Experimental and theoretical investigations of low-velocity collisions into granular material
Authors: Drilleau Melanie , Murdoch Naomi , Sunday Cecily , Nguyen Gautier , Thuillet Florian , Gourinat Yves 
Published: October 2020

In order to study the mechanical properties of a small body surface, accelerometers can be used to record the acceleration profile during landing and rebounding of a surface package. However, given the low gravity environment, the behaviour of the grains can be different to under terrestrial gravity [e.g., 1].Here we present data from experiments of low-velocity impacts of projectiles of various shapes into different types of granular material in both normal and reduced gravity. Then, to further investigate the dynamics, we employ a collisional model in order to fit the experimental data using an empirical force law involving both a hydrodynamic drag force term and a static resistance force term. The contributions of these two force terms are discussed and compared for the different configurations.In terms of collision frequency with the Earth, the most threatening NEOs are the smaller ones (< 1 km diameter). They are also the less well known, because ground observations cannot provide adequate information at these sizes. The JAXA Hayabusa 1 and 2 missions [2, 3], and the NASA OSIRIS-REX [4] missions, have highlighted the complex histories experienced by small bodies. The unintuitive nature of the asteroid response to the Small Carry-on Impactor experiment of the Hayabusa 2 mission [5] also demonstrated the importance of interacting directly with the surface in order to constrain the mechanical properties. The upcoming ESA Hera mission (launch planned for late 2024; [6]) to the binary asteroid Didymos is going to be a great opportunity to study an even smaller asteroid of 150 m diameter.This work is carried in the framework of the NEO-MAPP project (Near Earth Object Modelling and Payloads for Protection), which aims to provide significant advances in our understanding of the response of NEOs to external forces. In the context of the Hera mission, one goal of the NEO-MAPP project [6] is to better understand the physics of low-velocity collisions, and the mechanical properties of the regolith that composes the surface of asteroids. These points are crucial for understanding the evolution of small body surfaces, the main processes that have shaped the surfaces, and for the design and operations of space missions.Our objective is to use experimental data to obtained in both static and low-gravity trials, to develop a theoretical framework to describe low velocity collisions into granular material under varying gravity conditions.Following the previous work of [1], low-velocity impact experiments were performed using two different projectile shapes (spherical and cubic) and four different granular surface materials (quartz sand of 1.5 mm, 5 mm, and 10 mm glass beads). The projectiles, shown in Figure 1, weigh approximately 1 kg and are made of aluminium. The cubic projectile can be oriented to fall on a face, an edge or a corner. The experiments were performed under both Earth and reduced-gravity conditions (~0.02 &#8211; 0.1g effective gravitational acceleration), using a static laboratory set-up and an Atwood-type drop tower [7], respectively. During all trials, in-situ accelerometers were mounted inside the projectiles to measure the in-situ acceleration profile during the impact.From the recorded data, we extracted three key parameters: the maximum acceleration, the final penetration depth, and the collision duration. Figure 2 shows the measurements for the experiments performed using the spherical projectile at several drop heights, giving a range of collision velocities. Similarly to [1], the data present a quadratic trend for the maximal acceleration. The post-collision penetration depth was found to be linearly-dependent on the collision velocity, and the collision duration was found to be independent of the collision velocity, at least for collision velocity larger than 0.4 m s-1.We employ a collisional model [8] to fit the data, using an empirical force law involving both a hydrodynamic drag force term (h(z)) and a static resistance force term (f(z)). To fit the data, one must assume specific forms for h(z) and f(z), typically that they are constants [9]. However, the scatter of the data makes this assumption difficult to consider. Instead, we reformulate the force-law model into a linear differential equation in kinetic energy [10]. This solution provides a natural way to experimentally measure f(z) and h(z), with no assumptions about the functional form of these terms. The contributions of these two terms are discussed and compared considering the different projectiles, surface materials and gravity conditions. The behaviour of the granular medium in interaction with the probe, in a very low gravity environment, is an important opportunity for mathematical models combining the dynamics of shocks, frictions&#160;and deformations. These models can find applications in various fields of engineering.&#160;[1] Murdoch, N., et al. &#8220;An experimental study of low-velocity impacts into granular material in reduced gravity.&#8221;&#160;Monthly Notices of the Royal Astronomical Society,&#160;468 (2), p. 1259-1272 (2017).[2] Fujiwara, Akira, et al. "The rubble-pile asteroid Itokawa as observed by Hayabusa." Science 312.5778 (2006): 1330-1334.[3] Watanabe, Sei-ichiro, et al. "Hayabusa2 mission overview." Space Science Reviews 208.1-4 (2017): 3-16.[4] Lauretta, D. S., et al. "OSIRIS-REx: sample return from asteroid (101955) Bennu." Space Science Reviews 212.1-2 (2017): 925-984.[5] Saiki, T., Imamura, H., et al. &#8220;The Small Carry-on Impactor (SCI) and the Hayabusa2 Impact Experiment&#8221;. 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