astronomy and trapping of dust by coherent vortices in the...

Astron. Astrophys. 356, 1089–1111 (2000) ASTRONOMYAND

ASTROPHYSICS

Trapping of dust by coherent vortices in the solar nebula

P.H. Chavanis1 Laboratoire de Physique Quantique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France2 Istituto di Cosmogeofisica, Corso Fiume 4, 10133 Torino, Italia

Received 13 July 1999 / Accepted 11 February 2000

Abstract. We develop the idea proposed by Barge & Sommeria(1995) and Tanga et al. (1996) that large-scale vortices present inthe solar nebula can concentrate dust particles and facilitate theformation of planetesimals and planets. We introduce an exactvortex solution of the incompressible 2D Euler equation andstudy the motion of dust particles in that vortex. In particular, wederive analytical expressions for the capture time and the masscapture rate as a function of the friction parameter. Then, westudy how small-scale turbulent fluctuations affect the motionof the particles in the vortex and determine their rate of escape bysolving a problem of quantum mechanics. We apply these resultsto the solar nebula and find that the capture is optimum nearJupiter’s orbit (as noticed already by Barge & Sommeria 1995)but also in the Earth region. This second optimum corresponds tothe transition between the Epstein and the Stokes regime whichtakes place, for relevant particles, at the separation betweentelluric and giant planets (i.e. near the asteroid belt). At theselocations, the particles are efficiently captured and concentratedby the vortices and can undergo gravitational collapse to formthe planetesimals.

Key words: accretion, accretion disks – hydrodynamics – turbu-lence – planets and satellites: general – solar system: formation– solar system: general

1. Introduction

Many astrophysical objects, ranging from young stars to mas-sive black holes, are surrounded by widespread gaseous disks.The existence of a primordial disk around the sun was conjec-tured by Kant (1755) and Laplace (1796) in the18th centuryto explain the quasi-circular, coplanar and prograd motion ofthe planets in the solar system. Such protoplanetary disks haverecently been observed with the Hubble Space Telescope inthe Orion nebula around stars less than one million years old.These gaseous disks can be considered as a by-product of thestar formation: after the gravitational collapse of a molecularcloud, about99% of the initial angular momentum is spread inan extended disk, while99% of the mass forms the star, whoseinternal structure is hardly affected by rotation.

Whenever it has been possible to observe rotating, turbulentfluids with good resolution, it has been seen that individual, in-tense vortices form (Bengston & Lighthill 1982; Hopfinger etal. 1983; Dowling and Spiegel 1990). One of the most strikingexamples is Jupiter’s Great Red Spot, a huge vortex persistingfor more than three centuries in the upper atmosphere of theplanet. These coherent vortices are well reproduced in numer-ical simulations (McWilliams 1990, Marcus 1990) and labora-tory experiments (Van Heist & Flor 1989, Sommeria, Meyers& Swinney 1988) of two-dimensional turbulence and their or-ganization can be explained in terms of statistical mechanics(Miller 1990, Robert & Sommeria 1991, Sommeria et al. 1991,Chavanis & Sommeria 1998)1. It seems therefore natural to ex-pect their presence in accretion disks also (Dowling & Spiegel1990, Abramowicz et al. 1992, Adams & Watkins 1995).

However, accretion disks are exceptional among rotatingturbulent objects in the strong shears that these bodies are be-lieved to possess and this shear might lead to rapid destructionof any structures that tend to form. This objection has been over-ruled by the numerical simulations of a two-dimensional flowin an external Keplerian shear by Bracco et al. (1998,1999) foran incompressible flow and Godon & Livio (1999a,b,c) for acompressible flow. Although cyclonic fluctuations are rapidilyelongated and destroyed by the shear, anticyclonic vortices formand persist for a long time before being ultimately dissipatedby viscosity. Naturally, this does not prove that coherent struc-tures must form on disks, but this strengthens the argument thatdisks are likely to follow the norm of rotating, turbulent bodies.Other numerical results (Hunter & Horak 1983) and experimen-tal work (Nezlin & Snezhkin 1993) comfort this point.

Coherent vortices in circumstellar disks can play an impor-tant role in the transport of dust particles and in the processof planet formation. Planets are thought to be formed from thedust grains embedded in the disk after a three-stage process:(i) in a first stage, microscopic particles suspended in the gas

1 It can be noted that the statistical mechanics of two-dimensionalturbulence is very similar to the theory of “violent relaxation” (Lynden-Bell 1967) by which galaxies achieve an equilibrium state. It is fasci-nating to realize that despite their very different physical nature, two-dimensional vortices and stellar systems share some common features.This analogy is developed in detail in Chavanis et al. (1996) and Cha-vanis (1996,1998a,1999).

1090 P.H. Chavanis: Trapping of dust by coherent vortices in the solar nebula

stick together on contact due to electrostatic or surface forces.When they reach sufficient sizes, they begin to sediment in themid-plane of the disk due to the combined effect of the gravityand the friction with the gas. When settling dominates, a parti-cle can grow by sweeping up smaller ones (Safronov 1969) andmay easily reach sizes of several centimeters in a few thousandorbital periods (Weidenschilling & Cuzzi 1993). Bigger aggre-gates (> 100cm) are more difficult to form on relevant timescales because of collisional fragmentation (ii) When the layerof sedimented particles is sufficiently dense, the gravitationalinstability is triggered and the layer crumbles into numerouskilometer-sized bodies, the so-called “planetesimals” (Safronov1969, Goldreich & Ward 1973). (iii) The subsequent evolution ismarked by planet’s growth due to the accumulation of planetes-imals in successive collisions. This stage is well reproduced bydynamical models (Safronov 1969, Barge & Pellat 1991, 1993).When the solid core becomes sufficiently massive, it can accretethe surrounding gas and a giant planet, like Jupiter, is formed.

However, the above scenario faces two major problems. Re-cent studies have shown that circumstellar disks are relativelyturbulent and that small-scale turbulence strongly reduces thesedimentation of the dust particles in the ecliptic plane (Wei-denschilling 1980, Cuzzi et al. 1993, Dubrulle et al. 1995). Forparticles of relevant size, the density of the dust layer is not suf-ficient to overcome the threshold imposed by Jeans instabilitycriterion. Therefore, the formation of the planetesimals, i.e. thepassage from cm-sized to km-sized particles, is not clearly un-derstood. In addition, it seems difficult, with the above model,to build up sufficiently massive cores in less than one millionyears before the gas has been swept away by the solar windduring the T-tauri phase (Safronov 1969, Wetherill 1988).

Both difficulties are ruled out if we allow for the presence ofvortices in the disk. Their existence was first proposed by VonWeizäcker (1944) to explain the regularity of the planet distribu-tion in the solar system: the famous Titius-Bode law2. This ideahas been reintroduced recently by Barge & Sommeria (1995)and Tanga et al. (1996) who demonstrated that anticyclonic vor-tices in a rotating disk are able to capture and concentrate dustparticles. The capture is made possible by the action of the Cori-olis force which pushes the particles inward. These results aresupported by a dynamical model which integrates the motionof the particles in the velocity field produced by a full Navier-Stokes simulation of the gas component (Bracco et al. 1999,Godon & Livio 1999c). It is found that the particles are veryefficiently captured and concentrated by the vortices. This isinteresting because, without a confining mechanism, cm-sizedbodies would rapidly fall onto the sun due to the inward drift as-sociated with the velocity difference between gas and particles.Inside the vortices, the density of the dust cloud is increased by alarge factor which is sufficient to triggerlocally the gravitationalinstability and facilitate the formation of the planetesimals or thecores of giant planets. This trapping mechanism is quite rapid

2 An account of Von Weizs̈acker’s theory can be found in Chan-drasekhar (1946). Note that the idea of vortices in the solar systemgoes back to Descartes (1643).

(a few rotations) and can reduce substantially the time scale ofplanet formation.

In this paper, we present a simple analytical model for thecapture of dust particles by coherent vortices in a Kepleriandisk. This model is directly inspired by the numerical studiesof Barge & Sommeria (1995) and Tanga et al. (1996) and theirmain results are recovered and confirmed. One interest of ourapproach is to provide analytical results (leading to quantitativepredictions) and to isolate relevant parameters which prove tobe particulary important in the problem. In Sect. 2, we introducean exact solution of the incompressible 2D Euler equation ap-propriate to our sudy. This is an elliptic vortex with uniform vor-ticity matching continuously with the azimuthal Keplerian flowat large distances. We consider deterministic trajectories of dustparticles in that vortex and derive analytical expressions for thecapture time and the mass capture rate as a function of the fric-tion parameter. We find that the capture is optimum for particleswhose friction parameter is close to the disk angular velocity.In Sect. 3, we investigate the effect of small-scale turbulence onthe motion of the particles. Their trajectories become stochasticand their motion must be described in terms of diffusion equa-tions. We estimate the diffusion coefficient and determine thetypical length on which the particles are concentrated in the vor-tices. In Sect. 4, we evaluate the rate of particles which diffuseaway from the vortices due to turbulent fluctuations. An ex-plicit expression for the “rate of escape” is obtained by solvinga problem of quantum mechanics, namely a two-dimensionaloscillator in a box. In appendix A, we give some details aboutthe construction of the vortex solution and in appendix B, weextend Toomre instability criterion (Toomre, 1964) to the caseof a turbulentrotating disk.

