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

2. Deterministic motion of a particle in a vortex

2.1. The solar nebula model

We assume that the solar nebula is disk-shaped and is in hydrostatic equilibrium in the vertical direction. If the mass of the disk is much less that the solar mass (), then its self-gravity can be neglected compared with the sun's attraction. In that case, the disk has approximately Keplerian rotation

where r is the distance from the sun and cm³/(g.s²) the constant of gravity. For thin disks, the vertical component of the solar gravity is:

The condition of hydrostatic equilibrium implies that the vertical pressure gradient is precisely balanced by , i.e.:

If the local temperature is independent of z, then the vertical density profile of the gas is

The half-thickness (or scale height) of the disk is given by

where is the sound speed. Other plausible temperature profiles, e.g., adiabatic gradient in the z direction yield similar results.

We assume also that the gas is turbulent. Turbulence is necessary to explain the dynamical evolution of the protoplanetary disk and its cooling consistent with cosmochemical data (Dubrulle 1993). However, its origin is not well understood and remains controversial. Various mechanisms for inducing global nebula turbulence have been proposed. Lin & Papaloizou (1980) and Cabot et al. (1987a,b) suggested that thermal convection drives nebula turbulence. However, the applicability of their results depends on the presence of abundant micron-sized dust to provide substantial thermal opacity; thus, they are questionable when significant particle growth has already occurred and the thermal opacity has decreased. Dubrulle (1992,1993) suggested that the Keplerian shear is the main engine of the turbulence. This was already pointed out by Von Weizsäcker and further discussed by Chandrasekhar (1949) in view of the very small viscosity of the disk: "The successive rings of gas in the medium will have motions relative to one another, and turbulence will ensue". This apparently obvious claim is actually far from straightforward to support because the Keplerian shear is stable with respect to linear, infinitesimal, perturbations which are usually relied upon to induce turbulence. However, Dubrulle & Knobloch (1992) point out that it might be unstable to nonlinear finite amplitude perturbations, a property shared by most of the shear flows commonly met in engineering or laboratory experiments. The simplest example is the plane Couette flow, a plane parallel stream of constant shear. This analogy is contested by Balbus et al. (1996) who did not evidence nonlinear instabilities in Keplerian disks at the respectable Reynolds numbers they achieved numerically  ³. These authors have suggested, in contrast, that turbulence in Keplerian disks could be produced by a powerful MHD instability (Balbus & Hawley 1991). The numerical simulations of Bracco et al. (1998) suggest that MHD turbulence can form magnetized vortices able to capture charged particles. However, in the case of the disk that is supposed to have spawned the solar system, it is thought that the matter was too cool to be ionized. The recourse to magnetic effects to render the disk turbulent is therefore suspect and the problem of whether Keplerian disks are turbulent or not remains open. Recently, Lovelace et al. (1999) discovered a linear nonaxisymmetric instability in a thin Keplerian disk which can lead to the formation of Rossby vortices. This may open new perspectives for hydrodynamical turbulence in accretion disks.

In any case, the solar nebula must have been turbulent at least during its formation from the collapsing protostar, because of velocity discontinuities as the infalling matter struck the disk. The infall probably did not stop suddenly but decayed over some interval. Therefore, the initial conditions in the disk were turbulent and this is sufficient to form vortices that survive for many rotation periods of the disk (Bracco et al. 1998,1999; Godon & Livio 1999a,b,c). The appearance of large-scale coherent vortices in rotating flows is due to the presence of the Coriolis force and the bimodal nature of turbulence (Dubrulle & Valdettaro 1992). At small scales, the influence of the rotation is negligible and the turbulence is homogeneous and isotropic. Energy cascades toward smaller and smaller scales up to the dissipation length. Turbulent diffusion is important and can accelerate the mixing of dust particles. At larger scales, the Coriolis force becomes dominant and the gas dynamics is quasi two-dimensional. On these scales, the energy transfers are inhibited or even reversed; this is a manifestation of the celebrated "inverse cascade" process (see, e.g., Kraichnan & Montgomery 1980) in two-dimensional turbulence. There is therefore the possibility of formation and maintenance of coherent vortices (McWilliams 1990). These vortices can form either by a large-scale instability (Frisch et al. 1987, Dubrulle & Frisch 1991, Kitchatinov et al. 1994) or by the successive mergings of like-sign vortices, as observed in the simulations of Bracco et al. (1998,1999) and Godon & Livio (1999a,b,c). Due to the strong Keplerian shear, only anticyclonic vortices can survive this process. Initially, the vortices have size and typical vorticity . Their velocity is less that the sound speed and the flow can be considered as incompressible. The merging ends up when the Mach number reaches unity, i.e. . Bigger vortices radiate density waves and do not maintain (see Barge & Sommeria 1995).

