  The core-envelope analytic models of Negi et al. (1989) have been found to be
consistent with both the mechanisms of glitch generation in pulsars, namely -
(i) the starquake model and (ii) the vortex unpinning model. In Negi (2019),
the author has been able to reproduce the observed values of glitch healing
parameter, $ G_h (= I_{\rm core}/I_{\rm total}; G_h$ represents the fractional
moment of inertia of the core component in the starquake mechanism of glitch
generation) for the Crab as well as for the Vela pulsar. In another study, by
using the models discussed in Negi et al. (1989), the author has obtained the
minimum value of fractional moment of inertia of the crust about 7.4\% and
larger for all the values of masses in the range - $1M_\odot - 1.96M_\odot$
considered for the Vela pulsar (Negi 2020a). The latter study of the author is
found to be consistent with the recent requirement (on the basis of vortex
unpinning model of the glitch generation) which refers: The minimum fractional
crustal moment of inertia of the Vela pulsar should be about 7\% for a mass
higher than about 1$M_\odot$. However, an important study which requires
investigation of pulsational stability and gravitational binding of the models
of Negi et al. (1989) has not been carried out so far. The present paper deals
with such a study of the models (Negi et al. 1989) for all permissible values
of the compactness parameter $u (\equiv M/a$, total mass to size ratio in
geometrized units) and compressibility factor $Q$ (defined in Tolman's VII
solution as: $ x = r^2/K^2 = r^2/a^2Q)$. It is seen that the configurations
remain pulsationally stable and gravitationally bound for all permissible
values of $u (\leq 0.25)$ and $Q$ ($0 < Q \leq 1.2$).
