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C values close to 3.00, the possibility of temporary satellite capture must definitely be considered together with that of temporary Trojan-type librations. For nearly all Jupiter group comets, however, one finds Cs 2.97; therefore, satellite captures become rather unlikely. Such C values fall into the range typical for Trojan librations (as low as C = 2.673 for 1208 Troilus, among the known Trojan planets). Comets, however, may pass through ao = 1 in most cases, of course, without getting caught into temporary oscillations of a within the narrow boundaries 0.95 s a s 1.05 required for Trojan-type librations (Rabe, 1970). Because the Jupiter group comets as well as the Trojans are able, practically by definition, to experience crossovers through ao = 1, it should, of course, be expected that the C distributions of both groups will overlap. On the other hand, all known Trojans vary their osculating a values within the rather narrow limits 0.97 s a s 1.03, whereas the Jupiter comets attain a values as small as 0.44 (P/Encke) and as large as 1.38 (P/Oterma). For the eccentricities, the observed upper limit e = 0.15 for the Trojans has already been mentioned, whereas the comets considered here have e values up to 0.85. Only the inclinations I have similar distributions in both groups, with a very few being somewhat larger than 30° in each category. On the basis of their significantly different a and e distributions, however, it would not be surprising to find rather different C distributions, too, with a similarly narrow overlap as in a and e. As repeatedly mentioned, though, there is an extensive overlap of the two C distributions, suggesting a close dynamical affinity of the two groups of bodies. The principal purpose of this paper is the computation and presentation of a sufficiently large number of individual C values, as well as of some related quantities that are of interest with regard to possible conjectures concerning a dynamical relationship between Trojans and Jupiter group comets. For the 15 numbered Trojan planets, the elements needed in equations (1) and (2) have been taken from the Leningrad Ephemeris volume for 1971, except for those of the not yet listed newest member 1749 Telamon, which are given in Minor Planet Circular 3019. For 38 comets of the Jupiter group rather approximate but for the present purpose sufficiently accurate elements were taken from Marsden's (1967) tabulation of such comets with a values presently (~1965) inside Jupiter's a' = 1. Three other comets were added, without any attempt to achieve completeness: P/Oterma (Astron. J. 66, 248), P/Schwassmann-Wachmann 1 (Astron. J. 66, 268), and P/Slaughter-Burnham (Acta Astron. 18, 419). Finally, the quite exceptional minor planet 944 Hidalgo was also included here for the sake of comparison. For the total of 57 objects, the computed C values are listed in decreasing order in table I, together with the “crossover parameter” Yo from equation (4). Of the basic elements a, e, and I, only e has been listed, under the heading e1, so that it may be compared with the eo given in the last column, which is the largest possible value of eat crossover in connection with a = 1. Because equation (3) has to be satisfied also when Y = Yo, this maximum eo would occur only in conjunction with I = 0, whereas the also possible maximum I (a = 1, b = 0) = Yo would TABLE I.—The Jacobi Constant C and Related Quantities for Trojans and

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TABLE I.—The Jacobi Constant C and Related Quantities for Trojans and Selected Jupiter Group Comets–Concluded