We expect therefore that, after some evolution, the disk be filled with anticyclonic vortices of typical size H, the disk thickness. Vortices are expected to form throughout the nebula and there is no reason, a priori, to beleive that certain regions of the disk should be excluded. However, two vortices at comparable distance from the sun will approach each other (due to differential rotation) and finally merge. Therefore, we do not expect more than one (or a few) vortices at each radial distance. On the other hand, two successive vortices should be separated by a distance comparable to their size R. Since and H scales like a power law of the distance to the sun ( in the standard model considered below), the distribution of vortices should be consistent with an approximate geometric progression of the planetary positions (Barge & Sommeria 1995). As elucidated by Graner & Dubrulle (1994), the Titius-Bode law reflects more the general properties of scale invariance in the solar nebula than any particular physical process.

2.2. The vortex model

We consider the motion of a dust particle in a vortex located at a distance from the sun. For convenience, we work in a frame of reference rotating with constant angular velocity and we denote by the velocity field of the gas in that frame. The dust particle is subjected to the attraction of the sun and to a friction with the gas that we write where is the particle velocity. We shall come back to this expression and to the value of the friction coefficient in Sect. 2.5. Since we work in a rotating frame, apparent forces arise in the system. These are the Coriolis force and the centrifugal force . All things considered, the particle equation writes:

This is an ordinary second order differential equation coupled to the gas motion. The case of a time-dependent velocity field produced by the Navier-Stokes simulation of a random initial vorticity field superposed on the Keplerian rotation has been investigated by Bracco et al. (1999) and Godon & Livio (1999c) who observed the capture of the dust particles by anticyclonic vortices. The case of a static velocity field was first considered by Barge & Sommeria (1995) and Tanga et al. (1996) with different vortex profiles. Barge & Sommeria (1995) assume that the velocity field inside the vortex is made of concentric epicycles while it matches continuously the Keplerian flow at large distances. This epicyclic model is motivated by the underlying idea that the particles, when sufficiently concentrated, will retroact on the vortex and will force the gas to follow their natural motion. Tanga et al. (1996) consider a more complex streamfunction describing an ensemble of small vortices corresponding to Rossby waves corotating with the flow. This velocity field is obtained as a self-similar solution of the linearized equations of motion governing the large-scale dynamics of a turbulent nebula. These two models lead to qualitatively similar results indicating that dust particles are efficiently trapped into coherent anticyclonic vortices. However, the models differ in the details: in Tanga et al. (1996), the particles sink to the center of the vortices with no limit while in Barge & Sommeria (1995), the spiralling motion ends up on an epicycle. In that case, the friction force cancels out and the epicyclic motion is an exact solution of the particle Eq. (6).

We must note, however, that the vortex constructed by Barge & Sommeria (1995) is relatively ad hoc and does not satisfy the fluid equations rigorously. In this article, we introduce an exact vortex solution of the incompressible 2D Euler equation and study analytically the motion of dust particles in that vortex. Since the vortices of the solar nebula are small compared with the radial distance (we have typically , see formula (44)) the last term in Eq. (6) can be expanded to first order in the displacement . This is the so-called "epicyclic approximation". Introducing a set of cartesian coordinates centered on the vortex and such that the y-axis points in the direction opposite to the sun, the particle Eq. (6) reduces to:

At sufficiently large distance from the vortex, the velocity field is a simple shear:

obtained as a first order expansion of the Keplerian velocity around . Its vorticity is .

It remains now to specify the velocity field inside the vortex. We can construct an exact solution of the incompressible 2D Euler equation by taking an elliptic patch of uniform vorticity . This solution is well-known by model builders (see, e.g., Saffman 1992)  ⁴ but, to our knowledge, it has never been applied in an astrophysical context. Therefore, we give a short description of its construction in Appendix A. In the vortex, the velocity field writes

where is the aspect ratio of the elliptic patch ( are the semi-axis in the x and y directions respectively). Outside the vortex, the velocity field is given by Eq. (A.26) of Appendix A. At large distances, we recover the Keplerian shear (9)(10).

