AN OUTER PLANET BEYOND PLUTO AND THE ORIGIN OF THE TRANS-NEPTUNIAN BELT ARCHITECTURE

Cold and hot classical TNOs with different physical properties. In general, whereas cold classical TNOs are fainter (smaller) and exhibit mostly red colors, their counterparts in the hot population are intrinsically brighter (larger) with a wide variety of colors. Apparent correlations of colors and sizes with inclinations have been found (Tegler & Romanishin 2000; Levison & Stern 2001; Trujillo & Brown 2002; Doressoundiram 2003 and references therein; McBride et al. 2003). Although noteworthy, these properties are not caused by mechanisms related to impacts (Morbidelli & Brown 2004). Distinct color, size, and inclination distributions can be interpreted as evidence that cold and hot classical TNOs formed at distinct places in the primordial planetesimal disk (Gomes 2003b). That is, the cold classical population would have formed in situ, thus representing relics of the planetesimal disk between about 35 AU and the original edge of the disk. The hot population would consist of planetesimals that formed in the inner solar system (15–35 AU) and were later deposited in the classical region via resonant interactions (Gomes 2003b). Intriguingly, the superposition of cold and hot populations is apparent only among classical bodies;

Orbital excitation of cold (in eccentricities) and hot (both in eccentricities and inclinations) classical populations. Classical TNOs present an unexpected excitation in their eccentricities and inclinations (Figures 1 and 3) that cannot be attributed to gravitational interactions with Neptune (assuming the current solar system architecture) or to mutual planetesimal gravitational stirring (Morbidelli 2005). The excitation of eccentricities also reveals an apparent lack of low-e TNOs beyond about 45 AU (Figure 3). This remarkable feature is probably not due to observational biases (Morbidelli & Brown 2004; Morbidelli 2005). Moreover, studies of the long-term evolution of classical TNOs have confirmed that the outer skirts of the trans-Neptunian belt are stable at low eccentricities, so the absence of low-e TNOs beyond about 45 AU is unexpected (Holman & Wisdom 1993; Duncan et al. 1995; Kuchner et al. 2002; Lykawka & Mukai 2005c).

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Resonant TNOs. Prominent resonant populations that are stable over the age of the solar system are found throughout the entire trans-Neptunian region (Figures 1–3; Lykawka & Mukai 2007a; LM07b). To account for the population of 5:2-resonant TNOs, Chiang et al. (2003) showed that a disk of planetesimals with initially excited orbits is necessary prior to planet migration. This result was recently confirmed (Hahn & Malhotra 2005). In addition, the origin of long-term resonant bodies in the scattered disk strongly suggests that the ancient trans-Neptunian belt was already excited in both its eccentricities and inclinations when the Neptunian resonances passed through the belt (Figure 3; Lykawka & Mukai 2007a). A similar condition would also explain the formation of all 3:2-resonant TNOs (Wiegert et al. 2003).

Scattered TNOs. There is a substantial population of TNOs evolving on orbits that undergo scattering by Neptune (Figure 2). Scattered TNOs are thought to be primordial, and not sustained by TNOs coming from unstable regions of the trans-Neptunian belt (Morbidelli 2005).

Detached TNOs (Figure 2). This population is intriguing because the sole presence of the giant planets cannot explain their origin (Gomes et al. 2007). This is particularly evident in the case of extreme objects such as (148209) 2000 CR₁₀₅ (a = 224.6 AU; q = 44.1 AU) and (90377) Sedna (a = 525.6 AU; q = 76.2 AU), and low-i detached TNOs. Indeed, the great majority of detached bodies produced by resonances have i > 30° (Lykawka & Mukai 2004; Gomes et al. 2005a; Gallardo 2006a; Lykawka & Mukai 2006). Of all currently identified detached TNOs (nine objects) in LM07b, only one member is in resonance (8:3 resonance). In addition, all members but one have i < 30°.

Very high i TNOs (objects with i > 40°; LM07b). There are only three members known thus far, i.e., 2004 DG₇₇ (i = 47.6°), Eris (i = 44.0°), and 2004 XR₁₉₀ (i = 46.7°), because severe observational biases are involved. That is, most surveys focus near the ecliptic. Because very high i TNOs spend an extremely small time of their orbits near the ecliptic, their discovery is discriminated against. For a survey probing a sky region near the ecliptic, the probability of discovering a TNO with i = 40° is approximately 20–50 times smaller than finding one with i ≈ 0 (Trujillo and Brown 2001; see also Jones et al. 2005). The apparent fraction of very high i TNOs is about 0.5% (3 out of 600–650 TNOs with longer-arc orbits), so we estimate the intrinsic fraction to be roughly ∼10–25%. Because the results of simulations predict that 1% of objects would possess i > 40° (LM07b, and references therein), resonances or the sole perturbation of Neptune cannot produce this subclass.

The outer edge of the trans-Neptunian belt at 48 AU. The outer edge is characterized by the absence of near-circular TNOs beyond about 48 AU and the abrupt decrease in the number of TNOs with R, revealing a dearth of these objects at ∼47–50 AU (Jewitt et al. 1998). Despite the observational capability to detect sufficiently large TNOs beyond 48 AU, several observations did not find them, thus supporting the existence of the edge (Gladman et al. 1998; Jewitt et al. 1998; Allen et al. 2001; Gladman et al. 2001; Trujillo et al. 2001a; Bernstein et al. 2004; Morbidelli & Brown 2004; Larsen et al. 2007). Simulations of the orbital evolution of objects in near-circular orbits around 45–55 AU have shown that this region is stable over the age of the solar system (e.g., Brunini 2002). The 2:1 resonance is also unable to produce the edge (Duncan et al. 1995; Lykawka & Mukai 2005c). Furthermore, a cold thin disk composed of bodies larger than 185 km beyond 48 AU and with a similar size distribution as in the classical region can be ruled out with 95% confidence for any disk with inclination ≤ 1° to the invariable plane (Allen et al. 2002). Alternative explanations for the edge such as extreme size/albedo effects (e.g., maximum size/albedo decreases with R), drop in e-distribution with R, and steeper ecliptic-plane surface density variation with R have been ruled out (Trujillo & Brown 2001; Trujillo et al. 2001a). Possible explanations for the edge include external perturbations such as massive planetesimals, passing stars, or UV photoevaporation (Ida et al. 2000a; Brunini & Melita 2002; Adams et al. 2004); inward radial migration of bodies during the assembly of <1 km sized planetesimals (Weidenschilling 2003); or the outward transportation of bodies by the 2:1 resonance (Levison & Morbidelli 2003).

Current low total mass of the trans-Neptunian belt. Estimates of the total mass are on the order of 0.1 M_⊕ (Jewitt et al. 1998; Gladman et al. 2001; Bernstein et al. 2004; Chiang et al. 2007). Stern (1995) and Stern & Colwell (1997) demonstrated that 100 km sized TNOs are unable to form via accretion in the current outer solar system in timescales comparable to the formation of Neptune. Indeed, the formation of Neptune is likely to preclude the growth of ≥50 km sized TNOs because the planet induces eccentricities larger than 0.01 (i.e., higher collisional velocities), which favor nonaccreting collisions. This problem can be solved assuming that the trans-Neptunian belt carried much more mass in the past. Accretion/collisional models indicate that this would require at least ∼10–30 M_⊕ in an annulus from 35 AU to 50 AU (about 100 times more massive than now) to account for the formation of TNOs with sizes of 100–1000 km (Davis & Farinella 1997; Stern & Colwell 1997; Kenyon & Luu 1998; Kenyon & Luu 1999; Kenyon & Bromley 2004a). The formation of large rubble piles and satellites around planetary-sized TNOs requires a much more active collisional environment than now, and hence a more massive primordial belt (Jewitt & Sheppard 2002; Sheppard & Jewitt 2002; Brown et al. 2006b). If the ancient trans-Neptunian belt was more massive in the past, we have a new problem: because collisional grinding cannot account for all the mass loss, how was 99% of the mass lost?

In summary, a successful model for the trans-Neptunian region must explain (1) the excitation of the primordial planetesimal disk, (2) distinct classes of TNOs and their orbital structure, physical properties, and intrinsic ratios, (3) the creation of an edge at about 48 AU, and (4) the missing mass of the trans-Neptunian belt.

Several models have addressed the origin and orbital evolution of TNOs, but none has reproduced detailed observations of all dynamical classes or provided insightful predictions. However, we note that the intent of several models below was not to explain the entire trans-Neptunian belt structure. We included them because a more comprehensive model probably requires two or more mechanisms.

2.2.1. The Scattered Disk Model

2.2.2. The Passing Star Model

2.2.3. The Large Planetesimal Model

2.2.4. The Resonance Sweeping Model

2.2.5. The Solar Companion Model

2.2.6. The Packed-Ice Giants Model

2.2.7. The Nice Model

The most comprehensive planet formation models are based on the accretion of planetesimals and can be divided into three main stages: planetesimal formation, runaway growth, and oligarchic growth (Kenyon 2002; Kokubo & Ida 2002; Rafikov 2003; Goldreich et al. 2004b). First, there is the build-up of kilometer-sized bodies. After this stage, the largest bodies in the swarm of planetesimals grow much faster than their smaller neighbors. Finally, several large planetesimals (or planetary embryos) are obtained, a few of which can eventually collide to form larger planets or the cores of giant planets (Pollack et al. 1996; Kenyon & Luu 1999; Goldreich et al. 2004a; Kenyon & Bromley 2004a; Rafikov 2004).

