The 2 - D structure of dusty disks around Herbig Ae / Be stars I . Models with grey opacities

In this paper the two-dimensional structure of protoplanetary disks around Herbig Ae/Be stars is studied. This is done by constructing a self-consistent model based on 2-D radiative transfer coupled to the equation of vertical hydrostatics. As a simplifying assumption a grey opacity is used. It is found that the disk can adopt four different structures, dependent on the surface density distribution Σ(R) as a function of radius, i.e. on radial-and vertical optical depth of the disk. For the case of high to intermediate vertical optical depth, the temperature and density structures are in agreement with the simple " disk with inner hole " model of Dullemond, Dominik & Natta (2001, henceforth DDN01). At large radii the disk adopts a flaring shape as expected, and near the dust destruction radius (located at about 0.5AU for most Herbig Ae stars) the disk is superheated and puffs up. The region directly behind this " puffed-up inner dust wall " is shadowed, as predicted by DDN01. For the case of intermediate to low vertical optical depth, but still high radial optical depth, the 2-D models show that the shadow can cover the entire disk. For such competely self-shadowed disks the inner rim emission in the near infrared constitutes the dominant part of the SED, since the flaring component in the mid-and far infrared is suppressed by the self-shadowing effect. When the disk is optically thin even in radial direction, it becomes unshadowed again because the inner rim can no longer block the stellar light. Such disks have relatively weak infrared excess compared to the stellar flux. Finally, for disks that flare at intermediate radii, but become too optically thin at large radii, the outer parts once again become shadowed. But this time the shadowing is caused by the flaring part of the disk, instead of the inner rim. The disk then consists of a bright inner rim, a shadow, a flaring part and finally a (dim) shadowed outer part. Different observational methods of determining the size of the disk (e.g. from the SED, from continuum mapping or from CO mapping) may yield different results.