Formation of Kuiper-belt binaries through multiple chaotic scattering encounters with low-m

a r X i v :a s t r o -p h /0504060v 1 4 A p r 2005

Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 2February 2008

(MN L A T E X style ?le v2.2)

Formation of Kuiper-belt binaries through multiple chaotic

scattering encounters with low-mass intruders

Sergey A.Astakhov,1?

Ernestine A.Lee 2,3

?and David Farrelly 2?

1

John von Neumann Institute for Computing,Forschungszentrum J¨u lich,D-52425J¨u lich,Germany

2Department

of Chemistry and Biochemistry,Utah State University,Logan,UT 84322-0300,USA

3FivePrime Therapeutics,951Gateway Boulevard,South San Francisco,CA 94080,USA

Submitted 2004November 7;Accepted 2005April 4

ABSTRACT

The discovery that many trans-neptunian objects exist in pairs,or binaries,is proving invaluable for shedding light on the formation,evolution and structure of the outer Solar system.Based on recent systematic searches it has been estimated that up to 10%of Kuiper-belt objects might be binaries.However,all examples discovered to-date are unusual,as compared to near-Earth and main-belt asteroid binaries,for their mass ratios of order unity and their large,eccentric orbits.In this article we propose a common dynamical origin for these compositional and orbital properties based on four-body simulations in the Hill approximation.Our calculations suggest that binaries are produced through the following chain of events:initially,long-lived quasi-bound bina-ries form by two bodies getting entangled in thin layers of dynamical chaos produced by solar tides within the Hill sphere.Next,energy transfer through gravitational scat-tering with a low-mass intruder nudges the binary into a nearby non-chaotic,stable zone of phase space.Finally,the binary hardens (loses energy)through a series of rela-tively gentle gravitational scattering encounters with further intruders.This produces binary orbits that are well ?tted by Kepler ellipses.Dynamically,the overall process is strongly favored if the original quasi-bound binary contains comparable masses.We propose a simpli?ed model of chaotic scattering to explain these results.Our ?ndings suggest that the observed preference for roughly equal mass ratio binaries is probably a real e?ect;that is,it is not primarily due to an observational bias for widely sepa-rated,comparably bright objects.Nevertheless,we predict that a sizeable population of very unequal mass Kuiper-belt binaries is likely awaiting discovery.

Key words:celestial mechanics -methods:N-body simulations -minor planets,asteroids -Kuiper Belt -binaries:general -scattering

1INTRODUCTION

Binary systems occupy a special place in astronomy.First and foremost they allow the determination of binary partner masses which,in the case of stars,proved critical in the early development of the mass-luminosity relation (Eddington 1924;Noll 2003).In fact,the majority of all stars seem to be members of gravitationally bound multiplets with most be-ing members of binaries (Heggie &Hut 2003).The dynam-ics and energetics of star clusters,in particular,are strongly in?uenced by the presence of binaries.This is because the largest single contributor to the total mechanical energy of the cluster may be the internal energy (binding energy)of

?

E-mail:s.astakhov@fz-juelich.de;www.astakhov.newmail.ru ?E-mail:ernestine.lee@https://www.wendangku.net/doc/476076417.html, ?E-mail:david.farrelly@https://www.wendangku.net/doc/476076417.html,

binaries (Hills 1975;Inagaki 1984;Janes 1991;Heggie &Hut 2003).Not surprisingly the dynamics of binary systems and their formation mechanism are a subject of continuing in-terest in stellar dynamics (Heggie 1975;Hills 1975,1983a,b,1990;Mikkola &Valtonen 1992;Mardling 1995a,b;Quinlan 1996;Heggie &Hut 2003).

Binaries are also relatively common in planetary physics;these include the Earth-Moon and Pluto-Charon systems (Christy &Harrington 1978;Canup &Asphaug 2001;Canup 2004,2005)as well as a substantial num-ber of asteroid binaries in the main belt.Asteroid bina-ries usually contain partners with mass ratios in the range m r ～10?3?10?4where m r =m 2/m 1and m 1and m 2are the masses of the primary and secondary binary partners respectively.Objects with such asymmetric masses are most often thought of as “asteroids with satellites”(Merline et al.2003)rather than as “binaries”although there exists no

2Astakhov,Lee and Farrelly

Table 1.Proposed Kuiper-belt binary formation mechanisms.

meaningful quantitative distinction between these de?ni-tions.The recent discovery of binary trans-neptunian ob-jects (TNOs)is particularly signi?cant because these objects allow for the direct determination of TNO masses thereby

opening up an important window into the origin,evolution and current composition of the Kuiper-belt (KB)(Stewart 1997;Toth 1999;Luu &Jewitt 2002;Noll 2003;Gladman 2005).It is quite remarkable that recent systematic searches (Trujillo &Brown 2002;Noll 2003;Schaller &Brown 2003)seem to suggest that up to 10%of Kuiper-belt objects might exist as gravitationally bound pairs (Burns 2004).

Kuiper-belt binaries (KBBs)are notable for their unusual compositional and orbital properties as com-pared to main-belt asteroids:the most salient are (Noll 2003)(i )large mutual orbits,(ii )highly ec-centric mutual orbits and (iii )mass ratios of order unity (Margot 2002;Veillet et al.2002;Durda 2002;Weidenschilling 2002;Goldreich,Lithwick &Sari 2002;Noll 2003;Schaller &Brown 2003;Osip,Kern &Elliot 2003;Altenho?,Bertoldi &Menten 2004;Funato et al.2004;Burns 2004;Noll et al.2004a,b;Takahashi &Ip 2004;Nazzario &Hyde 2005).However,in each case certain caveats apply (Noll 2003;Noll et al.2004a,b).Firstly,by some measures KBB orbits are indeed large;e.g.,for main-belt asteroids with satellites the semimajor axis of the mu-tual orbit (a )is typically about an order of magnitude larger than the radius of the primary asteroid (r A )whereas for KBBs a/r A ～50?500(Merline et al.2003;Noll et al.2004b).Curiously,however,if the semimajor axis is com-pared to the radius of the binary’s Hill sphere (R H )then

Formation of Kuiper-belt binaries3

main-belt asteroid binaries and KBBs have rather similar semimajor axes,i.e.,expressed as a percentage a/R H～2?5%(Noll2003;Noll et al.2004b).Secondly,although KBB orbits are noticeably eccentric as compared to the almost circular orbits of main-belt asteroid binaries,re-cent observations suggest that extremely large eccentricities (e>0.8)may be rarer than was initially thought(Noll2003; Noll et al.2004b).Finally,while the mass ratios of known KBBs are in the range m r～0.1?1it is also possible that this result is at least partially due to an observational bias for well-separated,equally bright objects(Burns2004).Nev-ertheless,the properties of KBBs are su?ciently striking to suggest that their formation mechanism di?ered consider-ably from that of main-belt or near-Earth asteroid binaries, which are thought to have been produced by physical colli-sions(Chauvineau,Farinella&Harris1995;Xu et al.1995; Durda et al.2005).

Four di?erent pathways(Weidenschilling2002; Goldreich et al.2002;Funato et al.2004)–summarized in Table1–have been proposed to explain the formation and properties of KBBs which we will refer to as Paths 1–4:Path1involves physical collisions between two planetesimals within the Hill sphere of a larger Kuiper-belt object(Weidenschilling2002).The two bodies then accrete and remain gravitationally bound as a single object to the larger body,thereby producing a binary.This mechanism assumes that binary objects are primordial;Similar to Weidenschilling’s mechanism(Weidenschilling2002)Paths 2and3also proceed from the temporary gravitational cap-ture of two objects within the three-body Hill sphere.Now, however,the large third body is the Sun(Goldreich et al. 2002).In both Paths2and3the initial stage of binary formation is the formation of a transitory binary which will eventually ionize unless stabilization occurs?rst.Two stabilization mechanisms have been proposed;in Path2 stabilization results from dynamical friction through inter-actions with a sea of smaller(<10km)Kuiper-belt bodies. In Path3the transitory binary is stabilized by energy loss through gravitational scattering with a large third body–the L3channel.Both of these mechanisms emphasize the importance of the Sun-binary Hill sphere(Szebehely1967; Murray&Dermott1999).

