view of the previous history and subsequent evolution of a disc. However, in these considerations, a razor-thin disc was assumed, characterized by its surface density $\Sigma$. Already @1964ApJ...139.1217T pointed out that a disc with finite thickness leads to a reduction of the gravitational force in the mid-plane, where structure formation takes place. Therefore, the disc remains stable even below a $Q_0$-value of one and a revised critical value has to be determined with $Q_0 < Q_{\mathrm{crit}} < 1$, describing the unstable regime.\
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A self-gravitating gas disc with its isothermal vertical structure in hydrostatic equilibrium is given by the ${\, \mathrm{sech} }^2$ density profile [@1942ApJ....95..329S]. Since there is no analytical solution for the reduction factor of the potential in the mid-plane for this profile, a good approximation is achieved by using an exponential function instead [e.g. @1987ApJ...312..626E], for which the reduction factor deviates from the one of the ${\, \mathrm{sech} }^2$ profile with a maximum error of $14\%$ [@Kim:2002gl; @2006ApJ...649L..13K; @Kim:2007ek]. The critical value for the instability parameter was thus determined to $Q_{\mathrm{crit}} \simeq 0.647$ for an exponential profile approximation by @Kim:2007ek and for the ${\, \mathrm{sech} }^2$ profile to $Q_{\mathrm{crit}} \simeq 0.693$ in @Wang:2010wr. Moreover, the reduced self-gravity also 