The Bayesian approach to statistics allows to compute genuine probabilities for the credibility of a law, incorporating not only the goodness of its fitting of given observational data but also any a priori knowledge or preference on its validity. This permits to develop a statistically consistent formulation of what is known as Occam's razor, that a simpler law should be preferred over an unnecessarily complicated theory.
We develop such a practical approach and demonstrate its working by simulations with artificial data. We apply this method to the question, which kind of dependence of the star formation rate on gas surface density and galactocentric radius represents best the presently available observational data of galactic disks. As indicator for the SFR the H surface brightness is used, and the gas density g is obtained from both atomic and molecular gas. The SFR laws of the type are considered, both in its general form with two free parameters and restricted forms with one or no free parameter. This is done for twelve galaxies, including the Milky Way. For the individual objects as well as for the joint probabilities of the entire sample, one gets the largest probabilities for the linear 'Schmidt law' and for the more general law where the exponent x may be different for different galaxies. The probability for other forms including the quadratic Schmidt law , is much lower. In particular, the accuracy of the data presently rule out th need for any explicit dependence on the galactocentric radius, This finding agrees with the global studies of Donas et al. (1987) and Buat (1992) as well with the local one by Kennicutt (1989).
As the test example shows, one has to keep in mind that the given observational data are just one realization of the data, which may give rise to an appreciable amount of scatter in the Bayes factors. In our study, we do not separate the various sources of observational uncertainties and genuine scatter in the physical processes. The dependence of the results on which radial points are included, poses some problem. For several galaxies one finds that excluding a datum at the inner or outer edge may greatly improve the fit and thus causes the scatter of the value of in Table 3 (cf. Sect. 4.3). However, in the overall assessment, e.g. by the joint likelihood, inclusion of these 'disturbing' points does not alter the basic finding that it is the simple linear Schmidt law which is most compatible with all the data. Finally, this finding is also quite insensitive to changes of the CO-H2 conversion factor, and even the employment of a metallicity-dependent conversion prescription does not change the situation.
This might mean that the observational data is still so poor or so strongly dominated by genuine irregularities that all what one can say, is that the SFR increases about linearly with the gas density, i.e. that the inverse SF timescale is constant. Because of the assumption that the relative errors follow a normal distribution, our method would not detect an intrinsic radial dependence of , if it were compensated by a power-law dependence of the gas surface density g on radius. Among the galaxies in the sample, such a situation arises only once, with NGC 6946. The radial profiles of the (atomic and molecular) gas are shown in Fig. 7, where only NGC 6949 exhibits the straight line characteristic for a power-law. But since most galaxies follow rather closely an exponential decrease of - examples are NGCs 4254, 5194, 5457 - such a compensation is not common to all objects. This is a strong argument in favour that SFR and SF timescale do not depend on radial distance explicitly.
While this suggests that the projected SFR increases linearly with the gas surface density, does this really exclude a non-linear relationship between the SFR per volume and the volume density of interstellar matter? A counterexample may illustrate that this must not be the case: Consider a vertical column with cross-section A in the galactic disk. Suppose that all the gas is in the form of N clouds with a certain internal density but different volumes: . Star formation also occurs only in these clouds, with the SFR per volume . Summing over all clouds gives the SFR per unit area , and the surface density . Therefore, the SFR per unit area may be proportional to the gas surface density
even if , only provided that the internal density of the star forming clouds is constant. It goes without saying that a study of the relation between averaged quantities can never substitute spatially resolved studies.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998