the evidence of an early stellar encounter in edgeworth–kuiper belt
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Icarus 177 (2005) 246–255www.elsevier.com/locate/icaru
The evidence of an early stellar encounter in Edgeworth–Kuiper be
Hiroshi Kobayashia,∗, Shigeru Idab, Hidekazu Tanakab
a Graduate School of Environmental Studies, Nagoya University, E-125 Furo-cho, Chikusa-ku, Nagoya 464-8602, Japanb Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
Received 26 March 2004; revised 24 February 2005
Available online 20 April 2005
Abstract
We investigate the influence of a stellar fly-by encounter on the Edgeworth–Kuiper belt objects through numerical orbital calcin order to explain both mass depletion and high orbital inclinations of the classical Edgeworth–Kuiper belt (CEKB) objects, whsemimajor axis of 42–48 AU and perihelia beyond 35 AU. The observationally inferred total mass of the CEKB is∼1/10 Earth masseswhich is only∼0.02 of that extrapolated from the minimum-mass solar nebula model. The CEKB consists of bimodal populatiopopulation” with inclinationsi 0.2–0.6 radians and “cold population” withi 0.1. The observationally suggested difference in sizecolor of objects between the two populations may imply different origins of the two populations. We find that both the depletionmaterials in the CEKB and the formation of the hot population are accounted for by a single close stellar encounter with pericenteof 80–100 AU and inclination relative to the initial protoplanetary disk50–70. Such a stellar encounter highly pumps up eccentricof most objects in the CEKB and then their perihelia migrate within 35 AU. These objects would be removed by Neptune’s pertuafter Neptune is formed at or migrates to the current position (30 AU). Less than 10% of the original objects remain in stable orsmall eccentricities and perihelion distances larger than 35 AU, in the CEKB, which is consistent with the observation. We findi ofthe remaining objects are as large as that of the observed hot population. The only problem is how to stop Neptune’s migration a∼30 AU,which is addressed in a separate paper. The depletion by the stellar encounter extends deeply into∼30–35 AU, which provides the basisthe formation model for the cold population through Neptune’s outward migration by Levison and Morbidelli (2003, Nature, 426, 41The combination of our model with Levison and Morbidelli’s model could consistently explain the mass depletion, truncation atbimodal distribution ini, and differences in size and color between the hot and the cold populations in the CEKB. 2005 Elsevier Inc. All rights reserved.
Keywords: Kuiper belt objects; Orbits; Origin; Solar System
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1. Introduction
More than 900 objects are observed in the EdgewoKuiper belt (EKB). Edgeworth–Kuiper belt objects (EKBOs) are divided into three dynamical classes: resonantBOs, classical EKBOs (CEKBOs), and scattered EKB(Jewitt and Luu, 2000). Fig. 1 shows the distribution of eccentricitiese and inclinationsi of EKBOs as a function osemimajor axesa. Inclinations are scaled in units of radiawith respect to the ecliptic plane in this paper. The filcircles show EKBOs that have been observed over m
* Corresponding author. Fax: +81 52 789 3013.E-mail address: [email protected](H. Kobayashi).
0019-1035/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2005.02.017
ple oppositions as of March, 2004. The loci of orbits wperihelion distancesq of 30 and 35 AU are shown by thdotted and dashed lines, respectively. Most resonantBOs are in the 3:2 mean motion resonance (39.5 AU) withNeptune’s orbit, which are called “Plutinos.” The scatteEKBOs haveq of 30–35 AU and are considered to habeen strongly scattered by Neptune at30 AU (Duncan etal., 1995). The CEKBOs havea 42 AU andq 35 AU.
The CEKBOs are dynamically excited, having highe andi, but present planetary perturbations and resonances cbe responsible for the excitation (e.g.,Malhotra et al., 2000
Kuchner et al., 2002). The orbits should have been excitedby some process during formation stage of the Solar System.The inclination distribution further shows a puzzling feature.
early
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The evidence of an
Fig. 1. Eccentricities (top) and inclinations (bottom) of all EKBOs as a fution of semimajor axes. The loci of orbits withq at 30 AU (dot lines) and35 AU (dashed lines) are also plotted in top panels.
Fig. 2. The eccentricity—inclination distribution for CEKBOs.
