  We study the general relativistic collapse of neutron star (NS) models in
spherical symmetry. Our initially stable models are driven to collapse by the
addition of one of two things: an initially in-going velocity profile, or a
shell of minimally coupled, massless scalar field that falls onto the star.
Tolman-Oppenheimer-Volkoff (TOV) solutions with an initially isentropic,
gamma-law equation of state serve as our NS models. The initial values of the
velocity profile's amplitude and the star's central density span a parameter
space which we have surveyed extensively and which we find provides a rich
picture of the possible end states of NS collapse. This parameter space survey
elucidates the boundary between Type I and Type II critical behavior in perfect
fluids which coincides, on the subcritical side, with the boundary between
dispersed and bound end states. For our particular model, initial velocity
amplitudes greater than 0.3c are needed to probe the regime where arbitrarily
small black holes can form. In addition, we investigate Type I behavior in our
system by varying the initial amplitude of the initially imploding scalar
field. In this case we find that the Type I critical solutions resemble TOV
solutions on the 1-mode unstable branch of equilibrium solutions, and that the
critical solutions' frequencies agree well with the fundamental mode
frequencies of the unstable equilibria. Additionally, the critical solution's
scaling exponent is shown to be well approximated by a linear function of the
initial star's central density.
