Interiors of small bodies: foundations and perspectives

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Planetary and Space Science 51 (2003) 443–454

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Interiors of small bodies: foundations and perspectives

Richard P. Binzela ;∗, Michael A’Hearnb, Erik Asphaugc, M. Antonella Baruccid,Michael Beltone, Willy Benzf , Alberto Cellinog, Michel C. Festouh, Marcello Fulchignonid,

Alan W. Harrisi, Alessandro Rossij, Maria T. Zubera

aDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 01239, USAbAstronomy Program, University of Maryland, College Park, MD 20742, USA

cEarth Sciences Department, University of California, Santa Cruz, CA 95064, USAdLESIA, Observatoire de Paris a Meudon, 92195 Meudon, Cedex, France

eBelton Space Exploration Initiatives, Tucson, AZ 85716, USAfPhysikalisches Institut, University of Bern, Bern, Switzerland

gINAF-Osservatorio Astronomico di Torino, Pino Torinese 10025, ItalyhObservatoire Midi-Pyr5en5ees, Toulouse, France

iSpace Science Institute, 4603 Orange Knoll, La Canada, CA 91011, USAjISTI-CNR, CNR - Area della Ricerca di Pisa, Via Moruzzi 1, Pisa 56124, Italy

Received 29 November 2002; accepted 18 March 2003

Abstract

With the surface properties and shapes of solar system small bodies (comets and asteroids) now being routinely revealed by spacecraftand Earth-based radar, understanding their interior structure represents the next frontier in our exploration of these worlds. Principalunknowns include the complex interactions between material strength and gravity in environments that are dominated by collisions andthermal processes. Our purpose for this review is to use our current knowledge of small body interiors as a foundation to de<ne the sciencequestions which motivate their continued study: In which bodies do “planetary” processes occur? Which bodies are “accretion survivors”,i.e., bodies whose current form and internal structure are not substantially altered from the time of formation? At what characteristic sizesare we most likely to <nd “rubble-piles”, i.e., substantially fractured (but not reorganized) interiors, and intact monolith-like bodies? Fromafar, precise determinations of newly discovered satellite orbits provide the best prospect for yielding masses from which densities maybe inferred for a diverse range of near-Earth, main-belt, Trojan, and Kuiper belt objects. Through digital modeling of collision outcomes,bodies that are the most thoroughly fractured (and weak in the sense of having almost zero tensile strength) may be the strongest in thesense of being able to survive against disruptive collisions. Thoroughly fractured bodies may be found at almost any size, and becauseof their apparent resistance to disruptive collisions, may be the most commonly found interior state for small bodies in the solar systemtoday. Advances in the precise tracking of spacecraft are giving promise to high-order measurements of the gravity <elds determinedby rendezvous missions. Solving these gravity <elds for uniquely revealing internal structure requires active experiments, a major newdirection for technological advancement in the coming decade. We note the motivation for understanding the interior properties of smallbodies is both scienti<c and pragmatic, as such knowledge is also essential for considering impact mitigation.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Asteroids; Comets; Interiors

1. Introduction

Asteroids and comet nuclei (which we collectively re-fer to as small bodies) are emerging from the domain ofastronomers into the realm of geology and geophysics.

∗ Corresponding author. Tel.: +1-617-253-6486; fax: +1-617-253-2886.

E-mail address: [email protected] (R.P. Binzel).

Spacecraft have provided resolved images of 1P/Halley, 951Gaspra, 243 Ida, 253 Mathilde, 433 Eros, and 19P/Borrellywhile radar observations are allowing shape modeling fordozens of small near-Earth objects (NEOs) and main-beltasteroids. Thus, these bodies are no longer just “star-like”or dust/gas obscured points of light viewed through a tele-scope. They are becoming individual worlds that beg for de-tailed exploration, explanation, and geological/geophysicalunderstanding.

0032-0633/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0032-0633(03)00051-5

444 R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454

As our knowledge base and understanding of shapes andsurface characteristics advance, we are naturally drawn to“deeper” questions about the nature of the interior. Explor-ing and understanding the interior properties of small bod-ies is challenging from a classical geophysics perspectivebecause so many diIerent formation and evolutionary pro-cesses are at work. For example, objects that we considerto be “asteroid-like” formed in the more dense and hotterregions of the solar nebula inside the orbits of the newlyforming giant planets. More “comet-like” small bodiesacquired increasing volatile content as they formed in theregion of the giant planets themselves and even far beyond.Superimposed on this diversity of formation environmentsand compositions is the relentless role of collisions. Hereagain a diversity of outcomes is likely the rule. In the aster-oid belt collisional outcomes may be the single most dom-inant factor in controlling present day interior structures.At further heliocentric distances, collisions can dictate notonly the interior structure but can also modify the originalpristine state of the volatile content.The challenges and the complexities of understanding

small body interiors relative to the vacuum of our knowledgeis what draws us to their scienti<c study. We also recognizethat the long-term survival of human civilization may alsodepend on our having suJcient knowledge of how to miti-gate a potential impact. In these early stages, learning aboutthese objects is the same thing as doing something aboutthem. Herein, we focus on the scienti<c motivation for theirstudy and organize our review by posing a set of fundamen-tal science questions to serve as a basis for assessing thecurrent foundation and limits of our knowledge as well asto identify prospects for advancement.

1.1. Terminology

Alas, in any emerging <eld there is often a proliferationof terms that are used to describe similar phenomena orstructures. In an eIort to promote convergence of terminol-ogy, we adopt (and extend, with new terms in quotations)the lexicon proposed by Richardson et al. (2002) to de-scribe the full range of possible interior states. Monolithsare essentially wholly intact units having a strength that iseIectively equal to the tensile strength (maximum force perunit area that a body can withstand before fracture or rup-ture occurs) of its constituent material. Bodies that are notmonoliths are referred to as aggregates whose categoriza-tion is described in terms of decreasing coherence (increas-ing number of boundaries or interfaces between units) andincreasing bulk porosity. A “primitive aggregate” describesa body for which the lack of connection between diIerentunits is the result of its primordial formation process, ratherthan the result of fractures propagated by collisions. Frac-tured bodies have a suJcient number of cracks or faults thattheir tensile strength is reduced, yet their original structureremains intact. Shattered bodies have interior structures that

are even more dominated by an abundance of joints andcracks. The relative tensile strength (de<ned as the eIec-tive tensile strength divided by the tensile of its constituentmaterial) may approach zero. As in the case of fracturedbodies, the original structure may remain mostly in place.Bodies that are shattered with rotated components havebeen thoroughly fractured (by collisions or tidal stresses)such that original units within the interior have been some-what displaced and reoriented. Rubble piles are bodies thathave been completely shattered and reassembled, where thenew structure may be completely disorganized relative tothe original. If the units somehow become attached or ce-mented to one another, the resulting structure is a coherentrubble-pile. (More generally, Richardson et al. denote anyaggregate structure that undergoes any re-attachment or ce-menting as a coherent aggregate.) A “thermally modi<edprimitive aggregate” might also increase or decrease in bulkporosity and in eIective strength. An increase in the strengthof contact between units could be described as a “lithi<edprimitive aggregate.”

