Seismic Waves in the Asteroid Environment - Impactor Momentum Paul Sánchez1and DanielJ. Scheeres2, 1Colorado Center for Astrodynamics Research, University of Colorado Boulder, 3775 Discovery Dr,Boulder, CO 80303, 2Ann and H.J. Smead Aerospace Engineering Sciences, University of Colorado Boulder, 3775Discovery Dr, Boulder, CO 80303 (diego.sanchez-lana@colorado.edu).

By means of numerical simulations we investigateimpact generated seismic wave transmission in granu-lar media under extremely low pressure. This mimicsthe conditions in the interior of asteroids. We find thatthe wave induced pressure depends on the momentumof the impactor. This is an update and continuation toour abstract from 2020 [1].

Introduction: Experimentally, it is known thatseismic waves speed in a granular medium is relatedto the overburden pressure (P ) applied to it [2, 3].The theory that has been developed is called Effec-tive Medium Theory (EMT) and is based in consid-ering the media as elastic, thus removing the difficul-ties presented by particle size and shape. This the-ory, based on the Hertzian laws for contacts, predictsa sound velocity that has a P

16 dependency, whereas

the experiments have found this is true only for highenough pressures and that at low pressures the depen-dency changes to P

14 . In fact, given that the sound

speed for P and S waves are related to the elastic con-stants of the material, in the long wave limit, these canbe written as:

vp =

√K + 4/3µ

ρ(1)

vs =

õ

ρ(2)

where K is the bulk modulus, µ is the shear modulusand ρ is the density of the system. However, most ex-periments have been limited to kPa and MPa pressuresas they have been carried out under Earth’s gravity.

Small planetary bodies (asteroids, comets and smallmoons) produce gravitational fields in the milli- andmicro-g regimes and, as a consequence, their interiorpressure vary from zero (on their surface) to just a fewPascals or tens of Pascals in their innermost regions.

Soft-Sphere DEM: The simulation program thatis used for this research applies a Soft-Sphere Dis-crete Element Method (SSDEM) [4], implemented asa computational code to simulate a granular aggre-gate [5, 6, 7, 8]. The particles, modelled as spheresthat follow a predetermined size distribution, inter-act through a soft-repulsive potential when in contact.This method considers that two particles are in contactwhen they overlap. When this happens, normal andtangential contact forces are calculated [9]. The formeris modelled by a hertzian spring-dashpot system and isalways repulsive, keeping the particles apart; the latter

is modelled with a linear spring that satisfies the localCoulomb yield criterion. The normal elastic force ismodelled as

~fe = knξ3/2n̂, (3)

the damping force as:

~fd = −γnξ̇n̂, (4)

and the cohesive force between the particles is calcu-lated as

~fc = −2πr21r

22

r21 + r22

σyyn̂ (5)

where r1 and r2 are the radii of the two particlesin contact, σyy is the tensile strength of this contact,which is given by a cohesive matrix formed by the(non simulated) interstitial regolith [10], and r̂12 is thebranch vector between the centres of these two par-ticles. Then, the total normal force is calculated as~fn = ~fe +~fd. In these equations, kn is the elastic con-stant, ξ is the overlap of the particles, γn is the damp-ing constant (related to the dashpot), ξ̇ is the rate ofdeformation and n̂ is the vector joining the centres ofthe colliding particles. This dashpot models the energydissipation that occurs during a real collision.

The tangential component of the contact force mod-els surface friction statically and dynamically. This iscalculated by placing a linear spring attached to bothparticles at the contact point at the beginning of thecollision [9] and by producing a restoring frictionalforce ~ft. The magnitude of the elongation of this tan-gential spring is truncated in order to satisfy the localCoulomb yield criterion |~ft| ≤ µ|~fn|.

Procedure: Here, we have chosen to use materialparameters close to those of basalt so that the resultsare relevant for asteroids; the grains are spherical so

Figure 1: Simulation setup of a randomly packed ag-gregate; the redder the particle, the greater its verticalspeed component (P=0.1 Pa).

1850.pdf52nd Lunar and Planetary Science Conference 2021 (LPI Contrib. No. 2548)

that the results can also be related to sound transmis-sion theory. We use 3120 grains with diameters be-tween 2-3 cm that follow a uniform, random distribu-tion for the randomly packed grains (RCP) (see Fig. 1)and 3570, 2.5 cm grains for the crystalline packing(HCP). Their density is 3200 kg m−3, Young modulusis 7.8×1010 N m−2, Poisson ratio is 0.25 [11]. Twocoefficients of restitution are used: 0.1(settling) and0.5 (wave transmission) to reduce the settling time.

