The Lagrange planetary equations are used to study to secular evolution of a small, eccentric satellite that orbits within a narrow gap in a broad, self-gravitating planetary ring. These equations show that the satellites secular perturbations of the
ring will excite a very long-wavelength spiral density wave that propagates away from the gaps outer edge. The amplitude of these waves, as well as their dispersion relation, are derived here. That dispersion relation reveals that a planetary ring can sustain two types of density waves: long waves that, in Saturns A ring, would have wavelengths of order 100 km, and short waves that tend to be very nonlinear and are expected to quickly damp. The excitation of these waves also transports angular momentum from the ring to the satellite in a way that damps the satellites eccentricity e, which also tends to reduce the amplitude of subsequent waves. The rate of eccentricity damping due to this wave action is then compared to the rates at which the satellites Lindblad and corotation resonances alter the satellites e. These results are then applied to the gap-embedded Saturnian satellites Pan and Daphnis, and the long-term stability of their eccentricities is assessed.