A common feature of Paths2and3is that immedi-ately before and after capture the binary partners are fol-lowing an essentially three-body orbit.That is,solar pertur-bations or tides are important.Thus the orbits are periodic, quasi-periodic or chaotic(Lichtenberg&Lieberman1992) and,in general,cannot be well described by orbital elements of the Kepler problem(Murray&Dermott1999).However, actual KBBs follow orbits which can be well?tted by Ke-pler ellipses(Veillet et al.2002;Noll2003;Osip et al.2003; Noll et al.2004a,b),i.e.,solar tides are relatively unimpor-tant and the orbits are approximately two-body in nature. Therefore,a mechanism for gradually transforming captured –and,therefore,bound–quasiperiodic three-body orbits into(almost)periodic two-body orbits is required.We term this general process“Keplerization”–i.e.,the gradual en-ergy loss of three-body binary orbits to produce orbits that are essentially Kepler ellipses.As three-body orbits lose en-ergy their semimajor axes undergo a steady reduction in size.In principle,Keplerization may occur directly during the capture process itself(e.g.,as posited to happen in L3)or it can happen more gradually after the initial capture event through,e.g.,continued dynamical friction over ex-tended timescales(Goldreich et al.2002).

Path4,proposed by Funato et al.(2004),involves ex-change reactions(Heggie&Hut2003)in which the smaller member of an already bound“asteroid-like”binary is dis-placed by a larger body during a three-body encounter.This mechanism is essentially a variant of binary star-intruder scattering(Hills1975,1983a,b,1990;Hut&Bahcall1990; Heggie&Hut2003).In this mechanism solar tides are ig-nored and both the pre-and post-collision binary(if it sur-vives)follow a two-body Keplerian ellipse.

While each of these four pathways is feasible,no con-sensus has yet emerged as to their relative importance or even whether a di?erent mechanism altogether might have operated(Burns2004).This is because each mechanism ad-mits at least one potentially serious drawback.The?rst three paths are sensitive to assumptions about the size distributions of Kuiper-belt objects(Funato et al.2004). Path1depends on the existence of more large bodies in the Kuiper-belt than seems to be consistent with observa-tions(Goldreich et al.2002;Funato et al.2004);The basis of Path2is dynamical friction which,to be e?ective,requires a much larger sea of small bodies than is predicted by current theories of planetesimal formation(Goldreich&Ward1973; Wetherhill&Stewart1993;Ra?kov2003;Bernstein et al. 2004;Funato et al.2004;Kenyon&Bromley2004).Fur-ther,it is unclear how and why dynamical friction should select preferentially for roughly equal mass ratio binaries. This assumes,of course,that the apparent preference for mass ratios of order unity is not an observational artefact (Burns2004).

The L3channel(Goldreich et al.2002),Path3,depends on relatively rare close encounters between three large ob-jects and,based on the estimates of Goldreich et al.(2002), it is unclear if these occurred often enough to produce the current population of KBBs.In this regard it is interesting to note that recent calculations suggest that,if the Kuiper-belt lost its mass through collisional grinding,then an or-der of magnitude more binaries were likely present in the primordial Kuiper-belt than has otherwise been thought (Petit&Mousis2004).This illustrates that the number and size distributions of primordial KB objects might be subject to revision(Stern2002;Bernstein et al.2004; Kenyon&Bromley2004;Petit&Mousis2004;Elliot et al. 2005).A second drawback to Path3is that,as origi-nally proposed,it provides no explicit mechanism–be-yond possible post-encounter dynamical friction–for pro-ducing the approximately two-body Keplerian binary or-bits which are actually observed(Veillet et al.2002;Noll 2003;Schaller&Brown2003;Osip et al.2003;Noll et al. 2004a,b).It is unlikely,as our simulations con?rm,that a single collision between a quasi-bound binary(i.e.,one that is following a three-body Hill trajectory)and a massive in-truder will result in a two-body Keplerian binary orbit.Fi-nally,Path4leads almost exclusively to binary eccentricities e 0.8and very large semimajor axes which may approach R H itself(Funato et al.2004).However,moderate eccentric-ities,in the range0.25 e 0.82,and semimajor axes that are only a few percent of R H seem to be the rule(Noll2003; Osip et al.2003;Noll et al.2004a,b).

In the model described herein the?rst step is the

4Astakhov,Lee and Farrelly

formation of long-lived quasi-bound binaries through the recently proposed mechanism of chaos-assisted capture (CAC,Astakhov et al.2003;Astakhov &Farrelly 2004;Trimble &Aschwanden 2003)in the Hill approximation (Szebehely 1967;H′e non &Petit 1986;Murray &Dermott 1999;Heggie &Hut

2003).In CAC particles become tem-porarily caught-up in thin chaotic layers within the

Hill sphere of,

in this

case,the

Sun-binary system.

These chaotic layers,which separate regular from asymptotically hyper-bolic (scattering)orbits,are the result of the perturbation of the Sun on the motion of the two bodies making up the bi-nary.Because the orbits are chaotic any quasi-bound (tran-sitory)binary which is formed will eventually break apart.However,the lifetimes of these quasi-bound objects can be su?ciently long that,in the interim,stabilization can occur.Since the chaotic layers in the Hill problem (Sim′o &Stuchi 2000;Astakhov et al.2003;Astakhov &Farrelly 2004)lie adjacent to regular Kolmogorov-Arnold-Moser (KAM,Zaslavsky 1985;Lichtenberg &Lieberman 1992)regions,then,in principle,even relatively weak perturbations can switch the binary into such KAM regions and thereby lead to permanent capture.

Of course,the precise nature of the stabilization mech-anism is the crux of the matter.We propose that capture and subsequent hardening of three-body Hill orbits (i.e.,Keplerization)(Hills 1975,1983a,b,1990;Quinlan 1996;Heggie &Hut 2003)proceed through chaotic gravitational scattering encounters with low-mass intruders which happen to transit the Hill sphere.Once a binary has been captured initially then further encounters with intruders are proba-ble –about once every 3,000-10,000years (Weidenschilling 2002)–and this can eventually produce binaries whose mu-tual orbits are Kepler ellipses.Empirically we ?nd that the process is most e?cient for equal mass binaries and rela-tively low-mass intruders having ～2%or less of the total binary mass.Incidentally,we argue that CAC is also the dy-namical basis of the twin mechanisms proposed empirically by Goldreich et al.(2002).

To model the statistics of capture we compute cap-ture probabilities in four-body (Sun,binary,intruder)Monte Carlo scattering simulations for various binary mass ratios in the Hill approximation (Scheeres 1998).Our simulations reveal a clear preference for equal mass binaries to survive multiple subsequent encounters as they harden through in-truder scattering.

The paper is organized as follows;Section 2introduces the equations of motion for the three-and four-body prob-lem in the Hill approximation and provides a brief overview of how chaos can assist capture into stable three-body orbits (Astakhov et al.2003;Astakhov &Farrelly 2004).Monte Carlo simulations in the four-body Hill approxima-tion (Scheeres 1998)are described in Sec.3.A simpli?ed (reduced dimensionality)model of chaotic low-mass intruder scattering is introduced in Sec.4to explain our results.To study the dynamics in this model we employ the Fast Lya-punov Indicator for several mass ratio and binary eccen-tricity combinations.A more detailed comparison with the other formation models summarized in Table 1is made in Sec.5.Conclusions are in Sec.6.

points demark the gateways into the capture zone.All units are scaled Hill units.

