Icarus147, 205–219 (2000)

doi:10.1006/icar.2000.6414, available online at http://www.idealibrary.com on

Comparative Study of Mean-Motion Resonancesin the Trans-Neptunian Region

M. D. Melita and A. Brunini

Facultad de Ciencias Astronomicas y Geofısicas, Universidad Nacional de La Plata, Paseo del Bosque, (1900) La Plata, ArgentinaE-mail: melita@fcaglp.unlp.edu.ar

Received November 5, 1999; revised March 17, 2000

In this work we are interested in the present dynamical struc-ture of the trans-neptunian region. It is known that at moderate tohigh eccentricities, stable orbits lie close to an exterior-mean-motionresonance with Neptune (NMMR). We study some NMMRs underdifferent points of view. Intrinsic probabilities of collision and dy-namical diffusion time-scales using frequency-map analysis havebeen computed. We have found that collisions and gravitationalencounters by themselves would not produce remarkable differ-ences between the number of objects orbiting in each resonance atpresent. However, frequency-map analysis reveals a much more ro-bust region at the 2 : 3 NMMR than at the other NMMRs. Naturallythe net orbital effect of the encounters can be enhanced differentlyin each individual NMMR due to differences in size of the stableniches, allowing the populations in the more unstable regions toevaporate sooner. We also study how certain evolutionary models,related with the orbital expansion of the outer planets during theirformation stage, could result in resonant populations with a notice-ably different primordial number of members. Finally, our resultsare discussed with reference to the present observational evidenceand to our current understanding of the formation of the outer SolarSystem. c© 2000 Academic Press

Key Words: Kuiper belt objects; resonances; comets, dynamics;comets, origin; celestial mechanics; origin, Solar System.

1. INTRODUCTION

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summarized as follows (see Fig. 1). There is a large populationfewandMR.anylina-

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The Kuiper belt is a comet reservoir that extends beyondorbit of Neptune and would reach up to∼103 AU or more.The first hypotheses about the existence of such a beltproposed by Edgeworth (1949) and independently by Ku(1950). Fern´andez (1980) put forward the idea that the Kuibelt is the source of the Jupiter family of comets, which woexplain their typical low-inclinations orbits. This hypothesissupported by recent long-term numerical integrations, whhave also found that the centaurs may be a transitional pbetween those two groups of objects (Levinson and Dun1997); however, actual confirmation of this hypothesis mustfor the availability of further observational constraints.

More than a hundred trans-neptunian objects have beencovered at present. Their main dynamical characteristics ca

2

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ldschaseanait

dis-be

at the 2 : 3 NMMR (neptunian mean-motion resonance), aobjects can be found close to other higher order NMMRs,recently a few others have been located close to the 1 : 2 NMObjects with well-determined orbits and not associated withmajor resonance have moderate to low eccentricities and inctions, with median values of∼0.06◦ and∼3◦, respectively. Also,only three of the observed objects can be found with semi-maxes between 36 and 39 AU. It should be noticed that the ordetermination of these low moving objects is very uncertaSince the discovery of the first, Kuiper belt orbit (KBO) (Jewand Luu 1993) only a few opposition dates have past, whrepresents only a very small portion of an orbit. Recent obsetions have allowed the re-assignment of two objects previouassociated with the 2 : 3 NMMR and with the 1 : 2 NMMR (MP33752, MPC 34332).

The estimated physical radii of the discovered objects rabetween∼40 to∼350 km. Accretion studies have determinthat for those objects to grow up to those sizes, the innershould have been 30–50 times more massive than it is obseat present (Stern 1996, Davis and Farinella 1997); howevercording to other studies those values might be somewhat oestimated (Kenyon and Luu 1999).

According to studies of the formation of the outer Solar Stem, the exchange of angular momentum between the foing proto-planets and the surrounding planetesimals would hproduced a remarkable proto-planetary migration. For the thoutermost planets, it would have been produced in the outwdirection (Fern´andez and Ip 1984, 1996). These results hbeen confirmed by recent studies using a more accurate numcal procedure (Brunini and Fern´andez 1999, Hahn and Malhotr1999). Also, as a by-product of thisradial migration stageofthe planets, the trans-neptunian region would have been invby a number of big planetesimals of lunar size or more (St1991). This invasion would certainly aid the putative masspletion in the region, through the orbital excitation sufferby the native bodies (Morbidelli and Valsecchi 1997, Meland Brunini 1999), which would have driven them to a cosional cascade (Davis and Farinella 1997, Stern and Col1997).

5

0019-1035/00 $35.00Copyright c© 2000 by Academic Press

All rights of reproduction in any form reserved.

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206 MELITA AND

FIG. 1. Present distribution of objects in the trans-neptunian regi(a) Histogram of the number of objects as a function of their semi-major a(b) Eccentricity vs semi-major axis for the KBOs. Open circles indicate objeobserved in three or more oppositions.

The outward radial migration would have enabled proNeptune to capture a great number of KBOs into its exteMMRs. (Malhotra 1995, from now on, M95). This hypothesnaturally explains the peculiar orbital characteristics of Pl(Malhotra 1993). Nevertheless Malhotra’s results would sgest that a remarkable population of objects would exist at1 : 2 NMMR, which is still the subject of debate (Jewittet al.1998).

In some basic models of the planetary migration (M9Brunini and Melita 1998) the expansion of the proto-planetorbits is introduced by means of a law for the evolution of tsemi-major axis which produces a smooth evolution. Thispoorly realistic approximation because the proto-planetarydial migration would have occurred as a consequence ofexchange of angular momentum with the surrounding planeimals, an exchange that is produced through gravitational cencounters and physical collisions. Therefore, it is expectedthe semi-major axis evolution would occur in a discontinuo

fashion (Brunini and Fern´andez 1999, Hahn and Malhotra 1999)Also, the masses of the proto-planets are considered constant

BRUNINI

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equal to the present masses of the outer planets. But actuthe proto-planets would reach their present mass through thecretion of minor bodies as a consequence of physical collisioprocesses that occurs during the planetary migration proc(Brunini and Fern´andez 1999).

