## 2. Deterministic motion of a particle in a vortex## 2.1. The solar nebula modelWe 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 The condition of hydrostatic equilibrium implies that the vertical pressure gradient is precisely balanced by , i.e.: If the local temperature is
independent of 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
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
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 ## 2.2. The vortex modelWe 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 We must note, however, that the vortex constructed by Barge &
Sommeria (1995) is relatively 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)
where is the aspect ratio of the
elliptic patch ( are the semi-axis in
the The matching conditions between the elliptic vortex and the
Keplerian shear require that ,
and The solution (11)(12) is valid for
, implying
. The vortex is ## 2.3. The capture timeBefore 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
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 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 defined as the 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
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 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 rateIn 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 ## 2.5. Application to the solar nebulaWe shall assume that the solid particles are spherical, of radius
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 where According to the results of Sect. 2.3, the capture time
(measured in rotation periods) is
optimal when . Since
is a function of and in the Epstein (outer) zone, it is at:
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:
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.
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
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 |