Scheeres et al.: The Fate of Asteroid Ejecta 527

The Fate of Asteroid Ejecta

D. J. Scheeres The University of Michigan

D. D. Durda Southwest Research Institute

P. E. Geissler University of Arizona

The distribution of regolith on asteroid surfaces has only recently been measured directly by in situ observations from spacecraft. To the surprise of many researchers, most of the classical predictions 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 significant insight 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 to the recent history of such studies, and to indicate the current gaps in our understanding. The chapter begins with a discussion of the initial conditions of ejecta fields generated from impacts on the asteroid surface. Then the relevant physical laws and forces affecting asteroid ejecta, in orbit and on the surface, are reviewed and the basic dynamical equations of motion for ejecta are stated. Some general results and constraints on the solutions to these equations are given, and a classification scheme for ejecta trajectories is given. Finally, recent studies of asteroid ejecta are reviewed, showing the application of these techniques to asteroid science.

1. INTRODUCTION and strings of secondaries. The importance of erasure mecha- nisms, such as seismic shaking (Greenberg et al., 1994) and Whether the debris ejected from impacts on asteroids electromagnetic forces (Lee, 1996), that compete with dy- escapes or reimpacts has important implications for the namical effects to shape the surfaces of asteroids has re- erosion of asteroids, retention and distribution of regolith, cently been emphasized by detailed studies of Eros by dispersal or reaccretion of fragments after catastrophic dis- NEAR Shoemaker. ruptions, and the formation of temporary satellites and per- Rapid advances are expected in our understanding of im- manent moons. Asteroids present complex dynamical envi- pact cratering on diverse objects through in situ experiments ronments because of their low gravitational accelerations, such as NASA’s Deep Impact mission to Comet Tempel 1, nonspherical shapes, complex geological makeup, and di- numerical simulation of ejecta trajectories that employ real- verse rotation states. Additionally, critical parameters related istic shape and gravity models and consider third-body and to the flux and size distribution of impactors and the result- nongravitational forces, and geological evidence from anal- ing initial ejecta fields are only poorly known. Thus the ysis of spacecraft data. The fundamental motivation for the physics and dynamics of regolith processes are complicated study of asteroid regoliths arises from the meteoritics com- and not fully understood. Finally, physical observations of munity and the interpretations of (primordial) asteroid rego- asteroids are only now approaching the resolution necessary liths as observed in the meteorite database. Future motiva- to seriously constrain and delineate between competing tion for ejecta studies will include the necessity of providing theories of the asteroid environment, making the study of a complete mechanical understanding of the asteroid envi- asteroid ejecta a timely endeavor. ronment. Ultimately, a detailed understanding of ejecta dy- Within the last decade we have obtained closeup pictures namics will be crucial to characterize the safety of the or- of asteroids Gaspra, Ida and Dactyl, Mathilde, and Eros to bital environment about asteroids for rendezvous missions, supplement earlier images of the martian moons. Radar and landed operations on asteroid surfaces, and other close- telescopic observations have revealed the shapes and rota- proximity operations. A key issue of concern for any surface tion states of many more objects. Morphological indications operation on an asteroid, such as sampling, will be the trajec- of regolith on these asteroids include blocks, landslides, tories of ejecta disturbed and lofted into orbit during rou- buried craters, and color variations. Observational tests of tine operations, as disturbed regolith may reimpact on the dynamical theories include nonuniform regolith, ejecta block surface with speeds on the order of the surface escape speed distributions and asymmetric crater ejecta blankets, rays, long after being dislodged (Scheeres and Asphaug, 1998).

527 528 Asteroids III

The study of asteroid ejecta is intimately tied to the study of binary asteroid systems relative to the solar tide, jovian of impacts on asteroids [see the reviews by Asphaug et al. perturbations, and collisions in a series of papers, based on (2002) and Holsapple et al. (2002)], transient and long-term the earlier studies of the dynamics of the “Hill problem” orbital dynamics close to asteroids, and the study of natu- (Hénon, 1969). Hamilton and Burns (1991a,b) investigated ral and artificial asteroid satellites (Merline et al., 2002). the limits of stable motion about an asteroid, with a spe- Additionally, the dynamics of asteroid ejecta have many cial interest given to the safety of the planned Galileo flybys similarities to the dynamics of cometary ejecta. In general, of asteroids Gaspra and Ida. These papers, taken together, methods devised for the study of each area will have ap- provide a clear picture of the stability of asteroid binaries, plication to both areas. and place constraints on the stability of asteroid ejecta that Out of necessity, this chapter brings together several di- move relatively far from the asteroid. Such studies are still verse areas of asteroid and dynamical science. Ideally, this continuing, and additional progress in understanding the dy- chapter will serve as a starting point for future investiga- namics of trajectories far from an asteroid have been made tions into the dynamics of ejecta from the surfaces of small (Richter and Keller, 1995; Hamilton and Krivov, 1997). bodies. As such, we have injected many topics into the Chauvineau et al. (1993) and Scheeres (1994) initiated chapter, in some instances without detailed descriptions of the study of dynamics in the near-asteroid environment, the background theory or its development. Thus, to supple- studying the motion of particles and ejecta close to rotating ment the work described herein, we suggest that the follow- ellipsoids. These early studies showed that the near-asteroid ing textbooks be referenced: Melosh (1989) for an intro- orbital environment was fundamentally different from the duction to the basic principles and physics of impacts, environment found in the vicinity of a planet or larger satel- Murray and Dermott (1999) for an introduction to the or- lite. Studies along these lines have continued with the de- bital dynamics of natural bodies, and Szebehely (1967) for tailed analysis of specific asteroid shapes (Geissler et al., an introduction to advanced orbital dynamics theory. 1996; Petit et al., 1997; Scheeres et al., 1996, 1998a, 2000a) and the theoretical analysis of motion in generalized models 2. A BRIEF HISTORY of asteroid gravity fields (Scheeres, 1999). Now this area of study has its first precision set of data with the results of the The study of asteroid ejecta has been considered in the NEAR Shoemaker mission to asteroid 433 Eros (Yeomans books Asteroids and Asteroids II, distributed among chap- et al., 2000; Miller et al., 2001), the fruits of which are already ters on asteroid regoliths (Cintala et al., 1979; Housen et al., being published (Thomas et al., 2001, Robinson et al., 2001). 1979b; Veverka and Thomas, 1979; McKay et al., 1989) and asteroid satellites (van Flandern et al., 1979; Weidenschil- 3. EJECTA GENERATION ling et al., 1989). Yet the study of ejecta and satellite dy- namics about asteroids was only fully validated with the The study of asteroid regolith mechanics and the dynam- discovery of Dactyl in orbit about Ida (Belton et al., 1996). ics of impact ejecta fields must first concern itself with the In recent years the relevance of this topic has continued to mechanics of impact ejecta generation. Weidenschilling et grow, with the rapid rate at which asteroid satellites have al. (1989) noted that there are tight constraints on ejecta been discovered (see the chapter by Merline et al., 2002) speed before all ejecta immediately escape from the asteroid and recent realizations from the NEAR Shoemaker mission and into heliocentric space. Indeed, early estimates on rego- that the small-scale structure of asteroid surfaces are not lith depth (or lack thereof) on smaller asteroids predicted well understood (Veverka et al., 2001; Cheng et al., 2001). little, if any, retained regolith. This view of asteroid surfaces Initial studies of asteroid ejecta, and orbital dynamics has changed with recent observations of asteroids from space- about asteroids, assumed that their dynamical environment craft and radar. In the following we review some basic re- was analogous to, and directly scalable from, planetary sat- sults on the ejecta fields resulting from impact events, with ellite dynamics (van Flandern et al., 1979; Weidenschilling an emphasis on the implications of these models for the ini- et al., 1989). However, Weidenschilling et al. (1989) already tial conditions of an ejecta fragment field. noted that the potential for complex dynamics close to aster- oids existed and proposed that further studies on this topic 3.1. Mathematical Models and be done. Prior to this, Dobrovolskis and Burns (1980) had Scaling-Law Predictions already performed detailed ejecta trajectory analysis for the asteroids Phobos and Deimos, noting that ejecta trajectories One approach to the understanding of generation and were strongly influenced by the rotation state and gravity redistribution of regoliths on small bodies is through theo- field. For these bodies, however, the Mars tidal force is so retical modeling of impacts. These models have been guided strong that there is no direct analogy between those works by observations of crater ejecta and regolith on the Moon and the evolution of ejecta about asteroids. and by the results of numerous laboratory impact experi- Detailed dynamical studies of orbital motion about aster- ments. Detailed models of regolith emplacement and evolu- oids has blossomed since the publication of Asteroids II. tion on small bodies, incorporating quantitative treatments Initial studies focused on the stability of binary asteroids. of cratering rates and ejecta thickness, were considered by Chauvineau et al. (1990a,b, 1991) investigated the stability Housen et al. (1979a,b) and Housen (1981). Those models Scheeres et al.: The Fate of Asteroid Ejecta 529

