  We first stress that any spherically symmetric galactic model whose
integrated mass profile $M (r) \to 0$ as $r \to 0$ is physically consistent
close to the centre only provided that the circular velocity $v_c (r) \to 0$
and the gravitational field $g (r) \to 0$ as $r \to 0$. Next, we apply such a
statement to a broad class of five-parameter spherical galactic models, which
includes most of those used in astrophysics and cosmology. In particular, we
discover that the Jaffe and Hernquist models can only be trusted for $r \gtrsim
0.2 \, R_e$, while the NFW model cannot describe the central region either of
regular galaxy clusters or of pure dark matter halos, thereby failing to
predict any central cusp.