In this work we study a number of mechanisms, candidato produce a noticeable difference in the number of objepopulating some exterior NMMRs. These mechanisms inclu(1) the different dynamical diffusion-time-scales in each reonant region and (2) physical collisions and gravitational ecounters between the members of each population. (3) Finain the framework of the radial migration hypothesis of the plaets, we have explored the effects of varying the proto-planetmasses during the radial migration, the time-scales of the proplanetary orbital expansion, and a random component inproto-planetary semi-major axes evolution caused by closecounters and physical collisions with surrounding planetesimaThe goal of this evolutive investigation is to understand beter some mechanisms that have a major impact in shapingprimordial population of the Kuiper belt; where primordial imeant as the stage right after the planetary orbital distributis already in much of its present form. As we shall see, tknowledge of the primordial distribution makes it possibleput some constraints on cosmogonic theories. The primorddistribution can be estimated from the present unbiased dtribution, where the dynamical effects must be taken into acount. Naturally, at present the orbital distribution of the tranneptunian region is very poorly determined, some time shpass until a reliable unbiased distribution of KBOs becomavailable.

In Section 2 we briefly review the frequency-map-analysmethod and we apply it to some NMMRs. In Section 3 we compute intrinsic probabilities of collision and differential velocitiefor some NMMRs. In Section 4 we study how some primordidifferences can arise between the populations of the NMMin the framework of the radial migration scenario of the protplanets. Finally, in Section 5 we discuss our results regardour knowledge of the present structure of the Kuiper belt aalso regarding our current knowledge of the formation of thouter planets.

2. DYNAMICAL STABILITY

2.1. Frequency Map Analysis

We have used the frequency-map-analysis method as induced by Laskar (1993) to make a comparative study of tdynamical stability in the trans-neptunian region.

In the inner Kuiper belt the long-term stable regions at modate to high eccentricities are associated with Neptune’s exteMMRs (Duncanet al. 1995). As recently shown by Nesvorn´yand Ferraz-Mello (1997) for the asteroidal problem, the surviv

.andof a considerable population of objects inside a resonance de-pends on how robust is the region, i.e., how large and deep the

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MEAN-MOTION RESONAN

stable niche is. Therefore a comparative study of the stabilitNeptune’s MMRs can be done by comparing the corresponfrequency maps.

The frequency-map-analysis method, as we have implented it, can be summarized as follows. First a complex signfiltered out of high frequencies (McClellanet al.1973, Grenez1983) to eliminate aliasing effects that arise at the samplThe resulting signal corresponds to the averaged problemthe analytic theory (Carpinoet al.1987). Then the values of thfrequencies with the highest contribution are extracted fromfiltered signal using the frequency modified Fourier transfomethod (FMFT) (Sidlickovsky and Nesvorn´y 1997). The sameprocedure is repeated over a running window. A particularin the signal is chosen, usually one that permits a good decomsition, i.e., one well isolated. The dispersion,σ f , of the values ofthis particular frequency through all the windows is computThen a diffusion coefficient,ε, can be obtained as

ε = log

(σ f

f0

),

where f0 is the value of the frequency in the first running widow. The diffusion of the proper frequency is assumed to brandom walk process and then a dynamical time-scale,D−1, canbe defined as (Nesvorn´y and Ferraz-Mello 1997)

log(D−1) = ε − log(1),

where1 is the time-interval between each running window.This procedure has been applied to a great number of p

cles in a grid. Each particle’s orbit is integrated using a secorder multi-step symplectic map (Wisdom and Holman 19implemented in our own code, where the perturbations offour major planets are included, with a time-step of 0.5 yrthe planets and 5 yr for the particles. These time-step valuesmaller than those used by Morbidelli (1997) to study diffuswith a similar integration procedure.

The signal analyzed was that of the elements

h = esin($ )(1)

k = ecos($ ),

wheree is the eccentricity and$ the longitude of the perihelion. Runs of 50 Myr were performed. The frequencies wcomputed over a running window of length 10 Myr—whicis a time-scale greater than any secular period found inregion—at intervals of1 = 5 Myr. The particles with a valueof log(D−1)≤−9.39 would be stable over the age of the SoSystem. Using a color-scale (or gray-scale) we plot the valulog(D−1) as a function of the initial values of the semi-majaxis and the eccentricity. A check on the accuracy of the met

was done by computing the diffusion of the planets, which incases gave a value of log(D−1)<−10.24.

CES IN THE KUIPER BELT 207

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2.2. 2 : 3 NMMR

For the study of the 2 : 3 NMMR a grid of 961 particlewas integrated comprising 31 initial eccentricities uniformdistributed between 0 and 0.35 and 31 initial semi-major axeuniformly distributed betweena2/3− δa anda2/3+ δa, wherea2/3= (3/2)2/3aN , aN being the initial semi-major axis oNeptune andδa= 0.5 AU.

The argument of the perihelion and the longitude of the nof all the particles are set equal to those of Neptune. The inmean anomaly of the particles,λ, is such that the resonant ang

σ = 3λ− 2λN −$N,

is equal to 180◦, whereλN and$N are the mean longitude anthe longitude of the perihelion of Neptune, respectively. Tchosen value ofσ is the location of the center of libration ithe restricted three-body problem (Malhotra 1996). The iniinclination is set equal for all the particles.

In Fig. 2 we show the maps corresponding to the 2 : 3 NMMfor three different initial inclinations. A map of the secular reonances inside the 2 : 3 NMMR for the planar-restricted-thrbody problem can be found in Morbidelli (1997). We can sthat at low inclinations this map is remarkably reproduced. Tunstable region at small eccentricities corresponds to the chregion originated by the overlap of secondary resonances. Ato moderate eccentricities, the borders of the mean-motiononance are close to the region where the Kozai—which ocwhen the precession rate of the perihelion stops—and theresonances overlap also causing instability. At an eccentrici∼0.25, the Kozai resonance is at the center of the NMMR. Tpeculiar orbit of Pluto lies in this region which offers a knowprotection mechanism to ensure the long-term stability. Theter of the resonance appears as a robust region. The locatiorbits with a dynamical time-scale slightly smaller than thathe Solar System’s age, which could be the source of Jupifamily of comets, can also be seen in this diagram. All thfeatures are described in detail by Morbidelli (1997).