predicted that for the smallest asteroidal bodies (D < 10 km), highest speeds between one-sixth and one-tenth the impact nearly all impact ejecta escapes and such objects should speed, so that some portion of the crater ejecta may be re- have only thin (on the order of 1 mm) coatings of commi- tained on the surface of the asteroid. nuted debris. With increasing asteroid size, more ejecta is retained and regoliths are predicted to be thicker, on the 3.2. Laboratory Experiments order of hundreds of meters for asteroids ~100 km in diam- eter and larger. In Veverka et al. (1986) the minimum diam- The results of laboratory-scale impact experiments can eter for regolith retention was estimated to be 20 km for provide a useful guide in understanding the generation of icy bodies and 70 km for rocky bodies. Now that the sur- regoliths on asteroids and estimating the amount of debris faces of several asteroids have been imaged at resolutions retained on their surfaces or ejected to escape. Indeed, allowing small-scale surface features to be examined in mathematical models as described above rely in part on data detail, regolith thickness and spatial distribution may be from these experiments. There is extensive literature on more directly estimated as a function of asteroid size, shape, laboratory experiments of impactors, reviewed by Holsapple and rotation state. et al. (2002). From these laboratory experiments essential, The volume of ejecta material and the mass of the largest basic relations for ejecta volumes, speeds, ejecta field orien- fragments excavated from an impact crater on an asteroid tations, and asymmetries have been established. Fundamen- may be assumed to scale with the crater size, although it tal results from these areas can be found in Gault et al. must be recognized that factors including target surface (1963) and are reviewed by Fujiwara et al. (1989). gravity, porosity, layering and structure in the target, and Recent evidence suggests that C- and F-type asteroids impact angle may all play a role in complicating the predic- have remarkably low densities (in the range of 1.2–1.8 g cm3) tions of simple scaling relations. The simplest estimates for and that high porosity probably plays a significant role in ejecta volume may be made by scaling from the apparent limiting the excavation of debris from, and the disruption of diameters of craters (e.g., Lee et al., 1996). Photoclinometry terrain surrounding, craters on these bodies. One possible ex- applied to fresh craters on Ida (Sullivan et al., 1996) indi- planation for this is given in Housen et al. (1999), where cates that a diameter D to depth h ratio D:h is ~6:5. Craters they used a centrifuge and impacts into porous, highly crush- on Eros show similar diameter:depth ratios (Veverka et al., able silicate materials to experimentally simulate crater 2000). Assuming that craters are spherical segments with formation and ejecta deposition on low-density, porous as- depth h ~ D/6.5 and diameter D, their volume is V ~ 0.06 D3. teroids. Their results show that the ratio of ejecta mass (mate- With ejecta volumes estimated in this fashion, the total rial deposited outside the crater rim) to crater mass (crater vol- volume of material ejected from craters larger than 0.5 km ume multiplied by initial target density) steadily decreases diameter on Ida amounts to ~500 km3, which would amount as the target porosity increases. This is due to ejecta speeds to a regolith layer ~130 m thick, if retained and distributed being very low in porous materials, and the fact that much evenly over the ~3800-km2 surface area of Ida. If Ida re- of the crater volume is formed by compression of the target sponds to impacts as a strong, competent object, then ejecta material as opposed to excavation. Large craters on porous would have escaped the surface to space and such an esti- asteroids should exhibit only minor ejecta deposits; centri- mate of regolith thickness is not valid. However, if craters fuge experiments at 250 g in material with porosity 70%, of this size formed in the gravity regime, preexisting rego- the conditions of the formation of largest impact crater on lith might be present, in which case the estimate is a lower Mathilde (Karoo, 33 km diameter), indicates that only 10% limit to regolith depth. As pointed out in Hartmann (1978), of the crater mass will be ejected outside the crater. (For a “Regolith begets regolith.” competing explanation of the Mathilde craters found using From observations of blocks on Ida and a review of pre- numerical experimentation, see the next section.) vious work of blocks on the rims of lunar craters, Moore Finally, recent laboratory work has shown mineral-spe- (1971) and Lee et al. (1996) derive a general relationship cific comminution. This hints that some mineral compo- between the largest ejecta block size, L, and crater diam- nents isolated as chondrules or phenocrysts may be ejected eter, D, for craters in rocky targets, L ~ 0.25 D0.7, where L at different speeds, which might be of interest in segregat- and D are in meters. This ejected material ranges in size ing mineralogically distinct portions of regolith (Hörz et al., from the largest ejecta blocks down to dust-sized particles, 1985; Durda and Flynn, 1999). with a cumulative mass distribution expressed as N(>m) = Cm–b, where b commonly ranges between 0.8 and 0.9, C is 3.3. Numerical Experiments a normalizing constant, and m is the cumulative mass frac- tion. A fraction of this material is jetted from the impact site Hydrocode experiments incorporate gravity and what is at high speed or spalled from the near-surface interference known about the fracture mechanics of rock into numerical zone of the growing crater at speeds up to half the impact simulations of the contact, compression, and excavation speed of the projectile (Melosh, 1989, p. 73). With typical stages of an impact. Hydrocode simulations can be used to main-belt impact speeds of ~5 km/s, most of this spalled model impacts at scales far too large to be directly accessible material immediately escapes the target asteroid. Excava- through laboratory experimentation (Benz and Asphaug, tion flow speeds are much lower, however, with even the 1999). 530 Asteroids III

Using such hydrocode techniques, Asphaug et al. (1996) jectile and the target [see Geissler et al. (1996) for a spe- modeled the formation of craters on Ida ranging in size from cific application], g is the gravitational acceleration, and α is 60 m to 8 km, and compared the ranges of ballistic ejecta a scaling exponent that depends on the target properties but from these craters with the area of seismically disturbed falls in the range of 3/7 to 3/4. Similar arguments (Housen regolith surrounding them. The results confirm that ballis- et al., 1983) suggest that ejecta follow a power-law speed α α tically emplaced ejecta deposits around small craters on Ida, distribution with an exponent ev = 6 /(3 – ). When the and by inference, on asteroids of similar size and compo- target has substantial cohesive strength, or in the limit of sition, should be diffuse and widespread, so that bright halos small impactors or small target sizes, the volume of the around small craters could be due to seismic disturbance crater scales linearly with the mass of the impactor. This is of surrounding regolith rather than continuous ejecta blan- the strength regime, exemplified by shooting at boulders kets. Larger-scale impacts create a “megaregolith”-like zone with a rifle. Ejecta velocities tend to be higher than those of of intense fracturing within a depth approaching one cra- gravity-dominated craters but the volume of material exca- ter diameter below large craters and appear to be able to vated is much lower. As the size of the impact increases, deposit a significant amount of debris in irregular blankets even cohesive materials gradually transition to the gravity around them. regime due to the scale- and strain-rate-dependence of mate- Numerical experiments also give insight into the role that rial strength. The transition between strength- and gravity- high porosity may have in modifying the resultant impact dominated impacts depends on the strength and gravity of ejecta field. In contrast to the laboratory results described the target, but for asteroids that are a few tens of kilometers above, Asphaug (2000) shows that high porosity can lead to in diameter the transition is expected to occur at crater di- high ejecta speeds, which also matches the observed lack of ameters in the range of 10 to 1000 m. It may seem surpris- impact blocks on the Mathilde surface. This is due to energy ing that the relatively weak gravitational grasp of an asteroid confinement in the immediate crater fracture zone, due to the could control the formation of kilometer-sized craters. The inability of the porous asteroid material to efficiently trans- transition to gravity-controlled cratering is aided by the mit the impact energy though the entire body. In Asphaug presence of regolith and rubble left by earlier impacts, and (2000) it was shown that the crater Karoo on Mathilde by fragmentation of even strong targets by shock waves that would have launched almost all its ejecta faster than 30 m/s, precede the crater excavation (Asphaug and Melosh, 1993; sufficient for escape, if Mathilde was 50% porous. Nolan et al., 1996).

