The Tully-Fisher relation is an empirical relationship between the luminosity of a spiral/irregular galaxy and its rotational velocity. It is interesting to note that even though a variant of the Tully-Fisher relation was first used by Öpik opik22 to find the distance to the Andromeda galaxy (Messier 31), it took 55 years to fully realize its potential as a major component of the extragalactic distance scale [\protect\astronciteTully and Fisher1977]. The global neutral hydrogen 21cm linewidth corrected for inclination and receiver broadening is usually used as a measure of a galaxy's rotational velocity while CCD detectors are used to measure the galaxy's total light and inclination.
The physical basis of the relation is not understood because the process of galaxy formation remains for the most part a mystery. Simple ``back-of-the-envelope'' arguments [\protect\astronciteJacoby et al.1992] based on too many assumptions (spherical symmetry, constant surface brightness and constant mass-to-light ratio) can be used as a plausibility argument for the relation, but a thorough explanation will have to wait for better models of galaxy formation. The Tully-Fisher relation highlights two remarkable properties of galaxies. First, the mass-to-light ratio of galaxies must remain constant over a broad range in luminosity since the relation remains tight over at least 7 magnitudes (a factor of 600 in luminosity!). The dispersions for the Tully-Fisher relations in the B, R and I bands are 0.37, 0.31 and 0.28 mag respectively. Second, there must be a conspiracy between the disks (luminous matter) and halos (dark matter) of galaxies such that both components ``know'' exactly how much each one should contribute to the mass within a characteristic optical radius as a function of total galaxy mass.
Since there is a good correlation between HI linewidth and galaxy luminosity, it is reasonable (and important!) to ask whether rotation velocities measured from optical emission lines follow a similar relationship. After all, since the HI distribution in disk galaxies typically extends twice as far in galactocentric distance as the optical emission, optical emission might not sample the full velocity width of disk galaxies. Moreover, rotational and turbulent motions both contribute to the intrinsic HI 21cm linedwidth. High luminosity galaxies are rotationally-supported whereas low luminosity dwarf galaxies are supported by turbulence. A major advantage of optical rotation curves is the possibility of isolating the rotational component based on its characteristic ``S'' shape which eliminates the need for a turbulence correction to the velocity width. Figure shows that rotation velocities measured using the H optical emission line ( = 6562 Å) are very well-correlated with HI linewidths and that internal kinematics measured from optical emission lines should follow the same Tully-Fisher relation. Recent simultaneous observations of [OII] and H [\protect\astronciteVogt et al.1993] indicate that both lines yield the same rotation velocities.
In order to determine whether the internal kinematics of intermediate redshift galaxies deviate significantly from the local Tully-Fisher relation, it is important to define the local locus of that relation in the best way possible. Two methods were used to do so. The first method is based on the absolute calibration of the B-band Tully-Fisher relation [\protect\astronciteJacoby et al.1992][\protect\astroncitePierce and Tully1992]. This calibration based on 15 local galaxies is given by
where is the total B magnitude corrected to a face-on inclination and for internal absorption. is the HI linewidth corrected for inclination and turbulence according to the prescription of [\protect\astronciteTully and Fouqué1985]:
where = sin , is the measured HI width at 20 of peak intensity, is the expected 20 width due to turbulence (38 km s) and is the characteristic transition width from ``horned'' to Gaussian-shaped profiles (120 km s). Thus, for a given absolute magnitude, was obtained by finding the roots of eqs. and with RTSAFE [\protect\astroncitePress et al.1986], and was then converted to H using a linear regression fit (solid line, Figure ) to () versus for 204 galaxies [\protect\astronciteMathewson et al.1992].
The above method suffers from two drawbacks: (1) it is indirect, and (2) it does not give any information on the dispersion of the local relation. These drawbacks are eliminated by using the H rotation velocities and total magnitudes of 1355 local galaxies observed by Mathewson et al. mathew92 to define the locus of the local Tully-Fisher relation as described in the next section. This approach allows one to tackle a possible morphological dependence of the Tully-Fisher relation and the wide range of properties of local galaxies.