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Astron. Astrophys. 332, 1099-1122 (1998)

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8. Disk model

Based on the assumptions outlined above a model for a protostellar accretion disk has been calculated assuming a mass of the central star of one solar mass and an accretion rate of [FORMULA] yr. This corresponds to a late stage of the viscous evolution of the accretion disk around a solar type star when the rate of mass infall from the parent molecular cloud has become small and such material is added only to the far outer disk region. Then there develops a nearly stationary state in the inner parts of the accretion disk the structure of which is described by the set of equations of Sect.  6.1.

A model is calculated for the radius regime 40 AU [FORMULA]. The stellar radius [FORMULA] is determined by assuming a luminosity of [FORMULA] and an effective temperature of [FORMULA] K for a newly born star of one solar mass (e.g. Stahler 1983).

First the equations for the stability limits of all dust components important for the opacity have been solved simultaneously with the disk equations to determine the radius where the last grains of these dust components disappear. For instance we solved Eq. (18) simultaneously with the disk Eqs. (63), (64) to obtain the inner radius where forsterite disappears. This was done for water ice, enstatite, forsterite, iron, and corundum, and for the limit where conversion of the amorphous silicate grains to crystalline silicate grains is completed. For iron and forsterite this is a somewhat delicate problem since both species disappear in the region where the stability limits of both condensates intersect each other and it is not a priori known which one of the two disappears first. We calculated both, the limit of disappearance of silicate assuming that iron disappears first, and vice versa. The result with the higher temperature for destruction of both components is the correct one. In the model based on our above assumptions on the dust opacity the iron survives the forsterite, see Fig. 19, but if the opacity of aluminium compounds would be higher than that one used in the present calculation the forsterite might survive the iron. In all other cases the stability limits of the dominating absorbers are well separated in T and there is no doubt with respect to the order of disapperance of the dust components. The disk equations then where solved in the intervals between the stability limits considering the opacities of all components which do exist in that interval. Numerical experience showed that close to the stability limits of the dominating absorbers it is advantageous to prescribe the fraction f of the condensible material condensed in the dust and then using the corresponding P -T relations (like Eq. (18) for forsterite, for instance) simultaneously with the disk equations to solve for the radius s. This was done for [FORMULA]. In the remaining part of the intervals between the stability limits of two important absorbers the disk equations where solved for a sufficiently dense s -grid and iterating for the degree of condensation f of the dust species.

The opacity in the region between the annealing of the amorphous silicates and the destruction of aluminium compounds was calculated as


where [FORMULA] denotes the fraction of all Si condensed into enstatite and (or) forsterite, [FORMULA] the fraction of the iron condensed in iron grains and [FORMULA] the fraction of aluminium condensed into the aluminium compounds. These fractions are calculated for the chemical equilibrium state. Strictly speaking, the Rosseland mean opacity of a mixture cannot be obtained by simply adding the Rosseland mean opacities of the individual components. Since (i) dust opacities are rather smooth and (ii) usually the opacity of one of the components strongly dominates the opacity of the mixture, Eq.  (72) is nearly correct for such temperature and pressure conditions where the opacity is dominated by one component and can be taken as a smooth interpolation procedure in the transition regions.

The resulting P -T stratification in the midplane of the accretion disk is shown in Fig. 16. This curve shows four distinct temperature plateaus each corresponding to a strong reduction of the opacity related to the following processes:

  1. vapourisation of ice mantles on grains at about 150 K 5,
  2. crystallisation of the dirty silicate dust inherited from the molecular cloud at about 800 K,
  3. vapourisation of iron grains and chemisputtering of forsterite, which both occur at about 1 430 K, and
  4. chemisputtering of corundum grains at about 1 840 K.

More precise limits are listed in Table 3 together with stability limits for some other dust components which are less important for the opacity.

