## Abstract

High-resolution images of the surface of asteroid 433 Eros revealed evidence of downslope movement of a loose regolith layer, as well as the degradation and erasure of small impact craters (less than ∼100 meters in diameter). One hypothesis to explain these observations is seismic reverberation after impact events. We used a combination of seismic and geomorphic modeling to analyze the response of regolith-covered topography, particularly craters, to impact-induced seismic shaking. Applying these results to a stochastic cratering model for the surface of Eros produced good agreement with the observed size-frequency distribution of craters, including the paucity of small craters.

The Near Earth Asteroid Rendezvous (NEAR) Shoemaker mission to the asteroid 433 Eros revealed a heavily cratered surface, covered with a veneer of loose regolith and peppered with numerous boulders (Fig. 1A). This regolith layer displays evidence of downslope movement in several forms (*1*–*4*): debris aprons at the base of steep slopes, bright streaks of freshly exposed material on crater walls, the pooling of regolith in topographic lows, a large number of degraded craters, and a scarcity of craters less than ∼100 m in diameter. One plausible explanation for these phenomena is seismic reverberation of the asteroid after impact events, which is potentially capable of destabilizing slopes, causing regolith to migrate downslope, and degrading or erasing small craters (*1*–*5*).

Impact-induced seismic shaking of an asteroid in the 1 to 100 km size range is an attractive mechanism for three reasons. First, the small volume of the asteroid keeps the concentration of seismic energy high even after the seismic energy injected by an impact has completely dispersed throughout the body (*6*). Second, the very low surface gravity of the asteroid (surface gravity *g*_{a} ≈ 10^{–3} to 10^{–5} *g*_{earth}) permits small seismic accelerations to destabilize material resting on slopes (*6*), where destabilization begins at 0.2 to 0.5 *g*_{a} for loose regolith (*7*). Third, S-type asteroids such as Eros—composed of silicate rock, residing in a vacuum, and having an extremely low moisture content—should have very low seismic energy attenuation rates (*8*).

We illustrate the first and second points above by equating the seismic energy injected by an impactor, which is a small fraction of the impactor's kinetic energy [seismic efficiency η ≈ 10^{–3} to 10^{–5} (*9*)], with the seismic energy necessary to produce accelerations that exceed 1 *g*_{a} throughout the asteroid and destabilize all slopes on the surface. We then solved for the minimum diameter of a stony projectile, traveling at a typical asteroid impact speed [*ν*_{p} = 5 km s^{–1} (*10*)], that meets this condition (Fig. 1B) (*11*). For an asteroid the size of Eros (mean diameter, ∼17 km), the impactor size necessary to achieve global seismic accelerations of 1 *g*_{a} is quite small, ∼ 2 m (0.5 to 10 m) in diameter: far smaller than the size of impactor that would disrupt the asteroid (1.1 to 1.6 km) (*12*, *13*).

The availability of sufficient seismic energy, however, is not enough to show the efficacy of this mechanism: This requires a detailed analysis. The precursor for our work, like that of Cheng *et al.* (*5*), is a classic study of the seismic effects of impacts on lunar surface topography (*14*). Building upon this study, we investigated the process through five modeling steps: (i) finding the typical seismic reverberation signal generated by an impact, (ii) using this signal to synthesize generic impact seismograms for Eros, (iii) computing the response of regolith-covered slopes to these seismic vibrations, (iv) applying this downslope motion to the degradation and erasure of impact craters, and finally, (v) using these results to produce a model for the cratering record on the surface of the asteroid.

First, we found the typical seismic reverberation signal generated by an impact by using the finite differencing *SALES-2* hydrocode (*11*, *15*). Briefly accelerating individual cells in a numerical mesh at the impact point, we then monitored the resulting motion at selected seismograph points (Fig. 2A). The goal was to obtain the seismic reverberation signal from an impact into a highly fractured, heterogeneous target (Fig. 2B) typical of an asteroid body (*16*). This modeling showed that the frequency response of a fractured medium is primarily dependent upon the inherent fracture spacing rather than on the impactor diameter. We therefore chose a fracture structure that produced results consistent with impacts into the upper lunar crust (*14*, *17*), which like Eros is highly fractured from long impact exposure (*8*). The resulting seismograms (Fig. 2B) had a frequency spectrum generally between 1 and 100 Hz, with a peak at about 10 to 20 Hz.

In principle, the same finite differencing methods could be used for the second step, the synthesis of generic seismograms for Eros, but the asteroid is too large for this method to be practical at sufficient frequency resolution. Instead, we took advantage of Eros's highly fractured global structure (*4*, *18*). Lunar seismic studies show that the dispersion of seismic energy in a fractured, highly scattering medium is a diffusion process, such that the seismic energy density ≤ in a fractured asteroid should obey the equation (*8*) (1) where *t* is the time, *K*_{s} is the seismic diffusivity, ▽^{2} is the Laplacian operator, ω is the seismic frequency, and *Q* is a dissipation parameter (seismic quality factor). We solved Eq. 1 in Cartesian coordinates (*11*) and approximated the initial spatial energy distribution as a delta function—reasonable because the impactors considered here are much smaller than the target asteroid. This solution was used to obtain a mean or “global average” seismic energy profile as a function of time for each impactor diameter (0.5 to 500 m). Local seismic effects, such as enhanced vibration close to the impact site or unusual effects due to the irregular shape of Eros, were not included.

