Our observational knowledge of the trans-Neptunian population1 is very recent. The first object, Pluto, was discovered in 1930, but unfortunately this discovery was not quickly followed by the detection of other trans-Neptunian objects. Thus, Pluto was thought to be an exceptional body - a planet - rather than a member of a numerous small body population, of which it is not even the largest in size. It was only in 1992, with the advent of CCD cameras and a lot of perseverance, that another trans-Neptunian object - 1992 QB1 - was found [86]. Now, 13 years later, we know more than 1,000 trans-Neptunian objects. Of them, about 500 have been observed for at least 3 years. A timespan of 3 years of observations is required in order to compute their orbital elements with some confidence. In fact, the trans-Neptunian objects move very slowly, and most of their apparent motion is simply a parallactic effect. Our knowledge of the orbital structure of the trans-Neptunian population is therefore built on these ~ 500objects.

Before moving to discuss the orbital structure of the trans-Neptunian population, in the next subsection, a brief overview of the basic facts of orbital dynamics is given. The expert reader can move directly to Sect. 1.2.

Neglecting mutual perturbations, all bodies in the Solar System move on an elliptical orbit relative to the Sun, with the Sun at one of the two foci of the ellipse. Therefore, it is convenient for astronomers to characterize the relative motion of a body by quantities that describe the geometrical properties of its orbital ellipse and its instantaneous position on the ellipse. These quantities are usually called orbital elements.

The shape of the ellipse can be completely determined by two orbital elements: the semi-major axis a and the eccentricity e (Fig. 1). The name eccentricity comes from e being a measure of the distance of the focus from the center of the ellipse, in units of semi-major axis' length ("eccentric" means xThere is no general consensus on nomenclature, yet. In this work, I use "trans-Neptunian population" to describe the collection of small bodies with semi-major axis (or equivalently orbital period) larger than that of Neptune, with the exception of the Oort cloud (semi-major axis larger than 10,000 AU).

aphelion

Fig. 1. Keplerian motion: definition of a, e and E

aphelion

Fig. 1. Keplerian motion: definition of a, e and E

"away from the center"). The eccentricity is therefore an indicator of how much the orbit differs from a circular one: e = 0 means that the orbit is circular, whereas e =1 means that the orbit is a segment of length 2a, the Sun being at one of the extremes. Among all "elliptical" trajectories, the latter is the only collisional one, if the physical radii of the bodies are neglected. A semi-major axis of a = to and e =1 denotes parabolic motion, while the convention a < 0 and e > 1 is adopted for hyperbolic motion. I will not deal with these kinds of unbounded motion in this chapter and will therefore concentrate, hereafter, on the elliptic case. On an elliptic orbit, the closest point to the Sun is called the perihelion, and its heliocentric distance q is equal to a(1 — e); the farthest point is called the aphelion, and its distance Q is equal to a(1 + e).

To denote the position of a body on its orbit, it is convenient to use an orthogonal reference frame qi,q2 with origin at the focus of the ellipse occupied by the Sun and q1 axis oriented towards the perihelion of the orbit. Alternatively, polar coordinates r, f can be used. The angle f is usually called the true anomaly of the body. From Fig. 1, using elementary geometrical relationships, it can be seen that

where E, as Fig. 1 shows, is the angle subtended at the center of the ellipse by the projection - parallel to the q2 axis - of the position of the body on the circle that is tangent to the ellipse at perihelion and aphelion. This angle is called the eccentric anomaly. The quantities a, e and E are enough to characterize the position of a body in its orbit.

From Newton equations, it is possible to derive [28] the evolution law of E with respect to time, usually called the Kepler equation:

where _

is the orbital frequency, or mean motion, of the body, m0 and mi are the masses of the Sun and of the body, respectively, and Q is the gravitational constant; t is the time and t0 is the time of perihelion passage. Astronomers like to introduce a new angle

called the mean anomaly, as an orbital element that changes linearly with time. M also denotes the position of the body in its orbit, through equations (3) and (2).

To characterize the orientation of the ellipse in space, with respect to an arbitrary orthogonal reference frame (x, y, z) centered on the Sun, one has to introduce three additional angles (see Fig. 2). The first one is the inclination, i, of the orbital plane (the plane that contains the ellipse) with respect to the (x, y) reference plane. If the orbit has a nonzero inclination, it intersects the (x, y) plane in two points, called the nodes of the orbit. Astronomers

distinguish between an ascending node, where the body passes from negative to positive z, and a descending node, where the body plunges towards negative z. The orientation of the orbital plane in space is then completely determined when one gives the angular position of the ascending node from the x axis. This angle is traditionally called the longitude of ascending node and is usually denoted by Q. The last angle that needs to be introduced is the one characterizing the orientation of the ellipse in its plane. The argument of perihelion, u, is defined as the angular position of the perihelion, measured in the orbital plane relative to the line connecting the central body to the ascending node.

