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

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5. Global evolution of the low-mass disk

In order to produce planetesimals the initial disk must apparently be less massive and more extended than the one considered in the previous section. Therefore we consider a scenario where the 1 [FORMULA] star is surrounded by a disk with the initial surface density of the gas given by

[EQUATION]

The first term ensures that there is some mass up to very large distances from the star. The second term corresponds to the central concentration of the mass and sets the location of the evaporation radius. The total mass of the gaseous disk is equal to 0.023 [FORMULA] and an angular momentum is equal to [FORMULA] g cm2 s-1. Overall, the disk described by this model has about an order of magnitude less mass than the disk described by the previous model; therefore, we call it a low-mass scenario. Our calculations, starting from the low-mass initial conditions, are carried out identically to those started from the high-mass initial conditions. Figs. 2, 3, 4, and 5 summarize the low-mass disk evolution for values of dimensional viscosity parameter [FORMULA] equal to [FORMULA], [FORMULA], [FORMULA], and [FORMULA] respectively. This range covers all values of [FORMULA] that are conceivable in the context of protoplanetary disks.

[FIGURE] Fig. 2. Summary of the evolution of gas and solids for the low-mass initial conditions scenario with [FORMULA]. See Fig. 1 for legend.

[FIGURE] Fig. 3. Summary of the evolution of gas and solids for the low-mass initial conditions scenario with [FORMULA]. See Fig. 1 for legend.

[FIGURE] Fig. 4. Summary of the evolution of gas and solids for the low-mass initial conditions scenario with [FORMULA]. See Fig. 1 for legend

[FIGURE] Fig. 5. Summary of the evolution of gas and solids for the low-mass initial conditions scenario with [FORMULA]. See Fig. 1 for legend.

The most important result of the low-mass model calculation is that such a model leads to the survival of solid material. This manifests itself by the emergence of the converged, nonvanishing surface density distribution of solids (panel (b) of Figs. 2-5). It is interesting that although the converged radial distribution of the solid material depends on the value of [FORMULA], the total mass of solids in a disk is about the same, independent of the value of [FORMULA], and approximately equal to the initial mass of solids in the disk (panel (e) of Figs. 2-5). Thus, the evolution of the low-mass disk results in the reshuffle of solids within the disk, but not to their loss into the star. Of course, some solids, those initially located close to the evaporation radius, are lost, but, given the initial mass distribution (40), they constitute a small percentage of the total solid material, which is predominantly located in the outer disk. Once particles in the outer disk grow to the size of maximum radial velocity, their characteristic travel time to the evaporation radius is longer than the characteristic coagulation time so they manage to stop their radial movement before reaching the destruction zone.

The value of [FORMULA] determines the radial distribution of solids: the smaller the value of [FORMULA], the broader the distribution of solids. This is because particles suspended in a more vigorously turbulent disk (larger value of [FORMULA]) have larger inward radial velocities and consequently are locked into planetesimals closer to the star than particles in a less turbulent disk. The evaporation radius is located between 1 AU and 2 AU for all values of [FORMULA], the outer limit of converged [FORMULA] is about 10 AU for [FORMULA] and [FORMULA], about 30 AU for [FORMULA], and about 60 AU for [FORMULA].

By the time the surface density of solids attains convergence, the particles coagulate to about [FORMULA] cm, or planetesimal sizes (panel (c) Figs. 2-5). In disks characterized by smaller values of [FORMULA], and thus a more extended distribution of solids, the range of sizes, from [FORMULA] cm at the evaporation radius to [FORMULA] cm at the outer limit, can be found. This is because the coagulation process is less efficient at larger radii. These solids will continue to increase their sizes, but will not change their radial position, as they are already large enough to have a negligible radial motion. The thickness of the solid particles sub-disk, given by Eq. (21) depends both on the particle size and on the value of [FORMULA]. Examining panels (d) of Figs. 2-5, it can be seen that the smaller the value of [FORMULA], the relatively thinner the particle sub-disk. This is intuitively easy to understand, as a more turbulent (larger value of [FORMULA]) disk inhibits sedimentation of particles. In any case, by the time the surface density of solids attains convergence, the particles form a layer [FORMULA] - [FORMULA] times thinner than the surrounding gaseous disk.

Note that the converged radial distribution of solids does not vary monotonically. There are bulges of matter near the evaporation radius as well as at the outer limit of [FORMULA]. These bulges are present in all cases, but become narrower for smaller values of [FORMULA]. Their existence is a consequence of an intricate and nonlinear interdependence between advection and coagulation, modulated by changing gas conditions and the character of turbulence (the value of [FORMULA]).

Consider the region outside the evaporation radius, [FORMULA]. The efficiency of coagulation is a decreasing function of the radius [see Eq. (38)] so, shortly after evolution starts, the particle size is also a decreasing function of the radius. As particles grow they acquire inward radial velocities in excess of the gas inflow velocity. There exists a particular size, [FORMULA] cm, for which the radial velocity of a particle is fastest. For as long as the particles near the evaporation radius are all smaller than [FORMULA], the coagulation rate is actually impeded as the density of particles decreases due to monotonically decreasing advection velocity. Particles pass [FORMULA] and are lost. This process will continue, and may, as in the case of the scenario considered in Sect. 4, bleed the disk of all solids, unless particles bigger than [FORMULA] start arriving at [FORMULA]. For this to occur, the disk must be large enough. With the appearance of particles larger than [FORMULA], an advection velocity is no longer monotonically decreasing; instead, particles at [FORMULA] move slower than more remote particles, which leads to a fast increase of particle density at [FORMULA] and corresponding increase in the coagulation rate. The two phenomena, increasing density and enhanced coagulation rate, form a self-feeding loop resulting in a very rapid growth of particles and formation of the bulge in the solids surface density near [FORMULA]. The formation of this bulge also signals the "freezing" of the total mass of the solids in the disk, inasmuch as no more solids are lost to the vapor zone in subsequent disk evolution. They will either be captured by the bulge or come to rest by themselves at larger radii. Notice that it is a radial squeezing due to particle dynamics, rather than vertical squeezing due to sedimentation, that is primarily responsible for establishing the bulge and keeping particles from falling into the vapor zone.

The same mechanism is responsible for the abrupt drop in [FORMULA], which we associate with the outer limit of solid matter distribution, as well as the presence of the [FORMULA] bulge at the location of this drop. The outer region of the disk is characterized by the slowest coagulation rate, so particles there can travel relatively long distances before acquiring a size larger than [FORMULA]. Once they grow to [FORMULA], they slow down, allowing particles that trail them to catch up and form the bulge in the fashion described above. With time, the entire region beyond the outer bulge is swept clean of particles.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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