ON THE ORIGIN OF THE HIGH-PERIHELION SCATTERED DISK: THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES

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1 ON THE ORIGIN OF THE HIGH-PERIHELION SCATTERED DISK: THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES RODNEY S. GOMES 1, TABARE GALLARDO 2, JULIO A. FERNA NDEZ 2 AND ADRIA N BRUNINI 3 1 GEA/OV/UFRJ and ON/MCT Ladeira do Pedro Antoˆnio, 43, Rio de Janeiro, RJ, Brazil, rodney@ov.ufrj.br 2 Departamento de Astronomıá, Facultad de Ciencias, Igua 4225, Montevideo, Uruguay 3 Facultad de Ciencias Astrono micas y Geofıśicas, Universidad de La Plata, 1900 La Plata, Argentina, and Instituto Astrofıśico de La Plata, CONICET (Received: 8 June 2005; revised: 5 October 2005; accepted: 8 October 2005) Abstract. We study the transfer process from the scattered disk (SD) to the high-perihelion scattered disk (HPSD) (defined as the population with perihelion distances q > 40 AU and semimajor axes a > 50 AU) by means of two different models. One model (Model 1) assumes that SD objects (SDOs) were formed closer to the Sun and driven outwards by resonant coupling with the accreting Neptune during the stage of outward migration (Gomes 2003b, Earth, Moon, Planets 92, ). The other model (Model 2) considers the observed population of SDOs plus clones that try to compensate for observational discovery bias (Ferna ndez et al. 2004, Icarus, in press). We find that the Kozai mechanism (coupling between the argument of perihelion, eccentricity, and inclination), associated with a mean motion resonance (MMR), is the main responsible for raising both the perihelion distance and the inclination of SDOs. The highest perihelion distance for a body of our samples was found to be q ¼ 69:2 AU. This shows that bodies can be temporarily detached from the planetary region by dynamical interactions with the planets. This phenomenon is temporary since the same coupling of Kozai with a MMR will at some point bring the bodies back to states of lower-q values. However, the dynamical time scale in high-q states may be very long, up to several Gyr. For Model 1, about 10% of the bodies driven away by Neptune get trapped into the HPSD when the resonant coupling Kozai-MMR is disrupted by Neptune s migration. Therefore, Model 1 also supplies a fossil HPSD, whose bodies remain in non-resonant orbits and thus stable for the age of the solar system, in addition to the HPSD formed by temporary captures of SDOs after the giant planets reached their current orbits. We find that about 12 15% of the surviving bodies of our samples are incorporated into the HPSD after about 4 5 Gyr, and that a large fraction of the captures occur for up to the 1:8 MMR (a 120 AU), although we record captures up to the 1:24 MMR (a 260 AU). Because of the Kozai mechanism, HPSD objects have on average inclinations about 25 50, which are higher than those of the classical Edgeworth Kuiper (EK) belt or the SD. Our results suggest that Sedna belongs to a dynamically distinct population from the HPSD, possibly being a member of the inner core of the Oort cloud. As regards to 2000 CR 105, it is marginally within the region occupied by HPSD objects in the parametric planes ðq; aþ and ða; iþ, so it is not ruled out that it might be a member of the HPSD, though it might as well belong to the inner core. Key words: Edgeworth Kuiper belt, scattered disk, Kozai, comets: dynamics Celestial Mechanics and Dynamical Astronomy (2005) 91: Ó Springer 2005

