Abstract
We analyze the three-dimensional geometry of solar flares that show so-called interacting flare loops in soft X-ray,
hard X-ray, and radio emission, as previously identified by Hanaoka and Nishio. The two flare loops that appear
brightest after the flare are assumed to represent the outcome of a quadrupolar magnetic reconnection process, during
which the connectivity of magnetic polarities is exchanged between the four loop footpoints. We parameterize the
three-dimensional geometry of the four involved magnetic field lines with circular segments, additionally constrained
by the geometric condition that the two pre-reconnection field lines have to intersect each other at the onset of the
reconnection process, leading to a 10 parameter model. We fit this 10 parameter model to Yohkoh Soft and Hard
X-Ray Telescopes (SXT and HXT) data of 10 solar flares and determine in this way the loop sizes and relative
orientation of interacting field lines before and after reconnection. We apply a flare model by Melrose to calculate the
magnetic flux transfer and energy released when two current-carrying field lines reconnect to form a new
current-carrying system in a quadrupolar geometry. The findings and conclusions are the following. (1) The
pre-reconnection field lines always show a strong asymmetry in size, consistent with the scenario of newly emerging
small-scale loops that reconnect with preexisting large-scale loops. (2) The relative angle between reconnecting
field lines is nearly collinear in half of the cases, and nearly perpendicular in the other half, contrary to the antiparallel
configuration that is considered to be most efficient for magnetic reconnection. (3) The angle between interacting field
lines is reduced by ~10 deg-50 deg after quadrupolar reconnection. (4) The small-scale flare loop experiences a
shrinkage by a factor of 1.31+/-0.44, which is consistent with the scaling law found from previous electron
time-of-flight measurements, suggesting that electron acceleration occurs near the cusp of quadrupolar
configurations. (5) The large-scale loop is found to dominate the total induction between current-carrying loops,
providing a simple estimate of the maximum magnetic energy available for flare energy release because of current
transfer, which scales as ΔEI~1029.63(r2/109 cm)(I2/1011A)2 (with r2 the curvature radius and I2 the current of
the large-scale loop) and is found to correlate with observed flare energies deduced from soft X-ray and hard X-ray
fluxes. Most of the energy is transferred to small-scale loops that have one-half of the large-scale current (I1=I2/2).
(6) The quadrupolar reconnection geometry provides also a solution of Canfield's dilemma of the offset between the
maximum of vertical currents and the HXR flare loop footpoints. (7) The quadrupolar geometry provides not only a
framework for interacting double-loop flares, but it can also be considered as a generalized version of (cusp-shaped)
single-loop flares.