Astron. Astrophys. 356, 795-807 (2000)

3. Radiation transfer model

The approach adopted by XAD (see also Xilouris et al. 1997,1999) is to simulate the optical/NIR appearance of edge-on spirals by creating a model galaxy consisting of an exponential stellar disk, a bulge and an exponential dust distribution. A pixel-to-pixel comparison is made between the real and simulated object in order to fit scale-heights (z-direction) and scale-lengths (radial direction) to both the stellar and dust disks and to determine the optical depth through the centre of the galaxy (as seen face-on). In addition to radiation absorption, the model takes account of photon scattering in the disk by adopting the Henyey-Greenstein phase function (Henyey & Greenstein 1941; Bohren & Huffman 1983). Our confidence in the model is significantly boosted by the fact that the output parameters are highly consistent across several optical and NIR wavebands. Thus for NGC 891, the radial scale-length for the grain distribution is calculated to be 8.1 kpc in the V-band whilst a determination in the remaining filters (K,J,I,B) strays by only 5-10% from this value. Simulations for 7 edge-on galaxies indicate, in all cases, an extinction lane consistent with Galactic-type reddening. It is important to emphasize that our modelling of the dust lane vastly transcends simplistic screen models (which have often been applied in the past) and takes full account of mixing between stars and dust. Admittedly, one obvious limitation to our simulation is that no account is taken of dust clumping or indeed spiral structure within the disk. Kylafis et al. (1999) have tested the influence of spiral structure on the radiation transfer in NGC 891 and found that it produces almost no change to the large-scale properties inferred from the model. Clumping, on the other hand, may well `hide' large amounts of dust so that it cannot be detected by a fit to the large-scale extinction. We postpone a discussion on the effects that clumping might have on our results until later (Sect. 4.2).

Since we shall be concentrating on the V-band when making a comparison between extinction in NGC 891 and the submm emission detected by ABR, we reproduce in Table 1 the relevant parameters fitted by XAD at this wavelength. Using the tabulated parameters, we can construct a synthetic map of V-band optical depth (), which, after smoothing to the same spatial resolution as the m SCUBA map ( FWHM), can be compared directly with the submm image. In Figs. 1 and 2, we display profiles of along the minor and major axes of the galaxy model after the smoothing process. The corresponding m cross-sections are also shown. In all cases, a width of ( beam FWHM) is used to sample the emission/optical-depth perpendicular to the profile direction. The submm profile, in Fig. 1, is the mean of several transects across the major axis taken at various distances from the galactic nucleus. The average profile has been normalized, a long with the curve, to unity at zero z-height. We can see that the observed and model profiles are extremely similar in this direction (a deconvolved size of or 610 pc for both FWHM). To some extent, this similarity may be attributed to the point spread function (PSF) being quite large with respect to the intrinsic dust layer. However, by artificially varying the scale-height of , before smoothing, we have assured ourselves that the thickness of the absorption lane must lie within 30% of the submm width. For z-heights of (900-2300 pc), there is an excess at m over what would be expected purely from the extinction model. The SCUBA beam is principally gaussian but slight wings in the PSF may contribute to this observed excess. In fact, until a more careful comparison is made between the high-z `emission' and the PSF wings (Alton et al. 1999a), we are not in a position to attribute this excess to the `vertical' dust chimneys evident in optical images of NGC 891 (Howk & Savage 1997; Dettmar 1990).

Fig. 1. Submm profile along the minor axis of NGC 891 compared with the corresponding extinction model of Xilouris et al. (1998). The solid curve and markers denote the average profile at m measured by SCUBA. The dashed line represents the V-band optical depth () that is predicted from radiation transfer modelling, after convolving to the same spatial resolution as the submm data. The dotted line shows the point spread function (PSF) measured for the submm data. All curves have been normalized to a value of unity at zero z-height.

Fig. 2. Submm profile along the major axis of NGC 891 (solid line) compared with the corresponding extinction model of Xilouris et al. 1999 (dashed line). The profile in optical depth () was created only after the extinction model had been convolved to the same spatial resolution as the submm data (this explains why here is a factor 4 or so lower than the opacity derived directly from optical images). Positive distances along the major axis correspond to the northeast half of the disk. photon errors in the m profile are 2 mJy/beam.

Table 1. Properties of the grain and stellar disks in NGC 891. The Right Ascension (R.A.) and Declination (Dec.) are given by the B1950.0 nuclear position recorded by Sukumar & Allen (1991). The remaining parameters are derived from a V-band radiation transfer simulation carried out by Xilouris et al. (1998). and denote the exponential scale-height of the stars and dust respectively. Similarly, and represent the exponential scale-length of the stellar and dust disks. is the V-band optical depth through the centre of the disk, if the galaxy were to be viewed exactly face-on. is the inclination of the disk with respect to the plane of the sky.

For profiles along the major axis, we plot the optical depth given by the model against the observed m surface brightness (Fig. 2). It should be noted that when the comparison is made here, it is after the opacity model has been smoothed to the same resolution as the submm image (thus is a factor of 4 lower than the optical depth directly inferred from the original optical image). The correspondance along the major axis is not as good as that for the minor axis and can only be described as fair. Due to its simplicity, we clearly cannot expect the XAD model to reproduce the local fluctuations in the submm profile. However, on average, the curve still appears to be somewhat more extended than the m emission. In fact, if we could choose a curve that would match the behaviour of SCUBA data perfectly, the dust model would have a scale-length of 5.3 kpc as opposed to 8.1 kpc derived by XAD. Under such circumstances the dust layer would possess the same radial fall-off as the V-band stars. The ratio between optical depth and m surface brightness, averaged over the profiles in Fig. 2, is as follows:

where is the submm surface brightness, in Jy/beam, and the uncertainty represents the standard deviation of a linear fit between and .

Although the major axis fit between submm emission and optical attenuation is only fair, we will assume that in both cases the same population of grains manifests itself. Certainly, simulations of dust bathed in interstellar radiation fields suggest that the classical `big grains', which are responsible for extinction (size m), increasingly dominate the emission process at wavelengths beyond m (Desert et al. 1990). And indeed, observationally, Bianchi et al. (1998) have recently established a convincing correlation between optical extinction and submm emission in the nearby spiral NGC 7331. For NGC 891, we therefore fix the m surface brightness with respect to the visual optical depth, , according to Eq. 1. By doing this, we will be able to derive the dust emissivity at m. This calculation is carried out in the next section.

© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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