The matching conditions between the elliptic vortex and the Keplerian shear require that , and q be related according to (see Appendix A):

The solution (11)(12) is valid for , implying . The vortex is anticyclonic () and is oriented with its major axis parallel to the shear streamlines. It can be shown to be stable to infinitesimal two-dimensional disturbances. For (circular vortices), and for (infinitely elongated vortices), . Therefore, is in the range . In a rotating disk, we expect that , corresponding to a Rossby number of order 1. This value is achieved by vortices with aspect ratio , in good agreement with the model of Tanga et al. (1996). The epicyclic vortex considered by Barge & Sommeria (1995) corresponds to and . These values do not satisfy the matching condition (13) with the Keplerian shear so the epicyclic vortex is not an exact solution of the incompressible Euler equation. It may be used, however, as an approximate solution if we take into account a coupling between the particles and the gas in the spirit of a two-fluids model.

2.3. The capture time

Before going into mathematical details, we recall the argument of Tanga et al. (1996) which shows very simply why particles are trapped by anticyclonic vortices in a rotating disk. Introducing a system of polar coordinates whose origin coincides with the vortex center, the radial component of Eq. (6) reads:

where we have used the epicyclic approximation. The first term is a centrifugal force due to the rotation of the vortex (not to be confused with the centrifugal force due to the rotation of the disk) and the second term is the Coriolis force. The drag term in Eq. (6) forces the particle velocity to approach the fluid velocity, i.e. . For cyclonic vortices (), both centrifugal and Coriolis forces are positive and push the particles outward. For anticyclonic vortices (), the Coriolis force pushes the particles inward and comes in conflict with the centrifugal force which is always directed outward. If the vortex rotates rapidly, the centrifugal force prevails over the Coriolis force and the particle is expelled. In the other case, the Coriolis term dominates and the particle is captured. We conclude that only slowly rotating anticyclonic vortices can capture dust particles. This trapping process is specific to rotating fluids; in ordinary simulations of two-dimensional turbulence (without Coriolis force) the particles never penetrate the vortices. Note that the epicyclic acceleration (last term in Eq. (14)) is always directed outward. This term is responsible for a flux of particles toward the sun (if the gravitational force dominates, i.e. for ) or toward the outer nebula (if the centrifugal force dominates, i.e. ). However, this flux is quite small and does not affect the overall trapping process (see Tanga et al. 1996).

We now give a more precise analysis of the trapping process and derive an explicit expression for the capture time of the particles as a function of their friction parameter . To that purpose, we notice that Eqs. (7) (8) with the velocity field (11) (12) form a linear system of coupled differential equations. We seek therefore a solution of the form:

where and are complex numbers. Substituting into Eqs. (7) (8), we obtain a linear system of algebraic equations:

There are nontrivial solutions only if the determinant of this system is zero. We are led therefore to a fourth order polynomial equation in :

By definition, we will say that a particle is light or heavy whether or respectively. This terminology will take more sense in the sequel (see in particular Sect. 2.5). We now consider the asymptotic limits and of Eq. (19).

For (light particles), the four roots of Eq. (19) are

The first solution is rapidly damped and will not be considered anymore. The second solution describes the rotation of the dust particles in the vortex:

The particles follow ellipses of aspect ratio q and move with angular velocity . In fact, for , the drag term in Eq. (6) implies so, in a first approximation, the particles just follow the vortex streamlines (their angular velocity coincides with the angular velocity of the fluid particles, see Appendix A). In addition, they experience a drift toward the center due to the combined effect of the friction and the Coriolis force. They reach the vortex center in a typical time:

defined as the capture time . Note that for light particles, increases linearly with .