Fig. 2 shows the distribution ofe andi of CEKBOs, whichhavea between 42 and 48 AU andq larger than 35 AU. Theiri are distributed up to 0.6 while theire are smaller than 0.2because of the constraint onq by crossing with Neptune’orbit. Higher-inclination CEKBOs spend less time nearecliptic plane.Brown (2001)proposed that this bias can bcorrected for the subset of low-eccentricity objects (CEBOs) discovered near the ecliptic plane, by multiplyingnumber of the objects by the factor of sini. In Fig. 3, weshow the corrected distribution of CEKBOs obtained frthe subset of objects discovered with the latitudes up
degrees (near ecliptic plane). The inclination distribution isdivided into two inclination-populations, i.e., the cold popu-lation (i ∼ e) and the hot population (i e) (Brown, 2001).stellar encounter 247
Fig. 3. The number of CEKBOs in each 0.025 radian corrected thromultiplying the factor sini.
As a result of the correction, the number of objects in thepopulation is of the same order of that in the cold populat
In the minimum-mass solar nebula model, solid materof about 10 Earth-masses are expected to have existed i50 AU. But, the total mass of CEKBOs is observationaestimated as∼0.1M⊕ by Trujillo et al. (2001), indicatingthat CEKBOs may have been severely depleted. Furtmore, no CEKBOs have been detected beyond 50 AUcept for Sedna, which may be an object of another dynamclass), though current observations with largest teleschave ability to observe them. These depletion could betributed to the same process that excited the orbits ofleftover CEKBOs.
The observed size distribution of CEKBOs is given bypower-law asdN ∝ r−p dr , wheredN is the number of objects with radius betweenr andr + dr , andp is estimatedto be 3.5–4 (e.g.,Jewitt and Luu, 2000). In this power-lawsize distribution, the largest objects (radii of∼100 km) dom-inate the total mass of CEKBOs. In order to depletetotal mass, large objects have to be removed. One posity is collisional disruption through their mutual collisionHowever, collisional timescale between the objects withradius of 100 km is estimated to be 1012 years, because otheir small number density(Durda and Stern, 2000). Thus,destructive collisions cannot be responsible for the strdepletion in EKB.
Several authors addressed the excitation of CEKBPetit et al. (1999)suggested that hypothetical Earth-size ptoplanets scattered by Neptune can pump upe andi of CEK-BOs.Thommes et al. (2002)proposed that if Neptune is bornear Jupiter’s orbit and is scattered outward by Jupitering its gas accretion stage, Neptune can pump upe andi ofCEKBOs until its orbital eccentricity is damped by dynam
cal friction from the CEKBOs. These two models, however,account for only the cold population (i ∼ e). Nagasawa andIda (2000)andNagasawa et al. (2002)showed that secular
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248 H. Kobayashi et al. / I
resonances may sweep the CEKB region during the nedepletion to pump upi highly. This model accounts for thhot population (i e). It seems difficult to explain the bimodal inclination distribution of CEKBOs with one procesFurthermore, none of these models accounts for signifidepletion in the CEKB and its truncation at∼50 AU.
The most promising model for formation of Plutinossweeping 3:2 mean motion resonance due to an outwarbital migration of Neptune (e.g.,Malhotra, 1993 1995; Ida etal., 2000b). The outward migration by∼7 AU can pump upe up to 0.3, which explains large eccentricities of PlutinThis outward migration affects CEKBOs, too.
Levison and Morbidelli (2003)investigated transportation of objects from inner regions to the CEKB regiontemporal trapping to the 2:1 mean motion resonanceNeptune’s outward migration, assuming hypothetical dtruncation at30 AU. The migrating 2:1 resonance tranports objects that are originally ata 30 AU, exciting e
andi moderately. Since the migration that is caused by stering of planetesimals by Neptune, the migration stopthe planetesimal disk edge ata 30 AU (Gomes et al.2004). As fluctuation of the migration occasionally releasobjects, objects are distributed in the CEKB. The 2:1 renance is at48 AU at present, so that the objects aretransported beyond 50 AU. Thus, their model can expthe cold population, depletion in the CEKB compared wthe minimum-mass solar nebula model, truncation ofCEKB at∼50 AU, and termination of Neptune’s migratioat 30 AU.
Gomes (2003)suggested that the migration of Neptucan also form the hot population. However, the hot polation in the author’s result hase 0.2 in a = 44–48 AU(e 0.1 in a = 42–44 AU), which would rather belong tthe scattered EKBOs and may not be consistent withservation. If this model is combined with the Levison aMorbidelli model, the hot and cold populations come frothe same region. It may be inconsistent with the observaally suggested differences in size(Levison and Stern, 2001and color(Trujillo and Brown, 2002)between the hot ancold populations. However, note that the color differencould also be attributed to redding by cosmic ray andlisional resurfacing (e.g.,Gil-Hutton, 2002), so that it doesnot necessarily imply different original locations. We wdiscuss this issue in detail in Section3.
