Download Plasma Diagnostics and Hydrodynamic Evolution of Solar Flares

Plasma Diagnostics and
Hydrodynamic Evolution of
Solar Flares
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 Fulbright 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-Suárez who 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
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
1.6
2 Theory
2.1
41
Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.1.1
Continuum emission . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.1.1.1
Thermal Bremsstrahlung . . . . . . . . . . . . . . . . .
45
Emission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.1.2.1
Atomic Structure . . . . . . . . . . . . . . . . . . . . .
50
2.1.2.2
Modelling Emission Line Flux in the Corona . . . . . .
52
2.1.2
xi
CONTENTS
2.2
2.1.3
Contribution Functions & Emission Measures . . . . . . . . . . .
56
2.1.4
CHIANTI Atomic Database . . . . . . . . . . . . . . . . . . . . .
58
2.1.5
Radiative Loss Function . . . . . . . . . . . . . . . . . . . . . . .
58
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
3.1
3.2
3.3
3.4
75
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
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
Hinode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.3.1
X-Ray Telescope (XRT) . . . . . . . . . . . . . . . . . . . . . . .
91
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
99
Multiple EUV Grating Spectrograph-A (MEGS-A) . . .
4 Thermal Properties of Solar Flares Over Three Solar Cycles
103
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.1
4.3
The GOES Event List . . . . . . . . . . . . . . . . . . . . . . . . 108
Background Subtraction Method . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1
Previous Background Subtraction Methods . . . . . . . . . . . . 112
4.3.2
Temperature and Emission measure-Based Background Subtraction (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
5.3
5.4
5.2.1
SDO/AIA Measurements . . . . . . . . . . . . . . . . . . . . . . 146
5.2.2
GOES/XRS Measurements . . . . . . . . . . . . . . . . . . . . . 152
5.2.3
RHESSI Measurements . . . . . . . . . . . . . . . . . . . . . . . 153
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.1
The GOES Temperature Bias . . . . . . . . . . . . . . . . . . . . 156
5.3.2
The RHESSI Temperature Bias . . . . . . . . . . . . . . . . . . . 162
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
6.5
6.4.1
Comparing Observed and Modelled Cooling Times . . . . . . . . 185
6.4.2
Inferring Heating During Decay Phase . . . . . . . . . . . . . . . 187
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7 Conclusions and Future work
199
7.1
Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3
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
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) . . . . . . . . . . . . . . . . . . . .
1.2
5
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. . . . .
1.3
7
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) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Modelled temperature and density of the solar atmosphere with height
above the photosphere. (Aschwanden, 2004) . . . . . . . . . . . . . . . .
1.5
9
11
Observed spectrum of the Sun’s emission. At wavelengths longer than
103 Å, 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). . . . . . . . . . . . . .
1.6
13
Full disk images taken with SDO/AIA of the photosphere at 4500 Å (top),
the chromosphere & transition region at 304 Å (left) and the corona at
193 Å (right). Images courtesy of helioviewer.org . . . . . . . . . . . . . .
1.7
Image of granulation on the photosphere. Courtesy of the National Optical Astronomy Observatory. . . . . . . . . . . . . . . . . . . . . . . . .
1.8
14
15
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
17
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 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 Å 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: Magnetogram 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. . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
27
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 tearingmode 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 Å and
0.5–4 Å passbands. d) GOES emission measure. . . . . . . . . . . . . . .
2.1
37
Model solar spectrum from 1 – 400 Å created using the CHIANTI atomic
physics database (Landi et al., 2012). The influence of both emission
lines and continua can clearly be seen. . . . . . . . . . . . . . . . . . . .
2.2
43
Contributions from free-free, free-bound, and two-photon continua to the
solar spectrum in the range 1 – 300 Å (EUV/X-ray regime). Calculated
with CHIANTI by Raftery (2012). . . . . . . . . . . . . . . . . . . . . .
2.3
Diagrams of a) free-bound and b) free-free emission processes. Adapted
from Aschwanden (2004). . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
44
45
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). . . . . . . . . . . .
2.5
49
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). . . . . . . . . . . . . . . . . . .
