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Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

Abstract– In the absence of global turbulence, solid particles in the solar nebula tend to settle toward the midplane, forming a layer with enhanced solids/gas ratio. Shear relative to the surrounding pressure-supported gas generates turbulence within the layer, inhibiting further settling and preventing gravitational instability. Turbulence and size-dependent drift velocities cause collisions between particles. Relative velocities between small grains and meter-sized bodies are typically about 50 m s−1 for isolated particles; however, in a dense particle layer, collective effects alter the motion of the gas near the midplane. Here, we develop a numerical model for the coupled motions of gas and particles of arbitrary size, based on the assumption that turbulent viscosity transfers momentum on the scale of the Ekman length. The vertical distribution of particles is determined by a balance between settling and turbulent diffusion. Self-consistent distributions of density, turbulent velocities, and radial fluxes of gas and particles of different sizes are determined. Collective effects generate turbulence that increases relative velocities between small particles, but reduce velocities between small grains and bodies of decimeter size or larger by bringing the layer’s motion closer to Keplerian. This effect may alleviate the “meter-size barrier” to collisional growth of planetesimals.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

The initial stage of planetesimal formation involved coagulation of microscopic dust grains in the solar nebula. For very small particles, collisions occurred at low velocities due to thermal motions, and surface forces were sufficient to ensure sticking. However, the further evolution of grain aggregates and their growth to macroscopic objects is still controversial. Many mechanisms have been proposed for the production of planetesimals (Dominik et al. 2007; Blum and Wurm 2008 and references therein). The viability of any given mechanism depends on the sizes of bodies that can be produced by coagulation in the nebular environment. This in turn depends on the stickiness and impact strength of grain aggregates, and the properties of the nebula, e.g., the strength of turbulence, if present. Some suggested mechanisms rely on large scale turbulence to form planetesimals by concentrating particles of selected sizes, in the range of millimeters (Cuzzi et al. 2001) or meters (Johansen et al. 2006). The source of global turbulence with the required properties is not clear. Such models will not be discussed here; the present paper addresses phenomena associated with dynamics of particles and gas in a relatively quiescent nebula or a midplane “dead zone” where magneto-rotational instability is ineffective (Gammie 1996).

Even in a laminar nebula, the dynamical interaction of gas and particles can be surprisingly complex. Thermal pressure in the gas partially supports it against solar gravity, causing it to rotate at slightly less than the Kepler velocity, VK. The deviation from VK is

  • image(1)

where P is the gas pressure, R the heliocentric distance, ρg the gas density, and ΩK the Kepler frequency. Solid particles lack this pressure support, and in the absence of gas would pursue Keplerian orbits. The mismatch implies that solid bodies cannot be at rest with respect to the gas. They are subject to drag forces that alter their orbits, generally causing them to decay. The rate of inward drift depends on a particle’s size, or more precisely, on its characteristic time scale of response to the drag force. The behavior of individual particles was first described by Whipple (1972), and more generally by Adachi et al. (1976) and Weidenschilling (1977). A small particle with large area/mass ratio is constrained to move with the angular velocity of the gas; the residual component of solar gravity causes inward drift at a terminal velocity that increases with size. A very large body is decoupled from the gas and moves at the Kepler velocity. The resulting “headwind” causes it to spiral inward at a rate that decreases with size, while the transverse velocity relative to the gas asymptotically approaches ΔV. In the transition between these regimes, the radial velocity reaches a peak value that is equal to ΔV, attained by a body with response time equal to the Kepler time (1/ΩK). For most low-mass models of the nebula, this response time corresponds to a body of about 1 m in size; the gas density and ΩK vary together in such a manner that the critical size is not sensitive to heliocentric distance. The peak speed depends on the nebular parameters (Weidenschilling 1977), but is generally of order 50 m s−1 for a typical low-mass nebular model.

The variation of the drift rate with size means that any two particles of different sizes have some drag-induced relative velocity. In any realistic ensemble of non-identical particles, this effect leads to collisions, even in the absence of turbulence. If the particles are distributed sparsely, so that their presence does not affect the motion of the surrounding gas, then their impact velocity is a function only of their sizes. Contour plots of relative velocities versus size for different nebular models have been presented by Weidenschilling and Cuzzi (1993), Blum and Wurm (2008), and Weidling et al. (2009). These show common features of relative velocity between unequal bodies increasing with the size of the larger, up to a “ridge” with a maximum near meter size, and a narrow “valley” of low velocity for nearly equal sizes. The height of the ridge is about ΔV, i.e., tens of meters per second. If both bodies are much larger than meter size, their relative velocities are lower because radial velocities decrease while their transverse velocities approach the limiting value; but small particles still impact large ones at nearly the highest speed. This behavior has inspired a great deal of theoretical and experimental study of impacts of individual particles and aggregates over this range of velocities (Blum and Wurm 2000; Dominik et al. 2007; Langkowski et al. 2008 and references therein). Surface forces are effective for coagulation of small particles at low impact speeds, but most experimental results suggest that collisions of aggregates at speeds of tens of meters per second are likely to result in erosion or disruption of the target, rather than net mass gain. The increasing relative velocities as larger bodies approach the ridge suggests that growth would stall at sizes of centimeters to decimeters. In addition, turbulence adds a velocity component between equal-sized bodies, filling in the valley. Only at much larger sizes, of order of a kilometer, are relative velocities typically smaller than escape velocities, so that gravitational binding becomes effective. This behavior gives rise to the so-called “meter size barrier,” which has been invoked as an argument against collisional coagulation as a mechanism of planetesimal formation (e.g., Youdin 2004). Another problem with bodies of meter size is the rapid decay of their orbits. A typical radial velocity of 50 m s−1 corresponds to one AU per century, which could result in loss of potential planetesimals into the Sun, unless they can decouple from the gas by growing to much larger sizes on a time scale shorter than that of orbital decay.

