**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

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.