  The modeling of many neutron star observables incorporates the microphysics
of both the stellar crust and core, which is tied intimately to the properties
of the nuclear matter equation of state (EoS). We explore the predictions of
such models over the range of experimentally constrained nuclear matter
parameters, focusing on the slope of the symmetry energy at nuclear saturation
density $L$. We use a consistent model of the composition and EoS of neutron
star crust and core matter to model the binding energy of pulsar B of the
double pulsar system J0737-3039, the frequencies of torsional oscillations of
the neutron star crust and the instability region for r-modes in the neutron
star core damped by electron-electron viscosity at the crust-core interface. By
confronting these models with observations, we illustrate the potential of
astrophysical observables to offer constraints on poorly known nuclear matter
parameters complementary to terrestrial experiments, and demonstrate that our
models consistently predict $L<70$ MeV.
