x_{\mathrm{min}}= 5.86$ pc inside, for the densest regions, employing an effective resolution of $2^8$–$2^{13}$ grid cells in all three direction. At each resolution level, we ensure that the Jeans length is resolved by at least 18 grid cells (see Section \[subsec:Numerical considerations\]).

![Local disc instability parameter: the solid line represents the numerical solution, according to the vertical ${\, \mathrm{sech} }^2$ density profile (equation \[eq:Q\_sech2\]). The disc is unstable between radius $R = 0-10.517$ kpc and the stable part is illustrated by the blue shaded region. The exponential profile approximation (dashed line), gives a slightly more stable disc and therefore results in a smaller unstable region (see Section \[subsec:Local disc instability parameter\]). The thin disc approximation, for $z_0 = 0$, with Toomre’s $Q_0$ (dotted line) (equation \[eq:Toomre\_Q\]), overestimates the instability dramatically, in intensity and with respect to the extent of the unstable region. \[Q\]](f3.eps){width="84mm"}

Disc model {#subsec:Disk model}
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We perform simulations of a massive gas disc $M_{\mathrm{disc}}=2.7 \times 10^{10} \; \mathrm{M_{\sun}}$ embedded in a spherical dark matter halo that within $16$ kpc has a mass of $M_{\mathrm{DM}} = 1.03 \times 10^{11} \; \mathrm{M_{\sun}}$ following the @Burkert:1995jr density profile with a scalelength $a_{0} = 4$ kpc. The gas temperature is $10^4$ 