detailed description of the scaling procedure). Let the initial density distribution be characterized by a total mass ${{M_*}}$ and a characteristic radius ${r_*}$. We rescale the field equations so that the dimensionless source term is the same in Newtonian, MOND and dMOND simulations. We also require that the Second Law of Dynamics, when cast in dimensionless form, is independent of the specific force law considered, and this leads to fix the time unit. As a result, Newtonian and MOND simulations have the same time unit ${t_{\rm *n}}={r_*}^{3/2} (G {{M_*}})^{-1/2}$, while the natural time unit in dMOND simulations is ${t_{\rm *d}}={r_*}(G {{M_*}}{a_0})^{-1/4}$. Note that MOND simulations are characterized by the dimensionless parameter $\kappa=G {{M_*}}/{r_*}^2{a_0}$, and scaling of a specific simulation is allowed provided the value of $\kappa$ is maintained constant. So, simulations with lower $\kappa$ values describe lower surface-density, weaker acceleration systems; dMOND simulations represent the limit case $\kappa \ll 1$, while Newtonian ones describe the regime with $\kappa \gg 1$. With the time units fixed, the corresponding velocity and energy units are ${v_{\rm *n}}\equiv{r_*}/{t_{\rm *n}}$, ${v_{\rm *d}}\equiv{r_*}/{t_{\rm *d}}$, ${E_{\rm *n}}={{M_*}}{v_{\rm *n}}^2$, and ${E_{\rm *d}}={{M_*}}{v_{\rm *d}}^2$ (see Table 1 for a summary).

Initial conditions and analysis of the simulations {#secic}
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