Cd

Fig. 32. Disk location and LHB timing. The histogram reports the average dynamical lifetime of massless test particles placed in a planetary system with Jupiter, Saturn, and the ice giants on nearly circular, coplanar orbits at 5.45, 8.18, 11.5, and 14.2 AU, respectively (marked as black triangles on the plot). The dynamical lifetime was computed by placing 10 particles with e = i = 0 and random mean anomaly in each semi-major axis bin. Each vertical bar in the plot represents the average lifetime for those 10 particles, after having removed stable Trojan cases. The "lifetime" is defined as the time required for a particle to encounter a planet within one Hill radius. A comparison between the histogram and the putative lifetime of the gaseous nebula [72] suggests that, when the nebula dissipated, the inner edge of the planetesimal disk had to be about 1-1.5 AU beyond the outermost ice giant. The time at which Jupiter and Saturn crossed their 1:2 mean-motion resonance, as a function of the location of the planetesimal disk's inner edge, is shown with filled dots. From [64]

8 10 12 14 Heliocentric Distance (AU)

Fig. 32. Disk location and LHB timing. The histogram reports the average dynamical lifetime of massless test particles placed in a planetary system with Jupiter, Saturn, and the ice giants on nearly circular, coplanar orbits at 5.45, 8.18, 11.5, and 14.2 AU, respectively (marked as black triangles on the plot). The dynamical lifetime was computed by placing 10 particles with e = i = 0 and random mean anomaly in each semi-major axis bin. Each vertical bar in the plot represents the average lifetime for those 10 particles, after having removed stable Trojan cases. The "lifetime" is defined as the time required for a particle to encounter a planet within one Hill radius. A comparison between the histogram and the putative lifetime of the gaseous nebula [72] suggests that, when the nebula dissipated, the inner edge of the planetesimal disk had to be about 1-1.5 AU beyond the outermost ice giant. The time at which Jupiter and Saturn crossed their 1:2 mean-motion resonance, as a function of the location of the planetesimal disk's inner edge, is shown with filled dots. From [64]

with the mass (6 x 1021 g) estimated from the number and size distribution of lunar basins that formed around the time of the LHB epoch [75].

As discussed in [114], however, the planetesimals from the distant disk - which can be identified as "comets" - were not the only ones to hit the terrestrial planets. The radial migration of Jupiter and Saturn forced the secular resonances v6 and v16 to sweep across the asteroid belt [58], exciting the

Fig. 33. Planetary migration and the associated mass flux toward the inner Solar System from a representative simulation of [64]. Top: the evolution of the four giant planets. Each planet is represented by a pair of curves - the top and bottom curves are the aphelion and perihelion distances, respectively. Jupiter and Saturn cross their 1:2 mean-motion resonance at 880 My. Bottom: the cumulative mass of comets (solid curve) and asteroids (dashed curve) accreted by the Moon. The comet curve is offset, so that the value is zero at the time of 1:2 resonance crossing. The estimate of the total asteroidal contribution is very uncertain but should be roughly of the same order of magnitude as the cometary contribution. However, it should occur over a longer time-span. From [64]

Fig. 33. Planetary migration and the associated mass flux toward the inner Solar System from a representative simulation of [64]. Top: the evolution of the four giant planets. Each planet is represented by a pair of curves - the top and bottom curves are the aphelion and perihelion distances, respectively. Jupiter and Saturn cross their 1:2 mean-motion resonance at 880 My. Bottom: the cumulative mass of comets (solid curve) and asteroids (dashed curve) accreted by the Moon. The comet curve is offset, so that the value is zero at the time of 1:2 resonance crossing. The estimate of the total asteroidal contribution is very uncertain but should be roughly of the same order of magnitude as the cometary contribution. However, it should occur over a longer time-span. From [64]

eccentricities and the inclinations of asteroids. The fraction of the main belt population that acquired planet-crossing eccentricities depends quite crucially on the orbital distribution that the belt had before the LHB, which is not well known. The asteroid belt could not be a massive, dynamically cold disk at the time of the LHB. If it were, essentially all of the asteroids would have been ejected onto planet-crossing orbits, the bombardment of the Moon would have been orders of magnitude more intense than that recorded by the LHB [114], and the few asteroids surviving in the belt after the secular resonance sweeping would have an orbital distribution inconsistent with that currently observed [58]. Presumably, the asteroid belt underwent a first phase of dynamical depletion and excitation at the time of terrestrial planet formation [142,181] and then a second dynamical depletion at the time of the LHB. If, at the end of the first phase, the orbital distribution in the belt was comparable to the current one, then the secular resonance sweeping at the time of the LHB would have left —10% of the objects in the asteroid belt [64]. Assuming this figure, the pre-LHB main belt contained roughly 5 x 10-3 Me (10 times its current mass) and the total mass of the asteroids hitting the Moon was comparable to that of the comets (see Fig. 33). However, slight changes in the pre-LHB asteroid distribution, and the migration rate of Jupiter and Saturn (also highly variable from simulation to simulation, depending on the chaotic evolution of Neptune), can change this result for the asteroidal contribution to the Lunar cratering rate by a factor of several. In conclusion, the model in [64] cannot state whether asteroids or comets dominated the impact flux on the terrestrial planets. What it can say, however, is that the asteroidal contribution came later and more slowly than the cometary contribution (see Fig. 33), possibly erasing much of the signature of the cometary bombardment.

