## 3. Stochastic motion of a particle in a vortex## 3.1. The diffusion equationDue to small-scale turbulence, the motion of a particle in a vortex
is not deterministic but The transport equation governing the evolution of the dust surface density inside a vortex can be written: where is the mean velocity of
the particles and where corresponds to the pure rotation of the particles in the vortex and is their drift toward the center. In the case of light particles (), is equal to the velocity of the vortex while in the case of heavy particles (), is equal to the epicyclic velocity (see Sect. 2.3). When (58) is substituted into (57), the diffusion equation takes the form: The first term in the right hand side is a pure diffusion due to small-scale turbulence and the second term is a drift toward the vortex center due to the combined effect of the Coriolis force and the vortex (anticyclonic) rotation. Eq. (59) illustrates the bimodal nature of turbulence: the diffusion is due to small-scale fluctuations hardly affected by the rotation of the disk (3D turbulence) and the trapping process is a consequence of the Coriolis force and the existence of coherent structures in the disk (2D turbulence). Since we are mainly interested in orders of magnitude and in order
to avoid unnecessary mathematical complications, we shall consider
from now on that the vortices are circular with typical radius
For an initial condition consisting of a Dirac function centered at , there is a well-known analytical solution of Eq. (60): where The equilibrium solution of the Fokker-Planck Eq. (60) satisfies the condition: expressing the balance between the diffusion and the drift. It corresponds to a Gaussian density profile of the form: This is also the solution of Eq. (61) for . In practice, the equilibrium distribution (63) is established for . Then, the particles are concentrated in the vortices on a typical length: ## 3.2. Vertical sedimentationEq. (60) is similar to the diffusion equation used by Dubrulle et al. (1995) to describe the sedimentation of the dust particles and determine the sub-disk scale height (see also Weidenschilling 1980). The drift toward the vortex center is replaced in their study by the drift toward the ecliptic plane due to gravity. For light particles, [see Eq. (33)] and the two equations coincide up to numerical factors. This implies that the particles are concentrated in the vortices on a lenght comparable with the sub-disk scale height determined by Dubrulle et al. (1995) [see their Sect. 3]. It is relatively straightforward to include the vertical
sedimentation of particles in our study, although we shall specialize
in the following on their where and are the component of in the directions perpendicular and parallel to the disk rotation vector . Integrating Eq. (66) on the vertical direction returns Eq. (60) for the surface density. ## 3.3. Diffusivity of dust particlesIt remains now to specify the value of the diffusion coefficient
appearing in Eq. (60). In general, the turbulent viscosity of the
gas is written in the form where
is the sound speed and
a non dimensional parameter which
measures the efficiency of turbulence (Shakura & Sunyaev 1973).
Following current nebula models, .
When the turbulence is generated by differential rotation,
(Dubrulle 1992) and we shall take
this value for numerical applications. Since the disk height is
(see Eq. (5)), the turbulent
viscosity can be written or,
alternatively, where According to Dubrulle et al. (1995), the diffusivity of the particles can be written: is a function of the friction parameter. The reduction factor of Völk et al. (1980): depends on the size of the largest eddies of turbulence and on the systematic velocity of the dust grains. In the vortices, is equal to the drift velocity . This result is expectable since light particles mainly follow the streamlines of the gas. Consequently, their diffusivity tends to the gas turbulent viscosity . For , since there is no coupling with the gas. For , Eqs. (71) and (72) give the same result, so we can use these expressions in the whole range of friction parameters. Note, however, that the diffusion approximation is not strictly valid for heavy particles, so Eq. (72) must be taken with care. Substituting Eqs. (33)(34) and (71)(72) in Eq. (64), the concentration length can be written explicitly: As expected, the concentration is optimum for . In that case, the particles are distributed over a length . Lighter and heavier particles are less concentrated. When the drift is negligible and there is no concentration (). ## 3.4. Application to the solar nebulaWe now apply these results to the case of the solar nebula and show how the vortex scenario can make possible the formation of planetesimals at certain preferred locations of the disk. It is generally beleived that planetesimals formed by the gravitational instability of the particle sublayer (Safronov 1969, Goldreich & Ward 1973). The dispersion relation for an infinite uniformly rotating sheet of gas is (see, e.g., Binney & Tremaine 1987): where is the unperturbed surface
density of the particles and their
velocity dispersion. The particle sub-layer (which behaves as a very
compressible fluid) will be unstable provided that
, for some Once instability is triggered, the system crumbles into numerous
planetesimals of order 10 km in size. Moreover, the growth time of
density perturbation is predicted to be short, of the order of an
orbital period. In addition, the instability criterion gives the
impression that its operation does not require any sticking mechanism.
