Plasma Diagnostics and Hydrodynamic Evolution of Solar Flares

Home , List of solar deities, Solar transition region, Standard Solar Model

Daniel F. Ryan, B. A. (Mod.) School of Physics University of Dublin, Trinity College

A thesis submitted for the degree of PhilosophiæDoctor (PhD)

2014 ii Declaration

I, Daniel F. Ryan, hereby certify that I am the sole author of this thesis and that all the work presented in it, unless otherwise referenced, is entirely my own. I also declare that this work has not been submitted, in whole or in part, to any other university or college for any degree or other qualiﬁcation.

The thesis work was conducted from October 2009 to October 2013 under the supervision of Dr. Peter T. Gallagher at Trinity College, University of Dublin.

In submitting this thesis to the University of Dublin I agree to deposit this thesis in the University’s open access institutional repository or allow the library to do so on my behalf, subject to Irish Copyright Legislation and Trinity College Library conditions of use and acknowledgement.

Name: Daniel F. Ryan

Signature: ...... Date: ...... Summary

Solar flares are among the most powerful events in the solar system with the ability to damage satellites, disrupt telecommunications and produce spectacular aurorae. They are believed to occur when energy is rapidly released from highly stressed magnetic fields in the solar corona. Part of this energy heats the coronal plasma to millions of kelvin resulting in plasma flows and electromagnetic emission, among other things. However, despite decades of research, the evolution of these eruptive events is still not fully understood. In this thesis, we examine the thermo- and hydrodynamic evolution of solar flares and develop plasma diagnostics to better study them. To date, the study of the thermo- and hydrodynamic evolution of solar flares has been dominated by studies of single or small samples of events. In this thesis we develop an automatic background subtraction algorithm for GOES/XRS observations, the Temperature and Emission measure-Based Background Subtraction (TEBBS). This allows the thermal properties of large numbers of solar flares to be analysed quickly and accurately, which permits flares to be studied in a statistically meaningful way. As part of this work, we analyse over 50,000 flares in the period 1980–2007 and create an online database of flare thermal properties for use by the solar physics community. The TEBBS method is then used in subsequent studies of ensembles of solar flares. The first compares the peak temperatures of 149 flare DEMs (Differential Emission Measure distributions) calculated using SDO/AIA with those determined with GOES/XRS and RHESSI using the isothermal assumption. It is found that the isothermal assumption leads to overestimates of the DEM peak temperature in GOES/XRS and RHESSI observations and hence the resulting isothermal temperature biases are quantified. We also find from a discrepancy between predicted and observed RHESSI biases that accurate flare DEMs must be determined by simultaneous fitting EUV (SDO/AIA) and SXR (GOES/XRS and RHESSI) fluxes by an appropriately parameterised function, e.g. an asymmetric bi-Gaussian.

Finally, GOES/XRS and SDO/EVE are used to chart the cooling of 72 flares and the observations are compared to a simple hydrodynamic flare cooling model. The model is found to provide a well-defined lower limit to the observed cooling time of a flare, but does not well fit the distribution. The discrepancies between the model and observations are assumed to be due to additional heating which is then compared to the flares’ overall thermal energies. It is found that the heating required is physically plausible, typically making up about half of the thermally-radiated energy as determined by GOES/XRS. This suggests that the energy released during a flare’s decay phase is just as significant as that released during its impulsive phase.

The work outlined in this thesis sheds light on coronal plasma diagnostics and the thermo- and hydrodynamic evolution of solar flares. It demonstrates the importance of examining an ensemble of events in order to put the detailed results of single event studies into context and also give statistical significance to such results. The results outlined here would be useful in finding new ways of testing more advanced hydrodynamic flare models and developing a more comprehensive understanding of the evolution of solar flares.

To my brother, Cormac, the greatest example of perseverance and triumph

and

to my Father per ardua ad astra

Acknowledgements

Firstly, I would like to acknowledge the Irish Research Council, the Ful- bright Association and NASA’s Living With a Star Targeted Research and Technology Program for funding the research contained in this thesis.

I would like to thank my supervisor, Prof. Peter Gallagher for giving me the opportunity to do a PhD. Thanks for his invaluable guidance, support and understanding throughout these four years. Thanks also to Dr. Ryan Milligan and Dr. Phil Chamberlin, both of whom supervised me during my times at NASA/GSFC and continued to support me since becoming involved in my research.

I would like to thank my collaborators at NASA/GSFC: Dr. Brian Dennis, Richard Schwartz, Kim Tolbert, and Dr. Alex Young for their help and support and for making me at home while I was in America. In addition, thanks to Dr. Markus Aschwanden at LMSAL for his invaluable insight and encouragement during our collaboration as well as Aidan O’Flannagain for his very helpful contribution to that same work.

I would also like to thank Dr. David Pérez-Suárezwho helped so much with creating the TEBBS website as well as Dr. Shaun Bloomfield for his willingness to help whenever asked.

Many thanks to all the members of the Astrophysics Research Group during the time I was there for the great atmosphere and support which made being a PhD student such as pleasure.

Last but not least, thanks to my close friends and my family, my parents for raising me and my brothers for being my brothers.

Publications

Refereed

1. Ryan, D. F., O’Flannagain, A. M., Aschwanden, M. J., Gallagher, P. T. The Compatibility of Flare Temperatures Observed with AIA, GOES and RHESSI Solar Physics, 289, 2547, 2014

2. Ryan, D. F., Chamberlin, P. C., Milligan, R. O., Gallagher, P. T. Decay Phase Cooling and Heating of M- and X-class Solar Flares, Astrophysical Journal, 778, 68, 2013

3. Bloomfield, D. S., Gallagher, P. T., Maloney, S. A., Pérez-Suárez,D., Higgins, P. A., Carley, E. P., Long, D. M., Murray, S. A., O’Flannagain, A., Ryan, D. F., and Zucca, P. A Comprehensive Overview of the 2011 June 7 Solar Storm, Astronomy & Astrophysics, in review, 2012

4. Ryan, D. F., Milligan, R. O., Gallagher, P. T., Dennis, B. R., Tolbert, A. K., Schwartz, R. A., Young, C. A. Thermal Properties of Solar Flares Over Three Solar Cycles Using GOES X-ray Observations, Astrophysical Journal Supplemental Series, 202, 11, 2012

ix 0. PUBLICATIONS

x Contents

Publications ix

List of Figures xv

List of Tables xxix

Glossary xxxi

1 Introduction 1 1.1 Internal Structure ...... 4 1.2 The Solar Atmosphere ...... 11 1.2.1 The Photosphere ...... 12 1.2.2 The Chromosphere & Transition Region ...... 13 1.2.3 The Corona ...... 16 1.3 The Sun’s Magnetic Field & the Solar Cycle ...... 19 1.4 Active Regions ...... 26 1.5 Solar Flares ...... 30 1.5.1 The CSHKP Flare Model ...... 33 1.6 Thesis Outline ...... 38

2 Theory 41 2.1 Atomic Physics ...... 42 2.1.1 Continuum emission ...... 44 2.1.1.1 Thermal Bremsstrahlung ...... 45 2.1.2 Emission Lines ...... 48 2.1.2.1 Atomic Structure ...... 50 2.1.2.2 Modelling Emission Line Flux in the Corona ...... 52

xi CONTENTS

2.1.3 Contribution Functions & Emission Measures ...... 56 2.1.4 CHIANTI Atomic Database ...... 58 2.1.5 Radiative Loss Function ...... 58 2.2 Hydrodynamics ...... 60 2.2.1 Plasma Kinetic Theory ...... 60 2.2.2 Equations of Hydrodynamics ...... 65 2.2.3 Flare Cooling Models ...... 67 2.2.4 The Cargill Flare Cooling Model ...... 69

3 Instrumentation 75 3.1 Geostationary Operational Environmental Satellite (GOES) ...... 76 3.1.1 The X-Ray Sensor (XRS) ...... 77 3.1.2 Deriving Thermal Plasma Properties Using GOES/XRS . . . . . 80 3.2 Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) . . 85 3.2.1 The RHESSI Instrument ...... 86 3.2.2 Deriving Thermal Plasma Properties Using RHESSI ...... 89 3.3 Hinode ...... 91 3.3.1 X-Ray Telescope (XRT) ...... 91 3.4 Solar Dynamics Observatory (SDO) ...... 94 3.4.1 Atmospheric Imaging Assembly (AIA) ...... 95 3.4.2 EUV Variability Experiment (EVE) ...... 98 3.4.2.1 Multiple EUV Grating Spectrograph-A (MEGS-A) . . . 99

4 Thermal Properties of Solar Flares Over Three Solar Cycles 103 4.1 Introduction ...... 104 4.2 Observations ...... 107 4.2.1 The GOES Event List ...... 108 4.3 Background Subtraction Method ...... 111 4.3.1 Previous Background Subtraction Methods ...... 112 4.3.2 Temperature and Emission measure-Based Background Subtrac- tion (TEBBS) ...... 116 4.4 Results ...... 125 4.5 Discussion ...... 133 4.6 Conclusions & Future Work ...... 138

xii CONTENTS

5 Comparison of Multi-Instrument Temperature Observations 143 5.1 Introduction ...... 144 5.2 Data Analysis ...... 146 5.2.1 SDO/AIA Measurements ...... 146 5.2.2 GOES/XRS Measurements ...... 152 5.2.3 RHESSI Measurements ...... 153 5.3 Discussion ...... 156 5.3.1 The GOES Temperature Bias ...... 156 5.3.2 The RHESSI Temperature Bias ...... 162 5.4 Conclusions ...... 167

6 Decay Phase Cooling & Inferred Heating of Solar Flares 171 6.1 Introduction ...... 172 6.2 Observations & Data Analysis ...... 174 6.2.1 Flare Sample ...... 174 6.2.2 Observing Flare Cooling ...... 175 6.3 Modelling ...... 179 6.4 Results & Discussion ...... 185 6.4.1 Comparing Observed and Modelled Cooling Times ...... 185 6.4.2 Inferring Heating During Decay Phase ...... 187 6.5 Conclusions ...... 192

7 Conclusions and Future work 199 7.1 Principal Results ...... 200 7.2 Future Work ...... 202 7.2.1 Applying TEBBS to Future Studies ...... 202 7.2.2 Improving TEBBS Algorithm ...... 203 7.2.3 Extending the TEBBS database ...... 204 7.2.4 Constraining the High-Temperature Tails of DEMs ...... 204 7.2.5 Testing More Advanced Hydrodynamic Flare Models ...... 207 7.3 Conclusion ...... 210

A GOES Saturation Levels 211

xiii CONTENTS

B Cooling Derivations 213 B.1 Cooling due to Conduction ...... 213 B.2 Cooling due to Radiation ...... 216

References 221

xiv List of Figures

1.1 Diagram showing the various branches of the pp-chain and their occurrence rates. (Carroll & Ostlie, 1996) ...... 5 1.2 Cut-away cartoon of the Sun’s interior showing the core, the radiative zone and the convection zone. It also shows the diﬀerent layers of the atmosphere, the photosphere, the chromosphere, and the corona. . . . .7 1.3 Plots of temperature and pressure (top panel) and density and cumulative mass (bottom panel) as a function of distance from the Sun’s centre as predicted by the Standard Solar Model. Adapted from Carroll & Ostlie (1996) ...... 9 1.4 Modelled temperature and density of the solar atmosphere with height above the photosphere. (Aschwanden, 2004) ...... 11 1.5 Observed spectrum of the Sun’s emission. At wavelengths longer than 103 A,˚ the spectrum closely resembles a blackbody of ∼5770 K, corresponding to the photosphere. The deviation at short wavelength is due to high energy emission from the upper layers of the Sun’s atmosphere (chromosphere and corona). (Aschwanden, 2004)...... 13 1.6 Full disk images taken with SDO/AIA of the photosphere at 4500 A(˚ top), the chromosphere & transition region at 304 A(˚ left) and the corona at 193 A(˚ right). Images courtesy of helioviewer.org ...... 14 1.7 Image of granulation on the photosphere. Courtesy of the National Op- tical Astronomy Observatory...... 15 1.8 The Sun’s corona in visible light during a total solar eclipse. This reveals its complex and highly non-spherical structure, largely determined by the solar magnetic ﬁeld...... 17

xv LIST OF FIGURES

1.9 Top: Sunspot number with time over the past 400 years taken from averages of observations from around the globe (blue). It can be seen that the sunspot number has an approximate 11-year periodicity. Prior to 1750 (red), observations were sporadic. This period includes the Maunder Minimum (1650–1700) when there appeared to be almost no sunspots at all. Bottom: ‘Butterﬂy diagrams’ for the period 1870 –2010. This shows the the total sunspot area in equally spaced latitude bands (as a percentage of the latitude band area) as a function of time. From this it can be seen that at the beginning of each solar cycle sunspots emerge at high latitudes (∼30o). But as time goes on, they emerge at lower and lower latitudes. Courtesy of NASA...... 20

1.10 Number of solar flares per month (B-, C-, M-, and X-class) as a function of time as recorded in the GOES (Geostationary Operational Environ- mental Satellite; Section 3.1) flare list for the period 1980–2008. The flare class refers to the order of magnitude of the peak flux in the 1– 8 A˚ GOES channel: 10−7 W m−2 (B-class) to 10−4 W m−2 (X-class). Note the approximate 11-year periodicity, just as in the sunspot cycle in Figure 1.9. Data courtesy of NOAA...... 22

1.11 Diagram of the Sun’s initially dipolar magnetic field (left) being wound up by differential rotation into a quadrupolar field (center), eventually leading to the emergence of magnetic field at low latitudes via the α-Ω effect and creating active regions and sunspots (right) (Carroll & Ostlie, 2006)...... 23

1.12 Diagram of the α-Ω eﬀect with time increasing from the bottom schematic to the top. Adapted from Babcock (1961)...... 24

1.13 Three images taken by Hinode/SOT of a flaring active region. Top: Mag- netogram showing the photospheric line of sight magnetic field strength polarity. Middle: Sunspot in the photosphere taken in the G-band. Bot- tom: Ca II image showing the chromosphere and flaring arcade. Images courtesy of JAXA/NASA...... 27

xvi LIST OF FIGURES

1.14 Cross-section of the magnetic topology of a sunspot. The magnetic field (arrow-headed lines) can be seen emerging through the photosphere (black horizontal line with no arrow-head) and up into the solar atmosphere. The magnetic field spreads out above the photosphere due to the reduced pressure of the tenuous atmosphere. (Parker, 1955). . . . . 28 1.15 Coronal loops imaged by TRACE...... 29 1.16 A diagram of the progression of magnetic reconnection...... 31 1.17 A diagram of magnetic islands forming in a current sheet due to a tearing- mode instability. (Aschwanden, 2004)...... 32 1.18 Diagram of the standard flare model. Adapted from Dennis & Schwartz (1989)...... 34 1.19 Time profiles of an M1.8 flare which occurred on 2002 April 10 at 19:00 UT. a) RHESSI count rate in the 6–12 keV, 12–25 keV and 25– 50 keV ranges. b) GOES temperature. c) GOES flux in the 1–8 A˚ and 0.5–4 A˚ passbands. d) GOES emission measure...... 37

2.1 Model solar spectrum from 1 – 400 A˚ created using the CHIANTI atomic physics database (Landi et al., 2012). The inﬂuence of both emission lines and continua can clearly be seen...... 43 2.2 Contributions from free-free, free-bound, and two-photon continua to the solar spectrum in the range 1 – 300 A˚ (EUV/X-ray regime). Calculated with CHIANTI by Raftery (2012)...... 44 2.3 Diagrams of a) free-bound and b) free-free emission processes. Adapted from Aschwanden (2004)...... 45 2.4 Bohr model of the atom showing a positive nucleus of protons and neutrons surrounded by electrons of discrete energies/orbits described by the principal quantum number, n. (Suchocki, 2004)...... 49 2.5 Diagram of an electron decaying from an upper atomic orbit to a lower one with the emission of a photon with an energy equal to the diﬀerence between the two levels. (Raftery, 2012)...... 51 2.6 Schematic of the various excitation (top row) and de-excitation (bottom row) processes which can occur in the corona. Adapted from Aschwan- den (2004)...... 53

xvii LIST OF FIGURES

2.7 Contributions functions of He I, (584.33 A),˚ O V (629.73 A),˚ Mg X (524.94 A),˚ Fe XVI (360.75 A),˚ and Fe XIX (592.23 A).˚ These were calculated with 9 −3 the CHIANTI software (Section 2.1.4) using density, ne = 5×10 cm , coronal abundances and the ionisation equilibria of Mazzotta et al. (1998). Taken from Raftery (2012)...... 57 2.8 Calculations of the radiative loss function, Λ(T ), compiled from various studies. (Aschwanden, 2004)...... 59 2.9 The Maxwell-Boltzmann distribution, showing the distribution of velocities among particles in a gas or plasma in thermal equilibrium. (Inan & Golkowski, 2011) ...... 61

2.10 Volume element, dxdvx, in position/velocity phase space showing the ways in which particles can enter and leave that volume element. Parti- cles entering/leaving the volume at side ‘3’ and ‘4’ move in or out of the spatial range x – x + dx due to their position. Particles entering/leaving at sides ‘1’ or ‘2’ are accelerated or decelerated in or out of the range

vx – vx + dvx by an external force, e.g. the Lorentz force. Also shown are particles accelerated/decelerated into the volume element via collisions. This picture is very useful in deriving the Boltzmann equation which describes how the velocity distribution evolves with time. (Inan & Golkowski, 2011) ...... 63

3.1 Diagram of GOES satellite ...... 76 3.2 Schematic of the GOES-8 XRS. (Hanser & Sellers, 1996)...... 77 3.3 Response functions against wavelength for the long and short channels of the XRS for the first 12 GOES satellites. (White et al., 2005) . . . . 78 3.4 Lightcurves on the long (red) and short (blue) XRS channels from the GOES-15 satellite for a period of three days in August 2013. Flares can be seen as spikes in the lightcurves on top of a background level of approximately B4 GOES-class. The variation in solar activity can be seen by comparing August 11 which exhibits many flares, while August 12, apart from the large M1-class flare, shows very little activity. Courtesy of SolarMonitor.org ...... 79

xviii LIST OF FIGURES

3.5 Relationship between temperature and XRS flux ratio as determined by Thomas et al. (1985)...... 82 3.6 Relationship between temperature-dependent part of the XRS response and temperature as determined by Thomas et al. (1985)...... 83 3.7 Image of the RHESSI satellite. Courtesy of NASA...... 85 3.8 Schematic of the RHESSI instrument. Left: the RHESSI rotation modulation collimators (RMCs). Right: the RHESSI spectrometer including the nine germanium detectors (GeDs). Taken from Hurford et al. (2002). 87 3.9 RHESSI spectrum taken around the peak of a C3.0 flare which occurred on 2002 March 26. The observations are denoted by the crosses which are fitted with thermal (solid line), non-thermal (dashed line), and background components (dot-dashed line). The feature around 10 keV in the background component is due to the excitation of a germanium line in the germanium detectors themselves. The bottom panel shows the residuals of the fit. (Raftery et al., 2009)...... 90 3.10 Illustration of the Hinode satellite and main components: XRT, EIS, and SOT (OTA and FPP). Figure courtesy of NASA...... 92 3.11 A simple schematic showing the use of grazing incidence in a telescope such as Hinode/XRT. Each mirror is arranged at a slight angle to the path of the incoming X-rays. As the X-rays are successively reflected by each mirror, their trajectories are increasingly altered from their original ones until the X-rays can be directed onto the focal point of the telescope. 93 3.12 Response functions as a function of temperature for the various filters on Hinode/XRT...... 93 3.13 Illustration of the Solar Dynamics Observatory highlighting its three instruments: the Atmospheric Imaging Assembly (AIA); the EUV Vari- ability Experiment (EVE); and the Helioseismic and Magnetic Imager (HMI). Courtesy of NASA...... 94 3.14 Image of one of AIA’s primary mirrors. Each half of the mirror has a different reflective coating to reflect a different passband. The mirror contains a hole in the middle through which the secondary mirror reflects the light onto the CCD (Cassegrain design). (Lemen et al., 2012). . . . 95

xix LIST OF FIGURES

3.15 The arrangement of the AIA telescopes and the diﬀerent passbands

within them. The top and bottom halves of the primary mirror of each

telescope have different coatings to reflect different passbands. The ex-

ception is the top half of telescope 3’s primary mirror, which is coated to

reﬂect broadband UV containing the 1600 A,˚ 1700 A,˚ and 4500 A˚ chan-

nels. These channels are then separated by a ﬁlter wheel just in front

of the CCD. The guide telescopes can be seen above each of the main

telescopes (to the right of each number label) and help with image sta-

bilisation. (Lemen et al., 2012)...... 97

3.16 Cross-section AIA’s telescope 2. (Lemen et al., 2012)...... 97

3.17 Temperature responses of the six EUV AIA channels. (Lemen et al., 2012). 98

3.18 Diagram of SDO/EVE with each instrument labelled: Solar Aspect

Monitor (SAM); Multiple Extreme-ultraviolet Grating Spectrograph A

(MEGS-A); Multiple Extreme-ultraviolet Grating Spectrograph B (MEGS-

B); Extreme-ultraviolet SpectroPhotometer (ESP)...... 99

3.19 Diagram of the SDO/EVE MEGS-A optical layout. The light enters

the door and passes through the ﬁlter which only transmits light in the

range 6–37 nm. It is then reﬂected and refracted oﬀ the A grating. This

disperses and focusses the light onto the CCD detector, thus creating

the 0.1 nm spectral resolution...... 100

xx LIST OF FIGURES

3.20 Top Panel: a sample solar spectrum with the spectral range of SDO/EVE MEGS-A highlighted in white. Bottom panel: the same sample solar spectrum as it would appear on the MEGS-A CCD. The top left quadrant represents the 6–18 nm range of the spectrum transmitted by the A1 slit. The top right quadrant shows higher-order photons (harmonics) in the range 18–37 nm transmitted by the A1 slit. The reason that the A1 and A2 slits focus light onto diﬀerent halves of the CCD is that these higher order photons would cause inaccuracies in the 18–37 nm section of the spectrum. The bottom right quadrant shows the sample spectrum above in the range 18–37 nm transmitted by the A2 slit. The image of the solar disk in the bottom left quadrant is created by the Solar Aspect Monitor (SAM). To generate the ﬁnal MEGS-A spectrum free of higher order artifacts, the A1 spectrum in the top left quadrant is combined with the A2 spectrum in the bottom right quadrant...... 101

4.1 X-ray lightcurves of an M1.0 solar flare observed by GOES. a) X-ray flux in each of the two GOES channels (0.5–4 A;˚ dotted curve and 1–8 A;˚ solid curve). b) The derived temperature curve. c) The derived emission measure curve. The vertical dotted and dashed lines denote the defined start and end times of the event, respectively. The vertical red, black and green lines mark the times of the peak temperature, peak 1–8 A˚ flux, and peak emission measure, respectively...... 109

4.2 Schematic of a flare X-ray lightcurve showing how the total flux detected by, for example, the GOES XRS, is divided into constituent components. (Adapted from Bornmann 1990). The total flux (solid line) is the sum of the flux from the flare plus the solar background (divided by the dashed line). The pre-flare flux, however, is the sum of the background component and the quiescent component of the flaring plasma (e.g., the associated active region)...... 111

xxi LIST OF FIGURES

4.3 GOES lightcurves and associated temperature and emission measure profiles for a B7 flare which occurred on 1986 January 15. The profiles in Figures 4.3a–4.3d are not background-subtracted. The profiles in Fig- ures 4.3e–4.3h have had the pre-flare flux in each channel subtracted, while Figures 4.3i–4.3l show the profiles obtained using the TEBBS method. The error bars represent the uncertainty quantified via the range of background subtractions found acceptable by TEBBS...... 113

4.4 GOES XRS lightcurves from 1986 January 15 06:35–10:55 UT. The start and end times of the B7 ﬂare shown in Figures 4.3 and 4.5 as deﬁned by the GOES event list are marked by the dashed and dot-dashed vertical lines respectively...... 117

4.5 Short channel flux versus the long channel flux for the 1986 January 15 B7 flare (solid curve). The grey shaded area in the bottom left hand corner represents the possible combinations of background values from each channel for this event. The orange line represents a linear least- squares fit to the latter five sixths of the rise phase (duration). The first sixth is excluded because significant increases are often not seen directly after the GOES start time as can be observed from the fit’s proximity to the minimum of the data...... 119

4.6 Sample background space for 1986 January 15 ﬂare. The black shaded areas illustrate the range of values which pass a given background test, while the hashed regions denote background values which fail: a) the hot ﬂare test; b) the increasing temperature test; c) the increasing emission measure test; and d) points which passed all three, or failed one or more. 121

4.7 Temperature and emission measure profiles for the 1986 January 15 flare for all possible background combinations. The left column shows profiles which passed all three tests, while the right column shows profiles which failed one or more tests...... 122

xxii LIST OF FIGURES

4.8 2D histograms of peak temperature, peak emission measure, and total radiative losses, as a function of peak long channel flux, for all selected GOES events between 1980 and 2007. The data in each column have had different background subtractions applied: no background subtracted (left), pre-flare flux subtracted (middle), and TEBBS (right). Overplot- ted on panels c and f are relationships derived by different studies: Garcia & McIntosh (1992, long-dashed), Feldman et al. (1996b, three-dotted- dashed), Battaglia et al. (2005, short-dashed), Hannah et al. (2008, dot- dashed) and this work (Equations 4.1, 4.2 and 4.4; solid). Arrow heads mark events which are upper or lower limits due to XRS saturation and point in the directions that the true values would have been located. The crosses mark events for which flux values are a lower limit and derived properties are only rough estimates due to saturation. See Appendix A for more detail. N.B. 806 events in panel b extend beyond the vertical plot range to ≈80 MK...... 126 4.9 2D histograms of peak emission measure and total radiative losses as a function of peak temperature for all selected GOES events between 1980 and 2007. The data in each column have had different background subtraction methods applied: no background subtracted (left), pre-flare flux subtracted (middle), and TEBBS (right). Arrow heads mark events which are upper or lower limits due to XRS saturation and point in the directions that the true values would have been located. The crosses mark events for which flux values are a lower limit and derived properties are only rough estimates due to saturation. See Appendix A for more detail. N.B. 806 events in panels b and e extend beyond the horizontal plot range to ≈80 MK...... 132

5.1 Temperature-response functions for the seven coronal EUV channels of SDO/AIA, according to the status of Dec 2012. The GOES 1-8 A˚ and 0.5-4 A˚ channels are also shown (in arbitrary ﬂux units), as well as thermal energy of the lowest ﬁttable RHESSI channels at 3 keV and 6

keV. The approximate peak temperature range of large ﬂares (Tp ≈ 5−20 MK) is indicated with a thatched area...... 148

xxiii LIST OF FIGURES

5.2 Gaussian DEM ﬁts of the 149 M- and X-class ﬂares analysed with SDO/AIA.149

5.3 GOES/XRS versus SDO/AIA peak temperatures Tp (top left panel) and

peak emission measures EMp (bottom left panel). The over-plotted solid line in each of these panels represents the 1:1 relationship while the dashed line represent the average GOES/AIA ratio of the distribution. N.B. They are not ﬁts. The ﬂare peak times refer to the GOES long

channel peak time tGOES and coincides with the times tAIA of SDO/AIA measurements within the used time resolution of ≈ 1 min. See the histogram of time diﬀerences in top right panel, which has a mean and

standard deviation of (tGOES − tAIA) = 27 ± 26 s...... 151

5.4 RHESSI versus SDO/AIA peak temperatures Tp (top left panel) and

channel peak time tGOES and coincides with the times tAIA of AIA measurements within the used time resolution of ≈ 1 min. See the histogram of time diﬀerences in top right panel, which has a mean and

standard deviation of (tRHESSI − tAIA) = 23 ± 25 s...... 154 5.5 Top: The ﬁlter ratio of the GOES 0.5-4 A˚ to the 1-8 A˚ channel is shown for an isothermal DEM (thick curve) and for Gaussian DEM distribu-

tions with Gaussian widths of log10(σT ) = 0.1, ..., 1.0. The ﬁlter ratio is

B4/B8 = 0.31 for an isothermal DEM with a peak at Tp = 10 MK. For a

Gaussian DEM with a width of σT = 0.5 (dashed curve), the correspond-

ing isothermal ﬁlter-ratio corresponds to a temperature of Tp = 17 MK,

which deﬁnes a temperature bias of qGOES = Tiso/TσT = 1.7. Bottom: The temperature bias of multi-thermal DEMs with a peak temperature

at Tp(σT ) compared with the temperature Tiso of isothermal DEMs is shown as a function of the temperature and for a set of Gaussian widths

σT ...... 158 5.6 Solid lines: Numerically determined GOES temperature biases for DEM

widths of σT = 0.1 and σT = 0.9. (As in Figure 5.5). Dashed lines: Corresponding curves calculated with Equation 5.6...... 161

xxiv LIST OF FIGURES

5.7 Top: Three simulated RHESSI thermal bremsstrahlung photon spectra generated using Equation 5.9 (Brown, 1974; Dulk & Dennis, 1982). The

bottom curve is an isothermal spectrum with a temperature of Tiso = 10 MK. The top (dashed curve) is a multi-thermal spectrum with a peak

temperature of TMT = 10 MK and a Gaussian width of log10(σT ) = 0.5. And the middle curve is an isothermal spectrum that has the same ﬂux

ratio qF = F6/F12 = 3.7, which is found for Tiso = 53 MK. This corre-

sponds to a temperature bias of qRHESSI = TRHESSI /TAIA = 5.3. Bot- tom left: The RHESSI ﬂux ratio of isothermal and multi-thermal spectra

is shown as a function of the DEM peak temperature, Tp, for Gaus-

sian DEM distributions with Gaussian widths of log10(σT ) = 0.1, ..., 1.0.

The ﬂux ratio, qF = 3.7, corresponding to the case shown in the top panel is marked with dashed line. Bottom right: The temperature bias,

qRHESSI = Tiso/Tp, of isothermal DEMs with a peak temperature at Tp

is shown as a function of the peak temperature, Tp, and for a set of Gaus-

sian widths, σT . The case with a temperature bias of qRHESSI = 5.3 of the spectrum shown in the top panel is indicated with a dashed line. . . 164 5.8 Solid lines: Numerically determined RHESSI temperature biases for

DEM widths of σT = 0.1 and σT = 0.9. (As in Figure 5.7). Dashed lines: Corresponding curves calculated with Equation 5.11...... 166

6.1 Cooling track for the 2010-Nov-06 M5.5 flare which began at 15:28 UT. a) Background-subtracted GOES temperature profile. Peak is marked by the vertical line. b) – h) Lightcurves of sequentially cooler Fe lines ranging from 15.8 MK to 2 MK observed by SDO/EVE MEGS-A. The peak of each lightcurve is also marked by a vertical line. i) Combined cooling track obtained by plotting the time of the peak of each profile (including GOES temperature profile) with its associated peak temperature. The resultant cooling time is the duration of this cooling track. . 176 6.2 Histograms showing the non-linear (panel a) and linear (panel b) coefficients of the second-order polynomial fits to the observed cooling profiles of the 72 M- and X-class flares in this study (Equation 6.1)...... 178

xxv LIST OF FIGURES

6.3 Relationships between density and Fe XXI line ratios, 12.121 nm/12.875 nm, (14.214 nm + 14.228 nm)/12.875 nm, and 14.573 nm/12.875 nm, calculated using CHIANTI v7. (Milligan et al., 2012) ...... 180 6.4 Hinode/XRT observations of 22 ﬂares within this study, plotted on a

log10-scale. The blue lines trace out the plane-of-sky measured loop lengths obtained via the ‘point-and-click’ method. Where unclear, the axis along which the loops should be measured was determined with the aid of SDO/AIA observations. These lengths were then used for comparison with the RTV-predicted values (Figure 6.5)...... 182 6.5 Comparison of RTV-predicted flare loop half-lengths with those measured with Hinode/XRT. Most of the data points are scattered around the 1:1 line (over-plotted). N.B. It is not a fit...... 184 6.6 Comparison of Cargill-predicted cooling times with observed cooling times. The 1:1 line is overplotted for clarity. This shows the that Cargill-predicted cooling time provides a lower bound to a flare’s observed cooling time...... 186 6.7 Heating during the decay phase as a function of the difference between the observed and Cargill-predicted cooling times for 38 M- and X-class flares. The line over-plotted is the best fit to the data (see Equation 6.7). 188 6.8 Histograms showing the required total heating during the decay phase of 38 M- and X-class flares to account for the difference between the Cargill-

predicted and observed cooling times (excess cooling time). a) Log10 of total decay phase heating. b) Total decay phase heating normalised by the total energy radiated by the ﬂare as measured by GOES. c) Total decay phase heating divided by the thermal energy at the beginning of the cooling phase...... 190

7.1 Top: Typical Gaussian such as those used for parameterising ﬂare DEMs in Chapter 5. Bottom: Bi-Gaussian with a certain standard deviation to the left (lower temperature) of the peak, and a smaller standard deviation to the right (higher temperature) of the peak. Such a function may be useful in better parameterising ﬂare DEMs...... 205

xxvi LIST OF FIGURES

7.2 Flare DEM (black curve) inferred by Graham et al. (2013) (Figure 4 from that paper) from Hinode/EIS observations and the regularised inversion technique of Hannah & Kontar (2012). The grey shaded area represents the uncertainty limits of the DEM while the coloured lines represent the measured line intensities divided by the contribution functions, indicating maximum possible emission measure. Note the there is a much more rapid fall-off in the high temperature tail of the DEM (black line), suggesting that an asymmetric parameterisation, such as the bi-gaussian in Figure 7.1 may be suitable to flare DEMs...... 206 7.3 Simulated representations of a multi-stranded coronal loop at different resolutions. Note how at low resolutions the loop can appear monolithic, but multi-stranded at high resolutions. (Aschwanden, 2004) ...... 208 7.4 Figure taken from Warren & Doschek (2005) showing how simulated lightcurves of unresolved strands (dotted lines), when convolved (thick lines), can well-approximate observed lightcurves (thin lines). This is shown for GOES/XRS long and short channels, and the Fe XXV, Ca XIX and S XV lines observed with Yohkoh/BCS...... 209

xxvii LIST OF FIGURES

xxviii List of Tables

1.1 GOES ﬂare classiﬁcations ...... 30

4.1 Values for B8-T relationship (Equation 4.5) ...... 134

4.2 Values for edge in B8-EM distribution (Same form as Equation 4.2) . . 136

4.3 Values for EM-B8 relationship (Equation 4.3) ...... 137

6.1 Wavelengths and temperatures of bandpasses and emission lines used in measuring cooling rates ...... 175 6.2 Events used in this study with observed and model-predicted cooling times and other thermodynamic properties ...... 196

A.1 GOES Saturation Levels ...... 212

xxix GLOSSARY

xxx GOES Geostationary Operational Environ- mental Satellite

HMI Helioseismic and Magnetic Imager onboard SDO

HXR Hard X-Ray Glossary IR Infra-Red JAXA Japanese Aerospace Exploration Agency

LASCO Large Angle and Spectrometric AIA Atmospheric Imaging Assembly on- Coronagraph onboard SOHO board SDO LHS Left Hand Side AR Active Region MDI Michelson Doppler Imager onboard BCS Bragg Crystal Spectrometer onboard SOHO Yohkoh MEGS Multiple EUV Grating Spectrograph CCD Charge Coupled Device onboard SDO/EVE

CME Coronal Mass Ejection MEGS-P MEGS Photometer onboard SDO/EVE COR1(2) Inner (Outer) Coronagraph onboard STEREO MHD Magnetohydrodynamic(s) NAOJ National Astronomical Observatory CVRMSD Coeﬃcient of Variation of the Root- of Japan Mean-Square Deviation NASA National Aeronautics and Space Ad- DEM Diﬀerential Emission Measure ministration

DRM Detector Response Matrix NOAA National Oceanographic and Atmo- EBTEL Enthalpy-Based Thermal Evolution spheric Adminitration of Loops NSC Norwegian Space Centre

EIS Extreme-ultraviolet Imaging Spec- OLS Ordinary Least-Squares trometer onboard Hinode OSO Orbiting Solar Observatory

ESA European Space Agency RHESSI Reuven Ramaty High Energy Solar ESP EUV SpectroPhotometer onboard Spectroscopic Imager SDO/EVE RHS Right Hand Side

EUV Extreme-UltraViolet RMC Rotating Modulation Collimators

EVE Extreme-ultraviolet Variability Ex- RMSD Root-Mean-Square Deviation periment onboard SDO RTV Rosner, Tucker, Vaiana

FOV Field Of View SAA South Atlantic Anomaly

GeD Germanium Detectors onboard SAM Solar Aspect Monitor onboard RHESSI SDO/EVE

xxxi GLOSSARY

SDO Solar Dynamics Observatory SXR Soft X-ray

SMEX NASA SMall EXplorer missions SXT Soft X-ray Telescope

SOHO SOlar and Heliospheric Observatory TEBBS Temperature and Emission measure- SOT Solar Optical Telescope Based Background Subtraction

SSM Standard Solar Model TRACE Transition Region And Coronal Ex- plorer STEREO Solar TErrestrial RElations Observa- tory UV UltraViolet

STFC Science and Technology Facilities XRS X-Ray Sensor onboard GOES Centre XRT X-Ray Telescope onboard Hinode SXI Soft X-ray Imager

xxxii Chapter 1

Introduction

In this chapter a general overview of solar flares is provided along with physical concepts associated with them. The chapter begins with an introduction to the Sun itself, its interior structure and atmosphere. It then discusses the Sun’s magnetic field and its link to solar activity, before moving onto active regions – the areas from which flares most commonly originate. Finally, solar flares themselves are discussed in detail in terms of observations and the models we use to describe them. The chapter concludes with an overall outline of this thesis itself, describing the problems addressed and the methods used to tackle them.

1 1. INTRODUCTION

The Sun has long been recognised as one of the most inﬂuential factors for life on Earth.

Its relation to the seasons was key to marking the passage of time and the growth of crops. This made it vital to the survival of early civilisations and instilled humanity with a deep curiosity of the Sun’s nature and behaviour. Over recent centuries this curiosity has led to an increasingly accurate scientiﬁc view of the Sun. Whereas early civilisations depicted the Sun as a god, we now recognise it as a star at the centre of our solar system. And whereas we once built great structures in the Sun’s honour such as Newgrange in Ireland (Ray, 1989) and Chankillo in Peru (Ghezzi & Ruggles, 2007), we have more recently built telescopes, observatories, and even satellites, which have given us an unprecedented understanding of our nearest stellar neighbour.

With this understanding has come the realisation that the Sun-Earth connection is more than simply that of light and gravity. This stemmed from the observation that the number of dark regions on the Sun’s surface, known as sunspots (Section 1.4), was often quite high at times of increased auroral activity. Observations of sunspots stretch back over two millennia to ancient China and Greece. However, a convincing connection between them and the Earth had never been made. This changed in 1852 when Edward

Sabine (Sabine, 1852) hypothesised that the Sun could produce magnetic eﬀects at the Earth, including aurora, at times of high sunspot number. Seven years later,

Richard Carrington (Carrington, 1859) and Richard Hodgson independently observed an intense and short-lived burst of light from a sunspot region on the Sun. These were the first recorded observations of a solar flare (Section 1.5). The following day, the world witnessed intense auroral activity as far south as Hawaii, as well as vast magnetic disturbances across the globe and huge disruption on the world’s telegraph systems. We now know that this was caused by a coronal mass ejection (CME) – a vast expulsion of hot ionised gas and magnetic field often associated with solar flares.

These can interact with the Earth’s magnetic ﬁeld channelling radiation down to the atmosphere and causing continent-wide electrical currents to ﬂow in the Earth.

2 It was realised that the effects of these ‘solar storms’ were not confined to spectacular auroral displays. Modern electrical technologies could be directly and detrimentally affected. Since the 19th century, we have become ever more dependent on such technologies. And although the 1859 Carrington flare is still the biggest observed to date, we have become increasingly vulnerable to solar activity in the decades since. In 1989, a CME caused the entire power network of Quebec, Canada, to be knocked out for nine hours. Fourteen years later in 2003, a similar event occurred in Scandinavia. Satellites beyond the protective layer of the Earth’s atmosphere can be irreparably damaged by intense solar flare radiation. And flights must sometimes be redirected from the poles where radiation is typically channelled by the Earth’s magnetic field during a CME impact.

