The origin of the Kuiper Belt high–inclination population

Rodney S. Gomes*GEA/OV/UFRJ and ON/MCT, Rua General Jose Cristino, 77, CEP 20921-400 Rio de Janeiro, RJ, Brazil

Received 4 June 2002; revised 31 October 2002

Abstract

I simulate the orbital evolution of the four major planets and a massive primordial planetesimal disk composed of 104 objects, whichperturb the planets but not themselves. As Neptune migrates by energy and angular momentum exchange with the planetesimals, a largenumber of primordial Neptune-scattered objects are formed. These objects may experience secular, Kozai, and mean motion resonances thatinduce temporary decrease of their eccentricities. Because planets are migrating, some planetesimals can escape those resonances while ina low-eccentricity incursion, thus avoiding the return path to Neptune close encounter dynamics. In the end, this mechanism produces stableorbits with high inclination and moderate eccentricities. The population so formed together with the objects coming from the classicalresonance sweeping process, originates a bimodal distribution for the Kuiper Belt orbits. The inclinations obtained by the simulations canattain values above 30° and their distribution resembles a debiased distribution for the high-inclination population coming from the realclassical Kuiper Belt.© 2003 Elsevier Science (USA). All rights reserved.

Keywords: Kuiper Belt; resonance; Solar System formation

1. Introduction

The discovery of the first member of the Kuiper Belt in1992 (Luu and Jewitt, 1993) confirmed what scientists wereexpecting to find since (Kuiper, 1951; Edgeworth, 1943)first predicted a disk of small bodies beyond Neptune. Lateron, however, further discoveries of these distant objectsintroduced increasing puzzles rather than solutions. Themain problem concerns the unexpected observed high-incli-nation orbits and the probable existence of two distinctpopulations in the classical belt (Levison and Stern, 2001;Trujillo and Brown, 2002; Brown, 2001). The first tentativeexplanation for the Kuiper Belt (KB) orbital structure waspresented by Malhotra (1993, 1995), who conjectured that aprimordial planetesimal disk forced the major planets tomigrate (Fernandez and Ip, 1984) and, in particular, Nep-tune’s outward migration swept most planetesimals by res-onance trapping. The final planetesimal orbital configura-tion thus obtained could nicely explain Pluto’s eccentricity.

This theory also predicted that Kuiper Belt objects (KBOs)should be found inside other mean motion resonances, suchas the 1:2 resonance. The classical belt, located between the2:3 and 1:2 resonances with Neptune, would be populatedby moderately eccentric KB orbits. Later KBO discoverieswould not confirm the expected amount of KBOs in themain resonances. Another problem that Malhotra’s ideawould gradually face concerns the high inclination of bothplutinos and classical KBOs (CKBOs). In Malhotra (1998),the author shows that some high-inclination plutinos couldbe formed through the resonance sweeping process. Morerecently, Gomes (2000) showed that in fact Pluto-like ob-jects could be formed at the 2:3 resonance with Neptune bythe resonance sweeping process alone; however, this pro-cess does not account for the right amount of these high-inclination objects. While this work was being done, newdiscoveries of classical KBOs would also show surprisinglyhighly inclined orbits. As new observational evidence wasalready showing a quite excited Kuiper Belt, Morbidelli andValsechi (1997) and Petit et al. (1998) proposed that a largeplanetesimal originally located just beyond Neptune with alifetime of several million years could have excited theorbits of the KBOs. At that time this theory would repro-

* Fax: �(55)(21)25898972.E-mail address: rodney@on.br

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duce the high eccentricities and inclinations of the KBOsbetter than Malhotra’s theory, which would anyway betteraccount for the plutinos. However, with increasing evidencethat highly inclined orbits should account for the greaterproportion of the classical KBOs, the large planetesimaltheory was also found not to adequately explain such highinclinations.

The importance of the highly inclined orbits in the clas-sical Kuiper Belt was more effectively established after thework of Brown (2001), Levison and Stern (2001), andTrujillo and Brown (2002). The first predicts a debiasedproportion of the high-inclination objects to the low-incli-nation ones, showing that highly inclined orbits are in factthe majority. Levison and Stern (2001) predict a bimodaldistribution for the CKBOs, with the high-inclination pop-ulation being correlated with lower magnitudes.1 A discus-sion of the association of inclinations with color is alsocarried out by these authors. Trujillo and Brown (2002)predicts a correlation of high inclinations with lower B-Rcolors. A little before, also motivated by the fact that Nep-tune and Uranus at their current positions would not be ableto accrete mass to their current size, Thommes et al. (1999)proposed that Neptune and Uranus were originally formedin the Jupiter–Saturn region and, after the two largest majorplanets quickly accreted their gaseous mass, the orbits ofUranus and Neptune were excited so that Neptune’s aph-elion temporarily reached the present KB location, thusproducing an excited KB. Dynamical friction from the plan-etesimal disk then acted to circularize the orbits of Uranusand Neptune, placing them near their present positions. Thistheory also predicts a bimodal inclination distribution forthe KB. Although this interesting idea can account forseveral explanations concerning the primordial Solar Sys-tem formation, it still fails to explain orbits sufficientlyhighly inclined for the classical Kuiper Belt.

The idea that the major planets migrated in a primordialsolar system due to energy and angular momentum ex-change with planetesimals in a disk was suggested by Fer-nandez and Ip (1984), who were particularly concerned withthe formation of Uranus and Neptune by mass accretionfrom planetesimals. Their simulations also showed that pro-toplanets would have migrated while accreting mass andthis process motivated several works thereafter concerningthe final orbital distribution of the outer planets, asteroids(Gomes, 1997; Liou and Malhotra, 1997) and more specif-ically the Kuiper Belt (Malhotra, 1995; Hahn and Malhotra,1999). Numerical simulations undertaken so far for themigration of planets inside a planetesimal disk are basicallyof two distinct types. Either the migration process is simu-lated by an artificial force imposed on the planets and onetracks the effect of this migration on massless planetesimals(Malhotra, 1995; Gomes, 1997) or the planetesimals are

considered massive objects and one tracks the effect of theplanetesimal disk on the planets and possibly the counteref-fect on the planetesimals (Hahn and Malhotra, 1999). Un-fortunately, since the planetesimal disk is simulated with arelatively small number (at most 103) of planetesimals withrelatively large masses, the resulting migration is conse-quently very nonuniform, thus preventing some importantdynamical processes such as resonance trapping. Thus, ei-ther the planetesimals are taken as the fuel for migration orthey are just the objects suffering the migration conse-quences, but never both together. To overcome this prob-lem, the idea is to go one step further in reality by includinga larger number of smaller planetesimals in the disk, how-ever long numerical integration may take for this.

