  The core momentum distribution of a weakly-bound neutron-neutron-core exotic
nucleus is computed within a renormalized zero-range three-body model, with
interactions in the s-wave channel. The halo wave-function in momentum space is
obtained by using as inputs the two-body scattering lengths and the two-neutron
separation energy. The core momentum densities are computed for $^{11}$Li,
$^{14}$Be $^{20}$C and $^{22}$C. The model describes the experimental data for
$^{11}$Li, $^{14}$Be and to some extend $^{20}$C. The recoil momentum
distribution of the $^{20}$C from the breakup of $^{22}$C nucleus is computed
for different two-neutron separation energies, and from the comparison with
recent experimental data the two-neutron separation energy is estimated in the
range $100\lesssim S_{2n}\lesssim 400$ KeV. The recoil momentum distribution
depends weakly on the neutron-$^{20}$C scattering length, while the matter
radius is strongly sensitive to it. The expected universality of the momentum
distribution width is verified by also considering excited states for the
system.
