,
where 
is a small factor which
accounts for the negative radial pressure gradient (Adachi et al. 1976).
This small perturbation causes the solid particles to experience a loss
of energy and angular momentum, even if considering circular orbits.
However, solid (metric) particles approaching a growing protoplanet in
the presence of (weak) gas drag can become trapped in mean-motion
resonances (Weidenschilling & Davis 1985), at which point their orbital
migration effectively stops, and their eccentricities reach an equilibrium
between resonance pumping and drag-induced damping. As a solid particle
approaches a resonance, perturbations from the planet can excite the
particle's eccentricity and increase its semi-major axis. If the drag force
is not very strong, these resonant perturbations will be able to compensate
for the particle's orbital evolution, and the particle will evolve into a
stable equilibrium orbit determined by the relative strengths of the drag
and the resonance. Therefore, in an outer resonance where a particle
revolves slower than a planet or planets, gain and loss of orbital momentum
as well as of orbital energy may be kept in equilibrium. The time scale for
this equilibrium state depends on the values for the planetary masses, for
the eccentricity of the planetary orbits, for the particles' radii and
on the order of the resonance. These values can be determined by
numerical simulations. Resonant capture can produce cometary belts beyond
Saturn but also between Jupiter and Saturn (Lecar et al. 1992). Resonances
discussed in this paper are not classical gravitational resonances but
gas-driven gravitational ones. The cosmogonic implications (formation
of the outer planets) of this process have been mainly considered by
Patterson (1987) and Beaugé et al. (1994) although little consideration
has been given to the role of this mechanism in the formation of minor
bodies in the outer Solar System. The most remarkable cosmogonic result in
this field is the conclusion (Beaugé et al. 1994) that the formation
mechanism originally proposed by Patterson (1987) is plausible and may well
explain the quasi-resonant configuration of the actual outer planets.
The phenomenon of exterior resonance trapping induced by gas drag has been studied from a strictly theoretical point of view in the framework of the planar restricted three-body problem (star, protoplanet and solid particle). The first analysis was made by its discoverers, Weidenschilling & Davis (1985), and they concluded that trapping is only possible when the falling body experiences an exterior commensurability relation with the perturbing protoplanet. In this scenario, the secular and simultaneous decrease in both the semi-major axis and eccentricity of the falling body induced by the drag force is balanced by an equally secular and simultaneous increase in the same orbital elements driven by the gravitational contribution of the protoplanet. A detailed analysis of this process has been carried out by Beaugé & Ferraz-Mello (1993), finding the conditions for trapping, as well as the final stable orbits of the falling bodies. Although in our present work we consider the more general case of two growing protoplanets but still coplanar, we will now summarize theoretical results relevant for our current study from Beaugé & Ferraz-Mello (1993) as well as numerical results from Beaugé et al. (1994). If we assume that the ratio between the orbital periods of the protoplanet and the falling body lies close to a generic (p + q) / p (p and q are small integers with p < 0) exterior mean-motion resonance:
(a) For a given resonance, trapping can occur only for particles larger than a critical size. In general, larger bodies will be preferentially trapped at distant resonances. Therefore, exterior resonance trapping resulting from gas drag is able to induce mass segregation in an evolving protoplanetary disk.
(b) This trapping mechanism is extremely sensitive to the initial conditions. Falling from outside a certain resonance and having a size less than the critical for capture does not ensure that capture indeed occurs as initial orbital elements play a key role in the trapping process. The object may cross that given resonance and become trapped by another one with smaller radial distance.
(c) The final orbits of the trapped bodies are of two types:
-libration and corotation. For -libration orbits,
accumulation occurs only in the radial direction but not in the
azimuthal angle and the eccentricity changes in the range
0.03-0.12. In corotation orbits, coherent motions appear with
particles being clustered about |p| evenly spaced phases of the
azimuthal angle and the eccentricity is a fixed universal quantity,
depending only on the order of the resonance.
(d) There is a rivalry among different resonances and no particular commensurability dominate the rest and control the accretion and dynamics of the system.
In our calculations, we investigate a model composed of the Sun, an existing proto-Jupiter (and also proto-Saturn) and a planar extended ring of solid bodies embedded in the primordial solar nebula. Our model neglects two-body interactions between solid bodies. By including the combined effects of gravitational perturbations and gas drag we have simulated the orbital evolution of primordial solid material and obtained some insight into the role that gas-induced resonance trapping could have played in minor body formation at the Jupiter-Saturn region. These minor bodies include the Jovian-class comets, in particular C/1999 S4 LINEAR, and possibly some Centaurean objects. If only proto-Jupiter is included, our numerical results should match those from the theory (Beaugé & Ferraz-Mello 1993) and numerical experiments (Beaugé et al. 1994). However, results from models including two protoplanets are presented here for the first time.
Copyright ESO 2002