## 3. Velocity dispersions from integrated-light spectraThe Since the CCF is a kind of mean spectral line, the above property also hold for CCFs. The CCF of an integrated-light spectrum of a globular cluster is the convolution of the CCF of the spectrum of a typical globular cluster star by the projected velocity distribution. Since the CCFs have a Gaussian shape, an estimate of the projected velocity dispersion in the integration area of a globular cluster is given by the quadratic difference, where is the width of the Gaussian fitted to the cluster CCF and is the average width of the obtained for a sample of standard stars, as representative as possible of the cluster stars which contribute most to the integrated light. We do not use this formula directly in the present study. In order to derive the cluster projected velocity dispersion, we carry out a large number of numerical simulations. The results of these simulations confirm, however, the validity of Eq. 1 (see Sect. 3.1.1). ## 3.1. Numerical simulationsIn order to simulate the integrated-light spectrum and the CCF obtained for globular cluster, we proceed in several steps. In the first row of Fig. 2, the integrated-light spectrum (left) and the CCF (right) obtained for the cluster Bo 218 are displayed. Corresponding simulation results are displayed in the last row, while the intermediate steps of the simulation are illustrated in the middle row of this figure.
1. The simulation input parameter is the cluster velocity dispersion (). The simulation starts with a high signal-to-noise spectrum of a standard star of appropriate spectral type, e.g., a K2 giant. To simulate the Doppler line broadening, this spectrum is convolved with a Gaussian function of standard deviation equal to the input velocity dispersion . A portion of the standard-star spectrum, before and after convolution, is shown in the panel (c) of Fig. 2. 2. In general, the spectral lines of the convolved standard-star spectrum do not have the same depths as those of a given cluster spectrum, mainly because of possible metallicity difference. The next step is thus to adjust the depth of the standard-star spectral lines. This is done by scaling the convolved standard-star spectrum so that its CCF is of the same depth as the cluster CCF. (With our cross-correlation technique, there is a clear relationship between the average depth of the spectral lines of a spectrum and the depth of the spectrum CCF). In Fig. 2 for example, the spectra and the CCFs displayed in the middle row are linearly scaled so that the convolved spectrum CCF (the broad CCF in panel d) be of the same depth as the cluster CCF (in panel b). A noiseless template of a cluster spectrum - for one particular velocity dispersion - results from the second step. 3. Random noise is added to this template spectrum to simulate the photon counting and CCD readout noises of observed cluster spectra. A portion of the template spectrum for Bo 218 (c) and of one example of simulated noisy spectrum are displayed in panel (e) of Fig. 2. We then cross-correlate the simulated noisy spectrum and derive the radial velocity () and the sigma () of the resulting CCF. This third step is repeated a large number of times (100 to 150 times) to observe the influence of the noise on and . Panel (f) of Fig. 2 shows three examples of simulated CCFs taken at random. The comparison of the upper and the lower panels of this figure shows how well our simulations reproduce the integrated-light spectrum and the CCF obtained for Bo 218. In order to derive the best estimates of the projected velocity
dispersion () in the clusters, we proceed as
follows. For each cluster, a set of different input velocity
dispersions (), varying by step of 0.5 or 1 km
s
A similar distribution is derived for each cluster, and a Gaussian is fitted to each of them. The resulting means and sigmas, which provide the most probable cluster velocity dispersions () and their uncertainties, are given in column (11) of Table 1. ## 3.1.1. GeneralizationThis section presents a generalization of the simulation results
(which can be skipped by less concerned readers). For each of the 9
clusters, simulations are carried out for a set of input velocity
dispersions (). In the simulations, a particular
cluster is characterized by the spectrum S/N ratio and by the depth
( where Fig. 4 displays, for each cluster, the sigma () of the CCFs of the observed spectrum as a function of the corresponding estimates of the velocity dispersion () resulting from the simulations. Eq. (1) is illustrated in this figure by the continuous line, and the good agreement with the simulation results indicates that this equation is a valid model.
## 3.2. ResultsFor each observation, Table 1 lists the cluster identification from Battistini et al. (1980) in column (1), and from Sargent et al. (1977) in column (2), the cluster apparent V magnitude in column (3), the cluster [Fe/H] in column (4), the date of the observation in column (5), the exposure time in column (6), the signal-to-noise ratio of the integrated-light spectrum in column (7), the heliocentric radial velocity in column (8), the depth of the CCF in column (9), the sigma of the CCF in column (10), and the projected velocity dispersion in column (11). The cluster apparent V magnitudes are taken from Battistini et al. (1987), and [Fe/H] are taken from Huchra et al. (1991). The radial velocity errors given in column (8) of Table 1 are
computed using Eq. (2). This equation provides an estimate of the
error due to the spectrum noise (photon counting and CCD readout
noises) but does not take into account the uncertainty of the
zero-point corrections computed from measurements of the night-sky
emission lines. Consequently, the radial velocity errors smaller than
0.3 km s Two cluster spectra are too noisy to provide useful velocity dispersion estimate. The CCF obtained for Bo 147 is as narrow as the CCFs obtained for individual standard stars, and much narrower that the value expected from its absolute magnitude. Consequently, Bo 147 is almost certainly a foreground star. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |