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

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6. Discussion

In this paper we have studied the global evolution of solids in protoplanetary disks. We have demonstrated that it is possible, under certain assumptions, to approach the problem of how a planet-building material is spatially distributed, from physical principles rather than phenomenologically, like in the minimum-mass solar nebula model (Hayashi et al. 1985). The basic problem is as follows: given some initial state, presumably chosen to reflect conditions at the onset of protoplanetary disk evolution, what is the spatial distribution of solids after they accumulate into planetesimals? On that issue, and under assumptions stated in Sect. 1, we have obtained the following results.

(1) The outcome is sensitive to the assumed initial conditions. The shape of the initial distribution of the gas and dust around the star makes a big difference in the ultimate location of the planetesimal swarm. Moreover, it is plausible, and has been demonstrated in Sect. 4, that some, otherwise perfectly reasonable, initial distributions of matter, lead to the extinction of all solid material from the disk. Therefore our results seem to contradict the widespread but unsubstantiated believe that planetary systems are the natural and unavoidable product of the star formation process. Instead, it appears that whether or not solid planetary cores can emerge from the star formation process depends on how this formation process proceeds. If the viscous stage starts from a relatively compact disk, solids are lost and no solar-system-like planets could form. If, on the other hand, the viscous stage begins with the disk extending up to large distances from the star, solid planetesimals, and subsequently planets/cores will develop. We may point out that observations of T Tauri stars (for a review, see Strom & Edwards 1993) suggest the existence of extended disks that, according to our calculations, are likely to form planetesimals. However, it remains uncertain whether these disks reflect the condition of a circumstellar material at the onset of the viscous stage, or rather during this stage after the original disk spread significantly.

(2) Of four basic processes governing the evolution of solids, advection, coagulation, sedimentation, and evaporation, the interplay between the first two is the single most important factor in determining the outcome. Advection draws solids toward the star, and coagulation first enhances, then inhibits, and ultimately stops the advection. The fate of particles is decided by the race between advection and coagulation. Sedimentation helps increase the coagulation rate, but is superseded in importance by advectional compression at the crucial stage of the inner bulge formation. Evaporation sets the inner limit of the distribution of planetesimals. Since there is a bulge in the number density of planetesimals at the location of the evaporation radius, one can speculate that the location of the innermost planet (in our case the innermost icy planet or the planet with the icy core) is determined by the evaporation radius.

(3) If the extended initial mass distribution is assumed (like the one considered in Sect. 5) solids evolve into planetesimals. After times shorter than [FORMULA] yr the radial distribution of planetesimals' number density settles and can only change on a much longer timescale by processes not considered in our model [like mutual gravitational interactions between planetesimals proposed first by Safronov (1968)]. Note that [FORMULA] yr is the time required for [FORMULA] to converge everywhere ; however, this convergence is not uniform and can be achieved on a time scale as short as [FORMULA] yr in the innermost disk. The radial extent of the planetesimal swarm depends on the assumed value of [FORMULA], but the total mass of the swarm does not. In fact, the final mass of solid material locked into planetesimals is about equal to the initial mass of solids distributed among all [FORMULA] cm particles. That means that the total mass of the solid constituent of planets is fixed by the mass of dust in the initial disk. Regardless of the value of [FORMULA], the number density of planetesimals has an abrupt outer limit resulting from advectional compression. This leads to the prediction that planetary systems end abruptly. This is certainly true of our solar system, where the mass of all objects in the Kuiper belt is estimated to be only a fraction of Earth's mass (Jewitt & Luu 1995).

We believe that our model captures the essence of processes leading to the distribution of icy planetesimals. There are certain assumptions in our model that reflect the present-day state of knowledge on the topic of protoplanetary disks, and, if revised, can change our conclusions. Most importantly, we envision that the protoplanetary disk indeed undergoes viscous evolution, and it is during that viscous stage, which is further assumed not to be accompanied by any infall of material, that the evolution of solids take place. Furthermore, we model the viscous disk to be powered by uniform turbulence, characterized by a constant value of [FORMULA]. In principle, our method does not depend on the fact that the underlying gaseous disk is described by a viscous model. We can, as well, couple the solid component to the gaseous component powered by a different mechanism (for example, magnetic torque), although such a disk should still be turbulent, because it is turbulence that makes coagulation possible in our model. In addition, we assume that small solid particles are initially homogeneously distributed throughout the gas, changing this assumption may produce different results.

Within these assumptions, we have made a number of approximations. We believe that important aspects of our results are not artifacts of these approximations. Following Cuzzi et al. (1993) we consider the protoplanetary disk as a two-phase fluid flow, the mixture of the gas and the "fluid" of solid particles. The fluid approximation is valid for small ([FORMULA]) particles because of their strong coupling to the gas, and for large ([FORMULA]) particles because they move on near-keplerian orbits in the midplane thus constituting a "cold," two-dimensional fluid. The motion of intermediate-sized ([FORMULA]) particles cannot be consider two-dimensional as they undergo damped vertical oscillations around the midplane (Nakagawa et al. 1986). For such particles the fluid description may not be fitting for their behavior in the vertical direction. Nevertheless, as the characteristic decay time for intermediate-sized particles oscillations is short (of the order of [FORMULA]), the thickness of the particle subdisk is primarily controlled by turbulent diffusion and should not be strongly influenced by oscillations of individual particles. Overall, we feel that the fluid approach provides at least a qualitatively correct description of solid particles in the protoplanetary disk.

