4. Submillimeter emissivity
As mentioned in Sect. 2, the m waveband is completely dominated by cold dust (15-20K) at all positions in the disk. In fact, the Rayleigh-Jeans tail of the warm grain component contributes about 8% to the flux density in this filter. It possible, therefore, to rewrite Eq. 1 approximately as follows:
where , the number of cold grains, comprises the bulk of interstellar dust in NGC 891. In order to be able to use the ratio (Eq. 2) in this relation, we have to refer to the definition of optical depth, namely:
where is the dust column density (grains per unit surface area of galaxy), , as before, represents the geometrical grain cross-section and constitutes the emissivity, or extinction efficiency, in the V-band.
If we can derive the flux density per unit surface area, in Eq. 3 becomes and we will be in a position to substitute , from Eq. 4, into Eq. 3. To do this we use , the m surface brightness (in Jy/beam). Dividing by the beam area (in m2), and multiplying by to convert from Jansky to Wm-2Hz-1, we obtain the flux density per unit surface area (as required). This can be substituted into the left-hand side of Eq. 3 thus:
where has now replaced .
To derive directly from Eq. 7, we must have some knowledge of . For `classical' grains of radius m, which are responsible for optical extinction in the diffuse ISM of the Milky Way, lies between 1 and 2 (Whittet 1992; Alton 1996; Spitzer 1978). The interstellar grains of NGC 891 are unlikely to be much different in this respect, given that extinction within the disk corresponds extremely well to a Galactic reddening law (Sect. 3). Thus we use in Eq. 7 to infer:
The grain emissivity within the Milky Way has usually been estimated at shorter FIR wavelengths (e.g. Hildebrand finds ). It is difficult, therefore, to compare Galactic data with Eq. 8 unless we assume some sort of wavelength behaviour for the dust emissivity. Fortunately, a large amount of recent observational evidence seems to be converging on an emissivity, for the general ISM of non-active spirals , which varies as where the spectral index for m. Indeed, rigorous FIR spectral measurements of Milky Way dust, using both COBE and balloon-borne instruments, imply a spectral index between 1.5 and 2.0 for the diffuse ISM in the Galaxy (Masi et al. 1995; Reach et al. 1995). Likewise, the relatively high flux density at m compared to emission at m, for galaxies such as NGC 891 and NGC 7331, is indicative of an index (Bianchi et al. 1998). Accordingly, we plot in Fig. 3 the dust emissivity for NGC 891 (as a ratio ) assuming a wavelength dependency (). Estimates for the FIR emissivity of Milky Way dust, as derived by Hildebrand (1983), Casey (1991), Draine & Lee (1984) and Bianchi et al. (1999b) (see Sect. 2), are annotated on this plot. We also add the emissivity measured by Agladze et al. (1994) for `astronomical-type' silicates tested at 20K under laboratory-controlled conditions.
The FIR grain emissivity in NGC 891 appears to be somewhat higher than most estimates of the same property in the Milky Way. In particular, the `canonical' values of Draine & Lee (1984) and Hildebrand (1983) are about a factor of 2-3 lower than the quantity we have derived here. The discrepancy is admittedly smaller for an emissivity law following , but both the Draine and Lee model and the Bianchi determination are consistent with and should, therefore, strictly speaking, be compared with the steepest line in Fig. 3. Similarly, the lowest values of , as given by Hildebrand, are derived under the assumption that . Therefore, the agreement in this case can only be described as marginal. In this respect, the NGC 891 emissivity seems more consistent with the Casey (1991) observations (although why there should be such a difference between the Hildebrand (1983) and Casey (1991) measurements, when both apply to Galactic reflection nebulae, is not altogether clear). Although the FIR emissivity we predict for NGC 891 is somewhat higher than most values used for the Galaxy, it is still much lower than the inference from laboratory-controlled experiments. Indeed, the values of , implied by Agladze et al., are astonishingly high, although the authors themselves do not offer an explanation as to why there is such an incongruence with astronomical observations.
It would certainly be something of a surprise if interstellar grains in NGC 891 possess a submm emissivity grossly different from Milky Way dust (particularly since XAD find that extinction within NGC 891 is characterized by a Galactic reddening law). The velocity fields and distribution of neutral gas within the disks of both systems has been noted by several authors as being strikingly similar (Guelin et al. 1993; Garcia-Burillo et al. 1992; Scoville et al. 1993). However, it is possible that NGC 891 may display a somewhat more active halo with `thick disks' of both neutral and ionized gas extending to several kpc above the midplane (Dettmar 1990). Since this extended disk only contains a few percent of the total dust in NGC 891 (Howk & Savage 1997; Alton et al. 1999a), the activity at the disk-halo interface should not necessarily imply any general pecularities for the dust properties of the main disk. Although the error bar in Fig. 3 represents only a 30% standard deviation in the fit between the submm and opacity measurements, the total uncertainty in our estimate of is quite likely to be about a factor of 2 (due to % error in both and ). Moreover, measurements of FIR emissivity for Galactic grains are likely to be uncertain by an order of magnitude once extrapolated to submm wavelengths (Hughes et al. 1993). Therefore we should not be unduly worried by the factor 2-3 difference between the emissivity derived for NGC 891 and estimates of the corresponding property for the Milky Way. It is interesting to note that grain models which successfully reproduce the extinction law within our own Galaxy, can display radically different submm emissivities. For example, the fluffy agglomerates of silicate and carbon, that have been postulated by Matthis & Whiffen (1989), satisfy the near-infrared to ultraviolet reddening law in much the same way as the Draine & Lee (1984) model. However in comparison to the latter, they possess an emissivity nearly 3 times larger at .
