Electron flux maps of solar flares: a regularization approach to RHESSI imaging spectroscopy

Transcription

1 Electron flux maps of solar flares: a regularization approach to RHESSI imaging spectroscopy Anna Maria Massone CNR-INFM LAMIA Genova Italy massone@ge.infm.it

2 People involved In collaboration with : Michele Piana Dipartimento di Informatica Università di Verona Italy Marco Prato Dipartimento di Matematica Pura ed Applicata di Modena e Reggio Emilia Italy Gordon Emslie Dep. of Physics Olahoma State University US Gordon Hurford Space Sciences Laboratory University of California at Bereley US Richard Schwartz NASA Goddard Space Flight Center Greenbelt US Eduard Kontar Dep. of Physics & Astronomy The University Glasgow UK

3 RHESSI Mission Reuven Ramaty High Energy Solar Spectroscopic Imager The RHESSI mission has been launched by NASA on February in order to understand the high-energy processes at the core of the solar flare phenomena : A solar flare is the most energetic explosion in the solar system. The energy released during a flare is typically on the order of erg per second ~10 19 KW. Large flares can emit up to erg of energy. Significant electromagnetic emission particularly in the X-ray range. Scientific objective: to study the processes of electron and ion acceleration in solar flares through the hard X-ray and gamma-ray radiation that the accelerated particles produce

4 RHESSI observations High spatial resolution X-ray images arc-sec-quality images image restoration: CLEAN MEM forward-fitting High energy resolution X- and γ-ray spectra 1 ev spectral resolution energy range: 1 ev 17 MeV Imaging Spectroscopy High-resolution spectroscopy at each point of the X-ray image

5 Photon space vs Electron space X-ray emission: Bremsstrahlung e e + H X-ray RHESSI spectra/images/imaging spectroscopy 1. Photon spectra electron spectra inversion of the Bremsstrahlung eq. 2. Photon images local photon spectra local electron spectra Is it possible to build images in the electron space?

6 RHESSI Hardware nine Ge detectors nine bi-grid collimators planar array of equally spaced X-ray-opaque slats separated by transparent slits nine different widths for the slit

7 RHESSI Geometry θ is the incident angle between the photons and the collimator axis duration of a complete RHESSI rotation: 4s As the spacecraft rotates imaging information is encoded as rapid time-variations of the detected flux

8 Modulation profiles point source smaller intensity different θ more off-axis source distribution bigger source distribution real profile

9 For each detector: Roll angle and aspect phase roll angle α: defines the grid direction with respect to a coordinate system y sun { x y sun sun} aspect phase β : measures the position of a reference point near the source with respect to the spacecraft axis α β Spacecraft axis x sun β α q t α β t α β q Data stacing

10 Data stacing I RHESSI data are light curves i.e. photon-induced counts recorded while the collimators rotate counts counts counts For each time point a roll angle and a phase are defined roll angle deg phase deg

11 Data stacing II For different rotations different phases correspond to the same roll angle The counts corresponding to the same roll and phase bin are staced in a histogram 32 roll bins 12 phases

12 Data stacing III For each roll bin the count profile as a function of the phase bin is fitted by a Fourier series If A and B are the first two Fourier components the complex number whose real part is A and imaginary part is B is called visibility

13 Formal definition: Visibilities I A RHESSI visibility is a complex observable number that can be derived from RHESSI data and which represents a measurement of a single Fourier component of the source distribution measured at a specific spatial frequency and energy- and time-range. where V u v q j = x y uv spatial frequency components xy point in the source Ixy;ε Vuv;q observed visibility at energy q Dq j ε Detector Response Matrix DRM D q j ε 2πi ux+ vy photon spectrum at point xy and energy ε I x y ε e ε dxdy

