ISSI Visiting Scientists Programme

**SCIENTIFIC RATIONALE
OF THE PROJECT**

RHESSI imaging-spectroscopy [5], 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 [3], 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 [4];

*• *the electron energy modification rate, which can be
empirically determined from the change of the hard X-ray spectrum with position
in the source [2] 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 [1],
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 g_{bp}(**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 g_{bp}(**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.*

**REFERENCES**

[1] Craig, I. J. D. & Brown, J. C., 1986, *Inverse Problems in
Astronomy*, Adam Hilger, London.

[2] 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.

[3] Kosugi T. et al., 1991, The Hard X-ray Telescope (HXT) for
the SOLAR-A mission, *Sol. Phys.* **136**, 17.

[4] 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.

[5] Lin, R.P. et al., 2002, The Reuven Ramaty High Energy Solar
Spectroscopic Imager (RHESSI). *
Sol. Phys.*
**210**, 31.

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