Resonantly-forced Eccentric Ringlets: Relationships between Surface Density, Resonance Location, Eccentricity and Eccentricity-gradient

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.