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Kuiper Belt: Dynamics

Alessandro Morbidelli Observatoire de la Cote ˆ d’Azur Nice, France

Harold F. Levison Southwest Research Institute Boulder, Colorado

CHAPTER

1. Historical Perspective 2. Basic Orbital Dynamics 3. Orbital and Dynamical Structure of the Trans-Neptunian Population 4. Correlations Between Physical and Orbital Properties

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6. Ecliptic Comets 7. The Primordial Sculpting of the Trans-Neptunian Population 8. Concluding Remarks Bibliography

5. Size Distribution of the Trans-Neptunian Population and Total Mass

T

he name Kuiper Belt is generically referred to a population of small bodies, the orbits of which have a semimajor axis—and hence orbital period—larger than those of Neptune. It can be viewed as a second Asteroid Belt, but located at the outskirts of the solar system. The Kuiper Belt objects—having formed at large distances from the Sun—are rich in water ice and other volatile chemical compounds and have physical properties similar to those of comets. Indeed, the existence of the Kuiper Belt was first deduced from observations of the Jupiter-family comets, a population with short orbital periods and small to moderate orbital inclinations, of which the Kuiper Belt is the source. In 14 years since the discovery of the first object, about 1200 Kuiper Belt objects have been detected. Of these, ∼700 objects have been observed for more than 2 years, a necessary condition to compute their orbital parameters with significant precision. The results of this detailed observational exploration of the Kuiper Belt structure have provided several surprises. Indeed, it was expected that the Kuiper Belt preserved the pristine conditions of the protoplanetary disk. But it is now evident that this picture is not correct: The disk has been affected by a number of processes that have altered its original structure.

The Kuiper Belt may thus provide us with a large number of clues to understand what happened in the outer solar system during the primordial ages. Potentially, the Kuiper Belt might teach us more about the formation of the giant planets than the planets themselves. And, as in a domino game, a better knowledge of giant planets formation would inevitably boost our understanding of the subsequent formation of the solar system as a whole. Consequently, Kuiper Belt research is now considered a top priority of modern planetary science.

1. Historical Perspective Since its discovery in 1930, Pluto has traditionally been viewed as the last vestige of the planetary system—a lonely outpost at the edge of the solar system, orbiting beyond Neptune with a 248 year period. Pluto receives very little light from the Sun (being almost 40 times farther from the Sun on average than the Earth) and thus it is very cold. The view was that it was a distant, isolated, and unfriendly place, with nothing of substance beyond it. Pluto itself has always appeared to be an oddity among the planets. Traditionally, the planets are divided into two

C 2007 by Academic Press. All rights of reproduction in any form reserved. Encyclopedia of the Solar System 2e 

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590 Encyclopedia of the Solar System main groups. The first group, the terrestrial planets, formed in the inner regions of the solar system where the material from which the planets were made was too warm for water and other volatile gases to be condensed as ices. These planets, which include the Earth, are small and rocky. Farther out from the Sun, the cores of the planets grew from a combination of rock and condensed ices and captured significant amounts of nebula gas. These are the jovian planets, the giants of the solar system; they most likely do not have solid surfaces. But, then there is Pluto, unique, small (its radius is only ∼1180 km, only two thirds that of the Earth’s Moon) and made of a mix of rock and frozen ices. The planets formed in a disk of material that originally surrounded the Sun. As the Sun formed from the collapse of its parent molecular cloud, it faced a problem. The cloud had a slight spin and as it collapsed, the spin rate had to increase in order to conserve angular momentum. The cloud could not form a single star with the amount of angular momentum it possessed, so it shed a disk of material that contained very little mass (as compared with the mass of the Sun), but most of the angular momentum of the system. As such, the planets formed in a narrow disk structure; the plane of that disk is known as the invariable plane. But, then there is Pluto, unique, having an orbital inclination of 15.6◦ with respect to the invariable plane. The orbits of the planets are approximately ellipses with the Sun at one focus. As the planets formed in the original circumsolar disk, they tended to evolve onto orbits that were well separated from one another. This was required so that their mutual gravitational attraction would not disrupt the whole system. (Or to put is another way, if our system had not formed that way, we would not be here to talk about it!) But, then there is Pluto, unique, having an orbit that crosses the orbit of its nearest neighbor, Neptune. So, the historical view was that Pluto was an oddity in the solar system. Unique for its physical makeup and size as well as its dynamical niche. But, this view changed in September 1992 with the announcement of the discovery of the first of a population of small (compared to planetary bodies) objects orbiting beyond the orbit of Neptune, in the same region as Pluto. Since that time, over 1000 objects with radii between a few tens and ∼1000 km have been discovered. One object, 136199 Eris (previously known under the provisional designation 2003 UB313 ), even turned out to be 10% larger than Pluto. Moreover, a modeling of the detection efficiency of the performed surveys suggests that there are approximately 1,000,000 objects larger than a few tens of kilometers occupying this region of space, approximately between 30 and 50 AU from the Sun. There are almost certainly many more smaller ones. As discussed in more detail in the following sections, these objects likely have a similar physical makeup to that of Pluto, and many have similar orbital characteristics. Thus, in the last decade, Pluto has been transformed from an oddity, to the found-

ing member of what is perhaps the most populous class of objects in the planetary system. The discovery of the Kuiper Belt, as it has come to be known, represents a revolution in our thinking about the solar system. First predicted on theoretical grounds and later confirmed by observations, the Kuiper Belt is the first totally new class of bodies to be discovered in the solar system since the first asteroid was found on New Year’s day, 1801. Its discovery is on a par with the discovery of the solar wind and the planetary magnetospheres in the 1950s and 1960s, and it has radically changed our view of the outer solar system. Speculation on the existence of a trans-Neptunian disk of icy objects dates back over 90 years. In the early 1900, Campbell, Aitken, and Leuschner considered the possibility of trans-Neptunian planets and speculated on the orbital distribution of small bodies in the outer planetary system. In the 1940s and early 1950s, a more comprehensive approach to the problem was made independently by Kenneth Edgeworth and Gerard Kuiper. They noticed that if one were to grind up the giant planets and spread out their masses to form a disk, then this disk would have a very smooth distribution, with a density that slowly decreases as the distance from the Sun increases. That holds until Neptune, at which point there is an apparent edge beyond which there was thought to be nothing except tiny Pluto. Edgeworth and Kuiper suggested that perhaps this edge was not real. Perhaps the disk of planetesimals (i.e., small bodies, potentially precursors of planet formation) that formed the planets extended past Neptune, but the density was too low or the formation times too long to form large planets. If so, they argued, these planetesimals should still be there in nearly circular orbits beyond Neptune. Unfortunately, Edgeworth’s contribution was overlooked until recently, and thus this disk has come to be known as the Kuiper Belt. The idea of a trans-Neptunian disk received little attention for many years. The objects in the hypothetical disk were too faint to be seen with the telescopes of the time, so there was no way to prove or disprove their existence. Comet dynamicists showed that the lack of detectable perturbations on the orbit of Halley’s comet limited the mass of such a disk to no more than 1.3 Earth masses (M⊕ ) if it was at 50 AU from the Sun. However, the idea was resurrected in 1980 when Julio Fernandez proposed that a cometary disk beyond Neptune could be a possible source reservoir for the short-period comets (those with orbital periods <200 years). Subsequent dynamical simulations showed that a comet belt beyond Neptune is the most plausible source for the low inclination subgroup of the short-period comets, named the Jupiterfamily comets. This work led observers to search for Kuiper Belt objects. With the discovery of the first object, 1992 QB1 by D. Jewitt and J. Luu, the Kuiper Belt ceased to be a speculation and became a concrete entity of the solar system.