2 110 RODNEY GOMES ET AL. 1. Introduction The growing number of trans-neptunian objects (TNOs) discovered since 1992 has uncovered a very complex dynamical structure. The first discovered TNOs were in non-resonant, low-eccentricity, low-inclination orbits, plus a few others in mean motion resonances (MMR) with Neptune, in particular the 2:3. The objects in this resonance have higher eccentricities and inclinations and in some cases approach or cross Neptune s orbit. Even though TNOs are in dynamically stable niches, there is nevertheless a slow diffusion process going on that generates transient populations like the Centaurs whose orbits lay in the region of the Jovian planets. As well as there are Centaurs scattered inwards, it was also expected to find bodies scattered outwards on very eccentric orbits and with perihelia beyond Neptune s orbit. The discovery of 1996 TL 66 (Luu et al., 1997) observationally confirmed the existence of such a population. This new class of bodies were designated Scattered Disk Objects (SDOs), and they are usually defined as those with perihelion distances q > 30 AU and semimajor axes a > 50 AU. From numerical integrations over 4 billion year, Duncan and Levison (1997) were able to reproduce such a scattered disk from TNOs strongly perturbed by close encounters with Neptune. The sample of discovered SDOs has raised to nearly 70 objects (end of 2002). The inventory of the trans-neptunian region comprises now several classes of objects. Jewitt et al. (1998) distinguish the following groups: (1) The classical belt composed of objects in non-resonant orbits with semimajor axes in the range 42 < a < 48 AU in low-inclination and low-eccentricity orbits; (2) Objects in MMR with Neptune, like the Plutinos in the 2:3 resonance (the largest resonant group), 1:2 and others. Resonant objects have in general large eccentricities and inclinations and some of them approach or even cross Neptune s orbit. Yet, the resonance prevents them from having close encounters with Neptune. (3) The Scattered Disk (SD) as described above. As we will see below, there is a fourth group that has been called the Extended Scattered Disk (ESD) whose dynamical characteristics will be addressed in this paper. Trujillo et al. (2001) have estimated that the number of SDOs with radii greater than 50 km is about bodies and the total mass of about 0.05 M, if a differential power-law size distribution of exponent q ¼ 4 is assumed. From an independent sky survey, Larsen et al. (2001) arrived at a similar figure for the SDO population. If we now assume that the differential size distribution keeps the same exponent q ¼ 4 down to a typical comet radius R ¼ 1 km, the total population of SDOs is estimated to be

3 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 111 NðR > 1kmÞ¼810 9 ð1þ Therefore, the SDO population turns out to be quite substantial, comparable in size to that of the classical belt. It may be large enough to keep the current population of JF comets in steady state (Trujillo et al., 2001). Since SDOs can diffuse either to the planetary region or to large heliocentric distances, it may also be a potential source of Oort cloud comets. The latter problem has recently been addressed by Fernández et al. (2003, 2004). The establishment that the distant objects 2000 CR 105 and 1995 TL 8 have their perihelion distances beyond the dynamical domains of the SD (Gladman et al. 2002), where TNOs interact with Neptune via gravitational encounters, have furnished evidence about the existence of a new population of objects, belonging to an ESD. Its total population is hard to estimate because of the difficulty to detect and identify by orbital tracking objects with so large perihelion distances. However, the detection of few objects implies that there must be a large population of TNOs in the ESD. Three recently discovered objects, that seem to have their perihelia beyond 40 AU, reinforces these evidences. Possible dynamical origins of this ESD, including gravitational perturbations by an as yet undiscovered tenth planet beyond Pluto (Brunini and Melita, 2002), long term diffusive chaos, perturbations by primordial embryos or passing stars, were also discussed by Gladman et al. (2002). In this paper, we investigate the dynamical processes that can raise a SDO perihelion due solely to the major planets perturbations. This is done through numerical integrations according to two different models. For the first one, we consider the migration scenario where planets are started in compact orbits and planetesimals that interact with the planets are initially distributed in a disk (Gomes, 2003a, b). This model is described in Section 2.1. Section 2.2 describes the second model for which we consider the major planets at their present positions and test particles coming from the sample of discovered SDOs plus clones. We plan to compare both models and analyze their similarities and differences. The objects from both models that attain perihelion distances above 40 AU and semimajor axes above 50 AU will hereafter be named High-Perihelion Scattered Disk Objects (HPSDOs), belonging to a High-Perihelion Scattered Disk (HPSD), thus avoiding the nomenclature ESD, which may be used for another probable population of TNOs, for which there is surely one and maybe two representatives. Section 3 describes the orbital distribution of the HPSDOs as coming from both models. Section 4 investigates the dynamical processes that produce an object belonging to the HPSD. Section 5 analyzes the transfer efficiency to the HPSD and finally some conclusions are drawn in Section 6.