Accretion models agree that planet formation is an inefficient process, despite the different parameters and assumptions used in these models (Gladman 2005). In the late stages of planet formation, giant planets clean their neighborhoods by scattering disk planetesimals that move through the system, many of which are massive. A huge number of planetesimals carrying a substantial mass have been scattered by the planets (Fernández & Ip 1984, 1996; Pollack et al. 1996, Jewitt 1999; Petit et al. 1999). In addition, a natural consequence of these scattering events is planet migration (Fernández & Ip 1984; Hahn & Malhotra 1999; Gomes et al. 2004). In the end, the fate of scattered planetesimals is one of the following: (a) ejection from the solar system; (b) collision with a planet or the Sun; (c) placement in the scattered disk; or (d) placement at very large distances (formation of the Oort cloud) (Oort 1950; Brasser et al. 2006).

Several lines of evidence in the present solar system support the existence of a substantial population of massive planetesimals in the past: Pluto and other multiple systems, the high tilts of Uranus and Neptune, the retrograde orbit of Triton, and the discovery of very large TNOs (e.g., Eris). The origin of the Pluto system is better understood as a giant impact of embryos during the early solar system (Stern 1992, 1998; Canup 2005; Stern et al. 2006; Weaver et al. 2006). The formation of Eris and (136108) 2003 EL₆₁ satellite systems and the tilts of Uranus and Neptune are also explained by giant collisions during the late stages of planet formation (Stern 1991 and references therein; Brunini & Melita 2002; Brunini et al. 2002 and references therein; Barkume et al. 2006; Brown et al. 2006b; Lee et al. 2007). Further evidence includes the Neptunian satellites Triton and Nereid. The former was possibly captured from the trans-Neptunian region because of its retrograde orbit (Luu & Jewitt 2002; Agnor & Hamilton 2006). Nereid may also be a captured TNO (Brown 2000; Schaefer & Schaefer 2000). In summary, the above examples are best explained by giant impact or close encounter events of massive planetesimals, which must have existed in large numbers to permit such events (Stern 1991; Brown 2002). Indeed, Stern (1991) predicted hundreds of planetary bodies with ∼0.1–1.0 M_⊕ and a few very large ones with ∼1–5 M_⊕ during the early history of the solar system.

Finally, we note that current observations are still incomplete. For instance, the existence of still other Pluto-sized objects in the trans-Neptunian region is predicted (Trujillo et al. 2001a). This expectation has been fulfilled recently, in particular with the discovery of planetary-sized TNOs such as Eris, 2003 EL₆₁, and (136472) 2005 FY₉ (Brown et al. 2004, 2005a; Brown et al. 2006b). Thus, it is perfectly possible that planetary bodies larger than Pluto or Eris exist in more distant resonances or far away in the scattered disk (Jewitt 2003; Gladman 2005).

The results of Lykawka & Mukai (2007a) suggest that the trans-Neptunian belt had a substantial population of planetesimals in excited orbits before planet migration (Figure 3) and the radius of the planetesimal disk was at least 45–50 AU. However, planetesimals in very excited orbits (above the gray region in Figure 3) or in near-circular and low-i orbits may have existed. Nevertheless, nonresonant objects with q < 37 AU are likely to be removed from the solar system through scattering by Neptune, implying that even if very excited planetesimals existed, they are long gone. Therefore, we think that the pre-migration ancient trans-Neptunian belt extended to 50 AU or more and was composed of objects with eccentricities and inclinations ranging from near zero (up to ∼45–50 AU) to the excited values illustrated by the gray region in Figure 3. This hypothesis is supported by the current distribution of TNOs in which we observe classical TNOs in near-circular orbits up to 45 AU and an excited portion (45–50 AU) overlapping with the gray region. This would imply that the original cold planetesimal disk suffered an excitation from outside prior to planet migration, leading to the observed perturbed orbital distribution (Figure 3). We propose that a massive planetesimal could be the main agent that excited the disk.

Here, we argue that the orbital history of a massive planetesimal with tenths of M_⊕ can explain the orbital structure in the trans-Neptunian belt. This massive body was likely scattered by one of the giant planets and managed to survive in the scattered disk. We refer to this body as the "outer planet," "trans-Plutonian planet," or "planetoid," denoted by a _P subscript. We built the model intending to self-consistently satisfy the main constraints discussed in Section 2.1.

For a given orbital configuration, the planetoid should be massive enough to efficiently perturb the outer regions of the trans-Neptunian belt and truncate it at 48 AU. However, this excitation should not last over excessive timescales, otherwise the inner trans-Neptunian belt would be excessively depleted. Likewise, for a certain fixed timescale, the outer planet should be located in a convenient orbital configuration (neither too close nor too far away) for its perturbation to affect classical TNOs.

More importantly, because trans-Plutonian planets in near-circular and low-inclination orbits have been ruled out (Morbidelli et al. 2002), our hypothetical planet must be inclined (>10°) and relatively distant (e.g., >60–70 AU). A Neptune-scattering mechanism probably provided such orbital conditions for this planet. However, because gravitational scattering occurs at approximately fixed positions in space (Gladman et al. 2002), the perihelion of the outer planet could remain near the orbit of Neptune over prohibitively long periods of time, thus destroying the entire trans-Neptunian belt. Alternatively, dynamical friction could raise the planetoid's perihelion before the disruption of the belt (Del Popolo et al. 1999). However, this process could lead to a near-circular and very low i orbit, which would be in conflict with the observational constraint above.

Investigations of resonances in the scattered disk offer an appealing possibility. First, the strongest and dominant resonances beyond 50 AU are distributed mainly at a < 250 AU and have the lowest argument s, in particular those of type r:1 or r:2 (i.e., 3:1, 4:1, ..., 24:1; 5:2, 7:2, ..., 35:2) (Gallardo 2006a, 2006b; Lykawka & Mukai 2006, 2007c). Furthermore, the increase in perihelion during resonance captures is particularly important for the resonances cited above (Figure 4). The capture probability in r:1 or r:2 resonances is also higher than in other resonances in the scattered disk (Lykawka & Mukai 2007c). In conclusion, we postulate that after being scattered outwards by a giant planet, the outer planet interacted with a Neptunian r:1 or r:2 resonance. Therefore, capture in a strong scattered disk resonance represents an excellent mechanism to modify the outer planet's orbit, helping to protect the classical region from disruption.

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We first explored the influence of outer planets in fixed orbits, varying the initial orbital and mass parameters of the planet in systems with the giant planets at their current orbits. The main goal was to see if a massive planet with an orbit near a resonance location in the scattered disk could reproduce the excitation in the classical region and form the edge at 48 AU. The time span of the simulations was 4 Gyr. A summary of the runs is given in Table 1.

We found that an outer planet, provided that it is massive enough and possesses a convenient perihelion distance, is able to disrupt the local cold disk at 48 AU, in particular the removal of objects with e < 0.1. This was possible for planetoids with 50 AU < q_P < 56 AU and having i_P < 20–25° and a few tenths of M_⊕ when located within about 90 AU. At larger distances, more massive outer planets yielded better results (0.5–1.0 M_⊕) (Figure 5). Planetoids with smaller perihelia yielded too much excitation in the trans-Neptunian belt, whereas those with larger perihelia and/or higher inclinations did not produce an edge at all. The disruption of the cold disk beyond 48 AU occurred on timescales of 0.5–1 Gyr for planetoids at 63–88 AU (3:1, 4:1, and 5:1 resonances), although better removal of objects with low e requires more than 2–3 Gyr. In the 10:1 resonance, these timescales were approximately twice as long. Another outcome of these runs was an excitation of inclinations for objects orbiting the outer planet crossing region.

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We also tested the presence of a less massive planetary body in the 2:1 resonance using M_P = 0.01–0.05 M_⊕ over 1.5 Gyr (SIM 1–3, Table 1). The planetoids excited local planetesimals to large eccentricities and i ∼10–20° in only a few hundred Myr, creating narrow and sharp peaks in a–e and a–i element space distributions. Nevertheless, 2:1-resonant planetoids either led to too little perturbation in the classical region and beyond or excessively depleted the inner classical region. Thus, a Moon-like or more massive body in the 2:1 resonance cannot satisfy the trans-Neptunian belt observational constraints discussed previously (Section 2.1).

A drawback of static planets is that the observed orbital distribution in the classical region is not reproduced (e.g., Figure 5). Indeed, in general, particles survived billion of years in near-circular orbits within 50 AU. When the classical region is excited by the planet (usually when q_P < 50 AU), the final distribution is incompatible with observations. In conclusion, static planets cannot form the edge of the trans-Neptunian belt and at the same time reproduce the orbits of TNOs in the classical region. A similar conclusion was found by Melita et al. (2004). However, one could argue that the fine-tuning of simulations or a much larger number of runs with higher resolution could yield results that are in agreement with observations. However, this is not the case, as we will show in Section 5.3.

Considering the difficulties of static outer planets in consistently explaining the excitation of the classical region and the trans-Neptunian belt edge, we decided to investigate systems with planet migration. The main goal was to verify whether a planetoid initially crossing the classical region could provide the appropriate excitation to the latter and remove bodies beyond 48 AU while migrating outwards. When not mentioned, the simulation time span was 100–200 Myr. A summary of the runs is provided in Table 2.

The giant planets are assumed to have formed in a more compact configuration than at present, in line with migration models (Morbidelli et al. 2007 and references therein). We assumed the planetoid had initial perihelion q_P0 < (a_N0 + 10 AU), where a_N0 is the initial semimajor axis of Neptune before migration. Thus, the planetoid represented a "recent" Neptune-scattered body. We set a_P0 as a free parameter using arbitrary values beyond 40 AU (sometimes close to the initial position of a resonance of interest) and 10° ≤ i_P0 ≤ 30° and tested different masses for the outer planet. We performed simulations in which, along with the giant planets, the planetoid was forced to migrate outwards obeying a predefined radial displacement (in some cases according to the position of a distant and strong r:1 resonance with Neptune). We imposed an artificial force to induce planet migration as described in Malhotra (1995) and Hahn & Malhotra (2005). For simplicity, we neglected disk interactions and other stochastic phenomena during migration, so most outer planets ended the simulations with perihelia around 50–80 AU. We stress that the intent of this simple migration model was to test the viability of the scenario in which a planetoid initially perturbs the classical region and is later transported outwards, ending the migration phase in a stable distant orbit near/in a resonance in the scattered disk. Thus, the final outcomes acquired by the outer planet here resemble the initial conditions conjectured in Section 5.1.