AlthoughLevison and Morbidelli (2003)did not addressin detail the formation of the hot population and the inittruncation of the protoplanetary disk near 30 AU that thmodel relies on, we will show that a stellar fly-by encounwith a protoplanetary system prior to Neptune’s migrathas a potential to explain both of them. It would also accofor the size and color differences. In general, stars are bomembers of an open cluster. Stellar clusters would evapoon timescales longer than 108 years(Kroupa, 1995, 1998).
This evaporation is caused by gravitational interactions be-tween stars, so that many stars experience close encountewith other stars during the evaporation. The probabilityP of177 (2005) 246–255
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a stellar encounter with pericenter distanceD in an opencluster with number densitynstar and velocity dispersionvstar is
P ∼ nstarSvstarteva 0.5
(nstar
103 pc−3
)(vstar
2 kms−1
)
(1)×(
D
100 AU
)2(teva
3× 108 years
),
where S and teva are the cross section of a stellar ecounter and the evaporation timescale of an open clurespectively. For a dense open cluster, like TrapeziumOrion, nstar is 104 stars pc−3 (Hillenbrand and Hartmann1998). The velocity dispersion is a few km s−1 (Binney andTremaine, 1987). If the Sun was born in a relatively dencluster, it may be likely that the Sun experienced a steencounter withD ∼ 100 AU before the cluster evaporatioPlanetary systems would be affected by stellar encounduring their formation stage, in particular, in the outergions of the protoplanetary disk (e.g.,Ida et al., 2000aKobayashi and Ida, 2001). If a parabolic encounter ofsolar-mass star with pericenter distance∼80–100 AU is con-sidered,e of most objects are pumped so highly in the CEKthat their orbits are destabilized by Neptune after it migrato 30 AU. The depletion by the stellar encounter extedeeply into∼30 AU. We will show thati of the remain-ing objects (10% of original bodies) are as large as thoof the hot population. Thus, a stellar encounter can fothe hot population and truncate the disk ata ∼ 30 AU asLevison and Morbidelli (2003)assumed. Moreover, the hpopulation comes from beyond 40 AU in our model, it woualso be consistent with the size and color difference betwthe hot and cold populations if self-gravitational instabilmodel for planetesimals in a laminar nebula(Safronov, 1969;Goldreich and Ward, 1973)is employed. The only diffi-culty is how to terminate Neptune’s migration at∼30 AU. Inour model, the CEKBOs are depleted by scattering by Ntune, which leads to further migration to the regions bey30 AU (Gomes et al., 2004). Orbital decay of highe objectscaused by gas drag could solve this problem, which wiladdressed in a separate paper.
In Section2, through numerical integration of a stellar ecounter with a planetesimal disk, we investigate the effeca stellar encounter on a planetesimal disk to evaluate thpletion of objects in the CEKB and high orbital inclinatioof remaining objects. We summarize and discuss our rein Section3.
2. Simulations of a stellar encounter with aplanetesimal disk
2.1. Model and basic equation
rs We model a planetesimal disk as a system of non-self-gravitating, collisionless particles that initially have coplanar
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circular orbits around a primary (host) star. This assumpis valid because their two-body relaxation time and mcollision time are much longer than the effective encountime in a stellar encounter that is comparable to Kepleperiod at pericenter distanceD of the encounter (for example, it is∼103 years forD ∼ 100 AU). We also neglechydrodynamical gas drag, because the damping time duthe drag [107–108 (r/100 km) years at 40 AU, wherer isa physical radius of an object] is much longer than the eftive encounter time∼103 years. Note that the gas drag coube important for orbital evolution on longer timescales athe stellar passing if eccentricities are pumped up to0.3 ata = 40–50 AU. We will investigate the gas drag effect aftestellar encounter in a subsequent paper. The equation otion of a planetesimal in the heliocentric frame (the frawith the primary star at the origin) is
(2)d2rdt2
= −GM1
| r |3 r + GM2
| R − r |3 (R − r) − GM2
| R |3 R,
whereM1 andM2 are masses of the primary and the passstars,r andR are position vectors of the planetesimal athe passing star. The first term in the r.h.s. is the gravitatiforce by the primary star, and the second and third termsdirect and indirect perturbing forces of the passing star.
We scale length with pericenter distanceD of the stel-lar encounter, mass with the primary star massM1, and timewith Ω−1
kep whereΩkep(=√
GM1/D3) is the Keplerian angular velocity ata = D. Equation(2) is then transformed to
(3)d2rdt 2
= − r| r |3 − M∗(r − R)
| r − R |3 − M∗R| R |3 ,
whereM∗ = M2/M1, r = r/D, R = R/D, and t = Ωkept .the orbits of the passing star have four parameters of encters, i.e., inclination (i∗) relative to the initial planetesimadisk, eccentricity (e∗), and argument of perihelion (ω∗) ofthe orbit of the passing star, and the scaled passing star(M∗). The parameters of the stellar orbit are explainedFig. 4. We calculate changes ine andi of planetesimals, integrating Eq.(3) for various values of the parameters.
