## 3. Coagulation and evaporation processesSo 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 ## 3.1. CoagulationWe assume that the mass distribution of particles at any given
radial location To calculate , we consider , which is the average of the square of the relative velocity between two particles at the point of collision, and we use . Expanding we obtain 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 Using (6) and (9) we can identify with and, if the particles are assumed to have the same size, Eq. (35) reduces to 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 and Eq. (34) transforms into where we used (37) to eliminate . 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. EvaporationThe 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 , and are converted into vapor. On the other hand, the vapor condenses into solid particles wherever it finds itself in a environment. The value of depends on parameters of the gaseous disk and thus slowly changes; however we assume that K, which is a reasonable value for all our models. In our calculations we treat vapor as particles with . ## 3.3. Computational approachEquation (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. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |