The Fate of Asteroid Ejecta

Scheeres et al.: The Fate of Asteroid Ejecta 527

527

The Fate of Asteroid Ejecta

D. J. ScheeresThe University of Michigan

D. D. DurdaSouthwest Research Institute

P. E. GeisslerUniversity of Arizona

The distribution of regolith on asteroid surfaces has only recently been measured directlyby in situ observations from spacecraft. To the surprise of many researchers, most of the classicalpredictions for the distribution of asteroid impact ejecta have not rung true, with regoliths appear-ing to be geologically active at small scales on asteroid surfaces. This indicates that significantinsight into geological processes on asteroids may be inferred by detailed studies of the distribu-tion of impact ejecta on asteroids. This chapter has been written to support these future investiga-tions, by trying to identify and clarify all the important elements for such a study, to point tothe recent history of such studies, and to indicate the current gaps in our understanding. Thechapter begins with a discussion of the initial conditions of ejecta fields generated from impactson the asteroid surface. Then the relevant physical laws and forces affecting asteroid ejecta, inorbit and on the surface, are reviewed and the basic dynamical equations of motion for ejecta arestated. Some general results and constraints on the solutions to these equations are given, and aclassification scheme for ejecta trajectories is given. Finally, recent studies of asteroid ejecta arereviewed, showing the application of these techniques to asteroid science.

1. INTRODUCTION

Whether the debris ejected from impacts on asteroidsescapes or reimpacts has important implications for theerosion of asteroids, retention and distribution of regolith,dispersal or reaccretion of fragments after catastrophic dis-ruptions, and the formation of temporary satellites and per-manent moons. Asteroids present complex dynamical envi-ronments because of their low gravitational accelerations,nonspherical shapes, complex geological makeup, and di-verse rotation states. Additionally, critical parameters relatedto the flux and size distribution of impactors and the result-ing initial ejecta fields are only poorly known. Thus thephysics and dynamics of regolith processes are complicatedand not fully understood. Finally, physical observations ofasteroids are only now approaching the resolution necessaryto seriously constrain and delineate between competingtheories of the asteroid environment, making the study ofasteroid ejecta a timely endeavor.

Within the last decade we have obtained closeup picturesof asteroids Gaspra, Ida and Dactyl, Mathilde, and Eros tosupplement earlier images of the martian moons. Radar andtelescopic observations have revealed the shapes and rota-tion states of many more objects. Morphological indicationsof regolith on these asteroids include blocks, landslides,buried craters, and color variations. Observational tests ofdynamical theories include nonuniform regolith, ejecta blockdistributions and asymmetric crater ejecta blankets, rays,

and strings of secondaries. The importance of erasure mecha-nisms, such as seismic shaking (Greenberg et al., 1994) andelectromagnetic forces (Lee, 1996), that compete with dy-namical effects to shape the surfaces of asteroids has re-cently been emphasized by detailed studies of Eros byNEAR Shoemaker.

Rapid advances are expected in our understanding of im-pact cratering on diverse objects through in situ experimentssuch as NASA’s Deep Impact mission to Comet Tempel 1,numerical simulation of ejecta trajectories that employ real-istic shape and gravity models and consider third-body andnongravitational forces, and geological evidence from anal-ysis of spacecraft data. The fundamental motivation for thestudy of asteroid regoliths arises from the meteoritics com-munity and the interpretations of (primordial) asteroid rego-liths as observed in the meteorite database. Future motiva-tion for ejecta studies will include the necessity of providinga complete mechanical understanding of the asteroid envi-ronment. Ultimately, a detailed understanding of ejecta dy-namics will be crucial to characterize the safety of the or-bital environment about asteroids for rendezvous missions,landed operations on asteroid surfaces, and other close-proximity operations. A key issue of concern for any surfaceoperation on an asteroid, such as sampling, will be the trajec-tories of ejecta disturbed and lofted into orbit during rou-tine operations, as disturbed regolith may reimpact on thesurface with speeds on the order of the surface escape speedlong after being dislodged (Scheeres and Asphaug, 1998).

528 Asteroids III

The study of asteroid ejecta is intimately tied to the studyof impacts on asteroids [see the reviews by Asphaug et al.(2002) and Holsapple et al. (2002)], transient and long-termorbital dynamics close to asteroids, and the study of natu-ral and artificial asteroid satellites (Merline et al., 2002).Additionally, the dynamics of asteroid ejecta have manysimilarities to the dynamics of cometary ejecta. In general,methods devised for the study of each area will have ap-plication to both areas.

Out of necessity, this chapter brings together several di-verse areas of asteroid and dynamical science. Ideally, thischapter will serve as a starting point for future investiga-tions into the dynamics of ejecta from the surfaces of smallbodies. As such, we have injected many topics into thechapter, in some instances without detailed descriptions ofthe background theory or its development. Thus, to supple-ment the work described herein, we suggest that the follow-ing textbooks be referenced: Melosh (1989) for an intro-duction to the basic principles and physics of impacts,Murray and Dermott (1999) for an introduction to the or-bital dynamics of natural bodies, and Szebehely (1967) foran introduction to advanced orbital dynamics theory.

2. A BRIEF HISTORY

The study of asteroid ejecta has been considered in thebooks Asteroids and Asteroids II, distributed among chap-ters on asteroid regoliths (Cintala et al., 1979; Housen et al.,1979b; Veverka and Thomas, 1979; McKay et al., 1989) andasteroid satellites (van Flandern et al., 1979; Weidenschil-ling et al., 1989). Yet the study of ejecta and satellite dy-namics about asteroids was only fully validated with thediscovery of Dactyl in orbit about Ida (Belton et al., 1996).In recent years the relevance of this topic has continued togrow, with the rapid rate at which asteroid satellites havebeen discovered (see the chapter by Merline et al., 2002)and recent realizations from the NEAR Shoemaker missionthat the small-scale structure of asteroid surfaces are notwell understood (Veverka et al., 2001; Cheng et al., 2001).

Initial studies of asteroid ejecta, and orbital dynamicsabout asteroids, assumed that their dynamical environmentwas analogous to, and directly scalable from, planetary sat-ellite dynamics (van Flandern et al., 1979; Weidenschillinget al., 1989). However, Weidenschilling et al. (1989) alreadynoted that the potential for complex dynamics close to aster-oids existed and proposed that further studies on this topicbe done. Prior to this, Dobrovolskis and Burns (1980) hadalready performed detailed ejecta trajectory analysis for theasteroids Phobos and Deimos, noting that ejecta trajectorieswere strongly influenced by the rotation state and gravityfield. For these bodies, however, the Mars tidal force is sostrong that there is no direct analogy between those worksand the evolution of ejecta about asteroids.

Detailed dynamical studies of orbital motion about aster-oids has blossomed since the publication of Asteroids II.Initial studies focused on the stability of binary asteroids.Chauvineau et al. (1990a,b, 1991) investigated the stability

of binary asteroid systems relative to the solar tide, jovianperturbations, and collisions in a series of papers, based onthe earlier studies of the dynamics of the “Hill problem”(Hénon, 1969). Hamilton and Burns (1991a,b) investigatedthe limits of stable motion about an asteroid, with a spe-cial interest given to the safety of the planned Galileo flybysof asteroids Gaspra and Ida. These papers, taken together,provide a clear picture of the stability of asteroid binaries,and place constraints on the stability of asteroid ejecta thatmove relatively far from the asteroid. Such studies are stillcontinuing, and additional progress in understanding the dy-namics of trajectories far from an asteroid have been made(Richter and Keller, 1995; Hamilton and Krivov, 1997).

Chauvineau et al. (1993) and Scheeres (1994) initiatedthe study of dynamics in the near-asteroid environment,studying the motion of particles and ejecta close to rotatingellipsoids. These early studies showed that the near-asteroidorbital environment was fundamentally different from theenvironment found in the vicinity of a planet or larger satel-lite. Studies along these lines have continued with the de-tailed analysis of specific asteroid shapes (Geissler et al.,1996; Petit et al., 1997; Scheeres et al., 1996, 1998a, 2000a)and the theoretical analysis of motion in generalized modelsof asteroid gravity fields (Scheeres, 1999). Now this area ofstudy has its first precision set of data with the results of theNEAR Shoemaker mission to asteroid 433 Eros (Yeomanset al., 2000; Miller et al., 2001), the fruits of which are alreadybeing published (Thomas et al., 2001, Robinson et al., 2001).

