## 5. Quantitative models for AFGL2688In order to model the IR continuum emission we observe in AFGL2688, we have used a code which solves the radiative transfer problem for circumstellar dust shells in axial symmetry, and which was first described by Collison & Fix (1991; hereafter CF). The code treats absorption, reemission and scattering of radiation by grains in a torus of arbitrary optical depth. Scattering is presently assumed to be isotropic in the model, which is probably not as realistic as forward peaked scattering. Only a single grain size is used presently: multiple grain size models will be constructed in the future. We do not propose to generate models which explain every aspect of AFGL2688's spectral energy distribution (SED) and morphology. Instead, we present two models that demonstrate the gross morphological and SED differences between a model that is optically thin ( 0.4) in the mid-IR and a model that is optically thick in the mid-IR ( 2.4). ## 5.1. Description of modelThe code is implemented as described by CF, with a few modifications. We have employed the slightly improved iteration scheme as described by Granato & Danese (1994). The density distribution in the envelope has been modified, and is now described as where is the dust grain number density at
radius In addition to specifying the structure of the dustshell with the A,B,C,D and E parameters, we need to set the density of the dustshell by specifying an equatorial optical depth at a wavelength . For the density distribution given by Eq. 2, the value of is now given by where and are the inner and outer radii of the envelope, is the ratio , and is the mass absorption coefficient of the dust at wavelength . The function is the incomplete gamma function, for which accurate numerical solutions are straightforwardly obtained. The full radiative transfer equation is solved following the formalism of CF. As pointed out by Granato & Danese, when a density discontinuity is used, such as that at the boundary of the bicone cavities in our model of AFGL2688, it is important to set the boundary conditions on the points of the latitude grid on both sides of the discontinuity. For the current density distribution, the total mass of the envelope is given by which can be solved straightforwardly to a high degree of accuracy
(better than 1 part in 10 We initially assumed a total luminosity for AFGL2688 of 1.4
10 We have three criteria we can use to judge the success of our AFGL2688 models. Firstly, we can compare the shape of the model nebula with the observations presented in Figs. 1 through 4 (Figs. 11 and 12). Secondly, we can compare radial profiles from the model with observations (Figs. 10, 11 and 12). Thirdly, we can compare the model SED with the observed photometry from the optical to the IR, where all the fluxes are integrated over the entire nebula. In Fig. 13, we plot observations from Ney et al. (1975), Latter et al. (1993), Justtanont et al. (1996) and Kleinmann et al. (1978). It should be noted in passing that AFGL2688 is known to have a strong feature in its spectrum, peaking close to 38 µm, which may be attributable to a carbon based dust grain or MgS (Cox 1993), and the emission from this feature strongly affects the SED between about 20 and 60 µm. We will not attempt to fit this feature in our models.
We use the same geometric parameters for both the optically thin and optically thick models. The value of 1+A (the equator-to-pole density contrast) is 4, the value of D (the inner bicone density factor) is 0.025 and the value of 1+E (the superwind mass-loss rate increase) is 20. The value of B (the geometric dilution factor) is 2. The value of C (sharpness of the superwind turn on) is 3. The grain size used for this model was 0.1 µm. The opening angle of the biconical outflow is arbitrarily set to be 9° in this model, following the results of Latter et al. (1993). ## 5.2. Optically thin modelWe present in Fig. 11 a model for AFGL2688 which is optically thin
in the mid-IR. Specifically, at a wavelength of 10.0
µm the optical depth is 0.4 in the equatorial
plane. The inner radius of the torus was found to be 5000 stellar
radii (3.3 10 ## 5.3. Optically thick modelIn Fig. 12 we present an optically thick model. In this model we
find the inner radius of the torus to be only 1600 stellar radii (1.4
10
The only significant difference between the optically thick model
and observations is at very long wavelengths (2600
µm), where we predict a very small source,
slightly elliptical and of order 3" in size. In contrast, the
continuum regions detected by Yamamura et al. (1995) at 2.6mm and by
Knapp et al. (1994) at 3.6cm are much larger with radii greater than
15" for the emission regions. We also underestimate the flux at these
long wavelengths, detecting only about 50% of the flux seen by
Yamamura et al., and a similar fraction of that seen by Knapp et al.
One possible explanation for the model's underestimate is that our
pre-superwind mass loss rate is too low or varies in intensity. It is
also possible that some fraction of the extended emission seen at
3.6cm is due to molecular line emission included within the spectral
bandpass, although there are no obvious candidates for a molecule
heavy enough and abundant enough to generate the required emission at
these wavelengths. Nevertheless, our purpose here in presenting our
current modelling efforts is to demonstrate that the AFGL2688 torus
## 5.4. Constraints on modelsOur model contains many parameters, some of which are well
constrained, and others less so. Here we describe our model's
sensitivity to some of the important parameters. The lower density
biconical sections described by and D are
important. Our observations of H ## 5.5. Derived physical parametersIn Table 2, we list some physical parameters derived from our model. As stated before, we stress that these are not intended to be definitive, but to give a reasonable indication of the likely values, consistent with the model we have qualitatively sketched out in the previous section. All masses are derived assuming a gas-to-dust ratio of 200 by mass, and a constant outflow velocity of 22.4km/sec throughout the nebula. The inner radius in this model implies that large scale mass loss stopped about 190 years ago. The model contains a total mass of nearly 5M of material, but the mass distribution beyond the supershell has not yet been well constrained. The mass required in the supershell alone, however, suggests that the progenitor must have had a fairly high mass, toward the high end of the mass range for AGB stars.
© European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |