$Q_0 < 1$. $Q_0$ can be obtained from the unmodified dispersion relation $\omega_0^2$, for equation (\[eq:modified\_dispersion\_relation\_generalized\]) with $z_0 = 0$. The global minimum of the dispersion relation determines the fastest growing $\lambda_{f_0}$ (equation \[eq:fastest\_growing\_wavelength\_thin\_disk\]) and therefore is the ’last’ possible wavelength which could grow if an unstable disc turns over to stability (from $\omega^2 < 0$ to $\omega^2 \geq 0$). The transition is given by $\omega_0^2(\lambda_{f_0}) = 0$ and by rearranging, it leads to the expression given in equation ($\ref{eq:Toomre_Q}$) with $Q_0 = 1$. In a similar way we proceed with the modified dispersion relation (equation \[eq:modified\_dispersion\_relation\_generalized\]) for the ${\, \mathrm{sech} }^2$ density profile and with the fastest growing wavelength we have $$\label{eq:disp_rel_fastest_grow_wavelength_sech2}
\omega_{\mathrm{sech^2}}^2 = \kappa^2 - \frac{4 \upi^2 G \Sigma}{\lambda_{f_{\mathrm{sech^2}}}} F_{\mathrm{sech^2}} + \frac{4 \upi^2 c_{\mathrm{s}}^2}{\lambda_{f_{\mathrm{sech^2}}}^2} = 0.$$ We insert $\lambda_{f_{\mathrm{sech^2}}}$ (equation \[eq:fastest\_growing\_wavelength\_sech2\]) to get $$\omega_{\mathrm{sech^2}}^2 = \kappa^2 -  \frac{\upi^2 G^2 \Sigma^2}{c_{\mathrm{s}}^2} \left( \frac{2 \; F_{\mathrm{sech^2}}}{A_{\mathrm{sech^2}}} - \frac{1}{A_{\mathrm{sech^2}}^2} \right)   = 0.$$ Now, we substitute the classical parameter $Q_0$ (equation \[eq:Toomre\_Q\]) and get $$\label{eq:Q_sech2}
Q_{\mathrm{sech^2}} = Q_0 \times C_{\mathrm{sech^2}} = 1,$$ with the constant proportionality factor $$C_{\mathrm{sech^2}} = \frac{A_{\mathrm{sech^2}}}{\sqrt{2 \; F_{\mathrm{sech^2}} \; A_{\mathrm{sech^2}} - 1}} \simeq 1.437.$$ There is thus a simple linear relation between the thin disc approximation and a 