  The hindrance phenomenon of heavy-ion fusion cross sections at deep
subbarrier energies often accompanies a maximum of an astrophysical $S$ factor
at a threshold energy for fusion hindrance. We argue that this phenomenon can
naturally be explained when the fusion excitation function is 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 $S$ factor provides a convenient tool to analyze the deep
subbarrier hindrance phenomenon, even though the $S$ factor may have a strong
energy dependence for heavy-ion systems unlike that for astrophysical
reactions.