For increasing inclinations, the stable region at the ceof the NMMR becomes more unstable. It should be notithat the Kozai resonance widens with the inclination and it moverlap with Saturn’s perihelion secular resonance, which ispresent at the center of the MMR. The increasing instabilitsmall eccentricities is due to the presence of the node seresonance (Gallardo and Ferraz-Mello 1998).

The positions of the discovered objects are also plotteFig. 2. Most of them tend to lie in the long-term stable regioHowever, some remarks should be noted. First, the observejects do not correspond exactly with these maps becauseangular variables and those corresponding to the particlesto build the map do not coincide in general. Moreover, thmaps are shown as a function of initial values, so that theserved objects would be oscillating with some finite amplituin semi-major axis, eccentricity, and inclination about the sho

allposition. So we should bear in mind that objects lying just out-side the borders of the long-term stable regions may actually

208 MELITA AND BRUNINI

FIG. 2. Frequency maps of the 2 : 3 and the 1 : 2 NMMRs. Open dots indicate the location of the observed objects in the region. The initial inclinations are

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be in very stable orbits. Finally, and more importantly, thebital determination of most of the trans-neptunian objects isuncertain at present because most of them have been obsonly at a very few opposition dates. Approximately 70% ofbodies known at present have been discovered in the pastyears, a time interval that represents only about 1% of theibital periods. At this point we cannot be certain about the orof the objects lying in the middle of very large unstable zonThey might be in a rapid transitional phase coming from oregions, they may also be escapees from the stable niches,close encounters or physical collisions (as fragments), ororbits might be very poorly determined and they actually liea more stable region of the phase space.

2.3. 1 : 2 NMMR

For the study of the 1 : 2 NMMR a grid of 1681 particlewas integrated comprising 41 initial eccentricities uniform

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distributed between 0 and 0.405 and 41 initial semi-major auniformly distributed betweena1/2 − δa anda1/2+ δa, wherea1/2= (2/1)2/3aN , andδa= 0.8 AU. To obtain greater detaia more dense grid (51× 51) was used to build the map corrsponding to an initial inclination of 10◦.

The phase angle in this case deserves particular attentiois known that the 1 : 2 exterior NMMR has asymmetric ceters of libration, whose location is a function of the eccentric(Message 1958, Beaug´e 1994, Morbidelliet al.1995, Malhotra1996). Also, the center of libration can be shifted due toperturbations of the other planets. Hence, we have performnumerical search for the center of libration for each particlethe grid. We have numerically integrated, using the same mdescribed above, a set of 181 particles with the same arguof the perihelion and longitude of the node as Neptune and invalues ofσ in the interval [0◦, 360◦] for each eccentricity and

slysemi-major axis in the grid. The time length of each of theseintegrations was 3× 105 yr, which is always greater than the

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MEAN-MOTION RESONAN

period of libration. From this set we choose the particle withminimum libration amplitude as the initial condition of the paticles in the map. For those particles that do not librate, the invalue ofσ was set to the one with the minimum rotation rate

Although no secular resonance with Neptune can be foinside the borders of the 1 : 2 NMMR, the existence of the asmetric modes of libration are a great source of chaos, parlarly when the objects approach the separatrix between t(Morbidelli et al. 1995), although in the restricted three-boproblem there is no discernible chaotic zone for eccentricbelow 0.25 (Malhotra 1996). A more detailed study of the proerties of the asymmetric librators under the perturbations orest of the major planets is needed, since the stability propeof each of them can be different.

Maps corresponding to the 1 : 2 NMMR are also shownFig. 2. As it can be seen, in this case we do not find such a rostable niche as in the former case. The stability regions apto be very disaggregated, but the whole region becomes mstable for increasing inclinations. As suggested by D. Nesvoy(personal communication, 1999) a check on this result hasperformed by building a frequency map close to the centethe resonance for low inclinations, where the initial phase anwere taken from the location of the center of libration predicby the analytic theory of the restricted three-body problema qualitative similar picture was obtained.

We would expect a smaller number of members orbiting in1 : 2 NMMR than in the 2 : 3 NMMR. Three of the four objecdetected so far in that region are in a zone characterizedvery low dynamical time-scales; thus there is a chance thatmay have not been in that region very long. The other onea smaller eccentricity and lies at a stable region at the bordthe resonance, hence being a candidate for a primordial ob

2.4. 3 : 5 NMMR

The initial conditions in this case are chosen in the followway. We have used a grid composed of 961 particles coming 31 initial eccentricities uniformly distributed between 0 a0.35 and 31 initial semi-major axes uniformly distributed btweena3/5− δa anda3/5+ δa, wherea3/5= (5/3)2/3aN , whereδa= 0.5 AU. Again, the argument of the perihelion and the logitude of the node of all the particles are set equal to thosNeptune. The initial phase angle

σ = 5λ− 3λN −$N

is searched in a way similar to the 1 : 2 NMMR case because fthe restricted three-body problem we know that the locatiothe center of libration also depends on the eccentricity (Malh1996). The corresponding map for an initial inclination of 1◦ isshown in Fig. 3.

