Two methods have been developed to determine corotation equilibrium points for the elliptical restricted 3-body problem. The simpler one is a first order (in the small parameter) method which has however no limitation for the planet or planetesimal eccentricities since it does not assume any truncation of the disturbing function. In this way, corotation points can be determined for any planetary eccentricity and any resonance and the results certainly show an improvement with respect to previous analytical methods (Beaugé & Ferraz-Mello). Moreover, corotation points for interior resonances are also determined by this first order method. Notwithstanding the improvement we obtain through this method, some corotation points found in numerical simulations cannot be reproduced by it. This is the case of the (exterior) extended corotation points (Mothé-Diniz & Gomes 2000) which are only obtained with higher order (in the small parameter) theory. A generic method, which is basically a numerical iterative algorithm, is thus developed to determine corotation points not only for any planetary orbit but also for any planetary mass. In this way, extended corotation points could be reproduced. It is noteworthy that both methods work for any nonconservative force acting on the massless body and we did tests with drag models both proportional to the velocity absolute value and to its square. By presenting both methods, we aimed at showing all the limitations imposed by a first order theory, although not truncated in the perturbing function. This point was not very clear from previous analytical works on this subject (Beaugé 1998; Beaugé et al. 1999). We must note however that the generic method must be in the future preferred to the first order one since, besides being much more accurate, it also presented a computing time comparable to that of the first order method.
Our results basically confirm that corotation resonances occur usually for a limited range of drag coefficients or, other things being constant, for a limited range of planetesimal sizes. For any resonance, given a non null planetary eccentricity, there is a minimum size for which trapping is possible and this will occur in the corotation regime. There is also a maximum size below which trappings always occur in the corotation regime. Above this limit, trapping will take place always in the libration regime. The range of sizes for which corotation trapping is possible will thus be a function of planetary eccentricity (increasing with it) and the specific resonance (usually increases for resonances closer to the planet). For a standard nebular model, planetesimals sizes for which corotation trapping occurs, considering a Jupiter-like planet (distance from the star and mass) extends from a few meters to near 1 km. It is interesting to note that for this size range gas drag is very efficient in inducing a fast orbital decay for planetesimals, thus a resonance trapping by a proto-planet can halt this fast decay. Moreover, because corotation resonance induces a fast accumulation of equal sized particles into common equilibrium points, with approximations taking place with very low relative speed, these points may work to produce larger planetesimals from smaller ones, which will then suffer a much slower drag decay.
For some resonances and planetary mass, the so called extended corotation resonance can take place. Its main characteristic is that there is no lower limit for drag rate (or upper limit for planetesimal size) below (above) which the regime is changed to libration. In this sense, the accretion process could be in principle carried on until a planetary embryo was formed. The problem is that this kind of resonance was only observed for some higher order resonances (4:7, 7:9), which are associated to a generally lower capture probability and also overlaps with more important lower order resonances inducing possible high velocity impacts among objects in different resonances.
Important (low j) first order interior resonance trappings have been found only for high planetary eccentricities with Jupiter sized planets. For modest (Jupiter-like) planetary eccentricity, only interior resonance closer to the planet (high j) are possible. In this case, planetary mass must be increasingly lower so as to avoid chaos. In all cases, however, trapping probability was found in the numerical examples to be very low, what limits the applicability of interior resonance trapping for mass accretion.
The importance of corotation resonance trapping seems thus limited in stopping the decay and promoting the accretion of meter-1 km sized planetesimals in exterior resonances with protoplanets in order to form larger objects that are subject to strong enough mutual gravitation forces to help in their further accretion process. In this sense, this mechanism is particularly useful if we consider the recently ressurected gravitational instability model (Boss 1997) for the formation of Jupiter and possibly Saturn. If the first planet is to be formed by the more conventional accretion theory no resonance trapping can be claimed to start meter sized planetesimals accretion. Now one question naturally rises: what is the fate of planetesimals that would be formed at corotation points and after gaining mass pass to the libration regime? For this case, there is no accumulation point. Since trapped planetesimals have higher eccentricities than nontrapped ones, a natural consequence, first noted by Weidenschilling & Davis (1985), would be the high velocity disrupting impacts of trapped planetesimals producing smaller fragments that would naturally escape resonance (larger drag rate) and eventually accrete to the planet. Yet it is not obvious that high eccentricities for planetesimals trapped in (the same) resonance will generally produce high velocity collisions. Because orbits of trapped planetesimals cannot be considered random, this point deserves a deeper investigation, which we are presently addressing. In advance it seems now clear that collisions between bodies of similar sizes (suffering similar drag forces) will occur in relatively low velocities, increasing however for higher planetary eccentricity.
The methods developed in this paper helped us to determine precisely the real limits of influence of a corotation resonance trapping. Our main goal, for which these methods work as a tool, is to determine the real influence of resonance trapping of planetesimals with proto-planets in a forming planetary system. Does this phenomenon favor accretion or hamper it? We plan to have a fairly precise answer to this question with our ongoing research.
Acknowledgements
D. E. Santos Jr. is grateful to CAPES for the scholarship.
Copyright ESO 2001