Interplanetary billiards
 
The ability to amplify a spacecraft’s heliocentric energy or dramatically change the direction of its trajectory by a planetary encounter was used to send Mariner 10 via Venus to within 1500 km ofMercury. The same principle was used to make Pioneer 11 and Voyagers 1 and 2 pass by Saturn after a Jupiter flypast, ultimately enabling Voyager 2 to visit Uranus and Neptune. The principle, sometimes called interplanetary billiards, is illustrated as follows.
Let a spacecraft follow a Hohmann elliptic transfer from Earth orbit to Jupiter orbit
 

Figure 1. (a) Hohmann transfer orbit of a spacecraft from Earth to Jupiter, the spacecraft entering Jupiter’s sphere of influence. (b) How, in the hyperbolic jovicentric orbit of the spacecraft within Jupiter’s sphere of influence, the spacecraft’s velocity V at A is effectively the exit velocity V at B.
(figure 1(a)). Let its aphelion velocity be VA. The circular heliocentric velocity in Jupiter’s orbit is V_J which is greater than V_A. Hence, from Jupiter’s point of view, the planet effectively overtakes the spacecraft with a relative velocity V = V_J − V_A, the spacecraft entering Jupiter’s sphere of influence at a point A close to Jupiter’s orbit (figure 1(b)). Once within Jupiter’s sphere of influence, the spacecraft makes a hyperbolic flight past Jupiter, exiting from Jupiter’s sphere of influence at B with the same velocity V but now effectively in the opposite direction to which it entered at A. Its new velocity relative to the Sun is
V_Θ= V + V_J = 2V_J − V_A > V
thus enabling the spacecraft to travel into the outer reaches of the Solar System without any additional expenditure of rocket fuel. It is almost as if the spacecraft ‘billiard ball’ has rebounded from themoving Jupiter ‘billiard ball’, gaining speed in the process. Careful planning of the exact point of entry of the spacecraft into the Jovian sphere of influence can produce almost any desired subsequent trajectory even to sending the spacecraft far above and below the plane of the Solar System for detailed study of the Sun’s polar regions.
This extreme sensitivity of the orbit of a spacecraft subsequent to slight changes in its Jovian flypast is another example of chaos but one where the concept is used to produce a desired effect.