4 Determination of redshifts and velocity dispersions

The accurate determination of the line-of-sight central velocity dispersions of the galaxies is critical to FP analysis. A number of different methods exist for determining the velocity dispersion $\sigma$ and the radial velocity cz. These include the Fourier quotient method (Sargent et al. 1977), the Fourier cross-correlation method (Tonry & Davis 1979), the Fourier difference method (Dressler 1979), and the Fourier fitting method (Franx et al. 1989).

For this study we used the Fourier cross-correlation method, as implemented in the IRAF package rv.fxcor which is based on the method of Tonry & Davis (1979). The spectra of the galaxy and the stellar template are cross-correlated in Fourier space, and the resultant maximum peak is fitted by a smooth symmetric function. The width and pixel shift of the peak are measures of $\sigma$ and the galaxy redshift (in km s^-1), found by comparison with the known radial velocity of the template. An indicator of the accuracy of the resulting value of $\sigma$ is the r value (Tonry & Davis 1979).

All spectra were rebinned, using onedspec.dispcor, to linear logarithmic wavelength coordinates. Total flux was conserved, and the same dispersion parameters were used for all spectra from all runs, resulting in spectra with logarithmic wavelength bins of $\Delta \ln \lambda = 6.46 \ 10^{-5}$. A cubic spline was fitted to the continuum for all spectra (template stars as well as galaxies). This fit was subtracted from the spectrum to flatten it; the resulting spectrum has zero mean in the continuum. The spectra were then Fourier filtered before the correlation. Data points outside the selected sample region were zeroed, and the ends of the region (12.5% on each end of the spectrum) were apodized with a cosine bell. A ramp function was used as the filter. The parameters of the filter were adjusted to find the best combination. After the galaxy and template spectra had been thus prepared, the two sets of spectra were cross-correlated. In most cases the best region for cross-correlation was found to be 4900 - 5800  $\mbox{\AA}$. This choice excludes the H$\beta$ and Na D lines.

The galaxy spectra were Fourier cross-correlated in fxcor against each standard star in turn. The observed FWHM of the cross-correlation peak was transformed into a value for $\sigma$by direct calibration with broadened template spectra, using the preocedure outlined by Baggley (1996). The spectrum of each template star was convolved with Gaussians of various known widths in the range 0 - 700 km s^-1, and the resulting broadened spectra were run in fxcor (with the same parameters) against the original template spectrum, giving the FWHM of the cross-correlation peak in each case. A calibration curve of this FWHM width versus the broadening $\sigma$ for Gaussians of different widths was produced for each template star observation, by linear interpolation between the FWHM values from fxcor. The galaxy FWHM values were then converted into values for $\sigma$ for the galaxy by reading off the calibration for that particular template star. An example of a calibration curve is shown in Fig. 2, for HD 194071, observed during run S8. In this case the resolution was approximately 100 km s^-1, and it can be seen from the figure that below this value of $\sigma$ it is more difficult to find an accurate determination of velocity dispersion.

For every galaxy there is a set of different values for $\sigma$ and cz, each pair of values the result of running the galaxy against a different stellar template spectrum. The values obtained by Fourier cross-correlation show small systematic differences depending on which stellar template is used. Template stars from all runs were used, a total of 45 observations of 20 different stars (as listed in Table 3). The rms difference between the estimates for cz and $\sigma$ from different template stars was typically < 1% in cz and $\sim$ 4% in $\sigma$.

Figure 2: The calibration curve for an observation of the standard star HD 194071. The points represent the FWHM from fxcor for the broadened template spectra compared with the unbroadened original template spectrum. The curve is found from linear interpolation between the points

Figure 3: A comparison of the S/N per $\mbox{\AA}$ of the spectra with the value of the Tonry & Davis r parameter from the cross-correlation, which is a measure of the ratio between the the peak height and the noise, and is therefore an indicator of the accuracy of the results

In Fig. 3 the mean r value from the cross-correlation analysis of a spectrum is shown plotted against the S/N of that spectrum. From this plot it can be seen that the scatter is larger than would be expected if the S/N were the only factor affecting the r value. It could be that r is also sensitive to a mismatch between the features of the galaxy and those of some of the template spectra. Dalle Ore et al. (1991) found no systematic variation of the width of the cross-correlation curve with spectral type, and showed that errors in the velocity dispersion owing to spectral type mismatch are negligibly small. J$\o$rgensen et al. (1995) found that template stars of the spectral type G8 - K3 result in significantly better fits than stars of types K4 - K5. For our sample data, we do not find this to be true in general. For a particular galaxy, certain templates work better than others, but we find that the best set of templates is different for each galaxy. We therefore combined the results for the velocity dispersion for each galaxy in such a way as to minimize the effect of the template mismatch problem. First, the mean of all the results from different templates was taken. Then the 2$\sigma$ outliers were excluded and the mean was taken again, to give the final result. The redshift for each galaxy was estimated by calculating the mean observed heliocentric velocity from all the templates. For both $\sigma$ and cz, the standard deviations from the means were calculated.

To check the zero-point for the redshift determinations, the spectrum of each radial velocity standard star was cross-correlated with the spectra of all the other standard stars, resulting in estimates of relative velocity. The radial velocity standards were run against each other in fxcor using the same scripts with the same parameters as used for the galaxies. For each star the mean of the estimates of heliocentric redshift was found, and compared with the known value of radial velocity for that star. The rms difference between the mean estimated value and the known value was about 20 km s^-1, which is therefore the accuracy of the zero-point of the radial velocities we determined.

Since a variation was observed among the separate determinations from each of the template spectra, estimates of the systematic errors for both redshift and velocity dispersion have been obtained from the rms scatter of the results from different templates. These error estimates are listed together with the results in Tables 5, 6, and 7.