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Fig. 8. The dynamical lifetime for small particles in the Kuiper belt derived from 4 billion year integrations [38]. Each particle is represented by a narrow vertical strip of color, the center of which is located at the particle's initial eccentricity and semi-major axis (the 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 x 109 years. Dark regions are particularly unstable on short 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 (a, e) elements of the Kuiper belt objects with well-determined orbits are also shown as green dots. Large dots are for i < 4°, small dots otherwise elements of the known Kuiper belt objects. Big dots refer to bodies with i < 4°, consistent with the low inclination at which the stability map has been computed. Small dots refer to objects with larger inclination and are plotted only for completeness.

As can be seen in the figure, the Kuiper belt has a complex dynamical structure, although some general trends can be easily explained.

Stability Limits Imposed by Close Encounters with Neptune

Most objects with perihelion distances less than ~ 35 AU are unstable. This is because they pass sufficiently close to Neptune that they are destabilized during the encounters. In fact, in these cases, Neptune's gravity is no longer a "small perturbation" relative to that of the Sun. The regularity of the oscillation of the orbital elements is broken. The semi-major axis suffers jumps at each encounter with the planet, and the eccentricity has correlated jumps to keep the perihelion distance roughly constant (more precisely, to conserve the Tisserand parameter, see Sect. 2). Through one encounter after another, the object wanders over the (a, e) plane: the object is effectively a member of the Scattered disk. Consequently, the q = 35 AU curve can be considered as the approximate border between the Kuiper belt and the Scattered disk, in the 30-50 AU semi-major axis range. The real border, however, has a more complicated, fractal structure, illustrated by the boundary between the black and the yellow regions in Fig. 8.

Not all bodies with q < 35 AU are unstable. The exception is those objects in mean-motion resonances with Neptune. These objects, despite approaching (or even intersecting) the orbit of Neptune at perihelion, never approach the planet to short distance. This happens because the resonance plays a role protecting against close encounters.

The stabilizing role of a mean-motion resonance can be understood in simple, qualitative terms. For instance, Fig. 9 illustrates the mechanism for the case of Pluto (2:3 mean-motion resonance). The trajectory of Pluto is shown in the figure in a frame that rotates with Neptune. Pluto moves in a clockwise direction when further from the Sun than Neptune and moves in a counter-clockwise direction when closer to the Sun. In the figure, an object with Pluto's eccentricity and exactly at Neptune 2:3 mean-motion resonance would have a trajectory that is a double-lobed structure oriented as in Fig. 9a. The configuration shown in the figure will remain fixed only if the object's semi-major axis is exactly equal to that characterizing the location of the resonance. For an object with semi-major axis slightly displaced, the double-lobed structure will slowly precess in the rotating frame.

If the semi-major axis of the object is slightly larger than that corresponding to the exact location of the resonance, the double-lobed trajectory will slowly precess toward that shown in Fig. 9b. If the precession continued indefinitely, eventually the trajectory would pass over the location of Neptune and a close encounter or a physical collision would occur. However, because the new trajectory is no longer symmetric with respect to Neptune, the object receives its largest acceleration (am) from Neptune when in or near the upper lobe. At this point, the object is leading Neptune in its orbit, and thus, it is slowed down in its heliocentric motion. Consequently, its semi-major axis decreases.

When the semi-major axis of the object becomes smaller than that corresponding to the exact location of the resonance, the situation reverses. Now the double-lobed trajectory slowly precesses in the opposite direction. The configuration of Fig. 9a is restored, and then the trajectory continues to pre-cess toward the configuration of Fig. 9c. In this case, the object gets its largest acceleration when it is near perihelion and is trailing Neptune in their orbits (near the lower lobe of the trajectory). Thus, the object's orbital velocity is increased, increasing its semi-major axis.

When the semi-major axis of the object again becomes larger than the exact resonant value, the precession of the double-lobed trajectory reverses

Fig. 9. The dynamics of an object in the 2:3 mean-motion resonance with Neptune. The double-lobed curve represents the orbit of an object with the eccentricity of Pluto. The coordinate frame rotates counterclockwise at the average speed of Neptune. Thus, Neptune (dot labeled "N") is stationary in this figure. The location of the Sun is labeled "S". A) The orbit of an object whose semi-major axis is equal to that characterizing the exact location of the resonance. The gravitational perturbations of Neptune cancel out because of the symmetry in the geometry. Thus, this orbit does not precess in the rotating frame. B) If the symmetry is broken, there is a net acceleration because of Neptune. Here, the strongest perturbation (am) is at the upper lobe. The object is leading Neptune at this lobe, so the net acceleration will decrease its semi-major axis. C) The strongest perturbation is in the lower lobe. Consequently, the object's semi-major axis has to increase. D) The orbit of an object that librates in the resonance. Courtesy of H. Levison

