Tests for the numerical calculations employed to compute the characteristic densities S_i(R) in Eq. (12). The case of a self-gravitating Schuster-Plummer profile provides the analytic expression for i = 7/2 given in Eq. (D.6). The analytic and numerical expressions are shown as indicated in the inset (marked as i = 3.5). The differences between them (ΔS_i/S_i and ΔS_i/maxS_i) are also included as the red dotted and dashed lines. The cases i = 3 and i = 4 are shown for reference.