Rediscussion of Maxwell's ADAMS Prize Essay on the stability of Saturn S rings

Abstract

This paper, which rediscusses the classical problem of the stability of Saturn's rings, indicates and removes three errors in Maxwell's Adams Prize Essay for 1856. Maxwell's two uniformly dense models with collisions neglected on the one hand and incompressibility assumed on the other hand are supplemented by three additional models. These three models are based on a Maxwellian velocity distribution for the particles. The first model neglects collisions; the second assumes adiabatic compression; and the third, isothermal co' pression. The first new model replaces Maxwell's first model and gives a lower limit on the mean density of 0.06 g , below which stability of the rings against self-gravitating condensations seems assured. Maxwell's second model (incompressible fluid) gives an upper limit on the mean density of 0.38 g , above which linearized theory predicts instability. The unstable mode is a purely radial disturbance consisting of alter- nating circular condensations and rarefactions. The situation is not clear at densities between 0.06 and 0.38 g . In the final section we first obtain an approximate velocity ellipsoid for the ring particles and equate the rates at which energy is transferred to and away from this distribution. This steady state analysis gives us a relation between the accommodation coefficient and the vertical ring thickness.