The synthesis of a generic seismogram begins with the fraction of impactor energy that is converted to seismic energy [using η = 10^{–4} (*9*)], then the division of this energy into frequency components in accordance with power spectra obtained from the hydrocode simulations. Moving through time, we used the solution to Eq. 1 to simulate the buildup of seismic energy by diffusion and the loss of seismic energy by attenuation for each frequency component, which were then combined using inverse Fourier analysis to produce a final seismogram level (Fig. 2, C and D). To be conservative, the assumed values of *Q* = 2000 and *K*_{s} = 0.25 km^{2} s^{–1} are about half of the values derived for the fractured lunar crust (*8*).

Next, we applied these synthetic seismograms to a numerical model of regolith resting on a variety of slopes under Eros gravity conditions. This was done using the Newmark slide-block method (*14*), which approximates the motion of a mobilized regolith layer by modeling the motion of a rigid block resting on an inclined plane (*11*). Computing the accelerations imparted to the block by gravity and the seismically shaken slope yielded an overall block (layer) displacement, with both hopping and sliding permitted. Although the model can include a regolith-layer shear strength (cohesion), as a first approach we assumed the layer was a uniform, noncohesive, Coulomb material, such as dry sand. This slide-block modeling showed that for Eros, global downslope motion on all slopes (2° to 30°) begins at impactor diameters of about 1 to 2 m, agreeing well with our previous analytical calculation of ∼2 m. Figure 3A shows the resulting downslope volumetric flux per impact *q*_{i} as a function of slope ▽*z* (where *z* is elevation) for a portion of the impactor diameter range (5 to 50 m).

The motion represented in Fig. 3A is typical of nonlinear, disturbance-driven, downslope flow and is described by (*19*) (2) We used Eq. 2 to fit each downslope flux curve and determine the downslope diffusion constant per impact, *K*_{i} (Fig. 3B, solid circles), and the critical slope, *S*_{c} (*20*). These fits yielded a set of diffusion constants *K*_{i} as a function of impactor diameter *D*_{p} that follows power-law relationships that fall into two regimes: *D*_{p} = 1 to 4 m, where sliding occurs in stick-slip fashion, and *D*_{p} > 4 m, where sliding occurs in continuous fashion (Fig. 3B, solid lines).

Next, we applied this downslope motion to the degradation and erasure of impact craters. If the slope is small, Eq. 2 becomes nearly linear with respect to slope (*q*_{i} ≈ *K*_{i} ▽*z*). This linearization permits us to use the results from the previous step in a model of topographic modification, described by a diffusion equation in terms of elevation *z* (*21*) (3) where *K*_{d} is a downslope diffusion constant per unit time. We solved Eq. 3 in cylindrical coordinates (*11*) and gave the initial topography the shape of an axially symmetric, fresh impact crater (*22*) with a depth to diameter (*d*/*D*) ratio of 0.2. This solution simulates the degradation (filling) of a crater because of seismic shaking (Fig. 3C), given our downslope diffusion constants. The most important term in the solution is a relaxation term (4) *h* is the mobilized regolith layer thickness, *k* is the spatial wave number, and *n* is the number of impacts. In our modeling, the crater was considered to be erased after six 1/*e* decays, *R* = *e*^{–6} = 0.0025, which gave a *d/D* ratio of 0.0005.

Our Bessel function form of the initial crater shape consists of a very narrow (Gaussian) range of spatial wave numbers *k* that peak at *k*_{0} = 4/*D*. The point at which a crater becomes erased can be approximated by substituting *k*_{0} for *k* in the relaxation term (Eq. 4) and equating the arguments –6 = –*Kh*(4/*D*)^{2} to give (5) for the erased state. When applied to the solution to Eq. 3, this corresponds to a *d/D* ratio of 0.0041: at least twice as flat as what could reasonably be counted from NEAR images, *d/D* = 0.01 (*4*). Equation 5 thus permits an assessment of crater seismic damage, as downslope diffusion accumulates over time (through multiple impacts) until final crater erasure.

Finally, we used these results to model the evolution of the crater size-frequency distribution on Eros and show how seismic modification changes the overall crater population (*11*). This model uses Monte Carlo techniques (*23*) to populate a surface with craters as a function of time, allowing them to be obliterated by the effects of subsequent impacts: super-positioning, blanketing by impact ejecta, and seismic shaking. The modeled impactor population matches that of the main asteroid belt (*24*), where Eros has spent most of its lifetime (*25*). The resulting crater size-frequency distributions are shown in Fig. 4, in which the observed (*3*, *4*) and modeled distributions are in good agreement at a Main Belt surface age of 400 ± 200 million years (My) (*26*).

The reduced numbers of small craters is a result of seismic erasure, causing lower equilibrium values than would otherwise be expected. This equilibrium point is a sensitive function of the assumed thickness of the mobilized regolith layer *h* (in Eq. 5). By varying this parameter, we found a best fit corresponding to *h* ≈ 0.1 m, with actual values for *h* perhaps as high as a few meters. This thickness is considerably less than the estimates of an average regolith thickness of 20 to 40 m from the NEAR observations (*4*). We infer from this that much of the regolith layer possesses a depth-dependent porosity and cohesion gradient, perhaps due to compaction from seismic shaking. This would produce lower porosity and higher cohesion with increasing depth (*4*). Such a gradient was observed for the lunar regolith, causing the regolith to preferentially slide at shallow critical depths (*14*).

This modeling produces good agreement with the empirical observations, but there is considerable uncertainty with regard to the asteroid's actual seismic and regolith properties. We have based our results on values appropriate to the one impact-generated environment that has been studied in detail: the upper lunar crust. Even with these uncertainties, however, this work constrains these properties and effectively demonstrates the ability of impact-induced seismic shaking of Eros to destabilize slopes, cause regolith to migrate downslope, and degrade or erase small craters.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/306/5701/1526/DC1

Materials and Methods