1. Regularized Mean and Accelerated 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)

2. From X-rays to energetic particles From X-rays to energetic particles pre-RHESSI X-ray spectra Thermal X-rays Non-thermal X-rays We want to know about particles as much as possible!

3. Constrains on accelerated electron spectra Constrains 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

4. Energy 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

5. Energy dependent photon spectral index Interval 3 (peak of the flare) Spectral index evolution:

6. 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

7. 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:

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

9. Mean Electron Spectrum: Temporal evolution Temporal evolution of the Regularized Mean Electron Spectrum (20s time intervals) 1 2 3 4 5 12 3 54 RHESSI Lightcurves 3-12keV; 12-25keV; 25-50keV; 50-300keV

10. Accelerated (injected) Electron Spectrum Temporal evolution of the Regularized Accelerated Electron Spectrum (20s time intervals) Accelerated (injected) electron spectrum for a thick-target model: 1254 3

11. There are some “odd” spectra  low energy cut-off in accelerated spectra  invalidity of purely collision transport

12. The gap in electron spectrum or albedo ? Electron spectral index Using regularization approach electron flux spectrum has been inferred (Piana, 2003) August 20, 2002 M-class flare at 8:26 UT shows 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

13. Conclusions 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.