The changing environmental conditions beyond Earth’s atmosphere which give rise to these eﬀects are known collectively as space weather. Space weather is monitored by various agencies across the world, including America’s National Oceanographic and

Atmospheric Administration (NOAA), which issue reports and warnings for any indi- viduals, companies or government agencies whose operations depend on space weather.

However, a comprehensive understanding of how forms of solar activity such as ﬂares and CMEs are initiated, and how they are connected to Earth, remains elusive.

As well as a source of space weather, the Sun is an ideal laboratory for studying fields such as plasma physics, magnetism, thermodynamics, shocks, fluid physics, atomic physics, etc. in extreme conditions impossible to recreate on Earth. It also gives us a very well observed example with which to compare astrophysical models such as those of stellar evolution and solar system formation. Therefore, the study of the Sun and solar activity is vital for both improving our predictions of space weather and better understanding the fundamental physics which drives our Universe. In this thesis, we examine the thermal processes of solar flares, develop new analysis techniques, better understand the observations obtained with new satellite observatories, and compare

3 1. INTRODUCTION models of thermal processes with new observations.

1.1 Internal Structure

Over the centuries, we have used astronomical observation to build up an increasingly comprehensive view of the Sun and its physical properties. It has a mass of

1.99×1030 kg, a radius of 6.96×108 m, a luminosity of 3.84×1026 W, and a surface temperature of ∼5,800 K. Its chemical composition at its surface is (by mass) hydrogen

(71%), helium (27%) and metals (2%) of which the most abundant are oxygen, carbon, iron, neon and nitrogen respectively (Phillips, 1992). By particle number this trans- lates to hydrogen (91%), helium (9%) and metals (0.1%). The Sun is believed to be approximately half way through its life with an age of 4.6×109 years. This is estimated from the oldest meteorites found to date with the assumption that they were formed around the same time as the Sun. Therefore this estimate of the Sun’s age is very uncertain.

The Sun is powered by nuclear fusion at its core which converts hydrogen into helium. This is made possible by the extremely high temperatures and densities (107 K and 105 kg m−3, respectively) and the process of quantum tunnelling which allows the protons to overcome the repulsive Coulomb barrier so that the attractive strong nuclear force can take aﬀect. This leads to the net result of

1 4 + 4 1H →2 He + 2e + 2νe + energy (1.1)

1 4 where 1H and 2He represent hydrogen and helium nuclei respectively with the subscript denoting the number of protons and the superscript denoting the number of nucleons

(protons+neutrons). Meanwhile e+ represents a positron (antimatter equivalent of an electron) and νe represents an electron neutrino. The vast majority of reactions in the Sun (99%) occur via the pp-chain. Figure 1.1 shows the three diﬀerent branches of

4 1.1 Internal Structure

Figure 1.1: Diagram showing the various branches of the pp-chain and their occurrence rates. (Carroll & Ostlie, 1996)

5 1. INTRODUCTION this chain and the occurrence rate of each. The other 1% of reactions occurs via the

CNO-cycle which is dominant in more massive stars.

The energy liberated by these reactions stems from the mass difference between the four hydrogen nuclei and the helium nucleus (4.8×10−29 kg). Einstein’s mass-energy equation, E = mc2, thus implies that 4.3×10−12 J of energy is liberated by each reaction chain. Although a small amount of this energy is carried away by the neutrinos, most of it goes directly into the generation of gamma-ray photons, γ, which give the Sun the luminosity we see today. By dividing the Sun’s luminosity by the energy per reaction we can estimate the number of reactions per unit time. This comes to 9×1037 s−1. This has been experimentally verified via the detection of solar neutrinos, neutral particles with tiny but non-zero masses produced via Equation 1.1. Early experiments measuring solar neutrinos from different steps of the pp-chain (Davis 1994; SuperKamiokande; SAGE;

GALLEX) consistently found less than half the expected number of neutrinos. This was explained by the concept of neutrino oscillation, whereby neutrinos can switch between the diﬀerent ﬂavours (electron, muon and tauon) on their way from the Sun to the

Earth. This was experimentally veriﬁed in 1998 by the SuperKamiokande experiment in Japan.

The radius, mass, luminosity, chemical composition, surface temperature, age, and nuclear reactions are all key boundary conditions for determining the internal structure of the Sun. This is done via what is known as the Standard Solar Model (SSM; see review by Bahcall et al., 1982). The SSM is based on a combination of assumptions and physical principles. The assumptions include that the Sun: is spherically symmetric; is driven by nuclear reactions at its core; has its energy transported from the core to the surface predominantly via radiation or convection; can only change its chemical composition through nuclear reactions at the core; and is in hydrostatic equilibrium,

(i.e. neither signiﬁcantly expanding or contracting). From these assumptions, a number of diﬀerential equations can be written which are the basis of the SSM. These include

6 1.1 Internal Structure

Figure 1.2: Cut-away cartoon of the Sun’s interior showing the core, the radiative zone and the convection zone. It also shows the diﬀerent layers of the atmosphere, the photosphere, the chromosphere, and the corona.

7 1. INTRODUCTION the equation of hydrostatic equilibrium (outward pressure is balanced by gravitation), mass conservation (mass as a function of distance from the core) and the luminosity gradient equation (luminosity corresponds to the rate of energy produced by nuclear reactions in the core). These are outlined in Carroll & Ostlie (1996).

The SSM predicts a highly structured interior, outlined by the cartoon in Figure 1.2.

As shown, the interior can be divided into three zones depending on the production and transport of energy. These are the core, radiative zone and convective zone. (The

ﬁgure also shows various features of the solar atmosphere. See Section 1.2.) Figure 1.3

(Carroll & Ostlie, 1996) shows temperature and pressure (top panel) and density and cumulative mass (bottom panel) as a function of distance from the Sun’s centre, which are useful is discussing each zone. The predictions in this figure can be tested via helioseismology. This involves measuring the periods of certain modes of oscillation on the Sun’s surface which correspond to sound waves travelling through the Sun before being refracted back to the surface. As different modes penetrate to different depths, the sound speed, and hence the temperature and density, as a function of depth can be inferred.

The core is deﬁned as the region where nuclear reactions occur. The high temperatures mean that all atoms are completely stripped of their electrons, creating a fully ionised plasma. At the centre of the core, the temperature and density are ∼15 MK

5 −3 4 −3 and ∼1.5×10 kg m . These drop to ∼8 MK and ∼4×10 kg m by around 0.25 R . This is not enough to sustain nuclear burning and the reaction rate drops to almost zero. This is deﬁned as the lower boundary of the radiative zone which itself extends to 0.7 R . Like the core, the radiative zone rotates as a solid and the primary mode of energy transport is radiation. Over the width of the radiative zone the temperature and densities drop to 0.04 MK and 2×102 kg m−3 respectively. Because of the high densities in the radiative zone, photons are repeatedly scattered, resulting in a very short mean free path (∼9×10−5 m−5). This means the photons undergo a random-walk on their

8 1.1 Internal Structure

Figure 1.3: Plots of temperature and pressure (top panel) and density and cumulative mass (bottom panel) as a function of distance from the Sun’s centre as predicted by the Standard Solar Model. Adapted from Carroll & Ostlie (1996)

9 1. INTRODUCTION journey to the edge of the radiative zone, which can take up ∼105 years (Mitalas &

Sills, 1992).

At 0.7 R the temperature becomes low enough that some nuclei can capture electrons. This vastly increases the opacity, κ, of the plasma as photons are absorbed, reionising the plasma. This simultaneously makes radiation a much less eﬃcient energy transport mechanism and greatly increases the temperature gradient. In accordance with the Schwarzchild criterion (Schwarzchild, 1906), this makes convection the dominant energy transport mechanism and thus deﬁnes the lower boundary of the convective zone.

The Schwarzchild criterion determines when convection becomes favourable. To understand this concept, consider a bubble of plasma within an ambient medium of the same substance. Assume that the bubble is perturbed upwards with a velocity much slower than the sound speed but fast enough that it does not exchange any energy with its surroundings. This means that that at each point the bubble instantaneously equalises its pressure with that of its surroundings via adiabatic expansion/contraction.

The rate at which the temperature changes with height due to this adiabatic expansion is called the adiabatic temperature gradient. If the bubble is cooler (and thus denser) than its surroundings when it has equalised its pressure, it will sink back to its original position. If however it is hotter (less dense), it will experience a buoyant force and will continue to rise. Thus Schwarzchild’s criterion states that convection is favourable where the temperature gradient of the star is less than the adiabatic temperature gradient:

dT dT > (1.2) dr dr ad where the LHS (left-hand side) is the star’s physical temperature gradient and the RHS

dT 1 T dP (right-hand side) is the adiabatic temperature gradient given by dr ad = 1 − γ P dr ,

10 1.2 The Solar Atmosphere

Figure 1.4: Modelled temperature and density of the solar atmosphere with height above the photosphere. (Aschwanden, 2004)

where P is pressure and γ is the adiabatic invariant. (See Carroll & Ostlie 1996 for a derivation.)

At the upper boundary of the convective zone, the temperature and density have dropped to photospheric values of ∼5,800 K and 2×10−4 kg m−3. This is the boundary between the Sun’s interior and its atmosphere. The solar atmosphere is also highly stratiﬁed and is the region from which solar ﬂares erupt. We will now discuss this region in detail in the following section.

1.2 The Solar Atmosphere

The solar atmosphere is typically divided into four zones: the photosphere, chromosphere, transition region, and corona. Figure 1.4 shows the temperature and density proﬁles of the solar atmosphere with height above the photosphere. The diﬀerent ther-

11 1. INTRODUCTION modynamic conditions in each layer drastically aﬀect their topologies as well as the physical processes dominant there.

1.2.1 The Photosphere

The photosphere is the visible ‘surface’ of the Sun. It is only a few hundred kilometres thick and has a typical number density of 1017 cm−3. Its temperature varies from

6,000 K at its base to 5,000 K at its upper boundary (Figure 1.4). It is defined as the region where the optical depth at 500 nm (visible yellow light) is two thirds. The optical depth describes how much of an incident light beam is transmitted through a slab of material without being absorbed, reflected or scattered. It is defined as

−τ I = I0e (1.3)

where τ is the optical depth, I0 is the incident intensity of the beam, and I is the intensity of the beam after passing through the material. Therefore, the photosphere is the region where >50% of the incident light escapes directly out into space. This means it is the region from which the Sun radiates the vast majority of its energy as visible and infra-red radiation (hence its name, meaning ‘light sphere’).

The spectrum of the photosphere closely resembles that of a blackbody with a temperature just below 5,800 K (Figure 1.5). This is deﬁned by the Planck function.

2 2hc 1 −2 −1 −1 Bλ(T ) = 5 [W m sr m ] (1.4) λ exp hc − 1 λkB T

Dark absorption features, known as Fraunhofer lines, can be seen throughout the photosphere’s spectrum. These are caused by ions absorbing photons at speciﬁc wavelengths. The energy gained by the ions puts them into a higher energy state, known as an excited state (Section 2.1.2.1). As each absorption feature corresponds to a spe-

12 1.2 The Solar Atmosphere

Figure 1.5: Observed spectrum of the Sun’s emission. At wavelengths longer than 103 A,˚ the spectrum closely resembles a blackbody of ∼5770 K, corresponding to the photosphere. The deviation at short wavelength is due to high energy emission from the upper layers of the Sun’s atmosphere (chromosphere and corona). (Aschwanden, 2004).

cific excitation state of a specific ion of a specific element, it is possible to deduce the quantities of the different elements present (i.e. their abundances).

The top panel of Figure 1.6 shows an image of the photosphere taken by SDO/AIA at 4500 A.˚ At this scale the photosphere is rather featureless. At smaller scales it can be seen that the photosphere is characterised by uneven, mottled granulation (Figure 1.7), caused by convective motions below. The most notable features of the photosphere are the dark sunspots, visible in Figure 1.6. These are locations where intense magnetic

ﬁelds have broken through the surface. They appear dark because the magnetic ﬁeld suppresses convection causing them to be cooler than the surrounding photosphere,

∼3,000–4,000 K instead of ∼5,800 K. (See Section 1.4.)

1.2.2 The Chromosphere & Transition Region

The chromosphere is usually invisible to the naked eye because of the brightness of the photosphere below. However, it can be seen as a thin red ring and prominences

13 1. INTRODUCTION

Figure 1.6: Full disk images taken with SDO/AIA of the photosphere at 4500 A(˚ top), the chromosphere & transition region at 304 A(˚ left) and the corona at 193 A(˚ right). Images courtesy of helioviewer.org

14 1.2 The Solar Atmosphere

Figure 1.7: Image of granulation on the photosphere. Courtesy of the National Optical Astronomy Observatory.

around the Sun during a total solar eclipse. Its red appearance is due to Hα emission

(6562.8 A)˚ caused by the de-excitation of neutral hydrogen which forms due to the lower temperatures and densities in the low chromosphere. The topology of the chromosphere is characterised by spicules, small jet like structures that shoot up into the transition region and corona before fading away. The number density ranges from 1016 cm−3 at the top of the photosphere to 1011 cm−3 at the base of transition region (Figure 1.4).

Initially the temperature falls radially just as in the photosphere. However it quickly reaches a temperature minimum of ∼4,500 K, before inexplicably starting to rise again.

At the upper boundary, the temperature has risen to ∼25,000 K, several times that of the photosphere (Figure 1.4).

Directly above the chromosphere is a very thin stratum called the transition region, so named because of the extraordinary changes that occur there. While the density continues to drop (1011 to 109 cm−3), the temperature increase begun in the chromosphere

15 1. INTRODUCTION rapidly accelerates. Over about 100 km, the temperature rises two orders of magnitude from 25,000 K to 1 MK (106 K). This appears to violate the laws of thermodynamics and is known as the coronal heating problem. The reason for this is still uncertain.

Acoustic and magnetic (Alfv´en)waves, spicules, and nanoﬂares have all been suggested as the cause. However, no consensus has yet been reached.

The left panel of Figure 1.6 shows the upper chromosphere and lower transition region taken at 304 A.˚ The bright regions just below the disk centre (known as plage) coincide with the sunspot locations seen in the photosphere and are typically found near active regions (Section 1.4). Also just visible above and to the left of disk centre is a long dark thin structure running diagonally toward the top left of the image.

This is known as a ﬁlament which is a region of photospheric material suspended above the chromosphere. It appears dark because it is much cooler than the rest of the chromosphere. When these ﬁlaments appear on the limb (edge of the Sun), they appear bright in contrast to the dark background and are then known as prominences.

Examples of prominences can be seen around the limb in Figure 1.6, especially near the solar north pole.

1.2.3 The Corona

The uppermost layer of the Sun’s atmosphere is the corona. It is very faint and tenuous with typical densities of 107 cm−3 and temperatures of ∼2 MK. In visible light (‘white light’) it is six orders of magnitude dimmer than the photosphere. Therefore, like the chromosphere, it is usually only visible during a solar eclipse, as shown in Figure 1.8.

The light seen here is not emitted, but reﬂected from the photosphere via free electrons.

This is known as Thompson scattering. The corona’s dimness is therefore indicative of its low density while the visible structures highlight its density variations. Some of these features include helmet streamers which are clearly visible at “2 o’clock” and “8 o’clock”.

They are so named because they resemble German First World War helmets. Helmet

16 1.2 The Solar Atmosphere

Figure 1.8: The Sun’s corona in visible light during a total solar eclipse. This reveals its complex and highly non-spherical structure, largely determined by the solar magnetic ﬁeld.

streamers are regions of relatively dense plasma contained by the coronal magnetic

ﬁeld. They have been dragged out into almost triangular structures by the solar wind, a steady stream of particles released into the solar system, or heliosphere. There are two types of solar wind, fast (∼700 km s−1) and slow (∼400 km s−1). The fast solar wind in typically associated with coronal holes (discussed below). Also visible in Figure 1.8 are polar streamers, long thin straight features emanating from the poles which outline the ‘open’ polar magnetic ﬁeld lines.

Nowadays the corona is more typically examined with a coronagraph. This instrument creates an artiﬁcial eclipse by placing an occulting disk in front of a camera.

Depending on the size of the the disk, it is possible to image diﬀerent regions of the corona. The more light is blocked out from close to the Sun, the further out into the increasingly faint corona one can see. Examples of coronagraphs include LASCO onboard

SOHO (Brueckner et al., 1995) and COR1 and COR2 onboard STEREO (Thompson

17 1. INTRODUCTION et al., 2003). The development of coronagraphs has greatly improved our knowledge of the outer corona and allowed us to track features such as CMEs further out into the heliosphere than ever before.

Coronal abundances relative to hydrogen can be measured via the ratio of emission lines (Section 2.1.2). However the abundance of hydrogen itself cannot be measured in this way as hydrogen is completely ionised in the corona. Therefore absolute coronal abundances are uncertain. However it has been noticed that there is a discrepancy between the coronal and photospheric relative abundances of some elements. Elements with a ﬁrst ionisation potential (FIP; energy required to remove an atom’s outermost electron), of less than 10 eV, which are partly ionised in the photosphere, appear to be enhanced in the corona. These include K, Na, Al, Ca, Mg, Fe, and Si. This was ﬁrst noticed for the cases of Mg, Al and Si by Pottasch (1964a,b). Meanwhile elements with

FIPs greater than 10 eV (C, H, O, N, Ar, Ne) show no such enhancement. Sulphur, with a FIP of ∼10 eV is an intermediate case. This phenomenon is known as the FIP eﬀect. Because the absolute hydrogen abundance is uncertain, it is unclear whether the

FIP eﬀect is due to an enhancement of low FIP elements in the corona or a depletion of high FIP elements in the photosphere. The FIP bias is the factor by which the low

FIP elements appear to be enhanced. This has been found to diﬀer for diﬀerent regions and appears to be related to how long the plasma is contained in the corona. It has been calculated to be in the range 1–7 for Active Regions (Carmichael, 1964; McKenzie

& Feldman, 1992), 8–16 for old Active Regions (Dwivedi et al., 1999; Feldman, 1992;

Feldman et al., 2004; Widing & Feldman, 1992; Young & Mason, 1997), and 1–10 solar

ﬂares (Fludra et al., 1990; Sterling et al., 1993; Sylwester, 1988). Meanwhile the FIP bias associated with coronal holes, which do not contain plasma but accelerate it out- wards in the fast solar wind, is much closer to unity. The exact cause for the FIP eﬀect remains unclear. Nonetheless its existence is important to consider when accounting for coronal abundances which are so important in coronal plasma diagnostics.

18 1.3 The Sun’s Magnetic Field & the Solar Cycle

The corona’s high temperature means that it emits strongly at Extreme Ultra-Violet

(EUV) and X-ray wavelengths. The right panel of Figure 1.6 shows the corona viewed at 193 A.˚ This further highlights the global inhomogeneity of the corona. There exists a region of particularly intense emission in the same location as the sunspots in the photosphere. This implies that plasma contained by the magnetic field above sunspots is very dense and/or very hot. There also exist additional regions of intense emission near the east (left) and west (right) limbs with no apparent associated sunspots. Throughout the corona, large loop-like structures can be seen which reveal the topology of the magnetic field. In contrast, dark regions of reduced emission can be seen near the top and bottom of the solar disk. These are known as coronal holes and are located at regions of ‘open’ magnetic flux. This magnetic configuration does not contain the coronal plasma, but accelerates it into the heliosphere. Thus the plasma does not emit strongly and appears as a void in the corona. As mentioned above, these regions are often associated with the fast solar wind.

EUV and X-ray images of the corona most strikingly reveal the importance of the corona’s magnetic ﬁeld and its topology. In the next section, we highlight how that magnetic ﬁeld is formed, the causes of its topology, and its consequences for solar activity.

1.3 The Sun’s Magnetic Field & the Solar Cycle

The Sun’s magnetic field is at the heart of its activity and is an integral part of the solar system. It is believed to be created by a dynamo effect at the tachocline, the boundary where the liquid convective zone interacts with the solid radiative zone. This basic principle is also believed to be responsible for the terrestrial magnetic field. However, the Sun’s magnetic field bears little resemblance to the smooth, well-behaved dipolar

ﬁeld of the Earth. Instead the Sun’s ﬁeld varies wildly from dipolar, to quadrupolar and

19 1. INTRODUCTION

Figure 1.9: Top: Sunspot number with time over the past 400 years taken from averages of observations from around the globe (blue). It can be seen that the sunspot number has an approximate 11-year periodicity. Prior to 1750 (red), observations were sporadic. This period includes the Maunder Minimum (1650–1700) when there appeared to be almost no sunspots at all. Bottom: ‘Butterﬂy diagrams’ for the period 1870 –2010. This shows the the total sunspot area in equally spaced latitude bands (as a percentage of the latitude band area) as a function of time. From this it can be seen that at the beginning of each solar cycle sunspots emerge at high latitudes (∼30o). But as time goes on, they emerge at lower and lower latitudes. Courtesy of NASA.

20 1.3 The Sun’s Magnetic Field & the Solar Cycle back to dipolar over the course of 11 years. This corresponds to the sunspot cycle (also known as the solar cycle), ﬁrst observed by Schwabe (1843). This can be seen in the top panel of Figure 1.9 which shows observations of sunspot number over the past 400 years. The bottom panel of Figure 1.9 shows the total sunspot area in equally spaced latitude bands as a function of time. The percentage of the area of each latitude band occupied by sunspots is designated by colour. This plot shows that the preferential emergent latitude of sunspots migrates from ∼30o down to the equator over the course of the solar cycle. By the end of this 11-year activity cycle, the magnetic polarity of the Sun’s global magnetic ﬁeld has been switched. The process repeats itself and the original orientation is restored. This reveals that there is also an approximate 22-year magnetic solar cycle, known as the Hale cycle (Hale & Nicholson, 1925).

The frequency of many forms of solar activity, including solar ﬂares and CMEs, also follow the 11-year solar cycle. This can be seen in Figure 1.10 which shows the number of solar ﬂares per month (black) observed over the past 30 years with the Geostationary

Operational Environmental Satellites (GOES; Section 3.1). It can be seen that ﬂares have the same periodicity as sunspot number, shown in red. This highlights that there is a fundamental link between the solar cycle and solar activity.

The details of the 22-year Hale cycle remain unclear. However, the current consensus is that many of its basic characteristics can be explained by the model of Babcock

(1961). In this model, the solar dynamo creates a basically dipolar ﬁeld, as in the Earth.

However in the convection zone, the gas pressure dominates the magnetic pressure.

This is represented by a high plasma-β value which is the ratio of the gas to magnetic pressure:

Pgas nkBT β = = 2 (1.5) Pmag B /8π

The various terms are: n, the number density; kB, the Boltzmann constant; T , tem-

21 1. INTRODUCTION

Figure 1.10: Number of solar flares per month (B-, C-, M-, and X-class) as a function of time as recorded in the GOES (Geostationary Operational Environmental Satellite; Section 3.1) flare list for the period 1980–2008. The flare class refers to the order of magnitude of the peak flux in the 1–8 A˚ GOES channel: 10−7 W m−2 (B-class) to 10−4 W m−2 (X-class). Note the approximate 11-year periodicity, just as in the sunspot cycle in Figure 1.9. Data courtesy of NOAA.

22 1.3 The Sun’s Magnetic Field & the Solar Cycle

Figure 1.11: Diagram of the Sun’s initially dipolar magnetic field (left) being wound up by differential rotation into a quadrupolar field (center), eventually leading to the emergence of magnetic field at low latitudes via the α-Ω effect and creating active regions and sunspots (right) (Carroll & Ostlie, 2006).

perature; and B, the magnetic field strength. The equation is given in cgs units. A high plasma-β value means that the motion of the magnetic field is determined by the motion of the plasma. In this situation, the magnetic field is said to be ‘frozen in’ to the plasma. This fact combined with the differential rotation of the convective zone causes the originally dipolar field to dragged out into a quadrupolar field (Figure 1.11).

The daily rate of diﬀerential rotation, ω, depends on the latitude φ via the following empirical equation.

ω(φ) = A + Bsinφ + Csin2φ (1.6) where A, B, and C are constants. Their values depend on the method used to measure the rotation. Newton & Nunn (1951) found A = 14.38, B = 0 and C = −2.77 degrees per day, while Snodgrass & Ulrich (1990) derived the currently accepted values of

A = 14.7, B = −2.4 and C = −1.88 degrees per day.

23 1. INTRODUCTION

Figure 1.12: Diagram of the α-Ω eﬀect with time increasing from the bottom schematic to the top. Adapted from Babcock (1961).

24 1.3 The Sun’s Magnetic Field & the Solar Cycle

After many rotations, the magnetic field is laid down on top of itself, thus amplifying the field and creating a flux rope. In equilibrium, the internal and external pressure of the flux rope must balance so that (in cgs units)

B2 P = P + [Ba] (1.7) e i 8π

where Pe is the gas pressure outside the flux rope, Pi is the pressure inside the flux rope, and B2/8π is the magnetic pressure inside the flux rope, where B is the magnetic

ﬁeld strength. The magnetic pressure outside the ﬂux rope is assumed to be negligible.

This equation directly implies that the gas pressure inside the flux rope is less than that outside. Thus the flux rope becomes buoyant and begins to rise, being twisted by the Coriolis force and small-scale turbulent motions as it does so. This known as the alpha-omega (α-Ω) effect (Parker, 1955). See Figure 1.12.

The ﬂux rope eventually emerges through the photosphere creating bipolar active regions (AR; Section 1.4). ARs are always oriented so that the trailing polarity is opposite to the polarity at the pole of that hemisphere. This means that the orientation of

AR polarities is opposite in opposite hemispheres (Hale’s polarity law). Over the lifetime of the ARs, the leading polarity tends to migrate equatorwards while the following polarity migrates poleward. The cumulative eﬀect of this behaviour in thousands of

ARs over the course of the solar cycle is thought to cause the magnetic polarities at the poles to switch. Meanwhile, the leading polarities in each hemisphere are believed to cancel each other out as they near the equator. This process is predicted to rearrange the global ﬁeld back into a poloidal conﬁguration with an opposite magnetic polarity.

This process takes roughly 11 years and explains the sunspot cycle. The process is repeated and the original global polarity is recovered, thus accounting for the 22-year

Hale cycle.

25 1. INTRODUCTION

1.4 Active Regions

Active Regions (ARs) are locations where intense magnetic fields emerge through the photosphere and into the solar atmosphere. They are important for space weather because events such as solar flares and CMEs tend to erupt from these locations. ARs manifest themselves differently in different layers of the atmosphere. Figure 1.13 shows three different views of the same active region taken with the Hinode Solar Optical

Telescope (SOT; Tsuneta et al., 2008). The top panel is a magnetogram showing the line of sight photospheric magnetic ﬁeld. The white regions represent regions of strong positive polarity (out of the Sun) and the black regions show regions of strong negative polarity (into the Sun). Two circular regions of opposite polarity can be seen interacting causing a complex boundary between them. This boundary is known as a magnetic neutral line. The middle panel shows an image of the photosphere taken in the visible G-band. Two sunspots can be seen corresponding to the areas of strong magnetic polarity. The sunspots have two distinct regions, the dark central umbra and the lighter outer penumbra. These sunspots are formed because the intense magnetic ﬁelds suppress convection resulting in the umbra having a cooler temperature

(∼4,000 K) than the surrounding photosphere (∼5,800 K). Meanwhile the penumbra has an intermediate temperature (∼5,000 K). A diagram of the magnetic structure of a sunspot can be seen in Figure 1.14. As the magnetic field extends into the tenuous atmosphere it spreads out in a fan-like structure due to the reduced pressure. The umbra corresponds to the inner region of this ‘magnetic fan’ at the photospheric level, while the penumbra corresponds to the outermost parts of the magnetic field which very quickly fan out horizontally. Many of the magnetic field lines of each sunspot will form loops in the corona which will link back to the other sunspot due to their opposite polarity. Some of these loops will be occupied by hot, dense coronal plasma which will emit at EUV and X-ray wavelengths. These are known as coronal loops and can be

26 1.4 Active Regions

Figure 1.13: Three images taken by Hinode/SOT of a flaring active region. Top: Mag- netogram showing the photospheric line of sight magnetic field strength polarity. Middle: Sunspot in the photosphere taken in the G-band. Bottom: Ca II image showing the chromosphere and flaring arcade. Images courtesy of JAXA/NASA.

27 1. INTRODUCTION

Figure 1.14: Cross-section of the magnetic topology of a sunspot. The magnetic ﬁeld (arrow-headed lines) can be seen emerging through the photosphere (black horizontal line with no arrow-head) and up into the solar atmosphere. The magnetic ﬁeld spreads out above the photosphere due to the reduced pressure of the tenuous atmosphere. (Parker, 1955).

imaged by instruments such as the Transition Regions and Coronal Explorer (TRACE;

Handy et al., 1999) and SDO/AIA (Figure 1.15). Other ﬁeld lines may connect with other regions of opposite polarity further aﬁeld.

In the photosphere, the magnetic field is also ‘frozen-in’ to the plasma. Therefore, as the active region evolves, the magnetic fields can become sheared and twisted over time by the differential rotation as well as the small-scale convective motions of the plasma. This places a lot of stress on the magnetic field in the corona where magnetic pressure dominates (low plasma-β). This increases the likelihood that the field will release some of that pressure as a solar flare. Such an event can be seen in the bottom panel of Figure 1.13. This image is taken in the wavelength of Ca II line which is mostly produced in the chromosphere. This image shows a long bright arcade of flaring loops suspended above the chromosphere. Note that no direct evidence of this flare is seen in either of the first two images indicating the complex stratification of the solar

28 1.4 Active Regions

Figure 1.15: Coronal loops imaged by TRACE.

atmosphere and of ARs. It also suggests that the main eruptive processes of solar flares begin above the photosphere and chromosphere. Therefore the key to understanding solar flares is understanding the condition of the magnetic field in the corona. This can be seen in part as coronal loops, such as in Figure 1.15. However, this only shows the magnetic field lines which happen to be occupied by dense emitting plasma. Dark regions in Figure 1.15 do not represent an absence of magnetic field, but rather an absence of emitting plasma. To date there is no satisfactory method which allows us to image the magnetic field in the corona. Magnetohydrodynamic (MHD) models can help us guess the coronal magnetic topology but understanding the coronal magnetic

ﬁeld, and hence solar ﬂares, remains elusive.

29 1. INTRODUCTION

Table 1.1: GOES ﬂare classiﬁcations.

GOES Class Peak ﬂux in the 1–8 A˚ channel (W m−2) X ×10−4 M ×10−5 C ×10−6 B ×10−7 A ×10−8

1.5 Solar Flares

Solar ﬂares are among the most powerful events in the solar system, releasing up to

1032 ergs (1025 J) in hours or even minutes (Emslie et al., 2012). They are intense bursts of electromagnetic radiation from the corona and are often associated with other events.

These include coronal mass ejections (CMEs) – expulsions of vast volumes of plasma and magnetic ﬁeld into the heliosphere – as well as solar energetic particles (SEPs) – charged particles accelerated through the solar system to near-relativistic speeds along open magnetic ﬁeld lines.

Solar ﬂares are categorised by GOES class. This is based on the peak emission of the

ﬂare in the 1–8 A˚ channel of the Geostationary Operational Environmental Satellite X-

Ray Sensor (GOES/XRS; Section 3.1.1). GOES class ranges from 10−8 – 10−4 W m−2.

A letter represents each dex, A, B, C, M, and X, in ascending order (Table 1.1).

Therefore each GOES class is ten times greater than the previous one. The letter is followed by a number which designates the coefficient. Thus an M3.7 flare has a peak flux in the 1–8 A˚ band of 3.7×10−5 W m−2.

Solar flares are believed to be caused by magnetic reconnection, a process whereby stressed and sheared magnetic fields rapidly reorganise themselves into a lower energy configuration. Magnetic reconnection in flares is still not fully understood, despite numerous models and numerical simulations. However, many of the main aspects are common to most of the models and are described below.

30 1.5 Solar Flares

Figure 1.16: A diagram of the progression of magnetic reconnection.

Magnetic reconnection occurs when oppositely directed magnetic field lines are arranged roughly anti-parallel with each other (Figure 1.16). Some models only require that the field lines be misaligned. This is called component reconnection. The efficiency of the reconnection then depends on the angle between the field lines, with the most efficient configuration being anti-parallel. In order for there to be a continuous gradient from positive to negative field strength, there must exist a region where the field is very small or zero. Normally the corona has a low plasma-β value which does not allow plasma to diffuse across magnetic field lines. However this very weak magnetic

field strength creates a region of high plasma-β, known as the diffusion region. This is associated with a perpendicular current sheet (out of the page in Figure 1.16) due to the Lorentz force. The conditions in the diffusion region allow the magnetic field to reorient itself from, say, vertical opposing field lines (left of Figure 1.16) to highly curved horizontal opposing field lines (right of Figure 1.16). The curvature of these

field lines creates a high magnetic tension which accelerates the field lines and plasma away from the diffusion region. Thus magnetic energy is converted into kinetic energy which is believed to be capable of driving a solar flare.

31 1. INTRODUCTION

Figure 1.17: A diagram of magnetic islands forming in a current sheet due to a tearing- mode instability. (Aschwanden, 2004).

Early attempts to model this process in 2D included the Sweet-Parker (Parker, 1963;

Sweet, 1958) and Petschek (Petschek, 1964) models. The Sweet-Parker model assumed a long thin diﬀusion region which implied a reconnection rate which was too slow for

flares. The Petschek model assumed a much smaller square-like diffusion region which increased the reconnection rate by three orders of magnitude. However this model, along with Sweet-Parker, implied a steady-state process and so was also unsuitable for describing flares (Aschwanden, 2004).

A more likely model of ﬂare reconnection is that of tearing-mode (Kliem, 1995; Lee

& Fu, 1986; Priest & Forbes, 2000) and coalescence instabilities (Leboef et al., 1982). A tearing-mode instability occurs when the diffusion region becomes too long. An instability occurs if the magnetic diffusion timescale is longer than that of the Alfvéntransit time. Thus once an Alfvéndisturbance occurs, magnetic diffusion cannot restore the system quickly enough and an instability is triggered. This leads to the creation of

‘magnetic islands’ in the current sheet (Figure 1.17). A coalescence instability then completes the task of reducing the current sheet by recombining these magnetic is-

32 1.5 Solar Flares lands. This process liberates some of the free magnetic energy which then becomes available for driving a solar ﬂare. Theoretical applications of tearing-mode instabilities to ﬂares have been carried out in many studies including Heyvaerts et al. (1977); Kliem

(1990); Sturrock (1966). Further studies have carried out examinations of magnetic reconnection in 3D. Here the topologies become far more complex, but many of the basic principles of the 2D models remain. (For a more detailed discussion of these models see Aschwanden 2004; Priest & Forbes 2000.)

1.5.1 The CSHKP Flare Model

The standard model of a solar ﬂare is known as the CSHKP model after the authors of the principal studies upon which it is based (Carmichael, 1964; Hirayama, 1974; Kopp

& Pneuman, 1976; Sturrock, 1966). Figure 1.18 shows a diagram of a flaring coronal loop which illustrates the processes of this model. The flare is assumed to be driven by a sizeable and sudden release of energy via magnetic reconnection. The energy release site can be seen marked ‘a’. The topology illustrated in Figure 1.18 is highly simplified.

In reality, this coronal loop may be part of an arcade of loops or may even be made up of a number of smaller entangled strands. As a result, the energy released may not be a single reconnection event but a rapid series of smaller events whose reconnection rate varies.

Whatever its nature, the energy release accelerates electrons and ions to near- relativistic speeds. These spiral along the magnetic ﬁeld lines, emitting gyrosynchotron radio emission as they do so (‘b’ in Figure 1.18). This can cause direct heating of the coronal plasma via joule heating and shocks. However, due to the tenuous densities in the the corona, most of the charged particles proceed relatively unhindered until they reach the transition region and chromosphere. These locations are called the footpoints. A small fraction (∼10−5) of the particles in the beam undergo bremsstrahlung, or

‘braking’ radiation (Section 2.1.1). This occurs via collisions between electrons and ions

33 1. INTRODUCTION

f f a b

d c

Figure 1.18: Diagram of the standard ﬂare model. Adapted from Dennis & Schwartz (1989).

34 1.5 Solar Flares which result in the emission of a photon. In the case of ﬂares, this leads to non-thermal hard X-ray (HXR) emission at the footpoints via the thick target model (Brown, 1971)

(‘c’ in Figure 1.18). However the majority of the charged particles lose their energy via Coulomb collisions which heats the ambient plasma to 10–40 MK. The dominant particles in these processes are thought to be the electrons. This is because the number of gamma rays predicted by the ions are greater than those observed. Meanwhile, the low energy protons cannot on their own account for the observed HXR emission.

However for years there was a recognised number problem in assuming that the electrons were dominant. The number problem meant an unphysically high density was required in the electron beam to account for the HXR emission. However, Brown et al.

(2009) claim the necessary density can be reduced to physical levels by including local re-acceleration of electrons as they pass through the HXR source itself. In addition to beam heating, the chromosphere can be heated by thermal conduction fronts which travel down the loop from the energy release site to the footpoints (‘f’ in Figure 1.18).

Once the chromospheric plasma has been heated, it conducts heat downward to the cooler plasma below which can cause Hα footpoint brightenings. However, the rate of heating is usually so great that the plasma cannot radiate its energy away quickly enough. This causes it to expand back into the coronal loop where the pressure is much lower (‘d’ in Figure 1.18). This is commonly known as chromospheric evaporation.

However, it should more correctly be called chromospheric ablation, as evaporation implies a change of state, which does not occur in flares. However, throughout this thesis we will refer to it as chromospheric evaporation in line with convention. Chromospheric evaporation can be classed as explosive with upflow velocities of ∼200 km s−1 (Milligan et al., 2006a) or gentle, with plasma upflow velocities of ∼10–100 km s−1 (Milligan et al.,

2006b). Although the heating necessary for chromospheric evaporation is assumed to come from electron beam heating, Yokoyama & Shibata (1998) showed that it could be produced by thermal conduction front heating as well. This may be important in

35 1. INTRODUCTION

ﬂares which do not show strong HXR emission.

Because the ﬂaring plasma is at millions of kelvin, it emits thermally at soft X-rays

(SXR) and EUV (‘e’ in Figure 1.18). Figure 1.19 shows SXR and HXR lightcurves of a typical M1.8 ﬂare which occurred 2002 April 10 at 19:00 UT. Panel a shows RHESSI

HXR observations (Section 3.2) in the ranges 12–25 keV and 25–50 keV. It can be seen that these peak ﬁrst as expected from the standard ﬂare model. Panel b shows the temperature derived from the GOES/XRS SXR observations in Panel c (Section 3.1).

Because the plasma is heated by Coloumb collisions around the same time as the bremsstrahlung, the temperature peaks shortly after the HXRs. Panels c and d show the SXR emission in the two GOES passbands (1–8 A˚ and 0.5–4 A)˚ and the emission measure derived from these respectively. The emission measure is deﬁned as EM =

R 2 nedV , where ne is the electron density and V is the flare volume. Therefore it is related to the amount of emitting material in the flare, and its rise and fall loosely plots the chromospheric evaporation if the volume does not change significantly. Both the SXR emission and emission measure increase in response to the temperature rise.

The SXR reaches its peak after the temperature when a balance between the now falling temperature and still increasing emission measure is reached. Shortly afterwards, the emission measure reaches its maximum before also starting to decrease towards ‘pre-

ﬂare’ values.

