The spatial, spectral and polarization properties of solar flare X-ray sources Natasha Louise Scarlet Jeffrey, M.Sci. Astronomy and Astrophysics Group School of Physics and Astronomy Kelvin Building University of Glasgow Glasgow, G12 8QQ Scotland, U.K. arXiv:1412.8163v7 [astro-ph.SR] 6 Mar 2015 Presented for the degree of Doctor of Philosophy The University of Glasgow March 2014 This thesis is my own composition except where indicated in the text. No part of this thesis has been submitted elsewhere for any other degree or qualification. Copyright c 2014 by Natasha Jeffrey 17th March 2014 For my parents, James and Catherine Jeffrey. Abstract X-rays are a valuable diagnostic tool for the study of high energy accelerated electrons. Bremsstrahlung X-rays produced by, and directly related to, high energy electrons accelerated during a flare, provide a powerful diagnostic tool for determining both the properties of the accelerated electron distribution, and of the flaring coronal and chromospheric plasmas. This thesis is specifically concerned with the study of spa- tial, spectral and polarization properties of solar flare X-ray sources via both modelling and X-ray observations using the Ramaty High Energy Solar Spectroscopic Imager (RHESSI). Firstly, a new model is presented, accounting for finite temperature, pitch angle scattering and initial pitch angle injection. This is developed to accurately infer the properties of the acceleration region from the observations of dense coronal X-ray sources. Moreover, examining how the spatial properties of dense coronal X-ray sources change in time, interesting trends in length, width, position, number density and ther- mal pressure are found and the possible causes for such changes are discussed. Further analysis of data in combination with the modelling of X-ray transport in the photo- sphere, allows changes in X-ray source positions and sizes due to the X-ray albedo effect to be deduced. Finally, it is shown, for the first time, how the presence of a photospheric X-ray albedo component produces a spatially resolvable polarization pat- tern across a hard X-ray (HXR) source. It is demonstrated how changes in the degree and direction of polarization across a single HXR source can be used to determine the anisotropy of the radiating electron distribution. Contents List of Tablesv List of Figures vi Preface xii Acknowledgements xiv 1 Introduction1 1.1 The Sun, its atmosphere and solar flares . 1 1.2 Electron and ion interactions the solar atmosphere . 6 1.2.1 Coulomb collisions . 6 1.3 Solar flare X-rays: bremsstrahlung . 9 1.3.1 Bremsstrahlung produced by a single accelerated electron . 9 1.3.2 Bremsstrahlung X-rays from a solar flare . 10 1.3.3 Electron-ion versus electron-electron bremsstrahlung . 13 1.3.4 Thermal bremsstrahlung . 13 1.3.5 Non-thermal bremsstrahlung . 14 1.4 Solar flare X-rays: photon interaction processes . 15 1.4.1 Thomson scattering . 15 1.4.2 Compton scattering . 16 1.5 Solar flare X-rays: observations . 19 1.5.1 X-ray temporal evolution of a solar flare . 20 1.5.2 The X-ray and gamma-ray solar flare energy spectrum . 20 1.5.3 X-ray imaging of a solar flare . 23 CONTENTS ii 1.5.4 Solar flare X-ray and gamma ray polarization . 28 1.5.5 X-rays from the photosphere and albedo emission . 31 1.6 Current X-ray telescopes and X-ray imaging methods . 37 1.6.1 RHESSI: instrument overview . 38 1.6.2 RHESSI imaging . 38 1.6.3 RHESSI spectroscopy and polarimetry . 42 2 The variation of solar flare coronal X-ray source sizes with energy 45 2.1 Introduction to the chapter . 45 2.2 Electron collisional transport in a cold plasma . 48 2.3 Electron transport in a hot plasma with collisional pitch angle scattering 53 2.3.1 The Fokker-Planck Equation and coefficients . 53 2.3.2 Steady-state solution . 55 2.3.3 High velocity limit . 55 2.3.4 Cold plasma limit . 55 2.3.5 Conversion to the electron flux distribution . 56 2.3.6 Derivation of the stochastic differential equations . 57 2.3.7 The low-energy limit . 61 2.4 Simulations . 63 2.4.1 Simulation input, boundary and end conditions . 64 2.4.2 Gaussian fitting and the determination of the source length FWHM 65 2.4.3 Numerical results . 70 2.5 Discussion and conclusions . 80 3 The temporal and spatial evolution of solar flare coronal X-ray sources 83 3.1 Introduction to the chapter . 83 3.1.1 Past studies of coronal loop spatial properties . 84 3.2 Chosen events with coronal X-ray emission . 85 3.2.1 Lightcurves for each event . 