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.1 .08 .06 .04 .02 0 300 200 100 0

i (Deg)

Much of the story of the Kuiper Belt to date involves the distribution of the orbits of its members. In this section, we present a brief overview of the important aspects of the orbits of small bodies in the solar system. [For a more detailed discussion, see Solar System Dynamics: Regular and Chaotic Motion.] The most basic problem of orbital dynamics is the twobody problem: a planet, say, orbiting a star. The orbit’s trajectory is an ellipse with the Sun at one of the foci. Energy, angular momentum, and the orientation of the ellipse are conserved quantities. The semimajor axis, a, of the ellipse is a function of the orbital energy. The eccentricity, e, of the ellipse is a function of the energy and the angular momentum. For a particular semimajor axis, the angular momentum is a maximum for a circular orbit, e = 0. These two-body orbits are known as Keplerian orbits. A Keplerian orbit is characterized by its semimajor axis and eccentricity, as well as by three angles that describe the orientation of the orbital ellipse in space. The first, known as the inclination, i, is the angle between the angular momentum vector of the orbit and some reference direction for the system. In our solar system, the reference direction is usually taken as the angular momentum vector of the Earth’s orbit (which defines the ecliptic plane, the reference plane), but it is sometimes taken to be the angular momentum vector of all the planetary orbits combined (which defines the invariable plane). The point where the orbit passes through the reference plane in an “upward” direction is called the ascending node. Thus, the second orientation angle of the orbit is the angle between the ascending node and some reference direction in the reference plane, as seen from the Sun. In our solar system, the reference direction is usually taken to be the direction toward the vernal equinox. This angle is known as the longitude of the ascending node, . The third and final orientation angle is the angle between the ascending node and the point where the orbit is closest to the Sun (known as perihelion), as seen from the Sun. It is called the argument of perihelion, ω. Another useful angle, known as the longitude of perihelion,  ω, is defined to be ω + . The first-order gravitational effect of the planets on one another is that each applies a torque on the other’s orbit, as if the planets were replaced by rings of material smoothly distributed along their orbits. This torque causes both the longitude of perihelion,  ω, and the longitude of the ascending node, , to rotate slowly, a motion called precession. For a given planet, the precession of  ω, is typically dominated by one frequency. The same is true for , although the dominant frequency is different. The periods associated with these frequencies range from 4.6 × 104 to 2 × 106 years in the outer planetary system. This is much longer than the orbital periods of the planets (164 years for Neptune).

e

2. Basic Orbital Dynamics

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5 4 3 2 1 0 300 200 100 0

0 time (years)

FIGURE 1 The temporal evolution of the orbit of the first Kuiper Belt object found, 1992 QB1 . As described in the text, the eccentricity, e, and inclination, i, oscillate, while the longitude of the ascending node, , and the longitude of perihelion  ω = ω +  circulate.

The orbit of a small object in the solar system, when it is not being strongly perturbed by a close encounter with a planet or is not located near a resonance (see later), is usually characterized by slow oscillations of e and i and a circulation (i.e., continuous change) in  ω and . The variation in the eccentricity is coupled with the  ω precession and the variation in the inclination is coupled with the  ω precession. Figure 1 shows this behavior for the first discovered Kuiper Belt object, 1992 QB1 . The behavior of objects that are in a resonance can be very dramatic. There are two types of resonances that are known to be important in the Kuiper Belt. The most basic is known as a mean-motion resonance. A mean-motion resonance is a commensurability between the orbital period of two objects. That is, the ratio of the orbital periods of the two bodies in question is a ratio of two (usually small) integers. Perhaps the most well-known and important example of a mean-motion resonance in the solar system is the one between Pluto and Neptune. As noted earlier, one of the unique aspects of Pluto’s orbit is that when Pluto is at perihelion, it is closer to the Sun

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592 Encyclopedia of the Solar System than Neptune. Normally, this configuration would, sooner or later, lead to close encounters between the two planets that would eventually scatter Pluto away. However, close encounters do not occur because Pluto is locked in a meanmotion resonance where it goes around the Sun twice every time Neptune goes around three times. So, every time Pluto crosses the trajectory of Neptune, the giant planet is always in one of three specific locations on its orbit, all very far away from the crossing point. This resonance is known as the 2:3 mean-motion resonance. The other type of resonance that is important in the Kuiper Belt is called a secular resonance. There are actually two types of secular resonances. The first, which was discussed earlier, is a resonance between the precession rates of the longitudes of perihelion. As discussed, this can lead to changes in eccentricity. These resonances are identified by the Greek letter v with a numbered subscript that indicates the resonant planet (1 for Mercury through 9 for Pluto). In the Kuiper Belt, the perihelion secular resonance with Neptune, or v8 , is most important. The other type of secular resonance occurs when the small body’s nodal precession

rate is the same as for a planet. This type of resonance can cause significant changes in the inclination of the orbit of the small body. These resonances are identified by v1x , where x is the number of the resonant planet. For example, the nodal resonance with Neptune is the v18 . The dynamical structure of the Kuiper Belt has been sculpted by a combination of mean-motion and secular resonances and by the evolution of these resonances during the formation of Uranus and Neptune. We come back to this issue in Sections 3 and 7.

3. Orbital and Dynamical Structure of the Trans-Neptunian Population Figure 2 shows the distribution of the objects with semimajor axis larger than 30 astronomical units (AU) whose orbits have been determined from observations spanning over at least 3 years. A glance at the figure reveals that the orbits of the trans-Neptunian objects can be very diverse. The majority

FIGURE 2 The distribution of the objects with well-determined orbits, as to February 1, 2005. The upper and lower panels show respectively the inclination and the eccentricity vs. semimajor axis. Two different semimajor axis scales are used to illustrate the Kuiper Belt (left panels) and the scattered disk (right panels) distributions. Red dots correspond to the scattered disk, magenta dots to the extended scattered disk, blue dots to the classical Kuiper Belt, and green dots to the resonant populations. The big, crossed circle denotes the orbit of Pluto. The vertical lines labeled 3:4, 2:3, and 1:2 mark the location of the corresponding mean-motion resonances with Neptune. The two dotted curves on the lower panels correspond to perihelion distances q = 30 AU and q = 35 AU on the left, and q = 30 AU and q = 38 AU on the right. These curves approximately bound the scattered disk orbital distribution.

