Astron. Astrophys. 324, 32-40 (1997)

4. Comparison between extinction and gas column density

4.1. versus the column density of total gas (HI+H₂)

As discussed in Paper II, a correlation between dust column density and gas column density is expected given the tight connection between heavy elements in gas phase and in grains (Spitzer 1978) provided that the metallicity does not vary a lot among galaxies. Indeed, it has been shown (see Roberts & Haynes 1994 for a review) that the variation in metallicity becomes important only for galaxies fainter than . In our sample, we have only 9 galaxies with and 9 galaxies with .

Then, according to the relative distribution of the dust, gas and stars, a physical link between the extinction and the gas column density may or may not exist: if the dust and gas are well mixed in a slab in front of the stars the extinction is proportional to the optical depth and to the gas column density when one assumes a constant value for ; if the dust gas and stars are approximately uniformly mixed in galaxy disks, this also implies some link between the extinction and the gas column density but the relation between these two quantities depends on the exact distribution of each component (dust, gas and stars). On the other hand if the dust distribution is very different from the stellar distribution, for example when that the dust is concentrated in a few opaque dust lines covering only a small portion of the stellar disk, or when there is a dust hole in the inner part of a galaxy disk where most of the stellar light comes from, the extinction may not depend on the average dust column density. In this case the extinction correction based on the gas column density will be unreliable.

In the following we correlate the extinction calculated in the previous section with the gas column density for galaxies in our sample. In the literature, the gas column density is usually estimated from the HI and CO flux divided by a disk area estimated from optical diameters of galaxies (Donas et al. 1987; Buat et al. 1989). We shall take the same approach since the majority of the target galaxies in this study are unresolved in CO and HI.

Young et al. (1995) have estimated that the CO distribution can be described by an exponential disk with a scale length equal to one tenth the optical diameter at the 25th magnitude (the in RC3). On the other hand, the ratio of the scale length of the B distribution to is 0.15 (Peletier et al. 1994). Hence we can adopt a ratio of the B to CO radius equal to 1.5 to estimate the molecular column density. Using the to CO conversion factor of Strong et al. (1988) leads to:

and to a mean column density given by:

where is the total CO flux in Jy km/s and the optical diameter in arcmin. Recent observations of the Milky Way and of nearby spirals, however, suggest a lower to CO conversion factor, of the order of = 10²⁰  mol cm (Lequeux 1995), thus would be overestimated by a factor of two. Therefore calculated using the above formula is a crude estimate of the real molecular gas column density of the target galaxies: the optical to CO disc scale length ratio of 1.5 is an average value, and in a given galaxy might be wrong up to a factor of two. Furthermore the determination of the molecular gas surface density is based on the universality of the CO to H₂ conversion factor, while it is well known that it might easily change from galaxy to galaxy (even in the bright and massive spirals) by a factor of two (Boselli et al 1996).

On the other hand, the atomic gas extends over a region much more extended than the optical disk: the isophotal HI diameter has been found to be about twice the optical one with often a rather flat distribution very different from that of the optical light (e.g. Bosma 1981, Cayatte et al. 1994). The HI column densities are therefore obtained by dividing the HI mass by an area with a diameter of . We caution that this is only a very crude estimate, in particular for galaxies located in clusters which are known to be HI deficient. This deficiency has been shown to reduce the effective HI diameters and gas densities, a factor directly influencing any estimate of a mean HI column density based on a uniform choice for the HI diameter (Cayatte et al. 1994).

Fig. 2 is a plot of the extinction v.s. the column density (i.e. the face-on surface density divided by cos(i) with i the inclination angle) of the total gas (HI+H₂). A weak but significant linear correlation, with r=0.48, is found for the 51 galaxies with CO detections. In order to exploit also the information carried by the upper limits of both variables, a generalized Kendall's rank correlation statistical analysis (Isobe & Feigelson 1986) is carried out for the entire sample (79 sources). Again a significant correlation, with a probability for the null-hypothesis of 4 , is found.