4 112 RODNEY GOMES ET AL. 2. The Computed Samples We have analyzed the dynamical evolution of two different samples of SDOs whose fundamental difference lies in the way that the initial conditions are generated. The details are as follows: 2.1. MODEL 1: THE MIGRATION MODEL This model is based on a numerical integration of the four major planets on initial compact orbits and 10 4 planetesimals which perturb the planets but not themselves. The initial semimajor axes for the planets are 5.65, 8.2, 11.5 and 13.8 AU, with zero eccentricities and near zero inclinations. The particles are distributed as r 1 in a disk extending from 14 to 26 AU with zero eccentricities and near zero inclinations and total mass equal to 75 M. For this model we used the MERCURY package (Chambers, 1999) for the integrations. Due to close encounter interactions with the planetesimals the planets migrate (Fernández and Ip, 1984), with Neptune ending at around 31 AU. 1 The migrating Neptune traps external planetesimals in MMR driving them outward along with Neptune, in agreement with the resonance sweeping mechanism devised by Malhotra (1993, 1995). At 350 million years, the surviving planetesimals (170) are cloned so as to get a tenfold increase in the sample (170! 1700) and the integration was restarted for the Solar System age. In this cloning process the total mass of the planetesimals is not changed. Note that at the cloning time, the planetary migration has virtually stopped so the sample of 1700 test bodies essentially corresponds to a solar system that already attained its current configuration. Although in the end we obtain planetesimals distributed in the classical EK belt, resonant population and SD, we will concentrate here only on this last population. For compatibility with Model 2, we normalize the results on semimajor axes and perihelia with respect to Neptune s present position at 30 AU MODEL 2 : THE OBSERVED POPULATION MODEL Model 2 is based on the real SDOs population. We integrated numerically 76 massless real objects plus 399 clones generated in order to compensate for an 1 The use of a truncated disk is warranted by the results in Gomes et al. (2004) which show that a disk with an outer edge beyond 30 AU would unavoidably take Neptune to that edge. On the other hand, we are not here much concerned on the very right extension and mass of the planetesimal disk, but just on the effects of the migration process on the objects scattered by Neptune.

5 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 113 Figure 1. Perihelion distance versus semimajor axis for the last 1 billion years of integration for Model 1, at each 1 million years. Below the distribution of real objects is plotted with the symbol x. Figure 2. Perihelion distance versus semimajor axis of all the objects of Model 2 plotted every 50 Myr. The cluster of points detached to the right and to the top of the panel represents object 2000 CR 105.

6 114 RODNEY GOMES ET AL. observational bias in semimajor axis. The number of clones for each real SDO was chosen in order to approximately follow an empirical distribution law of semimajor axis given by a 2, that was found to fit the bias-corrected sample of real SDOs. Each clone has the same orbital elements from its progenitor but different initial mean longitude. The integration started with the present outer solar system from Jupiter to Neptune and ended after 5 Gyr of evolution. We used the EVORB integrator for this model, which is described in Fernández et al. (2002, 2004). More details and some results of this integration can be found in Fernández et al. (2003, 2004). 3. The High-Perihelion Disk Objects In Figure 1, we can see the distribution of the SD population of Model 1 for the last 1 Gyr of integration time (points taken at 1-Myr intervals). We can compare it with the same distribution but for the SDOs from Model 2 in Figure 2. We can see in the latter that the perihelion distance q ¼ 36 AU indicates a dynamical limit. Objects with lower q are subject to a strong diffusion in orbital energy due to gravitational encounters with Neptune, and may reach the Oort cloud or end up in hyperbolic orbits. This region corresponds to the SD population whose semimajor axes can extend to Oort cloud distances. This population was studied by Fernández et al. (2003, 2004). We find that for q > 40 AU (see, e.g. Figure 2) the changes in q are much slower, and in general are associated to MMR. The dynamical lifetime in this region is very long, in some cases comparable to the age of the solar system. The objects with perihelion distances in the range 36 < q < 40 AU belong to a transition zone between the diffusion lane to the Oort cloud or interstellar space and the HPSD. The highest perihelion distance for a body from our samples was found to be q ¼ 69:2 AU (see Figure 1), which shows that some SD bodies can be detached from the planetary region by solely dynamical interactions with the outer planets, without need to invoke cosmogonic causes or external perturbers, as for instance close stellar passages. Figures 3 and 4 show the distribution of the orbital inclination with semimajor axis for both models. Even though Model 2 appears to show a general trend in the mean inclination of HPSDOs to increase with increasing semimajor axis, this trend is not so remarkable in Model 1 that produces HPSDOs that extend to larger a. Figures 5 and 6 show the distribution of the orbital inclination with perihelion distance for Models 1 and 2. We note the coupling between the perihelion distance with the orbital inclination of the objects, once trapped in

7 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 115 Figure 3. Distribution of semimajor axes and inclinations for the last 1 billion years of integration for the HPSD of Model 1. Figure 4. Distribution of semimajor axes and inclinations for the HPSD of Model 2.