In Hahn and Malhotra (1999), the authors simulate aplanetesimal disk extending from 10 to 50 AU and initialplanets with semimajor axes differing from present valuesby 0.2, �0.8, �3.0, and �7.0AU, respectively, for Jupiter toNeptune. Taking four different mass values for the disk,they conclude that 50 Earth masses is the one that bestexplains the final planetary positions after migration. Theynote, however, that migration was still going on when theintegration was stopped 5 � 107 years, so the planets arepredicted to stop somewhat further if the integration iscarried longer. In these simulations, the small number oflarge planetesimals prevented the resonance-trapping phe-nomenon due to a too noisy migration. So the planetesimalsacted as fuel to drive the planets but they could not see whatwould be the countereffect on the planets of trapped plan-etesimals. The authors computed the back torque of thetrapped planetesimals on the planets analytically and con-cluded that migration would slow down as the amount oftrapped mass increased.

Although many works followed Malhotra’s pioneeringidea that the Kuiper Belt was sculpted by the effect ofplanetary migration on primordial planetesimals, I under-stood that the simple migration mechanism could againyield new insight into the Kuiper Belt problem if moredeeply explored. In particular, the mutual dynamics of plan-ets and a sufficiently high number of planetesimals to in-duce an adiabatic migration to the planets was not ade-quately understood. With this in mind, in order to check theeffect of a disk composed of many planetesimals on themigration process, I started some numerical integrations ofthe orbits of the four major planets and a planetesimal diskcomposed of 10,000 objects, using the MERCURY integra-tor that considers the gravitational effect of the planetesi-mals on the planets (Chambers, 1999). In one example, Iassumed a planetesimal disk of 60 Earth masses extendingfrom 20 to 45 AU, with a surface density distribution asr�1.5. I found that this disk was effective enough to bringNeptune (initially at 18 AU) to 45 AU after 1.4 � 108 years.This result might be somewhat surprising as in this casethere is real trapping of planetesimals into the main reso-nances with Neptune, mainly the 2:3 resonance. The idea isthat although, in the beginning, the trapped planetesimals

1 Levison (2002) (personal communication), on the other hand, nowclaims that a recent reanalysis of the data shows that the i–H correlation isno longer apparent.

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hamper the outward migration of Neptune as predicted byHahn and Malhotra (1999), when (and if) the planetesimals’eccentricities succeed in getting so high as to make theplanetesimals’ orbits cross Neptune’s orbit, these planetes-imals become fuel to drive the planets and thus succeed indisplacing Neptune to the outer edge of the disk. An extraoutcome of these integrations concerned the orbits of someplanetesimals that seemed to be in quite stable orbits withmoderate eccentricities located in a “displaced” KuiperBelt. Because Neptune had migrated all the way to the outeredge of the disk, these stable objects would necessarily bepast Neptune-scattered planetesimals. This finding moti-vated me to undertake a series of systematic integrations tocheck whether objects would be inserted into the KuiperBelt in an integration for which Neptune would stop near itscurrent position. With this in mind, I propose here a ratherartificial model based on a two-component disk that canmake the planets stop at their approximate current positions,create a large number of primordial Neptune-scattered ob-jects, and check whether these planetesimals would even-tually be decoupled from Neptune’s close encounters. Thismodel and the results are described in Section 2. Section 3presents some extra simulations based on a massless-parti-cle model. In Section 4, other simulations based on a two-step series of runs better demonstrates the process of pro-duction of high-inclination main KBOs. Section 5 addressesthe case of plutinos and stable scattered objects. Typicalexamples of how a high-inclination KBO is created arepresented in Section 6, and Section 7 gives final conclusionsand discussions.

2. Results for Simulation I

For this set of simulations, I used the software packageMERCURY (Chambers, 1999), specifically the hybrid sym-plectic/Bulirsch–Stoer integrator, which can deal with mas-sive small objects that perturb the big bodies (planets) butnot themselves. I performed a total of seven runs whosebasic aspects are a main disk for all cases composed of 104

planetesimals, with its inner edge between 14 and 18 AUand its outer edge between 26 and 28 AU, with a densitydistribution as r�1. The mass of each planetesimal wasassumed to be in the range from 0.13 � 10�7 to 0.15 �10�7 solar mass (around twice Pluto’s mass), giving a totaldisk mass between 43 and 50 Earth masses. A much lessmassive disk starting just outside the inner one and extend-ing to around 50 AU was considered. This disk was com-posed of 250 to 500 objects, each object with the same massas those in the inner disk. The initial eccentricities andinclinations were assumed to be random and very small(0.5° at most for inclinations), other orbital elements ran-dom. Initial semimajor axes for the planets were 5.4 to 5.45AU for Jupiter, 8.6 to 8.7 AU for Saturn, Neptune at 0.5 AUinside the inner edge of the disk, and Uranus at 1.5 to 2 AUinside Neptune’s position. In four of the seven cases, I also

considered a large planetesimal (considered as a big body inMERCURY’s nomenclature) with masses in the range fromMars to half an Earth mass, the planetesimal being placed inthis case at the inner edge of the inner disk, Neptune 2 AUinside the planetesimal’s position, and Uranus with thesame relative position with respect to Neptune as for thecases without the large planetesimal. Other orbital elementswere randomly chosen, however, with eccentricities equalto zero and at most 0.5° for the planet’s inclination. Plan-etary masses were the same as the present ones.

