IAA Planetary Defense Conference PDC May 2017, Tokyo, Japan

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1 5th IAA Planetary Defense Conference PDC May 2017, Tokyo, Japan IAA-PDC COHESIVE STRENGTH OF DIDYMOS, THE TARGET OF THE AIDA MISSION Yun Zhang (1), Derek C. Richardson (2), Olivier S. Barnouin (3), Patrick Michel (4), Stephen R. Schwartz (4, 5), Ronald-Louis Ballouz (2), Junfeng Li (1) (1) School of Aerospace Engineering, Tsinghua University, Beijing , China. Phone: (2) Department of Astronomy, University of Maryland, College Park, MD 20742, United States. (3) JHUAPL, Laurel, MD 20723, United States. (4) Université Côte d Azur, Observatoire de la Côte d Azur, CNRS, Lagrange Laboratory, CS 34229, Nice Cedex 4, France. michelp@oca.eu (5) School of Earth and Space Exploration, Arizona State University, 781 E. Terrace Rd, Tempe, Arizona, 85283, United States. srs@oca.eu Keywords: Asteroids, dynamics; Asteroids, rotation; geological processes; cohesion. Abstract For a planetary defense mission, it is crucial to understand the structural strength of a hazardous asteroid, which has a strong influence on the asteroid s response to most mitigation techniques, before taking action. In this study, to support the proposed Asteroid Impact & Deflection Assessment (AIDA) mission (planned for fall 2022 encounter, which would be the first kinetic impactor experiment carried out at real asteroid scales), we investigate the plausible physical properties of the mission s target, the near-earth binary asteroid Didymos. The intriguing feature of this binary system is that its primary spins very fast, indicating the existence of cohesive forces in the primary s structure. By representing the primary as a self-gravitating granular aggregate rotating at the current observed spin rate, the critical cohesive strength of the body with various possible bulk densities is numerically explored. The results show that the Didymos primary rubble-pile model needs cohesion of ~6.5 Pa for the nominal bulk density (2.1 g/cc). 1. Introduction Growing evidence suggests that asteroids larger than a few hundred meters in diameter are gravitational aggregates, i.e., they are rubble-pile asteroids for which gravity is the principal force holding the body together (Richardson et al., 2002). However, because the gravity is so small on these small bodies, other forces may also have a significant role on the mechanics and dynamics of asteroids, among which Van der Waals cohesive forces could well be a dominant force (Scheeres et al., 2010). The mechanism of interparticle van der Waals cohesive force is associated with molecular or atomic polarization effects. The cohesive strength arising from van der Waals forces could range from several pascals to thousands of

2 pascals for particles with size on the order of several micrometers (Gundlach and Blum, 2015). Given that the regolith sample of asteroid Itokawa returned by the Hayabusa spacecraft has a size distribution of ~10 to ~100 μm (Nakamura et al., 2011), it is appropriate to consider that these micro-sized interstitial grains could provide significant cohesive interactions between neighbouring meter-sized boulders throughout the whole body (Sánchez and Scheeres, 2014). The existence of cohesive forces can improve the strength of rubble-pile asteroids and reduce their chances of breakup by rotational centrifugal or tidal forces (Holsapple, 2007; Sánchez and Scheeres, 2014). This could be a convincing explanation for the current observed fast-rotating asteroids (i.e., Rozitis et al., 2014; Polishook et al., 2016). The primary of the target of the proposed Asteroid Impact & Deflection Assessment (AIDA) mission, the near-earth binary asteroid Didymos, is a well-known fast rotator, whose spin period is 2.26 h. This asteroid could possess cohesion in its structure. To gain a better understanding of the effect of cohesion and to support the AIDA mission, we use a high-efficiency Soft-Sphere Discrete Element Method (SSDEM) code, pkdgrav, to investigate the effect of cohesion on the structural stability and dynamic behavior of the Didymos primary and secondary. 2. Methodology 2.1 SSDEM with cohesion In a rubble-pile asteroid, each constituent particle can interact with its surroundings through short-range interactions (e.g., mechanical contact force, van der Waals force) and long-range interactions (e.g., gravity). Within the high-efficiency parallel tree code framework, pkdgrav (Stadel, 2001), a soft-sphere discrete element model including five components along the normal, tangential, rolling and twisting directions is used for computing particle contact forces (Schwartz et al., 2012; Zhang et al., 2017). Figure 1 presents the directions of the forces and torques acting on particle i generated by the contact with particle j. The linear spring-dashpot normal contact force FN and the tangential stick-slip force FS are given by, F ˆ N kn x n CNu n, 1 (1) F min k δ C u, F δ δ, S S S S t S N S S and the rolling and twisting torques, MR and MT are given by M M R T k δ C ω, if k δ M M δ δ, if k δ M R R R R R R R,max R,max R R R R R,max k δ C ω, if k δ M M δ δ, if k δ M T T T T T T T,max T,max T T T T T,max The cohesive force model used in this study is based on the granular bridge idea proposed by Sánchez and Scheeres (2014, 2016). With the assumption that there,. (2) 1 The min() function means that the vector quantity of least magnitude is the one chosen.

