Higher order mass aggregation terms in a nonlinear predator-prey model maintain limit cycle stability in Saturn's F ring

TL;DR

This work extends predator–prey modeling of Saturn's F ring by incorporating a generic nonlinear mass growth term $k M^{n}$ to account for higher-order aggregation, coupling mean mass $M(t)$ (prey) with the squared velocity dispersion $V(t)$ (predator) under Prometheus-driven forcing. Using time-series, phase portraits, and stroboscopic maps, coupled with an eigenvalue stability analysis of the Jacobian, the study identifies two distinct dynamical regimes: a fixed-point attractor and a limit-cycle regime, with a boundary in the $(k,n)$ parameter space indicating a quasi-periodic Hopf bifurcation. The results demonstrate that limit-cycle oscillations, which mirror observed clumping cycles in the F ring, can persist despite nonlinear higher-order mass terms, providing a mechanistic explanation for the cyclic dynamics. Overall, the paper offers a robust framework linking nonlinear coagulation processes to the observed ring structures and highlights the parameter ranges that sustain cyclic stability, aligning with Cassini-era observations of clumps, knots, and spokes.

Abstract

We consider a generic higher order mass aggregation term for interactions between particles exhibiting oscillatory clumping and disaggregation behavior in the F ring of Saturn, using a novel predator-prey model that relates the mean mass aggregate (prey) and the square of the relative dispersion velocity (predator) of the interacting particles. The resulting cyclic dynamic behavior is demonstrated through time series plots, phase portraits and their stroboscopic phase maps. Employing an eigenvalue stability analysis of the Jacobian of the system, we find out that there are two distinct regimes depending on the exponent and the amplitude of the higher order interactions of the nonlinear mass term. In particular, the system exhibits a limit cycle oscillatory stable behavior for a range of values of these parameters and a non-cyclic behavior for another range, separated by a curve across which phase transitions would occur between the two regimes. This shows that the observed clumping dynamics in Saturn's F ring, corresponding to a limit cycle stability regime, can be systematically maintained in presence of physical higher order mass aggregation terms in the introduced model.

Figures (6)

• Figure 1: An image of Saturn and its rings. The zoomed image shows the F ring with its nearest moons: Prometheus (inside ring) and Pandora (outside ring). Image credits: NASA/JPL/Space Science Institute.
• Figure 2: Time series plots and phase plots of $M$ and $V$, showing distinct dynamic behavior for $k=0.15, n=1.2$ (up) & $k=0.6, n=1.3$ (bottom). Parameters: $T_{syn}=112$ periods, $v_{esc}=0.5$ m/s, $M_{0}=2\times10^{9}g$, $A_{0}=\tau=0.1$, $\epsilon=0.6$ and initial conditions $M(0)=4.5\times10^{9}$g and $V(0)=3 m^{2}/s^{2}$.
• Figure 3: Stroboscopic phase space plots of $(M(t_{N}),V(t_{N}))$ at $t=N T_{syn}$ for $k=0.15, n=1.2$ (left) & $k=0.6, n=1.3$ (right) for $4$ different initial conditions denoted by circular dots. On the left, all orbits reduce to a single point (red x), while on the right, all orbits fall into a closed cycle path.
• Figure 4: The real part of an eigenvalue of the Jacobian in Eq.7 ($A_{0}=0$), for parameters set to same values as in Fig. 2, plotted in navy blue. The green plane corresponds to the $(n,k)$ plane or $Re(\lambda)=0$.
• Figure 5: A scatter plot of $(k,n)$ values (gray dots) corresponding to $Im(\lambda)<0$ for $A_{0}=0$. The black line is a regression fit of the values at the edges of the scatter plot corresponding to $Im(\lambda)=0$. The region above the line (white region) corresponds to $Im(\lambda)>0$