth encounter rotates $\mathbf{v}_\infty$ by up to 40.8°, taking $\alpha$ to 111° and $a$ to about 0.85. The spacecraft is now on a period commensurate with nothing in particular, so it has to be placed on one; the nearest resonance available is chosen and the orbit trimmed to it with a small manoeuvre.

The second encounter takes $\alpha$ to 70°, and $a$ to 1.58 — the 1:2 resonance, a two-year orbit. The third is the one that matters: from that orbit, at that pump angle, the encounter is the one that throws the spacecraft outwards.

Three flybys of one planet, at an encounter speed that never changed, and a heliocentric semi-major axis that went from two-thirds of an astronomical unit to whatever the final turn produces. The propellant spent was the trim manoeuvres, tens of metres per second against the several kilometres per second the same change in orbit would have cost as a burn.

Leverage: buying $v_\infty$ with a small burn #

Since $v_\infty$ cannot be changed by a flyby and everything depends on it, a way to change it cheaply is worth a great deal.

There is one. A small burn far from the planet, near the aphelion of the return orbit, changes the heliocentric orbit slightly — and because the encounter geometry at the next flyby is the difference between the spacecraft’s velocity and the planet’s, a small change in the first can produce a much larger change in the second.

The lever is the ratio of the heliocentric speed at the burn to the encounter speed. A burn of a few tens of metres per second at aphelion, on an orbit returning to the Earth, can change $v_\infty$ at the next Earth encounter by several hundred. That is $\Delta v_\infty$-leveraging, and it is what a “deep-space manoeuvre” in a mission profile usually is. Galileo’s two-year Earth-to-Earth leg carried exactly such a burn, and it is the reason the second Earth flyby delivered the energy for Jupiter.

The same plane drawn at other arrival speeds shows how much of the tour design is decided before the first flyby.

The costs nobody puts on the plot #

Three of them, and each has cancelled or reshaped a real mission.

Time. Every resonant return costs whole years. Galileo took six years to Jupiter and Cassini nearly seven to Saturn, against about two and four for a direct transfer. A tour buys propellant with schedule.

Radiation and thermal design. A trajectory to the outer solar system by way of Venus goes inwards first, to 0.7 astronomical units, where the solar flux is twice what it is at the Earth. Cassini flew with its high-gain antenna used as a sunshade for the first two years, which meant no high-rate communication at all for that stretch of the cruise. A spacecraft designed for Saturn has to survive Venus, and the mass of the thermal design it needs to do so comes out of the science payload — so the propellant a tour saves is partly spent again on insulation. The launch window. The sequence only works if the planets are in the right places at the right times, and the alignments recur on the beat frequency of the synodic periods involved. Galileo’s window recurred every thirteen months and Cassini’s roughly every year, but a five-body sequence can have a window that recurs once a decade or once a century. There is a limit to the leverage and it is worth naming. The manoeuvre changes $v_\infty$ by an amount roughly proportional to $\Delta v$ times the ratio of the heliocentric speed to the encounter speed, so it works best when the encounter speed is small — which is exactly when the flybys are already effective. Where a mission most wants to raise $v_\infty$, at the outer end of a tour, the leverage is weakest. Every mechanism in this rung is strongest where it is least needed, which is a fair summary of why interplanetary trajectory design is hard.

The mission that did exactly this, first #

The technique’s first use was also the cleanest demonstration of the resonant return, and it is worth setting out because everything in this rung is visible in one flight plan.

Mariner 10 launched in November 1973 for Mercury, by way of Venus. The Venus flyby was the first gravity assist ever flown to a planet: it lowered the spacecraft’s perihelion enough to reach Mercury’s orbit, which a direct launch could not have done with the vehicle available.

What happened after the first Mercury encounter is the part this essay is about. The trajectory was arranged so that the spacecraft’s heliocentric period came out at almost exactly twice Mercury’s — 176 days against 88 — which is a 1:2 resonance, and therefore a return to the same point at the same time as the planet. The spacecraft encountered Mercury again in September 1974 and a third time in March 1975, on the same orbit, having spent essentially no propellant to arrange either return.

Three encounters with one planet, from one launch, at no cost beyond the attitude-control gas that eventually ran out and ended the mission. The phrase in this essay’s title is not a figure of speech; it is a flight plan from 1973.

There was a cost, and it is a good illustration of what a resonance fixes and what it does not. The period is locked, so the spacecraft arrives when the planet does — but Mercury’s rotation is itself locked to its orbit in a 3:2 spin–orbit resonance, so at every 176-day return the same hemisphere was in daylight. All three encounters photographed the same side of the planet, and 55 per cent of Mercury’s surface remained unmapped for the next thirty-three years, until an orbiter arrived that was not tied to a resonant return.

