Mem. S.A.It. Suppl. Vol. 16, 84c© SAIt 2011

Memorie della

Supplementi

Asteroids: equilibrium shapes of rotatinggravitational aggregates

C. Comito1,2, P. Tanga2, P. Paolicchi3, D. Hestroffer4, A. Cellino1,

D. Richardson5, and A. Dell’Oro1

1 Osservatorio Astronomico di Torino, via Osservatorio 20, 10025 Pino Torinese (TO),Italia

2 Observatoire de la Cote d’Azur – Laboratoire Cassiope - UMR CNRS 6202, Boulevardde l’Observatoire, F-06304 NICE Cedex 4, France, e-mail: [comito;tanga]@oca.eu

3 Universita di Pisa – Dipartimento di Fisica, largo Bruno Pontecorvo 3, 56127 Pisa, Italia4 Observatoire de Paris – Institut de Mecanique Celeste et de Calcul des Ephemerides -

UMR CNRS 8028, 77 avenue Denfert-Rochereau, F-75014 Paris, France5 Department of Astronomy, University of Maryland, College Park, MD 20742, USA

Abstract. Many evidences lead us to think that a significant portion of asteroids are ag-gregates of boulders kept together by gravity, with little or no contribution from cohesiveforces. The standard hydrodynamics results for a self-gravitating and rotating cohesionlessbody would impose it to stay close to one of the well-known equilibrium sequences ofMacLaurin and Jacobi; yet, when we compare the shapes of known asteroids, we find thebulk of the shapes far away from the classical equilibrium case. In this work an analysisof the preferential shapes that a gravitational aggregate tends to assume is presented. Ourapproach considers the evolution of a variety of initial shapes, free to evolve towards theirown shape of equilibrium. The results show that actual asteroid shapes are consistent withthe evolution, under the action of gravity alone, of aggregates of free components tendingtowards states of minimum free energy.

Key words. Asteroids: shapes – Numerical simulations – Solar System evolution

1. Introduction

Asteroids (W. F. Bottke Jr., A. Cellino, P.Paolicchi, and R. P. Binzel, in ASTEROIDS III2002) are the remnants of the primitive phasesof Solar System formation, when asteroid-sized objects were in the process of building uptoday planets by mutual encounters. In the re-gion between the orbits of Mars and Jupiter thisaggregation process could not produce an ob-

Send offprint requests to: Comito C., Tanga P.

ject of planetary size due to the strong dynam-ical influence of a rapidly growing Jupiter dis-turbing the stability of the orbits of the nearbysmall bodies. The result is a collection of bod-ies no larger than about 500km in diameter (ex-cept the largest one, Ceres, of about 950km)down to dust-sized particles, the bulk of whichorbits the Sun in a ”belt” between the orbitsof Mars and Jupiter. Several observational evi-dences suggest that many medium-to-large as-teroids are not monolithic in nature, but rather

C. Comito et al.: Gravitational aggregates in equilibrium 85

“rubble piles” of fragments sticking togetherby simple gravity. The size of the fragmentscould be in the 50-200m range. The charac-teristic diameter distribution of the asteroids:dn ∝ D≈−2.5dD, is easily obtained from the-ory as the equilibrium distribution for a highlycollisional population where the bodies are fre-quently shattered into smallest ones by en-counters with moderately smaller impactors(Dohnanyi 1969). In the main asteroid beltthe presence of so-called “asteroid families”,groups of asteroids of similar orbital parame-ters and spectroscopic properties, can be ob-served. In many cases, the orbits of most ofthe bodies in each family are compatible tothe disruption of a ”parent body” by a catas-trophic event. Such events can be reconstructedby numerical and semi-analytical models thatshow how fragments partly re-aggregate intoseveral bodies, i.e. the currently observed fam-ily members. One of the most simple and ro-bust tests for this scenario, is provided by thedistribution of the rotation period of asteroids,showing an abrupt cut so that virtually all ob-jects larger than ∼100m have a rotation periodalways larger than ≈2h. This threshold wouldcorrespond to the critical value beyond whicha cohesionless aggregate begin losing materialfrom the surface due to the centrifugal accel-eration at the equator. Images of asteroids col-lected by space probes, such as those of 253Mathilde or the Mars moon Phobos, show bigcraters whose size is comparable to the diam-eter of the object itself. The impact that gen-erated them would have completely disruptedthe object, unless a large part of the impactenergy were dissipated by the propagation inhigly porous, energy-absorbing material, com-patible with a rubble pile model.

The low bulk densities of many candi-date rubble-pile asteroids with respect to thoseof meteorite rocks of similar spectroscopicproperties implies that large voids are presentin the interior of the bodies, with typicalvalues for the macroporosity in the 25-45%range of the total volume (D.T. Britt et al., inASTEROIDS III 2002).

