  One of the major difficulties encountered in modeling core collapse
supernovae is obtaining an accurate description of the transport of neutrinos
through the collapsed stellar core. The behavior of the neutrino distribution
function transitions from an LTE distribution in the center of the core to a
non-LTE distribution in the outer regions of the core. One method that has been
recently employed in order to model the flow of neutrinos in 2-D models is the
gray approximation. This approximation assumes that the neutrino distribution
can be described by a function that is parameterized in terms of a neutrino
temperature and a neutrino chemical potential. However, these parameters must
be assumed. Furthermore, the parameters will also differ between the LTE and
NLTE regions. Additionally, within the gray approximation the location at which
the neutrino distribution function transitions from LTE to NLTE must be
assumed. By considering a series of models where the LTE/NLTE decoupling point
is varied we show that the outcome of the numerical models is critically
sensitive to the choice of the decoupling point when the gray approximation is
employed. We also examine the effects of the neutrino--electron scattering
(NES) rate which is difficult to correctly formulate within the gray
approximation. We show that NES has a dramatic affect on the the overall
neutrino heating rate and the dynamics of the model. This result conflicts with
the results of high resolution multi-group models which model neutrino
transport without employing the gray approximation.
