A map of the transfers that are free

The rung below made the Tisserand parameter the number that survives an encounter: a comet that passes Jupiter comes away with every orbital element changed, and one particular combination of them unchanged, which is enough to recognise the comet afterwards.

Turn that around and it stops being a classification tool and becomes a constraint on what a spacecraft can do. If the combination cannot change during a flyby, then the set of orbits reachable by flybys alone is a curve, and the whole business of designing a tour of a planetary system is the business of getting onto a useful curve and walking along it.

From an invariant to a speed #

The Tisserand parameter with respect to a planet of semi-major axis $a_p$ is

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

and it is conserved through an encounter because it is, up to a constant, the Jacobi integral of the restricted three-body problem written in the spacecraft’s osculating elements.

What makes it a design tool is the identity

$$v_\infty^2 = (3 - T)\,v_p^2,$$

relating it to the speed of the spacecraft relative to the planet, far from the planet, in units of the planet’s own orbital speed. A conserved $T$ is a conserved magnitude of the relative velocity, and that is what a gravity assist actually does to a trajectory: it rotates the relative velocity vector and leaves its length alone.

So an encounter has exactly two degrees of freedom: how far to turn the relative velocity, which the flyby altitude sets, and about which axis, which the approach geometry sets. The first is bounded — a spacecraft cannot pass through the planet — and the second is free.

The pump angle, and why the plane is the right one #

Resolve the relative velocity into a component along the planet’s velocity and a component perpendicular to it. The angle between the two vectors is the pump angle $\alpha$, and it is the coordinate everything is done in.

With $V = v_\infty/v_p$,

$$\frac{a}{a_p} = \frac{1}{1 - V^2 - 2V\cos\alpha}, \qquad 1 - e^2 = \frac{a_p}{a}\left(1 + V\cos\alpha\right)^2,$$

which gives a one-parameter family of orbits for each $V$. Substituting them back into $T$ returns exactly $3 - V^2$: the contour and the family of reachable orbits are the same curve, arrived at from opposite ends.

Drawing that family in the plane of perihelion against aphelion is what makes the picture useful, for three reasons.

Both quantities are what a mission cares about. A flyby of an inner planet needs the perihelion to reach it; a flyby of an outer one needs the aphelion. Reading a trajectory’s usefulness off a plot of $a$ against $e$ requires arithmetic; reading it off this one does not.

The constraint that the orbit crosses the planet’s is a corner. Every accessible orbit has $r_p \le a_p \le r_a$, so the whole of every contour lies in the quadrant to the left of and above the point $(1,1)$, and the corner is the planet.

And a resonance is a straight line. An orbit in a $p{:}q$ resonance with the planet has a fixed semi-major axis, and $r_p + r_a = 2a$ — so every resonant orbit lies on a line of slope $-1$, and the intersections of a contour with those lines are the finitely many places a tour can stop.

What a resonant return is for #

The picture so far says which orbits are reachable. It does not say whether the spacecraft can get back to the planet, and without that the reachability is theoretical.

After a flyby the spacecraft is on a heliocentric orbit that crosses the planet’s. Crossing is not enough: the planet has to be there. If the new orbital period is an arbitrary fraction of the planet’s, the two will not coincide again for a very long time.

If the new period is a rational multiple — the spacecraft goes round $q$ times while the planet goes round $p$ — then after $p$ planetary years both are back where they started and the encounter repeats. That is a resonant return, and it is why the diagonals on the map are the useful places to be.

The resonance also constrains what the return flyby can do. Arriving on the same resonance means arriving with the same pump angle, so the second encounter starts where the first ended and can rotate the vector further in the crank direction — which changes the inclination rather than the shape of the orbit. That is how a spacecraft is put into a polar orbit about the Sun: repeated flybys of the same planet on the same resonance, each cranking the relative velocity a little further out of the ecliptic — which is the manoeuvre a propulsive plane change cannot afford, with the pump angle never changing at all.

What one flyby is worth #

The step along a contour that a single encounter delivers is bounded by the turn angle,

$$\sin\frac{\delta}{2} = \frac{1}{1 + r_p v_\infty^2/\mu},$$

with $r_p$ the closest approach and $\mu$ the planet’s gravitational parameter. Two things follow immediately.

A massive planet is worth far more. Jupiter’s $\mu$ is three hundred times the Earth’s, so at the same $v_\infty$ and the same number of radii, Jupiter turns the velocity vector by a large angle and the Earth by a small one.

