Supplementary Information Bennu & Ryugu: Diamonds in the sky Tapan Sabuwala1, Pinaki Chakraborty1& Troy Shinbrot2
1Fluid Mechanics Unit, Okinawa Institute of Science & Technology Graduate University, Onna-son, Okinawa JAPAN
2Department of Biomedical Engineering, Rutgers University, Piscataway, NJ USA 1. Analytic model
Figure S1 – Forces acting on a block of debris resting on the surface of a rotating asteroid.
To obtain the critical slope of the surface of a rotating asteroid, we consider the forces acting on a block of debris of massmlocated on the surface at an angle of elevation, = atan / , where( , )are radius and height in cylindrical coordinates (Fig. S1). The forces, as sketched in the inset in Fig. S1, depend on the block’s angle, , and the angle of the surface, , both with respect to the horizontal. Here, is the friction coefficient, the rotation rate of the asteroid, and the gravitational acceleration, which is assumed to be centrally directed. The net normal and tangential forces acting on the block are
= + − 2 [1]
= − + 2 − + [2]
At equilibrium, we set = 0, and obtain an expression for the corresponding critical slope of the surface, , from [1] and [2],
= tan−1 − ++ −+ μr,zr,z . [3]
Here , = 2‹ 2
‹2 is the ratio between centrifugal and gravitational forces and = 2+ 2 is the distance from the surface to center. ‹ and ‹are respectively the radius and gravity at the equator, so at the equator itself = ‹, = 0, and the ratio of outward to inward forces is ‹= 2 ‹‹.
2. Simulations
We conduct simulations using the discrete element method (DEM) as implemented in the freely available package LIGGGHTS1. In DEM, particles are idealized as soft spheres that interact with each other through a parametrized interaction potential2. The motion of particles, subject to interaction forces acting on them, is tracked in time using numerical integration. We model normal interaction between particles using a damped Hertzian spring while tangential interactions follow Coulomb’s friction law. The geometric and material parameters used are listed below.
Particle diameter 0.1 ± 0.005 c
Particle density 2.7 g/cc
Young’s modulus 94 MPa
Poisson’s ratio 0.17
Coefficient of sliding friction 0.6 Coefficient of restitution 0.5 Coefficient of rolling friction 0.2
We assume North-South symmetry and model one half of the spinning asteroid with a frictionless base at the equator. As in previous works3,4,5, our simulations employ a two-layer model: a stiff core in the interior, where the stiffness may be attributed to cohesive forces, surrounded by non-cohesive debris particles. Observations bear out this scenario, suggesting that the surface regolith that sits atop the core is largely non-cohesive and mobile6. The core is comprised of particles whose positions and velocities are updated at each timestep as per the prescribed angular velocity. The mass of the core is computed assuming a porosity of 50%, as given in Ref.
[7].
2.1 Influence of deposited mass
Figure S2 – Effect of deposited mass on simulated asteroid shapes for ‹= 0.7. (a) Results for a final mass of deposited to core particles given by / = 0.75and (b) / = 1.3. The exterior surfaces are as defined in Fig. 1(d) of the main text, and beneath each case we show the relative radius divided by the equatorial radius as a function of latitude, as in Fig’s 2(e)-(f) in the main text.
As mentioned in the main text, we have performed additional simulations to confirm that the effect reported is robust. In the main text, we report results for simulations with / = 1, where is the mass of deposited particles at the end of the simulation and is the constant mass of the core. In Fig. S2, we present additional results with fewer ( / = 0.75) and more ( / = 1.3) deposited particles. In these simulations, all other parameters are identical as in the main text, and as before we use a spherical core that rotates at the same constant rate as before. Fig. S2, which includes plots of relative radius vs. latitude, indicates that increased elevation at poles and equator and decreased elevation at mid-latitudes appears to be insensitive to / . 2.2 Effect of core shape
Figure S3 – Effect of core shape on simulation results. (a) Ellipsoidal core with aspect ratio 0.9, (b) spherical core.