Goldreich & Ward (1973) state that "...the fate of planetary
accretion no longer appears to hinge on the stickiness of the surface
of dust particles". This is very attractive because sticking
mechanisms are relatively However, Weidenschilling (1980), followed by Cuzzi et al. (1993)
and Dubrulle et al. (1995), realized that this simplisitic picture was
ruled out if the primordial nebula was turbulent. Indeed, turbulence
reduces considerably the vertical sedimentation of the dust particles
and prevents gravitational instability. According to Eqs. (41)
and (42), the velocity threshold imposed by the instability criterion
(78) is of order throughout the
nebula. Even if the nebula as a whole was perfectly laminar, the
formation of a dense layer of particles (considered as a heavy fluid)
would create a turbulent shear with the overlying gas (see, e.g..,
Weidenschilling & Cuzzi 1993). When turbulence is accounted for,
the velocity dispersion of the particles can be estimated by
and easily reaches several
There is therefore a major problem to form planetesimals by
gravitational instability in a turbulent disk. As suggested by Barge
& Sommeria (1995), the presence of vortices in the disk can solve
this problem. Indeed, by capturing and concentrating the particles,
the vortices can increase locally the surface density of the dust
sublayer and initiate the gravitational instability. Let us first
discuss the The amplification is maximum for
and takes the value . As we have
seen in Sect. 2.5, this corresponds to particles of size
cm and bulk density
in the regions of the Earth and
Jupiter. This enhancement is sufficient to satisfy the instability
criterion (78)
On Table 1, we report, as a function of the heliocentric distance, the minimum size of the particles which satisfy this criterion (see also Figs. 5 and 9). These results indicate that particles must have grown up to some centimeters to trigger the gravitational instability. Therefore, sticking processes are needed to reach this range of sizes (recall that this was claimed to be not necessary in the initial Safronov-Goldreich-Ward scenario).
In conclusion, by allowing a local enhancement of the particle
surface density, the vortices can favour the formation of
planetesimals by gravitational instability. This rehabilitates the
Safronov-Goldreich-Ward theory at certain preferred locations of the
disk (i.e. inside the vortices) and for sufficiently large
(decimetric) particles. A sufficient enhancement is achieved simply by
the horizontal with (see Sect. 2.5). The maximum amplification, reached by particles with , is a little bit larger than the previous value (recall however that is not known precisely). Gravitational instability will be possible for particles whose friction parameter satisfies See also Table 1 and Figs. 5, 10 for the same criterion expressed in terms of the size of the particles.
If we now take into account both the concentration effect and the capture process, we obtain an amplification with a maximum value . The range of particles which can collapse is enlarged: but, even in this optimistic situation, the particles must have reached relatively large sizes to trigger the gravitational instability (see Table 1). Of course, if the vortex lifetime is increased, smaller particles have the possibility to collapse since the vortex captures more mass. In fact, this is not completely correct because the previous results assume that the escape of particles due to turbulent fluctuations can be neglected. This is not always the case (in particular for small particles) and this problem is now considered in detail. © European Southern Observatory (ESO) 2000 Online publication: April 17, 2000 |