These arguments are based on the drag-induced velocities of isolated particles. Settling increases the spatial density of the midplane particle layer, and may render it greater than the density of the nebular gas. If the particles are coupled to the gas by drag forces, they can cause the gas within the layer to move with them and approach Keplerian rotation despite its pressure support. This effect could be expected to decrease the drag-induced relative velocities between particles of different sizes, i.e., to lower the height of the “ridge.” On the other hand, the shear flow can generate turbulence, which also contributes to relative velocities. The goal of this paper is to model these processes and determine their effects on the collisional evolution of particles in the solar nebula, and implications for planetesimal formation. Specifically, we develop a numerical model for the density and vertical structure of a layer of particles of arbitrary size, which is determined by a balance between settling and shear-driven turbulence. The absolute velocities of particles of different sizes, and their relative impact velocities, are evaluated.

Collective Particle–Gas Interactions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

An analytic solution for the coupled velocities of particles and gas in the solar nebula was derived by Nakagawa et al. (1986). Radial and transverse velocity components of both the particles and the gas are expressed as functions of a single parameter that depends on the particle size and the solids/gas mass ratio. The inward flux of particles is balanced by an equal and opposite outward flux of gas. The particles also drag the gas, causing its azimuthal velocity to increase. As the mass loading is increased, e.g., by settling of particles toward the midplane, the gas approaches Keplerian motion. Such a result would eliminate the meter-size barrier, and would also allow planetesimals to form by gravitational instability. However, the Nakagawa et al. model assumes that momentum is only exchanged locally between the particles and gas by drag forces. There is no transport by turbulent viscosity; hence, this result is valid only for a purely laminar flow. Actually, turbulence is inevitable in or near a dense particle layer, even if the nebula as a whole is laminar. It was recognized by Goldreich and Ward (1973) that if a particle layer was dense enough to rotate at the Kepler velocity, the shear flow in the boundary layer between it and the pressure-supported gas would be unstable, and would become turbulent. Weidenschilling (1980) showed by scaling arguments that the shear instability would include the particle layer itself, as well as the boundary layer. Numerical modeling of two-phase fluid dynamics by Cuzzi et al. (1993) and recent direct integrations of coupled systems of particles and gas by Johansen et al. (2006) confirmed that such shear flows are subject to Kelvin-Helmholtz instability that generates turbulence. Thus, the Nakagawa et al. solution is limited to the early stage of particle settling, or must be modified in order to apply it to dense particle layers.

A semi-analytic model that explicitly includes turbulence was developed by Sekiya (1998), who considered a layer of small particles that are strongly coupled to the gas due to their large area/mass ratio. He assumed that the onset of shear-driven turbulence would cause diffusion, inhibiting further settling. The particle layer would attain an equilibrium density profile such that it would be marginally unstable to turbulence at all distances from the midplane. The principal criterion for this instability is the dimensionless Richardson number. Consider a shear flow in which both the density ρ and horizontal velocity V vary with vertical distance Z. If two fluid elements at different values of Z are interchanged, work done against the gravity field must be extracted from the kinetic energy of the flow. If the change in kinetic energy exceeds that of the potential energy, the excess is able to drive turbulence, and the flow is unstable. The Richardson number is usually defined as

  • image(2)

where gz is the Z-component of the gravity field and ρ is the density. Usually it is assumed that ρ is the sum of the gas density ρg and the spatial density of particles ρp, with the density gradient due to vertical variation in particle concentration. That assumption is valid for particles small enough to be strongly coupled to the gas, but needs to be modified if they are large. The flow is unstable if Ri < 0.25. Sekiya (1998) assumed that V corresponded to the gas velocity due to mass loading derived by Nakagawa et al. (1986) in the limit of small particles; this is a good assumption as long as the turbulence is weak and the particles smaller than centimeter-sized. He derived vertical density profiles such that the Richardson number had the critical value at all Z, including the self-gravity of the particle layer in the calculation of gZ. This approach is valid for perfect coupling of particles and gas, but breaks down for larger particles, and cannot treat the decimeter- or meter-sized bodies that are of interest for collisional modeling.