All the properties in Figure 1.19 exhibit an initial impulsive phase dominated by energy release and heating processes causing rapid increases in their time profiles. From the end of the impulsive phase to when the system returns to ‘pre-flare’ conditions is known as the decay phase. Although energy release and heating events may continue beyond the impulsive phase, the decay phase is dominated by cooling processes which lead to a gradual fall in emission, temperature, and emission measure. Note that properties in Figure 1.19 peak in the following order: HXR, temperature, SXR, and emission measure. This is the characteristic behaviour predicted by the standard flare

36 1.5 Solar Flares

Figure 1.19: Time profiles of an M1.8 flare which occurred on 2002 April 10 at 19:00 UT. a) RHESSI count rate in the 6–12 keV, 12–25 keV and 25–50 keV ranges. b) GOES temperature. c) GOES flux in the 1–8 A˚ and 0.5–4 A˚ passbands. d) GOES emission measure.

37 1. INTRODUCTION model. And although flares in reality are much more complicated than described here, often involving multiple loops at different stages of evolution, this behaviour still occurs in the majority of flares. It has been noted in numerous observations (e.g. Battaglia et al., 2009; Fludra et al., 1995) and numerical models (e.g. Aschwanden & Tsiklauri,

2009; Fisher et al., 1985a).

1.6 Thesis Outline

This thesis examines the hydrodynamic and thermodynamic evolution of solar ﬂares which is determined by the interplay between energy input and cooling mechanisms.

This evolution is fundamental to how the energy which drives solar ﬂares is transferred and dissipated through the solar corona. We explore a number of areas within this

field including flare cooling rates and thermodynamic scaling laws. In addition, this thesis explores the temperature distribution within flares. This distribution, known as the differential emission measure (DEM; Chapter 2), has implications both for the interpretation of flare observations and flare thermo- and hydrodynamic evolution. It is therefore key to understanding the fundamental behaviour of solar flares.

To date, the study of thermo- and hydrodynamic evolution of solar ﬂares has been dominated by studies of single or small samples of events. While many of these have been helpful in unveiling the physics of these eruptive events, they can have drawbacks.

For example, comparisons of their results with models or other observations can be difficult as it may not be possible to say whether discrepancies are due to a different interpretation of the fundamental physics or simply the nature of the different events studied. In this thesis, we examine ensembles of solar flares in order to put the results of such studies into context and reveal the global nature of solar flares. In doing so, we present new automatic plasma diagnostic techniques which allow dozens, hundreds, or even thousands of solar flares to be analysed quickly and accurately. This facilitates

38 1.6 Thesis Outline a better understanding of how the fundamental physics of flares behaves throughout the global flare distribution. These techniques are then used to examine a host of solar flare properties including thermodynamic scaling laws, multi-thermal temperature distributions, and hydrodynamic evolution.

In Chapter 2 we outline the theory which underpins this research. It begins by discussing the atomic physics responsible for the electromagnetic emission we observe from the corona. It then discusses the basics of hydrodynamics which explains the behaviour of ﬂaring plasma as it evolves. This allows models of ﬂare evolution to be developed which can then be tested with observations.

In Chapter 3 we describe the various instruments used in this study along with techniques for determining thermal properties of the emitting plasma from their observations.

In Chapter 4 we develop a new automatic background subtraction algorithm for observations from GOES/XRS (Chapter 3). Known as the Temperature and Emission measure-Based Background Subtraction (TEBBS), it allows the thermal properties of thousands of observed ﬂares to be analysed quickly and accurately. TEBBS is then used to examine the relationships between thermodynamic properties for over 50,000

ﬂares between the years 1980–2007. This makes it the largest and most reliable study of solar ﬂare thermal properties to date.

In Chapter 5 we present a comparison of peak temperatures of ﬂare diﬀerential emission measure distributions (DEMs; Chapter 2) as determined with SDO/AIA,

GOES/XRS, and RHESSI. This allows us to explore the biases and pitfalls of traditional temperature diagnostic methods which assume that the flaring plasma is isothermal. As these instruments have different temperature sensitivities, they also allow us to explore the nature of the DEM itself and better understand the temperature distributions within an ensemble flares.

In Chapter 6 we examine the cooling of 72 M- and X-class ﬂares and compare the

39 1. INTRODUCTION observations with the predictions of a simple hydrodynamic model (Cargill et al., 1995).

It is one of the few studies which observes and models the hydrodynamic evolution of a significant number of flares. Furthermore, this comparison also allows us to infer the heating rates during the flare decay phases. We are thus able to determine the importance of decay phase heating for the overall energetics of solar flares.

In Chapter 7 we summarise the work outlined in this thesis and conclude by discussing ways in which future studies can improve upon and further the work described here.

40 Chapter 2

Theory

In this chapter, the theoretical constructs that underpin the research contained in this thesis are introduced. The first section describes the nature of the interactions between electrons and ions in the corona which cause the electromagnetic emission we detect at Earth. Understanding these processes is vital to interpreting observations of the solar corona and unravelling the physics that drives solar flares. The second section discusses the concepts and equations of hydrodynamics. This framework allows us to probe the forces and energies involved in plasma flows in the corona. We then use these principles to develop and discuss models to explain the thermal evolution of solar flares.

41 2. THEORY

2.1 Atomic Physics

Our primary method of examining the Sun is measuring the electromagnetic emission it produces. However, making sense of these observations requires an intricate understanding of how that emission is created. This thesis is primarily concerned with thermal emission of solar ﬂares in the corona. The coronal temperature ranges from

∼2 MK in quiet regions to ∼40 MK in the hottest ﬂares. Therefore coronal thermal emission is dominated by EUV and X-rays.

The corona is optically thin and therefore cannot be modelled as a blackbody as in the case of the photosphere. What’s more, the corona is highly ionised due to the high temperatures found there. Hydrogen and helium are fully ionised while heavier elements are at least partially ionised. Thus the spectrum of the corona is dominated by emission lines from these ions with contributions from two-photon, free-free and free-bound continua. This can be seen in Figure 2.1 which shows a modelled CHI-

ANTI spectrum (Section 2.1.4) for an active region in the wavelength range 1–400 A˚

(EUV/X-ray regime). It is dominated by a myriad of spiky features (emission lines) which are superimposed on more slowly varying continua. The strengths of each emission line/continuum is dependent on the physical conditions of the emitting plasma.

Therefore, an intricate understanding of the processes responsible for this emission can help us infer these physical conditions. This is important not only for studying spectrally resolved observations, such as those made with SDO/EVE (Section 3.4.2), but also spectrally unresolved ones (‘broadband’), such as those made by GOES/XRS (Sec- tion 3.1.1). This is because the underlying spectrum will determine the measurements made by these broadband instruments. This also includes so-called ‘narrowband’ ﬁlters on instruments such as SDO/AIA (Section 3.4.1). Although these may be centred on the wavelength of a single emission line, they nonetheless include contributions from other emission lines and continua. This further underlines the importance of a

42 2.1 Atomic Physics , 2012). The et al. ˚ A created using the CHIANTI atomic physics database (Landi Model solar spectrum from 1 – 400 Figure 2.1: inﬂuence of both emission lines and continua can clearly be seen.

43 2. THEORY

Figure 2.2: Contributions from free-free, free-bound, and two-photon continua to the solar spectrum in the range 1 – 300 A˚ (EUV/X-ray regime). Calculated with CHIANTI by Raftery (2012).

comprehensive knowledge of emission processes when studying the Sun.

2.1.1 Continuum emission

Continuum emission is so called because photons are produced over a wide continuous wavelength range. This is in contrast to line emission (Section 2.1.2) which forms within very precise narrow wavelength ranges. There are three main continuum mechanisms in the EUV/X-ray regime of the solar corona. These are free-free, free-bound and two- photon, which are named according to the relationship between the particles both before and after the interaction which produces the emission. The contributions from each process can be calculated using CHIANTI (Section 2.1.4) and are shown in Figure 2.2.

It can be seen that the two dominant processes are free-bound and free-free, with the two-photon continuum providing only minor contributions throughout the spectrum.

Free-bound emission occurs when an electron and ion become bound, as seen in

Figure 2.3a. The bound conﬁguration has a lower energy than that of the two free

44 2.1 Atomic Physics

Figure 2.3: Diagrams of a) free-bound and b) free-free emission processes. Adapted from Aschwanden (2004).

particles and the excess energy can be released as a photon. In a thermal plasma, there is a continuous distribution of particles’ kinetic energies, deﬁned by the Maxwell-

Boltzmann distribution. Therefore, a continuous distribution of photon energies is produced, resulting in the free-bound continuum seen in Figure 2.2. This process can also occur in reverse if the electron gains enough energy to escape the ion. If this energy is provided by interaction with a photon, then this process absorbs, rather than emits.

However this is rare due to the low densities of the corona.

Free-free emission occurs when two unbound particles scatter and remain unbound.

Figure 2.3b shows this process between a free electron of electric charge −e, and an ion of charge +Ze. As the trajectory of the electron is altered by the electric ﬁeld of the ion, it emits a photon resulting in a loss of kinetic energy. This is known as bremsstrahlung radiation (German for ‘braking’). Once again, the continuous distribution of particle energies results in the free-free continuum in Figure 2.2. This thermal bremsstrahlung process is the most common interaction in the solar corona and is dominant throughout most of the EUV/X-ray regime. Therefore it is discussed below in more detail.

2.1.1.1 Thermal Bremsstrahlung

As mentioned above, thermal bremsstrahlung involves the emission of a photon via the scattering of an electron and an ion. As the ion is so much heavier than the electron, it can be assumed that only the electron is aﬀected by the interaction. Therefore, the

45 2. THEORY energy of the resultant photon is equal to the diﬀerence between the initial and ﬁnal kinetic energies of the electron (Eγ = Ei − Ef ).

Such emission in the corona emanates from an entire population of electrons with various initial energies. Therefore the bremsstrahlung process is related to the photon energy spectra which we observe at Earth. Following Aschwanden (2004), consider an electron such as that in Figure 2.3b. Let it have a mass me and charge −e, moving at a velocity v, through a volume of stationary ions of ion density, nion, each with charge

+Ze. The total power, Pi(v, ν), emitted by the electron due to collisions with the ions in the frequency range ν – ν + dν, is given by

Pi(v, ν) = nionvσr(v, ν) (2.1)

where σr(v, ν) is the radiation cross-section given by

16 Z2e6 Z bmax db σr(v, ν) = 2 3 2 3 mec v bmin b 16 Z2e6 π √ 2 −1 = 2 3 2 g(ν, T ) [cm erg Hz ] (2.2) 3 mec v 3

Here, c is the speed of light in a vacuum and b is the impact parameter. This is de-

ﬁned as the perpendicular distance between the ion and the electron’s initial trajectory

(Figure 2.3b). g(ν, T ) is the gaunt factor which is approximately unity in the corona.

A plasma such as the corona is made up of a sea of electrons, with electron density, ne. Therefore, the total bremsstrahlung power emitted by the plasma per unit volume, per unit frequency, per unit solid angle (i.e. volume emissivity, ν) can be calculated by integrating the total bremsstrahlung power of one electron (Equation 2.1) over the plasma’s electron velocity distribution, f(v):

n Z = n e P (v, ν)f(v)dv (2.3) ν ν 4π i

46 2.1 Atomic Physics

Here, nν is the refraction index of the plasma. For a thermal plasma, the velocity distribution is given by the Maxwell-Boltzmann distribution.

2 1/2 m 3/2 mv2 f(v)dv = e v2 exp − dv (2.4) π kBT 2kBT

where kB is the the Boltzmann constant and T is temperature. Substituting this into Equation 2.3 and integrating gives

n n E = 5.4 × 10−39Z2n ion e g(ν, T ) exp − γ [erg s−1 cm−3 Hz−1 rad−2] (2.5) ν ν 1/2 T kBT

where Eγ = hν is the photon energy and h is Planck’s constant.

The ﬂux measured at Earth, F , is related to the volume emissivity in the following manner:

Z F ∝ νdV (2.6) V where V is the volume of the emitting source. Substituting Equation 2.5 into the above relation gives the photon ﬂux at Earth as a function of photon energy. Further realising that, ﬁrstly, in the corona the ion and electron densities are roughly equal (nion ≈ ne), secondly, hydrogen is by far the most abundant element (Z ' 1), and thirdly, the gaunt factor is approximately unity (g(ν, T ) ' 1), gives

Z 1 E F () ∝ exp − γ n2dV [keV s−1 cm−2 keV−1] (2.7) 1/2 e V T kBT

R 2 If this plasma is assumed to be isothermal and V nedV is deﬁned to be the emission measure, EM, then Equation 2.7 can be rewritten as

1 E F () ∝ exp − γ EM [keV s−1 cm−2 keV−1] (2.8) 1/2 T kBT

47 2. THEORY or

F () ∝ G(T )EM [keV s−1 cm−2 keV−1] (2.9) where G(T ) is the contribution function, which contains all the temperature dependent terms. This is discussed in more detail in Section 2.1.3.

This form of the equation is very useful. It shows that the photon energy spectrum of thermal bremsstrahlung emission is dependent on only two parameters: temperature and emission measure. This means that given an observed thermal bremsstrahlung spectrum, the temperature and emission measure of the emitting source can be inferred by ﬁtting it with Equation 2.8. This is precisely the sort of method used when analysing RHESSI observations. If the plasma is assumed to have two temperature regimes, Equation 2.8 can be expanded to include two terms, each a function of different temperatures and emission measures, T1, EM1, and T2, EM2. However, as the multi-thermal nature of the plasma increases, temperature becomes a continuous function and the diﬀerential emission measure (DEM) must replace the emission measure.

This is discussed in more detail in Section 2.1.3.

2.1.2 Emission Lines

While continuum emission contributes signiﬁcantly, the solar spectrum in the EUV/X-ray regime is dominated by a myriad of emission lines (Figure 2.1). These are very narrow well-deﬁned wavelength intervals of intense emission. Unlike the continua which are caused by free-free and free-bound processes, emission lines are cause by bound-bound processes, whereby an electron gains or loses energy while remaining bound to the ion.

48 2.1 Atomic Physics

Figure 2.4: Bohr model of the atom showing a positive nucleus of protons and neutrons surrounded by electrons of discrete energies/orbits described by the principal quantum number, n. (Suchocki, 2004).

49 2. THEORY

2.1.2.1 Atomic Structure

Emission lines stem from the quantum mechanical nature of the atom. The Bohr model

(Figure 2.4) depicts the atom as a central nucleus of protons and neutrons surrounded by electrons of discrete energies contained by the Coulomb force (also known as orbits or levels). The angular momentum, L, in each orbit is quantised, such that L = n~, where ~ = h/(2π) is the reduced Planck constant, and n = 1, 2, 3, ... is the principal quantum number describing the orbit (Figure 2.4). The electrons usually occupy the least energetic orbit available. However, if the electron gains energy through a collision with another particle or absorption of a photon, it may jump to a higher orbit. This is called excitation. An excited state is energetically unfavourable and eventually the electron will fall back to a lower orbit. This is known as de-excitation. One of the ways an atom can de-excite is via the emission of a photon with an energy equal to the difference between the two energy levels (Figure 2.5). If the electron is excited with enough energy it can escape the atom altogether. This is called ionisation. The degree of ionisation is the number of electrons which have been removed from the neutral atom. This is equivalent to the difference between the number of electrons and protons in the ion. The ionisation state is designated, X+m, where X is the element symbol and m denotes the number of electrons removed. In astrophysics, Roman numerals are often used instead of m. However in this nomenclature I denotes the neutral atom while II denotes the first ionisation state and so on. For example, Fe I (≡ Fe) denotes neutral iron, while Fe IX (≡ Fe+8) denotes iron with eight electrons removed. The reverse process to ionisation, where a free electron is captured by an ion, is called recombination. This is what causes the free-bound continuum discussed above.

For a hydrogen-like ion (an ion with one electron), the wavelength of a photon emitted when an electron decays from an upper to a lower one can be described by the

50 2.1 Atomic Physics

Figure 2.5: Diagram of an electron decaying from an upper atomic orbit to a lower one with the emission of a photon with an energy equal to the diﬀerence between the two levels. (Raftery, 2012).

Rydberg formula.

1 1 1 −1 = RM 2 − 2 [m ] (2.10) λ nl nu where nu and nl are the principal quantum numbers of the upper and lower orbits, respectively, and RM is the Rydberg constant given by

2 4 Z µe −1 RM = 2 3 [m ] (2.11) 8ε0h c

Here, Z is the number of protons in the nucleus, µ is the reduced mass of the electron and nucleus, e is the electron charge, ε0 is the electric permittivity of free space and c is the speed of light in a vacuum. Note that RM is a function of Z. For the hydrogen atom, the transition from the n = 2 to n = 1 orbits (known as the Lyman-α transition) will result in a photon of wavelength 1215 A.˚ However for singly ionised helium, for which Z2 = 4, the same transition will result in a wavelength of 304 A.˚ For ions with a greater numbers of electrons, the equivalent of the Rydberg formula becomes increasingly complicated.

In addition to the principal quantum number, an electron’s energy state is also described by three other quantum numbers: the orbital angular momentum quantum

51 2. THEORY number, l = 0, ..., n − 1, in integer steps; the projected angular momentum quantum number (also known as the magnetic quantum number), ml = −l, ..., l, in integer steps; and the spin quantum number, ms = −~/2, +~/2. The combination of these four quantum numbers describes a unique energy state in accordance with the Pauli Exclusion

Principle. This adds a plethora of additional energy levels within the framework of the principle orbits. These additional energy levels drastically increase the number of possible transitions, and hence photon energies which can be emitted by an ion. All of these must be taken into account in order to satisfactorily model the emission coming from the corona.

2.1.2.2 Modelling Emission Line Flux in the Corona

Consider a plasma containing an elemental species of a given ionisation state, X+m, with number density, nion. The rate of photon emission due to a transition from an upper energy state, u, to a lower energy state, l, is proportional to both the density of ions in the upper energy state, nu, and the transition probability.

To calculate nu we must account for all the ways in which an ion can be both excited and de-excited, and the probabilities of those occurring. The top row of Figure 2.6 shows the three ways an ion can be excited. The ﬁrst two are collision with an electron and collision with a proton. The rate of collisional excitation via an electron and proton for a single ion depends on the densities of free electrons (ne) and protons (np). They are

e p written as neCl,u and npCl,u, respectively. The ﬁnal excitation process is stimulated absorption, i.e. the absorption of a photon. The rate of stimulated absorption for a given ion is dependent on the density of photons with an energy equal to the diﬀerence between the two energy levels and is written simply as Bl,u. The bottom row of Figure 2.6 shows the four ways in which an ion can be de-excited.

The ﬁrst two again are collision with an electron or proton. These de-excitations do not cause the emission of a photon. Similar to before, the rates of collisional de-excitation

52 2.1 Atomic Physics

Figure 2.6: Schematic of the various excitation (top row) and de-excitation (bottom row) processes which can occur in the corona. Adapted from Aschwanden (2004).

e p are written as neCu,l and npCu,l. The final two de-excitation process are stimulated emission and spontaneous emission. Both of these processes result in emission of a photon with energy equal to the difference between the energy levels (as well as the incident photon, in the case of stimulated emission). The rates of stimulated and spontaneous emission for a single ion are Bu,l and Au,l, respectively. Au,l is often called the Einstein coefficient of spontaneous emission.

Under the assumption of statistical equilibrium, the rate of excitations equals the rate of de-excitations. This gives

X e X p X nlneCl,u + nlnpCl,u + nlBl,u = l l l

Each collisional term has been scaled by and summed over the density of ions in other levels l, while each stimulated/spontaneous term has been scaled by and summed over the density of ions in lower energy levels in order to get the total number of transitions

53 2. THEORY per unit volume per unit time to and from state u. For clarity, the location of each term in the equation is the same as its corresponding diagram in Figure 2.6.

The full treatment of Equation 2.12 is somewhat unwieldy but can be simpliﬁed by employing some physically justiﬁed assumptions of the corona. The corona is optically thin and so photons rarely interact with ions. Therefore it is assumed that Bl,u =

Bu,l = 0. Furthermore it assumed that ionising collisions are primarily due to electrons p p and so Cl,u = Cu,l = 0. Finally, because the corona is so tenuous the rate of collisions is typically much lower than the rate of spontaneous emission. Therefore it can be

e assumed that Cu,l = 0 and that all excitations are from the lowest possible energy level, known as the ground state. This also means the density of ions in the ground P state is roughly the density of the ion in question, i.e. l nl ≈ ng ≈ nion, where subscript g denotes the ground state. These assumptions imply that the energy level, u, is populated by electron collisions with ions in their ground state, g, and then depopulated via spontaneous emission. This reduces Equation 2.12 to

e X nionneCg,u = nu Au,l (2.13) l

This is known as the coronal approximation.

The energy emitted per unit volume per unit time by the plasma as the result of a certain transition from level u to l is the volume emissivity, u,l. This is given by

u,l = nuAu,lhνu,l (2.14)

where hνu,l is the energy of a photon released due to the transition. The ﬂux due to this emission line measured at Earth is given by

1 Z 1 Z Fu,l = 2 u,ldV = 2 nuAu,lhνu,ldV (2.15) 4πR V 4πR V

54 2.1 Atomic Physics where R is 1 AU. Here we have assumed that the corona is optically thin, as in the coronal approximation and have therefore not included any absorption term. Although measuring nu is impractical, it can be written in terms of properties which in principle can be measured.

nu nion nel nH nu = ne (2.16) nion nel nH ne

Here, nu/nion is the fraction of the given ions in energy level u, nion/nel is the fraction of atoms of the given element in the ionisation state in question, nel/nH is the abundance of the given element relative to hydrogen, nH /ne is the ratio of hydrogen nuclei to free electrons, and ne is the electron density. However, from Equation 2.13, nu/nion = e P neCg,u/ k

Z e 1 Cg,u nion nel nH Fu,l = 2 hνu,lAu,l P dV (2.17) 4πR V k

e The collisional coeﬃcient, Cg,u, is obtained from integrating the collisional cross- section, σg,u(v), with the plasma particle velocity distribution.

Z e Cg,u = σg,u(v)f(v)vdv (2.18)

The collisional cross-section is given by (Mariska, 1992)

2 πa0Ωg,u(E) σg,u = (2.19) glE

where a0, the Bohr radius, is the typical radius of an electron’s orbit in the ground state of hydrogen, Ωg,u is the collisional strength for the given transition from the ground

2 state, g, to u, E = 0.5mev is the kinetic energy of the colliding electron, and gl is a statistical weight due to the principle of detailed balance. The velocity distribution

55 2. THEORY of a thermal plasma is given by the Maxwell-Boltzmann distribution (Equation 2.4).

Using Equations 2.19 and 2.4 to evaluate Equation 2.18 gives (Mariska, 1992)

8.63 × 10−6 Z ∞ E Ce = Ω (E) exp − dE [cm−3 s−1] (2.20) g,u 3/2 g,u glkBT Eg,u kBT

The ﬂux due to the emission line observed at Earth (Equation 2.17) can now ﬁnally be rewritten using Equation 2.20.

hν 8.63 × 10−6Ω n Z n 1 hν u,l g,u H ion √ u,l 2 −3 −1 Fu,l = 2 exp − nedV [ergs cm s ] 4πR gl ne V nel T kBT (2.21)

2.1.3 Contribution Functions & Emission Measures

It is often convenient to express Equation 2.21 in terms of temperature-dependent and temperature-independent parts. The temperature-dependent term is known as the contribution function, G(T ). It is given by

n 1 hν G(T ) = ion √ exp − u,l (2.22) nel T kBT

If we assume the plasma is isothermal, then the contribution function can be placed outside the integral and we can rewrite Equation 2.21 as

Fu,l = kG(T )EM (2.23) where k represents all the constants in Equation 2.21. The EM term in Equation 2.23 is the emission measure and is given by

Z 2 EM = nedV (2.24) V

56 2.1 Atomic Physics

Figure 2.7: Contributions functions of He I, (584.33 A),˚ O V (629.73 A),˚ Mg X (524.94 A),˚ Fe XVI (360.75 A),˚ and Fe XIX (592.23 A).˚ These were calculated with the CHIANTI soft- 9 −3 ware (Section 2.1.4) using density, ne = 5×10 cm , coronal abundances and the ionisation equilibria of Mazzotta et al. (1998). Taken from Raftery (2012).

Here, the electron density, ne, can be a function of position within the emitting volume, i.e. the density may not be homogeneous. Figure 2.7 shows examples of the contribution functions of ﬁve emission lines in the corona: He I, (584.33 A),˚ O V (629.73 A),˚ Mg X

(524.94 A),˚ Fe XVI (360.75 A),˚ and Fe XIX (592.23 A).˚

In the above paragraph we assumed that the plasma was isothermal. If however the plasma is multi-thermal, we must replace the emission measure with the diﬀerential emission measure (DEM) which describes the temperature distribution throughout the emitting plasma. It is given by

dEM dV DEM = = n2 (2.25) dT e dT

57 2. THEORY

This would mean that Equation 2.21 would have to be rewritten as

Z dEM Fu,l = k G(T ) dT (2.26) T dT

2.1.4 CHIANTI Atomic Database

In order to model the solar spectrum accurately, all the signiﬁcant emission lines and continua must be reproduced. To tackle this huge task, we, in this thesis, use the

CHIANTI atomic physics database (Dere et al., 1997) which is available through Solar

SoftWare (SSW). First established in 1996, CHIANTI has become widely used by the solar physics community to analyse observed spectra. CHIANTI is periodically updated and the current version is used in this thesis (version 7; Landi et al., 2012).

As well as the mathematical tools outlined in Sections 2.1.1 and 2.1.2, CHIANTI requires additional information to model the solar spectrum. This includes the elemental abundances and the ionisation fractions of each element. In this thesis, we use the coronal elemental abundances of Feldman et al. (1992) and the ionisation equilibria of

Mazzotta et al. (1998) unless otherwise stated. In addition, a DEM is needed. This can be entered manually from observation or chosen from sample DEMs typical of diﬀerent coronal regions (quiet sun, active region, ﬂare etc.). Alternatively, the plasma can be assumed to be isothermal which corresponds to a delta-function DEM. A comparison between multi-thermal and isothermal DEMs is the subject of Chapter 5.

2.1.5 Radiative Loss Function

Treatment of single lines is useful for comparison with spectrally resolved observations from instruments such as SDO/EVE (Section 3.4.2). However for spectrally unresolved observations from instruments such as GOES/XRS and SDO/AIA (Sections 3.1.1 and

3.4.1), it is often more appropriate to deal with the summed intensities of all lines and continua in a given temperature interval, T – T + dT . The rate at which energy is

58 2.1 Atomic Physics

Figure 2.8: Calculations of the radiative loss function, Λ(T ), compiled from various studies. (Aschwanden, 2004).

radiated per unit volume from this temperature interval by a plasma of a given density is the radiative loss rate, ER(T, n). It is given by

2 ER(T, n) = neΛ(T ) (2.27) where Λ(T ) the temperature dependent term, known as the radiative loss function.

Figure 2.8 shows radiative loss functions calculated by various studies. The diﬀer- ence between them is often due to a diﬀerent assumption of the elemental abundances

(e.g. photospheric or coronal). Rosner et al. (1978) derived an analytical expression for the radiative loss function in the form of a six piece power-law parameterisation. This is used in Chapter 6 of this thesis.

59 2. THEORY

2.2 Hydrodynamics

Having developed an understanding of how emission from a plasma relates to its physical conditions, we must next understand the nature and evolution of the plasma itself.

This is the purpose of hydrodynamics, which is the study how a fluid flows and interacts with time. Hydrodynamics is based on fundamental conservation laws of mass, momentum and energy. It is linked with thermodynamics which considers the flow of heat in a medium and the work done due to that heat flow. In this section we shall discuss the origins and relevance of the basic equations of hydrodynamics. We then use this framework to describe flare cooling models which will be used in Chapter 6 of this thesis. We begin with plasma kinetic theory, which is the basis of much of plasma physics.

2.2.1 Plasma Kinetic Theory

Kinetic theory treats a plasma or gas in terms of the motions of the individual constituent particles. These particles are described in terms of their position vector, rˆ = xî + yˆj + zkˆ, and velocity vector,v ˆ = vxî + vyˆj + vzkˆ, where î, ˆj and kˆ are the unit vectors in the x, y, and z directions, respectively. This creates a six dimensional phase space in which a particle in three spatial dimensions is described by six coordi- nates. The distribution of particles with various velocities through space is described by the velocity distribution function, f(ˆr, v,ˆ t). There are many different possible velocity distribution functions, but perhaps the most important is the Maxwell-Boltzmann distribution (Figure 2.9). This was used earlier in this chapter and has the general form of

2 f(ˆr, v,ˆ t) = f(v) = Ce−av (2.28)

60 2.2 Hydrodynamics

Figure 2.9: The Maxwell-Boltzmann distribution, showing the distribution of velocities among particles in a gas or plasma in thermal equilibrium. (Inan & Golkowski, 2011)

where v = |vˆ|, and a = m is a constant representing the distribution’s width (with m 2kB T the mass of each particle, kB the Boltzmann constant, and T the plasma temperature), a 3/2 while C = n π (where n is number density) is a normalisation factor equal to the velocity number density at the peak. (See Inan & Golkowski 2011 for derivation of these constants.) The peak of the distribution gives the most probable velocity of particles in the plasma. However, it can be seen that there are some particles with velocities much smaller and bigger than this value. Although this distribution is only one of many possibilities, it is so important because it describes the velocity distribution of a gas or plasma in thermal equilibrium.

Although the position and velocity of each individual particle is impossible to measure, or even to model, quantities such as temperature, density, thermal energy etc. can be derived from the velocity distribution function. This makes plasma kinetic theory very useful in relating the microscopic behaviour of distributions of particles to the macroscopic hydrodynamic quantities we measure.

61 2. THEORY

The number density, n(ˆr, t), can be obtained by integrating over all velocities:

Z ∞ n(ˆr, t) = f(ˆr, v,ˆ t)d3v (2.29) −∞

3 where d v = dvxdvydvz is the velocity element. Other macroscopic quantities are related to averages weighted by the distribution. In other words, an average quantity, gav(ˆr, t), can be obtained by:

1 Z g (ˆr, t) = g(ˆr, v,ˆ t)f(ˆr, v,ˆ t)d3v (2.30) av n(ˆr, t)

3 For example, the thermal energy of the plasma, 2 kBT is simply a measure of the 1 2 average kinetic energy of the particles h 2 mv i (where angle brackets denote an average). Therefore, the thermal energy of the plasma and hence the temperature can be obtained by:

3 1 1 m Z k T = mv2 = v2f(ˆr, v,ˆ t)d3v (2.31) 2 B 2 n(ˆr, t) 2

The average momentum etc. of the particles can be obtained similarly.

The next step is to understand how the velocity distribution evolves with time, which will help us to understand the hydrodynamic and thermodynamic evolution of the plasma as a whole. The evolution of the velocity distribution is determined by the Boltzmann equation. Consider a volume element in position/velocity phase space over the time interval, dt. This time interval must be long relative to the time for an interaction between particles to be completed, yet short enough that the particles will only undergo one interaction. For simplicity, consider only the position and velocity along the x-axis, x, vx. The volume element can be represented as a rectangle of width between x and x + dx, and height between v and v + dv (Figure 2.10). The total number of particles in this 2D volume element is f(x, vx, t)dxdvx. Particles can move

62 2.2 Hydrodynamics

Figure 2.10: Volume element, dxdvx, in position/velocity phase space showing the ways in which particles can enter and leave that volume element. Particles entering/leaving the volume at side ‘3’ and ‘4’ move in or out of the spatial range x – x+dx due to their position. Particles entering/leaving at sides ‘1’ or ‘2’ are accelerated or decelerated in or out of the range vx – vx + dvx by an external force, e.g. the Lorentz force. Also shown are particles accelerated/decelerated into the volume element via collisions. This picture is very useful in deriving the Boltzmann equation which describes how the velocity distribution evolves with time. (Inan & Golkowski, 2011)

into this phase space in two ways. They can move into the spatial range, x – x + dx via a change in position, or they can be accelerated into the velocity range v – v + dv by an external force, F . In the case of a plasma, this would be the Lorentz force,

F = q[Eˆ +v ˆ × Bˆ], where q is the electric charge of the particle, Eˆ is the electric ﬁeld, and Bˆ is the magnetic ﬁeld. In addition, particles can move or be accelerated in or out of the volume element via collisions with other particles. The net gain/loss of particles in the volume element via collisions is denoted as ∂f . ∂t coll

The total change in the number of particles in the 2D volume element in the time

63 2. THEORY dt, is

∂ [f(x, v , t)dxdv ] = ∂t x x

[f(x, vx, t)vx − f(x + dx, vx, t)vx]dvx ∂f + [f(x, vx, t)ax(x, vx, t) − f(x, vx + dvx, t)ax(x, vx + dvx, t)]dx + ∂t coll (2.32)

∂vx where ax = ∂t is acceleration in the x-direction. The ﬁrst term on the RHS is the net gain/loss of particles moving in and out of the volume element spatially (horizontally in

Figure 2.10) and the second term is the net gain/loss of particles accelerating in or out of the phase volume (vertically in Figure 2.10). Dividing both sides by dxdvx, recalling that ax = Fx/m in the x-direction, and rearranging gives the Boltzmann equation in one dimension.

∂f ∂f Fx ∂f ∂f + vx + = (2.33) ∂t ∂x m ∂vx ∂t coll

Extending to three dimensions gives

∂f Fˆ ∂f +v ˆ · ∇rf + · ∇vf = (2.34) ∂t m ∂t coll where

∂ ∂ ∂ ∇ ≡ ˆi + ˆj + kˆ r ∂x ∂y ∂z ∂ ∂ ∂ ∇v ≡ ˆi + ˆj + kˆ ∂vx ∂vy ∂vz

If it is assumed that the force is the Lorentz force, and that the plasma is collision-

64 2.2 Hydrodynamics less, the Boltzmann equation reduces to the Vlasov equation,

∂f q +v ˆ · ∇ f + [Eˆ +v ˆ × Bˆ] · ∇ f = 0 (2.35) ∂t r m v which is very important in describing the evolution of plasmas.

2.2.2 Equations of Hydrodynamics

The fundamental equations of hydrodynamics can be derived directly from the Boltz- mann equation by taking its moments. The ith moment involves multiplying the velocity distribution in the Boltzmann equation by velocity to the ith power, vi, i.e.

i ˆ i ∂v f i F i ∂v f +v ˆ · ∇rv f + · ∇vv f = (2.36) ∂t m ∂t coll

The zeroth moment (v0) leads directly to the conservation of mass (or charge), otherwise known as the continuity equation.

∂ n(ˆr, t) = −∇ · [n(ˆr, t)ˆu(ˆr, t)] (2.37) ∂t whereu ˆ(ˆr, t) is the average velocity of the particles and we have dropped the spatial subscript, r from the ∇ for convenience. It states that the rate of change of density is given by the divergence of the ﬂow of particles. (For derivation, see Inan & Golkowski

2011.)

The ﬁrst moment (v1) leads to the conservation of momentum when both sides are multiplied by the particle mass, m. This is also known as the momentum transport equation. For a given particle species, i (e.g. electrons), it is given by,

duˆ mn = −∇p + n Fˆ + Sˆ (2.38) i dt i ij

65 2. THEORY

where ∇p is the pressure gradient, and Sˆij represents momentum gained or lost due to collisions with particles of other species, j (e.g. protons, ions, neutrals etc.). This equation states that the net rate of change of the momenta of the particles in species i is caused by the pressure gradient (ﬁrst term, RHS), the external force (e.g. the

Lorentz force; second term, RHS), and collisions with particles of other species. This equation assumes the plasma is isotropic, i.e. contains no shearing motions. To relax this assumption, the pressure, p, must be replaced to the pressure tensor, Ψ. (See Inan

& Golkowski 2011 for derivation.)

The second moment of the Boltzmann equation (v2) gives the conservation of en-

1 ergy, when both sides are scaled by 2 nm. This is also known as the energy transport equation and will be used a number of times in this thesis.

∂[ 1 nmhv2i] 1 2 = −∇( nmhv2iuˆ) + p∇ · uˆ + ∇ · qˆ + Sˆ (2.39) ∂t 2 coll whereq ˆ is the heat-ﬂow vector, which can include various heating and cooling processes, e.g. conduction, and even radiation. (See Inan & Golkowski 2011 for derivation.) Since

1 2 2 nmhv i, the average kinetic energy of the particles, is also the thermal energy of the 3 1 plasma, it can be re-expressed as 2 kBT , or even γ−1 p where γ is the adiabatic constant, or ratio of speciﬁc heats. Therefore the energy transport equation can also be expressed as

1 ∂p 1 = − ∇(puˆ) + p∇ · uˆ + ∇ · qˆ + Sˆ (2.40) γ − 1 ∂t γ − 1 coll

The energy transport equation states how the thermal energy density of a plasma can change with time. The LHS represents the rate of change of thermal energy density while the RHS represents all the diﬀerent ways in which the energy can change. The

ﬁrst term represents energy added or removed due to addition or removal of particles via ﬂows. The second term represents energy gained or lost due to contraction or

66 2.2 Hydrodynamics expansion of the plasma. The third term represents energy gained or lost due to heat-flow processes such as external heating, conduction, radiation etc. While the final term represents energy gained or lost due to collisions with particles of different species.

Once again these forms of the energy transport equation assume the plasma is isotropic.

Replacing the pressure term in Equation 2.39 with the pressure tensor, Ψ, would relax this assumption.

These equations are the bedrock of hydrodynamics and are essential for understanding the evolution of plasma in ﬂaring loops. In the rest of this chapter we use these laws to discuss and outline ﬂare cooling models.

2.2.3 Flare Cooling Models

Since the corona is a magnetised plasma, it is subject not only to hydrodynamics, but magnetohydrodynamics (MHD). MHD is concerned with motions and evolution of electrically conducting and magnetised fluids. It combines the Maxwell Equations of electromagnetism with the hydrodynamics equations discussed above. Since the electric and magnetic fields are coupled, these differential equations must be solved simultaneously. For example, the magnetic field could exert a Lorentz force on the charged particles in the plasma, which would in turn alter their motion and hence the electric field, thus altering the original magnetic field itself. Solving these equations in a system such as the Sun can often only be done numerically and requires vast amounts of computing power. This means that full 3D MHD treatments are slow and difficult.

The problem can be simpliﬁed by making some physically motivated assumptions.

In this thesis we are concerned with the evolution of ﬂaring coronal plasma after the initial energy release. This coronal plasma is a low-β plasma in which the plasma is frozen onto the magnetic ﬁeld at all times except during process of magnetic reconnection.

After magnetic reconnection, the magnetic ﬁeld is typically arranged in loop-like structures whose ﬁeld strength and topology are not seen to vary quickly with time. This

67 2. THEORY means that the ﬂare loops can be modelled as isolated rigid curved cylinders of plasma.

What’s more, the magnetic field does not allow the plasma to move perpendicular to the axis of the magnetic field. Therefore, the only spatial coordinate which must be considered is that along the axis of the magnetic field. By confining the plasma to one dimension, we approximate the effect of the magnetic field without having to explicitly solve for it. Thus the problem is reduced from 3D magnetohydrodynamic problem to a 1D hydrodynamic one and allows us to use the equations outlined in Section 2.2.2.

This approach drastically cuts down on computing time and allows higher resolution simulations to be carried out.

There have been several studies aimed at hydrodynamic ﬂare modelling over the decades (e.g. Antiochos, 1980; Antiochos & Sturrock, 1978; Bradshaw & Cargill, 2005;

Cargill, 1993; Doschek et al., 1983; Fisher et al., 1985b; Klimchuk, 2006; Klimchuk

& Cargill, 2001; Moore & Datlowe, 1975; Reeves & Warren, 2002; Sarkar & Walsh,

2008; Warren, 2006; Warren & Winebarger, 2007). These include both 1D and 0D hydrodynamic models. Although 1D models are much more wieldy than their 3D counterparts, they nonetheless still require a lot of computing power. 0D models reduce the problem further by treating ﬁeld-aligned averages of the hydrodynamic properties

(e.g. temperature, density, pressure etc.). This means that simulations can be run in seconds or minutes instead of days or even weeks. Dealing with field-aligned averages is justified by the fact that hydrodynamic properties are fairly constant throughout the loop, except near the interface of the corona and transition region which is characterised by steep gradients. However, despite the sizeable simplifications of 0D models, they have been found to compare favourably with their 1D counterparts (Klimchuk et al.,

2008). The 0D model used in this thesis is that of Cargill et al. (1995) and is now discussed in detail.

68 2.2 Hydrodynamics

2.2.4 The Cargill Flare Cooling Model

The Cargill model (Cargill et al., 1995) is an easy-to-use analytical 0D hydrodynamic model which describes the cooling of flare plasma during its decay phase. It is based on conductive and radiative cooling timescales derived from the energy transport equation discussed above. In this equation we can separate the heat-flow term (third term) into conductive and radiative components as well as an additional heating term for unknown heating processes such as magnetic reconnection etc. By assuming that the plasma is confined to the axis of the magnetic field, s, and that there are no shearing motions within the plasma, the energy transport equation (Equation 2.40) can be reduced to one dimension and written as

1 ∂p 1 ∂ ∂u ∂ = − (pu ) − p s − F − n2Λ(T ) + S + h (2.41) γ − 1 ∂t γ − 1 ∂s s ∂s ∂s c coll

In this equation, γ is the adiabatic constant, p is pressure, us is the plasma ﬂow ve-

2 locity (along the axis of the magnetic field, s), Fc is the conductive heat flux, n Λ(T ) is the radiative loss rate (Equation 2.27, where we have dropped subscript ‘e’ for convenience), Scoll is collisional energy loss rate between different particle species (e.g. between electrons and protons), and h is the heating rate per unit volume.

The Cargill model makes a number of assumptions to simplify Equation 2.41.

Firstly, it is assumed that no energy is lost or gained through ﬂows, and that the

flare loop does not appreciably expand or contract during the decay phase. This causes the first two terms on the RHS to vanish. Secondly, energy exchanged between particles of different species via collisions is assumed to be negligible, i.e. Scoll = 0. It is further assumed that the plasma is isothermal and obeys the ideal gas law, p = nkBT , where kB is the Boltzmann constant and T is temperature. Although this assumption is not necessarily well justified in flares, it is often the only assumption possible and is common throughout the literature. The conductive term (third term RHS) is given

69 2. THEORY

∂T by the divergence of the heat ﬂux, Fc, which in turn is given by Fc = −κ ∂s , where ∂T κ is the thermal conductivity and ∂s is the temperature gradient. In this treatment 5/2 −6 Spitzer conductivity is assumed, i.e. κ = κ0T where κ0 ≈ 10 is a constant. Mean- while the radiative loss rate, n2Λ(T ), can be approximated in the range 106 – 107 K the parameterisation of Rosner et al. (1978), i.e., n2Λ(T ) = nζ χT α erg cm3 s−1, where

ζ = 2, χ = 1.2 × 10−19 and α = −1/2.

These assumptions allow us to rewrite Equation 2.41 as

1 ∂nk T ∂ ∂T B = − κ T 5/2 − χn2T −1/2 + h [ergs cm−3 s−1] (2.42) γ − 1 ∂t ∂s 0 ∂s

This equation states that the ﬂare plasma’s evolution is determined by the balance between conductive and radiative losses and additional heat introduced to the system, for example, via additional magnetic reconnection.

The conductive cooling timescale can be calculated from Equation 2.42 by neglecting heating and radiative losses. Rearranging Equation 2.42 under these assumptions gives

∂ ∂T k ∂T [κ T 5/2 ] = B n [ergs cm−3 s−1] (2.43) ∂s 0 ∂s γ − 1 ∂t

This assumes that density, n, is constant in time. Although this is partially justiﬁed by the fact that the emission measure (∝ n2) varies slower than the temperature, it is nonetheless a weakness of the model. Integrating both sides of Equation 2.43 with respect to s, from −L to L where L is the loop half-length gives

2k nL dt = B ds [s] (2.44) 5/2 κ0(γ − 1) T

Integrating the LHS from 0 to τc, the characteristic conductive timescale, and the RHS

70 2.2 Hydrodynamics again from −L to L, gives

4k nL2 τ = B [s] (2.45) c 5/2 κ0(γ − 1) T

Finally, assuming that the plasma is monatomic, (γ = 5/3), gives

nL2 τ = 4 × 10−10 [s] (2.46) c T 5/2

The radiative timescale is similarly derived, i.e. by neglecting conductive losses and heating in Equation 2.42, giving

1 ∂T χnζ T α = nk [ergs cm−3 s−1] (2.47) γ − 1 B ∂t

Again this assumes density is constant. Rearranging gives

k dt = B T −αdT [s] (2.48) (γ − 1)χnζ−1

Integrating the LHS from 0 to τr, the characteristic radiative cooling timescale, and the RHS from 0 to T gives

k T 1−α τ = B [s] (2.49) r (γ − 1)(1 − α)χ nζ−1

Finally, recalling from Rosner et al. (1978) that χ = 1.2 × 10−19, ζ = 2 and α = −1/2, and assuming the plasma is monatomic, (γ = 5/3), gives

T 3/2 τ = 3.45 × 103 [s] (2.50) r n

These timescales are only useful in calculating a total cooling time if we know the relation between them and the temporal evolution of temperature, i.e. given an initial

71 2. THEORY temperature and cooling timescale, what will the temperature be after time, t? This was derived for conductive cooling by Antiochos & Sturrock (1978) (evaporative case) and found to be

t −2/7 T (t) = T0 1 − [K] (2.51) τc

where T0 is the temperature at the start of the cooling phase and T (t) is the temperature after time, t. Meanwhile, Antiochos (1980) derived the corresponding relation for radiative cooling:

3 t T (t) = T0 1 − [K] (2.52) 2 τr

These relations are derived in Appendices B.1 and B.2, respectively.

A total cooling time can now be derived if it is assumed that the ﬂare only cools by either conduction or radiation at any one time and that there is no additional heating.

This final assumption is the least well justified of the entire model and will be discussed later in this thesis. If the conductive timescale is initially shorter than the radiative timescale, the flare is assumed to cool purely by conduction from its initial temperature,

T0, to temperature, T∗, in time, t∗, when the two timescales become equal. From then the flare is assumed to cool purely radiatively to the final temperature, TL, which takes additional time, t∗∗. It should be noted here that t∗ and t∗∗ are different from τc and

τr. The former are the periods when the ﬂare cools purely by conduction and radiation respectively. The latter are characteristic timescales of the cooling processes. The time, t∗, at which the timescales become equal is given by

τr0 7/12 t∗ = τc0[( ) − 1] [s] (2.53) τc0 where subscript ‘0’ denotes the value of that property at the start of the cooling phase.

72 2.2 Hydrodynamics

Meanwhile the temperature at this time, T∗, is given by

τr0 −1/6 T∗ = T0( ) [K] (2.54) τc0

Thus calculating t∗ from Equation 2.53 and t∗∗ from Equations 2.54 and 2.52 gives the total cooling time, ttot (= t∗ + t∗∗), as

" 7/12 # 5/12 " 1/6 # τr0 2τr0 τc0 τc0 TL ttot = τc0 − 1 + 1 − [s] (2.55) τc0 3 τr0 τr0 T0

If, however, the radiative cooling timescale is initially shorter than the conductive one, radiation is assumed to be the dominant cooling mechanism throughout the entirety of the ﬂare. This is because as temperature falls, radiation becomes more eﬃcient relative to conduction. In this case t∗ and T∗ need not be calculated and the the total cooling time can be calculated directly from Equation 2.52, giving

2τr0 TL ttot = 1 − [s] (2.56) 3 T0

The Cargill model is a very simple, easy-to-use analytical model. However, its simplicity gives rise to a number of limitations. It assumes that at any one time cooling occurs via either conduction or radiation, with an instant switch between the two when their cooling timescales are equal. This assumption may be an acceptable approximation when either the conductive or radiative timescales are much longer than the other. However, it is certainly not valid when the two timescales are similar. In addition, this model does not account for enthalpy-based cooling. This type of cooling is most signiﬁcant towards the end of a ﬂare when the temperature is low and no longer supports the plasma against gravity. Therefore these equations are not suitable for modelling plasma cooling below ∼1–2 MK.

Furthermore, the Cargill model treats the ﬂare as a monolithic loop. Other 0-D

73 2. THEORY models (e.g. Klimchuk et al., 2008; Warren, 2006) employ the idea that ﬂaring loops are comprised of many smaller strands, each heated and cooled at diﬀerent times.

However, some recent studies (e.g. Aschwanden & Boerner, 2011; Peter et al., 2013) have examined coronal loop cross-sections at high resolution and found no discernible structure. This implies that such strands are below a resolution of 0.2 arcsec or that a monolithic treatment may be justified. If the flaring loops are indeed made of sub- strands they would have the same orientation. And since the plasma is frozen onto the magnetic field lines and diffusion across them is minimal, reducing the multiple strands to one spatial dimension along the axis of the magnetic field, as in the Cargill model, is somewhat justified via symmetry. One could argue something similar for multiple loops in a flaring arcade. However, this treatment implicitly assumes that all the loops/strands are being heated and cooled simultaneously which results in the average behaviour of all the loops/strands being modeled. Despite this restriction, such an approach can still be useful in examining flare hydrodynamics, especially since the additional free parameters introduced by multi-strand modelling are very unwieldy when modelling a large number of flares.

Finally, the Cargill model does not account for heating, which has been suggested can continue well into the decay phase (e.g. Jiang et al., 2006; Warren, 2006; Withbroe,

1978). Thus this assumption is not very well justiﬁed and would be expected to produce shorter predicted cooling times than those observed. Despite the above limitations, the

Cargill model contains much of the physics believed to be responsible for ﬂare cooling, and quantifying how well it simulates observations is important for better understanding

ﬂare evolution.

74 Chapter 3

Instrumentation

As well as understanding the physical mechanisms responsible for emission from the

Sun, it is vital to understand how such emission is recorded by observational instruments. This chapter provides a detailed discussion of the various instruments used in this thesis which include imagers and spectrographs ranging across the EUV and X-ray regimes. In addition, data reduction techniques are examined which can be used to determine the thermal properties of the emitting plasma. A comprehensive understanding of these topics is essential to correctly interpret the observations and accurately deduce the physical processes occurring on the Sun.

75 3. INSTRUMENTATION

Figure 3.1: Diagram of GOES satellite

3.1 Geostationary Operational Environmental Satellite (GOES)

The Geostationary Operational Environmental Satellites (GOES) are a series of satellites operated by the National Oceanographic and Atmospheric Association (NOAA)

(Figure 3.1). They are designed primarily for monitoring terrestrial weather, providing information on severe storms, winds, ocean currents etc. Since the launch of GOES-1 in 1975 there has always been at least one GOES satellite in operation. Currently, the primary GOES satellite is GOES-15.

As well as monitoring terrestrial weather, each GOES satellite has a suite of instruments for monitoring space weather. Among these is the X-Ray Sensor (XRS) which measures X-ray emission from the Sun. The GOES/XRS has become a vital tool in solar physics as no other instrument has been self-consistently and continuously monitoring the Sun for such a long period of time.

76 3.1 Geostationary Operational Environmental Satellite (GOES)

Figure 3.2: Schematic of the GOES-8 XRS. (Hanser & Sellers, 1996).

3.1.1 The X-Ray Sensor (XRS)

The GOES/XRS measures spatially integrated solar X-ray flux every three seconds (two seconds for GOES-14 and 15). This is done in two wavelength bands, or channels: the long channel (1–8 A);˚ and the short channel (0.5–4 A).˚ The XRS is a dual ion chamber instrument. Chamber A (short channel) is filled with xenon gas and its aperture is covered by a 20 mm thick beryllium filter. Chamber B contains argon and its aperture is covered by a 2 mm thick beryllium filter. Figure 3.2 shows a schematic of the GOES-8

XRS taken from Hanser & Sellers (1996) which outlines in detail the working of the instrument. The X-ray radiation entering the chambers ionises the inert gases inside, which is translated into a current by the strong voltage across each chamber. The amount of current created per unit ﬂux of incident radiation is described as a function of wavelength by the transfer functions, G(λ). Figure 3.3 shows the transfer functions of the long and short channels of the ﬁrst twelve GOES satellites. It can be seen that

77 3. INSTRUMENTATION

Figure 3.3: Response functions against wavelength for the long and short channels of the XRS for the ﬁrst 12 GOES satellites. (White et al., 2005)

they have changed very little between satellites. This fact along with the GOES/XRS’s longevity has made it a vital standard reference by which to compare past and present observations.

The current in each channel, Ai, is related to the incident ﬂux and can be expressed as (Thomas et al., 1985)

Z ∞ Ai = Ci Gi(λ)F (EM, T, λ)dλ (3.1) 0

Here, F (EM, T, λ) is the incident ﬂux as a function of emission measure of the emitting plasma, EM, temperature of the emitting plasma, T , and wavelength of the incident radiation, λ. C is a constant near unity while the subscript ‘i’ denotes either the long channel (i = 8) or the short channel (i = 4), and Gi(λ) is the wavelength-dependent transfer function of the given channel.

From the measured current it is possible to make an estimate, Bi, of the incident

78 3.1 Geostationary Operational Environmental Satellite (GOES)

Figure 3.4: Lightcurves on the long (red) and short (blue) XRS channels from the GOES- 15 satellite for a period of three days in August 2013. Flares can be seen as spikes in the lightcurves on top of a background level of approximately B4 GOES-class. The variation in solar activity can be seen by comparing August 11 which exhibits many ﬂares, while August 12, apart from the large M1-class ﬂare, shows very little activity. Courtesy of SolarMonitor.org

79 3. INSTRUMENTATION wavelength-averaged ﬂux by dividing Equation 3.1 by the wavelength-averaged transfer function, Gi (Thomas et al., 1985).

Bi = Ai/(CiGi) (3.2)

Figure 3.4 shows XRS data from GOES-15 for the period 11–13 August 2013. The red line shows the long channel flux while the blue line shows short channel flux. Two large flares can be seen, a C8-class around 22:00 UT on August 11 and an M1-class around 10:00 UT of August 12. A few other C-class events and several B-class events can be seen on August 11, while August 12 is much quieter with a fairly constant background around B4 level and a few small B-class events. This shows that solar activity varies drastically over both hours and days in addition to the 11-year solar cycle discussed in Section 1.3.

3.1.2 Deriving Thermal Plasma Properties Using GOES/XRS

Although GOES/XRS only measures X-ray ﬂux in two passbands, techniques have been developed to derive properties of the X-ray emitting plasma from the ratio of the short and long channels. These properties include temperature, T , emission measure, EM, and the total radiative loss rate, dLrad/dt. Here we outline the technique of Thomas et al. (1985).

First, it must be assumed that the radiated ﬂux is emitted thermally by an isothermal plasma. Thus, the ﬂux can be expressed as

F (EM, T, λ) = EM × f(T, λ) (3.3) where f(T, λ) is the ﬂux per unit emission measure as a function of the temperature

80 3.1 Geostationary Operational Environmental Satellite (GOES) and wavelength. From this, Equation 3.2 can be rewritten as

Bi = EM × bi(T ) (3.4)

where bi(T ) is the temperature-dependent part of the XRS response, given by

Z ∞ bi(T ) = Gi(λ)f(T, λ)dλ/Gi (3.5) 0

Thus if the ratio of the measured ﬂuxes in each channel, R(T ), is taken, the emission measure terms drop out and the ratio becomes solely a function of temperature.

R(T ) = B4/B8 = b4(T )/b8(T ) (3.6)

The emission measure can then be calculated by inverting Equation 3.4

EM = Bi/bi(T ) (3.7)

Although this can be done using either GOES channel, it is conventionally done using the 1 –8 A˚ passband.

Different studies have employed slightly different methods to determine the relationship between temperature (and emission measure) and the flux ratio. Thomas et al. (1985) folded model X-ray spectra of isothermal plasmas at various temperatures through the response of the GOES-1 XRS. From this, the XRS flux ratio and the temperature-dependent part of the XRS response were found as a function of temperature (Figures 3.5 and 3.6, respectively). These relationships were then approximated by simple monotonic polynomial fits of the form

T = 3.15 + 77.2R − 164R2 + 205R3 [MK] (3.8)

81 3. INSTRUMENTATION

Figure 3.5: Relationship between temperature and XRS ﬂux ratio as determined by Thomas et al. (1985).

and

55 −2 2 −4 3 55 −2 3 10 b8 = −3.86 + 1.17T − 1.31 × 10 T + 1.78 × 10 T [10 W m cm ] (3.9) where temperature is in units of MK. Hence they determined the emission measure to be

55 55 −3 EM = 10 Bi/(10 bi(T )) [cm ] (3.10)

White et al. (2005) furthered this work by taking updated model spectra which took into account more modern coronal abundance estimates and folding them through the

XRS transfer functions of each of the ﬁrst twelve GOES satellites (and later for more recent satellites.) The spectra were calculated using CHIANTI (Landi et al., 1999, 2002) assuming the ionisation equilibria of Mazzotta et al. (1998) and a constant density of

82 3.1 Geostationary Operational Environmental Satellite (GOES)

Figure 3.6: Relationship between temperature-dependent part of the XRS response and temperature as determined by Thomas et al. (1985).

83 3. INSTRUMENTATION

1010 cm−3. Although this latter assumption may not the case in solar flares, White et al. (2005) justified it by calculating the spectra of isothermal plasmas at 10 MK with densities of 109, 1010 and 1011 cm−3 and showed that there were no significant differences between them. From these results, tables of temperatures and emission measures for different flux ratios were generated. Values not explicitly included in the tables can be found via interpolation as the dependencies between temperature/emission measure and the flux ratio are monotonic. These dependencies were found to be similar to

Thomas et al. (1985). Throughout this thesis the method of White et al. (2005) is used unless otherwise stated.

Once the temperature and emission measure are known, it is possible to determine the radiative loss rate across all wavelengths, ER, of the full volume of emitting plasma observed by GOES. This is given by

−1 ER = EM × Λ(T ) [ergs s ] (3.11) where Λ(T ) is the radiative loss function. (N.B. The radiative loss rate presented here is integrated over the ﬂare volume. This is diﬀerent from how it was discussed in

Chapter 2 where is what presented as per unit volume.) The radiative loss rate can be calculated by the GOES software using CHIANTI and the methods of Cox & Tucker

(1969). It should be noted that although this is determined across all wavelengths, it only accounts for emission from the SXR-emitting volume observed by GOES/XRS.

Emission from other regions of the plasma, such as the cooler chromospheric footpoints, is not accounted for.

Having calculated the radiative loss rate, the total radiated energy over a given period (e.g. a ﬂare) can be obtained by simply integrating over time.

Z τ Lrad = ER(t)dt [ergs] (3.12) 0

84 3.2 Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI)

Figure 3.7: Image of the RHESSI satellite. Courtesy of NASA.

3.2 Reuven Ramaty High-Energy Solar Spectroscopic Im-

ager (RHESSI)

RHESSI (Lin et al., 2002) was launched by NASA in 2002 as part of its Small Explorer series (SMEX). Its aim was to study particle acceleration and energy release in solar

ﬂares through the examination of X-ray and γ-ray emission. To achieve this RHESSI was designed with both signiﬁcant spectroscopic and imaging capabilities and a full Sun

ﬁeld of view (FOV). RHESSI’s energy sensitivity ranges from 3 keV to 17 MeV, with an energy resolution of 1 keV up to 100 keV, increasing to ∼5 keV at 5 MeV. RHESSI’s spatial resolution is energy-dependent, with 2.3 arcsec up to 100 keV using RHESSI’s

85 3. INSTRUMENTATION

ﬁnest grids, 7 arcsec up to 400 keV, and 36 arcsec up to 15 MeV.

3.2.1 The RHESSI Instrument

RHESSI is comprised of nine rotating modulation collimators (RMCs; left of Figure 3.8) which rotate relative to the Sun due the rotation of the spacecraft. Each RMC contains a pair of widely separated grids which direct the incoming photons onto one of nine germanium detectors (GeD) (one for each RMC) which make up the RHESSI spectrometer (Smith et al., 2002) (right of Figure 3.8). RHESSI rotates around the Sun-satellite axis once every 4 seconds. As it does so, the grids in the RMCs periodically occult different parts of the solar disk, causing the observed photon count rate to be modulated.

Spatial information on the emitting region can be obtained from this modulation and hence images constructed. For more information on RHESSI imaging, see Lin et al.

(2002) and Hurford et al. (2002).

The GeDs are cryogenically cooled to ∼75 K to ensure that no electron-hole pairs are in the conduction band. This means that (barring quantum eﬀects such as leakage current) the only way an electron-hole pair is created in the conduction band is through the interaction of an X-ray or γ-ray photon with the GeD. Once this occurs, a current is produced by the high voltage across the GeD whose amplitude depends on the energy of the incoming photon.

The process of detecting a single pulse and then returning to a detection-ready state takes a ﬁnite amount of time (∼40 µs, known as dead time). Therefore, if two photons interact with a GeD within this period, they will be counted as a single detection with the combined energy of the two photons. This is known as pulse pileup.

In order to identify pulse pileup, the current produced by each detection is split in the onboard electronics into a fast-shaping and slow-shaping channel. The fast-shaping channel produces a triangular pulse of 800 ns which gives it greater ability to diﬀeren- tiate between single- and double-photon detections. However, the signal is noisy and

86 3.2 Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI)

Figure 3.8: Schematic of the RHESSI instrument. Left: the RHESSI rotation modulation collimators (RMCs). Right: the RHESSI spectrometer including the nine germanium detectors (GeDs). Taken from Hurford et al. (2002).

not well suited for determining the energy of the incident photon. This is the purpose of the slow-shaping channel which takes 8 µs to peak. If a second pulse is detected in the fast-shaping channel before the slow-shaping channel peaks, then both events are discarded. However, if the second pulse is not detected in the fast-shaping channel until after the slow-shaping channel peaks, only the second photon is rejected as the ﬁrst peak accurately represents the ﬁrst photon. However, rejecting such photons can bias the observed spectrum and therefore must be corrected for later in the data analysis software.

In large flares, the dead time can become a significant percentage of the characteristic timescale between photon detections due to the enhanced photon flux. This significantly increases pulse pileup. To avoid this and to prevent damage to the GeDs,

RHESSI is equipped with two aluminium attenuators which reduce the number of photons reaching the GeDs. These can be used individually or together. Each has a small thin section in the centre to ensure that there are always some lower energy

87 3. INSTRUMENTATION photons reaching the GeD. However, when the thicker attenuator is used, either alone or in combination with the thinner one, RHESSI can no longer reliably detect photons below 6 keV.

Due to the cost and weight restrictions of the SMEX programme, RHESSI was not designed with comprehensive shielding to protect the detectors from energetic particles.

This non-solar background signal is aﬀected by three main sources:

• the spacecraft’s transit through the South Atlantic Anomaly (SAA) where the

Earth’s magnetic ﬁeld concentrates energetic particles

• smooth modulations due to changes in geomagnetic latitude (and thus cosmic ray

ﬂux) during the spacecraft’s orbit

• occasional periods of electron precipitation from the outer radiation belt when

the spacecraft is at its highest geomagnetic latitudes.

The non-solar background signal generated by these processes as well as that due to instrumental eﬀects can bias the derived photon spectrum. Therefore it must be subtracted to ensure accurate data analysis. This is typically done by subtracting an averaged spectrum derived from a quiet non-ﬂaring period.

Although RHESSI measurements are predominantly dependent on incident photons, the resulting measured count rate is not necessarily representative of the population of those photons. There are several causes for this:

• absorption in the mylar blankets, cryostat windows, and grids

• Compton scattering into and out of the detectors

• Compton scattering oﬀ the Earth’s atmosphere. This can dominate the ﬂare

count rate in the rear segments below 100 keV

• noise in the electronics

88 3.2 Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI)

• resolution degradation due to radiation damage

• the low-energy cutoﬀ imposed by the electronics.

In order to accurately reconstruct the incident photon spectrum, it is necessary to understand what effect the above processes have on the measured count rate. These effects are represented in the detector response matrix (DRM). This is a two-dimensional matrix whose diagonal entries describe the efficiency of RHESSI detecting the incident photons at their original energies, while the off-diagonal entries represent the likelihood of measuring a photon at a different energy (usually lower). Thus the count rate as a function of energy, CR(E), can be expressed as,

CR(E) = DRM(E) · PR(E) + BG(E) (3.13) where PR(E) is the incident photon rate and BG(E) is the background photon rate, both as a function of energy. This equation can then be inverted to recover the incident photon spectrum from the measured count rate.

3.2.2 Deriving Thermal Plasma Properties Using RHESSI

Figure 3.9 shows a RHESSI spectrum taken over the course of one minute around the peak of a C3.0 flare which occurred on 2002 March 26. The crosses denote the observations while the different line-styles denote different components of the model fit applied to the data. The solid line shows the isothermal component of the fit which is dominated by thermal bremsstrahlung. This fit directly results in an estimation of the temperature and emission measure of the emitting plasma (Equation 2.8). The two bumps at 6.8 keV and 8 keV deviating from the thermal bremsstrahlung component are due to the inclusion of emission line complexes. In the typical flare temperature range of 10 – 25 MK, the 6.8 keV iron line complex is dominated by Fe XXV while the 8 keV iron-nickel complex is dominated by Fe XXV and Fe XXVI lines. For an

89 3. INSTRUMENTATION

Figure 3.9: RHESSI spectrum taken around the peak of a C3.0 flare which occurred on 2002 March 26. The observations are denoted by the crosses which are fitted with thermal (solid line), non-thermal (dashed line), and background components (dot-dashed line). The feature around 10 keV in the background component is due to the excitation of a germanium line in the germanium detectors themselves. The bottom panel shows the residuals of the fit. (Raftery et al., 2009).

90 3.3 Hinode in-depth discussion of these features, see Phillips (2004).

The deviation from the thermal fit at high energies (>10 keV) is a result of a non- thermal component represented by the dashed line. This is due to bremsstrahlung emission produced by non-thermal electrons accelerated by, for example, magnetic reconnection. This is present because the spectrum is taken near the peak of the flare and energy release has clearly not yet ceased. Finally, the dot-dashed line represents the background component which accounts for non-flaring emission as well as non-solar sources of background such as those listed in Section 3.2.1. The prominent feature around 10 keV is due to the excitation of a germanium line in the detector itself.

3.3 Hinode

Hinode (Kosugi et al., 2007) is a Japanese mission designed to study the heating mechanisms and dynamics of the active solar corona (Figure 3.10). It was developed and launched by the Japanese Aerospace Exploration Agency (JAXA) in partnership with

National Astronomical Observatory of Japan (NAOJ), the UK’s Science & Technology

Facilities Council (STFC) and NASA. It is operated in co-operation with the Euro- pean Space Agency (ESA) and the Norwegian Space Centre (NSC). It was launched in 2006 into a quasi-circular sun-synchronous orbit along the Earth’s day/night termi- nator, allowing almost continuous observation of the Sun. The instruments onboard

Hinode include the Solar Optical Telescope (SOT), the Extreme-ultraviolet Imaging

Spectrometer (EIS), and the X-Ray Telescope (XRT).

3.3.1 X-Ray Telescope (XRT)

Hinode/XRT (Golub et al., 2007; Kano et al., 2008) produces broadband images of the Sun at wavelengths of 0.2–20 nm. Normal incidence telescopes are useless for this range due to X-rays’ tendency to pass through, or be absorbed by, materials rather

91 3. INSTRUMENTATION

Figure 3.10: Illustration of the Hinode satellite and main components: XRT, EIS, and SOT (OTA and FPP). Figure courtesy of NASA.

than reflected. Therefore in order to focus the X-rays Hinode/XRT was designed as a grazing incidence telescope (Figure 3.11). The mirrors are aligned almost parallel to the path of the incoming X-rays which makes reflection favourable. Because the path of the X-rays is only altered slightly by such a shallow reflection, it is necessary to use a series of mirrors in order to sufficiently focus the X-rays and obtain the required spatial resolution.

With the use of this technology, Hinode/XRT can achieve a spatial resolution of

1 arcsec. In addition, it has a maximum field of view of 34×34 arcsecs but can also focus on several smaller regions of interest simultaneously. Its time cadence depends on the observing program used but is typically on the order of seconds. Hinode/XRT has numerous filters which allow temperature analyses to be conducted via flux ratios taken between different filters. They have quite wide temperature responses, but all peak around 8–13 MK (Figure 3.12). These filters allow Hinode/XRT to image features in the corona at temperatures in the range 1–30 MK. The temperature sensitivity,

92 3.3 Hinode

Figure 3.11: A simple schematic showing the use of grazing incidence in a telescope such as Hinode/XRT. Each mirror is arranged at a slight angle to the path of the incoming X-rays. As the X-rays are successively reﬂected by each mirror, their trajectories are increasingly altered from their original ones until the X-rays can be directed onto the focal point of the telescope.

Figure 3.12: Response functions as a function of temperature for the various ﬁlters on Hinode/XRT.

93 3. INSTRUMENTATION

Figure 3.13: Illustration of the Solar Dynamics Observatory highlighting its three instruments: the Atmospheric Imaging Assembly (AIA); the EUV Variability Experiment (EVE); and the Helioseismic and Magnetic Imager (HMI). Courtesy of NASA.

spatial resolution and time cadence of Hinode/XRT make it the most ideal instrument available for directly imaging hot (∼10 MK) X-ray- and EUV-emitting ﬂare plasma.

3.4 Solar Dynamics Observatory (SDO)

The Solar Dynamics Observatory (SDO; Pesnell et al., 2012) is the ﬁrst mission of

NASA’s Living With a Star program. This program aims to explore the Sun’s variability and its aﬀect on the Earth and the solar system at large. It was launched in

94 3.4 Solar Dynamics Observatory (SDO)

Figure 3.14: Image of one of AIA’s primary mirrors. Each half of the mirror has a different reflective coating to reflect a different passband. The mirror contains a hole in the middle through which the secondary mirror reflects the light onto the CCD (Cassegrain design). (Lemen et al., 2012).

February 2010 and now sits in a geostationary orbit above White Sands, New Mex- ico. This is particularly useful for transferring the vast amount of data it gathers back down to Earth (∼2 terrabytes per day). SDO is comprised of three instruments: the

Atmospheric Imaging Assembly (AIA); the EUV Variability Experiment (EVE); and the Helioseismic and Magnetic Imager (HMI). Figure 3.13 shows a diagram of the SDO satellite with the various instruments labelled.

3.4.1 Atmospheric Imaging Assembly (AIA)

The Atmospheric Imaging Assembly was designed to image the full solar disk in 10

UV and EUV channels (4500, 1700, 1600, 335, 304, 211, 193, 171, 131, and 94 A)˚ with unprecedented spatial and temporal resolution (≤1.5 arcsec with a pixel size of

0.6 arcsec, and 10–12 s respectively). This allows it to image the dynamics of chromosphere, transition region, and corona as never before. SDO/AIA is comprised of four

95 3. INSTRUMENTATION

Cassegrain telescopes. The top and bottom halves of each primary mirror are coated with different reflective layers corresponding to two different passbands (Figure 3.14).

Figure 3.15 shows a head on view of the telescopes, labelled 1 to 4, along with the diﬀerent passbands associated with each.

Figure 3.16 shows a cross-section of telescope 2. The light enters through the aperture door on the left hand side and passes through a ﬁlter which blocks unwanted infra-red (IR), visible, and UV radiation. It then hits the primary mirror which reﬂects light within the two passbands of interest onto the secondary mirror. The secondary mirror focusses the light back through the hole in the middle of the primary mirror

(Figure 3.14) and onto the CCD. Just in front of the CCD is a shutter which controls the exposure time, and a filter wheel which switches back and forth between the two passbands. Telescope 3 is slightly different in this regard. The coating on the top half of its primary mirror, marked ‘UV’ in Figure 3.15, reflects broadband UV emission.

Thus, in addition to 171 A˚ filter for transmitting the light reflected by the lower half of the primary mirror, the filter wheel contains three additional filters to isolate the three

UV channels, 1600 A,˚ 1700 A,˚ and 4500 A.˚ On top of the schematic in Figure 3.16 is the

Guide Telescope which is used to help image stabilisation. Also labelled are the baﬄes which protect the CCD from errant charged particles and scattered light which could otherwise contaminate the images.

Figure 3.17 shows the temperature response functions of each of AIA’s six EUV channels. It can be seen that several of the channels have quite wide temperature responses (e.g. 171 A),˚ while others are even double-peaked (e.g. 94 A).˚ Therefore, although AIA ﬁlters are often quoted as being sensitive to the peak temperature of their response functions, great care must be taken in interpreting the temperature of the imaged plasma. This is because an inhomogeneous diﬀerential emission measure distribution could result in strong emission from temperatures other than that of the response function peak.

96 3.4 Solar Dynamics Observatory (SDO)

Figure 3.15: The arrangement of the AIA telescopes and the different passbands within them. The top and bottom halves of the primary mirror of each telescope have different coatings to reflect different passbands. The exception is the top half of telescope 3’s primary mirror, which is coated to reflect broadband UV containing the 1600 A,˚ 1700 A,˚ and 4500 A˚ channels. These channels are then separated by a filter wheel just in front of the CCD. The guide telescopes can be seen above each of the main telescopes (to the right of each number label) and help with image stabilisation. (Lemen et al., 2012).

Figure 3.16: Cross-section AIA’s telescope 2. (Lemen et al., 2012).

97 3. INSTRUMENTATION

Figure 3.17: Temperature responses of the six EUV AIA channels. (Lemen et al., 2012).

3.4.2 EUV Variability Experiment (EVE)

The EUV Variability Experiment onboard SDO (EVE; Woods et al., 2012) was designed to monitor solar EUV irradiance to help better understand its eﬀect on the

Earth’s upper atmosphere. It is comprised of several instruments (Figure 3.18). The primary instruments are the Multiple EUV Grating Spectrograph A and B channels

(MEGS-A and MEGS-B) which together measure solar EUV spectra in the range 6–

105 nm. To provide in-ﬂight calibration, SDO/EVE has the EUV SpectroPhotometer

(ESP) which measures broadband EUV irradiance from 0.1–39 nm, and the MEGS-

Photometer (MEGS-P) which measures solar emission at the 121.6 nm hydrogen line.

In addition, SDO/EVE also has the Solar Aspect Monitor (SAM) which assists in pointing. Despite SDO/EVE’s primary objective of helping understand the solar EUV irradiance on the Earth’s upper atmosphere, it is also well suited to studying solar

98 3.4 Solar Dynamics Observatory (SDO)

Figure 3.18: Diagram of SDO/EVE with each instrument labelled: Solar Aspect Mon- itor (SAM); Multiple Extreme-ultraviolet Grating Spectrograph A (MEGS-A); Multiple Extreme-ultraviolet Grating Spectrograph B (MEGS-B); Extreme-ultraviolet SpectroPho- tometer (ESP).

ﬂares.

3.4.2.1 Multiple EUV Grating Spectrograph-A (MEGS-A)

MEGS-A (Hock et al., 2012a) measures spatially integrated solar spectral irradiance from 6 to 37 nm with a resolution of 0.1 nm. MEGS-A is an 80o grazing incidence off-Rowland circle spectrograph. Its optical layout is shown in Figure 3.19. It has two vertically aligned aperture slits, A1 and A2, each 20 µm wide and 2 mm tall. The light transmitted through these slits is then focussed and dispersed onto a CCD detector by a concave reflective diffraction grating. The light is refracted by the grating’s grooved upper layer both before and after it is reflected by the bottom later. Since the angle of refraction is wavelength-dependent, the light is dispersed and spectral resolution achieved. The CCD is not curved and therefore crosses rather than follows the curvature path defined by the concave diffraction grating (hence ‘off-Rowland circle’) and has been

99 3. INSTRUMENTATION

Figure 3.19: Diagram of the SDO/EVE MEGS-A optical layout. The light enters the door and passes through the filter which only transmits light in the range 6–37 nm. It is then reflected and refracted off the A grating. This disperses and focusses the light onto the CCD detector, thus creating the 0.1 nm spectral resolution.

positioned to optimise spectral resolution across the entire MEGS-A wavelength range.

To prevent inaccurate counting statistics caused by higher order wavelength photons

(i.e. harmonics), MEGS-A1 and A2 are covered with ﬁlters. The A1 ﬁlter (Zr (280 nm)

/ C (20 nm)) transmits light from 6–18 nm while the A2 (Al (200 nm) / Ge (20 nm) / C

(20 nm)) ﬁlter transmits 17–37 nm. Since the two slits focus on diﬀerent halves of the

CCD detector, higher order photons are eliminated in the measurements (Figure 3.20).

100 3.4 Solar Dynamics Observatory (SDO)

Figure 3.20: Top Panel: a sample solar spectrum with the spectral range of SDO/EVE MEGS-A highlighted in white. Bottom panel: the same sample solar spectrum as it would appear on the MEGS-A CCD. The top left quadrant represents the 6–18 nm range of the spectrum transmitted by the A1 slit. The top right quadrant shows higher-order photons (harmonics) in the range 18–37 nm transmitted by the A1 slit. The reason that the A1 and A2 slits focus light onto diﬀerent halves of the CCD is that these higher order photons would cause inaccuracies in the 18–37 nm section of the spectrum. The bottom right quadrant shows the sample spectrum above in the range 18–37 nm transmitted by the A2 slit. The image of the solar disk in the bottom left quadrant is created by the Solar Aspect Monitor (SAM). To generate the ﬁnal MEGS-A spectrum free of higher order artifacts, the A1 spectrum in the top left quadrant is combined with the A2 spectrum in the bottom right quadrant.

101 3. INSTRUMENTATION

102 Chapter 4

Thermal Properties of Solar Flares Over Three Solar Cycles

To date, observations from the GOES/XRS have been used to study the thermal properties of solar ﬂares, but have been limited by a number of factors. These include the lack of a consistent background subtraction method capable of being automatically applied to large numbers of ﬂares. In this chapter, we develop such a method (the Temperature and

Emission measure-Based Background Subtraction; TEBBS) which preserves the characteristic evolution of solar ﬂares. TEBBS is successfully applied to over 50,000 solar

flares occurring over nearly three solar cycles (1980-2007), and used to create an ex- tensive catalogue of solar flare thermal properties. Using this, we confirm that the peak emission measure and total radiative losses scale with background-subtracted GOES X- ray flux as power-laws, while the peak temperature scales logarithmically. As expected, the peak emission measure shows an increasing trend with peak temperature, although the total radiative losses do not. The resulting TEBBS database is publicly available on

Solar Monitor (www.solarmonitor.org/TEBBS/). This work has been published in the

Astrophysical Journal (Ryan et al., 2012).

103 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

4.1 Introduction

To date, the study of solar flares has been predominantly focussed on single events or small samples of events. While such studies have furthered our understanding of the physics of these particular flares, they are fundamentally limited since they cannot, with any certainty, explain the global behaviour of solar flares. In contrast, only the study of large-scale samples can give an insight as to whether findings of given studies are particular to individual events or characteristic of many. This can allow constraints to be placed on global flare properties and give a greater understanding of the fundamental processes which drive these explosive phenomena.

That said, large-scale studies of solar flare properties have been few in number over the past decades. Such a study was performed by Garcia & McIntosh (1992) who used GOES/XRS to examine 710 M- and X-class flares. They noted a sharp linear lower bound in the relationship between emission measure and GOES class. However, this study mainly focussed on categorising types of very high-temperature flares and examined whether these flares approached or exceeded this emission measure lower bound. Thus the authors did not focus on the main flare distribution.

A deﬁnitive example of a large-scale study of solar ﬂare thermal properties was conducted by Feldman et al. (1996b). They combined results from three previous studies

(Feldman et al., 1995, 1996a; Phillips & Feldman, 1995) to investigate how temperature and emission measure vary with respect to GOES class for 868 ﬂares, from A2 to X2.

Their work used temperatures derived using the Bragg Crystal Spectrometer (BCS) onboard Yohkoh. These temperature values were convolved with the corresponding GOES data to derive values of emission measure. They found a logarithmic relationship between GOES class and temperature, and a power-law relationship between GOES class and emission measure, with larger ﬂares exhibiting higher temperatures and emission measures. However, temperature and emission measure were derived at the time of the

104 4.1 Introduction peak 1–8 A˚ ﬂux and so are likely to be less than their true maxima. Furthermore, BCS temperatures have been found to be higher than those measured by GOES (Feldman et al., 1996b), and using these values to calculate GOES emission measure will give lower values than if GOES was used consistently.

