Kuiper belt dynamics
Kuiper belt dynamics studies the orbital motion of Kuiper belt objects. In a broad sense, the Kuiper belt can be defined as the collection of the small bodies that orbit the Sun beyond Neptune. With this definition, the Kuiper belt is also named "trans-Neptunian population". In a more restrictive sense, the Kuiper belt includes only trans-Neptunian objects whose orbital period is shorter than twice Neptune's orbital period, or, in other words, objects that are within the 1:2 resonance with Neptune. This is usually referred to as the "classical" Kuiper belt.
Kuiper Belt and Solar System Dynamics
The distribution of the orbits within the Kuiper belt reflects not only the dynamics in the framework of the current Solar System but also the dynamical events that characterized the early evolution of the Solar System. In this sense, the Kuiper belt orbits can help us to elucidate the dynamical history of the primordial Solar System.
G. Kuiper (1951) predicted the existence of a disk of icy bodies just beyond Neptune. This disk was supposed to be the remnant of a primordial accretion disk that was not able to sustain the coagulation process until the formation of full-mass planets. This disk was expected to be composed of asteroid-sized icy objects, in roughly circular and co-planar orbits, that never experienced important gravitational effects from the giant planets. Soon after the first trans-Neptunian object was discovered in 1992 (Luu and Jewitt 1993), it became evident that many objects have elongated orbits (high eccentricities) and/or large orbital inclinations relative to the invariable plane (the plane that contains the system barycenter and is perpendicular to its angular momentum vector). These unexpected orbits cannot be explained by ongoing dynamical processes, but are necessarily a consequence of the evolution of the early Solar System. Thus, below we give a brief description of primordial Solar System dynamics.
The most plausible scenario for the formation of the Solar System planets assumes that the primordial solid components (planetesimals) coagulated little by little into larger bodies until the planets were formed. But this mechanism did not happen in a uniform way throughout the original disk. In particular, the accretion timescale increases rapidly with the distance from the Sun. Consequently, at some early time, already formed giant planets should have co-existed with a disk of planetesimals located just outside the outermost planet's orbit. After the disappearence of the gas of the proto-planetary disk, the gravitational interactions of the planets and planetesimals should have induced a migration on the planets (i.e. a change of their orbital semi-major axes), while the planetesimals were scattered away. This process may have happened in a smooth way (Hahn and Malhotra 1999) or in a more stochastic way, where also the planets happened to have close encounters with each other (Thommes et al., 1999 and the so-called "Nice model": Tsiganis et al 2005; Morbidelli et al., 2005; Gomes et al., 2005). In both models, planetesimals scattered by the planets could reach large distances from the Sun, all the way up to the Oort cloud. However, achieving large distances is not enough to populate the Kuiper belt. Some mechanism has to decrease the orbital eccentricities of the objects (i.e. raise their perihelion distance) so that the latter do not suffer further encounters with the giant planets. These mechanisms are usually related to resonances in the Solar System. A brief tutorial of resonances follows.
- Mean motion resonance: Two objects are in mean motion resonance (MMR) when the ratio of their orbital periods can be represented by the ratio of two small integers. This is equivalent to say that two objects in resonance will be aligned with the Sun at specific points of their orbits. Resonance configurations tend to enhance the mutual gravitational interaction between the resonant objects. This can either help to stabilize or to destabilize the orbits.
- Secular resonance: Secular resonances do not involve directly the motion of the bodies along their orbits, but the variations of the orbits themselves. Orbits that are not perfectly circular, or are not exactly in the reference plane, have preferential directions in space, defined by the direction of the perihelion and by the line defined by the intersection of the orbital plane with the reference plane (the line of nodes). When the difference between the perihelia or the nodes of two interacting orbits oscillates around a fixed angle, the objects are said to be in secular resonance. These resonances are very powerful in producing large oscillations of the eccentricity and inclination.
