In order to visualize the global effects of solid particles
decoupling from the gas, it is convenient to calculate the
evolution of single-sized, noncoagulating particles. This is a
hypothetical evolutionary scenario, by itself not likely to be
pertinent to establishing the architecture of an emerging planetary
system. Nevertheless, it illustrates the difference in time
evolution between the gas and the solids. It provides a valuable
point of reference for more realistic calculations.
In the case of 
, computation of space-time distribution of 
is necessary to establish a particle velocity field
(see §3.2.) Here, however, we assume that 
and velocity of particles are known beforehand.
The ensemble of identical particles can be treated within the
framework of hydrodynamics, or as a ``fluid" with ``pressure"
and ``viscosity" provided by coupling between particles and turbulent eddies of the gas. By analogy with the gas, the
evolution equation can be obtained from mass conservation
supplemented by the 
component of the equation of motion.
The Reynolds averaged continuity equation for fluid of
single-sized particles in the absence of sources is
Substituting 
from equation (14) into equation (17) and integrating over the z coordinate we obtain the equation for time evolution of 
, the surface density of solid particles
One can also define 
as the accretion rate of a solid material through a disk.
As in the case of the gas, the mass flux of solid particles is the
sum of convective and diffusive fluxes.
Unlike equation (8), which directs the evolution of the gas, equation (18) is not a diffusive-type equation. Instead, it is an advection-diffusion equation.
Because equations (8) and (18) are of different types, a single numerical method that would concurrently advance both of them in time is difficult to contrive. Taking advantage of our assumption that the evolution of the gas is not affected by the
presence of solid particles, we first solve equation (8)
using an implicit scheme, then obtain the space-time distribution of 
by methods described in §3, and finally solve equation (18) using the operator splitting method. In such a method the advective term in (18) is treated
by numerical method of characteristics, whereas an implicit scheme
is applied to the diffusion term.
yr, 
yr, and
yr) for SL particles. The left-hand panels correspond to the
evolutionary scenario with 
, whereas the right-hand panels
correspond to the scenario with 
. Dotted lines show
the surface density of the gas 
. Solid lines show
the surface density of 1 cm particles, dashed lines show the surface
density of 10 cm particles, and dash-dotted lines show the surface
density of 
cm particles.
In general, particles can be divided into ``long-lived" (LL), characterized by evolutionary time scales comparable to or longer than the evolutionary time scale of the gas, and ``short lived'' (SL), characterized by time scales shorter then the lifetime of the gaseous disk. The small particles, which are strongly coupled to the gas and thus evolve on an approximately viscous time scale, and large particles, which are decoupled from the gas and move toward the central star very slowly, are LL particles. The medium particles have radial velocities larger than the radial velocity of the gas, they accrete quickly onto the central star, and therefore are SL particles. The terms LL and SL refer to an average lifetime of a particle. Because of the existence of a random component in particle motion, few individual SL particles may survive for a long time, and some individual LL particles may have a short lifetime.
Fig. 4 shows an evolution of the surface density for LL
particles. Our computation starts at t=0 with a disk of 0.245
and an angular momentum of 
g
cm 
s 
. Evolutionary scenarios with 
and
are calculated. The initial surface density of solid
material is 
to reflect that
about 1% of disk mass is in the form of solid particles. We follow the evolution of LL particles up to 
yr. Solids with a radius
of 
cm represent small particles, and solids with a radius
of 
cm represent large particles.
If the entire mass of the solid material is concentrated in
particles smaller than 0.1 cm, then solids are perfectly coupled to
the gas and the radial distribution of solids follows the radial
distribution of the gas as indicated on Fig. 4 by dotted
lines. For the purpose of the present discussion we refer to such
a distribution of solids as the putative distribution (it differs
from the distribution of gas by a factor of 100). Now assume,
however, that the solid material is concentrated into 0.1 cm
particles. It is clear from Fig. 4 that such solids slowly
decouple from the gas. First, let's discuss the 
scenario. At 
yr the decoupling of small particles from the
gas is not yet pronounced. The decoupling becomes noticeable at
yr, when solids located at the outer edge of the disk
spread outward at the slower rate than the gas.
This, combined with the fact that the inward radial
velocity of solids is slightly larger than the radial velocity of
the gas (see Fig. 1), causes the surface density of solids
within the inner 10 AU to be enhanced with respect to the putative
distribution. At 
yr the difference between the surface
density of small particles and putative surface density increases.
Within the inner 
AU the mass density of particles is enhanced with respect to the putative density, but ouward of 
AU the mass density of particles is actually depleted as compared to the putative density. The 
shows a similar behavior of small particles.
Now assume that the solid material is concentrated into 
cm
particles. Examination of Fig. 4 reveals that such large
particles are largerly decoupled from the gas. In the 
scenario the radial distribution of surface density
of such particles in the inner disk shows relatively little change during 
yr of evolution. Therefore, a large difference between the mass density of large particles and the putative mass density develops with time in the inner portion of the disk. In the
outer portion of the disk large particles evolve faster than the
putative mass density, again resulting in build up of a significant difference between both distributions with time. Overall the radial distribution of large particles steepens with time, whereas
the putative mass distribution flattens. In the 
scenario, after some initial time, large particles evolve faster
than the putative mass distribution. This is because the 
disk evolves more slowly than the 
disk and its gas density remains high for much longer. That, in turn,
makes the drag force on large particles stronger and longer lasting, thus enabling them to evolve faster. Again, a large difference between the mass density of large particles and the putative mass density develops with time.
Fig. 5 shows an evolution of the surface density for SL
particles. Our gas computation starts at t=0 with a disk of 0.245
and an angular momentum of 
g
cm 
s 
. Evolutionary scenarios with 
and
are calculated. To eliminate the influence of
arbitrary initial conditions on the evolution of SL particles,
for which evolutionary time scales are comparable to 
yr, we
start our particle computation at 
yr with
and we
follow their evolution up to 
yr. Solid particles with
radii 1 cm, 10 cm, and 
cm are all short-lived and their
evolution is shown on Fig. 5. If solid material is
concentrated in such particles, it decouples quickly from the gas
and is lost to the central star in a relatively short period of time.
The case of 10 cm particles offers the extreme example.
Such particles are lost in less than a 
yr
regardless of the magnitude of 
. Solid material
concentrated into 
cm particles is lost in about 
yr,
whereas solids concentrated into 1 cm particles are all accreted in
a 
yr. None of these SL particles survives
in an average sense, until 
yr. As can be seen from Fig. 5 radial distribution of the surface density for SL particles can be quite complicated. Consider, for example, 
cm particles during the evolution characterized by 
. Although most of them accrete quickly onto the central star, those initially located at the outer edge of the disk linger there for a relatively long time before starting their inward movement. This rather intricate
behavior is the result of large radial and temporal changes in the
aerodynamic regime encountered by particles.