In parallel, we apply these theoretical results to the solarnebula and make speculations about its actual structure. For rel-evant particles going from10 cm to100 cm in size, we remarkthat the transition between the Stokes and the Epstein regimes(at which the gas drag law changes) corresponds precisely tothe transition between telluric (inner) and giant (outer) planets.Moreover, in each zone there is a preferred location where thecapture of dust by vortices is optimum. For particles of densityρs ∼ 2 g/cm3 and size∼ 30 cm (a typical prediction of graingrowth models), this is near the Earth orbit in the Stokes (inner)zone and between Jupiter and Saturn in the Epstein (outer) zone.For a broader class of parameters, the preferred locations coverthe whole region of telluric and giant planets with a depletionnear the asteroid belt. In this model, the asymmetry between tel-luric and giant planets is explained by the fact that the vorticesare bigger in the outer zone, so they capture more mass. Insidethe vortices, the surface density of the dust particles is increasedby a factor100 or more sufficient to triggerlocally the gravi-tational instability. More precisely, we study how the surfacedensity enhancement depends on the size of the particles. Weshow that particles must have reached at least centimetric sizesto collapse and form the planetesimals. Smaller particles dif-fuse away from the vortices on account of turbulent fluctuationsand are not sufficiently concentrated. This implies that stickingprocesses are necessary to produce large particles. These re-

P.H. Chavanis: Trapping of dust by coherent vortices in the solar nebula 1091

sults rehabilitate the Safronov-Goldreich-Ward scenario in lo-calized regions of the disk (i.e., inside the vortices) and forsufficiently large (decimetric) particles. Preliminary results ofthis work were presented in Chavanis (1998b).

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 hy-drostatic equilibrium in the vertical direction. If the mass of thedisk is much less that the solar mass (< 0.1M ), then its self-gravity can be neglected compared with the sun’s attraction. Inthat case, the disk has approximately Keplerian rotation

Ω(r) =(GM

r3

)1/2(1)

wherer is the distance from the sun andG = 6.672 10−8

cm3/(g.s2) the constant of gravity. For thin disks, the verticalcomponent of the solar gravity is:

gz ' −GMr2

× zr

= −Ω2z (2)

The condition of hydrostatic equilibrium implies that the verticalpressure gradient is precisely balanced bygz, i.e.:

∂p

∂z= −ρgΩ2z (3)

If the local temperatureT = mkpρg

is independent ofz, then thevertical density profile of the gas is

ρg = ρ0e−z2/H2 (4)

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

H = cs/Ω (5)

wherecs ∼√

kTm is the sound speed. Other plausible temper-

ature profiles, e.g., adiabatic gradient in thez direction yieldsimilar results.

We assume also that the gas is turbulent. Turbulence is nec-essary to explain the dynamical evolution of the protoplane-tary disk and its cooling consistent with cosmochemical data(Dubrulle 1993). However, its origin is not well understoodand remains controversial. Various mechanisms for inducingglobal nebula turbulence have been proposed. Lin & Papaloizou(1980) and Cabot et al. (1987a,b) suggested that thermal con-vection drives nebula turbulence. However, the applicability oftheir results depends on the presence of abundant micron-sizeddust to provide substantial thermal opacity; thus, they are ques-tionable when significant particle growth has already occurredand the thermal opacity has decreased. Dubrulle (1992,1993)suggested that the Keplerian shear is the main engine of theturbulence. This was already pointed out by Von Weizsäckerand further discussed by Chandrasekhar (1949) in view of thevery small viscosity of the disk: “The successive rings of gasin the medium will have motions relative to one another, and

turbulence will ensue”. This apparently obvious claim is actu-ally far from straightforward to support because the Keplerianshear is stable with respect to linear, infinitesimal, perturbationswhich are usually relied upon to induce turbulence. However,Dubrulle & Knobloch (1992) point out that it might be unstableto nonlinear finite amplitude perturbations, a property shared bymost of the shear flows commonly met in engineering or labo-ratory experiments. The simplest example is the plane Couetteflow, a plane parallel stream of constant shear. This analogy iscontested by Balbus et al. (1996) who did not evidence nonlin-ear instabilities in Keplerian disks at the respectable Reynoldsnumbers they achieved numerically3. These authors have sug-gested, in contrast, that turbulence in Keplerian disks could beproduced by a powerful MHD instability (Balbus & Hawley1991). The numerical simulations of Bracco et al. (1998) sug-gest that MHD turbulence can form magnetized vortices ableto capture charged particles. However, in the case of the diskthat is supposed to have spawned the solar system, it is thoughtthat the matter was too cool to be ionized. The recourse to mag-netic effects to render the disk turbulent is therefore suspect andthe problem of whether Keplerian disks are turbulent or not re-mains open. Recently, Lovelace et al. (1999) discovered a linearnonaxisymmetric instability in a thin Keplerian disk which canlead to the formation of Rossby vortices. This may open newperspectives for hydrodynamical turbulence in accretion disks.

In any case, the solar nebula must have been turbulent at leastduring its formation from the collapsing protostar, because ofvelocity discontinuities as the infalling matter struck the disk.The infall probably did not stop suddenly but decayed over someinterval. Therefore, the initial conditions in the disk were turbu-lent and this is sufficient to form vortices that survive for manyrotation periods of the disk (Bracco et al. 1998,1999; Godon &Livio 1999a,b,c). The appearance of large-scale coherent vor-tices in rotating flows is due to the presence of the Coriolis forceand the bimodal nature of turbulence (Dubrulle & Valdettaro1992). At small scales, the influence of the rotation is negligibleand the turbulence is homogeneous and isotropic. Energy cas-cades toward smaller and smaller scales up to the dissipationlength. Turbulent diffusion is important and can accelerate themixing of dust particles. At larger scales, the Coriolis force be-comes dominant and the gas dynamics is quasi two-dimensional.On these scales, the energy transfers are inhibited or even re-versed; this is a manifestation of the celebrated “inverse cas-cade” process (see, e.g., Kraichnan & Montgomery 1980) intwo-dimensional turbulence. There is therefore the possibility offormation and maintenance of coherent vortices (Mc Williams1990). These vortices can form either by a large-scale instabil-ity (Frisch et al. 1987, Dubrulle & Frisch 1991, Kitchatinov etal. 1994) or by the successive mergings of like-sign vortices,as observed in the simulations of Bracco et al. (1998,1999) andGodon & Livio (1999a,b,c). Due to the strong Keplerian shear,only anticyclonic vortices can survive this process. Initially, the

3 See also the 2D simulations by Godon & Livio (1999a) start-ing from finite perturbations and showing that turbulence is not self-sustained.


vortices have sizeR � H and typical vorticityΩ. Their ve-locity v ∼ ΩR is less that the sound speedcs = ΩH and theflow can be considered as incompressible. The merging ends upwhen the Mach numberMa = vcs reaches unity, i.e.R ∼ H.Bigger vortices radiate density waves and do not maintain (seeBarge & Sommeria 1995).

We expect therefore that, after some evolution, the disk befilled with anticyclonic vortices of typical sizeH, the disk thick-ness. Vortices are expected to form throughout the nebula andthere is no reason, a priori, to beleive that certain regions of thedisk should be excluded. However, two vortices at comparabledistance from the sun will approach each other (due to differ-ential rotation) and finally merge. Therefore, we do not expectmore than one (or a few) vortices at each radial distance. Onthe other hand, two successive vortices should be separated bya distance comparable to their sizeR. SinceR ∼ H andHscales like a power law of the distance to the sun (r5/4 in thestandard model considered below), the distribution of vorticesshould be consistent with an approximate geometric progres-sion of the planetary positions (Barge & Sommeria 1995). Aselucidated by Graner & Dubrulle (1994), the Titius-Bode lawreflects more the general properties of scale invariance in thesolar nebula than any particular physical process.

2.2. The vortex model

We consider the motion of a dust particle in a vortex located at adistancer0 from the sun. For convenience, we work in a frame ofreference rotating with constant angular velocityΩ ≡ Ω(r0)andwe denote byu(r, t) the velocity field of the gas in that frame.The dust particle is subjected to the attraction of the sun−GMr3 rand to a friction with the gas that we write−ξ(v − u(r, t))wherev = drdt is the particle velocity. We shall come back tothis expression and to the value of the friction coefficientξ inSect. 2.5. Since we work in a rotating frame, apparent forcesarise in the system. These are the Coriolis force−2Ω ∧ v andthe centrifugal forceΩ2r. All things considered, the particleequation writes:

d2rdt2

= −ξ(drdt

− u(r, t))

− 2Ω ∧ drdt

+(

Ω2 − GMr3

)r (6)

This is an ordinary second order differential equation coupledto the gas motion. The case of a time-dependent velocity fieldu(r, t) produced by the Navier-Stokes simulation of a randominitial vorticity field superposed on the Keplerian rotation hasbeen investigated by Bracco et al. (1999) and Godon & Livio(1999c) who observed the capture of the dust particles by an-ticyclonic vortices. The case of a static velocity fieldu(r) wasfirst 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 con-centric epicycles while it matches continuously the Keplerianflow at large distances. This epicyclic model is motivated bythe underlying idea that the particles, when sufficiently con-centrated, will retroact on the vortex and will force the gas tofollow their natural motion. Tanga et al. (1996) consider a more

complex streamfunction describing an ensemble of small vor-tices corresponding to Rossby waves corotating with the flow.This velocity field is obtained as a self-similar solution of thelinearized equations of motion governing the large-scale dy-namics of a turbulent nebula. These two models lead to qualita-tively similar results indicating that dust particles are efficientlytrapped into coherent anticyclonic vortices. However, the mod-els differ in the details: in Tangaet al.(1996), the particles sinkto the center of the vortices with no limit while in Barge & Som-meria (1995), the spiralling motion ends up on an epicycle. Inthat case, the friction force cancels out and the epicyclic motionis an exact solution of the particle Eq. (6).

We must note, however, that the vortex constructed by Barge& Sommeria (1995) is relativelyad hocand does not satisfy thefluid equations rigorously. In this article, we introduce an exactvortex solution of the incompressible 2D Euler equation andstudy analytically the motion of dust particles in that vortex.Since the vortices of the solar nebula are small compared withthe radial distancer0 (we have typicallyR/r0 ∼ 0.04, seeformula (44)) the last term in Eq. (6) can be expanded to firstorder in the displacementr−r0. This is the so-called “epicyclicapproximation”. Introducing a set of cartesian coordinates(x, y)centered on the vortex and such that they-axis points in thedirection opposite to the sun, the particle Eq. (6) reduces to:

d2x

dt2= −ξ

(dx

dt− ux

)+ 2Ω

dy

dt(7)

d2y

dt2= −ξ

(dy

dt− uy

)− 2Ωdx

dt+ 3Ω2y (8)

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

ux =32Ωy (9)

uy = 0 (10)

obtained as a first order expansion of the Keplerian velocityaroundr0. Its vorticity isωK ≡ ∂xuy − ∂yux = − 32Ω.