To deal with such a huge number of objects, two sim-plifications have been considered. First, I assume that anobject is discarded from the integration when its distancefrom the Sun is above 500 AU (this number is usually takenas 1000). This accelerates the cleaning of objects and thusreduces the total integration time. Although this procedureprobably underestimates the final planetary radial displace-ment, it has no more influence on the basic mechanism thatproduces the high-inclination objects than any other simpli-fying assumption here considered. The other simplificationis a step length of 1 year, which is not ideal for the jovian/saturnian region but anyway is accurate enough for theNeptune region, where the basic dynamical resonance pro-cesses and escape from Neptune’s close encounter takeplace (Kuchner et al., 2002). In one of the seven runs Irepeated the integration with a step length of 0.2 yr for 107

yr. At this time about two thirds of the migration hadalready proceeded. The semimajor axes for the 1-year steplength and for the 0.2-year step length at 108 years are, forthe planets from Jupiter to Neptune, respectively 5.24, 9.54,16.69, 25.53 and 5.22, 9.59, 16.61, 25.32, all values inastronomical units. Considering that the initial semimajoraxes for the planets were 5.45, 8.7, 14.0, and 15.5, all valuesin astronomical units, we notice that the radial displace-ments of Jupiter and Saturn are somewhat underestimatedfor the larger step length, but for Uranus and Neptune, therelative differences are quite modest.

I assumed Neptune and Uranus started further inside thanin previous works (Malhotra, 1995; Gomes, 1997). In allcases the planets evolved smoothly with no mutual closeencounters. Starting further inside can be probably closer toreality if one wants to explain the formation of Uranus andNeptune by mass accretion (Levison and Stewart, 2001). Infact, I might even start these planets in the Jovian–Saturnianregion as in Thommes et al. (1999), but, for this, a one-yearstep length would not be acceptable since the process ofescaping from the Jupiter–Saturn region, which is essentialin the Thommes et al. scenario, might be jeopardized. Thisidea is further discussed in the last section. As a reference,Fig. 1 shows the temporal evolution of Neptune’s semima-jor axis for one of the runs.

Fig. 2, panels A–D, shows the distribution of the eccen-tricities, inclinations, and semimajor axes for the classicalKuiper Belt after 108 years of integration, including allseven runs. For uniformity, semimajor axis values are nor-malized so that zero and unity represent the 2:3 and 1:2

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resonances with Neptune. In the simulations, the classicalbelt is considered to be composed of objects whose perihe-lion is at least 2 AU outside Neptune’s aphelion. Although2 AU may place the planetesimal in a still precariouslysteady orbit, I kept this value since some real objects haveperihelions this close to Neptune. Panels C and D presentthe results for the real KB objects with observations in twoor more oppositions as listed in Minor Planet Center elec-tronic pages (available at http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html). I plotted a smaller number of real ob-jects for inclination lower than 5° to create a more unbiasedsample according to estimates that the high-inclination pop-ulation is four times as numerous as the low-inclinationpopulation (Brown, 2001). This procedure is justified by thefact that the simulated disk was not constructed to yield theright proportion of objects from both populations. Panels A,B present the results for the objects from the simulations.Blue dots refer to objects coming from the outer diskwhereas red ones come from the inner disk. The blue objectsare only moderately excited, as in previous results fromresonance sweeping simulations (Malhotra, 1995; Hahn andMalhotra, 1999)). Red and blue objects thus constitute twodifferent populations with different origins, the red onescoming from inner regions of the primordial solar systemand attaining larger final inclinations, thus forming what isusually called a (dynamically) hot population, whereas theblue dots stand for the (relatively) cold population. Theseresults thus explain the bimodal inclination distribution ofthe classical Kuiper Belt, including the correlation of mag-nitudes with inclinations (Levison and Stern, 2001), as thehot population will be composed of larger objects comingfrom an inner region of the primordial disk where a highersurface density allowed a deeper accretional evolution (see,however, footnote 1). Also, a correlation of inclinations

with colors (Trujillo and Brown, 2002) can thus be ex-plained by different origins. It must be noted that we shouldnot expect a perfect coincidence of the simulated data withthe real objects, since there are details which could not beincluded in the simulation. These details include the righttemporal evolutions of the planets’ orbits and the right (or atleast a better) distribution of planetesimals in the disk, asdistances and sizes are concerned, and also the consider-ation of the gravitational action of the disk on itself. Theinclusion of some or all of these assumptions would yielddifferent evolutions of the locations of secular resonances,for instance, which is a basic aspect at the production ofhigh-inclination objects (see Section 6). On the other hand,the fact that every run here presented could produce high-inclination objects in the Kuiper Belt is anyway an evidenceof the robustness of the mechanism that creates these ob-jects.

The total number of evaders2 from Neptune’s close per-turbation into the CKB at t � 108 years corresponds toabout 0.2% of the total inner disk mass, which gives anestimate of the total mass for the hot population, whichvaries from 0.06 to 0.16 Earth masses, depending on therun, thus about the same order of magnitude as the presentKB mass. The mass coming from the exterior disk, even inthis artificially little dense outer disk, is in principle largerthan that coming from the inner disk but it is expected tocollisionally evolve (Davis and Farinella, 1997; Kenyon andLuu, 1999). In this way, I only plotted in Fig. 2 randomlychosen subsets of the final complete sets from the coldpopulation in order to match the results for the proportion ofobjects in each population estimated by Brown (2001). Fig.3 shows the distribution of the number of objects for incli-nation bins both for the simulation (solid line) and for thereal KB. Normalizations were introduced to make, for thesimulation, the hot population four times as numerous as thecold one. For the real objects, I followed Brown’s (Brown,2001; Kuchner et al., 2002) method to create a debiasedinclination distribution. I chose the objects from the CKBdiscovered at latitudes below 1° and multiplied the numberof objects with inclination inside the bin centered at Ic by sinIc. The similarity between both distributions is clear, with,however, still more inclined orbits for the simulations.

The large planetesimals were thrown away into hyper-bolic orbits in less than 1.1 � 107 years for two of the fourcases, with no practical effect on the final results. In theother two cases, the planetesimals were still present in theintegrations after 108 yr in apparently stable configurations(one was near the 2:3 resonance and the other near the 1:2resonance with Neptune). Fig. 4 shows the a � e and a �

2 This term will hereafter mean the objects that escape Neptune closeencounter perturbations into fairly stable orbits. This definition excludesobjects that are discarded from the integrations by collisions with the Sunor any planet or by attaining a semimajor axis larger than 500 AU. Thisterm also excludes objects that enter the close encounter regions of theother major planets.

Fig. 1. Temporal evolution of Neptune’s semimajor axis for one of theruns.