The giant planets and the outer planet migrated according to

$$\begin{equation} a_n (t) = a_n - \Delta a_n \exp ({{{ - t}/\tau }}), \end{equation} \tag{ 1 }$$

where Δa_n (in AU) is the total radial displacement of the planet (indicated by the subscript n), a_n is the current planet's semimajor axis, and τ is the migration timescale. We prepared three sets where Δa_n = {−0.2, 0.8, 3.0, 7.0}, {−0.2, 0.9, 4.0–4.3, 9.9–10.2}, and {−0.2, 0.9, 4.5–5.0, 12.0–13.0} for Jupiter (J), Saturn (S), Uranus (U), and Neptune (N), respectively (i.e., Neptune started at 23.1, ∼20, and 17.1–18.1 AU). We tested various migration timescales, but preferentially used τ = 5–15 Myr. These values are constrained by several past studies (Hahn & Malhotra 1999; Gomes 2000; Ida et al. 2000b; Melita & Brunini 2000; Friedland 2001; Chiang et al. 2007).

In general, the inclusion of a migrating planetoid resulted in the simultaneous formation of resonant populations (<50 AU), excitation of the classical region (mostly in eccentricities), and the disruption of the disk beyond about 48 AU well before the long-term stage (Section 5.1). Moreover, as the outer planet moved to large distances, the stability of the classical region was also guaranteed. A didactic example is shown in Figure 6, and some of the best runs are illustrated in Figure 7. Fast migrations (i.e., using smaller τ) yielded too little excitation in the disk, whereas incursions over several Myr at <50 AU caused too much perturbation in the region. Finally, even for final orbits as close as a_P = 60 AU, low-mass outer planets (≤0.2 M_⊕) had difficulties in correctly reproducing the classical region excitation and the creation of the trans-Neptunian belt edge.

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Although a migrating outer planet could explain some of the main constraints posed by observations, the formation of hot classical TNOs is problematic. We performed a few simulations to investigate the formation of the hot classical component following the approach of Gomes (2003b). We set two cold disks, i.e., 18.6–30.1 AU and 27.6–30.1 AU, with 9900 particles each, and evolved these systems over 200–250 Myr using the same model parameters of some of the best runs (Table 2). Hot classical particles were obtained in most cases. However, the efficiency is quite low at less than 0.1–0.2%. Also, the smallest eccentricities of hot objects appear to be somewhat higher than those of hot classical TNOs. We also noticed that the efficiency depends on the migration timescale. In test simulations, we found that the production of hot classical objects decreases with increasing migration timescales. Thus, because most of the best runs have τ ≥ 5 Myr, the low efficiency problem cannot be solved using models with fast migration in our scenario.

Another serious issue is the formation of long-term resonant populations beyond 50 AU. In fact, none of the runs produced any obvious resonant populations in the 9:4 (a = 51.7 AU), 5:2 (a = 55.4 AU), and 8:3 (a = 57.9 AU) resonances. The reason for this negative result is easy to understand. The formation of TNOs with 9:4, 5:2, and 8:3 resonances requires that a population of planetesimals had e > 0.1–0.3 before the resonances swept the region (Figure 3; Lykawka & Mukai 2007a). In Figure 6, when the resonances are passing through the classical region at 5 Myr (panel (b)), the eccentricities are less than 0.05. Later on, although the disk acquires e ∼ 0.1–0.12 near the resonances at 20 Myr (panel (c)), it is still quite below the minimum necessary to account for observed resonant populations (Figure 3). In other runs we noted very similar outcomes, some of which had even smaller stirred eccentricities. In summary, there are two major issues. First, the excitation of the trans-Neptunian belt occurs too late when distant resonances have already swept the disk. Second, the resulting excitation is not high enough inside 50 AU, even in the case of M_P > 0.5 M_⊕ or longer timescales for passages through the disk by the planetoid.

If the planetesimal disk were initially excited at an appropriate level, would the planetoids preclude resonant capture in resonances beyond 50 AU? We re-ran a few runs of migrating outer planets using excited disks with broad initial eccentricities beyond 45 AU. The outcomes after 100 Myr indicate that 5:2 and other resonant bodies beyond 50 AU are obtained.

In most of the successful runs discussed above, the perihelia of the outer planet were within 50–60 AU. In such circumstances, the planetoid could threaten the stability of 3:2, 5:3, 7:4, 2:1, and other observed resonant populations over 4 Gyr. That is, eccentric resonant bodies near aphelion could encounter the trans-Plutonian planet during its perihelion approach. These perturbations are likely to dislodge bodies from the resonances in question. Because these bodies will no longer be protected from close encounters with Neptune by the libration mechanism (e.g., Peale 1976; Malhotra 1998; Murray & Dermott 1999), they will be ultimately scattered by the giant planet for large eccentricities (q < 37 AU). For lower eccentricities, former members of resonances will become fossilized in orbits near their parent location, a_res. Because the majority of TNOs in the 3:2, 2:1, 5:2, and other resonances are known to be stable over the age of the solar system (i.e., with orbital motion deeply inside the resonances), could they survive under the perturbing influence of a massive outer planet as considered above?

To determine the survival of the resonant populations of interest, we initially evolved 1000 particles inside the 3:2, 2:1, and 5:2 resonances over 4 Gyr with initial i < 30°, and 0.05 < e < 0.36, 0.1 < e < 0.42, and 0.35 < e < 0.45, respectively. These ranges cover the observed orbital distributions. Angular orbital elements were chosen at random within 0–360°. Only the four giant planets were considered in this particular investigation. We then obtained three populations stable over the age of the system for the 3:2 (166 particles), 2:1 (147 particles), and 5:2 (260 particles) resonances within about the same eccentricity and inclination ranges. Finally, we checked the evolution of the stable resonant populations with the inclusion of a massive outer planet (see also Table 1).

First, we report on the survival of 3:2- and 2:1-resonant populations for outer planets with q_P = 50–60 AU. As expected, resonant bodies were strongly perturbed by the planet (Figure 8). The 3:2 resonance was depleted at ∼30–90% levels (Table 3). In particular, the destabilization of very large e-3:2 bodies (>0.25–0.3) was evident, often leading to their total disruption. In the case of the 2:1 resonance, the situation was much more severe (Figures 8 and 9). In fact, the 2:1-resonant population was completely devastated in almost all runs and yielded resonant survivors with e < 0.26 (Table 3). Therefore, the general tendency was fast destruction of the 2:1-resonant population at high eccentricities (500 Myr to 1 Gyr timescales) and significant depletion of the 3:2, 5:3, and 7:4 resonances over 4 Gyr. These results remain valid even if the outer planet's inclination is as high as 40° because higher inclinations just increased the timescales for the dynamical depletion in these resonances (but still ≪4 Gyr). Thus, the existence of stable resonant populations provides an invaluable constraint on the perihelion of the trans-Plutonian planet. That is, in our simulations, a fraction of resonant objects can survive 4 Gyr if their aphelion, Q, is about 2–3 AU smaller than the outer planet's perihelion for M_P = 0.1–0.5 M_⊕. This condition translates into q_P > Q + 2 or + 3 AU.

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We conclude that the observed 3:2- and 2:1-resonant populations can survive over the age of the solar system only if q_P > 53 AU and q_P > 67 AU, respectively (Figures 8 and 9 and Table 3). More importantly, a planetoid with perihelion around 50–60 AU cannot create the outer edge of the trans-Neptunian belt and excite the classical region without destroying most (if not the whole) of the 2:1-resonant population. Moreover, even if a few percent of 2:1-resonant TNOs could survive 4 Gyr, their maximum eccentricities would be in clear conflict with observations (Figure 9 and Table 3). It turns out that resident outer planets as shown in panels (a)–(c) of Figure 7 cannot hold for 4 Gyr because their final perihelia are too small. In contrast, outer planets that acquired q_P > 70–80 AU after migration could allow the survival of resonant populations in the trans-Neptunian belt (<50 AU; see Figure 7, panel (d)). These results also rule out resident outer planets as proposed by Matese & Whitmire (1986), Harrington (1988), Brunini & Melita (2002), Melita & Williams (2003), and Melita et al. (2004) because they generally have 50 AU < q_P < 60 AU.

The survival of 5:2-resonant TNOs offers another important constraint. We tested the long-term survival of 2:1- and 5:2-resonant populations with planetoids analogous to those in Figure 7 (see also Table 1). In general, we found that 5:2-resonant bodies can survive 4 Gyr if q_P is about the same or slightly greater than their aphelia (i_P = 10–40° and 0.3–1.0 M_⊕). This result implies that any resident trans-Plutonian planet must have q_P > 80 AU, otherwise we would not observe stable 5:2 resonants with their present orbital distributions (Figures 9 and 10). This result is also independent of the semimajor axes (60–140 AU) used for the outer planet. Consequently, the unseen planet at ∼70 AU proposed by Brown et al. (2004) to explain Sedna's orbital characteristics is also ruled out.

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We executed 300 runs of pre-migration systems with scattered planetoids over tens of Myr to provide a convenient e–i excitation in the outer part of the planetesimal disk (Sections 5.2 and 6). The wandering of the outer planets in the semimajor axis was never greater than 5 AU over the same timescales as a result of their perihelia ∼5–10 AU away from the initial position of Neptune.