[FIGURE] Fig. 16. Temperature-pressure stratification in the midplane of an accretion disc model with [FORMULA] and [FORMULA]. The dotted lines show the limit lines for vapourisation of H2 O ice resp. decomposition and vapourisation of the indicated dust components and the limit lines for dissociation of H2 and ionisation of H. The dashed part of the P -T stratification corresponds to the unstable branch of the solution in the region where multiple solutions of the stationary disk equations exist.


Table 3. Stability limits according to chemical equilibrium calculations for some important solids in the protoplanetary disk.

The dashed part of the P -T stratification shown in Fig. 16 in the inner region of the disk corresponds to a multiple solution of the stationary disk equations which is due to the strong opacity increase with increasing temperature in the region of partial hydrogen ionisation. There exist two singular radii at [FORMULA] AU and [FORMULA] AU between which the disk equations have three different solutions of which only the lower and upper one are stable. Between the two singular radii there necessarily occurs a discontinuity in the disk structure where the solution jumps from the low temperature branch of the solution valid for the cold outer part of the accretion disk to the high temperature branch of the solution valid for the hot inner part of the accretion disk close to the star. This jump can occur at any point somewhere between the two singular points. Its position cannot be specified within the frame of the approximation of a stationary disk. For time dependent models the discontinuity in the solution for the stationary model corresponds to wave fronts running to and fro in the region between the inner and outer singular points which in case of cataclysmic variables are the well known disk instabilities leading to the phenomenon of dwarf nova outbursts (see, e.g., the review by Cannizzo 1993). In the case of protoplanetary accretion disks this instability is thought to be responsible for the FU Ori outbursts (e.g., Hartmann & Kenyon 1996). Between the two singular radii of the stationary solution the disk temperature can either be low ([FORMULA] K) or high ([FORMULA] K). Which of the two states is realised at a specific radius and instant depends on the history of the disk evolution and can only be determined from time dependent models. A simple test calculation following essentially the approximations described in Ruden & Pollak (1991) and Bell & Lin (1994) showed that for most of the time the jump occurs near the outer singular point and that only a few and short excursions to the inner singular point occur during the evolution of the disk. It is important to note that in the inner disk region between the two singular points the disk matter repeatedly switches two and fro between a hot state where the matter is ionised and a cold state where molecules easely form and some solid state species like corundum are stable.

Fig. 17 shows the radial distribution of the temperature. The plateaus corresponding to the disappearance of an important absorber can clearly be recognised. The plateau related to the transition from amorphous to crystalline silicate dust is quite extended between [FORMULA] AU and [FORMULA] AU. The variation of the fraction of crystallised material calculated from Eq. (70) is shown in Fig. 18. The drop in opacity associated with progressing crystallisation keeps the temperature in this region at a rather constant level of [FORMULA] K and spreads the whole transition across an extended radius interval. The temperature in this region between the present positions of Mars and Venus is much lower than what one would obtain if annealing is not considered. This unexpectedly low temperature in the region of terrestrial planets certainly has strong implications for the composition of planets to be formed in this region.

[FIGURE] Fig. 17. Radial variation of the midplane temperature in a protoplanetary accretion disk model with [FORMULA] and [FORMULA]. The dashed part indicates the unstable branch of the solution in the region where multiple solutions of the stationary disk equations exist.

[FIGURE] Fig. 18. Radial variation of the degree of condensation f for silicon in the silicates, of aluminium in corundum (and in some other aluminium compounds), and of iron in solid iron grains in the protoplanetary accretion disk model with [FORMULA] and [FORMULA]. The dashed line shows the degree of conversion of the initially amorphous silicate into crystalline silicate.