- Kozai resonance: Named after Kozai (1962), this resonance occurs when the direction of the perihelion of an orbit oscillates around a fixed direction with respect to the line of nodes. An interesting feature of the Kozai resonance is that the eccentricity and inclination of an orbit show large coupled oscillations.
The present orbital configuration of the Kuiper belt reflects mostly the primordial planet-disk interactions. Some features, however, are a consequence of the present orbital architecture of the giant planets. The most noteworthy one is the deficit of objects in the region with semi-major axes between 40 and 43 AU. This is caused by the action of secular resonances. In this region the perihelion of an object precesses with the same frequency as Neptune's, causing a great increase in the object's eccentricity and the subsequent removal of the object due to encounters with the planet, on a relatively short time scale. An inclination type secular resonance (the node precesses with Neptune's node) also appears in this region, causing an important inclination increase. This does not cause the removal of the object, since a sole inclination increase cannot induce close encounters with the planet. Once at high inclination, particles can survive at this region, as shown in Figure 1.
Trans-Neptunian PopulationsIt is instructive to classify the trans-Neptunian objects (TNO's) into several groups according to their orbits. A brief description of these populations is given below (see Figures 1 and 2 for reference).
Classical Kuiper belt:
The first trans-Neptunian objects discovered (after Pluto, see below) belonged to what is presently known as the classical Kuiper belt. This corresponds to objects with orbits beyond Neptune but not beyond its 1:2 resonance and which are presently not being scattered by Neptune and not in any resonance. It is also understood that there are two populations in the classical Kuiper belt that statistically differ in physical properties and orbits. These are usually referred to as the "cold" and "hot" populations. Objects belonging to the cold population have low inclination (below 4⁰-5⁰), red surfaces and diameters below ~200 km. The hot population objects have orbital inclinations distributed in a wide range of values from 0⁰ to ~50⁰. Their largest members are as big as 2500 km in diameter and are more neutral in color than the cold population objects. These orbital/ physical differences suggest a different origin for these two populations.
The hot classical belt population:
Most recent formation scenarios argue for an outward transport of the hot population from a primordial planetesimal disk. This transportation is accomplished through the scattering of planetesimals by Neptune during the early evolution of the Solar System, described above. Some resonant perturbations can cause oscillations of the eccentricity/perihelion of the planetesimals. These perturbations include mean motion resonances, Kozai resonance and secular resonances. Lifting the planetesimal perihelion while Neptune is still migrating may cause a release of the orbit from resonance, so that the planetesimal can be deposited in a fairly stable region of the Kuiper belt, however having a large inclination (increased by successive close encounters with the planets) and a relatively low eccentricity, thus forming the Kuiper belt hot population (Gomes 2003).
The cold classical belt population:
It is still disputable whether the cold population was transported like its high inclination counterpart or just has a local origin. The most obvious reason to support the latter claim is that the cold population differs a lot from the hot population; in contrast, scenarios that transport the cold population into its current region from an original location closer to the Sun, require that the cold population formed not very far from the hot population, hardly explaining the physical differences observed between the two populations.
On the other hand, there are good reasons to expect that the cold population did not form where it is now: (1) The present Kuiper belt (in special its cold population) total mass is not larger than 0.1 Earth mass, possibly around 0.03 Earth mass. (2) An original mass as small as that of the current cold population would not have been enough to build the largest bodies in the cold KB. (3) If the cold KB was more massive in the past (in order to be able to build the objects observed) it is unlikely that this mass could have been lost in a collisional grinding process, or by dynamical ejection. (4) If Neptune had migrated through a massive disk extended into the current Kuiper belt, it would not have stopped at 30 AU (its current location) but it would have continued its outwards drift up to ~48 AU (Gomes et al. 2004). All these arguments suggest that the current KB region never hosted a massive population of planetesimals and the current cold belt objects formed elsewhere, where the mass concentration was high.
