What is the mathematical model used for the “capture” problem?
We used a model called the “Restricted Three-Body Problem.” This model goes all the way back to Lagrange and makes the approximation that we have two large bodies (e.g., Sun and Jupiter) moving on a circular orbit and a third particle which is influenced by the two larger bodies, but they are not influenced by it. This is a very good model in the Solar System because the planets are quite well separated from each other. So, while the planets do influence each others motion, to a first approximation we can think of them as only being influenced by the sun. Thus, a moon is influenced by the planet and the Sun but does not influence them much, being so much smaller. At the end of the 19th century Hill used a similar (but not identical) model to explain the motions of the moon in the Sun-Earth-Moon system. For small bodies getting captured by Jupiter or any of the giant planets this is an excellent first approximation. Poincare recognized that this problem contains the seeds of chaos in 18