Another model with turbulence is that of Youdin and Chiang (2004), who considered radial transport of particles and gas due to turbulent viscosity. They pointed out that a vertical gradient in spatial density or mass loading of particles would cause the denser region closer to the midplane to rotate faster than the less dense region at larger Z. This velocity gradient with turbulent stress would transfer angular momentum upward, causing the gas to flow inward near the midplane and outward at higher levels. The turbulent stress due to the velocity gradient is

  • image(3)

where inline imaget is the turbulent viscosity and VΦ the azimuthal velocity of the gas. The radial velocity induced by this stress is proportional to the gradient of the turbulent stress in the Z direction:

  • image(4)

Conservation of angular momentum requires the net radial mass flux to be zero, but the greater density in the inflowing region results in a net inflow of solid matter, balanced by net outflow of gas. As Sekiya (1998) did, Youdin and Chiang (2004) assumed that turbulent diffusivity was that value needed to balance settling with turbulent diffusion. They also assumed that the particles were small enough to be bound to the gas, and neglected any radial motion of particles relative to it. Consequently, their model did not include any direct exchange of momentum between the particles and the gas. For a layer of larger particles the vertical transport of angular momentum due to shear can still be significant, and must be added to the direct exchange of momentum between particles and gas due to drag. Such a model incorporating both phenomena is described below.

Ideally, one would wish to calculate the full spectrum of particle-gas interactions directly, with no approximations. Such an effort would include the actions of gas drag on the particles, the back-reaction of particles on the gas, generation and transport of turbulence, and direct gravitational attraction between particles of all sizes. This is not yet feasible, particularly for systems with multiple sizes of particles. Most analytic models have assumed perfect particle-gas coupling, i.e., the limit of very small particles. The computational fluid dynamical model of Cuzzi et al. (1993) treated larger bodies, but considered only to a single particle size for each case, with parameterizations for turbulence and gravitational effects. Champney et al. (1995) used the same model to treat a bimodal population with equal masses of particles 20 and 120 cm in diameter, but did not consider any other size. There have been a few attempts at direct many-particle integrations with gravity and drag forces. Tanga et al. (2004) included gravitational forces between particles subjected to gas drag, but omitted turbulence and effects of particles on the gas flow. Johansen et al. (2006) confirmed the onset of Kelvin-Helmholtz instability in 2-D integrations with particle-gas coupling, but without self-gravity. All such N-body simulations to date have involved only bodies of uniform size; it is not yet numerically feasible to treat systems of particles of varied sizes with very different response time scales.

Basic Concepts

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

In this section, we describe the assumptions and methodology used in a numerical model for the behavior of systems of particles of arbitrary size interacting with the nebular gas. Our goal is to construct self-consistent distributions of particle concentrations and shear-generated turbulence as functions of distance from the midplane, and to evaluate the consequent relative velocities between particles of different sizes. The code is an improved and more general version of the one described by Weidenschilling (2006), hereafter Paper 1. Here, we review some basic concepts used in the model.

Ekman Length

If one considers a disk rotating at angular rate ΩK within a fluid, scaling arguments give a length scale for the thickness of the turbulent boundary layer, called the Ekman length (Goldreich and Ward 1973; Cuzzi et al. 1993), given by the expression

  • image(5)

In the right-hand expression, we take the turbulent viscosity νt to be (ΔV)2/ΩKRe*, where Re* is a critical Reynolds number (cf. equation 21 of Cuzzi et al. 1993); a value of Re* = 100 is assumed for the simulations performed in this paper. Here, ΔV is the actual velocity difference between the particle-rich midplane and the particle-free pressure-supported gas far from the midplane (this may be less than the deviation of the gas from the Kepler velocity, as the midplane gas does not attain Keplerian rotation even at high particle mass loading). The Ekman length is a characteristic scale for momentum transport in the boundary layer. In the numerical model, we assume that basic properties of the gas flow, such as the radial and azimuthal velocity, are smoothed out by mixing and momentum transfer on this scale.

Response Time and Stokes Number

The time scale for a particle to respond to the force of gas drag is te = mV/FD, where m is its mass, V its velocity relative to the gas, and FD the drag force. Particles smaller than the mean free path of a gas molecule are subject to the Epstein drag law: for a spherical particle of diameter d and material density ρs,

  • image(6)

where c is the mean thermal velocity of the gas molecules (nearly equal to the sound velocity). In typical solar nebula models, this drag law applies to bodies smaller than about decimeter size. Larger bodies are in the continuum flow regime, for which the Stokes drag law is appropriate:

  • image(7)

where μ is the molecular viscosity of the gas.

The response of a particle to turbulence depends on the Stokes number, defined as St = ω te, where ω is the turnover frequency of the largest eddies. The turbulence is assumed to have a Kolmogorov spectrum from the largest eddies down to an “inner scale” at which molecular viscosity damps the smallest eddies. A particle in turbulence with eddy velocity Vturb responds by developing random velocity fluctuations of magnitude

  • image(8)

(Völk et al. 1980; Weidenschilling 1984; Cuzzi and Weidenschilling 2006). It is often assumed that ω in the definition of St is equal to ΩK, the Kepler frequency. According to Sekiya (1998), this is the case at the onset of weak turbulence, where the eddy frequency is imposed by the rotation of the system. However, Cuzzi et al. (1993) argue that in strong turbulence produced by shear in a rotating system, ω >> ΩK. The Rossby number Ro relates the eddy frequency in the boundary layer to the rotation frequency of the system, so that ω = 2 RoΩK. As described below, we interpolate between these values as a function of the Richardson number, which determines the strength of the turbulence.

