  The success or failure of the neutrino-transport mechanism for producing a
supernova in an evolved massive star is known to be sensitive not only to the
mass of the iron core that collapses, but also to the density gradient in the
silicon and oxygen shells surrounding that core. Here we study the systematics
of a presupernova core's "compactness" (O'Connor & Ott 2011) as a function of
the mass of the star and the physics used in its calculation. Fine-meshed
surveys of presupernova evolution are calculated for stars from 15 to 65 Msun.
The metallicity and the efficiency of semiconvection and overshoot mixing are
both varied and bare carbon-oxygen cores are explored as well as full
hydrogenic stars. Two different codes, KEPLER and MESA, are used for the study.
A complex interplay of carbon and oxygen burning, especially in shells, can
cause rapid variations in the compactness for stars of very nearly the same
mass. On larger scales, the distribution of compactness with main sequence mass
is found to be robustly non-monotonic, implying islands of "explodability",
particularly around 8 to 20 Msun and 25 to 30 Msun. The carbon-oxygen (CO) core
mass of a presupernova star is a better, though still ambiguous discriminant of
its core structure than the main sequence mass.
