8 The cluster formation history

One of the goals of this paper is to study the influence of the interaction between M 51 and its companion on the cluster formation in the region of the inner spiral arms. To this purpose we compare the observed age distribution of the clusters with predictions for a constant cluster formation rate. This comparison is hampered by two effects: (a) the disruption of clusters and (b) the fading of clusters below the detection limit. The disruption time of the clusters in the inner spiral arms of M 51 is given by Eq. (4). We see that only clusters with an initial mass larger than about 10⁴ $M_{\odot }$ will survive more than 40 Myrs.

Figure 13 shows the histogram of the formation rates of the detected clusters, in number per Myr, for 140 clusters with $\mbox{$M_{\rm cl}$ }> 10^4$ $M_{\odot }$ and with an energy distribution that is fitted with an accuracy of $\mbox{$\chi^2_{\nu}$ }\le 3.0$. The sample of 111 clusters with $\mbox{$M_{\rm cl}$ }> 10^4$ $M_{\odot }$ and with an age uncertainty of $\Delta \log(t) < 0.25$, and the full sample of 285 clusters with $\mbox{$M_{\rm cl}$ }> 10^4$ $M_{\odot }$, not shown here, have a distribution very similar to the one shown in Fig. 12.

Figure 13: The formation rates of the observed clusters with an initial mass of $M_{\rm i} > 10^4$ $M_{\odot }$  in number per Myr, for clusters with an accuracy of the fit of the energy distribution of $\mbox{$\chi^2_{\nu}$ }\le 3.0$. The $1 \sigma$ uncertainty due to the Poisson statistics in each bin ( $\sigma = 1/\sqrt (N)$) is indicated. The diagram shows that the formation rate of the observed clusters decreases with age. This is due to the disruption of the older clusters.

All three samples show about the same characteristics.
(a) There is a general trend of a decrease in the cluster formation rate towards increasing age.
(b) There is a steep drop around $\log(t)\simeq 7.5$. This drop is the result of the concentration of clusters at ages $\log(t)=7.45$ that was apparent in the mass versus age diagrams of Fig. 10, and was explained in Sect. 6.
(c) There is no clear evidence for a peak in the cluster formation rate near $\log(t)\simeq 8.6$, which is the time of the interaction of M 51 with its companion, and which is the age of the huge starburst in the nucleus of M 51. There is a hint of a small peak around $\log(t) \simeq 8.1$. However this peak is only $2 \sigma$ high. Its reality has to be verified with a larger sample of clusters.
(d) The peak in the last bin of 5 Gyr contains all the clusters ages older than 3 Gyr, because the cluster models that we used for fitting the energy distributions do not go beyond 5 Gyr.
We have checked that these characteristics are not the result of the binning process: the same features appear for different choices of the binning parameters.

The general decrease in the formation rate of the observed clusters is partly due to the evolutionary fading of the clusters and partly due to the disruption of clusters. In Sect. 6 and in Fig. 10 we have shown that clusters with an initial mass of 10⁴ $M_{\odot }$ fade below the detection limit when they are older than $\log(t)=8.2$. This can explain the decrease in the formation rate of clusters older than about 150 Myrs. However, clusters with $\mbox{$M_{\rm cl}$ }>10^4~\mbox{$M_{\odot}$ }$ and younger than about 100 Myr should still all be detectable. The fact that we see a decrease in the apparent cluster formation rate must thus be due to the disruption of clusters (unless for some unknown reason the cluster formation rate has been steadily increasing from 100 Myrs up to now, which we consider unlikely).

It is interesting that the CFR in the inner spiral arms of M 51, at a distance of about 1 to 3 kpc from the nucleus, does not show any evidence for a peak at the age of the strongest interaction of M 51 with its companion galaxy. The closest approach occurred about 250-400 Myrs ago according to the dynamical models of Barnes (1998) or 400-500 and 50-100 Myrs ago according to Salo & Laurikainen (2000). The only significant peak occurred at $\log(t) \simeq 7.4$. However we attributed this peak to the large changes in the energy distributions of the models in the age range of $6.5 < \log(t) < 7.5$, and not to a real increase in the CFR.