The reduction of ISAAC data was performed in a standard way using the IRAF data reduction package. The reduction included dark subtraction and flat-fielding (using a normalized flat-field created from internal flat observations) on the two-dimensional array. Sky subtraction was performed pairwise, followed by a rejection of cosmic ray hits and bad pixels. The spectra were then corrected for tilt and slit curvature by tracing the peak of the stellar spatial profile along the dispersion direction and fitting a polynomial to the function of displacement versus wavelength. The corresponding procedure was also applied to the orthogonal direction. Single integrations were combined by shifting-and-adding, including a rejection of highest and lowest pixels. The rejection could not be applied to observations of [WS99]2, because they consisted of only a limited number of single frames. The object spectra were then extracted from user defined apertures. A linear fit to the background (below $\approx$ 7% of the peak intensity for all clusters) on both sides of the object spectrum was subtracted.

The spectra were wavelength calibrated using observations of arc discharge lamps. An atmospheric calibrator (B5V) was observed and reduced in the same way as the target and used to divide out the atmospheric absorption features from the spectra.

$\begin{figure} \par\includegraphics[angle=360,width=15.57cm,clip]{ms1901f2.ps} \includegraphics[angle=90,width=16cm,clip]{ms1901f3.ps}\end{figure}$

Figure 2: ISAAC Spectra of the three clusters [WS95]355, [W99]15 and [W99]2 (top to bottom, black) with best fits (grey). The bottom plot shows the residuals of the fit to spectrum [W99]15 for the best fit and for two fits with lower and higher velocity dispersions. This gives a visual impression of how much the velocity dispersion affects the spectral features.

3.2 Reduction of the optical echelle data

Reduction of the UVES data was performed within the echelle spectra reduction environment of IRAF. Reduction included bias subtraction, flat-fielding and extracting the spectrum from user supplied apertures. A background is fit to neighboring regions and subtracted. Also here, the background was below $\approx$ 7% of the peak intensity for all clusters. This step includes a bad pixel rejection. The full moon was fairly close to the Antennae during the integrations, and its scattered light introduced a solar spectrum into the data. However, this contribution is very effectively removed during background subtraction.

Wavelength calibration required the identification of many lines in the ThAr-spectra for each of the echelle orders which were identified using a line table available from the ESO-UVES web pages. A dispersion function was then fit and applied to the object data. This data set contains each order of the spectrum in a separate channel. These separate channels are combined into one final spectrum covering the total wavelength range. This involves re-gridding of the wavelength axis into equidistant bins and averaging in the overlapping edges of the orders. The spectral resolution obtained is $R \approx$ 38000 across all orders.

3.3 Estimating the velocity dispersions

For each spectrum (both the optical and near-IR spectra), we estimated the $\sigma$ of the broadening function (assumed to be a Gaussian) which best fit the cluster spectrum in the following way. The stellar spectrum (the template spectrum) was broadened with Gaussian functions of variable $\sigma$ in velocity space. The resulting set of spectra were then compared with the cluster spectrum. The best fit was determined by evaluating $\chi ^2$ and then search for the minimum of the function $\chi ^2$ ( $v_{\rm r}$ , $\sigma$ ) using a simplex downhill algorithm for the tour through parameter space.

3.3.1 Unique character of the near-IR estimates

Obviously, the template spectrum has to be a good overall match to the cluster spectrum, otherwise an erroneous velocity width will be derived for the cluster. For a star cluster that formed $\sim$ 10 Myrs ago, late K through early M supergiants are expected to provide the largest contribution to the flux at 2.3 $\mu$ m. However, according to population synthesis models (e.g., Leitherer et al. 1999; Sternberg 1998), there will be a non-negligible contribution to the flux from hot main sequence stars. Since the stars are hot (O and B-type stars), this "diluting continuum'' will be an essentially featureless continuum which solely decreases the equivalent width of the CO band-heads. This has the effect of shifting the apparent dominating stellar type towards higher effective temperatures. Starting out with a template spectrum with weak CO features leads to very low velocity dispersions, the opposite is the case if an M5I star (strong band-heads) is used, with vast differences in the results (a few km s^-1 to up to about 30 km s^-1).