Richardson Number and Turbulence

The Richardson number was defined as seen in the section Collective Particle–Gas Interactions section, above. This definition is straightforward if the flow involves a single phase with variable density, e.g., if the particles are strongly coupled to the gas so that they move with it. For less than perfect coupling, i.e., large particles, the particles have less of a stabilizing effect on the gas. In that case, the density in Equation 2 should not be simply ρ = ρg + ρp; instead we use ρ ρg + ρp/(1 + St) to account for the lower diffusivity of the particles. If the particles have a range of sizes, then the effective density gradient may include a variation in the size distribution (and effective value of St) with Z.

To compute the velocity of the shear-driven turbulence, we assume that Vturb at any value of Z is a function of the total velocity difference between the local gas and that of the pressure-supported gas in the particle-free region far from the nebular midplane, with an additional dependence on the Richardson number:

  • image(9)

Here, ΔV0 is the deviation from Keplerian rotation of particle-free gas, and ΔV(Z) is the deviation for particle-laden gas at elevation Z. The radial velocity of the gas, Vrg, is negligible at large Z, but it can be significant in or near the particle layer, and contributes to the shear that drives the turbulence. As described in Paper 1, we take the dependence on Richardson number to be

  • image(10)

for Ri < 0.25. This has the desired properties that the turbulent viscosity matches the value in the simulations of Cuzzi et al. (1993) in the limit of fully developed turbulence as Ri approaches zero, and goes to zero at the critical value Ri = 0.25. At larger values of Ri, F(Ri) = 0; i.e., the flow is stable.

The eddy time scale is also assumed to vary with the strength of the turbulence. In order to match the arguments of Sekiya (1998) and Cuzzi et al. (1993) for the eddy time scales in weak and strong turbulence, we adopt the relation (cf. Paper 1)

  • image(11)

which gives ω ΩK at the onset of turbulence when Ri = 0.25, increasing to ω[RIGHTWARDS ARROW]Ro ΩK as Ri [RIGHTWARDS ARROW] 0. In these simulations we assume the Rossby number Ro = 40. For fully developed turbulence, the chosen parameters yield a turbulent eddy velocity of about 10% of the local velocity of the gas relative to its particle-free pressure-supported value. The values of Vturb and ω determined in this way are subject to averaging over the Ekman length, as described below.

Modeling Procedure

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

The algorithms of the modeling code were described in Paper 1, but will be briefly described in this section. The particle layer is modeled in a half-space, assuming symmetry about the nebular midplane. This region is divided into a series of levels (typically 100 or more) sufficient to resolve the vertical structure of the layer. At the start of a timestep, the spatial density of particles of a given size is known or specified. The radial and transverse velocity components of the particles and gas are computed in each level as a first approximation, using the laminar solution of Nakagawa et al. (1986). These are expressed as a function of a coupling parameter = (1 + ρp/ρg)/te. If there is more than a single particle size, their effects are assumed to be additive. The Ekman length is evaluated from the resulting value of ΔV in the midplane. As we expect the momentum in the gas flow to be transported on this scale, the radial and transverse velocity components in each level are recalculated as the running average of the values in all levels within a distance LE. If this averaging changes the midplane value of ΔV, the estimate of the Ekman length may be altered; in that case, the averaging is repeated with the new value until the solution converges.

The Richardson number can then be computed, and the turbulent velocity and eddy frequency ω are evaluated for each level (Equations 9–11). The turbulence is allowed to diffuse. The turbulent velocity generated at each level is assumed to decay exponentially on the scale of LE. If the value of Vturb propagated from another level exceeds the locally generated value, the larger velocity is substituted. The turbulent stress tensor, PZΦ, is then evaluated (Equation 3), and the corresponding changes in the gas radial velocity are added. Note that the net gas radial velocity at each level is the sum of two effects: the exchange of momentum with the particles, and the vertical flux of momentum due to turbulent viscosity and the gradient of azimuthal velocity with Z.

At this point, values for the radial and azimuthal velocity of the gas have been found for each level. The corresponding systematic particle velocities are then found by solving the equations of motion for drag-perturbed orbital evolution. These were derived by Weidenschilling (1977) for pressure-supported gas with no radial velocity, and used in Paper 1. In this paper we generalize the equations of motion for radial motion of the gas by explicitly including the difference between particle and gas radial velocities, rather than just the particle velocity, in the expression for the radial component of the drag force. As one might expect, an inward (outward) velocity component of the gas increases (decreases) the rate of orbital decay of the particles; the magnitude of the difference depends on the dimensionless parameter teΩK.