3.4. Strength and Gravity Regimes 3.5. Impact Statistics and Asteroidal Erosion

Two of the most important questions regarding regolith Erosion of asteroids competes with regolith generation/ generation and ejecta escape are (1) how much ejecta is retention and can sometimes yield important constraints on created during any given impact, and (2) at what speeds the the ages of small gravitationally bound objects (e.g., Geissler ejecta are launched. Calculation of crater volumes and the et al., 1996). A knowledge of impactor size distributions, velocities of the ejecta expelled relies on scaling the out- impact collision probabilities, and impact speeds is needed comes of laboratory and field experiments to vastly different in order to estimate ejecta generation and escape rates and sizes and gravitational accelerations. The theoretical basis timescales for the creation and destruction of ejecta blocks. for such scaling is dimensional analysis, introduced by Hols- These quantities are in general poorly known and consti- apple and Schmidt (1982) and Housen et al. (1983) (see tute the greatest uncertainties in erosion rate calculations. review by Holsapple, 1993). This approach groups variables The size-frequency distribution is well constrained by into dimensionless ratios to reduce the complexity of arbi- telescopic observations only for the largest asteroids (e.g., trary expressions relating impactor and target properties to Van Houten et al., 1970). Smaller size ranges must be esti- crater volumes or ejecta speeds. In cases where the number mated by (1) extrapolation of power laws for the observed of variables is one greater than the number of dimensions asteroids, (2) collisional models predicting the production (mass, length, time), the expression can be reduced to a of small fragments, and (3) observations of the size distri- simple power-law relationship between the variables. bution of the craters produced by these small impactors on By this analysis, the volume of a crater produced by an asteroid and planetary surfaces. Simple calculations assum- impactor of a given mass scales neither with the energy nor ing that the impact efficiency is independent of target size the momentum of the impact, but varies according to a yield a differential power law index of –3.5 (Dohnanyi, power law with an exponent between 1 (momentum) and 1969), i.e., the relation between the number of fragments 2 (energy). For strengthless targets, or in the case of large n and their radius r should follow the power law dn(r) ~ impacts in the gravity regime, the volume V of a crater r–3.5. However, Galileo observations of small craters on created by an impactor of diameter D and speed U is given Gaspra and Ida suggest a much steeper size distribution for by (Holsapple and Schmidt, 1982) small impactors in the main belt. The differential power- law index for small impactors on these objects (<175 m V(D) = Ag–α U2α D3 – α (1) diameter) is estimated to be near –4.0 (Belton et al., 1992; Chapman et al., 1996a,b). For such steep size distributions where A is a constant that includes the densities of the pro- (indexes >4), infinite mass is found in the smallest frag- Scheeres et al.: The Fate of Asteroid Ejecta 531

ments. Thus some minimum projectile size must exist, be- 10–6 yr–1, or a mean time between impacts of ~370,000 yr. low which a steep size distribution no longer applies. This Ejecta blocks launched from the surface of Eros may not be cut-off size threshold is poorly constrained by observations, dynamically stable for such long times, so Merline et al.’s and determines whether asteroid erosion is dominated by (1999, 2001) result of not finding any orbiting boulder big bites taken during large impacts, or if asteroid surfaces larger than 10 m does not, unfortunately, constrain the time are mainly sandblasted by particles that are centimeter-scale of formation of the last 200-m-scale crater on Eros. or smaller. An upper limit to this minimum impactor size The fraction of ejecta that escapes from an asteroid dur- is ~1 m, the size of a projectile that would produce craters ing a given impact depends on the target size and strength in the diameter range of 10 to 100 m (the resolution limit and the size of the impact. For both the strength and gravity of the Galileo observations). regimes, the mass eroded per mass of impactor is indepen- The rate and efficiency of impacts depends upon the size, dent of the size of the impactor (Geissler et al., 1996). location, and orbit of the target asteroid. The intrinsic col- Impacts into strong targets impart ejecta with speeds much lision probability and the distribution of impact velocities greater than the escape velocity of a typical asteroid. Al- for any specific target can be calculated by integrating these though only a small fraction of ejecta reaches escape speed quantities over the population of asteroids on intersecting during gravity-dominated cratering, erosion of a gravitation- trajectories. Many estimates of intrinsic collision probabili- ally bound rubble or sand pile is much more efficient than ties and impact velocities have been made using both direct that of a coherent object of similar size. Because craters numerical integration (Marzari et al., 1996, 1997; Dahlgren, created in soft targets are much larger than corresponding 1998) and statistical methods (Wetherill, 1967; Greenberg, strength-regime craters, the total volume of ejecta that es- 1982; Farinella and Davis, 1992; Bottke et al., 1994; Vedder, capes in this case can be much greater than the volume of 1996; Dell’Oro and Paolicchi, 1998). For main-belt aster- material excavated by a similar impact into a strong target. oids, collision probabilities are typically on the order of For example, Geissler et al. (1996) found that the mass 10–18 km–2 yr–1, and impact velocity distributions are broad, eroded (ejected and escaped) from a soft Dactyl (made of non-Gaussian, and often contain spikes. For the purposes sand) per unit mass of impactor should be at least 36× of evaluating impact efficiency, a value between the mean greater than that eroded from a Dactyl made of solid rock. impact velocity and the root mean square impact velocity should be used, depending on the target crater scaling expo- 3.6. Surface Launch Conditions nent α (equation (1)). As an example we calculate the rate of production of For studying the subsequent motion of ejecta the most 10-m-scale ejecta blocks on Eros. Merline et al. (1999, crucial item is its initial position and velocity relative to the 2001) completed a search for satellites around Eros during asteroid surface. Assume that the crater is measured from the NEAR Shoemaker flyby and rendezvous and, at a 70% a nominal vector r0 on the asteroid surface, that the aster- confidence level, found no objects near the asteroid with oid surface normal vector at that point is nˆ z, and that there a diameter larger than 10 m, and a 95% confidence level are two orthogonal unit vectors nˆ x and nˆ y tangent to the for diameters greater than 20 m. Estimates of the rate of asteroid surface. The location of a single ejecta, as measured production from impacts on Eros and knowledge of their from the asteroid center of mass, is dynamical lifetimes could thus be compared with the ob- δ served lack of such objects to constrain in an iterative fash- r = r0 + r (2) ion the present impact rate on Eros. Bottke et al. (1995) give δ =++ an intrinsic collision probability for near-Earth asteroids r xnˆˆˆxyzyn zn (3) (NEAs) hitting other NEAs of ~15 × 10–18 km–2 yr–1 at a most probable relative speed of ~18 km s–1. Multiplying this where |δr| << |r| in general. The ejecta velocity relative to the by the cross-sectional area of Eros and by the number of crater site is then specified as impactors capable of making 200-m and larger craters [the =++δβ βλβλ minimum crater size capable of producing 10-m ejecta VeeV (rn ) cos ˆ sin cos nˆ xsin sin nˆ y(4) blocks, according to the ejecta-block scaling law of Lee et al. (1996)] on Eros yields the block production rate. A where the angle β and λ define the direction of the velocity 2 sphere with the same 1106-km surface area of Eros would vector relative to the crater normal and the ejecta speed Ve have a radius of r = 9.4 km, so r2 = 88 km2 (the factor of π will depend on its position within the crater. Nominal as- in the cross-section is not required since it is included in sumptions are that β = 45°, and that λ ∈ [0,360]°. the instrinsic collision probability). Estimates based on For some dynamical computations the ejecta velocity strength regime cratering in soil (Holsapple, 1993) indicate must be transformed into an inertially oriented frame. Then that projectiles roughly 2–10 m in diameter are capable of the asteroid rotational velocity vector Ω must be introduced producing the requisite craters. The NEA population may 9 Ω contain ~2 × 10 objects of this size (Neukum et al., 2001; VI = Ve + × r (5) Rabinowitz et al., 2000; Ivanov et al., 2002); herein lies the Ω greatest uncertainty in such calculations. Multiplying, Pi × Note that is not necessarily constant and can have a sig- A × N = 15 × 10–18 km–2 yr–1 × 88 km2 × 2 × 109 = 2.7 × nificant time variation for bodies in nonuniform rotation. 532 Asteroids III