For (heavy particles), the four roots of Eq. (19) are:

The second solution describes again a damped rotation of the particles but with a different trajectory:

The particles follow ellipses with aspect ratio 2 and move with angular velocity . This is natural since heavy particles have the tendency to decouple from the gas and reach a pure epicyclic motion. However, due to a slight friction, they sink progressively in the vortex with a characteristic time

For heavy particles, the capture time increases like . Very heavy particles can even leave the vortex. Formally, this possibility is taken into account in the first solution (25) which corresponds to open trajectories. The motion of heavy particles is therefore more complicated and demands a numerical integration of the particle equation inside and outside the vortex. This study will not be undertaken in that paper. We shall only consider the case of closed trajectories described by Eqs. (27)(28).

For intermediate values of , the capture time and the angular velocity of the particles are plotted on Figs. 1 and 2 (for the particular value ). We see that presents an optimum at . Moreover, the asymptotic expressions (24) and (29) agree reasonably well with the exact solution for all the values of the friction parameter. Therefore, considering the intersection of the asymptotes, we find that the capture time is minimum for

and we have:

Fig. 1. Plot of vs for vortices with aspect ratio . The dash lines correspond to formulae (24) and (29) valid for light () and heavy () particles.

Fig. 2. Plot of (angular velocity of the particles) vs for vortices with aspect ratio . Light particles () move with the vortex velocity and heavy particles () with the epicyclic velocity .

According to Eqs. (24) and (29), the condition for particle trapping () requires that . This implies that the vorticity must be in the range [see Eq. (13)]:

Particles are ejected from rapidly rotating anticyclones in agreement with the qualitative discussion of Tanga et al. (1996) recalled at the begining of this section. In the following, we shall specialize on the value of q which minimizes the capture time . We find . As noted already, this value corresponds to , i.e. a Rossby number of order 1. For these vortices, the optimal friction parameter is and the corresponding capture time is of the order of one rotation period. For , we have

and for

Light particles move with angular velocity (the vortex velocity) and heavy particles move with angular velocity (the epicyclic velocity). These results should be compared with the settling time of the dust particles in the equatorial plane (see, e.g., Nakagawa et al. 1986). Light particles are settled exponentially with a characteristic time comparable with Eq. (33). On the other hand, heavy particles undergo overdamped oscillations around the midplane with a period of the order of and a settling time similar to expression (34). We shall come back on the analogy between the vortex trapping and the dust settling in Sect. 3.1.

The models of Barge & Sommeria (1995) and Tanga et al. (1996) lead to qualitatively similar results. In the model of Tanga et al. (1996), the particles sink to the center of the vortex on a typical time which also presents a minimum when is of order . Moreover, for light particles, increases linearly with like in Eq. (33). The model of Barge & Sommeria (1995) is physically different since the particles do not really reach the vortex center but end up on an epicycle after a time . However, the general tendency is the same. Light particles () mainly follow the streamlines of the gas and stop on epicycles close to the vortex edge (like in case (a) of their Fig. 1). Optimal particles with friction parameter reach deeper epicycles, almost at the center of the vortex (case (b)). Heavy particles () take a long time to connect an epicycle and can even escape from the vortex since their motion is nearly unaffected by the friction drag. This is also a possibility in our model and in Tanga et al. (1996). Therefore, the three vortex models give relatively similar results even if their physical contents are different. The recent numerical simulations of Godon & Livio (1999c) for a compressible flow also show a capture optimum when the drag parameter is of the order of the orbital frequency.

2.4. The mass capture rate

In addition to the capture time, an important aspect of the problem concerns the mass capture rate (Barge & Sommeria 1995). This quantity can be estimated as follows. First, we have to determine the capture cross section of the vortex as a function of the friction parameter . Without the effect of the drift, a particle with impact parameter would cross the vortex in a typical time , where is the Keplerian velocity. The particle will be captured by the vortex if, during that time, the deflection due to the drift is precisely of order . Light particles () follow the edge of the vortex and, for them, . On the other hand, heavy particles () keep their rectilinear motion and enter directly the vortex so that, for them, . The critical parameter is given by the condition . Moreover, for optimal particles with , we expect that . Regrouping all these results, we obtain

These results agree with the asymptotic behaviour of the function determined numerically by Barge & Sommeria (1995) in their model.