Fig. 9. The dynamics of an object in the 2:3 mean-motion resonance with Neptune. The double-lobed curve represents the orbit of an object with the eccentricity of Pluto. The coordinate frame rotates counterclockwise at the average speed of Neptune. Thus, Neptune (dot labeled "N") is stationary in this figure. The location of the Sun is labeled "S". A) The orbit of an object whose semi-major axis is equal to that characterizing the exact location of the resonance. The gravitational perturbations of Neptune cancel out because of the symmetry in the geometry. Thus, this orbit does not precess in the rotating frame. B) If the symmetry is broken, there is a net acceleration because of Neptune. Here, the strongest perturbation (am) is at the upper lobe. The object is leading Neptune at this lobe, so the net acceleration will decrease its semi-major axis. C) The strongest perturbation is in the lower lobe. Consequently, the object's semi-major axis has to increase. D) The orbit of an object that librates in the resonance. Courtesy of H. Levison once more. The trajectory goes back to the configuration of Fig. 9a and then to that of Fig. 9b, and the cycle repeats indefinitely. Each cycle is called a libration. Over a full libration cycle, the pattern drawn by the object's dynamics in the frame co-rotating with Neptune is that illustrated in Fig. 9d.

Therefore, the mean-motion resonance exerts on the object a restoring torque that reverses the precession of its double-lobed trajectory before a close encounter can occur. This of course happens only if the object is not too far from the exact resonance location, otherwise the precession is too fast, and the magnitude of the restoring torque is not sufficient. The limiting distance from the exact resonance location within which the restoring torque is effective defines the resonance width.

The analytic computation of resonance widths is detailed in [128]. This calculation, however, overestimates the width of the region where resonant objects are stable over the Solar System's age. In fact, the situation is complicated by the interaction between the libration motion inside the resonance and the precession motion of the orbits of the object and of the perturbing planet. A detailed exploration of the stability region inside the two main mean-motion resonances of the Kuiper belt, the 2:3 and 1:2 resonances with Neptune, has been done in [136,137]. Its results are beyond the scope of this chapter.

Secular Resonance Instabilities

In Fig. 8, one can see that the dark region extends significantly below the q = 35 AU line for 40 < a < 42 AU (and also for 35 < a < 36 AU). The instability in these regions is due to the presence of a secular resonance, such that dw/dt ~ dwN /dt, where w is the perihelion longitude of the object and that of Neptune.

This resonance forces large variations in the eccentricity of the trans-Neptunian object, so that - even if the initial eccentricity is zero - the perihelion distance eventually decreases below 35 AU, and the object enters the Scattered disk [82,126].

The destabilizing effect of a secular resonance between the longitude of perihelia can be understood in easy qualitative terms. Consider a simple case where the orbits of the object and of two planets are in the same plane. The presence of two planets is necessary, otherwise the planetary orbit would be a fixed, non-precessing ellipse. The orbit of the small body also precesses under the planets' perturbations. The left plot in Fig. 10 shows the long-term trajectories of these objects in a fixed frame. The middle plot shows the same system in a frame that rotates with the precession rate of the small body. Note that the orbit of the small body (the outermost orbit) is, in this frame, a fixed ellipse. If the precession rates of the planetary orbits are different from that of the small body, the trajectories of the two planets in the rotating frame are still, on average, axisymmetric, and thus, the small body experiences no long-term torques. However, if one of the planets precesses at the same rate as the small body, as in the right plot in Fig. 10, its long-term trajectory is also a fixed ellipse in the rotating frame, and it is no longer axisymmetric. Thus, the small body feels a significant long-term torque, which can lead to a significant change in its eccentricity (which is related to the angular momentum).

The location of secular resonances in the Kuiper belt has been computed in [98]. This work showed that this secular resonance is present only at small inclination. Large inclination orbits with q > 35 AU and 40 < a < 42 AU are

Fig. 10. The dynamics of a secular resonance. Three orbits are shown in each panel. The inner two are planets, which are shown as black lines. The outer orbit (gray line) is for a small object. The orbits of each object are ellipses, and the ellipses are precessing due to the mutual gravitational effects of the planets. Left: The orbits of the objects over a period of time that is long compared to the precession time of the orbits. Here, we are looking in a fixed, non-rotating reference frame. Each orbit sweeps out a torus of possible positions. Center: The same as in the left plot, except that we are looking in a frame that rotates at the precession rate of the small outer body. Thus, its orbit is again an ellipse. This panel shows the geometry if no secular resonance exists. Note that the trajectories of the planets look axisymmetric. Therefore, there is no net torque on the outer small object. Right: Same as the middle plot, except that the outer object is in a secular resonance with the inner planet, i.e. both orbits precess at the same rate. As a result, the outer object no longer sees an axisymmetric gravitational perturbation from the inner planet. Indeed, it feels a significant torque. Courtesy of H. Levison therefore stable. Indeed, Fig. 8 shows that many objects with i > 4° (small dots) are present in this region. Only large dots, representing low-inclination objects, are absent.