2.2. Numerical results of a stellar encounter
Regarding the method of numerical integration of a slar encounter, we followIda et al. (2000a)and Kobayashiand Ida (2001). We integrate orbits of 10,000 particleseach run. The particles are distributed with surface numdensityns ∝ a−3/2. The orbits of planetesimals are initialcircular and coplanar. We integrated Eq.(3), using a fourthorder predictor–corrector scheme.
Fig. 5 shows final orbital elements of particles as asult of each stellar encounter with (a)i∗ = 30, (b) 45,(c) 60, (d) 90, and (e) 150, in the case ofω∗ = 90,
e∗ = 1, andM∗ = 1. Prograde encounters (0 i∗ < 90 )cause steep increase ine and i of the particles in outer re-gion of a/D 0.3. The pumped-upi are distributed up tostellar encounter 249
-
-
s
Fig. 4. Encounter configuration in the frame centered at the primarywith massM1. The orbit of the passing star with massM2 is characterizedby pericenter distanceD, eccentricitye∗, inclination i∗ and argument ofperihelionω∗. If length and mass are scaled byD andM1, the encounterparameter areM∗ (= M2/M1), e∗, i∗, andω∗.
Fig. 5. Orbital eccentricitiese and inclinationsi of particles pumped-up bya passing star, as a function of scaled semimajor axisa/D, in the case ofω∗ = 90, e∗ = 1, andM∗ = 1. Orbital inclination of the passing star(a) 30, (b) 45, (c) 60, (d) 90, and (e) 150.
∼ 1 radian. In the outer regions, pumped-upe and i de-pend not only on the initial radial positiona but also on theazimuthal position of planetesimals, because encountelocity is comparable to Keplerian velocity there. Thereinclinations are distributed widely at each radial positiMost objects havee larger than 0.2–0.3. Such orbits haperihelia so close to Neptune’s orbit that they will bemoved by Neptune’s perturbation, resulting in significdepletion of CEKBOs. ForD ∼ 100 AU, inclinations of ob-jects witha between 42 and 48 AU are as large as thosthe observed hot population. Since the perturbation duretrograde encounters (90 i∗ 180) is relatively weak,
D 60 AU is required for the inclinations to be comparableto the observed level of the hot population. In the retrogradeencounters, however, most of objects witha between 42 and
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250 H. Kobayashi et al. / I
48 AU havee < 0.2, not resulting in significant reduction othe amount of CEKBOs.
Trujillo et al. (2001)estimated the number of CEKBOwith r 100 km as∼4×104, wherer are radii of CEKBOs.Assuming that the size distribution is in proportion tor−4,the total mass of CEKBOs is evaluated as∼0.1M⊕. In theminimum-mass solar nebula model, the total mass of smaterial between 42 and 48 AU is about 6M⊕. Thus, theCEKBOs are expected to have been depleted to∼1–2% ofthe original amount.
Fig. 6shows the retention ratio (Ne/N0) and the fractionof the hot population (Ne,h/Ne) as a function of (a)D, (b) i∗,and (c)ω∗, whereN0 and Ne are the numbers of objecwith 42 AU < a < 48 AU andq > 35 AU before a stellaencounter and after the encounter, andNe,h is the number ofthe hot population objects (i > 0.2) in Ne. The observationally inferred estimate suggests thatNe/N0 ∼ 0.02. Takinginto account uncertainty in the estimation, we look for ecounters withNe/N0 < 0.1. Note that the parameter rangwhere bimodal distribution is formed (Ne,h/Ne 0.5) arelimited. Even in such ranges, the obtained bimodal distrtions are quite different from the observed distribution.example, the cold population (i < 0.2) in these cases is distributed mostly ini 0.1–0.2, which is inconsistent with thobserved distribution (seeFig. 2). Hence, we assume that thcold population is formed by other process (e.g.,Levison andMorbidelli, 2003). Thus,Ne,h/Ne > 0.5 andNe/N0 < 0.1are required.
