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Astron. Astrophys. 321, 379-388 (1997)

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3. Velocity dispersions from integrated-light spectra

The projected velocity dispersions can be derived from the broadening of the cluster CCFs. This broadening results from the Doppler line broadening present in the integrated-light spectra because of the random spatial motions of the stars. Since the spectrograph slit used is large compared to the the M 31 globular clusters apparent size, a very large number of stars contributes significantly to the integrated light. Quantitatively, recent numerical simulations (Dubath et al. 1994, 1996a) show that statistical errors, which can be very important for integrated-light measurements of some Galactic globular clusters because of the dominance of a few bright stars, are negligible in the present case. An integrated light spectrum of an M 31 globular cluster is well approximated by the convolution of the spectrum of a typical globular cluster star with the projected velocity distribution.

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 [FORMULA] in the integration area of a globular cluster is given by the quadratic difference,

[EQUATION]

where [FORMULA] is the width of the Gaussian fitted to the cluster CCF and [FORMULA] is the average width of the [FORMULA] 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 simulations

In 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.

[FIGURE] Fig. 2a and b. Illustration of the numerical simulations in the case of the cluster Bo 218. This figure displays: a the observed integrated-light spectrum of the cluster; b its cross-correlation function (CCF); c the original and the convolved standard-star template spectrum; d their corresponding CCFs, one arbitrarily shifted for the purpose of display; e the convolved template spectrum (again) and one example of this spectrum with additional noise, simulating the observed cluster spectrum (panel a); and f three examples of CCFs of noisy convolved spectra taken at random and arbitrarily shifted, simulating the cluster CCF (panel b). This figure shows only a tiny fraction of the spectra ([FORMULA] 1/50 of their total wavelength range).

1. The simulation input parameter is the cluster velocity dispersion ([FORMULA]). 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 [FORMULA]. 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 [FORMULA] - 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 ([FORMULA]) and the sigma ([FORMULA]) 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 [FORMULA] and [FORMULA]. 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 ([FORMULA]) in the clusters, we proceed as follows. For each cluster, a set of different input velocity dispersions ([FORMULA]), varying by step of 0.5 or 1 km s-1 around the expected cluster "true" velocity dispersion (first evaluated with Eq.  1) are considered. For each [FORMULA], 100 or 150 simulations are carried out and the relative number of times that the width of the simulated CCFs is consistent with the width of the observed CCF is reported. Consistent means here that the sigma of the simulated CCF must be within [FORMULA] 0.2-0.5 km s-1 (depending on the different cases) of the sigma of the observed CCF. Fig. 3 shows, in the case of Bo 218, the distribution of these numbers as a function of the input velocity dispersions ([FORMULA]) of the simulations. This distribution is a kind of probability distribution; the best estimate of the "true" cluster velocity dispersion is given by the [FORMULA] which leads most often, through the simulations, to a CCF consistent with the observed one. For example, among 100 simulations carried out with [FORMULA] = 16 km s-1, 47 have a width consistent with the width of the observed CCF obtained for Bo 218, while none of the 100 simulations with [FORMULA] = 13 km s-1 is successful in reproducing the observed CCF.

[FIGURE] Fig. 3. Distribution of the number of time that the simulations produce a CCF consistent - in terms of width - with the observed CCF as a function of the input velocity dispersion ([FORMULA]), in the case of Bo 218. For each [FORMULA], 100 simulations are considered. The continuous line represents a Gaussian fitted to the distribution.

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 ([FORMULA]) and their uncertainties, are given in column (11) of Table 1.

3.1.1. Generalization

This 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 ([FORMULA]). In the simulations, a particular cluster is characterized by the spectrum S/N ratio and by the depth (D) of the cluster CCF. Therefore in general, the input parameters are the spectrum S/N, the CCF depth D, and the input velocity dispersion [FORMULA]. For each set of input parameters, 100 to 150 simulated CCFs are computed, and distributions of the resulting radial velocities and CCF sigmas are thus produced. The standard deviations of these distributions - [FORMULA] ([FORMULA]) for the radial velocities and [FORMULA] ([FORMULA]) for the sigmas - provide estimates of the uncertainties due to the spectrum noise. The following formula,

[EQUATION]

where C and [FORMULA] are two constants, is fitted to the results of the simulations. With [FORMULA] and [FORMULA], this equation gives in km s-1, (1) [FORMULA] ([FORMULA]) with an accuracy of order of 10%, and (2) [FORMULA] ([FORMULA])) with an accuracy of about 20%. It can then be used to estimated the uncertainties due to the spectrum noise in a more general way, for any set of parameters S/N, D, and [FORMULA]. Equation  2 is a generalization of Eq. (3) of Dubath et al. (1990) to broad CCFs. The constant C is lower in the present paper than in this previous study because of the larger wavelength range, and consequently larger number of spectral lines, taken into account in the cross-correlation process (C scales roughly with the square root of the number of lines).

Fig. 4 displays, for each cluster, the sigma ([FORMULA]) of the CCFs of the observed spectrum as a function of the corresponding estimates of the velocity dispersion ([FORMULA]) 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.

[FIGURE] Fig. 4. Relation between the sigmas ([FORMULA]) of the observed CCFs and the best estimates of the velocity dispersions ([FORMULA]) resulting from the simulations. Each point displays the result for one of the 9 M 31 clusters. The continuous line represents Eq. (1), whose validity is thus confirmed by the results of the simulations.

3.2. Results

For 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 [FORMULA] 0.3 km s-1 are probably underestimated. The [FORMULA] errors given in column (8) of Table 1 are the square root of the quadratic sum of the errors due to the noise (estimated using Eq. (2)), and of an instrumental error of 0.25 km s-1, derived from standard-star measurements.

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.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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