  We demonstrate that observations of glitches in the Vela pulsar can be used
to investigate the strength of the crust-core coupling in a neutron star, and
suggest that recovery from the glitch is dominated by torque exerted by the
re-coupling of superfluid components of the core that were decoupled from the
crust during the glitch. Assuming that the recoupling is mediated by mutual
friction between the superfluid neutrons and the charged components of the
core, we use the observed magnitudes and timescales of the shortest timescale
components of the recoveries from two recent glitches in the Vela pulsar to
infer the fraction of the core that is coupled to the crust during the glitch,
and hence spun up by the glitch event. Within the framework of a two-fluid
hydrodynamic model of glitches, we analyze whether crustal neutrons alone are
sufficient to drive the glitch activity observed in the Vela pulsar. We use two
sets of neutron star equations of state (EOSs), both of which span crust and
core consistently and cover a range of the slope of the symmetry energy at
saturation density $30 < L <120$ MeV. One set produces maximum masses
$\approx$2.0$M_{\odot}$, the second $\approx$2.6$M_{\odot}$. We also include
the effects of entrainment of crustal neutrons by the superfluid lattice. We
find that for medium to stiff EOSs, observations imply $>70\%$ of the moment of
inertia of the core is coupled to the crust during the glitch, though for
softer EOSs $L\approx 30$MeV as little as $5\%$ could be coupled. No EOS is
able to reproduce the observed glitch activity with crust neutrons alone, but
extending the region where superfluid vortices are strongly pinned into the
core by densities as little as 0.016fm$^{-3}$ above the crust-core transition
density restores agreement with the observed glitch activity.