Chaotic Diffusion in the Kuiper Belt

Figure 8 also shows the presence of narrow bands with brown colors, crossing the yellow stability domain. These bands correspond to orbits that become Neptune-crossing only after billions of years of evolution. What is the nature of these weakly unstable orbits?

It has been found [137] that these orbits are, in general, associated either with high-order mean-motion resonances with Neptune (i.e., resonances for which the equivalence kdX/dt = kNdXN/dt holds only for large values of the integer coefficients k, kN) or three-body resonances with Uranus and Neptune (which occur when kdX/dt + kNdXN/dt + kudAy/dt = 0 occurs for some integers k, kN, and ku).

The dynamics of objects in these resonances is chaotic. The semi-major axis of the objects remains locked at the corresponding resonant value, while the eccentricity of their orbits is slowly modified. In an (a, e)-diagram like Fig. 11, each object's evolution leaves a vertical trace. This phenomenon is called chaotic diffusion. Eventually, the growth of the eccentricity can bring the diffusing object above the q = 35 AU curve. These resonances are too

Fig. 11. The evolution of objects initially at e = 0.015 and semi-major axes distributed in the 36.5-39.5 AU range. The dots represent the proper semi-major axis and the eccentricity of the objects - computed by averaging their a and e over 10 My time intervals - over the age of the Solar System. They are plotted in gray after the perihelion has decreased below 32 AU for the first time. Labels Nk : kN denote the k : kN two-body resonances with Neptune. Labels kNN+kuU+k denote the three-body resonances with Uranus and Neptune, corresponding to the equality kNAn + kuAu + kA = 0. Reprinted from [137]

Fig. 11. The evolution of objects initially at e = 0.015 and semi-major axes distributed in the 36.5-39.5 AU range. The dots represent the proper semi-major axis and the eccentricity of the objects - computed by averaging their a and e over 10 My time intervals - over the age of the Solar System. They are plotted in gray after the perihelion has decreased below 32 AU for the first time. Labels Nk : kN denote the k : kN two-body resonances with Neptune. Labels kNN+kuU+k denote the three-body resonances with Uranus and Neptune, corresponding to the equality kNAn + kuAu + kA = 0. Reprinted from [137]

weak to offer an effective protection against close encounters with Neptune, unlike the low-order resonances considered above. Thus, once above this critical curve, the encounters with Neptune start to change the semi-major axis of the objects, which leave their original resonance and evolve - from that moment on - in the Scattered disk.

Notice from Fig. 11 that some resonances are so weak that, despite forcing the resonant objects to diffuse chaotically, they cannot reach the q = 35 AU curve within the age of the Solar System. Therefore, these objects are "stable" from the astronomical point of view.

Notice also that chaotic diffusion is effective only for selected resonances. The vast majority of the simulated objects are not affected by any macroscopic diffusion. They preserve their initial small eccentricity for the entire age of the Solar System. Thus, the current moderate/large eccentricities and inclinations of most of the Kuiper belt objects cannot be obtained from primordial circular and coplanar orbits by dynamical evolution in the framework of the current orbital configuration of the planetary system. 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. Also, the overall mass deficit of the Kuiper belt cannot be due to objects escaping through resonances, because most of the inhabited Kuiper belt is stable for the current planetary architecture. Therefore, all these intriguing properties of the Kuiper belt's structure must, instead, be explained within the framework of the formation and primordial evolution of the Solar System. This will be the topic of Sect. 4.

1.4 Note on the Scattered Disk

We have seen above that the bodies that escape from the Kuiper belt and decrease their perihelion distance below 35 AU, without being protected by a low-order mean-motion resonance, enter the Scattered disk.

Their subsequent evolution has been studied in detail in [107]. It was found that the median dynamical lifetime is — 50 My, the typical end-states being transport toward the inner Solar System (and eventual ejection from the Solar System because of an encounter with Jupiter or Saturn; see Sect. 2), a collision with a planet or outward transport toward the Oort cloud (see Sect. 3). This result suggests that the Scattered disk could be a population of transient objects, which is sustained in steady state by a continuous flux of objects escaping from the Kuiper belt. In this case, the Scattered disk would be, relative to the Kuiper belt, what the population of Near Earth Asteroids is, relative to the main asteroid belt.

However, [39] showed that about 1% of the Scattered disk objects can survive on trans-Neptunian orbits for the age of the Solar System. This leads to the possibility that the current Scattered disk is the remnant of a — 100 x more massive primordial structure, which presumably formed when the planets removed the left-over planetesimals from their formation regions. In this case, the Scattered disk would not be in steady state, and it would have no direct relationship with the Kuiper belt.

How can we discriminate between these two hypotheses on the origin of the Scattered disk? In the first case, if the Scattered disk is sustained in steady state by the objects leaking out of the Kuiper belt, the number ratio between the Kuiper belt and Scattered disk populations must be large. Indeed:

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