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I distributions including all runs, with the objects comingfrom the long-lived large planetesimal runs represented bycrosses, whereas all the other objects represented by trian-gles. We notice that there is no basic difference between thefinal distribution for the runs affected by the long-livedlarge planetesimals and those not affected. A possible dif-ference would be an object with a � 0.75 and e � 0.1 (seeFig. 4, panel A) coming from a long-lived large planetesi-mal run. This object looks quite alone in this region. Its pastdynamics includes a passage by the 1:2 resonance, where itlost some of its eccentricity and was later displaced from theresonance. This single example cannot, however, prove thata run without a large planetesimal could not also producesuch an object. In conclusion, the four major planets can bythemselves create the high-inclination population. A largeplanetesimal might perhaps give some extra help in produc-ing such objects. The large planetesimals, however, pose a

difficulty, which is nevertheless not conclusive, since wehave poor statistics. This concerns the fact that large plan-etesimals that help in producing members for the hot pop-ulation may get a final stable orbit in the neighborhood orinside the present CKB. Such a large object would havealready been observed.

For one of the runs (with no large planetesimal) I con-tinued the integration up to 109 years. The average eccen-tricity for the high-inclination KBOs decreased from 0.19 to0.15, the average inclination increased from 15.5° to 16.7°,and the total mass decreased from 0.11 to 0.06 Earthmasses. For the last 8 � 108 years, we notice basically novariation in these numbers, as shown in Fig. 5. Thus onewould expect in Fig. 2, for a later time, higher inclinations,lower eccentricities, and about half of the total mass. Thismass is still consistent with the present KB mass. This samerun was continued to a total of 4.5 � 109 years but now

Fig. 2. Distributions of semimajor axes, eccentricities, and inclinations for the real KB objects (panels C, D) and those coming from the simulations (panelsA, B). Red dots stand for objects coming from the inner disk, whereas blue dots come from the outer disk. In all panels, the semimajor axis is normalizedso that zero and unity represent the 2:3 and 1:2 resonances with Neptune.

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considering only the objects (now assumed to be massless)from the inner disk that were left after 109 yr (the outer diskparticles were discarded). There were a total of 49 suchobjects at 109 yr and 19 objects survived for the SolarSystem age. Note also that for this last run I assumed a lessapproximate discarding distance equal to 103 AU. Fig. 6shows the distribution of the semimajor axes, eccentricities,and inclinations for these objects after 4.5 � 109 yr exceptone whose orbital elements are a � 104.1 AU, e � 0.668,and I � 11.2°. There are 7 resonant objects in this figure;one in the 4:5, three in the 2:3, two in the 3:5, and one in the1:2 resonance. These objects are easily identifiable sincethey are the closest to the vertical lines that stand for eachresonance. Note, however, that those near the 2:3 resonanceline with e � 0.1 are not in resonance (no resonance anglelibrates) and also the one near the 3:5 resonance line withthe smallest eccentricity shows no 3:5 resonance libratingangle. The one near the Neptune-crossing line has beenstable in the 3:5 resonance for nearly the Solar System age.If we define the scattered objects as those with semimajoraxis above the 1:2 resonance nominal semimajor axis, wehave four members in this group. In the classical KuiperBelt, we would have 10 objects, including those in the 3:5resonance and those two near the 2:3 resonance. This cor-responds to 0.05 Earth masses in the CKB. These resultsseem compatible with the present proportion of observedscattered objects with respect to hot (I � 5°) CKB objects,which is roughly 1/3. This number must, however, be un-derestimated due to observational bias. The number of sim-ulated scattered objects would also be probably a little

higher if we considered a larger discarding distance in theintegrations.

My last concern in this section will be to check fromwhich part of the inner disk the evaders come from. Againconsidering a normalization by which the inner edges of theinner disks are represented by zero and the outer edges byunity, I plot in Fig. 7 the number of objects from normalizedsemimajor axes bins that ended in the classical Kuiper Belt,showing that about half of the evaders come from the mostexterior fifth part of the disk. No correlation between theinitial semimajor axis and final orbital elements was noticedfor the CKB.

Fig. 4. Distribution of semimajor axes, eccentricities, and inclinations forthe runs affected by long-lived large planetesimals (crosses) and thosewithout large planetesimals or with short-lived large planetesimals (trian-gles).

Fig. 3. The continuous line stands for the distribution of the number ofobjects for inclination bins for the simulations and the dotted line stands fora debiased inclination distribution for the real KB. All results refer to 108

yr after the beginning of the integration.

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Fig. 5. Time evolution of the total mass, average eccentricity, and average inclination for the hot population produced by one of the runs that was extendedto a total of 109 yr of integration time.

Fig. 6. Final distribution of semimajor axes, eccentricities, and inclinationsfor the run of Fig. 5, now extended for the Solar System age, consideringonly the hot particles without mass.

Fig. 7. The horizontal axis represents a normalized semimajor axis, wherezero stands for the inner edge of (any) inner disk and unity stands for theouter edge of the inner disks. The vertical axis gives the total number ofobjects from all the seven runs that were injected into the classical KuiperBelt at 108 yr.

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3. Simulation II

In this section I perform two similar tests to better dem-onstrate the production of the KB high-inclination popula-tion. Now I use a model with a massless disk and make theplanets migrate though a fictitious force included in theintegrator codes. For the first test, I still use MERCURYwith the same step length and discarding distance as inSection 2. Jupiter and Saturn are considered in their presentpositions and are not subject to nonconservative forces.Uranus is started 0.7 AU inside its present position andNeptune 2 AU inside its present position. The two outer-most planets are then made to linearly migrate to theirpresent positions for 108 years. A 1-AU-wide disk with 104

objects is placed just beyond Neptune’s starting position.This choice is based on the fact suggested by Fig. 7 thatmost objects that become KBOs come from the outer por-tion of the inner disk. Although the simulations in Section 2are more realistic in the sense that planetary migration ismore realistically simulated, here there is a gain in reality inthe sense that planets stop with their right present orbitalelements (a, e, I), so the details of the dynamics imposed bythe four planets on the massless objects for the tail of themigration are better simulated.