Again we see that the region of regular orbits is very disaggated inside the borders of the resonance. Most of the obse

objects lie at very low eccentricity on the long-term stable rgions. Bearing in mind that these objects have very poorly

CES IN THE KUIPER BELT 209

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FIG. 3. Frequency map of the 3 : 5 NMMR. Open dots indicate the locatof the observed objects in the region. The initial inclination is indicated.

termined orbits, the existence of a numerous population ateccentricities pose a challenge to our current understandinthe formation of the outer Solar System. Simulations of themation of Uranus and Neptune which predict their formationa realistic time-scale (Brunini and Fern´andez 1999) show thathe accretion process is very inefficient. As a by-product, so50% of the aggregated mass of those planets would have invthe Kuiper belt, exciting the orbits of the bodies formed primdially in the region (Morbidelli and Valsecchi 1997, Melita anBrunini 1999). Some of those invading objects could be foundday in the region forming a scattered disk (Levinson and Dun1997), from which one object would have been discoveredfar, 1996 TL66 (Trujillo et al.1997); naturally the scattered diswould also be composed of KBOs whose orbits are also excand fall into the gravitational control of Neptune. The excitatiproduced by the invading objects would have been instrumein bringing the primordial KBOs to a collisional cascade,ducing its total mass from 30–50M⊕ to its estimated presenvalue of∼0.1 M⊕ (Stern and Colwell 1997, Davis and Farinel1997). In fact, the absence of observed objects with semi-maxes between 36 and 39 AU may be a fossil record of thisvasion; however, this feature is naturally explained by the renance sweeping hypothesis (M95). Auto-consistent simulatof the invasion of planetesimals into the trans-neptunian reg

e-de-(Melita and Brunini 1999) would leave practically no bodiesin quasi-circular orbits; however, numerical experiments using

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210 MELITA AND

more realistic initial mass distributions are needed to clarify tpoint. Nevertheless we can be certain that it is very unlikely thfurther evolution after the invasion could have circularized thoorbits. As the observed objects are very probably the largesthe distribution, it could be hypothesized that their orbits habeen circularized by dynamical friction with the smallest bodin the distribution. We can roughly estimate the time-scale awhich the eccentricity of a body of massM would be reducedby this process by setting

de

dt= −4πρ log(1+32)G2 m(M +m)

1

e2v3K

, (2)

where we have only considered the secular contribution ofdynamical friction force due to a constant number density dtribution, ρ, of objects of massm. We have approximated threlative velocityv0 asevK , wherevK is the local Keplerian ve-locity and3= RHv

20/(G (M +m)), whereRH is Hill’s radius

of the large body andG is the gravitational constant (for the dynamical friction force, see, for example, Binney and Trema1987 and for the expression of eccentricity variation rate dto a tangential force, see, for example, Danby 1962). Assing primordial parameters, i.e., total mass of 50M⊕ uniformlydistributed between 35 and 50 AU, at a heliocentric distanc42.5 AU, the time-scale,τDinFric, for the dynamical friction dueto bodies of radius of 1 km to circularize the orbit of a body ophysical radius of 150 km from an initial value of 0.5 is greaterthan the age of the Solar System,τDinFric≈ 1.4× 1010 yr.

3. PHYSICAL COLLISIONS AND CLOSEGRAVITATIONAL ENCOUNTERS

A simple test can be performed to estimate if physical cosions and close gravitational encounters can have very diffeeffects when acting at the different resonances. We have cputed the intrinsic probabilities of collision using the methodMarzariet al. (1996).

It should be noticed that this method relies on numerical ingrations. Then the peculiar characteristics of the resonant oare taken into account.

Values of the intrinsic probability of collision and the moprobable relative velocity for some of Neptune’s exterior MMRare given in Table I. These values are in good agreementthe ones obtained by Davis and Farinella (1997), computedanother method. It is apparent that these values are quite sim

TABLE IIntrinsic Probabilities of Collision, p, and the Most Probable

Relative Velocities, 〈δv〉, for Some Exterior Neptune Mean-MotionResonances

2 : 3 3 : 5 1 : 2

〈δv〉 (km s−1) 0.64 0.4 0.3

p (10−22 km−2 yr−1) 1.58 3.01 2.33

BRUNINI

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TABLE IIValues of the Total Variation

of the Semi-major Axis, ∆aMig,for Each Planet

Planet 1aMig (AU)

Jupiter −0.2Saturn 0.8Uranus 3Neptune 7

to one another. So the collisional dynamics should be veryilar at each resonance as well. If a resonant population whave been cleared by a collisional cascade, it would be verylikely to find a large population at the 2 : 3 NMMR, where ois believed to exist (Jewittet al.1998).

However, it should be noticed that the net effect of collisioand gravitational encounters inside each resonant region cvery different due to the differences in the structure of the phspace. Since in the 2 : 3 NMMR the stable niche is much lathan in the other two NMMRs investigated, where the stazones are much smaller and they are surrounded by regcharacterized by a small dynamical time-scales, the perturbneeded to expel an object from the stable zone is much grin the former than in the latter cases. Hence, even if relavelocities and intrinsic probabilities of collisions are of the saorder in all three of the resonant regions, the evolutionary efof these perturbations is not the same.

4. THE RESONANCE CAPTURE MODEL

The dynamical evolution inside the mean motion resonancan lead to substantial differences in the number of objectbiting in those regions. It is of great interest also to investigif primordial differences can be produced during the formatof the outer planets. We have based our study on the radiagration scenario (Fern´andez and Ip 1984, 1996, Brunini anFernandez 1999), which has been modeled as in M95.

4.1. Planetary Migration

In M95, the radial expansion of the proto-planetary orbitintroduced as

a(t)=af −1aMige−t/τ , (3)

wherea(t) is the semi-major axis as a function of time,af isits present value, andτ is the time-scale in which the radiaexpansion occurs. Values of the total variation of the semi-maxis1aMig are given in Table II.

It should be noticed that this model was designed originto explain the peculiar orbital characteristics of Pluto (Malho1993). The semi-major axes’ exponential variation can be

plained by the fact that the number of massive objects surround-ing a proto-planet decrease very rapidly with time once it has

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MEAN-MOTION RESONAN

achieved a certain critical mass. This kind of behavior has bconfirmed by recent simulations of proto-planets evolving ebedded in a swarm of planetesimals (Brunini and Fern´andez1999, Hahn and Malhotra 1999, Kokubo and Ida 1995).

