4 Discussion and conclusions

The recent discovery of the Jovian-class comets (Mumma et al. 2001b; Kawakita et al. 2001), most likely formed in the Jupiter-Saturn region, raises the question of how they were formed. It has been concluded (Franklin et al. 1989; Murray & Dermott 1999) that low-eccentricity, low-inclination orbits between Jupiter and Saturn are unlikely to survive longer than 10⁷ years, therefore any object formed in that region during the early stages of the formation of the outer Solar System was ejected long ago. Similar calculations for the Saturn-Uranus zone (Gladman & Duncan 1990; Holman & Wisdom 1993) gave an upper limit for the survival time scale in that region of 10⁸ years. From a dynamical point of view, the Jovian-class comets consist of objects formed in the Jupiter-Saturn region, in a time scale likely shorter than 10⁸ years, that were ejected towards a stable region beyond Neptune. Objects in these region are long-lived but not indefinitely stable and they eventually fall inwards crossing the Giant Planets region. The purpose of this paper has been to investigate a plausible scenario of the formation of the comet C/1999 S4 LINEAR, the first member of this class: trapping of solid material in resonances induced by gas drag. We have shown that growing protoplanets can confine a disk of metric sized dust particles. From an orbital dynamics point of view, our results suggest an evolutionary scenario with solid material being trapped at different distances from the growing protoplanets and the trapping distances gradually changing as the protoplanets keep growing. This may translate into a continuous spectrum in the physical properties of the hypothetical minor bodies formed. On the other hand, the mechanism discussed here is size-selective, with bodies of certain size preferentially trapped by certain resonances. The orbital changes due to gas drag are monotonic in nature; as the orbital energy diminishes there is a secular decrease in the semi-major axis of the bodies. The orbital decay of each particle continues until it reaches a commensurability strong enough for the resonant disturbing function to be comparable to the drag. Because of the sensitivity of the trapping mechanism to initial conditions, trapping may not necessarily take place at the first such resonance each body encounters. The body may well continue its orbital decay until it reaches another strong resonance, where conditions for equilibrium are favourable. This explains the gaps in size found in Fig. 4. In principle, if these bodies were the building blocks of Jovian-class comets and asteroids, some of them may be very uniform in structural composition with blocks only of a certain size contributing to the final body. It is a plausible scenario when the growing proto-Jupiter was the only planet present in the outer Solar System. This is, however, unlikely as the resonances overlap when two planets are considered (see Figs. 3 and 4) and fragments of different size may appear after collisions.

Very recently, Wurm et al. (2001) have pointed out that current models of km-sized planetary building blocks, or planetesimals, by collisional accretion require unrealistically low collision velocities or ad hoc assumptions about sticking in order for growth to occur. Once cm-sized agglomerates have formed, collision velocities for typical nebula models increase by orders of magnitude to several tens of m/s (Weidenschilling & Cuzzi 1993). Therefore the impact velocities are so high that the smaller agglomerates should be completely disrupted and the larger bodies should be eroded or cratered. However, our results suggest that if the scenario presented in our paper is true, the collisional velocities for bodies trapped in a gas-induced resonance could be much lower, in fact lower than a few m/s. This is particularly favourable for the Jupiter-alone case where resonant rings do not overlap. As the relative velocity, $V_{\rm r}$, in a gas-induced resonance is low the total impact energy in a two-body collision

$$% E = \frac{1}{2} \frac{m_{1} m_{2}}{m_{1} + m_{2}} V^{2}_{\rm r},$$ (1)