Both conditions are satisfied by encounters withD =80–100 AU in the case ofi∗ = 60 andω∗ = 90. As seenin Fig. 6(a), close encounters withD 100 AU satisfyNe/N0 0.1. ForD 130 AU, excitation ofe is so weak(e < 0.2) that depletion does not occur. In addition, becathe objects beyond 50 AU are scattered to the CEKB regNe/N0 actually takes values>1. In the case ofD 80 AU,stellar encounters excite objects so strongly that planeaccretion would be inhibited even ina 20 AU. Kobayashiand Ida (2001)derived the boundary radius (aplanet) outsidewhich planet formation is inhibited by disruptive collisiowith high relative velocities as
(4)aplanet 20
(d
200 km
)1/2(F
2
)1/4(D
80 AU
)5/4
AU,
whereF = (M∗ + 1)/M2∗ (F = 2 atM∗ = 1). If a close en-counter withD < 80 AU occurs in theM∗ = 1 case, Neptunecannot form even if it was formed at20 AU before itsoutward migration. In addition, the transportation procproposed byLevison and Morbidelli (2003)may not workafter an encounter withD 80 AU, because pumped-upeat a 30 AU are too high for the objects to be capturedsweeping mean motion resonance due to Neptune’s mtion after the stellar encounter. After encounters withD =80–100 AU,Ne,h/Ne 1. Hence, in the case ofi∗ = 60
andω∗ = 90 , the encounter withD = 80–100 AU results inthe depletion and the formation of CEKB without preventingNeptune formation.177 (2005) 246–255
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Fig. 6. The residual ratioNe/N0 and the fraction of the hot populatioNe,h/Ne. Panel (a) showsNe/N0 (filled circles) andNe,h/Ne (open cir-cles) as a function ofD after the stellar encounter withi∗ = 60 andω∗ = 90. Panels (b) and (c) show the dependence oni∗ andω∗, respec-tively. The ratios are not plotted fori∗ < 40 in panel (b) and forω∗ < 30in panel (c) because ofNe = 0.
The dependence oni∗ andω∗ in the case ofD = 100 AUare shown inFig. 6(b) and 6(c), respectively. For smai∗ < 30, depletion is too strong. In the case ofi∗ 45,though most objects havei larger than 0.2,i is pumped uptoo highly (∼1), which is inconsistent with the observehot population. Fori∗ 70, on the other hand, depletiois too weak. After stellar encounters withω∗ 30, no ob-jects remain in the CEKB region in our simulation. In tcase ofω∗ 30, on the other hand, encounters satisfy bNe/N0 0.1 andNe,h/Ne > 0.5. Hence, the encounter witi∗ 50–70, ω∗ 30–90, andD = 80–100 AU resultsin the requited depletion and excitation ofi as large as tha
of the observed hot population.Fig. 7 shows thate and i of planetesimals in the CEKBregion (42 AU< a < 48 AU andq > 35 AU) after stel-
early
isthe
hthe
EKB
),rib-
of-
rved
nce
f
r
The evidence of an
lar encounters withi∗ = 60 and ω∗ = 90. In the caseof D 120 AU, the excitation by a stellar encounterstrong in the region, because this region corresponds toouter region ofa/D 0.3 in Fig. 5. Inclinations increaseup to 0.6 for 80 AU< D < 120 AU. Close encounters wit80–120 AU can form the hot population. Furthermore,
Fig. 7. Filled circles show distributions of eccentricitiese and inclinationsi of planetesimals pumped-up by a passing star in the region of C(a = 42–48 AU). The parameters arei∗ = 60, ω∗ = 90, e∗ = 1, andM∗ = 1. We set pericenter distanceD of the passing star as 80 AU (a100 AU (b), 120 AU (c), and 140 AU (d) in each panel. Dots show distutions ofe andi of observed CEKBOs.
Fig. 8. Filled circles show distributions of eccentricitiese and inclinationsiof planetesimals pumped-up by the stellar encounter withD = 80 AU, as a
function of semimajor axisa. The other encounter parameters arei∗ = 60,ω∗ = 90, e∗ = 1, andM∗ = 1. Dots show distributions ofe and i of ob-served CEKBOs.stellar encounter 251
encounters with 80–100 AU also explain the depletionCEKBOs (seeFig. 6). In Fig. 8, we also compare our result (after the stellar encounter withD = 80 AU, i∗ = 60,andω∗ = 90) with the observed CEKBOs ina–e anda–i plane. The excited objects with 42 AU< a < 48 AU andq < 35 AU have inclinations as large as those of the obseobjects in the hot population.
Figs. 9 and 10show the retention ratio (Ne/N0) and thefraction of the hot population in CEKBOs (Ne,h/Ne) in caseof M∗ = 0.5 and 0.3. We denote the encounter distaat which the ratioNe/N0 = 0.1 by D1 and that at whichNe,h/Ne = 0.5 by D2. Fig. 9 (Fig. 10) shows thatD1 is70 AU (40 AU) andD2 is 82 AU (60 AU) in the case oi∗ = 60, ω∗ = 90, andM∗ = 0.5 (0.3).D is required to besmaller thanD1 andD2 in order to satisfyNe/N0 < 0.1 andNe,h/Ne > 0.5 (seeFigs. 6a, 9a, and 10a). However, sinceaplanet is needed to be larger 20 AU in Eq.(4), the encounte
Fig. 9. Same one asFig. 6, except forM∗ = 0.5.
