calculated as $R_{out}=3.5 R_S$, where $R_S$ is the characteristic scale-length of the Hernquist density profile (following Tonini et al. 2011; this is consistent with other semi-analytic models, for instance GalICS, Hatton et al. 2003). $R_S$ is not well constrained from observations and its relation to other physical parameters is quite uncertain, depending on the formation mechanisms of bulges (for instance, are bulges formed in merger events, or are they formed through secular evolution from the disk?). For this reason, we introduce a random scatter in the value $R_S$, only costrained to be at most half of the disk scale-length $R_D$, which is reasonable for spiral galaxies.

The width of a spectral line is in principle obtained by averaging the velocity dispersion over all the stars; in models, that calculate theoretical velocity dispersion profiles, the equivalent quantity is the mass-averaged velocity dispersion, calculated over radial bins out to the bulge outer limit: $$\sigma = \frac{\sum_n M_n \ \sigma_n}{\sum_n M_n}~,
\label{massweight}$$ where $M_n$ is the bulge mass in the $nth$ shell (out to the bulge outer limit), and $\sigma_n$ is the total velocity dispersion in the centre of the bin, calculated from Eq. (\[sigma\]).

Selection of the model galaxies
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In the 