The sub-Keplerian rotation of the gas implies that the net radial mass flux of particles should be negative (i.e., inward). Conservation of momentum implies that the net flux of gas should be equal in magnitude and opposite in sign (outward). The equations of Nakagawa et al. guarantee this flux balance for the laminar case, but the non-localized gas flux resulting from the averaging procedure in the turbulent case does not ensure that the fluxes of particles and gas are equal. To ensure a physically plausible result, we evaluate the particle mass flux and the two components of the radial gas flux, due to vertical shear and reaction from the particles. Summing over all levels, the shear flux should sum to zero. This condition is met to a good approximation; the magnitude of the shear flux is much smaller than the reaction flux. We hold the turbulent shear-induced gas velocities constant, while adjusting the reaction component to ensure a physically plausible solution: If the magnitude of the total gas reaction flux is smaller (larger) than the particle flux, we multiply the reaction component of the gas radial velocity by a factor slightly larger (smaller) than unity, and solve again for the corresponding particle velocities. This procedure is iterated until the mass fluxes match to a desired precision, taken here to be within 5%.

At this point, we know the turbulent velocity and eddy time scale at each level; the diffusion coefficient depends on these quantities and on the Stokes number, as described in Paper 1. There is a mass flux in the Z-direction proportional to the concentration gradient, and a systematic settling velocity dependent on the particle size (or te). The computed settling velocity includes the self-gravity of the particle layer. The net particle flux, upward or downward, is evaluated, and the particles redistributed. The entire process is repeated for each timestep until the particle layer reaches a steady state, in which the spatial density of particles of each size remains constant.

The improvements since Paper 1 include the revision of the equations of motion to include explicitly the radial motion of the gas, and iteration to achieve balance of the mass fluxes of gas and particles. In the earlier paper, attempts to use local values for the Richardson number and turbulent stress at each level led to numerical instabilities, and weighted averaged values had to be used instead. Improvements to the algorithms (and correction of some coding errors) now allow local values to be used. These changes have altered some of the results for layers with particles of a single size; we present these before proceeding to multiple sizes.

Results for Single Size Particles

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

The nebular parameters assumed here are identical to those in Paper 1. All results shown are for a heliocentric distance R of 3 AU. The surface density of the gas at that distance is 833 g/cm2, varying as R−1; with the local temperature of 185 K, the resulting deviation from Keplerian rotation is 5220 cm s−1 for pressure-supported gas free from particles. The gas density ρg is 1.6 × 10−10 g/cm3 (10−6 bar pressure), with mean free path 12 cm. The solids/gas abundance ratio is taken to be 3.4 × 10−3, giving a surface density of solids 2.83 g/cm2. This is approximately 30% lower than the “total rock” abundance of Lodders (2003) for metal, silicates, and oxides, but is kept here for consistency with Paper 1. Particles are assumed to be spherical with density ρs = 2 g/cm3; this gives a response time te in seconds of 4.9 × 104 times their diameter in cm in the Epstein drag regime. As their behavior depends on te, the results can be applied easily to other assumed values of particle bulk density.

Figure 1 shows the vertical structure and density of a monodisperse layer for a range of particle sizes from 10−3 cm to 100 cm. For sizes less than approximately 1 cm, the maximum solids/gas ratio in the midplane is of order unity, with only weak dependence on particle size. The insensitivity to size in this regime is due to the variation of turbulent velocity with Richardson number; rather small changes in the density profile alter the rate of diffusion that counters settling. At sizes larger than a few centimeters the Stokes number exceeds unity, allowing particles to begin to decouple from the shear-generated turbulence and reach higher concentrations. The maximum solids/gas ratio in a layer of small particles in a steady state is of order 100 times solar abundance. Higher concentrations can only be attained as transient concentrations, e.g., within turbulent eddies (Cuzzi et al. 2001; Johansen et al. 2006), or by altering the nebular structure to increase the abundance of solids (Youdin and Chiang 2004) or decrease the amount of gas (Throop and Bally 2005). Solids/gas ratios much higher than unity are possible only for particles of decimeter size or larger.

image

Figure 1.  Computed equilibrium structures for layers of monodisperse particles of various diameters. The vertical density gradient in each case is maintained by a balance between settling and diffusion due to shear-induced turbulence. The assumed particle density is 2 g/cm3. Nebular parameters are for heliocentric distance of 3 AU, with gas density 1.6 × 10−10 g/cm3; the pressure-supported gas deviates from Keplerian motion by 52 m s−1.

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Figures 2–6 show the radial velocities of the gas and particles versus Z for particle sizes from 0.1 cm to 10 m (here radial velocities are defined as negative for inward motion, opposite to the convention used in Paper 1). Note that the scales vary; in particular, radial velocities are roughly proportional to the particle size for sizes less than a few decimeters. The behavior is similar for all cases in which the particles are strongly coupled to the gas (Stokes number << 1), i.e., at sizes smaller than a few centimeters (Figs. 2 and 3). The particle-gas coupling is never perfect, and radial drift of particles relative to the gas produces an outward flow of the gas comparable in magnitude to the radial motion due to turbulent stress. The separate contributions of the particle-gas momentum exchange and turbulent shear to the gas velocity are shown; their sum is the total radial velocity of the gas. The shear contribution decreases close to the midplane, where the velocity gradient goes to zero due to symmetry. The net flow of both particles and gas is inward near the midplane, and outward near the top of the particle layer. The net mass fluxes are inward for particles and outward for gas, but local fluxes can be in either direction depending on distance from the midplane. For m-sized or larger bodies (St > 1), particle-gas momentum exchange dominates the shear flow and the net gas velocity is outward, and solids move inward, at all values of Z.

image

Figure 2.  Radial velocities of particles and gas versus distance from the nebular midplane for an equilibrium layer of 0.1 cm particles. The separate contributions to the gas velocity due to momentum exchange with the particles (Reaction) and vertical transport by turbulent viscosity (Shear) are shown. The latter causes net outward velocity of the gas in the upper part of the layer; some particles are carried outward, although the net flux of solids is inward.

