5 Fitting the observed energy distributions to observed cluster models

   
5.1 The fitting procedure

We have fitted the energy distributions of the clusters with good photometry in four or more filters, viz. UBVRI, BVRI, UVRI and UBVR (see Table 1), with the energy distributions of the cluster evolutionary synthesis models, discussed above, using a three dimensional maximum likelihood method. The three fitting parameters are: the age of the cluster (t), the reddening (E(B-V)) and the initial mass of the cluster (M). For the clusters detected in only three filters we reduce the parameter space and make a two dimensional maximum likelihood fit. For objects not detected in a band, we adopted the magnitude lower limits (described in Sect. 3) to check which fits were acceptable.

   
5.2 Three dimensional maximum likelihood method

This method was applied to sources that are detected in at least four bands. We fitted the observed energy distributions with those predicted for the Starburst99 cluster models and the Frascati models (see Sect. 4.2). For each model-age we have two fitting parameters M and E(B-V). The cluster models are for an initial cluster mass of 10⁶ $M_{\odot }$. For other masses the flux simply scales $M/10^6~\mbox{$M_{\odot}$ }$. We have adopted an uncertainty in the model fluxes of 5 percent (0.05 mag) in all bands.

To reduce the range in masses in the parameter space, we make an initial guess of the cluster mass based on the observed magnitudes, since the distance to M 51 is known. By calculating the average difference in magnitude - weighted by the errors - between the theoretical energy distribution for 10⁶ $M_{\odot }$ and the observed one, we have a good first approximation of the mass of the cluster:

 

$$M_{\rm guess}(E(B-V),\tau) \simeq \frac{\sum_{\lambda} 1/\sigma_{... ...- m_{\lambda}^{\rm mod}(E(B-V),\tau)\right\}}{\sum_{\lambda}1/\sigma_{\lambda}}$$ (3)

with E(B-V) between 0.0 and 2.0 in steps of 0.02, where $\sigma$ is the uncertainty in the magnitude. This is a reasonable range, compared to the average colour excess for the bulge of 0.2 found by Lamers et al. (2002). With this initial guess we then make a three dimensional likelihood analysis, using a mass range from 1/1.5 to 1.5 times the initial mass estimate in steps of 0.004 dex. The extinction was varied between 0.0 and 2.0 in steps of 0.01 and the age was varied between 0.1 Myr and 1 Gyr for the Starburst99 models and between 10 Myr to 5 Gyr for the Frascati models.

For every fit we obtain a value for the reduced $\chi^2$, i.e. $\chi^2_{\nu}$ = $\chi^2$/$\nu$, where $\nu$ is the number of free parameters i.e. the number of the observed data points minus the number of parameters in the theoretical model. For a good fit, $\chi^2_{\nu}$ should be about unity. We checked that the fits were consistent with the faint magnitude limits of the filters in which the object was not detected. If not, the fit was rejected. The fit with the minimum value of $\chi^2_{\nu}$ was adopted as the best fit and the corresponding values of E(B-V), age and M_i were adopted. This method was applied for fits with the four sets of the Starburst99 models and with the Frascati models. Figure 8 shows some examples of the results of the fitting process.

To estimate the uncertainty in the determined parameters we use confidence limits. If $\mbox{$\chi^2_{\nu}$ }< \mbox{$\chi^2_{\nu}$ }(\min)+1$ then the resulting parameters, i.e. $\log (t)$, $\log (M_i)$ and E(B-V), are within the 68.3% probability range. So the accepted ranges in age, mass and extinction are derived from the fits which have $\mbox{$\chi^2_{\nu}$ }(\min) < \mbox{$\chi^2_{\nu}$ }< \mbox{$\chi^2_{\nu}$ }(\min)+1$. With this method we derived the ages, initial masses and extinction with their uncertainties of 602 clusters.

Figure 9 shows a histogram of number of clusters as function of E(B-V). We will use this distribution for the two dimensional maximum likelihood fit of clusters detected in three bands only. The redding is small: 90% of the clusters have a reddening lower than 0.40; 67% of the clusters have a reddening lower than 0.18 and 23% have no detectable reddening.

