ISSI Visiting Scientists Programme

__Proposed Research__

Application to Large Sample of Events

Our work for the last few years has focused on the
development of robust, rigorously-tested algorithms for solution of the
fundamental equation (1). We have explored different forms of the kernel
cross-section Q(ε,E), such as through including electron-electron bremsstrahlung
[6], and anisotropic emission [3, 4]. We have studied the solar
bremsstrahlung inverse problem in a rigourous mathematical framework [8]. We have analyzed both source-integrated spectra
[1, 2, 3, 4, 6, 9] and imaged spectra [5,7]. Our work has also been included in a
RHESSI-inspired book submitted to* Space Science Reviews *[10]. In
the process we have developed a long-overdue methodology which takes into
account that while images in different count energy bands are *a priori *
independent, they nevertheless are coupled through the physics of the emission
mechanism - electrons of energy E produce photons at all energies ≤ E. Of
necessity, we have studied, and eventually implemented, these techniques through
the study of a few, illustrative, events. It is now time to apply these proven
techniques to a large set of flares, both ourselves and through making the
techniques available to the larger community through the development of “userfriendly”
codes on the Solar SoftWare (SSW) tree. We have already done this for the
regularized inversion of* spatially integrated *spectra, and it is now time
to do the same for our recently-developed analysis of imaging spectroscopy date,
which utilizes the highly novel technique wherein the key photon ! electron
spectral inversion is performed in *visibility *space. (Visibilities are
two-dimensional spatial Fourier transforms of an image. They store “patterns” of
emission and are optimized to the manner in which *RHESSI *stores imaging
information). **
Analysis of the “electron flux images” that result from this unique application
have the highest potential to discover the physics of electron acceleration and
propagation in solar flares. **
Through the study of a large
sample of events, we hope to learn not only “statistical” truths, but also
explore the extremes.

Other Emission Processes

The correct inference of the electron spectrum F(E) depends not only on the quality of the data I(ε) but also on the correct form of the transformational kernel (emission cross-section) Q(ε,E). In the past we have explored various forms of Q(ε,E), including

• simplified, analytical forms for free-free emission (e.g., Kramers, Bethe-Heitler), in order to derive approximate analytic solutions that aid our overall understanding;

• exact, relativistic, numerical forms, aimed at deriving the most accurate forms of F(E);

• forms of Q that take into account the directivity of the bremsstrahlung process, and so can be used to probe the angular distribution (anisotropy) of the accelerated electrons;

• forms of Q(ε,E) that take into account both electron-ion and electron-electron bremsstrahlung. This allows extension of our results to moderately relativistic electron energies E.

There has recently, however, seen increasing interest in the
possibility that *free-bound *emission, especially on heavy ions such as
Fe, may play a significant role in flares. We therefore propose to add a term
representing this process. This may seem like a trivial undertaking, but the
fact the free-bound emission results in an emitted photon with an energy *
greater than *the exciting electron fundamentally changes the mathematical
character of the integral equation (1) (the lower limit changes from ε to ε − χ,
where χ is the ionization energy of the ion species in question).

Bivariate Analysis

For anisotropic forms of the cross-section Q, the emitted
hard X-ray spectrum depends not only on the *energy *spectrum F(E) but also
on the angular distribution of the emitting electrons. Interestingly, it turns
out that the highly differing dependencies of the cross-section on energy and
angle permit *both *the energy and angular distributions to be obtained,
even though the “data function” I(ε) is one-dimensional. We have made some
inroads on the formulation of this “bivariate” inversion problem, and we propose
to apply the technique to flare events, focusing on those for which the form of
F(E) inferred using an isotropic cross-section (or, equivalently, an isotropic
angular distribution of emitting electrons) contains “unusual” features that
suggest a bivariate approach is necessary.

References

[1] J. C. Brown, A. G. Emslie, G. D. Holman, C. M. Johns-Krull, E. P. Kontar,
R. P. Lin, A. M. Massone and M. Piana, “Evaluation of algorithms for
reconstructing electron spectra from their bremsstrahlung hard X-ray spectra,”
*The Astrophysical Journal*, **643**, 523-531 (2006).

[2] M. Prato, M. Piana, J. C. Brown, A. G. Emslie, E. P. Kontar, and A. M.
Massone, “Regularized reconstruction of the differential emission measure from
solar flare hard X-ray spectra,” *Solar Physics*, **237**, 61-83 (2006).

[3] E. P. Kontar & J. C. Brown, “Stereoscopic Electron Spectroscopy of Solar
Hard X-Ray Flares with a Single Spacecraft,” *The Astrophysical Journal*,
**653**, L149-L152 (2006).

[4] J. Kaˇsparov´a, E. P. Kontar, J. C. Brown, “Hard X-ray spectra and
positions of solar flares observed by RHESSI: photospheric albedo, directivity
and electron spectra,” *Astronomy & Astrophysics*, **466**, 705-712
(2007).

[5] M. Piana, A. M. Massone, G. J. Hurford, M. Prato, A. G. Emslie, E. P.
Kontar, & R. A. Schwartz, “Electron Flux Spectral Imaging of Solar Flares
through Regularized Analysis of Hard X-Ray Source Visibilities,”* The
Astrophysical Journal*, **665**, 846-855 (2007).

[6] E. P. Kontar, A. G. Emslie, A. M. Massone, M. Piana, J. C. Brown, & M.
Prato, “Electron-Electron Bremsstrahlung Emission and the Inference of Electron
Flux Spectra in Solar Flares,” *
The Astrophysical Journal*,
**670**, 857-861 (2007).

[7] Y. Xu, A. G. Emslie, & G. J. Hurford, “RHESSI Hard X-ray Imaging
Spectroscopy of extended sources and the physical properties of electron
acceleration regions in solar flares,” *The Astrophysical Journal*, **673**,
576-585 (2008).

[8] A. M. Massone, M. Piana, & M. Prato. “Regularized solution of the solar
Bremsstrahlung inverse problem: model dependence and implementation issues,” *
Inverse Problems in Science and Engineering*, **16**, 523-545 (2008).

[9] J. C. Brown, J. Kašparová, A. M. Massone, & M. Piana, “Fast Spectral
Fitting for Hard X-ray Bremsstrahlung from Truncated Power-Law Electron Spectra,”
*Astronomy & Astrophysics*, submitted (2008).

[10] E. P. Kontar, J. C. Brown, A. G. Emslie, W. Hajdas, G. D. Holman, G. J.
Hurford, J. Kašparová, P. C. V. Mallik, A. M. Massone, M. L. McConnell, M.
Piana, M. Prato, E. J. Schmahl & E. Suarez-Garcia. “Deducing Electron Properties
form Hard X-ray observations,” *Space Science Reviews*, submitted (2008).

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