Assistant Research Professor
This view of asteroid 433 Eros is part of an image mosaic taken in the early hours of October 26, 2000, during NEAR Shoemaker's low-altitude flyover of Eros. Taken while the spacecraft's digital camera was looking at a spot 8 kilometers (5 miles) away, the image covers a region about 800 meters (2600 feet) across. Rocks of all sizes and shapes are set on a gently rolling, cratered surface. Locally, fine debris or regolith buries the rocks. The large boulder at the center of the scene is about 25 meters (82 feet) across. The purpose of this research is to determine to what degree does impact-induced seismic shaking contribute to the modification and evolution of surfaces such as this one.
Impact-induced seismic vibrations have long been suspected of being an important surface modification process on small satellites and asteroids. In this study, we use a series of linked seismic and geomorphic models to investigate the process in detail. We begin by developing a basic theory for the propagation of seismic energy in a highly fractured asteroid, and we use this theory to model the global vibrations experienced on the surface of an asteroid following an impact. These synthetic seismograms are then applied to a model of regolith resting on a slope, and the resulting downslope motion is computed for a full range of impactor sizes. Next, this computed downslope regolith flow is used in a morphological model of impact crater degradation and erasure, showing how topographic erosion accumulates as a function of time and the number of impacts. Finally, these results are applied in a stochastic cratering model for the surface of an Eros-like body (same volume and surface area as the asteroid), with craters formed by impacts and then erased by the effects of superposing craters, ejecta coverage, and seismic shakedown. This simulation shows good agreement with the observed 433 Eros cratering record at a Main Belt exposure age of 400 +/- 200 Myr, including the observed paucity of small craters. The lowered equilibrium numbers (loss rate = production rate) for craters less than ~100 m in diameter is a direct result of seismic erasure, which requires less than a meter of mobilized regolith to reproduce the NEAR observations. This study also points to an upper limit on asteroid size for experiencing global, surface-modifying, seismic effects from individual impacts of about 70-100 km (depending upon asteroid seismic properties). Larger asteroids will experience only localized (regional) seismic effects from individual impacts.
Indications of downslope regolith motion on 433 Eros, imaged by the NEAR-Shoemaker spacecraft, in the form of: (A) bright steaks of freshly exposed material on a large crater wall, as the darker material moves downslope (MET 154409710, 14.79 W, 14.21 S, 2.67 m/pixel); (B) talus cones and debris avalanches emanating from a steep scarp (MET 132929106, 284.42 W, 41.31 N, 4.60 m/pixel); (C) a debris apron extending into a highly eroded crater from a rise at the top of the image, and a thick layer of regolith gently encroaching into the same crater from the bottom of the image (MET 153667920, 260.29 W, 14.89 S, 3.41 m/pixel); and (D) the collection of regolith (and scattered boulders) in topographic lows, several degraded (softened) craters, a few barely visible 'ghost' craters, and a general lack of small craters, particularly in the smooth areas on the right (MET 154251925, 24.81 W, 6.28 S, 2.10 m/pixel).
(Lower curves) Minimum stony impactor diameter necessary to cause 1 g accelerations throughout the volume of a stony asteroid of given diameter (destabilizing all regolith-covered slopes on the surface), for seismic frequencies f of 1 Hz (dashed), 10 Hz (dot-dashed), and 100 Hz (dot-dot-dashed). (Upper solid curves) Minimum stony impactor diameter necessary to cause disruption of a stony asteroid of the given diameter, calculated per Melosh and Ryan, 1997 (top) and Benz and Asphaug, 1999 (bottom). The region bounded by these curves, shown in gray, highlights the wide range of impactor sizes that can cause global seismic effects on an asteroid without disrupting it. This plot, however, does not include the important effect of seismic attenuation as the energy propagates throughout the asteroid volume.
Two examples of evidence for a joint and fracture structure underlying the regolith layer on 433 Eros, imaged by the NEAR-Shoemaker spacecraft, in the form of: (A) several structurally controlled, `square' impact craters (MET 132151598, 218.91 W, 16.64 S, 5.57 m/pixel); and (B) a network of criss-crossing ridges and grooves, with a few, small, structurally controlled craters (MET 136266921, 218.72 W, 42.00 N, 4.58 m/pixel).
