The planar /. body problem (pronounced "slash dot") is the gravitational interaction of a line mass (/) and a point mass (.). The force and torque on the line integrate exactly, facilitating analysis. The elongated asteroid Ida and its tiny moonlet Dactyl form a natural example.
To study the /. body problem we used an advanced form of numeric integration called Symplectic Integration to take advantage of the symmetry of the positions and velocities in our equations. This also helped manage the numerical instability that we found in our equations. We prove that such an instability is not an essential feature of the equations of the system but a numerical artifact. We combine this sophisticated programming approach with cluster computing to collect large amounts of data on this system that exhibits a rich array of behavior.
The /. body problem realizes the complexity of the 3-body problem with only 2 bodies. In parameter space, sequences of periodic orbits dot a background of chaotic orbits. We find known behaviors such a stabilization by gyroscopic motion and gravitationally stable orbits. Typically, the point and the line revolve (= orbit) in precessing ellipses, as expected for a perturbation of the classical 2-body problem. However, the line may also rotate (= spin) chaotically or periodically. Spin-orbit momentum transfer orbits can spin-up the line or unbind the point from the line, with applications in the statistics of asteroid rotation rates.
Though our computation grid has collected a large amount of data which containing the various periodic and chaotic behaviors, it has searched but a small portion of the available parameter space. Our research has given us a glimpse into the complex behavior of this system, and, from this, we can see that the system is ripe for further investigation.