Fig. 2. Diagram of the extinction in the B band versus the total gas (HI+ H2) column density . The arrows represent the upper limits.

4.2. Estimation of the extinction from the gas column density

Now we shall compare with the extinction estimated using the total gas column density. Assuming a Solar Neighbourhood blue-optical-depth-to-gas ratio of (atom^-1 cm²) (Savage & Mathis 1979), the face-on optical depth can be estimated from gas surface density (face-on) as following

The extinction can then be determined from the optical depth and the inclination angle, depending on the adopted radiative transfer model. Neglecting scattering effects, the `plane-parallel slab' model (Mihalas 1978), with stellar and dust layers of identical thickness, gives

where is the inclination angle (0 for face-on).

In Fig. 3 the extinction so calculated is compared with . The Kendall's rank correlation between the two variables is fairly strong (probability for the null-hypothesis of 3.7 ). Nevertheless, the slab model obviously over-estimates the extinction as compared to our "frequency converter" model.

Fig. 3. Diagram of versus the extinction estimated from total gas (HI+ H2) column density using a `slab' model.

A more realistic model is the so called 'Sandwich model' which includes different thickness of the stellar disk and the dust disk. Assuming that the stellar disk is twice as thick as the dust disk (Disney et al. 1989), one get

A further refinement is to take into account the scattering effect within the `Sandwich model'. To perform such calculations, we use our radiative transfer model presented in Paper I and II. Some relevant details about this model are given in the Appendix. The resulting is compared with in Fig. 4. The Kendall's rank correlation statistic gives a higher significance to the correlation in this plot (probability for the null-hypothesis of 1 ) compared to the one in Fig. 3. The extinction calculated from gas column density using this model, , matches the extinction estimated from the FIR, UV and optical radiations, , satisfactorily well: for the galaxies with an estimated from our "frequency converter" model and an available gas column density , we find a mean difference between and of , with a dispersion of 0.18 mag.

Fig. 4. Diagram of versus the extinction estimated from total gas (HI+ H2) column density using a 'Sandwich model' including scattering (Appendix).

4.3. versus column density of H₂ and HI separately

In this section we address the question whether the dust associated with the molecular gas or with the atomic gas dominates the extinction. In Fig. 5 and Fig. 6 plots of versus the column density of the H₂ gas and of the HI gas for the 79 galaxies in our sample are presented respectively. A significant Kendall's rank correlation between and is found for the entire sample, with a probability for null hypothesis of 7 10^-5, while the linear correlation coefficient for the detected data points (54 sources) is 0.43. On the other hand, and HI gas are not correlated: Kendall's rank correlation statistics tells that a probability for null hypothesis is as high as 0.38, and the linear correlation coefficient for the detected data points (63 sources) is only 0.15. These results are in good agreement with those in Paper II, in which we found significant correlation between optical depth and the H₂ surface density, but no correlation between and the HI surface density.

Fig. 5. Diagram of the extinction in the B band versus the H2 gas column density . The arrows represent the upperlimits.

Fig. 6. Diagram of the extinction in the B band versus the HI gas column density .

Can the absence of a correlation between and be due to the environment effects such as HI stripping, because many of galaxies in our sample are in clusters? The HI stripping mainly affects the external parts of the galaxies and therefore reduces the effective HI diameter: the normalization of the HI flux over twice the optical area induces an underestimate of the HI column density in HI deficient galaxies compared to the HI non-deficient ones. This problem is also relevant for the result in Paper II because most of galaxies studied in that paper for the dust/gas relation are in the Virgo cluster.

The effect of this bias has been investigated by estimating the HI deficiency of all the cluster galaxies of our sample using the formula of Haynes & Giovanelli (1984) for all morphological types. The UGC diameters necessary to estimate the HI mass of isolated galaxies has been obtained from following the RC3:

We thus obtain the formula:

where is the optical radius in kpc and the observed HI mass in solar units.

We split our sample in three sub-samples: (deficient cluster galaxies), (non deficient cluster galaxies) and non cluster galaxies. No correlation between and is found in any of the three sub-samples, nor in the entire sample.

