1 Introduction

During the last years, many interacting and merging galaxies were discovered to host large numbers of young star clusters that formed during the merging process (Holtzman et al. 1992; Whitmore et al. 1993; Meurer et al. 1995; Whitmore et al. 1997; Whitmore et al. 1999; Zepf et al. 1999). The overall properties of these clusters suggest that they could be the progenitors of the globular cluster populations seen in normal nearby elliptical and spiral galaxies (e.g., Zepf & Ashman 1993). Such an hypothesis also has the attractive implication that if ellipticals formed through the merger of two large spiral galaxies (as suggested in the popular hierarchical merging model of galaxy formation e.g., Kauffmann et al. 1993), these young clusters might evolve into the red (supposedly metal-rich) part of the globular cluster population of elliptical galaxies when the merger is complete (e.g. Schweizer 2001). However, this hypothesis needs to be tested by determining the characteristics of both individual young clusters and the cluster population as a whole.

Globular clusters have a typical mass of 1-2$\times$10⁵ $M_{\odot }$ and a mass function which is log-normal (e.g., Harris 1991). The population of clusters in NGC 4038/4039 ("the Antennae''), however, has a power law luminosity function, and the same shape is also suggested for the mass function (Whitmore et al. 1999; Zhang & Fall 1999). Masses determined from photometric data are as large as a few $\times$10⁶ $M_{\odot }$ for some of the clusters (Zhang & Fall 1999; Mengel et al. 2001) and the determined ages span a large range (Whitmore et al. 1999; Mengel et al. 2001).

At first glance, the population of globular clusters in the Milky Way and those in the Antennae seem to have vastly different ensemble characteristics. For example, the most massive clusters in the Antennae are at least a factor of a few more massive than those in the Milky Way. However, given the large number of clusters formed in the Antennae, it seems reasonable that the mass function is sampled up to a high upper mass, and moreover mass loss during the evolution of the young clusters can be expected to play a role. Evolution over a Hubble time might convert the power law cluster mass function into a log-normal one, if lower mass clusters were dissolved preferentially during the evolution. Dynamical models like those, for example, of Chernoff & Weinberg (1990) and Takahashi & Portegies Zwart (2000) provide theoretical support for the necessary differential evolution from a power-law mass distribution function into a log-normal distribution.

Dynamical cluster masses are less model dependent than those determined from photometry only, or, at the very least, have a different set of dependencies. The dynamical mass is derived from the stellar velocity dispersion in combination with the cluster size and light profile. While the photometric mass estimates depend on the model assumed for the star formation parameters (time-scale, age, IMF slope, lower and upper limiting masses, metallicity, etc.), the dynamical mass estimates rely only on the validity of the Virial equilibrium (i.e., the potential is due to the collective gravitational effects of individual stars - self-gravitating - and is changing only slowly with time), and the assumed constancy of the M/L ratio within the cluster. The comparison of photometric and dynamical cluster masses (cluster M/L) constrains the slope of the mass function and may reveal the presence or absence of low mass stars. The fraction of low mass stars influences the survival probability of the cluster during a few Gyrs of evolution, as clusters rich in low-mass stars are less prone to destruction (e.g., Takahashi & Portegies Zwart 2000).

More specifically, we have used high spectral resolution spectroscopy conducted at the ESO VLT to estimate the stellar radial velocity dispersion $\sigma$, and high spatial resolution imaging from archival HST images (Whitmore et al. 1999) to estimate the size scales (e.g., the projected half-light radius $r_{\rm hp}$). This results in $M_{\rm dyn}$ for individual clusters which can be estimated using the equation:

$$% M=\frac{\eta \sigma^2 r_{\rm hp}}{G}\cdot$$ (1)

Where $\eta$ is a constant that depends on the distribution of the stellar density with radius, the mass-to-light ratio as a function of radius, etc. The only assumption that goes into Virial relations of this form is that the cluster is self-gravitating and is roughly in equilibrium over many crossing times (i.e., it is not rapidly collapsing or expanding). The calculations of Spitzer (1987) indicate that for a wide range of models, $\eta$ $\approx$ 10. Under the assumption of isotropic orbits, $\eta$ differs from 3 (the pure Virial assumption) since the "gravitational radius" must be scaled to the projected half-light radius (or some measurable scale) and because the measurement of the velocity dispersion is not the central velocity dispersion but is a measurement which is weighted by the light profile. So differing mass profiles have different values of $\eta$ appropriate for estimating their dynamical masses. For example, globular clusters in the Milky Way exhibit a range of concentration parameters (logarithm of the ratio of the tidal radius to the core radius) in a King model (King 1966) of approximately 0.5 to 2.5 (e.g., Harris 1991). Over this range of concentrations, $\eta$ varies from about 9.7 to 5.6. Hence, accurately measuring the light profiles (and of course the assumption that the light profile traces the mass profile) is crucial in estimating the masses.

Recent attempts at estimating the dynamical masses of similar clusters in other galaxies has met with some success. Notably, Ho & Filippenko (1996a) and Ho & Filippenko (1996b) were able to estimate the masses of two luminous young star clusters in the blue compact dwarf starburst galaxy NGC 1705 (1705-1) and NGC 1569 (1569A). They measured velocity dispersions for 1705-1 and 1569A of 11.4 km s^-1 and 15.7 km s^-1 respectively. Under the assumption that $\eta$ = 10, Sternberg (1998) estimated that these clusters have masses of about 2.7 $\times$ 10⁵ $M_{\odot }$ for 1705-1 and 1.1 $\times$ 10⁶ $M_{\odot }$ for 1569A. More recently, Smith & Gallagher (2001) have estimated the dynamical mass of the luminous star-cluster in M 82, M 82-F. Assuming $\eta=10$, they find a mass of $1.2 \pm 0.1$$\times$ 10⁶ $M_{\odot }$. This mass estimate implies that the mass-to-light ratio of M 82-F is very extreme and requires either a very flat mass function slope or a lack of low mass stars for a Salpeter-like mass function slope. Interestingly, this small sample of clusters seems to require a range of IMFs to explain their mass-to-light ratios.

The purpose of the present paper is to derive masses for a small sample of young compact clusters in the Antennae galaxies - the nearest merger - and to use these mass estimates to constrain the functional form of the IMF in these clusters. Since it is important that the velocity dispersion be determined from the stellar component of the cluster, we have undertaken a program to observe extinguished clusters in the K-band, to take advantage of the strong CO absorption band-heads beyond 2.29 $\mu$m, and unextinguished clusters in the optical, to take advantage of the strong absorption of the Calcium Triplet (CaT) and other metal lines around 8500 Å. These features are strong in atmospheres of red supergiants which would be expected in large numbers in clusters with young ages ($\approx$6-50 Myrs) and thus particularly well-suited for estimating the masses of clusters like those in the Antennae. Given that previous investigators (e.g., Ho & Filippenko 1996a; Ho & Filippenko 1996b) have measured velocity dispersions around 15 km s^-1 suggests that to conduct such measurements would require a minimum resolution of $R \approx 9000$. Such resolutions are now available using ISAAC on VLT-UT1 which with its narrowest slit delivers $R \approx 9000$ and UVES on VLT-UT2 which delivers a resolution of $\approx$38000 with a 1 $^{\prime\prime}$ wide slit.