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image

Figure 3.  Structure of a layer of cm-sized bodies; compare with the case for millimeter-sized particles (Fig. 2). For all cases with particles closely coupled to the gas (St << 1), the structures are similar, but velocities of particles and gas are proportional to the particle size.

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image

Figure 4.  Structure of a layer of 10 cm bodies. The particles are not as strongly coupled to the gas.

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image

Figure 5.  For m-sized bodies, the Ekman length scale is comparable with the thickness of the particle layer. Gas velocities due to particle–gas reaction and turbulent shear are large and of opposite sign near the midplane.

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image

Figure 6.  For bodies of size 10 m, the net gas flow is outward, with radial velocity of about 15 m s−1 within the layer. The gas flow has little effect on the particle velocity due to the weak particle–gas coupling (St >> 1).

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In all of these cases, the Ekman length is of order 104 km. For small particles, this is much less than the thickness of the layer, and averaging gas motions over this distance has little effect. For meter-sized and larger bodies, LE is comparable with the thickness of the particle layer, and the localized particle-gas reaction of Nakagawa et al. (1986) breaks down due to turbulent transport of momentum in the Z-direction.

Figure 7 shows particle radial velocities versus size computed for an isolated particle in the nebula, and for particles moving collectively in a layer of equilibrium thickness. The collective results show both the velocity in the midplane at = 0 and the effective velocity averaged through the layer (total mass flux divided by surface density). The peak velocity is about 10 m s−1, i.e., the velocity of a meter-sized body in a layer of similar-sized bodies is reduced by about a factor of five from that of an isolated body. The principal cause of this reduction is the decrease in ΔV as the midplane gas is dragged by the particles to move at a rate nearer to the Kepler velocity. This effect is maximized at near meter size; smaller particles yield lower midplane densities for the layer, while larger ones are not as strongly coupled to the gas, and have less effect on the value of ΔV despite the higher densities in a layer of large bodies. The value of ΔV, which is approximately equal to the relative velocity between a small grain and a large body, approaches the value for particle-free gas when the dominant size is about 10 m.

image

Figure 7.  Velocities as functions of particle size. Solid line: radial velocity of an isolated particle. Dashed line: mean radial velocity (mass flux/surface density) for the entire layer. Dash-dot line: radial velocity of particles at = 0; decimeter-sized particles have higher collective velocities due to inflow of gas at the midplane, although the mean velocity is lower. Dash-dot-dot line: deviation of gas from Keplerian motion in the midplane (gas free value is 5400 cm s−1). Changes in slope near 50 cm are due to the transition from Epstein to Stokes drag law.

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We have compared our code with the results of Champney et al. (1995), who used different assumptions for nebular properties and particle density than those used in Paper 1 and the present paper. Using their values, our model yields good agreement with their values for the thickness and density of a layer of decimeter-sized particles, as well as the radial velocity of the gas a function of Z. The thickness of the layer in equilibrium varies roughly inversely with the assumed value of Re*, our best match to their result is found for Re* ≃ 40. The thickness of the layer and gas velocities are relatively insensitive to the assumed value of Ro.

Results for Multiple Particle Sizes

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

Any plausible natural system of particles in the solar nebula will have some distribution of sizes, resulting from a complex history of coagulation and disruption. It will depend on properties of the particles themselves, such as stickiness and impact strength, as well as nebular properties. We cannot sample all of the parameter space of multiple sizes, but will confine ourselves to a limited number of bimodal distributions, and a few mixtures of three or four particle sizes. First, we consider bimodal mixtures of particles of sizes 0.1 and 100 cm, in which the total surface density is kept constant. Figures 8–13 show results for varied proportions of these sizes, in which the mass distribution is either dominated by the small or large particles, or evenly divided. In each case, the meter-sized bodies are confined to a thin sublayer near the midplane; its thickness is not sensitive to their abundance, although its maximum density is roughly proportional to the surface density in meter-sized bodies.

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Figure 8.  Structure of a bimodal layer composed of 90% (by mass) millimeter-sized particles and 10% meter-sized bodies. The m-sized bodies form a thin, dense sublayer.

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Figure 9.  Radial velocities of particles and gas in the two-component layer, 90% millimeter-sized and 10% meter-sized bodies (see Fig. 8). Only the region near the meter-sized sublayer in the midplane is shown. At this scale, the radial velocity of the mm-sized bodies is indistinguishable from the total gas velocity; their net flux is outward due to entrainment in the high-velocity flow produced by the meter-sized bodies.

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Figure 10.  Structure of a layer composed of equal amounts (by mass) of millimeter- and meter-sized particles.