Once crystallisation has completed the midplane temperature in the disk rapidly increases to about 1 300 K between [FORMULA] AU and [FORMULA] AU and then slowly rises to [FORMULA] K between [FORMULA] AU and [FORMULA] AU when iron and olivine dust grains disappear. Both dust species disappear in the region in the P -T plane where their respective stability limit curves just intersect. The region of iron and silicate destruction is shown in more detail in Fig. 19. The pressure and temperature relation corresponding to the midplane of the disk crosses the limit lines for stability of iron and forsterite close to the point where both intersect. With our model parameters, the thermochemical data of Sharp & Huebner (1990), and with our assumptions with respect to the extinction coefficient the P -T relation first crosses the stability limit for forsterite and then the stability limit for iron. In our model there exists a region in the disk where iron exists as a solid but no silicates, a result which has already been found in other calculations (e.g. Grossman 1972, Lattimer et al. 1978) based on a somewhat different input physics for the structure of the solar nebula.

[FIGURE] Fig. 19. Pressure and temperature in the midplane of the disk in the region of iron and silicate grain destruction (dashed line). The full lines show the stability limits of forsterite and iron.

The destruction of iron and silicate grains is followed by a rapid rise of the temperature to approx [FORMULA] K and an extended temperature plateau until the last corundum grains are destroyed at [FORMULA] K at a distance of only [FORMULA] AU from the star. Closer to the star the temperature rapidly increases in the region of molecular dominated opacity and with the onset of [FORMULA] absorption the solution jumps to the high temperature branch.

Fig. 17 shows that there is a considerable overlap in the radius of existence for the low and high temperature branch of the solution of the disk equations between [FORMULA] AU and [FORMULA] AU. This regime extends nearly over the whole radius regime where corundum grains do exist on the lower branch of the solution. On the high temperature branch the midplane temperature exceeds [FORMULA] K and neither molecules nor dust exist on this branch of the solution. Between [FORMULA] AU and [FORMULA] AU there exists no dust if the disk is in this high temperature state. If the instability of the disk solution in this regime is really related to the observed FU Ori outbursts then during the viscous stage of the disk evolution the corundum and other aluminium dust grains occasionally are formed in this region as the disk switches from the high temperature branch of the solution to the low temperature branch. The dust material vapourised during the hot phase then probably condenses again into new dust grains, either on newly formed condensation nuclei or on grains which survived in colder regions of the disk and which are mixed by transport processes (drift, turbulent diffusion) into the previously hot zone. In this region, the calcium-aluminium rich dust grains obviously are strongly modified in their properties and size and in their extinction properties due to this repeated "thermal treatment" which in turn modifies the disk structure. Such effects can only be treated on the basis of time dependent models for the disk evolution which are out of the scope of this paper.

Fig. 20 shows the radial variation of the disk height. The assumption of a "thin" disk upon which our disk equations are based is obviously satisfied. The solution shows an abrupt change in the disk height at the position of the jump from the low to the high temperature branch of the solution. The region of the disk adjacent to this hump is shadowed from direct insolation from the protosun.

[FIGURE] Fig. 20. Radial variation of the disk height h of a protoplanetary accretion disk model with [FORMULA] and [FORMULA]. The labels point to regions of strong opacity changes by disappearance of the indicated dust component. The dotted line corresponds to [FORMULA].

Fig. 21 shows the result for the radial variation of the abundance of enstatite and forsterite dust in the disk according to chemical equilibrium. Minor silicon bearing dust components are neglected in this figure. Most of the silicon forms enstatite and only in a small zone in the inner part of the protoplanetary disk the forsterite is the dominating silicate dust component. The equilibrium mixture of enstatite and forsterite is different from the mixture assumed in the P94 dust model to enter the accretion disk from the parent molecular cloud. Considerable internal diffusion and restructuring processes are required to convert the dust mixture from the molecular cloud core into a mixture corresponding to a chemical equilibrium state. Whether this conversion is possible is an important question for the structure of the inner parts of the protoplanetary accretion disk which requires further detailed studies of the basic chemical and physical processes in a protoplanetary accretion disk.

[FIGURE] Fig. 21. Radial variation of the fraction of the silicon condensed into forsterite and enstatite and that remaining as SiO molecules in the gas phase. The ordinate is the percentage of the silicon bound in the solid.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998