To first order, the velocity dispersion determined from different stellar templates agrees if the CO equivalent widths match. This is achieved by adding a (positive or negative) continuum to the stellar spectrum, which dilutes or enhances the CO features.

To second order, there are two different indicators that can be used to constrain the matching stellar template. Both relate to the shape of the CO band-head, due to variations in that shape between the different supergiant spectral classes. The various rotational transitions are resolved in the ¹²CO(2-0) and the ¹²CO(3-1) band-head. We performed tests of our analysis technique by creating a simulated cluster spectrum with an input velocity dispersion $\sigma_{\rm in}$ , added noise (usually S/N = 15) and re-determined the velocity dispersion $\sigma_{\rm out}$ using different template star spectra. With the correct template (out = in), we obtained in 30 test runs an average $\sigma_{\rm out}$ = 14.7 $\pm$ 0.7 km s^-1 (with $\sigma_{\rm in}$ = 15.0 km s^-1). Performing the same test on an input spectrum that was diluted by a flat continuum (10%), but still fitting with the undiluted template, the velocity dispersion increased ( $17.9 \pm 1.7$ km s^-1), as expected. Fitting other stellar types (M0I, M1I) yields similar results. But the mismatch can be diagnosed by:

3.3.2 Unique character of the optical estimates

Examining the spectra in detail, it was obvious that they are composed of a mixture of light from supergiants and hot main sequence stars. This means that the spectra cannot be simply fit by broadening the stellar template, but that either the hot star contribution needs to be added to the template, as was done in the case of the ISAAC spectra, or that the contribution needs to be subtracted from the cluster spectrum.

Here, the procedure was different from that applied to ISAAC data, because the hot main sequence stars were not completely featureless over the analyzed wavelength range, but had some Paschen absorption features. Pa16, Pa15 and Pa13 lie slightly red-ward of the CaT wavelengths 8498 Å, 8542 Å, and 8662 Å, respectively. Since the widths of these lines are mainly due to line broadening in the atmosphere and rotation of the star itself and only an insignificant fraction of the width is due to the velocity dispersion of the star cluster, they cannot be used to estimate the velocity dispersion. Therefore this weak contribution was subtracted from each cluster before the velocity dispersion was estimated. The amount of subtraction was estimated by the strengths of the Paschen line absorption and the appropriate template for the subtraction was chosen for its ability to remove the contribution accurately. Given the S/N of the spectra and the weakness of the features removed, the final characteristics of the subtracted spectrum were not very sensitive to the exact template used.

The cluster [W99]2 merits some specific discussion since it has both ISAAC and UVES observations and also some peculiarities were noted compared to the other clusters observed with UVES. Figure 3 shows fits of an M3I spectrum to pieces of the spectrum of the cluster [W99]2. The latter had a 45% contribution of B2V stars subtracted. This contribution was determined by interactively subtracting the spectrum of the B2V star until the Paschen absorption features were reduced down to the noise level of the cluster spectrum. In addition, we fit the continuum of the cluster spectrum using a combination of a B2V and an M3I star spectrum, in which the relative contributions of the two template stars was the fit parameter. The best fitting combination again included a 45% contribution of B2V stars. Moreover, the relative contributions of M3I and B2V stars in I and K band agree with the determined contributions to the ISAAC and UVES spectra, respectively, for this cluster.

