3.2. Applying the TF Relation: A Few Details

Widely appreciated by practitioners of the TF relation, but often hidden to the wider astronomical public, are the careful correction procedures applied to the magnitudes and velocity widths that go into the TF relation. Probably the most important step is correction for projection of the disk on the plane of the sky. The observed velocity width is smaller by a factor sin (i), where i is the galaxy inclination, than the intrinsic value. Observers correct for this by esimating i from the apparent ellipticity of the galaxy disk. Modern CCD observations allow one to fit elliptical isophotes to the galaxy image; these isophotes typically converge to a constant ellipticity in the outer regions. When CCD surface photometry is not available (as is the case for many of the older infrared data), one simply takes = 1 - b / a, where a and b are the major and minor axis diameters of the galaxy obtained from photographic data. Whichever method is used, the inclination i is taken to be a function of . A typical formula employed is

(5)

where _max 0.8 is the ellipticity exhibited by an edge-on spiral. It is apparent that formulae such as equation (5) are at best approximations, hopefully valid in a statistical sense. However, they are usually the best we can do, and are certainly far better than doing nothing. Still, the inclination correction to the widths can go seriously awry at small inclinations, and most TF samples exclude galaxies with i 40°.

Another tricky detail of the TF relation is correcting for internal extinction. As a spiral galaxy tilts toward edge-on orientation, it becomes fainter. Since spirals are viewed at a range of orientations, it is important to correct for this effect. The most widely used correction is to brighten the raw magnitudes by an amount C_int x log(a / b), where C_int is the internal extinction coefficient. Studies have shown that C_int is bandpass-dependent, as one might expect. However, in the optical red (R and I bandpasses), the wavelength-dependence is very weak, and C_int 1 is a good approximation (Burstein et al. 1995; Willick et al. 1996, 1997). A controversial question is whether internal extinction depends on any galaxian property other than axial ratio. Giovanelli et al. (1995) argued that it is luminosity-dependent, but Willick et al. (1996) reached the opposite conclusion through a TF-residual analysis. This issue merits further consideration In the future.