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of the discovered objects are clustered in the 36–48 AU range, but several others form a “tail” structure extending beyond 50 AU. Their semimajor axes range up to several 100 AU. Their perihelion distances are generally between 30 and 38 AU (dotted curves in the bottom right panel of Fig. 2), so that on average the orbital eccentricities increase with semimajor axes. These objects are dynamically unstable because they suffer sufficiently close encounters with Neptune. At each encounter, they receive an impulse-like acceleration, which changes the semimajor axis of their orbits. The perihelion distance remains roughly constant during an encounter, so that the eccentricity changes together with the semimajor axis. Thus, under the scattering gravitational action of Neptune, these objects move in a sort of random walk in the region confined by the dotted curves. For this reason, this population of objects is now called the scattered disk. The name “disk” is justified, because the orbital inclinations, although large, are significantly smaller than 90◦ , giving this population a disk-like structure. Up to now, we have used “Kuiper Belt” to denote generically the population of objects with a > 30 AU. However, the existence of the scattered disk suggests that we should reserve the name “Kuiper Belt” for the population of objects that do not suffer encounters with Neptune and therefore have orbits that either do not significantly change with time, or do so very slowly. Adopting this definition, the objects of the Kuiper Belt are plotted with blue and green dots in Fig. 2, while the scattered disk objects are plotted in red. As one sees, scattered disk objects can also have a < 50 AU, provided that they have a small perihelion distance and are not in one of the most prominent mean-motion resonances with Neptune (indicated by the vertical lines labeled 3:4, 2:3 and 1:2 in Fig. 2). In Fig. 2, the scattered disk seems to be outnumbered by the Kuiper Belt bodies. However, the scattered disk objects are more difficult to discover, given that most of them have very elongated orbits and spend most of the time very far from the Sun. Accounting for this difficulty, astronomers have estimated that the scattered disk and the Kuiper Belt should constitute roughly equal populations. All solar system bodies should have accreted on quasicircular orbits. This is a necessary condition for small planetesimals being able to stick together and form larger objects. Indeed, if the eccentricities are large, the relative encounter velocities are such that, upon collisions, planetesimals do not grow, but fragment into smaller pieces. This consideration suggests that the scattered disk objects formed much closer to Neptune, on quasi-circular orbits, and have been transported outward by the scattering action of that planet. The fact that the scattering action is still continuing implies that the origin of the scattered disk does not necessarily require that the primordial solar system was different from the current one. However, recent observations have revealed that, in addition to the Kuiper Belt and the scattered disk, there is a third category of objects, represented with magenta dots

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in Fig. 2. Their orbital distribution mimics that of the scattered disk objects, but their perihelion distance is somewhat larger, so that they avoid the scattering action of Neptune. Their orbits do not significantly change over the age of the solar system. Among the objects with these orbital properties are 1995 TL8 (a ∼ 52 AU, q ∼ 40 AU), 2000 CR105 (a ∼ 225 AU, q ∼ 44 AU), 90377 Sedna (a ∼ 500 AU, q ∼ 76 AU), and the recently discovered 136199 Eris (a ∼ 67.5 AU, q = 38 AU, the largest Trans-Neptunian Object (TNO) known so far) and 2004 XR190 (a ∼ 57.4 AU, q ∼ 51 AU—exceptional for its inclination of about 45◦ ). For the previously listed reasons, these bodies also should have formed closer to Neptune on much more circular orbits, and presumably they have been transported outward through close encounters with the planet. However, given that they do not undergo close encounters now, their existence suggests that the solar system was different in the past (either the planetary orbits were different or the environment was different—rogue planets, passing stars, etc.), so that the scattered disk extended further out in perihelion distance during the primordial times. We will come back to this in Section 7. If we look at Fig. 2 more in detail (left panels), the Kuiper Belt can also be subdivided in a natural way in subpopulations. Several objects (green dots) are located in meanmotion resonances with Neptune. As explained in Section 2, the mean-motion resonances provide a protection mechanism, so that resonant objects can avoid close encounters with Neptune even if their perihelion distance is smaller than 30 AU, as it is in the case of Pluto. For this reason, resonant Kuiper Belt objects can be on much more elliptic orbits than the nonresonant ones, the eccentricities of the former ranging up to 0.35. The objects in the 2:3 mean-motion resonance with Neptune are usually called the Plutinos (because they share the same resonance as Pluto), while those in the 1:2 resonance are sometimes called twotinos. In Fig. 2, the resonant population seems to constitute a substantial fraction of the Kuiper Belt population. However, resonant objects are easier to discover because at perihelion they come closer to the Sun than the nonresonant ones. When accounting for this fact, astronomers estimate that, all together, the objects in mean-motion resonances constitute about 10% of the total Kuiper Belt population. The nonresonant Kuiper Belt objects (blue dots) are usually referred to as classical. This adjective is attributed because their orbital distribution is the most similar to what the astronomers were expecting, before the discoveries of trans-Neptunian objects began: that of a disk of objects on stable, low-eccentricity, nonresonant orbits. However, even the classical population has unexpected properties. Their eccentricities are moderate—a necessary condition to avoid encounters with Neptune, given that they are not protected by any resonant mechanism. Nevertheless, the eccentricities are definitely larger than those of the protoplanetary disk in which the objects had to form. Some mechanism

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594 Encyclopedia of the Solar System must have excited the eccentricities, making them grow from almost zero to the current values. The same is true, and even more striking, for the inclinations (top panel of Fig. 2). The inclinations are related to the relative encounter velocities among the objects, so that the Kuiper Belt bodies had to grow in a razor-thin disk. Despite this, the current inclinations range up to 30– 40◦ . Figure 2 gives the impression that large inclination bodies are a modest fraction among the classical objects. However, one should take into account that the discovery surveys have been concentrated near the ecliptic plane, so that large inclination bodies have a lower probability of being discovered than low inclination ones. Accounting for this selection effect, astronomers have computed that the real inclination distribution of the classical objects is bimodal (Fig. 3). There is a cluster of objects with inclination smaller than 4◦ and a second group of objects with a very distended inclination distribution. The former constitute what is now usually called the cold population and the latter the hot population. The adjectives “hot” and “cold” do not refer to physical temperature (it is always very cold out there) but to the encounter velocities inside each population, in an analogy with gas kinetic theory. The cold and the hot populations should contain roughly the same number of objects. The last striking property of the Kuiper Belt is its outer edge. Figure 2 shows that the belt ends at the location of the 1:2 mean motion resonance with Neptune. For several years, the astronomers suspected that this edge is only apparent, due to the fact that more distant objects are more difficult to discover. However, with an increasing statistical

FIGURE 3 The inclination distribution (in deg) of the classical Kuiper Belt after observational biases have been subtracted, according to the work of M. Brown. The points with error bars show the model-independent estimate constructed from a limited subset of confirmed classical belt bodies, while the smooth line shows a best fit bimodal population model. In this model ∼60% of the objects have i > 4◦ .