2

EQUATIONS OF MOTION IN THE

THREE-BODY AND FOUR-BODY HILL PROBLEMS

In our model the initial stages of binary formation in-volve temporary capture within the Sun-binary Hill sphere through two particles getting entangled in chaotic regions of phase space,i.e.,chaos-assisted capture (Astakhov et al.2003;Astakhov &Farrelly 2004).This process is well de-scribed in the three-body Hill approximation.2.1

Three-body Hill problem

The three-body vector equations of motion in the Hill ap-proximation are given by (Szebehely 1967;H′e non &Petit 1986;Murray &Dermott 1999;Scheeres 1998)¨ρ

+?×[2˙ρ+?×ρ]=?ρ+3a (a ·ρ)?ρ

3m 0

1/3

;in physical units:R H ～3·105km ～25a .

Formation of Kuiper-belt binaries5

In scaled Hill units(Szebehely1967;Murray&Dermott 1999)R H=(1/3)1/3and the radii of the binary partners r A～10?4.

2.2Chaos-assisted capture in Hill’s problem

In order for a transitory(quasi-bound)binary to form two bodies must come inside their mutual Hill sphere de?ned with respect to the Sun.The Lagrange saddle points L1and L2in Fig.1serve as gateways between the interior of the Hill sphere and heliocentric orbits.Figure2a shows a typical chaotic binary orbit obtained by integrating Hill’s equations (1)in two-dimensions.The orbit is trapped close to a peri-odic orbit for many periods before?nally escaping from the Hill sphere(not shown).The accompanying Poincar′e surface of section(SOS)in Fig.2b shows that the orbit is actually trapped in a chaotic layer separating a regular KAM island from a large region of hyperbolic scattering.Note that the orbits shown in Fig.2correspond to the mutual Hill orbit of the two binary partners and are independent of the particu-lar mass ratio of the binary partners(H′e non&Petit1986; Murray&Dermott1999).

Examination of Poincar′e surfaces of section in the Hill problem(Sim′o&Stuchi2000)–or,equivalently,the circu-lar restricted three-body problem(CRTBP)for small masses (Astakhov et al.2003)–reveals that,at energies above the Lagrange saddle points L1and L2all phase space is di-vided into three parts,one of which regular KAM orbits inhabit,chaotic orbits another,the third consists of direct scattering orbits or,in other words,hyperbolic orbits.All these di?er from each other in the behaviour of their or-bits.The chaotic orbits separate the regular from the hy-perbolic regions–see Fig.2b.As energy is increased above the saddle points the chaotic regions consist of relatively thin layers of chaos which cling to the progressively shrink-ing KAM tori(Sim′o&Stuchi2000;Astakhov et al.2003; Astakhov&Farrelly2004).Because incoming orbits cannot penetrate the regular KAM tori in2-dimensions(2D)and only exponentially slowly in3D(Lichtenberg&Lieberman 1992;Astakhov et al.2003),orbits entering the Hill sphere from heliocentric orbits must either enter chaotic layers or scatter out of the Hill sphere essentially immediately.

Binaries corresponding to chaotic orbits lying close to KAM tori can themselves follow almost regular orbits for very long times as shown in Fig.2a.In the presence of a source of dissipation or of other perturbations these orbits can easily be switched into the nearby,but otherwise impen-etrable,KAM regions.

It is remarkable that capture can occur if the binary de-creases or increases its energy during the encounter.This can happen because the topologies of stable KAM islands are, in general,a non-linear function of energy as is illustrated for four selected energies in Fig.2c.That is,if a perturba-tion causes a chaotic orbit to increase or decrease its energy suddenly it may,in either event,?nd itself captured in a nearby regular part of phase space.Alternatively,such per-turbations,though small,may cause the quasi-bound binary to break up more quickly if the size of the chaotic layer is suddenly reduced in favour of scattering regions.

Possible energy loss mechanisms which can sta-

bilize transitory binaries include physical collisions (Weidenschilling2002),gravitational scattering with other bodies and dynamical friction(Goldreich et al.2002). Ruling out physical collisions and dynamical friction for reasons already described leaves gravitational scattering with“intruder”objects which enter the binary Hill sphere. The e?ciency of this process depends critically on the intruder mass.For example,we?nd that the delicate chaotic binary orbits tend to be catastrophically desta-bilized by large intruders as, e.g.,in the L3mechanism (Goldreich et al.2002).Further,as Fig.2shows,the orbits of transitory binaries can be a very large fraction of the Hill radius whereas actual binary orbits are typically only a few percent of R H.Any KBB production mechanism must,therefore,explain how these large initial orbits are transmogri?ed into compact,Keplerian orbits.It is unlikely (as our simulations will show)that a single scattering encounter would propel the binary into such a?nal state directly.Rather,we propose that a series of gentle scattering encounters with low-mass intruders is necessary to capture and?nally produce Keplerian orbits.Gentle stabilization by low mass intruder scattering is possible because only a weak perturbation is needed to drive long-lived chaotic orbits into adjacent regular regions of phase space.

2.3Four-body Hill problem

The simulation of intruder scattering is facilitated by study-ing intruder scattering in the four-body(Sun,binary,in-truder)Hill approximation(Scheeres1998;Brasser2002; Scheeres&Bellerose2005).This approximation is appropri-ate given the mass ratios involved and the low to moderate eccentricities of typical KBB centre-of-mass orbits around the Sun.

The generalization of the Hill problem(Szebehely1967; H′e non&Petit1986;Murray&Dermott1999)to the case where three small bodies,with a mutual centre-of-mass,R c, orbit a much larger fourth body(the Sun,m0=1)on a near circular orbit(e.g.,three Kuiper-belt objects within their mutual Hill sphere)is due to Scheeres(1998).The total mass of the three bodies is de?ned by

μ=

3

j=1m j?1(2)

where R c≈a=(1,0,0)de?nes the motion of the three-body centre-of-mass along an almost circular orbit which de?nes the rotating frame.The vector equations of motion are(Scheeres1998)

¨ρ+?×[2˙ρ+?×ρ]=?ρ+3a(a·ρ)?(α1+α2)

ρ|ρ3?ρ2|3

?

ρ3?ρ1

|ρ3?ρ1|3

?α2

ρ3?ρ2

6Astakhov,Lee and Farrelly

Figure2.Non-linear dynamics of transitory binary formation in Hill’s three-body problem.Panel(a)shows a quasi-bound(chaotic) binary centre-of-mass orbit(green)temporarily trapped close to a periodic orbit(red)which lies at the centre of a stable KAM island. This trajectory eventually escapes from the Sun-binary Hill sphere(large black circle).Panel(b)is a composite showing the colour coded SOS(x?y hyperplane de?ned such that the momentum p x=0and the velocity dy

Formation of Kuiper-belt binaries

7

(iii)Intruder particles were ?red at each binary orbit with the phase of the binary chosen randomly.For each binary orbit 5000intruders were sent in and,each time,it was recorded whether the binary was stabilized or not.For each binary orbit this produces a capture probability,i.e.,the fraction of the 5000intruders that lead to stabilization in a single encounter with a binary.

(iv)Intruders were started isotropically with uniformly chosen random positions on a sphere of radius R H ,with

uniform random velocities |v | 5v H where v H is the Hill velocity (Goldreich et al.2002;Funato et al.2004;Goldreich,Lithwick &Sari 2004)v H ～?R H ,

(7)

with ?being the orbital frequency of the third body around the binary at distance R H .