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require e = 0. It is well known that I as well as e may undergo large variations during close approaches to Jupiter, so that the possibility of a near-zero inclination I after some approach cannot be excluded. Normally, though, I will not become very small; and, consequently, the related crossover eccentricity e satisfying equation (3) will be smaller than the listed maximum eo. No eo values have been computed for the 15 numbered Trojans, because it may be assumed that they move in stable libration orbits and do not experience close approaches to Jupiter. We know that their e and I values vary only rather moderately during the small long-period oscillations of a, so that the formal computation of eo for I = 0 would make no physical sense. If they would be computed and given, they would fall right into the systematic trend of this last column in table I, because the listing is in order of C, and eo and Yo are functions of C alone. To facilitate the separate recognition of Trojans and comets in table I, the Yo values of the Trojans have been omitted as well. It is seen from table I that even the numbered and presumably stable Trojans have C values that extensively overlap those of the Jupiter comets. It is very likely that most of the at least several hundred additional Trojans (van Houten et al., 1970) fall into about the same range, 2.67s C & 3.00, simply because of their probably stable association with the equilateral points. Of the 41 comets in table I, 34, or 83 percent, also have C values between 2.67 and 3.00. For those contemplated unstable or escaped Trojans with initial e values exceeding 0.15, the related C values could easily be as small as 2.5; therefore, there could be a complete overlap of the ranges of C values for the comets and Trojans. Of particular interest are the eo values of the 41 comets. Most of them are much smaller than the associated el values; therefore, in connection with the suggested Trojan origin of such comets, the related values of e for a = 1 should be in quite reasonable agreement with the e range that is permissible by dynamical considerations for librational motions of the Trojan type. First, the actual crossover values of e will normally be smaller than the maximum eo; and, second, on the theoretical side, we know that in the restricted Sun-Jupiter case, even stable librations of short period may involve e values near 0.64. In the one known case of temporary librations, for P/Slaughter-Burnham, such motion actually occurs with e - 0.52. When the possibility of Trojan origin for some Jupiter group comets was first suggested (Rabe, 1970), the detailed features of the reverse event of capture into libration for P/Slaughter-Burnham were interpreted as indicating a rather small probability of such capture for any given comet because, in most cases, the large perturbations in a during the required Jupiter approach will tend to overshoot the apparently necessary entry conditions. Such probability considerations have no bearing on the contemplated escapes from librations of an unstable nature. The only requirement seems to be that the original Trojan clouds had to be large enough to permit the formation and growth of condensations even near the fringes of librational stability. When Oort (1950) discussed the proposed existence of a very distant cloud of comets surrounding the solar system, he suggested that these bodies might actually be unstable escapees from the original minor planet belt between Mars and Jupiter. It appears now that, at least for the Jupiter comets, the escape from the two Trojan clouds provides a much simpler and more direct mechanism of asteroid transfer into cometary motion, without the need of moving these bodies first to the remote fringes of the solar system, and of then recapturing them in a complicated chain of dynamical events. It should be noted again, as in Rabe (1970), that the similarity of the anomalous distributions of the perihelion longitudes of the Trojans and of the Jupiter group comets lends further support to their proposed common origin. Also, the complete absence of retrograde orbits would automatically be accounted for by such an origin of these comets. The relatively large masses of some Trojans can no longer be considered as an argument against a common origin, because the van Houtens and Gehrels (1970) have found that the frequency of the Trojans increases greatly with decreasing magnitude.


Houten, C. J. van, Houten-Groeneveld, I. van, and Gehrels, T. 1970, Minor Planets and Related Objects. V. The Density of Trojans Near the Preceding Lagrangian Point. Astron. J. 75, 659-662.

Hunter, R. B. 1967, Motions of Satellites and Asteroids Under the Influence of Jupiter and the Sun. II. Asteroid Orbits Close to Jupiter. Mon. Notic. Roy. Astron. Soc. 136, 267-277.

Marsden, B.G. 1967, One Hundred Periodic Comets. Science 155, 1207-1213.

Oort, J. H. 1950, The Structure of the Cloud of Comets Surrounding the Solar System. and a Hypothesis Concerning Its Origin. Bull. Astron. Inst. Neth, 11, 91-110.

Rabe, E. 1970, Orbital Characteristics of Comets Passing Through the 1:1 Commensurability With Jupiter. Proc. IAU Symp. no. 45, Motion, Orbit Evolution and Origin of Comets. Leningrad.


Smithsonian Astrophysical Observatory

There has long been speculation as to whether comets evolve into asteroidal objects. On the one hand, in the original version of the Oort (1950) hypothesis, the cometary cloud was supposed to have formed initially from the same material that produced the minor planets; and an obvious corollary was that the main physical difference between comets and minor planets would be that the latter had long since lost their icy surfaces on account of persistent exposure to strong solar radiation (Öpik, 1963). However, following a suggestion by Kuiper (1951), it is now quite widely believed that, whereas the terrestrial planets and minor planets condensed in the inner regions of the primordial solar nebula, icy objects such as comets would have formed more naturally in the outer parts, perhaps even beyond the orbit of Neptune (Cameron, 1962; Whipple, 1964a). Furthermore, recent studies of the evolution of the short-period comets indicate that it is not possible to produce the observed orbital distribution from the Oort cloud, even when multiple encounters with Jupiter are considered (Havnes, 1970). We must now seriously entertain the possibility that most of the short-period orbits evolved directly from low-inclination, low-eccentricity orbits with perihelia initially in the region between, say, the orbits of Saturn and Neptune, and that these comets have never been in the traditional cloud at great distances from the Sun.

On the other hand, there is also the extreme point of view that comets completely disintegrate after only a few passages near the Sun. This feature was present in the original Whipple (1950) icy-conglomerate comet model, principally on account of the widespread assumption that the frequent and complete disappearance of comets was an observed fact. Twenty yr ago, 44 comets were known to have been observed at more than one perihelion passage, but 10 of these (i.e., 23 percent) were regarded as lost, having failed to appear at several of their recent returns. The number of more-than-oneappearance comets has now risen to 59; and 5, if not 6, of those lost have been found, reducing the proportion of those lost to only 7 or 8 percent. Two of the comets were found by accident, but the reduced percentage is mainly a demonstration of what can be done when modern computational and observational techniques are applied to the problem (Klemola, 1965; Kowal, 1970a,b; Marsden, 1963; Roemer, 1964, 1968; Schubart, 1965); and there is

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