## 2. Model of the accretion diskWe use in this paper the same semi-analytical model for a thin
stationary accretion disk as in Duschl et al. (1996, henceforth called
Paper II)
where Illumination due to the accreting star and/ or the disk-star boundary layer may lead to modifications of the disk structure, even to self induced warping (Pringle 1996). In the present investigation such effects are not accounted for. The inwards directed drift velocity of the disk material is Integrating this with respect to time yields the radial position of
a fixed gas parcel at each instant is the initial position of the parcel at . The vertical component of the gas velocity
accounting for the decrease of the height In the one zone approximation, the vertically averaged gradient of the vertical velocity is The basic parameters of the disk model used in our calculation are listed in Table 1.
## 2.1. OpacityIn our model calculation, the opacity of the disk material is determined for each time step simultaneously with the temperature in the central plane of the disk. The temperature dependent Rosseland mean opacity of the two main dust components (silicate and graphite) is calculated using the dust extinction model of Draine and Lee (1984) and Draine (1985) (see Paper I for more details). This is probably a strong oversimplification for the true absorption properties of dust in protoplanetary accretion disks since coagulation will modify the dust properties (e.g. Henning and Stognienko 1996), but this process presently cannot be coupled to our type of model calculation. For the Rosseland mean of the mass extinction coefficient of the gas, , we use the analytical approximations to tabular values given by Lin and Papaloizou (1985) and Bell and Lin (1994): (i) Once the whole dust has disappeared, the absorption is
dominated by molecules (mainly H (ii) In the temperature regime between K and K the H ions dominate the extinction. In this region the mass extinction coefficient can be approximated by We note the high power of the temperature in this case: a small
change in the temperature dramatically modifies the opacity. This
strong temperature dependence of is the cause of
the so called (iii) For higher temperatures and transitions of atoms and ions dominate the extinction of the gas. The mass extinction coefficient can be approximated in this case by Temperatures above K where this approximation becomes valid are usually not encountered in a protoplanetary accretion disk. For our set of accretion disk parameters (cf. Table 1) such high temperatures occur only at the smallest radii of the disk (Eq. 3) and in the transition layer to the stellar surface. Our model approximations for the disk break down in this transition zone (see, e.g., Duschl and Tscharnuter 1991). The total gas opacity is calculated from the interpolation formula which smoothly interpolates between the different cases. In
Fig. 1 the different opacity approximations are displayed for a
value of the mean density of 10
and are the mass density of the carbon and silicon dust in the gas-dust mixture and and are their extinction coefficients. This simple addition of the different opacities is surely not very precise but it should be sufficient for our present more explorative calculation. Fig. 2 shows as an example the temperature structure in the
inner part of the disk for AU resulting from
this opacity law. The dust vapourisation and the dissociation are
calculated here not by solving the corresponding rate equations but
from thermodynamic equilibrium as in Paper II. The disk equations with
the opacity law (13) have multiple valued solutions for
For time dependent models of an accretion disk it is known that the
very strong temperature dependence of the H
opacity (, Eq. 10), resulting in the
multi-valued solution for the stationary case, gives rise to an
unstable structure of the disk (Meyer and Meyer-Hofmeister 1981,
Duschl 1991). This instability, however, is not destructive to the
disk but rather gives rise a limit-cycle behaviour. This introduces an
inherent time dependence into the problem and thus leads beyond the
scope of our present work. While the basic features of the instability
and the time dependent disk evolution due to it have been investigated
and are understood fairly well, not very much is known how a proper
treatment of the chemical evolution influences the overall evolution
of such an unstationary accretion disk. In this context it is
important to note that not only steep opacity gradients due to
hydrogen ionisation/ recombination can give rise to an instability.
Also steep gradients due to destruction of molecular absorbers (e.g.,
H In the present models of the chemistry of the disk we perform the inwards integrations of the rate equations only down to a radius slightly above the critical radius at AU where the solution of the disk equations becomes singular. The calculation is stopped at this point since the high temperature branch of the solution is of no interest for the chemistry. The opacity law (13) is used to determine for each time step the
temperature from Eq. (3) and pressure from Eq. (4) which is equivalent
to determine the zeros of the two following functions The extremely non-linearity of these functions results in considerable numerical problems if coupled to the system of rate equations for the chemistry and dust destruction, but the numerical method presented in Paper I is well suited to handle such problems. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |