tokens: The effect is best understood by examining Fig. 6 . The particular explosion energy for which incomplete silicon burning mostly ceases inside the oxygen-burning shell is shown by the dashed line at 5.75 B in Fig. 6 ( top ). For explosion energy 1 B − 2 B below this value, incomplete silicon burning occurs extensively against the inner edge of the boundary, and for greater explosion energy, incomplete silicon burning does not occur at all within the oxygen-burning shell. Examining the peak temperature profile in Fig. 6 ( bottom ), we see a 'bump' of increased temperature at the inner edge of the shell boundary (location shown by the dashed line), indicating that material in this region is subject to some source of extra heating. We understand the cause of the 'bump' in the temperature profile against the outer edge of the oxygen-burning shell as follows: as the initial supernova shock travels through the composition boundary, a reverse shock is launched as a consequence of the change in the sign of the gradient in ρ r 3 at this point. The quantity ρ r 3 is important for the shock velocity in supernovae. As is described by Kompaneets ( 1960 ); Klimishin Gnatyk ( 1982 ); Bethe ( 1990 ); and Matzner McKee ( 1999 ), the shock velocity, U , can be approximated by Here β is parameter that varies with the properties of the material that the shock is propagating through, typically, 0.2 ≲ β ≲ 0.5. Nevertheless, for the our purposes the important point is that the shock velocity is inversely proportional to ρ r 3 , so when ρ r 3 is increasing, the shock must decelerate, and when ρ r 3 is decreasing, the shock accelerates. The inner ρ r 3 profile is shown for the 15 M ⊙ model in Fig. 7(a) . We see that there is a reversal in the sign of the gradient of the ρ r 3 profile at the oxygen-burning/carbon-burning shell interface. Such a rapid change in ρ r 3 , from a negative to positive gradient, leads to a rapid slowing of the shock velocity. This abrupt deceleration results in an increased density at this point, as interior matter piles up against the slowed mass. This has a chain effect on the matter behind it, and so on, which propagates as an ingoing shock. Figure 6. In the upper and middle plots, we show chromium and iron mass fractions, respectively, as a function of mass coordinate and explosion energy for the 15 M ⊙ model, as in Fig. 1 . The lower plot shows the peak temperature profiles for four models with explosion energy at which explosive silicon burning is affected by the reverse shock. Figure 7. The ρ r 3 profile (black line) and electron mole fraction, Y e (grey line) for the inner 10 M ⊙ of each of the 15 M ⊙ –80 M ⊙ progenitor models. The piston location is indicated by the dashed line. Note, Y , where η is the neutron excess. The reverse shock provides additional heating to the material immediately interior to the shell boundary, as can be seen in the temperature profile at various time intervals (Fig. 8 ). This results in an approximately
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