Moon Tours: Resonances, Pumping and the Endgame

v∞ and pump angle, resonant orbits, pump-down sequences, Tisserand graphs in a planetary system, and lowering v∞ to reach a moon

Cassini, 2004–2017

Over thirteen years at Saturn, Cassini made 127 targeted flybys of Titan. Almost every change to its orbit — its period, its inclination, the direction of its long axis, its visits to Enceladus, Rhea and Dione, and finally the 22 dives between Saturn and its rings — was made by Titan, not by its engine. Titan is the second-largest moon in the Solar System, and a single close pass could change Cassini's velocity by several hundred metres per second. The Cassini tour designers had a small number of tools: resonant orbits, the pump and crank angles, and a graph on which every orbit reachable from Titan is a single curve. This lesson builds those tools.

💡v∞ and pump angle

Treat the moon as a planet on a circular orbit of radius

a_M

and speed

v_M

, and each encounter as a patched-conic flyby (the unit on patched conics). The spacecraft's orbit about the planet is fixed entirely by its velocity relative to the moon,

\vec v_\infty

, at the encounter.

Its magnitude $v_\infty$ is unchanged by an unpowered flyby.
Its direction is described by the pump angle $\alpha$ , measured from the moon's velocity, and the crank angle $\kappa$ , out of the moon's orbit plane.

With crank zero, the spacecraft's planet-relative speed and orbit follow from the vector sum:

v^2 = v_M^2 + v_\infty^2 + 2 v_M v_\infty \cos\alpha,\qquad a = \left(\frac{2}{a_M} - \frac{v^2}{\mu}\right)^{-1}

Pump angle 0: the largest orbit for this v∞. 180°: the smallest. Changing the pump angle is the only thing an unpowered flyby can do — "pumping" the orbit up or down.

Resonant orbits. If the spacecraft's period is a ratio

N\!:\!M

of the moon's — the spacecraft completes

M

orbits while the moon completes

N

— then both are back at the same place after

N

moon periods, and the spacecraft meets the moon again. A resonant orbit is a free return ticket for the next flyby. For each resonance there is exactly one pump angle (at a given v∞), from the equation above with

a = a_M (N/M)^{2/3}

The first lab tabulates them for Ganymede. At v∞ = 6 km/s, the 10:1 resonance sits at α = 64°, 1:1 at 106°, 1:2 at 144°. One flyby at 100 km altitude can move the pump angle by up to 10.5° at that speed, so a tour steps from resonance to resonance.

\sin\frac{\delta_{\max}}{2} = \frac{1}{1 + r_{p,\min} v_\infty^2/\mu_M}

The largest pump-angle change per flyby. Big moons (Titan, Ganymede) and slow encounters give big steps; at 2 km/s one Ganymede pass can swing α by 57°.

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📐Worked Example: pumping down at Ganymede

After capture, raise perijove to 11 R_J at the first apojove and fly past Ganymede. The 200-day orbit crosses Ganymede's at v∞ = 7.81 km/s. The second lab chooses, at each flyby, the shortest resonance one pass at 100 km can reach (with

M \le 3

flyby 1 → 22:3 resonance, pump angle 81.2°, period 52.5 days;
flyby 2 → 11:3, 87.5°, 26.2 days;
flyby 3 → 7:3, 93.4°, 16.7 days;
flyby 4 → 5:3, 99.1°, 11.9 days;
flyby 5 → 4:3, 103.7°, 9.5 days — 351 days after the first flyby.

Five flybys and under a year, with no propellant. Starting from a 50-day capture orbit (as Juno did) instead: v∞ 7.11 km/s, four flybys, 165 days. But look at the v∞: 7.81 km/s on every line. Pumping changes the orbit, not the encounter speed.

💡Tisserand's parameter, inside a planetary system

The unit on patched conics used Tisserand's parameter with respect to a planet. Exactly the same invariant holds with respect to a moon, with the planet now playing the Sun's role:

T = \frac{a_M}{a} + 2\sqrt{\frac{a}{a_M}(1-e^2)}\cos i,\qquad \frac{v_\infty}{v_M} = \sqrt{3 - T}

Conserved through every flyby of that moon. An orbit's T with respect to a different moon gives the v∞ it would have there.

This is why tours hop between moons. To reach Europa slowly, do not arrive from the capture orbit: pump down at Ganymede until the orbit touches Europa's, and arrive with the v∞ that orbit has there. The third lab computes the cheapest possible rungs — Hohmann transfers between neighbouring moon orbits, which are also Tisserand's minimum:

Callisto → Ganymede: 1.22 km/s leaving, 1.41 km/s arriving;
Ganymede → Europa: 1.33 km/s leaving, 1.49 km/s arriving;
Titan → Enceladus: 3.71 km/s arriving — Titan is so far out that Enceladus cannot be reached slowly from it alone.

And the arrival speed decides everything: capturing into a 100 km orbit at Europa costs 0.82 km/s at v∞ = 1 km/s, 1.41 at 2 km/s, and 6.65 km/s at 7.8 km/s.

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💡The endgame: lowering v∞

Resonant flybys of one moon cannot lower v∞; flybys of other moons can, and so can small burns. In a v∞-leveraging transfer the spacecraft leaves a moon on a near-resonant orbit, makes a small burn near apoapsis (or periapsis), and meets the same moon again with a different v∞; the burn is leveraged by the flyby, so a few tens of metres per second can remove hundreds from v∞. Sequences of these — the 'endgame' of Campagnola and Russell — let JUICE approach Ganymede slowly enough to enter orbit around it, which is planned for December 2034 after a series of Ganymede and Callisto flybys.

🛰Europa Clipper: a tour instead of an orbit

Europa sits deep in Jupiter's radiation belts. An orbiter there would accumulate a lethal dose within months. Europa Clipper therefore never orbits Europa: it flies a tour of about four years with 49 Europa flybys, some as low as 25 km, plus 7 of Ganymede and 9 of Callisto that steer the orbit. Each flyby is followed by a long excursion out of the belts. The tour is designed so that the ground tracks of the flybys, on the side of Europa lit by the Sun, gradually cover almost the whole moon — a 'petal' pattern that rotates as the orbit's long axis is turned by successive flybys.

ALL THESE WORLDS ARE YOURS EXCEPT EUROPA. ATTEMPT NO LANDING THERE.

– Arthur C. Clarke, 2010: Odyssey Two, 1982. Europa Clipper, wisely, only flies past

💡The crank angle

Everything above was in the moon's orbit plane. Rotating v∞ out of that plane — changing the crank angle — tilts the spacecraft's orbit. Cassini used Titan to crank its inclination above 70° to study Saturn's poles and rings from above, and back down to zero to visit the icy moons, each time at constant v∞. A crank sequence and a pump sequence are designed together: the v∞ globe of Strange, Russell and Buffington maps every orbit a moon can give you as a point on a sphere of radius v∞.

↓But there is a problem

The tour has done its job: the spacecraft meets its target moon at under a kilometre and a half per second, and a modest burn will capture it. But the planet has not gone away. Around Europa it is a thousand times more massive than the moon and only 670,000 km off, pulling harder on one side of the orbit than the other. A spacecraft that settles into a beautiful low polar orbit around Europa finds its orbit stretching, week by week, until the low point of the ellipse reaches the ice.