It remains now to specify the velocity field inside the vor-tex. We can construct an exact solution of the incompressible2D Euler equation by taking an elliptic patch of uniform vor-ticity ω. This solution is well-known by model builders (see,e.g., Saffman 1992)4 but, to our knowledge, it has never beenapplied in an astrophysical context. Therefore, we give a shortdescription of its construction in Appendix A. In the vortex, thevelocity field writes

ux = − q2

1 + q2ωy (11)

uy =1

1 + q2ωx (12)

whereq = a/b is the aspect ratio of the elliptic patch (a, b arethe semi-axis in thex andy directions respectively). Outside

4 This solution was indicated to me by J.Sommeria.


the vortex, the velocity field is given by Eq. (A.26) of AppendixA. At large distances, we recover the Keplerian shear (9)(10).

The matching conditions between the elliptic vortex and theKeplerian shear require thatω, ωK andq be related accordingto (see Appendix A):

ωKω

=q(q − 1)1 + q2

(13)

The solution (11)(12) is valid forq > 1, implying0 < ωK/ω <1. The vortex isanticyclonic(ω < 0) and is oriented with itsmajor axis parallel to the shear streamlines. It can be shownto be stable to infinitesimal two-dimensional disturbances. Forq → 1 (circular vortices),ω → −∞ and forq → ∞ (infinitelyelongated vortices),ω → ωK . Therefore,ω is in the range] − ∞,− 32Ω]. In a rotating disk, we expect thatω ∼ −2Ω,corresponding to a Rossby number of order1. This value isachieved by vortices with aspect ratioq ' 4, in good agreementwith the model of Tanga et al. (1996). The epicyclic vortexconsidered by Barge & Sommeria (1995) corresponds toq =2 andω = − 52Ω. These values do not satisfy the matchingcondition (13) with the Keplerian shear so the epicyclic vortexis not an exact solution of the incompressible Euler equation.It may be used, however, as an approximate solution if we takeinto account a coupling between the particles and the gas in thespirit of a two-fluids model.

2.3. The capture time

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

d2ρ

dt2= ρ

(dθ

dt

)2+ 2Ωρ

dθ

dt− ξ

(dρ

dt− uρ

)+ 3Ω2ρ sin2 θ(14)

where we have used the epicyclic approximation. The first termis a centrifugal force due to the rotation of the vortex (not to beconfused with the centrifugal forceΩ2r due to the rotation of thedisk) and the second term is the Coriolis force. The drag term inEq. (6) forces the particle velocity to approach the fluid veloc-ity, i.e. dθdt ∼ ω. For cyclonic vortices (ω > 0), both centrifugaland Coriolis forces are positive and push the particles outward.For anticyclonic vortices (ω < 0), the Coriolis force pushes theparticles inward and comes in conflict with the centrifugal forcewhich is always directed outward. If the vortex rotates rapidly,the centrifugal force prevails over the Coriolis force and theparticle is expelled. In the other case, the Coriolis term domi-nates and the particle is captured. We conclude that only slowlyrotating anticyclonic vortices can capture dust particles. Thistrapping process is specific to rotating fluids; in ordinary simu-lations of two-dimensional turbulence (without Coriolis force)the particles never penetrate the vortices. Note that the epicyclicacceleration (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. fory < 0) or toward

the outer nebula (if the centrifugal force dominates, i.e.y > 0).However, this flux is quite small and does not affect the overalltrapping process (see Tanga et al. 1996).

We now give a more precise analysis of the trapping pro-cess and derive an explicit expression for the capture time ofthe particles as a function of their friction parameterξ. To thatpurpose, we notice that Eqs. (7) (8) with the velocity field (11)(12) form alinear system of coupled differential equations. Weseek therefore a solution of the form:

x = Xeλt (15)

y = Y eλt (16)

whereX,Y and λ are complex numbers. Substituting intoEqs. (7) (8), we obtain a linear system of algebraic equations:

− λ(λ+ ξ)X + 2Ω(

34

q

q − 1ξ + λ)Y = 0 (17)

(3Ω2

1q(q − 1)ξ + 2Ωλ

)X + (λξ + λ2 − 3Ω2)Y = 0 (18)

There are nontrivial solutions only if the determinant of thissystem is zero. We are led therefore to a fourth order polynomialequation inλ:

λ4 + 2ξλ3 + (ξ2 + Ω2)λ2 + 31 + qq(q − 1)ξΩ

2λ

+94ξ2Ω2

1(q − 1)2 = 0 (19)

By definition, we will say that a particle islight orheavywhetherξ > Ω or ξ < Ω respectively. This terminology will take moresense in the sequel (see in particular Sect. 2.5). We now considerthe asymptotic limitsξ → ∞ andξ → 0 of Eq. (19).

For ξ → ∞ (light particles), the four roots of Eq. (19) areλ = −ξ (double root) (20)

λ = ± 3Ω2(q − 1) i−

3Ω2

4ξ(q − 2)(2q + 1)q(q − 1)2 (21)

The first solution is rapidly damped and will not be consideredanymore. The second solution describes the rotation of the dustparticles in the vortex:

x = A cos[− 3Ω

2(q − 1) t]e−t/tcapt (22)

y =A

qsin

[− 3Ω

2(q − 1) t]e−t/tcapt (23)

The particles follow ellipses of aspect ratioq and move withangular velocity− 3Ω2(q−1) . In fact, for ξ → ∞, the drag termin Eq. (6) impliesdrdt ' u so, in a first approximation, the par-ticles just follow the vortex streamlines (their angular velocitycoincides with the angular velocity of the fluid particles, see Ap-pendix A). In addition, they experience adrift toward the center


due to the combined effect of the friction and the Coriolis force.They reach the vortex center in a typical time:

tcapt =4ξ

3Ω2q(q − 1)2

(q − 2)(2q + 1) (24)

defined as thecapture time. Note that for light particles,tcaptincreaseslinearly with ξ.

For ξ → 0 (heavy particles), the four roots of Eq. (19) are:

λ = −32

1 + q ± √1 + 2qq(q − 1) ξ (25)

λ = ±Ωi− (q − 3)(2q + 1)2q(q − 1) ξ (26)

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

x = A cos(−Ωt)e−t/tcapt (27)

y =A

2sin(−Ωt)e−t/tcapt (28)

The particles follow ellipses with aspect ratio2 and move withangular velocity−Ω. This is natural since heavy particles havethe tendency to decouple from the gas and reach a pure epicyclicmotion. However, due to a slight friction, they sink progressivelyin the vortex with a characteristic time

tcapt =2q(q − 1)

(q − 3)(2q + 1)1ξ

(29)

For heavy particles, the capture time increases likeξ−1. Veryheavy particles can even leave the vortex. Formally, this pos-sibility is taken into account in the first solution (25) whichcorresponds to open trajectories. The motion of heavy particlesis therefore more complicated and demands a numerical integra-tion of the particle equation inside and outside the vortex. Thisstudy will not be undertaken in that paper. We shall only con-sider the case of closed trajectories described by Eqs. (27)(28).

For intermediate values ofξ, the capture time and the angularvelocity of the particles are plotted on Figs. 1 and 2 (for theparticular valueq = 4). We see thattcapt presents an optimumatξ = ξopt. Moreover, the asymptotic expressions (24) and (29)agree reasonably well with the exact solution for all the values ofthe friction parameter. Therefore, considering the intersectionof the asymptotes, we find that the capture time is minimum for

ξopt '[

3(q − 2)2(q − 1)(q − 3)

]1/2Ω (30)

and we have:

tmincapt '[

8(q − 1)3q23(q − 3)(2q + 1)2(q − 2)

]1/2 1Ω

(31)

According to Eqs. (24) and (29), the condition for particletrapping (tcapt > 0) requires thatq > 3. This implies that thevorticity must be in the range [see Eq. (13)]:

− 52Ω < ω < −3

2Ω (32)

capt

tΩ

ξ 8( )ξ ( 0)

0

5

10

15

20

0 2 4 6 8 10

ξ/Ω

Light particlesHeavy particles

Optimal particles(ξ ∼ Ω)

Fig. 1.Plot of tcapt vsξ for vortices with aspect ratioq = 4. The dashlines correspond to formulae (24) and (29) valid for light (ξ → ∞)and heavy (ξ → 0) particles.

Ωp/

Ω

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

0 2 4 6 8 10

ξ/Ω

Heavy particles

Light particles

(epicyclic velocity)

(vortex velocity)

Fig. 2. Plot of Ωp (angular velocity of the particles) vsξ for vorticeswith aspect ratioq = 4. Light particles (ξ → ∞) move with thevortex velocity− 3Ω2(q−1) and heavy particles (ξ → 0) with the epicyclicvelocity−Ω.

Particles are ejected from rapidly rotating anticyclones in agree-ment with the qualitative discussion of Tanga et al. (1996) re-called at the begining of this section. In the following, we shallspecialize on the value ofq which minimizes the capture timetmincapt. We findq ' 4. As noted already, this value corresponds toω ' −2Ω, i.e. a Rossby number of order1. For these vortices,the optimal friction parameter isξopt = Ω and the correspond-ing capture timetmincapt =

83Ω is of the order of one rotation

period. Forξ > Ω, we have

tcapt ' 8ξ3Ω2 (light particles) (33)and forξ < Ω

tcapt ' 83ξ (heavy particles) (34)

Light particles move with angular velocity−Ω2 (the vortex ve-locity) and heavy particles move with angular velocity−Ω (the


epicyclic velocity). These results should be compared with thesettling time of the dust particles in the equatorial plane (see,e.g., Nakagawa et al. 1986). Light particles are settled exponen-tially with a characteristic timeξ/Ω2 comparable with Eq. (33).On the other hand, heavy particles undergo overdamped oscilla-tions around the midplane with a period of the order ofΩ−1 anda settling time2/ξ similar to expression (34). We shall comeback on the analogy between the vortex trapping and the dustsettling in Sect. 3.1.

