Space-time distribution of solid particles mean velocities

Equations (13) and (14) constitute the closed system for two unknowns, and , because can be eliminated using the relation with and can be eliminated using the relation with given by equation (6). With such an elimination, and keeping only terms linear in and , we obtain

 

and

 

The time evolution of and can be found for particles of any given size by solving the system of equations (15-16) as all coefficients in these equations depend on properties of the gas and thus are known functions of space and time. The last term in equation (16) is the exception as it depends on the density of the particles. The space-time evolution of particles density is discussed in §4, here we remark that calculating particle densities requires the knowledge of and . Therefore, in general, equations (15-16) have to be solved simultaneously with the particles density evolution equation (18). Note, however, that the last term in equation (16) vanishes if , which is consistent with estimation of by Canuto & Battaglia (1988 [4]) (see the begining of §2). In the bulk of our calculations presented in this paper we assume . This permits the discussion of particles mean velocities independently from the discussion of their densities. We have also performed calculations for , which requires the solution of a coupled velocity-density problem, and have found only expected, quantitative differences between the results for and the results for . Thus, it seems that although the condition is computationally advantageous, there is nothing physically unique about it.

  

Figure 1: Radial distributions of , the mean radial velocity of a solid particles; , the mean radial velocity of the gas; and , the mean relative transverse velocity between particles and the gas, respectively. The times shown are yr, yr, and yr. Particle have all the same radius cm. Solid lines indicate a positive velocity (an outward movement for radial velocities and particles faster then the gas for ), whereas dash-dotted lines indicate a negative velocity.

Let's consider the gas evolution scenario described in §2, which starts at t=0 with a disk of 0.245 and an angular momentum of g cm s . The evolution proceeds with dimensionless coefficient of viscosity . Our goal is to calculate the time evolution of and for particles of different sizes embedded in an evolving gaseous disk. The knowledge of particle mean velocities is important inasmuch as these velocities capture the bulk of particle dynamics. Beware, however, that the contribution of turbulent diffusion to particles dynamics is crucial for explanation of certain features of their evolution. First, we have found out that very small particles, with cm, can be considered perfectly coupled to the gas. Their mean velocities match the mean velocity field of the gas. On the other hand, very large particles, with cm, can be considered completely decoupled from the gas. Their mean velocities remain practically unchanged with time. Particles with sizes between these two extremes are partially coupled to the gas.

Fig. 1 shows time evolution of (mean radial velocity of the gas), (mean radial velocity of a particle), and (tangential component of mean relative velocity between a particle and the gas) for particles with cm. For the purpose of the present discussion we label such particles ``small." The tangential velocity of the gas is not shown in Fig. 1 as it always stays very close to the Keplerian velocity. As we have already mentioned in §2, is not always the accurate indicator of the mass flux. The distinctive feature of the mass flux distribution in the gaseous accretion disk is the existence of a stagnation radius that separates the region of gas accretion from the region of gas decretion. This stagnation radius moves outward during disk evolution. Examining Fig. 1 we notice that although has the same sense as the mass flux throughout the entire accretion zone and in the inner decretion zone, it is oriented inward in the outer decretion zone even so the mass flows outward there. This is because the transport of mass in the outermost disk, where the steep gradient of density is maintained, is dominated by turbulent diffusion. Inward convective mass flux is necessary to reduce the outward diffusive mass flux so that equation (6) is satisfied.

Small particles remain rather strongly coupled to the mean flow of the gas; is very small and . Throughout the accreting part of the disk the magnitude of remains within the same order of magnitude, about cm/s at yr, cm/s at yr, and finally cm/s at yr. In comparison, the Keplerian velocity in this region of the disk is in the range of - cm/s. The particles have super-gas (faster than the gas) mean tangential velocities and thus experience, on average, a tangential head wind. The particles have small super-gas mean radial velocities ( ) and thus experience, on average, a radial head wind as well.

  

Figure 2: Radial distributions of , , and , respectively. The times shown are yr, yr, and yr. Particles have a radius s=10 cm. Solid lines indicate a positive velocity (an outward movement for radial velocities and particles faster then the gas for ), whereas dash-dotted lines indicate a negative velocity.

Throughout the outer, decreting part of the disk, the particles are less coupled to the mean flow of the gas. The magnitude of changes by orders of magnitude and can be as large as cm/s at the outer edge of the disk. For times up to yr, the particles in the inner part of the gas decretion zone move on average outward with slightly sub-gas velocities, thus experiencing a radial tail wind. However, in the outer part of the gas decretion zone, the average radial velocity of small particles is directed inward, as is the average radial velocity of the gas. Nevertheless, mass fluxes of both particles and the gas are outward, as they are dominated by turbulent transport.

Fig. 2 shows time evolution of , , and for particles with s=10 cm. We label such particles ``medium." Medium particles are much less coupled to the mean flow of the gas than small particles: for medium particles is orders of magnitude larger than for small particles. There is not much change in the magnitude of throughout the disk evolution. Medium particles move on average with super-gas tangential velocities everywhere in a disk and any time during the disk evolution. Also, medium particles always move on average inward. Note that , thus these particles are ``fast" in their radial movement. With average radial velocities of the order of - cm/s, medium particles have evolutionary time scales comparable to yr. Therefore, handling of the medium particles encounters the problem described in §1. As their evolutionary time scale is comparable to the time the gaseous disk needs to forget the arbitrary initial conditions, their properties at later times may be sensitive to the initial conditions. To circumvent this problem we introduce medium particles into the gas at yr, instead of t=0 as we have done for the small particles. Also, we finish our calculations for medium particles at yr, as medium particles are severely depleted after this time. In fact, there would be no medium particles left in the disk after this time if not for the contribution of turbulent diffusion to the dynamics of particles.

  

Figure 3: Radial distributions of , , and , respectively. The times shown are yr, yr, and yr. Particles have a radius cm. Solid lines indicate a positive velocity (an outward movement for radial velocities and a particle faster than the gas for ), whereas dash-dotted lines indicate a negative velocity.

Fig. 3 shows time evolution of , , and for particles with cm. We label such particles ``large." Large particles are only weakly coupled to the mean gas flow: remains relatively large as particles move with almost a Keplerian velocity, faster than the gas that is pressure supported. Large particles, much like the medium particles, move with super-gas tangential velocities everywhere in a disk and at any time during the disk evolution. They also always move inward. Note, however, that conversely to the case of the medium particles, for yr, , so the radial movement of large particles is insignificant, and becomes more insignificant with time. Thus, large particles are ``slow" in their radial movement. They remain on Keplerian orbits drifting very slowly inward.

 

 

Figure 4: Radial distributions of surface density of solids at selected times ( yr, yr, and yr) for LL 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 0.1 cm particles and dashed lines show the surface density of cm particles.