ISSI Visiting Scientists Programme

**RHESSI SPECTRA:
**

**RECONSTRUCTION OF THE
ELECTRON SPECTRUM**

It is well known [3] that
the dominant X-ray emission mechanism during solar flares is collisional
bremsstrahlung of electrons with the ions of the solar plasma. The relation
between the measured optically thin

photon spectrum g(ε)
and the mean electron flux spectrum f(E)
is described by a Volterra integral equation of the first kind [4]:

where Q(ε,E)
is a suitable bremsstrahlung cross section for production of a photon of energy
ε
by an electron of energy E.

The linear inverse problem we are interested in, is the one to determine
f(E) from a noisy sampled
knowledge of g(ε).
In a thick-target interpretation, the knowledge of
f(E) provides the injected
electron flux spectrum by means of numerical differentiation.

If addressed naively, the solution of Equation (1) may lead to very unstable and
unphysical reconstructions of the electron spectra. A notable reduction of this
numerical instability is typically obtained by means of the following different
approaches:

*1) Parametric forward-fitting* [6]: a parametric form for
f(E) is inserted into the
integral at the right hand side of Equation (1) and the parameters are fixed by
minimizing the distance from the experimental data.

*2) Matrix inversion with data-adaptive binning* [8]: Equation (1) is
solved by standard square matrix row elimination and the electron spectrum is
derived locally from the photon spectrum exploiting the fact that:

The inversion of Equation (1) is then stabilized by adjusting Q through
judicious energy binning.

3) *Tikhonov regularization theory* [13]: a regularized solution of
Equation (1) is obtained by solving the Tikhonov minimum problem:

where A is the integral operator in Equation (1), L is an appropriate
differential operator and λ
is the so called regularization parameter providing an optimal trade-off between
stability (measured by the second term at the left hand side of Equation (3))
and fitting (measured by the first term at the left hand side of Equation (3)).
The computation of the minimum is performed by means of Singular Value
Decomposition (SVD) techniques [1,2].

The problem of the inversion of Equation (1) has been already widely carried out
by using the three methods separately [6,8,9,11,12], although a systematic
comparison of the effectiveness of these three approaches has never been
considered. We will perform such comparison by means of *blind* tests,
whereby physically significant synthetic photon spectra will be inverted without
knowing f(E) in advance.

Furthermore Equation (1) is a simplified model for the bremsstrahlung emission
process in which we use a solid-angle-averaged cross section depending only on E
and ε
(isotropic assumption) and a mean electron flux differential only in E. However,
in general, the correct cross section for bremsstrahlung photon emission depends
also on the emission angle (θ) between the incoming electron and the emitted
photon directions [5,7] and the electron flux is differential in E and in the
solid angle Ω
describing the incoming electron direction.

It follows that a more realistic model is given by:

and a more interesting problem is the one to recover a function of two variables
from the knowledge of a function of one variable (bivariate problem).

We have already faced the inversion of Equation (4) in the frame of the
regularization theory by assuming f(E,Ω)
separable in E and Ω
(i.e. f(E,Ω)=F(E)h(Ω))
and we have recovered F(E) by choosing a suitable form for the angular
dependence h(Ω)
[10].

Even in presence of strong constraints, this approach showed how the angular
dependence cannot be neglected: the use of anisotropic cross section yields to
recovered electron spectra significantly different from the ones reconstructed
by using the solid-angle-averaged cross section.

In this project we want to avoid the separability hypothesis and to recover f(E,Ω)
directly from the X-ray spectra by assuming the most general anisotropic form
for the integral kernel.

**REFERENCES**

[1] Bertero M., De Mol C. and Pike E. R. 1985, Linear Inverse Problems with
discrete data: I - General formulation and syngular system analysis, *Inverse
Problems ***1,** 300-330

[2] Bertero M., De Mol C. and Pike E. R. 1988, Linear Inverse Problems with
discrete data: II - Stability and regularization, *Inverse Problems*
**4,** 573-594

[3] Brown J. C. 1971, The deduction of energy spectra of non-thermal electrons
in flares from the observed dynamic spectra of hard X-ray bursts, *Solar
Physics* ** 18,** 489

[4] Brown J. C., Emslie A. G. and Kontar E. P. 2003, The determination and use
of mean electron flux spectra in solar flares, *ApJL* ** 595**,
L115-L117

[5] Gluckstern R. L. and Hull M. H. 1953, Polarization Dependence of the
Integrated Bremsstrahlung Cross Section, *Physical Review* **90**,
1030

[6] Holman G. D., Sui L., Schwartz R. A. and Emslie A. G. 2003, Electron
Bremsstrahlung Hard X-Ray Spectra, Electron Distributions and Energetics in the
2002 July 23 Solar Flare, *ApJL* **595**, L97

[7] Koch H. W. and Motz J. W. 1959, Bremsstrahlung Cross-Section Formulas and
Related Data, *Rev. Mod.* *Phys.* **31**, 920-955

[8] Johns C. M. and Lin R. P. 1992, The Derivation of Parent Electron Spectra
from Bremsstrahlung Hard X-ray Spectra, *Solar Physics* ** 137**,
121

[9] Massone A. M., Piana M., Conway A. and Eves B. 2003, A regularization
approach for the analysis of RHESSI X-ray spectra, *A&A* **405**,
325-330

[10] Massone A. M., Emslie A. G., Kontar E. P., Piana M., Prato M. and Brown J.
C. 2004, Anisotropic bremsstrahlung emission and the form of regularized
electron flux spectra in solar flares, *ApJ* **613**, 1233-1240.

[11] Piana M. 1994, Inversion of bremsstrahlung spectra emitted by solar plasma,
*A&A* **288**, 949-959

[12] Piana M., Massone A. M., Kontar E., Emslie A. G., Brown J. C. and Schwartz
R. A. 2003, Regularized electron flux spectra in the July 23, 2002 solar flare,
*ApJL* **595**, L127-L130

[13] Tikhonov A. N. 1963, On solving ill-posed problems and method of
regularization, *Dokl. Akad. Nauk.* *USSR* **153**, 501-504

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