Stability Analysis of Saturn's Rings with Differential Rotation

Abstract

A method is presented for calculating the potential due to an arbitrary thin gravitating ring by making use of cylindrical functions. Equilibrium configuration of the ring is defined in such a way that the centrifugal force exactly balances the self gravitation of the ring plus the gravitation due to the planet which is taken as spherical. By taking into account the differential rotation obtained in this manner mathematical analysis of the stability of motion about the equilibrium configuration is carried out by making use of the equations of hydrodynamics. Frequencies of the free oscillations are shown to be given as eigen values of a certain infinite matrix, and the eigen value problem is then solved numerically. The upper limit to the mass of the ring which is stable against axisymmetric perturbations is given for several model rings. If the density distribution of the ring is taken as a linear combination of the zeroth order Bessel functions of the first and second kind such that the density vanishes at the inner ( r = a ) and outer ( r = b ) boundaries, the limiting mass turns out to be 0.0386 M_s , 0.0109 M_s and 0.00209 M_s , for a / b = 0.2, a / b = 0.5 and a / b = 0.75, respectively, where M_s is the Saturn's mass., These values differ considerably from that obtained by Maxwell for a rigidly rotating ring where the effects of the edges are entirely neglected. Stability criterion for a different density distribution is also discussed.