3. EJECTA GENERATION

The study of asteroid regolith mechanics and the dynam-ics of impact ejecta fields must first concern itself with themechanics of impact ejecta generation. Weidenschilling etal. (1989) noted that there are tight constraints on ejectaspeed before all ejecta immediately escape from the asteroidand into heliocentric space. Indeed, early estimates on rego-lith depth (or lack thereof) on smaller asteroids predictedlittle, if any, retained regolith. This view of asteroid surfaceshas changed with recent observations of asteroids from space-craft and radar. In the following we review some basic re-sults on the ejecta fields resulting from impact events, withan emphasis on the implications of these models for the ini-tial conditions of an ejecta fragment field.

3.1. Mathematical Models andScaling-Law Predictions

One approach to the understanding of generation andredistribution of regoliths on small bodies is through theo-retical modeling of impacts. These models have been guidedby observations of crater ejecta and regolith on the Moonand by the results of numerous laboratory impact experi-ments. Detailed models of regolith emplacement and evolu-tion on small bodies, incorporating quantitative treatmentsof cratering rates and ejecta thickness, were considered byHousen et al. (1979a,b) and Housen (1981). Those models


predicted that for the smallest asteroidal bodies (D < 10 km),nearly all impact ejecta escapes and such objects shouldhave only thin (on the order of 1 mm) coatings of commi-nuted debris. With increasing asteroid size, more ejecta isretained and regoliths are predicted to be thicker, on theorder of hundreds of meters for asteroids ~100 km in diam-eter and larger. In Veverka et al. (1986) the minimum diam-eter for regolith retention was estimated to be 20 km foricy bodies and 70 km for rocky bodies. Now that the sur-faces of several asteroids have been imaged at resolutionsallowing small-scale surface features to be examined indetail, regolith thickness and spatial distribution may bemore directly estimated as a function of asteroid size, shape,and rotation state.

The volume of ejecta material and the mass of the largestfragments excavated from an impact crater on an asteroidmay be assumed to scale with the crater size, although itmust be recognized that factors including target surfacegravity, porosity, layering and structure in the target, andimpact angle may all play a role in complicating the predic-tions of simple scaling relations. The simplest estimates forejecta volume may be made by scaling from the apparentdiameters of craters (e.g., Lee et al., 1996). Photoclinometryapplied to fresh craters on Ida (Sullivan et al., 1996) indi-cates that a diameter D to depth h ratio D:h is ~6:5. Craterson Eros show similar diameter:depth ratios (Veverka et al.,2000). Assuming that craters are spherical segments withdepth h ~ D/6.5 and diameter D, their volume is V ~ 0.06 D3.

With ejecta volumes estimated in this fashion, the totalvolume of material ejected from craters larger than 0.5 kmdiameter on Ida amounts to ~500 km3, which would amountto a regolith layer ~130 m thick, if retained and distributedevenly over the ~3800-km2 surface area of Ida. If Ida re-sponds to impacts as a strong, competent object, then ejectawould have escaped the surface to space and such an esti-mate of regolith thickness is not valid. However, if cratersof this size formed in the gravity regime, preexisting rego-lith might be present, in which case the estimate is a lowerlimit to regolith depth. As pointed out in Hartmann (1978),“Regolith begets regolith.”

From observations of blocks on Ida and a review of pre-vious work of blocks on the rims of lunar craters, Moore(1971) and Lee et al. (1996) derive a general relationshipbetween the largest ejecta block size, L, and crater diam-eter, D, for craters in rocky targets, L ~ 0.25 D0.7, where Land D are in meters. This ejected material ranges in sizefrom the largest ejecta blocks down to dust-sized particles,with a cumulative mass distribution expressed as N(>m) =Cm–b, where b commonly ranges between 0.8 and 0.9, C isa normalizing constant, and m is the cumulative mass frac-tion. A fraction of this material is jetted from the impact siteat high speed or spalled from the near-surface interferencezone of the growing crater at speeds up to half the impactspeed of the projectile (Melosh, 1989, p. 73). With typicalmain-belt impact speeds of ~5 km/s, most of this spalledmaterial immediately escapes the target asteroid. Excava-tion flow speeds are much lower, however, with even the

highest speeds between one-sixth and one-tenth the impactspeed, so that some portion of the crater ejecta may be re-tained on the surface of the asteroid.

3.2. Laboratory Experiments

The results of laboratory-scale impact experiments canprovide a useful guide in understanding the generation ofregoliths on asteroids and estimating the amount of debrisretained on their surfaces or ejected to escape. Indeed,mathematical models as described above rely in part on datafrom these experiments. There is extensive literature onlaboratory experiments of impactors, reviewed by Holsappleet al. (2002). From these laboratory experiments essential,basic relations for ejecta volumes, speeds, ejecta field orien-tations, and asymmetries have been established. Fundamen-tal results from these areas can be found in Gault et al.(1963) and are reviewed by Fujiwara et al. (1989).

Recent evidence suggests that C- and F-type asteroidshave remarkably low densities (in the range of 1.2–1.8 g cm3)and that high porosity probably plays a significant role inlimiting the excavation of debris from, and the disruption ofterrain surrounding, craters on these bodies. One possible ex-planation for this is given in Housen et al. (1999), wherethey used a centrifuge and impacts into porous, highly crush-able silicate materials to experimentally simulate craterformation and ejecta deposition on low-density, porous as-teroids. Their results show that the ratio of ejecta mass (mate-rial deposited outside the crater rim) to crater mass (crater vol-ume multiplied by initial target density) steadily decreasesas the target porosity increases. This is due to ejecta speedsbeing very low in porous materials, and the fact that muchof the crater volume is formed by compression of the targetmaterial as opposed to excavation. Large craters on porousasteroids should exhibit only minor ejecta deposits; centri-fuge experiments at 250 g in material with porosity 70%,the conditions of the formation of largest impact crater onMathilde (Karoo, 33 km diameter), indicates that only 10%of the crater mass will be ejected outside the crater. (For acompeting explanation of the Mathilde craters found usingnumerical experimentation, see the next section.)

Finally, recent laboratory work has shown mineral-spe-cific comminution. This hints that some mineral compo-nents isolated as chondrules or phenocrysts may be ejectedat different speeds, which might be of interest in segregat-ing mineralogically distinct portions of regolith (Hörz et al.,1985; Durda and Flynn, 1999).

3.3. Numerical Experiments

Hydrocode experiments incorporate gravity and what isknown about the fracture mechanics of rock into numericalsimulations of the contact, compression, and excavationstages of an impact. Hydrocode simulations can be used tomodel impacts at scales far too large to be directly accessiblethrough laboratory experimentation (Benz and Asphaug,1999).

530 Asteroids III

Using such hydrocode techniques, Asphaug et al. (1996)modeled the formation of craters on Ida ranging in size from60 m to 8 km, and compared the ranges of ballistic ejectafrom these craters with the area of seismically disturbedregolith surrounding them. The results confirm that ballis-tically emplaced ejecta deposits around small craters on Ida,and by inference, on asteroids of similar size and compo-sition, should be diffuse and widespread, so that bright halosaround small craters could be due to seismic disturbanceof surrounding regolith rather than continuous ejecta blan-kets. Larger-scale impacts create a “megaregolith”-like zoneof intense fracturing within a depth approaching one cra-ter diameter below large craters and appear to be able todeposit a significant amount of debris in irregular blanketsaround them.

Numerical experiments also give insight into the role thathigh porosity may have in modifying the resultant impactejecta field. In contrast to the laboratory results describedabove, Asphaug (2000) shows that high porosity can lead tohigh ejecta speeds, which also matches the observed lack ofimpact blocks on the Mathilde surface. This is due to energyconfinement in the immediate crater fracture zone, due to theinability of the porous asteroid material to efficiently trans-mit the impact energy though the entire body. In Asphaug(2000) it was shown that the crater Karoo on Mathildewould have launched almost all its ejecta faster than 30 m/s,sufficient for escape, if Mathilde was 50% porous.

3.4. Strength and Gravity Regimes

Two of the most important questions regarding regolithgeneration and ejecta escape are (1) how much ejecta iscreated during any given impact, and (2) at what speeds theejecta are launched. Calculation of crater volumes and thevelocities of the ejecta expelled relies on scaling the out-comes of laboratory and field experiments to vastly differentsizes and gravitational accelerations. The theoretical basisfor such scaling is dimensional analysis, introduced by Hols-apple and Schmidt (1982) and Housen et al. (1983) (seereview by Holsapple, 1993). This approach groups variablesinto dimensionless ratios to reduce the complexity of arbi-trary expressions relating impactor and target properties tocrater volumes or ejecta speeds. In cases where the numberof variables is one greater than the number of dimensions(mass, length, time), the expression can be reduced to asimple power-law relationship between the variables.

