<i>N</i>-body Simulations of the Self-confinement of Viscous Self-gravitating Narrow Eccentric Planetary Ringlets

Abstract Narrow eccentric planetary ringlets have sharp edges, sizable eccentricity gradients, and a confinement mechanism that prevents radial spreading due to ring viscosity. Most proposed ringlet confinement mechanisms presume that there is one or more shepherd satellites whose gravitational perturbations keep the ringlet confined radially, but the absence of such shepherds in Cassini observations of Saturn’s rings casts doubt upon those ringlet confinement mechanisms. The following uses a suite of N -body simulations to explore an alternate scenario, whereby ringlet self-gravity drives a narrow eccentric ringlet into a self-confining state. These simulations show that, under a wide variety of initial conditions, an eccentric ringlet’s secular perturbations of itself causes the eccentricity of its outer edge to grow at the expense of its inner edge. This causes the ringlet’s nonlinearity parameter q to grow over time until it exceeds the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>q</mml:mi> <mml:mo>≃</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> threshold where the ringlet’s orbit-averaged angular momentum flux due to viscosity + self-gravity is zero. The absence of any net radial angular momentum transfer through the ringlet means that the ringlet has settled into a self-confining state, i.e . , it does not spread radially due to its viscosity, and simulations also show that such ringlets have sharp edges. Nonetheless, viscosity still circularizes the ringlet in time τ e ∼ 10 6 orbits ∼1000 yr, which will cause the ringlet’s nonlinearity parameter to shrink below the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>q</mml:mi> <mml:mo>≃</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> threshold and allows radial spreading to resume. Either sharp-edged narrow eccentric ringlets are transient phenomena or exterior perturbations are also sustaining the ringlet’s eccentricity. We then speculate about how such ringlets might come to be.