The models of Barge & Sommeria (1995) and Tanga et al.(1996) lead to qualitatively similar results. In the model of Tangaet al. (1996), the particles sink to the center of the vortex on atypical timetcapt(ξ) which also presents a minimum whenξ isof orderΩ. Moreover, for light particles,tcapt increases linearlywith ξ like in Eq. (33). The model of Barge & Sommeria (1995)is physically different since the particles do not really reachthe vortex center but end up on an epicycle after a time∼ 1/ξ.However, the general tendency is the same. Light particles (ξ �Ω) mainly follow the streamlines of the gas and stop on epicyclesclose to the vortex edge (like in case (a) of their Fig. 1). Optimalparticles with friction parameterξ ∼ Ω reach deeper epicycles,almost at the center of the vortex (case (b)). Heavy particles (ξ �Ω) take a long time to connect an epicycle and can even escapefrom the vortex since their motion is nearly unaffected by thefriction drag. This is also a possibility in our model and in Tangaet al. (1996). Therefore, the three vortex models give relativelysimilar results even if their physical contents are different. Therecent numerical simulations of Godon & Livio (1999c) for acompressible flow also show a capture optimum when the dragparameter 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 prob-lem concerns the mass capture rate (Barge & Sommeria 1995).This quantity can be estimated as follows. First, we have to de-termine the capture cross section of the vortex as a function ofthe friction parameterξ. Without the effect of the drift, a parti-cle with impact parameterη would cross the vortex in a typicaltime tcross ∼ R/uK , whereuK = 32Ωη is the Keplerian ve-locity. The particle will be captured by the vortex if, during thattime, the deflection due to the drift is precisely of orderη. Lightparticles (ξ → ∞) follow the edge of the vortex and, for them,vdrift ∼ R/tcapt. On the other hand, heavy particles (ξ → 0)keep their rectilinear motion and enter directly the vortex so that,for them,vdrift ∼ η/tcapt. The critical parameterηc is givenby the conditiontcross × vdrift ∼ ηc. Moreover, for optimalparticles withξ ∼ Ω, we expect thatηc ∼ R. Regrouping allthese results, we obtain

f(ξ) =ηcR

'(

Ωξ

)1/2(light particles) (35)

f(ξ) =ηcR

' ξΩ

(heavy particles) (36)

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

The mass capture rate can be estimated by assuming thatthe particles are carried to the vortex by the Keplerian shearuK = 32Ωy, and that they are continuously renewed by the in-ward drift (directed toward the sun) associated with the velocitydifference∆V between gas and particles (Barge & Sommeria1995). Another alternative, consistent with the simulations ofBracco et al. (1999), is that the particles are already concentratedby the turbulent fluctuations that accompany vortex formationin the early stages of the protoplanetary nebula. It is likely thatboth mechanisms come into play in the capture process. Con-sidering the first possibility, which can be studied analytically,we have:

dM

dt= 2

∫ ηc0

σduKdy =32σdΩR2f2(ξ) (37)

whereσd is the surface density of the dust particles outside thevortex. The total mass collected by the vortex during its lifetimeis

Mlife =32(Ωtlife)σdR2f2(ξ) (38)

The mass capture rate is proportional to the effective surface∼ f2(ξ)R2 of the vortex and is maximum for particles withξ ∼ Ω.

In conclusion, various vortex models and different physicalarguments show that the capture isoptimal for particles whosefriction parameter is close to the disk rotation. We shall nowconsider the implications of this result on the structure of thesolar nebula.

2.5. Application to the solar nebula

We shall assume that the solid particles are spherical, of radiusa and densityρs. The value of the friction parameterξ dependswhether the size of the particles is larger or smaller than themean free pathλ in the gas (see, e.g., Weidenschilling 1977).Whena < 94λ, we are in the Epstein regime and:

ξ =Ωσg2ρsa

(39)

whereσg is the gas surface density.Whena > 94λ, the situation is more complicated because,

in general, the friction parameter depends on the velocity differ-ence|v−u| between the particles and the gas. There is, however,a regime whereξ is independent of the particle relative velocity.This is the Stokes regime in which:

ξ =9σgΩλ8a2ρs

(40)

This regime is valid when the particle Reynolds numberRp =2aνg

|v − u|, whereνg = 12csλ is the kinematic viscosity of thegas, is smaller than1. This is the case for light particles whichmove typically with the gas velocityv ' u. For heavy particles


on the contrary, we haveRp > 1 and, rigorously speaking,the friction parameter depends on the relative velocity. The useof a more exact expression forξ would require a numericalintegration of the particle trajectory but the results should notdramatically differ from those obtained with expression (40).Since we are mainly interested in orders of magnitude, we willuse expression (40) for all particle sizes (witha > 94λ), but keepin mind this uncertainty for large particles.

We shall adopt a standard model of the solar nebula, follow-ing Cuzzi et al. (1993). It corresponds to a “minimum mass”circumstellar nebula with parameters:

Ω = 2π(

r

1A.U

)−3/2years−1 (41)

σd = 10(

r

1A.U

)−3/2g/cm2 (42)

σg = 1700(

r

1A.U

)−3/2g/cm2 (43)

H = 0.04(

r

1A.U

)5/4A.U (44)

λ = 1(

r

1A.U

)11/4cm (45)

We take1 year = 3.15 107 s and1 A.U = 1.49 1013 cm.For a given type of particles, the transition between the Ep-

stein and the Stokes regime is achieved at a specific distancefrom the sun given by:

rc =(

49

a

1cm

) 411

A.U (46)

We are particularly interested in particles of order10 cm insize. Indeed, smaller particles can grow by aggregation pro-cesses without the aid of vortices. However, when they reachdecimetric sizes, collisional fragmentation becomes prohibitiveand prevents further evolution (on relevant time scales). It istherefore at this range of sizes that the vortex scenario shouldcome at work and facilitate the formation of bigger structures,the so-called planetesimals. For particles between10 and100cm in size, we find that the critical radius (46) at which the gasdrag law changes lies in the range:

1.7A.U < rc < 3.9A.U (47)

that is to say just at the separation between telluric (inner) andgiant (outer) planets. This result is relatively robust becauseit depends only on a small power of the particles size and isindependent on their density. We can therefore wonder if thereis not a connection between the division of the solar system intwo 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 givefurther support to this idea.

0

1

2

3

4

5

6

-0.5 0 0.5 1 1.5

log(r)

ξ/ΩStokes zone Epstein zone

(inner) (outer)

Fig. 3. Evolution of the friction parameterξ/Ω as a function of thedistance to the sun for a given type of particles (the figure correspondsto particles with sizea = 30 cm and bulk densityρs = 2 g/cm3).The friction parameter is maximum atr = rc where the gas drag lawpasses from the Stokes to the Epstein regime. The conditionξ/Ω = 1determines two optimal regions in the disk.

Since the friction coefficient of a particle with sizea anddensityρs depends on parameters (likeσg, λ andΩ) which arefunctions of the distancer to the sun, the friction coefficientξ isitself a function ofr. Combining Eqs. (39) (40) with Eqs. (41)(43) (45), we obtain:

ξ

Ω=

1913a2ρs

r54 if r < rc (Stokes zone) (48)

ξ

Ω=

850aρs

r−32 if r > rc (Epstein zone) (49)

wherer is measured in A.U,a in cm andρs in g/cm3.According to the results of Sect. 2.3, the capture timeΩtcapt

(measured in rotation periods) is optimal whenξ/Ω = 1. Sinceξ/Ω is a function ofr with a maximum atrc, this criteriondeterminestwopreferred locations in the disk, one in each zone(see Fig. 3). In the Stokes (inner) zone, the optimum is at:

rin =(a2ρs1913

)4/5(50)

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

rout =(

850aρs

)2/3(51)

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

Ωtcapt =a2ρs717

r−5/4 (r < rin) (52)

Ωtcapt =5101a2ρs

r5/4 (rin < r < rc) (53)


t cap

tΩ

0

5

10

15

20

25

30

35

-0.5 0 0.5 1 1.5

log(r)

Epstein zone(giant planets)

Stokes(telluric

zoneplanets)

M

V E m JS

U

N

Fig. 4.Evolution of the capture timeΩtcapt as a function of the distanceto the sun for a given type of particles (a = 30 cm, ρs = 2 g/cm3). Wehave represented the present position of the planets. The capture timeis optimum near the Earth (in the Stokes zone) and between Jupiter andSaturn (in the Epstein zone).

log(

ξ/Ω

)

-4

-2

0

2

4

6

8

10

-6 -5 -4 -3 -2 -1 0 1 2 3

log(a)

5 A.U.

1 A.U.

Fig. 5. Variation of the friction parameter with the size of the particlesat r = 1 A.U andr = 5 A.U (we have takenρs = 2g/cm3). Thecapture is optimum for particles with friction parameterξ ∼ Ω. Thiscorresponds to decimetric sizes in the region of the planets.

Ωtcapt =2267aρs

r−3/2 (rc < r < rout) (54)

Ωtcapt =aρs319

r3/2 (r > rout) (55)

In order to make a numerical application, we assume that allthe particles have the same densityρs ∼ 2g/cm3 (the densityof a composite rock-ice material) and the same sizea ∼ 30 cm(a typical prediction of grain growth models). In principle, weshould consider a spectrum of sizes and densities but we choosethese particular values in order to illustrate at best the predic-tions of the vortex model. For these values, the optimum in theinner zone is atrin ∼ 1A.U , i.e. near the Earth orbit, and the op-timum in the outer zone is atrout ∼ 6A.U , i.e. between Jupiterand Saturn’s orbits. The transition between the Stokes and theEpstein regime occurs atrc ∼ 2.7A.U , i.e. near the asteroidbelt. For a broader class of parameters, the preferred locationscover the whole region of telluric and giant planets.