By this analysis, the volume of a crater produced by animpactor of a given mass scales neither with the energy northe momentum of the impact, but varies according to apower law with an exponent between 1 (momentum) and2 (energy). For strengthless targets, or in the case of largeimpacts in the gravity regime, the volume V of a cratercreated by an impactor of diameter D and speed U is givenby (Holsapple and Schmidt, 1982)

V(D) = Ag–α U2α D3 – α (1)

where A is a constant that includes the densities of the pro-

jectile and the target [see Geissler et al. (1996) for a spe-cific application], g is the gravitational acceleration, and α isa scaling exponent that depends on the target properties butfalls in the range of 3/7 to 3/4. Similar arguments (Housenet al., 1983) suggest that ejecta follow a power-law speeddistribution with an exponent ev = 6α/(3 – α). When thetarget has substantial cohesive strength, or in the limit ofsmall impactors or small target sizes, the volume of thecrater scales linearly with the mass of the impactor. This isthe strength regime, exemplified by shooting at boulderswith a rifle. Ejecta velocities tend to be higher than those ofgravity-dominated craters but the volume of material exca-vated is much lower. As the size of the impact increases,even cohesive materials gradually transition to the gravityregime due to the scale- and strain-rate-dependence of mate-rial strength. The transition between strength- and gravity-dominated impacts depends on the strength and gravity ofthe target, but for asteroids that are a few tens of kilometersin diameter the transition is expected to occur at crater di-ameters in the range of 10 to 1000 m. It may seem surpris-ing that the relatively weak gravitational grasp of an asteroidcould control the formation of kilometer-sized craters. Thetransition to gravity-controlled cratering is aided by thepresence of regolith and rubble left by earlier impacts, andby fragmentation of even strong targets by shock waves thatprecede the crater excavation (Asphaug and Melosh, 1993;Nolan et al., 1996).

3.5. Impact Statistics and Asteroidal Erosion

Erosion of asteroids competes with regolith generation/retention and can sometimes yield important constraints onthe ages of small gravitationally bound objects (e.g., Geissleret al., 1996). A knowledge of impactor size distributions,impact collision probabilities, and impact speeds is neededin order to estimate ejecta generation and escape rates andtimescales for the creation and destruction of ejecta blocks.These quantities are in general poorly known and consti-tute the greatest uncertainties in erosion rate calculations.

The size-frequency distribution is well constrained bytelescopic observations only for the largest asteroids (e.g.,Van Houten et al., 1970). Smaller size ranges must be esti-mated by (1) extrapolation of power laws for the observedasteroids, (2) collisional models predicting the productionof small fragments, and (3) observations of the size distri-bution of the craters produced by these small impactors onasteroid and planetary surfaces. Simple calculations assum-ing that the impact efficiency is independent of target sizeyield a differential power law index of –3.5 (Dohnanyi,1969), i.e., the relation between the number of fragmentsn and their radius r should follow the power law dn(r) ~r–3.5. However, Galileo observations of small craters onGaspra and Ida suggest a much steeper size distribution forsmall impactors in the main belt. The differential power-law index for small impactors on these objects (<175 mdiameter) is estimated to be near –4.0 (Belton et al., 1992;Chapman et al., 1996a,b). For such steep size distributions(indexes >4), infinite mass is found in the smallest frag-


ments. Thus some minimum projectile size must exist, be-low which a steep size distribution no longer applies. Thiscut-off size threshold is poorly constrained by observations,and determines whether asteroid erosion is dominated bybig bites taken during large impacts, or if asteroid surfacesare mainly sandblasted by particles that are centimeter-scaleor smaller. An upper limit to this minimum impactor sizeis ~1 m, the size of a projectile that would produce cratersin the diameter range of 10 to 100 m (the resolution limitof the Galileo observations).

The rate and efficiency of impacts depends upon the size,location, and orbit of the target asteroid. The intrinsic col-lision probability and the distribution of impact velocitiesfor any specific target can be calculated by integrating thesequantities over the population of asteroids on intersectingtrajectories. Many estimates of intrinsic collision probabili-ties and impact velocities have been made using both directnumerical integration (Marzari et al., 1996, 1997; Dahlgren,1998) and statistical methods (Wetherill, 1967; Greenberg,1982; Farinella and Davis, 1992; Bottke et al., 1994; Vedder,1996; Dell’Oro and Paolicchi, 1998). For main-belt aster-oids, collision probabilities are typically on the order of10–18 km–2 yr–1, and impact velocity distributions are broad,non-Gaussian, and often contain spikes. For the purposesof evaluating impact efficiency, a value between the meanimpact velocity and the root mean square impact velocityshould be used, depending on the target crater scaling expo-nent α (equation (1)).

As an example we calculate the rate of production of10-m-scale ejecta blocks on Eros. Merline et al. (1999,2001) completed a search for satellites around Eros duringthe NEAR Shoemaker flyby and rendezvous and, at a 70%confidence level, found no objects near the asteroid witha diameter larger than 10 m, and a 95% confidence levelfor diameters greater than 20 m. Estimates of the rate ofproduction from impacts on Eros and knowledge of theirdynamical lifetimes could thus be compared with the ob-served lack of such objects to constrain in an iterative fash-ion the present impact rate on Eros. Bottke et al. (1995) givean intrinsic collision probability for near-Earth asteroids(NEAs) hitting other NEAs of ~15 × 10–18 km–2 yr–1 at amost probable relative speed of ~18 km s–1. Multiplying thisby the cross-sectional area of Eros and by the number ofimpactors capable of making 200-m and larger craters [theminimum crater size capable of producing 10-m ejectablocks, according to the ejecta-block scaling law of Leeet al. (1996)] on Eros yields the block production rate. Asphere with the same 1106-km2 surface area of Eros wouldhave a radius of r = 9.4 km, so r2 = 88 km2 (the factor of πin the cross-section is not required since it is included inthe instrinsic collision probability). Estimates based onstrength regime cratering in soil (Holsapple, 1993) indicatethat projectiles roughly 2–10 m in diameter are capable ofproducing the requisite craters. The NEA population maycontain ~2 × 109 objects of this size (Neukum et al., 2001;Rabinowitz et al., 2000; Ivanov et al., 2002); herein lies thegreatest uncertainty in such calculations. Multiplying, Pi ×A × N = 15 × 10–18 km–2 yr–1 × 88 km2 × 2 × 109 = 2.7 ×

10–6 yr–1, or a mean time between impacts of ~370,000 yr.Ejecta blocks launched from the surface of Eros may not bedynamically stable for such long times, so Merline et al.’s(1999, 2001) result of not finding any orbiting boulderlarger than 10 m does not, unfortunately, constrain the timeof formation of the last 200-m-scale crater on Eros.

The fraction of ejecta that escapes from an asteroid dur-ing a given impact depends on the target size and strengthand the size of the impact. For both the strength and gravityregimes, the mass eroded per mass of impactor is indepen-dent of the size of the impactor (Geissler et al., 1996).Impacts into strong targets impart ejecta with speeds muchgreater than the escape velocity of a typical asteroid. Al-though only a small fraction of ejecta reaches escape speedduring gravity-dominated cratering, erosion of a gravitation-ally bound rubble or sand pile is much more efficient thanthat of a coherent object of similar size. Because craterscreated in soft targets are much larger than correspondingstrength-regime craters, the total volume of ejecta that es-capes in this case can be much greater than the volume ofmaterial excavated by a similar impact into a strong target.For example, Geissler et al. (1996) found that the masseroded (ejected and escaped) from a soft Dactyl (made ofsand) per unit mass of impactor should be at least 36×greater than that eroded from a Dactyl made of solid rock.

3.6. Surface Launch Conditions

For studying the subsequent motion of ejecta the mostcrucial item is its initial position and velocity relative to theasteroid surface. Assume that the crater is measured froma nominal vector r0 on the asteroid surface, that the aster-oid surface normal vector at that point is nz , and that thereare two orthogonal unit vectors nx and ny tangent to theasteroid surface. The location of a single ejecta, as measuredfrom the asteroid center of mass, is

r = r0 + δr (2)

δr xn yn znx y z= + +ˆ ˆ ˆ (3)

where |δr| << |r| in general. The ejecta velocity relative to thecrater site is then specified as

V V r n n ne e x y= + +( ) cos ˆ sin cos ˆ sin sin ˆδ β β λ β λ (4)

where the angle β and λ define the direction of the velocityvector relative to the crater normal and the ejecta speed Vewill depend on its position within the crater. Nominal as-sumptions are that β = 45°, and that λ ∈ [0,360]°.

