disc with finite thickness. The error of using $Q_0 = 1$ is $30.411 \%$. The line of neutral stability, $Q_{\mathrm{sech^2}} = 1$, corresponds to $Q_{0, {\mathrm{crit}}} \simeq 0.696$, which is very similar to the approximation from @Wang:2010wr with $Q_{0, {\mathrm{crit}}} \simeq 0.693$.\
For the exponential profile approximation we have $C_{\mathrm{exp}} = 1.546$ and hence an error of $7.585\%$ compared to $C_{\mathrm{sech^2}}$. In this case, the critical value is $Q_{0, {\mathrm{crit}}} \simeq 0.647$, consistent with what is found by @Kim:2002gl.

Time-scales {#subsec:Timescales}
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To get an estimation of the fragmentation time-scales, we calculate the time at which the amplitude of the fastest growing disturbance has increased by a factor of $\bmath{e}$. For negative $\omega^2$, the growth rate $p$ is $- p^2 = \omega^2$ and the amplitude of a perturbation grows with $\exp(p \; t)$ [@Binney:2011vb]. For an increase by a factor of $\bmath{e}$, the growth time-scale is $t = 1 / p$. For the thin disc approximation, $$t_{0} =\left(  \frac{ \upi^{2} G^{2} \Sigma^{2}}{ c_{\mathrm{s}}^{2}}  - \kappa^{2} \right) ^{-1/2},$$ and expressed by the $Q_0$ parameter $$\label{eq:t_0} 
t_{0} =\kappa^{-1} \left(  \frac{ 1}{ Q_{0}^{2}} - 1 \right) ^{-1/2}.$$ By taking the thickness with the ${\, \mathrm{sech} }^2$ profile into account, we get the general 