3 exists substantial micro-sized grains distributed around the gaps between particles i and j that form a granular bridge, the accumulative van der Waals forces transmitted through the granular bridge can be obtained by integration over an effective contact area. Introducing the shape parameter, β, to represent a statistical measure of the area where the interstitial grains have impact on and use a rectangular area to approximate this effective contact area A eff 4 R 2, (3) where the effective radius R = ri rj/(ri + rj), and ri and rj are the radii of the corresponding particles. The corresponding cohesive force can be modeled as FC ca ˆ eff n, (4) where the interparticle cohesive tensile strength c is designed to reflect the physical properties of the interstitial regolith, e.g., porosity, size distribution, surface energy etc. (Sánchez and Scheeres, 2014; Gundlach and Blum, 2015). Note that c is different from the macroscopic cohesive strength C used in the Mohr-Coulomb failure criterion (see Section 4), where the latter describes macroscopic properties of granular materials. Figure 1 Schematic of transmitted forces and torques at a contact in a local Cartesian coordinate system, where e ˆ x and e ˆ y are in the contact plane and e ˆ z is in the normal direction according to the right-hand rule. The dashed yellow circle on the contact plane denotes the actual contact area of particle i and j. We refer reader to Schwartz et al. (2012) and Zhang et al. (2017) for a detailed explanation of the meaning and determination of the parameters in the contact model described by Eqs. (1) and (2). Briefly, all the parameters can be designed according to the desired properties of simulated material. We use the same setup for all the parameters as those used in Zhang et al. (2017) for the simulations in this study, and leave the interparticle cohesive tensile strength c as an adjustable parameter.

4 2.2 Physical properties of Didymos Table 1 gives basic properties on the current dynamical and rotational state of Didymos determined from Earth-based radar and optical telescopic observations (Richardson et al., 2016). Figure 2 shows the surface slopes on the primary with the current radar shape model at the nominal bulk density of 2.1 g/cc. Since that slopes greater than 50 are well above any angle of repose of cohesionless terrestrial sands, the bulk density of the primary has to be higher than the nominal value; otherwise, cohesion should be present in its structure. Zhang et al. (2017) explored the critical bulk densities for several Didymos primary rubble-pile representatives (see their Table 4; with the critical bulk density, the corresponding rubble pile can exactly maintain its shape at the spin period of 2.26 h). The results show that the Didymos primary in certain configurations can remain geo-statically stable with a bulk density slightly lower than the maximum value allowed for the asteroid. In this study, we try to relax restrictions on the bulk density by adding cohesion, and find out the relation between the bulk density and the corresponding critical cohesive strength. Table 1 Didymos system basic properties. Primary diameter, D P km ± 10% Secondary diameter, D S Total system mass, M TOT ± km (5.278 ± 0.04) kg Component bulk density, ρ B 2100 kg/m 3 ± 30% Primary rotation period, T ± h Figure 2 Surface slopes on the Didymos primary from various perspectives based on the latest radar shape model and the parameters in Table 1.