A slow approach is worth more than a fast one. The turn falls as $v_\infty^2$, so a spacecraft arriving quickly is barely deflected. That is the fundamental tension of tour design: a low $v_\infty$ contour is easy to walk along and hard to reach, and a high one is easy to reach and nearly straight.

Cranking, and the mission that has no perihelion left to change #

The two degrees of freedom of an encounter are usually described as pumping and cranking, and the map draws only the first.

Pumping changes the pump angle, which changes the shape of the heliocentric orbit — its perihelion and aphelion — and moves the spacecraft along its contour. That is what the map shows.

Cranking rotates the relative velocity about the planet’s velocity vector, leaving the pump angle alone. The shape of the orbit is unchanged and its inclination changes, so a cranking sequence is a walk that stays at a single point on the map while going somewhere important.

The extreme case is a mission whose whole purpose is to leave the ecliptic. There is no propulsive route: raising a spacecraft’s inclination to ninety degrees from the Earth’s orbit costs a velocity change comparable to the Earth’s own orbital speed, which is far beyond any launcher. The route that exists is to go to Jupiter, use a single enormous flyby to crank the whole velocity out of the plane, and fall back through the inner solar system on a polar orbit — which is what Ulysses did, and which is the only way anyone has ever seen the Sun’s poles — where the polarity that doubles the activity cycle’s period reverses first.

Reading a real tour off the map #

The tour that made the method famous is Cassini’s: Venus, Venus, Earth, Jupiter, in that order, to reach Saturn on a launch vehicle that could not have sent it directly.

On the map, each of those is a change of contour rather than a walk along one, because they are flybys of different planets — and the Tisserand parameter is defined with respect to one planet at a time. That is the one thing the picture as drawn does not show, and the standard extension is to overlay the contours for several planets in the same $(r_p, r_a)$ plane. A trajectory is then a sequence of arcs, each on a contour belonging to whichever planet it is currently using, and the transitions happen where two planets’ contours intersect — each arc being one of the two-body problems a patched-conic trajectory is stitched from.

Where they intersect, the transfer is free. That is the whole content of a multi-planet tour, and it is why the sequences look arbitrary until the diagram is drawn.

Leveraging, and the burn that is worth taking #

The map also explains a manoeuvre that looks like a mistake.

A deep-space manoeuvre is a burn made far from any planet, usually near aphelion, between two flybys of the same body. On the map it moves the spacecraft from one contour to a neighbouring one. It costs propellant, so it should be avoided — and it is used constantly, because a small change of contour near aphelion can produce a large change in the next encounter’s geometry.

The reason is the same one that makes a burn worth more when moving fast, run backwards. Near aphelion the spacecraft is slow, so a given $\Delta v$ changes the velocity direction a great deal and the energy very little. That is precisely what is wanted when the objective is to arrive at the next encounter with a different pump angle rather than with more energy.

The technique is $v_\infty$ leveraging, and its figure of merit is the change in $v_\infty$ at the next encounter per unit of propellant spent — which can exceed one, because the planet is doing most of the work and the burn is only steering.

What the launch has to supply #

The tour begins somewhere, and the map says exactly what the launcher has to buy: a point on a contour, reached from the Earth.

Departure energy is quoted as $C_3$, the square of the hyperbolic excess speed leaving the Earth, and it is the quantity a launch vehicle’s performance curve is drawn against. A higher $C_3$ costs mass exponentially, through the same exponential that decides everything else about rockets — so the whole point of a tour is to accept a low $C_3$ and buy the rest of the energy from planets.

The consequence is that departure dates are scarce. The first flyby has to happen at a particular geometry, which means the spacecraft has to leave within a window set by the relative positions of two planets, which recurs at the synodic period.

The same map, inside a planetary system #

The construction is not specific to the Sun and the planets. Replace the central body with a giant planet and the perturbing body with one of its moons, and everything above holds unchanged — the invariant, the contours, the resonant lines, the pump and crank angles.

That is where the technique has done its most concentrated work. A spacecraft touring the Galilean moons is in exactly the situation the map is built for: four massive perturbers on nearly circular, nearly coplanar orbits, at spacings that put their contours across one another, and a spacecraft whose propellant budget cannot afford a single one of the required velocity changes propulsively.