In the definition of the orbital elements above, note that when the inclination is zero, u and M are not defined, because the position of the ascending node is not determined. Moreover, M is also not defined when the eccentricity is zero, because the position of the perihelion is not determined. Therefore, it is convenient to introduce the longitude of perihelion w = u + Q and the mean longitude X = M + u + Q. The first angle is well defined when i = 0, whereas the second one is well defined when i = 0 and/or e = 0.

In the absence of external perturbations, the orbital motion is perfectly elliptic. Thus, the orbital elements a, e, i, w, Q are fixed, and X moves linearly with time, with frequency given by (4). When a small perturbation is introduced (for instance the presence of an additional planet), two effects are produced. First, the motion of X is no longer perfectly linear. Correspondingly, the other orbital elements have short periodic oscillations with frequencies in the order of the orbital frequencies. Second, the angles w and Q start to rotate slowly. This motion is called precession. Typical precession periods in the Solar System are of the order of 10,000-100,000years. Correspondingly, e and i have long periodic oscillations, with periods of the order of the precession periods.

The regularity of these short and long periodic oscillations is broken when one of the following two situations occur: (i) the perturbation becomes large, for instance when there are close approaches between the body and the perturbing planet, or when the mass of the perturber is comparable to that of the Sun (as in the case of encounters of the Solar System with other stars) or (ii) the perturbation becomes resonant. In either of these cases, the orbital elements a, e, i can have large nonperiodic, irregular variations.

A resonance occurs when the frequencies of X,w or Q of the body, or an integer combination of them, are in an integer ratio with one of the time frequencies of the perturbation. If the perturber is a planet, the perturbation is modulated by the planet orbital frequency and precession frequencies. We speak of mean-motion resonance when kdX/dt = k'dX'/dt, with k and k' integer numbers and X' denoting the mean longitude of the planet. We speak of linear secular resonance when dw/dt = dw'/dt or dQ/dt = dQ'/dt, prime variables referring again to the planet. Other types of resonances exist in more complicated systems (non-linear secular resonances, three-body resonances, Kozai resonance etc.). Resonant motion will be discussed more specifically in

Sect. 1.3, when reviewing the dynamical properties of some traras-Neptunian sub-populations.

The trans-Neptunian population is "traditionally" divided into two subpopulations: the Scattered disk and the Kuiper belt. The definition of these sub-populations is not unique, with the Minor Planet center and various authors often using slightly different criteria. Here I propose a partition based on the dynamics of the objects and their relevance for the reconstruction of the primordial evolution of the outer Solar System, keeping in mind that all bodies in the Solar System must have been formed on orbits typical of an accretion disk (e.g. with very small eccentricities and inclinations).

I call the Scattered disk the region of the orbital space that can be visited by bodies that have encountered Neptune within a Hill's radius,2 at least once during the age of the Solar System, assuming no substantial modification of the planetary orbits. The bodies that belong to the Scattered disk in this classification do not provide us any relevant clue to uncover the primordial architecture of the Solar System. In fact their current eccentric orbits might have been achieved starting from quasi-circular ones in Neptune's zone by pure dynamical evolution, in the framework of the current architecture of the planetary system.

I call the Kuiper belt the trans-Neptunian region that cannot be visited by bodies encountering Neptune. Therefore, the non-negligible eccentricities and/or inclinations of the Kuiper belt bodies cannot be explained by the scattering action of the planet on its current orbit, but they reveal that some excitation mechanism, which is no longer at work, occurred in the past (see Sect. 4).

