Regularized Mean and Accelerated Regularized Mean and Accelerated Electron Flux Spectra in Solar Flares Electron Flux Spectra in Solar Flares

Eduard P. Kontar University of Glasgow

Michele Piana, Anna Maria Massone (INFM, UdR di Genova), A. Gordon Emslie (The University of Alabama

in Huntsville), and John C. Brown (University of Glasgow)

From X-rays to energetic particlesFrom X-rays to energetic particles pre-RHESSI X-ray spectra

Thermal X-raysNon-thermal X-rays

We want to know about particles as much as possible!

Constrains on accelerated electron spectraConstrains on accelerated electron spectra

And Mean electron spectra => injected (accelerated) spectra

2) From X-rays spectra => mean electron spectra:

3) Compare with WIND/3DP electron spectra at 1AU(time of arrival suggests free propagation)

1) Energy dependent spectral index analysis

Energy dependent photon spectral indexEnergy dependent photon spectral index

The derivative error calculated from noisy data set:

We will look for a function f(x) close to a given data set so that

While the second derivative has a minimum norm

Than the derivative error has much better behaviour

where

Energy dependent photon spectral indexEnergy dependent photon spectral index

Interval 3 (peak of the flare)

Spectral index evolution:

From X-rays to electron spectrum From X-rays to electron spectrum

To find an electron spectrum is to solve a least square problem:

X-ray spectrum is a convolution of a electron flux and cross-section:

and contains valuable information on electron spectrum via a system of linear equations:

where

But this problem is ill-posed and has no unique solution !

Additional constraints are needed to find a meaningful solution

Tikhonov regularization (Tikhonov,1963)Tikhonov regularization (Tikhonov,1963)

Constrained minimum problem can be solved using Lagrange multiplier method:

subject to

The solution of this problem is well-behaved and unique !

The constraints naturally follow from the physics of the problem, For example, thick-target mean electron flux is related to injected spectrum (Brown and Emslie,1988):

Leading to the following constraint:

Mean Electron Spectrum:2002-Apr-14 22:25 UT flareMean Electron Spectrum:2002-Apr-14 22:25 UT flare

Mean Electron Mean Electron Spectrum: Temporal evolutionSpectrum: Temporal evolution

Temporal evolution of the Regularized Mean Electron Spectrum (20s time intervals)

1234 5

1 2

3

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RHESSI Lightcurves3-12keV;12-25keV;25-50keV;50-300keV

Accelerated (injected) Electron Accelerated (injected) Electron SpectrumSpectrum

Temporal evolution of the Regularized Accelerated Electron Spectrum (20s time intervals)

Accelerated (injected) electron spectrum for a thick-target model:

1 2 54

3

There are some “odd” spectraThere are some “odd” spectra

low energy cut-off in accelerated spectra invalidity of purely collision transport

The gap in electron spectrum or albedo ?The gap in electron spectrum or albedo ?

Electron spectral indexUsing regularization approach electron flux spectrum has been inferred (Piana, 2003)

August 20, 2002 M-class flare at 8:26 UTshows very flat spectra (spectral index 1.2) in the range 20-40 keV

Monte-Carlo modelled albedo correction (Bai & Ramaty, 1978)

Without correction the spectral index is too small to be explained by collisional losses

ConclusionsConclusions

Regularized inversion gives us model-independent (without assumptions on functional shape of the spectra) mean electron flux independent and can detect features not predicted by current models.

Provides us with information about high energy part of the spectrum above maximum photon energy.

In case of collisional transport of electrons, accelerated electron spectrum can be obtained.