## 1. IntroductionMany astrophysical objects, ranging from young stars to massive black holes, are surrounded by widespread gaseous disks. The existence of a primordial disk around the sun was conjectured by Kant (1755) and Laplace (1796) in the century to explain the quasi-circular, coplanar and prograd motion of the planets in the solar system. Such protoplanetary disks have recently been observed with the Hubble Space Telescope in the Orion nebula around stars less than one million years old. These gaseous disks can be considered as a by-product of the star formation: after the gravitational collapse of a molecular cloud, about of the initial angular momentum is spread in an extended disk, while of the mass forms the star, whose internal structure is hardly affected by rotation. Whenever it has been possible to observe rotating, turbulent fluids
with good resolution, it has been seen that individual, intense
vortices form (Bengston & Lighthill 1982; Hopfinger et al. 1983;
Dowling and Spiegel 1990). One of the most striking examples is
Jupiter's Great Red Spot, a huge vortex persisting for more than three
centuries in the upper atmosphere of the planet. These coherent
vortices are well reproduced in numerical simulations (McWilliams
1990, Marcus 1990) and laboratory experiments (Van Heist & Flor
1989, Sommeria, Meyers & Swinney 1988) of two-dimensional
turbulence and their organization can be explained in terms of
statistical mechanics (Miller 1990, Robert & Sommeria 1991,
Sommeria et al. 1991, Chavanis & Sommeria 1998)
However, accretion disks are exceptional among rotating turbulent objects in the strong shears that these bodies are believed to possess and this shear might lead to rapid destruction of any structures that tend to form. This objection has been overruled by the numerical simulations of a two-dimensional flow in an external Keplerian shear by Bracco et al. (1998,1999) for an incompressible flow and Godon & Livio (1999a,b,c) for a compressible flow. Although cyclonic fluctuations are rapidily elongated and destroyed by the shear, anticyclonic vortices form and persist for a long time before being ultimately dissipated by viscosity. Naturally, this does not prove that coherent structures must form on disks, but this strengthens the argument that disks are likely to follow the norm of rotating, turbulent bodies. Other numerical results (Hunter & Horak 1983) and experimental work (Nezlin & Snezhkin 1993) comfort this point. Coherent vortices in circumstellar disks can play an important role in the transport of dust particles and in the process of planet formation. Planets are thought to be formed from the dust grains embedded in the disk after a three-stage process: (i) in a first stage, microscopic particles suspended in the gas stick together on contact due to electrostatic or surface forces. When they reach sufficient sizes, they begin to sediment in the mid-plane of the disk due to the combined effect of the gravity and the friction with the gas. When settling dominates, a particle can grow by sweeping up smaller ones (Safronov 1969) and may easily reach sizes of several centimeters in a few thousand orbital periods (Weidenschilling & Cuzzi 1993). Bigger aggregates () are more difficult to form on relevant time scales because of collisional fragmentation (ii) When the layer of sedimented particles is sufficiently dense, the gravitational instability is triggered and the layer crumbles into numerous kilometer-sized bodies, the so-called "planetesimals" (Safronov 1969, Goldreich & Ward 1973). (iii) The subsequent evolution is marked by planet's growth due to the accumulation of planetesimals in successive collisions. This stage is well reproduced by dynamical models (Safronov 1969, Barge & Pellat 1991, 1993). When the solid core becomes sufficiently massive, it can accrete the surrounding gas and a giant planet, like Jupiter, is formed. However, the above scenario faces two major problems. Recent studies have shown that circumstellar disks are relatively turbulent and that small-scale turbulence strongly reduces the sedimentation of the dust particles in the ecliptic plane (Weidenschilling 1980, Cuzzi et al. 1993, Dubrulle et al. 1995). For particles of relevant size, the density of the dust layer is not sufficient to overcome the threshold imposed by Jeans instability criterion. Therefore, the formation of the planetesimals, i.e. the passage from cm-sized to km-sized particles, is not clearly understood. In addition, it seems difficult, with the above model, to build up sufficiently massive cores in less than one million years before the gas has been swept away by the solar wind during the T-tauri phase (Safronov 1969, Wetherill 1988). Both difficulties are ruled out if we allow for the presence of
vortices in the disk. Their existence was first proposed by
Von
Weizäcker (1944) to explain the regularity of the planet
distribution in the solar system: the famous Titius-Bode law
In this paper, we present a simple analytical model for the capture
of dust particles by coherent vortices in a Keplerian disk. This model
is directly inspired by the numerical studies of Barge & Sommeria
(1995) and Tanga et al. (1996) and their main results are recovered
and confirmed. One interest of our approach is to provide analytical
results (leading to quantitative predictions) and to isolate relevant
parameters which prove to be particulary important in the problem. In
Sect. 2, we introduce an exact solution of the incompressible 2D
Euler equation appropriate to our sudy. This is an elliptic vortex
with uniform vorticity matching continuously with the azimuthal
Keplerian flow at large distances. We consider deterministic
trajectories of dust particles in that vortex and derive analytical
expressions for the capture time and the mass capture rate as a
function of the friction parameter. We find that the capture is
optimum for particles whose friction parameter is close to the disk
angular velocity. In Sect. 3, we investigate the effect of
small-scale turbulence on the motion of the particles. Their
trajectories become stochastic and their motion must be described in
terms of diffusion equations. We estimate the diffusion coefficient
and determine the typical length on which the particles are
concentrated in the vortices. In Sect. 4, we evaluate the rate of
particles which diffuse away from the vortices due to turbulent
fluctuations. An explicit expression for the "rate of escape" is
obtained by solving a problem of quantum mechanics, namely a
two-dimensional oscillator in a box. In appendix A, we give some
details about the construction of the vortex solution and in appendix
B, we extend Toomre instability criterion (Toomre, 1964) to the case
of a In parallel, we apply these theoretical results to the solar nebula
and make speculations about its actual structure. For relevant
particles going from 10 cm to 100 cm in size, we remark that the
transition between the Stokes and the Epstein regimes (at which the
gas drag law changes) corresponds precisely to the transition between
telluric (inner) and giant (outer) planets. Moreover, in each zone
there is a preferred location where the capture of dust by vortices is
optimum. For particles of density
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