2. Science questions

For small bodies, the answers to fundamental sciencequestions can be greatly dependent on factors such as: size,composition, presumed formation location, homogeneityof the original body (in terms of both composition andstructure), thermal history, collisional history, and dynami-cal (orbital) history. Here, we frame our science questions ina sequence that principally addresses objects from larger tosmaller sizes, but in reviewing current answers and provid-ing perspective we attempt to take into account all of thesefactors. Fundamentally we wish to understand in what cases“primitive aggregates” may still be found where the transi-tions may reside between bodies that have been and havenot been aIected by “planetary processes.” We also wish toknow how to distinguish between those objects, unaIectedby planetary processes, which retain primitive properties intheir interior, i.e., “accretion survivors,” and those that arehopelessly modi<ed by 4.5 billion years of collisional andthermal evolution. In order to achieve this, we take it as agiven that the modi<cation processes and their eIects mustbe well understood and therefore intensively studied.

2.1. At what size scales do “planetary processes” shapeinternal structure?

Loosely, we de<ne “planetary processes” as those that al-low some degree of internal diIerentiation which may notnecessarily culminate in core formation. Within the outersolar system, it remains only speculation on the degreeto which the largest trans-Neptunian objects (TNOs) mayhave undergone some diIerentiation. Pluto–Charon have themost precisely determined diameters and densities (2350±50 km, 1250± 50 km, 1:99± 0:07, and 1:66± 0:15, respec-

R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454 445

tively; Tholen and Buie, 1997) and models for their interiors(McKinnon et al., 1997) suggest some diIerentiation intoa rock/ice core. The possible diIerentiation of Pluto (andCharon) may also be the result of the formation and tidalevolution of this binary system. We note with excitementthat Pluto–Charon are not unique as binary objects in theouter solar system (e.g., 1998 WW31; Veillet et al., 2002).The upper size limit for which volatile-rich objects have

likely undergone any substantial diIerentiation remains un-constrained. (Kissel et al., 1986) did not <nd any evidenceof anomalous 26Mg in comet 1P/Halley, which may sim-ply indicate that comets formed after all the initial 26Al haddisappeared. Measurements of the ortho- to para-hydrogenratio in water lines and other hydrogenated species suggestthat comets have been preserved at low temperatures, con-sistent with the analysis by Lewis (1971) that shows thatobjects smaller than 300 km should have internal tempera-tures of less than 25 K. The presence of the S2 molecule isalso considered as a clue that comet nuclei have been ther-mally preserved at very low temperatures.Apart from primordial heating that may be experienced

by volatile-rich bodies, substantial thermal evolution (andconsequently internal evolution) can occur as a result oforbital evolution. The outcomes can be as fundamentallyimportant as the classical case of planetary diIerentiation.For example, a body dynamically evolving from the Kuiperbelt into an orbit typical for a Jupiter-family comet experi-ences both a seasonal thermal wave due to each perihelionpassage, which ultimately couples to a secular increase inthe interior temperatures. The details of this process are verysensitive to unknown parameters of cometary nuclei, butrepresentative calculations suggest that the seasonal thermalwave propagates tens of meters below the surface (e.g., com-pare Podolak and Prialnik, 1996; Benkhof and Boice, 1996for the case of P/Wirtanen’s orbit). These are the depths towhich gaseous transport is expected to take place assumingappropriate porosity. Below this depth, temperature gradu-ally increases to come into equilibrium with the tempera-ture expected for a given semi-major axis but thermal con-ductivity in these layers is slower and the time scale mightwell be longer than the time scale for dynamical evolution.This thermal evolution can lead to diIerentiation of ices,phase changes in the ices (most notably a phase changefrom amorphous to crystalline ice at temperatures around130–150 K), and formation of mantles due to the release ofvolatiles (Weissman 1990).

We also point out that thermal stresses can lead to theloss of the body. Weissman (1980) found that 10% of dy-namically new comets from the Oort cloud split, while 4%of returning comets and 1% of short period comets splitduring any given encounter. These breakups tend to oc-cur long before perihelion, so there is no known mecha-nism for explaining these random splitting events. Samaras-inha (1999) has proposed expanding volatiles, propagatingfrom the sun-warmed exterior to the interior of a coarselyporous comet, in order to explain the disruption of comet

LINEAR (C/1999 S4) into a power-law distribution of frag-ments (cumulative slope 1.74; MNakinen et al., 2001) priorto perihelion.Thermal stresses may also aIect asteroids, but in a dif-

ferent way. Besides the reradiation forces (Yarkovsky andYORP) which provide gentle non-gravitational thrusts andspins to asteroids, as discussed (above/below) thermal ex-pansion was postulated by Dombard and Freed (2002) asa mechanism for the production of pervasive fractures onEros, as it moved from its place of origin in the main beltinto near-Earth space.For bodies formed in the transition region between