The aggregate nature could suggests, as afirst order approximation, that the classic hy-drodynamic theory of equilibrium of fluid bod-

Fig. 1. Distribution of asteroids shapes of semi-axes a b c for which axes lenght ratios are known,compared to the Jacobi and MacLaurin sequences,adapted from Tanga et al. I (2009)

ies will well describe the general trend of ac-tual shapes. If this was the case, the expecteddistribution of asterois shapes would clusteraround the MacLaurin and Jacobi ellipsoidalsequences, but the observations appear to be incontradiction with the hypothesis, as illustratedin fig. 1.

Several different models have been used todescribe the behaviour of asteroids interiors us-ing e.g. elastic media with yielding conditions,such as Mohr-Coulomb or Drucker-Prager cri-teria (Holsapple 2004). This approach justify,in general, the overall range of shapes observedamong the asteroids. The primary problem thatelasto-plastic models encounter is the absenceof a clear “original” no-stress shape over whichto calculate the distortions, as the process thatbuilt up the asteroid has probably rather beena chaotic sequence of accretion and fragmen-tation.

Here we present a simple, alternativemodel for a rotating aggregate which showshow observed asteroid shapes can still be ac-counted for in the frame the equilibrium shapespredicted by the classical hydrodynamic the-ory.

86 C. Comito et al.: Gravitational aggregates in equilibrium

2. Potential for cohesionlessaggregates

2.1. Rigid body energy

If we assume an aggregate body in rigid rota-tion, its energy will be the sum of the gravita-tional energy U and of the rigid body rotationalenergy ErotRB. A rigid body rotation state canbe mantained only if the object is at equilib-rium, otherwise it will begin a process of re-shaping. This process implies the appearanceof internal motion of the composing fragmentswhich will have the secondary effect of dis-sipating energy because of internal frictions.Once at rest in an equilibrium shape, the to-tal energy of the body will again be of the formU + ErotRB. As this process will conserve angu-lar momentum L, it allows us to define

EL = U + ErotRB (1)

where the quantities depend only on the shapeof the body, and the angular velocityω is takensuch that L assumes a given fixed value.

It appears natural to assume that reshapingshall occur in the direction minimizing suchquantity E, i.e. following −∇shapespaceE.

Assuming for simplicity a body of uniformand constant density ρ that has an overall ellip-soidal shape defined by the semiaxes a1, a2, a3(with a3 being in the same direction as L) theU and ErotRB quantities can be explicitly calcu-lated (Chandrasekhar 1969):

U = −25πGρMa1a2a3

∫ ∞

0

du∆

(2)

ErotRB =110

M(a21 + a2

2)ω2 (3)

where

∆ =

√(a2

1 + u)(a22 + u)(a2

3 + u) (4)

in which we enforce the constraints

a1a2a3 = constant (5)ω = I−1L (6)

Taking care of working with appropriate”normalized” L and ω quantities

L2

=L2

GM3 3√

a1a2a3(7)

ω2=

ω2

πGρ(8)

we can obtain a description of the state of anobject independent of its actual mass and vol-ume.

2.2. Ellipsoidal case

In the classical incompressible fluid case, aself-gravitating body at equilibrium accomo-dates to an equipotential surface, always or-thogonal to the local (gravitational + cen-tripetal) force. This model assumes an absenceof resistance to internal motions (reshaping),which is not the case for real rocky objects: instandard earth conditions, for example, a pileof sand can sustain a slope angle of about 30◦(Holsapple 2001). For ellipsoidal shapes, thisresults in the classical MacLaurin sequence ofspheroids (or ”flattened” spheres) and Jacobisequence of triaxial ellipsoids. Only one shapeis possible in the MacLaurin case for each fixedvalue of L, and only one Jacobi shape is possi-ble for each value of L above ≈0.304, with highangular momentum shapes (above L ≈ 0.4) be-ing unstable. In figures 2 and 3, the plot of theEL over ellipsoidal shapes is shown for L = 0and 0.3, along with a normalized plot of thegradient.

If a body is free to evolve from a start-ing non-equilibrium ellipsoidal configurationin rigid rotation, it will migrate on the (α2 =

a2/a1); (α3 = a3/a1) plane of constant L alongthe direction of −∇α2;α3 EL towards the min-ima of the Maclaurin or the Jacobi shapes.As, however, the E plot appears quite flat ina large region around the classical equilib-rium configuration, a small amount of inter-nal friction can provide stability to shapes withnon-equipotential surfaces (Holsapple 2004;Sharma et al. 2009).

3. Gravitational aggregatessimulations

3.1. pkdgrav

To test the model and see which range ofshapes is potentially compatible with a grav-

C. Comito et al.: Gravitational aggregates in equilibrium 87

-3.5e+12

-3e+12

-2.5e+12

-2e+12

-1.5e+12

-1e+12

EL = 0,0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a2/a1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a3/a

1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

a3/a

1

a2/a1

grad E_{L = 0,0}

Fig. 2. Above: E0 (arbitrary scale) - Below: normal-ized −∇α2 ;α3 E0

itational aggregate, we used pkdgrav, a N-body gravitational simulator that uses finite-sized spherical particles with collisions. It hasbeen extensively used in the past for dynami-cal simulations of protoplanetary rings, proto-planet accretion and and equilibrium shapes ofaggregates (Richardson et al. 2000, 2005). Thesoftware is built on a parallel architecture andimplements a tree-code hierarchization of theparticle distribution. It is one of the most effi-cient N-body simulators.

