On the ADAMS Prize Problem for 1856.
Allan F. Cook, F. A. Franklin · The Astronomical Journal · 1961
The question of the dynamical stability of Saturn's rings has a direct and immediate bearing upon all discussions of struc- ture and particle size. The famous essay of ~. C. Maxwell considers in elegant detail the dynamical stability of a ring of zero thickness in which collisions are not important. His calculation for the case of finite thickness of the ring is in error and we have modified it to arrive at a criterion of dynamical stability increasing Maxwell's upper limit on the density by more than an order of magnitude. Consideration of the effect of collisions leads to the following results: First, the accommodation coefficient for interparticle collisions must exceed some minimum value (about one-half), or input to random motions from friction would exceed output to radiation of heat generated in the particle surfaces by impacts, and the ring would expand in thickness. Second, the efficiency of the above input diminishes as the ring becomes thinner so that an equilibrium between the two processes may be expected at much greater random velocities and thickness of the ring than from thermal motions alone. The collapse to a thickness corresponding to thermal motions will only occur if the accommodation coefficient is approximately unity. Third, the combination of an accommodation coefficient in the above range and of viscosity operates to stabilize the rings against infinitesimal disturbances provided the density does not exceed approximately one-fifth that of Saturn. This work was supported in part by the Air Force Cambridge Research Center through a contract with Harvard University.