Multiple Gravity-Assist Trajectory Design

Chaining Lambert legs and flybys, resonant legs, searching sequences and dates, and testing the design against real dynamics

Pasadena, 1986

Galileo is ready to go to Jupiter on a Space Shuttle in May 1986, boosted by a liquid-hydrogen Centaur upper stage powerful enough to send it there directly in about two years. Then Challenger is lost, and the decision follows that a fuelled Centaur will never again ride in the Shuttle's payload bay. The only upper stage left, the solid-fuel IUS, cannot give Galileo anything like the energy Jupiter needs. The mission designers find a way anyway: launch towards Venus, the wrong direction, fly past it to fall back to Earth, fly past Earth into an orbit that returns two years later, fly past Earth again — and only then head for Jupiter. VEEGA: Venus–Earth–Earth Gravity Assist. Galileo launches on 18 October 1989 and arrives at Jupiter on 7 December 1995. Six years instead of two. But it gets there.

For a successful technology, reality must take precedence over public relations, for nature cannot be fooled.

– Richard Feynman, appendix to the Rogers Commission report on the Challenger accident, 1986

The multiple gravity-assist (MGA) problem. Choose a sequence of bodies

P_0, P_1, \ldots, P_n

and encounter dates

t_0 < t_1 < \cdots < t_n

. Then:

Each leg $P_{k-1}(t_{k-1}) \to P_k(t_k)$ is a Lambert problem around the Sun, giving $\vec v_{\infty,k}^{+}$ at its start and $\vec v_{\infty,k}^{-}$ at its end.
At each intermediate body the incoming and outgoing $\vec v_\infty$ must be joined by a flyby: equal magnitudes and a reachable turn angle (unpowered), or a periapsis burn (powered).
The cost adds the launch ( $C_3$ or the departure Δv), every flyby burn or deep-space manoeuvre, and the arrival (capture Δv or arrival $v_\infty$ ).
Constraints include minimum flyby altitudes, maximum $C_3$ for the launcher, total flight time, and arrival conditions.

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📐Worked Example: rebuilding Galileo's VEEGA

Using the real event dates and astropy's built-in ephemeris:

Earth → Venus (18 Oct 1989 → 10 Feb 1990): launch $C_3 = 15.6$ km²/s² — far below the ≈77 km²/s² a direct Jupiter transfer needs.
Venus flyby: $v_\infty$ in 6.21, out 5.99 km/s, turn 33.2°, patched periapsis altitude 15,770 km. Galileo actually passed about 16,000 km above Venus. The 221 m/s magnitude mismatch is the cost of an approximate ephemeris and dates rounded to the day — the real trajectory needed no such burn.
Earth → Earth (8 Dec 1990 → 8 Dec 1992): the two positions are only 0.50° apart. Lambert's plane is undefined; this is not a Lambert leg at all, but a resonance.
Earth → Jupiter (8 Dec 1992 → 7 Dec 1995): needs $v_\infty = 8.92$ km/s at Earth; the spacecraft arrives with 8.83. Jupiter arrival $v_\infty = 5.64$ km/s.

💡Resonant legs

When a leg returns to the same planet after a whole number of its orbital periods, the two positions coincide (or nearly so) and Lambert's problem degenerates. Instead, the leg is designed through its period: a 2-year orbit has

a = 2^{2/3}

AU, which fixes the heliocentric speed at 1 AU (34.86 km/s) and therefore, for a given

|\vec v_\infty|

, the angle between

\vec v_\infty

and the planet's velocity (61.6° for Galileo's 8.83 km/s). That angle defines a cone of possible directions; the designer picks the one that sets up the next flyby. Resonant legs — Earth–Earth, Venus–Venus, Mercury–Mercury — appear in almost every modern tour.

Searching for a tour. The design space is a sequence (discrete) and a set of dates (continuous), with a highly non-convex cost. Methods, in roughly historical order:

Grid searches over pairs of dates, as in the second lab — fine for one or two flybys, hopeless for more.
Pruning: discard dates whose legs need impossible turns or exceed a C₃ limit before exploring further.
Tisserand graphs to propose promising sequences before any dates are chosen.
Global optimisers — differential evolution, genetic algorithms, monotonic basin hopping — over all dates at once, often with a deep-space manoeuvre allowed on each leg (the 'MGA-1DSM' model). ESA's GTOP database collects benchmark problems (Cassini, Messenger, Rosetta) on which such methods are compared; ESA's open-source pykep library implements the models.

