**3.1. The TF Relation and Galaxy Structure**

Many workers have attempted to ``explain'' the TF relation on the
basis of physical principles and models of galaxy formation.
While these attempts can claim some modest successes, it is probably
fair to say that a true explication of the TF relation remains elusive.
One can argue heuristically that something like the TF relation
must exist: Assuming that luminosity is proportional to mass,
and that a virial relation *v*^{2} ~ *GM / R* holds
for spirals,
it follows that *L*
*M*
*Rv*^{2}. If one further notes that
spirals have characteristic surface brightnesses *I* *L / R*^{2}
that varies little from galaxy to galaxy, then *R* *L*^{1/2}, and
it follows that *L*
*v*^{4}. This was indeed the power-law
exponent (i.e., *b* 10)
originally found by the Aaronson group, and the argument
seemed reasonable to them
(Aaronson et
al. 1979).

However, quite a few loose ends remain. First, as noted above, contemporary measures of the TF slope suggest that the exponent is closer to 3 than to 4. The aperture and wavelength dependences noted above tell us that the TF slope is not determined strictly by idealized dynamics, but depends also on the details of the distribution - in both space and wavelength - of the starlight emitted by the galaxy. Furthermore, while a number of theoretical approaches can approximately predict the TF slope, no realistic model has successfully accounted for its rather small (~ 0.3 mag; see below) intrinsic scatter (Eisenstein & Loeb 1996).

Another, more fundamental, problem is that the TF relation is evidently connected with the phenomenon of flat rotation curves (RCs) exhibited by most spiral galaxies. Were the RCs not flat, there would be no well-defined rotation velocity, and one would expect the TF relation to require a very specific type of velocity width measurement. In fact, a well-defined TF relation is found regardless of the specific algorithm for measuring rotation velocity (although slight variations of slope and zero point arise as a result of algorithmic differences). Whether one measures H I profile widths, asymptotic rotation velocities, ``isophotal'' rotation velocities (Schlegel 1996), or maximum rotation velocities, basically similar TF relations result. Because the origin of flat rotation curves is connected with the nature of dark matter, it follows that we cannot fully understand the TF relation until we understand how galaxies form in their dark matter halos.