SSDEM Discrete Seismology Models with Applications to the Apophis 2029 Close Earth Encounter
Authors: DeMartini Joseph , Murdoch Naomi , Garcia Raphael , Richardson Derek 
Published: July 2024

The 2029 close Earth encounter of Near-Earth Asteroid 99942 Apophis presents a unique opportunity to study the dynamics, bulk properties, and interior structure of a potential rubble-pile asteroid [1]. Numerical models &#8211; including Finite Element Methods (FEM) and Discrete Element Methods (DEM) &#8211; are essential to constrain dynamical outcomes of the encounter and support planned and potential science missions to Apophis [2, 3].The primary consensus from dynamical models of the encounter indicate that the most visible physical outcome of the close approach will be a change in the rotation state of Apophis [4, 5], and that Apophis will pass outside of the Earth&#8217;s Roche lobe and will thus not suffer a catastrophic disruption in this encounter if it is a rubble pile. DEM-only models [9, 10] point to a small or negligible and primarily elastic change of the body&#8217;s envelope.Importantly, all of the dynamics, FEM, DEM, and combined models indicate that any physical changes to the structure or surface of Apophis as a result of the close encounter will be small [6, 7, 8, 9]. The small stress variations and elastic nature of the encounter open the door for a seismic study using DEM, where the individual particles act as seismic stations to track the propagation of waves through the body. Results from such models may be critical when determining the kinds of instruments or observations a mission should prioritize.We present here an architecture for performing seismic investigations with DEM. We will also present new, preliminary soft-sphere DEM (SSDEM) results regarding the presence, timing and depth of seismic activity induced in Apophis during the close approach.For the simulations presented here, we use the parallelized, N-body gravity and SSDEM tree code PKDGRAV to model gravitational and contact forces between discrete, spherical particles [10]. The SSDEM in PKDGRAV allows particles to slightly interpenetrate at the point of contact, using a Hooke&#8217;s law restoring spring force to model the material&#8217;s stiffness and apply damping and friction forces for particles in contact [11].The most important parameters in modeling the propagation and attenuation of seismic waves in SSDEM are the choice of the spring constant in the normal direction (kn), and the damping parameters, respectively. The spring constant is akin to a Young&#8217;s modulus [10] and governs the wave speed in our models. We choose a kn equivalent to a Young&#8217;s modulus between 5-15 MPa (depending on particle radius) in our models, matching the values estimated by the Rosetta lander on comet 67P [12]. The energy-damping parameters influence wave attenuation and are chosen in conjunction with the frictional parameters to match our desired angle of repose for asteroid regolith: ~35 degrees [11]. We validate our seismic investigation method by modeling P-waves in simple 1-, 2-, and 3-dimensional structures: a particle chain, a grid, and a roughly spherical, densely packed rubble pile, respectively. We track the wave speed and attenuation in our models and compare them to the simulation input parameters mentioned above, to develop an interpretation from our discrete approach to the typical continuum-mechanics framework.For the full Apophis encounter models, we follow the method of our previously published work, as described in [10]. Apophis is modeled as a 350-meter-diameter, cohesionless, self-gravitating granular aggregate of ~10,000 spherical particles, with shape matching the best-fit, radar-derived shape model [13], and Earth is a single sphere.Each PKDGRAV particle can be used as a seismic station in our models, with velocities measured at every timestep (~0.038 s). We identify seismic sources inside the body from peaks in velocity profiles. Our preliminary analysis indicates that the quaking on Apophis will be shallow, with ~50% of sources occurring at a depth of ~30 m or less (see Fig. 1 bottom). Furthermore, our simulations show that most sources begin ~2 h after closest approach and persist for a period of ~2 h (Fig. 1 top). This period starting 2 h after close approach is the time when Earth&#8217;s gravitational influence on Apophis is becoming negligible compared to the local gravity and rotational forces and Apophis is settling into its new post-encounter equilibrium rotation state. Our results imply that it may be this change in rotational forces that induces seismicity in the near-subsurface of Apophis. These models also strongly support the inclusion of an in-situ seismic instrument [14] on any potential missions that will arrive at Apophis prior to its close approach in April 2029.Acknowledgments: This work was supported in part by the Chateaubriand Fellowship Program, CNES, and by NASA FINESST Award 80NSSC21K1531. DEM simulations were carried out on the deepthought2 and Zaratan supercomputing clusters administered by the University of Maryland Division of Informational Technology.References: [1] Binzel, R. et al. (2021) Planet. Sci. and Astrobio. Decadal Survey 2023-32, 53 (4) eid 045. [2] DellaGiustina, D. N. et al. (2023) Planet. Sci. Journal, 4 (10), 22 pgs. [3] Morelli, A. C. et al. (2023) arXiv 2309.00435. [4] Pravec, P. et al. (2014) Icarus 233, 48-60. [5] Benson, C. J. et al. (2023) Icarus 390, id 115324. [6] Kim, Y. et al. (2023) Mon. Not. Royal Astr. Soc. 520 (3), 3405-3415. [7] Yu, Y. et al. (2014) Icarus 242, 82-96. [8] DeMartini, J. V. et al. (2019) Icarus 328, 93-103. [9] Hirabayashi, M. et al. (2021) Icarus 365, id 114493. [10] Schwartz, S. R. et al (2012) Granular Matter 14, 363-380. [11] Zhang, Y. et al (2017) Icarus 294, 98-123. [12] M&#246;hlmann, D et al. (2018) Icarus 303, 251-264. [13] Brozovi&#263;, M. et al. (2018) Icarus 300, 115-128. [14] Murdoch, N. et al. (2024) EPSCFigure 1. Top: Frequency of seismic events per hour. The region of significant tidal influence is bracketed by the green and red dashed lines. Time of closest approach is marked with a black dashed line. Bottom: Frequency of seismic events at varying depths, with bins ~15 m wide.