A scattered outer planet represents an excellent stirring mechanism. In addition to satisfying the required excitation for the formation of distant resonant TNOs on reasonable timescales (typically tens of Myr), an interesting and unexpected feature is that the perturbed disks presented orbital distributions very similar to observations (nonresonant populations), especially those observed in the 40–50 AU region (Figure 11).

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For reference, we show some outcomes of stirred disks in pre-migration systems for different initial masses and orbital parameters of the outer planet in Figure 12. Although the heliocentric location of perturbations in the disk depends essentially on the initial position of the outer planet, the shape and degree of excitement of the disk depends on the outer planet's mass and time span.

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In certain large-scale simulations conducted in the migration phase, the planetesimal disk was modeled with a few thousand particles following an R⁻¹ distribution. In these simulations, outer planets of 0.4 and 0.5 M_⊕ were transported outwards following the location of the 6:1 and 9:1 resonances, respectively. The giant planets and the outer planet were forced to migrate using τ = 10 Myr (Table 4). We also confirmed that the results are independent of the timing of the onset of KR for the outer planet (SIM1, 2 in Table 4). We integrated all migrating systems for 100 Myr. A few outcomes after planet migration are illustrated in Figure 13. In the long-term stage, we created clones of each object up to 54 AU in high-resolution runs, obtaining an excited planetesimal disk composed of several thousand particles (up to 17,000). We evolved the system of four giant planets plus the outer planet plus the disk over 4 Gyr.

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Unsurprisingly, the migration of Neptune was responsible for the dynamical depletion of the inner region of the disk until 39 AU, confirming the results of previous studies (Morbidelli & Brown 2004). The removal of objects in this region occurred through the triple action of Neptune's gravitational scattering (quickly clearing a region up to 35 AU), overlapping of resonances giving rise to unstable chaotic behaviors, and the capture of local bodies by strong first-order sweeping resonances (e.g., 5:4, 4:3, 3:2). Therefore, the lack of TNOs in this particular region is evidence of the outward migration of Neptune, supporting our current understanding of the theory of planet migration.

7.2.1. Cold and Hot Classical Populations

To compare our results with current observations, we considered cold (i ≤ 5°) and hot (i > 5°) classical populations as nonresonant particles with a < 50 AU and q > 37 AU at the end of simulations. The results confirm the standard picture for the origin of cold and hot classical TNOs (Section 2.1), although it differs in some points, as discussed below.

The cold population was probably formed in situ beyond a ∼ 37 AU and was perturbed early by the planetoid (Sections 6 and 7.2.2). The final eccentricities of classical bodies were remarkably similar to observed values (cf. Figures 14 and 3). Our obtained median e and i of cold populations were 0.056–0.077 and 2.4–3.2°, respectively, comparable to the observed values of 0.056 and 2.4°. Furthermore, we found that some local (initially cold) classical bodies acquired i ∼ 5–15°. In addition, recalling that resonance excitation is active in the classical region (Nesvorný & Roig 2001; Lykawka & Mukai 2005a; LM07b), possibly a significant part of the observed cold population was promoted to the hot one, especially in the 5–15° range of inclinations.

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The hot population should have originated from two distinct sources, i.e., the local planetesimal disk and the inner solar system, between ∼15–35 AU. Interestingly, hot classical TNOs formed by the excitation of local planetesimals (mechanism above) should be moderately inclined (5–15°), whereas bodies that entered the classical region from inner regions would constitute hot classical TNOs with moderately high inclinations (10–35°) (Figure 15).

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Consequently, distinct physical properties (i.e., color and size distributions) would become more evident among classical TNOs with low inclinations and those with i > 10–15°. Indeed, this finding is supported by the distribution of inclinations with spectral slopes of classical TNOs (see Figure 4 of Chiang et al. 2007) in which the differences in these distributions become clearer for classical TNOs with higher inclinations.

Finally, the above picture also offers an explanation for the intrinsic fraction of cold and hot populations, estimated at unity (Morbidelli & Brown 2004). This fraction is in serious conflict with Gomes' scenario (Gomes 2006 and references therein) because of the too-low efficiency to produce hot classicals. According to our results, an important fraction of cold classical objects was promoted to the hot population, helping to balance the estimated ratio of the populations. The ratios of cold to hot classical bodies from the large-scale simulations can vary as much as 0.7–5.0.

7.2.2. An Excited Classical Region

In our scenario, despite the shortage of high-i classical bodies (>15–20°), the planetoid can explain the excitation of eccentricities and inclinations in the classical region quite well, including the lack of low-e objects beyond 45 AU (Figures 13 and 16). This excited orbital distribution is a consequence of the perturbation from the outer planet during the pre-migration phase (see Figures 11–13; see also Section 6.1). The time span necessary to reproduce the observed excitation was about 50–100 Myr for M_P = 0.3–0.5 M_⊕, although it can be as small as 20–30 Myr in the case of more massive planetoids (M_P = 0.7–1.0 M_⊕). Finally, the perturbation of the planetoid on the region during the migration phase was not significant.

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In summary, our results suggest that the global excitation of the classical region was created during the first tens of Myr after planet formation (4.5 Gyr ago) and ended when planet migration ceased. Consequently, the excitation of the classical region would provide an important constraint on the orbital evolution and elementary properties of the planetoid.

We identified objects locked in various resonances in the trans-Neptunian region (Figures 14, 16, and 17). The great majority of the resonant population was originally distributed between 30 and 50 AU during resonance capture. After 4 Gyr, the main occupied resonances, sorted by distance from the Sun, are: 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 9:5, 11:6, 2:1, 13:6, 11:5, 9:4, 7:3, 5:2, 8:3, 11:4, and 3:1. We also found 4 Gyr members in all of these resonances. More importantly, the stable resonant populations in the scattered disk agree well with the existence of long-term 9:4-, 5:2-, and 8:3-resonant TNOs (see also Section 5.2). Symmetric and asymmetric 2:1-resonant TNOs were also reproduced (see Murray-Clay & Chiang 2005 for details about the 2:1 resonance).

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The model is able to reproduce resonant TNOs quite well in eccentricities and inclinations (Figure 16 and Table 5). Moreover, the eccentricities of scattered disk resonant TNOs were better matched for disks without bodies beyond ∼51–54 AU in low- to intermediate-eccentricity orbits. Nevertheless, we stress that the scattered disk resonant structure is still not well known (i.e., low number statistics), so only future observations will better constrain model results. Concerning inclination distributions, our resonant objects had typically i < 20–25°, whereas the observed high-i component of 3:2-resonant TNOs (i > 25°) was obtained only in specific runs via the mechanism proposed in Gomes (2003b).

Concerning resonance properties, the distribution of the amplitudes of resonant angles (libration amplitudes), which are a measure of Neptune to TNOs' relative distance during resonant motion (e.g., Murray & Dermott 1999), is in good agreement with those derived from observations (Figure 18). Libration amplitudes ranged from a few to 120–160° for most occupied resonances (Table 5). In addition, stable populations in the 9:4, 5:2, and 8:3 resonances yielded a wide range of libration amplitudes, which is compatible with values determined from TNOs. Lastly, particles experiencing KR were found among resonant populations in the 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 2:1, 7:3, 5:2, 8:3, and 3:1 resonances. The relative fraction of Kozai librators inside these resonances is reasonably compatible with observations, except for the underabundance in the 5:3 and 2:1 resonances (compare Table 5 with Table 6 of LM07b).

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7.3.1. Stability of 5:2, 3:1, and Other Distant Resonant Populations

7.3.2. Neptune Trojans

There is a clear population of scattered objects at the end of the simulations (Figure 17), found in typical eccentric and low-to-high-inclination orbits (<50°), as a result of gravitational encounters with Neptune (mainly) and the planetoid. This is in agreement with observations. Although their identification is more difficult, scattered bodies are also found in the classical region (Figure 14). Even with the inclusion of an outer planet, the formation of scattered objects over billions of years proceeds in a very similar fashion to that proposed by Duncan & Levison (1997). For instance, the fraction of scattered bodies is on the order of 1–2% of the original population on Neptune-encountering orbits 4 Gyr ago.

Interestingly, the early excitation of the trans-Neptunian belt allowed the region beyond 40 AU to contribute significantly as a source of scattered bodies. Excluding objects leaving the main resonances (<10%), about half of the scattered TNOs would have originated somewhere around 40–50 AU, <5% in the path transversed by the migrating Neptune (∼20–30 AU), and the remaining (a few tens of percent) from the region sculpted by Neptune over Gyr, namely 30–40 AU. Table 6 shows the statistics of scattered populations during their evolution as compiled from several independent runs.

Finally, scattered bodies in orbits close to the outer planet suffered significant excitation in inclinations, leading to i = 40–50° (Figures 17, 19, and 20). The model also produced analogs of Eris (Section 8.3).

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A large population of detached objects was obtained in several simulations (Figures 17, 19, and 20). Detached bodies originated mostly from Neptune-scattered orbits and later acquired larger perihelia and/or inclinations caused by perturbation from the outer planet over typically several hundred Myr. At the end of 4 Gyr, the bulk of the detached population resulted in q = 40–60 AU and i < 60° (the two were weakly correlated), a result compatible with observational constraints (Morbidelli & Levison 2004). We observed this tendency in all runs with M_P ≤ 0.5 M_⊕. In runs in which larger values of M_P were used, the distribution of perihelia tended to reach higher values, revealing a more irregular pattern (Figure 20). More massive planetoids were also more efficient in creating detached objects with larger semimajor axes and perihelia. Alternatively, detached bodies could also have originated from the effect of the planetoid's perturbation on originally cold objects of the disk, which would have acquired higher eccentricities but still kept q > 40 AU.

