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

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4. Global evolution of the high-mass disk

First, we applied our model to initial conditions sometimes considered fiducial by modelers of gaseous disks. The 1 [FORMULA] star is surrounded by a disk with an initial gas surface density given by


Thus, the initial distribution of the gas is practically constant, equal to about 8540 g cm-2, between the inner radius assumed to be at 0.036 A.U. and the radius of about 15 AU. At larger distances there is practically no gas. The total mass of the gaseous disk is equal to 0.245 [FORMULA] and angular momentum is equal to [FORMULA] g cm2 s-1. In such a disk a relatively large amount of the gas is concentrated relatively close to the star; therefore, we call it a high-mass, high-concentration scenario. These initial conditions correspond closely to the "standard case" considered by Ruden & Pollack (1991) and are identical to those we have used in Paper I. We further assume that the dimensional viscosity parameter [FORMULA] is equal to [FORMULA], the fiducial value assuming that the disk's turbulence is driven by thermal convection.

Of course, the assumed initial surface density profile is arbitrary. Fortunately, the specific form of the profile does not influence the subsequent evolution of the gas, inasmuch as the process governed by Eq. (12) is diffusive in nature and the details of initial distribution are forgotten after a time short in comparison with the evolutionary timescale. Taking advantage of the "short memory" of the gaseous component, we introduce solid particles into the calculation only after the passage of [FORMULA] yr during which the gas evolves alone to get rid of the arbitrariness in the initial condition. Upon introduction, the solid particles all have the same size, [FORMULA] cm, and the surface density of the solid material constitutes 1% of the contemporaneous gas surface density to account for cosmic abundance.

The calculations are carried out for up to [FORMULA] yr, a period of time equal, within an order of magnitude, to observationally deduced lifetimes of protoplanetary disks (Strom & Edwards 1993). Fig. 1 shows the summary of the disk evolution starting from the high-mass initial conditions described above. The design of Fig. 1, as well as the design of subsequent figures in Sect. 5, is the following: panel (a) shows the evolution of [FORMULA], panel (b) shows the evolution of [FORMULA], panel (c) shows the changes in s, panel (d) shows the changes in [FORMULA], and panel (e) shows the evolution of the total mass of both gaseous and solid, components of the disk. Although these quantities are calculated and available at any given time during the evolution of the disk, the figures show only three snapshots: at [FORMULA] yr, or at the very beginning of the evolution; at [FORMULA] yr, or at the midway of particle evolution; and at [FORMULA] yr, when the surface density of solids has converged.

[FIGURE] Fig. 1. Summary of the evolution of gas and solids for the high-mass initial conditions scenario with [FORMULA]. Panels a to d show the surface density of the gas, [FORMULA] ; the surface density of solid particles, [FORMULA] ; the particle size, s ; and the thickness of solids sub-disk, [FORMULA], respectively, as functions of r at selected times. On panel (a) the values of [FORMULA] are divided by a factor of [FORMULA] to put it in the same order of magnitude as [FORMULA]. Various times are labeled by different line styles: the dotted line denotes [FORMULA] yr, the dashed line denotes [FORMULA] yr, and the solid line denotes [FORMULA] yr. The time elapses from the moment when solid particles are introduced into the disk. Panel e shows the time evolution of the total mass of the disk, the dash-dot line represents the mass of the gaseous disk divided by a factor of [FORMULA], and the dash-dot-dot-dot line represents the mass of the solids sub-disk.

The most important result of the high-mass model calculation is that such a model leads to a complete loss of all solids into the star. This can be seen from panels (b) and (e) on Fig. 1. A failure of this seemingly reasonable model to lock any solids into bodies that can permanently orbit the star is somehow disappointing, but readily understandable from concomitant actions of the processes occurring in a disk. The disk evolving from the high-mass initial conditions remains relatively hot for a long time. Therefore the ice/vapor interface is located beyond 10 A.U. for up to [FORMULA] yr. However, on a timescale of only [FORMULA] yr, particles located beyond this interface coagulate to sizes of the order of 1-10 cm, decouple from the gas, and start moving inward. Their inward movement is swift (see Paper I for details) and the distance to the evaporation radius is short, so particles have no time to further grow by coagulation and thus stop their motion toward the destruction zone. The end result is that particles never grow to sizes bigger than about 10 cm (see Fig. 1c), and eventually they are all destroyed, leaving a purely gaseous disk behind (see Fig. 1b). Note that because particles have no chance to grow to larger sizes, they remain quite well coupled to the turbulent eddies (but not to the large-scale motion of the gas), and therefore are distributed throughout the entire thickness of the disk, so [FORMULA] always remains comparable to H (see Fig. 1d).

A disk evolving from the high-mass initial conditions does not lead to the formation of planetesimals. Two features of this model, its high surface density and compactness, are responsible for destruction of all solids. The high surface density translates into the high temperature and the outward location of the evaporation radius. The compactness of the disk translates into a short distance between the outer edge of the disk and the evaporation radius. Once particles grow to the size of maximum radial velocity (1-10 cm, see Paper I), their travel time to the evaporation radius is shorter than the characteristic coagulation time.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998