4.2. Resonance Capture

Due to planetary migration, the inner Kuiper belt would habeen swept by Neptune’s exterior-mean-motion resonancethe KBOs are captured into these resonances at the migrthen an accumulation of objects is produced in those zonethis effect.

Capture into resonance is a very well-studied phenomeparticularly referred to the tidally driven resonant configuratiofound between some of the satellites of the major planets (for example, Peale 1976).

In general, resonance capture is possible when the orbitwo bodies are approaching one another as a consequenceaction of a dissipative force (Dermottet al. 1988). The transi-tion from a nonresonant orbit to a resonant one is a processsensitive to the initial conditions, the nature of the resonato which the orbits are approaching and to the speed at wthe orbits are migrating due to the dissipation. A probabiof capture can be defined in the frame of the restricted thbody problem, under certain conditions, i.e., that the resonais isolated and that the evolution is sufficiently slow such tan adiabatic invariant can be defined for the system (Hen1981). According to the adiabatic invariant theory, particlesare initially orbiting inside the inner libration region of the phaspace would invariably remain bounded in the resonance wits orbit approaches the resonance due to the action of the dpative force. Thus, for those objects the probability of capturequal to the unity. For particles initially orbiting in the exterilibration region, a probability of capture is defined on the baof the ratio between the areas enclosed in the phase spathe initial orbit and the one corresponding to the libration reg(for further details see Henrard 1981).

The probability of capture as a function of eccentricitysome Neptune exterior MMRs is shown in Fig. 4. Hence, this a critical limit for the initial eccentricity,ecrit, of the test parti-cle. If the test particle approaches the resonance with an ectricity smaller thanecrit (i.e., it is already orbiting in the innelibrating region), then the capture is certain. It can be seenin Fig. 4 that for values greater thanecrit, the probability of cap-ture decreases very rapidly. Values ofecrit for some of Neptune’sexterior-mean-motion resonances are given in Table III.

On the other hand, the adiabaticity condition can be put a(da

dt

)Mig

¿ 1aRes

TLib, (4)

where (da/dt)Mig is the rate of change of the planetary semmajor axis due to the migration,1aRes is the width of the resonance, andTLib is its libration period.

The values ofecrit and of the probability of capture presentedthis section have been obtained following Dermottet al.(1988).

CES IN THE KUIPER BELT 211

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FIG. 4. Probability of capture as a function of eccentricity for some exteNMMRs.

4.3. The dependence on the Speed of the Migration

From the adiabaticity condition (Eq. (4)) the value of the mimum initial value ofτ in the orbital expansion model (Eq. (3for which the capture in a given resonance is possible,τmin, canbe estimated.

τminÀ TLib1aMig

1aRes.

The initial value is the most relevant because most of thejects are captured in their early history, when the proto-planeresonances cover a wider region expanding at a greater sp

Libration periods as a function of the initial eccentricitysome of Neptune exterior MMRs were computed numericaParticles with eccentricities ranging from 0.0 to 0.25 at the exacresonant location were integrated for a time span of 3× 105 yrusing the second order symplectic map under the perturbaof the four major planets. The choice of initial angular vaables is the same as described in Section 2. Then, thefrequency of the signal corresponding toσ is computed usingthe FMFT method. Interpolations of the numerical valuestained are shown in Fig. 5. It should be noticed that in the cof the 1 : 2 NMMR and the 3 : 5 NMMR the values obtainagree reasonably well with those obtained by Malhotra (19for the restricted three-body problem and for the 2 : 3 NMMwith those obtained by Gallardo and Ferraz-Mello (1998).

TABLE IIIValues of ecrit and Maximum Values of (da/dt)Mig (in 10−5 AU/yr)

That Allow Objects To Be Captured for Initial Eccentricities onthe Order of 0.05, for Some Neptune Exterior-Mean-Motion Reso-nances

2 : 3 3 : 5 1 : 2

ecrit 0.0518 0.033 0.061

in(da/dt)Mig 2.93 0.10 0.096

D

io

e

n

o

m

t

i

h

ehlhhr

for

ste-seechtion,

ionion

tsanles

the

er,soci-

as

ions

lem-ratesshus,nt

212 MELITA AN

FIG. 5. Period of libration as a function of eccentricity for some exterNMMRs.

The resonance width was computed by averaging the smajor axis variation over a running window of length equala libration period. It should be noticed that the values obtaiare not in general equal to the amplitude corresponding tolibration frequency in a Fourier decomposition. It contains ctributions due to other interactions such as the ones withother planets.

The maximum speeds at which the capture is possiblesome NMMRs are shown in Fig. 6. Values ofτmin as a functionof eccentricity are shown in Fig. 7. In Table III the maximuvalues of (da/dt)Mig that allow capture in each resonance agiven for eccentricities on the order of 0.05.

Therefore, from Eq. (4) we can know whether an objecgoing to be captured in a given resonance at the end ofplanetary evolution depending on the speed of the migratIn fact, for values ofτ smaller than∼106 yr there would becapture in the 2 : 3 NMMR while it would not be possible at t1 : 2 NMMR if the eccentricity of the particles is below∼0.05when the resonance reaches them.

We have performed a series of numerical simulations whthe Kuiper belt is represented by 120 massless particles. Tare perturbed by the four major planets who are themseself-interacting and have a tangential force applied, so that torbits evolve according to Eq. (3). This radial displacementbeen introduced in the model by means of a tangential fo

FIG. 6. Maximum semi-major axis variation rate (da/dt)Max as a functionof eccentricity for some exterior NMMRs.

BRUNINI

r

mi-toedthen-the

for

re

istheon.

e

reese

veseirasce,

FIG. 7. Minimum timescale that allow capture,τMin , as a function of ec-centricity for some exterior NMMRs.