sample, it turned out that this is not true. It has been shown that more distant objects should have been discovered by now, unless either (1) the Kuiper Belt population steeply decays in number beyond 48–50 AU or (2) the maximal size of the objects beyond this limit is much smaller than that in the observed Kuiper Belt. For various reasons, astronomers tend to favor hypothesis (1): the existence of a physical outer edge of the Kuiper Belt. An important issue is to understand which of the orbital properties discussed earlier is due to the dynamical processes that are still occurring in the Kuiper Belt or not. For instance, do the eccentricities and the inclinations slowly grow due to some dynamical phenomenon? Are the low eccentricity objects beyond 48 AU unstable? If these are the cases, then the existence of large eccentricities and inclinations, as well as the outer edge of the Kuiper Belt could be simply explained. In the opposite case, these properties—like the existence of the extended scattered disk—reveal that the solar system was different in the past. Dynamical astronomers have studied in great detail the dynamics beyond Neptune, using numerical simulations and semianalytic models. Figures 4 and 5 show maps of the dynamical lifetime of trans-Neptunian bodies on a wide range of initial semimajor axes, eccentricities, and inclinations. These maps have been computed numerically, by simulating the evolution of thousands of massless particles under the gravitational perturbations of the giant planets. The latter have been assumed to be initially on their current orbits. Each particle was followed until it suffered a close encounter with Neptune. Objects encountering Neptune, would then evolve in the scattered disk for a time of order ∼108 years, until they are transported by planetary encounters into the inner planets region, or are ejected to the Oort cloud or to interstellar space. This issue is described in more detail in Section 6. In Fig. 4, the colored strips indicate the length of time required for a particle to encounter Neptune as a function of its initial semimajor axis and eccentricity. The initial inclination of the particles was set equal to 1◦ . Strips that are colored yellow represent objects that survive for the length of the simulation, 4 × 109 years, the approximate age of the solar system. As can be seen in the figure, the Kuiper Belt can be expected to have a complex structure, although the general trends are readily explained. Objects with perihelion distances less than ∼35 AU (shown as a red curve) are unstable, unless they are near, and presumably librating about, a mean-motion resonance with Neptune (Section 2). Indeed, the results in Fig. 4 show that many of the Neptunian mean-motion resonances (shown in blue) are stable for the age of the solar system. Objects with semimajor axes between 40 and 42 AU are unstable. This is presumably due to the presence of three overlapping secular resonances that occur in this region of the solar system: two with Neptune and one with Uranus.

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FIGURE 4 The dynamical lifetime for small particles in the Kuiper Belt derived from 4 billion year integrations by M. Duncan, H. Levison, and M. Budd. Each particle is represented by a narrow vertical strip of color, the center of which is located at the particle’s initial eccentricity and semimajor axis (initial orbital inclination for all objects was 1◦ ). The color of each strip represents the dynamical lifetime of the particle. Strips colored yellow represent objects that survive for the length of the integration, 4 × 109 years. Dark regions are particularly unstable on these timescales. For reference, the locations of the important Neptune mean-motion resonances are shown in blue and two curves of constant perihelion distance, q, are shown in red. The orbital distribution of the real objects is also plotted. Big dots correspond to objects with i < 4◦ , and small dots to objects with larger inclination. Remember that the dynamical lifetime map has been computed assuming i = 1◦ .

Indeed, secular resonances appear to play a critical role in ejecting particles from this region of the Kuiper Belt. This can be better seen in Fig. 5, which is an equivalent map, but plotted relative to the initial semimajor axis and inclination for particles with initial eccentricity of 0.01. Also shown are the locations of the Neptune longitude of perihelion secular resonance (in red) and the Neptune longitude of the ascending node secular resonance (in yellow). It is important to note that much of the clearing of the Kuiper Belt occurs where these two resonances overlap. This includes the low inclination region between 40 and 42 AU, which is indeed depleted of bodies (compare with Fig. 2). The Neptune mean-motion resonances are also shown (in green). It is interesting to compare the numerical results to the current best orbital elements of the known Kuiper Belt objects. This comparison is also made in Fig. 4, where the observed objects with good orbital determination are overplotted with green dots. Big dots refer to bodies with I < 4◦ , consistent with the low inclination at which the sta-

bility map has been computed. Small dots refer to objects with larger inclination and are plotted only for completeness. The conclusion is that most observed objects (with the exception of scattered disk bodies) are associated with stable zones. Their orbits do not significantly change over the age of the solar system. Thus, their current excited eccentricities and inclination cannot be obtained from primordial circular and coplanar orbits in the framework of the current planetary system orbital configuration. Likewise, the region beyond the 1:2 mean-motion resonance with Neptune is totally stable. Thus, the absence of bodies beyond 48 AU cannot be explained by current dynamical instabilities. Therefore, it is evident that the orbital structure of the Kuiper Belt has been sculpted by mechanisms that are no longer at work, but presumably were active when the solar system formed. The main goal of dynamical astronomers interested in the Kuiper Belt is to uncover these mechanisms and from them deduce, as far as possible, how the solar system formed and early evolved.

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596 Encyclopedia of the Solar System FIGURE 5 The dynamical lifetime for test particles with initial eccentricity of 0.01 derived from 1 billion year integrations by M. Duncan, H. Levison, and M. Budd. This plot is similar to that of Fig. 4 except that coordinates are semimajor axis and inclination (instead of semimajor axis and eccentricity) and a different color table was used for the solid bars. In addition, the red and yellow curves show the locations of Neptune longitude of perihelion secular resonances (v8 ) and the Neptune longitude of the ascending node secular resonances (v18 ), respectively. The green lines show the location of the important Neptune mean motion resonances.

4. Correlations Between Physical and Orbital Properties The existence of two distinct classical Kuiper Belt populations, called the hot (i > 4◦ ) and cold (i < 4◦ ) classical populations, could be caused in one of two general manners. Either a subset of an initially dynamically cold population was excited, leading to the creation of the hot classical population, or the populations are truly distinct and formed separately. One manner in which we can attempt to determine which of these scenarios is more likely is to examine the physical properties of the two classical populations. If the objects in the hot and cold populations are physically different, it is less likely that they were initially part of the same population. The first suggestion of a physical difference between the hot and the cold classical objects came from the observation that the intrinsically brightest classical belt objects (those with lowest absolute magnitudes) are preferentially found with high inclination. Figure 6 shows the distribution of the classical objects in an inclination vs. absolute magnitude diagram. As one sees, for an absolute magnitude H > 5.5, there is a given proportion between the number of objects discovered in the cold and the hot populations respectively. This ratio is completely different for H < 5.5, where cold population objects are almost absent. All the biggest classical objects, such as, for instance, 50000 Quaoar, 20000 Varuna, 19521 Chaos, 28978 Ixion, 2005 FY9 , and 2003 EL61 have inclinations larger than 5◦ . Their median inclination is 12◦ . It has been argued that this is a result

of an observational bias because the brightest objects have been discovered in wide field surveys not confined around the ecliptic, which are thus more likely to find large inclination objects than the deep ecliptic surveys that detected the fainter bodies. However, a recent survey for bright objects, which covered ∼70% of the ecliptic, found many hot classical objects but few cold classical objects, confirming that the effect illustrated in Fig. 6 is real. The second possible physical difference between hot and cold classical Kuiper Belt objects is their colors. With the name “color” astronomers generically refer to the slope of the spectrum of the light reflected by a trans-Neptunian object at visible wavelengths, relative to that of the light emitted by the Sun. “Red” objects reflect more at long than at short wavelengths, while “gray” objects have a more or less uniform reflectance. Colors relate in a poorly understood manner to objects’ surface composition. It has been shown and repeatedly confirmed that, for the classical belt, the inclination, and possibly the perihelion distance, is correlated with color. In essence, the low inclination classical objects tend to be redder than higher inclination objects. More interestingly, colors naturally divide into distinct red and gray populations at precisely the location of the divide between the inclinations of the hot and cold classical objects. These populations differ at a 99.9% confidence level. Interestingly, the cold classical population also differs in color from the Plutinos and the scattered objects at the 99.8 and 99.9% confidence level, respectively, while the hot classical population appears identical in color to these other populations. The possibility remains, however, that the colors of the objects, rather than being markers of different populations,