The issue of which population dominated the impact rate can be solved by looking for constraints on the Moon. In [102], analysis of Lunar impact melts indicated that at least one of the projectiles that hit the Moon, and probably more, had a chemistry inconsistent with carbonaceous chondrites or comets. In [162], it was found that the impact melt at the landing site of Apollo 17 was caused by a projectile of LL-chondritic composition. These results imply that the bombardment was dominated by asteroids typical of the inner belt.

In [159], the comparison of size distributions of the craters formed at the time of the LHB on Mercury, Mars, and the Moon allowed the calculation of the ratios of the impact velocities on these planets, leading to the conclusion that most projectiles had a semi-major axis between 1 and 2AU. Comets never acquire such a small semi-major axis during their evolution, so this argument again favors a dominant contribution from the inner main belt. More recently, [160] found that the crater size distribution on the lunar highlands is consistent with the size distribution of objects currently observed in the main belt.

Taken altogether, these results point with little doubt to asteroids being the dominating (or, possibly, latest-arriving) projectile population for the terrestrial planets at the time of the LHB. However, they do not imply that the asteroids triggered the LHB. On the contrary, the result in [160] implies that the LHB was triggered by a distant disk of comets (as in [64]), for the reasons explained below.

The remarkable match between the size distributions of craters and the main belt asteroids, pointed out in [160], implies that - at the LHB

time - asteroids were ejected from the main belt onto planet-crossing orbits in proportions independent of their size .11 Only the sweeping of secular resonances can give a size-independent ejection throughout the main belt. At the time of the LHB, the gas disk was already totally dissipated. Thus, secular resonance sweeping could only be caused by the radial displacement of Jupiter and Saturn. Now, even assuming that the entire LHB on the terrestrial planets was caused by asteroids, from the mass hitting the Moon at that time [75] and the collision probability typical of NEAs with the Moon, one can easily compute that the total asteroid mass on planet-crossing orbits was about 0.01 M®. This mass was too small to cause a significant migration of the giant planets. In conclusion, a more massive disk - which could only be trans-Neptunian - had to trigger and drive planet migration. Comets mandated the bombardment, and asteroids executed it.

A Note on the Trojans and the Satellites of the Giant Planets

To validate or reject a model, it is important to look at the largest possible number of constraints. Two populations immediately come to mind when considering the LHB scenario proposed in [64]: the Trojans and the satellites of the giant planets. Is their existence consistent with this scenario?

Jupiter has a conspicuous population of Trojan objects. These bodies, usually referred to as "asteroids," follow essentially the same orbit as Jupiter, but lead or trail that planet by an angular distance of ~ 60°, librating around the Lagrange triangular equilibrium points. The first Trojan of Neptune was recently discovered [23]; and detection statistics imply that the Neptune Trojan population could be comparable in number to that of Jupiter, and possibly even ten times larger [25].

The simulations in [64,172] led to the capture of several particles on long-lived Neptunian Trojan orbits (2 per run, on average, with a lifetime larger than 80My). Their eccentricities, during their evolution as Trojans, reached values smaller than 0.1. These particles were eventually removed from the Trojan region, but this is probably an artifact of the graininess of Neptune's migration in the simulation, because of the quite large individual mass of the planetesimals [71].

Jovian Trojans are a more subtle issue that is described in detail in [133]. There is a serious argument in the literature against the idea that Jupiter and Saturn crossed their 1:2 mean-motion resonance: if the crossing had happened, any pre-existing Jovian Trojans would have become violently unstable, and Jupiter's co-orbital region would have emptied [58,124]. However, the dynamical evolution of a gravitating system of objects is time reversible. Thus, if the original objects can escape the Trojan region when it becomes unstable, other bodies can enter the same region and be temporarily trapped. Consequently,

11 unlike the current Near Earth Asteroids (NEAs) which, escaping from the belt because of size-dependent non-gravitational forces, have a size distribution significantly steeper than that of the main belt population.

a transient Trojan population can be created if there is an external source of objects. In the framework of the scenario in [64], the source consists of the very bodies that are forcing the planets to migrate, which must be a large population given how far the planets must migrate. When Jupiter and Saturn move far enough from the 1:2 resonance that the co-orbital region becomes stable, the population that happens to be there at that time remains trapped. It then becomes the population of permanent Jovian Trojans still observable today.

