Final answer:
Planetary science, particularly a subfield of astronomy, is the branch of Earth science that seeks to understand the orbit of Saturn's moons, with significant contributions from the Cassini-Huygens mission.
Step-by-step explanation:
The branch of Earth science concerned with understanding the orbit of Saturn's moons is planetary science, which is a subfield of astronomy that involves the study of planets and their moons. This study includes the analysis of orbits and the physical and chemical characteristics of these celestial bodies. Planetary scientists utilize data from various missions like the Cassini-Huygens mission to Saturn, which provided detailed observations and contributed significantly to our understanding of Saturn's moons, such as Titan, known for its similarities to Earth despite frigid temperatures, and Iapetus with its distinctive two-toned hemispheres.
The Cassini mission was a cooperative project between NASA and the European Space Agency launched in 1997 and deposited an entry probe on Titan in 2005. Further, in the broader field of Earth science, this study complements our understanding of Earth and the entire solar system, illustrating the interconnectedness of celestial mechanics with planetary geology and atmospheric sciences.