  The Skyrme model and its generalisations provide a conceptually appealing
field-theory basis for the description of nuclear matter and, after its
coupling to gravity, also of neutron stars. In particular, a specific Skyrme
submodel, the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) Skyrme model,
allows both for an exact field-theoretic and a mean-field treatment of neutron
stars, as a consequence of its perfect fluid property. A pure BPS Skyrme model
description of neutron stars, however, only describes the neutron star core, by
construction. Here we consider different possibilities to extrapolate a BPS
Skyrme neutron star at high baryon density to a description valid at lower
densities. In the exact field-theoretic case, a simple effective description of
the neutron star crust can be used, because the exact BPS Skyrme neutron star
solutions formally extend to sufficiently low densities. In the mean-field
case, on the other hand, the BPS Skyrme neutron star solutions always remain
above the nuclear saturation density and, therefore, must be joined to a
different nuclear physics equation of state already for the outer core. We
study the resulting neutron stars in both cases, facilitating an even more
complete comparison between Skyrmionic neutron stars and neutron stars obtained
from other approaches, as well as with observations.