14 + = y x vy ux πi j j dxdy ε e ε y x I ε q D q v u V 2 = ε de E ε Q E y x F y x N πr ε y x I ; 4 1 ; 2 = q de E q K E v u W q v u V ; ; + = y x vy ux πi dxdy e E y x F y x N E v u W 2 ; : ; j j E Q q D R E q K ε ε ε π = 4 1 : 2 Count Count Electron Visibilities I Electron Visibilities I Electron Visibilities

15 Count Electron Visibilities II q V u v; q = W u v; E K q E de The relation between the measured count visibilities and the electron visibilities is described by a Volterra integral equation of the first ind Visibility inversion problem: determine the electron visibilities Wuv;E from the observed count visibilities Vuv;q

16 q Visibility inversion AW u v; q = W u v; E K q E de AW = V Solution strategy: Tihonov regularization method The Tihonov Method: 1. Solve the minimum problem 2 2 AW -V + λ W = min fidelity smoothness 2. Fix λ by means of some optimality criterion The Singular Value Decomposition of the operator A gives the set of triplets: { σ } N ;u v = 1 The regularized solution: Wλ = N = 1 σ σ g v 2 + λ u

17 The algorithm For each detector and each uv point: construct the count visibility spectrum count visibility vs count energy apply regularized inversion to obtain an electron visibility spectrum electron visibility vs count energy For each detector and each electron energy: construct the electron visibilities Selected flare: 2002 February 20-11:02:08 11:14:20 UT Time range selected: 11:06:02 11:06:34 UT Count energy range selected: ev Count energy binning selected: 4 ev

18 RHESSI data: count visibilities Fixed energy channel [qq+ q] Detector 1 u 1 1 v 1 1 ReVu 1 1 v 1 1 ;q ImVu 1 1 v 1 1 ;q u 32 1 v 32 1 ReVu 32 1 v 32 1 ;q ImVu 32 1 v 32 1 ;q Detector 9 u 1 9 v 1 9 ReVu 1 9 v 1 9 ;q ImVu 1 9 v 1 9 ;q u 32 9 v 32 9 ReVu 32 9 v 32 9 ;q ImVu 32 9 v 32 9 ;q

19 Visibility-based based count maps Imaging from visibilities: Maximum Entropy Method MEM ev ev ev ev ev ev ev ev ev ev

20 Count visibility spectra For each detector and each uv point: construct the count visibility spectrum count visibility vs count energy Spatial frequencies u i v i ReVu i v i q 1 ImVu i v i q 1 ReVu i v i q N ImVu i v i q N

21 Electron visibility spectra Spatial frequencies u i v i + j j j j V u v ; q = W u v ; E K q E de i i q apply regularized inversion to obtain an electron visibility spectrum electron visibility vs count energy i Electron visibilities reach energies higher than photon visibilities thans to bremsstrahlung i ReWu i v i E 1 ImWu i v i E 1 ReWu i v i E M ImWu i v i E M Electron energy range ev

22 Electron visibilities Fixed energy channel [EE+ E] Detector 1 u 1 1 v 1 1 ReWu 1 1 v 1 1 ;E ImWu 1 1 v 1 1 ;E u 32 1 v 32 1 ReWu 32 1 v 32 1 ;E ImWu 32 1 v 32 1 ;E Detector 9 u 1 9 v 1 9 ReWu 1 9 v 1 9 ;E ImWu 1 9 v 1 9 ;E u 32 9 v 32 9 ReWu 32 9 v 32 9 ;E ImWu 32 9 v 32 9 ;E

23 Visibility-based based electron maps Imaging from visibilities: Maximum Entropy Method MEM ev ev ev ev ev ev ev ev ev ev

24 Visibility-based based electron maps Imaging from visibilities: Maximum Entropy Method MEM ev ev ev ev ev ev ev ev ev ev

25 Photon maps vs electron maps ev ev ev ev ev

26 Photon maps vs electron maps ev ev ev ev ev

27 Photon maps vs electron maps ev ev ev ev ev

28 Photon maps vs electron maps ev ev ev ev