  The collective dynamics of objects moving through a viscous fluid is complex
and counterintuitive. A key to understanding the role of nontrivial particle
shape in this complexity is the interaction of a pair of sedimenting spheroids.
We report experimental results on two discs settling at negligible Reynolds
number ($\simeq 10^{-4}$), finding two classes of bound periodic orbits, each
with transitions to scattering states. We account for these dynamics, at
leading far-field order, through an effective Hamiltonian in which
gravitational driving endows orientation with the properties of momentum. This
leads to a precise correspondence with the Kepler problem of planetary motion
for a wide range of initial conditions, and also to orbits with no Keplerian
analogue. This notion of internal degrees of freedom manifesting themselves as
an effective inertia is potentially a more general tool in Stokesian driven
systems.