where m₁ and m₂ are the masses of the colliding bodies, is also low, making smooth sticking possible. The relative velocity is $V_{\rm r} \sim 0.5 G (M/a^{3})^{1/2} \Delta a$ for two bodies separated by $\Delta a = a_2 - a_1$, where M is the mass of the Sun, and G is the gravitational constant. In a typical resonance presented in this paper $\Delta a$ is very negligible therefore collisions are more similar to actual mergers than impacts, at least in the single planet case, where resonances do not overlap. This environment is unique in producing weakly gravitationally bound cometesimal aggregates. As heat dissipation in such an encounter is very negligible, the physicochemical structure of the building blocks could remain virtually unaltered, preserving protosolar abundances. For the single planet configuration, building blocks of the cometesimal objects are very uniform in size. If two planets are considered, then a relatively wide size spectrum is expected as collisions between bodies located in overlapping resonances may occur. The size spectrum found in C/1999 S4 LINEAR suggests that this comet was formed beyond Saturn when proto-Saturn was already present in the Solar System and therefore it may not be a genuine representative of the very primordial generation of minor bodies that formed soon after Jupiter. However, according to the recent constraints on the time scales of the formation of the giant planets presented by Hersant et al. (2001) and based on D/H measurements in the Solar System, the comet C/1999 S4 LINEAR may have formed when the Solar System was a few Myr old.

Figure 4: Same calculations as in Fig. 3. The figures show the orbital elements (semi-major axis and eccentricity) of the particles after they are captured in the gas-induced resonances. (upper panel) Eccentricity as a function of time of the trapped particles for two different simulations, (Sun+Jupiter and Sun+Jupiter+Saturn). The two resonances found in the Jupiter-alone case shift outward when Saturn is also included. Note the additional resonances that appear when both Jupiter and Saturn are considered. (lower panel) Radius of the particle as a function of the orbital elements. The gas-induced resonant trapping is rather selective as regards particle size. See text for further explanations.

The results of our calculations compare well with those from Beaugé et al. (1994) although such a comparison is only possible in the Jupiter case. At the end of their collisional simulation only three bodies remain with orbital elements: a₁ = 8.73 AU, e₁ = 0.076, a₂ = 10.03 AU, e₂ = 0.083, a₃ = 10.91 AU, e₃ = 0.111. In our non-collisional simulation we obtain two remaining rings, Fig. 3, with averaged orbital elements: a₁ = 8.7 AU, e₁ = 0.08, a₂ = 11.4 AU, e₂ = 0.13, in spite of using different nebula model, integration techniques and initial conditions. However, in our case the final orbits of the trapped bodies are of one type, $\sigma$-libration, although they also found corotation. This can be explained as a result of the non-collisional nature of our calculations (our particles do not grow or fragment) as well as different initial conditions (particle size).

On the other hand, it has become evident that the primordial Earth was totally degassed and that the volatile elements of the biosphere, including water vapor, were brought later by comets. The existence of these first generation of comets may have major implication on the rise of life on Earth. The low D/H ratio in Earth's oceans could reflect the contribution of Jovian-class comets, which would have provided most of the mass (Delsemme 1998, 2000; Morbidelli et al. 2001), as the D/H ratio in the oceans is very close to the mean value of the D/H ratio of the water inclusions in carbonaceous chondrites (for a recent and detailed discussion of this problem and its constraints see Hersant et al. 2001). In this scenario, asteroids and the comets from the Jupiter-Saturn zone were the first water deliverers, when the Earth was less than half its present mass. A late impact phase of icy planetesimals known as "Late Heavy Bombardment" may have been triggered by the formation of Uranus and Neptune (Levison et al. 2001).

Recently, Tegler & Romanishin (1998) have suggested the existence of two distinct populations of Centaurean objects. These authors found that some of the Centaurs have colors similar to the average colours of C-type asteroids. The other group includes some of the reddest objects known in the Solar System, similar (but redder) to D-type asteroids. However, Luu & Jewitt (1996) found diversity of colours rather than two colour populations although their photometry is less accurate. If the bimodal color distribution is confirmed, grey Centaurs (the Chiron and Chariklo group) may have formed in the Jupiter-Saturn region within the scenario discussed in this paper and red Centaurs (the Nessus and Pholus group) may have appeared beyond Uranus and Neptune. To confirm or deny this hypothesis, D/H ratio measurements for both families of objects should be carried out.

