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In this paper, we derive scaling relations between photometric observable quantities and disc galaxy rotation velocity  V  rot  or Tully–Fisher relations (TFRs). Our methodology is dictated by our purpose of obtaining purely photometric, minimal‐scatter estimators of  V  rot  applicable to large galaxy samples from imaging surveys. To achieve this goal, we have constructed a sample of 189 disc galaxies at redshifts  z  &lt; 0.1 with long‐slit Hα spectroscopy from Pizagno et al. and new observations. By construction, this sample is a fair subsample of a large, well‐defined parent disc sample of ∼170 000 galaxies selected from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7). The optimal photometric estimator of  V  rot  we find is stellar mass  M  ★  from Bell et al., based on the linear combination of a luminosity and a colour. Assuming a Kroupa initial mass function (IMF), we find: log [ V  80 /(km s −1 )] = (2.142 ± 0.004) + (0.278 ± 0.010)[log ( M  ★ /M ⊙ ) − 10.10], where  V  80  is the rotation velocity measured at the radius  R  80  containing 80 per cent of the  i ‐band galaxy light. This relation has an intrinsic Gaussian scatter  dex and a measured scatter σ meas = 0.056 dex in log  V  80 . For a fixed IMF, we find that the dynamical‐to‐stellar mass ratios within  R  80 , ( M  dyn / M  ★ )( R  80 ), decrease from approximately 10 to 3, as stellar mass increases from  M  ★ ≈ 10 9  to 10 11  M ⊙ . At a fixed stellar mass, ( M  dyn / M  ★ )( R  80 ) increases with disc size, so that it correlates more tightly with stellar surface density than with stellar mass or disc size alone. We interpret the observed variation in ( M  dyn / M  ★ )( R  80 ) with disc size as a reflection of the fact that disc size dictates the radius at which  M  dyn / M  ★  is measured, and consequently, the fraction of the dark matter ‘seen’ by the gas at that radius. For the lowest  M  ★  galaxies, we find a positive correlation between TFR residuals and disc sizes, indicating that the total density profile is dominated by dark matter on these scales. For the highest  M  ★  galaxies, we find instead a weak negative correlation, indicating a larger contribution of stars to the total density profile. This change in the sense of the correlation (from positive to negative) is consistent with the decreasing trend in ( M  dyn / M  ★ )( R  80 ) with stellar mass. In future work, we will use these results to study disc galaxy formation and evolution and perform a fair, statistical analysis of the dynamics and masses of a photometrically selected sample of disc galaxies.

Calibrated Tully–Fisher relations for improved estimates of disc rotation velocities