For some dynamical computations the ejecta velocitymust be transformed into an inertially oriented frame. Thenthe asteroid rotational velocity vector ΩΩΩΩΩ must be introduced

VI = Ve + ΩΩΩΩΩ × r (5)

Note that ΩΩΩΩΩ is not necessarily constant and can have a sig-nificant time variation for bodies in nonuniform rotation.

532 Asteroids III

4. EJECTA DYNAMICS

Once the ejecta has left the asteroid surface, it becomessubject to one of the more strongly perturbed environmentsthat can be found in the solar system. Any serious study ofasteroid ejecta must start with dynamical models that cap-ture the main elements of these perturbations, since eachof them can skew the global nature of ejecta dynamics intosignificantly different evolutions than would be expectedfrom the simple application of two-body orbital dynamics.

4.1. General Force Models

For the general study of dynamics about asteroids onemust, at the onset, determine which force perturbations willbe significant for the system. Due to the wide variety ofshapes, sizes, densities, rotation states, and orbits found forasteroids, this determination must usually repeated for eachnew asteroid.

4.1.1. Gravity fields. Several approaches to the model-ing of asteroid gravity fields are available. In general, themost accurate formulations for a gravity field are sphericalharmonic expansions where the gravity coefficients aremeasured from spacecraft radiometric tracking (Yeomans etal., 2000; Miller et al., 2001). Despite the high accuracyof these fields, they are in general inapplicable to the studyof ejecta motions that arise from the surface of an asteroiddue to the divergence of the expansion within the circum-scribing sphere surrounding the asteroid (the circumscrib-ing sphere is the sphere of minimum radius, centered at theasteroid center of mass, that encloses the asteroid). A modi-fication to this technique using ellipsoidal harmonics isavailable (Garmier and Barriot, 2001) that decreases theregion of divergence to within the circumscribing ellipsoidthat fits about the body. Even this, however, does not com-pletely eliminate the problem, as there will still be signifi-cant regions of divergence when close to or within this cir-cumscribing ellipsoid (Garmier et al., 2002).

To overcome this, recourse is usually made to the knownclosed-form gravitational potentials, a class that includes thesphere, the general ellipsoid (Danby, 1992), the tetrahedron(Werner, 1994), and a general polygonal shape (Werner andScheeres, 1997). The main restriction to these potentials isthat the mass density is assumed to be constant, or at thevery least is constrained to follow a very specific mathemat-ical variation (which in general may not be physical). Themain approaches to gravity field modeling have used collec-tions of point mass gravity potentials, collections of tetra-hedron gravity potentials (making up a single, polyhedralshape), and the use of the simple ellipsoidal shape model.

The point mass (or mascon) approach consists of tak-ing a defined shape model of the asteroid and populatingits interior by a distribution of point masses, properly scaledto yield the correct total mass. This approach can lead toregions of poor gravity field computation on the surface andis inefficient if a high resolution is desired (Werner andScheeres, 1997). The polyhedron approach takes a polygon

shape model and computes the gravity potential (and itsattendant partials) directly from this model. Even thoughthe individual computations needed to compute the poly-hedron gravity field are more involved than those used forthe mascon approach, the overall efficiency of a polyhe-dral gravity field computation is often better than a masconapproach, since the mascon approach must sum over theentire volume of the body while the polyhedron approachmust only sum over the surface elements of the body. Thepolyhedron approach also has the advantage of giving adirect indication of whether the point is inside or outside theasteroid. The ellipsoid model is useful for situations when aprecision model of a gravity field is not needed. Its advan-tages are that it is relatively simple to code, has no singu-larities (such as are found for collections of point masses),and can be specified based on light-curve analysis alone.

In the following, the gravitational force potential is speci-fied as U(r) where r is the position vector relative to theasteroid-fixed frame. The gravitational attraction acting ona particle is ∂U/∂r. The potential U is often split into themain contribution of the monopole (µ/r) plus the perturba-tion contribution (R) as

U rr

R r( ) ( )= +µ(6)

However, on an asteroid surface the perturbation contribu-tion can often compete with the main contribution, and thusthis form is only used for notational convenience.

When relatively far from an asteroid, MacCullagh’s for-mula can be used to approximate the gravity field of a gen-eral asteroid with a reasonable degree of accuracy (Danby,1992)

R rR

r

C

C

o( )cos

cos cos

=− +µ δ

δ λ

2

3

202

222

13

2

3 2

(7)

where C20 and C22 are gravity coefficients, δ is the decli-nation, and λ is the body-fixed longitude. This formulationassumes that the coordinate system is aligned with the in-ertial axes, with the maximum inertia axis along z and theminimum axis along x. In many cases, consideration of justthis contribution to the gravity field of the asteroid canadequately capture the major departures of orbit dynamicsfrom the simple Keplerian case. It must be noted that equa-tion (7) cannot be used on the surface of the asteroid orwithin the circumscribing sphere, as it will give nonphysi-cal values of potential and acceleration.

4.1.2. Rotation state. There are two classes of rota-tional motion that must be considered. The first, simplest,and most common is asteroid rotation about its maximumaxis of inertia. It is well known that this is a stable rota-tional end state for an asteroid when dissipation of energy istaken into account, as it provides the minimum energy rota-tional state for a given value of angular momentum (Burns


and Safronov, 1973). To completely specify the rotationaldynamics for a uniformly rotating asteroid requires the rota-tional velocity vector and a phase angle for the asteroid.The rotational angular momentum of the asteroid will besubject to solar torques and nongravitational effects, and canbe altered by impact events or planetary flybys. Still, overlong periods of time it is acceptable to treat such a rotationstate as a constant.

More interesting, but rarer, are cases where the aster-oid has a nonuniform rotation state, usually correspondingclosely to the general solution to Euler’s equations for atorque-free rotating body. Examples of such bodies includethe asteroids Toutatis, Mathilde, and Alinda. Most, if notall, of the asteroids observed to have a nonuniform rotationstate are slow rotators, which makes physical sense as thetime to relax to uniform rotation scales with the generalizedrotation period cubed. Thus, a body such as Toutatis has apredicted relaxation time longer than the age of the solarsystem (Harris, 1994). In modeling ejecta dynamics abouta nonuniform rotator the most efficient modeling approachis to use the classical solution for rotational dynamics in atorque-free environment (MacMillan, 1960); a summary ofsuch an application is given in Scheeres et al. (1998a).

In the following we specify the asteroid angular veloc-ity vector as ΩΩΩΩΩ. For the case of a uniformly rotating aster-oid, ΩΩΩΩΩ is constant in both the asteroid-fixed frame and inan inertially oriented frame since it is aligned with the aster-oid’s total rotational angular momentum vector. Whenmodeling an asteroid with a nonuniform rotation state, thevector ΩΩΩΩΩ is no longer aligned with the asteroid’s rotationalangular momentum vector, but has a precession and nuta-tion relative to this vector. If we model the asteroid non-uniform rotation using the solution for torque-free motion,the angular velocity vector ΩΩΩΩΩ is a periodic function of timein the asteroid-fixed frame, i.e., an observer sitting on theasteroid tracing out the path of this rotational velocity vectorwould see that it repeats itself exactly after a characteristicperiod (which is a function of the body’s moments of iner-tia, rotational energy, and rotational angular momentum).The implications of this are discussed in greater detail inScheeres et al. (1998a).

4.1.3. Solar effects. When far from the asteroid a par-ticle must contend with strong perturbations from the solargravity and radiation pressure. For precision computation,detailed models of the ejecta shape and interaction withsolar radiation could be developed if desired. This level ofdetail is not always necessary for understanding the basiceffect of the solar radiation pressure on the ejecta.

The solar gravity and radiation pressure forces are de-rived from a force potential written as

Vd rS S= −

+µ β1

(8)

where µS is the gravitation parameter of the sun, d is theasteroid position vector from the Sun, r is the ejecta positionvector from the asteroid, and β is the ratio of solar radiation

pressure force to solar gravity force acting on the ejecta. Areasonable assumption for ejecta motion relative to the as-teroid is |r| << |d|, leading to the simplified potential

Vd

dd r

dr r d r

SS= −

− × −

× − ×( )+µ β( )

ˆ

ˆ

11

1

12

32

2(9)

4.1.4. Other nongravitational forces. For specific pur-poses, other nongravitational forces may also be modeled.This is especially true for the modeling of comets, wherethere is significant gas pressure that emanates from the nu-cleus surface (Weeks, 1995; Scheeres et al., 1998b). Therehas been speculation in the past on an outgassing environ-ment for asteroids as well, but evidence for this has not beendetected to date, and the perturbations that would result fromsuch outgassing would be very small. Another nongravi-tational force that has been recently considered in manycontexts is the Yarkovsky effect (Bottke et al., 2000, 2002;Rubincam, 2000), which essentially consists of a thermalimbalance on a body. Since ejecta will have definite shapesand rotations, the Yarkovsky effect may be able to modifya particle’s orbital dynamics if it falls into a long-term stableorbit. There have been no studies performed on the appli-cation of this force to ejecta dynamics to date, however.

