Observational Tests of a Double Loop Model for Solar Flares

S. J. Hardy , D. B. Melrose , H. S. Hudson, PASA, 15 (3), 318
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Application of model

We report a detailed investigation of the loop configurations in which the occurrence of solar flares is preferred. The general approach is graphical, in that the phase space to be explored is four dimensional - the location of two footpoints may be fixed, leaving two footpoints to be varied, and there is a further parameter which is the ratio of the currents in the two loops. There is a degeneracy in these parameters as the scaling of the system is arbitrary, reducing the five free parameters to four.

One idea is to search for the maxima of the energy release according to equation (1). However, this proves unfruitful as the most favourable conditions are often those where one of the loops are longest. Thus there is no global maximum, as tex2html_wrap_inline353 increases as one of the footpoints is moved toward infinity, and the other footpoints remain fixed..

A more productive method of looking for geometries which are favourable for energy release through flares involves fixing three of the footpoints and the ratio of currents between the loops, and then plotting tex2html_wrap_inline353 as a function of the location of the fourth footpoint. Here we assume that maximal current transfer occurs. This leads to plots of the type shown in Figure 2, where footpoints 1+ and 1- are at (-1,0) and (1,0) respectively, footpoint 2- has been fixed at (1.12,-0.42), and footpoint 2+ is allowed to vary. Those regions which are light correspond to positions for which there is a net release of magnetic energy if a flare occurs. The squares denote the footpoints of positive polarity, and the triangles denote the footpoints of negative polarity.

  figure79
Figure 2: Plots of the fractional energy release due to maximal current redistribution between two loops. The axes represent two dimensional spatial coordinates on the solar surface, with arbitrary units. Positive polarity footpoints are represented by squares, and negative polarity footpoints are represented by triangles. In all three plots one initial current loop between (-1,0) and (1,0) is fixed with current tex2html_wrap_inline399, and the second initial current loop has a single fixed negative polarity footpoint at (1.12,-0.42), a positive polarity footpoint which varies over the plot, and a current tex2html_wrap_inline403 carried between them. The current ratio, tex2html_wrap_inline405 of the three plots is 3,1, and 1/3 respectively. The actual values in the plot are scaled by tex2html_wrap_inline413. Note that for all three current ratios, energy is released for parasitic topologies such as that seen in Figure 4. The second square marks the position of the 2+ footpoint corresponding to maximum energy release.

The dark regions of Figure 2 correspond to those configurations which have higher magnetic energy in their final state. For maximal current transfer, this final state is shown in Figure 3. Thus, the dark regions of Figure 2 correspond to configurations where reconnection of three loops to form two loops is favourable.

  figure91
Figure 3: Sketch of the model of maximal current transfer between the loops, assuming tex2html_wrap_inline417.

  figure98
Figure 4: Images of a C9.1 flare at 23:35, 10 April 1993, exhibiting parasitic magnetic topology. This figure is adapted from Figure 3 of Hanaoka (1997). In panel (a) the Yohkoh SXR image of the flare around the peak of the flare is shown, with microwave contours overlaid in white, and HXR contours overlaid in black. In panel (b) the Yohkoh SXR image in the decay phase of the flare is shown. In panel (c) the longitudinal magnetogram from Kitt Peak Observatory at 19:00 on Apr 10 is shown with the microwave contours overlaid. Panel (d) is a sketch of the geometry suggested by the both the displayed images and the microwave polarization data. For further details see Hanaoka (1997).

Parasitic topology

A C9.1 flare event displaying ``parasitic magnetic topology'' is shown in Figure 4. This flare was analysed in detail by Hanaoka (1997). The observations show an emerging flux loop at the right-hand footpoint of the long loop. This emerging loop interacts with the existing loop appearing to cause a flare. Many flares exhibit this behaviour, and several of these were analysed recently by Nishio et al. (1997) and Hanaoka (1997). We explore this class of events using the model of M97. In particular, we take the geometry suggested in Figure 4 as the base configuration for our exploration, and extrapolate from these results an observational test of the model of M97.