Furthermore, the detached populations are comparable to or a few times larger than the final population of scattered objects, which is in agreement with unbiased observational estimates (Gladman et al. 2002; Allen et al. 2006; Chiang et al. 2007). It is worth noting that simulations using only the four giant planets (static and migrating) result in ratios of scattered to detached populations higher than 4.0 (see Table 6). Finally, similar to the formation sites of scattered objects, there was no obvious preference for the source region of detached bodies.

The outer planet's influence could account for the existence of detached TNOs with low i (Figures 19 and 20), a feature not achieved in scenarios that rely just on resonant interactions (Gomes et al. 2005a; Lykawka & Mukai 2006; LM07b). We also report the production of some extreme bodies with a = 500–800 AU and usually 40 AU < q < 50 AU. The model also produced analogs of typical detached TNOs such as 2004 XR₁₉₀, 2000 CR₁₀₅, and Sedna, although the Sedna population was small (see Section 8.3).

Finally, the long-term residence of the planetoid can leave observable orbital signatures in the scattered disk. For instance, scattered and detached objects with orbits near that of the outer planet acquired the largest inclinations and perihelia (Figures 19 and 20). In conclusion, a better characterized orbital distribution of TNOs in the scattered disk may reveal the perturbing signatures of the outer planet.

Unlike standard models (e.g., Duncan & Levison 1997; LM07b), the trans-Plutonian planet can produce very high i objects, predicting a non-negligible fraction of TNOs with i > 40° beyond 50 AU. In particular, for M_P ≤ 0.5 M_⊕, scattered and detached bodies acquired inclinations up to 50° and 60°, respectively (Sections 7.4 and 7.5). More massive outer planets can excite the inclinations of some objects to values as high as 90°. The i excitation has a broad distribution in the semimajor axis, with a peak near the location of the outer planet (Figures 19 and 20). Another mechanism that can promote TNOs to the very high i subclass is the KR (Section 7.3). Therefore, it is possible that the outer planet perturbation and the KR mechanism may together explain this highly inclined subpopulation.

We discuss the production of analogs of very high i TNOs in Section 8.3.

During simulations, the outer planet was very effective in truncating the trans-Neptunian belt at approximately 48 AU, thus reproducing not only the absence of low-e bodies, but also the abrupt decrease of the number density of TNOs with R beyond 45 AU (e.g., Figures 13, 16, and 21). This finding is connected to planetoid-induced orbital excitation at 40–50 AU (Figures 11 and 12), a feature that arose before planet migration, when the planetoid was still on a typical Neptune-scattered orbit. Because these features exhibited shifting in heliocentric distance according to the planetoid's initial location, planetoids at 60–80 AU, with q_P = 20–30 AU and i_P < 15°, yielded the best outcomes before migration. This takes into account uncertainties in the initial position of Neptune at 15–20 AU.

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The location of the outer edge depends on the mass, initial orbital conditions, and timing of the perturbation of the planet (Figure 12). The subsequent migration of the planetoid plays a minor role. In summary, the outer edge of the trans-Neptunian belt was possibly created during the first tens of Myr of solar system existence, before planet migration. Thus, similar to the conclusions of Section 7.2.2, the outer edge would tell us about the origin and orbital history of the trans-Plutonian planet.

This scenario could solve the problem of the missing mass in the trans-Neptunian belt and offers two particular advantages. First, the planetesimal disk was dynamically depleted during pre-migration stirring (scattering of objects by Neptune) and remained active over the age of the solar system. For instance, in general, 15%–40% of the planetesimals remained in the system after 4 Gyr for disks of ∼51–54 AU radius. Second, with the outer planet's perturbation, a large fraction of planetesimals initially on very cold orbits were excited to levels at which random velocities render fragmentation rather than accretion (Stern & Colwell 1997; Kenyon & Bromley 2004a). Random velocity is given by

$$\begin{equation} v_{{\rm rnd}} = \sqrt {v_{{\rm disp}}^2 + v_{{\rm esc}}^2 }, \end{equation} \tag{ 2 }$$

where v_disp and v_esc are the dispersion and the planetesimal's escape velocities, respectively. The former is given by

$$\begin{equation} v_{{\rm disp}} = v_K \sqrt {e^2 + i^2 }, \end{equation} \tag{ 3 }$$

where v_K = 29.8 a^−1/2 (km s⁻¹) is the Keplerian velocity, and the orbital elements refer to the planetesimal.

By using the eccentricities and inclinations of excited planetesimals found in our pre-migration systems (e.g., Figures 11 and 12), Equation (2) yields catastrophic collisional velocities. Consequently, the trans-Neptunian belt experienced quite intense collisional grinding during this early period, and possibly after migration as well (Figure 13). In other words, the outer planet was the smoking gun enhancing total mass loss in the disk.

By combining dynamical depletion (60–85%) (this work) and collisional grinding (92–97%) (Kenyon & Bromley 2004a), we obtain (0.15–0.4) (0.03–0.08) ≈ 0.5–3% as the remainder of the original belt mass. This is certainly an upper limit, though. An overabundance of stable resonant populations results in the underestimation of dynamical depletion fractions. That is, stochastic migration (not modeled here) suggests a smaller probability of resonance capture (e.g., Zhou et al. 2002). Moreover, collisional grinding estimates are based on planetesimal disks under much less excited conditions because no external perturbers such as the outer planet proposed here were considered in these studies.

Neptune's migration is fed by planetesimals that can be scattered (transferring angular momentum), which follows the relation $\dot a_N \propto \eta (t)$, where η(t) is a function describing the total mass in planet-encountering orbits. This behavior has been observed in simulations with massive disks (Levison et al. 2007). Past studies also claimed that Neptune should migrate beyond 30 AU for massive disks extending out to 50 AU or beyond (Gomes et al. 2004). To solve this problem, Gomes et al. (2004) proposed that Neptune would have stopped at 30 AU because it reached the original planetesimal disk outer edge at ∼30–35 AU.

In our scenario, could Neptune stop at 30 AU in a 50–60 AU sized disk with embedded planetoids? First, we considered only the four giant planets in compact orbital configurations. We prepared disks composed of 10,000–20,000 equal-mass bodies with an inner edge at ∼10–20 AU and extended to 50–60 AU (27 runs). In 88 runs, we included a 0.5 M_⊕ planetoid in diverse initial orbits (e.g., near Neptune or already eccentric with i = 10° at 50–70 AU). Finally, in 16 other runs we included 5–10 planetoids embedded in each disk. In all of these runs, we used massive disks with objects in near-circular orbits and i ≈ 0. All disks followed an R^−g distribution with g = 3/2 ± 1/2 and total mass corresponding to 1, 0.9, 0.8, 0.5, 0.3, 0.25, and 0.1 times the minimum mass solar nebula (MMSN). The surface densities used at 40 AU were σ = 0.14, 0.125, 0.11, 0.07, 0.04, 0.035, and 0.014 g cm⁻², respectively. We considered less massive disks (<1 MMSN) as an expected result of collisional grinding during the pre-migration epoch (see Kenyon et al. 2007 and references therein). The outcomes of these simulations resulted in Neptune migrating over 30 AU for most 1 MMSN disks. However, we found that in almost all 0.9–0.5 MMSN disks, Neptune reached ∼25–30 AU after a few hundred Myr (Figure 22). Conversely, the giant planet ended at smaller orbital radii for less massive disks. These results are valid for disks with and without planetoids.

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Therefore, provided that the disks evolved collisionally before Neptune's migration, acquiring total masses <1 MMSN, Neptune can stop at 30 AU without requiring the disk to be truncated at 30–35 AU. It is worth noting that this can be true even in 50–60 AU sized 1 MMSN disks suffering run-out of feeding planetesimals (damped migration), which can be caused by asymmetries in the planetesimal disk. In addition, a close encounter or giant collision with a massive ancient planetesimal during migration could also change the migration behavior of the planet, leading to damped migration or even forcing the planet to stop migrating (Ida et al. 2000b; Gomes et al. 2004; Murray-Clay & Chiang 2006; Chiang et al. 2007; Levison et al. 2007). Such giant impacts involving large planetesimals and giant planets are the most accepted theory to explain the obliquities of Neptune and Uranus (Brunini et al. 2002; Lee et al. 2007). In support of these hypotheses, Neptune acquired 30 AU in one of our 1 MMSN disks due to the lack of planet-encountering planetesimals (Figure 22). In addition, giant collisions of planetoids with Uranus and Neptune occurred in two other runs within the first 100 Myr. Due to the impact events, the planets were dislodged by ∼0.5–1 AU, significantly affecting their migration behavior. Finally, mutual resonance crossings between giant planets also altered the migration evolution of Uranus and Neptune due to quick radial displacements of both planets.

Resonant configurations are a natural outcome after all dissipating forces are gone (e.g., gas drag, tidal effects, disk torques). Indeed, planets, satellites, and other minor bodies have a preference for commensurabilities in the solar system (Dermott 1973). Giant planets are in near-mutual resonances, namely 2:5 (Saturn–Jupiter), 1:3 (Uranus–Saturn), and 1:2 (Neptune–Uranus) (Malhotra 1998). Prominent resonance structures do exist in the asteroid belt and the trans-Neptunian region (Nesvorný et al. 2002; LM07b). Pluto is in 3:2 resonance with Neptune. Resonant configurations have been found in extrasolar systems too (Udry et al. 2007). Therefore, as discussed in Section 6, the possibility of the outer planet occurring inside a distant resonance with Neptune is in line with the above reasoning.