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FIGURE 6 The inclination of the classical Kuiper Belt objects as a function of their absolute magnitude. The horizontal dashed line at i = 4◦ separates the cold from the hot population. The vertical dotted line is plotted at H = 5.5. The distribution on the left side of the dotted line is clearly different from that on the right-hand side. The largest classical objects are all in the hot population.

are actually caused by the different inclinations. For example, it has been suggested that the higher average impact velocities of the high inclination objects could cause large-scale resurfacing by fresh water ice and carbonaceous materials, which could be gray in color. However, a similar color-inclination trend should be observed also among the plutinos and the scattered disk objects, which is not the case. A careful analysis shows that there is no clear correlation between average impact velocity and color. In summary, the significant color and size differences between the hot and cold classical objects imply that these two populations are physically different in addition to being dynamically distinct.

5. Size Distribution of the Trans-Neptunian Population and Total Mass As briefly described in Section 1, the disk out of which the planetary system accreted was created as a result of the Sun shedding angular momentum as it formed. As the Sun condensed from a molecular cloud, it left behind a disk of material (mostly gas with a little bit of dust) that contained a small fraction of the total mass but most of the angular momentum of the system. It is believed that the initial solid objects in the protoplanetary disk were pebble-sized, of the order of centimeters in size. These objects formed larger objects through a process of accretion to form asteroids and comets, which in turn accreted to form planets (or the cores of the giant planets which then accreted gas directly from the solar nebula). Understanding this process is one of the main goals of astronomy today.

There are few clues in our planetary system about this process. We know that the planets formed, and we know how big they are. Unfortunately, the planets have been so altered by internal and external processes that they preserve almost no record of their formation process. Luckily, we also have the Asteroid Belt, the Kuiper Belt, and the scattered disk. These structures contain the best clues to the planet formation process because they are regions where the process started, but for some reason, did not run to completion (i.e., a large planet). Thus, the size distribution of objects in these regions may show us how the processes progressed with time and (hopefully) what stopped them. The Kuiper Belt and the scattered disk are perhaps the best places to learn about the accretion process. Because the size of the object is not a quantity that can be easily measured (one needs to make hypotheses on the intrinsic reflectivity of the objects, or albedo), and the absolute magnitude is readily obtained from the observations, astronomers generally prefer the absolute magnitude distribution, instead of the size distribution. The magnitude distribution is usually given in the form Log N(< H) ∝ H a , a > 0 where N is the cumulative number of objects brighter than absolute magnitude H. The slope of this distribution, a, contains important clues about the physical strengths, masses, and orbits of the objects involved in the accretion process. For example, there are two extremes to the accretion process. If two large, strong objects collide at low velocities, then the amount of kinetic energy in the collision is

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598 Encyclopedia of the Solar System FIGURE 7 The cumulative magnitude distribution of the cold population (red) and of the hot population and scattered disk (green) according to a recent analysis by G. Bernstein and collaborators. A turnover of the magnitude distribution is detected around H ∼ 10. The slope of the magnitude distribution is very uncertain beyond this limit.

small compared to the amount of energy holding the objects together. In this case, the objects merge to form a larger object. If two small, weak objects collide at high velocities, then the energy in the collision overpowers the gravitational and material binding energies. In this case, the objects break apart, forming a large number of much smaller objects. In realistic models of the Kuiper Belt with a range of sizes and velocities, we expect small objects to fragment and large objects to grow. This produces a size distribution with a ∼ 0.4−0.5 at small sizes and a much steeper slope at large sizes where accretion is important. Statistics of discoveries of Kuiper Belt objects (Fig. 7) suggest that the absolute magnitude distribution is indeed very steep for H < 9 (approximately equivalent to a diameter D > 100 km), with a ∼ 0.6−0.7, and then turns over toward a significantly shallower slope. Interestingly, the hot and the cold classical population seems to have two different values of a in the steep part. More precisely, the hot population and the scattered disk have a shallower magnitude distribution than the cold population (Fig. 7). This is consistent with the fact that the largest bodies are all in the hot population, and yet the hot and cold populations and the scattered disk contain roughly the same number of bodies bigger than 100 km. The value of a in the shallow part of the magnitude distribution beyond H ∼ 10 is very uncertain. Only few surveys with the most powerful telescopes could probe this region, but they have discovered very few objects. The results are therefore affected by small number statistics. It is possible that a < 0.5 in some magnitude range. In fact, in the

Asteroid Belt, the magnitude distribution is wavy, and the canonical values of 0.4–0.5 of a is only a mean value. It is possible that the magnitude distribution in the Kuiper Belt is wavy as well, and that the range 10 < H < 14 corresponds to the very shallow part of one of these waves. It is possible to integrate under the magnitude distribution shown in Fig. 7 in order to estimate the total mass in the Kuiper Belt between 30 and 50 AU. Such an integration with limits between R = 1 km and 1200 km (the approximate radius of Pluto) and assuming a density of 1 g cm−3 , shows that the total mass is a few hundredths of an Earth mass. Given the uncertainties, it is possible that the mass is of order of 0.1 M⊕ , but not significantly larger. As with many scientific endeavors, the discovery of new information tends to raise more questions than it answers. Such is the case with the preceding mass estimate. Edgeworth’s and Kuiper’s original arguments for the existence of the Kuiper Belt were based on the idea that it seemed unlikely that the disk of planetesimals that formed the planets would have abruptly ended at the current location of the outermost known planet. An extrapolation into the Kuiper Belt (between 30 and 50 AU) of the current surface density of nonvolatile material in the outer planets region predicts that there should originally have been about 30 M⊕ of material there. However, as stated previously, our best estimate is over 200 times less than that figure! Edgeworth’s and Kuiper’s argument is not the only indication that the mass of the primordial Kuiper Belt had to be significantly larger in the past. Models of collisional accretion show that it is not possible for objects with radii

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greater than about 30 km to form in the current Kuiper Belt, at least by pairwise accretion, over the age of the solar system. The current surface density of solid material is too low to accrete bodies larger than this size. However, the models show that objects the size of 1992 QB1 could have grown in a more massive Kuiper Belt (provided that, as already said in Section 3, the mean orbital eccentricities of the accreting objects were much smaller than the current ones). A Kuiper Belt of at least several Earth masses is required in order for 100 km sized objects to have formed. The same applies even more strongly to the accretion of Pluto and Charon. For those two bodies to have grown to their current sizes in the trans-Neptunian region, there must have originally been a far more massive Kuiper Belt. A massive and dynamically cold primordial Kuiper Belt is also required by the models that attempt to explain the formation of the observed numerous binary Kuiper Belt objects. Therefore, the general formation picture of an initial massive Kuiper Belt appears secure, and understanding the ultimate fate of the 99% (or 99.9%) of the initial Kuiper Belt mass that appears to be no longer in the Kuiper Belt is a crucial step in reconstructing the history of the outer solar system.

