  The first detection of a binary neutron star merger has opened the brand new
era of multimessenger astronomy. This historic detection has been instrumental
in providing constraints on the tidal polarizability of neutron stars. In turn,
the tidal polarizability has been used to impose limits on stellar radii and
ultimately on the equation of state (EOS). The tidal polarizability is also
sensitive to the second tidal Love number k2. It is the main purpose of this
work to perform a detailed study of k2 which, for a given compactness
parameter, encodes the entire sensitivity of the tidal polarizability to the
EOS. In particular, we examine the role that the crustal component of the EOS
plays in the determination of k2. A set of realistic models of the equation of
state that yield an accurate description of the properties of finite nuclei and
support neutron stars of two solar masses is used. Given that the tidal
polarizability scales as the fifth power of the compactness parameter, a
universal relation exists among the tidal polarizability and the compactness
parameter that is highly insensitive to the underlying EOS. Thus, besides an
extraction of the tidal polarizabilities, a measurement of the individual
stellar masses is also required to impact the mass-radius relation. However, we
observe a strong sensitivity of k2 to the EOS, particularly to the contribution
from the inner crust. Although by itself the tidal polarizability can not
contribute to the determination of the mass-radius relation, future detections
of binary neutron star mergers are poised to provide significant constraints on
both the tidal polarizabilities and masses of the individual stars, and thus
ultimately on the mass-radius relation. Yet, subleading corrections to the
tidal polarizability are encoded in the second Love number k2 which displays a
large sensitivity to the entire (crust-plus-core) EOS.
