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Astron. Astrophys. 356, 1089-1111 (2000)

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3. Stochastic motion of a particle in a vortex

3.1. The diffusion equation

Due to small-scale turbulence, the motion of a particle in a vortex is not deterministic but stochastic . Turbulent fluctuations produce some kicks which progressively deviate the particle from its unperturbed trajectory. This is similar to what happens to a colloidal particle in suspension in a liquid (Brownian motion). An individual fluctuation has a minute effect on the motion of the particle, but the repeated action of these fluctuations gives rise to a macroscopic process of diffusion. The effect of turbulence on the sedimentation of dust particles in the protoplanetary nebula has been considered in detail by Dubrulle et al. (1995). We use a similar approach to study the effect of turbulence on the capture of dust by large-scale vortices.

The transport equation governing the evolution of the dust surface density inside a vortex can be written:

[EQUATION]

where [FORMULA] is the mean velocity of the particles and D their diffusivity. The mean velocity [FORMULA] is given by the deterministic model of Sect. 2.3. According to Eqs. (22)(23) or (27) (28), it can be written:

[EQUATION]

where [FORMULA] corresponds to the pure rotation of the particles in the vortex and [FORMULA] is their drift toward the center. In the case of light particles ([FORMULA]), [FORMULA] is equal to the velocity of the vortex while in the case of heavy particles ([FORMULA]), [FORMULA] is equal to the epicyclic velocity (see Sect. 2.3). When (58) is substituted into (57), the diffusion equation takes the form:

[EQUATION]

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 R. In this approximation, we can restrict ourselves to axisymmetric solutions for which the advection term in Eq. (59) cancels out. We are led therefore to study the Fokker-Planck equation:

[EQUATION]

For an initial condition consisting of a Dirac function centered at [FORMULA], there is a well-known analytical solution of Eq. (60):

[EQUATION]

where M is the total mass of particles contained in the vortex. This formula shows that the relaxation time is equal to the capture time [FORMULA] not affected by turbulence.

The equilibrium solution of the Fokker-Planck Eq. (60) satisfies the condition:

[EQUATION]

expressing the balance between the diffusion and the drift. It corresponds to a Gaussian density profile of the form:

[EQUATION]

This is also the solution of Eq. (61) for [FORMULA]. In practice, the equilibrium distribution (63) is established for [FORMULA]. Then, the particles are concentrated in the vortices on a typical length:

[EQUATION]

3.2. Vertical sedimentation

Eq. (60) is similar to the diffusion equation

[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 [FORMULA] toward the ecliptic plane due to gravity. For light particles, [FORMULA] [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 [FORMULA] comparable with the sub-disk scale height [FORMULA] 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 horizontal accumulation in vortices. Introducing the volume density [FORMULA] and using Eqs. (60) and (65), we obtain:

[EQUATION]

where [FORMULA] and [FORMULA] are the component of [FORMULA] in the directions perpendicular and parallel to the disk rotation vector [FORMULA]. Integrating Eq. (66) on the vertical direction returns Eq. (60) for the surface density.

3.3. Diffusivity of dust particles

It 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 [FORMULA] where [FORMULA] is the sound speed and [FORMULA] a non dimensional parameter which measures the efficiency of turbulence (Shakura & Sunyaev 1973). Following current nebula models, [FORMULA]. When the turbulence is generated by differential rotation, [FORMULA] (Dubrulle 1992) and we shall take this value for numerical applications. Since the disk height is [FORMULA] (see Eq. (5)), the turbulent viscosity can be written [FORMULA] or, alternatively, [FORMULA] where R is the vortex radius. More precisely, assuming a power law spectrum [FORMULA] for the gas turbulence (with [FORMULA]), we have

[EQUATION]

According to Dubrulle et al. (1995), the diffusivity of the particles can be written:

[EQUATION]

where

[EQUATION]

is a function of the friction parameter. The reduction factor of Völk et al. (1980):

[EQUATION]

depends on the size [FORMULA] of the largest eddies of turbulence and on the systematic velocity [FORMULA] of the dust grains. In the vortices, [FORMULA] is equal to the drift velocity [FORMULA].