Table 1.Minimum sizes of dust particles (in cm) which can trigger thegravitational instability (ρs = 2g/cm3).

Planets r(A.U) λ aopt aconcmin acaptmin amin

Telluric planets :Mercury 0.387 0.07 17 14 7 3V enus 0.723 0.4 25 20 11 5Earth 1 1 31 25 13 6Mars 1.52 3 40 33 17 8

Asteroid belt : 3 20 61 50 15 3

Giant planets :Jupiter 5.20 93 36 23 7 1Saturn 9.52 492 14 9 3 0.5Uranus 19.2 3364 5 3 1 0.2Neptune 30.0 11523 3 2 0.5 0.1

capt

t

Ω

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80

a

1 A.U.

5 A.U.

Fig. 6. Variation of the capture time with the size of the the particlesat r = 1 A.U andr = 5 A.U (ρs = 2g/cm3). The capture time isminimum for particles with radiusa ∼ 30cm.

Alternatively, we can determine, at each heliocentric dis-tance, the size of the particles which are preferentially con-centrated by the vortices. The results are indicated on Table 1(see also Figs. 5 and 6). They show that the optimal sizes lie be-tween1 and50 cm in the region of the planets (for a bulk densityρs = 2g/cm3). Such particles are concentrated in the vorticesafter only one rotation period. By contrast, the capture time forparticles of1µm in size (the initial size of the dust grains in theprimordial nebula) isΩtcapt ∼ 109 (we have a similar charac-teristic value for their settling time). This exceeds the lifetime ofa circumstellar disk by many orders of magnitude. These results(see also Sects. 3.4 and 4.5) indicate that sticking of particlesup to cm-sized bodies is an indispensable step in the process ofplanet formation.

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

MlifeM⊕

= 8.916 10−4(Ωtlife)rf2(ξ) (56)


M lifeM earth

0

1

2

3

4

5

0 10 20 30 40 50 60 70 80

a

5 A.U.

1 A.U.

Fig. 7. Variation ofMlife with the size of the particles atr = 1 A.Uandr = 5 A.U (ρs = 2g/cm3).

M lifeM earth

0

1

2

3

4

5

6

-0.5 0 0.5 1 1.5

Epstein zone(giant planets)

log(r)

Stokes(telluric planets)

zone

Fig. 8. Variation ofMlife with the heliocentric distance for particleswith sizea = 30cm and bulk densityρs = 2g/cm3.

where we have introduced the Earth massM⊕ = 5.976 1027 gas a normalization factor. Atr = 1A.U and for optimal par-ticles with sizea ∼ 30 cm and bulk densityρs ∼ 2g/cm3(see Table 1), we haveξ ∼ Ω leading torf2 ∼ 1. Since thevortex lifetime is not accurately known, it makes sense to cali-brate its value so as to satisfyMlife(1A.U) = M⊕. This yieldsΩtlife ' 1122, corresponding to∼ 180 rotation periods, inagreement with the estimate of Barge & Sommeria (1995) basedon the value of the diskα-viscosity and with the numerical sim-ulations of Godon & Livio (1999a,b,c). Fig. 7 shows how themassMlife depends on the size of the particles at a given posi-tion of the solar nebula. We can also determine how it varies withthe heliocentric distance for a given type of particles. The resultsare reported on Fig. 8 (full line) for particles witha ∼ 30 cm andρs ∼ 2g/cm3. The captured mass presents a global maximumnear Jupiter’s orbit and a plateau in the Earth’s region. Theseresults complete the work of Barge & Sommeria (1995) whofirst noticed the existence of an optimum near Jupiter. How-ever, they did not consider the Stokes regime in their article andmade the wrong statement that “inside Jupiter’s orbit, particleconcentration occurs in an annular region at the vortex periph-ery [because the friction parameterξ is larger]”. In reality, the

friction parameter decreases anew when we enter the Stokes do-main atr < rc (see Fig. 3). This allows the existence ofanotheroptimum near the Earth orbit. Therefore, the mass collected bythe vortices in the inner zone is larger that one would obtainwithout this change of regime (the dash line in Fig. 8 would befound if the Epstein law was used in the whole disk). This tran-sition may explain the division of the solar system in two groupsof planets. Moreover, the vortex scenario explains naturally thedisymmetry between these two groups: the mass capture rate islarger in the outer zone simply because the vortices are larger.Indeed, the captured mass is not only proportional tof2 (whichwould give the symmetrical dotted line of Fig. 8) but also tothe productσdR2 which increases linearly withr. This effectmay explain the difference in size and mass between telluricand giant planets. Moreover, the mass captured by the vorticesin the outer zone should be larger than these estimates sincethey intercept in priority the matter drifting toward the sun. Onthe other hand, the intermediate region at∼ 3 A.U should befurther depleted due to its proximity with the global maximum.

In conclusion, we find that there are two locations in the pri-mordial nebula where the trapping of dust by vortices is optimal.These locations belong to the Stokes and to the Epstein zonesand fall near the Earth and Jupiter’s orbit respectively. The zoneof transition of the gas drag law is consistent with the positionof the asteroid belt which marks the separation between telluricand giant planets. The asymmetry between the two groups ofplanets may be related to the size of the vortices which are big-ger in the outer zone and therefore capture more mass. The exactvalues ofrin, rc androut depend on the spectrum of size of theparticles, which is not well-known, but the W-shape of Figs. 4and 8 is generic and agrees with the global structure of the solarsystem.

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 vortexis not deterministic butstochastic. Turbulent fluctuations pro-duce some kicks which progressively deviate the particle fromits unperturbed trajectory. This is similar to what happens to acolloidal particle in suspension in a liquid (Brownian motion).An individual fluctuation has a minute effect on the motion ofthe particle, but the repeated action of these fluctuations givesrise to a macroscopic process of diffusion. The effect of turbu-lence on the sedimentation of dust particles in the protoplanetarynebula has been considered in detail by Dubrulle et al. (1995).We use a similar approach to study the effect of turbulence onthe capture of dust by large-scale vortices.

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

∂σd∂t

+ ∇(σd〈v〉) = ∇(D∇σd) (57)

where〈v〉 is the mean velocity of the particles andD theirdiffusivity. The mean velocity〈v〉 is given by the deterministic


model of Sect. 2.3. According to Eqs. (22)(23) or (27) (28), itcan be written:

〈v〉 = V − 1tcapt

r (58)

whereV corresponds to the pure rotation of the particles in thevortex and−r/tcapt is their drift toward the center. In the caseof light particles (ξ � Ω), V is equal to the velocity of thevortex while in the case of heavy particles (ξ � Ω), V is equalto the epicyclic velocity (see Sect. 2.3). When (58) is substitutedinto (57), the diffusion equation takes the form:

∂σd∂t

+ ∇(σdV) = ∇(D∇σd + σd

tcaptr)

(59)

The first term in the right hand side is a pure diffusion due tosmall-scale turbulence and the second term is a drift towardthe vortex center due to the combined effect of the Coriolisforce and the vortex (anticyclonic) rotation. Eq. (59) illustratesthe bimodal nature of turbulence: the diffusion is due to small-scale fluctuations hardly affected by the rotation of the disk (3Dturbulence) and the trapping process is a consequence of theCoriolis force and the existence of coherent structures in thedisk (2D turbulence).

Since we are mainly interested in orders of magnitude andin order to avoid unnecessary mathematical complications, weshall consider from now on that the vortices are circular withtypical radiusR. In this approximation, we can restrict our-selves to axisymmetric solutions for which the advection termin Eq. (59) cancels out. We are led therefore to study the Fokker-Planck equation:

∂σd∂t

= ∇(D∇σd + σd

tcaptr)

(60)

For an initial condition consisting of a Dirac function centeredatr0, there is a well-known analytical solution of Eq. (60):

σd =M

2πDtcapt(1 − e−2t/tcapt)e− (r−r0e

−t/tcapt )2

2Dtcapt(1−e−2t/tcapt ) (61)

whereM is the total mass of particles contained in the vor-tex. This formula shows that the relaxation time is equal to thecapture timetcapt not affected by turbulence.

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

D∇σd + σdtcapt

r = 0 (62)

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

σd = σd(0)e− r22Dtcapt (63)

This is also the solution of Eq. (61) fort → +∞. In practice,the equilibrium distribution (63) is established fort ∼ tcapt.Then, the particles are concentrated in the vortices on a typicallength:

ld =√

2Dtcapt (64)

3.2. Vertical sedimentation

Eq. (60) is similar to the diffusion equation

∂nd∂t

=∂

∂z

(D∂nd∂z

+Ω2

ξndz

)(65)

used by Dubrulle et al. (1995) to describe the sedimentationof the dust particles and determine the sub-disk scale height(see also Weidenschilling 1980). The drift toward the vortexcenter is replaced in their study by the drift−Ω2ξ z toward theecliptic plane due to gravity. For light particles,tcapt ∼ 8ξ3Ω2[see Eq. (33)] and the two equations coincide up to numericalfactors. This implies that the particles are concentrated in thevortices on a lenghtld comparable with the sub-disk scale heightHd determined by Dubrulle et al. (1995) [see their Sect. 3].

It is relatively straightforward to include the vertical sedi-mentation of particles in our study, although we shall specializein the following on theirhorizontalaccumulation in vortices. In-troducing the volume densityρd(r, z, t) = 1Hdnd(z, t)σd(r, t)and using Eqs. (60) and (65), we obtain:

∂ρd∂t

= ∇{D∇ρd + Ω

2

ξρd

(38r⊥ + z

)}(66)

wherer⊥ andz are the component ofr in the directions per-pendicular and parallel to the disk rotation vectorΩΩΩ. IntegratingEq. (66) on the vertical direction returns Eq. (60) for the surfacedensity.