   
5.3 Two dimensional maximum likelihood method

For the clusters observed in only three bands, it is not possible to make a three dimensional maximum likelihood fit. To reduce the parameter space we adopted the probability distribution of E(B-V) in the range of E(B-V) between 0.0 and 0.4, as shown in Fig. 9. For every value of E(B-V) between 0.0 and 0.40 (in steps of 0.02) and for every age of the model cluster, we first determine the mass of the cluster by means of Eq. (3). The results of the three dimensional fits have shown this is a good approximation. This results in a maximum likelihood age of the cluster for every value of E(B-V).

The $\chi^2_{\nu}$ which comes out from the two dimensional maximum likelihood method is used to distinguish between the accepted and rejected fits. For the parameter $\nu$ we used a value of 3-1=2, because we only fit the age of the cluster. The mass of the cluster comes from the scaling of the magnitudes to the best-fit model. A fit is accepted if it agrees with the lower magnitude limits in the filters where it was not measured. To determine the age of the cluster, we average the ages, weighted by the probability that each particular value of E(B-V) occurs, derived by normalizing the distribution in Fig. 9 in the range of 0.0 < E(B-V)<0.40. The error in the age is determined by calculating the value of the standard deviation $\sigma$ of the age, again weighted with the probability that the value of E(B-V) occurs. The value of E(B-V) is determined by using the one which belongs to the model with the age closest to the average age. The initial mass of the cluster is then calculated by Eq. (3) for the adopted value of E(B-V). With this method we derived the age, initial mass and the extinction of 275 clusters, using both the Starburst99 models and the Frascati models.

   
5.4 Starburst99 or Frascati models?

We have compared the energy distributions of the objects with five sets of models: four sets of Starburst99 models, with $Z=Z_{\odot}$ or $2 ~ Z_\odot$ and $\mbox{$M_{\rm up}$ }=30$ or 100 $M_{\odot }$, and the Frascati models with $Z=Z_{\odot}$ and $\mbox{$M_{\rm up}$ }=25$ $M_{\odot }$. Based on the lack of [OIII] emission we adopt the models with $\mbox{$M_{\rm up}$ }=25$ or 30 $M_{\odot }$. The models with $\mbox{$M_{\rm up}$ }=100$ $M_{\odot }$ are only used later to check the influence of this adopted upper limit on our results.

Figure 9: Histogram of the values of E(B-V) from the fits with observations in BVRI and UBVRI. The extinction is small: 90% of the clusters have $\mbox{$E(B-V)$ }<0.40$.

We have compared the ages derived from fitting the observed SEDs with the Starburst99 models and the Frascati models, for objects measured in at least four filters and with fits of $\mbox{$\chi^2_{\nu}$ }< 3.0$. We found that in the age range of 10 to 800 Myr there is a reasonable correlation between the results of the SB99 models and the Frascati models, with the Frascati models giving ages systematically about 0.4 dex smaller than the SB99 models. In this age range we adopt the SB99 models, because they are based on more reliable evolutionary tracks, better stellar atmosphere models and because the nebular continuum is included. We find that in this mass range the fits with the SB99 models have smaller $\mbox{$\chi_{\nu}^2$ }$ than the fits with the Frascati-models. For objects with ages above about 700 Myrs ( $\log (t) > 8.85$) the Frascati-models give the most reliable fits with smaller $\mbox{$\chi_{\nu}^2$ }$ than the SB99 models. This is probably because the lower mass limit of the SB99 models is 1 $M_{\odot }$, and the Frascati models go down to 0.6 $M_{\odot }$. Moreover the SB99 models do not go beyond 1 Gyr. Based on this comparison we adopt the SB99 fits for ages less than 700 Myrs and the results of the Frascati models for older ages.

   
5.5 Uncertainties in the derived parameters

The determination of the cluster parameters, age, mass and extinction, by means of the two or thee dimensional maximum likelihood fitting method, i.e. the 3(2)DEF-method, results not only in the values of $\log(\mbox{$M_{\rm cl}$ })$, $\log (t)$ and E(B-V) but also in their maximum and minimum acceptable values. The best-fit values are not necessarily in the middle of the minimum and the maximum values. If we define the uncertainties in the parameters, $\Delta E(B-V)$, $\Delta \log(\mbox{$M_{\rm cl}$ })$ and $\Delta \log(t)$ as half the difference between the maximum and mimimum values, we find that 30 percent of the clusters have $\Delta E(B-V)<0.08$, 50 percent have $\Delta E(B-V)<0.11$ and 70 percent have $\Delta E(B-V)<0.15$. For the uncertainties in the mass determination we find $\Delta \log(\mbox{$M_{\rm cl}$ })<0.20$, <0.33 and <0.55 for 30, 50 and 70 percent of the clusters respectively. For the uncertainties in the age determination we find $\Delta \log(t)<0.23$, <0.39 and <0.69 for 30, 50 and 70 percent of the clusters respectively. Taking the values for 50 percent of the clusters as representative, we conclude that the uncertainties are $\Delta E(B-V)\simeq 0.11$, $\Delta \log(\mbox{$M_{\rm cl}$ }) \simeq 0.33$ and $\Delta \log(t) \simeq 0.39$.

   
5.6 Contamination of the cluster sample by stars?

To estimate the number of stars that may contaminate our cluster sample we use the stellar population in the solar neighbourhood. From the tabulated stellar densities as a function of spectral type (Allen 1976) we derived the number of stars brighter than a certain value of M_V per pc³. Using the mass density of $0.13~ \mbox{$M_{\odot}$ }$pc^-3, we derived the number of stars per unit stellar mass. The results are listed in Table 2. The total stellar mass of the observed region of M 51 (see Fig. 2) is estimated to be about 1/20 of the total mass of $5\times 10^{10}$ $M_{\odot }$ in the disk of that galaxy (Athanassoula et al. 1987), i.e. about $2.5\times 10^{9}$ $M_{\odot }$. This implies that we can expect the following numbers of stars brighter than M_V<-6.5 per spectral type in the observed region: 4 (O), 13 (B), 6 (A), 6 (F), 6 (G), <1 (K) and <1 (M). So we can expect of the order of 40 bright stars with M_V<-6.5 in the observed region of M 51. However, these are all massive young supergiants of which the vast majority will be in clusters! So the number of bright stars outside clusters, that may contaminate our sample of clusters will be considerable smaller, and we expect it to be smaller than about 20 out of the total sample of 877. Moreover, we have shown above, from the lack of O[III] emission, that the clusters in M 51 have a shortage of massive stars with M>30 $M_{\odot }$. We can expect this effect also to occur for the massive field stars. This would reduce the number of expected contaminating stars even further. Tests have shown that a considerable fraction of possibly remaining stellar sources will be eliminated by the requirement that their energy distribution should fit that of cluster models within a given accuracy. Based on all these considerations, we conclude that contamination of our cluster sample with very massive stars of M_V<-6.5 outside clusters is expected to be negligible.

 

 

Table 2: Numbers of stars above a certain absolute magnitude limit in the solar neighbourhood.

Limit	O	B	A	F	G	K	M	Total
MV < -5.5	-8.3	-7.6	-8.0	-7.4	-7.4	-8.5	-8.5	-6.9
MV < -6.5	-8.8	-8.3	-8.6	-8.6	-8.6	<-9.5	<-9.5	-7.9
MV < -7.5	-9.8	-9.3	-9.6	-9.6	-9.6	<-10.5	<-10.5	-8.9

(1) The data are in log (Number/per $M_{\odot }$).
(2) The number of stars with M_V<-7.5 was estimated to be about 10 times smaller than for M_V<-6.5 because of the small number
of very massive stars formed, and their short evolution time.