The orbit of a spacecraft is primarily determined by the gravity of a single large central body like the sun, Earth, or, in the case of the Cassini spacecraft, Saturn. If the central body was the only gravitational body in the vicinity, then the spacecraft would follow an elliptical path around the central body with constant orbital energy and angular momentum. In our solar system, however, most of the large natural bodies have smaller natural bodies orbiting them. For instance, the sun has the nine major planets (and lots of smaller material) around it, whereas Earth has the moon and Saturn has nine major moons (along with many minor moons plus its rings).
When a spacecraft in orbit about a primary body comes close to a moon that is orbiting the same primary body, there is an exchange of orbital energy and angular momentum between the spacecraft and the moon. The total orbital energy remains constant, so if the spacecraft gains orbital energy then the moon's orbital energy decreases. Orbital period, which is the time required to complete one orbit about the primary body, is proportional to orbital energy. Therefore, as the spacecraft's orbital period increases (the slingshot effect), the moon's orbital period decreases.
But because the spacecraft is much, much smaller than the moon, the effect on the spacecraft's orbit is much greater than on the moon's orbit. For example, the Cassini spacecraft weighs about 3,000 kilograms, whereas Titan, the largest of Saturn's moons, weighs about 1023 kilograms. The effect on Cassini is thus about 20 orders of magnitude greater than the effect on Titan is.
As an example, consider what happens when Cassini comes close to Titan during one of its orbits around Saturn. As the spacecraft approaches, Titan's gravity increases the velocity of Cassini relative to itself and, more importantly, changes the direction of Cassini's velocity. As Cassini leaves Titan's neighborhood, its velocity relative to the moon decreases to its original value, but the direction of the velocity continues to change in the same direction. The result is that Cassini's velocity relative to Saturn has been altered. Depending upon the geometry of the flyby, the Saturn-relative velocity can be modified either in direction or magnitude or both. If the magnitude of the Saturn-relative velocity changes, then the orbital energy of Cassini changes with the resulting change in the orbital period. Changes in either the magnitude or the direction of its Saturn-relative velocity modify Cassini's angular momentum, which results in a change in orbital orientation.
If the spacecraft flies behind the moon then the effect is an increase in the velocity (and orbital energy) of the spacecraft relative to the primary body, which gives the appearance of a slingshot throwing the spacecraft into a larger orbit. Because navigation can control where the spacecraft flies by the moon, we can also fly it in front of the moon, thereby decreasing its velocity (and orbital energy). Moreover, flying above or below the moon can change the direction of the spacecraft's velocity resulting in a change only in the orientation (and angular momentum) of the orbit. Of course, all of these changes cause an inverse change in the energy and angular momentum of the moon, but its larger mass results in changes so small that they are undetectable among all the other forces that affect the moon's orbit.
It is interesting to note that spacecraft undergoing a gravity assist are only copying an effect that occurs on a regular basis in our solar system. Comets that reside in the outer reaches of the solar system are often thrown into the inner solar system by the major planets, primarily Jupiter.
Answer originally published September 27, 2004.