2.6
51
Schematic of the various excitation (top row) and de-excitation (bottom
row) processes which can occur in the corona. Adapted from Aschwanden (2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
53
LIST OF FIGURES
2.7
Contributions functions of He I, (584.33 Å), O V (629.73 Å), Mg X (524.94 Å),
Fe XVI (360.75 Å), and Fe XIX (592.23 Å). These were calculated with
the CHIANTI software (Section 2.1.4) using density, ne = 5×109 cm−3 ,
coronal abundances and the ionisation equilibria of Mazzotta et al. (1998).
Taken from Raftery (2012). . . . . . . . . . . . . . . . . . . . . . . . . .
2.8
Calculations of the radiative loss function, Λ(T ), compiled from various
studies. (Aschwanden, 2004). . . . . . . . . . . . . . . . . . . . . . . . .
2.9
57
59
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. 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) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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) . . . .
3.4
78
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xviii
79
LIST OF FIGURES
3.5
Relationship between temperature and XRS flux ratio as determined by
Thomas et al. (1985). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
82
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 Variability 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). . . .
xix
95
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 exception is the top half of telescope 3’s primary mirror, which is coated to
reflect broadband UV containing the 1600 Å, 1700 Å, and 4500 Å 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). . . . . . . . . . . . . . . . . . . . . . . .
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 (MEGSB); 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 Å; dotted curve and 1–8 Å;
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 Å 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 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
4.4
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 leastsquares 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). Overplotted on panels c and f are relationships derived by different studies: Garcia
& McIntosh (1992, long-dashed), Feldman et al. (1996b, three-dotteddashed), Battaglia et al. (2005, short-dashed), Hannah et al. (2008, dotdashed) 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 Å and
0.5-4 Å 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
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
5.5
Top: The filter ratio of the GOES 0.5-4 Å to the 1-8 Å 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
5.6
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.
xxiv
. . . . . . . . . . . 161
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 TM T = 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
5.8
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 Cargillpredicted 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.
7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
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
Glossary
AIA
Atmospheric Imaging Assembly onboard SDO
AR
Active Region
BCS
Bragg Crystal Spectrometer onboard
Yohkoh
CCD
Charge Coupled Device
CME
Coronal Mass Ejection
GOES
Geostationary Operational Environmental Satellite
HMI
Helioseismic and Magnetic Imager
onboard SDO
HXR
Hard X-Ray
IR
Infra-Red
JAXA
Japanese
Agency
LASCO
Large Angle and Spectrometric
Coronagraph onboard SOHO
LHS
Left Hand Side
MDI
Michelson Doppler Imager onboard
SOHO
MEGS
Multiple EUV Grating Spectrograph
onboard SDO/EVE
Aerospace
Exploration
MEGS-P MEGS
Photometer
SDO/EVE
COR1(2) Inner (Outer) Coronagraph onboard
STEREO
CVRMSD Coefficient of Variation of the RootMean-Square Deviation
onboard
MHD
Magnetohydrodynamic(s)
NAOJ
National Astronomical Observatory
of Japan
NASA
National Aeronautics and Space Administration
DEM
Differential Emission Measure
DRM
Detector Response Matrix
NOAA
EBTEL
Enthalpy-Based Thermal Evolution
of Loops
National Oceanographic and Atmospheric Adminitration
NSC
Norwegian Space Centre
Extreme-ultraviolet Imaging Spectrometer onboard Hinode
OLS
Ordinary Least-Squares
OSO
Orbiting Solar Observatory
ESA
European Space Agency
RHESSI
ESP
EUV SpectroPhotometer onboard
SDO/EVE
Reuven Ramaty High Energy Solar
Spectroscopic Imager
RHS
Right Hand Side
EUV
Extreme-UltraViolet
RMC
Rotating Modulation Collimators
EVE
Extreme-ultraviolet Variability Experiment onboard SDO
RMSD
Root-Mean-Square Deviation
RTV
Rosner, Tucker, Vaiana
FOV
Field Of View
SAA
South Atlantic Anomaly
GeD
Germanium
RHESSI
SAM
Solar Aspect
SDO/EVE
EIS
Detectors
onboard
xxxi
Monitor
onboard
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
SOT
Solar Optical Telescope
Temperature and Emission measureBased Background Subtraction
SSM
Standard Solar Model
TRACE
STEREO Solar TErrestrial RElations Observatory
Transition Region And Coronal Explorer
UV
UltraViolet
STFC
Science and Technology Facilities
Centre
XRS
X-Ray Sensor onboard GOES
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 individuals, 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 translates 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
4 11 H →42 He + 2e+ + 2νe + energy
(1.1)
where 11 H and 42 He 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
and ∼1.5×105 kg m−3 . These drop to ∼8 MK and ∼4×104 kg m−3 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 > dr dr ad
(1.2)
where the LHS (left-hand side) is the star’s physical temperature gradient and the RHS
= 1 − 1 T dP ,
(right-hand side) is the adiabatic temperature gradient given by dT
dr ad
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
I = I0 e−τ
(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.