More recently, Battaglia et al. (2005) studied the correlation between temperature and GOES class for a sample of 85 flares, ranging from B1 to M6 class. Although the values reported gave a flatter dependence than Feldman et al. (1996b), the large scatter in the data led to a very large uncertainty, making the two relations comparable. In contrast to Feldman et al. (1996b), Battaglia et al. (2005) accounted for solar background and extracted the flare temperature at the time of the HXR burst as measured by RHESSI, rather than at the time of the soft X-ray (SXR) peak. However, any discrepancies expected to be caused by these differences were not discernible in view of the large uncertainties.

Larger statistical samples were studied by Christe et al. (2008) and Hannah et al.

(2008) who investigated the frequency distributions and energetics of 25,705 microﬂares

(GOES class A–C) observed by RHESSI from 2002 to 2007. From those events for which an adequate background subtraction could be performed (6,740) a median temperature of ∼13 MK and emission measure of 3×1046 cm−3 were found. Hannah et al. (2008), in particular, looked at the temperature derived from RHESSI observations as a function of (background-subtracted) GOES class, and found similar trends to the works of Feld- man et al. (1996b) and Battaglia et al. (2005). However, their analysis only included events of low C-class and below.

While these studies have provided some insight into the global properties of solar

flares, they each have their limitations. In particular they lack a standard method of isolating the flare signal from the solar and instrumental background contributions. Pre- vious background subtraction methods have often been performed manually. Others, such as setting the background to the flare’s initial flux values, or fitting polynomials

105 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES between the flux values at the start and end of the flare, often exaggerate noise and do not preserve characteristic temperature and emission measure evolution. Therefore, the accurate separation of flare signal and background limits the number of events that can be analysed. For example, Battaglia et al. (2005), in accounting for solar background, were only able to compile a sample of 85 events. Although a larger dataset would not have reduced the range of scatter, it would have better revealed the variations in the density of points within the distribution. This would have allowed a fit to be more tightly constrained and thereby reduced the uncertainties. Conversely, Feldman et al.

(1996b), with a sample of hundreds of ﬂares, did not attempt to account for the solar background at all, which can bias smaller events as the background makes up a greater contribution to the overall ﬂux.

Few attempts have been made to develop automated background subtraction techniques for GOES observations. Bornmann (1990) developed a method to determine whether a given background subtraction preserves characteristic temperature and emission measure evolution without checking manually, i.e., that temperature and emission measure both increase during the rise phase of ﬂares (Section 1.5.1). This method used the polynomials of Thomas et al. (1985) which relate temperature and emission measure to the ratio of the short and long GOES channels, R = B4/B8 (Section 3.1.2). However, White et al. (2005) have since improved on this by assuming more modern spectral models (CHIANTI 4.2 Landi et al., 1999, 2002) and taking into account the diﬀerences between coronal and photospheric abundances (Section 3.1.2), requiring the tests of Bornmann (1990) to be updated.

In this chapter, we build upon the work of Bornmann (1990) and develop an automatic background subtraction method for GOES observations, the Temperature and

Emission measure-Based Background Subtraction (TEBBS), which preserves the characteristic behaviour of solar ﬂares (Section 1.5.1). We then use this method to study the thermal properties of solar ﬂares using GOES observations over nearly three solar

106 4.2 Observations cycles. In Section 4.2, we discuss the GOES event list from which our ﬂare sample has been taken. In Section 4.3 we describe previous background subtraction methods for

GOES observations and outline how we have improved upon the work of Bornmann

(1990). In Section 4.4 we use this method to improve upon previous statistical studies by deriving ﬂare properties such as peak temperature and emission measure for ﬂares in the GOES event list and examining the relationships between them. These scaling laws are important for understanding the hydro- and thermodynamic evolution of solar

flares as different theoretical models may predict different relationships (e.g. Rosner et al., 1978). Moreover, accurately quantifying the thermal properties of solar flares is important in itself, for example, in better understanding thermal energy in the context of global energy partition within solar flares (e.g. Emslie et al., 2012). Understanding this is fundamental to understanding how solar flares occur. In Sections 4.5 and 4.6 we discuss the results and provide some conclusions.

4.2 Observations

The observations used in this chapter have been made by the X-Ray Sensors onboard the first twelve GOES satellites (Section 3.1). These observations allow the thermal properties of solar flares to be studied using the methods of White et al. (2005) (Sec- tion 3.1.2). The GOES series has been in continuous operation since the mid-1970s during which time a GOES solar flare catalogue has been compiled, known as the

GOES event list. Despite a number of limitations, the GOES event list is the most substantial resource available for large-scale statistical studies of solar ﬂares, and has therefore been utilised in this study. In the following section the strengths and limitations of the GOES event list are highlighted and the sample used in this study is discussed.

107 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

4.2.1 The GOES Event List

In order for a solar flare to be included in the GOES event list, it must satisfy two criteria4.1: firstly, there must be a continuous increase in the one-minute averaged X- ray flux in the long channel for the first four minutes of the event; secondly, the flux in the fourth minute must be at least 1.4 times the initial flux. The start time of the event is defined as the first of these four minutes. The peak time is when the long channel

flux reaches a maximum and the end of an event is defined as the time when the long channel flux reaches a level halfway between the peak value and that at the start of the flare.

The flare start and end times determined by these definitions do not always agree with those identified manually. An example of this can be seen in Figure 4.1. It shows a typical M1.0 flare that occurred on 2007 June 2 which exhibits the characteristic X-ray, temperature and emission measure behaviour discussed in Section 1.5.1. Figure 4.1a shows the X-ray fluxes in the two GOES channels. The event list start and end times are marked by the vertical dotted and dashed lines, respectively. The start time of the GOES event is a couple of minutes before the onset of the flare. Nonetheless, this start time satisfies the event list criteria and highlights a drawback in the event list definitions. Another drawback is associated with the event list end time. It can clearly be seen that the decay of the flare in Figure 4.1a continues for over half an hour after the event list end time. This means that properties depending on the decay time or duration of the flare, such as total radiative losses, will be systematically underestimated.

However, these definitions also help reduce the number of ‘double-flares’ in the event list, i.e., two flares being incorrectly labeled as one. This can happen when one flare occurs on the decay of another thereby preventing the full-disk integrated flux reaching half the peak value of the first flare. Having searched the event list between 1991 and

2007 we found 1,865 out of 34,361 events (5.4%) contained points between their peak

4.1http://www.swpc.noaa.gov/ftpdir/indices/events/README

108 4.2 Observations

Figure 4.1: X-ray lightcurves of an M1.0 solar flare observed by GOES. a) X-ray flux in each of the two GOES channels (0.5–4 A;˚ dotted curve and 1–8 A;˚ solid curve). b) The derived temperature curve. c) The derived emission measure curve. The vertical dotted and dashed lines denote the defined start and end times of the event, respectively. The vertical red, black and green lines mark the times of the peak temperature, peak 1–8 A˚ flux, and peak emission measure, respectively.

109 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES and end times which satisﬁed the event list start criteria. Of these, the second ﬂare was recorded in the event list in 236 cases.

A further weakness of the event list is that its criteria do not locate small events

(e.g., B-class) at times of high background ﬂux or during large ﬂares (e.g., M-class).

This is because a small flare will not cause the full-disk integrated X-ray flux to increase to 1.4 times the initial value when that initial value is more than an order of magnitude greater than the flare itself. Therefore, although one would expect to always find more small events, the event list actually contains fewer around solar maximum when large events are more frequent and the background is often at the C1 level or higher.

The GOES event list for the period 1980 to 2007 was used in this study. Data from the 1970s were not included due to their poor quality and because many GOES events from this period were erroneously tagged. This means that a total of 60,424 events, from B-class to X-class, were considered. Of these ∼9,000 were excluded as they were badly observed. These events can be catagorised in the following ways: no data available; erroneously included in event list (i.e., did not satisfy the event list definitions); contain data drop-outs; unphysical, discontinuous lightcurves; short channel approaches the lower detection threshold leading to unphysical flux ratios and hence derived properties; ‘double flares’ (or even ‘triple flares’); and flares dominated by ‘bad’ data points. ‘Bad’ points are marked as such by the GOES software because of the instrument states reported by telemetry (e.g. because of gain changes) or because they are outliers from surrounding data. The identification of these ‘bad’ points is justified by simultaneous observations from other GOES spacecraft. In addition to the exclusion of these events, an appropriate background subtraction could not be found for a small fraction of flares (∼1%). The size distribution of events discarded was very similar to that of the original dataset. This shows that excluding these events did not bias the results of this study. After removing all the events described above, 50,703 remained.

110 4.3 Background Subtraction Method Flux

Flare Flux

Total Flux

Quiescent Flux

Preflare Flux Background Flux

0 Time Figure 4.2: Schematic of a flare X-ray lightcurve showing how the total flux detected by, for example, the GOES XRS, is divided into constituent components. (Adapted from Bornmann 1990). The total flux (solid line) is the sum of the flux from the flare plus the solar background (divided by the dashed line). The pre-flare flux, however, is the sum of the background component and the quiescent component of the flaring plasma (e.g., the associated active region).

4.3 Background Subtraction Method

As the GOES lightcurves do not include any spatial information, they contain contributions not only from the flare but also from all non-flaring plasma across the solar disk. In addition, the lightcurves include non-solar contributions such as instrumental effects which vary between the individual X-Ray Sensors. These various background contributions can cause significant artifacts when deriving flare properties. Therefore, it is imperative to isolate the flare signal from these contributions, particularly for weaker events. In this section we outline the limitations of previous background subtraction methods before the TEBBS method itself is developed, and the ways in which it improves upon previous studies are highlighted.

111 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

4.3.1 Previous Background Subtraction Methods

The schematic in Figure 4.2 shows how a hypothetical GOES lightcurve is divided into

flare and background components. The two limiting cases in calculating the boundary between the two are to assume that either the total flux is dominated by the flare, thereby performing no background subtraction, or that the background is equal to the flux near the beginning of the event (‘pre-flare’ flux). The first assumption may be valid for events which are orders of magnitude above the background level, but is clearly incorrect for weaker events. The second assumption may be incorrect as there may be significant flare emission before the flare detection algorithm reports the start time. An example of the first method can be found in Feldman et al. (1996b) in which no background was subtracted. An example similar to the second method can be used in the GOES workbench4.2 which allows the background to be calculated as a line

(polynomial or exponential) between the ﬂux values at the start and end times of a

ﬂare.

GOES observations of a B7 flare which occurred 1986 January 15 at 10:09 UT are shown in Figure 4.3. In the first column the background is set to zero, while in the second column the background is set to the pre-flare flux. The top row (Figures 4.3a and 4.3e) shows the non-background-subtracted lightcurves with the background levels overplotted as horizontal lines. The second row (Figures 4.3b and 4.3f) shows the lightcurves after background subtraction. (N.B. Since the background in the left column is zero, Figures 4.3a and 4.3b are the same.) The third and fourth rows show the temperature and emission measure profiles derived from the lightcurves in the second row. An acceptable temperature profile has been obtained in Figure 4.3c which peaks at 8 MK around 10:13 UT. However the corresponding emission measure (Figure 4.3d) decreases at the time of the flare. This is at odds with the characteristic behaviour seen in Figures 1.19 and 4.1 and discussed in Section 1.5.1. The cause of this behaviour is

4.2http://hesperia.gsfc.nasa.gov/rhessidatacentre/complementary_data/goes.html

112 4.3 Background Subtraction Method

Figure 4.3: GOES lightcurves and associated temperature and emission measure profiles for a B7 flare which occurred on 1986 January 15. The profiles in Figures 4.3a–4.3d are not background-subtracted. The profiles in Figures 4.3e–4.3h have had the pre-flare flux in each channel subtracted, while Figures 4.3i–4.3l show the profiles obtained using the TEBBS method. The error bars represent the uncertainty quantified via the range of background subtractions found acceptable by TEBBS.

113 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES that the background flux component in Figure 4.3d dominates the emission measure evolution of the flare, thereby making it impossible for properties to be derived accurately. Conversely, by subtracting the pre-flare flux (Figures 4.3e–4.3h) significant artifacts are introduced to both the temperature and emission measure profiles. This is because this background subtraction causes the flux ratio at the beginning of the

ﬂare to be comprised of two small numbers, which exaggerates the noise and leads to large discontinuities when folded through the temperature and emission measure calculations.

A more accurate approach would be to assume that the ﬂare ﬂux may also contain some contribution from the quiescent plasma from which it originates (Figure 4.2).

This assumption was the basis for the background subtraction method developed by

Bornmann (1990). This technique applies three tests to a given combination of long and short channel background values to determine whether a given choice of background levels produces physically meaningful results. These are the increasing temperature test, the increasing emission measure test (together known as the increasing property tests), and the hot flare test. The increasing property tests assume that both temperature and emission measure exhibit an overall increase during the rise phase. In these tests, background levels are selected and a preliminary subtraction is made. The relationship between the long and short channel fluxes during the rise phase is approximated with a linear fit of the form, B4 = mB8 + c, where m is the slope, c is the intercept, and B4 and B8 are the short and long channel fluxes, respectively. From these fitted values, the temperature and emission measure for each point along the rise phase are calculated using the polynomials of Thomas et al. (1985) (Equations 3.8 and 3.9) and compared to their previous value. If overall increases in these parameters are observed, then the background subtraction is said to have passed the increasing property tests.

In the hot ﬂare test of Bornmann (1990), the background temperature is calculated

B B by plugging the ratio of the background ﬂuxes, RB = B4 /B8 , into the temperature

114 4.3 Background Subtraction Method polynomial of Thomas et al. 1985 (Equation 3.8). In order to pass, this temperature must be less than the background-subtracted ﬂare temperature at all times during the

ﬂare. This helps prevent unphysical temperatures/emission measures being derived if the short channel approaches the detection threshold.

The tests of Bornmann (1990) were the ﬁrst attempt to isolate a GOES ﬂare signal from the background contributions based on the validity of the results produced. How- ever, they have some drawbacks. The tests use the simple parameterisations of Thomas et al. (1985) to calculate temperature and emission measure. Since then, White et al.

(2005) have devised tables from updated detector responses from GOES-1 to GOES-12 and plasma source functions that take into account the marked diﬀerences in temperature and emission measure when derived using coronal rather than photospheric abundances.

Additionally, Bornmann’s tests do not take into account the GOES instrumental temperature threshold which stands at 4 MK. This is because such a temperature would correspond to a ﬂux ratio of R = 1/100 which is beyond the sensitivity of the XRSs used in this study. This means that these tests may not always identify the background combinations which lead to unphysical proﬁles.

Another shortcoming lies in the linear fit to the rise phase used in the increasing property tests. When demonstrating the method, Bornmann (1990) did not include the beginning of the rise phase because significant flux increases are often not observed there (e.g., Figure 4.1a) and can affect the fit’s accuracy. However, this leaves the beginning of the rise phase untested, which is the period most likely to exhibit spikes or discontinuities due to an unsuitable background subtraction. If these spikes are big enough, they can easily be mistaken for the true peaks and produce unreliable results.

In the next section the TEBBS method is described in detail and the ways in which it improves upon the above-mentioned shortcomings of the Bornmann tests are discussed.

115 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

4.3.2 Temperature and Emission measure-Based Background Sub- traction (TEBBS)

TEBBS has been developed as a progression from the Bornmann tests and improves upon them in a number of ways. Firstly, explicit temperature and emission measure values are calculated using White et al. (2005), allowing the evolution of these properties to be more accurately analysed. Secondly, extra criteria have been added to the hot flare test so that the minimum background-subtracted flare temperature must be greater than the instrumental temperature threshold of 4 MK. Similarly the maximum background-subtracted temperature must be less than the upper limit of the White et al. (2005) tables, i.e., <100 MK (corresponding to a flux ratio <1). This upper limit is much higher than any GOES temperatures found by previous studies and helps to identify possible background subtractions which produce discontinuities in the derived thermal properties. Thirdly, another criterion has been added to the increasing property tests requiring that any temperature/emission measure value taken from the early rise phase which is not used in the linear fit, must be less than the peak taken from the rest of the rise phase. This helps to remove possible flare signals which show spikes at the beginning of the temperature/emission measure profiles which may not be identified by the original Bornmann tests.

Both TEBBS and the Bornmann method assume a constant background level in each channel. This may not necessarily be the case, especially when the ﬂare occurs during the decay of an earlier event. This was deemed to be a rare enough occurrence that it would not introduce any signiﬁcant errors (236 out of 34,361 events between

1991 and 2007, i.e., 0.7% – see Section 4.2.1). Moreover, as the peak ﬂux and peak temperature occur near the beginning of an event, the slope of the background would have a negligible eﬀect.

The assumption underlying TEBBS is that the boundary between flare and background flux lies somewhere between zero and the pre-flare flux. Bornmann (1990)

116 4.3 Background Subtraction Method

Figure 4.4: GOES XRS lightcurves from 1986 January 15 06:35–10:55 UT. The start and end times of the B7 ﬂare shown in Figures 4.3 and 4.5 as deﬁned by the GOES event list are marked by the dashed and dot-dashed vertical lines respectively.

117 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES justified this by assuming the presence of a quiescent flux component from the flare plasma. However, there are a number of reasons why the background may not be well represented by the pre-flare flux. For example, if the recorded start time of the flare is later than the true start time, the flare flux will have already risen considerably, thereby causing the pre-flare flux to be much higher than the actual background. This can occur in the case of a badly labelled flare or if the flare occurs on the decay phase of another or at a time of high background flux. This would cause the flare to not be seen in the XRS data until its flux dominated that from the rest of the solar disk. This was the case for the B7 flare on 1986 January 15 discussed above. GOES lightcurves from an extended period around the flare (06:45–10:55 UT) are shown in Figure 4.4. The start and end times of the B7 flare as defined by the GOES event list are shown as the vertical dashed and dot-dashed vertical lines respectively. It can be seen that this flare occurred on the decay phase of an M-class flare which began around 06:50 UT. Because of this high pre-flare flux, the initial evolution of the B7 flare was not readily detectable in the XRS data. Therefore the pre-flare flux is not an accurate approximation of the background and thus only a certain fraction should be subtracted.

In order to apply the TEBBS method to this or any other flare, the first step is to define a sample space of possible background combinations. The range in each channel is between zero and the pre-flare flux. The grey region in the bottom left of

Figure 4.5 shows the sample background space for the 1986 January 15 ﬂare. This sample space is divided into twenty equally linearly separated discrete values in each

B B channel (B8 ,B4 ), thereby creating four hundred possible background combinations. Results were found to be independent of this binning and so twenty was chosen to minimise computational time while ensuring that the background space was adequately sampled. Each background combination is then subtracted, thus creating four hundred sets of background-subtracted lightcurves as possibilities for the ﬂare signal. It is to these lightcurves that the hot ﬂare test and increasing property tests are applied.

118 4.3 Background Subtraction Method

Figure 4.5: Short channel flux versus the long channel flux for the 1986 January 15 B7 flare (solid curve). The grey shaded area in the bottom left hand corner represents the possible combinations of background values from each channel for this event. The orange line represents a linear least-squares fit to the latter five sixths of the rise phase (duration). The first sixth is excluded because significant increases are often not seen directly after the GOES start time as can be observed from the fit’s proximity to the minimum of the data.

119 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

The first test to be applied is the hot flare test. The minimum temperature, Tmin, of each lightcurve is calculated. Any background combinations corresponding to a temperature profile with Tmin ≤ 4 MK are discarded. Tmin is then compared to the background

B B temperature, TB, calculated using the background values, (B8 ,B4 ). If Tmin ≤ TB then that background combination is discarded. Furthermore, should the flux ratio at any point be greater than or equal to unity (i.e., T ≥ 100 MK) the background combination is also discarded. The background combinations of the 1986 January 15 flare which passed (solid region) and failed (hashed region) the hot flare test are shown in

Figure 4.6a. From this panel it can be seen that the number of possible background combinations has already been halved.

Next, the increasing property tests are applied. As in the Bornmann tests, the short channel ﬂux during the rise phase is ﬁtted with a linear function of the form,

B4 = mB8 + c, so as to reduce the influence of fluctuations in the data (orange line, Figure 4.5). Such a fit is justified by the fact that 90% of flares in this study have a

Pearson correlation coefficient greater than 0.85 for their rise phases and 95% have a value greater than 0.75. Following Bornmann (1990), the first sixth of the rise phase duration is not included in the linear fit. This is because significant increases are often not observed directly after the GOES event list start time (e.g. Figure 4.1) which can affect the accuracy of the fit. This can be seen for the orange line fit in Figure 4.5 which, although was not derived using the first sixth (duration) of the rise, still begins very close to the point from where the flare initially evolves (black line). However, because of this, the initial unfitted section of the rise phase is later tested independently.

Using the linearly approximated values, the evolution of the temperature and emission measure during the rise phase is calculated for all background-subtracted lightcurves.

Each value is compared with its preceding one and the percentage of times when the temperature/emission measure increases is calculated. Background combinations which result in proﬁles exhibiting a total increase less than a certain threshold are discarded.

120 4.3 Background Subtraction Method

Figure 4.6: Sample background space for 1986 January 15 ﬂare. The black shaded areas illustrate the range of values which pass a given background test, while the hashed regions denote background values which fail: a) the hot ﬂare test; b) the increasing temperature test; c) the increasing emission measure test; and d) points which passed all three, or failed one or more.

121 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

Figure 4.7: Temperature and emission measure profiles for the 1986 January 15 flare for all possible background combinations. The left column shows profiles which passed all three tests, while the right column shows profiles which failed one or more tests.

This threshold was chosen heuristically to be the maximum from all four hundred background combinations minus seven. For example, if the maximum number of rise phase temperature increases from all background combinations is 77%, the threshold is 70%. If this threshold leaves no background combinations which pass all three tests, it is iteratively reduced in steps of five percent until at least one is found. Finally, the initial part of the rise phase is tested. To do this, the temperature and emission measure profiles for the whole rise phase are calculated. Any profiles that show a peak in the

‘un-ﬁtted’ section of the rise phase greater than the peak found in the ‘ﬁtted’ section are discarded. Figure 4.6b and 4.6c show the background combinations which passed (solid region) and failed (hashed region) the increasing temperature and increasing emission measure tests respectively.

122 4.3 Background Subtraction Method

Having completed these tests, only background combinations which pass all three are deemed suitable. This leaves a small distribution of allowed background combinations, shown as the solid region in Figure 4.6d. The temperature and emission measure proﬁles corresponding to each of these background combinations are shown in Figure 4.7

(smoothed for illustrative purposes). The left column shows the profiles corresponding to background combinations which passed all three tests, while the right column shows profiles corresponding to combinations which failed one or more tests. It can be seen that all the profiles in the left column are well behaved and are more conducive to calculating peak values and peak times. In contrast, many of the profiles in the right column exhibit discontinuities and spikes, particularly near the beginning of the

flare. It is impossible to calculate useful peak values and times from these profiles, either automatically or manually. There do appear to be some acceptable temperature profiles in the right column. However their corresponding emission measure profiles are unphysical. The converse is true for the apparently acceptable emission measure profiles.

The TEBBS method was applied to the 1986 January 15 B7 flare and the resulting time profiles are shown in the third column of Figure 4.3 (Panels i – l). The background levels were chosen from the combination closest to the centre of the pass distribution in Figure 4.6d. This was defined by requiring that the chosen background combination included the median allowed long channel value. This extracted a subset of possible allowed background combinations which can be represented as a column or vertical line in the solid region in Figure 4.6d. The short channel background value was then defined as the median value in this subset. The long and short values of the chosen background combination are also shown as the horizontal dashed and dot-dashed lines in Figure 4.3i. The background-subtracted fluxes can be seen in Figure 4.3j. The error bars mark the uncertainty in the background subtraction which is taken as the range of the pass distribution in Figure 4.6d. The TEBBS temperature and emission

123 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES measure profiles are shown in Figures 4.3k and 4.3l, respectively. Both of these profiles show smooth rise and decay phases. Note that the temperature profile does not have a discontinuity as in Figure 4.3g. Furthermore, the emission measure evolution is no longer dominated by the background contribution as in Figure 4.3d nor exhibits spikes or discontinuities as in Figure 4.3h. The error bars in these panels represent the range of acceptable temperature and emission measure profiles seen in the left column of

Figure 4.7. Note that the uncertainties at the beginning of the flare are largest. This is expected as the flux ratio during the early rise phase is made of smaller numbers than the rest of the flare. Therefore, a slight inaccuracy in the background subtraction can cause a more significant change in temperature and emission measure. The beginning of the flare also shows an unexpectedly high temperature of &6 MK. This can be explained by the fact that the start time defined by the GOES event list was probably after the actual start time due to the high emission from the M-class flare which preceded it (see

Figure 4.4). This would imply that the flaring plasma was initially cooler than 6 MK but by the time the flare emission could be detected over that of the M-class flare, the plasma had already been heated substantially. Figure 4.3 shows that TEBBS has performed a successful, automatic background subtraction, superior to either of those performed with the other two methods discussed in Section 4.3.1.

Having successfully tested TEBBS on other flares chosen at random, the method was applied to the 50,703 selected flares in the GOES event list from 1980 January 1 to 2007 December 31. The specific background combinations were chosen in the same way as for the 1986 January 15 flare. The associated plasma properties (temperature, emission measure, radiative loss rates, and total radiative losses) were derived for each

ﬂare. Uncertainties on the plasma properties for each event were calculated from the corresponding range of allowed TEBBS background subtractions as was done in Fig- ures 4.3j–4.3l. Values from ‘bad’ data points and neighbouring data points were ignored due to their tendency to produce unreliable spikes when folded through the temper-

124 4.4 Results ature and emission measure calculations. The resulting TEBBS database is publicly available on Solar Monitor4.3. The statistical relationships between the above derived properties are discussed in the next section.

4.4 Results

Peak temperature, peak emission measure, and total radiative losses each as a function of peak long channel ﬂux are shown as a density of points in Figure 4.8. Each column shows distributions obtained using each of the three background subtraction methods discussed in Section 4.3. Uncertainties on each data point are omitted for clarity.

The relationship between peak flare temperature and peak long channel flux is shown in Figures 4.8a–4.8c. While the non-background-subtracted distribution in Fig- ure 4.8a displays some trend of larger flares exhibiting higher temperatures, there is a

flattening of the distribution below the C1 level. This is due to the influence of the background contributions which become highly significant at low fluxes. In addition, a horizontal edge at ∼5 MK is also seen which is due to the instrumental detection limit.

There is more scatter in the pre-flare background-subtracted distribution in Figure 4.8b, with events of all classes showing temperatures in excess of 25 MK up to a temperature of ∼80 MK (beyond the range of the plot axis). By subtracting the pre-flare flux, the value of the flux ratio at the beginning of the flare can become erroneously large due to dividing one small number by another. This can lead to spuriously high temperature values when folded through the temperature calculations which can often be greater than the real peak temperature (see Section 4.3). Many of the high temperature values

(>25 MK) in Figure 4.8b, particularly those corresponding to low peak long channel

ﬂuxes, have been taken from such spikes early in the ﬂare. In contrast, the TEBBS method performs background subtractions which tend not to cause such temperature spikes. As a result the distribution in Figure 4.8c shows much less scatter.

4.3http://www.solarmonitor.org/TEBBS/

125 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES f and c 80 MK. ≈ (1996b, three-dotted-dashed), et al. extend beyond the vertical plot range to b (2008, dot-dashed) and this work (Equations 4.1, 4.2 and 4.4; solid). Arrow et al. (2005, short-dashed), Hannah 2D histograms of peak temperature, peak emission measure, and total radiative losses, as a function of peak long channel et al. Battaglia heads mark events whichhave are been upper located. or Thedue lower crosses to limits mark saturation. due events See for to Appendix which XRS A flux saturation for values and are more a detail. point lower in N.B. limit the 806 and events directions derived in that properties panel the are only true rough values estimates would Figure 4.8: flux, for all selectedapplied: GOES no events background betweenare subtracted 1980 relationships (left), derived and by pre-flare 2007. different flux studies: The subtracted Garcia data (middle), & in McIntosh and (1992, each TEBBS long-dashed), column (right). Feldman have had Overplotted different on background panels subtractions

126 4.4 Results

In order to examine the relationship between peak temperature and peak long channel flux, a number of methods were used. First, the Kendall tau correlation coefficient was calculated which is non-parametric, i.e., it does not assume a pre-defined model for the data. It is based on the rank of the data points rather than the values themselves, making it more suited to distributions with significant outliers or scatter than other correlation coefficients (e.g. the Pearson linear coefficient). The Kendal tau correlation coefficient for Figure 4.8c was found to be 0.42 which represents a statistically significant correlation. Next, the relationship was quantified using linear regression.

Ordinary least squares (OLS) was not used because of three characteristics of the data: the presence of several outliers which produced non-normal behaviour between an OLS regression fit and the observations (i.e. the residuals were not normally distributed); data truncations due to observational cutoffs below B1 level and 4 MK; the underlying power-law number distribution of the observations, i.e. the greater number of smaller events relative to larger ones. To address these characteristics, the methods of robust statistics were used. The basic assumption in OLS is that the residuals are normally distributed and the solution to the problem is calculated by minimising the sum of the squared residuals. However in this case, the sum is replaced by the median of the squared residuals. This results in an estimator that is resistant to the outliers by finding the narrowest strip covering half the observations (Rousseeuw, 1984). To account for the population distribution, the regression analysis was weighted using the flux values themselves, with smaller events weighted less than larger events. The truncation in the data was handled using the method of Bhattacharya et al. (1983). In this method a slope for the distribution is chosen such that the weighted residuals greater and less than any given x-value are balanced. However, because of truncation, data points are only compared if the residual of one point (difference between fitted and observed) is less than the difference between the observed value of the other point and the truncation limit. For more information see Bhattacharya et al. 1983. The form of the fit

127 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES resulting from this method is given by

T = α + βlog10B8 [MK] (4.1)

This form was chosen because it uses a logarithmic relationship between temperature and long channel ﬂux such as those found by both Feldman et al. (1996b) and Battaglia et al. (2005). The values of α and β were found to be 34±3 and 3.9±0.5 respectively.

Finally, the goodness of this fit was examined by using a modified, robust R2 statistic which quantifies the variance in the data explained by the model. Whereas the usual

R2 value is based on the mean-squared-error, the modified robust R2 statistic is based on the median (consistent with the robust fitting method used above). It also accounts for the degrees of freedom in the fitting and the uncertainties on each data point. It was found that the modified robust R2 value for the above fit was 0.62. This value is lowered by the structure in the distribution, at least caused in part by the instrumental truncations below B-class and 4 MK. Nonetheless, this value still implies that the

Equation 4.1 is a suitable ﬁt to the distribution.

The relationship between peak emission measure and peak long channel ﬂux is shown as a density of points in Figures 4.8d–4.8f. Large amounts of scatter are found below the M1 level in Figures 4.8d and 4.8e which is not seen to the same degree in the TEBBS distribution in Figure 4.8f. The unusually high emission measures in

Figures 4.8d and 4.8e have been recorded from erroneous features such as those in

Figures 4.3d and 4.3h. There is a well-deﬁned lower edge in all three of the distributions.

A similar feature was found by Garcia (1988) and Garcia & McIntosh (1992). This edge is a natural consequence of the way emission measure is calculated, which is approximated by Equation 3.7. The temperature-dependent term in this equation asymptotically tends to zero, which means it varies very little at high temperatures.

As a result, emission measure becomes directly proportional to long channel ﬂux causing

128 4.4 Results the observed edge, which corresponds to high temperatures. This feature was also seen by Garcia & McIntosh (1992) but not explained.

To examine the correlation between peak emission measure and peak long channel

flux, the Kendall tau coefficient of the TEBBS distribution in Figure 4.8f was calculated and found to be 0.8, implying a significant correlation. In order to compare our results with those of previous studies, a power-law relationship between the two properties, such as those found by Garcia & McIntosh (1992), Battaglia et al. (2005), and Hannah et al. (2008), was applied to the data. The fit was performed linearly in log10-log10 space using the same linear regression method used for the relationship between peak temperature and long channel flux. It was then re-expressed in power-law form as

γ δ −3 EM = 10 B8 [cm ] (4.2)

The values of γ and δ were found to be 53±0.1 and 0.86±0.02 respectively. This relationship is expressed in the inverse as

ζ −2 B8 = ηEM [W m ] (4.3) where η and ζ were found to be 10−61±1 and 1.15±0.02 respectively. The modiﬁed robust R2 value for the above model was found to be 0.73, implying a good ﬁt.

Total radiative losses as a function of peak long channel flux are shown as a density of points in Figures 4.8g–4.8i. All three distributions clearly show an increasing trend with peak long channel flux. The similarity between the three distributions suggests that total radiative losses are not as sensitive to background subtraction as either peak temperature or peak emission measure. This is to be expected since peak values are taken from single points which can be very sensitive to erroneous spikes caused by inappropriate treatment of the background. However, total radiative losses are integrated over the flare duration. Therefore, if a flare contains erroneous temperature

129 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES or emission measure spikes, their contribution to the total radiative losses will not be as significant if the rest of the flare is ‘well-behaved’. For small flares, however, the effect of these erroneous values would be expected to have a greater influence. The ‘turn-up’ at A- and B-class levels in Figure 4.8h is consistent with this. This distribution was found to have a high Kendall tau correlation coefficient of 0.73. It was then fit using the same method as used for the emission measure - long channel flux relationship. The resulting fit was expressed in the form:

ε θ Lrad = 10 B8 [ergs] (4.4)

This form is justified by the high Pearson correlation coefficient in log10-log10 space (found to be 0.8). The values for ε and θ were found to be 34±0.4 and 0.9±0.07 respectively. The modified robust R2 statistic was found to be 0.71, implying that

Equation 4.4 well represents the distribution.

Distributions of peak emission measure and total radiative losses as a function of peak temperature are shown in Figure 4.9. Each column corresponds to distributions obtained from the same background subtraction methods as in Figure 4.8.

Peak emission measure as a function of peak temperature is shown as a density of points in Figures 4.9a–4.9c. A clear relationship between the two properties is not apparent in the non-background-subtracted distribution in Figure 4.9a. A horizontal edge from 5-12 MK and just above 1049 cm−3 is exhibited with the majority of ﬂares located just below this edge. Any relationship between these two properties is even less clear in Figure 4.9b. Very large scatter extends beyond the range of this plot to ∼80 MK.

The artifacts introduced into both the temperature and emission measure proﬁles by each of the respective background subtraction methods (such as those in Figures 4.3g and 4.3h) have once again exacerbated the scatter. A more discernible trend with less scatter is revealed by the use of TEBBS in Figure 4.9c. This distribution clearly

130 4.4 Results shows that ﬂares with hotter peak temperatures have greater peak emission measures.

However, there seems to be an edge to this distribution at low temperatures which may also be explained by a limit of Equation 3.7.

Total radiative losses as function of peak temperature is displayed as a density of points in Figures 4.9d–4.9f. Although the TEBBS distribution in Figures 4.9f appears comparable to the non-background-subtracted distribution in Figure 4.9d, it displays less scatter than seen in the pre-ﬂare subtracted distribution in Figure 4.9e which has data points extending beyond the range of the plot axis to ∼80 MK. No clear trend between peak temperature and total radiative losses is discernible in any of Figures 4.9d–

4.9f, although Figures 4.9d and 4.9f do show a tendency for higher temperature ﬂares to have greater total radiative losses. This implies there is no strong relationship between these properties. This is despite the fact that total radiative losses are a function of both temperature and emission measure.

131 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES 80 MK. ≈ extend beyond the horizontal plot range to e and b 2D histograms of peak emission measure and total radiative losses as a function of peak temperature for all selected Figure 4.9: GOES events between 1980background and subtracted 2007. (left), pre-flareor The lower flux data limits subtracted in due (middle),events each to for and column XRS which TEBBS have saturation flux had (right). andfor values different point more are Arrow detail. background in a heads the subtraction N.B. lower mark directions methods 806 limit that events events applied: and in the which derived panels no true are properties values upper are would only have been rough located. estimates due The to crosses saturation. mark See Appendix A

132 4.5 Discussion

4.5 Discussion

The TEBBS distributions in Figures 4.8 and 4.9 consistently show the least scatter and most discernible trends between properties derived from GOES/XRS observations. The non-background-subtracted and pre-flare subtracted distributions show a higher number of artifacts such as edges and anomalously high values. This shows that TEBBS is a superior method of automatically subtracting background than either of the other two methods, first because it successfully separates the flare signal from the background contributions, and second, produces fewer artifacts in doing so. However, there still may be biases in the distributions derived using TEBBS. Such biases may be due to the fact that TEBBS uses full-disk integrated observations. A comparison between temperature and emission measure profiles produced in this study and those derived from spatially resolved instruments could further highlight how reliable the TEBBS results are and be used to quantify any systematic biases. Spatially resolved observations could be taken from instruments which observe in similar wavelength bands to the XRS, such as the Soft X-ray Telescope (SXT) onboard Yohkoh, the Soft X-ray Im- ager (SXI) onboard GOES-12 and GOES-13, or Hinode/XRT. Furthermore, it must be acknowledged that several necessary assumptions were used in calculating the plasma properties, as must be done any time these properties are derived using GOES observations. In this study, coronal abundances (Feldman et al., 1992), a constant density of

1010 cm−3, the ionisation equilibrium from Mazzotta et al. (1998), and an isothermal plasma were assumed. It has been shown that coronal iron abundances during flares can reach over eight times the photospheric level (Feldman et al., 2004). Meanwhile coronal densities obtained from high-temperature density-sensitive ratios were found to be above 1013 cm−3 by Phillips et al. (2010) and in the range 1011–1012 cm−3 by Ryan et al. (2013) (See Chapter 6). Using either of these assumptions in the calculation of the flaring plasma properties could affect the results. However, this was not done here

133 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

Table 4.1: Values for B8-T relationship (Equation 4.5)

Study β κ Sample Size Feldman et al. (1996b) 0.185 -9 868 Battaglia et al. (2005) 0.33±0.29 -12 85 +0.04 +1 TEBBS 0.26−0.01 -9−2 50,703

in order to remain consistent with previous studies.

With these caveats in mind, the distribution in Figure 4.8c was compared with the studies of Feldman et al. (1996b) and Battaglia et al. (2005). These studies found a logarithmic correlation between peak temperature and peak long channel ﬂux. In both papers, the relationship was expressed in the form:

βT +κ −2 B8 = 3.5 × 10 [W m ] (4.5)

The values of β and κ from these studies can be found in Table 4.1. Corresponding values from this study were calculated by rearranging Equation 4.1 into the form of

Equation 4.5 and are also included in Table 4.1 for comparison. The Feldman et al.

(1996b) and Battaglia et al. (2005) relations are also overplotted on Figure 4.8c as the three-dotted-dashed and short-dashed lines respectively. The distribution of this study reveals predominantly lower temperatures for a given long channel ﬂux than both previous studies. There is closer agreement with Feldman et al. (1996b) than Battaglia et al. (2005) for B-class events but beyond this, lower temperatures are obtained. This can be explained by Feldman et al. (1996b) using the BCS to calculate temperature. In that paper, it was stated that temperatures obtained with the BCS agreed with those from GOES below 12 MK but above this point were higher on average by a factor of 1.4.

To investigate this, the mean peak temperature of all flares in our sample of M-class or greater was computed and found to be 16 MK. This flux threshold was chosen for two reasons. Firstly, the difference between background-subtracted (this study) and

134 4.5 Discussion non-background-subtracted (Feldman et al. 1996b) results are negligible in this regime.

Secondly, of these events, 95% had peak temperatures greater than 12 MK. This was compared to the mean Feldman temperature obtained for these events by plugging their long channel peak flux into the fit of Feldman et al. (1996b). The Feldman mean temperature was found to be 20.9 MK which differs from that of this study by a factor of 1.3. This is lower than the value quoted by Feldman et al. (1996b). This is because they measured temperature at the the time of the long channel peak which would be expected to occur after the temperature peak. Assuming that the temperature peaks before the long channel flux, the difference in ratios would imply that a flare’s temperature (M-class or greater) drops by 10% before the long channel peak. This implication is supported by this study, as part of which the temperature at the time of the long channel peak was also calculated. It was found that between the temperature and long channel peaks, a flare’s temperature drops on average by 10%–11% for flares greater than or equal to M-class and 9%–10% for all flares.