Schematic view of the analogy between the upper lunar crust (investigated via the Apollo seismic experiments) and a proposed fractured asteroid structure by Sullivan et al., 1996. If each began as monolithic rock, exposure to similar impactor populations should produce similar fracture structures within each: (1) a thin, comminuted regolith layer on the surface; (2) a highly fractured mixture of rock and regolith beneath (a `megaregolith' layer); and (3) a decreasing gradient of fractured bedrock below. In the case of the upper lunar crust, this fracture structure extends to depths of about 20-25 km, but in the case of asteroids, this fracture structure should extend throughout the body. Based on Fig. 4 of Dainty, et al., 1974.
Examples of one artificial and two natural impact seismograms recorded by the Apollo 12 seismic experiment (ALSEP) long-period (LP) instrument in 1969. Note the smooth, teardrop-shaped, seismic amplitude envelopes indicative of a diffusion process, and the long `coda' tails (long duration vibrations) indicative of an extremely low seismic attenuation rate--primarily due to a near zero moisture content and vacuum conditions. Based on Fig. 2 of Latham et al., 1970.
A plot of the normalized seismic energy density for three minutes following an impact for (dotted) an impact and receiver located in the centers of perpendicular faces on a cubical target and (solid) an impact and receiver located on opposite corners of a cubical target. Note the transient seismic energy peak in the dotted curve as compared to the gentle build-up and decay of seismic energy shown in the solid curve.
(Lower curves) Minimum stony impactor diameter necessary to cause 1 g accelerations throughout the volume of a stony asteroid of given diameter (destabilizing all regolith-covered slopes on the surface), for seismic frequencies f of 1 Hz (dashed), 10 Hz (dot-dashed), and 100 Hz (dot-dot-dashed). In this case, an estimate of seismic attenuation has been included (compare to Fig. 2), such that each seismic frequency has a finite distance over which it will be effective, with lower frequencies penetrating further than higher frequencies. Note that a single impact will produce a seismic frequency spectrum containing all of these frequencies (1-100 Hz). (Upper solid curves) Minimum stony impactor diameter necessary to cause disruption of a stony asteroid of given diameter, calculated per per Melosh and Ryan, 1997 (top) and Benz and Asphaug, 1999 (bottom). The region bounded by these curves, shown in gray, continues to show a wide range of impactor sizes that can cause global seismic effects on small to medium sized asteroids without disrupting them. However, this analytical calculation does point to an upper asteroid size limit of about ~100 km for global seismic effects from impacts. Larger asteroids will experience localized seismic effects only.
(Left background) This diagram illustrates the basic components of an impact seismic source, modeled in cylindrical coordinates as part of a hydrocode simulation. Nodal velocity vectors are shown during the initial acceleration of two selected mesh points, one downward and axial (-z direction) and one radial and on the surface (r direction). (Left foreground) The resulting pressure contours in the hydrocode mesh after 0.04 sec. showing a hemispherical, expanding body (P) wave, and an advancing surface or Rayleigh (R) wave. (Right) Theoretical (dashed) and hydrocode produced (solid) surface seismograms at 0.5 km distance from an impact into a homogeneous rock half-space, showing a weak body (P) wave arrival at 0.25 sec. and a strong surface (Rayleigh) wave arrival at 0.5 sec. Based on Fig. 2 of Richardson et al., 2004.
(A) Cross-sectional view of an axially-symmetric hydrocode mesh showing the pressure contours produced by seismic waves propagating through a `fractured,' 1-km-diameter, spherical, rock target following an impact. The wave propagation is a mix of unreflected, reflected, and multiply reflected wave-fronts, such that the propagation of seismic energy is beginning to approach the behavior of a diffusion process. (B) Hydrocode produced surface seismograms at 90 deg. away (half-way around the spherical model surface) from an impact into this 'fractured,' 1-km-diameter, spherical, rock target. Note that although the vibrations are extremely mixed in nature, produced by multiple wave-front arrivals (both body and surface waves), an amplitude `envelope' can be discerned in each, particularly in the vertical motion. Based on Fig. 2 of Richardson et al., 2004.