As a confirmatory addendum to the preceding paragraph, we note that Dunne et al. (1999) estimate the submm mass-absorption coefficient in NGC 891 () by using our reported values of m flux density (ABR) and fixing the dust mass via depletion arguments rather than observed extinction (a constant fraction of interstellar metals are assumed to be bound up in grains). The Dunne et al analysis yields m2kg-1 which, for grains of radius m and material density 3000 kgm-3, corresponds to , i.e. almost exactly the value we derive in Eq. 8. 2
As mentioned previously, one cause for concern in our derivation of submm emissivity is the fact that the opacity simulations of XAD do not account for dust clumping. This will tend to underestimate the amount of dust causing extinction within the disk. Witt et al. (1999; also Witt priv. comm.) have analysed V-K excesses from spirals disks using homogeneous and clumpy-phase models and found that the latter typically require 50% more dust mass for the same level of extinction. Similarly, the Monte Carlo radiative transfer simulations of Bianchi et al. (1999b), which assign dust clumps according to the distribution of molecular gas, indicate that homogeneous models, such as XAD, will underestimate the dust mass by a factor of 2 for spiral galaxies view ed edge-on. We have attempted to assess the effects of clumping in NGC 891 ourselves, by estimating the amount of dust in gas clouds too optically thick to be detected satisfactorily by the XAD homogeneous model. Our reddest waveband is K, therefore our technique becomes insensitive to environments with (). Within the Galaxy, this corresponds to a column density of cm-2 (Bohlin et al. 1978) which is close to the column density within most giant molecular clouds (GMCs) [see for example Larson (1981), who found cm-2 / for cloud sizes of L=0.05-100pc]. It is possible, then, that we may `miss' a sizeable fraction of grains residing in GMCs. By comparing relative emission strengths detected from 12CO and 13CO molecules in various gas clouds, Polk et al. (1988) estimate that 50% of molecular gas resides in Galactic GMCs and the remainder constitutes a more diffuse medium surrounding GMCs. This information, then, suggests that the XAD model will `overlook' about 50% of the H2 phase and, in total, perhaps 30% of grains associated with the (HI+H2) gas phase of NGC 891. Since the submm is optically thin, any increase in the amount of dust inferred from extinction would decrease in Eq. 8. Indeed, a 50-100% rise in , due to clumping, will yield a value of - for NGC 891. This is much closer to extrapolations of the Galactic emissivity into the submm waveband (). Intriguingly, if dust clumps are preferentially located towards the centre of NGC 891 (where the molecular gas is known to be concentrated; Guelin et al. 1993), a correction for clumping would steepen the profile in Fig. 2 to a shape more akin to the submm distribution.
The submm emissivity we have derived for NGC 891 (Eq. 8) can now be used, in conjunction with our SCUBA images, to create dust maps for the disk. In the following section, these maps are employed to derive the gas-to-dust mass ratio along the major axis. Before embarking on this part of the analysis, however, we recap the method we have used to find dust masses in NGC 891. In lieu of extrapolating the FIR emissivity estimated for the Milky Way to submm observations of NGC 891, we have taken the visual optical depth calculated from radiation transfer modelling (XAD), and effectively constrained the number of grains, , appearing in Eq. 3. This technique relies on knowledge of the extinction efficiency in the V-band, . However, this quantity is relatively well known (), compared with the near order of magnitude uncertainty in assumed for the Milky Way (Hughes et al. 1993; Hughes et al. 1997). Fixing , allows us to use Eq. 3 to derive the m emissivity from the m flux densities in our SCUBA maps.
where a and are the radius and material density of the interstellar grains, respectively, and D is the distance to NGC 891 (see, for example, Hildebrand (1983) for derivation of Eq. 9). The determination of will inevitably rely on assumptions about the grain properties (typically m and kgm-3 are applied), and, furthermore, some uncertainty is introduced by D appearing as the square in Eq. 9. However, the evaluation of is greatly improved by our newly-acquired knowledge of . Moreover, when we consider the neutral gas in NGC 891, we can expect the gas-to-dust mass ratio to be independent of distance. As a final note, we underline the fact that the blackbody intensity in Eq. 9, , varies linearly with temperature because we are in the Rayleigh-Jeans tail of the greybody curve. This means that will change by as little as over the suggested temperature range of 15-20K.
It is important to point out that, whilst we use Eq. 9 to infer the dust distribution from our m map of NGC 891, the overall dust mass has already been normalized to . This is because we have substituted into Eqs. 6-8 in order to determine . Thus, whilst our SCUBA map provides the detailed spatial information concerning the dust distribution, the total amount of dust will not be any different from what we would have directly inferred from the optical depth model of XAD.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000