6 An empirical calibration of X

6.1 The nearby sample of resolved galaxies

Table 4 contains the best-estimated values of X for a sample of 14 well-studied nearby galaxies. Table 4 shows that X spans the range $0.6 \leq X \leq 10\times 10^{20}$ mol cm^-2 (K km s^-1)^-1for galaxies of different morphological type.

The ratio between ¹²CO(1-0) line emission and the molecular hydrogen column density strongly depends on several physical properties of the ISM such as the UV radiation field, the metallicity and the cosmic ray density (Lequeux et al. 1994; Kaufman et al. 1999), which are known to vary from galaxy to galaxy.

We empirically quantify these dependences by plotting in Fig. 2. (left panels) the relationship between X, the H$\alpha +[$NII]EW and the metallicity ($12 + \log ($O/H)) for the 14 nearby galaxies.

Figure 2: The relationship for the template sample of nearby galaxies between the X conversion factor from CO line intensity to H2 column density and a) the H$\alpha +[$NII]EW, b) the metallicity index $12 + \log ($O/H), c) the H band luminosity and d) the absolute B magnitude. The dotted-dashed line is the best fit to the data; the dotted line is the best fit given by Arimoto et al. (1996).

The relationship with the H$\alpha +[$NII]EW (taken as a SFR tracer), if any, is ill-defined and that with metallicity is quite clear. In galaxies with a low metallicity and a strong UV radiation field (high H$\alpha +[$NII]EW) (both go together in general) the X conversion factor is a factor of $\sim$20 higher than in quiescent, high metallicity galaxies such as the Milky Way. These two relationships can be used in principle to determine a more accurate value of the X conversion factor once the metallicity and/or the H$\alpha +[$NII]EW is known. Metallicity measurements are available for only a minority of galaxies, while H$\alpha +[$NII]EW exist for a few hundred. However there is a well-known anticorrelation (correlation) between H$\alpha +[$NII]EW (metallicity) and luminosity (Gavazzi et al. 1998; Zaritsky et al. 1994) in normal galaxies. It is reflected here by a strong relation between X and the H luminosity (Fig. 2c) or the B absolute magnitude (Fig. 2d). The best fits to the data are given in Table 5. The slope of the fits are consistent with those found by Arimoto et al. (1996), but significantely steeper than that found by Wilson (1995) for the X vs. $12 + \log ($O/H) relation (see Table 5). This difference in slope with Wilson is probably due to the fact that our sample includes many metal rich spiral galaxies with low values of X ( $X \leq 10^{20}$ mol cm^-2 (K km s^-1)^-1) not present in the Wilson's sample. Our intercept for the X vs. $12 + \log ($O/H) relation is consistent with that of Arimoto et al. (1996). Our intercept in the X vs. M_B relation is lower since Arimoto et al. (1996) includes all the objects rejected here whose Xvalue is probably overestimated due to the low spatial resolution of the CO observations (>100 pc).

The relationships given in Table 5 between X and L_H and/or M_B are de facto empirical calibrations for a luminosity-dependent X conversion factor. Given the large uncertainty in the determination of X in the nearby sample of galaxies, it is difficult to quantify the resulting accuracy in the molecular gas mass estimated using the relationships given in Table 5. We should also remind that even inside a given object X might change by a factor of $\sim$10 from the diffuse medium to the core of GMCs (Polk et al. 1988); it is thus difficult to estimate a single value of X representing the entire galaxy. We can however conclude that the adoption of the relations given in Table 5 should remove the first-order systematic effect with luminosity in the estimate of the molecular hydrogen content of galaxies using CO data. The use of a standard X conversion factor as those generally used in the literature (X=2.3- $2.8\times 10^{20}$ mol cm^-2 (K km s^-1)^-1) overestimates the molecular gas mass by a factor of $\sim$2-3 in massive galaxies of $L_H \sim 10^{11}~L_{H\odot}$, or $M_B \sim {-}20.5$ mag, while underestimates M(H₂) in low mass objects of $L_H \sim 10^{9}~L_{H\odot}$, or $M_B \sim {-}17$ mag as those observed in this work by a factor of $\sim$2. The relationship between X and $12 + \log ($O/H) might be used to estimate the radial distribution of molecular hydrogen in galaxies mapped in CO with available measurements of the metallicity gradient.

6.2 An alternative method

An alternative technique for determining the molecular gas content can be pursued by assuming a metallicity-dependent dust to gas ratio and determining the dust mass using far-IR or submillimetric continuum data. The dust to gas ratio is then determined in regions with no CO emission, hence supposed to be strongly dominated by HI. In regions with CO emission, the excess dust emission with respect to this ratio indicates the mass of H₂. This technique has been succesfully applied to M 51, NGC 891, NGC 4565 and to some nearby irregular galaxies such as the Magellanic Clouds (Guélin et al. 1995; Guélin et al. 1993; Neininger et al. 1996; Israel 1997).

