Investigating Vertical Structure Formation in Saturn’s Rings with N-body simulations

Introduction and Scope of Work:

Imaging of Saturn’s rings by the Cassini spacecraft revealed fine detailed structure of its dense rings [e.g., Spitale & Porco 2010, Tiscareno et al. 2019], including the existence of i) irregular vertical structures (IVS’s) on the edge of the A- and B-rings, ii) propeller structures created by embedded moonlets, and iii) mountain-like wall structures at the edges of rings from fully gap-opening satellites (Fig. 1). We have been investigating these vertical structures in order to better understand their formation mechanisms, to provide better quantitative constraints on the physical properties of ring particles, and to construct scaling relationships of structure properties with ring particle and moonlet properties. Furthermore, these physical features of Saturn’s rings provide a natural laboratory to explore the relationship between dense rings, embedded moonlets, and exterior satellites.  

Figure 1. (a) An example of an Irregular Vertical Structures (IVS) in the B-ring outer edge [Spitale  & Porco 2010]. (b) An embedded moonlit forming a propeller (named Blériot) on the unlit side of the rings [Tiscareno et al. 2019]. (c) Cassini image of the Keeler gap, showing Daphnis and the vertical structure of density waves induced on the edge casting shadows on the ring.

We use the N-body code pkdgrav to directly simulate the formation of i) IVS’s and ii) propeller structures from embedded moonlets. pkdgrav has the ability to model dense granular regimes by incorporating detailed multi-contact, frictional, and cohesive forces between particles through a Soft-Sphere Discrete Element Method (SSDEM) [Schwartz et al. 2012, Zhang et al. 2018, Ballouz et al. 2017]. For modelling IVS’s, we also use the N-body code epi_int [Hahn et al. 2025] to generate initial conditions for modeling the edge of the B-ring that is influenced by the Mimas 2:1 Linblad resonance [Spitale & Proco 2010]. 

While there have been previous simulation studies of vertical structure using hydrodynamic codes, which capture the scale of the phenomenon, and N-body simulations, which directly model the ring particle interaction, few prior studies have been able to directly model vertical structure formation at the appropriate physical scale for direct comparison with observations. Here, we leverage the highly-parallel nature of pkdgrav to directly simulation the formation of vertical structures with physically-realistic ring particle sizes (1-10 m). For the presentation, we will present progress in modeling the formation of IVSs and propeller structures from embedded moonlets. Here, we highlight some progress in the modeling of propeller structures.

Simulation Setup and Preliminary Results

We simulate a patch of particles with a differential size frequency distribution defined by a power law with exponent -2, and a radius range of 0.5 to 1.5 meters. We vary the bulk density of individual particles, and set a nominal value of 0.6 g/cm³. The radius of the embedded moonlet is 50-m. We vary the normal, 𝜀_n, and tangential, 𝜀_t, coefficients of restitutions. The friction parameters are chosen such that particle shave an angle of friction of ~ 32°. The simulations are initialized by creating a uniform density ring patch with a prescribed dynamical optical depth (𝜏_dyn = 1.0) set in the B-ring (semi-major axis = 100,000 km). The azimuthal and radial extents of the patch are approximately 6 km and 1 km, respectively, which is approximately 85 x 15 embedded moonlet hill radii. The total number of ring particles simulated is N = 3.6 million. The simulations run for approximately 50 orbital periods, which is sufficient to reach a steady state configuration.

Fig. 2 shows examples of two simulations with different 𝜀_n and 𝜀_t. In both cases, we see the formation of propeller structures that extend to 4-5 km along the azimuthal direction. For the case with more dissipative collisions, we noted the formation of irregular dense vertical structures at the edges of the gaps (magenta box in Fig. 2b) that have a saw-tooth pattern, which have been observed in different regions of Saturn’s rings. Fig. 3 shows a closer look at some of these structures.

Figure 2. End-stage of embedded moonlet simulations with patch size of 1.1 km (radial) by 6 km (azimuthal). The propeller structure is evident and contained within the confines of the simulation box. a, case with less dissipative inter-particle collisions, 𝜀_n = 0.8, 𝜀_t = 0.8. b, case with less dissipative inter-particle collisions, 𝜀_n = 0.5, 𝜀_t = 0.5.

Figure 3. We highlight one of the regions with irregular saw-tooth structures in the simulation results shown in Fig. 2b. These structures are more pronounced for the case with more dissipative collisions between ring particles.

We analyzed the case shown in Fig. 2b case further to study how the structure seen in the simulations might control the appearance of the ring as seen by Cassini. We sub-divided the ring patch into a grid made of 20 m × 20 m cells and calculate the maximum height, H, of each cell, normalized by the embedded moonlet’s radius, r_m. We find that parts of the vertical structure have H/r_m > 1. Fig. 4 shows that the height of the ring patch is largest adjacent to the accretion tubes.

Figure 4. Sub-dividing the simulation shown in Fig. 2b into 20 m × 20 m cells, we measured the height range, H, normalized by the embedded moonlet’s radius (r_m) in each cell to better compare with Cassini observations of Saturn’s dense rings. 

Acknowledgements

This work is supported by the NASA Cassini Data Analysis program through grant 80NSSC23K0220. This work also made use of a NASA High-End Computing allocation (SMD-24-30244393)

 

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