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252 H. Kobayashi et al. / I
Fig. 10. Same one asFig. 6, except forM∗ = 0.3.
distance is further restricted to
(5)D D3 = 80
(F
2
)1/5
AU.
Fig. 11showsD1, D2, andD3 as functions ofM∗ in the caseof i∗ = 60 andω∗ = 90. ForD1 andD2, we plot the resultsfor M∗ = 0.3, 0.5, and 1. In the gray region, all of the aboconstraints onD are satisfied and the stellar encounterexplain the strong depletion of CEKBOs and the formatof the hot population without preventing Neptune formatiFig. 11shows that an encounter withM∗ 0.4 cannot ex-plain observed CEKBOs. We also examined the values oD1andD2 for other values ofi∗. In the range of 40 < i∗ < 70,the gray region is almost same as that inFig. 11. Espe-cially, in i∗ = 50–70, the encounter forms the hot poplation with moderate inclinations. On the other hand, tho
encounters withi∗ = 40 –50 satisfy the conditions, mostperturbed objects have inclinations smaller than 0.3. Wheni∗ is larger than 70 degrees,D1 is much smaller than that177 (2005) 246–255
Fig. 11. The encounter distancesD1 (solid line) andD2 (dashed line) whichis at Ne/N0 = 0.1 andNe,h/Ne = 0.5, respectively, andD3 (dotted line)which is ataplanet= 20 AU in Eq. (5).
in i∗ = 60. Then, no encounter satisfies bothD < D1 andD > D3. Furthermore, some encounters are too weak toalize Ne/N0 < 0.1 for anyD for i∗ > 70. In the case ofi∗ < 40, objects with high inclinations have high eccetricities and such objects do not belong to CEKBOs. Tdepletion ofω∗ is almost same as theM∗ = 1 case. Encounters withω∗ 30 deplete CEKBOs too strong. In the caof ω∗ 30, the gray region is almost same asFig. 11. As aresult, stellar encounters withM∗ 0.4, i∗ = 50–70, andω∗ 30 can deplete CEKB strongly and form the hot poulation without preventing Neptune formation. The probility of such an encounter is reduced by the factor∼1/10due to constraint oni∗ andω∗, if M∗ andD are in the grayregion ofFig. 11.
3. Summary and discussion
We investigated the depletion and formation of thepopulation in the classical Edgeworth–Kuiper belt by a paing stellar encounter. We numerically integrate orbits oparticulate disk that encounters a hypothetical star in aabolic orbit. Our results are summarized as follows:
(1) In general, a stellar encounter pumps up eccentricitie
and inclinationsi of objects strongly in the outer regioof a planetesimal disk. If an encounter with the stemass of a solar mass and the pericenter distanceD of80–100 AU is considered, most objects in CEKB aquire so highe that their perihelia (q) migrate to the in-side of 35 AU. After Neptune is formed at or migratesthe current position (30 AU), these highe objects will
be removed by Neptune’s perturbation. A small fractionof objects have smalle and henceq > 35 AU so thatthey remain stable in the CEKBOs. The remaining ob-
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The evidence of an
jects generally havei (e) as large as the observed hpopulation.
(2) A stellar encounter with the stellar mass of a solar mD of 80–100 AU,i∗ (inclination of the passing star reative to the initial planetesimal disk) 50–70, andω∗(argument of perihelion of the passing star)30 resultsin depletion of CEKBOs down to∼5% of the initialamount by Neptune’s perturbation and enough frac(1) of the hot population.
(3) A close stellar encounter dynamically excites planetmals and planetary formation is inhibited by disruptcollisions with high relative velocities between plaetesimals. A stellar encounter with stellar massM2 0.4M inhibits the formation of Neptune, when thencounter deplete CEKBOs strongly. Hence, a steencounter withM2 0.4M, D = 60–100 AU, i∗ =50–70, andω∗ 30 results in the depletion of CEKBOs and high fraction of the hot population withoinhibiting formation of Neptune.