In principle, one can argue against our previous discussion and say that our present work considers the possibility that the building blocks of the unusual comet C/1999 S4 LINEAR formed in the Uranus-Neptune region, migrated inward due to gas drag, were captured into an exterior mean motion resonance with the growing proto-Jupiter and/or proto-Saturn and therefore there is somewhat of a disconnect between the calculations and the conclusions of this study as it is conceivable that the cometesimals captured in a resonance would undergo enough heating to be consistent with the chemistry discussed by Mumma et al. (2001b) but this is not discussed in the paper. However, this apparent inconsistency can be easily explained by considering in a more detailed way the dynamical behaviour of infalling material during the early stages of the formation of the outer Solar System. The survival time scale for centimetric to metric size particles in the Jupiter-Saturn region before the formation of the Giant Planets is less than a few thousand years (e.g. Weidenschilling 1977; Cuzzi et al. 1993; de la Fuente Marcos & Barge 2001), therefore the solid material must be continuously replenished with particles from the outermost regions of the primordial nebula. Taking this fact into consideration all the solid material considered in our model has not been formed in the Uranus-Neptune but beyond that region. We choose 20 AU as initial condition to speed up the calculations but the solid material was likely formed beyond that distance. This choice does not affect any of our conclusions. On the other hand, the papers by Mumma et al. (2001b) and Kawakita et al. (2001) are based in the spectroscopic analysis of the organic volatile composition and the NH₂ respectively of C/1999 S4 LINEAR. In both cases, they are not providing any information about the chemical composition of the solid (non-volatile) component of the building blocks of the comet, to do so physical sampling is needed. When a solid particle coming from the outer regions of the primordial nebula spirals inwards as a result of gas drag, its surface is likely to be covered by ices that become depleted as the particle follows its path towards the proto-Sun. When the particle becomes trapped in one of the suggested resonances and is reprocessed by collisional coagulation to be part of a larger body some heating of the material is expected that can eventually produce chemical anomalies but local volatiles can also be trapped during the event. The surface of the newly formed body is in some way new and its composition is likely very different from the primordial of the smaller building blocks. The reprocessing has taken place at a temperature of about 200 K. As Jupiter and Saturn are being formed, the physical environment surrounding the trapping regions is gas rich and the ices condensed from locally processed nebular gas should reflect the local composition, not the primordial one (for recent works on this topic see (e.g.) Rodgers & Charnley 2002; Charnley & Rodgers 2002). Later, when the newly formed object becomes dynamically unstable and is ejected outwards, the modified volatile composition is preserved under new frozen material. Thousands of million years later, when the cometary object begins its journey back to the Giant Planets region the process is inverted and the outermost frozen material is depleted as the temperature increases making the inner (and older) layers available for spectroscopic study.

Our results and discussion are not meant to provide a unique or complete model of the formation of Jovian-class objects. Instead we simply point out a plausible scenario for the formation of these objects. If, as proposed in this paper, the Jovian-class objects (and possibly some Centaurs and irregular satellites of Jupiter and Saturn) are the debris left by Jupiter and Saturn, the study of this new cometary population will increasingly yield new and unique insights into the conditions that existed in the solar nebula at the formation epoch of Jupiter and Saturn.

Acknowledgements

We thank Dr. C. Beaugé and Dr. S. J. Aarseth for valuable discussions about the theoretical and numerical details of their work. We thank the Department of Astrophysics of Universidad Complutense of Madrid for providing excellent computing facilities at the Centro de Proceso de Datos in Moncloa. Part of the computations described in this paper have been performed on the SGI Origin 2000 of the "Centro de Supercomputación Complutense'' through the UCM project "Dinámica Estelar y Sistemas Planetarios'' (CIP 454). We would like to thank the referee, L. Dones. His suggestions have lead to a substantially improved paper. In preparation of this paper, we made use of the NASA Astrophysics Data System and the ASTRO-PH e-print server.