T , which is related to the semi-major axis evolution as (see,example, Danby 1962)(

da

dt

)Mig

= 2 V a2

GM¯T, (5)

where (da/dt)Mig is taken from Eq. (3),V is the velocity mag-nitude of the proto-planet, andM¯ is the mass of the Sun. Thidissipative force has been introduced in the symplectic ingrators as another perturbation (for details on this method,Cordeiro 1994). The total time of integration is indicated in eacase. As we are interested in the consequences of the migraruns of a simulated time-length of a fewτ have been done.

In Fig. 8, final histograms of the number of bodies as a functof the ratio of periods with Neptune at the end of the simulatare shown for cases with different values ofτ . It can be seenthat increasing values ofτ result in a greater number of objeccaptured in the 1 : 2 NMMR, particularly for values greater th2 Myr, and that practically only the background level of particis at that location whenτ = 0.5 Myr.

According to recent surveys (Trujilloet al. 1997) the intrin-sic ratio between the number of objects in the 2 : 3 NMMR,so-called Plutinos, and the rest of the KBOs would be∼0.38.According to the latest information of the Minor Planet Centthe present apparent ratio between Plutinos and objects asated with the 1 : 2 NMMR is∼4/50. If we assume a standard bifactor of 0.3 for this ratio (for details, see Trujilloet al. 1997)we can conclude that the intrinsic ratio between the populatof the 2 : 3 NMMR and the 1 : 2 NMMR would be∼0.27. In oursimulations, that kind of ratio is obtained for values ofτ on theorder of 1–2 Myr, which would imply a very short time-scafor the formation of the planets. However, given that the dynaical evolution does make differences between the survivalin the 2 : 3 NMMR and the 1 : 2 NMMR, then the further loof objects in the latter resonance should be accounted for. Tvalues ofτ on the order of 5–10 Myr would be more consiste

with the observations, a conclusion independently obtained byHahn and Malhotra (1999) on other grounds.

MEAN-MOTION RESONANCES IN THE KUIPER BELT 213

FIG. 8. Number of objects as a function of the commensurability of periods with Neptune at the end of the simulation for runs with different time-scalesτ ,u

tho

tllt

p

and

Inttedhe, i.e.,red.

the

for a model with smooth-varying planetary semi-major axes. The total sim

It is apparent that at increasing values ofτ , as the populationsin the 1 : 2 NMMR location increases, the populations at the oresonances decrease. The cause of this phenomenon is twFirst, as there is more objects captured in the 1 : 2 NMMR, whis the first one to sweep the belt, there would be less objecbe captured by the NMMRs coming behind, which we castealing effect. Also, as the 1 : 2 NMMR sweeps throughKuiper belt, objects that are not permanently captured intoresonance do experience some orbital excitation due to tem

captures or the effect of secular resonances that move athe MMR. Hence, the initial eccentricity of the particles to b

lated time is indicated in each case.

erfold.

ichs toa

hetheoral

captured by the MMRs that are trailing behind is greater,if those values exceed that ofecrit for the corresponding MMR,the probability of capture is greatly reduced (see Fig. 4).Fig. 9 the number of objects at the resonant locations is ploas a function of time. Also plotted as a function of time is taverage eccentricity in the zone adjacent to the resonancethe average eccentricity of the particle candidates to be captuThese plots correspond to a run withτ = 5 Myr where most ofthe radial migration occurs before 15 Myr and hence most of

longecapture occurs before this time-scale. For the case of the 3 : 5NMMR the capture rate decays noticeably when it reaches the

D

itnnt,a

dc

oc

s

eee

It isanden-

the

ition

tri-ajorne.

en

im-ateac-

inemi-can

ctsof

to-nterednncy

at

FIG. 10. Semi-major axis evolution of Uranus and Neptune for a model

214 MELITA AN

FIG. 9. (a) Number of objects as a function of time for a simulation wτ = 5 Myr in some resonant regions. Vertical lines indicate the time when an iresonance enters the zone already swept by an outer one. (b) Mean ecceof the bodies orbiting in the adjacent resonant region (bodies’ candidatescaptured). Horizontal lines indicate the values of the critical eccentricitiesec,of the corresponding NMMRs. Notice that at increasing times the stochfluctuations tend to grow due to the decreasing number of objects.

region already swept by the 1 : 2 NMMR at about 3 Myr, anpractically vanishes when the mean eccentricity of the partito be captured increases to a value close toecrit at about 5 Myr.In the case of the 2 : 3 NMMR, which is one of the innermones, the mean eccentricity of the particles in the region adjato the MMR exceeds the value ofecrit at about 4 Myr coming aplateau in the number of objects in the MMR location. A furthdecrease can be observed at about 7 Myr when the 2 : 3 NMreaches the region already swept by the 1 : 2 NMMR.

4.4. Possible Consequences of the Noise Produced by CloEncounters and Physical Collisions during the RadialMigration of the Outer Planets

We have seen that for certain values ofτ , there is capturein the 2 : 3 NMMR while there is not in the 1 : 2 NMMR (seFig. 8). The time-scale for the migration such that this effwould occur implies a time-scale of formation and an extrem

short evolution of the outer planets (τ < 1 Myr).

BRUNINI

hnertricityo be

stic

itles

stent

erMR

e

ctly

The migration is not a smooth and continuous process.produced by the angular momentum exchange of collisionsclose encounters with other minor bodies; hence, it is esstially constituted by noise. Therefore, for certain values ofamplitude,Anoise, and frequency,fnoise, of this noise, capture ina given resonance would not be possible. The capture condcan be put as:

Anoise× fnoise¿ 1a

TLib. (6)

We have devised a very simple model for the noise where aangular function is superimposed on the exponential semi-maxis evolution (Eq. (3)) only in the orbits of Uranus and NeptuThe amplitude of the noise also decreases exponentially as

Anoise(t)= Anoise(0)e−t/τnoise, (7)

whereτnoise= 1/ fnoiseis the period of the triangular function. Atypical semi-major axis evolution of Uranus and Neptune givby this model is shown in Fig. 10.