To categorize the observed trans-Neptunian bodies into the Scattered disk and Kuiper belt, one can refer to previous works on the dynamics of trans-Neptunian bodies in the framework of the current architecture of the planetary system. For the a < 50 AU region, one can use the results by [38] and [103], who numerically mapped the regions of the (a, e, i) space with 32 < a < 50 AU, which can lead to a Neptune encountering orbit within 4 Gy. Because dynamics are reversible, these are also the regions that can be visited by a body after having encountered the planet. Therefore, according to the definition above, they constitute the Scattered disk. For the a > 50 AU region, one can use the results in [107] and [39], where the the evolutions of the particles that encountered Neptune in [38] have been followed for another 4 Gy time-span. Although the initial conditions did not cover all possible configurations, one

2The Hill's radius is given by the formula RH = ap(mp/3)1/3, where mp is the mass of the planet relative to the mass of the Sun and ap is the planet's semi-major axis. It corresponds approximately to the distance from the planet of the Lagrange equilibrium points L1 and L2.

can reasonably assume that these integrations cumulatively show the regions of orbital space that can be visited by bodies transported to a > 50 AU by Neptune encounters. Again, according to my definition, these regions constitute the Scattered disk.

Figure 3 shows the (a, e, i) distribution of the trans-Neptunian bodies, which have been observed during at least three oppositions. The bodies that belong to the Scattered disk according to my criterion are represented as red dots. The Kuiper belt population is in turn subdivided into two subpopulations: the resonant population (green dots) and the classical belt (blue dots). The former is made of objects located in the major mean-motion resonances with Neptune (essentially the 3:4, 2:3 and 1:2 resonances, but also the 2:5 - see [23]), while the classical belt objects are not in any noticeable resonant configuration. Mean-motion resonances offer a protection mechanism against close encounters with the resonant planet (see Sect. 1.3). For this

Fig. 3. The orbital distribution of multi-opposition trans-Neptunian bodies, as of Aug. 26, 2005. Scattered disk bodies are represented in red, Extended Scattered disk bodies in orange, classical Kuiper belt bodies in blue and resonant bodies in green. We qualify that, in the absence of long-term numerical integrations of the evolution of all the objects, and because of the uncertainties in the orbital elements, some bodies could have been mis-classified. Thus, the figure should be considered as an indicative representation of the various subgroups that compose the trans-Neptunian population. The dotted curves in the bottom left panel denote q = 30 AU and q = 35 AU; those in the bottom right panel q = 30 AU and q = 38 AU. The vertical solid lines mark the locations of the 3:4, 2:3 and 1:2 mean-motion resonances with Neptune. The orbit of Pluto is represented by a crossed circle

Fig. 3. The orbital distribution of multi-opposition trans-Neptunian bodies, as of Aug. 26, 2005. Scattered disk bodies are represented in red, Extended Scattered disk bodies in orange, classical Kuiper belt bodies in blue and resonant bodies in green. We qualify that, in the absence of long-term numerical integrations of the evolution of all the objects, and because of the uncertainties in the orbital elements, some bodies could have been mis-classified. Thus, the figure should be considered as an indicative representation of the various subgroups that compose the trans-Neptunian population. The dotted curves in the bottom left panel denote q = 30 AU and q = 35 AU; those in the bottom right panel q = 30 AU and q = 38 AU. The vertical solid lines mark the locations of the 3:4, 2:3 and 1:2 mean-motion resonances with Neptune. The orbit of Pluto is represented by a crossed circle reason, the resonant population can have perihelion distances much smaller than those of the classical belt objects. Stable resonant objects can even have Neptune-crossing orbits (q < 30 AU) as in the case of Pluto (see Sect. 1.3). The bodies in the 2:3 resonance are often called Plutinos, because of the similarity of their orbit with that of Pluto. According to [168], the Scattered disk and the Kuiper belt constitute roughly equal populations, while the resonant objects, altogether, make up about 10% of the classical objects.

In Fig. 3, the existence of bodies on highly eccentric orbits with a > 50 AU can be seen. These objects do not belong to the Scattered disk according to my definition (orange dots). Among them are 2000 CR105 (a = 230 AU, perihelion distance q = 44.17 AU and inclination i = 22.7°), Sedna (a = 495 AU, q = 76 AU), 2004 XR190 (a = 57.4AU, q = 51 AU) and the current size-record holder 2003 UB313 (a = 67.7AU, q = 37.7 AU but i = 44.2°; diameter equal to 2400± 100km [18]), although for some objects the classification is uncertain for the reasons explained in the figure caption. Following [56], I call these objects Extended Scattered disk objects for three reasons. (i) They are very close to the Scattered disk boundary. (ii) Bodies of the sizes of the three objects quoted above (300-2000 km) presumably formed much closer to the Sun, where the accretion timescale was sufficiently short [153]. This implies that they have been transported in semi-major axis space (e.g. scattered), to reach their current locations. (iii) The lack of objects with q > 41 AU and 50 < a < 200 AU should not be due to observational biases, given that many classical belt objects with q > 41 AU and a < 50 AU have been discovered (see Fig. 6). This suggests that the Extended Scattered disk objects are not the highest eccentricity members of an excited belt beyond 50 AU. These three considerations indicate that in the past the true Scattered disk extended well beyond its present boundary in perihelion distance. The reason for this is still under debate. Some ideas will be presented in Sect. 4.