4. EJECTA DYNAMICS shape model and computes the gravity potential (and its attendant partials) directly from this model. Even though Once the ejecta has left the asteroid surface, it becomes the individual computations needed to compute the poly- subject to one of the more strongly perturbed environments hedron gravity field are more involved than those used for that can be found in the solar system. Any serious study of the mascon approach, the overall efficiency of a polyhe- asteroid ejecta must start with dynamical models that cap- dral gravity field computation is often better than a mascon ture the main elements of these perturbations, since each approach, since the mascon approach must sum over the of them can skew the global nature of ejecta dynamics into entire volume of the body while the polyhedron approach significantly different evolutions than would be expected must only sum over the surface elements of the body. The from the simple application of two-body orbital dynamics. polyhedron approach also has the advantage of giving a direct indication of whether the point is inside or outside the 4.1. General Force Models asteroid. The ellipsoid model is useful for situations when a precision model of a gravity field is not needed. Its advan- For the general study of dynamics about asteroids one tages are that it is relatively simple to code, has no singu- must, at the onset, determine which force perturbations will larities (such as are found for collections of point masses), be significant for the system. Due to the wide variety of and can be specified based on light-curve analysis alone. shapes, sizes, densities, rotation states, and orbits found for In the following, the gravitational force potential is speci- asteroids, this determination must usually repeated for each fied as U(r) where r is the position vector relative to the new asteroid. asteroid-fixed frame. The gravitational attraction acting on 4.1.1. Gravity fields. Several approaches to the model- a particle is ∂U/∂r. The potential U is often split into the ing of asteroid gravity fields are available. In general, the main contribution of the monopole (µ/r) plus the perturba- most accurate formulations for a gravity field are spherical tion contribution (R) as harmonic expansions where the gravity coefficients are µ measured from spacecraft radiometric tracking (Yeomans et U()r =+R ()r (6) r al., 2000; Miller et al., 2001). Despite the high accuracy of these fields, they are in general inapplicable to the study However, on an asteroid surface the perturbation contribu- of ejecta motions that arise from the surface of an asteroid tion can often compete with the main contribution, and thus due to the divergence of the expansion within the circum- this form is only used for notational convenience. scribing sphere surrounding the asteroid (the circumscrib- When relatively far from an asteroid, MacCullagh’s for- ing sphere is the sphere of minimum radius, centered at the mula can be used to approximate the gravity field of a gen- asteroid center of mass, that encloses the asteroid). A modi- eral asteroid with a reasonable degree of accuracy (Danby, fication to this technique using ellipsoidal harmonics is 1992) available (Garmier and Barriot, 2001) that decreases the region of divergence to within the circumscribing ellipsoid 3 that fits about the body. Even this, however, does not com- 2 − 2 δ + µR C20 1 cos pletely eliminate the problem, as there will still be signifi- R()r = o 2 (7) r3 cant regions of divergence when close to or within this cir- 2 δλ 32C22 cos cos cumscribing ellipsoid (Garmier et al., 2002). To overcome this, recourse is usually made to the known δ closed-form gravitational potentials, a class that includes the where C20 and C22 are gravity coefficients, is the decli- sphere, the general ellipsoid (Danby, 1992), the tetrahedron nation, and λ is the body-fixed longitude. This formulation (Werner, 1994), and a general polygonal shape (Werner and assumes that the coordinate system is aligned with the in- Scheeres, 1997). The main restriction to these potentials is ertial axes, with the maximum inertia axis along z and the that the mass density is assumed to be constant, or at the minimum axis along x. In many cases, consideration of just very least is constrained to follow a very specific mathemat- this contribution to the gravity field of the asteroid can ical variation (which in general may not be physical). The adequately capture the major departures of orbit dynamics main approaches to gravity field modeling have used collec- from the simple Keplerian case. It must be noted that equa- tions of point mass gravity potentials, collections of tetra- tion (7) cannot be used on the surface of the asteroid or hedron gravity potentials (making up a single, polyhedral within the circumscribing sphere, as it will give nonphysi- shape), and the use of the simple ellipsoidal shape model. cal values of potential and acceleration. The point mass (or mascon) approach consists of tak- 4.1.2. Rotation state. There are two classes of rota- ing a defined shape model of the asteroid and populating tional motion that must be considered. The first, simplest, its interior by a distribution of point masses, properly scaled and most common is asteroid rotation about its maximum to yield the correct total mass. This approach can lead to axis of inertia. It is well known that this is a stable rota- regions of poor gravity field computation on the surface and tional end state for an asteroid when dissipation of energy is is inefficient if a high resolution is desired (Werner and taken into account, as it provides the minimum energy rota- Scheeres, 1997). The polyhedron approach takes a polygon tional state for a given value of angular momentum (Burns Scheeres et al.: The Fate of Asteroid Ejecta 533

