“The competition was fierce, but collegial,” said Jonathan I. Lunine, a Cassini scientist and a professor of planetary science and physics at the University of Rome Tor Vergata.

Still, there were trade-offs. Nobody got everything, but everybody got a good many things. “We try to satisfy as many people as possible,” said Cassini’s mission planning engineer, John C. Smith of the Jet Propulsion Laboratory, who, with Mr. Buffington, is responsible for designing the tour. “We have to kill at least two and sometimes three birds with one stone.”

One of the fundamental tools for adjusting the trajectory of a large manufactured object in space — the essence of orbital mechanics — is the gravity assist. As a spacecraft approaches a planet or moon, gravity grips it and flings it in a different direction. In the 1970s and ’80s, NASA used the gravity assist technique to enable the tiny Voyager 2 to complete its “grand tour” of the outer planets of the solar system. Voyager 2 employed four gravity assists. The Cassini Solstice mission alone will require 56.

The popular analogy for the gravity assist is “slingshot,” but that term makes today’s orbit designers grit their teeth. “It’s a lot more sophisticated than that,” said David Seal, Cassini’s mission planning supervisor. “We can do a lot of things to get pretty much any trajectory we want.”

Image The bright arc within Saturn’s G-ring, truncated by the shadow of the planet, in an image taken by a camera aboard the Cassini spacecraft. Credit... JPL/NASA

A better analogy, he said, is two ice skaters in a hockey rink: a little girl and her father. The little girl is Cassini, small and fast; Dad is slow but strong. When the little girl reaches Dad at the red line, they clasp hands and Dad rotates. He can fling his daughter farther down the ice toward the far goal, toss her at right angles into the boards, send her back where she came from or let her go off at an angle.

In Cassini’s case, Dad, aptly perhaps, is Titan, Saturn’s largest moon. Bigger than the planet Mercury, it is the only thing in the Saturn system, besides Saturn, with enough gravity to make radical changes in the spacecraft’s trajectory every time it flies by.