The slope of the relationship of Battaglia et al. (2005) appears closer to that of the

fit found by this study. However, the relationship consistently gives temperatures which are 3–4 MK higher. This discrepancy in the intercept is due to Battaglia et al. (2005)’s use of RHESSI to obtain the temperatures. The value of T in that study was calculated as either the temperature of an isothermal fit or the lower of two temperatures in a multi-thermal fit to a RHESSI spectrum. Battaglia et al. (2005) compared these temperatures to GOES temperature, TG, and a relationship was derived, given by:

T = 1.12TG + 3.12 [MK] (4.6)

Substituting this into Equation 4.5 and rearranging into the form of Equation 4.1,

+198 +17.3 values for α and β are found to be 28−15 and 2.7−1.3 respectively. Although these values are similar to those found for this study (α = 33 ± 3; β = 3.9 ± 0.5), the large

135 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

Table 4.2: Values for edge in B8-EM distribution (Same form as Equation 4.2)

Study γ δ Sample Size Garcia & McIntosh (1992) 53.04 0.83 710 TEBBS (Eqn 4.2) 53±0.1 0.86±0.02 50,703 TEBBS (lower edge) 53.4 0.96 50,703

uncertainties mean that little statistical signiﬁcance can be assigned to this similarity.

This highlights that a more comprehensive study than that of Battaglia et al. (2005), such as this one, was needed to more precisely understand the statistical relationships between the thermal properties of solar ﬂares.

Next, the emission measure distribution in Figure 4.8f was compared with the work of Garcia & McIntosh (1992). In that paper, the edge to this distribution was addressed

(also discussed here in Section 4.4) and a ﬁt to this lower bound was quoted from a previous paper (Garcia, 1988). This was of the same form as Equation 4.2. Garcia’s values are shown in Table 4.2 and the ﬁt corresponding to them is overplotted on

Figure 4.8f as the long-dashed line. However, this relationship does not ﬁt the lower bound of this distribution very well. In fact it seems to better ﬁt the distribution itself being as it is so similar to the values found for Equation 4.2 (shown in the second row of

Table 4.2). A rough fit to the edge in this study shows it is much better formulated by the parameters shown in the third row of Table 4.2. The discrepancy may be because the sample of Garcia & McIntosh (1992) was insufficient to reveal the actual limit of this relationship. However, it may be also be due to the fact that methods different from those of White et al. (2005) were used to calculate temperature and emission measure

(e.g. Thomas et al., 1985). The credibility of this limit is important as it suggests a well-defined minimum amount of material emitting in the GOES passbands produced by a flare of a certain long channel peak flux. Although this limit is due to the nature of Equation 3.7, the result should be compared with statistical studies using other instruments to confirm whether it is a breakdown in the validity of the temperature and

136 4.5 Discussion

Table 4.3: Values for EM-B8 relationship (Equation 4.3)

Study η ζ Sample Size Battaglia et al. (2005) 3.6×10−50 0.92±0.09 85 Hannah et al. (2008) 1.15×10−52 0.96 6,740 TEBBS 1×10−61±1 1.15±0.02 50,703

emission measure calculations of White et al. (2005) or has any physical signiﬁcance.

This distribution was also compared to the work of Battaglia et al. (2005) and

Hannah et al. (2008) who found correlations between RHESSI emission measure and background-subtracted GOES long channel peak ﬂux. These relations were expressed in the same form as Equation 4.3. The values found by these studies are displayed in

Table 4.3 along with the values from this study for comparison. These previous ﬁts are also overplotted on Figure 4.8f as the short-dashed and dot-dashed lines respectively.

These fits are steeper than our distribution. The relationship of Hannah et al. (2008) however, agrees well at B- and C-class which was the range on which that study focussed. The difference in slope can not be due to the different sensitivities of GOES and

RHESSI as Hannah et al. (2008) showed that GOES emission measure is consistently a factor of two greater than that obtained from RHESSI. The diﬀerence in slope may therefore be attributed to the extension of the distribution to M- and X-class. How- ever, it may also have been aﬀected by the fact that Hannah et al. (2008) calculated the emission measure at the time of the peak in the RHESSI 6–12 keV passband rather than the peak emission measure, as in this study. The relationship of Battaglia et al.

(2005) gives consistently lower emission measures than both the TEBBS distribution and relationship of Hannah et al. (2008). This can be explained by Battaglia et al.

(2005) measuring the emission measure at the time of the hardest HXR peak which tends to occur early in the ﬂare before the SXR and emission measure peaks.

The distribution of peak emission measure as a function of peak temperature in

Figure 4.9c shows that hotter ﬂares have larger peak emission measures. Feldman

137 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES et al. (1996b) found a power-law relationship between these two properties expressed by

EM = 1.7 × 100.13T +46 [cm−3] (4.7)

However, the Pearson correlation coefficient between temperature and the log of emission measure in Figure 4.9c was calculated to be 0.3. This implies that these properties are in fact not well linearly correlated despite an apparent trend of hotter flares having higher emission measures. Hannah et al. (2008) also examined the relationship between emission measure and temperature for A–low C-class flares and found no correlation.

If the Figure 4.9c distribution is examined more closely, there does not appear to be any relationship between the two properties within the range Hannah et al. (2008) studied. Indeed, the Pearson correlation coeﬃcient for C1.0-class and below is only

0.2, which supports Hannah et al. (2008) ﬁndings. However further examination of this relationship is necessary to draw ﬁrmer conclusions.

4.6 Conclusions & Future Work

An automatic background subtraction method for GOES/XRS observations, TEBBS, has been presented which determines the background subtraction based on the validity of the results it produces. This allows the properties of the flaring plasma to be more accurately derived. It can be systematically applied to any number of flares, removing many of the inconsistencies that can be introduced when manually defining a background level. This makes it a particularly suitable method for conducting large-scale statistical studies of solar flare characteristics. TEBBS was found to produce fewer spurious artifacts in the derived temperature and emission measure profiles for both individual events (Figure 4.3) and in large statistical samples (Figures 4.8 and 4.9).

This led to more reliable relationships being derived between ﬂare plasma properties

138 4.6 Conclusions & Future Work which can in turn place constraints on the ‘allowed’ values of properties for a ﬂare of a given GOES magnitude.

TEBBS was successfully applied to 50,703 flares from B-class to X-class, making it the largest study of the thermal properties of solar flares to date. It was found that peak temperature scales logarithmically with peak long channel flux as described by

Equation 4.1. Meanwhile, peak emission measure and total radiative losses scaled with peak long channel flux as power-laws given by Equations 4.2 and 4.4. Uncertainties were calculated for these derived relations, unlike previous studies. The exception to this was Battaglia et al. (2005) who provided uncertainties for their slopes only. The uncertainties derived using TEBBS were nonetheless smaller than those of Battaglia et al. (2005) and include uncertainties on the intercepts as well as slopes. In addition, an apparent well-defined lower limit on the peak emission measure for a given peak long channel flux was found. Although previously seen by Garcia & McIntosh (1992), their fit to this edge differed from the findings of this study. This lower limit was shown to be due to methods used to calculate emission measure. It was determined that this should be further investigated using other instruments to determine if it is physical or due to a breakdown in the validity of the way in which emission measure is calculated from GOES/XRS observations.

Peak emission measure and total radiative losses were also examined as a function of peak temperature. It was found that ﬂares with high peak temperatures also have high peak emission measures (in agreement with Garcia 1988 and Garcia & McIntosh

1992). However, the derived correlation was relatively weak. Similarly, it was also found that flares of a given peak temperature could exhibit a large range of radiative losses with no clearly defined trend. Although both a constant density and a fixed coronal abundance were assumed in this study, both have been shown to vary during individual events (e.g. Graham et al., 2011). A follow-up analysis of how changes in these variables might affect the derived properties, particularly in conjunction with

139 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES hydrodynamic simulations, may lead to more reasonable correlations.

This compilation of solar ﬂare properties represents a valuable resource from which to conduct future large-scale statistical studies of ﬂare plasma properties. For example,

Stoiser et al. (2008) derived analytical predictions of temperature and emission measure in response to electron beam and conduction driven heating and compared the results to

RHESSI observations of 18 microﬂares. They found an order of magnitude discrepancy between conduction driven emission measures predicted by the Rosner-Tucker-Vaiana

(RTV; Rosner et al. 1978) scaling laws and observation. This seemed to suggest that electron beam processes dominated. However, they noted that RHESSI’s high temperature sensitivity (&10 MK) means that the observed temperatures may not have well represented the conduction value of the microﬂares, thus explaining the discrepancy.

The fact that the GOES/XRS has a sensitivity to lower temperatures than RHESSI makes the TEBBS database ideal for exploring this possibility. Since the RTV scaling laws and electron beam heating models are widely used to understand and model solar ﬂares, it is important to examine disagreements between their predictions and observation.

Another example of the use of RTV scaling laws in understanding ﬂares is Aschwan- den et al. (2008). They used these laws to derive theoretical (EM ∝ T 4.3) and observed

(EM ∝ T 4.7) scaling laws between peak temperature and emission measure for solar and stellar ﬂares. However, as part of their study, results from previous studies such as

Feldman et al. (1996b) and Feldman et al. (1995) were included which did not account for background issues. TEBBS can therefore also be used to examine these scaling laws with greater statistical certainty and therefore provide more clarity on the discrepancies between theory and observation. As the scaling laws derived by Aschwanden et al.

(2008) apply to solar and stellar ﬂares, conclusions drawn from TEBBS can be extended to stellar ﬂares as well.

TEBBS can be used to examine a wide range of ﬂare characteristics, such as ther-

140 4.6 Conclusions & Future Work modynamic evolution and, in light of the work of Stoiser et al. (2008), even flare loop topologies. Stoiser et al. (2008) used the time delay between the peaks in flare temperature and emission measure as a timescale for chromospheric evaporation. They then compared this to different flare loop and chromospheric evaporation models. This further highlights the range of diverse possible uses of the TEBBS database. And combined with the fact that it is also the largest database of thermal flare plasma properties to date, means it will provide a valuable resource for future solar flare research.

The work outlined in this chapter was conducted in collaboration with Ryan O.

Milligan, Peter T. Gallagher, Brian R. Dennis, A. Kim Tolbert, Richard A. Schwartz, and C. Alex Young, and has been published in Astrophysical Journal Supplements

(Ryan et al., 2012).

141 4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES

142 Chapter 5

Comparison of Multi-Instrument Temperature Observations

Having developed plasma diagnostic techniques such as TEBBS, the next step is to explore the nature of our observations with respect to instrumental biases, etc. This is vital for understanding what the observations reveal about the physics of solar flares. In this chapter we compare multi-thermal DEM peak temperatures (SDO/AIA) with those determined using the isothermal assumption (GOES/XRS, RHESSI). AIA finds an average DEM peak temperature at the time of the GOES long channel peak of 12.0±2.9 MK and Gaussian DEM widths of 0.50±0.13. Meanwhile GOES finds a mean temperature of 15.6±2.4 MK which is higher by a factor of 1.4±0.4. We show that this is due to the isothermal assumption in the GOES calculations. From isothermal fits to photon spectra at energies of 6–12 keV of 61 of these events, RHESSI finds the temperature to be still higher than AIA by a factor of 1.9±1.0. We find that this is partly due to the isothermal assumption. However, RHESSI only samples the DEM’s high-temperature tail which is not well-constrained by AIA, thus causing extra discrepancies. We conclude that self-consistent flare DEM temperatures require simultaneous fitting of EUV and SXR fluxes. This work has been published in Solar Physics (Ryan et al., 2014).

143 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

5.1 Introduction

In the previous chapter we discussed plasma diagnostic techniques and developed

TEBBS to allow us to automatically isolate the flare signal in GOES/XRS observations and hence more accurately calculate thermal flare properties. The next step is to explore the nature of these observations in the context of the assumptions made, instrumental biases, etc. This allows us to better understand what these observations reveal about the physics of solar flares. In this chapter we focus on temperature measurements.

The temperature of the solar corona is one of its most fundamental characteristics. It affects the nature of its physical processes and properties such as radiation, conduction, waves, shocks, the plasma-β, hydrodynamics etc. One of the most notable phenomena which encompasses many of these processes is solar flares. As discussed in Section 1.5.1, flares are believed to occur when energy stored in stressed magnetic

fields is suddenly released, causing, among other things, a rapid heating of the flare plasma. Temperature measurements play a vital role in better understanding these eruptive events. Observational studies of flare energy budgets (e.g. Emslie et al. 2012), thermodynamic properties (e.g. Feldman et al. 1996b, Ryan et al. 2012), hydrodynamic scaling laws (e.g. Rosner et al. 1978, Aschwanden et al. 2008), flare cooling (e.g. Raftery et al. 2009, Ryan et al. 2013) as well as many others all depend on temperature measurements. In order to perform these measurements, an array of satellite instruments has been developed. Among these are GOES/XRS, RHESSI, and SDO/AIA. These instruments are sensitive to most of the temperature range in which coronal flare plasmas are typically found, 0.5–20 MK (AIA), ∼4–40 MK (GOES) and ∼7–100 MK (RHESSI).

However, in order to understand both the context and limitations of temperature measurements made with these instruments, it is important to know how they compare and the cause of any discrepancies. Previous studies have compared the tempera-

144 5.1 Introduction ture measurements of GOES and RHESSI and typically found that RHESSI exhibits systematically higher temperatures. Battaglia et al. (2005), discussed in Chapter 4, computed RHESSI peak temperatures of 85 flares in the range B- to M-class from an isothermal fit (or two isothermal fits) combined with a non-thermal fit. The temperature of the isothermal fit (or cooler isothermal fit), T1, was compared to the GOES temperature, TG. The GOES temperature was found to be systematically lower. A loose relationship was found and fitted by T1 = 1.12TG + 3.12. This implies that for flares with temperatures of 10–25 MK, typical of M- and X-class flares (as revealed in Chapter 4), RHESSI gives higher temperatures than GOES by 4–6 MK. McTiernan

(2009) compared RHESSI and GOES temperature measurements of the non-ﬂaring Sun from 2002–2006. He found that the average RHESSI temperature was 6–8 MK while the average GOES temperature was 4–6 MK. This is broadly consistent with Battaglia et al. (2005). These measurements of McTiernan (2009) result in a temperature ratio of TRHESSI /TGOES = 1.4±0.2. In the same year, Raftery et al. (2009) examined the temperature evolution of a C1.0 ﬂare with several instruments including GOES and

RHESSI. The maximum RHESSI temperature was found to be ∼15 MK, while the maximum GOES temperature was found to be 10 MK. This results in a temperature ratio of TRHESSI /TGOES = 1.5 which is slightly higher than previous studies. However, the RHESSI maximum temperature was found to occur ∼4 minutes before the GOES maximum. Taking these temperature measurements simultaneously would lower this ratio, bringing it more into line with previous studies.

In this chapter, we calculate the GOES and RHESSI temperatures of an ensemble of M- and X-class ﬂares using the isothermal assumption (as in previous studies). We compare them to the peak temperature of the diﬀerential emission measure distribution

(DEM; Section 2.1.3) calculated with AIA, as per Aschwanden & Shimizu (2013) and

Aschwanden et al. (2013). In doing so, we explore the eﬀect of the traditional isothermal assumption on the GOES and RHESSI temperature measurements and quantify the

145 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS resulting bias. In Section 5.2, we discuss the instrumentation, observations and data analysis of AIA, GOES and RHESSI. In Section 5.3 we devise theoretical predictions of the eﬀect of the isothermal assumption on GOES and RHESSI temperatures as compared to DEMs of various widths. These predictions are then compared to the discrepancies between the AIA DEM peak temperatures and those from GOES and

RHESSI. Finally in Section 5.4 we provide our conclusions.

5.2 Data Analysis

5.2.1 SDO/AIA Measurements

During the ﬁrst two years of the SDO mission (May 2010 – March 2012) 155 M- and

X-class flares were observed by AIA (Section 3.4.1). These were analysed in a single- wavelength study at 335 A˚ by Aschwanden (2012). Multi-wavelength studies using all six coronal filters (94, 131, 171, 193, 221, and 335 A)˚ have analysed the spatio-temporal parameters of these flares (Aschwanden et al., 2013), as well as their temperatures and

DEMs (Aschwanden & Shimizu, 2013).

In this chapter we utilise the AIA DEM analysis method of Aschwanden & Shimizu

(2013) which reconstructs a ﬂare DEM at a given time from the background-subtracted

flare fluxes in the six AIA coronal channels. The main steps in this method outlined in that paper are now summarised here. The background-subtracted flux in a given coronal AIA channel centred on wavelength, λ, at a certain time, t, can be expressed in terms of the DEM and the response function of that channel, Rλ(T ). This is expressed in both integral and discrete summation form in Equation 1 of Aschwanden & Shimizu

(2013):

Z dEM(T, t) X F (t) − F (t ) = R (T ) = EM (T, t)R (T )∆T . (5.1) λ λ b dT λ k λ k k k

146 5.2 Data Analysis

Here, Fλ(t) is the measured flux in channel λ at time, t, tb is the time at which the dEM(T,t) background flux is taken, dT is the differential emission measure which is a function of time, t, and temperature, T . On the RHS of the equation, EM(T, t) is the emission measure at a given temperature, T , and time, t, and ∆T is the width of each temperature bin. By using this equation and a parameterisation of the DEM, the flux for a given AIA channel can be reproduced, but only if the DEM parameterisation is appropriate. Aschwanden & Shimizu (2013) assumed a single Gaussian DEM parameterisation, characterised by three parameters: the DEM peak emission measure EMp, a DEM peak temperature Tp, and a Gaussian width log10 (σT ). This was expressed mathematically by Equation 2 of Aschwanden & Shimizu (2013):

2 dEM(T ) −[log10(T ) − log10(Tp)] = EMp exp 2 . (5.2) dT 2σT

The most likely values for these three properties of the DEM can be then be found by minimising the combined residuals, σdev, between predicted and observed ﬂuxes in all six coronal AIA channels, i.e. (Equation 3 of Aschwanden & Shimizu 2013)

" #1/2 1 X 2 σdev = (ffit,λ − fobs,λ) (5.3) nλ λ where nλ is the number of coronal channels, i.e. 6, and ffit,λ and fobs,λ, are the ﬁtted and observed background subtracted AIA ﬂuxes in channel λ. In this way, the DEM is reconstructed.

In this study, this method was initially applied to the AIA observations of all the

155 M- and X-class ﬂares at the time of the peak in the GOES 1–8 A˚ ﬂux. Successful

DEM fits were not found for five of these flares while sufficient GOES observations were not available for one additional flare. Thus 149 of the 155 M- and X-class flares were analysed as part of this study.

147 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

171 10-24 304 193 211 -25 ] 10 -1 131 pix

-1 GOES 1-8 A s 5 10-26 335 94

10-27 GOES 0.5-4 A

Response [DN cm 10-28

10-29

10-30 RHESSI 3 keV RHESSI 6 keV 105 106 107 108 Temperature T[K]

Figure 5.1: Temperature-response functions for the seven coronal EUV channels of SDO/AIA, according to the status of Dec 2012. The GOES 1-8 A˚ and 0.5-4 A˚ channels are also shown (in arbitrary flux units), as well as thermal energy of the lowest fittable RHESSI channels at 3 keV and 6 keV. The approximate peak temperature range of large flares (Tp ≈ 5 − 20 MK) is indicated with a thatched area.

148 5.2 Data Analysis

51 57 140 63 3310 62 10049 69 54 73 50 53 9124 64 125144143 85 8388 61 3 29 ]) 146 129 74112 9287 36

-3 13 25 5289103 14198 142123 110114 79 24145126 3555 78137122 381856 9943 [cm 91 5039 72 20 86 300 p 425 44 6519471 5884 46 93 12126 11910640 1071161052251 11517 8108 9414 128147 211270 11127 27 15 28 9037 11314813010966 71777 102 49 2 118 1174148 68 138 45 104139 1341331659131 97 7567 95135 111 82 80 13660 4 76 31 120 101 32 48 8134 96 132 6 Emission measure log(EM 47 23

46 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Temperature log[T(MK)]

Figure 5.2: Gaussian DEM ﬁts of the 149 M- and X-class ﬂares analysed with SDO/AIA.

149 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

A key point in comparing temperature measurements from different instruments is the simultaneity of observations. Since the GOES/XRS long channel peak time is defined using 1-min averaged time profiles, the simultaneity between SDO/AIA and

GOES/XRS is (tGOES − tAIA) ≈ 0.5 ± 0.5 min. Another decisive criterion is the temperature coverage. From the SDO/AIA response functions shown in Figure 5.1 we can see that the AIA ﬁlters have their primary or secondary peaks in the range from

> < log10(T ) ∼ 5.8 (131 A)˚ to log10(T ) ∼ 7.3 (94, 131, 193 A),˚ which is the temperature range where a DEM distribution can be reliably obtained. A display of all 149 single-Gaussian

ﬁts to the AIA data is shown in Figure 5.2. The peak emission measures, integrated over the total ﬂare volume, are found in the range of log10(EMp) = 47.0−50.5, with a mean

−3 and standard deviation of log10(EMp) = 49.2 ± 0.6, in units of cm . The ﬂare peak temperatures are found in the range of Tp = 5.6 − 17.8 MK, with a mean and standard deviation of Tp = 12.0 ± 2.9 MK. The Gaussian half widths are found in the range

0.5 of log10(σT ) = 0.50 ± 0.13, which corresponds to a temperature factor of 10 ≈ 3.2. Since a single-Gaussian function has only 3 parameters, the DEM ﬁtting to 6 coronal

filters is a very robust procedure and we are confident that the peak emission measure and peak temperature are, for the most part, accurately retrieved. However, in a few cases it may not yield an acceptable χ2-value of the fit (see Table 2 in Aschwanden

& Shimizu 2013). The next best option would be a 4-parameter function. Such a function could comprise of two semi-Gaussians joined together at the DEM peak with two diﬀerent widths, σT 1 at the low-temperature side, and σT 2 at the high-temperature side (e.g. as used in Aschwanden & Alexander 2001). While SDO/AIA may not provide

> suﬃcient temperature coverage to constrain the high-temperature side at T ∼ 20 MK, RHESSI could provide strong constraints in this high-temperature tail. On the other hand, RHESSI does not have suﬃcient temperature coverage to constrain the peak temperature on the low-temperature side of the DEM, as we will see in Section 5.3.2.

150 5.2 Data Analysis

TGOES/TAIA= 1.4_1.4+ 0.4 tGOES-tAIA= 27+27_ 26 Median = 1.3 15

GOES temperature T[MK] 5

N = 149

1 0 1 10 -100 -50 0 50 100 AIA temperature T[MK] tGOES-tAIA [s]

52 ]) -3 log(EMGOES)-log(EMAIA)=-0.1+)=-0.1_ 0.4 [cm p 51 Median =-0.2

47 N = 149 GOES flare peak emission measure log(EM 46 46 47 48 49 50 51 52 -3 AIA flare peak emission measure log(EMp [cm ])

Figure 5.3: GOES/XRS versus SDO/AIA peak temperatures Tp (top left panel) and peak emission measures EMp (bottom left panel). The over-plotted solid line in each of these panels represents the 1:1 relationship while the dashed line represent the average GOES/AIA ratio of the distribution. N.B. They are not fits. The flare peak times refer to the GOES long channel peak time tGOES and coincides with the times tAIA of SDO/AIA measurements within the used time resolution of ≈ 1 min. See the histogram of time differences in top right panel, which has a mean and standard deviation of (tGOES −tAIA) = 27 ± 26 s.

151 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

5.2.2 GOES/XRS Measurements

The GOES temperatures were calculated using measurements made by the XRSs onboard the GOES-14 and -15 satellites (Section 3.1.1 – 3.1.2). The XRS channels have temperature sensitivities in the range ∼4–40 MK, as seen in Figure 5.1. It is therefore blind to the cooler coronal plasma which dominates the response functions of several of the AIA filters. However, it is well suited to observing the peak temperatures of M- and X-class flares which, as we saw in Chapter 4, GOES typically finds to be between

10–25 MK (Figure 4.8). To ensure the accuracy of the GOES temperatures, a background subtraction must be performed to remove the inﬂuence of non-ﬂaring plasma.

This was done using the TEBBS method which we developed in Chapter 4.

The GOES temperatures and emission measures resulting from the TEBBS analysis were found for the same 149 M- and X-class flares observed by SDO/AIA. The top right panel of Figure 5.3 is a histogram of the time difference between the GOES long channel peak, tGOES, and the AIA measurements, tAIA. This reveals that the average difference is 27±26 s. Thus the condition for the simultaneity of measurements is satisfied.

The top left panel of Figure 5.3 shows the GOES temperature of each event plotted against the peak DEM temperature found with SDO/AIA. A positive correlation is evident. Comparison with 1:1 line (solid line) reveals that the GOES temperatures are systematically higher than those found with AIA. The GOES temperatures range from

∼10–24 MK with a mean and standard deviation of 15.6±2.4 MK. This corresponds to an average ratio and standard deviation of TGOES/TAIA of 1.4 ± 0.4 (dashed line). N.B. Neither the solid nor dashed lines represent ﬁts to the data. Despite this, the dashed line agrees visually with the distribution which, despite a few ﬂares with particularly low AIA temperatures relative to GOES, shows that the vast majority of points lie within a factor of two of the average.

The bottom panel of Figure 5.3 shows the GOES emission measure as a function of the DEM peak emission measure as calculated with SDO/AIA. This distribution also

152 5.2 Data Analysis shows a positive correlation with well conﬁned scatter. The GOES emission measures range from 1048.6–1050.5 cm−3 with a mean and standard deviation of 1049.1±0.4 cm−3.

Once again the mean is represented by the dashed line. As before neither the solid nor dashed line represent ﬁts to the data. Nonetheless, by comparing the dashed line to the 1:1 line (solid line), it can be seen that GOES emission measures are systematically

−0.1±0.4 lower than the AIA values and imply an average ratio of EMGOES/EMAIA = 10 , or ∼0.8. There are a few events which show a deviation from the trend at low AIA emission measures. Most of these events have greater uncertainties for the DEM ﬁts

(see Table 2 Aschwanden & Shimizu 2013) and tend to have unusually low DEM peak temperatures (<10 MK). They may therefore be less reliable. However, this study is focussed on establishing the typical relationship between SDO/AIA, GOES, and

RHESSI temperatures in M- and X-class flares which is not significantly affected by these outliers.

5.2.3 RHESSI Measurements

RHESSI (Section 3.2) is capable of producing solar X-ray spectra in the range 3 keV to

17 MeV, with a spectral resolution of ∼1 keV in the range 3–100 keV (Smith et al., 2002).

X-rays of energies 3–∼25 keV are generally thermal bremsstrahlung (Section 2.1.1.1) originating from solar plasma at temperatures of ∼7 MK and above. By making the assumption that this plasma is isothermal, a temperature and emission measure can be produced by ﬁtting a model thermal spectrum to RHESSI observations (Section 3.2.2).

Of the 149 M- and X-class events used in this study, 61 were well observed by

RHESSI. The remaining events occurred during RHESSI’s passage through the South

Atlantic Anomaly or the shadow of the Earth. A further number of events occurred during the annealing process of RHESSI’s detectors, carried out in January and Febru- ary of 2012. For the remaining events we performed systematic ﬁtting procedures to the spectra for the twelve second time interval surrounding the time of peak GOES

153 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

TRHESSI/TAIA= 1.9_1.9+ 1.0 tRHESSI-tAIA= 23+23_ 25

Median = 1.6 8

10 6

4 RHESSI temperature T{MK] 2 N = 61

1 0 1 10 -100 -50 0 50 100 AIA temperature T(MK) tRHESSI-tAIA [s]

log(EMRHESSI)-log(EMAIA)=-0.9_)=-0.9+ 1.4 ]) -3 51 Median =-0.8 [cm p

47 RHESSI emission measure log(EM N = 61

46 46 47 48 49 50 51 52 -3 AIA emission measure log(EMp [cm ])

Figure 5.4: RHESSI versus SDO/AIA peak temperatures Tp (top left panel) and peak emission measures EMp (bottom left panel). The over-plotted solid line in each of these panels represents the 1:1 relationship while the dashed line represent the average GOES/AIA ratio of the distribution. N.B. They are not fits. The flare peak times refer to the GOES long channel peak time tGOES and coincides with the times tAIA of AIA measurements within the used time resolution of ≈ 1 min. See the histogram of time differences in top right panel, which has a mean and standard deviation of (tRHESSI −tAIA) = 23±25 s.

154 5.2 Data Analysis emission. In order to ensure that only the thermal component was included in the

fitting process, the energy range of the fit was set to 5–20 keV for all intervals. As all of the studied flares were M- or X-class, RHESSI’s aluminium attenuators were automatically moved in front of the grids to protect the germanium detectors during periods of peak flux. This meant that the only valid spectra to be used for background subtraction were those taken during adjacent night intervals, when solar emission was occulted by the Earth. However, as the count rate during these events was so high above quiet-sun background, the requirement for subtraction was vastly diminished. The top right panel of Figure 5.4 shows a histogram of the difference between measurement times of the RHESSI and AIA observations. Once again it peaks below one minute within uncertainty demonstrating that the requirement for measurement simultaneity is satisfied.

From the 61 ﬂare spectra analysed, the temperatures were found to have a mean and standard deviation of TRHESSI = 21 ± 10 MK. This value is higher than both the GOES-derived (15.6±2.4 MK), and AIA DEM peak values (12.0±2.9 MK). The average ratio between the RHESSI and GOES temperatures was found to be TRHESSI /TGOES = 1.3±0.7, which agrees very well with Battaglia et al. (2005), McTiernan (2009) and

Raftery et al. (2009). The RHESSI temperatures are plotted against the AIA DEM peak temperatures in the top left panel of Figure 5.4. Comparison with the average temperature ratio (dashed line) with the 1:1 line (solid line) conﬁrms that RHESSI exhibits higher temperatures with an average temperature ratio of TRHESSI /TAIA = 1.9±1.0. However, no clear increasing trend is visible in this distribution suggesting that the RHESSI temperature is not closely related to the DEM peak measured with

AIA, but rather skewed towards the high-temperature tail of the DEM. Once again, it is important to note that the over-plotted solid and dashed lines do not represent ﬁts to the data.

A similar scenario is seen in the bottom panel of Figure 5.4 which shows RHESSI

155 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS emission measures as a function of AIA peak DEM values. Here the mean RHESSI emission measure was found to be roughly 1049.0 cm−3. This is systematically lower than both the GOES and AIA values which have averages of 1049.1 and 1049.2 cm−3 respectively, and corresponds to an emission measure ratio of EMRHESSI /EMAIA = 10−0.9±1.4 = 0.13.

5.3 Discussion

5.3.1 The GOES Temperature Bias

In order to understand the discrepancies between the DEM peak temperatures obtained with SDO/AIA and GOES/XRS, we have to investigate the eﬀect of multi-thermal

DEMs on the GOES filter ratio. The standard GOES temperature and emission measure inversions (Thomas et al., 1985; White et al., 2005) are based on the assumption of an isothermal plasma, which corresponds to a δ-like DEM. SDO/AIA has 6 coronal channels that constrain the DEM, and we assume here that a Gaussian DEM distribution (in log10T-space) fitted to these fluxes yields an acceptable approximation of the peak emission measure and temperature of the true DEM.

For the GOES/XRS response functions, we use the simple expressions from the original fits of Thomas et al. (1985) (Section 3.1.2). As previously stated, updated and more complicated expressions specified with separate sets of polynomial coefficients for each of the GOES spacecraft are given in White et al. (2005). However, these are expected to yield very similar results. Summarising what was said in Section 3.1.2, the temperature-dependent part of the GOES long channel response function, b8(T ), can be fitted with a third-order polynomial with temperature, T , in units of MK (Equa- tion 3.9),

55 −2 2 −4 3 55 −2 3 10 b8(T ) = −3.86+1.17T −1.31×10 T +1.78×10 T [10 W m cm ]

156 5.3 Discussion

The temperature itself can be expressed as a function of the ratio of the GOES short

(B4) and long (B8) channel ﬂuxes, R(T ) = B4(T )/B8(T ). This is equivalent to the ratio of the temperature dependent parts of the response functions, R(T ) = b4(T )/b8(T ). The relation between temperature and the GOES ﬁlter ratio is then given by (Equa- tion 3.8),

T (R) = 3.15 + 77.2R − 164R2 + 205R3 [MK]

Using Equations 3.9 and 3.8, the emission measure, EM, can then be derived from the measured long channel ﬂux (Equation 3.10).

−3 EM = B8/b8(T ) [cm ]

The GOES ﬁlter ratio as a function of the temperature can easily be inverted from

Equation 3.8 by numerical interpolation of R-values for a ﬁxed temperature array,

T (Ri), in the range of 0 < R < 1. This GOES ﬁlter ratio R(T ) is shown in Figure 5.5 (curve labelled ‘isothermal’ in top panel) and varies from R(T = 4 MK) ≈ 0.01 to

R(T = 40 MK) ≈ 0.66.

We can calculate the GOES ﬁlter ratio for Gaussian DEM distributions (in log10T) with particular values for the Gaussian width σT . This is done by convolving the Gaussian DEM distributions (Equation 5.2). The GOES short and long channel ﬂuxes

55 can be then directly computed with the GOES response functions ρ8(T ) = b810 and

ρ4(T ) = ρ8(T ) × R(T ),

Z dEM(T ) B = ρ (T )dT (5.4) 4 dT 4

157 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

0.7 Isothermal

0.6

8 0.5 /B 4 Multi-thermal

0.4 σ =1.00 σT=0.90 σT T=0.80 0.3 σ T=0.70 σ T=0.60

GOES filter ratio R=B 0.2 σ T=0.50 σ T=0.40 0.1 σ T=0.30 σ =0.20 σT=0.10 0.0 T 17 MK 1 10 100 Temperature T[MK]

5 σ =1.00 σT=0.90 σT T=0.80 ) T σ σ 4 T=0.70

/T( σ T=0.60

GOES σ

=T T=0.50 T 3 σ T=0.40

2 σ T=0.30 qT=1.7 σ T=0.20 σ =0.10 σT 1 T=0.00 Isothermal GOES temperature bias q

0 1 10 100 Temperature T[MK]

Figure 5.5: Top: The filter ratio of the GOES 0.5-4 A˚ to the 1-8 A˚ channel is shown for an isothermal DEM (thick curve) and for Gaussian DEM distributions with Gaussian widths of log10(σT ) = 0.1, ..., 1.0. The filter ratio is B4/B8 = 0.31 for an isothermal DEM with a peak at Tp = 10 MK. For a Gaussian DEM with a width of σT = 0.5 (dashed curve), the corresponding isothermal filter-ratio corresponds to a temperature of Tp = 17 MK, which

deﬁnes a temperature bias of qGOES = Tiso/TσT = 1.7. Bottom: The temperature bias of multi-thermal DEMs with a peak temperature at Tp(σT ) compared with the temperature Tiso of isothermal DEMs is shown as a function of the temperature and for a set of Gaussian widths σT .

158 5.3 Discussion

Z dEM(T ) B = ρ (T )dT (5.5) 8 dT 8

From this, the GOES filter ratios, R(T, σT ), for any arbitrary temperature width, σT , can be obtained. These multi-thermal GOES filter ratios are shown in Figure 5.5 (top panel) for a range of widths, σT = 0.1, ..., 1.0. The slope of the filter ratio progressively

ﬂattens for larger thermal widths σT . For instance, the GOES ﬁlter ratio R(T = 10

MK, σT = 0) ≈ 0.11 for an isothermal DEM at a temperature of Tp = 10 MK, but increases to R(T = 10 MK, σT = 0.5) ≈ 0.31 for a Gaussian width of σT = 0.5 (marked with a dashed line in Figure 5.5 top panel). Consequently, if we make the assumption of an isothermal plasma, as it is done in the standard application of GOES- derived temperatures, we would infer from the same observed ﬁlter-ratio of R = 0.31 an isothermal temperature of TGOES = 17 MK. We would thus overestimate the peak

DEM temperature by a factor of qGOES = TGOES/Tp(σT = 0.5) = 1.7.

These temperature bias factors, qGOES = TGOES/Tp(σ), are computed for a number of Gaussian widths in the range of log10(σT ) = 0.1, ..., 1.0 in Figure 5.5 (bottom panel). From these calculations we see that the GOES temperatures are generally overestimated

< for ﬂare peak temperatures of Tp ∼ 22 MK, while they are underestimated above this critical value. The critical value Tcrit ≈ 22 MK is related to an inversion point in the GOES isothermal ﬁlter ratio function R(T ). The overestimation can be as large as a

> factor of 4 for low ﬂare temperatures near T ∼ 4.0 MK and for broad multi-thermal

DEMs with a Gaussian width of σT ≈ 1.0.

This temperature bias, qGOES, can approximately be ﬁtted by

0.9σT TGOES 22 qGOES = ≈ , (5.6) Tp Tp,MK

where all variables have the same meaning as above except for Tp,MK which is the

159 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS peak temperature of the Gaussian DEM in units of megakelvin. This equation was found empirically from the more rigorously determined curves in the bottom panel of Figure 5.5 which were calculated using Equations 5.5 and 5.4. Figure 5.6 shows the numerically calculated GOES temperature bias as a function of temperature for

σT = 0.1 and σT = 0.9 (solid lines), taken from Figure 5.5. Over-plotted as the dashed lines are the corresponding curves from Equation 5.6. It can be seen that the solid and dashed lines are very similar, which validates Equation 5.6. For the particular data set of 149 M- and X-class ﬂares observed with SDO/AIA in this study, we measured a mean DEM peak temperature of TAIA = 12.0±2.9 MK and Gaussian DEM half widths of log10(T ) = 0.50 ± 0.14. From this we predict (with Equation 5.6) a mean GOES temperature bias of

pred qGOES = 1.4 ± 0.3 (5.7)

When rounded to one decimal place, this precisely matches the observed GOES to AIA temperature ratio

obs qGOES = 1.4 ± 0.4 (5.8)

pred obs The residuals between observed and predicted temperature ratios (qGOES −qGOES) were found to be independent of temperature, suggesting that these averages well represent the overall distribution.

From these results, we conclude that GOES overestimates the peak temperature of large GOES ﬂares (M- and X-class) on average by 40%. Only for ﬂare temperatures around Tp ≈ 20 MK does the GOES temperature match the DEM peak temperature. pred For our sample we predict GOES temperatures with a mean of TGOES = qGOES ×

TAIA = 16.2 ± 2.1 MK, which also agrees with the observed temperature range of

TGOES = 15.6 ± 2.4.

160 5.3 Discussion

Figure 5.6: Solid lines: Numerically determined GOES temperature biases for DEM widths of σT = 0.1 and σT = 0.9. (As in Figure 5.5). Dashed lines: Corresponding curves calculated with Equation 5.6.

161 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

5.3.2 The RHESSI Temperature Bias

The temperature-dependent response functions (Figure 5.1) show that the temperature range of AIA filters covers DEM peak temperatures of TAIA ≈ 0.5 − 20 MK. RHESSI covers TRHESSI ≈ 7 − 140 MK, if we associate the fitted thermal energies of ≈ 6 − 12 keV with the DEM peak temperatures. This means that SDO/AIA and GOES/XRS can constrain the peak of flare DEMs well for flare temperatures of Tp ≈ 4 − 20 MK, while RHESSI applies thermal fits to the high-energy tail of the DEM distribution, but cannot constrain the peak of the DEM well. RHESSI fits to the thermal spectrum are often made with the assumption of an isothermal DEM. However, the RHESSI data clearly show evidence that all flare DEMs cover a broad temperature range and therefore should be fitted with a multi-thermal DEM model (e.g. Aschwanden 2007). In the following we will investigate the discrepancy in flare DEM peak temperatures resulting from isothermal RHESSI fits in the 6–12 keV range and multi-thermal (Gaussian) DEM

ﬁts obtained with AIA.