and Safronov, 1973). To completely specify the rotational pressure force to solar gravity force acting on the ejecta. A dynamics for a uniformly rotating asteroid requires the rota- reasonable assumption for ejecta motion relative to the as- tional velocity vector and a phase angle for the asteroid. teroid is |r| << |d|, leading to the simplified potential The rotational angular momentum of the asteroid will be subject to solar torques and nongravitational effects, and can 1 be altered by impact events or planetary flybys. Still, over 1 −×−drˆ µβ− d long periods of time it is acceptable to treat such a rotation = S()1 + VS (9) state as a constant. d 1 2 rr×−3() drˆ × More interesting, but rarer, are cases where the aster- 2d2 oid has a nonuniform rotation state, usually corresponding closely to the general solution to Euler’s equations for a torque-free rotating body. Examples of such bodies include 4.1.4. Other nongravitational forces. For specific pur- the asteroids Toutatis, Mathilde, and Alinda. Most, if not poses, other nongravitational forces may also be modeled. all, of the asteroids observed to have a nonuniform rotation This is especially true for the modeling of comets, where state are slow rotators, which makes physical sense as the there is significant gas pressure that emanates from the nu- time to relax to uniform rotation scales with the generalized cleus surface (Weeks, 1995; Scheeres et al., 1998b). There rotation period cubed. Thus, a body such as Toutatis has a has been speculation in the past on an outgassing environ- predicted relaxation time longer than the age of the solar ment for asteroids as well, but evidence for this has not been system (Harris, 1994). In modeling ejecta dynamics about detected to date, and the perturbations that would result from a nonuniform rotator the most efficient modeling approach such outgassing would be very small. Another nongravi- is to use the classical solution for rotational dynamics in a tational force that has been recently considered in many torque-free environment (MacMillan, 1960); a summary of contexts is the Yarkovsky effect (Bottke et al., 2000, 2002; such an application is given in Scheeres et al. (1998a). Rubincam, 2000), which essentially consists of a thermal In the following we specify the asteroid angular veloc- imbalance on a body. Since ejecta will have definite shapes ity vector as Ω. For the case of a uniformly rotating aster- and rotations, the Yarkovsky effect may be able to modify oid, Ω is constant in both the asteroid-fixed frame and in a particle’s orbital dynamics if it falls into a long-term stable an inertially oriented frame since it is aligned with the aster- orbit. There have been no studies performed on the appli- oid’s total rotational angular momentum vector. When cation of this force to ejecta dynamics to date, however. modeling an asteroid with a nonuniform rotation state, the vector Ω is no longer aligned with the asteroid’s rotational 4.2. Equations of Motion angular momentum vector, but has a precession and nuta- tion relative to this vector. If we model the asteroid non- The measured force parameters and models define the uniform rotation using the solution for torque-free motion, dynamical problem of motion in the vicinity of the asteroid. the angular velocity vector Ω is a periodic function of time Depending on the force parameters, the character of motion in the asteroid-fixed frame, i.e., an observer sitting on the in these equations will take on a variety of forms. Specifi- asteroid tracing out the path of this rotational velocity vector cally, for smaller asteroids the regions where solar and grav- would see that it repeats itself exactly after a characteristic ity field perturbations are important can coincide, leading period (which is a function of the body’s moments of iner- to very complicated dynamics. For larger asteroids these tia, rotational energy, and rotational angular momentum). regions of influence do not coincide, making it possible to The implications of this are discussed in greater detail in distinguish between a far-field regime dominated by solar Scheeres et al. (1998a). effects and a close-field regime dominated by asteroid grav- 4.1.3. Solar effects. When far from the asteroid a par- ity and rotation. Of course, a single ejecta trajectory can ticle must contend with strong perturbations from the solar transition between these regimes as it passes from apoapsis gravity and radiation pressure. For precision computation, to periapsis and back again. detailed models of the ejecta shape and interaction with The relevant equations of motion in an inertially oriented solar radiation could be developed if desired. This level of frame for the ejecta relative to the asteroid can be stated as detail is not always necessary for understanding the basic (Scheeres et al., 2001) effect of the solar radiation pressure on the ejecta. The solar gravity and radiation pressure forces are de- ∂V()r rived from a force potential written as r = I (10) I ∂ rI 1 − β V = µ (8) SSdr+ µ µβ V()r =+R ()r +S drˆ ×− r d2 where µ is the gravitation parameter of the sun, d is the (11) S 1 µ asteroid position vector from the Sun, r is the ejecta position S rr×−3 () drˆ ×2 vector from the asteroid, and β is the ratio of solar radiation 2 d3 534 Asteroids III

ˆ where d = |d|, d = d/d, and rI denotes that the position vec- 5. THE DYNAMICAL FATE OF EJECTA tor is referenced to an inertial, nonrotating frame. These equations are entirely general and only incorporate a few The equations of motion reviewed above can lead to assumptions (noted above). Note that it is necessary to have extremely complicated motion that cannot in general be a solution for the motion of the asteroid relative to the Sun; solved analytically. However, there are many insights to be however, it is generally sufficient to use a Keplerian orbit had from the study of these equations, both numerically for the computation of d and dˆ . Exceptions occur when the using the full models and analytically using suitably simpli- asteroid has a close encounter with a planet, but this would fied models. In all such investigations it is important to also require the addition of the tidal effect of that planet remember the guiding dynamical questions: What is the dy- on the motion of a particle and on the rotation state of the namical evolution of an impact ejecta field, what fraction of asteroid, situations we do not directly discuss here (Chauvi- an ejecta fragment field will escape, what fraction will reim- neau and Mignard, 1990b; Scheeres et al., 2000b). pact, and what fraction will be captured in a transient orbit? 4.2.1. Perturbation formulation. In the course of ana- lyzing motion about an asteroid, it is often convenient to 5.1. Stability of Synchronous Motion use the constants of motion of the two-body problem in order to characterize the strength and effect of the perturba- First, a special note must be given on the stability of syn- tions acting on the ejecta. The classical orbital elements can chronous (1:1) motion about an asteroid. In the past, many be defined as the semimajor axis, a, the eccentricity, e, the authors have made a tacit assumption that synchronous or- inclination, i, the longitude of the ascending node, Ω, and bits about an asteroid would follow the basic pattern found the argument of periapsis, ω. Frequently, the true or mean for geosynchronous orbits (such as described in Kaula, 2000, anomaly of the orbit, f or M respectively, are used to re- p. 54) with two stable, synchronous orbits and two unstable, place the classical sixth orbit element of the time of peri- synchronous orbits. Application of these assumptions lead apsis passage. The variation of these constants due to orbital to predictions for the stability of orbital motion about aster- perturbations are generally specified using the Lagrange oids and to the tidal evolution of asteroid satellites that are planetary equations with a perturbation function. An ex- not valid. In the following, “stable” means that a trajectory tended discussion of these equations can be found in Brou- close to the synchronous orbit will remain close to it for wer and Clemence (1961). For our system, the general force arbitrarily long periods of time, while “unstable” means that perturbation potential can be given as V(r) – µ/|r|. a trajectory close to the synchronous orbit will rapidly leave 4.2.2. Asteroid-fixed frame. For the analysis of ejecta its vicinity. motion close to the asteroid surface it is more convenient In Scheeres (1994) it is shown that synchronous orbits to shift the equations into an asteroid-fixed frame. In doing about asteroids are unstable in general. For an asteroid so we must allow for the fact that the asteroid is rotating whose shape is spheroidal, or for Earth, we find four syn- with an angular velocity vector Ω with respect to inertial chronous orbits, two of which are hyperbolically unstable space, so the equations of motion relative to the asteroid and two of which are linearly stable. An analogy can be have the form made with the restricted three-body problem, where for a small mass ratio we find three synchronous orbits that are ∂V()r hyperbolically unstable and two orbits (the so-called equi- r +×+Ω rr2Ω ×+××=Ω Ω r (12) ∂r lateral points) that are linearly stable. Now, as is well known, in the restricted three-body problem these stable If the asteroid is uniformly rotating, Ω = 0,Ω is constant, equilateral points become unstable if the mass ratio between and the equations simplify. On the other hand, if the asteroid the primaries is increased to greater than ~0.1 (cf. Szebehely, is in nonprincipal axis rotation, then the vectors Ω and Ω 1967). A similar phenomenon occurs in the asteroid prob- are time periodic. lem, where the stable synchronous orbits become unstable if These equations of motion have no integrals of motion in the body’s shape is sufficiently elliptic (more precisely, this general. However, for motion close to an asteroid we can involves both the asteroid’s rotation rate and its ellipticity). often disregard the solar perturbation terms. Then, if the aster- Furthermore, the instability timescale of these orbits is on oid is in uniform rotation, equation (12) is time-invariant the order of the rotation period of the asteroid, and hence and a Jacobi integral exists operate very quickly. Thus, particles placed near a 1:1 reso- nance with a rotating asteroid will in general either impact 1 1 µ with or escape from the asteroid, usually in a matter of J =×−×××−−r rrr()()Ω Ω R (13) 2 2 r hours or days at most.

This integral is often helpful in constraining and under- 5.2. Final Outcomes for Ejecta standing the limits on motion near the asteroid surface. In application, this integral can be used just as the Jacobi inte- We can delineate several distinct final outcomes for gral is used in the restricted three-body problem (Szebehely, ejecta trajectories whose initial conditions lie beneath or on 1967) and was used extensively for the analysis of Phobos the surface of an asteroid. Using a Keplerian dynamics (Dobrovolskis and Burns, 1980). model applied to a spherical asteroid there are three dis- Scheeres et al.: The Fate of Asteroid Ejecta 535

Class I Class II

q1

Class III

q1, q2, . . .