6. Ecliptic Comets As described in Section 1, the current renaissance in Kuiper Belt research was prompted by the suggestion that the Jupiter-family comets originated there. We now know that there are mainly two populations of small bodies beyond Neptune: the Kuiper Belt and the scattered disk. Which one is the dominant source of these comets? To answer this question, we need to examine a few considerations on the origin of the scattered disk. We have seen in Section 3 that the bodies in the scattered disk have intrinsically unstable orbits. The close encounters with Neptune move them in semimajor axis, until they either evolve into the region with a < 30 AU or reach the Oort cloud at the frontier of the solar system. In both of these cases, the bodies are removed from the scattered disk. Despite this possibility of dynamical removal, we still observe scattered disk bodies today. How can this be? There are a priori two possibilities. The first one is that the scattered disk population is sustained in a sort of steady state by the bodies escaping from the Kuiper Belt. This means that on a timescale comparable to that for the dynamical removal of scattered disk bodies, new bodies enter the scattered disk from the Kuiper Belt. For example, a similar situation occurs for the population of near-Earth asteroids (NEAs). NEA dynamical lifetimes are only of a few million years because they intersect the orbits of the terrestrial planets. Nevertheless, the population remains roughly constant because new asteroids enter the NEA population

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from the Main Asteroid Belt at the same rate at which old NEAs are eliminated. The second possibility is that the scattered disk that we see today is only what remains of a much more numerous population that has been decaying in number since planetary formation. Numerical simulations show that roughly 1% of the scattered disk bodies can survive in the scattered disk for the age of the solar system. Thus, the primordial scattered disk population should have been about 100 times more numerous. Which of these possibilities is true? In the first case, we would expect that the Kuiper Belt is much more populated than the scattered disk. For instance, the Asteroid Belt contains about 1000 times more objects than the NEA population, at comparable sizes. However, observations indicate that the scattered disk and the Kuiper Belt contain roughly the same number of objects. Thus, the second possibility has to be true. Scattered disk objects most likely formed in the vicinity of the current positions of Uranus and Neptune. When these planets grew massive, they scattered them away from their neighborhoods. In this way, a massive scattered disk of about 10 M⊕ formed. What we see today is just the last vestige of that primordial population, which is still decaying in number. The fact that the scattered disk is not sustained in steady state by the Kuiper Belt, but it is still decaying, implies that the scattered disk provides more objects to the giant planet region (a < 30 AU) than it receives from the Kuiper Belt. Thus, the outflow from the scattered disk is more important than the outflow from the Kuiper Belt. This implies that the scattered disk, not the Kuiper Belt, is the dominant source of Jupiter family comets. A significant amount of research has gone into understanding the dynamical behavior of objects that penetrate into the a < 30 AU region from the scattered disk. These studies show that the encounters with the planets spread them throughout the planetary system. These objects are usually called ecliptic comets, even if at large distances from the Sun they typically do not show any cometary activity. The distribution of these objects as predicted by numerical integrations is shown in Fig. 8. The ecliptic comets that get close to the Sun become active. When their semi-major axis is smaller than that of Jupiter, they are called Jupiter-family comets. It is somewhat surprising that about a third of the objects leaving the scattered disk in the simulations spend at least some of their time as Jupiter-family comets. The Jupiter-family comets that we see today are, in majority, small, R <10 km.  However, if our understanding of the size-distribution of these objects is correct (see Section 5), we should expect to see a 100 km sized Jupiter-family comet about 0.4% of the time. What a show that would be! Those ecliptic comets between Jupiter and Neptune are called the Centaurs (only the largest of which are observable). The simulations predict that there are ∼106 ecliptic

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FIGURE 8 The surface number density (on the plane of the ecliptic) of ecliptic comets as determined from numerical integrations by M. Duncan and H. Levison. There are approximately 106 comets larger than about 1 km in radius in this population.

comets larger than about 1 km in radius currently in orbits between the giant planets.

7. The Primordial Sculpting of the Trans-Neptunian Population In the previous sections, we have seen that many properties of the Kuiper Belt cannot be explained in the framework of the current solar system: 1. The existence of the resonant populations 2. The excitation of the eccentricities in the classical belt 3. The coexistence of a cold and a hot population with different physical properties 4. The presence of an apparent outer edge at the location of the 1:2 mean-motion resonance with Neptune 5. The mass deficit of the Kuiper Belt 6. The existence of the extended scattered disk population These puzzling aspects of the trans-Neptunian population reveal that it has been sculpted when the solar system was different, due to mechanisms that are no longer at work. Like detectives on the scene of a crime, trying to reconstruct

what happened from the available clues, the astronomers try to reconstruct how the solar system formed and evolved from the traces left in the structure of the Kuiper Belt. Planet migration has been the first aspect of the primordial evolution of which the astronomers found a signature in the Kuiper Belt. Once the planets formed and the gas disappeared, the planetesimals that failed to be incorporated in the planets’ cores had to be removed from the planets’ vicinity by the gravitational scattering action of the planets themselves. If a planet scatters a planetesimal outward, the latter gains energy. Because of energy conservation, the planet has to lose energy, moving slightly inward. The opposite happens if the planet scatters the planetesimal toward the inner solar system. A planet is much more massive than a planetesimal, thus the displacement of the planet is infinitesimal. However, if the number of planetesimals is large, and their total mass is comparable to that of the planet, the final effect on the planet is not negligible. This is a general process. We now come to what should have happened in our solar system. Numerical simulations show that only a small fraction of the planetesimals originally in the vicinity of Neptune was scattered outward: About 1% ended up in the scattered disk, and 5%, in the Oort cloud. The remaining 94% of the planetesimals eventually were scattered inward toward Jupiter. The latter, given its large mass, ejected from the solar system almost everything that came to cross its orbit. Thus, the net effect was that Neptune took energy away from the planetesimals and moved outward, while Jupiter gave energy to them and moved inward. Numerical simulations show that Saturn and Uranus also moved outward. Following Neptune’s migration, the mean-motion resonances with Neptune also migrated outward, sweeping the primordial Kuiper Belt until they reached their present position. During this process, some of the Kuiper Belt objects swept by a mean-motion resonance could be captured into resonance. Once captured, these bodies had to follow the resonance in its migration, while their eccentricity had to steadily grow. Thus, the planetesimals that were captured first, ended up on very eccentric resonant orbits, while those captured last could preserve a small eccentricity inside the resonance. Numerical simulations show that this process produces an important population of resonant bodies inside all the main mean-motion resonances with Neptune. To reproduce the observed range of eccentricities of resonant bodies, Neptune had to migrate more than 7 AU, thus starting not further than 23 AU (see Fig. 9). The existence of resonant bodies in the Kuiper Belt thus provides a strong indication that planet migration really happened. However, as Fig. 9 also shows, several important properties of the Kuiper Belt cannot be explained by this simple model invoking resonance sweeping through a dynamically cold, radially extended disk. The eccentricity of the classical belt is only moderately excited, and the inclination remains very cold. The planetesimals are only relocated, from the