A second test is performed with the swift_rmvs3 inte-grator (Levison and Duncan, 1994). It is similar to themassless run with Mercury with the planets having the sameinitial conditions and migration parameters as in that exam-ple. The disk is, however, a little different as it contains4000 objects and it has its edges at 29.5 and 30 AU. Thischoice avoids initial Neptune Trojans that were inadver-tently created in the previous MERCURY simulation. The

results from both simulations are summarized in Fig. 8,which shows the distribution of the semimajor axes withperihelion distances. The results from MERCURY are rep-resented by small crosses and those from SWIFT by trian-gles. We notice that the distributions from both integratorsare fairly similar. Fig. 9 shows the e � I distribution forboth integrators, where I kept the same meaning for thesymbols as in Fig. 8. Again the distributions are similar.Now when we compare the simulations with massless ob-jects with those of Section 2, the results for the massive diskmodel show average higher inclinations and larger perihe-lion distances. The average inclination for the case of mass-less planetesimals (putting together the results from MER-CURY and SWIFT) is around 10°, with, however, someobjects attaining an inclination around 30°, as shown in Fig.9. Anyway, the tests undertaken in this section proved therobustness of the process that creates high-inclinationKBOs, as it is fairly independent of the migration model.Moreover, the test with SWIFT proves that we are notobtaining an artifact of MERCURY’s integrator.

4. Simulation III

The simulations performed in this section are motivatedby the results of the previous sections that suggest how ahigh-inclination KBO is created. This process can thus bedivided into two steps. First, low-eccentricity incursions forthe planetesimals are produced, as usually occurs also whenplanets are not migrating. Second, the migration processdisplaces the object from its otherwise returnable path while

Fig. 8. Distribution of semimajor axes with perihelion distances for thesimulations with massless objects and planetary migration induced byartificial forces. Crosses stand for the results from MERCURY integrator,whereas triangles stand for those with the SWIFT integrator.

Fig. 9. Distribution of eccentricities with inclinations for the simulationswith massless objects and planetary migration induced by artificial forces.Crosses stand for the results from the MERCURY integrator whereastriangles stand for those from the SWIFT integrator.

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the planetesimal is in a low-eccentricity mode. This cancause an eventual placement of the object in a more stableconfiguration. Based on this idea, I performed a pair oftwo-step runs. I integrated the planets and massless particlesfor 108 years with no migration. Planets’ initial coordinatesare either as in the runs in Section 3 or, for the second case,Uranus and Neptune are started midway from that positionto their present position. For both cases, I add 100 masslessobjects that are placed so that they rapidly become Neptune-scattering. The orbital elements of these objects are trackedin time (each 104 years), showing several low-eccentricityincursions for the 108-yr integration. A few of these objectsat their low-eccentricity incursions are chosen as seeds fromwhich clones are created (100 cloned for each seed) andnew numerical integrations are performed with the clonesand the planets started at their positions for the time atwhich the seed was taken, now with the planets migrating.For the first case, the integration time is 108 yr and for thesecond case, 5 � 107 yr, all linear migration so that theplanets stop at their present position.

Fig. 10 shows the a � e plot of all objects coming fromcase I at every 104 yr for a total integration time of 108 yr.The vertical lines denote the 2:3 and 1:2 resonance posi-tions. The curves stand for Neptune crossing orbits andperihelion 4 AU outside Neptune. We notice many low-eccentricity incursions below the q � aN � 4AU line. Twopoints represented by the filled circles are chosen from thisdistribution and 100 clones are generated from either one.Fig. 11 shows the final a � e distribution coming from the

numerical integration of the clones and the planets migrat-ing to their present position. The initial positions of theclones are indicated by the large square and triangle andtheir final positions are represented by the equivalent smallsymbols. Fig. 12 and 13 show the results for case II. Here,three seed points are chosen, one of them beyond the 1:2resonance. We notice in Fig. 13 that on average moreevaders are created in case II. Here the low-eccentricityincursions occur for the planets closer to their present po-sitions. This strengthens the idea that these objects aregenerated during the tail of planetary migration. In partic-ular, for one seed in case II more than 50% of the initialclones ends up with q � 34AU. Because these objects arenear secular resonances many of them may become Neptune

Fig. 11. Distribution of semimajor axes, eccentricities and inclinationsfrom the 100 clones created from the filled circles in Fig. 10, after 108 yrintegration of migrating planets (to their current positions) and the masslessclone objects. Large squares and triangles stand for the initial coordinates,whereas small symbols refer to final coordinates associated with the initialcoordinates represented by the equivalent large symbols.

Fig. 10. Small dots represent a � e points coming from the static numericalintegration including all major planets and massless particles, with Uranusand Neptune 0.7 and 2 AU inside their present positions. This integrationwas carried out for 108 yr and points are plotted at every 104 yr. Large filledcircles are chosen points from which clones are created for a numericalintegration with migration. Vertical lines stand for the 2:3 and 1:2 reso-nances and the curves stand for the Neptune crossing orbits limit and orbitswith perihelion at 4 AU beyond Neptune.

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crossers in the Solar System age. Another interesting pointis that a few squares are displaced to the middle region ofthe CKB from a point beyond the 1:2 resonance. In case II,another seed was considered for a point at 173 AU withperihelion 4 AU beyond Neptune. At the end of the inte-gration, the highest q obtained was 6 AU outside Neptune.While this result is far from conclusive, it suggests thatobjects such as 2000 CR105 can be created in this way. Wejust have to have a deeper low-eccentricity incursion at highsemimajor axes. This is a possibility, as found in otherworks. In particular, Gladman et al. (2002) find a low-eccentricity incursion at 88 AU in the 1:5 resonance withNeptune, where q got as high as 50 AU. It seems possiblethat such a low-eccentricity incursion during a still migrat-ing process would create a high-q large-semimajor-axisscattered object such as 2000 CR105. In the followingsection I analyze the results coming from the simulation inSection 2 in what concerns the creation of stable scatteredobjects.

It must be noted that the results in this section are verysuggestive as to how an evader can be created. In this senseit confirms the results of the two previous sections. As allintegrations are done with SWIFT it is also an extra test forindependent integrators. On the other hand, as the results areobtained in a rather artificial way, it is not a good test toestimate the number of evaders that would be created, asdone in Section 2. A deeper analysis using this device mayyield a clearer insight on which objects can be produced(where can they be placed?) and where they come from

(where is the original low-eccentricity incursion located?).A quantitative analysis may also be possible.

