The Formation of the Oort Cloud

To explain the formation of the Oort cloud, it is intuitive to invoke the mechanism described in the previous section for the origin of LPCs, but "played" in "reverse mode." Imagine an early time when the Oort cloud was still empty and the giant planets' neighborhoods were full of icy planetesimals. The scattering action of the planets dispersed the planetesimals throughout the Solar System. Some were moved onto eccentric orbits with large semi-major axis, but with perihelion distance still in the planetary region. Those that reached a semi-major axis of ^10,000 AU started to feel a galactic tide strong enough to modify their orbit on a timescale of one orbital period. During the scattering process, these planetesimals remained relatively close to the ecliptic plane, so that their inclination relative to the galactic plane 1 was ~ 120°. Because of their large e and 1, the effect of the tide on the evolution of e, I was large. The planetesimals with Q between 90° and 180° (or, symmetrically, between 270° and 360°) had their eccentricity decreased. This lifted their perihelion distances beyond the planets' reach, so that they could not be scattered any more: they became Oort cloud objects. The precession of Q and the occasional passage of rogue stars randomized the planetesimals' distribution, resulting in the Oort cloud structure that is inferred from observations of LPCs.

This scenario, originally proposed in [104], was first simulated in [42,43] using a Monte Carlo method to represent the effects of repeated, uncorrelated encounters of the planetesimals with the giant planets and passing stars (the role of the galactic tide was not yet taken into account). The first simulation of Oort cloud formation using direct numerical simulations and accounting for the galactic tide was done in [36]. To save computing time, however, the simulations were started with comets already on low inclination, high eccentricity orbits: initially having a = 2,000 AU and q uniformly distributed between 5 and 35 AU.

The formation of the Oort cloud has recently been revisited in [33] (see also [32]), using more modern numerical simulation techniques. The authors started with more realistic initial conditions, assuming planetesimals initially distributed in the 4-40 AU zone with small eccentricities and inclinations. The giant planets were assumed to be on their current orbits, and the migration of planets in response to the dispersion of the planetesimals (see Sect. 4) was neglected. The evolution of the planetesimals was followed for 4 Gy, under the gravitational influence of the four giant planets, the galactic tide (both radial and disk components - see (11), (12)), and passing stars. Both the tide and the statistics of passing stars were calibrated using the current galactic environment of the Sun. A stellar density of 0.041M0/pc3 was assumed, with stellar masses distributed in the range 0.11-18.24 M0 and relative velocities between 1.7 and 158kms-1 (with a median value of 46kms~1). In total, the simulation described in [33] recorded ^50,000 stellar encounters within 1 pc of the Sun in 4 Gy. In the following discussion of Oort cloud formation, I mostly refer to the results of this work.

Figure 21 shows an example of the evolution of a comet from the neighborhood of Neptune to the Oort cloud. Through a sequence of encounters, the object is first scattered by Neptune to larger semi-major axis, while keeping the perihelion distance slightly beyond 30 AU (typical of Scattered disk bodies). After about 700My, the random walk brings the body's semi-major axis to ~ 10,000 AU. At this time the galactic tide starts to be effective, and the perihelion distance is rapidly lifted above 45 AU. Neptune's perturbations cease to be important and further changes in semi-major axis are due to the effects of distant stellar encounters. When the body starts to feel the galactic tide, its inclination relative to the galactic plane is 120°. As the perihelion distance is lifted (the eccentricity decreases), the inclination decreases towards 90°.9 A stellar passage causes a sudden jump of Z to 65° just before t =1 Gy. This allows the effect of the tide to become more pronounced, bringing the perihelion distance of the object beyond 1,000 AU and the inclination Z up to 80°. This configuration is reached at t = 1.7Gy, when cZ is 180°. From this time onward, the galactic tide reverses its action, decreasing q and Z. In principle, the action of the galactic tide is periodic, so that the object's perihelion should be decreased back to planetary distances. However, the jumps in a, q, Z caused by stellar encounters break this reversibility. The oscillation of q becomes more shallow and the object never returns to the planetary region within the age of the Solar System. Notice finally that during this evolution,

9Notice that, for the dynamical evolution forced by the galactic disk tide, the decrease of I from 120 to 90° is equivalent to an increase from 60 to 90°, in agreement with what has been said in the previous section on the anti-correlation of the evolutions of eccentricity and inclination.