8 116 RODNEY GOMES ET AL. the region q > 40 AU. As it will be discussed in the following section, this behavior is due to the action of the Kozai mechanism. Object 1995 TL 8 (initial q ¼ 40:1 AU, a ¼ 52:6 AU, i ¼ 0:24 ) is an exception. It remains for all the integration time outside from any resonance, and its orbital parameters show very small variations. In particular, by contrast to the rest of HPSDOs, it has a very small inclination (the positions of 1995 TL 8 and its four clones are concentrated at the lower left corner of Figure 6, detached from the rest). 4. The Kozai mechanism and the MMR Several examples are given for the formation of the HPSD. The first example refers to Model 1, while there was still some planetary migration and it is associated to the creation of fossilized objects. This example concerns the 2:5 resonance with Neptune. Figure 7 shows the evolution of the orbital elements of the planetesimal during the formation process. Figures 8 and 9 plot level curves for the Kozai mechanism p in the parametric plane: argument of the perihelion (x) versus X ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 e 2 Þ, considering the data for the simulated particle. To obtain these equipotential curves, we follow the semianalytical procedure given in Thomas and Morbidelli (1996). We consider circular planar orbits for all planets from Jupiter to Neptune. This reduces the problem of computing the average Hamiltonian to a double integral in the mean longitudes, for the non-resonant problem, which is solved semianalytically, as in Thomas and Morbidelli (1996). For the resonant problem, with libration amplitude equal to zero, there will be a fixed relation between the mean longitudes of planet and particle, which reduces the problem to a single integral. Considering a finite libration amplitude, we turn again to a double integral where the mean longitudes are still related through the resonant angle, which is made to vary according to a sinusoid of the given amplitude. So Figures 8 and 9 are constructed with the same parameters except that for Figure 8 the constraints given by the 2:5 resonance with Neptune (mean longitudes relation) are considered in the averaging process. The value of the semimajor axis for the particle and planets are the same for both figures. Figure 10 shows the path of the particle that can be compared to the theoretical Kozai diagrams. These three last figures show that this planetesimal was trapped with a high perihelion in a stable position during the planetary migration phase. The particle was initially trapped into the 2:5 + Kozai mechanism with Neptune which made the eccentricity decrease following equipotential curves on the diagram of Figure 8. However it could not find its way back to the planetary crossing regime because it was released from resonance during migration while in the low-eccentricity orbit,

9 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 117 Figure 5. Distribution of perihelion distances and inclinations for the last 1 billion years of integration for the HPSDOs of Model 1. following, in the end, a flat curve with a low eccentricity (high X) on the non-resonant diagram of Figure 9. Figure 11 shows our next example, still from Model 1, concerning a planetesimal trapped into the 1:24 resonance with Neptune. This example took place closer to the end of the integration when there was virtually no migration. In this respect, it resembles all the examples associated to Model 2, given below. In this case, the conservative status of the planet particle dynamics allows for the particle to return in the end to its Neptune-crossing regime. This is the example associated to the largest semimajor axis found in our simulations (see Figure 1). Figures show three examples concerning Model 2. In Figure 12, the particle suffers a very slow diffusion in semimajor axis evolving near the 1:8 resonance until it is captured into this resonance at t ¼ 1:6 Gyr. The Kozai mechanism starts almost immediately and after an excursion to high values the perihelion distance diminishes having an encounter with Neptune and leaving the resonance at t ¼ 2:4 Gyr. A diffusion to the Oort cloud follows. In Figure 13, the particle also suffers a slow diffusion in semimajor axis until it is captured into the 1:11 resonance at t ¼ 1:38 Gyr. Again, the Kozai mechanism starts almost immediately. The particle in Figure 14 is quickly

10 118 RODNEY GOMES ET AL. Figure 6. Distribution of perihelion distances and inclinations for the HPSDOs of Model 2. captured into the 1:6 resonance as shown by the evolution of the critical angle. In this case the Kozai mechanism only appears 400 Myr after the capture into the 1:6 resonance with Neptune. After analyzing the orbital evolution of SDOs we can make the following scheme for the particles that enter to the HPSD. Initially the perihelion distances are small enough for the perturbations of Neptune to dominate the evolution of the particle which shows a diffusion in orbital energy. As the particle s semimajor axis and eccentricity get ever greater values, it could reach a region where the diffusion in energy is slow enough that a capture into a MMR 2 is possible. This will be a high-order resonance because of its large semimajor axis. This capture into a high-order MMR is possible because of the high values of the eccentricity. We can assume that the resonant terms in the disturbing function are proportional to e k where k is the order of the resonance. Consequently, if the eccentricity is low, e.g. e 0:1, a tenth order resonant term will be negligible. But, for moderate to high 2 We keep the term capture into a resonance although it must be noted that the trapping mechanism here is quite different from the one associated to the second fundamental model for resonance (Henrard and Lemaitre, 1983) for which this nomenclature is usually employed.