Given that observational biases become more severe with increasing perihelion distance and semi-major axis, the currently known Extended Scattered disk objects may be like the tip of an iceberg, the emerging representatives of a conspicuous population, possibly outnumbering the Scattered disk population [56].

An important clue to the history of the early outer Solar System is the dynamical excitation of the Kuiper belt. While the eccentricities and inclinations of resonant and scattered objects are expected to have been affected by interactions with Neptune, those of the classical objects should have suffered no such excitation. Nonetheless, the confirmed classical belt objects have an inclination range up to at least 32° and an eccentricity range up to 0.2, significantly higher than expected from a primordial disk, even accounting for mutual gravitational stirring.

The observed distributions of eccentricity and inclination in the Kuiper belt are highly biased. High eccentricity objects have closer approaches to the Sun, and thus, they become brighter and are more easily detected. Consequently, the detection bias roughly follows curves of constant q. At first sight, this bias might explain why, in the classical belt beyond a = 44 AU, the eccentricity tends to increase with semi-major axis. However, the resulting (a, e) distribution is significantly steeper than a curve q =constant. Thus, the apparent relative under-density of objects at low eccentricity in the region 44 < a < 48 AU is likely to be a real feature of the Kuiper belt distribution.

High-inclination objects spend little time at the low latitudes3 at which most surveys take place, while low-inclination objects spend no time at the high latitudes where some searches have occurred. Using this fact, [16] computed a de-biased inclination distribution for classical belt objects (Fig. 4).

A clear feature of this de-biased distribution is its bi-modality, with a sharp drop around 4° and an extended, almost flat distribution in the 4-30° range. This plateau is required to fit the presence of objects with large inclinations. The bi-modality can be modeled with two Gaussian functions and suggests the presence of two distinct classical Kuiper belt populations, called hot (i > 4) and cold (i < 4) after [16].

Fig. 4. The inclination distribution (in degree) of the classical Kuiper belt, from [131]. 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 the best fit two-population model f (i)di = sin(i)[96.4 exp(-i2/6.48) + 3.6exp(-i2/288)]di [16]. In this model, ~60% of the objects have i > 4°

inclination

Fig. 4. The inclination distribution (in degree) of the classical Kuiper belt, from [131]. 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 the best fit two-population model f (i)di = sin(i)[96.4 exp(-i2/6.48) + 3.6exp(-i2/288)]di [16]. In this model, ~60% of the objects have i > 4°

3Latitude (angular height over a reference curve in the sky) and inclination should be defined with respect to the local Laplace plane (the plane normal to the orbital precession pole), which is a better representation for the plane of the Kuiper belt than is the ecliptic [41].

Physical Evidence for Two Populations in the Classical Belt

The co-existence of a hot and a cold population in the classical belt 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 one 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 [108] who noted that the intrinsically brightest classical belt objects (those with lowest absolute magnitudes) are preferentially found on high-inclination orbits. This conclusion has been recently verified in a bias-independent manner in [171], with a survey for bright objects which covered —70% of the ecliptic and found many hot classical objects but few cold classical ones.

The second possible physical difference between hot and cold classical Kuiper belt objects is their colors, which relate in an unknown manner to surface composition and physical properties. Several possible correlations between orbital parameters and color were suggested by [163] and further investigated by [34]. The issue was clarified by [170] who quantitatively showed that for the classical belt, the inclination is correlated with color. In essence, the low-inclination classical objects tend to be redder than higher inclination objects (see Fig. 5). This correlation has since been confirmed by several other authors [35,41]. Whether or not there is also a correlation between color and perihelion distance is still a matter of debate [35].

More interestingly, we see that the colors naturally divide into distinct low-inclination and high-inclination populations at precisely the location of the divide between the hot and cold classical objects. These populations differ at a 99.9% confidence level. In addition, 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 [170]. The possibility remains, however, that the colors of the objects, rather than being markers of different populations, are actually caused by the different inclinations. For example [157] has suggested that the higher average impact velocities of the high-inclination objects will cause large-scale resurfacing by fresh water ice, which could be blue to neutral in color. However, careful analysis has shown that there is no clear correlation between average impact velocity and color [165].

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.

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