The scenario presented here relies essentially on the capture of the outer planet in an r:1 (or r:2) resonance (Sections 3.3 and 6). In addition, according to the best results, the planetoid must have spent at least tens of Myr within about 60–80 AU before migration (see also Section 7.7). Which r:1 resonances were located within this distance range in the past? There is not yet consensus about Neptune's location at the beginning of planet migration. Thus, if the giant planet was initially located within 15–24 AU prior to migration, the best candidate r:1 resonances to trap the planetoid range from 6:1 to 12:1. Alternatively, those resonances or even farther r:1 resonances—in particular, similarly strong 13:1 and 14:1 resonances—could lock the planetoid after planet migration (Table 7; See also Section 9.2). What would be the likelihood of the capture of the planetoid into these resonances?

Recent studies using planet migration have shown that the capture probability of initially scattered bodies in the 9:4, 7:3, 5:2, 8:3, and 3:1 resonances are 0.5–1%, 0.5–1.2%, 2–3.5%, 0.5–1.2%, and 2–3%, respectively (Lykawka & Mukai 2007a). In supplementary simulations, we found capture probabilities of 1.5–2.6% and 1.3–2.2% in the 3:1 and 4:1 resonances, respectively (using τ = 1–10 Myr). Capture probabilities are linked to resonance strength and stickiness, both of which can be determined quantitatively (e.g., Gallardo 2006b; Lykawka & Mukai 2007c). Based on these quantities, we calculated capture probabilities in the 6:1, 7:1, ..., 14:1 resonances to be within a factor of ∼0.25–1 times that found for the 3:1 and 4:1 resonances, finding roughly 0.5–3% for the r:1 resonances considered in our scenario (smaller for farther resonances) (Table 7). These capture probabilities are quite low. Nevertheless, there are at least two important points favoring captures in an r:1 resonance. First, during pre-migration stages, the giant planets were in a more compact orbital configuration. Consequently, all of Neptune's resonances were mutually closer. For instance, for initial locations of Neptune at 15–20 AU, the r:1 resonances of interest are spaced only 5–10 AU apart (or about half if we include r:2 resonances). Therefore, a scattered planetoid would have traversed a distance range (e.g., 60–80 AU) with a higher density of r:1 and r:2 resonances. Second, the probability of capture in these high-order resonances is maximized at high eccentricity, a region of space in which resonance width is wider (e.g., Murray & Dermott 1999). Thus, because the outer planet was initially scattered by Neptune, its eccentric orbit would favor resonance capture (see also Figure 23).

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In conclusion, the capture of the outer planet in an r:1 (or r:2) resonance during the early solar system appears likely. Moreover, recalling that large populations of massive planetesimals have appeared in the past, the likelihood of this scenario becomes plausible and consistent.

On the basis of orbital distributions and the properties of objects at the end of our simulations (especially distant resonant populations and detached bodies around 50–60 AU), we found that the ancient trans-Neptunian belt appeared to have a radius of at least 50 AU. Moreover, disk planetesimals suffered eccentricity excitation with negligible radial changes. This disk size thus not only supports the formation of the observed distant resonant populations (Lykawka & Mukai 2007a), but also suggests that the planetesimal disk extended beyond the apparent outer edge at a ≈ 48 AU. Further observational evidence is the existence of two detached TNOs, 2003 UY₂₉₁ (a = 49.3 AU; q = 41.3 AU) and (48639) 1995 TL₈ (a = 52.6 AU; q = 40.0 AU). Interestingly, their stable orbits and proximity to the outer trans-Neptunian belt suggest that these objects represent the continuation of the classical region beyond 48 AU. Therefore, 2003 UY₂₉₁ and 1995 TL₈ could be members of the primordial planetesimal disk prior to eccentricity pumping. If true, the minimum size of the ancient trans-Neptunian belt would be about 53 AU.

The future identification of long-term populations in distant resonances can also constrain the extension of the primordial planetesimal disk. According to our scenario, we found that the number of bodies, orbital elements, and other properties in the 5:2, 8:3, 11:4, and 3:1 resonances were different for disks extended to ∼51–54 AU and 60 AU (see Table 5).

In more extended trans-Neptunian belts (>53 AU), local planetesimals perturbed by the outer planet acquired moderately eccentric orbits and remained after 4 Gyr, forming excited wings beyond 50 AU (Figures 11, 13, and 14). In principle, these orbital structures are detectable by surveys. However, the timescale for accretion growth is proportional to R³. Considering that the largest cold classical TNO at 45 AU has a diameter of about 600 km (assuming an albedo p = 0.05), its size at 55 and 65 AU would be a factor of 0.55 (330 km) and 0.33 (200 km) for the same growth time span, respectively. If planetesimals were perturbed by the planetoid during their formation era, the excited eccentricities and inclinations would lead to nonaccreting collisions (see Equation (2) and Section 7.8). Therefore, TNOs formed in situ beyond 48 AU may have suffered a growth cutoff. As a result, these distant local TNOs may be intrinsically smaller, perhaps with maximum diameters within 150–200 km, thus justifying their present nondetection by surveys (Trujillo et al. 2001a; Allen et al. 2002). Lastly, the characterization of an anomalous size distribution among some TNOs at a > 45–60 AU would also support the idea of interrupted growth. Indeed, the distribution of the largest (i.e., brightest) classical TNOs showing an apparent preference at a < 45 AU appear to support the above scenario (see also Lykawka & Mukai 2005b). The results of state-of-the-art collisional models and their further comparison with physical properties of TNOs provide additional evidence for external perturbers in stirring the primordial planetesimal disk (Kenyon et al. 2007).

In this section, we briefly discuss the production of some TNOs of interest according to our scenario.

(148209) 2000 CR₁₀₅ and (90377) Sedna. These objects are currently the most distant detached TNOs, and their origin is of great importance for understanding the history of the solar system (Gladman et al. 2002; Brown et al. 2004). The outer planet model produced several analogs of 2000 CR₁₀₅ in almost all runs, even with different initial conditions for the planet (Figures 19 and 20). The 2000 CR₁₀₅-like bodies were usually first scattered by Neptune and later detached from the giant planet's domain by external perturbation from the outer planet. Although the model had difficulties in creating Sedna-like bodies with less massive planetoids (≤0.5 M_⊕) at a_P < 160 AU (Figures 19 and 20), more "Sednas" were commonly obtained in runs using planetoids with M_P = 0.7 M_⊕ and 1 M_⊕, and/or located at a_P = 200–250 AU (the outermost boundaries for resonance interactions to work. See also Figure 20). The analogs of Sedna originated in two possible ways: (a) near-circular objects located at 120–210 AU, which were later scattered by the outer planet (Sedna's accretion would be possible at these distances; Kenyon & Bromley 2004b; Stern 2005); and (b) objects initially scattered by Neptune that were detached by the influence of the planetoid. In both cases, long timescales (>1 Gyr) were a common requirement. Lastly, we stress that our model is able to produce more extreme detached TNOs with q ∼100–200 AU and various inclinations. Alternative scenarios to produce Sedna-like objects include a passing star (Kenyon & Bromley 2004b; Morbidelli & Levison 2004), formation of the solar system in a dense star cluster (Brasser et al. 2006), massive (1–2 M_⊕) rogue planets that may have roamed the solar system at hundreds of AU in the past (Gladman & Chan 2006), and a solar companion model (Matese et al. 2005).

2004 XR₁₉₀. This is a very high i object in a detached orbit located close to the planetary system (a = 57.5 AU; q = 51.2 AU). The discovery of a TNO with such orbital properties was unexpected (Allen et al. 2006). In our model, analogs of 2004 XR₁₉₀ were obtained in several runs due to the planetoid's perturbation. A few are visible in Figures 19 and 20. It is noteworthy that all 2004 XR₁₉₀-like objects originated from Neptune-scattered orbits, suggesting that this TNO was a scattered body in the past. A similar orbital history has been suggested previously (Allen et al. 2006; Gladman & Chan 2006). Alternatively, being close to the 8:3 resonance, 2004 XR₁₉₀'s small e and very high i could arise through the KR, which is expected to induce such e–i anticorrelated configurations. However, the likelihood of being an 8:3-resonant TNO is only 4–5% (Allen et al. 2006; LM07b). On the other hand, the TNO could have left the 8:3 resonance in its very high i phase during planet migration (Gomes et al. 2007).

(136199) Eris. This planetary-sized very high i body is classified as a scattered TNO (LM07b). We obtained some analogs of Eris at a < 100 AU in our simulations (i = 40–45°). Although the outer planet excited inclinations, Eris-like objects managed to remain with q < 40 AU at the end of the runs (e.g., Figure 19). Therefore, a probable scenario for the history of Eris would be formation near Neptune, further scattering by the giant planet, and extra i excitation by the outer planet. It is worth noting that this agrees with the idea that a small fraction of Pluto-sized objects that were initially located near the giant planets were deposited in the trans-Neptunian belt (Stern 1991).

(134340) Pluto. The largest TNO known thus far in the 3:2 resonance was fully reproduced in our model. Pluto-like objects were obtained in substantial number in the 3:2 resonance, reproducing not only Pluto's orbital elements, but also its libration amplitude and the locking in KR over the age of the solar system (Figures 16 and 18). This supports the idea that Pluto was initially located in a near-circular orbit closer to the Sun during the capture by the sweeping 3:2 resonance (Malhotra 1995; Hahn & Malhotra 2005).

(136108) 2003 EL₆₁. Apart from the Pluto quadruple system, 2003 EL₆₁ is currently the only other known multiple system in the trans-Neptunian belt, with two satellites (Brown et al. 2005b, 2006b). In addition, 2003 EL₆₁ is currently locked in the 12:7 resonance (a = 43.1 AU) (LM07b). According to our results, there were four 12:7-resonant objects, but only two of them survived over 4 Gyr. However, the orbits of these objects are not compatible with that of 2003 EL₆₁. Interestingly, 2003 EL₆₁ appears to be part of a collisional family (Brown et al. 2007), supporting the giant impact theory for the creation of its satellite system. Therefore, we think that 2003 EL₆₁ was probably captured in the 12:7 resonance after the collision event; otherwise, it would have been dislodged from the resonance with such a quite energetic impact.