It is also a remarkable characteristic of the cold Kuiper belt that its outer border is just inside the 1:2 resonance with Neptune (see Figure 1). Several proposed transporting mechanism would naturally deposit the cold population interior to the 1:2 resonance, in agreement with the observations. Instead, if the cold population formed in situ, its outer edge near the 1:2 resonance with Neptune would just be a coincidence. A mechanism that can transport the cold population from a primordial location closer to the Sun, preserving its low inclination (Levison et al. 2008a) invokes an temporary large eccentricity phase of Neptune during an instability phase of the giant planet system that is probably related to the origin of the Late Heavy Bombardment of the Moon (i.e. the Nice model: Tsiganis et al., 2005; Morbidelli et al., 2005; Gomes et al., 2005). This mechanism can bring preferentially particles from the outer edge of the disk to the cold population, whereas the hot population would come more evenly from the full planetesimal disk. This might possibly explain the physical differences between the two populations. On the other hand, a local origin for the cold belt must resort to an effective collisional evolution model (for recent results on this subject, see Kenyon et al. 2008)
The Resonant Population:
Resonant objects are in mean motion resonance with Neptune. The most populated of such resonances with Neptune is the 3:2, and Pluto is the largest representative of this resonant population, whose members are generically named plutinos. Pluto is the second largest so far known trans-Neptunian object and is presently classified as a dwarf planet. Like some other plutinos, Pluto's perihelion distance is closer to the Sun than Neptune, but a collision with the ice planet is avoided by a protecting mechanism associated with the mean motion resonance coupled with the Kozai resonance, which places Neptune at a wide angle from Pluto when the dwarf planet is near its perihelion. The plutinos probably had a similar origin to the hot population, as they share the same physical properties. The difference is that the plutinos are still in resonance while the hot population is not. Two scenarios aim at explaining the origin of the resonant orbits of the plutinos. In a smooth migration scenario (Malhotra, 1995), as Neptune migrates outwards by interaction with the planetesimals disk, the resonances with the planet migrate outwards as well. In doing so, they sweep through the planetesimal disk. Consequently, many planetesimals got captured in resonances; after capture, they migrate with the resonance, increasing their orbital eccentricities, by a classical mechanics phenomenom called adiabatical invariance. In the Nice model, planetesimals get trapped in mean motion resonances while they are still being scattered by the planet, which at the time had a more eccentric orbit. The eccentricities of the resonant particles were initially large, but they could decrease due to secular interactions with the eccentric orbit of Neptune or due to the Kozai resonance.
The most distant of the scattered disk bodies found to date is 2000 astronomical units from the Sun at aphelion. But this same object is as close as 24 astronomical units at perihelion, revealing a highly eccentric orbit. More generally, the characteristic of scattered objects is an orbit with semi major axis larger than 50 AU and a perihelion distance not far beyond Neptune's orbit, so that strong perturbations are felt by the object when it passes at perihelion in conjunction with Neptune (scattering events). These perturbations are capable of modifying the orbit in a non-periodic way. Consequently, all scattered disk objects are, by definition, on unstable orbits, which can radically change in a relatively short period of time. Possibly, the orbital changes can lead to the dynamical removal of the object. It is expected, in fact, that the scattered disk population has decayed in time, since the early epochs of the solar system. The scattered disk is considered to be the main source of Jupiter family comets (Levison and Duncan 1997) .