3.3. Diffusivity of dust particles

It remains now to specify the value of the diffusion coefficientappearing in Eq. (60). In general, the turbulent viscosity of the

gas is written in the formν ∼ α c2sΩ wherecs is the sound speedandα a non dimensional parameter which measures the effi-ciency of turbulence (Shakura & Sunyaev 1973). Following cur-rent nebula models,10−4 < α < 10−2. When the turbulence isgenerated by differential rotation,α = 2 10−3 (Dubrulle 1992)and we shall take this value for numerical applications. Since thedisk height isH ∼ cs/Ω (see Eq. (5)), the turbulent viscositycan be writtenν ∼ αH2Ω or, alternatively,ν ∼ αΩR2 whereR is the vortex radius. More precisely, assuming a power lawspectrumE(k) ∼ k−γ for the gas turbulence (withγ ' 5/3),we have

ν =αΩR2√γ + 1

(67)

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

D = g(ξ)ν (68)

where

g(ξ) =(

ξ

ξ + ΩArctan(Bk0)

Bk0

)1/2(69)


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

Bk0 =k0vsξ + Ω

(70)

depends on the sizek−10 ∼√αH of the largest eddies of tur-

bulence and on the systematic velocityvs of the dust grains. Inthe vortices,vs is equal to the drift velocityr/tcapt.

For light particles (ξ � Ω), g(ξ) → 1 and

D =αΩR2√γ + 1

(light particles) (71)

This result is expectable since light particles mainly follow thestreamlines of the gas. Consequently, their diffusivity tends tothe gas turbulent viscosityν.

For heavy particles (ξ � Ω), g(ξ) → √ξ/Ω andD =

αΩR2√γ + 1

(ξ

Ω

)1/2(heavy particles) (72)

For ξ → 0, D → 0 since there is no coupling with the gas.For ξ ∼ Ω, Eqs. (71) and (72) give the same result, so we canuse these expressions in the whole range of friction parameters.Note, however, that the diffusion approximation is not strictlyvalid for heavy particles, so Eq. (72) must be taken with care.

Substituting Eqs. (33)(34) and (71)(72) in Eq. (64), the con-centration length can be written explicitly:

ldR

=(

16α3√γ + 1

)1/2(ξ

Ω

)1/2(light particles) (73)

ldR

=(

16α3√γ + 1

)1/2(Ωξ


As expected, the concentration is optimum forξ ∼ Ω. In thatcase, the particles are distributed over a lengthld ∼

√αR.

Lighter and heavier particles are less concentrated. When

ξ

Ω>

3√γ + 1

16α' 153 (75)

or

ξ

Ω<

(16α

3√γ + 1

)2' 4 10−5 (76)

the drift is negligible and there is no concentration (ld > R).


We now apply these results to the case of the solar nebula andshow how the vortex scenario can make possible the formationof planetesimals at certain preferred locations of the disk.

It is generally beleived that planetesimals formed by thegravitational instability of the particle sublayer (Safronov 1969,Goldreich & Ward 1973). The dispersion relation for an infiniteuniformly rotating sheet of gas is (see, e.g., Binney & Tremaine1987):

ω2 = k2c2d + 4Ω2 − 2πGσdk (77)

whereσd is the unperturbed surface density of the particlesandcd their velocity dispersion. The particle sub-layer (whichbehaves as a very compressible fluid) will be unstable providedthatω2 < 0, for somek. From Eq. (77), we obtain the criterionfor gravitational instability (Toomre, 1964):

cd <πGσd2Ω

= Vcrit (78)

Once instability is triggered, the system crumbles into nu-merous planetesimals of order10 km in size. Moreover, thegrowth time of density perturbation is predicted to be short, ofthe order of an orbital period. In addition, the instability cri-terion gives the impression that its operation does not requireany sticking mechanism. Goldreich & Ward (1973) state that“...the fate of planetary accretion no longer appears to hingeon the stickiness of the surface of dust particles”. This is veryattractive because sticking mechanisms are relativelyad hocand ill-understood. For these reasons (and also for a lack of al-ternatives), the Safronov-Goldreich-Ward scenario was nearlyuniversaly accepted as the key mechanism for forming planetes-imals. Therefore, a swarm of bodies of a few km in diameter wasa common starting point for numerical simulations of planetaryformation.

However, Weidenschilling (1980), followed by Cuzzi et al.(1993) and Dubrulle et al. (1995), realized that this simplisiticpicture was ruled out if the primordial nebula was turbulent.Indeed, turbulence reduces considerably the vertical sedimen-tation of the dust particles and prevents gravitational instabil-ity. According to Eqs. (41) and (42), the velocity threshold im-posed by the instability criterion (78) is of orderVcrit = 5cm/sthroughout the nebula. Even if the nebula as a whole was per-fectly laminar, the formation of a dense layer of particles (con-sidered as a heavy fluid) would create a turbulent shear with theoverlying gas (see, e.g.., Weidenschilling & Cuzzi 1993). Whenturbulence is accounted for, the velocity dispersion of the parti-cles can be estimated bycd ∼

√αcs and easily reaches several

metersper second (when a numerical value is needed, we shalltakecd = 5m/s). Therefore, the instability criterion (78) is notsatisfied (see appendix B for a justification of this criterion ina turbulent disk). Turbulence is responsible for too high veloc-ity dispersions or, alternatively, the surface density of the dustsublayer is not sufficient to trigger the gravitational instability.The density needs to be increased by a factorAc ∼ 100 ormore to overcome the threshold imposed by Toomre instabilitycriterion.

There is therefore a major problem to form planetesimalsby gravitational instability in a turbulent disk. As suggested byBarge & Sommeria (1995), the presence of vortices in the diskcan solve this problem. Indeed, by capturing and concentratingthe particles, the vortices can increase locally the surface densityof the dust sublayer and initiate the gravitational instability.Let us first discuss theconcentrationeffect. Inside a vortex, aninitial mass of orderσdπR2 is concentrated on a typical lengthld given by Eq. (64). The surface density is therefore amplified


σ dvor

t /σ d

) con

c(

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100a

5 A.U.

1 A.U.

Fig. 9.Enhancement of the dust surface density inside the vortices (dueto concentration) as a function of the size of the particles atr = 1A.Uandr = 5A.U (ρs = 2g/cm3).

by a factor(R/ld)2 depending on the size of the particles. UsingEqs. (73)(74), this amplification can be written explicitly(σvortdσd

)conc.

=3

16α

√γ + 1

Ωξ


(σvortdσd

)conc.

=3

16α

√γ + 1

(ξ

Ω

)1/2(heavy particles)(80)

The amplification is maximum forξ ∼ Ω and takes the valueAconc.max =

316α

√γ + 1 ' 150. As we have seen in Sect. 2.5, this

corresponds to particles of sizea = 30 cm and bulk densityρd = 2g/cm3 in the regions of the Earth and Jupiter. This en-hancement is sufficient to satisfy the instability criterion (78)5. For microscopic particles, on the contrary, there is no den-sity enhancement. In that case,ld ∼ 103R (in the region ofthe planets) and the particles rapidly diffuse away from the vor-tices (see Sect. 4.5). Gravitational instability will be possibleprovided that:

ξ

Ω<

316α

√γ + 1

1Ac

' 1.53 (81)

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 musthave grown up to some centimeters to trigger the gravitationalinstability. Therefore, sticking processes are needed to reach thisrange of sizes (recall that this was claimed to be not necessaryin the initial Safronov-Goldreich-Ward scenario).

In conclusion, by allowing a local enhancement of the par-ticle surface density, the vortices can favour the formation of

5 We can argue that whenld is comparable to the sub-disk thicknessHd, the instability criterion derived in the case of a thin disk is notapplicable anymore. However, according to Jeans criterion, a volumeelement of sizeHd and massMd is unstable if the velocity dispersionof the particlesc2d is less than their potential energy

GMdHd

. Sincecd ∼HdΩ andMd ∼ σdH2d , this condition returns the criterion (78).

σ dvor

t /σ d

) cap

t(

0

100

200

300

400

500

600

0 20 40 60 80 100a

1 A.U.

5 A.U.

Fig. 10. Enhancement of the dust surface density inside the vortices(due to capture) as a function of the size of the particles atr = 1A.Uandr = 5A.U (ρs = 2g/cm3).

planetesimals by gravitational instability. This rehabilitates theSafronov-Goldreich-Ward theory at certain preferred locationsof the disk (i.e. inside the vortices) and for sufficiently large(decimetric) particles. A sufficient enhancement is achievedsimply by the horizontalconcentrationof the dust layer in thevortex (a process similar to the vertical sedimentation). How-ever, this mechanism alone is not sufficient to produce enoughplanetesimals to form the planets. The vortices must alsocap-ture the surrounding mass. This is another source for densityenhancement. Returning to Eq. (38), we find that the averagesurface density of the particles collected by a vortex during itslifetime is( 〈σvortd 〉

σd

)capt.

=32π

(Ωtlife)f2(ξ) (82)

with Ωtlife ∼ 1122 (see Sect. 2.5). The maximum amplifica-tion, reached by particles withξ ∼ Ω, isAcapt.max = 32π (Ωtlife) '536 a little bit larger than the previous value (recall howeverthattlife is not known precisely). Gravitational instability willbe possible for particles whose friction parameter satisfies

ξ

Ω<

32π

(Ωtlife)1Ac

' 5.4 (83)

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

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

σvortdσd

=Mlifeσdπl2d

=(σvortdσd

)conc.

( 〈σvortd 〉σd

)capt.

(84)

with a maximum valueAmax = Aconc.max Acapt.max ∼ 105. The range

of particles which can collapse is enlarged:

ξ

Ω<

(9

32π

√γ + 1α

(Ωtlife)1Ac

)1/2' 28.6 (85)

but, even in this optimistic situation, the particles must havereached relatively large sizes to trigger the gravitational insta-bility (see Table 1). Of course, if the vortex lifetime is increased,


smaller particles have the possibility to collapse since the vor-tex captures more mass. In fact, this is not completely correctbecause the previous results assume that the escape of particlesdue to turbulent fluctuations can be neglected. This is not alwaysthe case (in particular for small particles) and this problem isnow considered in detail.