(Top row) Normalized vertical power spectra (for seismic motion) from a 4-m impactor striking a 1-km-diameter, homogeneous rock sphere (left) and fractured rock sphere (right) at 100 m sec^-1. The slow speed helps to maintains stability in a Lagrangian mesh for at least 3 seconds. (Bottom row) Normalized vertical power spectra (for seismic motion) from a 60-m impactor striking a 1-km-diameter, homogeneous rock sphere (left) and fractured rock sphere (right) at 100 m sec^-1. The smaller impactor produces an inherently higher frequency spectrum than that produced by the larger impactor, when no fractures are present. However, when fractures are present these spectra show that the blocks within the fractured mesh act as a crude band-pass filter to the injected signal, preferentially passing those frequencies with wavelengths near the harmonics associated with the typical fracture spacing (about 10-80 Hz).
(A) The first 6 minutes of a synthetic seismogram for the `far-side' of Eros following the strike of a 10-m stony impactor, showing an asymmetrical, mixed-phase, reverberation signal (compare to Fig. 5). (B) The corresponding seismic accelerations (gray) for the seismogram shown in (A). The two dashed lines indicate the approximate surface gravity magnitude (g = 5 mm sec^-2), indicating that seismic accelerations that exceed 1 g last for about 5 minutes following this impact. Based on Fig. 2 of Richardson et al., 2004.
(Left) A basic illustration of the Newmark slide-block model. The regolith layer resting on a slope is represented by a rigid block resting on a inclined plane. Forces on the rigid block include surface gravity (static loading), seismic accelerations (dynamic loading--applied by the inclined plane), and frictional forces(both static and dynamic). Ballistic launching of the block (layer) is also permitted and tracked in this model. (Upper right) Overall motion of an asteroid regolith layer (1 m depth) resting on a 10 deg. slope, under the seismic shaking conditions produced by a 10-m impactor on the `far-side' of an Eros-like asteroid. Note that the motion involves vertical hopping in addition to horizontal sliding in the asteroid's very low gravity field (~ 5 mm sec^-2). (Lower right) Close up view of the vertical motion of the inclined plane (dotted) and slide-block (solid), showing the vertical launching and `flight' of the block (regolith layer) in detail.
(Lower curves) A plot of the size of impactor necessary to produce greater than 1 g accelerations (vertical launching) of an h = 1 m thick model regolith layer resting on a slight 2 deg. slope, for a variety of asteroid sizes, seismic properties, and regolith types. For (A) we use 'nominal' seismic propagation conditions (eta = 10^-4, v_s = 3 km sec^-1, Q = 2000) and a non-cohesive regolith layer, while for (B) we used more restrictive seismic propagation conditions (eta = 10^-6, v_s = 3 km sec^-1, Q = 1000) and the cohesive regolith model described in Sec. 3.1. In both cases, a variety of seismic diffusivity values are tested, from K_s = 0.125 km^2 sec^-1 to K_s = 2.000 km^2 sec^-1, corresponding to mean free paths for scattering l_s of: 2.000 km (solid), 1.000 km (dashed), 0.500 km (dot-dashed), 0.250 km (dot-dot-dashed), and 0.125 km (dot-dot-dot-dashed). (Upper solid curves) Minimum stony impactor diameter necessary to cause disruption of a stony asteroid of given diameter, calculated per per Melosh and Ryan, 1997 (top) and Benz and Asphaug, 1999 (bottom). The region where global surface effects can occur from a single impact without disrupting the asteroid is shown in gray. This modeling agrees well with our previous analytical calculations (Fig. 7) and indicates an upper asteroid size limit of about 70-100 km for global seismic effects from impacts.
(A) Newmark slide-block model results for six impactor sizes, plotting volumetric flux per impact q_i (m^3 m^-2) as a function of slope gradient (delta-z) and displaying the non-linear relationship typical of disturbance-driven flow (Roering et al., 1999). (B) Downslope diffusion constants per impact K_i (m^3 m^-2 or simply m) plotted as a function of impactor size D_p where solid circles show the derived values and solid lines shows a linear least-squares fit to these points. The resulting power-law relationships fall into two regimes: D_p = 1-4 m, where sliding occurs in stick-slip fashion; and D_p > 4 m, where sliding occurs in hop-slip fashion. This plot is compared with the seismic `jump' distances reported in Greenberg et al., 1994 & 1996, which were estimated from surface velocities on a homogeneous, spherical hydrocode model following impact. Based on Fig. 3 of Richardson et al., 2004.