Class IV Class V q1

Fig. 1. The five classes of ejecta fate.

tinct classes of motion. First, if the orbital energy is nega- given a periapsis passage qi, if qi + 1 does not ensue, then tive then the ejecta will reimpact as periapsis is initially on the ejecta has escaped. Finally, if the sequence never ter- or beneath the asteroid surface. Second, if the energy is zero minates (i → ∞), then the ejecta is in a stable orbit about or positive the ejecta will escape. Third, a subset of the the asteroid. second class of escaping ejecta may eventually reimpact on Based on this understanding, we tender the following the asteroid after an extended period of time in orbit about classifications (see Fig. 1): Class I — Immediate reimpact: the Sun. We will discount this third class, however, as it is Ejecta reimpacts with the surface prior to first periapsis pas- practically indistinguishable from other impacts. sage. Class II — Eventual reimpact: Ejecta does not reim- As additional perturbations are considered, the possible pact at the first periapsis passage, but eventually reimpacts classes of motion expand. It is useful to use periapsis pas- in the future. Class III — Stable motion: Ejecta is placed sage relative to the asteroid to delimit between different into a long-term stable orbit about the asteroid. Class IV — classes of motion. At launch the ejecta are starting from an Eventual escape: Ejecta has at least one periapsis passage ≥ initial radius r0 q0, since in general the initial periapsis by the asteroid before it escapes. Class V — Immediate (q0) lies beneath the body’s surface. In the absence of per- escape: Ejecta escapes from the asteroid prior to its first turbations the next periapsis passage q1 will either equal q0, periapsis passage. and thus will be an impact, or will never occur, indicating Classes I and V are clear carryovers from the nonper- escape. When force perturbations are incorporated, or even turbed case, and we expect most ejecta to fall into these if nonspherical shapes are allowed, it becomes possible for two catagories. The fraction of ejecta that fall into classes II, multiple periapsis passages to occur. We denote these as a III, and IV is an important consideration for understanding series qi; i = 0, 1, 2, …. Associated with each periapsis pas- the formation of asteroid regolith and asteroid binaries. sage is the periapsis vector, qi, representing the periapsis Given a specific system it is relatively easy to find regions location in the asteroid-fixed space. If we denote the set of of ejecta initial conditions that fall into classes II and IV. ∈ points that constitute the asteroid body as B, then if qi B In terms of celestial mechanics and astrodynamics, it would the sequence stops and an impact has occurred. Conversely, appear to be very difficult to place a particle into class III 536 Asteroids III

(a) 6 m/s 5.3. Impact and Escape Conditions 7 m/s 60 8 m/s 9 m/s By applying analytical theories to the motion of ejecta 10 m/s Asteroid it is possible in many instances to immediately determine if 40 an individual ejecta particle will fall into class I or V, based only on its initial conditions. Using such determinations can 20 greatly decrease the amount of computational effort needed to evaluate the outcome of a high-resolution impact crater-

0 ing event. Also, such methods can directly compute the fraction of an ejecta fragment field that falls within these classes, and hence provides an indication of the fraction that –20 may reside in classes II–IV. In Scheeres et al. (1996, 1998a) a number of analytical results directly pertaining to the –40 computation of reimpact conditions and escape conditions are given. Specific results developed in these papers include

–60 the 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, –60 –40 –20 0 20 40 60 launched normal to the surface, will have sufficient energy to escape the asteroid. The guaranteed return speed is the (b) 8 m/s maximum speed an ejecta can have while still being ener- 10 m/s getically trapped by the zero-velocity curves surrounding 100 11 m/s 12 m/s the asteroid (see section 5.6). These methods have also been 13 m/s 14 m/s applied to the Eros dataset, which is definitive since all the Asteroid force model parameters have been measured (Yeomans et 50 al., 2000; Miller et al., 2001). For Eros the escape speeds range from 3.3 to 17.3 m/s over its surface. This large varia- tion is due to the combined shape/gravity field variation and

0 the rapid rotation rate of the asteroid. At the other end of the spectrum, the guaranteed return speeds computed over the surface of Eros range from 1 to 5 m/s, but are always less than the escape speed at any particular point. Many of –50 these ideas can be developed in additional detail, and can provide sharper conditions on the fraction of an ejecta frag- ment field that will immediately escape, or that will redistri- –100 bute itself on the asteroid.

5.4. Transient Classes –100 –50 0 50 100 Of particular interest are classes II–IV, as they define the Fig. 2. Effects of location and ejection speed on ejecta trajecto- space where interesting things can happen to an ejecta frag- ries launched from a uniformly rotating asteroid shape. (a) Launch ment field. If we draw a “spectrum” of outcomes, class III from the leading edge of an asteroid; because of the asteroid’s rota- will lie at the intersection between classes II and IV, as it is tion, the local escape speed is lower. (b) Launch from the trailing the limit of these cases. Thus, one of the fundamental ques- edge of an asteroid; because of the asteroid’s rotation, the local tions concerning the fate of asteroid ejecta is how a particle escape speed is higher. can be placed into one of these transient classes, and what its subsequent evolution will be. Additionally, if a fraction of an ejecta fragment field falls into class II or IV for an ex- due to the nature of these dynamical systems; however, tended period of time, there is a higher probability that addi- there are other physical forces that can cause a particle to tional perturbations or impacts may push it into class III, transition into a stable orbital motion, and these will be creating a binary asteroid. reviewed below. Figure 2 shows the trajectory evolution of 5.4.1. Problem of initial capture. The basic dynamical a number of different ejecta particles launched off an as- problem is how to generate class II and IV ejecta, and sub- teroid at different speeds. We note that changes in initial sequently transition these into class III ejecta. The real prob- speed or location on the asteroid can have dramatic conse- lem is not whether such trajectories exist, as we can firmly quences for the final state of the ejecta. establish the existence of trajectories that fall into class III. Scheeres et al.: The Fate of Asteroid Ejecta 537

