Origin of a maximum of the astrophysical S factor in heavy-ion fusion reactions at deep subbarrier energies

The hindrance phenomenonof heavy-ion fusion cross sections at deep subbarrier energies often accompanies a maximum of an astrophysical$ S$ factorat a threshold energyfor fusion hindrance. We argue that this phenomenoncan naturally be explained when the fusion excitation functionis fitted with two potentials, with a larger (smaller) logarithmic slope at energies lower (higher) than the threshold energy. This analysis clearly suggests that the astrophysical$ S$ factorprovides a convenient tool to analyze the deep subbarrier hindrance phenomenon, even though the $ S$ factormay have a strong energy dependence for heavy-ion systems unlike that for astrophysicalreactions.