  The relationships between solar flare parameters (total importance, time
duration, flare index, and flux) and sunspot activity (Rz) as well as those
between geomagnetic activity (aa index) and the flare parameters can be well
described by an integral response model with the response time scales of about
eight and thirteen months, respectively. Compared with linear relationships,
the correlation coefficients of the flare parameters with Rz, of aa with the
flare parameters, and of aa with Rz based on this model have increased about
6%, 17%, and 47% on average, respectively. The time delays of the flare
parameters to Rz, of aa to the flare parameters, and of aa to Rz at their peaks
in solar cycle can be predicted in part by this model (82%, 47%, and 78%,
respectively). These results may be further improved when using a cosine filter
with a wider window. It implies that solar flares are related to the
accumulation of solar magnetic energies in the past through a time decay
factor. The above results may help to understand the mechanism of the solar
cycle and to improve the solar flare prediction.