Specifically, families of unstable periodic orbits and equi- veloped (Scheeres et al., 1996, 1998a, 2000a) and, at the librium points (in the asteroid-fixed frame) exist close to least, can be used to establish the existence of transient the asteroid surface. Each of these orbits has a stable mani- orbits of extremely long duration. fold that asymptotically approach these special solutions. 5.4.3. Mechanisms for capture into stable orbits. There It can be shown that many of the stable manifolds of these are many mechanisms that have been hypothesized that lead objects intersect (or emanate from) the surface of the aster- to ejecta becoming captured into stable orbits. We will pro- oid, and hence provide exact initial conditions that lead to vide a very brief summary of these approaches. Issues of orbital capture (Scheeres et al., 1996). The problem with long-term stability and the lifetime of such orbits are not con- these solutions, of course, is that they are unstable and the sidered, but are discussed in more detail in Merline et al. set of initial conditions that leads to capture is vanishingly (2002). small. Thus, the real question is whether there are any signi- Direct initial condition generation. In this scenario a ficant regions of initial conditions that lead to long-term, parent asteroid is subject to an intense impact event, which trapped orbits about an asteroid. Again, the answer here shatters and disperses the original body into many frag- appears to be yes, but the proof is not as direct, and the full ments, all imparted with a range of speeds. In general, the extent of initial conditions that actually lead to such cap- smaller particles have higher speeds and the larger have ture has yet to be fully explored. slower speeds. Given such a random distribution of particle 5.4.2. Methods of analysis. There are several different positions and speeds it is probable that some of the frag- approaches to determining if a particle falls into one of the ments will be placed into mutually bound orbits as they transient classes. The first is direct numerical simulation of escape (indeed this postdisruption environment can even discretized elements of the ejecta field. This approach, used influence their motion during the short period when the by Geissler et al. (1996) in studying the evolution of ejecta asteroid disperses), leading to primitive binary systems. about Ida, provides definite results, subject to modeling Such bound orbits will have large eccentricities in general, assumptions used in setting up the computations, and allows but assuming long-term stability against impact and escape, for the use of a full perturbation model. It is limited by the energy dissipation (i.e., tidal effects with energy dissipation) finite number of ejecta that can be propagated and due to can cause the orbits to circularize over time, leading to the the discrete nature of each propagation. Indeed, in an actual types of stable binaries now being found. Hartmann (1979) ejecta field we expect a near continuum flow of particles, first suggested this scenario, which has been investigated which should in general lead to higher probabilities for cap- analytically by Weidenschilling et al. (1989), and more re- ture into transient dynamical situations. cently has been simulated by Durda (1996), Dorresoun- Analytically motivated approaches can give greater in- diram et al. (1997), and Michel et al. (2001). They have sight into the evolution of larger numbers of particles and found that small numbers of bound asteroid pairs do result, can model the ejecta field as a continuous flow in some but these studies have not addressed the long-term stability situations, at the cost of lost precision in the computed tra- and evolution of these pairs. jectories. In Scheeres and Marzari (2000) an averaging Mass shedding in tidal flybys. Additional mechanisms approach is used that provides analytical solutions to ejecta not involving impacts have also been suggested to increase evolution following ejection from the surface of a small the rotation rate of asteroids to the point of mass shedding. body. Their approach only incorporated solar radiation pres- Richardson et al. (1998) and Bottke et al. (1999) numeri- sure perturbations, but could be generalized to include other cally simulated the tidal disruption of asteroids modeled as effects. With this approach it is possible to rapidly compute “rubble piles” (see Richardson et al., 2002) composed of the evolution of ejecta fields, which allows for more pre- numerous equal-sized spherical components encountering cise estimates on the fraction of an ejecta fragment field Earth, and found that rotational spinup frequently induces that is injected into a transient class, potentially allowing debris to be cast off the primary bodies. In many cases, the for direct computation of probabilities of different outcomes shed fragments were found to go into initially bound orbits from a given ejecta field. around the progenitor. Bottke and Melosh (1996a,b) and Application of advanced understandings of dynamical Richardson et al. (1998) have shown that tidal disruption systems could also be used to evaluate the likely outcomes can create enough satellites in the NEA population to ex- for an ejecta fragment field. As mentioned earlier, the space plain the statistics of doublet craters seen on the terrestrial around an asteroid is filled with periodic orbits, both stable planets. and unstable, each of which have manifolds that can influ- Mutually impacting ejecta. A postimpact ejecta field ence the dynamical flow of an ejecta fragment field. Recent will have a distribution of particle sizes and speeds resem- advances in the application of dynamical systems theory to bling, in some aspects, a continuous field distribution. Thus, spacecraft trajectory design (Koon et al., 2000) could also it is likely that mutual impacts between elements of the be brought to bear on the evolution of asteroid ejecta, and ejecta field will ensue, and that these slow-velocity impacts provide qualitative descriptions of ejecta field flow that may will mutually alter the trajectory of the particles, in some allow for specific quantitative predictions in some cases. cases leading to capture orbits. Weidenschilling et al. (1989) Initial approaches to this sort of application have been de- considered this mechanism in an analytical argument, and 538 Asteroids III concluded that satellites formed from reaccreted ejecta are tion. Perhaps the most interesting results, related to the life- expected to be small and found in prograde orbits. Durda time and evolution of an asteroidal satellite, are found in and Geissler (1996) simulated impact ejecta fields to search Davis et al. (1996) and based on the work by Geissler et al. for such impacts, but did not find any that evolved into (1996), where it is posited that Ida and Dactyl may actually stable trajectories. Their approach used 1000 ejecta par- be in an equilibrium state, exchanging mass between the ticles, which may be too few to reliably find such outcomes. bodies, driven by impacts and ejecta field evolution on each This approach becomes more likely to yield stable trajec- body. As the statistics on asteroid binaries is improved, with tories when portions of the ejecta fields are captured into increasing numbers of detections, a firmer context for such transient orbits that may not reimpact for many orbits (clas- studies can be established and, most likely, real distinctions ses II–IV), since there will be a higher probability of mutual between different classes of binary systems will be found. impacts and repeated impacts that could yield stable trajec- tories. Such a long-term analysis has not been performed to 5.5. Reimpact Dynamics date, however. Rotational bursting. A novel idea for injection of ejecta An unanswered question involves the dynamics of a par- into stable orbits was posited in Giblin et al. (1998), based on ticle after it reimpacts on the asteroid surface. A distinction observations of laboratory impact events. In this scenario, should be made between high-energy secondary impacts specific particles in the ejecta field have large rotational that may occur in the fractions of a second after a primary velocities and are placed in tension. In some situations, impact (due to ricochets) and low-energy impacts that may these rotating fragments have been observed to spontane- occur immediately or months after the primary impact with ously “burst,” or disassemble into smaller fragments, shortly speeds less than or equal to surface escape speed. For the after ejection from the laboratory target. This situation, if second type of reimpacts, it can be hypothesized that colli- found in nature, creates a situation such as found in the para- sion with the surface may not be disruptive nor completely graph above on direct initial condition generation, and can inelastic, so that some amount of rebound energy will exist. plausibly lead to particles placed directly into stable orbits. If true, there are significant implications for the modeling External force perturbation. Underlying many of the of reimpact ejecta. This is especially interesting in light above mechanisms, and indeed a mechanism in itself, is the of the recent returns from the NEAR Shoemaker mission, effect of force perturbations on the trajectory. As mentioned which found that the asteroid surface at high resolution was earlier, the ejecta are subject to an extremely perturbed force dominated by ejecta blocks, with a paucity of craters, which environment, first from the asteroid gravity field, and sec- raises a host of scientific questions on the nature of the Eros ond from the solar radiation pressure and tidal perturbations. surface at centimeter scales (discussed at the end of this Any of these can place a particle into an orbit that persists chapter). The existence of transient dynamical behavior of for some time about the asteroid. Specifically, in Scheeres a reimpacting ejecta block has implications for the extent et al. (1998a) a particle orbit perturbed only by the aster- of downslope motion a particle will experience, and hence oid gravity field is described that has a “hang time” of over the degree of ponding at lows in the potential that will occur 100 d. In Fulle (1997) and Scheeres and Marzari (2000) prior to a particle settling on the surface. the effect of solar radiation pressure on a ejecta particle is This issue has been studied in an engineering application shown to capture regions of initial ejecta conditions into in the context of the settling time of a navigational aid de- bound orbits that do not reimpact for hundreds of days in ployed on the surface of an asteroid (Sawai et al., 2001). In some cases. A study that combines solar and asteroid gravity that study it was found that, even for relatively low coeffi- effects has not yet been performed, but may provide mecha- cients of restitution on the order of 0.1, settling times of nisms that could extend the lifetime of such transient orbits 10–20 min were common. This is ample time for a particle to multiple asteroid years. At these timescales it becomes to migrate toward potential lows on an asteroid’s surface. possible for small perturbation forces to influence the orbits, potentially leading to stable orbits. 5.6. Surface Forces and Dynamics 5.4.4. Long-term lifetime and evolution of captured orbits. Once in orbit about an asteroid, a particle is subject to a Finally, we must consider the environment that is felt by variety of perturbation forces that can cause orbital evolu- an impact ejecta once it comes to rest on the surface of an tion over long timespans. These effects include the aster- asteroid. The mechanical forces felt on the surface can be oid gravity field and tidal effects (Petit et al., 1997), the reduced to surface normal and transverse frictional forces solar tide (Chauvineau and Mignard, 1990a; Hamilton and acting on a particle. These are, in turn, defined by the aster- Burns, 1991a), solar radiation pressure (Hamilton and oid’s gravity field, surface, and rotation state. Recent inter- Burns, 1991b; Richter and Keller, 1995), four-body effects est in other forces acting on the asteroid surface have been (Chauvineau and Mignard, 1990b), and disruption by im- revived by the unexpected morphology of the Eros surface. pacts (Davis et al., 1996). The majority of these analyses Indeed, electromagnetic forces operating on small dust par- have only considered these perturbations in isolation. From ticles are being considered to explain some of the dust such studies, it is clear that long-term, stable orbits can exist ponding seen on Eros (Lee, 1996; Robinson et al., 2001). about asteroids, in some cases with minimal orbital evolu- In addition to these are occasional impulsive forces that may Scheeres et al.: The Fate of Asteroid Ejecta 539

