We present a study of the local B and K-band Tully-Fisher Relation (TFR) between absolute magnitude and maximum circular speed in S0 galaxies. To make this study, we have combined kinematic data, including a new high-quality spectral data set from the Fornax Cluster, with homogeneous photometry from the RC3 and 2MASS catalogues, to construct the largest sample of S0 galaxies ever used in a study of the TFR. Independent of environment, S0 galaxies are found to lie systematically below the TFR for nearby spirals in both optical and infrared bands. This offset can be crudely interpreted as arising from the luminosity evolution of spiral galaxies that have faded since ceasing star formation. However, we also find a large scatter in the TFR. We show that most of this scatter is intrinsic, not due to the observational uncertainties. The presence of such a large scatter means that the population of S0 galaxies cannot have formed exclusively by the above simple fading mechanism after all transforming at a single epoch. To better understand the complexity of the transformation mechanism, we have searched for correlations between the offset from the TFR and other properties of the galaxies such as their structural properties, central velocity dispersions and ages (as estimated from line indices). For the Fornax Cluster data, the offset from the TFR relates with the estimated age of the stars in the individual galaxies, in the sense and of the magnitude expected if S0 galaxies had passively faded since being converted from spirals. This correlation implies that a significant part of the scatter in the TFR arises from the different times at which galaxies began their transformation.
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