In our model, the cold population is not formed bystellar encounter. As mentioned in the introduction,Levisonand Morbidelli (2003)showed that the cold population wi∼0.1M⊕ can be formed through the transportation frothe inner region by 2:1 mean-motion resonance sweecaused by the outward migration of Neptune. In their mothe truncation of the planetesimal disk ata 30 AU is as-sumed prior to Neptune’s migration. Although the depletof CEKBOs by the stellar encounter occurs after the migtion of Neptune, the dynamical excitation ata 30 AU isequivalent to the truncation prior to Neptune’s migratioSince eccentricities are strongly excited ata 30 AU, Nep-tune’s 2:1 mean-motion resonance cannot capture obwith high e at a 30 AU, while it can capture objects aa 30 AU where eccentricity excitation is relatively sma
Therefore, the combination of our model with the Leson and Morbidelli model may consistently explain the mdepletion, truncation at 50 AU, bimodal distribution ini inthe classical Edgeworth–Kuiper belt. We propose the folling scenario. (1) The stellar encounter withD 80–100 AUpumps upe and i of objects witha 30 AU. Most ofthe excited objects acquiree 0.2 and their perihelia migrate to35 AU. A small fraction (0.1) of objects re-mains in stable orbits to form the hot population of CEKThe stellar encounter would occur within the timescalecluster evaporation, which is of the order of108 years.(2) The 2:1 mean-motion resonance sweeping due tooutward migration of Neptune transports objects ata 30 AU to CEKB up to the present position of the resona(a 48 AU). They form the cold population of CEKBO(Levison and Morbidelli, 2003). The outward migration oNeptune would occur after gas accretion of Jupiter. Thit would occur att ∼ 107–108 years (e.g.,Mizuno, 1980;
Bodenheimer and Pollack, 1986; Ikoma et al., 2000). In thisscenario, Neptune is needed to migrate after the stellar encounter, because the stellar encounter destroys the cold popstellar encounter 253
ulation if Neptune migration occurs prior to the encoun(3) Most of objects having orbits crossing with Neptunorbit would be removed by Neptune at the current posi(30 AU) on a longer timescale (∼109 years), resulting in significant depletion in CEKB.
In the above scenario, the two populations of CEKBoriginate from different regions, which may account for tsize and color differences between the two populationsare suggested by observations(Levison and Stern, 2001Trujillo and Brown, 2002). The hot population is formed ithe region of 40–50 AU, while the cold one is ata 30 AU.The mass of observed CEKBOs is comparable to thaprimordial planetesimals formed through self-gravitatioinstability at 30–50 AU. The sizes of the primordial planetimals r are given by(Safronov, 1969; Goldreich and War1973)
(6)r ∼ 200
(Σ
Σmin
)(a
50 AU
)1/2
km,
whereΣ andΣmin are the surface density of solid materand that in the minimum-mass disk model, given byΣmin 0.1(a/50 AU)−3/2 gcm−2. If the dynamical excitation andepletion by the stellar passage occurs in early stage, petesimal accretion in CEKB are mostly prevented, becacollisions would be disruptive after the stellar encounteprimordial planetesimals formed through self-gravitatioinstability, the formation timescale is comparable to thathe formation of dust layer in the solar nebula, which isthe order of 106 years(Nakagawa et al., 1981). This wouldallow CEKBOs to have their observed size prior to the slar encounter. Since the size of primordial planetesimain proportion toa1/2 [Eq. (6)], objects in the hot populatio(r ∼ 200 km at 50 AU) should be slightly larger than thosethe cold population (r ∼ 150 km at 30 AU), which may beconsistent with the observation(Levison and Stern, 2001.If Σ ∝ a−1, which might be more consistent with observtions thanΣ ∝ a−1.5, the size of primordial planetesimalsin proportion toa. This makes the difference in size betwetwo populations much larger than that in the minimum-msolar nebula model (Σmin ∝ a−1.5).
The different formation site (different temperature) coalso be responsible for the difference in color the obsetion (Trujillo and Brown, 2002). In the minimum-mass solar nebula model, the temperature and the pressure insolar nebula are given by 40× (a/50 AU)−1/2 [K] and4 × 10−11(a/50 AU)−13/4 [bar], respectively. The differences in the temperature and the pressure between dent locations would result in different compositions betwethe hot and the cold populations. To estimate this effecshould be further investigated how large difference in coposition is caused by the difference in birth location.