jolt asteroid regolith, due to impacts of other asteroids on uniformly rotating body this is just the Jacobi integral dis- the asteroid surface (Greenberg et al., 1994). cussed earlier. This gives a direct measure of the available The total acceleration that a particle feels when at rest energy that can be converted to kinetic energy (and hence on the surface of a rotating asteroid is easily dissipated) based on the location of a particle in the asteroid frame (Thomas, 1993). The effective potential en- ∂V ergy function of an asteroid is Nr=×+××−Ω Ω Ω r (14) ∂r 1 C()rrr=− (Ω × ) × (Ω × ) −U ()r (18) If the local surface normal is nˆ z, then the surface force is 2 split into a normal and tangential component Using this, the dynamical height of the asteroid surface can = Nzznˆ · N (15) be computed, a relative measure from a locally defined average gravity (Thomas, 1993). =− NNtzzN nˆ (16) On the surface of a uniformly rotating asteroid, this same effective potential energy can also be related to the mini- and the local slope of the system is defined as mum amount of energy a particle requires before it can escape from the asteroid [the guaranteed reimpact speed in N Scheeres et al. (1996)]. Specifically, the value of C(r) at φ= t arctan (17) the synchronous orbits (CR) defines the zero-velocity sur- Nz face that surrounds and encloses the asteroid in three-di- mensional space. The effective potential energy evaluated The surface slope can be related to the coefficient of fric- at this synchronous orbit defines the minimum energy that tion on the surface, µ, as µ ≥ tan φ (Greenwood, 1988). The a particle must have before it becomes possible to escape rotational dynamics of the body can take a significant role from the asteroid; i.e., a particle with Jacobi constant greater in modifying the surface environment, and may change the than this value could, theoretically, escape from the aster- stability and structure of motion on the surface. For bodies oid following a purely ballistic trajectory. If a particle has in complex rotation the slopes and surface forces are time- an energy less than this, and is within the zero-velocity periodic, and could potentially add sufficient “shaking” curve, then it is impossible for it to leave the vicinity of (physically realized by slowly varying slopes at each point the asteroid. This surface has also been referred to as the on the surface) to cause the surface to relax, reducing the Roche lobe, and was studied in the particular case of Phobos potential energy stored in local slopes. Any asteroid subject (Dobrovolskis and Burns, 1980), and more recently has to nonuniform rotation following a large impact or plan- been computed for Eros (Yeomans et al., 2000; Miller et al., etary flyby will have these time-periodic forces acting on 2001). Phobos was found to “fill” this minimum energy its surface, which could play a role in smoothing a surface surface, meaning that particles on its surface were prone to after an impact. This is distinguished from seismic shak- escape that body when given sufficient speeds. Conversely, ing, where the asteroid frequently feels small seismic events Eros lies entirely within this energy surface, although 56% due to the flux of impactors striking the asteroid (Greenberg of that asteroid’s surface lies within 1 km of this energy sur- et al., 1994). While the magnitude of shaking expected from face, the closest point lying only 90 m from the energy sur- impactors should be larger than from nonuniform rotation, face. Figure 3 shows the computed Eros Roche lobe pro- the nonuniform rotation will act continuously on the aster- jected into the Eros equatorial plane. oid over the time it takes for it to relax into uniform rotation. Estimates of this effect for Toutatis are given in Scheeres 6. CURRENT DATA CONSTRAINTS et al. (1998a). ON EJECTA EVOLUTION Finally, it should be noted that if an asteroid would ac- tually describe a figure of equilibrium (Weidenschilling, The interpretation and analysis of asteroid regolith and 1981), then the surface slope would be identically zero over the dynamics of impact ejecta will ultimately be constrained the entire body. In fact, deviations of surface slope from by in situ observations of asteroids. Historically, the analysis zero indicate deviations from a figure of equilibrium. Slope of Phobos and Deimos from spacecraft images has allowed distributions of asteroids have been measured from space- for a rather complete understanding of the regolith and dy- craft observations and from radar imaging of asteroids. namical environment of these bodies to be developed. Sev- Some bodies measured in this way, such as Toutatis and eral obvious indications of regolith on the martian moons Kleopatra, have uniformly low slopes that, at the least, could Phobos and Deimos were noted in Viking Orbiter pictures be indicators of their rotational and impact past (Scheeres (e.g., Thomas, 1979; Thomas and Veverka, 1980; Lee et al., et al., 1998a; Ostro et al., 2000). 1986) and recently scrutinized with new images from Mars A second parameter of interest for the surface environ- Global Surveyor (Thomas et al., 2000). The dynamics of ment is the effective potential, defined by the combined ejecta lofted from the martian moons are complicated by gravitational potential and rotational potential terms. For a tidal forces from nearby Mars as well as the rapid rotation 540 Asteroids III

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 high resolution by the Galileo spacecraft, indicate the existence of impact ejecta retained on their surfaces (e.g., Belton et al., 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 ejecta blocks, the largest size fraction of the regolith; (2) chutes and albedo streaks oriented down local slopes, interpreted as mass-wasting scars in regolith; (3) grooves, which may be the surface expression of deep-seated fractures partially filled by regolith; and (4) color/albedo variations associated with slopes and apparently fresh impact craters consistent with regolith maturity variations. Applications of basic mod- els of impact ejecta fields and their dynamics have been able to 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- Fig. 3. Eros Roche lobe computed from NEAR Shoemaker data. craft, Mathilde and Eros, have not fit as well with the ex- pected theory. Several of the observations made were, in essence, totally unexpected. For Mathilde this includes the and irregular shapes of these satellites (Dobrovolskis and size and extent of its craters, along with a lack of observed Burns, 1980; Davis et al., 1981). Changes in the moons’ depositional features (Veverka et al., 1997). Theories of semimajor axes due to orbital evolution should have pro- impact physics that describe this situation have been posited duced gradual variations in the effects of tides over time. that are consistent with its low measured density (Housen Using the asymmetric ejecta deposit from the giant crater et al., 1999; Davis, 1999; Asphaug, 2000). Still, the ability Stickney, Thomas (1998) deduced that the impact occurred of that asteroid to survive intact is surprising, as is its ex- while Phobos was slightly farther from Mars than it is at tremely slow rotation rate. Planned radar observations of present. The thick mantle of regolith on Deimos, presum- Mathilde will hopefully provide additional insight into this ably derived from a giant impact near the satellite’s south primitive body.

Fig. 4. Theoretical landing locations of ejecta launched from the giant crater Azzurra on Ida. This ejecta distribution, calculated considering Ida’s irregular shape and rapid rotation, provides a close match to the bright, relatively blue spectral unit found on the asteroid. The distinct color of Azzurra’s ejecta deposits suggests that relatively fresh, unweathered materials were excavated by the impact (see Geissler et al., 1996, for a more detailed description). Scheeres et al.: The Fate of Asteroid Ejecta 541

For Eros, detailed mapping of the largest ejecta blocks tempted to draw from a handful of S-type objects may not coupled with calculations of trajectories from candidate apply to asteroids elsewhere or asteroids of different types, source craters led Thomas et al. (2001) to suggest the Shoe- i.e., craters may look very different on metallic objects and maker Crater as the source of most of the ejecta blocks on extinct comets. Closeup observations of main-belt asteroids Eros. Moreover, the lack of large blocks associated with the will ultimately decide whether the dearth of small craters other giant impacts on Eros confirms that these house-sized on Eros is due to the lack of small projectiles in the main boulders are rapidly destroyed or buried. While some as- belt or due to postimpact surface processes. The observed pects of the distribution of ejecta blocks have been ex- depth, or lack, of regolith on smaller asteroids should di- plained, the observed lack of cratering at high resolution is rectly constrain the strength-to-gravity transition for impac- an outstanding puzzle. NEAR Shoemaker imaging of Eros tors. Direct experimentation, such as that planned for the (Veverka et al., 2000, 2001) revealed a surprising lack of Deep Impact mission (A’Hearn et al., 1999), will shed light small craters, and closeup imaging showed pools of fine on low-gravity cratering mechanics. Finally, continued de- sediment in topographic lows with no dust deposits on top velopment and advancement of the mathematical tools, of the boulders. This suggests that the regolith has been methods, and simulations need to understand regolith and shaken and stirred since deposition. 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