5. Results for plutinos and stable scattered objects

The results presented in this section refer to the simula-tion described in Section 2. As Neptune migrates, it trapsmany objects initially from the inner disk into the 2:3resonance. When Neptune gets near 30 AU the trappedobjects coming from the inner disk have become scattered.

Fig. 13. Distribution of semimajor axes, eccentricities, and inclinationsfrom the 100 clones created from the filled circles in Fig. 12, after 5 � 107

yr integration of migrating planets (to their current positions) and themassless clone objects. The large circle, square, and triangle stand for theinitial coordinates, whereas small symbols refer to final coordinates asso-ciated with the initial coordinates represented by the equivalent largesymbols.

Fig. 12. Small dots represent a � e points coming from the static numericalintegration including all major planets and massless particles, with Uranusand Neptune 0.35 and 1 AU inside their present positions. This integrationwas carried out for 108 yr and points are plotted at every 104 yr. Large filledcircles are chosen points from which clones are created for a numericalintegration with migration. Vertical lines stand for the 2:3 and 1:2 reso-nances and the curves stand for the Neptune-crossing orbits limits andorbits with perihelion at 4AU beyond Neptune.

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Some objects from the outer disk are also trapped into the2:3 resonance with Neptune and are kept there in the end ofthe integration. Fig. 14 shows the distribution of eccentric-ities with inclinations for the plutinos, where panel A refersto the simulations and panel B to the real objects. Small dotsin panel A stand for the cold population. I did not check allplutinos coming from numerical integrations for the 2:3librating angles; I just checked for the semimajor axis res-onance range and eccentricity below 0.36. So the number ofplutinos in panel A may be somewhat overestimated,mainly for those with high eccentricities. This proceduremay influence the total number of plutinos from the hotpopulation but not their a � e distribution. Interestingly, forthe plutinos, the hot population seems to account for almost

all the real objects. Although a combination of the twopopulations is reasonable, this figure alone suggests thatmost plutinos would have come from the inner regions ofthe primordial planetesimal disk. Thus, the formation ofmost plutinos would have followed a way in a sense oppo-site to that initially conjectured by Malhotra (1995), comingfrom highly eccentric Neptune-crossing orbits and circular-izing on their way to the 2:3 resonance. In particular, thismodel predicts that Pluto, represented by a triangle in Fig.5B, would be a primordial scattered object rather than aproduct of the resonance-sweeping process. These conclu-sions are based on the fact that the cold population, asindicated by Fig. 14, panel A, has a particularly low averageinclination. This may be an artifact of the unrealistic dy-namics of the tail of the migration, since the planets do notstop at their right present position and even for the cases forwhich the planetary semimajor axes are in the end closer totheir real values, the eccentricities for the simulations arealways much lower than those for the real planets. In fact,Malhotra (1998) shows some relatively high inclinations forplutinos using the resonance sweeping process, with anartificial force imposed on the planets. However, Gomes(2000) makes a comprehensive analysis of the possibility ofcreation of high-inclination plutinos through the resonance-sweeping process alone and although it is confirmed thatthese objects can really be created this way, the conclusionis that this is not achieved with the right inclination distri-bution from the real objects.

Fig. 15 shows the distribution of semimajor axes withperihelion distances both for the results from simulation I(circles) and for real scattered objects (crosses) with obser-

Fig. 15. Distribution of semimajor axis with perihelion distance for theobjects from Simulation I (dots) and real scattered objects (crosses), ob-served in more than one opposition.

Fig. 14. Distribution of eccentricities with inclinations for the plutinos.Panel A represent the objects coming from the simulation, with the largedots standing for the hot population and the small ones for the coldpopulation. In panel B, the real objects, with Pluto represented by a largetriangle.

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vations on more than one opposition. We notice that exceptfor 2000 CR105, located in the right upper corner of thediagram, the difference in both distributions is not remark-able. In fact, 2000 CR105 looks as if it did not belong toeither population (real or simulated), as if invoking a quitedifferent origin. As to the objects from the simulation, itwould be expected that their perihelion distance might in-crease on the average if the integration was carried on for alonger time. In particular, one of the simulated objects thatappear in Fig. 15 with semimajor axis around 60 AU andperihelion distance around 41.5 AU belongs to the run thatwas extended to 109 yr. This object has its origin in the 1:3resonance with Neptune and after 109 years, its periheliondistance was around 48 AU, suggesting that at the SolarSystem age, the perihelion distances for the simulationobjects as shown in Fig. 15 may be increased. It must benoted that for the case of the scattered objects, the role of themigration process may be not so essential as for the case ofthe classical belt, since the timescale of a low-eccentricityincursion can be of the order of the Solar System age.Gladman et al. (2002) show that clones from 2000 CR105can reach a perihelion distance lower than 40 AU if theirorbits are integrated for 5 � 109 years. This at least suggeststhat some scattered objects, including 2000 CR105 itself,might be created considering only conservative dynamics(non migrating planets) with, however, a low probability.The migrating process may contribute to increasing thisprobability to the right number. Of course, more integra-tions both with and without migration should be undertakento better elucidate this point.

6. The planetesimal–Neptune decoupling process

As Neptune migrates by energy and angular momentumexchanges with planetesimals, it brings along a large num-ber of primordial scattered objects. In a conservative dy-namics, there is no permanent escape (except for ejections)from Neptune’s close encounter perturbations. Some dy-namical processes like mean motion resonances, Kozai andsecular resonance may indeed bring some objects to a tem-porary low-inclination mode but the planetesimal will even-tually find its way back to the high-eccentricity mode andagain experience close approaches to Neptune (Malhotra etal., 2000). However, with migration, the resonant dynamicalstructure may change while the particle is in the low eccen-tricity mode so that it may not find its way back to higheccentricities and thus get trapped in a moderately low-eccentricity high-inclination stable orbit. Next I presentthree examples of this process, one for each resonance type.This classification only means that the major factor in thecreation of the evader was a mean motion resonance, theKozai resonance, or a secular resonance. In general, how-ever, more than one resonance process is present in thedynamics of the evader creation. The examples describedbelow were didactically chosen to show each of the three

processes working fairly independently but note that anaverage case is not so idealized. All the examples belowcome from the simulations in Section 2.