11 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 119 Figure 7. Orbital evolution of a planetesimal that was fossilized near the 2:5 resonance with Neptune during its migration. The particle was released from resonance while in the loweccentricity regime. Figure 8. Kozai level curves for the example of Figure 7, where the average Hamiltonian was computed considering the constraints given by the 2:5 resonance.

12 120 RODNEY GOMES ET AL. Figure 9. Same as Figure 8 but the average Hamiltonian does not consider the longitude constraints of the resonance. Figure 10. Path of the planetesimal orbit of Figure 7 in the X x plane. This is to be compared with Figures 8 and 9. In the beginning the particle was not in resonance and followed a path coming from Figure 9. Later when it got trapped into the 2:5 resonance it started to follow paths shown in Figure 8, having its eccentricity decreased. Being decoupled from the resonance while in low-eccentricity regime, it restarted to follow the flat configuration given by Figure 9 for the age of the Solar System.

13 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 121 Figure 11. Orbital evolution of a planetesimal temporarily trapped into the 1:24 MMR with Neptune. During the trapping time, the Kozai mechanism worked together to raise the perihelion up to around 64 AU. The particles was previously trapped into the 1:6 resonance. This is the example with the highest semimajor axis for a HPSD found in the simulations for Model 1. eccentricities, say e 0:5, the resonant term of the disturbing function is 10 7 stronger and it becomes important. That is why a capture into resonance can occur provided there are no strong perturbations by close encounters with Neptune. Once captured in a MMR, the semimajor axis will be locked at a fixed value. At this point no high variations in eccentricity can be expected essentially because of the small libration widths of the exterior resonances with Neptune (see for example Malhotra, 1996). If the resonant orbital evolution drives the inclination to a value big enough, the positive sign of _- typical of a TNO can switch to a negative value. As a consequence _-! _ X and, since _x ¼ _- _ X! 0, the Kozai mechanism appears. Similarly, even outside a resonance, if i grows enough, the sign of _- can also switch to a negative value, also triggering the Kozai mechanism. This situation cannot appear for low inclinations as is the case for 1995 TL 8 and its clones. This mechanism generates long excursions in eccentricity and inclination and can store the particle in a stable region in the HPSD for several Gyr.

14 122 RODNEY GOMES ET AL. Figure 12. Orbital evolution of a test particle captured into the 1:8 resonance and then following the Kozai mechanism. The test particle is a clone of 1999 RD 215 (initial orbital parameters: q ¼ 37:9 AU, a ¼ 118:0 AU, i ¼ 25:9 ). Eventually, due to the Kozai mechanism, the eccentricity grows again and the perihelion distance decreases, thus allowing the perturbations by Neptune to decouple the orbit from the resonance. It is also possible to see in some cases that the particle is temporarily decoupled from MMR when the eccentricity diminishes enough (the critical angle loses the libration mode) due to the sharp dropoff of the coefficient of the resonant term in the disturbing function (see, e.g. Figure 14 around Gyr). But due to the chaotic character of the evolution, the particle is eventually driven back to the resonance and the Kozai mechanism restarts, increasing once more the particle s eccentricity. 5. The Transfer Efficiency to the HPSD Figures 15 and 16 show for either model the fraction of surviving objects that are placed in the HPSD as a function of time. We find that both models roughly agree in the fraction of objects that are transfered to the HPSD during the Solar System age. This fraction can be estimated to be about at the end of the studied period (4.5 5 Gyr).