The bremsstrahlung spectrum F () as a function of the photon energy = hν was discussed in Section 2.1.1.1 of this thesis. It is given by (Brown 1974; Dulk & Dennis

1982),

Z exp (−/k T ) dEM(T ) F () = F B dT [keV s−1 cm−2 keV−1] (5.9) 0 T 1/2 dT

−39 −1 −2 −1 where F0 ≈ 8.1 × 10 keV s cm keV . This equation assumes the coronal electron density is equal to the ion density (ni = ne), the ion charge number Z ≈ 1, and neglects factors of order unity, such as from the Gaunt g(ν, T ). The dEM(T )/dT speciﬁes the DEM (n2dV ) in the element of volume dV corresponding to temperature range dT ,

dEM(T ) dT = n2(T ) dV (5.10) dT

162 5.3 Discussion

Here we use the same parameterisation of the DEM as in Section 5.3.1 (Equation 5.2).

This can be characterised by three parameters, DEM peak emission measure EMp,

DEM peak temperature Tp, and Gaussian width log10(σT ), all of which we obtained in Section 5.2.1. Inserting the DEM function (Equation 5.2) into the bremsstrahlung spectrum (Equation 5.9) we obtain an isothermal spectrum for σT 7→ 0, and a multi- thermal spectrum for σT > 0. As an example we show the isothermal (Tp = 10 MK) photon energy spectrum, F (ε), in the energy range of ε = 3 − 30 keV in Figure 5.7

(top panel). We see that the thermal spectrum falls oﬀ steeply, with a ﬂux ratio of

3 qF = F6/F12 = 10 between 6 keV and 12 keV.

Now we calculate a multi-thermal spectrum F (ε) for a Gaussian DEM with the same peak temperature Tp = 10 MK, but a Gaussian width of log10(σT ) = 0.5. This is shown in Figure 5.7 (top panel, dashed spectrum). It is much flatter and has a flux ratio of qF = F6/F12 ≈ 3.7 between 6 keV and 12 keV. This is more than two orders of magnitude smaller than the isothermal case. An isothermal fit to this flux ratio would correspond to a DEM peak temperature of Tiso = 53 MK because this temperature produces the same flux ratio of qF = F6/F12 ≈ 3.7 (Figure 5.7, top panel, thick solid line). Thus the assumption of isothermal DEMs leads to significant overestimates of temperature and emission measure. In the example here, the DEM peak temperature is overestimated by a factor of qRHESSI = Tiso/Tp(σT = 0.5)=(53 MK/10 MK)=5.3, and the DEM peak emission measure is underestimated by about a factor of 0.03. Since

RHESSI spectra are often ﬁtted with an isothermal spectrum, the obtained temperature virtually always overestimates the DEM peak temperature substantially.

Next we calculate the ﬂux ratios, qF = F6/F12, for a range of DEM Gaussian widths, log10(σT ) = 0.1, ..., 1.0, and show their dependence on the DEM peak temperature, Tp (Figure 5.7, bottom left panel). The ﬂux ratio is highest for an isothermal spectrum, but progressively decreases with broadening DEMs (i.e. larger Gaussian widths, σT ).

We also calculate the RHESSI isothermal temperature bias, qRHESSI = Tiso/Tp(σT ),

163 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

105

F 6 F12 σ Tp,mt=10 MK, T=0.5 100 Tiso=53 MK

10-5 RHESSI flux F(E) (arbitrary units) 10-10

E = 6 E =12 1 2 Tiso=10 MK 1 10 100 Energy E[keV]

15 σ 15 σT=0.90=1.00 σT=0.80 σT σ iso-thermal =0.70 T=0.30 T )

T σ T=0.60 σ ( p σ /T T=0.50 12 iso /F 6I =T

10 T

=F 10 σ =0.40 F T

σ =0.30 σ T T=0.40

5 qT=5.3 σ =0.50 5 σT σ RHESSI flux ratio q T=0.60 q =3.7 =0.20 σ F T T=1.00 multi-thermal

RHESSI temperature bias q σ T=0.10 Isothermal 0 0 106 107 108 106 107 108 Temperature T[MK] Temperature T[MK]

Figure 5.7: Top: Three simulated RHESSI thermal bremsstrahlung photon spectra generated using Equation 5.9 (Brown, 1974; Dulk & Dennis, 1982). The bottom curve is an isothermal spectrum with a temperature of Tiso = 10 MK. The top (dashed curve) is a multi-thermal spectrum with a peak temperature of TMT = 10 MK and a Gaussian width of log10(σT ) = 0.5. And the middle curve is an isothermal spectrum that has the same flux ratio qF = F6/F12 = 3.7, which is found for Tiso = 53 MK. This corresponds to a temperature bias of qRHESSI = TRHESSI /TAIA = 5.3. Bottom left: The RHESSI flux ratio of isothermal and multi-thermal spectra is shown as a function of the DEM peak temperature, Tp, for Gaussian DEM distributions with Gaussian widths of log10(σT ) = 0.1, ..., 1.0. The flux ratio, qF = 3.7, corresponding to the case shown in the top panel is marked with dashed line. Bottom right: The temperature bias, qRHESSI = Tiso/Tp, of isothermal DEMs with a peak temperature at Tp is shown as a function of the peak temperature, Tp, and for a set of Gaussian widths, σT . The case with a temperature bias of qRHESSI = 5.3 of the spectrum shown in the top panel is indicated with a dashed line. 164 5.3 Discussion between an isothermal fit and a multi-thermal DEM (Figure 5.7, bottom right). We see that the temperature overestimation can be up to a factor of qRHESSI ≈ 5 for narrowband DEMs with log10(σT ) = 0.25 and low flare temperatures of Tp ≈ 4 MK, and up to the same factor for broadband DEMs with log10(σT ) ≈ 0.5 − 1.0 for larger temperatures of Tp ≈ 10 − 12 MK, which are typically measured in flares.

Applying this model for the isothermal bias of spectral fits in the 6–12 keV energy range to the AIA flare measurements, we can predict the expected temperature range measured by RHESSI for the same set of M- and X-class flares. This isothermal temperature bias, qRHESSI = TRHESSI /Tp, shown in Figure 5.7 (bottom right panel), can approximately be represented by the simple relationship,

σ1/2 TRHESSI 60 T qRHESSI = ≈ (5.11) Tp Tp,MK where all variables have the same meanings as for Equation 5.6. This equation was found empirically from the more rigorously determined curves in the bottom right panel of Figure 5.7. Figure 5.8 shows the numerically calculated RHESSI temperature bias as a function of temperature for σT = 0.1 and σT = 0.9 (solid lines), taken from Figure 5.7. Over-plotted as the dashed lines are the corresponding curves from

Equation 5.11. It can be seen that the solid and dashed lines are very similar which validates Equation 5.11. For the 61 ﬂares observed by both SDO/AIA and RHESSI, we measured a mean DEM peak temperature of TAIA = 12.0±2.9 MK and Gaussian DEM half widths of log10(σT ) = 0.51 ± 0.14, from which we predict (with Equation 5.11) a mean RHESSI temperature of TRHESSI = 37.2 ± 6.1 MK, or a RHESSI isothermal temperature bias of qRHESSI = TRHESSI /TAIA of

pred qRHESSI = 3.3 ± 1.0 (5.12)

This is commensurate with the observed RHESSI to AIA temperature ratio (Figure 5.4,

165 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

Figure 5.8: Solid lines: Numerically determined RHESSI temperature biases for DEM widths of σT = 0.1 and σT = 0.9. (As in Figure 5.7). Dashed lines: Corresponding curves calculated with Equation 5.11.

top panel).

obs qRHESSI = 1.9 ± 1.0 (5.13)

There is not a very close agreement between the observed and predicted temperature ratios. However a very accurate prediction is not expected. This is because the high- temperature part of the DEM in the range of Tp ≈ 10–20 MK is not so well constrained with AIA, to which only the 193 A˚ line (with a Fe XXIV line) and the 94 A˚ ﬁlters are sensitive. Also the shape of the DEM function, for which we choose a simple symmetric

Gaussian, may not adequately describe the high-temperature tail of the DEM function.

This is supported by both the results of Graham et al. (2013) and our ﬁnding that the

pred obs residuals between observed and predicted temperature ratios (qRHESSI − qRHESSI )

166 5.4 Conclusions have a slight temperature dependence. The residuals are greater for lower peak DEM temperatures. For lower peak temperatures, the high-temperature part of the DEM sampled by RHESSI is further away (in temperature space) from the peak. Therefore, the prediction of a temperature bias requires a greater extrapolation of the Gaussian

DEM into the high-temperature tail. This can exaggerate any discrepancy between the high-temperature tail predicted by a Gaussian parameterisation and the ‘true’ high- temperature tail sampled by RHESSI. An asymmetric DEM function with a steeper fall-off at the high-temperature tail (e.g. Aschwanden & Alexander 2001) could bring the predicted RHESSI bias in better agreement with the observed RHESSI/AIA temperature ratio. Despite this, our model prediction of a substantial temperature overestimation by isothermal fits to the RHESSI spectra is consistent with the systematically higher measured RHESSI temperatures. Thus we conclude that self-consistent flare temperatures and emission measures require simultaneous fitting of EUV (AIA) and soft X-ray (GOES, RHESSI) fluxes with a suitably parameterised DEM distribution function.

5.4 Conclusions

In this chapter, the differential emission measures of 149 M- and X-class flares were calculated at the time of the GOES peak 1–8 A˚ flux using SDO/AIA. GOES temperatures and emission measures of these events were also calculated at the flare peak using an isothermal assumption (White et al. 2005) and compared to the peak temperatures and emission measures of the AIA DEMs. It was found that, on average, the GOES temperatures were a factor of 1.4±0.4 higher than the AIA DEM peak temperatures.

The temperatures and emission measures of 61 of these ﬂares were also calculated with

RHESSI using an isothermal ﬁt to the observed spectra between 5–20 keV. The RHESSI temperatures were found to be higher than both GOES and SDO/AIA. On average the

167 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

RHESSI temperatures were a factor of 1.9±1.0 higher than SDO/AIA and a factor of

1.3±0.7 higher than GOES. The ratio of RHESSI to GOES temperatures was found to agree with previous studies. Conversely, the GOES emission measures were typically lower than the AIA DEM peak emission measures, while the RHESSI emission measures were found to be lower still.

The eﬀect of the isothermal assumption on the calculation of the GOES temperatures was investigated. It was found that DEMs of greater widths (more multi-thermal) increasingly altered the relationship between DEM peak temperature and GOES ﬁlter ratio. For temperatures less than 22 MK, the isothermal assumption was predicted to result in higher derived GOES temperatures than multi-thermal DEMs peak temperatures. However, for temperatures greater than 22 MK, the isothermal assumption was predicted to lead to lower-derived GOES temperatures than the multi-thermal

DEMs. The resulting bias between temperatures derived from the isothermal assumption, TGOES, and a DEM of width of σT , was described by Equation 5.6. This resulted in a mean predicted isothermal bias for the events observed by AIA and GOES of 1.4±0.3. This agreed well to a precision of one decimal place with the observed

GOES/AIA temperature ratio of 1.4±0.4.

A similar analysis was performed on derived RHESSI temperatures. It was found that in the range 4–50 MK, the isothermal assumption was predicted to lead to higher derived temperatures than those obtained with multi-thermal DEMs. The discrepancy was described by Equation 5.11. This resulted in a mean predicted RHESSI isothermal bias for the 61 events observed by both SDO/AIA and RHESSI of 3.3±1.0. This is commensurate with the observed RHESSI-AIA temperature ratio of 1.9±1.0 but is not in close agreement. However a close agreement is not necessarily expected since the high temperature tail of the DEM in the RHESSI temperature range is not well constrained by SDO/AIA and a symmetric Gaussian may not be best suited to describing this high temperature tail. Therefore, in order to self-consistently obtain ﬂare temperatures,

168 5.4 Conclusions

EUV (AIA) and soft X-ray (GOES and RHESSI) ﬂuxes must be simultaneously ﬁtted with a suitably parameterised DEM distribution function, e.g. a bi-Gaussian.

The work outlined in this chapter was conducted in collaboration with Aidan M.

O’Flannagain, Markus J. Aschwanden, and Peter T. Gallagher and has been published in Solar Physics (Ryan et al., 2014).

169 5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS

170 Chapter 6

Decay Phase Cooling & Inferred Heating of Solar Flares

In this chapter, we ﬁnally focus more directly on the hydrodynamic evolution of ﬂares.

The cooling of 72 M- and X-class ﬂares is examined using GOES/XRS and SDO/EVE.

The observed cooling rates are quantified and the observed total cooling times are compared to predictions of an analytical 0D hydrodynamic model. It is found that the model does not fit the observations well, but provides a well defined lower limit on a flare’s total cooling time. The discrepancy between observations and the model is then assumed to be primarily due to heating during the decay phase. The heating needed to account for the discrepancy is quantified and found be ∼50% of the total thermally radiated energy calculated with GOES. This decay phase heating is found to scale with the observed peak thermal energy. It is predicted that decay phase heating is only a small fraction of the peak in small flares and thus the peak well approximates the total. However, in the most energetic flares such an approximation is not suitable as the decay phase heating inferred from the model can be several times greater than the observed peak thermal energy. This work has been published in the Astrophysical Journal (Ryan et al., 2013).

171 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

6.1 Introduction

Having discussed and developed plasma diagnostic techniques and explored the biases and limitations of temperature measurements, we now focus more directly on the hydrodynamic evolution of solar ﬂares. In Section 1.5.1 we saw that the standard solar

ﬂare model states that ﬂares are powered by a vast and rapid release of energy via the process of magnetic reconnection. This causes a rapid heating and expansion of the

flare plasma which is then believed to cool by conductive, radiative and enthalpy-based processes. However, the balance between cooling and heating in solar flares and the processes which determine this are still not fully understood. A greater insight into this interaction would allow us to better constrain the energy release mechanisms of solar flares.

In Section 2.2.3 we also saw that to date there have been many studies aimed at modelling the heating and cooling of solar ﬂares (e.g. Antiochos, 1980; Antiochos &

Sturrock, 1978; Bradshaw & Cargill, 2005; Cargill, 1993; Doschek et al., 1983; Fisher et al., 1985b; Klimchuk, 2006; Klimchuk & Cargill, 2001; Moore & Datlowe, 1975;

Reeves & Warren, 2002; Sarkar & Walsh, 2008; Warren, 2006; Warren & Winebarger,

2007). These include full 3D magnetohydrodynamic (MHD) models as well as 1D MHD models. 1D models assume that ﬂare loop strands are magnetically isolated and therefore only solve the MHD equations along the axis of the magnetic ﬁeld (e.g. Bradshaw

& Cargill, 2005). This is less computationally draining than the full 3D treatment and therefore allows a higher resolution, more useful for detailed comparison with observation (Section 2.2.3). 0D models have also been developed (e.g. Enthalpy-Based

Thermal Evolution of Loops, EBTEL; Klimchuk et al., 2008) which treat field-aligned average properties (Section 2.2.3). Although these models sacrifice some completeness, they are much faster to run, allowing an easier exploration of the dependence of results on different possible coronal property values. Although these models are valuable for

172 6.1 Introduction increasing our understanding of ﬂare heating and cooling, they nonetheless suﬀer from drawbacks. These can include arbitrary inputs of unobservable parameters such as heating function and number of loop strands.

As well as theoretically focussed papers, observational studies of ﬂare cooling are also numerous. Culhane et al. (1970) compared simple collisional, radiative, and conductive cooling models to observations of four ﬂares made with the fourth Orbiting

Solar Observatory (OSO-4). They found that collisional cooling was unphysical while conduction and radiation were equally plausible. Although they could not determine which was dominant, they did ﬁnd that for radiative cooling to dominate, the ﬂare den-

11 −3 sity would have to be high (&10 cm ) while conduction would require low densities (∼1010 cm−3) to dominate.

In contrast, Withbroe (1978) was able to compare the relative importance of cooling mechanisms. This study examined the diﬀerential emission measure (DEM) of a single

ﬂare using Skylab and hence determined that conductive and radiative losses were comparable. From discrepancies between observations and conductive and radiative cooling models, it was determined that ∼1031 ergs of additional heating must have been deposited after the ﬂare peak.

More recently, Jiang et al. (2006) examined loop-top sources in 6 ﬂares using

RHESSI. They found that the observed cooling rate was slightly higher than expected from radiative cooling, but signiﬁcantly lower than that expected from conduction. To account for this, they calculated that more than 1030 ergs of additional heating during the decay phase was necessary. This was greater than that seen during the impulsive phase. However they concluded that much of this discrepancy was more plausibly explained by suppressed conduction.

Raftery et al. (2009) also used RHESSI along with several other instruments to chart the thermal evolution of a single C1.0 ﬂare. They performed a best ﬁt to the observations using the EBTEL model and an assumed heating function to infer radiative

173 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES and conductive cooling proﬁles. Conduction was found to dominate initially while radiation dominated in the latter phases. It was found that no additional energy input after the ﬂare peak was required for observation and model to agree.

A common aspect of many flare cooling studies such as those mentioned above is that they focus on single or small numbers of events. This means they cannot say if their findings are anomalous or characteristic of flares. As a result, it is still unclear just how well cooling models describe ensembles of flares. In this chapter we aim to improve upon previous studies by observing the cooling profiles of 72 M- and X-class

ﬂares. This is done using observations from GOES/XRS and SDO/EVE. The observed cooling times are then compared to predictions made by the model of Cargill et al.

(1995), a simple, analytical 0D model. Although this model is highly simplified, it was chosen as a first step because it is quick and easy to apply to many flares. In Section 6.2 we describe our observations. In Section 6.3 we discuss the assumptions, limitations and equations of the Cargill et al. (1995) model and describe how we observationally calculated the required inputs. In Section 6.4 we compare the observed cooling times to those predicted by the model and quantify the discrepancy. We then infer the decay phase heating required to account for this difference. Finally we outline our conclusions in Section 6.5.

6.2 Observations & Data Analysis

6.2.1 Flare Sample

Observations for this study were taken from three instruments: the XRS onboard the

GOES-14 and 15 (Section 3.1.1); MEGS-A onboard SDO/EVE (Section 3.4.2.1); and the Hinode/XRT (Section 3.3.1).

The 72 M- and X-class ﬂares examined in this study were chosen via two criteria.

Firstly, their decay phases had to be temporally isolated from other ﬂares. This was

174 6.2 Observations & Data Analysis

Table 6.1: Wavelengths and temperatures of bandpasses and emission lines used in measuring cooling rates

Instrument Wavelength [nm] Temperature [MK] 0.05–0.4 – Short GOES/XRS >4 0.1–0.8 – Long Ion Wavelength [nm] Temperature [MK] Fe XXIV 19.20 15.8 Fe XXII 11.71 12.6 Fe XIX 10.83 10.0 Fe XVIII 9.39 7.9 Fe XVI 33.54 6.3 Fe XV 28.41 2.5 Fe XIV 26.47 2.0

determined from visual inspection of the GOES lightcurves. Secondly, the ﬂares had to be observed to cool to at least 8 MK with either the GOES/XRS or SDO/EVE

MEGS-A. A complete list of the ﬂares and their properties are listed in Table 6.2.

6.2.2 Observing Flare Cooling

The cooling of the flares in this study was charted by combining the peak of the GOES temperature profile with the peaks of lightcurves of various temperature-sensitive Fe lines observed by SDO/EVE MEGS-A. The GOES temperature was calculated using the TEBBS method which we developed in Chapter 4. The Fe lines used in this study along with their formation temperatures are listed in Table 6.1. These lines were chosen because in the conditions of a solar flare, they are dominant over neighbouring lines within the MEGS-A resolution and therefore minimally blended. Before extracting these lightcurves, a background subtraction was made to each observed flare spectrum.

The background spectrum was found by averaging the spectra within a quiet period before the flare start time. This period was determined for each flare by visual inspection of the GOES lightcurves. This helped ensure that the behaviour of the lightcurves was minimally contaminated by emission from non-flaring plasma. The irradiance observed

175 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.1: Cooling track for the 2010-Nov-06 M5.5 flare which began at 15:28 UT. a) Background-subtracted GOES temperature profile. Peak is marked by the vertical line. b) – h) Lightcurves of sequentially cooler Fe lines ranging from 15.8 MK to 2 MK observed by SDO/EVE MEGS-A. The peak of each lightcurve is also marked by a vertical line. i) Combined cooling track obtained by plotting the time of the peak of each profile (including GOES temperature profile) with its associated peak temperature. The resultant cooling time is the duration of this cooling track.

176 6.2 Observations & Data Analysis at the wavelength of each line in Table 6.1 was then summed with that within ±0.05 nm, i.e. the spectral resolution of MEGS-A. This was done for each spectrum taken during the ﬂare and hence ﬂare lightcurves were formed. A cooling track was then generated by plotting the peak time of each lightcurve against its associated formation temperature.

The cooling time was then given by the duration of this track. Despite the ﬁndings in

Chapter 5, we have assumed here that the ﬂare plasma is isothermal in order to remain consistent with the Cargill model. Therefore, the results must be taken with the caveat that the true temperature distributions within the ﬂares may be more complicated.

Figure 6.1 shows an example for an M5.5 ﬂare which occurred on 2010 November 06 at 15:27 UT. Figure 6.1a shows the GOES temperature curve while Figures 6.1b–6.1h show the lightcurves of the Fe lines measured by SDO/EVE. The vertical lines in each panel mark the peak time of that lightcurve. The lightcurves peak in order of descend- ing temperature. This is interpreted as being due to plasma cooling. Figure 6.1i shows the resulting cooling track, with each datum point representing the peak time and temperature associated with the lightcurves above. From this it can be seen that this ﬂare cooled from 17 MK to 2 MK over the course of 389 ± 10 seconds. The uncertainty comes from combining the time resolutions of GOES/XRS and SDO/EVE in quadrature.

In order to parameterise the flare cooling, each flare’s cooling profile was fit with a second-order polynomial of the form

2 T (t) = T0 + θt + µt [MK] (6.1)

where t is time since the start of the cooling phase in seconds, T0 is the temperature at the start of the cooling phase in megakelvin (i.e. GOES temperature peak), θ is the linear cooling coeﬃcient [MK s−1], and µ is the non-linear cooling coeﬃcient [MK s−2].

Figure 6.2a shows a histogram of the non-linear cooling coeﬃcients, µ. The distribution

−4 −2 is very narrowly peaked around zero with a full width half max of .10 MK s and

177 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.2: Histograms showing the non-linear (panel a) and linear (panel b) coefficients of the second-order polynomial fits to the observed cooling profiles of the 72 M- and X-class flares in this study (Equation 6.1).

178 6.3 Modelling a mean of -6×10−5 MK s−2. This implies that the majority of the flare cooling profiles are very linear. This agrees qualitatively with Raftery et al. (2009), whose observed cooling profile of a C1.0 flare was also quite linear. Figure 6.2b shows a histogram of the linear cooling coefficients, θ. Since the non-linear cooling coefficients are so small, the linear cooling coefficients approximate the cooling rates. The histogram ranges from

-1.5–0 MK s−1 and has a mean of -0.035 MK s−1. This implies that the SXR-emitting plasma of an average M- or X-class ﬂare cools at a rate of ∼3.5×104 K s−1. It should be noted that although the histogram in Figure 6.2b peaks at the bin centred on zero, all

ﬂares have non-zero linear cooling coeﬃcients. These parameterisations are used again in Section 6.4.2.

6.3 Modelling

To model the cooling observations discussed in the previous section, we used the analytical 0D hydrodynamic model of Cargill et al. (1995). The derivations, assumptions, and limitations of this model are outlined in Section 2.2.4. The model is based on the characteristic cooling timescales of conduction (τc; Equation 2.46) and radiation

(τr; Equation 2.50), derived from the energy transport equation (Equation 2.40). De- pending on which timescale is shorter at the start of the cooling phase (denoted with a subscript ‘0’), the total cooling time of a ﬂare can be predicted by Equation 2.55

(τc0 < τr0) or Equation 2.56 (τr0 < τc0).

The Cargill model requires three observable inputs. These are initial temperature,

T0, initial density, n0, and loop half-length, L, which is assumed to be constant. The initial temperature, T0, is that at the beginning of the observed cooling track, i.e., the GOES temperature peak. As stated in Section 6.2.2 this was calculated using the

TEBBS method. This measurement has two sources of uncertainty: one due to the instrument (Garcia, 1994, Section 7) and one due to the background subtraction (Ryan

179 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.3: Relationships between density and Fe XXI line ratios, 12.121 nm/12.875 nm, (14.214 nm + 14.228 nm)/12.875 nm, and 14.573 nm/12.875 nm, calculated using CHIANTI v7. (Milligan et al., 2012)

et al., 2012, Section 3.2). The total uncertainty in the initial temperature was found by combining these two uncertainties in quadrature.

The initial density, n0, was determined as per Milligan et al. (2012). This method uses CHIANTI (version 7; Landi et al., 2012) to determine the relationship between density and the ratios of three density dependent Fe XXI line pairs (12.121 nm/12.875 nm, (14.214 nm + 14.228 nm)/12.875 nm, and 14.573 nm/12.875 nm). Figure 6.3, taken from

Milligan et al. (2012), shows the theoretical relationships between each line ratio and density. In this study, only the first ratio was used as only these lines consistently exhibited increased emission due to the flares. This method is only valid for temperatures above 10 MK due to the formation temperatures of these lines. It is also not sensitive outside the range 1010–1014 cm−3. However, the Cargill model only requires the initial density, i.e. the density at the time of the peak GOES temperature. Since all the flares

180 6.3 Modelling in this study peak above 10 MK and were found to have densities within this range, the method used by Milligan et al. (2012) is suitable.

The uncertainty of the density measurements is due to the uncertainty in the

12.121 nm/12.875 nm line intensity ratio as measured by EVE. The ratio itself has two main sources of uncertainty. First, the instrumental uncertainty of the irradiance of the two lines. Second is the uncertainty due to the noise in the ratio time proﬁle.

The latter was evaluated as the standard deviation during the time period when the

ﬂare was hotter than 12 MK as determined with the GOES/XRS. This threshold was chosen as it ensured that the ﬂare temperature (accounting for uncertainty) was in the valid range of the Milligan et al. (2012) method. The total uncertainty in the ratio was determined from the standard propagation of errors of the two uncertainty sources.

This uncertainty was then transformed to density by propagating the ratio’s upper and lower limits through the Milligan et al. (2012) method. It should be noted that there are additional uncertainties associated with the modelled relationship between

Fe XXI line intensities and density. However, these are expected to be much smaller than the uncertainty sources discussed above. For more information on the modelling of the Fe XXI lines in CHIANTI see Section 4.7.1 of Dere et al. (1997), and references therein.

Ideally the loop half-length, L, would be measured by Hinode/XRT. Its temperature sensitivity, spatial resolution and time cadence make it the most ideal instrument available for directly measuring loop lengths of hot (>1 MK) X-ray- and EUV-emitting ﬂare plasma. It is better suited than SDO/AIA which is often saturated by M- and X-class

ﬂares and which has greater sensitivity to cooler coronal plasma (∼1 MK). However, of the 72 ﬂares included in this study, only 22 were well observed by Hinode/XRT. There- fore, loop lengths were determined using the RTV-scaling law (Rosner et al., 1978)

181 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.4: Hinode/XRT observations of 22 ﬂares within this study, plotted on a log10- scale. The blue lines trace out the plane-of-sky measured loop lengths obtained via the ‘point-and-click’ method. Where unclear, the axis along which the loops should be measured was determined with the aid of SDO/AIA observations. These lengths were then used for comparison with the RTV-predicted values (Figure 6.5).

182 6.3 Modelling given by

T 3 L = (1.4 × 103)−3 max [cm] (6.2) p

where p is pressure and Tmax is the maximum temperature in the loop. By assuming that the plasma is isothermal and obeys the ideal gas law, this can be rewritten in terms of temperature, T , and density, n, which can be calculated using GOES/XRS and SDO/EVE respectively.

1 T 2 L = 3 3 [cm] (6.3) kB(1.4 × 10 ) n

The Hinode/XRT measurements of the 22 well observed flares in this study were used to quantify the uncertainties of the RTV-predicted values Hinode/XRT. Figure 6.4 shows these observations plotted on a log10-scale. The blue lines represent plain-of-sky measurements which performed ‘by eye’ via the ‘point-and-click’ method. A more rigorous analysis attempting to account for projection effects is expected to alter the measured loop lengths only up to a factor of ∼2 which is sufficient for our purposes. In some of the images, diffuse regions of emission can be seen. However, because this is on a log10-scale, emission from these regions is thought to contribute very little relative to the bright loops. The different XRT filters used for each event can be found in Table 6.2.

These filters all peak between 8–13 MK. Their response functions (Figure 3.12) show contributions from plasma at temperatures below 1 MK of at least 2 orders of magnitude lower than the peak. This suggests that the images are not significantly contaminated by emission from lower temperature plasma which might otherwise affect the measured loop lengths. The one exception is the Al-mesh filter, but this was only used for one event and was not found to be an outlier. Where possible, SDO/AIA was used to help determine the axis of the magnetic field along which the loop length should be measured. For instance, a combination of AIA and XRT movies revealed that the long

183 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.5: Comparison of RTV-predicted ﬂare loop half-lengths with those measured with Hinode/XRT. Most of the data points are scattered around the 1:1 line (over-plotted). N.B. It is not a ﬁt.

axis of the ﬂares in the panels e, o, and t of Figure 6.4 were ﬂare arcades and the loop lengths themselves were actually along the shorter axis. Despite its usefulness in these instances, AIA’s sensitivity to cooler plasma and greater tendency to saturate made it less well suited to making the actual measurements than XRT. The loop half-lengths obtained from the XRT measurements were compared to the RTV-predicted values

(Figure 6.5) and a loose correlation around the 1:1 line was found.

The CVRMSD (coeﬃcient of variation of the root-mean-square deviation) of the distribution in Figure 6.5 was used to quantify the uncertainty of the RTV-predicted loop half-lengths. This was found to be 1.8, implying that the loop half-length and uncertainty is given by L = LRTV ± 1.8 · LRTV . This compares favourably with Aschwanden & Shimizu (2013) who compared RTV-predicted loop lengths with the length-scales of less intense ﬂares using SDO/AIA and found an uncertainty of ±1.6 · LRTV .

184 6.4 Results & Discussion

Having measured the initial temperature, initial density and loop half-length, the

Cargill-predicted cooling times were found from Equation 2.55 if τc0 < τr0 and Equa- tion 2.56 if τc0 > τr0. The uncertainties on these cooling times were calculated by ﬁrst rewriting Equations 2.55 and 2.56 in terms of the observed input properties (temperature, density and loop half-length) and then propagating their uncertainties by the standard error propagation rules. Having done this, we then compared the model- predicted cooling times with the observations discussed in Section 6.2.2.

6.4 Results & Discussion

6.4.1 Comparing Observed and Modelled Cooling Times

Figure 6.6 shows the comparison of Cargill-predicted and observed cooling times for 72

M- and X-class flares. The 1:1 line is over-plotted for clarity. It can clearly be seen that the Cargill model is consistent with observations at the shortest cooling times, but is not a good overall fit to the distribution. Upon closer inspection it was found that only 14 events (20%) had observed cooling times which agreed with Cargill within experimental error. Meanwhile 58 (80%) disagreed. Of those, only 1 was overestimated by Cargill. The remaining 57 were underestimated. Thus these results statistically prove that the Cargill model provides a lower limit to the time needed for a flare to cool. In addition, it was found in 52 flares (72%) that radiation dominated conduction for the entirety of the cooling phase. Conduction initially dominated radiation in only 20 flares (28%). This suggests that flares for which radiation is the dominant cooling mechanism (such as those examined by López Fuentes et al., 2007; McTiernan et al., 1993) are far more common than those in which conduction initially dominates

(e.g. Jiang et al., 2006; Moore & Datlowe, 1975; Raftery et al., 2009). Furthermore,

Culhane et al. (1970) concluded from examining a simple radiative cooling model that

ﬂare plasma cooling by radiation in a timescale of ∼500 s would exhibit high densities

185 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.6: Comparison of Cargill-predicted cooling times with observed cooling times. The 1:1 line is overplotted for clarity. This shows the that Cargill-predicted cooling time provides a lower bound to a ﬂare’s observed cooling time.

(1011–1012 cm−3). The average observed cooling time of the events in Figure 6.6 is 653 s and their average density is 1.4×1012 cm−3, which is very close to the conclusions of

Culhane et al. (1970). This appears to strengthen the claim that radiation is typically dominant over conduction throughout a ﬂare’s decay phase. However, this must be treated with caution as there is a lot of scatter in the distribution of observed cooling times in Figure 6.6, and so the mean is not a robust representation of the distribution.

To further quantify the discrepancy between predicted and observed cooling times

(henceforth referred to as the excess cooling time), the root-mean-square deviation

(RMSD) of the distribution was calculated. This was found to be 961 s. Normalizing this to the mean of the observed cooling times (653 s) gives the coeﬃcient of variation of the root-mean-square deviation (CVRMSD). This quantiﬁes the spread of the excess

186 6.4 Results & Discussion cooling times relative to the mean of the observed cooling times. The CVRMSD was found to be 1.47 indicating a large spread as is visually suggested in Figure 6.6.

If the Cargill model is adequately describing the cooling mechanisms of solar ﬂares, the excess cooling time suggests that there is additional heating occurring throughout the decay phase. Similar assumptions have been made in previous studies (e.g. Hock et al., 2012b; Jiang et al., 2006; Withbroe, 1978). In the following section we explore just how much additional heating energy is required to account for the excess cooling times and examine the distributions of these energies.

6.4.2 Inferring Heating During Decay Phase

For radiatively dominated ﬂares, the decay phase heating required to account for the excess cooling time can be determined from the following modiﬁed version of the energy transport equation

∂T 3k n = −χnζ T α + h [ergs s−1] (6.4) B 0 ∂t 0

where kB is Boltzmann’s constant, n0 is the density (assumed to be constant and equal to the initial density to remain consistent with the Cargill model), T is temperature, t is time, χ, ζ and α have the same values as implicitly used previously in this chapter and explicitly given in Section 2.2.4 (1.2 × 10−19, 2, and -1/2, respectively; Rosner et al. 1978), and h is the heating rate per unit volume. This equation states that the rate of change of thermal energy density (LHS) is determined by the radiative energy losses (1st term, RHS) and heating (2nd term, RHS). The total decay phase heating energy, H, can then be evaluated by integrating over time and multiplying by ﬂare volume, V , assumed to be constant.

Z ttot ∂T (t) −19 2 −1/2 H = V 3kBn0 + 1.2 × 10 n0T (t) dt [ergs] (6.5) 0 ∂t

187 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.7: Heating during the decay phase as a function of the difference between the observed and Cargill-predicted cooling times for 38 M- and X-class flares. The line over- plotted is the best fit to the data (see Equation 6.7).

This analysis was performed on 38 flares within our sample (marked in Table 6.2 by their non-zero values in the ‘Decay Ph. Energy’ column). These flares were chosen because the Cargill model implied that radiation was the dominant cooling mechanism throughout their decay phases, making Equations 6.4 and 6.5 valid. These flares were also seen to cool down to at least 6 MK so the majority of their cooling could be analysed. The rate of change of temperature, dT/dt, in Equation 6.5 was found by differentiating the second-order polynomial fits to the cooling profiles discussed in

Section 6.2.2. The ﬂare volume was calculated from the density and peak emission measure using the equation,

EM V = 2 (6.6) n0

The emission measure was calculated from the ratio of the GOES long channel ﬂux

188 6.4 Results & Discussion and temperature using the same assumptions and methods as described in Section 6.2.1

(TEBBS, Ryan et al., 2012; White et al., 2005). The total decay phase heating required to account for the excess cooling time was then calculated from Equation 6.5.

Figure 6.7 shows resultant energies as a function of the excess cooling time. The

Pearson correlation coefficient of the distribution was calculated in log10-log10 space and found to be 0.77, implying a statistically significant correlation. The following power-law was fit to the data,

H = 1026.73∆t1.06±0.24 [ergs] (6.7) where H is the total heating energy during the decay phase throughout the flare volume, and ∆t is the excess cooling time. The uncertainty on the exponent represents one standard deviation. This power-law quantifies, in very simple terms, the effect of heating during the decay phase on a flare’s cooling time.

Figure 6.8a shows a histogram of these energies which range from 2×1028 – 5×1030 ergs.

The ﬁndings of Withbroe (1978) and Jiang et al. (2006) ﬁt into the upper limit of this range. They inferred total decay phase heating of 1031 ergs for the 1973 September 7

ﬂare and >1030 ergs in the 2002 September 20 ﬂare, respectively. Although the energies in Figure 6.8a are plausible, further testing of the Cargill model is necessary to categor- ically prove whether they are correct. Nonetheless, from these heating calculations, it is possible to work out some implications of these energies being correct. This provides extra ways of testing the Cargill model’s accuracy.

Firstly, the distribution in Figure 6.8a was ﬁt with an exponential using the method of maximum-likelihood, resulting in

f(H) ∝ e−γ·H (6.8) where f(H) is the number of events as a function of total decay phase heating energy, H

189 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

Figure 6.8: Histograms showing the required total heating during the decay phase of 38 M- and X-class flares to account for the difference between the Cargill-predicted and observed cooling times (excess cooling time). a) Log10 of total decay phase heating. b) Total decay phase heating normalised by the total energy radiated by the flare as measured by GOES. c) Total decay phase heating divided by the thermal energy at the beginning of the cooling phase.

190 6.4 Results & Discussion

(in ergs), and γ = 1.7 (±0.3) × 10−30. Again, the uncertainty represents one standard deviation. An exponential ﬁt was chosen because the Kolmogorov-Smirnov test (Wall

& Jenkins, 2003, Chapter 5: Hypothesis Testing) implied it was best suited to the data.

However energy frequency distributions of solar ﬂares are often found to be power-laws

(e.g. Aschwanden, 2011) which may be due to self-organized criticality. In this case, we cannot rule out the possibility that selection eﬀects may have biased this distribution and including more events might reveal it to be more power-law-like. With this in mind, a power-law was also ﬁt to this distribution via the method of maximum-likelihood and found to be

f(H) ∝ H−0.6±0.1 (6.9)

Next, the values in Figure 6.8a were compared to the total thermally radiated energy (Figure 6.8b). CHIANTI was used to determine the spectra corresponding to the temperature and emission measure as calculated from GOES. The total radiated energy was then found by integrating over all wavelengths and over flare duration (Sec- tion 3.1.2). The distribution of decay phase energy normalised by the total GOES radiated energy is shown in Figure 6.8b and ranges from 0.2–0.9, peaking at 0.5. As previously stated, the Cargill model implies that radiation is the dominant loss mechanism for these flares. If this is true, Figure 6.8b suggests that the total heating during the decay phase typically makes up half of the flare’s total thermal energy budget.

The signiﬁcance of the total decay phase heating is further highlighted in Figure 6.8c where it has been normalized by the thermal energy at the ﬂare peak, calculated from the following equation.

Epeak = 3nkBT [ergs] (6.10)

The distribution has a negative slope and ranges from <1 to >7. This implies that

191 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES the total decay phase heating energy inferred from the excess cooling time can be several times greater than the thermal energy at the peak. This agrees with Jiang et al. (2006) who found that the inferred decay phase heating in the 2002 September 20 event was greater than the energy deposited during the impulsive phase. Such a result is significant as previous studies (e.g. Emslie et al., 2012) have used peak thermal energy as an estimate for the total thermal energy of a flare. To quantify the relationship between decay phase heating, H, and peak thermal energy, Epeak, a power-law was fit to the data and found to be

−2.7±0.4 1.1±0.1 H = 10 Epeak [ergs] (6.11)

This implies that the total decay phase heating as a fraction of the peak thermal energy is greater for larger values of the peak thermal energy. In the range explored here, the average total decay phase heating is ∼2.5 times the peak thermal energy. However, this is expected to be less for less energetic flares. Thus if the excess cooling times inferred from the Cargill model are to be believed, estimating a flare’s total thermal energy from its peak is valid for small flares, but not for the most energetic events.