(v)In order to choose the phase of the binary randomly intruder integrations were started at randomly chosen times t ∈(0,T H )along the binary orbit.In practice this has the e?ect of sending the intruder towards the binary at di?erent relative con?gurations of the binary partners in phase space.(vi)The integrations proceeded until either the binary broke up or it survived for a time t =10T H at which point it was considered captured.After the intruder escaped from a sphere of radius 3R H the integrations for the binary continued assuming m 3=0,i.e.,the usual three-body Hill equations (1)were used.

(vii)At the end of the integration each captured binary and its intruder were back integrated in time to ensure that the binary (in the four-body integration)broke up in the past.This helps protect against accidentally starting the binary-intruder trinary in already bound regions of phase space.

(viii)Integrations were stopped if particles came within a distance r A =10?4of each other.

(ix)These simulations were repeated ten times so as to be able to compute variances (error bars)for the capture probabilities.Each time a di?erent random number seed was used to guarantee that the simulations were independent.Thus,e.g.,Orbit 1may end up with capture probability 0.4±0.02,Orbit 2,0.23±0.018,and so on.

(x)Orbits in the cohort were then binned according to their assigned capture probabilities.This produced his-tograms showing the number-density of orbits in the cohort having capture probabilities in the various speci?ed inter-vals.Mainly the error bars were used (a)to establish that the calculations had converged and (b)to decide on a rea-sonable granularity for the probability bins.

(xi)The actual number of binary orbits included in the cohort was chosen so as to provide acceptable convergence given available computational resources.Convergence was judged acceptable when variances in capture probabilities were on the order of a few percent or less.

3.2Single intruder scattering

In these simulations,each binary orbit in the cohort ends up with a capture probability and an error bar (of the order of a few percent)attached to it.However,for di?erent mass ratios the same binary orbit in the cohort will,in general,have di?erent capture probabilities and error bars.Figure 3

Figure 3.Fraction of binary orbits (number-density distribu-tions)binned according to their individual capture probabilities as described in the text with α3=0.025;Mass ratios are (a)α1=α2and (b)α1=14α2(b).Smooth curves are best ?ts to normal distributions.The same set of 170chaotic Hill orbits in three-body centre-of-mass coordinates,appropriately rescaled,was used in both cases.The binary was considered captured only if the original binary partners remained bound to each other,i.e.,exchange reactions were ignored.In 98.5%of cases the average capture probability for any given Hill orbit was higher for equal mass binaries.

presents histograms showing the fraction of captured orbits binned according to their individual capture probabililities for two typical mass ratios.We tested that these histograms are robust to the actual number of binary orbits included in our cohort –e.g.,if 85rather than 170orbits are used the shapes of the histograms are essentially identical to those in Fig.3.The width of the bins is roughly commensurate with the average width of the error bars.

The distributions in Fig.3clearly indicate that the frac-tion of binary orbits peaks at a signi?cantly higher capture probability for equal mass ratio binaries than for unequal mass ratio binaries.This translates into a higher probability for capture in the equal mass case.Recall that the actual bi-nary orbits used in both cases are identical.Since the cohort of binary orbits used is the same,and the intruder masses and initial conditions are also the same,then the di?erence in the distributions is due entirely to the di?erent binary mass ratios employed.In fact the only parameter that was varied between the simulations shown in Fig.3was the mass ratio.

We also compared individual capture probabilities or-bit for orbit and found that in 98.5%of cases the aver-age capture probability for any given Hill orbit was higher in the equal mass simulation.Thus,at any ?xed Hill ra-dius (i.e.,the same total binary mass)a clear preference for the capture of same-sized binary partners is appar-ent.For comparison,the mass ratios m r of the four best

8Astakhov,Lee and Farrelly

characterized KBBs are～0.57(1998W W31,Veillet et al. 2002),～0.56((66652)1999RZ253,Noll et al.2004a),～0.55((58534)1997CQ29,Noll et al.2004b)and～0.34 ((88611)2001QT297,Osip et al.2003).These are signi?-cantly larger than the mass ratios of main-belt binaries for which,as noted,m r～10?3?10?4(Merline et al.2003).

The calculations for Fig.3took～1month using all nodes on a32-node Beowulf cluster.Interestingly the mass e?ect manifested itself quite directly in that the simula-tions for Fig.3b?nished about a week earlier than those for Fig.3a.This is because of the additional tests and inte-grations that needed to be performed to verify that an orbit had been captured permanently.

3.3Multiple intruder scattering

The probabilities in Fig.3are for permanently captur-ing already-formed quasi-bound binaries through a sin-gle intruder scattering event.However,the overall prob-ability of binary formation depends too on the probabil-ity of forming quasi-bound binaries in the?rst place.Be-cause Hill’s equations(1)are parameter free(after rescal-ing,Murray&Dermott1999)once two objects come within their mutual Hill sphere the probability of chaos-assisted quasi-binary formation becomes independent of their rela-tive masses.Nevertheless,the entry rate of bodies into each other’s Hill sphere–and thus the overall capture proba-bility–will depend on the actual masses and velocities of the objects involved.Uncertainties in the size and velocity distributions of contemporary and primordial Kuiper-belt objects means,however,that estimating Hill sphere entry rates is an open problem but one whose resolution should be aided by the discovery and characterization of further KBB objects(Petit&Mousis2004;Bernstein et al.2004; Kenyon&Bromley2004;Noll et al.2004b).Despite these uncertainties the mass e?ect is expected to persist because it is progressively enhanced by later scattering events.

After its initial capture through a single scattering en-counter with an intruder a binary is,typically,following a very large three-body orbit with a semimajor axis similar in size to that of the periodic orbit illustrated in Fig.2a.That is,though now bound,the binary partners are still strongly in?uenced by solar tides.In this subsection we demonstrate that essentially Keplerian two-body orbits can be produced through a series of subsequent scattering encounters with further small intruders.

That this is possible is demonstrated by the simulations underlying Fig.4.In these simulations,for each mass ratio,a cohort of1200randomly chosen Hill binary orbits(in the en-ergy range?2.1550scaled Hill time units.The?rst intruder was sent in at a random point along the binary orbit during the orbit’s Hill lifetime t∈(0,T H)(the binary orbit integration was itself started at t=0).If the binary survived to t=50the next intruder was sent in.If the binary survived that event then the next in-

truder was sent in,and so on.Thus,intruders were spaced50Figure4.Distributions of orbital elements,eccentricity,e,and semimajor axis,a(scaled by the Hill radius R H)after N enc=200 intruder scattering events per binary The?gure combines results for twenty mass ratios:m r=α2

Formation of Kuiper-belt binaries

9

still strictly solutions to Hill’s problem,their energy has been reduced su?ciently that,for all intents and purposes,they follow two-body Kepler ellipses,i.e.,

the orbits are well de-scribed by Keplerian orbital elements (Murray &Dermott 1999).

Experimental simulations for multiple intruder encoun-ters reveal that the main e?ect of increasing the number of encounters (N enc )is to reduce the ?nal value of the semi-major axis whereas the ?nal eccentricity distributions were quite robust to N enc .Thus we chose the single parameter N enc =200heuristically so that orbits ended up roughly in the observed semimajor axis range of 1998W W 31.There-fore,Fig.4represents a single parameter ?t to the semimajor axis of one of the most well characterized KBBs.However,no special attempt was made to re?ne the ?t.

