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4 Determination of AV and of the dust mass

  The determination of the extinction in both passbands requires comparing the unaffected light with the extinguished one. Assuming that the isophotes of the observed galaxies are intrinsically ellipses, it is possible to build a model image adjusting ellipses to the isophotes of the galaxy. This fitting is done with the isophote routines in the STSDAS package within IRAF. This model image represents the light not extinguished by the dust, since it is not affected by local intensity irregularities caused by the dust absorption and scattering. With this model image it is possible to construct extinction maps in both filters for each galaxy  
A_{\lambda} = - 2.5 \log \left[
\frac{F_{\lambda,\mathrm{obs}}}{F_{\lambda,\mathrm{model}}} \right]\end{displaymath} (1)
where F is the count level for each image pixel. The flux calibration and sky background are not relevant for the extinction maps, since the model incorporates them. In the galaxies with isophotes strongly affected by the dust (e.g. NGC 3489, NGC 5044), the first model was used to identify the dust distribution to be masked in further fittings, at least in the regions where the isophote routines do not easily converge.

The mean AV and AR values derived from the dust maps are listed in Table 2 as well as the mean (V-R) values. Note that the standard deviations are not due to errors, but due to the fact that the extinction is not uniform over the region used in the measurements. The particle density $N_{\mathrm{d}}$ of the dust cloud can be estimated by making simple assumptions about the physical properties of the dust grains together with the mean AV value inside the dust region, as follows.

The theoretical model to compute AV assumes that it is proportional to the optical cross section $C_{\mathrm{ext}}$weighted by the grain size distribution n(a), integrated over the grain sizes (a-<a<a+) along the dust cloud length l in the line of sight. Assuming n(a) to be the same for the whole cloud and using the definition of the efficiency factor $Q_{\mathrm{ext}}(a,\lambda)=C_{\mathrm{ext}}(a,\lambda)/\pi a^{2}$, i.e., the ratio between the extinction and the geometrical cross sections, $A_{\lambda}$ can be written (Spitzer 1978) as:  
A_{\lambda} = 1.086 \;l\; \int_{a_-}^{a_+} \; Q_{\mathrm{ext}}(a,\lambda)
\;\pi a^2 \; n(a) \;{\rm d}a. \end{displaymath} (2)
The efficiency factor is related to the grain sizes and composition and varies with wavelength. $Q_{\mathrm{ext}}$ must be parameterized according to the sizes and optical properties of the grains. In the optical region, assuming that the extinction is caused mainly by silicates (Mathis et al. 1977), $Q_{\mathrm{ext}}$ can be assumed to vary as (Goudfrooij et al. 1994c):
Q_{{V,\rm ext}} = \left\{ 
 ...rm m} \\  
 a_{Si} = 0.1 \; \mu {\rm m}. 
 \end{array}\end{array} \end{eqnarray} (3)
Since the extinction curves of elliptical galaxies in the optical region are similar to that of our Galaxy (Goudfrooij et al. 1994), we can also use the same grain size distribution. Mathis et al. (1977) demonstrated that the function which best reproduces the extinction curve of our Galaxy over a wide range of wavelengths is  
n(a) = n_0 \; a^{-3.5} \end{displaymath} (4)
N_{\rm d} = \int_{a_-}^{a_+} n(a)\;{\rm d}a .\end{displaymath} (5)
By means of Eq. (2) and Eq. (4) $N\rm _d$ can then be evaluated. Eq. (5) is used to compute the normalization factor n0 introducing $N\rm _d$ into Eq. (2). We measured the extinction AV as the mean value of the absorption features in the extinction maps and estimated the associated dust cloud size l from the colour map (V-R). Using these parameters and those previously derived in this section, we evaluate the mass density $\rho$ and the total mass $M_{\mathrm{dust}}$ of the dust cloud.

The derived dust mass ranges between 103 and 105 ${M}_{\odot}$. These values are comparable to the ionized gas mass derived in Paper I. The good correlation between $M_{\mathrm{dust}}$ and $M_{\mathrm{HII}}$ (Fig. 2) suggests that both ISM components are intrinsically related.

Table 2: Measured values of the extinction AV and AR, the colour index (V-R), the mean size l of the dust cloud, the dust mass and the cloud morphology for the galaxies of the sample. Morphology: SD - Small discs, F - Filamentary, RE - Regular extended; Scale relative to gas distribution (Paper I): $ \star $- larger, $\dagger$ - equal, $\ddagger$ - smaller. (Values inside parentheses are at our limit of detection)

{l l l l l l l} \hline \hline
&&&&&&\\ Galaxy & $\bar{A_{V}} \pm...
 ...9&0.615 $\pm$\space 0.04 &
$12.9^\star$\space & 5.10 & RE \\ \hline\end{tabular}

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