The approximation that the mass distribution of particles is narrowly peaked about the mean value artificially decreases the rate of coagulation because particles having the same size can coagulate only due to the difference in turbulent speed, and not the difference in the systematic speed stemming from particles having different sizes. However, for our purpose, only mechanisms leading to the growth of the largest particles, the ones that contain the bulk of the mass, are important. The encounters of these particles with the small particles, due to the difference in their regular velocities, does not increase the size of the large particle significantly. On the other hand, encounters between two large particles, due to the difference in their turbulent speeds, increase the size of the particle significantly. Thus, our approximation should give a coagulation rate of the right magnitude, which is the only accuracy we are seeking at this stage.

Our assumption of the perfect sticking coefficient assures the formation of planetesimals. In reality, sticking coefficient depends on the relative velocity between particles. However, because the bulk of the solid mass is concentrated in the largest, all about the same size, particles, the perfect sticking coefficient is a reasonable assumption for our purpose of studying the global evolution of solid mass, as the relative velocity between particles of the same size is minimized. Other approximations, such as ice-only solids, and neglecting the feedback of the solid component on the gaseous component have obviously only a minor effect on our results. The addition of other components of solids, such as "rock" and "metal" will not introduce any new insights in to the problem, although it would permit consideration of planetesimals closer to the star, for modeling the "terrestrial" zone.

Finally, we assess how the distribution of the solid mass calculated in Sect. 5 compares to that found in our solar system. Of course, we don't know what was the distribution of planetesimals in the solar nebula, we only know the present locations and masses of planets. From this data it is customary to "reproduce" the continuous surface density of solids. According to Hayashi et al. (1985) such a surface density of icy solids in the solar nebula has the form


This distribution extends from 2.7 AU (evaporation radius) to 36 AU. It is convenient to consider the mass of the disk interior to a given radius


The dotted line on Fig. 6 shows [FORMULA]. Our low-mass model yields [FORMULA] and we can calculate [FORMULA], which is represented on Fig. 6 by the solid line. Note that all models have similar total mass with our models having a little higher mass than Hayashi's model. At first, it seems that our models cannot yield a solar-system-like configuration. Indeed, mass distribution in models characterized by [FORMULA] and [FORMULA] do not extend far enough to account for the architecture of the solar system, and the mass distribution in the model characterized by [FORMULA] does not have enough mass within the inner 30 AU. Only mass distribution in the model characterized by [FORMULA] has the mass and almost the extension to account for the solar system, but its mass distribution [FORMULA] is markedly different from [FORMULA]. However, it is important to remember that [FORMULA] is an artificially constructed quantity, and as such is only one of many mass distributions from which planetary masses and locations can be produced. Assume that four giant planets in the solar system have cores of [FORMULA], [FORMULA], [FORMULA], and [FORMULA] respectively. From Fig. 6 we can find out that, according to [FORMULA], the mass constituting the core of Jupiter comes from the zone between 2.7AU and 8.3AU, the mass constituting the core of Saturn comes from the zone between 8.3AU and 16.8AU, the mass constituting the core of Uranus comes from the zone between 16.8AU and 23AU, and the mass constituting the core of Neptune comes from the zone between 23AU and 30AU. The phenomenological model has been constructed in such a fashion that the location of planets are obtained by taking the geometric average of the outer and the inner limit of the corresponding zone. This gives, [FORMULA] AU, [FORMULA] AU, [FORMULA] AU, and [FORMULA] AU. These location are not exactly where the planets actually are because Hayashi's model assumes different values for masses of the giant planets cores. Applying the same procedure to our [FORMULA] model we obtain "solar giant planets" at locations, [FORMULA] AU, [FORMULA] AU, [FORMULA] AU, and [FORMULA] AU, which in these qualitative terms is not much different from the actual locations. The most important difference is the excess of mass near the outer limit of the mass distribution. This causes "Neptune" to be too close to "Uranus." Keeping in mind that our model starts from arbitrary initial conditions and is not expected to actually reproduce the solar system, the excess of mass at 20-30 AU may nevertheless account for the mass lost from the plane of the ecliptic due to gravitational scattering of unaccreted planetesimals by planets that have already attained their final masses (Duncan et al. 1987). Overall, our model seems to account quite well for the large-scale character of our solar system while clearly indicating that planetary systems of different layouts are possible.

[FIGURE] Fig. 6. The plot of the disk's mass interior to a given radius versus the radius for different solid mass distributions. Solid lines represent mass distributions calculated starting from the low-mass initial conditions. Different solid lines are labeled by the value of [FORMULA]. The dotted line represents the mass distribution of the icy solid material in the solar system spread over the outer solar system according to the minimum-mass model.
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998