2003 UY₂₉₁ and (48639) 1995 TL₈. The outer planet model can reproduce the orbital elements of these detached TNOs quite easily (e.g., Figure 16). Our results suggest that both of these objects are part of the excited outer region of a primordial planetesimal disk of at least 53 AU radius (Section 8.2).

Several TNOs have been identified as binary or multiple systems (Noll et al. 2007). Interestingly, there is a statistically significant preference for binarity among classical TNOs with i < 10° compared with the other TNO populations (approximately a 3–4 times greater fraction) (Stephens & Noll 2006). Because close gravitational encounters are likely to disrupt weakly bound binaries, this preference can be interpreted as evidence that low-i classical TNOs suffered little perturbation from the giant planets, in contrast to the active dynamical evolution of high-i classical, resonant, scattered, and possibly detached populations. Alternatively, early conditions in the planetesimal disk may have favored binary formation in specific regions of the disk. Thus, more binary systems may have formed at about 35–50 AU than in the inner regions of the disk, so that the binarity preference among low-i classical TNOs would represent a primordial feature (Noll et al. 2006; Noll et al. 2007 and references therein). In any case, either possibility is consistent with the view that cold classical TNOs formed in situ (Section 2.1).

According to our findings (Sections 7.2 and 8.2), the bulk of the population of classical bodies with i < 10–15° formed in situ and extended to at least a = 50 AU. These bodies were perturbed solely by the outer planet before migration (e.g., Figure 11) and later survived for 4 Gyr essentially unaffected by any of the planets. Thus, the preference for binarity among low-i classical objects could result from the lack of continuous perturbation from the giant planets and the planetoid over 4 Gyr and/or formation in situ in the classical region, where supposedly more binary systems were formed during the pre-migration epoch. The binary TNOs 1995 TL₈ (a = 52.6 AU; i = 0.2°) and 2002 GZ₃₁ (a = 50.2 AU; i = 1.1°) provide additional evidence for our scenario. That is, both TNOs have orbital properties compatible with those obtained for objects at the excited wings around 50–53 AU in our simulations (Figures 12–14), suggesting that both TNOs are members of the primordial planetesimal disks that extended to 53 AU (see also Section 8.2).

In accordance with the results of Melita et al. (2004), we found that the strength of the planetoid's perturbation depends essentially on a_P, its relative longitudes with disk planetesimals, and M_P (e.g., Figure 12). According to our self-consistent simulations (Section 7.9), the dynamical evolution of a planetoid in a massive planetesimal disk is complex and stochastic. The planetoid's initially scattered orbit and the properties of the disk such as its total mass and size can lead to diverse evolutionary paths. If the disk is relatively small, the effect of dynamical friction is reduced, implying that the outer planet could survive a much longer time in a typical scattered disk orbit. In extended disks (e.g. 100 AU), dynamical friction is stronger and the timescales for circularization and flattening of the orbit (i near the plane of the disk) become smaller.

Figure 23 illustrates two representative examples of the evolution of scattered planetoids through massive disks. These bodies spent a few tens of Myr under the control of r:1 resonances during different stages of the pre-migration systems. Although not exhaustive, these examples show that the early capture of a large planetesimal in resonances would be possible.

Below, we provide evidence supporting the current existence of a trans-Plutonian planet as described in our scenario.

A prominent population of detached TNOs. To explain the detached population and, in particular, its large total population number (e.g., its intrinsic ratio to the scattered population is ≥1.0), a quite broad inclination distribution (currently from a few degrees to almost 50°), and peculiar members (e.g., 2004 XR₁₉₀, 2000 CR₁₀₅, Sedna), the perturbation of the planetoid over 4 Gyr is probably the most plausible mechanism (see Sections 2.1, 2.2, and 7.5). Furthermore, other distant and detached TNOs were reproduced only when the gravitational influence of the outer planet lasted over billions of years. Note also that even very massive outer planets (1 M_⊕) are apparently unable to explain all these features in timescales less than 1 Gyr (Table 6);

A population of TNOs with inclinations larger than 40°. As demonstrated in Section 2.1, the existence of a significant fraction of very high i TNOs cannot be explained according to previously published models. On the other hand, in our scenario, apart from gravitational scattering by Neptune, the planetoid provides an extra perturbation that, on average, increases the inclinations of objects in the whole trans-Neptunian region (see Figure 19). Finally, to better match the estimated intrinsic fractions of the very high i subclass, the outer planet's perturbation had to last longer than 3 Gyr (see also Sections 7.4–7.6);

Lack of long-term resonant populations beyond the 8:3 resonance. Although 3:1-, 7:2-, and 4:1-resonant populations stable over 4 Gyr (located beyond the 8:3 resonance) could occur in the trans-Neptunian region according to recent models (Hahn & Malhotra 2005; Lykawka & Mukai 2007a), no evidence exists of such populations (LM07b). Despite the low numbers of TNOs in the scattered disk, this intriguing feature is unexplained by other scenarios. However, here the absence of long-term resonant TNOs at the 3:1 (at high eccentricities) and greater resonances can be well explained by the gravitational perturbation of an outer planet with q_P = 80–90 AU in timescales comparable to the age of the solar system (e.g., see Sections 5.3 and 7.3.1).

Taking together all the aforementioned observational evidence, we conclude that a massive undiscovered trans-Plutonian planet is required to explain them consistently.

The possible capture in a distant resonance suggests a_P ∼ 100–175 AU near current locations of 6:1–14:1 resonances. It is noteworthy that large e–i variations and long-term resonant capture during the planetoid's orbital evolution would have been more likely to occur in those resonances (Lykawka & Mukai 2007c and references therein). However, the planetoid may have left its resonant configuration during the stochastic migration of Neptune, after dynamically diffusing at low eccentricity or by dynamical friction (Hahn & Malhotra 1999; Zhou et al. 2002; Gomes et al. 2007). Indeed, stochastic jumps in semimajor axis caused by continuous scattering may have allowed the planetoid to encounter resonances beyond 175 AU. Nevertheless, the presence of sufficiently strong resonances is limited at a ≤ 250 AU (Gallardo 2006a, 2006b; Lykawka & Mukai 2007c). Finally, as discussed in Sections 5.3 and 6, the model suggests q_P > 80 AU and i_P = 10–50°, although 20–40° were apparently more commonly obtained in self-consistent simulations.

Concerning the physical properties of the outer planet, ideally we would like to constrain these from the distributions of planetary-sized TNOs. However, the sample of objects with determined physical properties is still very small (Luu & Jewitt 2002; Delsanti & Jewitt 2006; Delsanti et al. 2006; Cruikshank et al. 2007).

Simultaneous measurements of reflected and emitted light from the object of interest permit us to calculate albedos and diameters using well-established formulas (Russell 1916; Jewitt & Sheppard 2002; Jewitt 2008),

$$\begin{equation} p\left({\frac{D}{2}} \right)^2 \Phi (\alpha) = 2.25 \times 10^{16} R^2 \psi ^2 10^{0.4({m_ * - m})}, \end{equation} \tag{ 4 }$$

where D is in kilometers, Φ(α) is the phase function, R is in AU, ψ is the geocentric distance (AU), and m is the apparent magnitude of the object (m_* represents that of the Sun). For the sake of simplicity, we assume that the phase function is negligible for our purposes.

A massive trans-Plutonian planet would very probably be a differentiated body with a rocky interior and layers composed of ices (e.g., De Sanctis et al. 2001; Merk & Prialnik 2006). In such a case, we can expect mean densities of ∼2–3 g cm⁻³. This density range is consistent with 2 g cm⁻³, 2.3 g cm⁻³, and 2.6 g cm⁻³ densities determined for Pluto, Eris, and 2003 EL₆₁, respectively (Stern 1992; Rabinowitz et al. 2006; Brown & Schaller 2007; Lacerda & Jewitt 2007). We then estimated the planetoid's diameter assuming a spherical shape, M_P = 0.3–0.7 M_⊕, and a mean density of 2–3 g cm⁻³, finding a D_P of ∼10,000–16,000 km. Using the equation of density as a function of diameter given in Jewitt (2008), we obtain 2.1–2.4 g cm⁻³ for a 10,000–16,000 km sized body.

Considering the planetoid's orbital properties given above (a_P ≥ 100 AU; q_P > 80 AU), we assumed an albedo p_P = 0.1–0.3 for the outer planet, which agrees with expected values for hypothetical distant icy planetoids (Stern 1991). For instance, a planet with q_P ≈ 100 AU would be too far to hold an atmosphere; so surface darkening would be expected (∼0.03–0.3). The lower albedos of these distant bodies may be a consequence of long exposure to space weathering because icy surfaces are expected to get darker from continuous UV irradiation and bombardment of solar and galactic ions (Thompson et al. 1987; Moroz et al. 1998; Cooper et al. 2003; Brunetto et al. 2006). Other potential reasons include very low temperatures that inhibit the sublimation of volatiles and negligible collisional resurfacing effects. Furthermore, our albedo assumption above also agrees with Sedna's albedo, which was constrained in the range 0.1–0.3 (Emery et al. 2007 and references therein). Sedna is the best-known representative of distant large icy bodies.

