and $\gamma$ stands for the cusp coefficient. In order to restrain the model to finite dimensions a complementary error function with cut-off radius $r_c$ and width $r_w$ is used. The velocity distribution is computed by solving the Eddington equation for the prescribed density profile.\
For each removed halo two replacements are created: one with $\gamma = 1.0$ and another with $\gamma = 0.6$. The cut-off radius and width are set to be linear functions of the virial radius of the original object:

$$r_c = r_{vir} - r_w$$

$$r_w = \frac {\sqrt{2} r_{vir}}{10}.$$

The scale radius is determined indirectly by assuming that the concentration of an average halo of mass $M_{vir}$ at redshift $z$ are given by the empirical relations found by [@2011ApJ...740..102K]:

$$c(M_{vir}, z) = c_0(z) \left(\frac{M_{vir}}{10^{12} h^{-1}  M_\Sun} \right)^{-0.075} \left( 1 + \left( \frac{M_{vir}}{M_0(z)}\right)^{0.26}\right).$$

With the concentration and virial radius set, the scale length can be determined.

$$r_s = \frac{r_{vir}}{c}$$

The two variants corresponding to shallow and steep dark matter density profile have by construct the same scale radius. In order to have the same mass enclosed in the same virial volumes with different cusp coefficients the scale velocity is adjust accordingly. The values of the characteristic quantities associated 