Dynamical transitions in the N-body granular problem to identify breakup limits of rubble-pile asteroids
Authors: Ferrari Fabio , Alessi Elisa Maria 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Introduction&lt;/strong&gt;&lt;br /&gt;Evidence from in-situ and remote observations supports the idea that asteroids are rubble piles, i.e., gravitational aggregates of loosely consolidated material (Chapman 1978; Hestroffer et al. 2019). However, no direct measurements of asteroids&amp;#8217; interior exist, and little is known about the mechanisms governing their formation and evolution. To date, only a handful of asteroids have been visited by space probes. Compared to remote surveys, these provided important and unprecedented data, although focused on very specific asteroids. Recently, NASA's OSIRIS-REx and JAXA's Hayabusa2 revealed unexpected features on the surfaces of respectively, asteroids Bennu and Ryugu (Lauretta et al. 2019; Watanabe et al. 2019) and showed that constitutive relations derived from Earth-based experiments, or from previous in-situ measurements can hardly scaled up and re-adapted to different asteroid scenarios (Arakawa et al. 2020; Ballouz et al. 2021). Not only limited by a lack of data, the understanding of asteroids&amp;#8217; properties is challenged at a fundamental level by their rubble-pile nature. This makes their dynamics subject not only to the complex N-body gravitational interactions between its constituents, but also to the laws of granular mechanics, which is one of the major unsolved problems in physics.&lt;br /&gt;To mitigate these limitations, we propose here the use of dynamical system theory to study these complex N-body/granular systems, where the dynamics of individual bodies are driven both by mutual gravity and contact/collision interactions. The goal of this work is to investigate the feasibility of using chaotic indicators to infer the dynamical properties and transitions of granular systems. In particular, we implement the methodology and apply it to systems of granular aggregates, in the context of rubble-pile asteroid dynamical scenarios.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Methodology&lt;/strong&gt;&lt;br /&gt;We develop here a new theoretical framework to support the qualitative and quantitative investigation of dynamical transitions in a complex N-body/granular system. With the term &quot;N-body/granular system&quot;, we refer to systems made of several fragments that interact mutually through self-gravity and contact/collisions. These fragments have finite density and irregular shape, and reproduce particles in a granular media. The theoretical framework is based on the global representation of the behavior of the system by using chaotic indices. We define here several chaotic indices, including Finite-Time Lyapunov Exponents and Lagrangian Descriptors (Mancho et al 2013), both customized to the problem studied here, and test their suitability to identify the qualitative and quantitative behavior of the N-body granular system. In particular, we focus on applications related to rubble-pile asteroid scenarios. We test the newly developed theoretical framework against high-fidelity numerical simulations, which are considered here as ground-truth model of the reality. This choice is motivated by the lack of real-world and full-scale data. Also, we highlight that theoretical models such as the continuum model provide major simplification of the granular nature of the system and thus are not sufficiently accurate to be considered as real-world models. Nonetheless, existing theoretical models provide a useful mean of comparison and will be referenced throughout the analysis in this paper.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Results&lt;/strong&gt;&lt;br /&gt;As a test case, we reproduce the dynamics of a spinning rubble-pile aggregate, within a large parameter space including a range of different bulk densities and spin rates. We use chaotic indicators to build stability maps, to identify transitions in the dynamical behavior of the aggregate. These are used to identify limiting values in the density and spin rate which make the aggregate either stable or unstable, i.e., to identify its breakup limits.&lt;br /&gt;Preliminary results show good agreement between predictions based on chaotic indices and our ground-truth model, based on high-fidelity numerical simulations. In particular, Finite-Time Lyapunov Exponents appear to be better suited to identify complex transient motion within the aggregate, while Lagrangian Descriptors are shown to provide faster identification of the aggregate's breakup limit dynamical transition. Most notably, transition maps are shown to evolve in time, following the complex and dissipative nature of the N-body/granular system. Finally, we compare predictions based on chaotic indicators with the theory of continuum, which identify disruption limits of gravitational aggregates based on the simple static relation &amp;#969;&lt;sub&gt;lim&lt;/sub&gt;=&amp;#8730;(G&amp;#961;&amp;#960;); where &amp;#969;&lt;sub&gt;lim&lt;/sub&gt; is the spin rate limit before breakup and &amp;#961; is the bulk density of the aggregate.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;324&quot; height=&quot;243&quot; /&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;307&quot; height=&quot;230&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;br /&gt;Figures show examples of transition maps using Finite-Time Lyapunov Exponent (upper figure, red line indicates FTLE=0) and Lagrangian Descriptor (lower figure), both based on the time evolution of the polar inertia of the system. In the Figures, the breakup curve for continuum is shown using a blue dashed line.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;br /&gt;Our results provides a proof-of-concept to support the conceptual validity of our hypothesis, and demonstrate the capability of our newly proposed methodology to identify dynamical transitions in N-body/granular systems. In particular, we applied it to the case of spinning rubble-pile asteroids, to investigate their rotational breakup. This work represents a first step towards the generalization of our theoretical framework to other dynamical scenarios involving the formation and evolution of rubble-pile asteroids.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;br /&gt;Chapman 1978, In Asteroids: An Exploration Assessment. NASA Conf. Publ. 2053.&lt;br /&gt;Hestroffer et al. 2019, A&amp;A Rev. 27&lt;br /&gt;Lauretta et al. 2019, Nature 568&lt;br /&gt;Watanabe et al. 2019, Science 364&lt;br /&gt;Arakawa et al. 2020, Science 368&lt;br /&gt;Ballouz et al. 2021, Mon Not R Astron Soc 507&lt;br /&gt;Mancho et al. 2013, Comm Nonlin Sci Num Sim 18&lt;/p&gt;