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10.2 Methods

10.2.1 Precision of and Corrections to Measured Quantities

The implementation of the Dn-sigma method requires the measurement of the central velocity dispersion and luminosity profile (or curve of growth) of each galaxy to sufficient precision that the measurement errors do not contribute to the intrinsic uncertainty of the distance estimator (sigma1.2 / Dn). LFBDDTW report a typical scatter of 23% in Dn at fixed sigma. Their single measurements of Dn range in precision from 5-12% and for sigma from 9-14% (these estimates were made from repeat measurements which were used to improve the precision of their measurements). Single measurements of this quality would contribute significantly to the scatter in Dn in the best observed clusters. A reasonable aim for future work is to determine Dn to 6%, and sigma to 5%, which in the absence of cosmological scatter would generate a distance determination with an uncertainty leq 9%.

If the peculiar motions of aggregates of galaxies (hypothesised to be at the same distance) are to be determined, then a measurement of recession velocity is required in addition to the parameters which enter into the distance estimator. (Of course, this can be obtained from the same observational data that was used to derive the velocity dispersions.) The measured galaxies must then be assigned to groups on the basis of position and recession velocity using an algorithm (e.g., Geller and Huchra 1983).

Observationally determined values of sigma and Dn need to be corrected as follows:

(1) For galaxies covering a range of distances the sigma values need to be corrected to a single linear aperture size. This is a statistical correction which, in the mean, accounts for the variation of velocity dispersion as a function of radius in individual galaxies. The corrections are determined using simulated large aperture measurements on the closest galaxies of a sample (see Dressler 1984). The correction is small, 5% between Virgo and Coma (Davies et al. 1987), and will be smaller still for a sample of distant ellipticals. In some cases where radial dispersion profiles are available for many galaxies in a cluster, global values, rather than central values, of the dispersion have been used to reduce the scatter in the Dn-sigma relation (Jacoby et al. 1990).

(2) The aperture magnitudes or luminosity profile must be corrected for absorption in the Milky Way. There is a straightforward and accurate prescription for this (Burstein and Heiles 1978) that is satisfactory for most of the sky. In regions where the absorption is large it may be difficult to determine a value with sufficient precision. However, the uncertainty in the correction may contribute to the apparent complexity of galaxy motions in the Centaurus region, where AB can be as high as 0.47 magnitudes (BFD; Lucey et al. 1989). By measuring diameters in redder bands (R and I), the size of this uncertainty can be reduced.

(3) The aperture magnitudes or luminosity profile must be corrected for K dimming, the cosmological surface brightness dimming, and other cosmological effects (that depend on the precise cosmological model chosen). These corrections are very small for the nearby samples of ellipticals so far studied and are discussed in Appendix A of LFBDDTW.

Finally LFBDDTW report that galaxies with extreme properties show increased scatter in the distance estimator and they did not use them. They eliminated galaxies having sigma < 100 km s-1 and effective surface brightness higher than B = 19.5 mag arcsec-2. Low velocity dispersion galaxies exhibit an increased diversity of physical properties. For example, they exhibit a larger spread in color and a higher frequency of emission lines, indicating that they do not form a homogeneous class of objects and the Dn-sigma method should not be applied to those galaxies.

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