This possibility has been tested with numerical simulations in [133]. Of the particles that were Jupiter or Saturn crossers during the critical period of Trojan instability, a fraction between 2.4 x 10~6 and 1.8 x 10~5 remained permanently trapped as Jovian Trojans. More importantly, at the end of the simulations, the distribution of the trapped Trojans in the space of the three fundamental quantities for Trojan dynamics - the proper eccentricity, inclination, and libration amplitude [125] - was remarkably similar to the current distribution of the observed Trojans, as illustrated in Fig. 34. In particular, this is the only model proposed so far that explains the inclination distribution of the Jovian Trojans. The origin of this distribution was considered to be the hardest problem in the framework of the classical scenario, according to which the Trojans formed locally and were captured at the time of Jupiter's growth [121].

10 20 libration amplitude

Fig. 34. Comparison of the orbital distribution of Trojans between the simulations in [133] and observations. The simulation results are shown as red circles and the observations as blue dots in the planes of proper eccentricity vs. libration amplitude (left) and proper inclination vs. libration amplitude (right). The distribution of the simulated Trojans is somewhat skewed toward large libration amplitudes, relative to the observed population. However, this is not a serious problem because a fraction of the planetesimals with the largest amplitudes would leave the Trojan region during the subsequent 4Gy of evolution [106], leading to a better match. The similarity between the two inclination distributions provides strong support for the LHB model in [64]

10 20 libration amplitude

Fig. 34. Comparison of the orbital distribution of Trojans between the simulations in [133] and observations. The simulation results are shown as red circles and the observations as blue dots in the planes of proper eccentricity vs. libration amplitude (left) and proper inclination vs. libration amplitude (right). The distribution of the simulated Trojans is somewhat skewed toward large libration amplitudes, relative to the observed population. However, this is not a serious problem because a fraction of the planetesimals with the largest amplitudes would leave the Trojan region during the subsequent 4Gy of evolution [106], leading to a better match. The similarity between the two inclination distributions provides strong support for the LHB model in [64]

The capture probabilities reported above allowed [133] to conclude that the total mass of the captured Trojan population was between — 4 x 10~6 and — 3 x 10~5 M®. Previous estimates from detection statistics [88] concluded that the current mass of the Trojan population is —10~4 M®. However, taking into account modern, more refined knowledge of the Trojans absolute magnitude distribution (discussed in [133]), mean albedo [52] and density [120], the estimate of the current mass of the Trojan population is reduced to 7 x 10~6 M®, which is consistent with the simulations in [133]. The bulk density of 0.8+01 gcm 3, measured for the binary Trojan 617 Patroclus [120] is an independent confirmation of the model of chaotic capture of Trojans from the original trans-Neptunian disk. In fact, this density is significantly smaller than any density measured so far in the asteroid belt, including for the most primitive objects, while it is essentially identical to the bulk densities inferred for the trans-Neptunian objects Varuna [89] and 1997 CQ29 [138].

In conclusion, the properties of Jovian Trojans are not simply consistent with the LHB model of [64]: they constitute a strong indication - if not a smoking gun - in support of the 1:2 mean-motion resonance crossing of Jupiter and Saturn, which is at the core of the model in [64].

I now briefly come to the satellites of the giant planets. As discussed above, the non-survivability of the regular satellite systems is one of the killing arguments against the exotic scenario proposed in [110]. Because Saturn, Uranus, and Neptune also have encounters with each other in the model of [64,172], it is important to look at the satellites' fates in this new framework. This issue has been addressed in [172]. The authors recorded all encounters deeper than one Hill-radius occurring in eight simulations. Then, they integrated the evolution of the regular satellite systems of Saturn, Uranus, and Neptune during a re-enactment of these encounters. They found that, in half of the simulations, all of the satellite systems survived the entire suite of encounters with final eccentricities and inclinations smaller than 0.05. The difference in comparison to the case of [110] is that, in the latter model, both ice giants had to have close and strong encounters with Jupiter or Saturn, whereas in the simulations of [64,172], encounters with Jupiter never occur, and encounters with Saturn are typically distant, with moderate effects. Thus, the survivability of the regular satellites is not a problem for the LHB model. However, the more distant, irregular satellites would not survive the planetary encounters. Thus, if the LHB model is correct, they must have been captured at the time of the LHB (see Sect. 6).

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