86 3.2.2 Imaging of each event . 88 CONTENTS iii 3.2.3 Spectroscopy of each event . 92 3.3 Spatial and spectral changes with time . 94 3.3.1 Emission measure and plasma temperature . 94 3.3.2 Loop width . 94 3.3.3 Loop length . 95 3.3.4 Loop radial position . 96 3.4 Corpulence, volume and other inferred parameters . 98 3.4.1 Loop corpulence . 98 3.4.2 Volume, number density, thermal pressure and energy density . 98 3.5 Summary and discussion . 100 3.5.1 Three temporal phases and suggested explanations for the obser- vations . 106 4 Solar flare X-ray albedo and the positions and sizes of hard X-ray (HXR) footpoints 112 4.1 Introduction . 112 4.2 The modelling of X-ray transport in the photosphere . 114 4.2.1 The modelling of a hard X-ray footpoint source . 115 4.2.2 X-ray transport and interaction in the photosphere . 115 4.2.3 Photoelectric absorption . 117 4.2.4 Compton scattering . 118 4.3 The position and sizes of backscattered and observed hard X-ray sources 120 4.3.1 The moments of the hard X-ray distribution . 121 4.3.2 Resulting brightness distributions . 121 4.3.3 Changes due to hard X-ray spectral index . 125 4.3.4 Changes due to hard X-ray primary source size . 125 4.3.5 Changes due to hard X-ray anisotropy . 126 4.4 Discussion and conclusions . 128 CONTENTS iv 5 Solar flare X-ray albedo and spatially resolved polarization of hard X-ray (HXR) footpoints 130 5.1 Introduction . 130 5.2 Defining the polarization of an X-ray distribution . 132 5.3 HXR footpoint bremsstrahlung polarization . 134 5.3.1 The radiating electron distribution . 134 5.3.2 The emitted primary X-ray photon distribution . 135 5.4 Photon transport in the photosphere and changes in hard X-ray polar- ization . 137 5.4.1 Monte Carlo simulation inputs . 137 5.4.2 Photoelectric absorption and hard X-ray polarization . 139 5.4.3 Compton scattering and hard X-ray polarization . 139 5.4.4 Updating photon polarization states . 140 5.5 Integrated distribution of hard X-ray polarization . 142 5.5.1 Hard X-ray polarization and electron directivity . 142 5.5.2 Hard X-ray polarization and the high energy cutoff in the electron distribution . 143 5.6 Spatial distribution of hard X-ray polarization . 145 5.6.1 Single Compton scatter for an isotropic unpolarised source . 145 5.6.2 Anisotropic source at a height of h=1 Mm (1".4) and size of 5" 149 5.7 Discussion and conclusions . 159 6 Conclusions and final remarks 161 A Calculating the photon stepsize 170 Bibliography 171 Bibliography 172 List of Tables 3.1 Table showing the main parameters of Flares 1, 2 and 3. 85 List of Figures 1.1 The changing number density and temperature structure of the solar atmosphere. 4 1.2 Yohkoh soft and hard X-ray images of a flare from the 13th January 1992. 5 1.3 Diagram of a Coulomb collision between an electron and an ion. 7 1.4 A polar diagram of the angular dependent e-i bremsstrahlung cross section. 11 1.5 Diagrams of Thomson and Compton scattering . 17 1.6 The full and differential Thomson and Compton scattering cross sections plotted against X-ray energy and scattering angle. 18 1.7 GOES and RHESSI lightcurves for a flare that occurred on the 20th September 2002. 21 1.8 An example X-ray and gamma ray solar flare spectrum. 22 1.9 Four different examples of solar flare X-ray source morphologies. 24 1.10 Changes in X-ray spatial parameters with energy, for both a chromo- spheric HXR footpoint (top) and coronal X-ray source (bottom). 27 1.11 The degree of polarization plotted against X-ray emission angle for dif- ferent energies. 29 1.12 The azimuthal X-ray emission angle plotted against the polar X-ray emission angle, showing the changing polarization angle with loop tilt. 30 1.13 Solar flare polarization measurements from RHESSI. 31 1.14 A cartoon of solar flare X-ray interactions in the photosphere, after being emitted from the chromosphere. 33 LIST OF FIGURES vii 1.15 X-ray albedo reflectivity and an X-ray spectrum with and without an albedo contribution, calculated using a Green's function. The figure also shows a real flare X-ray spectrum observed with RHESSI before and after albedo correction. 35 1.16 The varying magnitude of polarization for a completely isotropic source viewed at different locations on the solar disk due to the presence of a photospheric backscattered albedo component. 36 1.17 Measured X-ray anisotropy from RHESSI observations using two differ- ent methods that take advantage of the X-ray albedo component. 37 1.18 Diagram of the RHESSI grids and detectors.