1 Introduction

Resonance trapping occurs when the gain of energy and angular momentum of a particle due to resonant gravitational perturbations from a planet exactly compensates for the loss of energy and angular momentum due to an external dissipation (gas drag for instance). Since the discovery of this phenomenon for the case of planetesimals subject to gas drag by Weidenschilling & Davis (1985), several works have been done in an attempt to understand the dynamics as well as the implications of this process in the formation of the Solar System (see for instance, Beaugé & Ferraz-Mello 1993; Beaugé et al. 1994; Gomes 1995; Kary & Lissauer 1995). Weidenschilling & Davis suggested that fragments of collisions between planetesimals would pass through resonances and under the influence of gas drag be accreted by a growing planetary embryo. Patterson (1987) differently suggested that planetary formation could have processed at two-body exterior resonances due to capture and global accretion of planetesimals at resonance sites starting with Venus for the inner planets and Jupiter for the outer planets. Beaugé et al. (1994) did a numeric simulation envisaging the formation of a proto-Saturn by considering the joint effects of planetary perturbations, gas drag, mutual gravitation and collisions among a swarm of 1000 equal-radius planetesimals. They suggested that the mechanism proposed by Patterson could explain the quasi-commensurability of the planets for the outer solar system. During its evolution in inward spiraling orbits, a planetesimal can pass through all resonances and collide with the Sun or a planet, be sent away from the solar system by a close approach with the perturber or be trapped into a resonance with the planet. When a mean motion and a secular resonance occur simultaneously, we say that the body is trapped into a corotation resonance. Otherwise, a libration resonance takes place. There are important differences between the geometry of these two types of resonances. In the libration case, trapped planetesimals share the same average semi-major axis, but other orbital elements present a noticeable dispersal. This happens due to the circulation of the longitude of the perihelion. In other words, planetesimals trapped in libration resonance are distributed over all longitudes for a given time. On the other hand, equal-sizes planetesimals trapped in corotation resonances accumulate into a single point. In this case, the longitude of the perihelion librates around a fixed value (in respect with the perturber's perihelion direction). These points of accumulation have been understood as sites for accretion of small planetesimals during the process of planetary formation. In a recent work, Mothé-Diniz & Gomes (2000) studied the collisional dynamics of planetesimals subject to a gas drag dissipation. They observed that the relative collisional speed between the planetesimals at these points was very low, reinforcing the idea that corotation points are favorable places for accretion of small planetesimals.

There is usually a maximum radius of the planetesimal and a minimum one for which corotation trapping takes place. Above the maximum planetesimal radius, trappings always occur in libration resonance. Although this fact limits the range of applicability of corotation resonance trapping in planetesimals accretion, the very efficient mass accumulation process thus induced motivates the present study on a more comprehensive determination of corotation equilibrium points for any resonance, planetary orbit and mass. It must also be emphasized that the planetesimals sizes for which corotation resonances are most effective range from a few meters to a few kilometers (depending on the planet's mass, nebula's physical properties, etc.) Accretion at this size range is hardly explained through physical coalescence (valid for much smaller grains) or mutual gravitation (valid for large bodies). Moreover, this size range is that for which radial shift by drag is most effective, yielding a drag time scale of just a few thousand years (Lissauer 1993). A mechanism that can both halt the fast radial migration and provide a process of effective accretion may have had an important influence in the initial process of planetary formation.

The first work to systematically determine corotation points with gas drag was done by Beaugé & Ferraz-Mello (1993) who found these equilibrium points by developing a first order method based on mean equations, considering the case of a Stokes drag dissipation. There are however some limitations related to the first order averaging method used by Beaugé & Ferraz-Mello (1993). Due to an approximation in the classical Laplace expansion for the perturbing function, this method does not give good results for high (planetary or planetesimal) eccentricities. Also, they did not find these points for interior resonances. Before that, Weidenschilling & Davis (1985) claimed that trappings could only occur for exterior resonances. They explained that only for exterior resonance the effect of planetary resonance (move away the planetesimal semi-major axis from the planet) could counterbalance the effect of gas drag. Patterson (1987) also did not find them, but more recently, some numerical simulations have shown trappings of planetesimals for interior resonances (see for instance, Kary & Lissauer 1995). In particular, for the case of corotation resonances, these trappings are stable.

For libration points, in particular, Beaugé (1999) showed that some effects of short-period terms in the location of the equilibrium solutions that are not considered in the first order averaging method brings significant difference to the location of the points. More recently, Beaugé et al. (1999) developed a second order method to explain the discrepancy between the corotation equilibrium points given by the analytic methods and numerical simulations. The results obtained by this new model showed a considerable improvement as compared with the first order method. However, even using the second order method, Beaugé et al. (1999) did not found some types of trappings that are found in numerical simulations, as the extended corotation resonance (Mothé-Diniz & Gomes 2000) and trappings of particles into an interior resonance with the planet (Kary & Lissauer 1995; Mothé-Diniz & Gomes 2000).

In this paper, we use an iterative method to determine equilibrium points in corotation resonance. We show, by using a generic method (described in Sect. 3), that our results are in a very good agreement with those from numerical integration of the complete N-body equations. We also found, with the same method, the corotation points for all cases pointed out above. The paper is organized as follows. In Sect. 2, we present a first order method valid for any planet or planetesimal eccentricity. We show some results obtained by comparing the points given by the method with the ones given by numerical integration of the complete N-body equations. In Sect. 3, we present a generic method valid not only for any eccentricity, but also for any planetary mass. With this method we obtain accurate corotation points for both cases: extended resonances and interior resonances. We also vary some parameters as eccentricity and mass of the planet in order to know the effects of high eccentricities and large masses in the location of the points. Stability and instability of these points are also checked in order to know the limits of stable points. In the last section, we discuss the results obtained and draw some conclusions.