3.4 Estimating the sizes of the observed clusters

In order to make an estimate of the mass, it is necessary to have a measure of the cluster radius and profile shape. Whitmore et al. (1999) found that many of the young star clusters in images taken with the Wide Field Planetary Camera on HST (WFPC-2) of the Antennae were slightly-to-well-resolved and estimated their sizes. Their main aim was to determine the average radii of the clusters and to compare them to those observed in other galaxies, both to the young clusters in other mergers and to globular clusters. They found that the mean effective radius of the clusters in the Antennae is $r_{\rm eff}$ $\approx$ 4 pc. However, Whitmore et al. (1999) did not present their results on individual clusters. Brad Whitmore kindly provided us with their estimates of the effective radii for the clusters we observed with ISAAC and UVES. But given the importance of size and light profile to the mass determination, we additionally measured them ourselves from the HST I-band images.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{ms1901f4.eps}\end{figure}$

Figure 3: Displayed are parts of the normalized spectrum of cluster [W99]2 (black), together with the fit (red). The four pieces were fit separately, and the average value of the velocity dispersion is $\sigma$ = 14.2 km s^-1. It is worth noting that this is the cluster which was also observed with ISAAC, and to compare the results (see Fig. 2).

Similar to what Whitmore et al. (1999) did to make such estimates we reduced archival HST I-band images (see Whitmore et al. 1999 for the details concerning the images) using the drizzle technique (see the Drizzle Handbook available from STScI). Given the pixel sizes of the HST chips (0 $.\!\!^{\prime\prime}$ 101 pix^-1 for the Wide Field Camera arrays and 0 $.\!\!^{\prime\prime}$ 045 pix^-1 for the Planetary Camera array), and the distance to NGC 4038/4039, which implies a scale of 93 pc arcsec^-1, the factor of two improvement in the sampling of the HST point-spread-function (PSF) due to drizzling the images is critical in getting a robust estimate of the sizes. At the typical cluster size, $r_{\rm eff} \approx$ 4 pc, the subtended scale is only 0 $.\!\!^{\prime\prime}$ 043 or about one pixel of the Planetary Camera! For this analysis we used the ishape routine described by Larsen (1999) implemented in the BAOLAB data reduction package developed by Larsen. The ishape routines were especially designed to work more efficiently than typical deconvolution routines at determining the sizes of slightly extended objects. ishape convolves the user-provided PSF with the given cluster profile and determines the minimum $\chi ^2$ for a range of sizes using a simplex downhill algorithm to find the minimum. The PSF we selected for this convolution is one from the Tiny Tim program (Krist 1995) which Whitmore et al. (1999) argue persuasively is appropriate for these data. We fit each cluster with a range of provided profiles which include Gaussian profiles, King profiles of several concentration parameters (King 1966), and Moffat profiles of two different concentrations (Moffat 1967). The program produces a reduced $\chi ^2$ to measure the significance of the fit and various images (artificial cluster image, the original input image, an image of the residuals between original and fit, and an image of the weights applied to each pixel) to judge the quality of the fit. The code also provides a handy weighting procedure that enables the rejection of "outliers" in the radial cluster profile, thereby excluding, for example, neighboring stars and/or clusters.

Table 3: The cluster masses as they were derived from the ISAAC and UVES spectra (in column "Inst.'' indicated by I and U, respectively) in comparison with the supergiant template spectrum. The age was derived from the combination of $W_{\rm Br\gamma }$ , $W_{\rm CO}$ , and $W_{\rm CaT}$ . For cluster [W99]2, it agrees well with the age estimated from the UV spectrum of $7 \pm 1$ Myrs (Whitmore et al. 1999). The size is the projected half-light radius $r_{\rm hp}$ estimated as described in Sect. 3.4 (for the cluster marked with ^a, we could not obtain a satisfactory fit and used the value provided by B. Whitmore). $W_{\rm CO}$ was estimated from the ISAAC spectra between rest-frame wavelengths 2.2924 $\mu$ m and 2.2977 $\mu$ m, ([W99]2 also from 3D integral field spectroscopy), and $W_{\rm CaT}$ from the UVES spectra, according to the definition in Díaz et al. (1989). The stellar velocity dispersion $\sigma$ was determined as described in Sect. 3.3, for [W99]2 the average value of ISAAC and UVES measurements is given ( $\sigma = 14.0 \pm 0.8$ and $14.3 \pm 0.5$ km s^-1, respectively). $M_{\rm vir}$ is the Virial mass determined from equation 1 with the 1 $\sigma$ -uncertainties given in column $\sigma _M$ . The light-to-mass ratios were derived from the extinction corrected magnitudes and a distance modulus to NGC 4038/4039 of 31.41.
Cluster Inst. $W_{\rm CO}$ $W_{\rm CaT}$ Age $\sigma$ $r_{\rm hp}$ log $M_{\rm vir}$ $\sigma _M$ log L_V/ M log L_K/ M