For light particles ([FORMULA]), [FORMULA] and

[EQUATION]

This result is expectable since light particles mainly follow the streamlines of the gas. Consequently, their diffusivity tends to the gas turbulent viscosity [FORMULA].

For heavy particles ([FORMULA]), [FORMULA] and

[EQUATION]

For [FORMULA], [FORMULA] since there is no coupling with the gas. For [FORMULA], 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:

[EQUATION]

[EQUATION]

As expected, the concentration is optimum for [FORMULA]. In that case, the particles are distributed over a length [FORMULA]. Lighter and heavier particles are less concentrated. When

[EQUATION]

or

[EQUATION]

the drift is negligible and there is no concentration ([FORMULA]).

3.4. Application to the solar nebula

We 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):

[EQUATION]

where [FORMULA] is the unperturbed surface density of the particles and [FORMULA] their velocity dispersion. The particle sub-layer (which behaves as a very compressible fluid) will be unstable provided that [FORMULA], for some k. From Eq. (77), we obtain the criterion for gravitational instability (Toomre, 1964):

[EQUATION]

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 ad hoc and ill-understood. For these reasons (and also for a lack of alternatives), the Safronov-Goldreich-Ward scenario was nearly universaly accepted as the key mechanism for forming planetesimals. Therefore, a swarm of bodies of a few km in diameter was a common starting point for numerical simulations of planetary formation.

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 [FORMULA] 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 [FORMULA] and easily reaches several meters per second (when a numerical value is needed, we shall take [FORMULA]). Therefore, the instability criterion (78) is not satisfied (see appendix B for a justification of this criterion in a turbulent disk). Turbulence is responsible for too high velocity dispersions or, alternatively, the surface density of the dust sublayer is not sufficient to trigger the gravitational instability. The density needs to be increased by a factor [FORMULA] or more to overcome the threshold imposed by Toomre instability criterion.

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 concentration effect. Inside a vortex, an initial mass of order [FORMULA] is concentrated on a typical length [FORMULA] given by Eq. (64). The surface density is therefore amplified by a factor [FORMULA] depending on the size of the particles. Using Eqs. (73)(74), this amplification can be written explicitly

[EQUATION]

[EQUATION]

The amplification is maximum for [FORMULA] and takes the value [FORMULA]. As we have seen in Sect. 2.5, this corresponds to particles of size [FORMULA] cm and bulk density [FORMULA] in the regions of the Earth and Jupiter. This enhancement is sufficient to satisfy the instability criterion (78)  5. For microscopic particles, on the contrary, there is no density enhancement. In that case, [FORMULA] (in the region of the planets) and the particles rapidly diffuse away from the vortices (see Sect. 4.5). Gravitational instability will be possible provided that:

[EQUATION]

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).

[FIGURE] Fig. 9. Enhancement of the dust surface density inside the vortices (due to concentration) as a function of the size of the particles at [FORMULA] and [FORMULA] ([FORMULA]).

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 concentration of the dust layer in the vortex (a process similar to the vertical sedimentation). However, this mechanism alone is not sufficient to produce enough planetesimals to form the planets. The vortices must also capture the surrounding mass. This is another source for density enhancement. Returning to Eq. (38), we find that the average surface density of the particles collected by a vortex during its lifetime is

[EQUATION]

with [FORMULA] (see Sect. 2.5). The maximum amplification, reached by particles with [FORMULA], is [FORMULA] a little bit larger than the previous value (recall however that [FORMULA] is not known precisely). Gravitational instability will be possible for particles whose friction parameter satisfies

[EQUATION]

See also Table 1 and Figs. 5, 10 for the same criterion expressed in terms of the size of the particles.

[FIGURE] Fig. 10. Enhancement of the dust surface density inside the vortices (due to capture) as a function of the size of the particles at [FORMULA] and [FORMULA] ([FORMULA]).

If we now take into account both the concentration effect and the capture process, we obtain an amplification

[EQUATION]

with a maximum value [FORMULA]. The range of particles which can collapse is enlarged:

[EQUATION]

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.

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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