The Detached Population:
The detached objects also have very large average distances from the Sun but they differ from the scattered objects in that at their closest distance (perihelion) they are not close enough to Neptune to experience strong perturbations that would otherwise make their average distance from the Sun vary considerably. In other words, the detached objects are in stable orbits. Thus, in a sense we could differentiate a scattered object from a detached object by their perihelion distances. However, it is not a very good idea to fix a threshold value for the perihelion to distinguish detached from scattered objects. This is because being in a stable or unstable orbit may depend on other factors like being in resonance with Neptune. Perhaps the best way to distinguish these bodies is through a numerical simulation of the evolution of their orbits during a long enough time to check for possible large variations of the semimajor axis. Such simulations suggest that perihelion distances above 40 AU characterize detached objects, less than 35 AU characterize scattered objects, while in between these values we should check for its orbital behavior more carefully. The fact that in many cases it is difficult to distinguish a scattered object from a detached one may be a simple consequence of the fact that these orbits have a common origin. In fact the best explanation for the origin of a detached object is that it was once a scattered object that managed to get into a more stable orbit through resonance perturbations with Neptune. A good such example is TNO 2004 XR190, whose perihelion distance at 51.54 AU and semimajor axis at 57.62 AU defines it as a typical detached object. A good hypothesis for the origin of its orbit is that it was a past scattered object that became trapped in the 3:8 resonance with Neptune, had its perihelion distance increased by the Kozai resonance and escaped resonance while in low eccentricity (high perihelion), reaching a stable orbit near the 3:8 resonance with Neptune, with high perihelion distance and high inclination (Gomes 2011).
Sedna is a singular trans-Neptunian object that cannot be included in any of the populations so far defined. Its semimajor axis is at 529 AU, its perihelion distance at 76 AU and its inclination is 12°. Sedna's mean distance from the Sun associated to its remarkably high perihelion distance cannot be reproduced by the same process that probably created the detached population. The most likely scenario for its orbital generation also assumes that Sedna was in the past scattered by Neptune. But the mechanism responsible to raise its perihelion cannot be associated to the present orbital architecture of the Solar System. The most likely scenario to detach Sedna from the scattered disk supposes that while the Sun inhabited its primordial star cluster, Jupiter and Saturn were scattering remnant planetesimals that were not accreted to the gas giants (Brasser et al. 2006). Due to strong tidal perturbations from the primordial cluster some planetesimals had their perihelion increased and stayed like that until after the cluster dissipated. Another scenario invokes a still undiscovered planetary-mass solar companion (Gomes et al. 2006). This putative planet could have raised the perihelion of scattered objects and could still be perturbing these objects. There are other two objects that might also belong to Sedna's class. They are 2000 CR105 and 2004 VN112. The first one with a semi-major axis at 222 AU and a perihelion distance at 44 AU can have its orbit marginally explained by the resonance mechanism that creates the detached population but VN112 with semi-major axis at 348 AU and perihelion distance at 47 AU must preferably have its orbit explained by the same mechanism as Sedna.
The Origin of the Kuiper Belt
The best accepted scenario for the formation of most orbits of the trans-Neptunian population is based on an early transportation of ice planetesimals from a primordial disk, remnant of the primordial planetary formation. While the giant planets interacted with the planetesimals, the planets themselves migrated and the planetesimals were scattered by the planets producing orbits with large semi-major axes and perihelion near the planets orbit. The present scattered disk would thus be composed by the remnants of a previously much more numerous population of planetesimals, being scattered by the planets. They survived to present date due to chance or some temporarily active, protective resonant configuration. The resonant population would be formed by objects coming from a primordial scattered disk that managed to get into a mean motion resonance with Neptune and profit from a protection mechanism by the same resonance. The detached population would have been removed from close encounters with Neptune through resonant perturbations exerted by the ice planet, and managed to escape the resonance while Neptune was experiencing a slow, residual migration. The hot population in the classical Kuiper belt would have most likely experienced a past dynamical history similar to that of the detached population, although the characteristic of the past resonant episodes is more subtle. Another mechanism invoked to create the hot population considers collisions between pairs of scattered objects during the primordial phase of planetary migration (Levison et al. 2008b). This collision might have displaced the target orbit from the scattered disk into a stable orbit in the classical belt. So most of the TNO's must have had a common origin, except for the classical belt cold population, whose origin is still disputable between locally formed or outwardly transported.
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