4.2. Equations of Motion

The measured force parameters and models define thedynamical problem of motion in the vicinity of the asteroid.Depending on the force parameters, the character of motionin these equations will take on a variety of forms. Specifi-cally, for smaller asteroids the regions where solar and grav-ity field perturbations are important can coincide, leadingto very complicated dynamics. For larger asteroids theseregions of influence do not coincide, making it possible todistinguish between a far-field regime dominated by solareffects and a close-field regime dominated by asteroid grav-ity and rotation. Of course, a single ejecta trajectory cantransition between these regimes as it passes from apoapsisto periapsis and back again.

The relevant equations of motion in an inertially orientedframe for the ejecta relative to the asteroid can be stated as(Scheeres et al., 2001)

rV r

rII= ∂

∂( )

I

(10)

V r R rd

d r

dr r d r

S

S

( ) ( ) ˆ

ˆ

= + + × −

× − ×( )

µ µ β

µ

2

3

21

23

r(11)

534 Asteroids III

where d = |d|, d = d/d, and rI denotes that the position vec-tor is referenced to an inertial, nonrotating frame. Theseequations are entirely general and only incorporate a fewassumptions (noted above). Note that it is necessary to havea solution for the motion of the asteroid relative to the Sun;however, it is generally sufficient to use a Keplerian orbitfor the computation of d and d. Exceptions occur when theasteroid has a close encounter with a planet, but this wouldalso require the addition of the tidal effect of that planeton the motion of a particle and on the rotation state of theasteroid, situations we do not directly discuss here (Chauvi-neau and Mignard, 1990b; Scheeres et al., 2000b).

4.2.1. Perturbation formulation. In the course of ana-lyzing motion about an asteroid, it is often convenient touse the constants of motion of the two-body problem inorder to characterize the strength and effect of the perturba-tions acting on the ejecta. The classical orbital elements canbe defined as the semimajor axis, a, the eccentricity, e, theinclination, i, the longitude of the ascending node, Ω, andthe argument of periapsis, ω. Frequently, the true or meananomaly of the orbit, f or M respectively, are used to re-place the classical sixth orbit element of the time of peri-apsis passage. The variation of these constants due to orbitalperturbations are generally specified using the Lagrangeplanetary equations with a perturbation function. An ex-tended discussion of these equations can be found in Brou-wer and Clemence (1961). For our system, the general forceperturbation potential can be given as V(r) – µ/|r|.

4.2.2. Asteroid-fixed frame. For the analysis of ejectamotion close to the asteroid surface it is more convenientto shift the equations into an asteroid-fixed frame. In doingso we must allow for the fact that the asteroid is rotatingwith an angular velocity vector ΩΩΩΩΩ with respect to inertialspace, so the equations of motion relative to the asteroidhave the form

r r rV r

r+ × + × + × × = ∂

∂Ω Ω Ω Ω2

( )r (12)

If the asteroid is uniformly rotating, Ω = 0,ΩΩΩΩΩ is constant,and the equations simplify. On the other hand, if the asteroidis in nonprincipal axis rotation, then the vectors ΩΩΩΩΩ and Ωare time periodic.

These equations of motion have no integrals of motion ingeneral. However, for motion close to an asteroid we canoften disregard the solar perturbation terms. Then, if the aster-oid is in uniform rotation, equation (12) is time-invariantand a Jacobi integral exists

J r r rr

R= × − × × × − −1

2

1

2r ( ) ( )Ω Ω µ

(13)

This integral is often helpful in constraining and under-standing the limits on motion near the asteroid surface. Inapplication, this integral can be used just as the Jacobi inte-gral is used in the restricted three-body problem (Szebehely,1967) and was used extensively for the analysis of Phobos(Dobrovolskis and Burns, 1980).

5. THE DYNAMICAL FATE OF EJECTA

The equations of motion reviewed above can lead toextremely complicated motion that cannot in general besolved analytically. However, there are many insights to behad from the study of these equations, both numericallyusing the full models and analytically using suitably simpli-fied models. In all such investigations it is important toremember the guiding dynamical questions: What is the dy-namical evolution of an impact ejecta field, what fraction ofan ejecta fragment field will escape, what fraction will reim-pact, and what fraction will be captured in a transient orbit?

5.1. Stability of Synchronous Motion

First, a special note must be given on the stability of syn-chronous (1:1) motion about an asteroid. In the past, manyauthors have made a tacit assumption that synchronous or-bits about an asteroid would follow the basic pattern foundfor geosynchronous orbits (such as described in Kaula, 2000,p. 54) with two stable, synchronous orbits and two unstable,synchronous orbits. Application of these assumptions leadto predictions for the stability of orbital motion about aster-oids and to the tidal evolution of asteroid satellites that arenot valid. In the following, “stable” means that a trajectoryclose to the synchronous orbit will remain close to it forarbitrarily long periods of time, while “unstable” means thata trajectory close to the synchronous orbit will rapidly leaveits vicinity.

In Scheeres (1994) it is shown that synchronous orbitsabout asteroids are unstable in general. For an asteroidwhose shape is spheroidal, or for Earth, we find four syn-chronous orbits, two of which are hyperbolically unstableand two of which are linearly stable. An analogy can bemade with the restricted three-body problem, where for asmall mass ratio we find three synchronous orbits that arehyperbolically unstable and two orbits (the so-called equi-lateral points) that are linearly stable. Now, as is wellknown, in the restricted three-body problem these stableequilateral points become unstable if the mass ratio betweenthe primaries is increased to greater than ~0.1 (cf. Szebehely,1967). A similar phenomenon occurs in the asteroid prob-lem, where the stable synchronous orbits become unstable ifthe body’s shape is sufficiently elliptic (more precisely, thisinvolves both the asteroid’s rotation rate and its ellipticity).Furthermore, the instability timescale of these orbits is onthe order of the rotation period of the asteroid, and henceoperate very quickly. Thus, particles placed near a 1:1 reso-nance with a rotating asteroid will in general either impactwith or escape from the asteroid, usually in a matter ofhours or days at most.

5.2. Final Outcomes for Ejecta

We can delineate several distinct final outcomes forejecta trajectories whose initial conditions lie beneath or onthe surface of an asteroid. Using a Keplerian dynamicsmodel applied to a spherical asteroid there are three dis-


tinct classes of motion. First, if the orbital energy is nega-tive then the ejecta will reimpact as periapsis is initially onor beneath the asteroid surface. Second, if the energy is zeroor positive the ejecta will escape. Third, a subset of thesecond class of escaping ejecta may eventually reimpact onthe asteroid after an extended period of time in orbit aboutthe Sun. We will discount this third class, however, as it ispractically indistinguishable from other impacts.

As additional perturbations are considered, the possibleclasses of motion expand. It is useful to use periapsis pas-sage relative to the asteroid to delimit between differentclasses of motion. At launch the ejecta are starting from aninitial radius r0 ≥ q0, since in general the initial periapsis(q0) lies beneath the body’s surface. In the absence of per-turbations the next periapsis passage q1 will either equal q0,and thus will be an impact, or will never occur, indicatingescape. When force perturbations are incorporated, or evenif nonspherical shapes are allowed, it becomes possible formultiple periapsis passages to occur. We denote these as aseries qi; i = 0, 1, 2, …. Associated with each periapsis pas-sage is the periapsis vector, qi, representing the periapsislocation in the asteroid-fixed space. If we denote the set ofpoints that constitute the asteroid body as B, then if qi ∈ Bthe sequence stops and an impact has occurred. Conversely,

given a periapsis passage qi, if qi + 1 does not ensue, thenthe ejecta has escaped. Finally, if the sequence never ter-minates (i → ∞), then the ejecta is in a stable orbit aboutthe asteroid.

Based on this understanding, we tender the followingclassifications (see Fig. 1): Class I — Immediate reimpact:Ejecta reimpacts with the surface prior to first periapsis pas-sage. Class II — Eventual reimpact: Ejecta does not reim-pact at the first periapsis passage, but eventually reimpactsin the future. Class III — Stable motion: Ejecta is placedinto a long-term stable orbit about the asteroid. Class IV —Eventual escape: Ejecta has at least one periapsis passageby the asteroid before it escapes. Class V — Immediateescape: Ejecta escapes from the asteroid prior to its firstperiapsis passage.