For the Kozai example, this resonance induces an eccen-tricity decrease and an inclination increase to a scatteredprimordial planetesimal. Fig. 16 shows a typical example ofthis process for this case. Note that just around 5.5 � 107 yr,the planetesimal experiences a Kozai resonance character-ized by the slow variation of the argument of the perihelion(dots in upper panel). At the same time, the eccentricity(thick line in upper panel) and inclination (thin line inmiddle panel) suffer opposite variations typical of the Kozairesonance. This process is, however, discontinued beforeone complete cycle could be accomplished and at 6 � 108

years the planetesimal is already in a stable orbit with e �0.15 and I � 29° and q � aN � 7 AU. The nomenclatureKozai resonance is somewhat abusive since, for a reso-nance, one would expect a complete cycle of the periodicbounded variation of the argument of the perihelion. In allexamples that I could analyze there is just a slowing downof this variation, which is anyhow sufficient to help in thecreation of evaders from Neptune’s close encounters. Fig.16, lower panel, shows the time variation of the meanmotion of the planetesimal relative to Neptune’s mean mo-tion. We notice no trapping into any high-order mean mo-

Fig. 16. Example of the creation of a high-inclination KBO, where the mainmechanism is the Kozai resonance. Around 5.3 � 107 yr the temporalvariation of the perihelion argument, represented in the upper panel bydots, slows down. At that time, the eccentricity starts to decrease and theinclination (thin line in the middle panel) pumps up. At around 6 � 107 yr,the resonance is already broken and the perihelion distance (thick line,middle panel) is around 7 AU beyond Neptune in a safe configuration. Themean motion ratio in the lower panel shows no mean motion resonancetrapping during and after the Kozai mechanism.

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tion resonance with Neptune. After the decoupling process,the orbit stays between the 4:7 and 3:5 resonances.

Fig. 17 shows an example for the case of secular reso-nances. This is probably the most common case of evadercreation, perhaps the basic mechanism. In the upper panelthe dots refer to the difference between the planetesimal’sand Neptune’s longitude of the perihelion and the eccen-tricity is represented by the line. The main term for thevariation of the planetesimal eccentricity involving sin (�� �N) is e � mN n a eN f(�) sin (� � �N), where mN isNeptune’s mass and n, a, e, and � refer respectively to themean motion, semimajor axis, eccentricity and perihelionlongitude. The subscript N refers to Neptune and no sub-script to the planetesimal. f(�) is a function of Laplacecoefficients, which is negative for the considered region.This implies a negative time derivative of the planetesimaleccentricity for 0 � � � �N � 180°. A secular resonanceis keeping � � �N between 0° and 180° most of the time,inducing an average eccentricity decrease. The main decou-pling process occurs between 4.5 � 107 and 5.8 � 107

years, when the planetesimal’s perihelion, represented bythe thick line in the middle panel, goes from a Neptune-crossing configuration to near 8 AU away from Neptune’ssemimajor axis. Note that as the planets migrate, the loca-tion of the secular resonance also moves and the particle canthus temporarily stay in a secular resonance, experiencingan eccentricity decrease. If the resonance moves before theeccentricity can again be pumped up, an escape from aNeptune-crossing orbit has thus taken place. In the middle

panel, the thin line represents the planetesimal’s orbitalinclination. The lower panel shows the relative variation ofthe planetesimal’s mean motion. The horizontal lines standfor some important high-order resonances. There is no trap-ping into any of them. The oscillation around the 3:5 com-mensurability did not show an equivalent librating 3:5 res-onant angle.

Fig. 18 shows an example for the case of mean motionresonances. In the upper panel we notice a temporary trap-ping of the planetesimal into the 1:3 mean motion resonancewith Neptune, through the libration of the 1:3 resonant anglebetween 7.2 � 107 and 9.3 � 107 years. When the librationceases, we notice a considerable decrease of the eccentricity(thick line) and the planetesimal’s perihelion gets to around14 AU away from Neptune (thick line, middle panel). AKozai resonance is also active during the time the 1:3resonance lock is taking place, what also causes an inclina-tion increase (thin line, middle panel). The lower panelshows the time variation of the relative mean motion. Wenotice a slow withdraw from the resonance position. Asnoted in the beginning of this section, these examples havea didactic purpose; in general a decoupling process mayinclude two or all three resonance mechanisms.

Fig. 18. Example of the creation of a high-inclination KBO, where the mainmechanism is a mean motion resonance. The dots in the upper panelrepresents the 1:3 resonance angle (eccentricity type). When this anglestarts librating, the semimajor axis oscillates around a fixed value (inrespect with Neptune’s semimajor axis, lower panel) and the eccentricitystarts decreasing. When the resonance breaks up, the perihelion distance(thick line, middle panel) is above 12 AU beyond Neptune. A Kozairesonance is also effective during the 1:3 resonant angle libration, whichmakes the inclination (thin line, middle panel) be pumped up during theprocess. After the object leaves the resonance, the mean motion ratioslowly deviates from 3.

Fig. 17. Example of the creation of a high-inclination KBO, where the mainmechanism is related to secular resonances. The eccentricity decreaseswhenever � - �N (dots in upper panel) is between 0° and 180°. There isno trapping into mean motion resonances during this evolution. In themiddle panel, the thick line stands for the perihelion distance and the thinline stands for the inclination.

