5. Quantitative models for AFGL2688
In 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 model
The 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 r and latitude , the dust grain number density on the polar axis at the inner edge of the envelope, and A, B, C, D and E are user supplied constants. The wind of the AGB progenitor is assumed to have abruptly increased in strength by a factor 1+E at the termination of its AGB phase, as indicated in various other post-AGB stars (e.g. Skinner et al. 1994; Meixner et al. 1993, 1995). Prior to this stage the wind is spherically symmetric. As the star enters the `superwind' phase, the wind becomes toroidal, and the resulting toroidal `supershell' has a radius defined by Eq. (2). The equator to pole density contrast for the torus is a factor 1+A. The post-AGB fast wind has punched its way through the supershell, generating a low density bicone, and is the opening half-angle of the bicone. The dust grain number density at the inner edge of the envelope inside the bicone is actually D. The factor C determines how abruptly the mass-loss changed from an isotropic outflow at a modest rate to a toroidal superwind. The factor B is the usual geometrical dilution factor, equal to two for a constant mass-loss rate.
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 106). (Note that there are no solutions to Eqs. 3 or 4 for B=1.00, nor to Eq. 4 for B=3.00: however, alternative equations can be derived if necessary to allow for a solution for these cases.)
We initially assumed a total luminosity for AFGL2688 of 1.4 104 L , and a stellar effective temperature of 6500K, consistent with Cohen & Kuhi (1977). We then varied the luminosity until a satisfactory fit to the observations could be obtained. The star is assumed here to emit like a blackbody. The dust is assumed to be entirely amorphous carbon, because the mid-IR spectrum of AFGL2688 is almost featureless (Justtanont et al. 1996). Optical constants for amorphous carbon have been taken from Hoare (1990). A single grain size is used in this model, and we have tried a variety of values.
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 model
We 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 1016 cm), which at a distance of 1.2kpc corresponds to 1.8". We discuss the model's sensitvity to these parameters at the end of this subsection. This model predicts that at a wavelength of 3.4 µm the nebula should be teardrop shaped, with FWHM roughly 4.5", and with constant surface brightness over the central 3" or so. In the mid-IR the model predicts that we should see a slightly limb-brightened disk with radius roughly 3.6". Both the model's 3.4 µm and 9.8 µm predictions are in clear contradiction to the observations. The core of the IR nebula is considerably more compact than this model would suggest, which confirms that indeed the optical depth of the nebula must be very large. Further confirmation comes from Fig. 13, where we plot the SED of the model. The model flux is reasonably close to the observations at most wavelengths. However, the hump in the model SED around 1-3 µm is the reddened stellar photosphere: direct stellar radiation has not been shown in the maps in Fig. 11, because it greatly outshines the reflection nebulosity in the near-IR and optical. In a source with this low an optical depth, in the near-IR we would see a bright star, with a much fainter halo of nebulosity around it. To remedy the various deficiencies in this model, we must increase the optical depth at 10 µm significantly. This result stands irrespective of the angle from which the torus is viewed. As the system is viewed from progressively closer to the polar axis, the central star becomes brighter relative to the nebula, and the nebula becomes more annular. Thus a low optical depth in the mid-IR is entirely ruled out.
5.3. Optically thick model
In 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 1016 cm), which is 0.76" at 1.2kpc). The optical depth at 10.0 µm is now 2.4, and we see that the resulting morphologies are much closer to what is observed. The torus is no longer seen at any wavelength for which we have an image. Instead, the model nebula is elongated in the direction of the polar axes both in the mid- and near-IR. Comparison of the observed cross-cuts (Fig. 10b) and images (Fig. 1, 2, 3 and 4) with model predictions for the cross-cuts and images (both in Figs. 11 and 12) show fairly good agreement: the lobes steadily grow closer together, and the length of the nebula decreases, with increasing wavelength. The ratios of the brightnesses of the two lobes in the model are slightly larger than observed, but similar. All these details agree reasonably with the observations, although we should be wary of overinterpreting such details as these because, as we mentioned earlier, our model is assuming isotropic scattering and a single grain size, neither of which is likely to be realistic. Similarly, the appearance of horns in these model images should not be taken to be very significant - horns are produced relatively easily, and we do not regard the morphology we have used in our models as definitive. The SED produced by this model is shown in Fig. 14, and provides a fair fit to the observations. The scattering in the optical is somewhat excessive, and in the near-IR somewhat deficient due to our single grain size assumption. The model's central star is no longer directly seen in the SED, nor in the model images in agreement with observations. The length of the nebula at R is somewhat excessive (about 90" from end to end; Fig. 12), while at K it is somewhat too small (11" at the 1% level, compared with 16" in our observations). The dramatic reduction in size as we progress towards longer wavelengths would not occur if we adopted a distribution of grain sizes instead of a single grain size. The shape of the nebula at 3.4 µm is again similar to what we observe, but the size is much smaller. At 9.8 µm we generate a model with a length of about 4", a little smaller than observed. Note that many of the sharp gradients in our model images would be smoothed out if we were to convolve with a circular PSF of FWHM 1.2" (nearly 2r ), as appropriate to our observations.
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 must be optically thick at 10 µm, and that a superwind plus spherical AGB wind model can plausibly reproduce the observed general characteristics of the nebula. We will further investigate the physical structure of the AFGL2688 nebula in much more detail in a forthcoming paper.
5.4. Constraints on models
Our 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 H2 imply that a fast wind has indeed carved out cavities in the nebula. The `waist' seen in the optical and near-IR images demands an extremely large equatorial optical depth for this system, yet the very large reflection nebula requires a rather low optical depth in the direction of the lobes. The amount of scattered radiation, which fully determines the SED in the optical and near-IR, is very sensitive to the value of the parameters D (the inner bicone density factor) and (the bicone opening half-angle). The value of B (the density geometric dilution factor) we have left at 2, consistent with a constant mass loss rate. Any other value would imply a slowly time-varying mass loss rate. The value of (the supershell inner radius) has a strong effect on the colour of the dust emission spectrum, and (the toroidal supershell outer radius) significantly affects the separation of the reflection lobes, and both are well constrained by our data. The value of E (the superwind mass-loss rate increase) has a large effect on the size of the reflection nebula, and can be well constrained. Models without a large amplification in mass loss rate in the superwind tend to produce excessive flux in the far-IR, and require a larger total mass in the envelope to fit the SED. Such models also tend to produce a rather extended nebula in the 10 and 20 µm regions, compared with the fairly compact source which we observe at wavelengths as long as 19.2 µm. The value of A (1+A is the equator to pole density ratio) is not very well constrained. Varying A has some effect on the slope and width of the SED of the dust shell, but the effect is not particularly pronounced when the optical depth is so large. The values of A, D and are not independent, but the images and SED provide separate constraints on these three parameters. Finally, we note that the luminosity of our model is much larger than the 1.3 104 L determined by Cohen & Kuhi (1977) (corrected for the different distance which they assumed), but they pointed out that their luminosity estimate (which was based on the SED, under the assumption that the torus was optically thin in the mid-IR) was highly uncertain because of the unknown physical structure of the nebula. Because our model has a strong departure from spherical symmetry that allows radiation to escape easily in the polar directions, while in the equatorial direction we only see multiply reprocessed radiation emerging from the dust shell, the apparent luminosity of AFGL2688 is dependent on viewing angle.
5.5. Derived physical parameters
In 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.
Table 2. Physical parameters and results for models
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998