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FIGURE 9 Final distribution of the Kuiper Belt bodies according to a simulation of the sweeping resonances scenario by R. Malhotra. The simulation is done by numerical integrating, over a 200 million year timespan, the evolution of 800 test particles on initial quasi-circular and coplanar orbits. The planets are forced to migrate (Jupiter, −0.2 AU; Saturn, 0.8 AU; Uranus, 3 AU; Neptune, 7 AU) and reach their current orbits on an exponential timescale of 4 million years. Large solid dots represent “surviving” particles (i.e., those that have not suffered any planetary close encounters during the integration time); small dots represent the “removed” particles at the time of their close encounter with a planet. In the lowest panel, the solid line is the histogram of semimajor axis of the surviving particles; the dotted line is the initial distribution. Most of the initial mass of the Kuiper Belt is simply relocated from the classical belt to the resonant populations. The mass lost is only a small fraction of the total mass.

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classical belt to the resonances, and only a minority of them are lost, which cannot explain the mass depletion of the belt. Finally, the region beyond the 1:2 mean-motion resonance is unaffected by planet migration, and therefore the existence of an outer edge requires a different explanation. Four plausible models have been proposed so far to explain the formation of an outer edge: (1) the outer part of the disk was destroyed by the passage of a star; (2) it was photoevaporated by the radiation emitted by massive stars originally in the neighborhood of the Sun; (3) planetesimals beyond some threshold distance could not grow because of the enhanced turbulence in the outer disk which prevented the accumulation of solid material; and (4) distant dust particles and/or planetesimals migrated to smaller heliocentric distance during their growth, as a consequence of gas drag, thus forming sizeable objects only within some threshold distance from the Sun. The first two scenarios require that the Sun formed in a dense stellar environment, consistent with recent observations showing that stars tend to form in clusters which typically disperse in about 100 million years. The entire protoplanetary disk— both the gas and the planetesimal components—would be truncated by these mechanisms. However, the protoplanetary disks that we see around other stars, even in dense stellar associations, are typically much larger than 50 AU. Thus, the history of our proto–solar system disk was not typical. The third and the fourth scenarios, conversely, form

a truncated planetesimal disk out of an extended gaseous disk. Therefore, they are more consistent with observations, which are sensitive only to the gas and dust components, and do not detect the location of planetesimals. Whatever mechanism formed the edge, it is intriguing that the latter is now at the location of a resonance with Neptune, despite the fact that Neptune did not play any role in the edge formation. Is this a coincidence? Probably not. It may suggest that originally the outer edge of the planetesimal disk was well inside 48 AU, and that the migration of Neptune pushed somehow a small fraction of the disk planetesimals beyond the disk’s original boundary. These pushed-out planetesimals are now identified with the current members of the Kuiper Belt. The fact that the Kuiper Belt is mass deficient all over its radial extent (36–48 AU), in addition suggests that the original edge was inside 36 AU. In fact, if the original edge had been somewhere in the 36– 48 AU range, we would see a discontinuity in the current radial mass distribution of the Kuiper Belt, which is not the case. An edge of the planetesimal disk close to 30 AU also helps to explain why Neptune stopped there and did not continue its outer migration beyond this limit. Several mechanisms have been identified to push beyond the original disk edge a small fraction (of order 0.1%) of the disk’s planetesimals, and to implant them on stable Kuiper Belt orbits. They are described next. More mechanisms might be identified in the future.

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602 Encyclopedia of the Solar System As Neptune moved through the disk on a quasi-circular orbit, it scattered the planetesimals with which it had close encounters. Through multiple encounters, some planetesimals were transported outward on eccentric, inclined orbits. A small fraction of these objects still exist today and constitute the scattered disk. Occasionally, some scattered disk objects entered a resonance with Neptune. Resonances can modify the eccentricity of the orbits. If decreased, the perihelion distance is lifted away from the planet; the sequence of encounters stops, and the body becomes “decoupled” from Neptune like a Kuiper Belt object. If Neptune had not been migrating, the eccentricity would have eventually increased back to Neptune-crossing values—the dynamics being reversible—and the sequence of encounters would have restarted again. Neptune’s migration broke the reversibility so that some of the decoupled bodies managed to escape from the resonances and remained permanently trapped in the Kuiper Belt. These bodies preserved the large inclinations acquired during the Neptune-encountering phase, and they can now be identified with the “hot” component of the Kuiper Belt population. At the same time, while Neptune was migrating through the disk, its 1:2 and 2:3 resonances swept through the disk, capturing a fraction of the disk planetesimals as explained earlier. When the 1:2 resonance passed beyond the edge of the disk, it kept carrying its load of objects. Because the migration of Neptune was presumably not a perfectly smooth process, the resonance was gradually dropping objects during its outward motion. Therefore, like a farmer seeding as he advances through a field, the resonance disseminated its previously trapped bodies all along its way up to its final position at about 48 AU. This explains the current location of the outer edge of the Kuiper Belt. Because the 1:2 resonance does not significantly enhance the orbital inclinations, the bodies transported by the resonance preserved their initially small inclination and can now be identified with the cold component of the Kuiper Belt. This scenario, reproduced in numerical simulations, explains qualitatively the orbital properties of the transNeptunian population, but it has difficulties explaining why the hot and the cold classical populations have different physical properties. Indeed, the members of these two populations should have formed more or less in the same region of the disk, although they followed two different dynamical evolutions toward the Kuiper Belt. An alternative possibility is that the hot population formed as explained earlier, but the cold population formed in situ, where it is now observed. Thus, the formation places being well separated, the corresponding physical properties could be different. However, this model has difficulties explaining how the cold population lost most of its primordial mass. It has been proposed that the objects grind down to dust in a collisional cascade process, but the latter has not been shown to be really effective, and seems inconsis-

tent with a number of constraints, such as the existence of binary objects with large separations, or the total number of comet-sized bodies in the scattered disk. Moreover, if the cold population is local, one is faced again with the problem of explaining why the outer edge of the population is exactly at the location of the 1:2 resonance with Neptune. A further possibility may be offered by a recent model, on the evolution of the outer solar system, that has been developed in order to explain the origin of the so-called Late Heavy Bombardment (LHB) of the terrestrial planets. The latter is a cataclysmic period characterized by huge impact rates on all planets that occurred between 4.0 billion and 3.8 billion years ago, namely about 600 million years after planet formation. In this model, the giant planets are assumed to be initially on quasi-circular and coplanar orbits, with orbital separations significantly smaller than the current ones. In particular, Saturn is assumed to be closer to Jupiter than their mutual 1:2 resonance (they are now close to the 2:5 resonance). The planetesimal disk is assumed to exist only from about 1.5 AU beyond the location of the outermost planet, up to ∼35 AU, with a total mass of ∼35 M⊕ . With this setting, the planetesimals at the inner edge of the disk acquire Neptune-scattered orbits on a timescale of a few million years. Consequently, the migration of the giant planets proceeds at very slow rate, governed by the slow escape rate of planetesimals from the disk. This slow migration continues for hundreds of millions of years, until Jupiter and Saturn cross their mutual 1:2 resonance. This resonance crossing excites their eccentricities, which destabilizes the planetary system as a whole. The planetary orbits become chaotic and start to approach each other. Both Uranus and Neptune are scattered outward, onto large eccentricity orbits (e ∼ 0.3−0.4) that penetrate deeply into the disk. This destabilizes the full planetesimal disk and triggers the LHB. The interactions with the planetesimals damp the planetary eccentricities, stabilizing the planetary system once again, and forcing a residual short radial migration of the planets, which eventually reach final orbits when most of the disk has been eliminated. Simulations show that this model is consistent with the current orbital architecture of the giant planets of the solar system. In this model, objects can be implanted into the current Kuiper Belt during the large eccentricity phase of Neptune. In fact, the full Kuiper Belt is unstable at that time, so that it can be visited by objects that leave the original planetesimal disk when the latter is destabilized. When Neptune’s eccentricity is damped, the Kuiper belt becomes stable so that the objects which, by chance, are in the Kuiper Belt region at that time, become trapped forever. Because the large eccentricity phase of Neptune is short, the inclinations of these objects remain predominantly small, consistent with the cold population of the current Kuiper Belt. The objects with the largest inclinations, conversely, are captured later, during the final bit of Neptune’s migration, as explained before. As Fig. 10 shows, this model reproduces the structure

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FIGURE 10 The distribution of semimajor axes and eccentricities in the Kuiper Belt. Left panel: Result of a simulation based on the recent model on the origin of the LHB. Right panel: The observed distribution. The model reproduces fairly well the outer edge of the Kuiper Belt at the 1:2 resonance with Neptune, the characteristic shape of the (a, e) distribution of the classical belt, the scattered and the extended scattered disks, and the resonant populations. The vertical solid lines mark the main resonance with Neptune. The dotted curve denotes perihelion distance equal to 30 AU, and the dashed curve delimits the region above which only high inclination objects or resonant objects can be stable over the age of the solar system. The overabundance of objects above this curve in the simulation is therefore an artificial consequence of the fact that the final orbits of the giant planets are not exactly the same as the real ones.

of the Kuiper Belt remarkably well. It is also consistent with its low mass because in the simulations the probability of capture in the Kuiper Belt is roughly of 1/1000 (which predicts a final mass of 0.03 M⊕ ). Moreover, the Kuiper Belt objects with final low inclinations and those with final large inclinations are found to come predominantly from different portions of the original planetesimal disk (respectively outside and inside 29 AU), which can explain, at least at a qualitative level, the correlations with physical properties. We finally come to the issue of the origin of the extended scattered disk. Simulations show that, in the same process described earlier, bodies are also delivered to orbits with moderate semimajor axis and perihelion distance, like that of the extended scattered disk object 1995 TL8 and its companions. The origin of Sedna is probably different. The key issue is that bodies with comparably large perihelion distances (∼80 AU) but smaller semimajor axis (a < 500 AU) have never been discovered despite the more favorable observational conditions. Therefore, they probably do not exist. If this is true, and the population of extended scattered

disk bodies with large perihelion distance starts only beyond several hundreds of AUs, then an “external” perturbation is required. The best candidate is a stellar passage at about 1000 AU from the Sun, lifting the perihelion distance of the distant members of the primordial, massive scattered disk. Such a stellar encounter is very unlikely in the framework of the current galactic environment of the Sun, but it would have been probable if the Sun formed in a moderately dense cluster, as mentioned earlier.

8. Concluding Remarks At the time of the first edition of this encyclopedia, 60 objects had been discovered in the Kuiper Belt. Now, we know 20 times more objects, and our view of the Kuiper Belt has become much more precise. It is now clear that the trans-Neptunian population has been sculpted in the primordial phases of the history of the solar system, by processes that are no longer at work.

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604 Encyclopedia of the Solar System It has been argued that the explanation of the most important observed properties of the trans-Neptunian population require a “cocktail” with three ingredients: (1) a truncated planetesimal disk; (2) a dense galactic environment, favoring stellar passages at about 1000 AU from the Sun; and (3) the outward migration of Neptune with, presumably, a phase of large eccentricity of the planetary orbits. Some problems still remain open, and the details of some mechanisms have still to be understood, but the basic composition of the cocktail appears quite secure. This is a big step forward with respect to our understanding of solar system formation, before the discovery of the Kuiper Belt. What is next? The upcoming generation of telescopic surveys will probably increase by another order of magnitude the number of discovered trans-Neptunian objects with good orbits within a decade. Thus, in the third edition of the encyclopedia, we will probably have a different story to tell. It is unlikely that our view will totally change with the new discoveries (or at least we hope so!), but certainly there will be surprises. We are anxious to know more precisely the absolute magnitude distributions of the various subpopulations of the Kuiper Belt, their color properties, the real nature of the outer edge and its exact location, the orbital distribution of the extended scattered disk. This information will allow us to refine the scenarios outlined earlier, possibly to reject some and design new ones, in an attempt to read with less uncertainty the history of our solar system that is written out there.

Bibliography Bernstein, G. M., Trilling, D. E., Allen, R. L., Brown, M. E., Holman, M., and Malhotra, R. (2004). The size distribution of trans-Neptunian bodies. Astron. J. 128, 1364–1390. Brown M. (2001). The inclination distribution of the Kuiper Belt. Astron. J. 121, 2804–2814. Gomes R. S. (2003). The origin of the Kuiper Belt high inclination population. Icarus 161, 404–418. Gomes, R., Levison, H. F., Tsiganis, K., and Morbidelli, A. (2005). Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets. Nature 435, 466–469. Levison H. F., and Stern S. A. (2001). On the size dependence of the inclination distribution of the Main Kuiper Belt. Astron. J. 121, 1730–1735. Levison, H. F., and Morbidelli, A. (2003). The formation of the Kuiper Belt by the outward transport of bodies during Neptune’s migration. Nature 426, 419–421. Malhotra R. (1995). The origin of Pluto’s orbit: Implications for the solar system beyond Neptune. Astron. J. 110, 420–432. Morbidelli, A., Brown, M.E., and Levison, H. F. (2003). The Kuiper Belt and its primordial sculpting. Earth Moon and Planets 92, 1–27. Morbidelli, A., and Levison, H. F. (2004). Scenarios for the origin of the orbits of the trans-Neptunian objects 2000 CR105 and 2003 VB12 (Sedna). Astron. J. 128, 2564–2576. Trujillo C. A., and Brown M. E. (2001). The radial distribution of the Kuiper Belt. Astrophys. J. 554, 95–98. Trujillo C. A., and Brown M. E. (2002). A correlation between inclination and color in the classical Kuiper Belt. Astroph. J. 566, 125–128.