Classes I and V are clear carryovers from the nonper-turbed case, and we expect most ejecta to fall into thesetwo catagories. The fraction of ejecta that fall into classes II,III, and IV is an important consideration for understandingthe formation of asteroid regolith and asteroid binaries.Given a specific system it is relatively easy to find regionsof ejecta initial conditions that fall into classes II and IV.In terms of celestial mechanics and astrodynamics, it wouldappear to be very difficult to place a particle into class III

Fig. 1. The five classes of ejecta fate.

Class I Class II

q1, q2, . . .

Class III

Class IV Class Vq1

q1

536 Asteroids III

due to the nature of these dynamical systems; however,there are other physical forces that can cause a particle totransition into a stable orbital motion, and these will bereviewed below. Figure 2 shows the trajectory evolution ofa number of different ejecta particles launched off an as-teroid at different speeds. We note that changes in initialspeed or location on the asteroid can have dramatic conse-quences for the final state of the ejecta.

5.3. Impact and Escape Conditions

By applying analytical theories to the motion of ejectait is possible in many instances to immediately determine ifan individual ejecta particle will fall into class I or V, basedonly on its initial conditions. Using such determinations cangreatly decrease the amount of computational effort neededto evaluate the outcome of a high-resolution impact crater-ing event. Also, such methods can directly compute thefraction of an ejecta fragment field that falls within theseclasses, and hence provides an indication of the fraction thatmay reside in classes II–IV. In Scheeres et al. (1996, 1998a)a number of analytical results directly pertaining to thecomputation of reimpact conditions and escape conditionsare given. Specific results developed in these papers includethe computation of guaranteed reimpact speed and guaran-teed escape speed as a function of location on an asteroid.The guaranteed escape speed is the speed at which an ejecta,launched normal to the surface, will have sufficient energyto escape the asteroid. The guaranteed return speed is themaximum speed an ejecta can have while still being ener-getically trapped by the zero-velocity curves surroundingthe asteroid (see section 5.6). These methods have also beenapplied to the Eros dataset, which is definitive since all theforce model parameters have been measured (Yeomans etal., 2000; Miller et al., 2001). For Eros the escape speedsrange from 3.3 to 17.3 m/s over its surface. This large varia-tion is due to the combined shape/gravity field variation andthe rapid rotation rate of the asteroid. At the other end ofthe spectrum, the guaranteed return speeds computed overthe surface of Eros range from 1 to 5 m/s, but are alwaysless than the escape speed at any particular point. Many ofthese ideas can be developed in additional detail, and canprovide sharper conditions on the fraction of an ejecta frag-ment field that will immediately escape, or that will redistri-bute itself on the asteroid.

5.4. Transient Classes

Of particular interest are classes II–IV, as they define thespace where interesting things can happen to an ejecta frag-ment field. If we draw a “spectrum” of outcomes, class IIIwill lie at the intersection between classes II and IV, as it isthe limit of these cases. Thus, one of the fundamental ques-tions concerning the fate of asteroid ejecta is how a particlecan be placed into one of these transient classes, and whatits subsequent evolution will be. Additionally, if a fraction ofan ejecta fragment field falls into class II or IV for an ex-tended period of time, there is a higher probability that addi-tional perturbations or impacts may push it into class III,creating a binary asteroid.

5.4.1. Problem of initial capture. The basic dynamicalproblem is how to generate class II and IV ejecta, and sub-sequently transition these into class III ejecta. The real prob-lem is not whether such trajectories exist, as we can firmlyestablish the existence of trajectories that fall into class III.

–60

–40

–20

0

20

40

60

–60 –40 –20 0 20 40 60

6 m/s7 m/s8 m/s9 m/s

10 m/sAsteroid

–100

–50

0

50

100

–100 –50 0 50 100

8 m/s10 m/s11 m/s12 m/s13 m/s14 m/s

Asteroid

Fig. 2. Effects of location and ejection speed on ejecta trajecto-ries launched from a uniformly rotating asteroid shape. (a) Launchfrom the leading edge of an asteroid; because of the asteroid’s rota-tion, the local escape speed is lower. (b) Launch from the trailingedge of an asteroid; because of the asteroid’s rotation, the localescape speed is higher.

(a)

(b)


Specifically, families of unstable periodic orbits and equi-librium points (in the asteroid-fixed frame) exist close tothe asteroid surface. Each of these orbits has a stable mani-fold that asymptotically approach these special solutions.It can be shown that many of the stable manifolds of theseobjects intersect (or emanate from) the surface of the aster-oid, and hence provide exact initial conditions that lead toorbital capture (Scheeres et al., 1996). The problem withthese solutions, of course, is that they are unstable and theset of initial conditions that leads to capture is vanishinglysmall. Thus, the real question is whether there are any signi-ficant regions of initial conditions that lead to long-term,trapped orbits about an asteroid. Again, the answer hereappears to be yes, but the proof is not as direct, and the fullextent of initial conditions that actually lead to such cap-ture has yet to be fully explored.

5.4.2. Methods of analysis. There are several differentapproaches to determining if a particle falls into one of thetransient classes. The first is direct numerical simulation ofdiscretized elements of the ejecta field. This approach, usedby Geissler et al. (1996) in studying the evolution of ejectaabout Ida, provides definite results, subject to modelingassumptions used in setting up the computations, and allowsfor the use of a full perturbation model. It is limited by thefinite number of ejecta that can be propagated and due tothe discrete nature of each propagation. Indeed, in an actualejecta field we expect a near continuum flow of particles,which should in general lead to higher probabilities for cap-ture into transient dynamical situations.

Analytically motivated approaches can give greater in-sight into the evolution of larger numbers of particles andcan model the ejecta field as a continuous flow in somesituations, at the cost of lost precision in the computed tra-jectories. In Scheeres and Marzari (2000) an averagingapproach is used that provides analytical solutions to ejectaevolution following ejection from the surface of a smallbody. Their approach only incorporated solar radiation pres-sure perturbations, but could be generalized to include othereffects. With this approach it is possible to rapidly computethe evolution of ejecta fields, which allows for more pre-cise estimates on the fraction of an ejecta fragment fieldthat is injected into a transient class, potentially allowingfor direct computation of probabilities of different outcomesfrom a given ejecta field.

Application of advanced understandings of dynamicalsystems could also be used to evaluate the likely outcomesfor an ejecta fragment field. As mentioned earlier, the spacearound an asteroid is filled with periodic orbits, both stableand unstable, each of which have manifolds that can influ-ence the dynamical flow of an ejecta fragment field. Recentadvances in the application of dynamical systems theory tospacecraft trajectory design (Koon et al., 2000) could alsobe brought to bear on the evolution of asteroid ejecta, andprovide qualitative descriptions of ejecta field flow that mayallow for specific quantitative predictions in some cases.Initial approaches to this sort of application have been de-

veloped (Scheeres et al., 1996, 1998a, 2000a) and, at theleast, can be used to establish the existence of transientorbits of extremely long duration.

5.4.3. Mechanisms for capture into stable orbits. Thereare many mechanisms that have been hypothesized that leadto ejecta becoming captured into stable orbits. We will pro-vide a very brief summary of these approaches. Issues oflong-term stability and the lifetime of such orbits are not con-sidered, but are discussed in more detail in Merline et al.(2002).

Direct initial condition generation. In this scenario aparent asteroid is subject to an intense impact event, whichshatters and disperses the original body into many frag-ments, all imparted with a range of speeds. In general, thesmaller particles have higher speeds and the larger haveslower speeds. Given such a random distribution of particlepositions and speeds it is probable that some of the frag-ments will be placed into mutually bound orbits as theyescape (indeed this postdisruption environment can eveninfluence their motion during the short period when theasteroid disperses), leading to primitive binary systems.Such bound orbits will have large eccentricities in general,but assuming long-term stability against impact and escape,energy dissipation (i.e., tidal effects with energy dissipation)can cause the orbits to circularize over time, leading to thetypes of stable binaries now being found. Hartmann (1979)first suggested this scenario, which has been investigatedanalytically by Weidenschilling et al. (1989), and more re-cently has been simulated by Durda (1996), Dorresoun-diram et al. (1997), and Michel et al. (2001). They havefound that small numbers of bound asteroid pairs do result,but these studies have not addressed the long-term stabilityand evolution of these pairs.

