Exact and approximate expressions of energy generation rates and their impact on the explosion properties of pair instability supernovae

Energetics of nuclear reaction is fundamentally important to understand the mechanism of pair instability supernovae (PISNe). Based on the hydrodynamic equations and thermodynamic relations, we derive exact expressions for energy conservation suitable to be solved in simulation.We also show that some formulae commonly used in the literature are obtained as approximations of the exact expressions. We simulate the evolution of very massive stars of ~100-320 Mo˙ with zero- and 1/10 Zo˙, and calculate further explosions as PISNe, applying each of the exact and approximate formulae. The calculations demonstrate that the explosion properties of PISN, such as the mass range, the 56Ni yield, and the explosion energy, are significantly affected by applying the different energy generation rates.We discuss how these results affect the estimate of the PISN detection rate, which depends on the theoretical predictions of such explosion properties.