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7.2 Method

The current lack of a strong theoretical basis for the TF relations (see Sec. 7.3) prevents a precise definition of the optimum luminosity and rotational parameters which produce the minimum scatter and best calibration for the relations. For luminosity, the choices are: (i) the total magnitude for the galaxy, or (ii) a suitably defined aperture magnitude. The former offers the advantage of having a relatively simple definition, while the latter has been a necessity for infrared photometry where until recently single element detectors were state of the art. For the rotational velocity parameter the choices are: (i) the maximum of the rotation curve (Vmax), or (ii) the rotational velocity at a suitably chosen radius, perhaps corresponding to a particular aperture magnitude definition (e.g., Madore and Woods 1987).

Total magnitudes have been the choice for the luminosity parameter over visible wavelengths, even when derived through multiaperture photometry and template growth curves (e.g., de Vaucouleurs et al. 1976; Visvanathan 1981, 1983). The use of total magnitudes was motivated by the availability of the data, the desire for a uniform magnitude system, and the expectation that the mass-to-light ratio of galaxies is constant, or at least a smooth function of mass. The latter is of particular concern at optical wavelengths where the large color difference between the bulge and disk components of spiral galaxies suggests differences in the mass-to-light ratio. This results in a significant morphological type dependence for the TF relations at bluer wavelengths. Consequently, a TF relation based on blue photometry through small apertures is of limited use (see Pierce and Tully 1988). The numerous problems associated with multi-aperture photometry (e.g., centering errors, contaminating stars, the use of circular apertures and mean growth curves, etc.) have been discussed by Corwin (1979). With the introduction of CCDs, full growth curves using elliptical annuli can be easily constructed with contaminating regions (e.g., stars, cosmetic defects) suitably dealt with. The sky level for each image can also be easily estimated, provided the field of view is sufficiently large. Together, these advantages of CCD photometry can result in total magnitudes with a precision of ~ 5% (Pierce 1988, 1992).

Pierce and Tully (1988) found dispersions for the TF relations of 0.37, 0.31, and 0.28 mag for the B, R and I bands, respectively, from CCD photometry of a complete sample of galaxies in the Ursa Major Cluster. There is likely significant line-of-sight depth for this sample, since it spans about 8° on the sky, suggesting an intrinsic dispersion for the longer wavelength TF relations of leq 0.25 mag, or 12% in distance. Much of the improvement over previous work (typically with sigma ~ 0.5 mag) was attributed to an increased accuracy of inclination estimates obtained via ellipses fitted to galaxy isophotes. For example, they found a significant improvement in the dispersion for the IRTF relation using photometry from the literature, provided that the line-widths were corrected with the improved inclinations. Apparently, the redder wavelengths are preferable, though the B-band is still useful for comparative purposes.

Although the infrared aperture magnitudes have been very successful, they are not free of problems. The H-band magnitude (H-0.5) is defined to be the apparent H magnitude through an aperture size (A) given by log(A / D25b,i) = -0.5, where D25b,i is the B-band diameter of the galaxy at the 25 mag arcsec-2 isophote, corrected for internal and Galactic extinction, as well as projection. This definition allows the use of small apertures that are more tractable with traditional infrared photometers, and also reduces the effect of the sky signal to a manageable level. (The H-band sky is very bright: µH ~ 13.4 mag arcsec-2. Compare this with a mean H-band surface brightness, over D25b,i, of µH ~ 18 mag arcsec-2 for a moderate luminosity spiral and ~ 21 mag arcsec-2 for a typical dwarf irregular galaxy.) There are well-known observational techniques used in infrared astronomy to combat these high background levels. Nevertheless, nearby systems having low surface brightness and large angular size pose a special problem due to uncertainties in the sky level. In addition to the general problems associated with aperture photometry mentioned above, there is an additional difficulty in tying the infrared aperture size to the optical diameter (e.g., van den Bergh 1981; Aaronson et al. 1986). Significant systematic errors have been introduced into some of the IRTF distance estimates from biased B-band isophotal diameters, a consequence of using different catalogs of B-band diameters for different distance ranges (e.g., Cornell et al. 1987). CCD photometry is almost a necessity to provide a uniform and unbiased aperture definition, at which point aperture photometry itself becomes of limited value. The application of infrared imaging technology over the next few years should alleviate most of these concerns, provided the data reach a sufficiently faint surface brightness level for total magnitudes to be estimated. An example of the application of infrared array technology to the IRTF can be found in Peletier and Wilner (1991).

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