Mass shedding in tidal flybys. Additional mechanismsnot involving impacts have also been suggested to increasethe rotation rate of asteroids to the point of mass shedding.Richardson et al. (1998) and Bottke et al. (1999) numeri-cally simulated the tidal disruption of asteroids modeled as“rubble piles” (see Richardson et al., 2002) composed ofnumerous equal-sized spherical components encounteringEarth, and found that rotational spinup frequently inducesdebris to be cast off the primary bodies. In many cases, theshed fragments were found to go into initially bound orbitsaround the progenitor. Bottke and Melosh (1996a,b) andRichardson et al. (1998) have shown that tidal disruptioncan create enough satellites in the NEA population to ex-plain the statistics of doublet craters seen on the terrestrialplanets.

Mutually impacting ejecta. A postimpact ejecta fieldwill have a distribution of particle sizes and speeds resem-bling, in some aspects, a continuous field distribution. Thus,it is likely that mutual impacts between elements of theejecta field will ensue, and that these slow-velocity impactswill mutually alter the trajectory of the particles, in somecases leading to capture orbits. Weidenschilling et al. (1989)considered this mechanism in an analytical argument, and

538 Asteroids III

concluded that satellites formed from reaccreted ejecta areexpected to be small and found in prograde orbits. Durdaand Geissler (1996) simulated impact ejecta fields to searchfor such impacts, but did not find any that evolved intostable trajectories. Their approach used 1000 ejecta par-ticles, which may be too few to reliably find such outcomes.This approach becomes more likely to yield stable trajec-tories when portions of the ejecta fields are captured intotransient orbits that may not reimpact for many orbits (clas-ses II–IV), since there will be a higher probability of mutualimpacts and repeated impacts that could yield stable trajec-tories. Such a long-term analysis has not been performed todate, however.

Rotational bursting. A novel idea for injection of ejectainto stable orbits was posited in Giblin et al. (1998), based onobservations of laboratory impact events. In this scenario,specific particles in the ejecta field have large rotationalvelocities and are placed in tension. In some situations,these rotating fragments have been observed to spontane-ously “burst,” or disassemble into smaller fragments, shortlyafter ejection from the laboratory target. This situation, iffound in nature, creates a situation such as found in the para-graph above on direct initial condition generation, and canplausibly lead to particles placed directly into stable orbits.

External force perturbation. Underlying many of theabove mechanisms, and indeed a mechanism in itself, is theeffect of force perturbations on the trajectory. As mentionedearlier, the ejecta are subject to an extremely perturbed forceenvironment, first from the asteroid gravity field, and sec-ond from the solar radiation pressure and tidal perturbations.Any of these can place a particle into an orbit that persistsfor some time about the asteroid. Specifically, in Scheereset al. (1998a) a particle orbit perturbed only by the aster-oid gravity field is described that has a “hang time” of over100 d. In Fulle (1997) and Scheeres and Marzari (2000)the effect of solar radiation pressure on a ejecta particle isshown to capture regions of initial ejecta conditions intobound orbits that do not reimpact for hundreds of days insome cases. A study that combines solar and asteroid gravityeffects has not yet been performed, but may provide mecha-nisms that could extend the lifetime of such transient orbitsto multiple asteroid years. At these timescales it becomespossible for small perturbation forces to influence the orbits,potentially leading to stable orbits.

5.4.4. Long-term lifetime and evolution of captured orbits.Once in orbit about an asteroid, a particle is subject to avariety of perturbation forces that can cause orbital evolu-tion over long timespans. These effects include the aster-oid gravity field and tidal effects (Petit et al., 1997), thesolar tide (Chauvineau and Mignard, 1990a; Hamilton andBurns, 1991a), solar radiation pressure (Hamilton andBurns, 1991b; Richter and Keller, 1995), four-body effects(Chauvineau and Mignard, 1990b), and disruption by im-pacts (Davis et al., 1996). The majority of these analyseshave only considered these perturbations in isolation. Fromsuch studies, it is clear that long-term, stable orbits can existabout asteroids, in some cases with minimal orbital evolu-

tion. Perhaps the most interesting results, related to the life-time and evolution of an asteroidal satellite, are found inDavis et al. (1996) and based on the work by Geissler et al.(1996), where it is posited that Ida and Dactyl may actuallybe in an equilibrium state, exchanging mass between thebodies, driven by impacts and ejecta field evolution on eachbody. As the statistics on asteroid binaries is improved, withincreasing numbers of detections, a firmer context for suchstudies can be established and, most likely, real distinctionsbetween different classes of binary systems will be found.

5.5. Reimpact Dynamics

An unanswered question involves the dynamics of a par-ticle after it reimpacts on the asteroid surface. A distinctionshould be made between high-energy secondary impactsthat may occur in the fractions of a second after a primaryimpact (due to ricochets) and low-energy impacts that mayoccur immediately or months after the primary impact withspeeds less than or equal to surface escape speed. For thesecond type of reimpacts, it can be hypothesized that colli-sion with the surface may not be disruptive nor completelyinelastic, so that some amount of rebound energy will exist.If true, there are significant implications for the modelingof reimpact ejecta. This is especially interesting in lightof the recent returns from the NEAR Shoemaker mission,which found that the asteroid surface at high resolution wasdominated by ejecta blocks, with a paucity of craters, whichraises a host of scientific questions on the nature of the Erossurface at centimeter scales (discussed at the end of thischapter). The existence of transient dynamical behavior ofa reimpacting ejecta block has implications for the extentof downslope motion a particle will experience, and hencethe degree of ponding at lows in the potential that will occurprior to a particle settling on the surface.

This issue has been studied in an engineering applicationin the context of the settling time of a navigational aid de-ployed on the surface of an asteroid (Sawai et al., 2001). Inthat study it was found that, even for relatively low coeffi-cients of restitution on the order of 0.1, settling times of10–20 min were common. This is ample time for a particleto migrate toward potential lows on an asteroid’s surface.

5.6. Surface Forces and Dynamics

Finally, we must consider the environment that is felt byan impact ejecta once it comes to rest on the surface of anasteroid. The mechanical forces felt on the surface can bereduced to surface normal and transverse frictional forcesacting on a particle. These are, in turn, defined by the aster-oid’s gravity field, surface, and rotation state. Recent inter-est in other forces acting on the asteroid surface have beenrevived by the unexpected morphology of the Eros surface.Indeed, electromagnetic forces operating on small dust par-ticles are being considered to explain some of the dustponding seen on Eros (Lee, 1996; Robinson et al., 2001).In addition to these are occasional impulsive forces that may


jolt asteroid regolith, due to impacts of other asteroids onthe asteroid surface (Greenberg et al., 1994).

The total acceleration that a particle feels when at reston the surface of a rotating asteroid is

N r rV

r= × + × × − ∂

∂Ω Ω Ω (14)

If the local surface normal is nz , then the surface force issplit into a normal and tangential component

N Nz z= n · (15)

N N Nt z z= − n (16)

and the local slope of the system is defined as

φ = arctanN

Nt

z

(17)

The surface slope can be related to the coefficient of fric-tion on the surface, µ, as µ ≥ tan φ (Greenwood, 1988). Therotational dynamics of the body can take a significant rolein modifying the surface environment, and may change thestability and structure of motion on the surface. For bodiesin complex rotation the slopes and surface forces are time-periodic, and could potentially add sufficient “shaking”(physically realized by slowly varying slopes at each pointon the surface) to cause the surface to relax, reducing thepotential energy stored in local slopes. Any asteroid subjectto nonuniform rotation following a large impact or plan-etary flyby will have these time-periodic forces acting onits surface, which could play a role in smoothing a surfaceafter an impact. This is distinguished from seismic shak-ing, where the asteroid frequently feels small seismic eventsdue to the flux of impactors striking the asteroid (Greenberget al., 1994). While the magnitude of shaking expected fromimpactors should be larger than from nonuniform rotation,the nonuniform rotation will act continuously on the aster-oid over the time it takes for it to relax into uniform rotation.Estimates of this effect for Toutatis are given in Scheereset al. (1998a).

Finally, it should be noted that if an asteroid would ac-tually describe a figure of equilibrium (Weidenschilling,1981), then the surface slope would be identically zero overthe entire body. In fact, deviations of surface slope fromzero indicate deviations from a figure of equilibrium. Slopedistributions of asteroids have been measured from space-craft observations and from radar imaging of asteroids.Some bodies measured in this way, such as Toutatis andKleopatra, have uniformly low slopes that, at the least, couldbe indicators of their rotational and impact past (Scheereset al., 1998a; Ostro et al., 2000).

A second parameter of interest for the surface environ-ment is the effective potential, defined by the combinedgravitational potential and rotational potential terms. For a

uniformly rotating body this is just the Jacobi integral dis-cussed earlier. This gives a direct measure of the availableenergy that can be converted to kinetic energy (and henceeasily dissipated) based on the location of a particle in theasteroid frame (Thomas, 1993). The effective potential en-ergy function of an asteroid is

C r r r U r( ) ( ) ( ) ( )= − × × × −1

2Ω Ω (18)

Using this, the dynamical height of the asteroid surface canbe computed, a relative measure from a locally definedaverage gravity (Thomas, 1993).