4. The rate of escape

4.1. Formulation of the problem

The diffusion Eq. (60) is similar, in structure, with the Kramers-Chandrasekhar equation:

∂f

∂t=

∂

∂v

(D∂f

∂v+ ξfv

)(86)

introduced in the case of colloidal suspensions and in stellar dy-namics (Kramers 1940, Chandrasekhar 1943a,b). In this equa-tion, f(v, t) governs the velocity distribution of the particlesin the system. The first term in the r.h.s is a pure diffusion andthe second term is adynamical friction. These terms model theencounters between stars or the collisions between the colloidalparticles and the fluid molecules. Comparing Eqs. (60) and (86),we see that the position in (60) plays the role of the velocityin (86) and the capture timetcapt the role of the friction timeξ−1. In particular the friction force and the drift term arelinearin v andr respectively. Eq. (86) was used by Chandrasekhar(1943b) to study the evaporation of stars in globular clusters(this is similar to the Kramers problem for the escape of col-loidal suspensions over a potential barrier). Due to collisions,some stars may acquire very high energies and escape from thesystem (being ultimately captured by the gravity of nearby ob-jects). Similarly, in our situation, turbulent fluctuations allowsome dust particles to diffuse toward higher and higher radiiand finally leave the vortex (being eventually transported awayby the local Keplerian shear). In each case, the friction force orthe drift acts against the diffusion and can reduce significantlythe escape process.

We try now to evaluate the rate of particles that leave thevortex on account of turbulent fluctuations. To that purpose,we formulate the problem in terms of the density probabilityW (r, t) = W (|r|, t) that a particle located initially in the annu-lus between|r| = r0 andr0 + dr0 will be found in the surfaceelement aroundr at timet. According to Eq. (60), the time evo-lution of the probabilityW (r, t) is given by

∂W

∂t= ∇

(D∇W + W

tcaptr)

(87)

with initial condition

W (r, t) =δ(|r| − r0)

2πr0as t → 0 (88)

whereδ stands for Dirac’sδ-function. We assume that when theparticle reaches the vortex boundary at|r| = R, it is immedi-ately transported away by the Keplerian shear. In other words,we adopt the boundary condition:

W (r, t) = 0 for |r| = R for all t > 0 (89)

We call

J = −(D∇W + W

tcaptr)

(90)

the current of probability, i.e.Jdln̂ gives the probability thata particle crosses an element of lengthdl betweent and t +dt (n̂ is a unit vector normal to the element of length underconsideration).

We first introduce the probabilityp(r0, t)dt that a particlelocated initially in the annulus between|r| = r0 andr0 + dr0reaches for the first time the vortex boundary betweent andt+dt. According to what was just said concerning the interpretationof (90), we have:

p(r0, t) =∮

|r|=RJn̂dl = −

(2πDr

∂W

∂r

)r=R

(91)

The total probabilityQ(r0, t) that the particle has reached thevortex boundary between0 andt is

Q(r0, t) =∫ t

0p(r0, t′)dt′ (92)

Finally, we averageQ(r0, t) over an appropriate range of initialpositions in order to get the expectationQ(t) that the particlehas left the vortex at timet. We have

Q(t) =∫Q(r0, t)µ(r0)2πr0dr0 (93)

whereµ(r0) = µ(|r0|) governs the initial probability distribu-tion of the particles in the vortex. In terms of the functionQ,the rate of escape of the particles can be written

1tesc

=1

(1 −Q)dQ

dt(94)

As mentioned already, this problem is similar to the diffu-sion of colloidal suspensions over a potential barrier or to theevaporation of stars in globular clusters. As will soon becomeapparent, it reduces to solving a pseudo Schrödinger equationfor a quantum oscillator in a “box”. In this analogy, the rate ofescape appears to be related to the fundamental eigenvalue of thequantum problem. An explicit expression for the rate of escapecan be obtained in two limits: whenξ → 0 or ξ → ∞, the driftterm can be ignored and the Fokker-Planck equation reduces toa pure diffusion equation (Sect. 4.2). On the other hand, whenξ ' Ω, the drift term is dominant and a perturbation approachinspired by the work of Sommerfeld and Chandrasekhar canbe implemented to determine the ground state of the artificiallylimited quantum oscillator (Sects. 4.3 and 4.4).

4.2. The rate of escape when the drift term is ignored

When the drift term can be ignored (i.e. for very light or veryheavy particles, see inequalities (75)(76)), the Fokker-Planckequation (87) reduces to a pure diffusion equation

∂W

∂t= D∆W (95)


which has to be solved in a circular domain with boundary con-ditions (88) and (89). The solution of this classical problem is

W (r, t) =1

πR2

+∞∑n=1

1J21 (α0n)

J0

(α0n

r0R

)

×J0(α0n

r

R

)e−

Dα20nR2

t (96)

whereJm is the Bessel function of orderm and theα0n’s denotethe roots of the Bessel functionJ0. The probabilityp(r0, t)that a particle with an initial positionr0 has reached the vortexboundary betweent andt+ dt is [see Eq. (91)]:

p(r0, t) =2DR2

+∞∑n=1

α0nJ1(α0n)

J0

(α0n

r0R

)e−

Dα20nR2

t (97)

and the total probability that the particle has escaped during theinterval(0, t) is [see Eq. (92)]:

Q(r0, t) = 2+∞∑n=1

1α0nJ1(α0n)

J0

(α0n

r0R

)

×(

1 − e−Dα20n

R2t

)(98)

Averaging the foregoing expression over allr0’s in the range[0, R] with equal probabilityµ(r0) = 1/πR2 (this correspondsto an initially homogeneous distribution of particles in the vor-tex), we obtain

Q(t) = 4+∞∑n=1

1α20n

(1 − e−

Dα20nR2

t

)(99)

To sufficient accuracy, we can keep only the first term in theseries. The expectation that the particle has left the vortex attime t is therefore:

Q(t) ' 4α201

(1 − e−

Dα201R2

t

)(100)

Sinceα01 ' 2.40482..., this term represents∼ 70% of the valueof the series (99) and the approximation (100) is reasonable.

In conclusion, when the drift term is ignored, we find thatthe escape time is

tesc =R2

Dα201(101)

It corresponds, typically, to the time needed by the particles todiffuse over a distance∼ R, the vortex size. For light particles,using (71), we obtain explicitly

Ωtesc =√γ + 1α201α


and for heavy particles

Ωtesc =√γ + 1α201α

(Ωξ


We shall come back to these expressions in Sect. 4.5.

4.3. The effective Schrödinger equation

We now return to the general problem for the rate of escape whenproper allowance is made for the drift. We find it convenient tointroduce the notations

τ =t

tcaptand ρρρ =

1√2Dtcapt

r (104)

or, in words, the time is measured in terms of the capture timeand the distances are normalized by the concentration length(64). We let also:

w = 2DtcaptW (105)

and

ρ∞ =1√

2DtcaptR (106)

In terms of these new variables, the problem (87)(88) and (89)takes the form:

∂w

∂τ=

12∆w + ∇(wρρρ) (107)

w(ρρρ, τ) =δ(ρ− ρ0)

2πρ0as τ → 0 (108)

w(ρρρ, τ) = 0 for ρ = ρ∞ for all τ > 0 (109)

With the change of variables

w = ψe−ρ2/2 (110)

we can transform the Fokker-Planck Eq. (107) in a Schrödingerequation (with imaginary time) for a quantum oscillator:

∂ψ

∂τ=

12∆ψ + (1 − 1

2ρ2)ψ (111)

However, contrary to the standard quantum problem, Eq. (111)has to be solved in a bounded domain of sizeρ∞ with the bound-ary condition (109). In other words, our problem consists in de-termining the characteristic functions of a quantum oscillator ina “box”.

First, we notice that a separation of the variables can beeffected by the substitution

ψ = φ(ρ)e−λτ (112)

whereλ is, for the moment, an unspecified constant. This trans-formation reduces the Schrödinger equation in a second orderordinary differential equation:

d2φ

dρ2+

1ρ

dφ

dρ+ (2 + 2λ− ρ2)φ = 0 (113)

Let{φn} be the solutions of this differential equation satisfyingthe boundary conditionφn(ρ∞) = 0 andλn the correspondingeigenvalues. The eigenfunctions form a complete set of orthog-onal functions for the scalar product

〈fg〉 =∫ ρ∞

0f(ρ)g(ρ)2πρdρ (114)


This system can be further normalized, i.e.〈φnφm〉 = δnm.Any functionf(ρ) satisfying the boundary condition (109) canbe expanded on this basis, and we have the formula:

f(ρ) =∑n

〈fφn〉φn (115)

In particular:

δ(ρ− ρ0) =∑n

2πρ0φn(ρ0)φn(ρ) (116)

The general solution of the problem (107) (109) can be ex-pressed in the form

w(ρ, τ) =∑n

Ane−λnτe−ρ

2/2φn(ρ) (117)

where the coefficientsAn are determined by the initial condition(108), using the expansion (116) for theδ-function. We obtain:

w(ρ, τ) = e−(ρ2−ρ20)/2

∑n

e−λnτφn(ρ)φn(ρ0) (118)

Using the foregoing solution forw, the probability that a particleinitially located atρ0 leaves the vortex betweenτ andτ + dτ is[see Eq. (91)]:

p(ρ0, τ) = −πρ∞e−(ρ2∞−ρ20)/2

×∑n

e−λnτdφndρ

(ρ∞)φn(ρ0) (119)

The probabilityQ(ρ0, τ) that the particle has left the vortex attime τ is therefore [see Eq. (92)]:

Q(ρ0, τ) = −πρ∞e−(ρ2∞−ρ20)/2

×∑n

1 − e−λnτλn

dφndρ

(ρ∞)φn(ρ0) (120)

Finally, to obtainQ(τ), we have to take the average of the fore-going expression over the relevant range ofρ0. To make thesolution explicit, it remains to determine the eigenfunctionsφnand eigenvaluesλn of the quantum oscillator.