Mars and Jupiter (classically the asteroid belt), our in-sights into diIerentiation processes are substantially moreenhanced through the availability of meteorite sam-ples. Cooling rates from iron meteorites generally sug-gest bodies in the diameter range of about 100 km andlarger (though some rapid cooling rates suggest sizesdown to ∼ 20 km) were capable of undergoing sub-stantial diIerentiation in the early solar system (Haacket al., 1990; Mittlefehldt et al., 1998). Vesta, with a 500 kmdiameter and basaltic (igneous) crust (McCord et al., 1970),appears to be the only “intact” surviving diIerentiated par-ent body within the asteroid region. Spectroscopic colors,albedos, and radar reQectivities (GaIey et al., 1989; Ostroet al., 2000) are all consistent with objects such as 16 Psy-che and 216 Kleopatra being 100–200 km remnant ironcores of subsequently demolished parent bodies.Not all large asteroids appear to have undergone sub-

stantial diIerentiation, with the nearly 1000 km diameterCeres displaying spectral and albedo characteristics consis-tent with a relatively primitive (unheated) class of carbona-ceous chondrite meteorites (GaIey et al., 1989). Ceres andother similarly common “C-class” objects in the outer re-gions of the asteroid belt may have accreted slowly enoughfor radiogenic heat (principally from 26Al) to have been ad-equately dissipated without diIerentiation. Particularly im-portant for Ceres may have been the increasing abundanceof volatiles with increasing heliocentric distance. These ad-ditional volatiles may have been essential for quenching(through the heat of fusion for water ice) the diIerenti-ation process within Ceres (Grimm and McSween, 1989,1993). Certainly if accretion time scales and/or quenchingproved suJcient to arrest diIerentiation within the outerregions of the asteroid zone, “single” (non-binary) Tro-jan asteroids and TNOs also would be expected not to bediIerentiated.

2.2. Which bodies are “accretion survivors”, i.e., bodieswhose current form and internal structure are notsubstantially altered from the time of formation?

Here, we seek to understand which individual bodies orclasses of bodies are observable today that provide actualwindows back to the shapes and internal structures created

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in the early solar system. In other words, are there currentlyobservable remnants of the <nal outcomes of accretion?As we discuss in the following section, it seems likely

that virtually all asteroids have been thoroughly shattered bycollisions over the age of the solar system. Intriguingly andperhaps counter intuitively, collisions that severely fracturethe interior (without causing total disruption) may be essen-tial for a body’s long-term survival. While Ceres, by virtueof its size and Vesta by virtue of its thermal evolution maybe best termed as surviving “protoplanets”, asteroids in theintermediate size range (a few hundred km) are most likelyshattered survivors from the time of their accretion. Evi-dence abounds that such survival is stochastic, as evidencedby the widespread number of asteroid families that are al-most certainly the remnants of catastrophically disruptivecollisions of parent bodies having originals sizes of less than100 km to greater than 200 km (Hirayama, 1918; Zappalaet al., 2002; Tanga et al., 1999).Trojan asteroids at the L4 and L5 Lagrange points of

Jupiter have long been suspected of being primordial rem-nants retaining structures from the time of their formationbecause of their dynamical isolation and their inferred (fromextreme lightcurve variations) elongated or possible con-tact binary shapes (Cook, 1971; Hartmann and Cruikshank,1978; Cellino et al., 1985). Intuitively, the “binary argu-ment” is that a large pair (as opposed to “just” a primarywith a small satellite) is easiest and most likely to form inan early accretion period when relative encounter velocitiesare low. Any substantial amount of subsequent collisionalevolution is likely to disrupt or dislodge the binary pair,making binaries a telltale signature of survivors from theaccretion epoch. If this logic holds, then the discovery bydirect imaging showing the L4 object 617 Patroclus to bea ∼ 100 km binary pair (Merline et al., 2001) supports thepresumption of the largest Trojans being accretion survivors.This survivor size range inferred from binaries is consistentwith Trojan collisional evolution and population size distri-bution models by Marzari et al. (1997), who estimate thatthis transition occurs around 50–100 km.Perhaps of greatest interest to planetesimal studies is iden-

tifying the smallest objects that may be accretion survivors.While we are observationally challenged to resolve or evendetect km-sized bodies in the Trojan region, Kuiper Belt,and Oort cloud, comets in the inner solar system are infact small-sized outer solar system representatives that areclose enough for study. Their inferred strengths, or lackthereof, within their internal structure provides the clearestevidence for most comets being well described as accre-tion survivors having a “rubble-pile” structure (Weissman,1986). From its tidal encounter with Jupiter, prior to itsplunging demise, the separation of the components withincomet D/Shoemaker-Levy-9 has given the basis for esti-mating not only its internal strength (as the maximal tidalstress at periapse was no greater than 1000 bars), but alsoits density and probable structure (Asphaug and Benz, 1994,1996). Similarly, a low interior strength for comet LINEAR

(C/1999 S4) is suggested by the breakup of this comet farfrom any massive body. The presence of fragments of sizesof order 50–100 m in diameter indicates that this body en-tered the inner solar system as a surviving accretion aggre-gate.Thermal evolution of comets, primarily as a result of

perihelion passages, complicate the inferences we maymake on their formation process and interior structure fromremote-sensing observations (e.g., Keller et al., 1986; Bellet al., 2000) or direct impact experiments (A’Hearn, 1999).The thermal skin depth is related to the rotation period ofthe nucleus, its thermal conductivity and its density. Forpure water ice, the skin depth is only 20 cm and is probablylower in reality given the presence of an insulating layer ofporous dusty material and not ice. The presence of a mantleprobably makes the comet stronger against collisional dis-ruption perhaps by providing an additional layer structureto render even more ineJcient the propagation of impactshock waves.

2.3. At what characteristic sizes are we most likelyto Cnd: rubble-piles, substantially fractured (but notreorganized) interiors, and intact monolith-like bodies?

Understanding the complex interplay between strengthand gravity is the basis for answering this question, wherethe answers are not necessarily intuitive for inhabitants of agravity-dominated planet. In some ways, a small body maybe thought of as a planet having a lithosphere extending intoits deep interior. At present, we are generally left to inferthe nature of this lithosphere through measurements of thebulk density, through observations of collisional outcomes(both in the laboratory and asteroid families), through obser-vations of the consequences of tidal, rotational, or thermalstresses (cometary breakups), and through analytical andnumerical simulations. The current wisdom on the overallcollisional evolution of the asteroid belt (e.g., Davis et al.,1989, 1994, 1999, 2002) is that all sizable bodies are thor-oughly shattered (Melosh and Ryan, 1997). What eludes usis an understanding of the arrangement or coherency of theseinteriors. Whether TNOs are shattered or not is less clear.Although collisions are as frequent in the inner Kuiper beltas in the asteroid belt, they generally occur at signi<cantlylower velocities and thus are more likely to have their eIectscon<ned to the near-surface layers. Since the internal struc-ture is also unknown, the threshold energy for “shattering”throughout is also unknown. Davis and Farinella (2001) ar-gue that “many” of the TNOs are shattered.Bulk density measurements require a determination for

both the mass and the volume, parameters that have been his-torically diJcult to determine. Classically, asteroid masseshave been determined by their interactions with one another(e.g., Schubart and Matson, 1979; Viateau, 2000) whiletheir volumes are inferred from direct measurements andfrom modeling of their albedos and diameters (e.g., Tedescoet al., 1992). Apart from using spacecraft in Qyby mode to

