ISSI Visiting Scientists Programme
SCIENTIFIC RATIONALE OF THE PROJECT
RHESSI imaging-spectroscopy , a combination of imaging and high spectral resolution, is fundamentally new at X-ray wavelengths; it was not available with previous imaging instruments, such as the Yohkoh hard X-ray telescope , due to insufficient performance in terms of both spectral resolution, spatial resolution and Fourier-space coverage. Imaging spectroscopy is a powerful tool for the investigation of fundamental processes in solar flare physics such as:
• the motion of footpoints of magnetic loops created by bremsstrahlung of energetic electrons released in the flash phase of a flare, which is related to the rate of magnetic reconnection ;
• the electron energy modification rate, which can be empirically determined from the change of the hard X-ray spectrum with position in the source  and which plays a crucial role in understanding electron acceleration and transport processes;
• the space/time dependence of the photon spectral index, which provides insight into the physical structure (e.g., temperature, density) of the source.
The current approach to RHESSI imaging spectroscopy is based on comparing images generated at multiple energy bands. Because the images are generated independently, the resulting energy spectra obtained at each location do not have sufficient quality to support inversion. The aim of the current project is to address this problem using three strategies. In the first strategy, more effective image reconstruction methods, explored during the previous grant activity, will be applied to RHESSI counts. The methods will be extended to incorporate a priori knowledges of the spectral shapes. For example the spectra must decrease monotonically with energy at every spatial point. Although these are relatively weak constraints in terms of solar physics, they will significantly reduce the ‘noise’ in the resulting set of images and so dramatically improve the quality of the resulting spatially resolved spectra. In the second strategy, forward-fitting techniques will be applied to determine the shape, size and intensity parameters of sources directly at each energy. Although applicable only to the simpler source geometries, this technique effectively ‘focuses’ all the input data onto the determination of a small number of source parameters at each energy. This will yield high quality energy spectra across the resulting sources. Recent advances in exploiting a Fourier interpretation (visibilities) of the input data makes this a very promising approach. This technique has the additional advantage of generating statistically viable error estimates in the resulting parameter spectra. A further advantage is that in some cases, the source parameter spectra (e.g. size vs energy) can be compared directly to the predictions of theoretical models. The third, and most advanced, strategy is to consider the imaging spectroscopy problem as a linear inverse problem in three dimensions (two spatial and one spectral). The quantity at the core of this approach is the ‘local’ averaged flux spectrum F(y,E), where y is a 2-D spatial location and E is the electron energy. In accordance with the model-independent formulation , F(y,E) is related to the ‘local’ photon flux g(y,ε) through the equation
where the bremsstrahlung cross-section Q(ε,E) is considered isotropic and position-independent. The RHESSI instrument collects time-modulated counts which can be synthesized in two-dimensional ‘dirty’ maps of the flare by means of a back-projection operation. For a single collimator, the back-projected map gbp(x,ε) is related to the local photon flux through the integral equation
where A covers the extension of the 2D projection of the loop (i.e., of the 2D image involved) and K(x,y) is the Point Spread Function (PSF) of the collimator. By combining equation (1) and equation (2) a 3D model for imaging spectroscopy results:
Here the input data gbp(x,ε) are provided by a
set of ‘dirty’ back-projected maps corresponding to different photon energy
channels and the function F(y,E) is the desired local averaged electron
distribution. This approach exploits the fact that the RHESSI Point Spread
Function is spatially invariant (K(x, y) = K(x −
y)). In fact, a straightforward Fourier analysis shows that the local
electron spectrum can be recovered by solving two successive linear inverse
problems, where the computational effort can be significantly reduced by using
Fast Fourier Transform (FFT) applications.
The unique feature of this
approach is that the electron flux maps at different energies are constrained by
both spatial and spectral smoothness.
 Craig, I. J. D. & Brown, J. C., 1986, Inverse Problems in Astronomy, Adam Hilger, London.
 Emslie, A.G., Barrett, R.K. and Brown J.C., 2001, An Empirical Method to Determine Electron Energy Modification Rates from Spatially Resolved Hard X-Ray Data, ApJ 557, 921.
 Kosugi T. et al., 1991, The Hard X-ray Telescope (HXT) for the SOLAR-A mission, Sol. Phys. 136, 17.
 Krucker, S., Hurford, G.J. and Lin, R.P., 2003, Hard X-Ray Source Motions in the 2002 July 23 Gamma-Ray Flare, ApJ 595, L103.
 Lin, R.P. et al., 2002, The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).Sol. Phys. 210, 31.
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