I/ U [Å] [Å] [10⁶ yr] [km s^-1] [pc] [ $M_{\odot }$ ] [%] [ $L_{\odot}$ / $M_{\odot }$ ] [ $L_{\odot}$ / $M_{\odot }$ ]

[WS95]355 I $16.3 \pm 0.2$ - $8.5 \pm 0.3$ $21.4 \pm 0.7$ $4.8 \pm 0.5^a$ 6.67 12 ... 1.05

[W99]15 I $17.0 \pm 0.2$ - $8.7 \pm 0.3$ $20.2 \pm 0.7$ $3.6 \pm 0.5$ 6.52 16 0.89 1.09

[W99]2 I/ U $16.2 \pm 0.2$ ${6.4} \pm 1.0$ $6.6 \pm 0.3$ $14.2 \pm 0.4$ $4.5 \pm 0.5$ 6.31 12 1.54 2.00

[W99]1 U $17.5 \pm 1$ ${8.6} \pm 1.0$ $8.1 \pm 0.5$ $9.1 \pm 0.6$ $3.6 \pm 0.5$ 5.81 19 1.76 2.22

[W99]16 U $19 \pm 4$ ${9.9} \pm 1.0$ $10 \pm 2$ $15.8 \pm 1$ $6.0 \pm 0.5$ 6.51 15 0.50 1.20

3 Reduction and analysis

3.1 Reduction of the near-IR data

3.2 Reduction of the optical echelle data

3.3 Estimating the velocity dispersions

3.3.1 Unique character of the near-IR estimates

3.3.2 Unique character of the optical estimates

3.4 Estimating the sizes of the observed clusters

Cluster	Inst.	$W_{\rm CO}$	$W_{\rm CaT}$	Age	$\sigma$	$r_{\rm hp}$	log $M_{\rm vir}$	$\sigma _M$	log L_V/ M	log L_K/ M
	I/ U	[Å]	[Å]	[10⁶ yr]	[km s^-1]	[pc]	[ $M_{\odot }$ ]	[%]	[ $L_{\odot}$ / $M_{\odot }$ ]	[ $L_{\odot}$ / $M_{\odot }$ ]
[WS95]355	I	$16.3 \pm 0.2$	-	$8.5 \pm 0.3$	$21.4 \pm 0.7$	$4.8 \pm 0.5^a$	6.67	12	...	1.05
[W99]15	I	$17.0 \pm 0.2$	-	$8.7 \pm 0.3$	$20.2 \pm 0.7$	$3.6 \pm 0.5$	6.52	16	0.89	1.09
[W99]2	I/ U	$16.2 \pm 0.2$	${6.4} \pm 1.0$	$6.6 \pm 0.3$	$14.2 \pm 0.4$	$4.5 \pm 0.5$	6.31	12	1.54	2.00
[W99]1	U	$17.5 \pm 1$	${8.6} \pm 1.0$	$8.1 \pm 0.5$	$9.1 \pm 0.6$	$3.6 \pm 0.5$	5.81	19	1.76	2.22
[W99]16	U	$19 \pm 4$	${9.9} \pm 1.0$	$10 \pm 2$	$15.8 \pm 1$	$6.0 \pm 0.5$	6.51	15	0.50	1.20