The predictions and comparisons made here all assume that the total decay phase heating inferred from the Cargill model is reasonable. These predictions give further ways of testing the validity of the Cargill model via observations or more advanced modelling of decay phase heating.

6.5 Conclusions

In this chapter, the cooling phases of 72 M- and X-class solar ﬂares were examined with

GOES/XRS and SDO/EVE. The cooling proﬁles as a function of time were parameterised and typically found to be very linear. The average cooling rate was found to be

∼3.5×104 K s−1. These observations were compared to the predictions of the Cargill

192 6.5 Conclusions et al. (1995) model. Loop half-lengths needed by this model were calculated via the

RTV scaling law (Rosner et al., 1978). The uncertainty on this law was quantiﬁed by comparing the predicted lengths of 22 ﬂares within the sample with observations made by Hinode/XRT. The loop half-lengths predicted by RTV scaling law were typically within a factor of 3 of those seen in Hinode/XRT.

It was found that the Cargill model provides a well defined lower limit on flare cooling times, and the deviation from the model was quantified. The root-mean-square deviation between the observations and the model was found to be 961 s which was

1.47 times the mean observed cooling time. Furthermore, the Cargill model ﬁnds that radiation is the dominant loss mechanism throughout the cooling phase for 80% of

ﬂares. For the remaining 20%, Cargill ﬁnds that conduction dominates initially, before being superseded by radiation.

Next, the excess cooling time was assumed to be due to additional heating. The total decay phase heating required to account for the excess cooling time was inferred for 38 flares within the sample. The energies were found to be physically plausible, ranging from 2×1028 – 5×1030 ergs. The frequency distribution can be described by either an exponential with an exponent of −1.7(±0.3) × 10−30 or a power-law with an exponent of −0.6 ± 0.1. These total decay phase heating energies were found to be highly correlated with the excess cooling time and were fit with a power-law with an exponent of 1.06±0.24 and a scaling factor of 1026.73. It was also found that the total decay phase heating predicted from the Cargill model typically makes up about half of the thermally-radiated energy budget of the hot flare plasma. Finally, it was determined that if the decay phase heating inferred from the Cargill model is to be believed, then peak thermal energy is an acceptable estimate for the total thermal energy of small flares. However, this method will underestimate the thermal energy budget for the most energetic events.

In order to conﬁrm or refute the ﬁndings inferred using the Cargill model, compar-

193 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES isons with direct observations of the decay phase heating must be made for an ensemble of flares. This would further highlight the strengths and weaknesses of the Cargill model. In addition, including more temperature-sensitive lines in a similar analysis to this one, or performing fits of the full EVE observed spectrum would give more comprehensive observations of the temperature and density evolution of the flare plasma.

Studies comparing similar observations with results of more advanced hydrodynamic simulations would also help us better understand the thermodynamic evolution and energetics of ﬂare decay phases.

The work outlined in this chapter was conducted in collaboration with Phillip C.

Chamberlin, Ryan O. Milligan, and Peter T. Gallagher and has been published in the

Astrophysical Journal (Ryan et al., 2013).

194

Table 6.2: Events used in this study with observed and model-predicted cooling times and other thermodynamic properties

Date GOES GOES Observed Cargill T Peak T Peak T Min T Min Density Loop Half XRT Half XRT Filter Decay Ph. Start Time Class Cooling [s] Cooling [s] [MK] Time [s] [MK] Time [s] [1012 cm−3] Length [cm] Length [cm] Energy [ergs] 2010 May 05 17:13:00 M1.3 212 241 21.0 17:17:59 2.5 17:21:31 0.9 1.0 0.5 Al thick (...) 2010 Jun 12 00:30:00 M2.0 295 112 20.6 00:56:13 1.6 01:01:09 1.8 0.7 (...) (...) 1.2 2010 Jun 13 05:30:00 M1.0 267 55 15.4 05:37:12 7.9 05:41:39 1.2 0.7 (...) (...) (...) 2010 Oct 16 19:07:00 M3.2 115 122 17.3 19:11:44 2.5 19:13:40 1.2 1.0 1.2 Al mesh (...) 2010 Nov 06 15:27:00 M5.5 389 257 17.4 15:35:06 2.0 15:41:35 0.6 1.8 (...) (...) 3.4 2011 Feb 09 01:23:00 M1.9 112 154 16.9 01:30:18 2.5 01:32:11 0.9 0.8 (...) (...) (...) 2011 Feb 13 17:28:00 M6.6 422 113 20.4 17:34:30 2.5 17:41:32 1.6 1.0 1.7 Ti poly 4.2 2011 Feb 14 17:20:00 M2.3 128 50 17.9 17:24:44 2.5 17:26:52 3.0 0.5 0.4 Be thick 0.5 2011 Feb 15 01:44:00 X2.3 501 213 24.5 01:53:20 2.5 02:01:42 1.2 1.7 0.3 Be thin (...) 2011 Feb 16 01:32:00 M1.0 540 187 17.5 01:37:52 7.9 01:46:53 0.6 1.3 2.1 Ti poly (...) 2011 Feb 16 14:19:00 M1.7 118 66 15.9 14:24:44 6.3 14:26:43 1.3 0.7 0.9 Be thin 0.3 2011 Feb 18 12:59:00 M1.5 215 112 20.1 13:02:28 2.5 13:06:03 1.6 2.2 0.6 Be thick 0.5 2011 Mar 07 07:49:00 M1.6 196 88 21.4 07:52:21 7.9 07:55:38 1.5 0.3 (...) (...) (...) 2011 Mar 07 09:14:00 M1.8 157 77 17.6 09:19:01 7.9 09:21:38 1.6 0.4 (...) (...) (...) 2011 Mar 08 03:37:00 M1.5 2025 209 13.0 03:46:13 2.5 04:19:58 0.4 1.7 (...) (...) 8.2 2011 Mar 08 10:35:00 M5.4 386 85 20.2 10:41:02 2.5 10:47:28 2.1 0.7 0.5 Be thick 2.9 2011 Mar 09 23:13:00 X1.5 542 181 23.8 23:21:36 2.5 23:30:38 1.3 1.6 (...) (...) 13.4 2011 Mar 15 00:19:00 M1.1 140 82 19.6 00:22:20 2.5 00:24:40 2.1 0.8 0.5 Be thick (...) 2011 Mar 24 12:01:00 M1.0 341 107 16.7 12:06:01 2.5 12:11:43 1.2 0.7 0.8 Al med 0.6 2011 Jul 27 15:48:00 M1.1 959 48 14.0 15:59:24 7.9 16:15:23 1.1 1.0 (...) (...) (...) 2011 Aug 03 04:29:00 M1.7 136 60 19.7 04:31:39 6.3 04:33:55 2.3 0.9 (...) (...) 0.3 2011 Aug 04 03:41:00 M9.3 451 194 18.0 03:54:14 2.0 04:01:45 0.8 1.3 (...) (...) 7.2 2011 Aug 09 03:19:00 M2.5 1430 310 17.7 03:49:06 2.5 04:12:56 0.6 1.1 1.2 Al thick (...) 2011 Aug 09 07:48:00 X7.3 233 225 32.5 08:03:23 1.6 08:07:16 1.8 1.9 (...) (...) (...) 2011 Sep 07 22:32:00 X1.8 281 145 21.8 22:37:42 2.5 22:42:24 1.4 1.0 (...) (...) 6.4 2011 Sep 22 10:29:00 X1.4 2934 378 20.2 10:44:13 7.9 11:33:08 0.4 2.7 1.7 Al thick (...) 2011 Sep 24 17:19:00 M3.1 498 10 19.5 17:22:10 7.9 17:30:28 10.9 0.3 (...) (...) (...) 2011 Sep 25 02:27:00 M4.6 228 125 20.1 02:31:54 2.5 02:35:43 1.4 1.0 1.9 Al med + Al 0.9 2011 Sep 25 04:31:00 M7.4 1152 209 18.2 04:39:56 2.5 04:59:09 0.7 2.0 (...) (...) 13.5 2011 Sep 25 15:26:00 M3.8 325 108 15.7 15:32:44 2.5 15:38:09 1.1 0.7 0.3 Be thick 2.3 2011 Sep 25 16:51:00 M2.2 202 272 18.5 16:55:36 2.5 16:58:59 0.6 1.5 (...) (...) (...) 2011 Sep 28 13:24:00 M1.3 240 76 17.9 13:27:29 2.5 13:31:30 2.0 0.6 0.9 Al med 0.6 2011 Oct 02 00:37:00 M3.9 681 215 21.0 00:45:59 2.0 00:57:20 0.9 2.0 (...) (...) 5.2 2011 Oct 20 03:10:00 M1.6 1360 1825 32.4 03:15:25 7.9 03:38:06 0.2 10.9 (...) (...) (...) Date GOES GOES Observed Cargill T Peak T Peak T Min T Min Density Loop Half XRT Half XRT Filter Decay Ph. Start Time Class Cooling [s] Cooling [s] [MK] Time [s] [MK] Time [s] [1012 cm−3] Length [cm] Length [cm] Energy [ergs] 2011 Oct 21 12:53:00 M1.3 426 260 16.3 12:57:27 2.5 13:04:33 0.6 0.9 1.3 Al med (...) 2011 Oct 31 14:55:00 M1.1 2033 92 29.0 15:00:45 7.9 15:34:39 2.8 1.2 (...) (...) (...) 2011 Dec 25 18:11:00 M4.1 197 130 18.9 18:15:15 2.5 18:18:33 1.3 1.2 (...) (...) 1.2 2011 Dec 26 02:13:00 M1.5 787 151 17.2 02:22:06 2.5 02:35:13 0.9 1.3 (...) (...) 2.2 2011 Dec 29 21:43:00 M2.0 393 124 18.2 21:48:00 7.9 21:54:34 0.8 2.1 (...) (...) (...) 2012 Jan 17 04:41:00 M1.0 942 88 15.1 04:46:06 6.3 05:01:49 0.9 1.2 0.8 Al med 1.8 2012 Jan 18 19:04:00 M1.7 461 122 15.9 19:09:08 2.5 19:16:50 1.0 1.2 1.1 Al med 1.5 2012 Jan 19 13:44:00 M3.2 5318 50 14.3 15:14:51 7.9 16:43:30 1.1 0.9 1.0 Be thick (...) 2012 Jan 23 03:38:00 M8.7 1427 171 18.6 03:49:43 1.6 04:13:31 1.0 1.4 (...) (...) 23.4 2012 Feb 06 19:31:00 M1.0 2783 340 12.9 19:42:31 2.0 20:28:55 0.2 0.7 (...) (...) (...) 2012 Mar 05 02:30:00 X1.1 2499 142 19.1 03:52:33 6.3 04:34:12 0.9 1.4 (...) (...) 46.9 2012 Mar 06 12:23:00 M2.2 1369 128 18.3 12:36:54 7.9 12:59:43 0.8 1.4 (...) (...) (...) 2012 Mar 06 21:04:00 M1.4 365 126 18.0 21:06:18 7.9 21:12:23 1.0 0.7 (...) (...) (...) 2012 Mar 06 22:49:00 M1.0 321 62 19.0 22:52:31 2.5 22:57:53 2.6 0.5 (...) (...) 0.5 2012 Mar 14 15:08:00 M2.8 397 113 16.0 15:17:38 2.0 15:24:15 1.1 0.9 (...) (...) 1.9 2012 Mar 17 20:32:00 M1.4 264 184 19.0 20:37:41 2.0 20:42:06 1.0 1.1 (...) (...) (...) 2012 Mar 23 19:34:00 M1.0 172 138 18.5 19:39:35 7.9 19:42:28 1.0 0.7 (...) (...) (...) 2012 Apr 27 08:15:00 M1.1 616 103 14.8 08:22:32 2.5 08:32:48 1.1 1.3 (...) (...) 1.4 2012 May 07 14:03:00 M1.9 1410 312 14.6 14:18:50 2.5 14:42:20 0.3 2.0 (...) (...) 5.3 2012 May 08 13:02:00 M1.4 150 57 18.4 13:06:30 6.3 13:09:00 2.4 0.2 (...) (...) (...) 2012 May 10 04:11:00 M5.9 258 43 18.5 04:16:52 6.3 04:21:11 2.8 0.4 (...) (...) 2.6 2012 May 10 20:20:00 M1.8 328 45 16.3 20:23:23 2.5 20:28:51 2.8 0.3 0.9 Ti poly 1.1 2012 May 17 01:25:00 M5.1 1256 139 15.8 01:38:06 6.3 01:59:03 0.6 1.7 (...) (...) 12.4 2012 Jun 03 17:48:00 M3.4 231 97 14.9 17:54:26 2.0 17:58:17 1.2 0.7 (...) (...) 1.9 2012 Jun 09 16:45:00 M1.9 200 193 18.5 16:50:17 2.5 16:53:38 1.0 0.8 (...) (...) (...) 2012 Jun 10 06:39:00 M1.3 317 107 17.5 06:43:21 6.3 06:48:38 1.0 1.1 (...) (...) 0.7 2012 Jun 30 12:48:00 M1.1 130 64 17.3 12:51:33 6.3 12:53:43 1.6 1.0 (...) (...) 0.2 2012 Jun 30 18:26:00 M1.6 185 80 17.7 18:31:08 6.3 18:34:13 1.4 1.2 (...) (...) 0.4 2012 Jul 02 00:26:00 M1.1 351 174 17.2 00:33:41 2.0 00:39:33 0.8 1.7 (...) (...) 0.7 2012 Jul 02 19:59:00 M3.8 694 186 18.9 20:04:29 2.0 20:16:03 0.9 2.4 1.0 Al thick 5.3 2012 Jul 04 14:35:00 M1.3 138 84 18.6 14:39:24 7.9 14:41:43 1.2 1.9 (...) (...) (...) 2012 Jul 05 01:05:00 M2.5 208 206 19.2 01:09:34 7.9 01:13:03 0.6 2.0 (...) (...) (...) 2012 Jul 05 10:44:00 M1.8 150 565 19.9 10:47:23 7.9 10:49:53 0.1 1.5 (...) (...) (...) 2012 Jul 05 13:05:00 M1.2 1010 119 14.4 13:11:03 7.9 13:27:53 0.6 1.0 (...) (...) (...) 2012 Jul 06 01:37:00 M3.0 150 52 22.0 01:39:23 2.0 01:41:53 4.1 0.5 (...) (...) 0.8 2012 Jul 06 08:17:00 M1.6 187 421 17.8 08:23:06 7.9 08:26:14 0.3 1.8 (...) (...) (...) 2012 Jul 06 23:01:00 X1.1 178 239 26.7 23:07:05 2.5 23:10:04 1.2 2.6 (...) (...) (...) 2012 Jul 07 03:10:00 M1.2 472 77 20.0 03:13:11 7.9 03:21:04 1.1 0.2 (...) (...) (...) 6. DECAY PHASE COOLING & INFERRED HEATING OF FLARES

198 Chapter 7

Conclusions and Future work

The research throughout this thesis has aimed to better understand the thermo- and hydrodynamic evolution of solar ﬂares by analysing ensembles of events. In doing so, new plasma diagnostic techniques have been developed and questions about ﬂare thermodynamic scaling laws, multi-thermal temperature distributions, and hydrodynamic evolution have been investigated. Here we summarise the principal results of this thesis and outline how this work can be improved upon and furthered in future studies.

199 7. CONCLUSIONS AND FUTURE WORK

7.1 Principal Results

The main ﬁndings of this thesis are as follows:

1. A new automatic background subtraction method for GOES/XRS observations,

the Temperature and Emission measure-Based Background Subtraction (TEBBS),

has been developed (Chapter 4). It allows the thermodynamic properties of tem-

perature, emission measure, radiative loss rates and total radiative losses to be

calculated quickly and accurately for large numbers of ﬂares. It was shown that

this method performs better than other naive ways of automatically treating the

background in analysing both single-event and large sample studies. (See Ryan

et al., 2012)

2. The TEBBS method was used to compile a database of thermal properties of

over 50,000 ﬂares between 1980 and 2007 and made publicly available at www.

SolarMonitor.org/TEBBS/. This database was used to examine the relationships between numerous thermodynamic properties. These relationships were

quantiﬁed with ﬁts and, unlike previous studies, uncertainties were provided.

The relationships were found to be qualitatively similar to these of previous stud-

ies. However, the improved background subtraction revealed that ﬂares of given

GOES classes have higher temperatures and lower emission measures than previ-

ously found. (See Ryan et al., 2012)

3. The peak temperatures of 149 ﬂare DEMs determined with SDO/AIA at the

time of the GOES long channel peak were compared to those obtained with

GOES and RHESSI using the isothermal assumption in Chapter 5. Theoretical

modelling of the aﬀect of the isothermal assumption on these calculations was

then made and compared to the observations. It was found in the case of GOES

that the isothermal assumption causes an average overestimation of the DEM

peak temperature of 40%. Meanwhile, in the case of RHESSI, the model and

200 7.1 Principal Results

observations both revealed a greater overestimation than GOES but disagreed as

to how big that overestimation was. The observations showed an overestimation

of 90% while the model predicted an overestimation of 230%. (See Ryan et al.,

2014)

4. The discrepancy between the predicted and observed overestimations of the DEM

peak temperature by RHESSI using the isothermal assumption was concluded to

be due to the Gaussian characterisation of the DEM used in the ﬁtting of the

SDO/AIA ﬂuxes. RHESSI observes the high-temperature tail of the DEM and

a symmetric Gaussian (in log10T-space) may not be the most suitable parameterisation in this regime. This is supported by Graham et al. (2013) who found

asymmetric DEMs (in log10T-space) with much more rapid fall-oﬀs on the high- temperature tail. It was concluded that accurate determinations of ﬂare tem-

perature and emission measure must be made via simultaneous ﬁtting of EUV

(SDO/AIA) and SXR (RHESSI) ﬂuxes with a suitably parameterised function,

e.g. an asymmetric bi-Gaussian. (See Ryan et al., 2014)

5. The hydrodynamic evolution of 72 ﬂare decay phases were charted in Chapter 6.

The cooling time proﬁles were ﬁt with a polynomial and found to be predomi-

nantly linear with an average cooling rate of ∼3.5×104 K s−1. The cooling times

were then compared to the predictions of a simple analytical hydrodynamic model

(Cargill et al., 1995). It was found that ∼80% of ﬂares disagreed with the model

beyond uncertainty. However, the Cargill model was found to provide a well

deﬁned lower bound on a ﬂare’s cooling time. (See Ryan et al., 2013)

6. Finally, the discrepancies between the observed and predicted cooling times were

then assumed to be due to additional heating during the decay phase. This

implied that 80% of M- and X-class ﬂares exhibited signiﬁcant decay phase heating

while only 20% did not. The heating needed to account for the discrepancy was

201 7. CONCLUSIONS AND FUTURE WORK

calculated for a subsample of ﬂares and compared to the total energy radiated

throughout the ﬂare duration as calculated with GOES/XRS. It was found that

this heating typically made up 50% of the total radiated ﬂare energy suggesting

that the energy released during the decay phase signiﬁcantly contributes to the

overall energy budget of M- and X-class ﬂares. (See Ryan et al., 2013)

7.2 Future Work

This thesis has shed some light on the global behaviour of solar ﬂares. However, a comprehensive understanding of these eruptive events remains elusive. We conclude this thesis by discussing ways in which the diagnostics and results presented here can be improved upon in future studies and help further our knowledge of solar ﬂares.

7.2.1 Applying TEBBS to Future Studies

The TEBBS algorithm (and resulting database of thermal properties) is a highly versa- tile tool for studying thermal aspects of solar flares. It would be greatly helpful in any study which requires measurements of temperature, emission measure, and/or radiative loss rates of coronal flaring plasma. Although it was discussed extensively in Chapter 4, it also played a vital role in the studies outlined in Chapters 5 and 6, vindicating its value in supporting multi-instrument, multi-event studies. Because TEBBS makes the accurate analysis of large samples of GOES/XRS observations quick and easy, it allows extra time to analyse more events with other instruments. This allows the flare sample of a given study to be bigger and increases the statistical significance of the results. In addition, the sheer number of flares that TEBBS can analyse (>50,000 in the TEBBS database) could be useful in improving upon previous single-instrument studies. For example, Stoiser et al. (2008) developed models of chromospheric evaporation via thermal conduction and non-thermal electron beam heating in monolithic and filamented

202 7.2 Future Work

ﬂare loops. They tested their models by examining the time delay between the temperature and emission measure peaks. However, they only examined 18 events. Such an analysis conducted on ∼105 events would add much greater statistical certainty to their results. Furthermore, the longevity of the GOES mission which has spanned three solar cycles means that TEBBS would be useful for investigating any dependence on the distribution of ﬂare properties on the solar cycle (e.g. Aschwanden, 2011).

7.2.2 Improving TEBBS Algorithm

We have shown that the TEBBS algorithm is an improvement over other ways of automatically treating the background component. This was shown in Chapter 4 in a single event and also a large-scale statistical study. Its physically justiﬁed discrepancy with other instruments in other studies such as that outlined in Chapter 5 also added to the case. However, there are still ways in which the strengths and weaknesses of the

TEBBS method can be investigated and quantiﬁed. Solar emission in the wavelength range of the GOES/XRS (0.5–8 A)˚ could be modelled with ﬂaring and background components with known temperatures, emission measures, and radiative loss rates.

These could then be folded through the GOES/XRS response functions and theoretical lightcurves reproduced. TEBBS could then be applied to the lightcurves and the resulting TEBBS-derived thermal properties could be compared to the original input parameters. The discrepancies could also be compared to the TEBBS uncertainties so that their suitability could be quantified. Such a study could be carried out for simulated flares of different magnitudes and in different scenarios, e.g. isolated flares, small flares occurring on the decay of larger flares and vice versa, etc. This would help reveal the ways in which the TEBBS algorithm could be improved.

203 7. CONCLUSIONS AND FUTURE WORK

7.2.3 Extending the TEBBS database

Currently, the TEBBS database only spans the period 1980–2007. In order to make this resource as useful as possible, it should updated to the present day. This would allow the TEBBS database to be directly used in studies involving observations from more modern satellites such as SDO and Hinode. To do this a facility could be developed whereby up-to-date event lists of GOES flares could be periodically fed into the TEBBS algorithm and results uploaded to the database. This means that within a few years, over four solar cycles worth of analysed GOES flare observations would be included in the database. This would make the database more useful for examining any variations in flare properties over the course of the solar cycle e.g. Aschwanden (2011).

In Section 4.2.1, we discussed the pitfalls of the GOES event list to which TEBBS was applied here. These included the fact that many flares are often not included in this list and flare end times are often recorded well before the flare actually finishes. An improved automatic flare detection algorithm, designed to identify a greater percentage of flares, and accurately record their start and end times could be developed and linked to the TEBBS algorithm. This would increase the usefulness of TEBBS in statistical studies on subjects such as waiting-time distributions, as the flare sample would more reliably represent the true flare distribution.

7.2.4 Constraining the High-Temperature Tails of DEMs

In Chapter 5 we quantiﬁed the isothermal biases of GOES and RHESSI temperatures as compared with the peak temperatures of ﬂare DEMs determined with SDO/AIA.

These results have implications for all studies relying on GOES and/or RHESSI temperatures. Furthermore, it was concluded that in order to characterise flare DEMs sufficiently, simultaneous fitting of EUV (SDO/AIA) and SXR (GOES and RHESSI) was required. This work used a symmetric Gaussian to characterise the DEM and could be improved with the use of a more suitable function (e.g. an asymmetric bi-

204 7.2 Future Work

Figure 7.1: Top: Typical Gaussian such as those used for parameterising ﬂare DEMs in Chapter 5. Bottom: Bi-Gaussian with a certain standard deviation to the left (lower temperature) of the peak, and a smaller standard deviation to the right (higher temperature) of the peak. Such a function may be useful in better parameterising ﬂare DEMs.

205 7. CONCLUSIONS AND FUTURE WORK

Figure 7.2: Flare DEM (black curve) inferred by Graham et al. (2013) (Figure 4 from that paper) from Hinode/EIS observations and the regularised inversion technique of Hannah & Kontar (2012). The grey shaded area represents the uncertainty limits of the DEM while the coloured lines represent the measured line intensities divided by the contribution functions, indicating maximum possible emission measure. Note the there is a much more rapid fall-oﬀ in the high temperature tail of the DEM (black line), suggesting that an asymmetric parameterisation, such as the bi-gaussian in Figure 7.1 may be suitable to ﬂare DEMs.

206 7.2 Future Work

Gaussian; Figure 7.1) fitted to fluxes from all three instruments. This work could then be furthered by developing algorithms to automatically find the best fit DEM to the

SDO/AIA, GOES/XRS, and RHESSI fluxes. TEBBS could be used as part of this algorithm in determining the appropriate background-subtracted GOES fluxes. This could further work such as Graham et al. (2013) who determined asymmetric flare DEMs from

Hinode/EIS observations using a regularised inversion technique outlined by Hannah

& Kontar (2012) (Figure 7.2). Note how their flare DEM shown in Figure 7.2 shows much steeper fall-offs in the high temperature tail. However, if the electron population has a non-thermal component which can be represented by a κ-distribution, the peak of the bremsstrahlung spectrum can be moved to lower energies and have a less steep high-energy tail. This could affect the derived DEM and cause it to also have a lower peak temperature and less steep gradients in the high-temperature tail (Dud´ık et al.,

2012).

An additional improvement to such work could be the inclusion of SDO/EVE observations in the DEM fitting. Furthering this work would allow flare temperature distributions to be charted over time. This would be hugely valuable in observing the hydrodynamic evolution of flares and improve our ability to test hydrodynamic flare models.

7.2.5 Testing More Advanced Hydrodynamic Flare Models

The Cargill model, to which our cooling observations in Chapter 6 were compared, is highly simpliﬁed. One of the major drawbacks in this model was that it treats the

flare as a monolithic flux tube and cannot account for the complexities of observed solar flare behaviour. In addition, the model assumes that the flare density remains constant which is known from observations to be untrue (e.g. Milligan et al., 2012).

The work outlined in this thesis could therefore be both furthered and improved by comparing such results to more advanced 0D and even 1D hydrodynamic ﬂare models

207 7. CONCLUSIONS AND FUTURE WORK

Figure 7.3: Simulated representations of a multi-stranded coronal loop at diﬀerent resolutions. Note how at low resolutions the loop can appear monolithic, but multi-stranded at high resolutions. (Aschwanden, 2004)

(e.g. Bradshaw & Cargill, 2005, 2010; Klimchuk et al., 2008). These could include multi-strand models (e.g. Warren, 2006; Warren & Doschek, 2005) which model the

flaring loops as a bundle of unresolved sub-loops, or strands, each of which is heated at different times. Figure 7.3 shows this concept of multi-stranded coronal loops at several different resolutions. This reveals that coronal loops can appear monolithic at certain resolutions and multi-stranded at others. It has been shown by a number of studies (e.g. Warren, 2006; Warren & Doschek, 2005) that this approach can accurately account for the observed evolution of SXR and EUV lightcurves. Figure 7.4 (Warren

& Doschek, 2005) shows exactly this, where the dotted lines represent the simulated lightcurves of individual strands, the thick line represents their convolution and the thin line represents the observed lightcurves for numerous instruments. Despite the apparent success of multi-strand approach some high-resolution observations of coronal loops have failed to reveal this multi-stranded nature (e.g. Peter et al., 2013). Therefore,

208 7.2 Future Work

Figure 7.4: Figure taken from Warren & Doschek (2005) showing how simulated lightcurves of unresolved strands (dotted lines), when convolved (thick lines), can well- approximate observed lightcurves (thin lines). This is shown for GOES/XRS long and short channels, and the Fe XXV, Ca XIX and S XV lines observed with Yohkoh/BCS.

further testing of such models as well as attempts to observe coronal loops at higher resolutions are key to further understanding the hydrodynamic evolution of ﬂaring loops.

In addition to testing more advanced models, using alternative methods of charting the density evolution at lower temperatures and improving the observational temperature coverage by including more temperature sensitive emission lines in the analysis would allow us to better observe the hydrodynamic evolution of ﬂares. Alternatively, an algorithm such as the one suggested in the previous paragraph may also be useful in improving this study and allowing us to test more advanced hydrodynamic models.

209 7. CONCLUSIONS AND FUTURE WORK

Chapter 6 also quantified the heating during flare decay phases. However this relied on the assumption that the excess cooling time was solely due to heating. However, as the Cargill model is highly simplified, it is possible that inaccuracies in the model also contributed. To test the findings in this thesis, comparisons should be made with direct observations of decay phase energy release. This could be very difficult to do.

However, comparison of these results with more advanced models would also help give a better idea of the validity of the decay phase heating inferred here. An investigation of the energy partition between impulsive and decay phases would also be a possible direction for future work and could reveal more about the nature of energy release and dissipation in solar ﬂares. Comparison with RHESSI HXR observations may be useful in such a study.

7.3 Conclusion

The research presented in this thesis has examined a wide range of topics concerned with the thermo- and hydrodynamic evolution of solar flares. It has improved our understanding of this field by analysing ensembles of flares, not just single events. It has examined how our interpretation of observations can be affected by the thermodynamic nature of flares as well as shed new light on flare hydrodynamic evolution and even thermal energy partition. Finally, it has also developed plasma diagnostic tools to aid in future studies of flare evolution. The new insights revealed in this thesis as well as the tools developed in doing so has, and will continue to improve our overall understanding of these eruptive and potentially destructive events.

210 Appendix A

GOES Saturation Levels

During the period examined in Chapter 4 (1980–2007), there were 32 X-class flares in the GOES event list which saturated either the short channel or both channels. No events included in this period saturated the long channel without also saturating the short channel. Saturation of the GOES channels has different and important effects when deriving each of the flare plasma properties. If the short channel saturates but the long channel does not, then the derived temperature during the period of saturation is a lower limit because T ∝ FS/FL, to a first order approximation. However, derived emission measure during the same period is an upper limit because EM ∝ T −1, to a

ﬁrst order approximation. Likewise, since dLrad/dt ∝ EM, radiative loss rates (and total radiative losses) are also upper limits. If both channels saturate however, then it cannot be determined (without extrapolation of the lightcurves) whether properties derived during the saturation period are upper or lower limits since they are all functions of the ﬂux ratio.

Within the GOES event list for the period 1980–2007, only the XRSs on board

GOES-6, GOES-10, and GOES-12 were seen to saturate. Each channel in each XRS had diﬀerent saturation levels which can be seen in Table A.1. These values were taken from the GOES lightcurves of saturated events throughout the GOES event list.

211 A. GOES SATURATION LEVELS

Table A.1: GOES Saturation Levels

GOES Satellite Time Period Long Channel Short Channel (10−4 W m−2) (10−4 W m−2) GOES-6 06-Nov-1980 – 17-Dec-1982 (...) 1.8 GOES-6 24-Apr-1984 13 1.2 GOES-6 20-May-1984 – 24-Jun-1988 (...) 1.2 GOES-6 06-Mar-1989 – 02-Nov-1992 12 1.2 GOES-10 02-Apr-2001 – 15-Apr-2001 18 4.7 GOES-12 28-Oct-2003 – 7-Sep-2005 17 4.9

GOES-6 had a very long lifetime and as a result, the saturation levels of each channel were seen to degrade over time, also shown in Table A.1. (Dates are inclusive.)

212 Appendix B

Cooling Derivations

B.1 Cooling due to Conduction

In this appendix we derive how temperature evolves over time due to conductive cooling as per Antiochos & Sturrock (1978) (evaporative case). This is stated in this thesis as

Equation 2.51:

−2/7 T (t) = T0(1 + t/τc0)

where T (t) is the temperature after time t, T0 is the temperature at the start of the cooling period (t = 0), and τc is the conductive cooling timescale, given by Equa- tion 2.46. This relationship is used as part of the Cargill ﬂare cooling model (Cargill et al., 1995) which is outlined in Section 2.2.4.

To derive Equation 2.51, we start with the energy transport equation as given by

Equation 2.41.

1 ∂p 1 ∂ ∂u ∂ = − (pu ) − p s − F − n2Λ(T ) + S + h γ − 1 ∂t γ − 1 ∂s s ∂s ∂s c coll

This version of the equation assumes that the plasma is isotropic, isothermal, and is

213 B. COOLING DERIVATIONS conﬁned to the axis of the magnetic ﬁeld, s. Since we are deriving the temperature evolution due to conduction, we implicitly assume that this is the dominant cooling term. We can therefore neglect the radiative, collisional, and additional heating terms and rewrite the equation as

1 ∂p 1 ∂(pu ) ∂u ∂F = − s − p s − c (B.1) γ − 1 ∂t γ − 1 ∂s ∂s ∂s

where γ is the adiabatic constant, p is pressure, us is the average velocity of the particles along the axis of the magnetic field, and Fc is the conductive heat flux. Expanding the first term on the RHS and rearranging gives

1 ∂p ∂u ∂p ∂u ∂F + p s + u + p s = − c (B.2) γ − 1 ∂t ∂s s ∂s ∂s ∂s

By assuming that the plasma ﬂow velocity is small compared to the sound speed,

∂us ∂p ∂s → 0 and ∂s → 0. Using the ﬁrst of these consequences and also substituting in the ideal gas law, p = nkBT , Equation B.2 becomes:

1 p ∂n ∂n ∂F + u = − c (B.3) γ − 1 n ∂t s ∂s ∂s where n is number density.

Next we employ the continuity equation which was shown in Section 2.2.2 to express the principle of conservation of mass. Using assumptions already made in this derivation, we can rewrite the continuity equation from Equation 2.37 as:

∂n ∂(nu ) = − s (B.4) ∂t ∂s

Expanding the RHS and rearranging gives

∂n ∂n ∂u u = − − n s (B.5) s ∂s ∂t ∂s

214 B.1 Cooling due to Conduction

This form of the equation assumes that the cross-sectional area, A, does not vary

∂p appreciably along the loop. Substituting this into Equation B.3, recalling that ∂s ∼ 0, and rearranging gives

∂ 1 pu − F = 0 (B.6) ∂s γ − 1 s c

This implies that

1 pu − F = f (t) (B.7) γ − 1 s c enth

where fenth(t) is the enthalpy flux. Assuming this is negligible implies that the conductive flux is a product of the thermal energy density and the plasma flow velocity.

1 F = pu (B.8) c γ − 1 s or equivalently that the ﬂow velocity is the ratio of the conductive ﬂux and the thermal energy density

(γ − 1)F u = c (B.9) s p

We now return to Equation B.3 and make two further assumptions. The ﬁrst is that the conduction obeys Spitzer conductivity, and second, that the cross-sectional

5/2 ∂T area, A, does not change appreciably along the loop. This gives Fc = κ0T ∂s , where −6 κ0 = 10 . By using this relation, Equation B.9, and the ideal gas law (n = p/kBT ), we can rewrite Equation B.3 like so:

1 p ∂T ∂ ∂T ∂T = − (κ T 5/2 ) − κ T 3/2( )2 (B.10) γ − 1 T ∂t ∂s 0 ∂s 0 ∂s

215 B. COOLING DERIVATIONS

In this equation, all terms on the LHS are a function of the temporal derivative of temperature, while all terms on the RHS are a function of the spatial temperature derivative of temperature. Therefore both sides must be equal to a constant, −k2.

We can therefore employ the separation of variables technique where we rewrite T as

T0θ(t)φ(s). Here, T0 is the temperature at the top of the loop at the start of the cooling phase, θ(t) is the normalised temporal variation of temperature where θ(0) = 1, and

φ(t) is the normalised spatial variation of temperature along the loop where φ(0) = 1.

(s = 0 implies the loop top.) Substituting this into Equation B.10 and solving gives

d (θ(t)−7/2) = 1/τ (B.11) dt c where τ is the conductive cooling timescale given τ = 1 p , or alternatively, c c γ−1 7/2 2 κ0T0 k by Equation 2.46.

Integrating both sides and then multiplying both sides by T0 gives our ﬁnal result of Equation 2.51:

−2/7 T (t) = T0(1 + t/τc0)

B.2 Cooling due to Radiation

In this appendix we derive how temperature evolves over time due to radiative cooling as per Antiochos (1980) (static case):

(1 + α)t1/(1+α) T (t) = T0 1 − (B.12) τr0

where T (t) is the temperature after time, t, T0 is the temperature at the start of the cooling phase (t = 0), τr0 is the radiative cooling timescale at the beginning of the cooling period, and α is a constant. Using the radiative temperature/density scaling law

216 B.2 Cooling due to Radiation of Serio et al. (1991) and Jakimiec et al. (1992) (i.e. T ∝ n2), and the parameterisation of the radiative loss function of Rosner et al. (1978) (i.e. α = 1/2), Cargill et al. (1995) showed that this equation could be re-expressed as Equation 2.52 of this thesis:

3 t T (t) = T0 1 − 2 τr0

This relationship is used as part of the Cargill ﬂare cooling model (Cargill et al., 1995).

However, in this appendix we shall prove the validity of the original Antiochos (1980) expression (Equation B.12).

Once again we start from the energy transport equation (Equation 2.40). As we are deriving the eﬀect on the temperature evolution due to radiation, we implicitly assume that conductive, collisional and additional heating terms are negligible and can therefore can drop these terms from the equation. In addition, we assume that the plasma is initially static and isobaric and that the radiative timescale is much shorter than the sound speed. This means that over the radiative timescale, plasma ﬂows are negligible and density remains approximately constant in time, i.e. ∂n/∂t ∼ 0. These assumptions mean that the energy transport equation can be written as

1 ∂p = −n2Λ(T ) (B.13) γ − 1 ∂t where γ is the adiabatic constant, p is pressure, n is number density and Λ(T ) is the radiative loss function (Section 2.1.5). This form of the equation states that the dominant way in which the thermal energy density of the plasma can change is via radiative emission.

Let us assume that the radiative loss function can be parameterised over the temperature range of interest by a power-law of the form, Λ(T ) = λT −α, where λ and α

217 B. COOLING DERIVATIONS are constants. Therefore, the energy transport equation can be written as

1 ∂p = −n2λT −α (B.14) γ − 1 ∂t

Antiochos (1980) found that Equation B.12 was a solution of Equation B.14. There- fore to prove this we shall derive Equation B.14 from Equation B.12.

Before beginning, let us re-express the radiative timescale in terms of the parameterisation of the radiative loss function and the ideal gas law, p = nkBT . Although we have derived the radiative timescale in Equation 2.50, it can equivalently be deﬁned by the ratio of the thermal energy to the radiative energy loss rate:

p/(γ − 1) τ = (B.15) r n2Λ(T )

Substituting in the radiative loss function parameterisation and the ideal gas law gives

1 k T 1+α τ = B (B.16) r γ − 1 λ n or alternatively

1 k T 1+α n = B (B.17) γ − 1 λ τr

We now begin the proof by substituting Equation B.16 into Equation B.12 and rearranging to give:

1+α 1+α (γ − 1)λn T = T0 − (1 + α)t (B.18) 2kB

218 B.2 Cooling due to Radiation

Multiplying both sides by kB/((γ − 1)λn) and substituting in Equation B.17 gives

τr = τr0 − (1 + α)t (B.19)

where τr0 is the radiative timescale at the beginning of the cooling period.

Replacing τr with Equation B.16 and integrating both sides with respect to time and using the ideal gas law gives

1 1 d dn−2 dτ n−2 (pT α) + (pT α) = r0 − (1 + α) (B.20) γ − 1 λ dt dt dt

Since n and τr0 are not functions of time, the last term on the LHS and the ﬁrst term on the RHS vanish. Expanding the remaining derivative on the LHS and rearranging gives

1 p dT dp α + = −(1 + α)n2λT −α (B.21) γ − 1 T dt dt

Finally substituting the ideal gas law into the LHS and rearranging gives Equa- tion B.14:

1 ∂p = −n2λT −α γ − 1 ∂t

This is the result we set out to prove. It proves that Equation B.12 and hence

Equation 2.52 is valid representations of how a plasma cools over time due to radiation within the temperature range of the radiative loss function’s parameterisation. In the case of this study and Equation 2.52, this is ∼106 – 107 K.

219 B. COOLING DERIVATIONS

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