In Figure 2 we present three plots of the energy released between pre- and post-flare configurations within the model of M97; the positions of three of the footpoints, 1+, 1-, and 2-, are fixed in the geometry shown in Figure 4(d). The fourth footpoint, 2+, is allowed to vary within the plane. The three plots correspond to three different ratios, tex2html_wrap_inline405, where tex2html_wrap_inline399 is the current in the longer loop (loop 1). Clearly, on energetic grounds, a flare is allowed for all three values of r. Thus, based purely on geometry, the model of M97 is not sufficient to constrain the flare events to a particular current ratio. The model of M97 demonstrates that current transfer within a parasitic configuration leads to a reduction in the stored magnetic energy, and thus is a favourable configuration for flares.

There are further constraints we may investigate by restricting our attention to a single configuration of footpoints. In Figure 5 the location of all four footpoints is specified as shown and the energy released is plotted against the ratio of the currents in the two loops. A strong peak in the amount of energy released occurs when the smaller loop carries around twice the current of the long loop. The existence of a correlation between the ratio of currents and the energy release in a flare in parasitic totpologies is a strong prediction of the theory of M97, and is subject to experimental verification. For strong flares, current maps may be generated from magnetogram data and combined with X-ray and microwave data to determine the locations of the footpoints and the currents passing through them. These may then be used as inputs for the theory and the predicted energy release may be compared against the total energy released in the flare. Such a program is under way.

  figure111
Figure 5: Plot of the fractional energy released for the configuration shown as a function of the ratio of currents in the two initial loops. Though energy is released for any ratio, the amount of energy released is a strong function of the ratio, and thus stronger flares should be expected where the current in the smaller loop is approximately twice the current in the larger loop.

We note that Figure 5 corresponds only to a particular footpoint configuration. Other configurations that we have examined also show strong dependence on the ratio of currents in the loops.

  figure119
Figure 6: Plot of the fractional energy released, for the configuration shown in Figure 5, as a function of the ratio of currents in the two initial loops and of the proportion of the maximal current transferred. The energy released is a strong function of the ratio of currents for all but the smallest current transfers. Thus, larger energy releases should be expected for a stronger smaller loop regardless of the size of the current transfer.

Another effect that may be important is incomplete current transfer between the loops. Figure 6 shows a plot of the energy released as a function of both the current ratio, r, and the fraction of the total allowed current which is transferred. As noted in Section 2, incomplete current transfer is likely for a variety of reasons. Figure 6 demonstrates that the total energy release remains a strong function of the ratio of currents for all but the smallest current transfers.

We conclude that flares which exhibit parasitic magnetic topology, often associated with emerging flux loops, form a class of events which may be used to observationally test the theory of M97.

Formation of long flux tubes between active regions

A further application of this theory, suggested by M97, is to the formation of long flux tubes between active regions. The proposed mechanism is that an existing flux loop interacts with a flux loop in a network field element or an ephemeral active region, reconnecting to produce a longer loop. A sequence of these events would lead to the formation of a long loop connecting active regions. Figure 7 shows the energy release for two configurations with three fixed footpoints - the first loop from (-1,0) to (1,0); the second loop with the positive footpoint fixed at (0.7,0) in (a) and the negative footpoint fixed at (0.7,0) in (b). In both the plots, the currents in the initial loops are assumed equal. In Figure 7(a) we see that energy release is possible for essentially any configuration which makes the loop longer, but not for the configurations which make the loops shorter. In Figure 7(b) we see that if the footpoint of the ephemeral emerging flux loop which is between the loops is negative in polarity, energy release is negative for the majority of initial conditions. Thus, these long flux loops should appear as the result of interactions between loops where the footpoint of the emerging flux is of opposite polarity to the nearest footpoint of the existing loop. This claim of the theory of M97 may be tested using a combination of X-ray and magnetogram data.

  figure132
Figure 7: A plot of the fractional energy release for two configurations where the footpoint of one flux loop is between the two footpoints of another flux loop. These plots demonstrate that if an ephemeral flux loop appears in such a configuration the majority of favourable configurations for flares and current transfer involve increasing the size of the loop. Thus, this provides a mechanism for the creation of the long flux loops connecting active regions which are observed.

Largest energy release

There are various footpoint configurations and current ratios which lead to energy release. Animations of the fractional energy release for a large class of initial conditions have been calculated. They may be obtained from S.J.H. on request. The interested reader is directed to these animations for a more complete description of the predictions for energy release of equation (1).


Next Section: Conclusion
Title/Abstract Page: Observational Tests of a
Previous Section: Current redistribution model for
Contents Page: Volume 15, Number 3

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