Kuiper-belt binary semimajor axes –expressed as a fraction of the binary Hill radius –are generally compa-rable to those of asteroid binaries and Fig.4captures this ?nding.Figure 4also reproduces the moderate eccentrici-ties which seem to be a feature of actual KBBs (Noll et al.2004b).This ?nding contrasts with the extremely large ec-centricities (e >0.8)produced exclusively in exchange re-actions (Funato et al.2004).Finally,Fig.4demonstrates that equal mass binaries have signi?cantly higher survival probabilities than asymmetric mass binaries after multiple intruder encounters.

4

REDUCED MODEL OF INTRUDER SCATTERING

The simulations described so far indicate that roughly equal mass binaries have a statistically signi?cantly higher cap-ture probability than binaries with small mass ratios.How-ever,the origin of this e?ect is not directly apparent from the simulations.In this section we develop a reduced model of intruder scattering which provides an explanation of the mass e?ect discovered in the simulations.The origin of this e?ect is related to chaotic scattering of the intruder by the binary.

4.1Chaotic scattering and dwell times

Chaotic scattering involves a complex and sensitive depen-dence of some “output variable”(e.g.,scattering angle,interplay-or dwell-time,etc.)on an “input variable”(e.g.,impact parameter).For example,scattering in the CRTBP has been demonstrated to be chaotic (Boyd &McMillan 1993;Benet et al.1997;Macau &Caldas 2002;Nagler 2004);Benet et al.(1997)plotted scattering functions (out-put energy and trajectory length)as a function of impact parameter and found clear evidence of chaotic scattering.Similar results were found by Boyd &McMillan (1993)who demonstrated chaotic scattering in binary star –intruder scattering.However,the high dimensionality of the four-body problem,even in the Hill approximation,makes it dif-?cult to study a single output variable as a function of a single input variable.Nevertheless,we performed a series of exploratory simulations in which the dwell-time of the in-truder inside the Hill sphere was calculated as a function of binary mass ratio.The dwell-time is here de?ned to be the

Figure 5.Dwell-time distributions for intruder scattering for (a)m r =1and (b)m r =0.035.For each of the two mass ratios shown 5000low-mass intruder scattering events were simulated as described in Sec.3.The histograms show densities of intruders in various dwell-time ranges.Some dwell-times t >10occur but these have been omitted from the ?gure for clarity.For the sake of comparison,in these simulations,a single chaotic binary orbit was used but the results are similar if di?erent chaotic binary orbits are used.An identical series of intruders was used in both cases.The Hill lifetime of the binary orbit used is T H ≈17in scaled units.Intruders are binned according to the time they remain within the Hill sphere –their dwell-time.Average dwell-times are (a)2.1and (b)2.7scaled time units.

total time the intruder remains inside the Hill sphere in our Monte Carlo single-encounter scattering simulations.

Fig.5presents typical histograms of intruder dwell times for two mass ratios.The dwell-time distributions for the two cases di?er in that for unequal masses the dis-tribution has a longer tail corresponding to longer dwell-times.That is,the intruder has a greater chance of becoming caught up in a long-lived resonance (Heggie &Hut 2003)if the binary has very asymmetric mass partners.The average dwell times shown are quite typical and are fairly consistent between di?erent binary orbits.The unequal mass case typi-cally has an average dwell-time ≈20?30%longer than that for equal masses depending on the actual mass ratio used and the particular binary orbit selected.

In an attempt to understand this result and how (or if)it relates to the di?erential stabilization of equal and un-equal mass binaries we have developed a simpli?ed model of intruder scattering.The main reason why we resort to studying a simpli?ed problem is the di?culty in visualizing the dynamics and the structure of the 12-dimensional phase space in the full four-body Hill problem.We stress,however,

10Astakhov,Lee and Farrelly

that,in the following,we are only interpreting the dynamics exhibited in the full problem using reduced dimensionality dynamics.None of the results obtained in the previous sec-tion rely on this analysis.

Empirically it seems likely that the origin of the mass ef-fect in Figs.3and4can be traced to the observed di?erence in intruder dwell-or interplay-times(Shebalin&Tippens 1996).This is because the only consistent di?erence we were able to?nd between simulations which were otherwise iden-tical,except for having di?erent mass ratios,was in intruder dwell-time distributions.A similar e?ect has also been ob-served by Hills in simulations of star stellar-binary scattering (Hills1975,1983a,b,1990).However,Hills took the binary orbits to be bound,two-body Keplerian orbits rather than chaotic three-body orbits;i.e.,Hills’work does not contain the equivalent of solar tides which,in our model,are essen-tial for the formation of quasi-bound binaries in the?rst place.

The model we build has its foundations in the following observations;

(i)A separation in timescales exists between fast intruder scattering and the timescale of the mutual binary orbit.That is,chaos in the binary orbit caused by solar tides generally develops on a slower timescale than does the intruder scat-tering event.

(ii)The di?erence in dwell-times between equal and un-equal mass binaries is greatest for small intruders.

(iii)If simulations are done using zero-mass intruders then,clearly,no binary stabilization can occur.Neverthe-less,the intruder dwell-time e?ect persists.In this limit the problem reduces to the so-called restricted Hill four-body problem(RH4BP)(Scheeres1998).

(iv)Experimental simulations reveal that the dwell-time e?ect in the RH4BP exists even if the binary follows an el-liptical Keplerian orbit.These?ndings are based on exten-sive simulations in the four-body Hill problem and in several variants of the RH4BP in which di?erent solutions forρ(t) were used–see the discussion following equation(6). (v)As the binary hardens then solar tides become rela-tively less important and the problem reduces to the elliptic restricted three-body problem(ERTBP)(Scheeres1998).

4.2The Elliptic Restricted Three-body Problem Combining the observations in the previous subsection leads to the following set of assumptions:neglect solar tides,as-sume elliptical binary orbits and set the intruder mass to zero.After making these approximations the four-body Hill problem reduces to the elliptic restricted three-body prob-lem(Scheeres1998)for which the Hamiltonian is given by (Szebehely1967;Llibre&Pi?n ol1990;Mikkola&Valtonen 1992;Astakhov&Farrelly2004);

H e=

1

2(x2+y2+z2)+(1?μ′)x+

1

dt

=

(1+e cos f)2

2

(p x2+p y2+p z2)?(x p y?y p x)?

1?μ′

(1+x)2+y2+z2

?

μ′

x2+y2+z2

?

(1?μ′)x?

1

Formation of Kuiper-belt binaries11

larger relative velocity.While this might seem contradic-tory,equipartition of energy in a particle disk implies that the largest bodies will have the most circular coplanar or-bits,while the smaller bodies-here the intruders-will be more eccentric and inclined.Thus the relative veloc-ity of approach should indeed be larger for the small in-truders(Goldreich et al.2002;Wetherhill&Stewart1993; Stewart&Ida2000;Lewis&Stewart2002;Ohtsuki et al. 2002;Goldreich et al.2004).

Because the binary is actually following an unbound chaotic orbit it is easily disrupted.Even small intruders can, in principle,lead to early disruption of the binary(as com-pared to T H,the binary lifetime in the absence of intrud-ers).If,on the one hand,the intruder has a short dwell-time then it is in and out of the Hill sphere quickly and there is an opportunity for sudden energy transfer with minimal disruption of the binary.Granted,this energy transfer may destabilize the binary but,in the intruder velocity regime of interest,we?nd that,on average,this is not the case–again this result is quite similar to the?ndings of Hills(Hills 1975,1983a,b,1990).Note especially that ionization of the binary is possible–though not necessarily equally likely–for small as well as large intruders.This situation should be contrasted with the case of already bound(Keplerian)bina-ries when,in general,a comparably massive intruder may be needed to produce complete or“democratic”ionization (Heggie&Hut2003).

