Influence of Particle Shape in Discrete Modeling of Apophis’ 2029 Close Earth Encounter.

The upcoming close encounter of asteroid 99942 Apophis with Earth in 2029 presents a once-in-7000-years opportunity to study the dynamics, bulk properties, and interior structure of a potential rubble-pile asteroid as it passes deeply through Earth’s gravitational field. Numerical modeling—including via Discrete Element Methods (DEMs)—has helped to develop our understanding of the dynamics and physical outcomes of the tidal encounter between Apophis and Earth, including the expected change in the bulk shape and spin of the body, and predictions of potentially measurable surface and seismic outcomes due to the short period of natural tidal forcing. These models have helped to design missions to Apophis to ensure that we can make the most of the natural experiment that the Apophis encounter provides. 

 

In this talk, we will present new and ongoing DEM models of the full Apophis-Earth close encounter, making use of recent developments in modeling realistic particle shapes with both a “glued-sphere” approach and a level-set DEM (LSDEM) approach in the parallelized N-body gravity and soft-sphere DEM (SSDEM) code PKDGRAV. The glued-sphere method provides simpler spherical gravity and collision detection calculations but requires smaller timesteps and stiffer constituent particles. The level-set method provides a more realistic shape representation at the cost of increased memory requirements and the loss of precise gravitational torques (as we do not calculate polyhedral gravity). Here, we compare the results and performance of both techniques, and how these methods compare with previous spheres-only DEM models to get a clearer picture of the deformation, spin change, and seismic activity induced in Apophis during the close approach.

 

For the simulations presented here, we use PKDGRAV to model interparticle gravitational and contact forces between discrete, spherical particles. The SSDEM in PKDGRAV allows particles to slightly interpenetrate at the point of contact, using a Hooke’s law restoring spring force to model the material’s stiffness and apply normal and tangential damping forces, interparticle friction, and cohesive forces for particles in contact. The LSDEM defines nodes on the boundaries of the (irregularly shaped) particles, defined by the mathematical constructs of “level sets” to determine contours of distance from the true particle boundary, with an increasing number of nodes per particle improving the shape approximation. LSDEM particles are soft, as in SSDEM: when they contact, they are allowed to interpenetrate, and the restoring force is determined by the depth of penetration at each overlapping node on the particle boundary.

 

Modeling with irregular particle shapes (rather than independent spheres) allows us to increase the macroporosity of the resultant rubble pile while also increasing the body’s shear strength. This occurs naturally when packing irregular shapes due to the void spaces created by interlocking grains, and the physical strength of those interlocked structures, which cannot be replicated by spheres alone. The bonded aggregate SSDEM approach and the LSDEM approach allow us to create a high macroporosity regolith body—like those investigated in recent missions to rubble-pile asteroids Bennu, Ryugu, and Itokawa—that is also more resistant to reshaping or disaggregation than previous spheres-only models. 

 

Following the method of our previously published work, we represent Earth as a single, rigid sphere and Apophis as a cohesionless, self-gravitating, granular aggregate of irregular boulders several meters in radius. We also model the asteroid with different interior structure profiles: contact-binary; large single-core (ellipsoidal); and rubble throughout. We use the best-fit, radar-derived shape model to carve the appropriate shape of Apophis from a random cloud of constituents allowed to collapse in free space under self-gravity, subject only to gravitational and contact forces. 

 

The body is then placed under rotation. The primary source of uncertainty in the 2029 Apophis-Earth encounter is the orientation of Apophis at the time of close approach. In one subset of our comparative simulations, we align the spin axis with the intermediate body axis of Apophis, and model the encounter in the plane defined with a normal vector parallel to that spin axis. In this way, we can construct encounters with either the long axis or the short axis of Apophis directly aligned with Earth at perigee, which should bracket the expected deformation as the strongest and weakest encounter geometries, respectively. For the models investigating change in rotation state and seismic influence, we implement a spin axis chosen from the uncertainties in the current lightcurve data. In all cases, we choose the spin frequency to match the averaged effective spin period for its tumbling state and do not model the tumbling motion. The body is allowed to settle under its rotation until residual particle speeds in the body frame are much less than 1% of their expected peak values during the encounter simulations.

 

The encounters last for 10 simulated hours: ~5 h before closest approach and ~5 h after, in the non-rotating frame of the center of mass of Apophis. The lead time ensures there are no strong jolts from the sudden addition of the Earth’s gravity. The trailing time allows the body to settle into its post-encounter equilibrium state after Earth’s gravitational effects have diminished.

 

In addition to updates on the prior SSDEM results, and comparisons between different constituent particle geometries and interior structures, we have carried out an SSDEM analysis of the seismicity on Apophis in a high-resolution (time and particle number; relative to the newest set of models) encounter simulation. Each PKDGRAV particle exhibiting elastic motion can be used as a seismic station in our models, with velocities measured at every timestep. By finding peaks in velocity magnitudes, we identify seismic sources in the body. Our analysis indicates that the quaking on Apophis will be shallow and that most sources begin ~2 h after closest approach and persist for a period of ~2 h. Our results indicate that the seismic signals in our models would be measurable by an in-situ seismometer (like the one described in the abstract by Murdoch et al.) taking measurements on Apophis during and after the close Earth encounter.