7 The cluster initial mass function

The data in Fig. 10 and the detailed study of this distribution by Boutloukos & Lamers (2002) show that the clusters in the inner spiral arms of M 51 disrupt on a time scale of about tens of Myrs. In fact, these authors derived the dependence of the disruption time on the initial mass of the clusters in the inner spiral arms of M 51 as

$$\log t_{\rm disr} = \log t_4 ~+~\gamma \times \log (M_{\rm cl}/10^4~\mbox{$M_{\odot}$ })$$ (4)

with $\log t_4 = 7.64 \pm 0.22$ and $\gamma=0.62 \pm 0.06$ for the mass range of $3 \le \log (\mbox{$M_{\rm cl}$ }/ \mbox{$M_{\odot}$ }) \le 5.2$, where $M_{\rm cl}$ is the initial mass of the cluster. We see that clusters with an initial mass larger than 10⁴ $M_{\odot }$ survive $4\times10^7$ years. Clusters with an initial mass of only 10³ $M_{\odot }$ disrupt on a time scale of $1\times10^7$ yrs. This implies that the cluster initial mass function cannot be derived from the total sample of clusters, because the disruption will produce a strong bias towards the more massive clusters. However for the youngest clusters with ages less than about 10 Myr disruption is not yet an important effect and, therefore, these clusters can be used to derive the initial cluster mass function.

Figure 11: Histograms of the mass of clusters (in $\log M_{\rm cl}$) with an age of 10 Myrs or younger. The light shaded area shows the distribution of all 354 clusters. The dark shaded area shows the histogram of clusters with an accurate mass determination of $\Delta \log M_{\rm cl} < 0.25$. The steep part at $\log(M) < 3.0$ is due to the detection limit. The decrease at $\log(M) >3.0$ reflects the slope of the cluster IMF. (All masses have to be increased by a factor 2.1 if the lower limit of the stellar mass is 0.2 $M_{\odot }$ rather than the value of 1 $M_{\odot }$ that was adopted in the Starburst99 models.)

Figure 11 shows the resulting mass distribution of clusters with an age less or equal to 10 Myr for two samples of clusters. The first sample contains all 354 clusters younger than 10 Myr. The second sample contains 168 clusters in the same age range but with a mass determination of $\Delta \log(\mbox{$M_{\rm cl}$ }) \le 0.25$. Both samples show the same characteristics: a steep increase in number between $2.5 < \log \mbox{$M_{\rm cl}$ }<3.0$ and a slow decrease to higher masses. The steep increase is due to the detection limit or the disruption of the low mass clusters. The slow decrease reflects the cluster initial mass function (CIMF). The decrease indicates that the clusters are formed with an CIMF that has a negative slope of ${\rm d}\log (N)/{\rm d}\log (M)$, as expected.

If the CIMF can be written as a power law of the type

$$N(M)~{\rm d}M~ \sim ~ M^{\alpha}~{\rm d}M~~~~~{\rm for}~~\mbox{$M_{\rm min}$ }<M<\mbox{$M_{\rm max}$ }$$ (5)

then the normalized cumulative distribution will be

 

$$\Sigma (M)=A - B\times M^{-\alpha+1}~~~~~{\rm with}$$

$$A=B \times \mbox{$M_{\rm min}$ }^{-\alpha+1}$$

$$B=\left(M_{\rm min}^{-\alpha+1}~-~M_{\rm max}^{-\alpha+1}\right)^{-1}\cdot$$ (6)

Figure 12 shows the observed and predicted normalized cumulative distribution of the 149 clusters with an age less than 10 Myr and with a mass $\mbox{$M_{\rm cl}$ }>2.5\times 10^3~\mbox{$M_{\odot}$ }$ that are used for the determination of the CIMF. We eliminated the clusters with $\mbox{$M_{\rm cl}$ }<2.5\times 10^3~\mbox{$M_{\odot}$ }$ from the sample because Fig. 11 shows that the sample may not be complete for smaller masses. The cumulative distributions of the 84 clusters of $M>2.5\times 10^3~\mbox{$M_{\odot}$ }$, younger than 10 Myr, which are fitted to the models with an accuracy of $\mbox{$\chi^2_{\nu}$ }\le 3.0$, or of the 66 clusters with $\Delta \log(\mbox{$M_{\rm cl}$ }) \le 0.25$, not shown here, (where $\Delta \log(\mbox{$M_{\rm cl}$ })$ is the uncertainty in the mass determination), have the same shape as the distribution in Fig. 12. For these three samples we have determined the values of $\alpha$ and $M_{\rm min}$ by means of a linear regression under the reasonable assumption that $\mbox{$M_{\rm min}$ }\ll\mbox{$M_{\rm max}$ }$, which implies that $A \simeq 1$. The resulting values of $\alpha$ and $M_{\rm min}$ are listed in Table 3. The table shows that $\alpha \simeq 2.1$ for all three samples.

 

 

Table 3: The CIMF for clusters with t<10 Myr and $\mbox{$M_{\rm cl}$ }<2.5\times 10^3~\mbox{$M_{\odot}$ }$.

Sample	Nr	$\log \mbox{$M_{\rm min}$ }$	$\alpha$
All	149	3.49	2.12 $\pm$ 0.26
$\Delta \log \mbox{$M_{\rm cl}$ }< 0.25$	66	3.48	2.04 $\pm$ 0.41
$\mbox{$\chi^2_{\nu}$ }< 3.0$	82	3.46	2.16 $\pm$ 0.40

Figure 12: The cumulative mass distribution of 149 clusters with an age less than 10 Myrs and an inital mass of $\log (M) > 3.40$. The full line is the best power-law fits for an IMF with the value of $\alpha$ and $M_{\rm min}$ from Table 3. The dashed line is the fit with $\alpha =2.00$.

The predicted cumulative distribution with the derived values of $\alpha=2.12$ and $M_{\rm min}$ is shown in Fig. 12 (full line). The figure shows a slight underabundance of clusters in the range of $4.0 < \log (\mbox{$M_{\rm cl}$ }) < 4.3$ and a slight overabundance in the range of $\log (\mbox{$M_{\rm cl}$ })>4.3$. In fact, for the mass range of $3.5 < \log (\mbox{$M_{\rm cl}$ }) < 4.3$ a fit with $\alpha =2.00$, shown by dashed lines, fits the distribution excellently. We conclude that the CIMF of clusters younger than 10 Myr has a slope of $\alpha \simeq 2.1 \pm 0.3$ in the mass range of $3.0 < \log (\mbox{$M_{\rm cl}$ }) < 5.0$ and a slope of $2.00 \pm 0.05$ in the range of $3.0 < \log(\mbox{$M_{\rm cl}$ }) < 4.3~\mbox{$M_{\odot}$ }$.

The derived exponent of the cluster IMF is very similar to the value of $\alpha = 2.0$ of young clusters in the Antennae galaxies, as found by Zhang & Fall (1999). It shows that the IMF of clusters formed in the process of galaxy-galaxy interaction is very similar to the one of clusters formed in the spiral arms of a galaxy, long after the interaction. This mass distribution is also similar to that of giant molecular clouds (e.g. McKee 1999; Myers 1999). This may support the suggestion that the mass distribution of the clusters is determined by the mass distribution of the clouds from which they originate.