To confirm that the diamond shape is a self-organized phenomenon brought on by deposition and is not an incidental byproduct of flattening that previous work has shown is caused by centrifugation alone, we additionally started with an ellipsoidal (rather than spherical) core having the same aspect ratio as the final shape (pole-to- equator ratio = 0.9). We then deposited an equal mass of particles exactly as before to compare the final shapes on an ellipsoidal and on a spherical core. This comparison is shown in Fig. S3, yielding essentially indistinguishable results for the two cases. We therefore conclude that although surface flow produces both the diamond shape and a pole-to-equator ratio less than 1, the two phenomena are distinct, and deposition leads to a diamond shape with or without a reduced aspect ratio.
3. Comparison with shape of Ryugu
In the main text, we showed comparisons between the predicted and simulated shape profiles and Bennu.
Ryugu is also diamond-shaped, but is significantly more irregular, especially near the poles. This is evident from reconstructions of the two asteroids, as shown in Fig. S4(a)-(b). Bennu’s large scale profile changes little with viewpoint, while Ryugu has substantial depressions (examples abbreviated “void”) and prominences (“bulge”).
This lumpiness is quantified in Fig. S4(c)-(d), which compares the azimuthally averaged profile of Bennu and Ryugu: the grey area indicates 2 about the mean in both panels. Evidently at the poles themselves, there is little variation, but the surrounding voids and bulges produce quite large error bars.
Figure S4 – (a) Four views of Bennu, and (b) four views of Ryugu., both from [8]. Notice that Bennu is substantially more regular than Ryugu, which displays obvious near polar depressions and prominences (abbreviated “void” and “bulge”). Azimuthally averaged (c) Bennu profile, (d) raw Ryugu profile. Gray shows2 about the means, where the large error bars in panel (d) are measures of surface irregularities.
(e) Identical Ryugu data in which variation due to surface irregularities is reduced by shifting the rotation axis 5% of the equatorial diameter.
In Fig. S4(e), we show that we can artificially remove this effect by simply translating the axis of rotation a small distance, here by 5% of the equatorial diameter. This reduces the apparent uncertainty, and in Fig. S5 we show both the raw shape profile (panel (b): “Ryuguraw”) and the shape profile that results from this artificial translation (panel (c): “Ryugushift”). We also reproduce the Bennu case from the main text (panel (a) for comparison.
Evidently in all cases, the prediction from Eq. [1] and the simulation agree well away from the anomalous polar lumps. As expected, the lumps shown in Fig. S4(b) change the raw profile near the pole, and if these are artificially removed by the shift indicated, agreement is equally good everywhere on the asteroid.
Figure S5 – Comparison between Eq. [1] using = 0.999, = 0.4and azimuthally averaged simulation with (a) Bennu, (b) Raw Ryugu data, and (c) shifted Ryugu data.
References:
1 Kloss, C. et al., “Models, algorithms and validation for opensource dem and cfd–dem,” Prog. Comp. Fluid Dyn. 12 (2012)140–52.
2 Cundall, PA & Strack, ODL, “A discrete numerical model for granular assemblies,” Geotechnique, 29 (1979) 47–65.
3 Cheng, B. et al., “Reconstructing the formation history of top-shaped asteroids from the surface boulder distribution,”
Nature Astro. 5 (2021) 134-8.
4 Scheeres, DJ, “Landslides and mass shedding on spinning spheroidal asteroids,” Icarus 247 (2015) 1-17.
5 Walsh, KJ, Richardson, DC & Michel, P/ “Spin-up of rubble-pile asteroids: Disruption, satellite formation, and equilibrium shapes,” Icarus 220 (2012) 514-29.
6 Arakawa, M. et al., "An artificial impact on the asteroid (162173) Ryugu formed a crater in the gravity-dominated regime,"
Science 368 (2020) 67-71.
7 Barnouin, OS et al., “Shape of (101955) Bennu indicative of a rubble pile with internal stiffness,” Nature Geosci. 12 (2019) 247–52.
8 Bennu shape data are here:https://www.asteroidmission.org/updated-bennu-shape-model-3d-files/
Ryugu shape data are here: https://www.darts.isas.jaxa.jp/pub/hayabusa2/paper/Watanabe_2019/README.html