On the other hand if the intruder gets trapped in a rel-atively long-living resonance with the primaries then there exists a greater opportunity for disruption since the pertur-bation to the binary orbit is being applied for longer.That is,if the intruder gets caught up in a sticky chaotic layer its stay within the Hill sphere will be extended(in a reso-nance)as compared to if it had entered an asymptotically hyperbolic regime.While the lifetime of the resonance may still be much shorter than T H it will be signi?cantly longer than the dwell time of an intruder undergoing hyperbolic scattering.Such resonances have three main decay channels (Hills1983b;Heggie&Hut2003);(i)complete ionization, (ii)expulsion of the intruder and(iii)expulsion of one of the original binary partners(exchange).While channel(ii) is the most likely of the three for small intruders,channels (i)and(iii)can also occur because the binary orbit is not bound.Thus,as compared to sudden scattering,resonances (in the sense of long-lived trinary complexes(Heggie&Hut 2003))tend,on average,to reduce the capture probability.

In view of the foregoing we suggest that the relative sizes of the chaotic regions as compared to the hyperbolic re-gions will correlate directly with the intruder’s dwell-time. We argue that the relative sizes of these regions should, therefore,also correlate to capture probabilities.Chaos de-lays the intruder within the Hill sphere and so ampli?es the e?ect of the intruder on the binary orbit.Repeated inter-actions–and energy transfer–between the binary and the intruder tend to destabilize the fragile binary orbit.Impor-tantly,we are here comparing relative rather than absolute propensities for stabilization.Not all fast encounters are sta-bilizing nor do all resonances cause the binary to decay.

Thus,what requires examination is the relative size of chaotic versus hyperbolic regimes for various mass ra-tio/eccentricity combinations.The value of the ERTBP limit of the four-body Hill problem is that it allows one to visualize phase space directly;in this case using the Fast Lyapunov Indicator(FLI)as will now be described (Froeschl′e,Guzzo&Lega2000;Astakhov&Farrelly2004).

4.3Fast Lyapunov Indicator

The FLI is useful for mapping chaotic and regular regions of phase space when Poincar′e SOS cannot readily be computed (Froeschl′e et al.2000;Pilat-Lohinger,Funk&Dvorak 2003;Astakhov&Farrelly2004).Given an n-dimensional continuous-time dynamical system,

d x/dt=F(x,t),x=(x1,x2,...,x n),(11) th

e Fast Lyapunov Indicator is de?ned as(Froeschl′e et al. 2000)

F LI(x(0),v(0),t)=ln|v(t)|,(12) where v(t)is a solution of the system of variational equa-tions(Tancredi,S′a nchez&Roig2001)

d v

?x v.(13) Regularization(Stiefel&Scheifele1971;Aarseth2003) was used to integrate equations(11)and(13).

4.3.1Choice of initial conditions and energies in FLI

maps

To analyze the structure of phase space in our simpli?ed system,i.e.,in the ERTBP,we computed FLI maps in the planar(2D)circular and elliptical cases.In the circular limit a direct comparison with surfaces of section(in the Hill limit,μ′?1,Astakhov&Farrelly2004)can be made.We have previously demonstrated that in the planar CRTBP the FLI reproduces correctly all relevant features visible in the SOS (Astakhov&Farrelly2004).In particular,the chaotic layer, as it evolves with increasing energy,can be easily identi-?ed by high values of the FLI.Note also that FLI measure-ments make sense even for relatively short time integrations, and,even in2D,are much more e?cient than is the con-struction of SOS(Froeschl′e et al.2000;Pilat-Lohinger et al. 2003;Astakhov&Farrelly2004).

In the planar limit of the ERTBP initial conditions were generated randomly within the Hill radius on the surface of section(see Fig.2caption)taking,initially,f=π/2. This guarantees that the ERTBP initial conditions reduce to those of CRTBP(Astakhov&Farrelly2004).For a givenμ′and e all ERTBP initial conditions are,therefore,generated with identical initial energies(Astakhov&Farrelly2004).

A further issue in generating the FLI maps in Fig.6 concerns the choice of initial energy.Because we are com-paring di?erent masses and ellipticities it is not possible to work at exactly the same energy in each case.This is be-cause in the ERTBP the energies of the Lagrange points, for example,depend on ellipticity,mass and the instanta-neous value of the true anomaly,f.That is,energy(or the Jacobi constant,Murray&Dermott1999)is not conserved in the ERTBP.Thus,in order to make a fair comparison of the dynamics between these di?erent cases we need to con-struct a criterion of comparability;in practice,we selected initial values of the Jacobi constant such that the sizes of

12Astakhov,Lee and

Farrelly

Figure6.Fast Lyapunov indicator maps for the planar ERTBP and four eccentricity/mass ratio combinations.The origin is shifted so that the binary partners are located at(x,y)=(±0.5,0)in scaled units and here units are rescaled so that R H=1.Colour scale runs from blue to red to white with increasing FLI.Initial energies are as de?ned in the text.Blank regions correspond to directly scattering or hyperbolic trajectories which survive for less than～20binary periods.Areas that resemble shotgun-like patterns of points correspond to chaotic regions(Astakhov&Farrelly2004)(see the text).At the higher eccentricities scattering(hyperbolic)regions are noticeably larger for(c)(equal masses)than for(d)(unequal masses);this translates into enhanced capture probabilities for equal masses.

the gateways in the zero-velocity surface at time t=0(see

Fig.1)were comparable(Astakhov&Farrelly2004).Ap-

proximately,this equates the?uxes of incoming/outgoing

particles and facilitates a fair comparison of relative densi-

ties of chaotic versus hyperbolic orbits.The actual energies

used at t=0(f=π/2)were E=?1.725for m r=1and

E=?1.675for m r=0.05.Admittedly this is an ad hoc ap-

proach but we?nd that the general appearance of the FLI

plots is quite robust to the precise values of the initial Ja-

cobi constant used provided that the gateways have similar

sizes.

4.3.2Interpretation of FLI maps

Figure6shows FLI maps for di?erent mass ratios and ec-

centricities.It is easy to distinguish between chaotic regions,

directly scattering(hyperbolic)regions and–to the intruder

–inaccessible KAM regions(Astakhov&Farrelly2004).In

the chaotic regions nearby orbits tend to be dense and have

similarly large FLI values.This results in chaotic regions

having the appearance of a shotgun-like pattern–unlike in

a SOS,however,each point corresponds to the initial con-

dition of a single orbit coloured according to the?nal com-

puted value of its FLI.By contrast,in hyperbolic regions

orbits rapidly leave the Hill sphere and enter heliocentric

orbits.We chose20mutual orbital periods of the primaries

(in the ERTBP limit)as a cut-o?such that orbits which

survived for less than this cut-o?were not plotted.We ver-

i?ed that our results are not sensitive to the precise choice

of cut-o?time.The large solid regions in Fig.6correspond

to regular(KAM)regions.Elsewhere we have examined the

e?ect of ellipticity on chaos-assisted capture and the di?er-

ences in FLI maps for chaotic,hyperbolic and regular zones

was examined in detail in the ERTBP(Astakhov&Farrelly

2004).

In summary,in FLI maps hyperbolic regions manifest

themselves by their relative sparsity of points as compared

to chaotic and regular zones(Astakhov&Farrelly2004).

4.3.3E?ect of mass ratio and eccentricity on capture

probabilities

First we compare the situation for equal versus unequal

masses in the case of circular orbits,i.e.,Fig.6a and

6b.Clearly the observable sea of chaos is much larger for

frame(a)i.e.,equal masses,than for frame(b)i.e.,un-

equal masses.However,in both cases it is also true that

there are relatively few hyperbolic regions.For the case of

unequal masses the regular KAM regions are very large

but since KAM regions represent barriers in phase space

(Lichtenberg&Lieberman1992)these regions are inaccessi-

ble to the intruder.Thus in both Fig.6a and6b an intruder

is likely to become caught up in a chaotic zone thereby pro-

ducing a resonance.According to our earlier arguments this

will,on average,increase dwell-times and so tend to desta-

bilize the binary.This suggests that circular(or very low

eccentricity)binary orbits will be less likely to be captured.