5 2.3 Determination of critical interparticle cohesive strength An iteration procedure including two separate tests is used to determine the critical interparticle cohesive strength ccrit for a particular rubble-pile model. The procedure is similar to the interval halving method. During these tests, the simulated body is strictly forced to stay at the critical spin state, i.e., the spin period T = 2.26 h. In the first test (henceforth, the first stage cohesion test), the interparticle cohesive strength c is artificially decreased by a certain amount c every time interval t from a high enough value c0 until the simulated body globally disrupts. By examining the evolution of the state variables during a simulation (see Fig. 3 for an example), the critical interparticle cohesive strength ccrit can be narrowed down to a region [c1, c1+ c]. Then, the following test (henceforth, the second stage cohesion test) is carried out by reducing c from c1 by a smaller cohesive strength increment c/10. Other setups in the second stage cohesion test is the same as the first one. The value of ccrit can be determined though the above procedure with an uncertainty of c/10. In this study, c is set to 100 Pa and t is set to about s, so the error of estimation on ccrit is less than 10 Pa. In principle, the error can be reduced by further iterations. However, given that the critical macroscopic cohesive strength Ccrit is hundredths of the magnitude of ccrit (as shown in Section 4), excessive precision is cumbersome and useless. 3. Simulation Results Previous study has shown that the arrangement and size distribution of constituent particles have significant impact on the structural strength and failure behaviors of spinning rubble piles (Zhang et al., 2017). We take the hexagonal closest packing (HCP) configuration as the study case in this paper and leave the other five configurations for a future study. Table 2 summarizes the basic properties of the rubble-pile model and the material properties found in previous study (Zhang et al., 2017). Table 2 Basic properties of rubble-pile models with the nominal bulk density. Here initial B is the initial packing efficiency, N is the number of particles in the model. Rubble-pile Model N Particle radius (m) initial B Critical bulk density (g/cc) Friction angle (deg) HCP Figure 3 shows the evolution of axis ratios, packing fraction and the mean coordination number (CN, i.e., mean number of contacts per particle) during the second stage cohesion test where c ranges from 200 Pa to 110 Pa. The interparticle cohesive strength c plays an important role in the internal structure and dynamical behavior of the body. As c decreases, its shape becomes more oblate, and packing efficiency and coordination number gradually decrease.

6 Figure 3 Evolution of the axis ratios (top two frames), packing efficiency and (lower-middle frame), and coordination number (bottom frame) of the simulated body during the second stage cohesion test. The initial interparticle cohesive strength c is 200 Pa, which is then decreased by 10 Pa every s. The insets show the detailed information of a stable case (presented in the green rectangles, where c = 140 Pa) and an unstable one (presented in the red rectangles, where c = 130 Pa). The critical value ccrit is determined by checking the creep stability of the simulated body during the process. We define the terminology of creep stability used in this study as the ability of a rubble pile to hold its shape at a given spin state without any creep deformation. The creep behavior in a rubble-pile body can be diagnosed when gradual decreases in the internal packing efficiency and coordination number are observed (Zhang et al., 2017). Generally, for a granular solid body, when the states of the body is below the limit failure state, the readjustments or deformations at those states are within the elastic domain (Holsapple, 2004). Therefore, after stopping loading, the state variables of this body can be maintained in small ranges. As shown in the insets in Fig. 3, the values of axis ratios and packing efficiency vibrates around the equilibrium state in the stable

7 case (presented in the green rectangles). This is the situation where the loading stress is below the material limit failure load, the vibrations of particles due to the reconstruction process can be well damped out. On the contrary, in the unstable case (presented in the red rectangles in Fig. 3), creep deformation gradually develops and the packing efficiency and coordination number can continually decrease with time, where the whole structure may eventually disrupt. Therefore, the critical interparticle cohesive strength ccrit for the HCP model is identified as 140 Pa. Figure 4 Normal contact force network of the HCP model with different c. Top row: force chains of the whole body; middle row: force chains of Slice 1; bottom row: force chains of Slice 2. Note that the color scale for visualizing the force networks corresponds to the magnitude range of normal contact forces for each situation.

8 Figure 5 Tangential contact force network of the HCP model with different c. First row: force chains of Slice 1 (defined in Fig. 4); second row: force chains of Slice 2 (defined in Fig. 4). Note that the color scale for visualizing the force networks corresponds to the magnitude range of tangential contact forces for each situation. Figures 4 and 5 illustrate the distribution of the force networks in the HCP body with the interparticle cohesive strength c of 1000 Pa, 200 Pa and 130 Pa, respectively. Most of the cohesive force chains (the blue chains shown in Fig. 4) distribute parallel to the horizontal plane, which is the best direction to offset the centrifugal force caused by the rapid rotation. It is clear to see that crack can be initiated in the center even the cohesive strength is much higher than ccrit. An explanation for this is that the strong cohesion motivates the formation of agglomerates. As shown in Fig. 4, the external particles are well connected to the beneath particles in the sub-surface, so they are all together susceptible to forming larger agglomerates. The centrifugal forces acting on the surface particles will also pull the sub-surface particles outward. And then the surface and sub-surface particles work together to pull the more internal particles until the accumulative centrifugal forces can be balanced by the gravitational forces or crack occurs to cut off the agglomeration. Given that the spin rate of the simulated body is close to its spin limit, the agglomeration behavior can propagate to the central region, causing crack initiation. Along with the agglomeration process, most of the particles move outward, leading to volumetric dilation and causing crack extension along the axial