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7. Conclusions and discussion

The original ingenious idea described in Malhotra (1993)and Malhotra (1995) by which the present orbital configu-ration of the Kuiper Belt must be a consequence of aprimordial planetary migration was here revisited, assum-ing, however, a planetesimal disk with an order of magni-tude higher number of objects than in previous works. Iperformed numerical integrations of the planetary orbits ofthe giant planets started in a compact configuration justinside a primordial massive planetesimal disk simulated by104 objects. I found that a fraction of around 0.2% ofprimordial Neptune-scattered planetesimals can get rid ofNeptune’s close approach perturbations. This mechanismproduces a population of steady, fairly eccentric, and highlyinclined orbits that can account for the present observedhigh-inclination orbits of the Kuiper Belt. The average ec-centricity and inclination for the simulated population in-cluding all seven runs is 0.20 and 16.1°. There are, however,some simulated objects with inclination above 30°. In oneof the runs, I extended the integration to 109 years and, from108 years to 109 years, the average eccentricity dropped 0.04and the average inclination increased 1°. These valuesshould be expected for all other runs. The decrease of theaverage eccentricity is due to the loss of some candidates forthe high-inclination population which, at 108 years, hadtheir eccentricities still high enough to eventually return toa Neptune-crossing orbit. Also, the total number of surviv-ing bodies was reduced to half of the number at 108 years,but this number was quite stable for the last 8 � 108 years;thus a fraction of about 0.1% of the initial Neptune’s scat-tered planetesimals is expected to migrate to steadier re-gions to form present Kuiper Belt high inclination popula-tion. This accounts for around 0.05 Earth masses which iswithin the order of the estimated present KB mass (Kenyonand Luu, 1999). Still, for the same run, I extended theintegration for the Solar System age, considering, however,for this last integration only the particles from the inner diskthat were left after 109 years, now assumed massless. In theend, 10 objects were left in the CKB and four objects in thescattered disk. There were also 3 planetesimals in the 2:3resonance, one in the 4:5 resonance, and a last one in the 1:2resonance with Neptune.

The inclination distribution obtained by the simulationsresembles a debiased inclination distribution from the realKBOs, following Brown’s (Brown, 2001; Kuchner et al.,2002) debiasing method. It must be noted that the produc-tion of high-inclination objects is mostly due to the fact thatthese planetesimals come from a rather long history of closeencounters with Neptune, unlike the scenario of Thommeset al. (1999), where Neptune’s aphelion excursion inside theKuiper Belt is much shorter.

The objects that form the present KB high-inclinationpopulation must thus have their origin at a region around 25AU in a primordially denser part of the planetesimal disk.These objects are expected to have accretionally evolved

more than those initially 10 to 20 AU further out. Thus theseobjects are on average larger than those forming the low-inclined orbits. These smaller objects, on the other hand, areexpected to have been formed at their current positions or alittle closer to the Sun and evolved by resonance sweeping,as considered in previous works (Malhotra, 1993). Theselow-inclination objects are considered as just a small frac-tion of the initial population as estimated by Davis andFarinella (1997) and Kenyon and Luu (1999). Davis andFarinella (1997) also conclude that KB objects larger thanabout 100km are not significantly altered by collisions overthe age of the Solar System. Their results are, however,based on a population with the sine of the orbital inclina-tions not greater than 0.12. It is not certain that the relativelylarger objects from the high-inclination population obtainedin the simulations would survive the collisions with a denseprimordial low-inclination population. The primordial ini-tial cold population might be much less massive than sup-posed, in order not to cause erosion to the high-inclinationobjects. On the other hand, a too low-density outer disk mayimply an original disk not sufficiently massive to createlarge enough objects for the cold population (Kenyon andLuu, 1999). This point deserves a further future investiga-tion.

Like the classical belt, the same process also suggeststhat high-inclination plutinos may be evaders from a pri-mordial Neptune-scattered population. In particular, Pluto’s(e, I) pair is located well inside the simulated hot populatione � I distribution rather than inside the distribution comingfrom the resonance-sweeping process. Although evolutionin resonance lock can produce some plutinos with fairlyhigh inclinations (Gomes, 2000) this process could notgenerate the right proportion of highly inclined objects.Also, stable scattered objects were formed by this Neptune’sclose encounters decoupling mechanism, although no orbitlike that of 2000 CR105 was obtained. Extra simulations ofthe orbital evolution of primordial Neptune-crossing objectswith perturbing migrating or nonmigrating planets mayeventually show the possibility of production of such anorbit. In all, the main simulations described in Section 2suggest that the present scattered population, high-inclina-tion CKBOs and high-inclination plutinos may have had acommon origin as primordial scattered objects originallycoming from an inner denser region of the original plane-tesimal disk around 25AU.

Other simulations have been undertaken to give furtherevidence for the reality of the mechanism that producesevaders. They were based on a massless particles model,where the migrating effect is artificially imposed on theplanets. Also, I used both MERCURY and SWIFT codes forthese simulations. The results from these massless modelsimulations basically confirm the results for the massivedisk model, with, however, smaller values in average for theperihelion distance and inclinations. The SWIFT runs alsoproduced evaders with similar distributions as for the (mass-

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less) MERCURY runs, confirming that this is not an artifactof MERCURY integrator.

Again for the simulations with a massive planetesimaldisk, the positions of the planets after 108 years are similarto the present ones, with Neptune ending between 28.2 and32.7 AU depending on the run. Two of the large planetes-imals attained hyperbolic orbits out of the Solar System inless than 1.1 � 107 years. A third one remained between the3:4 and 2:3 resonance with Neptune after 3 � 108 years andseemed to be in a stable configuration. The fourth one wasjust beyond the 1:2 resonance with Neptune with a smalleccentricity at 108 years. The idea of introducing the largeplanetesimals, differently from Morbidelli and Valsechi(1997), was to help in creating evaders from the primordialscattered disk. Although some help in this sense wasachieved for the two cases in which the large planetesimallived longer, the conclusion is that they do not play anyessential role in producing objects for the hot population.On the other hand, these objects, whose primordial exis-tence is quite probable, would not jeopardize the final or-bital configuration, provided they were expelled from theSolar System at some not too late time.

A last word is now in order with respect to the planetaryinitial positions. I chose to start them with the classicalassumption by preserving the planets’ relative positions. Inone case, however, I started Neptune as close to the Sun as13.5 AU. Maybe this region was not yet dense enough toform these large planets. On the other hand, the results hereobtained are not in principle incompatible with Neptune andUranus being formed at the Jovian/Saturnian region (Thom-mes et al., 1999). In this case, after Neptune and Uranus hadescaped from all crossing orbits they would more gentlyevolve outward with decreasing eccentricities. This finalplanetary evolution would have the same characteristics asin the simulations here presented and would most probablyproduce the high-inclination population.

Acknowledgments

I thank CNPq and FAPERJ for supporting grants. I amindebted to R. Vieira Martins for allowing me to use hispersonal computers for numerical integration. Many thanksto the referees H. Levison and R. Malhotra for their carefuland helpful reviews.

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