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simultaneously measure mass and volume (e.g., Yeomanset al., 1997), perturbations of Mars as measured by theViking mission have yielded mass estimates for the largestasteroids (Standish and Hellings, 1989). In the future, theplanned GAIA astrometric mission is expected to providemasses for ∼100 main-belt asteroids based on astrometricmeasurements to detect mutual perturbations with unprece-dented precision. An alternative approach for density estima-tion is based on knowledge of shape and rotational periodsfor bodies which can be assumed to have equilibrium shapes(Farinella et al., 1981, 1982). The most obvious candidatesare the large bodies exhibiting short rotation periods andlarge lightcurve amplitudes, for which an overall rubble-pilestructure (coherent aggregate) has been long proposed. Alsoin this case, the observational challenge is to measure withhigh accuracy the overall shapes of these objects, and thiscan be best done currently by taking advantage of the ad-vancement in high-resolution techniques such as adaptiveoptics and speckle interferometry.Apart from the above possibilities, the existence and de-

tectability of small body satellites (e.g. Belton et al., 1995;Merline et al., 1999, 2001; Margot et al., 2002; Veillet et al.,2002; seeWeidenschilling et al., 1989 for a summary of pos-sible earlier detections) represents the breakthrough with thegreatest new potential for determiningmasses and estimatingbulk densities for a wide diversity of objects. Mass, volume,and density determinations for comets are extremely prob-lematic with density estimates for 1P/Halley falling withinthe range 0.2–1:5 g cm−3 (Sagdeev et al., 1988). Cometdensity estimates based on observed non-gravitational accel-eration (deduced from successive apparitions) and estimatesof the reaction force from outgassing have been calculatedby several authors but are sensitive to model assumptions.Perhaps, the best constraints on those assumptions are for19P/Borrelly (Soderblom et al., 2002) for which Farnhamand Cochran (2002) (Icarus, in press) deduce a density be-tween 0.3 and 0:8 g cm−3.Inferring something about the interior structure of small

bodies requires an additional step of being able to esti-mate the bulk porosity. Essential input to estimating thebulk porosity is knowledge of the “grain density” of theconstituent material, i.e., the average density of the solids(“grains”) that make up the body. Laboratory measurementsof primitive and diIerentiated meteorites (Consolmagno andBritt, 1998; Britt and Consolmagno, 2000) yield both thegrain density and the “microporosity” arising from frac-tures, voids, and pores on the scale of tens of microns. Brittet al. (2002) examined the bulk porosities for nearly 20 smallbodies, including Phobos and Deimos. Utilizing the graindensity and microporosity for the meteorite material mostconsistent with the composition inferred by telescopic spec-tra, these researchers derive “macroporosity” (large-scaleporosity) estimates that range from nearly zero to about70%. For the purpose of inferring large scale internal struc-ture, macroporosity constitutes one of the most fundamentalparameters that we seek to determine.

Britt et al. (2002) <nd their solutions distinguishable intothree rough groups. The <rst group, having eIectively zeromacroporosity is comprised by the three largest asteroids:Ceres, Vesta, and Pallas. With diameters near 1000, 500,and 500 km, respectively, these bodies appear to have suf-<cient self-gravity to eliminate any signi<cant macroporos-ity. Their inferred nature as protoplanets surviving over theage of the solar system certainly implies they have experi-enced substantial collisions such that their interiors are ei-ther fractured or shattered (in the lexicon of Richardson etal.). However, the lack of macroporosity and the preserva-tion of Vesta’s basaltic crust imply that these largest bodiesare not rubble-piles.A second group of objects has macroporosities ranging

from 15% to 25% over a diameter range from about 30 to300 km. Eros, the most thoroughly studied small body todate, falls in the middle of this range (macroporosity∼ 20%;Wilkison et al., 2002) and thereby provides the best oppor-tunity for insight. All indications are that Eros is a heavilyfractured or shattered body, but not one that was previouslydisrupted and reaccumulated as a rubble pile. Eros appearsto have a highly homogeneous structure because the oI-set between the center of mass from the gravity and shapemodels only amounts to 30–50 m. (Miller et al., 2002; Zuberet al., 2000; Thomas et al., 2002). Even with its irregularshape (diameter dimensions ranging from 9 to 32 km) andrelatively fast 5:27 h spin period, Eros’ interior is not in ten-sion due to rotation or precession wobble as the spin axisand principal body axis are aligned within 0:02◦ (Milleret al., 2002). Even though the interior is likely heavily frac-tured or shattered, surface expressions of structural features(ridges, grooves, etc.) demonstrate that still Eros retainssome signi<cant degree of tensile strength (Prockter et al.2002). Other evidence for structural coherence of Eros in-cludes its high mean density, twisted platform, clustered re-gions of surface slopes above the angle of repose, subduedor missing crater rims, and the continuity of long groovesand fractures (Zuber et al., 2000).A third group of objects, ranging in diameter between

about 50–250 km (setting aside Phobos and Deimos, whoseresidence within the Mars gravity well may make themspecial cases), have macroporosities in excess of 30%.These appear to be the best candidates for “rubble-pile” ob-jects that have been substantially disrupted and reaccretedsuch that their original internal arrangement has undergonemoderate or substantial displacement. Most bizarre is theinferred ¿ 70% macroporosity for 16 Psyche (Viateau,2000). Psyche, a ∼ 250 km diameter object, has an M-typereQectance spectrum that is interpreted as being analogousto very strong and very dense (7:4 gm cm−3 grain density)iron meteorites. A very high radar reQectivity for Psyche(Magri et al., 1999) provides the highest weight evidence foran iron meteorite analog. A composition of such extremelystrong material may be essential for maintaining such a highporosity, where for a value¿ 70% the volume is dominatedby voids greatly exceeding the abundance of holes in Swiss

