We measure the relation between galaxy luminosity and disk circular velocity ( the Tully-Fisher , or TF , relation ) , in the g , r , i , and z -bands , for a broadly selected sample of galaxies from the Sloan Digital Sky Survey , with the goal of providing well defined observational constraints for theoretical models of galaxy formation .
The input sample of 234 galaxies has a roughly flat distribution of absolute magnitudes in the range -18.5 > M _ { r } > -22 , and our only morphological selection is an isophotal axis-ratio cut b / a < 0.6 to allow accurate inclination corrections .
Long-slit spectroscopy from the Calar Alto and MDM observatories yields usable H \alpha  rotation curves for 162 galaxies ( 69 % ) , with a representative color and morphology distribution .
We define circular velocities V _ { 80 } Â by evaluating the rotation curve at the radius containing 80 % of the i -band light .
Observational errors , including estimated distance errors due to peculiar velocities , are small compared to the intrinsic scatter of the TF relation .
The slope of the forward TF relation steepens from -5.5 \pm 0.2 \hbox { mag } / \hbox { log } _ { 10 } { km } { s } ^ { -1 } in the g -band to -6.6 \pm 0.2 \hbox { mag } / \hbox { log } _ { 10 } { km } { s } ^ { -1 } in the z -band .
The intrinsic scatter is \sigma \approx 0.4 mag in all bands , and residuals from either the forward or inverse relations have an approximately Gaussian distribution .
We discuss how Malmquist-type biases may affect the observed slope , intercept , and scatter .
The scatter is not dominated by rare outliers or by any particular class of galaxies , though it drops slightly , to \sigma \approx 0.36 mag , if we restrict the sample to nearly bulgeless systems .
Correlations of TF residuals with other galaxy properties are weak : bluer galaxies are significantly brighter than average in the g -band TF relation but only marginally brighter in the i -band ; more concentrated ( earlier type ) galaxies are slightly fainter than average ; and the TF residual is virtually independent of half-light radius , contrary to the trend expected for gravitationally dominant disks .
The observed residual correlations do not account for most of the intrinsic scatter , implying that this scatter is instead driven largely by variations in the ratio of dark to luminous matter within the disk galaxy population .