8 Conclusions

In this paper the structure of passive circumstellar disks was theoretically investigated. The problem was defined in a mathematically "clean'' way: it is the problem of computing the temperature and density structure of rotating circumstellar matter around a star with a certain mass, radius and luminosity from basic principles of radiative transfer, radiative equilibrium and vertical hydrostatic equilibrium. The disk parameters that went into the calculation were the inner and outer radius, and the surface density distribution as a function of radius. The only mathematical approximation made here was the reduction of the hydrostatic equilibrium equations to 1-D vertical equations. From a physical point of view, many more approximations were made (related to dust-gas coupling, active accretion, dust opacities, etc.). But these were necessary to keep the problem clear of uncertain physics for now.

Four different kinds of solutions were found: a flaring disk, a self-shadowed disk, a transparent disk, and a flaring disk with self-shadowed outer region. These solutions are pictographically listed in Fig. 12. The numerical models described in this paper can be downloaded from a website: www.mpa-garching.mpg.de/PUBLICATIONS/DATA/radtrans/grey2d/.

The main conclusions are summarized as follows:

1.

A flaring disk around a Herbig Ae/Be star has a hot inner rim, a shadowed region behind it, and the usual flared geometry at large radii (Fig. 12A). These findings are in accordance with the predictions of Dullemond et al. (2001). But the effect of shadowing in suppressing the emission from the shadowed region is not as strong as was predicted in that paper.

2.

Disks with intermediate to low vertical optical depth but high equatorial optical depth can become entirely self-shadowed (Fig. 12C). The SED falls off more steeply at long wavelength than for flaring disks.

3.

Disks with equatorial optical depths that are smaller than unity are un-shadowed again, since the inner rim can no longer stop the stellar radiation. These disks are fully optically thin (Fig. 12D).

4.

The outer regions of flared disks can become shadowed if beyond a certain radius the surface density becomes too low. This time it is the flared part of the disk that casts the shadow (Fig. 12B). These outer parts do not contribute much to the SED, but may still be detectable using (sub-)millimeter interferometers. Measurements of the outer radius of a disk may therefore yield different results, depending on whether one uses SED-fitting, continuum mapping or CO mapping.

5.

In the case of self-shadowed disks, and the shadowed outer parts of flaring disks, the usual 1+1-D approach to disk modeling breaks down, and so does the approach used by CG97 and DDN01. A full 2-D approach, such as the one used in this paper, is then necessary. It might be that the SED is still relatively well described using a 1+1-D model or a DDN01-type model up to the self-shadowing radius, but this remains to be proven using a 2-D model with more realistic opacities.

6.

Bimodel solutions were not found. It seems that protoplanetary disks obey a kind of "flaring disk principle'': if the disk can flare, it will. Self-shadowed disks are therefore disks which cannot be made to flare.

Many of these conclusions will presumably still hold when more realistic opacities are included. But this will be the topic of the second paper in this series.

Acknowledgements

I wish to thank Carsten Dominik and Antonella Natta for their careful reading of the manuscript and many interesting remarks, Tom Abel for inspiring me during the debugging of the variable eddington tensor code, and Rens Waters and G.-J. van Zadelhoff for useful discussions. I acknowledge support from the European Commission under TMR grant ERBFMRX-CT98-0195 ("Accretion onto black holes, compact objects and prototars'').