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

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.

Fig. 2. The locations in the parameter space of the modes (black dots) and the contours within which 90 percent of the total integrated lies. The dashed lines refer to galaxies whose likelihood peak lies outside the region shown (NGC 4689) or whose 90 per cent region is partially cut

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

Fig. 3. The probabilities for two hypotheses for the SFR law, as computed for each galaxy by using all but one datum. Along the solid line the probabilities for the laws are equal, along the dashed lines they are 1 and 99 per cent or vice versa. The symbols designate: Milky Way (small filled square), 2841 (open circle), 4254 (filled circle), 4303 (open square), 4321 (delta), 4535 (nabla), 4654 (hourglass), 4689 (asterisk), 4736 (+), 5194 (smaller +), 6946 (x). For convenience, NGC 5457 is not shown, as its points cluster at much higher values (5,3)

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.

Fig. 4. The 90 per cent confidence region of the joint likelihood for all galaxies and how it depends on the CO-H2 conversion factor X. The solid line depicts the region computed with the standard value , short dashes indicated the regions for and 100, and long dashes refer to application of the metallicity-dependent prescription of Arimoto et al. (1996). Filled circles show the best parameters obtained for different values of . For the plus-sign the Galaxy and NGC 5457 are excluded, and the asterisk indicates the parameters for all galaxies if data of the rim points are removed (cf. Sect.  4.2)

4.3. Comments on individual galaxies

For several galaxies, some remarks are necessary:

• The Milky Way galaxy: Using all points, one gets a nearly circular confidence ellipse, as shown (cf. Fig. 2), and for the most probable parameter set (, ) a rather poor fit with a strong radial trend of the residuals. The results depend strongly whether one excludes the innermost and/or outermost points at and 1.25. This is evident already from Fig. 3, where eight of the ten data points cluster at the lower left edge of the diagram. The two remaining points, with much higher probabilities, result from taking away either the inner or the outer point. Excluding both, a much better fit could be obtained, without any radial trend of the residuals. The confidence region would become more elongated, like for the other objects, and the linear Schmidt law would now be well included. This law would be 25 times more probable than the one-parameter law .
• NGC 4254: The radial profiles of H I, CO, and H are very smoothly dropping with increasing radius. Thus a narrow confidence region is produced. Only H shows an increase in the inner 2 kpc. If one left out this central region, the goodness of fit for a non-linear Schmidt law could be improved considerably, with no radial dependence being necessary.
• NGC 4321 shows a central enhancement of the H brightness. Without this central point the linear Schmidt law would be roughly 25 times more probable than the non-linear Schmidt law.
• NGCs 4535 and 4654 have fairly large ellipses, mainly because of the small number of points (4 and 5).
• NGC 4689: From the seven data points one obtains a very strongly elongated confidence ellipse, the maximum probability density is outside the parameter space. The innermost point has a lower H brightness than the next one further out. If one removed this point, the ellipse would remain as elongated, but the maximum density would happen to be at the opposite boundary of the parameter region.
• NGC 4736 has the largest confidence region of the sample, thus nearly all SFR laws are equally probable. Its H brightness profile is characterized by an inner ring about 100 times brighter than the exterior part. To explain such a strong variation with the rather smooth H I and CO-profiles gives rise to a rather poor fit for any combination of the parameters.
• NGC 5194: eleven points yield a fairly large ellipse. While both H I and CO show a smooth radial profile, H exhibits a rather strong bump between 5 and 7 kpc, which is difficult to explain by a simple, systematic dependence on gas density or radius.
• NGC 5457: eleven data points with a very smooth H -profile yield by far the smallest confidence region, dominating thereby the joint probability distribution.
• NGC 6946: The eleven data points give a strong correlation between x and y, because accidentally the run of total gas surface density with radius follows a power law quite closely (cf. Fig. 7). The ratio of H and CO brightnesses is almost constant with radius, H showing more structure than the very smooth CO profile.

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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