Deriving frictional properties of regolith during small body landing

Introduction:

Understanding the properties of planetary regolith is essential to predict the response of a surface to interactions and correctly interpret the outcome of in-situ operations. In preparation for the interpretation of data from small body missions with surface interaction, we investigate models relating the dynamics of an impact into granular media and the mechanical properties of the surface material. The objective is to constrain frictional properties of regolith using data acquired during landing.

Methods:

Landing on small bodies can be modelled as low-velocity collisions into a granular material in low-gravity conditions. We use experimental data to assess the agreement between empirical scalings from collisional models and frictional properties. Our data are from past impact experiments [3,4] conducted in terrestrial (1g) and low-gravity conditions using a drop tower [5]. The experiments consist of releasing an instrumented spherical projectile from a controlled height into a granular material. From the acceleration profile, the velocity and position are derived by integration. Here, we focus on specific datasets obtained for glass beads and quartz sand of average grain sizes 1.5 mm and 1.8 mm, respectively. The experimental range of impact velocities is 0.2 – 1.2 m/s in 1g and 0.01-0.4 m/s in low-gravity. The average gravity level in the low-gravity experiments is 1.8 m/s² for the glass beads and 0.8 m/s² for the sand.

Results:

We consider the time series for acceleration, velocity, and position of the projectile during impact for various impact velocities. To describe the impact dynamics, the Poncelet collisional model is frequently used, including a drag force composed of two contributions:  quasi-static friction and hydrodynamic drag. In Katsuragi et al. [1] the quasi-static term is linear in depth (z), and in the framework of Sunday et al. [2], this component is modified to highlight a regime transition after a certain depth z₁, at which the quasi-static friction saturates. 

Here we consider a hydrodynamic drag coefficient h(z), and a depth-dependent quasi-static friction force f(z), giving the following equation of motion:

ma = mg – Fd, with Fd = h(z)v² + f(z)     (1)

We plot the total drag force F_d in Fig. 1 for the glass beads and the sand as a function of the velocity squared at different depths in the material. The slope of each linear fit is the hydrodynamic coefficient h(z), and the intercept is the quasi-static friction f(z). We represent the estimated quasi-static friction force against depth in Fig. 2.

 

 

In 1g, the quasi-static friction force increases linearly until the transition depth where it becomes roughly constant, consistent with [2]. Using this model, the transition depth z₁ and the saturated friction force f₀ can be determined (Fig. 2). As expected, the sand is more frictional than glass beads. The derived parameters (f₀, z₁) are also consistent with previous estimates using a different method [4].

In low-gravity, the quasi-static friction is considerably lower and approximately constant with depth, in agreement with previous studies showing that the quasi-static regime is significantly reduced in low-gravity [2-4]. The consequence is that we cannot identify a clear transition depth, nor fit the quasi-static force parameters.

Different empirical scalings from impact models have been proposed to relate the drag force coefficients to the angle of repose of the material. We show in Eq. 2 the scaling from [6] for the coefficient of friction (µ_f) using the coefficient of the quasi-static term k, and in Eq. 3 the scaling from [2] using the equivalent f₀/z₁ term. 

We find that the scaling of [6] (Eq. 2) overestimates µ_f. However, when using the model of [2] (Eq. 3) accounting for the regime transition, the 1g experimental results provide results consistent with measured values of µ_f. In low-g, the use of these scalings is no longer possible given the absence of the quasi-static friction regime. Therefore, to retrieve frictional properties in low-gravity, scalings based on the hydrodynamic component need to be used. 

Conclusions and perspectives:

We quantify the drag force in low velocity impact experiments into granular material in both 1g and low-gravity conditions. Our results show that the quasi-static friction is considerably lower in low-gravity and approximately constant with depth, in agreement with previous studies [2-4]. We also show that the model of Sunday et al. [2], provides the most reliable estimates of the material frictional properties in 1g. However, the same approach cannot be applied in low-gravity due to the absence of the quasi-static friction term. This analysis will be expanded to additional datasets, in order to further examine the influence of friction, grain size, and gravity levels and we will also address the challenge of retrieving mechanical properties from a single impact event. Finally, new experimental data will be collected from the future variable-gravity tower at ISAE-SUPAERO [8] and will bring us means to further investigate the mechanics of regolith in low-gravity.

Acknowledgements:

We acknowledge funding support from CNES (in the context of Hera and MMX rover/wheelcam), from the French ANR Tremplin-ERC ‘GRAVITE’, and from the European Research Council (ERC) GRAVITE project (Grant Agreement N°1087060). 

References:

[1] Katsuragi, H., & Durian, D. J., Nat. Phys., 2007. https://doi.org/10.1038/nphys583

[2] Sunday, C. et al., A&A, 2022. https://doi.org/10.1051/0004-6361/202142098

[3] Murdoch, N. et al., MNRAS, 2017. https://doi.org/10.1093/mnras/stw3391

[4] Murdoch, N. et al., MNRAS, 2021. https://doi.org/10.1093/mnras/stab624

[5] Sunday, C. et al., Rev. Sci. Inst., 2016. https://doi.org/10.1063/1.4961575

[6] Katsuragi, H., & Durian, D. J., PRE, 2013. https://doi.org/10.1103/PhysRevE.87.052208

[7] Kang, W. et al., Nat. Comm., 2018. https://doi.org/10.1038/s41467-018-03344-3

[8] Wilhelm, A. et al., EPSC-DPS, 1355, 2025.