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Astron. Astrophys. 319, 1007-1019 (1997)

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3. Coagulation and evaporation processes

So far the formalism derived in Sect. 2.2 is quite similar to what we have done in Paper I, with the notable addition of the vertical analysis, which permits the evaluation of solid particles' sub-disk thickness. In a hypothetical scenario of single-sized, noncoagulating, and nonevaporating particles, Eq. (33), which keeps track of radial advection and diffusion of particles, alone gives the time evolution of the surface density of solid particles. This has been done in Paper I. However, in reality, both coagulation and evaporation occur and cannot be ignored. Our method of incorporating these processes into our calculations relies on keeping Eq. (33) as the principal mathematical description of the global evolution of particles, but freeing its parameters from constraints of single-size and thermal indestructibility assumptions. Thus, our method can be characterized as solving a radial advection-diffusion problem modulated by coagulation, with the possibility of a cut-off by the evaporation. This method requires that the mass distribution of particles at any given radial location of a disk is narrowly peaked about a mean value particular for this location and a given time instant. Such an assumption may appear quite arbitrary; however, it has a reasonable physical justification as collisions with small particles do not significantly increase the size of a test particle and the bulk of the solid mass is concentrated in the largest, all about the same size, particles. This is supported by numerical simulations (Mizuno et al. 1988) of grain growth in protoplanetary disk which clearly show that although a broad size distribution is maintained, most of the mass is nevertheless always concentrated in the largest particles.

3.1. Coagulation

We assume that the mass distribution of particles at any given radial location r is narrowly peaked about the mean value [FORMULA], which corresponds to the size [FORMULA]. The goal is to evaluate the functional dependence of s on r and t. Particles are assumed to be spheres with a bulk density of [FORMULA]. The density of matter concentrated into solid particles is [FORMULA] and the particles' number density is [FORMULA]. The geometrical cross section for collision between two such particles is [FORMULA] and the mean time between collisions is [FORMULA] where [FORMULA] is the mean relative speed between particles. Assuming that particles stick to each other upon collision, the growth of particle mass [FORMULA] in unit time can be expressed as follow

[EQUATION]

To calculate [FORMULA], we consider [FORMULA], which is the average of the square of the relative velocity between two particles at the point of collision, and we use [FORMULA]. Expanding [FORMULA] we obtain

[EQUATION]

The first term on the right-hand side of (35) corresponds to the difference between average, large-scale velocities between two particles. In our case, where both particles are assumed to have the same size, this term vanishes. The last three terms on the right-hand side of (35) stem from particles having different, chaotic, small-scale velocities at the point of collision. These terms have to be expressed in terms of large-scale quantities. It can be shown using the relations given by Cuzzi at el. (1993) that

[EQUATION]

Using (6) and (9) we can identify [FORMULA] with [FORMULA] and, if the particles are assumed to have the same size, Eq. (35) reduces to

[EQUATION]

which is similar to an analogical expression obtained by Morfill (1985).

Because we assume the particles to be perfect spheres, the mass of the particle is given by [FORMULA] and Eq. (34) transforms into

[EQUATION]

where we used (37) to eliminate [FORMULA]. Equation (38) can be integrated over the period of time equal to the time step in the solid particle's evolution [Eq. (33)] to obtain the increase of the mean particle size at any radial location of the disk. In turn, the new particle size upgrades the coefficients of Eq. (33). Continuing this process, we manage to incorporate the effects of coagulation into the global evolution of solid particles' surface density.

3.2. Evaporation

The temperature of the gas in the disk decreases with distance from the star. The particle traveling inward will evaporate when it finds itself at the location where the ambient gas is sufficiently hot. Such a location defines an evaporation radius that depends on the composition of the particle. The evaporation radius decreases with time, as the entire disk cools down in the process of its diminishment. Therefore we have to consider in our computation three instead of two components: gas, solids, and the vapor of the material constituting solid particles under cool enough conditions. As we have assumed that solid particles are made up of water ice, they cannot exist for [FORMULA], and are converted into vapor. On the other hand, the vapor condenses into solid particles wherever it finds itself in a [FORMULA] environment. The value of [FORMULA] depends on parameters of the gaseous disk and thus slowly changes; however we assume that [FORMULA] K, which is a reasonable value for all our models. In our calculations we treat vapor as particles with [FORMULA].

3.3. Computational approach

Equation (12) is solved by means of an implicit scheme to find out the evolution of the gaseous component. The necessary opacity law is adopted according to formulas given by Ruden & Pollack (1991). The evolution of gas is computed independently from the evolution of particles (see assumptions in Sect. 1). At every time step the quantities needed for evaluating the change in the mass distribution of solids are calculated and the change itself is computed from Eq. (33) using the operator splitting method. In such a method the advective term in (33) is treated by the numerical method of characteristics, whereas an implicit scheme is applied to the diffusion term. The obtained distribution of solid material is then modified because of the existence of the evaporation radius, and the mass distribution of the vapor is calculated using the implicit scheme. Finally, the new particle size distribution is calculated before proceeding to the next time step.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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