4.4. The ground state of the quantum oscillator

The dependence of the eigenvaluesλn on the size of the “box”can be obtained from a procedure developed by Sommerfeld inhis studies of the Kepler problem and the problem of the rota-tor in the quantum theory with “artificial” boundary conditions(Sommerfeld & Welker 1938, Sommerfeld & Hartmann 1940).This was used later on by Chandrasekhar (1943b) to determinethe rate of escape of stars from globular clusters, from whichour study is inspired.

With the change of variables

x = ρ2 (121)

Eq. (113) becomes:

xd2φ

dx2+dφ

dx+

(12

+λ

2− x

4

)φ = 0 (122)

Another change of variables:

φ = fe−x/2 (123)

transforms Eq. (122) into Kummer’s equation

xd2f

dx2+ (1 − x) df

dx+λ

2f = 0 (124)

In the case of a “free” oscillator (ρ∞ → ∞), it is well-knownthat the eigenvalues areλn = 2n (with n = 0, 1, ...) and theeigenfunctions are proportional to the Laguerre polynomialsfn = AL

(0)n . In a bounded domain (ρ∞ < ∞), the eigenvalues

will be different but ifρ∞ is sufficiently large, we expect thatthe difference be small. Therefore, we expectλn ' 2n. Ac-cordingly, for values ofτ of the order of unity, or greater, thefirst term in the series (120) will provide ample accuracy. Thus

Q(ρ0, τ) ' −πρ∞e−(ρ2∞−ρ20)/2

× 1λ0

(1 − e−λ0τ )dφ0dρ

(ρ∞)φ0(ρ0) (125)

We have therefore to determine theground stateof our artifi-cially limited quantum oscillator. To that purpose, we expandfin a series:

f =∑n

anxn (126)

and substitute this expansion in Eq. (124). Identifying term byterm, we obtain the recursion formula:

an+1 =n− λ2

(n+ 1)2an (127)

which is easily reduced to

an =(n− 1 − λ2 )(n− 2 − λ2 )...(−λ2 )

(n!)2a0 (128)

We know already that the eigenvalueλ0 of our artificial quantumoscillator is very small. Keeping only terms of orderλ0 in theproducts (128), we obtain:

an '(n− 1)(n− 2)...1(−λ02 )

(n!)2a0

= − (n− 1)!(n!)2

λ02a0 = − a0

n!nλ02

(129)

The ground state function can therefore be written

f0(x) = a0(1 − λ02 χ(x)) (130)

with

χ(x) =+∞∑n=1

xn

n!n= Ei(x) − lnx− γ (131)

where

Ei(x) = P∫ x

−∞

et

tdt (132)


χ(x)

0 2 4 6 8 100

200

400

600

800

1000

1200

1400

x

Fig. 11.The functionχ(x).

is the Exponential integral andγ = 0.577215... the Euler con-stant. The functionχ(x) is plotted on Fig. 11. The fundamentaleigenvalueλ0 is determined by the conditionf0(ρ2∞) = 0 yield-ing

λ0 =2

χ(ρ2∞)(133)

Returning to the original eigenfunctionφ0(ρ), we have estab-lished that

φ0(ρ) = a0(1 − λ02 χ(ρ2))e−ρ

2/2 (134)

wherea0 is a normalizing factor. The final result can thereforebe expressed in the form:

Q(τ) = A(1 − e−λ0τ ) (135)with

A = −πρ∞λ0

e−ρ2∞/2

dφ0dρ

(ρ∞)eρ20/2φ0(ρ0) (136)

In the expression for the amplitude, the bar indicates an averageover the relevant range ofρ0. In practice,A is close to unity butits precise value determines to which accuracy the approxima-tion consisting in keeping only the dominant term in the series(120) is justified. For sufficiently largeρ∞’s, this will alwaysbe the case, so we shall take

Q(τ) ' 1 − e−λ0τ (137)This expression is consistent with the physical conditionQ(τ) → 1 asτ → ∞.


LetM0 denote the total mass of particles present in the vortexat timet = 0. If we assume no renewal from the outside then,on account of turbulent fluctuations, some particles will escapethe vortex and, according to Eq. (137), the mass will decay like:

M(t) = M0e−λ0t/tcapt (138)

The “evaporation” will take place on a typical time

tesc =1λ0tcapt (139)

whereλ0 is given by Eq. (133). Note that bothλ0 and tcaptdepend on the friction parameterξ of the particles. For verylight or very heavy particles,ρ∞ → 0 andλ0 ' 2/ρ2∞. Withthe definition (106) we find:

tesc =R2

4D(140)

in good agreement with the result (101) obtained when the driftterm is ignored. This shows that Eq. (139) can be used with goodaccuracy for all types of particles. Substituting Eqs. (73)(74) and(33)(34) in Eq. (139), we obtain explicitly:

Ωtesc =43ξ

Ωχ

(3√γ + 1

16αΩξ

)(light particles) (141)

Ωtesc =43

Ωξχ

(3√γ + 1

16α

(ξ

Ω

)1/2)(heavy particles) (142)

whereχ is the function defined by Eq. (131). Whenξ → ∞,Eq. (141) tends to

Ωtesc =√γ + 14α

' 200 (143)Very light particles are not concentrated in the vortices and theydiffuse away after only∼ 30 rotation periods. By contrast, foroptimal particles withξ ∼ Ω, the concentration reduces con-siderably the evaporation and we find

Ωtesc =43ξ

Ωχ

(3√γ + 1

16α

)∼ 1064 (144)

The particles are so much concentrated in the vortices (ld ∼0.1R) that the turbulent fluctuations are not sufficient to allowthem to reach the edge of the vortex. Therefore, their escape iscompletely negligible. Finally, forξ → 0, we have

Ωtesc =√γ + 14α

(Ωξ

)1/2(145)

For very heavy particles, the escape time goes to infinity becausethe particles are not coupled to the gas and therefore not affectedby diffusion. Recall, however, that our study is not well suitedfor very heavy particles. As discussed in Sect. 2.3, these particlesare more likely to cross the vortices without being captured. ThecurveΩtesc versusξ/Ω is potted on Fig. 12 (see also Fig. 13for the dependence ofΩtesc on the size of the particles). Theasymptotic regimes (143) and (145) are obtained for particleswith ξ/Ω > 153 and ξ/Ω < 4 10−5 respectively. For theseparticles,ld > R (see inequalities (75) and (76)), so there isno concentration. Therefore, the asymptotic limits (143) and(145) agree with the results (102) and (103) obtained when thedrift term is ignored. When the drift becomes efficient, i.e. forξ ∼ Ω, the escape time increases extremely rapidly and theparticles remain emprisoned in the vortices.

These results assume that there is no renewal of the particlesin the vortices. If we now take into account a capture processlike in Sect. 2.4, we are led to consider the balance equation:

dM

dt=

32σdΩR2f2(ξ) − 1

tescM (146)


tΩ

esc

0 50 100 150 200

500

1000

1500

2000

2500

3000

tΩ life

ξ/Ω

Fig. 12. Escape time of the particles as a function of their frictionparameter.

Ωt e

sc

-6 -4 -2 0 2

500

1000

1500

2000

2500

3000

log(a)

Ωt

1 A.U.

5 A.U.

life

Fig. 13.Escape time of the particles as a function of their size atr =1A.U andr = 5A.U (ρs = 2g/cm3).

The first term in the right hand side accounts for a flux of parti-cles inside the vortex (see Eq. (37)) and the second term for anexponential decay of the particle number due to “evaporation”(see Eq. (138)).

For particles which can experience gravitational collapse(see inequalities (81)(83) and (85)), the escape time is muchlonger than the vortex lifetime (see Fig. 12). In that case, evap-oration can be neglected and Eq. (146) reduces to Eq. (37). Themaximum mass captured by the vortex is determined by its life-time and the results of Sects. 2.5 and 3.4 are unchanged.

Consider, however, the idealistic situation whentlife → ∞.For sufficienlty large times (t > tesc), the vortex will achievea stationary distribution of dust particles obtained by settingdM/dt = 0 in Eq. (146). This gives a maximum mass

Mmax =32σd(Ωtesc)R2f2(ξ) (147)

which is now limited by the evaporation time (compare Eq. (147)with Eq. (38)). The density enhancement in the vortices (takinginto account the concentration effect) is(σvortdσd

)∞

=32π

(Ωtesc)f2(ξ)(R

ld

)2(148)

as represented on Fig. 14. We find that even if the vortex hadan infinite lifetime, particles with friction parameterξ/Ω > 30cannot trigger the gravitational instability. This result implies(see Fig. 5) that subcentimetric particles cannot form planetes-imals even in the most optimistic case. Sticking processes arenecessary to produce larger particles.

σ dvor

t /σ d

)(

8

0 20 40 60 80

25

50

75

100

125

150

175

200

ξ/Ω

Fig. 14. Enhancement of the dust surface density in the vortices as afunction of the friction parameter when evaporation due to small scaleturbulence is taken into account (tlife → ∞).

5. Conclusion

This article follows the numerical works of Barge & Sommeria(1995), Tangaet al. (1996), Bracco et al. (1999) and Godon& Livio (1999c) concerning the interaction between dust parti-cles and large-scale vortices in the solar nebula. The originalityof our study is to start from an exact and realistic solution ofthe fluid equations and to provide analytical results and rele-vant parameters for the trapping process. Moreover, we haveconsidered for the first time the effect of small-scale turbulentfluctuations on the motion of the particles and determined anexplicit expression for the escape time by solving a problem ofquantum mechanics. These theoretical results have been formu-lated in the context of Keplerian disks and plan

astronomy and trapping of dust by coherent vortices in the...

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