448 R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454

cheese. Belskaya and Lagerkvist (1996) note that thereare possibly other compositional interpretations for M-typeasteroids. If Pysche’s actual grain density is substantiallylower than 7:4 gm cm−3, the inferred value for the macro-porosity would be correspondingly lower. High materialstrength may not be a prerequisite for achieving 40–50%macroporosity values, as three of the other four objects inthis Britt et al. grouping have C-type spectral properties.These objects are interpreted as likely analogs to weakercarbonaceous chondrite meteorites. 253 Mathilde (50 kmdiameter) is a prototype example with an estimated macro-porosity near 40%. Weaker strengths for carbonaceous-likematerial may contribute to their likelihood of being sub-stantially disrupted into rubble-pile structures.There appears to be no characteristic size range (within the

present sample) for a transition between fractured/shatteredobjects and bodies whose rearranged interiors classify themas rubble piles. It appears that stochastic collision his-tory and basic composition play a greater role than size.However, a characteristic size does seem to be revealedbetween shattered/fractured bodies and monoliths hav-ing tensile strengths. Asteroid lightcurve studies pushingto progressively smaller sizes through measurements ofnear-Earth objects (e.g., Pravec et al., 2000) have revealed aset of remarkably fast rotating objects with periods as shortas a few minutes (as opposed to∼ 2 h for the previous shortperiod record). An analysis by Pravec and Harris (2000)<nds a transition near 100 m where fast rotations (periodsshorter than ∼ 2 h) are rare or absent for objects larger than∼ 100 m, although the mean tension ∼ R2� (!2 − G�)across their plane of maximum stress is generally miniscule.This transition may represent a barrier where ¿ 100 mbodies have such a high degree of fracturing (and a cor-responding lack of tensile strength) that self-gravity is thedominant force for their coherency. This transition size isalso found in numerical simulations which show that largeremnants from a catastrophic disruption of a larger parentbody are no longer monolithic fragments once the latterexceeds 100–300 m in radius (Benz and Asphaug, 1999).Below ∼ 100 m almost all objects observed to date rotate

faster than this limit, requiring their internal structure to begoverned by their tensile strength. But we note that this ten-sile strength still may be incredibly weak: for the ∼ 11 minperiod and ∼ 30 m diameter of 1998 KY26 (Ostro et al.,1998) the required tensile strength is ∼ 300 dyn cm−2

(presuming � ∼ 1:3 g cm−3 for this C-type), orders ofmagnitude weaker than snow. Nevertheless, it appears that∼ 100 m may represent a characteristic size transitionbetween “intact” strength dominated monoliths and evenweaker (eIectively strengthless?) fractured/shattered orrubble-pile bodies predominantly held together by gravity.

3. Perspectives

Having set out a fundamental set of science questionsand having brieQy outlined our current understanding, we

continue by examining ways in which we are likely to fur-ther improve our answers over the time scale of a decade ormore. Most interesting, of course, are the <ndings we cannotpredict—<ndings which completely revamp our current un-derstanding and dramatically change our science questionsand perspectives. We are well advised not to be presumptu-ous in our current knowledge. For example, an accountingof the diJculties and uncertainties arising in measuringa mass, deriving a volume, and interpreting a basic com-position highlights the challenge for determining a bulkporosity as a starting point for inferring interior structure.To our advantage, however, we have a number of vantagepoints for trying to improve our understanding of smallbody interiors.

3.1. Outside looking in—the view from afar

Measuring and de<ning shapes and spin states, detectingbinaries and satellites and measuring their orbital dimen-sions and periods, and inferring parent body interiors fromthe collisional remnants within asteroid families all provideopportunities for insights from a distant perspective. As dis-cussed in the previous section, the ∼ 100 m limit againstthe spin up of small bodies may be the most fundamentalinsight gained by rotation studies. The rate at which an ob-ject in a non-principal axis or “excited” state loses its ro-tational energy—i.e., the rate at which it relaxes toward itsground state (pure spin) is a strong function of the size ofthe object (Burns and Safronov, 1973) depending on the in-verse square of an eIective radius. Thus new insights mayperhaps be gained through unveiling a boundary size be-low which excited spin predominates. The clearest signa-ture of excited spin in lightcurve observations is the simul-taneous presence of two independent periodicities. Unfor-tunately, nature is not always kind and the independenceof the periodicities is not always clear. For example, in thecase of 1P/Halley the periodicities seen in the lightcurveare, to the limits of accuracy, all harmonically related tothe 7.4 day periodicity. Our certainty about the excitationof Halley’s spin state comes from spacecraft observationsof the orientations of the nucleus at the Vega and Giottoencounters, not from the lightcurve. If we depended onlyon the lightcurve information it is almost certain that wewould have convinced ourselves by now that Halley spinwas fully relaxed with a 7.4 day period. Numerical calcu-lations show why this can be the case. As the spin stateevolves for an elongated object torqued by jet activity it isfound that the evolution tends to get hung up whenever theprecession and rotation periods in the spin become commen-surable, i.e., harmonically related. In order to make full useof lightcurve information to analyze spin states of cometarynuclei, simultaneous information on nucleus shape and thedistribution (i.e., coma morphology—orientations and cur-vatures of molecular and dust jets) and strength of comaactivity is also required. For asteroidal bodies, the boundary

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size and rotation rate below which such “tumbling” occursprovides a measure of the material properties of interiors,primarily rigidity and speci<c dissipation factor (Q). Stillto be unraveled is the mystery of a separate population ofvery slow rotators (Harris, 2002). Functionally, this popu-lation <ts a distribution like N (¡f) ˙ f, rather than the3D Maxwellian form that <ts the main population of spinrates within the asteroid belt (Pravec and Harris, 2000).Insights into the interior structure and tidal dissipation ca-pabilities might be gleaned if this subpopulation is a rem-nant of massive unstable binaries that gravitationally dis-integrated, in the process using up most of the primary’sspin energy.After decades, even centuries, of speculation and dubi-

