Astron. Astrophys. 325, 1264-1279 (1997)

2. Model of the accretion disk

We 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) ¹ and in Paper I. The details of the model are described in Paper II. The resulting basic equations for the disk structure are

where s is the radial distance from the protosun in units AU, is the surface density, h the (half) thickness of the disk, M the mass of the protostar, the solar mass, the constant accretion rate in units of , and P and T are the pressure and temperature in the midplane of the disk, respectively. is the mass absorption coefficient and µ the mean molecular weight. The opacity is determined by the opacity of the dust and the gas (see Sect. 2.1).

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 t

is the initial position of the parcel at .

The vertical component of the gas velocity accounting for the decrease of the height h of the accretion disk with decreasing s is

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.

Table 1. Model parameters of the accretion disk used in the computation of disk structure

2.1. Opacity

In 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 H₂ O and TiO, see for instance the Figs. in Alexander 1975 or Sharp 1992). In the region where molecular extinction dominates the mass extinction coefficient of the gas can be approximated by

(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 viscous instability in accretion disks. Test calculations for the time evolution of a protoplanetary disk using this opacity law, indeed, showed this type of instability triggered by the steep temperature raise of to occur (cf. Bell and Lin 1994).

(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^-8 g cm^-3 together with the interpolated value of the total opacity. The total mass extinction coefficient of the disk material (dust + gas) finally is obtained by the simple summation

Fig. 1. Interpolation of the extinction coefficient. The dashed curves show the extinction by the indicated opacity sources for a mass density of g/ cm3. The full line is the interpolated extinction coefficient according to the prescription (12).

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 T and P in the radius interval AU AU which results from the steep increase of opacity in the region of hydrogen dissociation. If one starts at large s and moves inwards one first moves on the lower branch of the solution into the multiple valued region. In the stationary case there results a strong upwards jump in the temperature if one reaches the lower critical radius at 0.0728 AU where becomes singular. As one moves further inwards the solution then continues on the high temperature branch.

Fig. 2. Structure of a stationary accretion disk with opacity law (12) for the gas component.

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., H₂ O) can lead to similar consequences. We will address the question of the chemical evolution of unstationary accretion disks in a forthcoming paper.

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 F and G:

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

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