An analytical description of the disruption of star clusters in tidal fields with an application to Galactic open clusters

We present a simple analytical description of the disruption of star clusters in a tidal field. The cluster disruption time, defined as t(dis) = {dln M/dt}(-1), depends on the mass M of the cluster as t(dis) = t(0)(M/M-.)(gamma) with gamma = 0.62 for clusters in a tidal field, as shown by empirical studies of cluster samples in different galaxies and by N-body simulations. Using this simple description we derive an analytic expression for the way in which the mass of a cluster decreases with time due to stellar evolution and disruption. The result agrees very well with those of detailed N-body simulations for clusters in the tidal field of our galaxy. The analytic expression can be used to predict the mass and age histograms of surviving clusters for any cluster initial mass function and any cluster formation history. The method is applied to explain the age distribution of the open clusters in the solar neighbourhood within 600 pc, based on a new cluster sample that appears to be unbiased within a distance of about 1 kpc. From a comparison between the observed and predicted age distributions in the age range between 10 Myr to 3 Gyr we find the following results: (1) The disruption time of a 10(4) M-. cluster in the solar neighbourhood is about 1.3 +/- 0.5 Gyr. This is a factor of 5 shorter than that derived from N-body simulations of clusters in the tidal field of the galaxy. Possible reasons for this discrepancy are discussed. ( 2) The present star formation rate in bound clusters within 600 pc of the Sun is 5.9 +/- 0.8 x 10(2) M-. Myr(-1), which corresponds to a surface star formation rate of bound clusters of 5.2 +/- 0.7 x 10(-10) M. yr(-1) pc(-2). ( 3) The age distribution of open clusters shows a bump between 0.26 and 0.6 Gyr when the cluster formation rate was 2.5 times higher than before and after. ( 4) The present star formation rate in bound clusters is about half that derived from the study of embedded clusters. The difference suggests that about half of the clusters in the solar neighbourhood become unbound within about 10 Myr. ( 5) The most massive clusters within 600 pc had an initial mass of about 3 x 10(4) M-.. This is in agreement with the statistically expected value based on a cluster initial mass function with a slope of - 2, even if the physical upper mass limit for cluster formation is as high as 10(6) M-..