ous claims, asteroid satellites (binaries) have at last gaineda <rm footing. As referenced above, binaries have beenfound among practically every class of small body, includ-ing NEOs, main-belt asteroids, a Trojan, and several TNOs.The holy grail within binary systems, of course, is to havesuJcient measurements to accurately solve for the mass ofthe system (the most diJcult step toward a density determi-nation). In many cases, long-term measurements with largeaperture telescopes (often utilizing adaptive optics system)are essential for yielding the prize of well-determined or-bital parameters. Precise determination of the total volumeof the bodies (allowing for the presence of concavities) isalso a challenging step toward achieving density determina-tions. Given the rate at which new satellites and binaries arecurrently being discovered (e.g., Veillet et al., 2002 estimatethat 1% of TNOs may be binary), over the next decade thegreatest growth in the number of available density measure-ments will come from this area. Because the uncertaintiesthat make precise density determinations diJcult will per-sist, analyses for inferring interior properties (such as therigidity parameter Q=k2) will be best accomplished on a sta-tistical basis using a large sample.Still to be unraveled, or even con<rmed, is the appear-

ance that the “average” properties of binaries in diIerentorbital regions diIer remarkably. NEO binaries tend tohave similar-sized components, with moderate orbital sep-arations. TNO binaries are likewise close to equal sized,but with huge orbital spacings. Main-belt asteroid satellitestend to be smaller relative to their primaries (with one re-markable exception, 90 Antiope, that is a nearly equal mass,nearly contact binary). One might ascribe these diIerencesto observational selection eIects (e.g., only nearly equalsized, widely spaced binaries could possibly be detected inthe TNO region). However, if main-belt binary statisticswere like either NEO or TNO statistics, binary asteroidswould have been discovered a century or so ago from reg-ular telescopic observations. Thus, some diIerences mustbe real, and therefore call for explanation. We are onlyjust beginning to have a suJcient number of examples toattempt generalizations as to mechanisms of formation andevolution. At the moment we can oIer only speculations tostimulate new insights into the formation and evolution of

these bodies with implications for interior structure. Theseinsights include:

• The greater relative abundance of NEO binaries may bethe result of tidal disruptions (Ta la comet Shoemaker-Levy9). If correct, this argues that many NEOs may have veryweak rubble-pile structures that have been formed in therecent solar system.

• Main-belt binaries may be formed by collisional pro-cesses. A satellite such as Dactyl might plausibly havestarted out as a clump of unescaped ejecta from a majorimpact, with the orbit circularized and expanded by tidaldissipation before it could chance to reimpact the primary.Such an accumulated ejecta process for forming satelliteswould explain why satellites of main-belt asteroids tendto be small compared to their primary.

• TNO binaries may rely on the solar tide to provide the“second burn” to put collisionally disrupted componentsinto orbit about each other. In any case, the huge dimen-sion of the range of stable orbits is such that very littlesubsequent (tidal) evolution should have occurred.

• Time scales of tidal evolution may provide constraintson internal properties of rigidity and energy dissipation,and/or time since formation.

• The shape, spin and density of some asteroids (best de-termined for Eros, but almost certainly also the case forIda) imply Roche lobes so closely hugging the presentsurfaces of those asteroids that almost anything dislodgedfrom the surface would likely end up in orbit, at least fora while.

• For larger main-belt binaries, one can plausibly point backto the early solar system to explain their formation. Forexample, 90 Antiope might have formed as a blob ofnearly dispersed ejecta, from some super-large collision,satisfying the Jeans criterion for coalescence into a singlegravitationally bound system but with too much angularmomentum to exist as a single body. Such a blob wouldhave little choice but to settle into a nearly contact binarycon<guration of two similar mass components.

Having been produced by the catastrophic disruption oflarge parent bodies, asteroid families continue to present uswith observable outcomes of natural collision experiments.By studying families we are probing the pieces of smallbody interiors. Still not yet understood in terms of earlysolar system accretion and diIerentiation processes is whyasteroid families observed to date (Bus, 1999; Cellino et al.,2002) are remarkably uniform in their spectral properties,indicating little evidence for substantial diIerentiation. Thiscontrasts with the remnant metal core interpretation for largeM-type asteroids like 16 Psyche. Burbine et al. (1996) sug-gest an explanation that for families formed from diIerenti-ated parent bodies, the crust and mantle material may havebeen “battered to bits” over the age of the solar system—butthe extent of this “battering” is constrained by their re-maining a reasonable chance that Vesta’s basaltic crust has

450 R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454

survived. Probing the spectral properties to smaller andsmaller sizes may reveal the missing pieces of disrupteddiIerentiated parent bodies. Indeed the discovery of a small(15 km) basaltic asteroid located far from (and very un-likely related to) Vesta by Lazzaro et al. (2000) may bejust the tip of the iceberg for understanding the diversediIerentiation history of inner solar system bodies.More directly related to inferring interior strengths and

structures are the size and velocity distributions within fam-ilies (Cellino et al., 1999; Zappala et al., 2002). In princi-ple these are related to the energetics and geometry of theimpact and the strain-rate release of the fragments duringthe break-up and dispersion process. Tantalizingly, we nowview only the <nal state of this complex process. An addi-tional challenge now being faced is the extent to which theejection velocities of the fragments (inferred from their or-bital displacement from the center of mass) is aIected byother process which may spread their orbits. Understand-ing processes like the Yarkovsky eIect (Bottke et al., 2000)have a direct bearing on our being able to utilize the poten-tial information contained within asteroid families. Indeed,the observed size-proper elements relations seen in manyfamilies tend to suggest that properties of the original ejec-tion velocity <elds of the fragments are still recognizablein present families. In particular, the trend exhibited by thesize (or absolute magnitude) versus proper inclination plotsis something that is not expected to have been substantiallyinQuenced by the Yarkovsky eIect.