Next we examine the case of higher eccentricity,e=0.7,

shown in Fig.6c and6d.This case is actually the most

relevant to the real situation since the osculating eccentric-

ity of the chaotic binary is generally quite high.For equal

masses,shown in frame(c),very large hyperbolic regions

Formation of Kuiper-belt binaries13

are clearly visible whereas in frame(d)(unequal masses) the“non-regular”regions are primarily chaotic rather than hyperbolic.Incoming intruders will therefore encounter very di?erent environments between the two cases.If the binary partners are of equal mass,intruders will most likely enter hyperbolic regions and so will exit the Hill sphere promptly. In the case of unequal masses the battleground is of a very di?erent nature,resembling a?eld of sticky mud in which in-truders rapidly become bogged down.That is,they become temporarily trapped in trinary resonances which increases the probability that the binary will be disrupted rather than stabilized.This can be explained,in part,by noting that in the CRTBP with equal mass primaries(the Copenhagen problem,Szebehely1967;Benet et al.1997;Nagler2004), the zero velocity surface is symmetric in x and so the gate-ways into and out of the capture region are equivalent.This facilitates quick transits through the capture zone,whereas in the asymmetric mass system one gateway is smaller than the other constituting a bottleneck which hinders passage.

The FLI maps in Fig.6provide a qualitative explana-tion of the observed mass e?ect.The dwell-times of intruders within the Hill sphere are a sensitive function of whether in-truders enter hyperbolic or chaotic regions of phase space inside the Hill sphere.As system parameters change(total initial energy,ellipticity,etc.)the sizes and shapes of these regions alter in a highly nonlinear manner(see Fig.2c)as is characteristic of chaotic scattering.With increasing ellip-ticity the sizes of the chaotic versus the directly scattering regions decrease,especially in the case of equal mass binary partners.This translates into enhanced capture probabilities for equal masses.

Overall,based on a comparison of the four cases in Fig.6in the ERTBP,we conclude that intruder stabilization will be most e?cient for(i)equal masses and(ii)moderate to high eccentricity orbits.It is somewhat ironic that,in this mechanism,the existence of chaos in the binary orbit(pro-duced by solar tides)is critical to nascent KBB formation, whereas if intruders enter zones of chaotic(as opposed to hyperbolic)scattering then the result is destabilization of the binary.

However,the ERTBP model has its limitations.The assumption that the mutual chaotic binary orbit can be treated(from the point of view of the intruder)as an el-liptical Kepler orbit on short timescales is not valid if the actual binary orbit is itself extremely unstable on compara-ble timescales.Such three-body orbits tend to involve“near misses”of the binary partners and are less stable than are more“circular”three-body orbits(e.g.,see Fig.2a).Thus for very high instantaneous eccentricities(orbital segments involving near misses)the binary partners(in the full four-body problem)are di?cult to stabilize.This tends to re-duce the capture probability into very high eccentricity or-bits considerably.

The net e?ect of all of these considerations is to favor moderately elliptical binary mutual orbits when the binary partners have approximately equal masses.This is precisely what is found in our simulations and in actual observations of KBBs(Burns2004;Noll et al.2004a,b).5ALTERNATIVE FORMATION MODELS

Of the four paths mentioned in the Introduction and sum-marized in Table1extensive numerical simulations have only been reported for the exchange mechanism,Path4 (Funato et al.2004;Kolassa2004).In this section we present a brief comparison of our multiple scattering model with al-ternative theories and also report some model simulations which we have performed for Paths2and3(Goldreich et al. 2002).

5.1Exchange reactions

In the exchange reaction model(Funato et al.2004)the?rst step is the formation of a binary through a two-body dis-sipative encounter.This can happen in two ways;(i)tidal disruption of an object during a close encounter followed by accretion of the resulting fragments to produce a bi-nary;and,(ii)a“giant”impact in which coagulating de-bris after the collision produces a satellite orbiting a larger object.Main-belt asteroid binaries(and the Earth-Moon binary(Canup&Asphaug2001;Canup2004)and(with some important di?erences)possibly the Pluto-Charon bi-nary(Canup2005))are thought to have formed in this man-ner.Generally,the result is a binary with an extreme mass ratio and a tight,circular orbit.In the exchange reaction mechanism,the mass ratio of such proto-binaries in the KB is increased through later gravitational scattering encoun-ters with a third object originating(in the simulations)at in?nity.Funato et al.(2004)modelled this by performing ex-tensive scattering simulations between strongly asymmetric-mass binaries already following bound,compact,circular orbits and large incoming intruder masses.They found a propensity for the smaller member of the initial binary to be replaced by the intruder with the?nal binary then fol-lowing a large,highly eccentric Keplerian orbit.

While this mechanism is plausible and does produce binaries which are at least qualitatively similar to known KBB orbits a number of di?culties nevertheless remain.In the?rst place,this mechanism does not provide an ab initio explanation for why the the mass ratios of actual KBBs are of order unity.Instead,it simply provides a route by which a binary can increase its mass ratio.Because Funato et al. assumed that the primary member of the initial binary and the intruder had equal masses then,if exchange occurs,a binary of order unity mass ratio is guaranteed to result. There appears to be no compelling reason,at least based on the simulations reported(Funato et al.2004),to conclude that mass ratios of order unity should necessarily result from this process,only that exchange reactions can increase mass ratios.

In addition,exchange reactions seem to produce orbital eccentricities(and semimajor axes)larger than those typi-cally observed(Noll2003;Noll et al.2004b).However,this might simply be the result of the particular choice of masses made by Funato et al.(2004).Thus it would be interesting to extend these simulations to include a broader range of intruder masses so as to understand(a)the role of intruder mass in determining the probability of exchange and(b)the e?ect of intruder mass on post-exchange orbital properties of the binary.

14Astakhov,Lee and Farrelly

5.2Two-body collisions inside the Hill sphere The?rst mechanism

proposed for the production of trans-neptunian binaries is that of Weidenschilling(2002).In this model two bodies accrete after a mutual collision within the Hill sphere of a third,larger,body.They then remain bound as a single object orbiting the larger body,thereby produc-ing a binary.The main objections to this model have to do with the scarcity of large objects in the primordial KB. However,Petit&Mousis(2004)(see also Stern2002)dis-cuss this issue at length and conclude that the mechanism of Weidenschilling(2002)might have operated,possibly in tandem with other mechanisms.In the absence of detailed simulations it remains an open question whether this mech-

anism can provide quantitative agreement with the observed properties of KBBs.

5.3Capture through dynamical friction

This mechanism was originally proposed by Goldreich et al. (2002).Here we have shown that the dynamical basis for this model is chaos-assisted capture(Astakhov et al.2003; Astakhov&Farrelly2004).No detailed simulations of KBB formation or their orbital and compositional properties were presented by Goldreich et al.(2002).Because this mech-anism proceeds from similar assumptions to the one we propose,we performed a limited number of simulations in which multiple intruder scattering as an energy loss mecha-nism was replaced by dynamical friction,modelled as simple velocity dependent dissipation(Murray&Dermott1999; Astakhov et al.2003).We?nd that,dynamical friction pro-vides an e?cient route to binary capture and Keplerization, producing?nal orbital elements similar to those in Fig.4. However,it is not sensitive to the mass ratio of the initial quasi-bound binary.This is because the scaled Hill equations (1),even including dissipation,contain no parameters and so cannot depend on the masses of the binary partners.Thus, no mass e?ect is predicted.Of course,more sophisticated models of dynamical friction(Chandrasekhar1943;Binney 1977;Goldreich et al.2004;Just&Pe?n arrubia2004)might conceivably reveal such a preference.In addition,dynami-cal friction has been ruled out on the basis of estimates of planetesimal mass distributions although this issue may be worth revisiting(Funato et al.2004).

