We use a simple model of the dynamics of a narrow-eccentric ring, to put some
constraints on some of the observable properties of the real systems.In this
work we concentrate on the case of the `Titan ringlet of Saturn'.Our approach
is fluid-like, since our description is based on normal modes of oscillation
rather than in individual particle orbits. Thus, the rigid precession of the
ring is described as a global m=1 mode, which originates from a standing wave
superposed on an axisymmetric background. An integral balance condition for the
maintenance of the m=1 standing-wave can be set up, in which the differential
precession induced by the oblateness of the central planet must cancel the
contributions of self-gravity, the resonant satellite forcing and collisional
effects. We expect that in nearly-circular narrow rings dominated by
self-gravity, the eccentricity varies linearly across the ring. Thus, we take a
first order expansion and we derive two integral relationships from the
rigid-precession condition. These relate the surface density of the ring with
the eccentricity at the center, the eccentricity gradient and the location of
the secular resonance. These relationships are applied to the Titan ringlet of
Saturn, which has a secular resonance with the satellite Titan in which the
ring precession period is close to Titan's orbital period. In this case, we
estimate the mean surface density and the location of the secular resonance.Comment: Accepted for publication in Celestial Mechanics and Dynamical
Astronom
