9 Summary and conclusions

We have studied images of the inner spiral arms of the interacting galaxy M 51 obtained with the HST-WFPC2 camera in broad-band UBVRI and in the narrow band H$\alpha$ and [OIII] filters. The study can be summarized as follows:

1.

We found a total of 877 point-like objects, which are probably clusters. Many of the clusters are strong H$\alpha$ emitters, but none of the clusters, not even the youngest ones, have an excess of radiation in the [OIII] line at 5007 Å (F502N-filter). This suggests that the upper mass limit of the stars in the clusters is about 25 to 30 $M_{\odot }$.

2.

We have compared their energy distributions with those of Starburst99 cluster models (Leitherer et al. 1999) for instantaneous star formation with a stellar IMF of exponent 2.35, solar metallicity, a lower and upper stellar mass limit of 1 $M_{\odot }$ and 30 $M_{\odot }$ respectively. The energy distributions were also compared with those of the Frascati models (Romaniello 1998). For clusters younger than 700 Myr the results from the fitting with the Starburst99 models were adopted because these models are more accurate for young clusters and the fits of the energy distributions are better than those of the Frascati models. For older clusters the results from the fits with the Frascati models were adopted.

3.

For clusters that were observed in four or five bands a three dimensional maximum likelihood method was used to derive the properties of the clusters from the comparison between the observed and predicted energy distributions. The free parameters are the age t and E(B-V), which together determined the shape of the energy distribution, and the initial cluster mass $M_{\rm cl}$ which determines the absolute magnitude. For clusters that were not observed in all bands, the empirically derived lower magnitudes limits were taken into account.

4.

For clusters that were observed in only three bands the age and mass were derived in a two-dimensional maximum likelihood fitting of the energy distributions, with t and $M_{\rm cl}$ as free parameters. The observed probability distribution of E(B-V) was used as a weighting factor in the fitting procedure.

5.

The histogram of E(B-V) is strongly peaked at very small $E(B-V) \simeq 0$. All cluster have a reddening smaller than E(B-V)<1.0 and 67% of the clusters have E(B-V)<0.18.

6.

We have analysed the observed clusters also with cluster models of higher metallicity, $Z=2\times Z_{\odot}$. These higher metallicity models fit the observations considerably worse than the solar metallicity models. For instance, for solar metallicity models the energy distribution of 294 clusters can be fitted with an accuracy of $\mbox{$\chi^2_{\nu}$ }\le 1.0$ and 392 with $\mbox{$\chi^2_{\nu}$ }\le 3.0$. For models with twice the solar metallicity these numbers are respectively 138 and 217. So the energy distributions of the clusters support the adopted solar metallicity.

7.

The clusters have masses in the range of $2.5 < \log (\mbox{$M_{\rm cl}$ }) < 5.7$ and ages of $\log(t) >5.0$. These masses are the initial masses of the clusters, i.e. the current mass corrected for stellar evolution effects, but not corrected for evaporation or disruption. All derived masses have to be multiplied by a factor 1.3 if the lower mass of the stars is 0.6 $M_{\odot }$, instead of the adopted 1 $M_{\odot }$, and by a factor 2.1 if the lower mass is 0.2 $M_{\odot }$, as found for the Orion Nebula cluster.

8.

The distribution of the clusters in a mass-versus-age diagram shows the predicted lower limit due to the evolutionary fading of the clusters, including the dips at $\log (t) \simeq 6.8$ and 7.1. Three apparent concentrations at $\log(t)=6.7$, 7.2 and 7.45 are not real but due to the properties of the cluster models used.

9.

About 60% of the clusters are younger than 40 Myr. The number of older clusters is much less than expected for a constant cluster formation rate. This is partly due to the evolutionary fading of low mass clusters below the detection limit, and partly due to the disruption of the clusters.

10.

The cluster initial mass function (CIMF) was derived from the cumulative mass distribution of clusters younger than 10 Myr, for which disruption has not occured. The CIMF has a slope of $\alpha = 2.1 \pm 0.3$ in the range of $3.0 < \log (\mbox{$M_{\rm cl}$ }) < 5.0$ and $\alpha= 2.00 \pm 0.05$ in the range of $3.0 < \log (\mbox{$M_{\rm cl}$ }) < 4.5$ $M_{\odot }$, for $N(\mbox{$M_{\rm cl}$ }) \sim \mbox{$M_{\rm cl}$ }^{-\alpha}$. This slope is the same to that found in the interacting Antennae galaxies (Zhang & Fall 1999). Zhang and Fall deived a power law slope of the CIMF of $\alpha=1.95 \pm 0.03$ and $2.00 \pm 0.04$ for two cluster samples of the Antennae galaxies. The good agreement between these slopes and the one found by us suggests that $\alpha$ is about the same for cluster formation triggered by strong galaxy-galaxy interactions, such as presently going on in the Antennae, as for cluster formation that is not dominated by the interactions.

11.

The age distribution of clusters with $\mbox{$M_{\rm cl}$ }> 10^4$ $M_{\odot }$, is used to derive the history of the cluster formation rate (CFR). There is a general trend of a decrease of the formation rate of the observed clusters with age. It is unlikely that the real CFR has been increasing continuously from about 1 Gyr to the present time. The decrease of the CFR with age of clusters younger than about 100 Myr cannot be due to evolutionary fading, but it is due to the disruption of clusters. For clusters older than 200 Myr the decrease of the derived CFR could, at least partly, be due to evolutionary fading.

12.

There is no evidence for a peak in the CFR at about 400 Myr, which is the time of the interaction of M 51 with its companion and the age of the huge starburst in the nucleus.

In a forthcoming paper we describe the cluster formation as a function of location in a large part of M 51, using the same methods as used here (Bastian et al. 2002). The disruption of clusters in M 51, derived from the results of the study presented here, are described by Boutloukos & Lamers (2002).

Acknowledgements

H.J.G.L.M.L. and N.B. are grateful to the Space Telescope Scence Institute for hospitality and financial support during several stays. We thank Claus Leitherer for help and advice in the calculation of the cluster models. Support for the SINS program GO-9114 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. N.B. ackowledges a grant from the Netherlands Organization for Scientific Research. We thank the unknown referee for constructive comments that resulted in an improvement of this paper.