5.4The L3mechanism

Again,this mechanism was originally proposed by Goldreich et al.(2002)and is the most similar to the one which we present here.Although no simulations of L3were made by Goldreich et al.(2002)a number of issues can be raised in regard to it;(a)in L3,as originally proposed, roughly equal mass binaries result directly from the assump-tion that a transient binary undergoes a scattering encounter with a third body of comparable mass to the binary part-ners(which are assumed to be themselves of similar mass). That is,the preference for order unity mass ratios is an inescapable result of the foundational assumptions of the model;(b)unless multiple L3type encounters are being as-sumed,then hardening and Keplerization of the large tran-sitory binary must occur in a single encounter which seems unlikely.To test this we have performed simulations exactly Figure7.Histogram showing the survival probability of equal mass chaotic binaries(α1=α2)after N enc L3encounters with large intruders(α3=0.3,see equation(5)).1000initial binaries were taken with initial conditions for three-body Hill orbits and intruders originated as described in Sec.3.To generate statistics the simulation was repeated50times,each time choosing the phase of each binary randomly with respect to the intruder.

as for Fig.4except using a large intruder.The results are summarized in Fig.7and indicate that multiple large body encounters tend to disrupt the binary catastrophically.Fur-ther,examination of the actual orbital elements produced in these simulations con?rms that neither a single nor a small number of L3-type events is usually su?cient to produce Keplerian orbits.Instead,after a small number of such en-counters the orbits resemble those of the initial quasi-bound binary,e.g.,that shown in Fig.2.In these cases Keplerian orbital elements cannot meaningfully be de?ned.

Of course a hybrid of our model and L3is also possible. Certainly our simulations do not rule out the possibility of L3-type collisions with additional hardening being produced through collisions with smaller bodies;however,L3collisions are not necessary for capture or to produce roughly equal mass binaries.

6CONCLUSIONS

We have presented what are,to our knowledge,the?rst simulations of binary-intruder scattering in which the bi-nary orbit is itself chaotic.Previous simulations of binary star intruder scattering have focussed on the important but simpler case in which the initial binary already follows a bound Keplerian ellipse.Here the binary orbits used are long living chaotic orbits(in the Hill approximation)which cling for long periods to regular KAM structures in phase space.These transitory orbits are possible only if the Sun-binary Hill sphere is taken into account.The importance of the Hill sphere in temporarily capturing Kuiper-belt bi-naries was?rst noted by Goldreich et al.(2002)on empiri-cal grounds.The dynamical explanation for the results pro-posed here is chaos-assisted capture(Astakhov et al.2003; Astakhov&Farrelly2004;Holman et al.2004).

Once a proto-binary has formed within the Hill sphere then,if it is to survive,some stabilization mechanism must operate.Our simulations demonstrated that multiple low-mass intruder scattering can not only stabilize transitory binaries but also provides an e?cient route to binary hard-

Formation of Kuiper-belt binaries15

ening and Keplerization of the orbit.By adjusting a sin-gle parameter–the number of intruder-binary encounters–we were able to obtain excellent agreement with the orbital properties of known KBBs.In particular,our simulations predict that KBBs will have moderate eccentricities.This is in good agreement with recent observations(Noll2003; Noll et al.2004a,b)and contrasts with the exchange mech-anism of KBB formation which produces exclusively large eccentricities(Funato et al.2004;Kolassa2004).

Our simulations also lead to a striking preference for the production of binaries having roughly equal masses. Strongly asymmetric mass ratios–typical of main belt as-teroid binaries–are selected against,and,again this appears to be consistent with observations.The model we propose is not set up to produce only order unity mass ratio bina-ries;nevertheless,a preference for mass ratios of order unity emerges.

An important di?erence between our model and L3con-cerns how the binary hardens.Single encounters with large intruders are unlikely to produce the relatively compact(as compared to R H)Keplerian binary orbits that are actually observed.Moreover,we?nd that multiple encounters with large intruders tend,on average,to disrupt the binary.In-stead,the model we propose involves a sequence of low-mass intruder events rather than a single scattering encounter with a massive object.It turns out that this provides an e?cient mechanism for binary hardening and Keplerization. However,because we are unable to estimate the probabili-ties for the initial rate of production of quasi-bound binaries of varying mass ratios in the Hill sphere,it is unclear how strongly this mass e?ect will persist.Nevertheless,our simu-lations demonstrated that equal mass binaries have a much higher survival probability after multiple encounters than do asymmetrical mass ratio objects.Thus,we are con?dent that the mass e?ect we have found will not be washed out by the statistics of transitory binary formation.

Throughout we have assumed that the binary centre-of-mass follows a circular(heliocentric)orbit.However,in principle,we also need to consider the e?ect of introduc-ing eccentricity into the heliocentric binary orbits.This is because some observed KBBs have signi?cantly eccentric he-liocentric orbits with perihelia close to,or even inside,Nep-tune’s orbit(e.g.,Pluto-Charon,(47171)1999T C36and (26308)1998SM165).Typical KBB heliocentric elliptici-ties lie in the range～0.03?0.37(Noll2003).In our re-cent study of capture and escape in the elliptic restricted three-body problem(Astakhov&Farrelly2004)we found that the introduction of moderate heliocentric ellipticity re-duces,but does not entirely eliminate,the e?cacy of the chaos-assisted capture mechanism.While further study of this issue is clearly warranted,we expect that the results presented here will similarly carry over to the elliptic helio-centric case.

Of course,it is also possible that observed heliocen-tric TNO binary eccentricities are not primordial.Actually, this is suggested by the observation that massive bodies in eccentric orbits will tend to have large approach velocities apart from coincidences such as,e.g.,when perihelia are aligned.Because such cases are rare,relative velocities will tend to be large and this will serve to reduce the probabil-ity of capture into chaotic orbits.This conclusion is consis-tent with results for capture in the elliptic restricted three-body problem(Astakhov&Farrelly2004).These consider-ations suggest that,as with Paths1-3,binaries formed when the disk was massive and dynamically cold,i.e.,prior to the dynamical excitation and depletion of the Kuiper-belt(Duncan&Levison1997;Gomes2003;Morbidelli et al. 2004).

Finally,we note that Figs.3-5demonstrate that,while signi?cant,the selection for order unity mass ratio objects is not to the complete exclusion of smaller mass ratios.For example,the di?erences in dwell-time distributions are not so great as to eliminate the capture of small mass-ratio bi-naries entirely.Rather,Figs.3and4lead us to predict that a sizeable population of asymmetric mass KBBs is prob-ably awaiting detection.Heralds of this group of objects might already exist in the low mass ratio binary objects (47171)1999T C36(Trujillo&Brown2002;Altenho?et al. 2004)and(26308)1998SM165(Brown&Trujillo2002; Stern2002;Noll2003;Petit&Mousis2004). ACKNOWLEDGMENTS

This work was supported by grants from the US Na-tional Science Foundation through grant0202185and the Petroleum Research Fund administered by the American Chemical Society.All opinions expressed in this article are those of the authors and do not necessarily re?ect those of the National Science Foundation.S.A.A.also acknowl-edges support from Forschungszentrum J¨u lich.We thank Kevin Hestir for valuable suggestions and comments on the manuscript.We also thank two anonymous referees for help-ful and insightful suggestions.

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