The mass capture rate can be estimated by assuming that the particles are carried to the vortex by the Keplerian shear , and that they are continuously renewed by the inward drift (directed toward the sun) associated with the velocity difference between gas and particles (Barge & Sommeria 1995). Another alternative, consistent with the simulations of Bracco et al. (1999), is that the particles are already concentrated by the turbulent fluctuations that accompany vortex formation in the early stages of the protoplanetary nebula. It is likely that both mechanisms come into play in the capture process. Considering the first possibility, which can be studied analytically, we have:

where is the surface density of the dust particles outside the vortex. The total mass collected by the vortex during its lifetime is

The mass capture rate is proportional to the effective surface of the vortex and is maximum for particles with .

In conclusion, various vortex models and different physical arguments show that the capture is optimal for particles whose friction parameter is close to the disk rotation. We shall now consider the implications of this result on the structure of the solar nebula.

2.5. Application to the solar nebula

We shall assume that the solid particles are spherical, of radius a and density . The value of the friction parameter depends whether the size of the particles is larger or smaller than the mean free path in the gas (see, e.g., Weidenschilling 1977). When , we are in the Epstein regime and:

where is the gas surface density.

When , the situation is more complicated because, in general, the friction parameter depends on the velocity difference between the particles and the gas. There is, however, a regime where is independent of the particle relative velocity. This is the Stokes regime in which:

This regime is valid when the particle Reynolds number , where is the kinematic viscosity of the gas, is smaller than 1. This is the case for light particles which move typically with the gas velocity . For heavy particles on the contrary, we have and, rigorously speaking, the friction parameter depends on the relative velocity. The use of a more exact expression for would require a numerical integration of the particle trajectory but the results should not dramatically differ from those obtained with expression (40). Since we are mainly interested in orders of magnitude, we will use expression (40) for all particle sizes (with ), but keep in mind this uncertainty for large particles.

We shall adopt a standard model of the solar nebula, following Cuzzi et al. (1993). It corresponds to a "minimum mass" circumstellar nebula with parameters:

We take  s and cm.

For a given type of particles, the transition between the Epstein and the Stokes regime is achieved at a specific distance from the sun given by:

We are particularly interested in particles of order 10 cm in size. Indeed, smaller particles can grow by aggregation processes without the aid of vortices. However, when they reach decimetric sizes, collisional fragmentation becomes prohibitive and prevents further evolution (on relevant time scales). It is therefore at this range of sizes that the vortex scenario should come at work and facilitate the formation of bigger structures, the so-called planetesimals. For particles between 10 and 100 cm in size, we find that the critical radius (46) at which the gas drag law changes lies in the range:

that is to say just at the separation between telluric (inner) and giant (outer) planets. This result is relatively robust because it depends only on a small power of the particles size and is independent on their density. We can therefore wonder if there is not a connection between the division of the solar system in two groups of planets (telluric and gaseous) and the two regimes (Stokes and Epstein) of the gas drag law in the primordial nebula. In the following we show how the vortex scenario can give further support to this idea.

Since the friction coefficient of a particle with size a and density depends on parameters (like , and ) which are functions of the distance r to the sun, the friction coefficient is itself a function of r. Combining Eqs. (39) (40) with Eqs. (41) (43) (45), we obtain:

where r is measured in A.U, a in cm and in g/cm³.

According to the results of Sect. 2.3, the capture time (measured in rotation periods) is optimal when . Since is a function of r with a maximum at , this criterion determines two preferred locations in the disk, one in each zone (see Fig. 3). In the Stokes (inner) zone, the optimum is at:

and in the Epstein (outer) zone, it is at:

Fig. 3. Evolution of the friction parameter as a function of the distance to the sun for a given type of particles (the figure corresponds to particles with size and bulk density ). The friction parameter is maximum at where the gas drag law passes from the Stokes to the Epstein regime. The condition determines two optimal regions in the disk.

More generally, we can combine Eqs. (33) (34) with Eqs. (48) (49) to express the capture time as a function of the distance to the sun (for a given type of particles). We find a W-shaped curve (see Fig. 4) determined by the power laws:

Fig. 4. Evolution of the capture time as a function of the distance to the sun for a given type of particles (, ). We have represented the present position of the planets. The capture time is optimum near the Earth (in the Stokes zone) and between Jupiter and Saturn (in the Epstein zone).

