Dr. Ojakangas performed his post-doctoral research at the Lunar and Planetary Laboratory of the University of Arizona, where he studied the formation of the solar system, and the closely related problem of planetary ring dynamics. Saturn’s rings are composed of an enormous number of ice balls (more than a mole of them!) and as such can be treated using statistical mechanics as with a gas of molecules – except that each “molecule” is a ball of ice, ranging from centimeters in diameter to the size of small moons. Alternatively, they may be treated with the equations of fluid mechanics. The differential equations employed in these methods are extremely challenging to solve and the solutions are difficult to understand. Dr. Ojakangas’ research expanded upon a qualitative idea conceived by his advisor, Dr. Richard Greenberg, which greatly simplifies the problem. Ojakangas placed this concept within a precise mathematical framework using a phase space comprised of (1) the tangential velocity of ring particles relative to a local circular orbit, (2) half of the radial velocity relative to the circular orbit, and (3) the radial position (Figure 4). In this coordinate system, ring particles constitute a continuum that circulates at constant angular velocity about the r-axis, while moving back and forth along it. Though much simpler, this approach reproduces many of the important results in planetary ring dynamics, such as the effective viscosity of such rings, and their tendency to spread radially, or form into ringlets. Link here.