MODELS FOR THE MOTIONS OF FLARE LOOPS AND RIBBONS

We have found a conformal mapping which is valid for any magnetic boun dary condition at the photosphere and which can be used to determine t he evolution of an open, two-dimensional magnetic field configuration as it relaxes to a closed one. Solutions obtained with this mapping ar e in quasi-static equilibrium, and they contain a vertical current she et and have line-tied boundary conditions. As a specific example, we d etermine the solution for a boundary condition corresponding to a subm erged, two-dimensional dipole below the photosphere. We assume that th e outer edges of the hottest X-ray loops correspond to field lines map ping from the outer edges of the H alpha ribbon to the lower tip of th e current sheet where field lines reconnect at a Y-type neutral line w hich rises with time. The cooler H alpha loops are assumed to lie alon g the field lines mapping to the inner edges of the flare ribbons. Wit h this correspondence between the plasma structures and the magnetic f ield we determine the shrinkage that field lines are observed to under go as they are disconnected from the neutral line. During the early ph ase of the flare, we predict that shrinkage inferred from the height o f the H alpha and X-ray loops is close to 100% of the loop height. How ever, the shrinkage should rapidly decrease with time to values on the order of 20% by the late phase. We also predict that the shrinkage in very large loops obeys a universal scaling law which is independent o f the boundary condition, provided that the field becomes self-similar (i.e., all field lines have the same shape) at large distances. Speci fically, for any self-similar field containing a Y-type neutral line, the observed shrinkage at large distances should decrease as (Delta X/ X(R))(-2/3), where Delta X is the ribbon width and X(R) is the ribbon separation. Finally, we discuss the relation between the electric fiel d at the neutral line and the motions of the flare loops and ribbons.