A plausible interpretation for the results in Fig. 5 and Fig. 6, namely a correlation between and and non-correlation between and , is that the dust associated with the H₂ gas contributes dominantly to the extinction and the contribution from the dust associated with HI gas is relatively insignificant. Hence is insensitive to . This is hinted by the fact that for most of galaxies in our sample is at least a factor of few times higher than . Indeed, the mean is with a dispersion of 0.52 for the entire sample. Even if we consider only HI non deficient and non cluster galaxies in order to avoid the effect of the HI deficiency with a dispersion of 0.48. This value is similar to that found for a sub-sample of 29 galaxies, mainly located in Virgo, for which radial HI distributions are available (V. Cayatte, private communication): with a dispersion of 0.53. The relative H₂ richness of the present sample is probably due to the adopted FIR selection criterium, which favors CO rich galaxies (Boselli et al. 1996). However, in the case where the assumed X conversion factor, and thus the column density, is overestimated by a factor of two (see Sect. 4.1), the atomic and molecular gas surface densities are found similar. Nevertheless, as discussed in Paper II, the extinction deduced from our model is representative of the extinction occuring in regions with a high UV and FIR emission i.e. the inner disk. Given the exponential distribution of the molecular gas and the rather flat distribution of the atomic one, it is likely that the molecular phase dominates the atomic one in these regions. Therefore, in our sample galaxies the dust causing extinction is likely to be mainly associated to the molecular phase of the gas, therefore the extinction is found to correlate with the molecular content.

The above interpretation can be tested in the following way. If indeed the non-correlation between and is due to the dominance of , then for a subsample of the galaxies with high HI-to-H2 ratios there should be a v.s. correlation. To this aim we have first chosen non-cluster galaxies or cluster members which are non-HI-deficient (HIdef 0.3). This selection would ensure us to exclude galaxies with a truncated HI distribution. The selected galaxies must also exhibit an HI to H₂ ratio larger than 10. Given that the adopted linear sizes of the HI disks are a factor of 3 larger than that of the H2 disks (Sect. 4), this means that the HI surface density of these galaxies is more than 1.1 times of that of the H₂ gas. Once again, if the assumed X conversion factor is overestimated by a factor of two, the HI surface density of the selected galaxies is more than 2 times the molecular one. The sub-sample contains 11 galaxies and is presented in Table 1.

Table 1. Galaxies with .

The diagram of v.s. for this subsample is plotted in Fig. 7. Indeed a significant correlation is found in this plot with the linear correlation coefficient of r=0.84 for the 10 detected data and, for the entire subsample (11 sources), a probability of 0.02 for the null hypothesis is found from the generalized Kendall's rank correlation. In Fig. 8 we plot , estimated from the HI surface density using the `Sandwich+scattering' method (Appendix) and the optical depth to HI gas ratio of the SN, v.s. for the sample. The agreement between estimated from and is quite good. In above two plots, the four galaxies fainter than (NGC 4299, NGC 4383, NGC 3353 and NGC 5474) are marked by circled crosses. Except for NGC 3353, no significant difference in were found for these galaxies compared to the rest of the subsample. NGC 3353 has a large / ratio, resulted from its rather high FIR flux.

Fig. 7. versus the HI column density for galaxies with .

Fig. 8. versus the extinction estimated from HI gas column density, for galaxies with .

From the above results we conclude that for most of the galaxies in our sample the extinction is mainly due to the dust associated with the molecular gas as indicated by the good correlation between the extinction and the column density of the molecular gas, and by the high molecular to atomic gas column density ratio. On the other hand, for galaxies whose gas column density is largely dominated by the atomic gas, the extinction seems to be mainly caused by the dust associated with atomic gas. This is corroborated by recent sub-mm observation (Guélin et al. 1993, Neininger et al. 1996) where the dust emission at 1.2 mm in spiral disk is found to follow the dominant gas phase either atomic or molecular. Our results are also in agreement with those of Andreani et al. (1995) who found that the cold dust emission is probably associated to both the atomic and the molecular phases.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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