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Figure 11.  Radial velocities near the midplane sublayer for equal masses of millimeter- and meter-sized bodies. The radial velocity of the millimeter-sized particles is indistinguishable from the total gas velocity; their net mass flux is outward.

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Figure 12.  Structure of a layer with 10% millimeter-sized and 90% meter-sized particles.

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Figure 13.  Radial velocities near the midplane for 10% millimeter-sized and 90% meter-sized particles.

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The presence of the larger bodies does not have much effect on the vertical structure of the layer of small particles, other than that due to their decreasing abundance as more of the total mass is tied up in the meter-sized bodies. Turbulence produced in the dense sublayer of meter-sized bodies decays exponentially on the Ekman length scale, which is much less than the thickness of the layer of small particles. Thus, the presence of the large bodies does not have a significant effect on the scale height of small particles. Due to the reaction of the large bodies on the local gas, the net radial flux of gas in the dense sublayer is outward. Cuzzi et al. (1993) noted similar outward flows of gas in their simulations of layers of decimeter- and meter-sized particles, and speculated that small particles could be carried outward. We find that this effect can yield a net outward flux of small particles. The small particles entrained in the flow near the midplane are carried outward at speeds much greater than their inward drift rate in the rest of the layer. If the fractional mass in meter-sized bodies is more than about 1% of the total, the outward flux of millimeter-sized particles in the sublayer exceeds the inward flux in the rest of the layer, so that their net mass flux is outward. If the larger bodies are only 10 cm in size, a mass fraction of a few percent produces a similar reversal of the small particle flux.

Relative velocities are due to differential radial and transverse velocities for particles of different sizes, and random motions caused by turbulence. For the latter, we use the expressions from Weidenschilling (1984), assuming a Kolmogorov turbulence spectrum with the time scale for the largest eddies at any value of Z from Equation 11. Within the dense sublayer, shear produced by the presence of the larger bodies yields turbulent velocities that are significantly higher than those at larger values of Z. The midplane eddy velocity depends on the abundance of the large bodies, but is typically a few m s−1. The motions of the millimeter-sized particles are correlated within eddies, but the time scale of the smallest eddies (the inner scale of the Kolmogorov spectrum, typically ∼ 10−2/ω in these simulations) is shorter than their response times, so some relative motion occurs due to eddy crossing. This contrasts with the shear-induced turbulence well above the sublayer (or in a layer of only millimeter-sized particles), where the smallest eddy time scales exceed te, and correlated motions cause relative velocities to be much lower (Weidenschilling 1984; Cuzzi and Weidenschilling 2006). If the particles were not identical their relative velocities would be somewhat larger due to differential accelerations within eddies (Weidenschilling 1984), but still much less than their drift velocities. By contrast, the velocity of a small particle relative to a meter-sized body is significantly lower than the differences between radial and transverse drift velocities of isolated bodies, due principally to the decrease in ΔV due to the mass loading in the midplane. The net result of collective motions is to increase impact velocities between small (centimeter-sized) particles of comparable size, while decreasing velocities in collisions of small particles with meter-sized or larger bodies. The relative velocities are not very sensitive to the mass fraction in meter-sized bodies (Fig. 14).

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Figure 14.  Relative particle velocities in the midplane for layers composed of varied proportions of millimeter- and meter-sized bodies. The turbulent velocity of the gas is also plotted. The velocity of a small particle relative to a meter sized body is approximately 10 m s−1, with only slight dependence on the mass fraction. Relative velocities of equal-sized bodies are lower than the turbulent velocity. The millimeter-sized particles have mostly correlated motions within eddies. The meter-sized bodies are mostly decoupled from the turbulence because the eddy frequency ω is much greater than the Kepler frequency ΩK.

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Figure 15 shows the vertical structure of a layer with 25% of the mass each in bodies of 0.1, 1, 10, and 100 cm. The stratification by size is pronounced; at = 0, the mass fractions are 0.04, 0.07, 0.19, and 0.7, respectively. It appears that a continuous size distribution, e.g., a power law, would also be strongly sorted by size in the vertical direction. We can compare relative velocities at = 0 to the values that would be produced by differential drift of isolated particles. For sizes of 0.1 and 1 cm, their relative velocity is increased by a factor of ∼15 due to turbulence, while velocities of both 0.1 and 1 cm particles relative to those of 10 cm are doubled. However, velocities of millimeter-, centimeter-, and decimeter-sized bodies relative to meter-sized bodies are all decreased by about a factor of four below the individual values, due to the lower value of ΔV in the midplane.

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Figure 15.  Vertical distribution of particles in a layer composed of equal masses of bodies with sizes 0.1, 1, 10, and 100 cm. The four components have distinct scale heights.

