Jnj t B

where A and B are constants representing the amplitude of the libration and the phase, respectively. With p ^0.001, nJ ~ 0.52 year-1, the characteristic frequency is w = (27pnJ /4)V2 ~ 0.043 year 1, corresponding to a period 2n/w ~ 150 years, or about 10 times Jupiter's orbital period. The distribution of libration amplitudes, A, is very broad, with a mean near 30° [110,136]. In addition to wide excursions about the Lagrangian points in the orbital plane of the planet, the Trojans also occupy a broad distribution of inclinations, with a bias-corrected mean of 14° [69] to 17° [136]. As a consequence, the velocity dispersion amongst the Trojans (~5kms-1) rivals that amongst the main-belt asteroids. Collisions between Jovian Trojans are expected to be highly erosive.

Limited work on the long-term stability of Trojans at Jupiter suggests two loss mechanisms. There is a slow leak because dynamical chaos [97], with a timescale that depends on A (larger A being less stable). The more significant loss process is due to ejection from the Lagrangian clouds following collisions or near-miss interactions. Kilometer-sized and larger bodies are lost this way at a rate ~103My-1 [103], meaning that the observed population of small objects should vanish in a few x 102 My. That small Trojans remain is presumably a result of a collisional cascade, with the small objects being both lost and continually supplied from the break-up of larger bodies. Ejected Trojans pursue orbits that are scattered by the planets, quickly becoming indistinguishable from the orbits of Jupiter family comets. Up to —10% of the latter could be escaped Jovian Trojans [70,103]: the contributions from the Trojan swarms of other planets are unknown.

Several physical properties of the Trojans have been measured. The size distribution is a broken power law [70,136]. Objects with absolute magnitudes V(1,1,0) < 9.5 (corresponding to diameters > 84 km, for albedo 0.04) are well described by a differential power law index q = 5.5 ± 0.9 (Fig. 43). Those with 11 < V(1,1,0) < 14 (diameters 4.4 < D < 40 km, for the same albedo) instead have q = 3.0 ± 0.3 [70]. The index for the smaller objects is close to the nominal value expected for a system in collisional equilibrium [34], consistent with the idea that these smaller bodies are part of a collisionally-produced cascade. The steep slope of the large Trojans presumably reflects a "production function": at least, these big bodies seem unlikely to have been molded

Fig. 43. Brightness distribution of the Jovian Trojans , showing the break in the size distribution. Red points show the numbered Trojans. The roll-over above V(1,1,0) — 10 is due to observational incompleteness. The blue points are from [70], scaled to correct for the small area of the Trojan swarms observed in that survey. The difference in slope between the large and small objects is independent of the scaling. The radius scale at the top is computed on the assumption that the Trojans all have albedo 0.04. From [70]

Fig. 43. Brightness distribution of the Jovian Trojans , showing the break in the size distribution. Red points show the numbered Trojans. The roll-over above V(1,1,0) — 10 is due to observational incompleteness. The blue points are from [70], scaled to correct for the small area of the Trojan swarms observed in that survey. The difference in slope between the large and small objects is independent of the scaling. The radius scale at the top is computed on the assumption that the Trojans all have albedo 0.04. From [70]

much by energetic collisions. For comparison, the D > 100 km KBOs occupy a distribution with q ~ 4 [151]. Within the errors (~2<r), this is compatible with the size distribution of the larger Trojans, as might be expected if the latter were captured from the Kuiper belt [115].

As already noted, the optical color distribution of the Trojans is different from that of the KBOs and Centaurs because the Trojans lack ultrared matter. This could mean that there is no relation between the Trojans and the KBOs or Centaurs, or it could mean that the surfaces of the Trojans have been modified in some way by their exposure to sunlight (as have the surfaces of the Jupiter family comet nuclei, which very likely do come from the Kuiper belt). We prefer the latter explanation, but it does not tell us anything about the source of the Trojans, because the surface modification process could operate regardless of the origin of the bodies. Any object formed beyond the snow-line (perhaps originally at ~3AU or slightly closer) is expected to be icy and should evolve when heated to develop a surface mantle. In the same vein, the albedo distribution of the Trojans is very narrow compared to that of the KBOs and Centaurs [48] but more similar to the nuclei of Jupiter family comets (Fig. 34). This is probably also a result of surface modification on bodies that have been heated strongly by the sun but, again, we cannot use this information to specify the source of the Trojans. In terms of their spectra, the Trojans have steadfastly resisted every attempt to assess surface composition from observations taken in the optical and near infrared [37,42, 101]. The absence of features is consistent with the very dark surfaces of these bodies and suggests (but does not require) an organic-rich compositional nature [24]. Observations at thermal wavelengths have revealed features consistent with emission from silicates in three Trojans (624 Hektor, 911 Agamemnon and 1172 Aneas; [25]).

Lastly, the density of Trojan (617) Patroclus has been estimated from infrared observations [48] and from its dynamical system mass as p = 800+100 kgm+3 [105]. Although the authors of [105] cite this low density as evidence that Patroclus is a captured KBO, in fact low density is only evidence for a high mass fraction of ice and/or vacuum ("porosity") and cannot be diagnostic of the Trojan source. In fact, any object formed at any distance beyond the snow-line would be expected to have a high ice content and correspondingly low density. Simply put, "density is not destiny."

An accurate summary is that the available physical data on Trojans, from their surface colors [69] to their albedos [48] to the one measured density [105] are similar to the corresponding quantities reported for the nuclei of comets but not similar to those of the KBOs. The measured Trojan properties very probably reflect refractory surface mantles left behind following ancient mass-loss, but we cannot uniquely determine the source of the Trojans from the physical data. An interesting exercise for the readers of this article is to think of observations that could be taken to uniquely determine the source of the Trojans. I, for my part, will be trying to do exactly the same.

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