The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the set, students use the mathematical constant pi and Kepler's third law to find out when a Saturn orbiter will make its mission-ending plunge into the ringed planet.
The Cassini-Huygens mission—a joint endeavor of NASA, ESA (the European Space Agency), and the Italian Space Agency—was the first mission to orbit Saturn and explore its environs in detail.
The mission was conceived from the beginning as an international endeavor, in 1982, just after the two NASA Voyager
spacecraft flew past Saturn. The Voyager flybys whetted the appetites of planetary scientists for more in-depth exploration, particularly with regard to the mysterious moon Titan. Launched in 1997, Cassini toured the Saturn system after arriving there in 2004, performing a detailed, up-close study of the planet, its rings and moons.
The mission delivered ESA's Huygens probe to Titan in 2005, where it performed the first descent and landing on a world in the outer solar system. In complement to Huygens' dazzling revelations about Titan, the Cassini orbiter performed 127 of its own close flybys of Titan (with many more distant encounters).
By the end of its mission, the Cassini spacecraft had observed almost half of a Saturn year, which is 29 Earth years long. The four seasons of Saturn's year last about seven Earth years apiece, and upon Cassini's arrival at Saturn, the planet's northern hemisphere was just beginning to emerge from winter. Following its initial, four-year tour, Cassini's mission was extended two more years, to enable the spacecraft to observe changes -- particularly in the rings -- as Saturn reached equinox and the Sun shone edge-on to the rings. After equinox, Cassini was granted an additional seven-year extension. This enabled scientists to follow up on their earlier discoveries at Enceladus and Titan, and watch as summer sunlight came to the northern hemisphere of Saturn and its moons, while winter darkness embraced the south.
The findings of the Cassini mission revolutionized our understanding of Saturn, its complex rings, the amazing assortment of moons and the planet’s dynamic magnetic environment. The most distant planetary orbiter ever launched, Cassini started making astonishing discoveries immediately upon arrival and continues today. Icy jets shoot from the tiny moon Enceladus. Titan’s hydrocarbon lakes and seas are dominated by liquid ethane and methane, and complex pre-biotic chemicals form in the atmosphere and rain to the surface. Three-dimensional structures tower above Saturn’s rings, and a giant Saturn storm circled the entire planet for most of a year. Cassini’s findings at Saturn also fundamentally altered many of our concepts of how planets form around stars.
When Cassini ended, it left a rich scientific and engineering legacy.
In 2017, after more than 12 years at Saturn, the Cassini mission came to an end with a plunge into Saturn. The finale was designed to keep Cassini from impacting and possibly contaminating any of Saturn’s scientifically intriguing moons. First, mission operators planned a daring series of orbits that took Cassini closer to Saturn than ever before. Cassini used the gravity of Saturn’s moon Titan to alter its trajectory and fly into the gap between Saturn and its rings. It all began with a flyby of Titan on April 22, putting Cassini on a new orbital path whose first apoapsis was on April 23. Then, it completed 22 elliptical orbits with an average periapsis altitude of 63,022 km and an average apoapsis altitude of 1,274,828 km. A final flyby of Titan placed Cassini on a half-orbit trajectory for Saturn impact.
Use Kepler’s third law below to find approximately how many days each orbit took. Approximately what day did Cassini dive into Saturn’s atmosphere?
- µcb (gravitational parameter of Saturn) = 3.7931187 x 1016 m3/s2
- aSC = semi-major axis of Cassini's orbit
- TSC = orbital period of Cassini
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