  We systematically study the symmetry energy effects of the transition density
$n_{\rm t}$ and the transition pressure $P_{\rm t}$ around the crust-core
interface of a neutron star in the framework of the dynamical and the
thermodynamical method respectively. We employ both the parabolic approximation
and the full expansion, for the definition of the symmetry energy. We use
various theoretical nuclear models, which are suitable for reproducing the bulk
properties of nuclear matter at low densities, close to saturation density as
well as the maximum observational neutron star mass. Firstly we derive and
present an approximation for the transition pressure $P_{\rm t}$ and crustal
mass $M_{\rm crust}$. Secondly, we explore the effects of the Equation of State
(EoS) on a few astrophysical applications which are sensitive to the values of
$n_{\rm t}$ and $P_{\rm t}$ including neutron star oscillation frequencies,
thermal relaxation of the crust, crustal fraction of the moment of inertia and
the r-mode instability window of a rotating neutron star. We found that the
above quantities are sensitive mainly on the applied approximation for the
symmetry energy (confirming previous results). Furthermore, an additional
sensitivity also exists, depending on the used method (dynamical or
thermodynamical). The above findings lead us to claim that the determination of
the $n_{\rm t}$ and $P_{\rm t}$ must be reliable and accurate before they are
used to constrain relevant neutron star properties.