The planetoid possibly accreted near Uranus and Neptune during the pre-migration period (probably at the 10–20 AU region); so it would be composed mainly of CH₄, CO, or N₂, which are the most common outer solar system volatiles. Given its appreciable mass, the outer planet would probably have large reservoirs of icy species on the surface. For instance, recent spectral measurements support the existence of large amounts of CH₄, H₂O, and N₂ on the surface of planetary-sized TNOs (Brown et al. 2005a; Barkume et al. 2006; Licandro et al. 2006a; Licandro et al. 2006b; Rabinowitz et al. 2006; Dumas et al. 2007; Tegler et al. 2007; Trujillo et al. 2007). Besides, due to the orbital characteristics and large diameter of the planetoid, it should have retained its surface ice over the age of the solar system (Schaller & Brown 2007). Therefore, we should expect an inactive surface and the presence of heavily space-processed ice on the planetoid.

Finally, using Equation (4) with the derived ranges of D_P and p_P above, we found m_P ∼15–17 mag for a planetoid with q_P = 80–90 AU during perihelion approach. The predicted apparent magnitude is uncertain because the planetoid's orbital properties and assumed albedo cannot be more strictly constrained.

First, it is important to understand some physical quantities derived from surveys and the strategies used in the surveys. The cumulative luminosity function (CLF) provides the cumulative number of TNOs per sky surface area at a given limiting magnitude. The CLF is given by log [Σ(m)] = b(m − m₀), where Σ(m) is the number of objects per square degree brighter than the apparent magnitude m, b is a constant defining the slope of the curve, and m₀ is a constant for which Σ = 1 deg⁻². Recent studies suggest that b = 0.6–0.8, and m₀ ≈ 23 mag (Trujillo et al. 2001b; Bernstein et al. 2004). The size distribution is usually derived from the CLF, described by a power law with exponent u = 5b + 1 (Bernstein et al. 2004 and references therein).

A straightforward interpretation of the CLF implies smaller sky densities for brighter bodies (bigger TNOs), and larger sky densities for fainter ones (smaller TNOs). Therefore, large TNOs have a greater probability of being discovered in volume-limited surveys (i.e., wide-area surveys) than in flux-limited surveys. Regarding the outer planet, we estimate that there would be one object every ∼10⁴–10⁵ square degrees as bright as m_P ∼ 15–17 mag (at perihelion); hence, it would be rare in the sky.

Surveys usually concentrate the search near opposition when a TNO has the highest relative sky motion due to parallax. Indeed, the probability of discovery of large TNOs is also affected by their apparent rate of movement across the sky, which can be determined using

$$\begin{equation} \Theta \cong \frac{{148}}{{R + \sqrt R }}{\rm (arcsec}\;{\rm h}^{ - 1}), \end{equation} \tag{ 5 }$$

where R is in AU. For typical trans-Neptunian belt orbits (30–60 AU), TNOs move at a rate of 2–5 arcsec h⁻¹. Because the majority of surveys has used the strategy of finding TNOs that move at such apparent sky motions, they are sensible at most to 1.5 arcsec h⁻¹ (Brown et al. 2004). For example, when Eris and Sedna were discovered (the most distant TNOs discovered thus far), they were moving at similar critical rates, i.e., 1.42 and 1.75 arcsec h⁻¹, respectively (Trujillo & Brown 2003; Brown et al. 2005a). In our scenario, for an outer planet with q_P = 80–90 AU, we obtain Θ_P = 1.5–1.7 arcsec h⁻¹ during perihelion approach, near the lower limit discussed above. In addition, having a moderately large e_P (Table 7), the outer planet would also spend more time near aphelion during its orbit, thus implying smaller apparent rates (i.e., more difficult to detect). Only a few recent wide-area surveys have probed apparent motion below these critical values (Larsen et al. 2007).

Past wide-area surveys have searched for very bright bodies in the outer solar system (m < 17 mag). Tombaugh (1961) concluded that there were no objects brighter than m = 15.5 mag in the entire sky north of a declination −40°, and no bodies with m < 17.5 mag for any object near the ecliptic (see also Trujillo et al. 2001b). Kowal (1989) surveyed the sky near the ecliptic for objects with m < 20 mag and found no TNOs. However, photographic plates have poor sensitivity for slowly moving objects and can present surface defects (Trujillo et al. 2001b). Except for Pluto, only recent planetary-sized TNOs have been discovered by dedicated surveys (Brown et al. 2004, 2005a). For reference, Pluto, Eris, 2003 EL₆₁, 2005 FY₉, and Sedna had apparent magnitudes of 14, 18.8, 17.5, 16.8, and 21 at discovery, respectively (Brown et al. 2006b). Other wide-area surveys have not found any very bright TNOs within about 10° (Sheppard et al. 2000; Trujillo & Brown 2003; Jones et al. 2006; Larsen et al. 2007) (see also Figure 24).

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In addition, recalling that the planetoid would have an inclined orbit (20–40°), we stress that (a) high-i objects are approximately four times more likely to be discovered at ecliptic latitudes, β ≈ i, than in the ecliptic (β = 0°); (b) Σ(m) is smaller at higher β and drops off quickly (for instance, the sky density at 20° is about one order of magnitude smaller than that near the ecliptic); and (c) among high-i objects (20–40°), the fraction of an orbit spent near the ecliptic (β = 0–10°) is only ∼1.5–4% (Trujillo et al. 2001a). In general, because wide-area surveys have searched near the ecliptic, the discovery of objects with higher inclinations has been less likely, even in surveys probing very slowly moving objects (Tombaugh 1961; Luu & Jewitt 1988; Kowal 1989; Jewitt & Luu 1995; Jewitt et al. 1998; Sheppard et al. 2000; Gladman et al. 2001; Trujillo et al. 2001a, 2001b; Trujillo & Brown 2003; Jones et al. 2006; Larsen et al. 2007).

Therefore, a massive trans-Plutonian planet has probably escaped detection because it is currently either far from the Sun (with sky motion below survey sensibility) or away from the nodal crossing point with the ecliptic. Apart from the planet's properties, the probability of nondetection should be ∼10–15% because surveys avoid the region near the galactic plane (Chiang & Jordan 2002; Trujillo & Brown 2003). Other potential biases that may have prevented the detection of massive bodies in distant orbits are discussed in Bernstein & Khushalani (2000) and Horner & Evans (2002).

The planets, planetary embryos, and other TNOs formed in <100 Myr (Pollack et al. 1996; Kenyon 2002; Kenyon & Bromley 2004a). Uranus and Neptune formed within ∼10–20 AU. The planetesimal disk was dynamically very cold (nearly circular and i ≈ 0) and at least 53 AU in size.

During the late stages of planet formation, a large planetesimal (a planetoid with 0.3–0.7 M_⊕) was scattered by Neptune in a few Myr, experiencing an a_P ∼60–80 AU, q_P near the location of Neptune, and probably a moderate i_P (e.g., 10–15°).

The outer planet excited the outer region of the planetesimal disk for several tens of Myr. This produced the first hot classical TNOs (moderate i). At the same time, the outer planet's perturbation also disrupted the disk at 48 AU.

During planet migration, all resonant TNOs and the remaining hot classical TNOs were created.

During or just after the end of planet migration, the outer planet was captured by a distant r:1 resonance (e.g., 6:1). The outer planet likely evolved into a KR orbit, which forced the planet to decrease its eccentricity and increase the inclination in 100–200 Myr (q_P > 80 AU, i_P = 20–40°). The trans-Plutonian planet was detached from the solar system, remaining in a stable orbit at ∼100–175 AU (the exact location depends on the current specific r:1 resonance location).

The giant planets and the planetoid gravitationally sculpted the trans-Neptunian belt over billions of years. The belt's outer edge appeared as unstable objects left the region around 45–53 AU. Scattered and detached TNO populations were produced mainly during the long-term phase. Cold classical TNOs, formed in situ, represent relics of the local disk planetesimals. The region beyond 30 AU lost 99% of its initial total mass by dynamical depletion and enhanced collisional grinding.

• 1.  
  It explains the lack of TNOs up to 39 AU (Figures 13 and 14).
• 2.  
  It explains the dual nature of the classical region (cold and hot populations), including their orbital excitation and distinct physical properties. The final distributions in eccentricities are remarkably similar to observations (Figure 16).
• 3.  
  It reproduces the resonant structure in the entire trans-Neptunian region, including eccentricities, inclinations, dynamical properties (e.g., libration amplitudes), and KR members (Figures 14, 16, and 18 and Table 5).
• 4.  
  It reproduces the population of scattered TNOs with their distribution of orbital elements, including analogs of Eris (Figures 16 and 17).
• 5.  
  It produces a substantial population of detached TNOs, including analogs of 2004 XR₁₉₀, 2000 CR₁₀₅, and Sedna. The obtained detached population is in agreement with intrinsic fraction estimates for this population in the scattered disk (Figures 17, 19, and 20 and Table 6).
• 6.  
  It produces low-i detached TNOs (Figures 19 and 20).
• 7.  
  The very high i population can be produced by the simultaneous perturbation of the outer planet and the action of the KR mechanism inside resonances (Figures 19 and 20).
• 8.  
  It produces the trans-Neptunian belt outer edge at 48 AU without threatening stable resonant populations, namely those in the 3:2, 2:1, and 5:2 resonances (Figures 13, 14, and 16 and Table 3).
• 9.  
  It reproduces the abrupt decrease in number density of TNOs beyond 45 AU (Figure 21).
• 10.  
  The problem of the missing mass of the trans-Neptunian belt can be solved by the simultaneous action of enhanced collisional grinding induced by the planetoid and dynamical depletion.
• 11.  
  It explains Neptune's current orbit at 30.1 AU (Figure 22).

The following predictions are testable, so future observations will help to confirm and/or improve the model.

10.3.1. Outer Solar System

10.3.2. Tentative Properties of the Resident Trans-Plutonian Planet