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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