15 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 123 Figure 13. Orbital evolution of a test particle captured into the 1:11 resonance and then following the Kozai mechanism. The test particle is a clone of 1999 RZ 215 (initial orbital elements: q ¼ 30:9 AU, a ¼ 100:4 AU, i ¼ 25:5 ). It must be noted that for the migrating model, we already start with about 10% of bodies in the fossil HPSD created during migration, as commented in the previous section. The size of this fossil population will be a function of the initial parameters that drove the planetary migration. In general, a fast and/or irregular migration will give rise to a greater fossil population. Because in our migration model, the migration proceeded fairly fast, we might set an upper limit of 10% of the SD for this fossil population. Moreover, this population will be represented by objects with relatively small semimajor axes as compared to the temporary population of the HPSD. The reason for this is that the diffusion to low-eccentricity orbits, by the combined action MMR + Kozai, occurs much faster at the beginning of the integration for smaller semimajor axes, when migration is still present. When the MMR+Kozai mechanism for high semimajor axis starts to act, there is no more migration left to fossilize the objects. In conclusion, depending on the size of the fossil HPSD population, which itself depends on the conditions of planetary migration, the overall population of HPSDOs can be as high as about 20 25% of the SD population. The lifetime of objects in the HPSD region is large, comparable to the age of the solar system. As a result of this, the HPSD is continuously replenished, so the relative number of HPSDOs (with respect to the total SDOs at a given

16 124 RODNEY GOMES ET AL. Figure 14. Orbital evolution of a test particle captured into the 1:6 resonance that after 400 Myr is driven by the Kozai mechanism and stored for 1 Gyr with q 55 AU. The test particle is a clone of 2002 GB 32 (initial orbital parameters: q ¼ 37:6 AU, a ¼ 100:1 AU, i ¼ 15:0 ). time) increases with time (see Figures 15 and 16). In this regard it can be envisioned as a cometary reservoir, where objects from the SD are trapped, remaining there for long periods of time. It is interesting to stress that the average inclinations of HPSDOs are higher than those of SDOs. In actuality, no low-inclination orbits (i < 10 ) are obtained for our model HPSDOs, and most of these objects have inclinations above 20. The space distribution of HPSDOs, together with other populations in the outer solar system, are depicted in Figure Discussion The numerical integrations undertaken through two different models show some points of agreement that we can conclude as general features of the HPSD. These are: 1. Roughly 12 15% of the objects presently belonging to the SD must have a perihelion above 40 AU, thus belonging to the HPSD. If the SD

17 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 125 Figure 15. Fraction of HPSDOs (a > 50 AU) with respect to the total surviving population as a function of time, for Model 1. population is bodies down to a radius R ¼ 1 km, this means a HPSD population of 1:0 1: bodies in the same size range. 2. We obtain fairly high perihelia during the integration for the evolving planetesimals, the highest of all reaching around 70 AU. 3. The mechanism that creates these objects corresponds to an association of a high-order MMR with Neptune with the Kozai mechanism. This mechanism induces long-period, large-perihelion incursions of the particles. It also creates the correlation of perihelia with inclinations shown in Figures 5 and 6. The main differences between the two models are: 1. The migrating model also generates fossilized HPSDOs due to the escape of the particle from a resonance that had decreased the eccentricity of its orbit. The escape from the resonance is due to Neptune s migration. The escape occurs just when this eccentricity was low enough, creating highperihelion particles in stable orbits. 2. A second difference concerns the higher perihelia and semimajor axes attained by the particles in the migrating model. A possible interpretation

18 126 RODNEY GOMES ET AL. Figure 16. Fraction of HPDOs (a > 50 AU) with respect to the total surviving population as a function of time, for Model 2. Figure 17. Space distribution of the different populations of minor bodies beyond the planetary region. The units are AU. is that the migrating phase creates the appropriate initial conditions for the SDOs (yet with low perihelia) to eventually get trapped into a resonance associated to higher semimajor axes. Usually these resonances, as shown in Figures 1 and 2, allow the formation of higher perihelion

