4. Results for individual galaxies
4.1. The confidence regions
Fig. 2 shows for all twelve galaxies the confidence regions in the parameter space: Each line depicts the contour level enclosing the area within which 90 per cent of the contributions to is found. Their modes, i.e. the best values for the exponents x and y, are marked in the figure and are given in Table 1.
Table 1. The best parameters for fitting
The plot shows that the parameter modes cluster around a linear or quadratic dependence on gas density, and there is little indication in favour of an explicit dependence on radial distance: The horizontal line crosses or touches (for NGC 4321) the 90 per cent regions of all galaxies but the Milky Way galaxy.
Except for NGC 2841, the orientation of the confidence ellipses is quite similar. This is merely a consequence of the radial profile of the gas density: Let and . Since we test , a good fit must have: . Usually the gas density decreases outward (, see Fig. 7) which gives rise to the anti-correlation seen in Fig. 2.
4.2. Comparison of SFR laws
We consider eight laws for the SFR. Three have no free parameters, viz. the linear and quadratic dependence on gas density and . Four have one free parameter, such as . Finally there is the most general law with two free parameters: . Since all the laws are nested into the most general one, we can compute their mean likelihoods by integrating the likelihood mountain over the appropriate parameter space, weighted with the relevant prior distribution for the parameters.
Bayes' theorem (Eq. 1) applies only to mutually exclusive hypotheses. This means that in a more general law, e.g. , we must exclude all sub-hypotheses it contains, here: g and . If one considers a sub-hypothesis of a lower dimension as a separate hypothesis, one takes that into account in the parameter prior distribution by superposing a -distribution peaked at each parameter value characterising a sub-hypothesis. Thus, exclusion of the sub-hypotheses is simply done by integration with the unmodified parameter prior (cf. O'Hagan 1994).
Afterwards, the posterior probability of each law is computed by multiplying with the law's prior probability. Since we do not give any preference to any one of the laws, we assign the same prior value. We emphasise that in doing so, we actually treat the 'simpler' laws less favourably than we would be tempted from everyday practice, where one would give preference to a law with fewer free parameters. In Table 2 we compare, for the sake of convenience, only appropriately normalized Bayes factors, not the real s, for each galaxy.
Table 2. Individual Bayes factors for 8 models of the star formation rate SFR, depending on gas surface density g and galactocentric radius r. For each galaxy, the factors are normalized to the most probable model
In Sect. 2.2 has been shown to depend strongly on the outcome of a realization in the presence of appreciable noise. In order to check the robustness of our results we generated for each galaxy in the sample new data sets by leaving out every one datum at a time, i.e. from n data we make n sets of data points. Computing the values of for the two SFR-laws and gives an impression of the sensitivity due to the choice of data points. Table 3 provides the means and the dispersions of and of the best parameters for both star formation laws. In Fig. 3 the two values of obtained from each configuration are depicted.
Table 3. Results from multiple analyses of the data, leaving out one datum at a time: the averages and dispersions for and for the parameters
In half of the galaxies the scatter is lower than about 0.35 dex, thus we expect the real uncertainty for a given Bayes factor to be about a factor of two or three. The worst cases are the Milky Way galaxy and NGC 4321. For the Galaxy, the large scatter is due to the innermost and outermost points. Neglecting either one would improve the goodness of fit drastically. In NGC 4321 the large dispersion is due to the innermost point. We notice that removal of the innermost or the outermost datum point would strongly increase in eight of the twelve galaxies of our sample. This may reflect merely the fact that often the profile of the H surface brightness shows a steep drop towards the centre, and in some galaxies near the outer rim, too. For his analysis Kennicutt (1989) excluded the innermost 2 kpc, arguing that there the extinction in H II regions can be considerably higher than in the disk proper, or even the star formation law itself could be disturbed from its normal form. While this exclusion of rim points may strongly improve , tests show that the overall assessment of the SFR laws is not strongly influenced. For instance, the optimal parameter values for the joint mean likelihood - see Fig. 4 - are still within the 90 per cent confidence region, if the rim points are left out.
4.3. Comments on individual galaxies
For several galaxies, some remarks are necessary:
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998