Bλ (T ) =
2hc2
λ5 exp
1
hc
λkB T
[W m−2 sr−1 m−1 ]
(1.4)
−1
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 Å,
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 Å. 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 Å (top), the
chromosphere & transition region at 304 Å (left) and the corona at 193 Å (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 Å) 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 (Alfvén) 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 Å. 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 outwards 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 Å. 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:
β=
Pgas
nkB T
= 2
Pmag
B /8π
(1.5)
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 Å 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 + Csin2 φ
(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)
Pe = Pi +
B2
8π
[Ba]
(1.7)
where Pe is the gas pressure outside the flux rope, Pi is the pressure inside the flux
rope, and B 2 /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: Magnetogram 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
X
M
C
B
A
1.5
Peak flux in the 1–8 Å channel (W m−2 )
×10−4
×10−5
×10−6
×10−7
×10−8
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 Å channel of the Geostationary Operational Environmental Satellite XRay 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 Å 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 tearingmode 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 Alfvén transit
time. Thus once an Alfvén disturbance 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 nearrelativistic 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
e
f
f
b
a
c
d
b
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 Å and 0.5–4 Å) and the emission
measure derived from these respectively. The emission measure is defined as EM =
R 2
ne dV , 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 ‘preflare’ 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 Å and 0.5–4 Å 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 CHIANTI spectrum (Section 2.1.4) for an active region in the wavelength range 1–400 Å
(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 (Section 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
43
2.1 Atomic Physics
Figure 2.1: Model solar spectrum from 1 – 400 Å created using the CHIANTI atomic physics database (Landi et al., 2012). The
influence of both emission lines and continua can clearly be seen.
2. THEORY
Figure 2.2: Contributions from free-free, free-bound, and two-photon continua to the
solar spectrum in the range 1 – 300 Å (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 twophoton, 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 MaxwellBoltzmann 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) = nion vσr (v, ν)
(2.1)
where σr (v, ν) is the radiation cross-section given by
σr (v, ν) =
=
16 Z 2 e6
3 m2e c3 v 2
Z
bmax
bmin
db
b
Z 2 e6
16
π
√ g(ν, T )
3 m2e c3 v 2 3
[cm2 erg Hz−1 ]
(2.2)
Here, c is the speed of light in a vacuum and b is the impact parameter. This is defined 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):
ne
ν = nν
4π
Z
Pi (v, ν)f (v)dv
(2.3)
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.