In normal galaxies such as those in our sample the dust mass is dominated by the cold dust emitting in the far-IR with a peak at $\sim$200 $\mu$m. The determination of the total dust mass can be achieved provided that the 100-1000 $\mu$m far-IR flux and the cold dust temperature are known. Recent observations aimed at determining the spectral energy distribution in the far-IR of normal, quiescent galaxies indicate that their SED can be fitted by a modified Planck law $\nu$$^{\beta}$ $B_{\nu}(T_{\rm d})$, with $\beta=2$ (Alton et al. 2000). The total dust mass can be thus determined from the relation (Devereux & Young 1990):

$${M_{\rm dust}=CS_{\lambda}D^2({\rm e}^{a/T_{\rm dust}}-1)~{M_\odot}}$$ (4)

where C is a quantity which depends on the grain opacity, $S_{\lambda}$is the far-IR flux at a given wavelength (in Jy), D is the distance of the galaxy (in Mpc), $T_{\rm dust}$ is the dust temperature, and a is a quantity which depends on $\lambda$. The majority of our sample has only IRAS data at 60 and 100 $\mu$m. Given the strong contamination of the emission at 60 $\mu$m by very small grains, the 60 to 100 $\mu$m ratio cannot be used to measure $T_{\rm dust}$ (Contursi et al. 2001). The ISOPHOT data at 100 and 200 $\mu$m give a better measure of the dust temperature. $T_{\rm dust}$ determined for the ISOPHOT sample, as well as for 6 quiescent galaxies observed with ISOPHOT by Alton et al. (1998) consistently with Contursi et al. (2001), seems to be independent of the UV radiation field as traced by the H$\alpha +[$NII]EW, of the metallicity or of the total luminosity. The average value is $T_{\rm dust}=20.8\,\pm\, 3.2$ K, which we will assume also for galaxies without ISOPHOT measurements. We then estimate the dust mass of the sample galaxies using Eq. (4) with $C=1.27~M_{\odot}$ Jy^-1 Mpc^-2, consistent with Contursi et al. (2001), and a=144 K for $S_{\lambda}=S_{\rm 100\,\mu m}$ (Devereux & Young 1990).

The determination of the dust to gas ratio in a way consistent with that obtained in the solar neighbourhood, requires the estimate of the gas and dust surface densities, thus of the spatial distribution of dust and gas over the discs. Unfortunately only integrated HI and dust masses are available for our spatially unresolved galaxies. It is however reasonable to assume that the cold dust is as extended as the optical disc (Alton et al. 1998). The HI gas surface density is available only for a few galaxies in our sample from VLA observations (Cayatte et al. 1994). For these objects we observe a good relationship between the HI surface density $\Sigma$HI and the HI-deficiency parameter ( ${\rm HI}-{\rm def}$), (defined as in Sect. 5.1) (Fig. 3):

$$\log \Sigma {\rm HI} = 20.92 (\pm 0.17) - 0.65 (\pm 0.11) \times ({\rm HI}-{\rm def})~\rm {cm^{-2}}$$ (5)

which can be used to predict $\Sigma$HI for most of the galaxies of our sample with single dish HI observations.

Figure 3: The relationship between the HI surface density and the HI-deficiency parameter for the galaxies in common with Cayatte et al. (1994). The dashed line gives the best fit to the data.

The gas to dust ratio should depend on metallicity since the dust content is expected to be proportional to the metal content. A gas to dust vs. metallicity relation can be calibrated using the data available for the MW (Sodroski et al. 1994), the LMC (4 times solar; Koornneef 1982) and the SMC (10 times solar; Bouchet et al. 1985). This gives the relation:

$$\log({\rm gas/dust}) 10.207 (\pm 0.015) - 1.146 (\pm 0.024)$$ =

$$\times (12 + \log({\rm O/H})) + \log({\rm gas/dust})_{\odot}$$ (6)

where the gas to dust ratio is given relative to the solar neighborhood, estimated by Sodroski et al. (1994) at (gas/dust) $_{\odot}=160$. The metallicity can be predicted using the metallicity vs. H band luminosity relation shown in Fig. 4:

$$12 + \log({\rm O/H}) = 4.98 (\pm 0.15) + 0.37 (\pm 0.04) \times \log L_H.$$ (7)

The dispersion in the nearby sample is lower than in the other galaxies probably because of a more accurate determination of the metallicity, which has been measured from spectroscopic observations of single HII regions, while for the remaining galaxies it has been determined from long slit spectra integrated over their disc (drifting technique, Gavazzi et al. in preparation). The adopted fit is however consistent with the result of Gavazzi et al., based on a large sample of galaxies with spectroscopic measurements.

Figure 4: The relationship between the metallicity ( $12 + \log ($O/H)) and the H band luminosity. Filled dots are for the nearby galaxies, empty dots for the remaining objects with metallicity measurements. The dashed line gives the best fit to the data for the nearby sample.

Using Eqs. (6) and (7) we can predict the gas to dust ratio for a galaxy of any H luminosity. The molecular gas mass comes directly if we assume that the H₂ is homogeneously distributed over the optical disc and X is given by X=M(H $_2)_{\rm dust}$/I(CO). The assumption of a homogeneous, flat distribution for the molecular hydrogen component over the disc of galaxies, which is in contradiction with the observational evidence that the CO emission is generally centrally peaked (see Sect. 4.1), might introduce a systematic error in the determination of X. We remark however that the expected H₂ distribution is flatter than that of the CO emitting gas because of the observed decrease of the metallicity in the outer parts of galaxy discs.

Figure 5: Same as Fig. 2 but including values of X determined using the alternative method described in Sect. 6.2 for the unperturbed sample (small symbols). Small open dots are for galaxies detected at 100 $\mu$m and CO, small open triangles for galaxies undetected at 100 $\mu$m ( $\bigtriangledown$) or CO ($\triangle$). Filled symbols are for the ISOPHOT sample.

The values of X obtained for the sample galaxies are compared with those of the template galaxies in Fig. 5 (small symbols). In spite of the larger scatter, it is encouraging to see that the new values of X, at any given luminosity, metallicity and UV radiation field, are in rough agreement with those obtained for the template. The large uncertainty and systematic effects are not unexpected given the number of assumptions underlying the method. From Fig. 5 we conclude that the luminosity-dependent X conversion factor given in Table 5 is appropriate for estimating the molecular gas content of late-type galaxies from ¹²CO(1-0) line intensity measurements. From now on, the molecular gas content of the 266 sample galaxies, M(H₂), is estimated using the H band luminosity-dependent X value given in Table 5.