Fig. 21. An example of evolution of a comet from the vicinity of Neptune into the Oort cloud, from [33]. The top panel shows the evolution of the object's semi-major axis (red) and perihelion distance (blue). The bottom panel shows the inclinations relative to the galactic plane (green) and Solar System invariable plane (the plane orthogonal to the total angular momentum of the planetary system; in magenta)

Fig. 21. An example of evolution of a comet from the vicinity of Neptune into the Oort cloud, from [33]. The top panel shows the evolution of the object's semi-major axis (red) and perihelion distance (blue). The bottom panel shows the inclinations relative to the galactic plane (green) and Solar System invariable plane (the plane orthogonal to the total angular momentum of the planetary system; in magenta)

the inclination relative to the invariable plane is strongly changed. It is turned to retrograde, and then back to prograde values, as the longitude of galactic node Q precesses.

Not all particles follow this evolution, though. Those that interact closely with Jupiter and Saturn are mostly ejected from the Solar System. Those that have distant encounters with Saturn are transported more rapidly and further out in semi-major axis compared with the evolution shown in Fig. 21. The strength of the galactic tide increases with a; thus, for the comets that are scattered to a ~ 20, 000 y or beyond, the oscillation period of q and 1 is shorter than for the particle in Fig. 21.

Figures 22 and 23 give a global illustration of the Oort cloud formation process, showing snapshots of the (a, q) and (a, i) distributions of all planetes-imals at 0 (initial conditions), 1, 10, 100 My and 1, 4 Gy. The planetesimals in these plots are color-coded according to their initial position: Jupiter region objects are magenta; Saturn region objects are blue; Uranus region objects are green; Neptune region objects are red, and trans-Neptunian objects are black. Figure 22 shows that, after only 1 My, a Scattered disk is formed by Jupiter and Saturn, out of particles initially in the Jupiter-Uranus region. This Scattered disk differs from the current Scattered disk because most of

1 10 100 1000 10* 10s 1 10 100 1000 104 10B

Fig. 22. Scatter plot of osculating barycentric pericenter distance vs. osculating barycentric semi-major axis, at various times in the Oort cloud formation simulations of [33]. The points are color-coded to reflect the region in which the simulated comets formed. Each panel is labeled by the simulation time that it corresponds to

1 10 100 1000 10* 10s 1 10 100 1000 104 10B

Fig. 22. Scatter plot of osculating barycentric pericenter distance vs. osculating barycentric semi-major axis, at various times in the Oort cloud formation simulations of [33]. The points are color-coded to reflect the region in which the simulated comets formed. Each panel is labeled by the simulation time that it corresponds to its objects have q < 10 AU. Particles originally in Neptune's region or beyond have not yet been scattered. At 10 My, a signature of the galactic tide starts to be visible. The Oort cloud begins to form. Particles with a > 30,000, mostly from the Jupiter-Saturn region, have their perihelia lifted beyond the orbits of the planets. Neptune's particles start to populate the Scattered disk. From 100 My to 1 Gy, particles continue to enter the Oort cloud from the Scattered disk. The population of the Oort cloud peaks at 840 My, at which time 7.55% of the initial particles occupy the cloud. Objects from the Uranus-Neptune region gradually replace those from the Jupiter-Saturn zone. The latter have been lost during stellar encounters, as they predominantly occupied the very outer part of the Oort cloud (a > 30,000 AU). Because of the longer time over which the galactic tide has acted and to stellar encounters, the population of bodies with perihelion distances above 100 AU can have semi-major axes as low as 3,000 AU. The Oort cloud with a < 20,000 AU is usually called the inner Oort cloud, or Hills cloud from [80]. The last panel in Fig. 22, representing the

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