  By performing neutrino-radiation hydrodynamic simulations in spherical
symmetry (1D) and axial symmetry (2D) with different progenitor models by
Woosley & Heger (2007) from 12 $M_{\odot}$ to 100 $M_{\odot}$, we find that all
1D runs fail to produce an explosion and several 2D runs succeed. The
difference in the shock evolutions for different progenitors can be interpreted
by the difference in their mass accretion histories, which are in turn
determined by the density structures of progenitors. The mass accretion history
has two phases in the majority of the models: the earlier phase in which the
mass accretion rate is high and rapidly decreasing and the later phase with a
low and almost constant accretion rate. They are separated by the so-called
turning point, the origin of which is a change of the accreting layer. We argue
that shock revival will most likely occur around the turning point and hence
that its location in the $\dot M$-$L_\nu$ plane will be a good measure for the
possibility of shock revival: if the turning point lies above the critical
curve and the system stays there for a long time, shock revival will obtain. In
addition, we develop a phenomenological model to approximately evaluate the
trajectories in the $\dot M$-$L_\nu$ plane, which, after calibrating free
parameters by a small number of 1D simulations, reproduces the location of the
turning point reasonably well by using the initial density structure of
progenitor alone. We suggest the application of the phenomenological model to a
large collection of progenitors in order to infer without simulations which
ones are more likely to explode.
