Flare Loop Asymmetries
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 presented 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 an asymmetric loop.
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, in 1994, devoted part of his PhD thesis to this subject 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 early 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. Two examples of forward-fit maps are shown below,
along with Clean maps for comparison.
Figure 2 Visibility Forward-Fit maps (top) and Clean maps (bottom)
Some observed widths and fluxes
We have selected a sample of 26 double-component events and computed
FWHM and flux of each component. The figure below shows
the correlation of width and brightness for the pair of (flux,FWHM)
values in each event.
Figure 3. 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.
Brighter is broader
We have found that the brighter footpoint is 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.
It is also possible to extract information from these width-flux
asymmetries about the pitch angle distributions and the loss-cone
angles, but that's another story to be told later.
Acknowledgments
Without Gordon Hurford's invention of RHESSI-style visibilities,
this work would have been hugely difficult if not impossible. We also must
credit him for the new visibility forward-fit method, which has many potential
applications for indirect imaging by RHESSI.