The difference in color between the populations can abe explained by collisional resurfacing model (e.g.,Gil-
--
Hutton, 2002). In this model, the long-term irradiation bygalactic cosmic-ray results in loss of hydrogen and red-dens the surface. This “red surface” are peeled off through
carus
er-ted
loc-y of
es,ore
nhot
tionceite
nal
ionn
-del,the
ex-t bes inoreon-
ex-ed”
ring
4
re10–ing
EK-Wetuneforerateoc-
uchpite
ofis-theso-
ould
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e’she-ringAUsi-llar
thby
ffects
un-
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owur-eighityob-ith
as
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27,
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254 H. Kobayashi et al. / I
collisions with other objects. It may cause the color divsity. The collision rate between two objects is estimaasnσv ∼ nsσΩkep
√1+ (e/i)2, wheren, σ , v, andns are
the number density, the collisional cross section, the veity dispersion of objects, and the surface number densitobjects, respectively(Greenzweig and Lissauer, 1992). Asobjects in the hot population have high collisional velocitithe “red surface” would be peeled through collisions meffectively for the hot population. Sincee/i ∼ 1 in the coldpopulation ande/i 1 in the hot population, the collisiorate in the cold population is almost same as that in thepopulation. Hence this model reddens the cold popularather than the hot population. Even if the color differenwas not sufficiently caused by the difference in original sof the two population, it can be caused by the collisioevolution model.
The hot population can also be formed by the migratof Neptune(Gomes, 2003). However, their model results irelatively highe (e 0.1 for a = 42–44 AU ande 0.2 fora = 44–48 AU) in CEKB (a = 42–48 AU), which may be inconsistent with the observed distribution. In Gomes’ mothe hot population originates from the same region ascold population, although the difference in color can beplained by the collisional resurfacing, that in size may noeasily explained by the Gomes’ model. These correlationcolor and size have been observationally clarified yet. Mdetailed observations on size and color of the CEKBOs cstrain the origin of the hot population.
In our model, objects beyond 50 AU are dynamicallycited strongly. Some scattered EKBOs (so-called “extendscattered EKBOs) haveq 40 AU and largea. Recentlythese objects would form not through Neptune scattebut through stellar encounters (e.g.,Melita et al., 2005;Morbidelli and Levison, 2004; Kenyon and Bromley, 200).On the other hand, a stellar encounter withD ∼ 100 AU re-sults in objects with largeq that are more populous than aexpected from observations if the encounter occurs at100 Myr after the Oort cloud stared to form due to scatterof planetesimals by giant planets(Levison et al., 2004). Ifthe encounter occurs earlier, the distribution of scatteredBOs is consistent with that expected from observations.assume that the stellar encounter occurs prior to the Nepmigration. This means the stellar encounter occurs begas accretion of Jupiter, because Neptune would migdue to the accretion. Thus, the stellar encounter wouldcur before the Oort cloud formation by the scattering. Sa encounter is not so early because gas accretion of Juwould occur att 107.
Brown and Pan (2004)suggested that the directionthe total angular momentum vector of EKBOs is inconstent with that of the Solar System. They showed thatdifference is not explained by only perturbations and renances of planets. In our model, a stellar encounter w
cause the difference, because the effects of stellar encounters strongly depend on radial position in the planetesimaldisk. Thus, the stellar encounter model should consistently177 (2005) 246–255
r
account for orbital excitation, depletion of CEKBOs, the ogin of “extended” scattered EKBOs, and the inclined toangular momentum vector of EKBOs.
In our model, the depletion of CEKB relies on Neptunscattering of highe objects after Neptune migrates to tcurrent position (30 AU). If Neptune’s outward orbital migration is caused by planetesimal scattering, this cleawould cause further migration to the regions beyond 30(Gomes et al., 2004). A stellar encounter remove planetemals in the outer region of a planetesimal disk. After a steencounter withD ∼ 100 AU, however, planetesimals wia = 30–50 AU have large eccentricities but are boundSun. In the present paper, we have neglected gas drag efor orbital evolution of objects. Equation(1) shows that thepossibility for our Solar System to undergo stellar encoters withD ∼ 100 AU during nebula lifetime (∼107 years),which is much shorter than cluster evaporation time, is10% for the dense clusters withn = 103–104 pc−3. Theprobability is not negligible. Since gas density is very lat30 AU, the gas drag effects are generally negligible ding disk gas lifetime∼107 years. However, orbital decay duto the gas drag cannot be neglected for objects with horbital eccentricity, because of their high relative velocto the gas motion. The orbital decay would remove thejects highly excited by a stellar encounter. Most objects wa = 30–50 AU migrate to the inside of 30 AU due to gdrag. Neptune may stop at the current location (a 30 AU)in such a planetesimal disk, because Neptune scatterinobjects hardly occurs ata 30 AU. On the other hand, objects corresponding to the hot population having highi andlow e would not affected by the gas drag because they spmost of time in the regions over disk scale height wheredensity is extremely small. In the subsequent paper, wefurther investigate the orbital evolution of objects in higheccentric and/or inclined orbits due to the gas drag.
Acknowledgments
We thank H. Levison and A. Morbidelli for helpful comments and M. Melita for the valuable comments. We athank K. Nakazawa for useful scientific discussions.
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