The Jovian case is more favourable than the solar one in one respect and much worse in another. More favourable, because the moons are massive relative to their orbital speeds — the turn angle a single Ganymede encounter delivers is large, so a tour makes rapid progress. Much worse, because the tour is conducted inside a radiation environment that damages electronics on a schedule of its own, so flight time is not merely a cost but a hard budget, and the optimisation is against total dose rather than against propellant alone.

The result is tours with a hundred or more encounters designed as one object, in which the sequence of moons is chosen so that each flyby sets up the next and the whole walk terminates at the intended destination with the smallest possible remaining burn. Designing them is not a matter of reading the map by eye; it is a search over an enormous discrete space, and the map’s role is to bound that space to the sequences that are possible at all.

A contour diagram whose purpose is to say which transfers are free becomes, at that scale, a way of pruning a combinatorial search — which is the ordinary fate of a good geometric insight once the problems get large.

The map’s shape depends on the arrival speed and on how much of the plane is drawn, and both are worth reading once more at the high end.

What the map does not tell #

Time is not on it. The plot says which orbits are connected and says nothing about how long a walk takes. A resonant return on a 2:1 resonance costs two planetary years, and a tour of the Jovian moons has hundreds of encounters, each costing a fraction of an orbit — so the flight time is a separate optimisation that frequently dominates the design.

Inclination is projected away. The Tisserand parameter contains $\cos i$, so a given contour in the $(r_p, r_a)$ plane is really a surface, and the crank angle moves along the hidden direction. Reading a two-dimensional map of a three-dimensional constraint is safe as long as the inclination is small and is exactly wrong for the missions that are trying to change it.

And the planetary orbits are not circles. Every expression here assumes them so. For the Earth and Venus that is a small error; for Mercury, with an eccentricity of 0.21, it is not, and the contours become bands.

The map has two readings and both of them depend on a single number that the launch fixes, so it is worth drawing each of them at a second value of it.

How a tour ends #

The map says how to move between orbits at no cost. It has nothing to say about the last step, which is the expensive one, and the way the two interact shapes every tour that has been flown.

A spacecraft on a contour is on a heliocentric or planetocentric orbit that repeatedly encounters the perturbing body and never captures around anything. To end the mission in orbit about a destination, the relative velocity at arrival has to be removed, and removing it is a burn whose size is essentially $v_\infty$ — the very quantity the whole tour has conserved.

So the tour’s objective is not merely to reach the destination. It is to reach it on the lowest contour that is reachable, because the contour’s label is the capture cost. A sequence of flybys that arrives at the right place with a large $v_\infty$ has solved the geometry and not the problem.

That inverts the intuition the earlier sections build. A high-$v_\infty$ contour is easy to get onto and nearly straight, and it is also expensive to get off. A low-$v_\infty$ contour is hard to reach and strongly curved, and it is cheap to leave. The design problem is therefore a walk that steps down in $v_\infty$ where it can — which the map says is impossible without a burn, so the stepping down is done at the destination system’s own moons, each encounter reducing the relative velocity with respect to the planet at the cost of nothing.

The clearest example is arrival at a giant planet with a large satellite system. A single flyby of the outermost large moon on the way in removes a substantial fraction of the arrival energy, so the capture burn at the planet is smaller by that amount — and the manoeuvre is free. Both of the spacecraft that have orbited Jupiter used a moon encounter that way, and the saving is a large fraction of the propellant they carried.

The endgame is the same physics with the sign of the objective reversed: for the whole cruise the flybys are used to gain, and at arrival they are used to give back.

Where this ladder goes next #

This rung has turned a conserved quantity into a map, and turned mission design into the reading of it: along a contour is free, between contours costs, and the resonances are where a spacecraft is allowed to stop.

The rung above is the low-energy alternative. Near the collinear libration points the same three-body problem has invariant manifolds — tubes — along which a spacecraft can move between regions for almost nothing, and a tour built out of those has a different currency entirely: it costs time and station-keeping rather than propellant, and the tube that leads out of a neck is the base rung of it.

Beside it lies the same map used for the natural population rather than the artificial one: which comets can reach which regions by encounters alone, and why the Jupiter-family comets are confined to a band of Tisserand parameter that a spacecraft designer would recognise as a single contour.

And below it, the habit: plot the invariant, not the state. Six orbital elements change at every encounter and one combination does not, and a diagram whose axes are chosen so that the invariant is a curve turns an optimisation problem into a question about geometry.