19 THE ROLE OF THE KOZAI MECHANISM AND MEAN MOTION RESONANCES 127 orbits. Yet, the samples are not large enough to draw definite conclusions about whether there are significant differences between Models 1 and 2 in the highest perihelia and semimajor axes attained. The amount of fossilized objects created by the migrating phase is very model dependent. A faster and/or irregular migration tends to create more fossilized objects. In our case, the planetesimal disk was fairly dense, what causes a fast migration. A more reasonable model must be associated to a less dense disk (Gomes et al., 2004). Thus we may conclude that the number of fossilized objects obtained here must be taken as an upper limit to the potential fossil HPSD population. Thus a reasonable conclusion is that the ratio of the number of objects in the HPSD with respect to the total number of objects in the SD must range between 0.12 (no migration or very slow and regular migration) and 0.25 for a case of fast irregular migration. Presently there are four TNOs with a > 50 AU and q > 40 AU. We are next discussing their possible origin. Object 2000 YW 134, according to the integrations of Model 2, seemingly occupies a non-resonant orbit, although its semimajor axis is near the location of the 3:8 resonance with Neptune. Having an inclination of 19 :8, 2000 YW 134 can be possibly a fossil object that was detached from that resonance during Neptune s migration. Object 1995 TL 8 has a very low inclination (0:2 ) and is near the 3:7 resonance with Neptune although integrations in Model 2 also showed a stable orbit like 2000 YW 134. Although 1995 TL 8 could be a fossilized object, integrations in Model 1 did not show any fossil object with such a low-inclination. More work should be done to determine its formation mechanism. It must be also noted that observations favor the discovery of low-inclination orbits, thus its formation process can be rather exceptional. Also, if we take the lower perihelion limit for the HPSD to be 39 AU instead of 40 AU, other five objects would enter the HPSD, all with inclinations near or above 20, confirming the exceptional character of 1995 TL 8 inclination. Sedna is the observed Solar System object presently located the furthest of all from the Sun. Its orbit also presents the highest semimajor axis (532 AU) and perihelion (76 AU). These orbital elements leave Sedna far away from the HPSD population. In particular, we note that its inclination is too low (11:9 ) by contrast to the high-inclination orbits attained by the HPSDOs. In fact, through the Kozai mechanism which is responsible for the formation p of all HPSDOs, we can conclude that by keeping H ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 e 2 Þ cos i constant, Sedna would have a zero inclination for an eccentricity equal to 0:86. This corresponds to a perihelion still too high (73:4 AU) to explain its origin as a scattered object by Neptune that attained its perihelion by solely perturbations by the known planets. Sedna must thus belong to another family (Oort Cloud Inner Core) and its origin is probably

20 128 RODNEY GOMES ET AL. associated to external perturbers in the primordial Solar System (e.g. Brown et al., 2004, Morbidelli, A. and Levison, H. F., submitted). As discussed by Fernández and Brunini (2000), a tightly bound inner core of the Oort cloud with perihelia above 40 AU and semimajor axes from several hundreds to a few thousands AU could have been formed if the solar system was immersed at that time in a dense galactic environment. The case of 2000 CR 105 is more critical. Looking at Figures 1 and 2, we see that it occupies a marginal position in semimajor axis and perihelion distance for a HPSDO. The distributions of a versus i and q versus i, as shown in Figures 3 6, do not discard CR105 as a member of the HPSD. The argument on H above used for Sedna also places CR105 marginally as a candidate for the HPSD. It may however be as well a member of the same family as that of Sedna. The numerical simulations carried out in Model 2 have indicated that this object lies in a very stable dynamical zone, without dynamical connection to the SD population, which argues in favor of its emplacement in its current location at the beginnings of the solar system by dynamical effects that are not longer active, as for instance stellar perturbations or a putative massive planetoid. We note however that the uncertainties in the listed orbital elements of CR105 do not allow us to rule out the hypothesis that given the right correction to its orbital elements allowed by these uncertainties, a resonant orbit would be obtained. Although there are at present four real objects with a > 50 AU and q > 40 AU, from our numerical computations of Model 2 we find a total of twelve real objects that reach the HPSD region at different times. For example, 1999 HW 11 has at present q ¼ 39:2 AU but over all the numerical integration the perihelion oscillates between 34 and 44 AU due to the action of the 3:7 and the Kozai mechanism. References Brown, M. E., Trujillo, C. and Rabinowitz, D.: 2004, Discovery of a candidate inner Oort cloud planetoid, Astrophys. J., in press. Brunini, A. and Melita, M. D.: 2002, The existence of a planet beyond 50 AU and the orbital distribution of the classical Edgeworth Kuiper belt objects, Icarus 160, Chambers, J. E.: 1999, A hybrid sympletic integrator that permits close encounters between massive bodies, MNRAS 304, Duncan, M. J. and Levison, H. F.: 1997, A disk of scattered icy objects and the origin of Jupiter-family comets, Science 276, Fernández, J. A. and Brunini, A.: 2000, The buildup of a tightly bound comet cloud around an early Sun immersed in a dense galactic environment: numerical experiments, Icarus 145, Fernández, J. A., Gallardo, T. and Brunini, A.: 2002, Are there many inactive Jupiter family comets among the near-earth asteroid population?, Icarus 159,

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