1 Introduction

The interaction of galaxies triggers star formation as clearly seen in the young starbursts in interacting galaxies (for reviews see Kennicutt 1998; Schweizer 1998). One of the best examples is the Antennae system where the interaction is presently going on and star formation and cluster formation occurs on a very large scale in the region between the two merging nuclei (Whitmore et al. 1999).

The way the triggered star formation progresses through an interacting galaxy is best studied from a slightly older interacting system, where the close passage of the companion is over. In such a system one can hope to derive the age distribution of clusters as a function of location in the galaxy and thus measure the time sequence of the induced star and cluster formation as a function of location in the galaxy.

One of the best systems for this purpose is the interacting sytem of the Whirlpool galaxy (M 51) at a distance of $8.4 \pm 0.6$ Mpc (Feldmeier et al. 1997). The M 51 system consists of a grand design spiral galaxy (NGC 5194) interacting with its dwarf companion (NGC 5195). The almost face-on orientation of M 51 allows the observation of its structure in great detail with minimum obscuration by interstellar dust. The interaction of these two galaxies has been modelled by various authors, starting from the fundamental paper by Toomre & Toomre (1972). These authors derived an age of the closest passage to be $2 \times 10^8$ years ago. Later refined models, e.g. by Hernquist (1990), and reviewed by Barnes (1998), have improved this time estimate. The closest approach is now believed to have occurred about 250-400 Myrs ago. The best model is found for a distance of the pericenter of 17 to 20 kpc, for a mass ratio of $M/m \approx 2$ and a relative orbit which crosses the plane of M 51 under an angle of about 15 degrees. (For a discussion of the problems with this model and possible improvements, see Barnes 1998). Recently Salo & Laurikainen (2000) suggested that on the basis of N-body simulations that the M 51 system had multiple passages, with the two last at 50-100 Myrs and 400-500 Myrs ago.

Because of its relatively small distance from us, and the fact that we see the M 51 system face one, the system is ideally suited for the study of the progression of cluster formation due to galaxy - galaxy interaction. For this reason we started a series of studies of different aspects of the M 51 system based on HST-WFPC2 observations in six broad band and two narrow band filters.

The nucleus of M 51 was studied by Scuderi et al. (2002). They found that the core contains a starburst with an age of $410 \pm 140$ Myrs and a total stellar mass of about $2 \times 10^7 ~M_{\odot}$ within the central 17 pc. This age agrees with the estimated time of closest passage of the companion, so the starburst in the core is most likely due to the interaction with the passing companion. M 51 contains an unresolved nucleus with a diameter smaller than 2 pc and a luminosity of $2\times 10^6~ L_{\odot}$ (Scuderi et al. 2002).

The bulge, i.e. the reddish region with a size of $11 \times$ 16 arcsec $^2 = 460 \times 680$ pc², between the nucleus and the inner spiral arms, was studied by Scuderi et al. (2002) and by Lamers et al. (2002). The bulge is dominated by an old stellar population with an age in excess of 5 Gyrs. The HST-WFPC2 images of the bulge show the presence of dust lanes. The total amount of dust in the bulge is about $2.3 \times 10^3$ $M_{\odot }$ and the dust has about the same extinction law and approximately the same gas to dust ratio as our Galaxy (Lamers et al. 2002). This suggests a metallicity close to that in the solar neighbourhood. The HST-WFPC2 images of the bulge of M 51 show clearly that the dust is concentrated in structures in the forms of spiral-like dust lanes and a bar that reaches all the way down into the core. Most intriguingly is the discovery of about 30 bright and mainly blue point-like sources in the bulge that are aligned more or less along the spiral-like dust lanes. Lamers et al. (2002) have shown that these are most likely very young massive stars $20 < M_* < 150~\mbox{$M_{\odot}$ }$ with little evidence of associated clusters. This mode of star formation is the result of the peculiar conditions, in particular the destruction of CO molecules, of the interstellar clouds in the bulge of M 51.

In this study we focus on the clusters at a distance of about 1 to 3 kpc from the nucleus, i.e. near the inner spiral arms. The purpose of the paper is two-fold:
- (a) to determine the cluster intitial mass function, and
- (b) to determine the presence or absence of a starburst period that can be linked to triggering by the passage of the companion. This is not an easy task, because our data will show that the age distribution of the clusters is strongly affected by the disruption of clusters older than about 40 Myrs.

We study the clusters and their properties by identifying point-like sources in the HST-WFPC2 images and measuring their UBVRI magnitudes to obtain their energy distributions. The magnitudes indicate that the sources are clusters instead of single stars. Their energy distributions are compared to cluster evolutionary synthesis models to determine the age, mass and E(B-V) of these clusters.

In Sect. 2 we describe the observations and the data reduction. In Sect. 3 the selection of cluster candidates is discussed. In Sect. 4 we describe the cluster evolutionary synthesis models and in Sect. 5 the fitting procedure for the derivation of the cluster parameters is explained. The mass versus age distribution of the clusters is derived in Sect. 6. In Sect. 7 the initial mass function of the clusters is derived from the sample of clusters younger than 10 Myrs. The cluster formation history is studied in Sect. 8. The summary and conclusions are given in Sect. 9.