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Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

The mechanism(s) by which planetesimals formed is still enigmatic. Much of the uncertainty is due to lack of knowledge of the fundamental properties of the solar nebula, in particular the extent, strength, and duration of turbulence in the gas. The material properties of the particles that composed primitive planetesimals are also obscure; the viability of any mechanism involving collisional coagulation depends on their impact strength, i.e., such factors as the surface forces between individual grains, and the structure and porosity of their aggregates. Experimental studies strongly suggest that coagulation could readily produce aggregate bodies in the size range of centimeters to decimeters (Blum and Wurm 2008), but growth to sizes of meter scale or larger is more difficult. A common objection to formation of planetesimals by collisional coagulation is the magnitude of impact velocities induced by differential drift due to nebular gas drag. The magnitude of the deviation of pressure-supported gas from Keplerian motion is of order 50 m s−1 in typical nebular models, implying comparable relative velocities between small grains and large bodies. This figure has dominated arguments about coagulation efficiency, as well as experimental impact studies (Blum and Wurm 2008). Another objection to collisional growth is the rapid inward spiraling of meter-sized bodies; bodies must grow through this size range in order to avoid loss into the Sun. A radial velocity of 50 m s−1 corresponds to migration of 1 AU per century.

Velocities of this magnitude would occur for isolated bodies; i.e., if they are distributed too sparsely to affect the motion of nebular gas in their vicinity. If the nebula is not globally turbulent, or if there is a “dead zone” in which the ionization is too low to generate turbulence by magneto-rotational instability, solid particles can readily settle to the midplane, reaching densities comparable with, or exceeding, that of the gas. In that case, their interaction becomes mutual, with exchange of momentum and energy due to drag forces. Drift velocities and relative velocities do not depend only on the sizes of individual particles, but on the spatial density of all solids in their vicinity, and the size distribution, which determines the effectiveness of particle-gas coupling. It is not possible to represent impact velocities as a simple function of size, but there is a general tendency for collective effects to decrease the maximum impact velocities, which occur between small and large bodies.

This paper presents a heuristic numerical model for the behavior of particles of arbitrary size interacting with the local gas. It avoids some of the limitations of analytic models that assume perfect coupling of particles and gas, and/or neglect shear-generated turbulence and momentum transfer via turbulent viscosity. The results of the model depend to some extent on the relevant length scale (assumed to be the Ekman length), and parameters such as the Richardson and Rossby numbers; however, they are not very sensitive to their chosen values. The qualitative conclusions are plausible, i.e., collective effects that drag the midplane gas in more nearly Keplerian rotation decrease the peak relative velocities and orbital decay rates of bodies in the critical meter size regime. The present results suggest that these are reduced by about a factor of five; in that case the relevant velocity regime for impact experiments is on the order of 10 m s−1. It is not clear that this difference is a significant aid to collisional growth; some experiments suggest accretion efficiency increases with impact velocity in this range (Wurm et al. 2005). However, the time scale allowed for growth before excessive loss of solids by orbital decay is also lengthened by a similar amount.

For a single particle size, the interplay of direct exchange of momentum between particles and gas and vertical transport of momentum in the gas by turbulent shear results in complex variation with elevation for motions of both particles and gas. For small particles (Stokes numbers <1), mass fluxes are inward at lower elevations, and outward in the part of the layer farther from the midplane, although the net fluxes are inward for solids and outward for gas. For larger particles the net motions of solids and gas are inward and outward, respectively, at all elevations.

One notable result of models with more than one size of particles is a tendency for strong sorting by elevation, with larger particles confined to a thin sublayer in the midplane. If the size difference is substantial, e.g., millimeter- and meter-sized bodies, the scale height of the larger particles is much less than that of the smaller ones. The strong shear and turbulence produced by the sublayer does not propagate very far in the vertical direction, i.e., the Ekman length is much smaller than the equilibrium thickness of the layer of small particles. The presence of large bodies has little effect on the thickness of the layer as a whole. However, there may be a significant effect on particles of all sizes within the sublayer. There the reduction in ΔV diminishes velocities of impacts by small particles on large ones, while turbulence generated by the shear substantially increases relative velocities between the small particles. For some range of impact strength, small aggregates in the sublayer would be shattered by collisions with others of comparable size. The mass flux impacting the larger bodies would then be dominated by very small particles or individual grains. Their supply would be continually renewed by coagulation, settling, and turbulent diffusion in the thicker layer of small particles. Thus, meter-sized bodies might grow through this critical size range by accreting small particles, without encountering impactors of intermediate size.

A dense sublayer of meter-sized bodies is associated with a locally strong outflow of gas. Small particles with St << 1 are entrained in this flow. The outward radial velocity attained by small particles in the sublayer is comparable with the inward velocity of the large bodies. This is so much greater than the inward drift rate in the rest of the layer that the net mass flux of small particles is outward. This effect would allow some outward transport of solid material in the form of small particles, although the net flux of all solids is always inward (for a monotonic nebular pressure gradient). More detailed modeling is needed to determine whether this phenomenon could transport small grains over significant radial distances in the solar nebula.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References

Acknowledgments— I thank G. Wurm and A. R. Dobrovolskis for constructive comments on the manuscript. This research was supported by the NASA Planetary Geology and Geophysics program, Grant NNG05GJ31G. This is PSI Contribution No. 464.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Collective Particle–Gas Interactions
  5. Basic Concepts
  6. Modeling Procedure
  7. Results for Single Size Particles
  8. Results for Multiple Particle Sizes
  9. Discussion and Conclusions
  10. Acknowledgments
  11. Acknowledgments
  12. References
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