📐Worked Example: searching the Venus flyby

Fix Galileo's first Earth return (8 December 1990). Scan launch dates from 2 October to 11 December 1989 and Venus dates from 26 January to 1 March 1990 (432 pairs). For each pair, solve both legs and patch the Venus flyby; reject passes below 300 km; cost = launch

v_\infty

+ flyby burn. 421 pairs are flyable. The best: launch 7 November 1989, Venus 23 February 1990,

C_3 = 13.5

, a 14 m/s burn and a 17,200 km pass — total 3.691 km/s. Galileo's own dates sit in a slightly more expensive part of the same valley in this simplified model (C₃ 15.9, 128 m/s). Real constraints the grid ignores — the launch period of the Shuttle, the Earth-2 flyby, Jupiter arrival geometry, the probe's entry — decide the rest.

Mission	Launch	Sequence	Destination arrival
Mariner 10	Nov 1973	Earth–Venus–Mercury (×3)	Mercury, Mar 1974
Voyager 2	Aug 1977	Earth–Jupiter–Saturn–Uranus–Neptune	Neptune, Aug 1989
Galileo	Oct 1989	Earth–Venus–Earth–Earth–Jupiter	Jupiter, Dec 1995
Cassini	Oct 1997	Earth–Venus–Venus–Earth–Jupiter–Saturn	Saturn, Jul 2004
MESSENGER	Aug 2004	Earth–Earth–Venus–Venus–Mercury ×3	Mercury orbit, Mar 2011
Juno	Aug 2011	Earth–(deep-space burn)–Earth–Jupiter	Jupiter, Jul 2016
JUICE	Apr 2023	Earth–Moon–Earth (2024), then Venus and Earth flybys	Jupiter, 2031

🛰Juno: a deep-space manoeuvre plus a flyby

Juno launched in August 2011 onto an orbit that took it beyond Mars and back. Near aphelion, in 2012, it fired its engine twice; the burns set up an Earth flyby in October 2013 that sent it to Jupiter. This 'ΔV-EGA' pattern — a deep-space manoeuvre to shape an Earth return, then an Earth gravity assist — lets a smaller launch vehicle do the work of a larger one. Each leg is a Lambert arc, and the manoeuvre is the point where two of them are joined.

How wrong is the patched-conic tour? In the third lab the Mars 2020 Lambert arc, started six days after launch, is integrated with the other planets switched on:

Jupiter alone moves the arrival by 11,400 km; Jupiter and Saturn, 11,800 km; adding Venus, 15,600 km.
Adding Earth — still a few million kilometres away, still pulling — moves it by 1.94 million km: the patched conic's biggest omission is the residual pull of the departure planet.
One correction burn in the full model fixes it: 49 m/s on day 15, 102 m/s on day 60, 416 m/s on day 150. Correct early.

💡From patched conics to a flown trajectory

Patched conics are the design model: fast, transparent, accurate in orbital elements to a fraction of a percent, and good for searching millions of candidates. The chosen candidate is then re-optimised in a full-ephemeris, numerically integrated model — with GMAT (NASA, open-source), JPL's MONTE, NASA's Copernicus or ESA's tools — using the patched-conic solution as the initial guess, exactly as the shooting corrector of the lesson on rendezvous and intercept targeting used Lambert. Launch targets are computed in that full model, so a real launch does not need the 49 m/s correction above; trajectory-correction manoeuvres then remove launch dispersions and navigation errors, typically for a few to a few tens of metres per second.

↓But there is a problem

Galileo spent six years reaching Jupiter by way of Venus and the Earth twice; Cassini took almost seven to reach Saturn by way of Venus twice, the Earth and Jupiter. Their tours were built from exactly the pieces of this unit: Lambert legs, flybys, resonant returns, a full-model check. The patched conic got them there. It works because almost everywhere one body clearly dominates: inside a sphere of influence the planet, outside it the Sun. Not everywhere. About 1.5 million kilometres from the Earth, towards the Sun and just outside its sphere of influence, the Earth and the Sun share control of a spacecraft so evenly that it can keep pace with the Earth for years without drifting away. Observatories have watched the Sun from there since 1978. The patched conic cannot describe the place at all — and in 2023 India sent a solar observatory to it.