The change in particle number was required so thatthe height of the columns remained the same for allsimulations. The different particle arrangements im-plied different filling fractions and so, different bulkdensities. However, we chose to keep the particle den-sity the same and let the bulk density change as thismore closely represents realistic laboratory conditions.

The particles are contained in a box with a solidbottom, horizontal periodic boundary conditions anda moving top that allows us to impose a very well de-termined pressure to the system. Though the particlesinitially settle under Earth’s gravity, this is removed ata later stage to avoid pressure gradients. The height ofthe settled system is approximately 82 cm (see Fig. 1).

We will initiate the wave with the piston; we willchange its thickness so changing its mass, we will alsochange its impact velocity. The thickness of the pis-ton was set to: 2, 4 and 8 cm; the impact speed was:0.1, 0.2, 0.4, 0.8 and 1.6 m/s. The nominal overburdenpressure was kept at 1000 Pa to minimise the settlingtime and to keep the accuracy of the numerical method.The system is then divided into horizontal slices 5 cmthick to monitor energy transmission. The kinetic en-ergy of each slice is calculated in order to observe thewave transmission. Data is collected every 5×10−5s after the wave is started and this is done for 0.015s. Sound speed is measured for the peak of the wavewhen it reaches the 2nd slice from the bottom.

Results: Figs. 2 show semi-log plots of the pressureinduced wave Pw vs. the momentum of the moving top(piston). We have also included three trend lines ineach plot to make clear that the relationship betweenPw and piston momentum is not linear. Notice thatthough we have 15 data points for each plot only 7 areevident in them as some of them are so close that theyoverlap. Meaning that, as long as the momentum ofthe impactor is the same, Pw will be the same. So then,for systems with the same overburden pressure, wavesproduced by impactors with the same momentum willbe transmitted at the same speed. This is true for RCPand HCP systems. We also tried to do the same anal-ysis relating Pw to the energy of the impactor, but thisdidn’t render any evident relationship.

Unfortunately, from the fitting of our data, it is diffi-cult to discern which non-linear fitting curve should be

y=4831.6xR²=0.98341

y=62.567x2+3637.4xR²=0.99954

y=3191.9x1.1302R²=0.99907

0

50000

100000

150000

0.1 1 10 100

Indu

cedPressure[P

a]

PistonMomentum[kgm/s]

InducedPressurevs.Pistonmomentum(RCP)

Linear

Polynomial

Power

y=8267.6xR²=0.98389

y=102.89x2+6303.6xR²=0.99874

y=5015.2x1.17R²=0.99912

0

50000

100000

150000

200000

250000

0.1 1 10 100

Indu

cedPressure[P

a]

PistonMomentum[kgm/s]

InducedPressurevs.PistonMomentum(HCP)

LInear

Polynomial

Power

Figure 2: Wave-induced pressure vs. piston momen-tum for the RCP and HCP configurations. Three dif-ferent trend lines have been included as well as theirequations.

preferred. Both lines (polynomial and potential rela-tionships) diverge for higher momenta and so the truedependency should come from a theoretical analysisof seismic waves transmission in granular media. Thisand other results will be presented at the conference.

References: [1] P. Sanchez, et al. (2020) in Lunarand Planetary Science Conference Lunar andPlanetary Science Conference 1685. [2] K. Walton(1977) Geophysical Journal International 48(3):461ISSN 0956-540X doi. [3] P. J. Digby (1981) Journalof Applied Mechanics 48(4):803. [4] P. Cundall(1971) in Proceedings of the International Symposiumon Rock Mechanics vol. 1 129–136 -, Nancy. [5] P.Biswas, et al. (2003) Phys Rev E 68:050301(R). [6] P.Sánchez, et al. (2009) in Lunar and PlanetaryInstitute Science Conference Abstracts vol. 40 ofLunar and Planetary Inst. Technical Report 2228–+.[7] P. Sánchez, et al. (2011) The AstrophysicalJournal 727(2):120. [8] D. P. Sánchez, et al. (2012)Icarus 218(2):876 ISSN 0019-1035 doi. [9] H.Herrmann, et al. (1998) Continuum Mechanics andThermodynamics 10:189 ISSN 0935-117510.1007/s001610050089. [10] P. Sánchez, et al.(2014) Meteoritics & Planetary Science 49(5):788ISSN 1945-5100 doi. [11] B. Gundlach, et al.(2013) Icarus 223:479.

1850.pdf52nd Lunar and Planetary Science Conference 2021 (LPI Contrib. No. 2548)