3.2. Outside looking in—the digital view

Numerically simulating the properties of interiors, in par-ticular their response to impacts, presents a powerful toolfor applying and evaluating a variety of physical models. Inprinciple, the simulation of a body’s response to a collisionby means of numerical calculations is a straightforwardprocedure. Once the relevant elasto-dynamical conserva-tion equations have been appropriately transformed intonumerical code, initial conditions can be speci<ed and theoutcome of collisions computed. The entire procedure canbe checked in detail by comparing numerical results to labo-ratory impact experiments (e.g., Fujiwara et al., 1989; Benzand Asphaug, 1995; Ryan and Melosh, 1998; Ryan et al.,1999). In practice, however, the procedure is considerablymore complicated for several reasons: (1) the physics in-volved may not necessarily be described by simple laws(fractures, phase changes, rheology, etc.). (2) The equa-tions may be diJcult to make discrete. (3) The computingpower available may not be suJcient to allow for thedesired or needed resolution. Applying numerical modelsto real solar system bodies presents serious challenges, inparticular when trying to infer the initial conditions of abody’s interior that have lead to the morphological and/ordynamical properties observed today. We are dealing witha classical under-determined inverse problem that is greatly

in need of new constraints that can be best supplied by thetypes of experiments described in Section 3.4.Given these diJculties, and while we await measure-

ments providing new constraints, our present task is to un-derstand the properties of classes of models rather thanfocus on speci<c details. As an example we can exam-ine how pre-existing fractures and/or porosity on micro- ormacroscopic scales aIect the overall strength of a body orits ability to transmit shock waves (e.g., Asphaug et al.,1998). One of the most important perspectives these mod-els are giving us is that the eIective strength of a bodystrongly depends upon these characteristics. For example, abody large enough to be in the gravitational regime (greaterthan roughly 1 km), that is pre-fractured in a small num-ber of pieces without much void between them is weaker(in the sense that it is easier to disrupt) than a similar-sizedmonolith. On the other hand, the same body thoroughly shat-tered with large amounts of voids is more diJcult to dis-rupt than the corresponding monolith despite the fact thatthe shattered body is “weaker” in the sense that it has al-most no tensile strength! Thus, “survival of the weakest”may be the rule of the collisional evolution jungle as theprincipal eIect for a shattered and porous interior is to at-tenuate the stress wave generated in an impact. As a strongshock propagates through a porous target, the energy is in-eJciently transferred across the fracture interfaces and porespaces, eIectively localizing the energy dissipation. Com-paction may also be very eIective in dissipating the en-ergy in a porous target, as proposed for the large craterscomparable to the radius of Mathilde (Veverka et al., 1999;Housen et al., 1999). Thus, our current perspective is evolv-ing toward a view that the majority of small bodies areshattered survivors that become increasing diJcult to dis-rupt as sub-catastrophic collisions eIectively add strength byadding fractures and by possibly increasing porosity. Statedanother way, thoroughly shattered and porous interiors, evento the point of rubble-pile structures, are the longest-livedstate for small bodies and may be considered their naturalend state—thus predicting that when explored, almost allsmall body interiors will be found to be thoroughly shatteredand porous.Particularly relevant toward unveiling the current interior

structure of small bodies is how fragments from catastrophiccollisions reaccumulate themselves after the blow, when theimpact indeed proves energetic enough to catastrophicallydisrupt the target. Modeling results by Michel et al. (2001)suggest that the largest members of asteroid families arelikely rubble-piles, composed of fragments dispersed from acatastrophically disrupted parent that are able to reaccumu-late through their mutual gravitational attraction. If correct,this work points toward families as the best place to studythe rubble-pile end state for small body interiors. Naturecould be giving us a helping hand in resolving this ques-tion if there actually is a propensity for forming satelliteswithin asteroid families (Davis et al., 1996; Durda, 1996;Doressoundiram et al., 1997; Michel et al., 2001).

R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454 451

3.3. Outside looking in—the NEAR and near view

Although a model of the geologic structure can uniquelyde<ne the gravitational <eld of a body, a model of the grav-itational <eld cannot uniquely de<ne the geologic structurethat produced it. Even though we cannot achieve uniquesolutions, well-constrained solutions in terms of overall ho-mogeneity are achievable with orbiting missions, as demon-strated by NEAR at Eros and discussed in Section 2.3 above.Promising new approaches based on alternative representa-tions of the gravity potential (e.g., the direct calculation ofthe potential obtained by modeling the body as a polyhedron,Werner and Scheeres, 1997) could allow us to model in-teriors even for highly irregular shapes. Mixed approaches,using both spherical harmonics and polyhedral representa-tions (e.g., Scheeres et al., 2000) could help to discriminatethe presence of density layers with relative accuracies of theorder of 10−9.

New missions on the horizon promise the exploration ofinteriors for a wide variety of small bodies. On the pro-toplanet scale, the Dawn mission to Vesta and to Ceres(Russell et al., 2002) will reveal the presence or absenceof a core. CONTOUR’s intended close Qybys of comets2P/Encke and 73P/Schwassmann-Wachmann 3 (Bell et al.,2000) are examples for a low-cost mission to derive mul-tiple comet densities. Rosetta’s Qight path may include oneor more asteroid Qybys enroute to its cometary destination.Rosetta will send a lander to the comet’s surface to performin situ measurements of its physical and chemical proper-ties. The Rosetta spacecraft is planned to operate for 2 yearswithin a short distance from the comet in a parallel orbit,providing data on the evolution of a comet nucleus duringits approach to perihelion.Detailed examination of small body interiors from orbital

missions currently appears achievable through very accurateorbit determination by means of multiple frequency track-ing in conjunction with an accelerometer to reduce the ef-fects of non-gravitational perturbations on the orbit solu-tion (particularly critical in the low gravity <eld of a smallbody). These techniques will be demonstrated on the Bepi-Colombo mission to Mercury (Milani et al., 2001). The mul-tiple frequency X and Ka band link of BepiColombo willprovide tracking precision of the order of 10 cm in rangeand 10−5 mm s−1 in range rate (RMS values). Simulationsperformed for the BepiColombo mission show that, togetherwith the accelerometer data, this ultra-accurate tracking willallow the determination of the spacecraft position with anactual error of a few tens of centimeters. This will lead toan unprecedented accurate knowledge of the gravity <eldof a solar system body (beyond the Earth) and will <rmlyconstrain the presently unknown nature of Mercury’s core(Milani et al., 2001). As a comparison, the Eros gravity <eldderived by NEAR is deemed reliable up to degree and order6 (Miller et al., 2002) whereas a reliable <eld of Mercury upto degree 25 should be attainable. A similarly instrumentedmission around a small body could employ the same tech-

niques and yield an improved knowledge of the gravity <eld(in terms of spherical, or ellipsoidal harmonics). The resultwould be a better detection of lateral and internal inhomo-geneities in the body and set the appropriate reference sur-face (analogous to the Earth’s geoid) to be compared withother representation methods (as stated above).

