  For the spherical symmetric case, all quantities describing the relaxed dark
matter halo can be expressed as functions of the gravitational potential
$\Phi$. Decomposing the radial velocity dispersion $\sigma_r$ with respect to
$\Phi$ at very large and very small radial distances the possible asymptotic
behavior for the density and velocity profiles can be obtained. If reasonable
boundary conditions are posed such as a finite halo mass and force-free halo
centre the asymptotic density profiles at the centre should be much less steep
than the profiles obtained within numerical simulations. In particular cases
profiles like Plummer's model are obtained. The reasons of that seeming
discrepancy with respect to the results of N-body simulations are discussed.

Astrophysics