The following numerical experiments are not intended to sulate the conditions in the early Solar System but to investigthe effect of a semi-major axis evolution with a certain charteristic noise on the relative populations of objects capturedeach resonance; however, a certain resemblance with the smajor axes evolution obtained by Hahn and Malhotra (1999)be found.

In Fig. 11 we show the histograms of the number of objeas a function of the ratio of periods with Neptune at the endthe simulation for different characteristic noises. As the proplanet migrates, its exterior-mean-motion resonances encouobjects in very cold orbits. So, the maximum migration spethat allows capture has rough limits given in Table III. It cabe readily seen that a noise with an amplitude and frequebeyond the limit for the 1 : 2 NMMR practically eliminates th

with noise as described in Section 4.4.

pt)

a

p

ra-ce tonor-the

esti-esetionlesthe

dialrate.ledthe-

sep

lar

tionu-

iserba-and

n theual-ce

d or-e ofm the

anbyy if

re-uses

r

ioned

MEAN-MOTION RESONAN

FIG. 11. Number of objects as a function of the commensurability ofriods with Neptune at the end of the simulation for runs in which the planesemi-major axes evolve according to a characteristic noise (see Section 4.4total simulated time,T , the characteristic time-scale of the planetary semi-maxis evolution,τ , and the corresponding amplitude,Anoise, and frequency,fnoise,of the noise are indicated in each case.

population, while with a noise below the limit a substantial polation is formed. Moreover, a noise with characteristic amplitu

and frequency beyond the limit for the 2 : 3 NMMR practicalleliminates all resonant populations.

his

CES IN THE KUIPER BELT 215

e-ary. Thejor

u-dey

It should be noticed that this model for the planetary migtion has been implemented using a constant-tangential forthe orbit. Then, due to the absence of a perturbation on themal and orthogonal directions, there is no perturbation onnode and the perturbation on the line of apsides is undermated. Thus, it could be thought that the capture rate of thsimulations is overestimated given that a realistic perturbaon the proto-planetary orbits would affect the angular variabmore, which would tend to drive away the phase angle fromequilibrium position.

We devised a test for the accuracy of the model for the ramigration, particularly concerning the resonance captureIn the evolution of the proto-planetary orbits with our modenoise (Eq. (7)) we have introduced a Monte Carlo model fornormal,N, and the orthogonal,S, components of the perturbation. These are given by

N = ηN T(8)

S= ηS T,

whereT is given by Eq. (5), andηN andηS are random numbertaken from a uniform distribution in the interval [0, 10] to kethe N andS components in the same order of magnitude asT .The values ofηN andηS are changed each time that the triangufunction—constituent of the noise—changes its slope.

In Fig. 12 histograms of the number of objects as a funcof the ratio of periods with Neptune at the end of the simlation are shown for a case with radial migration with noin cases where the nontangential components of the pertutions have and have not been introduced. The amplitudefrequency of the noise chosen make capture possible only i2 : 3 NMMR. It is apparent that the results do no change qitatively. The cause for the lack of any remarkable differencan be explained since the perturbations by the normal anthogonal components of the force on the node and on the linapsides do not have a secular component, as can be seen froLagrange equations in the Gauss form (see, for example, D1962), which implies a great deal of cancellation, particularlthe perturbation is introduced as a constant. A further, morealistic test has been done by introducing the force which cathe radial migration as

T0 = ηNT sin

(2π

Ps+ φ

)N = ηNT0 (9)

S= ηST0,

whereT0 is the tangential force applied,Ps is a random numbetaken from a uniform distribution in the interval [0, 104] andφ is also a random number taken from a uniform distributin the interval [0, 2π ]. Again the random numbers are changeach time that the triangular function changes its slope. T

model has some physical basis which can be explained in the

2

planetarys lied.(

pe

16 MELITA AND BRUNINI

FIG. 12. Number of objects as a function of the commensurability of periods with Neptune at the end of the simulation for runs in which the theemi-major axes evolve according to a characteristic noise.Anoise=, fnoise=, τ = 5 Myr and the total simulated time is 50 Myr in all cases. (a) No Noise app

b) Noise originated only by a constant tangential force. (c) Noise originated by a force with three constant orthogonal components. (d) Noise originated by a force

s

a

u

v

tur-seeis

cita-

with three orthogonal components with a harmonic modulation. For detail

following way. The migration is mainly produced by gravittional close encounters, but the excursions in the semi-maxis by the proto-planet are not produced by a single encouwith a surrounding planetesimal, but it is the byproduct of scessive encounters with the same massive object which orecurrently until the minor body is ejected or falls into the gratational control of another proto-planet (Brunini and Fern´andez1999). This approximation attempts to simulate this process.

erturbations are modulated by a periodic function with perqual to a random sinodic period,Ps, and an arbitrary phase,φ.

see Section 4.4.

-ajornterc-

ccuri-

The

For a periodic function, the degree of cancellation in the perbation is smaller, which is revealed in Fig. 12, where we canthat the number of objects orbiting around the 2 : 3 NMMRslightly smaller.

4.5. The Dependence on the Varying Proto-Planetary Mass

As has been already discussed (Section 4.3), the orbital ex

iodtion suffered by the objects that are not captured in the outermostNMMRs is critical to determine the probability of capture in the

lanetary

MEAN-MOTION RESONANCES IN THE KUIPER BELT 217

FIG. 13. Number of objects as a function of the commensurability of periods with Neptune at the end of the simulation for a model with growing p

masses. The runs where the masses do not vary with time are included for comparison. For details see Section 4.5. The corresponding total simulated time and the

of

e

-o

jo

n

aeo

itth

a

w

is,ve: 3

hatof

nal-the

ith

u-Rthen theithblyns-vedofatesen-ase.rbitsed

characteristic time-scale,τ , are indicated in each case.

inner NMMRs. Naturally, the slower the pace at which the mexterior resonant zone travels and the greater the mass oproto-planet, the greater the excitation of the orbits. In this stion we investigate how the varying proto-planetary mass affthe capture rate in each NMMR due to this effect.

