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Spatial Structure and Coherent Motion in Dense Planetary Rings Induced by Self-Gravitational Instability

We investigate the formation of spatial structure in dense, self-gravitating particle systems such as Saturn's B-ring through local $N$-body simulations to clarify the intrinsic physics based on individual particle motion. In such a system, Salo (1995) showed that the formation of spatial structure such as wake-like structure and particle grouping (clump) arises spontaneously due to gravitational instability and the radial velocity dispersion increases as the formation of the wake structure. However, intrinsic physics of the phenomena has not been clarified. We performed local $N$-body simulations including mutual gravitational forces between ring particles as well as direct (inelastic) collisions with identical (up to $N\sim40000$) particles. In the wake structure particles no longer move randomly but coherently. We found that particle motion was similar to Keplerian motion even in the wake structure and that the coherent motion was produced since the particles in a clump had similar eccentricity and longitude of perihelion. This coherent motion causes the increase and oscillation in the radial velocity dispersion. The mean velocity dispersion is rather larger in a more dissipative case with a smaller restitution coefficient and/or a larger surface density since the coherence is stronger in the more dissipative case. Our simulations showed that the wavelength of the wake structure was approximately given by the longest wavelength $\hsλ{cr} = 4π^2 GΣ/κ^2$ in the linear theory of axisymmetric gravitational instability in a thin disk, where $G$, $Σ$, and $κ$ are the gravitational constant, surface density, and a epicyclic frequency.