It is interesting to note that the typi-cal space-filling ratio of a random packing

-2.8e+12

-2.6e+12

-2.4e+12

-2.2e+12

-2e+12

-1.8e+12

-1.6e+12

-1.4e+12

-1.2e+12

-1e+12

EL = 0,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a2/a1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a3/a

1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

a3/a

1

a2/a1

grad E_{L = 0,3}

Fig. 3. Above: E0.3 (arbitrary scale) - Below: nor-malized −∇α2 ;α3 E0.3

of equal-sized spheres is ≈ 36% (Song et al.2008), i.e. of the same range of the typicalmacroporosity of asteroidal bodies.

3.2. The simulated aggregates

We have created a set of rotating ellipsoidalshapes with varying axis ratios:

(α2;α3) ∈ {0.1, 0.2, 0.4, 0.6, 0.8, 1.0}2 (9)

Each object is composed by 1000 identical par-ticles. A rigid body rotation is imposed by as-

88 C. Comito et al.: Gravitational aggregates in equilibrium

signing to each particle a spin and, as a func-tion of its distance from the rotation axis (as-sumed to be along the a3 axis), the appropriatevelocity derived from an imposed value of thebody angular momentum L according to eq.(6).

As each object is composed by non-sticking particles that are free to change theirposition, under the action of gravity, modifica-tions of the shapes are expected as the N-bodycomputes the evolution.

Each object was left free to evolve inshape and to reach the equilibrium with theoverall process solely governed by gravitation,and with a small amount of energy dispersionvia reciprocal friction and inelastic bouncingamong the particles.

3.3. The aggregates equilibrium shapes

In fig. 4 and 5 we show the resulting paths onthe α2;α3 plane along with the distribution ofthe ending shapes for the L = 0 and 0.3 cases.We want to stress here a general property: thepaths follow the gradient lines quite well, yetthe resulting final shapes are dispersed in awide region instead of being clustered aroundthe MacLaurin point.

The explaination is straightforward. As thebody is driven by internal forces along thepaths of decreasing E, it is deformed towardsshapes that are more and more stable, up to apoint in which the free energies allotted to thesingle particles are no longer capable of over-coming the potential boundaries of particle in-terlocking. This can be easily seen by the factthat the boundary of the region where the finalshapes are distributed is characterised by thoseshapes that can be sustained by a cohesionlessmaterial with angle of repose of 5 to 7◦. Sincethis value appears to be consistent for differentangular momentum cases, it can be consideredas the typical angle of repose for cohesionlessaggregate of equal spheres.

As can be observed, the shape evolutiontends to drive most bodies towards prolateshapes with a2 = a3, i.e. the ”main” diagonalof the α2;α3 plane, and is in the region around

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

a3/a

1

a2/a1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a3/a

1

a3/a1

Fig. 4. Above: the evolution paths of the 36 objectswith axis ratios as in 9 in the L = 0 case super-imposed to normalized −∇α2 ;α3 E0 - Below: the finalresulting shapes (hollow circles) and the MacLaurinequilibrium shape (red circle)

the diagonal that most final shapes tends to befound.

4. Conclusions

If we compare (fig. 6) the diagram of finalshapes for the cases of L = 0 to 0.5 (in incre-ments of 0.1) to the distribution of real aster-oids, the similarity is obvious (Tanga et al. II2009). As a consequence, the shapes of realbodies can be justified in the context of theE model. In fact, if we measure by maximum

C. Comito et al.: Gravitational aggregates in equilibrium 89

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

a3/a

1

a2/a1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a3/a

1

a3/a1

Fig. 5. Above: the evolution paths of the 36 objectswith axis ratios as in 9 in the L = 0.3 case superim-posed to normalized −∇α2 ;α3 E0.3 - Below: the finalresulting shapes (hollow circles) and the MacLaurinequilibrium shape (red circle)

slope angle of the overall form, it appears thatthey are effectively near the classical equilib-rium shapes.

This is both a complement of previousstudies (Holsapple 2004) and a justification ofthe actual shapes which are perfectly compati-ble with a “rubble pile” gravitational aggregatewith even a mildly departure (5◦ slope) froma fluid system, arguably less than for irregularrocky or icy boulders.

One shold note that our conclusions arebased on the analysis of known axis ratios

only. This can be seen as a limitation, butour approach avoids poorly known parameters(such as the density) that could seriously af-fect the conclusions reached in previous papers(Holsapple 2004).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a3/a

1

a2/a1

Fig. 6. Final shapes for simulations using L = 0 ;0.1 ; 0.2 ; 0.3 ; 0.4 ; 0.5, with the Jacobi sequencein red (MacLaurin sequence coincides with the rightside of the box); cfr. fig. 1

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