3.4. Up close and personal—active experiments

However clever we are with models and data interpre-tation of gravity <elds, the interiors of small bodies willremain a mystery until we begin to probe inside directlythrough active experiments. Deep Impact (A’Hearn, 1999)will make the <rst such experiment in the most simple way:impacting a projectile of a known mass and velocity andobserving the consequences. The selection of 9P/Tempel-1as the Deep Impact target will directly address our perva-sive lack of direct and detailed knowledge of the strength,structure, and composition of cometary nuclei.Taking further steps toward directly studying small body

interiors will require applying the tools of modern geophys-ical prospecting: radar reQection and transmission tomogra-phy, seismo-acoustic waveform inversion, magneto-telluricimaging, and good old-fashioned drilling and blasting.Some of these are at present more feasible than others, butthe overall mission requirements involve spacecraft at leastcapable of rendezvous and in most cases landing. RadarreQection tomography may be the most cost-eIective toolfor learning about small body interiors, as it can be donefrom an orbiter without a lander. But, it is not likely toprovide unambiguous characterization in the absence ofground truth. Furthermore, a small body with metallicQecks (a chondrite) may be opaque to radar, as may onewhich is clay rich. Radar plus grenades might be a bettercombination, especially for probing near-surface geology(fault expressions, crater roots, cohesion of “ponded” ma-terials, etc.). Radar plus grenades plus a seismic networkmight reveal whole-body structural characteristics, andthe required landers would allow for radio transmissiontomography (much like the Rosetta CONCERT soundingexperiment) unless the material is too opaque or the bodytoo large. Seismology on small bodies might prove verychallenging if their near-surface layers are highly porousand strongly attenuative. Blasts are very poor seismogenicactivators to begin with; they may create holes in theregolith and little more. A deep penetrator with seismicthumper and several deep-anchored receivers may be re-quired for high-quality seismic imaging of larger objects—avery complex mission goal that may not be achievable forseveral decades. Due to the potential for strong attenua-tion of electromagnetic and seismic energy, the smallestobjects may be the best <rst candidates for seismology andradar. NEOs derived from cometary source material up toa few km across may also be good candidates for radar.Where seismic imaging is challenging, radio imaging maybe optimal, and perhaps vice versa. An ideal candidate may

452 R.P. Binzel et al. / Planetary and Space Science 51 (2003) 443–454

be those objects near the transition size of ∼ 100 m. Onemight <re large projectiles (Ta la Deep Impact) to revealnearly global consequences. One might try to spin up anobject, causing it to rotate so fast that it cracks or otherwiseQings itself apart. The list of creative possibilities goes onand on.Only the most simple entrees from the above menu of

possibilities for directly measuring small body interiors arelikely to <t within the cost envelope of Discovery class mis-sions within the United States. Low-cost (Discovery class)missions with focused science goals in this area are likely tobe competitive. More extensive and more expensive smallbody interior missions will <nd it increasingly diJcult tocompete for Qight selection in the face of many compellingmission opportunities within all of planetary science. Feasi-bility studies for future missions to NEOs are being fundedwithin ESA’s Aurora Program. The goal of these studies isto give preliminary <ndings relevant to preparing and devel-oping a mitigation plan. The governments of ESA’s membercountries will have the opportunity to select and to <nanceone of these missions. Among the mission concepts are thosethat deal with (i) the discovery and follow up of potentiallydangerous objects, (ii) the knowledge of their bulk compo-sition through the determination of the taxonomy of a largesample of the NEOs population, and (iii) the determina-tion of the bulk physical properties (mass, density, surfacecomposition, and microscopic and macroscopic roughness).Two other mission studies (named Don Quixote and Ishtar)are addressing understanding the detailed internal structureof select NEO’s. Don Quixote proposes to send penetratorswith seismometers to perform tomography of the asteroidtarget by monitoring the response to shooting a known massat the surface. Ishtar plans an in-orbit radar global surveyof few small asteroids to characterize the internal structuretogether with their bulk geophysical properties. Overall, thenatural evolution of our increasing science knowledge aboutsmall bodies and careful and rational (fact, not fear) assess-ment of the impact hazard provides the most likely pathby which extensive interior investigations will become areality.

3.5. Learning how to learn: the science role for mitigation

As we state in the Introduction, our motivations for thestudy of small body interiors are the science questions andthe diverse range of answers that can give us broad newinsights across the <eld of planetary science. We note thatnearly all of the answers gained from the science questionsare both fundamental and essential to the “practical” prob-lem of mitigation, should an NEO be discovered on a de-cidedly hazardous trajectory. Fortunately, the odds greatlyfavor that there is no sizable threatening object (or objects)on course for impact within the next century or more. Yet,prudence dictates that we should search so as to be sure andthat we should have an understanding of how to mitigatea hazardous object should it be necessary. One can argue

that the only object we need to understand in great detailis the one that will cross our path. We disagree. From thestandpoint of practicality, the science questions we addressand the methods we employ to answer them are essentialfor learning how to learn about the structure and interiorsof small bodies. Without a sound knowledge base gainedby the scienti<c study of these objects, we have no con-text within which to formulate realistic mitigation strategies.While it is unlikely that an actual mitigation will have tobe performed for generations, the advancement of our sci-enti<c understanding through the exploration of small bodyinteriors is certain.

4. Conclusions

Although the study of small body interiors is in itsinfancy, we are able to begin to formulate fundamentalscienti<c questions whose answers are as diverse as thesolar system small body population as a whole. We havethe opportunity to explore and understand objects rangingfrom diIerentiated protoplanets to unaltered remnant plan-etesimals. The same collision processes that are likely tohave thoroughly weakened and shattered interiors, in somecases creating rubble-piles, may have eIectively made theirtargets virtually impossible to destroy completely. Theseprocesses also may be responsible for creating satellites,whose orbital motions provide the <rst key for unlockingtheir interior secrets. Only the smallest bodies may physi-cally behave as monoliths, but their interior structures arelikely complex as well. Our spacecraft exploration of theseworlds is only just beginning, with steps toward increas-ingly sophisticated experiments falling within our vision.The scienti<c motivations for the understanding of smallbody interiors, perhaps fueled by a practical need to under-stand the requirements for impact mitigation, put us on anew and exciting threshold for exploration and discovery.

Acknowledgements

We thank the International Scienti<c Workshop Center ofthe Observatoire de Paris at Meudon for hosting an “InteriorStructures of Small Bodies” workshop 10–14 June 2002,which formed the basis for the synthesis and prospectuspresented here.

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