Irregular structure in planetary rings is often attributed to the intrinsic
instabilities of a homogeneous state undergoing Keplerian shear. Previously
these have been analysed with simple hydrodynamic models. We instead employ a
kinetic theory, in which we solve the linearised moment equations derived in
Shu and Stewart 1985 for a dilute ring. This facilitates an examination of
velocity anisotropy and non-Newtonian stress, and their effects on the viscous
and viscous/gravitational instabilities thought to occur in Saturn's rings.
Because we adopt a dilute gas model, the applicability of our results to the
actual dense rings of Saturn are significantly curtailled. Nevertheless this
study is a necessary preliminary before an attack on the difficult problem of
dense ring dynamics. We find the Shu and Stewart formalism admits analytic
stability criteria for the viscous overstability, viscous instability, and
thermal instability. These criteria are compared with those of a hydrodynamic
model incorporating the effective viscosity and cooling function computed from
the kinetic steady state. We find the two agree in the `hydrodynamic limit'
(i.e. many collisions per orbit) but disagree when collisions are less
frequent, when we expect the viscous stress to be increasingly non-Newtonian
and the velocity distribution increasingly anisotropic. In particular,
hydrodynamics predicts viscous overstability for a larger portion of parameter
space. We also numerically solve the linearised equations of the more accurate
Goldreich and Tremaine 1978 kinetic model and discover its linear stability to
be qualitatively the same as that of Shu and Stewart's. Thus the simple
collision operator adopted in the latter would appear to be an adequate
approximation for dilute rings, at least in the linear regime