On the surface of a uniformly rotating asteroid, this sameeffective potential energy can also be related to the mini-mum amount of energy a particle requires before it canescape from the asteroid [the guaranteed reimpact speed inScheeres et al. (1996)]. Specifically, the value of C(r) atthe synchronous orbits (CR) defines the zero-velocity sur-face that surrounds and encloses the asteroid in three-di-mensional space. The effective potential energy evaluatedat this synchronous orbit defines the minimum energy thata particle must have before it becomes possible to escapefrom the asteroid; i.e., a particle with Jacobi constant greaterthan this value could, theoretically, escape from the aster-oid following a purely ballistic trajectory. If a particle hasan energy less than this, and is within the zero-velocitycurve, then it is impossible for it to leave the vicinity ofthe asteroid. This surface has also been referred to as theRoche lobe, and was studied in the particular case of Phobos(Dobrovolskis and Burns, 1980), and more recently hasbeen computed for Eros (Yeomans et al., 2000; Miller et al.,2001). Phobos was found to “fill” this minimum energysurface, meaning that particles on its surface were prone toescape that body when given sufficient speeds. Conversely,Eros lies entirely within this energy surface, although 56%of that asteroid’s surface lies within 1 km of this energy sur-face, the closest point lying only 90 m from the energy sur-face. Figure 3 shows the computed Eros Roche lobe pro-jected into the Eros equatorial plane.

6. CURRENT DATA CONSTRAINTSON EJECTA EVOLUTION

The interpretation and analysis of asteroid regolith andthe dynamics of impact ejecta will ultimately be constrainedby in situ observations of asteroids. Historically, the analysisof Phobos and Deimos from spacecraft images has allowedfor a rather complete understanding of the regolith and dy-namical environment of these bodies to be developed. Sev-eral obvious indications of regolith on the martian moonsPhobos and Deimos were noted in Viking Orbiter pictures(e.g., Thomas, 1979; Thomas and Veverka, 1980; Lee et al.,1986) and recently scrutinized with new images from MarsGlobal Surveyor (Thomas et al., 2000). The dynamics ofejecta lofted from the martian moons are complicated bytidal forces from nearby Mars as well as the rapid rotation

540 Asteroids III

and irregular shapes of these satellites (Dobrovolskis andBurns, 1980; Davis et al., 1981). Changes in the moons’semimajor axes due to orbital evolution should have pro-duced gradual variations in the effects of tides over time.Using the asymmetric ejecta deposit from the giant craterStickney, Thomas (1998) deduced that the impact occurredwhile Phobos was slightly farther from Mars than it is atpresent. The thick mantle of regolith on Deimos, presum-ably derived from a giant impact near the satellite’s south

pole, has been subjected to significant postdepositional re-distribution, perhaps due to seismic shaking.

The morphology of surface features on Gaspra and Ida(and Ida’s satellite Dactyl), imaged at moderate to highresolution by the Galileo spacecraft, indicate the existenceof impact ejecta retained on their surfaces (e.g., Belton etal., 1992; Sullivan et al., 1996; Geissler et al., 1996). Mor-phological indications of regolith include (1) numerous iso-lated positive relief features, which appear to be ejectablocks, the largest size fraction of the regolith; (2) chutesand albedo streaks oriented down local slopes, interpretedas mass-wasting scars in regolith; (3) grooves, which maybe the surface expression of deep-seated fractures partiallyfilled by regolith; and (4) color/albedo variations associatedwith slopes and apparently fresh impact craters consistentwith regolith maturity variations. Applications of basic mod-els of impact ejecta fields and their dynamics have been ableto explain the observed regolith features on these bodies(see Fig. 4). While the explanation of Dactyl has been a chal-lenge, several reasonable ideas on its formation and evolu-tion do exist.

The asteroids imaged by the NEAR Shoemaker space-craft, Mathilde and Eros, have not fit as well with the ex-pected theory. Several of the observations made were, inessence, totally unexpected. For Mathilde this includes thesize and extent of its craters, along with a lack of observeddepositional features (Veverka et al., 1997). Theories ofimpact physics that describe this situation have been positedthat are consistent with its low measured density (Housenet al., 1999; Davis, 1999; Asphaug, 2000). Still, the abilityof that asteroid to survive intact is surprising, as is its ex-tremely slow rotation rate. Planned radar observations ofMathilde will hopefully provide additional insight into thisprimitive body.

Fig. 3. Eros Roche lobe computed from NEAR Shoemaker data.

Fig. 4. Theoretical landing locations of ejecta launched from the giant crater Azzurra on Ida. This ejecta distribution, calculatedconsidering Ida’s irregular shape and rapid rotation, provides a close match to the bright, relatively blue spectral unit found on theasteroid. The distinct color of Azzurra’s ejecta deposits suggests that relatively fresh, unweathered materials were excavated by theimpact (see Geissler et al., 1996, for a more detailed description).


For Eros, detailed mapping of the largest ejecta blockscoupled with calculations of trajectories from candidatesource craters led Thomas et al. (2001) to suggest the Shoe-maker Crater as the source of most of the ejecta blocks onEros. Moreover, the lack of large blocks associated with theother giant impacts on Eros confirms that these house-sizedboulders are rapidly destroyed or buried. While some as-pects of the distribution of ejecta blocks have been ex-plained, the observed lack of cratering at high resolution isan outstanding puzzle. NEAR Shoemaker imaging of Eros(Veverka et al., 2000, 2001) revealed a surprising lack ofsmall craters, and closeup imaging showed pools of finesediment in topographic lows with no dust deposits on topof the boulders. This suggests that the regolith has beenshaken and stirred since deposition. Seismic disturbancesdue to distant impacts could cause vertical redistribution ofthe regolith and perhaps erase the smaller craters (Green-berg et al., 1994). Alternatively, electromagnetic forces mayact on sunlit dust particles, causing them to rise and subse-quently fall, leading to ponding of small particles on the aster-oid surface (Robinson et al., 2001). Unknown is whetherthe surface morphology of Eros implies a dearth of smallprojectiles in the main belt, or is caused by postdepositionalmodification during Eros’ recent history as a NEA.

The radar asteroid dataset has also expanded greatly inrecent years. There are now many radar-derived asteroidshapes in existence, from which many insights can be ex-tracted (see Ostro et al., 2002). Each asteroid measured todate with this technique has some unique features that defya uniform classification. Even without direct measurementsof an asteroid’s mass or density, it is still possible to placeconstraints on its surface environment. First, all asteroidsmeasured by radar to date have exhibited no strong densityinhomogeneities, which would be observable by a mismatchbetween the observed center of rotation and the shape cen-ter of the body. Some asteroids have been observed to haveuniformly low slopes on their surfaces, indicating that theymay be covered with regolith, particularly asteroids Toutatisand Kleopatra (Scheeres et al., 1998a; Ostro et al., 2000).Other asteroids clearly have regions of exposed, monolithicmaterial; in particular, Golevka has regions of high slopeon its surface (up to 60°) consistent with bare rock (Hudsonet al., 2000). The recent observation of binary asteroids withthis technique make it possible to determine the mass ofthese objects in addition to their size, shape, and rotationstate (Margot et al., 2001). Once these observations areproperly reduced it should be possible to test a range ofspecific theories of satellite formation and evaluate the fateof asteroidal ejecta in these regimes.

7. FUTURE OBSERVATIONS

Future observations should answer some of the outstand-ing questions concerning ejecta capture and regolith reten-tion on asteroids. Substantial uncertainties about the popula-tion of impactors that collide with asteroids and the processof ejecta generation remain. The generalizations that we are

tempted to draw from a handful of S-type objects may notapply to asteroids elsewhere or asteroids of different types,i.e., craters may look very different on metallic objects andextinct comets. Closeup observations of main-belt asteroidswill ultimately decide whether the dearth of small craterson Eros is due to the lack of small projectiles in the mainbelt or due to postimpact surface processes. The observeddepth, or lack, of regolith on smaller asteroids should di-rectly constrain the strength-to-gravity transition for impac-tors. Direct experimentation, such as that planned for theDeep Impact mission (A’Hearn et al., 1999), will shed lighton low-gravity cratering mechanics. Finally, continued de-velopment and advancement of the mathematical tools,methods, and simulations need to understand regolith andejecta will enable a better understanding of this particularaspect of asteroid science.

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The Fate of Asteroid Ejecta