The alignment that comes round twice a century #

The other historical case shows the opposite constraint: not what a resonance locks, but what an ephemeris permits.

A trajectory that visits Jupiter, Saturn, Uranus and Neptune in sequence requires all four to be arranged so that each flyby delivers the spacecraft to the next. The geometry recurs on the beat of the four synodic periods, and the interval is about 175 years. The previous occurrence was in the early nineteenth century, during Jefferson’s presidency, and it was noticed in the mid-1960s that the next one fell in the late 1970s.

That is the whole reason two spacecraft launched within sixteen days of each other in 1977 and one of them is now the only object to have visited Uranus and Neptune. The mission was proposed as a Grand Tour, cut on cost grounds to a Jupiter–Saturn mission, and flown to the outer planets anyway once the first spacecraft’s Saturn encounter had been aimed to preserve the option.

Neither the alignment nor the resonance is something a designer chooses, and the two together define the shape of the field. A tour’s contours say what is reachable; the ephemeris says when; and the interval between opportunities is set by numbers — the planets’ periods — that nothing about the spacecraft can influence.

Both examples also show why a tour’s design is dated in a way that a direct transfer’s is not. A Hohmann transfer to Mars is available every twenty-six months for ever; a Grand Tour is available twice a century, and a resonant sequence is available only for the arrival conditions the first flyby happens to deliver. The design space is not a continuum but a timetable, and missing an entry on it can cost a generation.

That is also why a mission’s trajectory is frozen years before launch and why a slip of a few weeks can force a complete redesign rather than a delay.

It is worth noting what the two historical cases have in common beyond their dates. In each, the enabling fact was arithmetic about periods rather than anything about propulsion: a ratio of two orbital periods in the first case and a coincidence of four in the second. Neither could have been bought with a larger launch vehicle, and neither required one. The cheapest thing in spaceflight is a number that happens to come out right, and the whole of this rung is about how to find those numbers before committing to a design.

Where the model stops #

The construction above assumes circular coplanar planetary orbits, an instantaneous flyby at a point, and a two-body heliocentric leg. All three are wrong at the level that matters for a real design.

Planetary eccentricities mean $v_p$ is not constant and the contours move slightly with the encounter’s date — which for Mercury, whose eccentricity is 0.206, is not a small effect at all, and is one reason a Mercury tour is designed differently from a Venus one. It also means that the Tisserand parameter is not exactly conserved for a real planet, only nearly so; the drift per encounter is second order in the eccentricity and accumulates over a long tour, so a sequence designed on the graph has to be re-checked against the ephemeris before it can be believed. Inclinations mean the encounter geometry has a second angle — the crank angle, the rotation of $\mathbf{v}_\infty$ about the planet’s velocity — which the two-dimensional picture hides and which is what an out-of-plane target needs. And a real trajectory is optimised numerically against a full ephemeris — a fitted table whose own uncertainty is anisotropic — with the Tisserand construction used to generate candidate sequences rather than to fly them.

The aim point at each encounter is then the thing actually controlled: a flyby is targeted in the B-plane, and a tour is a sequence of B-plane targets each chosen to set up the next. The turn angle that this rung treats as a free parameter is, in flight, the consequence of an aim point delivered to within a few kilometres by a manoeuvre of centimetres per second.

That division of labour is worth stating: the graph does not produce trajectories, it produces shortlists. An automated search over flyby sequences with the resonance structure as the state space returns a few hundred candidates from a space of billions, and each is then handed to an optimiser.

The same two readings taken for a different starting orbit make the conservation law’s indifference to the starting point explicit.

Where this ladder goes next #

This rung has established the design space: a contour set by the arrival speed, a step size set by the planet’s pull, and a set of resonant stopping places along it.

The rung above is the multi-body tour — sequences involving several different planets, where each encounter moves the spacecraft to a different Tisserand contour because the invariant is defined with respect to one planet at a time. That is what makes a Venus–Earth–Earth–Jupiter sequence more capable than any number of Earth flybys, and it is why the graph search is over planets as well as resonances.

Beside it lies the satellite tour: the same construction applied inside a planetary system, where Cassini’s 127 Titan flybys walked its orbit around Saturn’s moons for thirteen years on almost no propellant.

And below it, the habit: when something is conserved, design in coordinates where it is a coordinate. The Tisserand graph is not a clever picture of gravity assists; it is the statement that a whole class of manoeuvres is confined to a level set, drawn so that the confinement is visible.