Astron. Astrophys. 325, 961-971 (1997)

6. The CO-H₂ conversion factor

There is agreement that CO is a good tracer for molecular hydrogen, but the value of the conversion factor (from velocity integrated CO intensity into molecular hydrogen surface mass density) - and whether a common factor is appropriate at all - is still rather unclear (cf. Bloemen 1989, Vila-Costas & Edmunds 1992, Boselli et al. 1995, Sodroski et al. 1995, Arimoto et al. 1996). The value from Gordon & Burton (1976) still serves as a widely used 'standard' value which seems to be appropriate at least for giant molecular clouds (Magnani & Onello 1995).

6.1. Changing X for the whole sample

We investigate the effect of changing this factor more closely, because the molecular gas is thought to be more intimately related to star formation than the atomic one (cf. Sect. 1). Hence, raising X a better fit might be expected. Fig. 4 shows the path of the peak of the likelihood mountain in the x -y -plane when changing the conversion factor from 0 (i.e. the gas responsible for star formation is only atomic hydrogen) to very large values (i.e. only molecular hydrogen). An increase by a factor 2 or a decrease by a factor 3 takes the maximum of the joint likelihood outside the confidence region for the standard factor. If one considered only molecular gas , the maximum becomes sharper, but still is rather close to the standard confidence region. On the other hand, if only atomic hydrogen were considered, i.e. , one needs an explicit radial dependence of the SFR. This is because in most galaxies the radial H I profile is flatter than that of the CO-surface brightness. This also shows that star formation seems more closely linked to the molecular gas than to the atomic gas.

Contrary to the expectation, we find formally a maximum of between one fourth and one third of the standard value of the conversion factor, although this is not a very pronounced one. is increased by about a factor of 4 which is not significant in view of the scatter seen in Table 3.

The Milky Way galaxy and NGC 5457 (M 101) are somewhat different from the other galaxies, as they are the only ones where the H I, CO, and H brightnesses are given for the same radial rings. Furthermore, obtaining the radially dependent data for the Galaxy requires additional assumptions than the more directly available data from external galaxies. Finally, NGC 5457 has a confidence ellipse very much smaller than that of any other object. The dramatically larger probability density makes this galaxy dominate in any kind of joint probability. But even if one excludes both objects from the sample, the confidence region remains of about the same size but is shifted somewhat towards the upper left, the maximum lying at the point marked with a plus-sign. This is still within the confidence region for the sample as a whole. There is a very strong overlap, thus the results are not critically dependent on the inclusion of NGC 5457 and its narrow peak.

6.2. NGC 5457

This galaxy has by far the smallest confidence ellipse in the x -y parameter space (cf. Fig. 2). This is partly due to the fact that data is given at 20 radial points, and partly because all the profiles look rather smooth. Thus, this galaxy is particularly well suited to study the influence of the conversion factor. Fig. 5 shows how the peak of the joint likelihood moves across the x -y -plane as X is changed. It is worth emphasizing that, just like for the joint likelihoods of the entire sample, changing between 0.5 and 10 one still remains within the 90 per cent confidence region. With the metallicity-dependent conversion factor the maximum is just outside the region, at and , i.e. with almost no radial dependence. is largest for , about a factor 3 larger than for the standard value . This means that the best fit is achieved for relating the SFR with the molecular gas rather than the atomic gas.

Fig. 5. Positions of the most likely parameters of the law in NGC 5457, as the CO-H2 conversion factor X is changed. The numbers denote , its ratio relative to the standard value; the kinks depict the ratios 0.03, 0.2, 0.3, 0.5, 2, and 5. The ellipse is the 90 per cent confidence ellipse as in Fig. 2. The triangle shows the position obtained by using the conversion recipe of Arimoto et al. (1996)

Instead of considering a variation of X one can introduce a weight factor w which describes the relative importance of the two gas species for the star formation process. Hence, the total gas density is substituted by the weighted mean:

Performing the integration over w (with a constant prior) results in the likelihood distribution shown in Fig. 6. The peak is situated at and , but a linear Schmidt law with no dependence on radius is still within the 90 per cent confidence contour. The linear Schmidt law is found to have the highest probability, though it is slightly outside the 90 per cent r egion for the parameter of the non-linear one: . This illustrates nicely Occam's razor: the parameter space for the non-linear Schmidt law is just too large.

Fig. 6. The confidence contours for 90, 50, and 25 per cent for the law in NGC 5457, with the CO-H2 conversion factor X being a nuisance parameter

6.3. A metallicity dependent conversion factor

From CO-observations of individual molecular clouds, Arimoto et al. (1996) derive a conversion factor that depends on the local oxygen abundance of the gas. Such a dependency would allow to take into account the claimed underestimate of the relative amount of molecular gas at the outskirts of the Milky Way galaxy (Lequeux et al. 1993, Sodroski et al. 1995) as well as in low-luminosity galaxies (Boselli et al. 1995). For 7 galaxies of our sample, including the Milky Way galaxy, radial abundance gradients are known. Taking the corresponding abundance data from Vila-Costas & Edmunds (1992), we re-analyze the objects of this subsample. The results are collected in Tables 5 and 6. Only for the Milky Way galaxy, the value of is significantly (factor 10) increased over that derived with the standard conversion factor. Likewise, Fig. 4 shows that the confidence region for the joint likelihood strongly overlaps with the one from the standard assumption. Thus, it seems that in general there is no urgent necessity to apply a more sophisticated conversion recipe: Table 6 does not significantly deviate from Table 4.

Table 5. Same as Table 1, but using the metallicity-dependent CO-H₂ prescription of Arimoto et al. (1996). The last column shows the relative increase in as compared with the standard case

Table 6. As Table 4, but using the metallicity-dependent CO-conversion recipe

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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