The Three-Body Problem, and the Great Inequality of Jupiter and Saturn

We propose to produce a self-contained analysis and reduction of the equations of motion of the much-studied three body problem, making application to the Great Inequality (GI) of Jupiter and Saturn. The GI is a result of the mutual gravitational attraction between these two planets. Their orbits are very nearly in resonance (integer ratio of periods), which greatly amplifies the effect of the perturbation away from two independent (elliptic) Keplerian orbits. We propose to present the self-contained analysis using modern mathematical notation and physical concepts, which will serve to make the topic more accessible to modern audiences, including for undergraduate lectures. The problem also readily presents itself to asymptotic techniques, and we intend to complete a multiple-scales analysis which will elucidate how the GI arises. Certain terms in the asymptotic expansions have a low frequency (related to the near-integer ratio of periods), and this gives rise to a small denominator which corresponds to the large displacements that Jupiter and Saturn exhibit away from their mean motions. This discrepancy puzzled astronomers and mathematicians for much of the 18th century.

Faculty Supervisor:

Alan Coley

Student:

Partner:

Indian Institute of Technology Tirupati

Discipline:

Mathematics

Sector:

Aerospace; Education

University:

Dalhousie University

Program: