4 Cluster evolution models

To determine the age, initial mass and E(B-V) of the clusters, we compare the observed energy distributions with energy distributions derived from theoretical cluster evolutionary synthesis models. We used two sets of models: the Starburst99 models for ages up to 1 Gyr, and the Frascati models for ages of 10 Myr to 5 Gyr. In this section the models are described.

4.1 Starburst99 cluster evolution models

To determine the age, initial mass and E(B-V) of the clusters, we compare the observed energy distributions with energy distributions derived from theoretical cluster evolutionary synthesis models. We used two sets of models: the Starburst99 models for ages up to 1 Gyr, and the Frascati models for ages of 10 Myr to 5 Gyr. In this section the models are described.

   
4.2 Starburst99 cluster evolution models

We compared the observed spectral energy distributions (SED) of our objects with the cluster evolutionary synthesis models of Leitherer et al. (1999), i.e. the Starburst99 models for instantaneous star formation. In these models, the stellar atmosphere models of Lejeune et al. (1997) are included. For stars with a strong mass loss the Schmutz et al. (1992) extended model atmospheres are used. The stellar evolution models of the Geneva group are included in these models. For our analysis we used the models of the Spectral Energy Distributions (SEDs).

In the cluster models the stars are assumed to be formed with a classical Salpeter IMF-slope of ${\rm d} \log (N) /{\rm d}\log (M) =-2.35$. The lower cut-off mass is $M_{\rm low} = 1~M_{\odot}$. We adopt two values for the upper cut-off mass: the standard value of $M_{\rm up} = 100~M_{\odot}$, and the value of $\mbox{$M_{\rm up}$ }= 30~ \mbox{$M_{\odot}$ }$. This last value is adopted because it is suggested by the lack of [OIII] emission (see Sect. 3.1). The total initial mass of all models is 10⁶ $M_{\odot }$. Leitherer et al. (1999) present models for 5 different metallicities, from Z= 0.001 to $Z=0.040 = 2 \times Z_{\odot}$. Observations of HII-regions have shown that the metallicity of the inner region of M 51 is approximately $2 ~ Z_\odot$ or slightly higher (e.g. Diaz et al. 1991; Hill et al. 1997). Therefore we adopt the models with Z=0.020 and 0.040. The almost solar metallicity agrees with the study of Lamers et al. (2002), who found that the extinction properties and the gas to dust-ratio in the bulge of M 51 is very similar to that in the Milky Way. So we use four sets of models, denoted by the pair $(Z,\mbox{$M_{\rm up}$ })$ of (0.02, 100), (0.02, 30), (0.04, 100) and (0.04, 30). For these combinations we calculated the SEDs of the cluster models, taking into account the nebular continuum emission.

To obtain the absolute magnitudes from the theoretical spectral energy distributions in the five broad-band filters and the two narrow-band filters we convolved the predicted spectra of the Starburst99 models with the WFPC2 filter profiles. The spectrum of Vega was also convolved with the filter functions in order to find the zero-points of the magnitudes in the VEGAMAG system. For each combination of Z and $\mbox{$M_{\rm up}$ }$ we calculated the SED of 195 models in the range of 0.1 Myr to 1 Gyr, with time steps increasing from 0.5 Myr for the youngest models to 10 Myr for the oldest models.

The adopted lower mass limit of 1 $M_{\odot }$ for the cluster stars has consequences for the derived masses of the clusters. If the IMF with a slope of -2.35 has a lower mass limit of 0.2 $M_{\odot }$, as for the Orion Nebula Cluster (Hillenbrand 1997), the derived masses of the clusters would be a factor 2.09 higher for an upper limit of 30 $M_{\odot }$. If the lower limit is 0.6 $M_{\odot }$, the mass of the clusters will be a factor 1.28 higher than those of the Starburst99 models for $M_{\rm up}=30$ $M_{\odot }$. We will take this effect into account in the determination of the cluster masses.

Some of the theoretical energy distributions in the WFPC2 wide-band filters used in this study are shown in Fig. 5, for clusters with an initial mass of 10⁶ $M_{\odot }$ at the distance of M 51. During the first 5 Myrs the UV flux remains high because most of the O-type stars have main sequence ages longer than 5 Myr. Between 5 and 10 Myr the O-type stars disappear. This results in a general decrease at all magnitudes, except in the I band where the red supergiants start to dominate. At later ages the flux decreases at all wavelengths as stars end their lives. After 900 Myrs only stars of types late-B and later have survived.

Figure 5: Examples of some characteristics of the theoretical energy distributions, in magnitude versus wavelength (Å) of the UBVRI filters (from left to right), for STARBURST99 clusters with an initial mass of 106 $M_{\odot }$ at the distance of M 51. The age of the cluster in Myrs is indicated. The evolutionary variation is discussed in Sect. 4.1.

Figure 5 shows that at any wavelength in the observed range the flux decreases with time. This fading of the clusters indicates that the detection limit of the clusters will gradually shift to those of higher mass as the clusters age. This is because the flux of a cluster (for a given stellar initial mass function) at any age and wavelength is proportional to the initial number of stars. The more massive the initial cluster, the brighter the cluster and the longer the flux will remain above the detection limit as the cluster fades due to stellar evolution. For any given detection limit, we can calculate the lower mass limit of the observable cluster as a function of age.

Figure 6 shows the relation between the initial mass of a cluster and its age when it fades below the detection limit, derived from the Starburst99 models. The R-magnitude (R) of a cluster of initial mass M_i without extinction at the adopted distance of 8.4 Mpc of M 51 (dm=29.62) is related to the absolute R-magnitude (M_R) predicted for the Starburst99 models of 10⁶ $M_{\odot }$ by

$$R(t)=M_R(t) + 29.62 - 2.5 \times \log \left(M/10^6\right)\cdot$$ (1)

So, for a given detection limit $R_{\rm lim}$ the critical mass $M_{\rm lim}$ of a cluster that can be detected at age t is given by

$$\log M_{\rm lim}(t)~=~6+0.4 \times(M_R(t)+29.62-R_{\rm lim}).$$ (2)

We have adopted the R magnitudes to calculate the observed lower limits of the distributions for two reasons: (a) all clusters were detected in the R band and (b) the effect of extinction is small in the R band. The resulting relation between $M_{\rm lim}$ and age is shown in Fig. 6 for detection limits of $R_{\rm lim}$= 22.0 and 23.0 mag. We will use these limits later, in Sect. 5, to explain the distribution of clusters in the mass versus age diagram.

Figure 6: The relation between the critical initial mass $M_{\rm lim}$ and age, derived for two values of the R-band magnitude limit; 22.0 and 23.0 mag. It is calculated from the Starburst99 cluster models, for a distance of 8.4 Mpc and no extinction. Only clusters with an initial mass higher than $M_{\rm lim}$ can be detected as faint as $R_{\rm lim}$ at time t. Alternatively, clusters with a mass $M_{\rm lim}$ can only be detected down to a magnitude $R_{\rm lim}$ when they are younger than the age t, indicated by these relations.

Figure 7 shows the variations of the colours of the Starburst99 models for solar metallicity and with an upper mass limit of 30 $M_{\odot }$. The figure shows that the colours do not change monotonically, but that there are peaks and dips, especially in the age range of $6.5 < \log(t) < 7.3$. These are related to the phases of the appearance of red supergiants that also produced the dips in the limiting magnitude curves of Fig. 6.

   
4.3 The Frascati models

Since the Starburst99 models only cover the age range up to 1 Gyr, we also used synthetic cluster models for older ages from the Frascati group. These "Frascati-models'' were calculated by Romaniello (1998) from the evolutionary tracks of Brocato & Castellani (1993) and Cassisi et al. (1994) using the HST-WFPC2 magnitudes derived from the stellar atmosphere models by Kurucz (1993). These models are for instantaneous formation of a cluster of solar metallicity stars in the mass range of 0.6 to 25 $M_{\odot }$, with a total initial mass of 50 000 $M_{\odot }$, distributed according to Salpeter's IMF. These models cover an age range of 10 to 5000 Myr. There are 48 models with timesteps increasing from 10 Myr for the youngest ones to 500 Myr for the oldest ones. We have increased the magnitudes of the Frascati models by -3.526 mag in order to scale these models to a total mass of 10⁶ $M_{\odot }$ in the mass range of 1.0 to 25 $M_{\odot }$; i.e. -3.25 mag correction for the conversion from $5\times 10^4$ to $1\times 10^6$ $M_{\odot }$, and 0.279 mag correction for the conversion from a lower stellar mass limit of 0.6 $M_{\odot }$ to 1.0 $M_{\odot }$. In this way the masses can be compared directly with those of the Starburst99 models.

The Starburst99 models are expected to be more accurate for the younger clusters, because they are based on the evolutionary tracks of the Geneva-group which include massive stars. The Frascati models are expected to be more accurate for the old clusters, because they include stars with masses down to 0.6 $M_{\odot }$.

Figure 7: The variations of colours with age of the Starburst99 models of Z=0.02 and an upper mass limit of 30 $M_{\odot }$. Notice the periods of rapid variability near the peaks and dips around $t \simeq 6$ and 16 Myrs, $\log (t) \simeq 6.8$ and 7.2.

   
4.4 The extinction curve

To determine the values of E(B-V) of the clusters we assume the mean galactic reddening law. Lamers et al. (2002) and Scuderi et al. (2002) have shown that the extinction law of respectively the bulge and the core of M 51 agree very well with the galactic law. The values of R_i=A_i/E(B-V) for the HST-WFPC2 filters have been calculated by Romaniello (1998). They are respectively 4.97, 4.13, 3.11, 2.41 and 1.91 for the U, B, V, R and I bands.

In the fitting of the energy distributions we did not take into account the disruption of clusters. We assumed that the stars contribute to the total energy distribution of the clusters until the cluster is completely dissolved and is no longer detectable as a point source. This is a reasonable assumption because we measured the magnitudes in a circle of 12 pc radius (see Sect. 2).

Figure 8: Example of some fits. The filled dots represent the observed energy distributions of objects measured in at least four bands, with their error bars in magnitude versus wavelength (in $\mu$m). In several cases the error bars are smaller than the size of the dots. Crosses indicate magnitude lower limits. The parameters of the fits are indicated in an array $(E(B-V),~\log(t),~\log(\mbox{$M_{\rm cl}$ }))$. The second array gives the uncertainties in these parameters. The line is the best fit obtained with the maximum likelihood method. Top figures: fits with $\mbox{$\chi^2_{\nu}$ }< 1$; bottom figures: fits with $\mbox{$\chi^2_{\nu}$ }\simeq 3$. Notice that the uncertainty of the fit parameters depends not only on $\chi _{\nu }^2$ but also strongly on the quality of the data. For instance, the errors in the magnitudes of the last two clusters are very small, so that the fit parameters are accurate, despite the high value of $\chi _{\nu }^2$.