9 direction (e.g., Fig. 4(2a)) and the radial direction (e.g., Fig. 4(3a)). The resulting agglomerates behaves like solid materials, where their relative slip motion can be blocked by the frictional resistance. As shown in Fig. 5, the cracks between these agglomerates carry the highest tangential forces. When the cohesion decreases, since the innermost particles in each agglomerate are subject to the highest accumulative centrifugal forces, the cohesive connections break from the inside out, and the cracks gradually propagate to the surface. The rubble-pile structure will inter an unstable phase when a local region reaches the material s limit failure state. 4. Critical macroscopic cohesive strength for Didymos primary The material properties of the HCP model can be obtained by analyzing the stress state of this rubble pile before the failure occurs. Macroscopic stress variables in a discrete element assembly can be assessed by homogenization and averaging methods based on Representative Volume Elements (RVEs) with the help of Voronoi tessellation (see Zhang et al., 2017 for details about how to carry out stress analyses on a simulated granular assembly). In this study, we use the Mohr-Coulomb failure criterion to check the limit failure state. The criterion defines a maximum shear stress that a granular material can withstand for a given compressive normal stress σn, C tan, (5) n where C is the macroscopic cohesive strength and is the angle of friction. With the help of Mohr s circle, Eq. (5) can be written as a function of the principal stress, sin 2 C cos, (6) where σ1 and σ3 are the maximum and minimum components of the principal stress, respectively. Figure 7 Stress distribution of RVEs on the shear-pressure plane for c = 140 Pa.Color denotes the distance from the aggregate mass center to the center-ofmass position of each RVE.

10 At the limit failure state, the stress variables satisfy the equality of Eq. (5). Figure 7 plots the stress state of each RVE in the critical cohesion (c = 140 Pa) case on the shear-pressure plane. The macroscopic cohesive strength C and the angle of friction can be determined from the intercept and slope of the best-fitting straight failure envelope, where C = 6.5 Pa and = 43 in the presented case. 5. Conclusions The internal structure and strength of asteroids significantly influence the impact processes on these small bodies and their subsequent collisional evolution. Our study provides some constraints on the possible physical properties of this binary system. The results show that the Didymos primary rubble-pile model can maintain its shape with a friction angle of ~43 at the current observed spin period (2.26 h) within the uncertainty of the observed bulk density (< 2.7 g/cc) without cohesion, but needs cohesion of ~6.5 Pa for the nominal bulk density (2.1 g/cc). The relation between the primary s bulk density and the critical cohesion (i.e., the minimum cohesion required to maintain the shape stable) is presented. References: Gundlach, B., Blum, J., Regolith grain size and cohesive strength of near-earth Asteroid (29075) 1950 DA. Icarus 257, Holsapple, K.A., Equilibrium figures of spinning bodies with self-gravity. Icarus 172 (1), Holsapple, K.A., Spin limits of Solar System bodies: From the small fast-rotators to 2003 EL61. Icarus 187(2), Michel, P., Richardson, D.C., Durda, D.D., et al., Collisional formation and modeling of asteroid families. Asteroids IV, Polishook, D., Moskovitz, N., Binzel, R.P., et al., A 2 km-size asteroid challenging the rubble-pile spin barrier A case for cohesion. Icarus 267, Richardson, D.C., Leinhardt, Z.M., Melosh, H.J., et al., Gravitational aggregates: evidence and evolution. Asteroids III, Richardson, D.C., Barnouin, O.S., Benner, L.A.M., et al., Dynamical and Physical Properties of Didymos. In: Proc. Lunar Planet. Sci. Conf. 47th, Rozitis, B., MacLennan, E., Emery, J.P., Cohesive forces prevent the rotational breakup of rubble-pile Asteroid (29075) 1950 DA. Nature 512, Sánchez, P., Scheeres, D.J., The strength of regolith and rubble pile asteroids. Meteoritics & Planetary Science 49(5), Sánchez, P., Scheeres, D.J., Disruption patterns of rotating self-gravitating aggregates: a survey on angle of friction and tensile strength. Icarus 271, Scheeres, D.J., Hartzell, C.M., Sánchez, P., et al., Scaling forces to asteroid surfaces: The role of cohesion. Icarus 210(2), Schwartz, S.R., Richardson, D.C., Michel, P., An implementation of the soft-sphere discrete element method in a high-performance parallel gravity tree-code. Granular Matter 14(3), Nakamura, T., Noguchi, T., Tanaka, M., et al., Itokawa dust particles: A direct link between S-type asteroids and ordinary chondrites. Science 333(6046),

11 Zhang et al. 2017, Icarus, under review.