In order to make a numerical application, we assume that all the particles have the same density (the density of a composite rock-ice material) and the same size  cm (a typical prediction of grain growth models). In principle, we should consider a spectrum of sizes and densities but we choose these particular values in order to illustrate at best the predictions of the vortex model. For these values, the optimum in the inner zone is at , i.e. near the Earth orbit, and the optimum in the outer zone is at , i.e. between Jupiter and Saturn's orbits. The transition between the Stokes and the Epstein regime occurs at , i.e. near the asteroid belt. For a broader class of parameters, the preferred locations cover the whole region of telluric and giant planets.

Alternatively, we can determine, at each heliocentric distance, the size of the particles which are preferentially concentrated by the vortices. The results are indicated on Table 1 (see also Figs. 5 and 6). They show that the optimal sizes lie between 1 and 50 cm in the region of the planets (for a bulk density ). Such particles are concentrated in the vortices after only one rotation period. By contrast, the capture time for particles of in size (the initial size of the dust grains in the primordial nebula) is (we have a similar characteristic value for their settling time). This exceeds the lifetime of a circumstellar disk by many orders of magnitude. These results (see also Sects. 3.4 and 4.5) indicate that sticking of particles up to cm-sized bodies is an indispensable step in the process of planet formation.

Fig. 5. Variation of the friction parameter with the size of the particles at and (we have taken ). The capture is optimum for particles with friction parameter . This corresponds to decimetric sizes in the region of the planets.

Fig. 6. Variation of the capture time with the size of the the particles at and (). The capture time is minimum for particles with radius cm.

Table 1. Minimum sizes of dust particles (in cm) which can trigger the gravitational instability ().

We can also study how the mass captured by the vortices varies throughout the nebula and how it depends on the size of the particles. According to Eqs. (38) (42) and (44), the mass captured by a vortex during its lifetime can be written:

where we have introduced the Earth mass as a normalization factor. At and for optimal particles with size  cm and bulk density (see Table 1), we have leading to . Since the vortex lifetime is not accurately known, it makes sense to calibrate its value so as to satisfy . This yields , corresponding to rotation periods, in agreement with the estimate of Barge & Sommeria (1995) based on the value of the disk -viscosity and with the numerical simulations of Godon & Livio (1999a,b,c). Fig. 7 shows how the mass depends on the size of the particles at a given position of the solar nebula. We can also determine how it varies with the heliocentric distance for a given type of particles. The results are reported on Fig. 8 (full line) for particles with  cm and . The captured mass presents a global maximum near Jupiter's orbit and a plateau in the Earth's region. These results complete the work of Barge & Sommeria (1995) who first noticed the existence of an optimum near Jupiter. However, they did not consider the Stokes regime in their article and made the wrong statement that "inside Jupiter's orbit, particle concentration occurs in an annular region at the vortex periphery [because the friction parameter is larger]". In reality, the friction parameter decreases anew when we enter the Stokes domain at (see Fig. 3). This allows the existence of another optimum near the Earth orbit. Therefore, the mass collected by the vortices in the inner zone is larger that one would obtain without this change of regime (the dash line in Fig. 8 would be found if the Epstein law was used in the whole disk). This transition may explain the division of the solar system in two groups of planets. Moreover, the vortex scenario explains naturally the disymmetry between these two groups: the mass capture rate is larger in the outer zone simply because the vortices are larger. Indeed, the captured mass is not only proportional to (which would give the symmetrical dotted line of Fig. 8) but also to the product which increases linearly with r. This effect may explain the difference in size and mass between telluric and giant planets. Moreover, the mass captured by the vortices in the outer zone should be larger than these estimates since they intercept in priority the matter drifting toward the sun. On the other hand, the intermediate region at should be further depleted due to its proximity with the global maximum.

Fig. 7. Variation of with the size of the particles at and ().

Fig. 8. Variation of with the heliocentric distance for particles with size cm and bulk density .

In conclusion, we find that there are two locations in the primordial nebula where the trapping of dust by vortices is optimal. These locations belong to the Stokes and to the Epstein zones and fall near the Earth and Jupiter's orbit respectively. The zone of transition of the gas drag law is consistent with the position of the asteroid belt which marks the separation between telluric and giant planets. The asymmetry between the two groups of planets may be related to the size of the vortices which are bigger in the outer zone and therefore capture more mass. The exact values of , and depend on the spectrum of size of the particles, which is not well-known, but the W-shape of Figs. 4 and 8 is generic and agrees with the global structure of the solar system.

© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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