To simulate the mass evolutionm(t) of the proto-planets during the radial expansion stage we have designed a simple m

m(t) = mf − (1m) exp(−t/τ ), (10)

wherem is the corresponding present planetary mass,1m isthe difference between the present and the initial value onplanetary mass, andτ is the same time-scale of the semi-maaxes evolution (Eq. (3)). The proto-planetary initial value walways chosen equal to an Earth mass.

Histograms of the number of bodies as a function of Neptuperiods’ commensurability at the end of the 50 Myr simulatioare shown in Fig. 13.

We see in Fig. 13 that a rapid evolution of the proto-planetorbit (τ = 2 Myr) do not make a noticeable difference betwethe simulation where its mass is kept constant and thewhere the mass varies according to Eq. (10). A slower elution (τ = 10 Myr) when Neptune’s mass is set equal topresent value from the beginning of the simulation allows1 : 2 NMMR not only to capture more objects but to exciteeccentricity beyond the limit that allows capture for the innresonances (see Fig. 13). Thus, a great deal of difference cseen between the number of objects captured in the 1 : 2 NMand in the 2 : 3 NMMR. When the planetary mass increases

time—up to its present value—the outer resonances naturcapture fewer objects and the perturbations produced would

stthe

ec-cts

del,

ther

as

e’sns

rynne

vo-tshee

ern beMRith

smaller. If a plausible time-scale for the orbital migrationapproximately∼τ = 10 Myr (Hahn and Malhotra 1999), thenaccording to this effect, the most distant 1 : 2 NMMR could habeen primordially equally (or less) populated than the inner 2NMMR.

5. DISCUSSION AND CONCLUSIONS

In this work we have studied a number of mechanisms twould contribute to shaping the present orbital distributionobjects in the trans-neptunian region.

The dynamical structure revealed by the frequency map aysis inside each of the NMMRs studied is very different andglobal picture obtained from them gives a good agreement wthe observations available at present.

The 2 : 3 NMMR shows a very robust stable niche, particlarly at low inclinations. In both the 3 : 5 and the 1 : 2 NMMthe regions comprising regular orbits inside the borders ofmean-motion resonance are small and very disaggregated. Icase of the 1 : 2 NMMR there is a mild tendency to grow wincreasing inclination. These characteristics explain remarkawell the distribution of objects as observed at present in the traneptunian region (see Fig. 1). A great number of the obserobjects lie inside 2 : 3 NMMR stable niche. However, a fewthem can be found on very unstable regions, being candidas objects escaped from the stable niche by collisions orcounters or, perhaps, they are bodies in a transitional phThere is a chance, then, that those bodies in unstable oare fragments of collisions. This hypothesis could be clarifi

allybein the future when their orbits are accurately determined andphysical information about them becomes available. It should

D

mrM

u

a

t

e

o

d

e

dr

um

u

i

are

n

ry

s

-

th–

ian

e of

n ofals.

ing

nce

ned

in a

g-

in a

992

218 MELITA AN

also be considered that a collisional scenario for the fortion of the Pluto–Charon binary (Stern 1992) predicts the pence of a great number of fragments inside the 2 : 3 NMand some of them may have recently escaped from theble niche by the action of the slow dynamical diffusion ovthe age of the Solar System or by the perturbations produby gravitational encounters or physical collisions with othPlutinos.

A substantial population of objects in the 1 : 2 NMMR is nexpected, at least at low inclinations, unless the primordial nber of bodies in the 1 : 2 NMMR region was very high. Thresult is consistent with present estimations of an intrinsic rbetween the Plutinos and the 1 : 2 NMMR populations lowthan unity (see Section 4.3).

The existence of a great number of bodies at low eccentricalso can be explained by the stable regions found just outhe 3 : 5 NMMR. A better understanding of the formation of touter Solar System is needed to know if the number of objobserved in very cold orbits is compatible with the scenariowhich the Kuiper belt is invaded by a number of massive objeat the time of the formation of the outer planets and also withcollisional history.

The noisy evolution of the migrating proto-planets playsimportant role in shaping the present inner structure of the Kubelt. Given the amplitude and frequency of the perturbatidue to close encounters and physical collisions suffered byproto-planets—particularly in the early stages—and depenon which mean-motion resonances fulfill the condition givenEq. (4), capture could occur in all resonances, in just somein none at all. Also, the mass variation of the migrating proplanets can have a major significance in shaping the primdial structure of the trans-neptunian region. Perhaps whreliable—regarding the orbital determination—unbiased disbution of objects in the Kuiper belt is constructed, some cstraints can be drawn upon the formation process of the oplanets, where the further dynamical evolution in the regshould be taken into account; i.e., given the intrinsic ratiotween two resonant populations and taking into account theferences that arise from the dynamical evolution, the primorratio can be derived, which can then be compared with diffeevolutionary models of the outer Solar System.

Also, a definitive observational confirmation of the existenof the scattered disk would give additional support to thepothesis of the invasion of massive objects in the Kuiper bOn the other hand, the giant planets migration scenario shbe reconciled with this invasion. It is very relevant when woit have occurred, referred to as the radial migration. If thegrating mean motion resonances find the KBOs already incited orbits—cause of the invasion—the probability of captwould be very small (see Fig. (4)). The numerical simulatioperformed so far (Hahn and Malhotra 1999, Melita and Brun1999) lack sufficient accuracy on the initial mass distribut

of planetesimals—a drawback naturally related with computlimitations—to give plausible results in this respect.

BRUNINI

a-es-R

sta-erceder

otm-istioer

itiessidehectsinctsits

anipernstheingby, orto-or-n atri-on-uterionbe-dif-ialent

cehy-elt.ouldldi-

ex-rensinion

ACKNOWLEDGMENTS

We acknowledge the financial support by PROFOEG (CONICET). Wevery grateful to Alessandro Morbidelli and David Nesvorn´y for useful com-ments. We thank P. Kabal and F. Wachlin for providing pieces of software.

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