(Left) Vertical cross-sections taken through a 200-m-diameter crater, shown in light-gray (horizon level shown in dark-gray, plotted at four different times and showing its gradual degradation and erasure by impact-induced seismic shakedown on an asteroid having the same volume and surface area as Eros. Complete erasure occurs at a crater age of about 30 Myr in a Main Belt impactor flux. A 20-m-crater under the same conditions, will have a life-time of about 300 kyr. Note the rapid initial degradation while the slopes are still relatively high, followed by a more gradual degradation as slopes flatten. (Right) A field of softened and degraded craters on the surface of Eros, showing a range of morphologies consistent with degradation by seismic shakedown. (MET 138807458, 154.34 W, 6.15 S, 5.12 m/pix)
(A) An inverted cumulative distribution curve showing the mean time between impacts from impactors of a given size (or greater) on a spherical asteroid with a surface area matching that of 433 Eros (1125 km^2). In this work we use an impactor population produced by the modeling described in O'Brien and Greenberg,2005 (solid), which has a cumulative log-log slope of 2.93 for impactor diameters D_p in the range 0.1 m < D_p < 95 m, and shallowing to a cumulative log-log slope of ~1 for impactor diameters D_p > 95 m. Also plotted are the similar impactor populations used by Belton et al., 1992 (dashed) and Greenberg et al., 1994 & 1996,(dotted) to model the cratering records on the asteroids 951 Gaspra and 243 Ida. (B) A plot of the simple 30x scaling-law used in this study for mapping impactor size to crater size (solid), showing it in comparison to strength scaling (Holsapple, 1993), gravity scaling (Holsapple, 1993; Melosh, 1989), and the hydrocode simulations for 951 Gaspra (dot-dashed) from Greenberg, 1993, and 243 Ida (dashed) from Greenberg, 1996. The gray bands represent a variety of target material types, from loose sand to competent rock. Note that the size of impactors used in my stochastic cratering model range from 0.667 m (producing 20 m craters) to 667 m (producing 20 km craters).
Example screens from the stochastic cratering model following 400 Myr of impacts on an Eros-like target body. The area shown in these displays is 34 km X 34 km (1156 km^2), with a model resolution of 20 m X 20 m. Craters are color coded by size (see color bar). The curves shown in the cumulative distribution plots are: geometric saturation (dashed), estimates of empirical saturation at 5% to 10% geometric saturation (dotted), modeled crater distribution (thick solid), and the observed crater distribution on 433 Eros (dot-dashed) from Chapman et al., 2002; Robinson et al., 2002. (top) Crater population after 400 Myr without seismic shaking included in the simulation. Note the empirical saturation level of small craters and a poor match with the observed distribution of craters on Eros. (bottom) Crater population after 400 Myr with seismic shaking included in the simulation. Note the excellent match between the modeled and observed crater distributions, particular with regard to small craters (less than 100 m diameters).
A relative size-frequency distribution plot of 433 Eros craters per square kilometer as a function of crater diameter, showing a favorable comparison between observed (Chapman et al., 2002; Robinson et al., 2002) and modeled values after 400 +/- 200 Myr (Main Belt exposure age). The low abundance of small craters is a result of seismic erasure, causing lower equilibrium values than would otherwise be expected (empirical saturation; (thin dashed line). Symbols are the same as those listed in Table 1 of Chapman et al., 2002. Based on Fig. 4 of Richardson et al., 2004.
Two examples of ponded deposits on 433 Eros, imaged by the NEAR-Shoemaker spacecraft. Note the marked difference in morphology between these ponds and the degraded craters shown Fig. 1(D) and Fig. 15, indicative of different formation processes. The ponds shown here are located on the low surface-gravity (2.5-3.0 mm sec^-2) 'nose' of the asteroid, which also spends a longer than average amount of time near the terminator (light/dark boundary). (A) A beautiful 100 m diameter ponded deposit containing an embedded 25 m boulder. Note the extremely flat surface containing a tiny (few-meter diameter) impact crater (MET 155888598, 179.04 W, 2.42 S, 0.55 m/pixel). (B) A smaller 75 m diameter ponded deposit. Note the difference between the smooth, fine-grained pond surface and the coarse, boulder strewn terrain surrounding the deposit (MET 155888731, 183.88 W, 3.21 S, 0.63 m/pixel).