  Various neutron star (NS) models correspond to an unrealistic mass range
$\leq 0.5M_\odot$ for the Vela pulsar, if the observational constraints of the
glitch healing parameter in the starquake model ($Q = I_{\rm core}/I_{\rm
total} \leq 0.2$), are imposed on these models. However, we show that these
observational constraints yield a realistic mass range for NS models,
corresponding to a core given by the stiffest equations of state (EOS) and the
envelope is characterized by the well known EOS of adiabatic polytrope, if the
continuity of the adiabatic speed of sound together with pressure,
energy-density, and the two metric parameters is assured at the core-envelope
boundary of the models and this boundary is worked out on the basis of the
`compatibility criterion' for hydrostatic equilibrium. The models yield the
surface redshift $z_R \simeq 0.6913$ and mass $M \simeq 2.153 M_\odot$ for the
``central'' weighted mean value, $Q = 0.12 \pm 0.07$, of the glitch healing
parameter of the Vela pulsar. These values of mass and surface redshift can
increase upto $M \simeq 2.196 M_\odot$ and $z_R \simeq 0.7568$ (which
represents an ultra-compact object (UCO; $z_R \geq 0.73$)) respectively for the
upper weighted mean value of $Q \simeq 0.19$. However, for the lower weighted
mean value of $Q \simeq 0.05$, the mass and surface redshift can decrease upto
the values of $M \simeq 2.052 M_\odot$ and $z_R \simeq 0.6066$ respectively.
The observation of the lower bound on the energy of a $\gamma$-ray pulse at
about 0.30 MeV from the Vela pulsar in 1984 is in excellent agreement with the
results of this study, provided this energy could be interpreted as the energy
of a gravitationally redshifted electron-positron annihilation radiation from
the star's surface.
