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

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1. Introduction

Recently, with the identification of a solar-like star showing evidence for planets circling around it (Mayor & Queloz 1995), our interest in understanding the formation of a planetary system on its largest scale has intensified and widened beyond the long-standing question of the origin of the solar system. It is therefore timely to attempt, on theoretical grounds and from an evolutionary point of view, a prediction of the large-scale properties of a planetary system around a solar-like star. Of particular interest is the spatial distribution of material making up a planetary system, as this is about the only information the present observations, based on the Doppler technique, can provide. To start addressing this problem we are developing a model that would keep track of circumstellar material as it evolves from the form of a protoplanetary disk to the form of a planetary system. In the first paper of the series (Stepinski & Valageas 1996; hereafter referred to as Paper I) we laid down the foundations of our model and developed a numerical method to study the effects of aerodynamic forces acting on solid particles entrained in a gaseous disk. We refer the reader to that paper for elucidation of the essential concepts underlying our approach.

In the current paper we take our model one step further by taking into consideration the processes of coagulation, sedimentation, and evaporation/condensation of solid particles. These processes, acting in addition to gas-solid coupling caused by aerodynamic forces, shape the radial distribution of solid material around the star, until such time when solids augment to planetesimal sizes and further evolution of solid material is dominated by mutual gravitational interaction between planetesimals. Thus, our model, in its present form, given some initial distribution of gaseous and solid matter, computes the evolution of these two components, and can predict the radial distribution of solid mass locked into the planetesimal swarm. Arguably, such a distribution should well approximate the radial apportionment of condensed components of the planets spread over the radial extent of the mature planetary system. This is because the process of accumulation of planetesimals into planets or planetary cores is thought to happen with minimum radial displacement.

The location of solid mass in the present-day solar system presents an inevitable test for our model, and the bulk of our calculations were carried out to determine what kind of initial conditions, if any, lead to the development of a planetesimal swarm consistent with the solid matter in the solar system. Indeed, we have found initial conditions leading to a configuration of solid matter in rough agreement with the large-scale architecture of the solar system. However, we have also found that the outcome is sensitive to initial conditions, as well as, in some cases, to the values of the free parameters characterizing our model. This opens the theoretical possibility of planetary system diversity. Additional diversity may result from the different quantities of gas that various planetary systems may subsequently add to some of their solid protoplanets. Note that, although we argue that our present model may predict the mass distribution of a condensed material in a nascent planetary system, it cannot predict the distribution of a whole planetary mass consisting of solid and volatile materials. Nevertheless, as solid protoplanets or cores constitute the backbone of a planetary system onto which volatile envelopes are subsequently added, modeling its structure is of a primary interest.

The major novelty of our work is its emphasis on the global, comprehensive treatment of the problem. This follows from our interest in attempting to establish the link between initial conditions that characterize a protoplanetary disk at the onset of star-disk formation, and the large-scale character of an ultimate planetary system. To achieve this goal we have to include all relevant physical processes. This, in turn, presented us technically and, to certain degree, conceptually with an intricate problem, which required major simplification in order to become tractable. Therefore, our handling of several processes, most notably coagulation, is less advanced than can be found in some published work (for a review see Weidenschilling & Cuzzi 1993) dedicated exclusively to the issue of coagulation and not addressing the evolution of solids globally. In order to make progress, we have assumed that the size distribution of particles at any given radial location of a disk is narrowly peaked about a mean value particular for this location and time instant. Such an approximation was first proposed by Morfill (1985). This allows us to keep track of the increase of the mean particle size alone, and frees us from daunting calculations required for computing the shift in the entire particle size distribution function. This approximation is important for the viability of our calculations. It also captures the essence of the coagulation process accurately enough, at least for our purpose, which is to keep track of solid's mass whereabouts regardless of how it is apportioned between particles of different sizes. Two other major simplifications characterize our current model. First, we concentrate on ice, the most abundant component of solid material, and disregard other compositional constituents such as "rock" and "metal." Hence, at present, we expect to model only the development of icy planetesimals, or outer zones of planetary systems. In the context of the solar system, we expect to model distribution of mass presently located in solid cores of giant planets. Second, unlike in the calculations by Cuzzi et al. (1993), which were devoted to the investigation of growth and sedimentation of solid particles at the fixed radial location, we assume that the evolution of the gaseous component remains unaffected by the changing character of the solid component. These last two assumptions provide the current model with much desired amenability; however, unlike the assumption about coagulation, they could be removed without jeopardizing the integrity of our method.

Our basic method of simultaneously keeping track of the evolution of gaseous and solid components of protoplanetary disks is described in Sect. 2. Separately, in Sect. 3, we describe our treatment of coagulation and evaporation processes, and offer a very brief depiction of our numerical technique. The next two sections are devoted to the presentation of results. In Sect. 4 the results for an initially high-mass, high-concentration disk are given, and in Sect. 5 an initially low-mass, low-concentration disk is examined for various values of dimensionless viscosiameter [FORMULA]. Finally, in Sect. 6, we present discussion and conclusions.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998