Flare Footpoint Aymmmetries

Schmahl, Pernak & Hurford

The hard X-ray domain is not usually where one would think of finding diagnostics of coronal magnetic fields, but the shape and cross sections of the flare loops that constrain high energy electrons contain useful information about magnetic fields not obtainable in any other way. One example of such information comes from HXR observations of the footpoints of flare loops, visible as elliptical features in flare images at energies where thick-target emission dominates. It has been known for over a decade, largely from HXT, but also radio, that footpoint fluxes are seldom equal, and that asymmetry is the rule rather than the exception. It is possible that the asymmetry is just due to an asymmetric injection of high energy electrons, but there are new RHESSI observations that seem to favor a symmetric injection..

Back in 1979, before hard X-ray imaging became a reality, Melrose and White offered up a trap/precipitation model that suggested that the flux asymmetry of hard X-ray footpoints in flaring loops would be caused by a magnetic asymmetry. In this model, magnetic convergence at the footpoints should mirror a fraction of streaming and spiraling electrons (green spirals in the figure below), and for electrons with small pitch angles (i.e moving more parallel to the magnetic field), where the mirror point lies in the chromosphere, hard X-ray emission would be emitted from regions whose area was that of the trapping magnetic flux tube.


Figure 1 Cartoon of trap/precipitation model for symmetric & asymmetric loops.

The footpoint (red) with stronger magnetic field mirrors a larger fraction of the electrons than the other footpoint, and so fewer thick-target hard X-rays are emitted from that footpoint. Thus hard X-ray footpoint flux should correlate negatively with magnetic field strength. As a function of position along the loop, the cross sectional area of the magnetic loop will be inversely proportional to the magnetic field strength, so the footpoint area ratio should equal the reciprocal of the magnetic footpoint field ratio, and hard X-ray flux should correlate positively with footpoint area.

The trap/precipitation scenario led many HXT workers to compare hard-X-ray footpoint flux ratios with magnetic fields determined from magnetograms. Taro Sakao wrote part of PhD thesis on this subject in 1994 at the University of Tokyo, and found that out of 5 flares that he studied, the footpoint with higher magnetic flux (B) was weaker in hard X-rays. This was a vindication of the trap/precipitation model. Later HXT studies by Goff et al. came to a different conclusion than Sakao. Out of 32 flares, only 14 showed the low-B, bright X-ray association. But these studies suffer from the fact that the magnetic fields were measured at the photospheric level, rather that at the chromospheric level where the hard X-rays were emitted. Also, with two different instruments, one has different cadences, resolutions, and pointing systems. So if hard X-ray imaging could determine the area of the footpoints, the asymmetry in both flux and area (inverse magnetic field) could be seen without ambiguity. Unfortunately this was not possible with HXT, nor, until recently, with RHESSI.

A new method of imaging

In 2006 it became possible to make quantitative area measurements with RHESSI data. The method makes use of "visibilities", which are the calibrated amplitudes and phases of the modulation profiles. (A discussion of RHESSI visibilities should be a subject of a future RHESSI Nugget.) To apply visibilities to imaging, one makes an image model with adjustable free parameters. A Fourier transform then provides model visibilities which can be compared with the observed visibilities. The free parameters are adjusted until a good (or the best possible) fit is obtained. This "visibility forward fit" technique is very fast--faster than Clean--and provides error bars derived from the count statistics and the estimated hardware uncertainties, all fed through the nonlinear fitting process. Some examples of maps are shown below.


Figure 2. Three examples of Visibility Fwd-fit (top row) and Clean for the same times and energies. Note the agreement of the two methods for the positions of the footpoints, but the frequent difference in the Visibility Fwd-fit and Clean sizes.

Example of fit to an amplitude profile

Here we show the amplitude profile of a double-component RHESSI flare as a function of both subcollimator (SC) and position angle (PA). The horizontal axis is SC+PA/180, so for example, for abscissa values between 6 and 7, the data are for SC 6 with PA between 0 and 180 degrees.

The bumps and wiggles are caused by the sources "beating" against each other as the collimator slats rotate from positions where both sources are visible to positions where only one is visible. The curve was fit by a model with 8 parameters describing two circular Gaussians. The χ2 was found to be about 1, where the sigmas are derived from count statistics and a 2% instrumental uncertainty.


Figure 3. Observed (x), fitted (-) and residual (squares) amplitudes for one double footpoint flare.

There are a number of ways to test the fit of such models to the data, and its uniqueness:

Some observed widths and fluxes

A sample of 26 double-component events FWHM plotted against flux shows that the brighter component is also broader, as predicted by Melrose & White, in the large majority of cases (23 out of 26). Relatively small error bars give us confidence in the result.

Figure 4. Flux and FWHM of components in double-footpoint flares

Each colored bar represents a separate flare or flare interval, with squares at the bar's end showing the flux and fwhm of that component. Crosses at the end of each bar show estimated 1-sigma errors in flux or fwhm for each component. Note that the slope is positive for all but 3 bars; only one (white bar) of those is significant at the 2 or 3 sigma level.

Our conclusions

We have found that the brighter footpoints are broader than the fainter conjugate footpoint in 23 of the 26 flare intervals studied. This is shown by the predominantly positive slopes of the line segments joining the footpoint parameters (flux, fwhm) in the above figure. This result validates the Melrose and White prediction, and it is in general agreement with Sakao's result, although the width-flux correlation is better (23/26) than his magnetic field-flux correlation (4/5), and significantly better than Goff et al's result (14/32). The possibility of an asymmetric injection as an explanation for flux asymmetry now seems to be unlikely on the basis of this strong correlation of footpoint flux and area, since if the injection were highly asymmetric in a random way, the correlation would be close to zero.

Systematics in the determination of areas must be carefully considered, although no one so far who has seen these results (at 3 meetings) has suggested any instrumental mechanism that might make brighter sources look broader to RHESSI.

Comparisons with magnetograms can check the sign of the width asymmetry. But there are often large horizontal (and perhaps vertical) gradients in photospheric magnetic fields that might make the correspondence with footpoint area less clear. Microwave observations can help estimate the degree of trapping to check the consistency of our results, although the microwave maps necessarily have different (usually lower) resolution, and are generated by higher energy electrons, which leads to other complications.

It is also possible to draw conclusions from these width-flux asymmetries regarding the pitch angle distributions and the loss-cone angles, but that's another story to be told later.