mass $M_{\rm disk}$ with initial scale-length $R_{\rm D_{\rm old}}$, the disk scale-length after angular momentum transfer is $$R_{\rm D}=R_{\rm D_{\rm old}} \left( 1+(1-f_{\rm x}) \frac{M_{\rm bulge}}{M_{\rm disk}} \right)~,
\label{rdafter}$$ with the fiducial value $f_{\rm x}=0.25$ indicated by Dutton et al. (2007). The new $R_{\rm D}$ represents a ’corrected’ disk scale-length, that takes into account the additional gravitational potential of the bulge. After the correction, all galaxies move onto the disk mass-disk scale length relation that holds for Sc galaxies (see Tonini et al. 2011). After definining a $R_{\rm D}$ that evolves with morphology, it then makes physical sense to adopt $r=2.2R_{\rm D}$ as our radius of choice to build the TF relation.

Integrated galaxy velocity dispersion
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From the mass distribution we can build velocity profiles for all galaxy components. The theoretical velocity dispersion is the sum of the contribution to the rotation curve by all components that are pressure-supported, namely the dark matter halo and the bulge: $$\sigma(r)=\sqrt{V^2_{\rm DM}(r)+V^2_{\rm bulge}(r)}~,
\label{sigma}$$ where each velocity term in the equation is determined by the mass profile of the dynamical component: $V_i^2(r) \propto GM_i(r)/r$ (see Tonini et al. 2011 for a detailed description). However this particular kind of output is not readily 