1/2 2
me 3/2 2
mv 2
f (v)dv =
dv
v exp −
π
kB T
2kB T
(2.4)
where kB is the the Boltzmann constant and T is temperature. Substituting this into
Equation 2.3 and integrating gives
ν = 5.4 × 10−39 Z 2 nν
Eγ
nion ne
g(ν,
T
)
exp
−
[erg s−1 cm−3 Hz−1 rad−2 ] (2.5)
kB T
T 1/2
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 ∝
ν dV
(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
F () ∝
V
T 1/2
Eγ
exp −
kB T
n2e dV
[keV s−1 cm−2 keV−1 ]
If this plasma is assumed to be isothermal and
R
V
(2.7)
n2e dV is defined to be the emission
measure, EM , then Equation 2.7 can be rewritten as
F () ∝
1
T 1/2
Eγ
exp −
EM
kB T
[keV s−1 cm−2 keV−1 ]
47
(2.8)
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
= RM
λ
1
1
− 2
nu
n2l
[m−1 ]
(2.10)
where nu and nl are the principal quantum numbers of the upper and lower orbits,
respectively, and RM is the Rydberg constant given by
RM =
Z 2 µe4
8ε20 h3 c
[m−1 ]
(2.11)
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 Å. However for singly ionised helium,
for which Z 2 = 4, the same transition will result in a wavelength of 304 Å. 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 and n C p , respectively. The final excitation process is stimulated
written as ne Cl,u
p l,u
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 and n C p . The final two de-excitation process are stimulated
are written as ne Cu,l
p u,l
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
nl ne Cl,u
+
l
X
l
X
p
nl np Cl,u
+
l
e
nu ne Cu,l
+
X
l
X
nl Bl,u =
l<u
p
nu np Cu,l
+
X
l<u
(2.12)
nu Bu,l +
X
nu Au,l
l<u
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 = 0 and that all excitations are from the lowest possible energy
assumed that Cu,l
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
nion ne Cg,u
= nu
X
Au,l
(2.13)
l<u
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 = nu Au,l hνu,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
Fu,l
1
=
4πR2
Z
V
1
u,l dV =
4πR2
Z
nu Au,l hνu,l dV
V
54
(2.15)
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 =
nu nion nel nH
ne
nion nel nH ne
(2.16)
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 =
P
e /
ne Cg,u
k<u Au,k . Thus Equation 2.15 can be rewritten as
Fu,l
1
=
4πR2
Z
V
e
Cg,u
nion nel nH
dV
k<u Au,k nel nH ne
hνu,l Au,l P
(2.17)
e , is obtained from integrating the collisional crossThe collisional coefficient, Cg,u
section, σg,u (v), with the plasma particle velocity distribution.
e
Cg,u
Z
=
σg,u (v)f (v)vdv
(2.18)
The collisional cross-section is given by (Mariska, 1992)
σg,u =
πa20 Ωg,u (E)
gl E
(2.19)
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
state, g, to u, E = 0.5me v 2 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)
e
Cg,u
8.63 × 10−6
=
gl kB T 3/2
∞
E
Ωg,u (E) exp −
dE
kB T
Eg,u
Z
[cm−3 s−1 ]
(2.20)
The flux due to the emission line observed at Earth (Equation 2.17) can now finally be
rewritten using Equation 2.20.
Fu,l
hνu,l 8.63 × 10−6 Ωg,u nH
=
4πR2
gl
ne
Z
V
hνu,l
nion 1
√ exp −
n2e dV
nel T
kB T
[ergs cm−3 s−1 ]
(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
hνu,l
nion 1
√ exp −
G(T ) =
nel T
kB T
(2.22)
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
EM =
n2e dV
(2.24)
V
56
2.1 Atomic Physics
Figure 2.7: Contributions functions of He I, (584.33 Å), O V (629.73 Å), Mg X (524.94 Å),
Fe XVI (360.75 Å), and Fe XIX (592.23 Å). These were calculated with the CHIANTI software (Section 2.1.4) using density, ne = 5×109 cm−3 , 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 Å), O V (629.73 Å), Mg X
(524.94 Å), Fe XVI (360.75 Å), and Fe XIX (592.23 Å).
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
DEM =
= n2e
dT
dV
dT
(2.25)
57
2. THEORY
This would mean that Equation 2.21 would have to be rewritten as
Z
G(T )
Fu,l = k
T
2.1.4
dEM
dT
dT
(2.26)
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
ER (T, n) = n2e Λ(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 difference 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,
r̂ = xî + y ĵ + z k̂, and velocity vector, v̂ = vx î + vy ĵ + vz k̂, where î, ĵ and k̂ 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 coordinates. The distribution of particles with various velocities through space is described
by the velocity distribution function, f (r̂, v̂, 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
f (r̂, v̂, t) = f (v) = Ce−av
2
(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 = |v̂|, and a =
m
2kB T
is a constant representing the distribution’s width (with m
the mass of each particle, kB the Boltzmann constant, and T the plasma temperature),
3/2
(where n is number density) is a normalisation factor equal to the
while C = n πa
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(r̂, t), can be obtained by integrating over all velocities:
Z
∞
f (r̂, v̂, t)d3 v
n(r̂, t) =
(2.29)
−∞
where d3 v = dvx dvy dvz is the velocity element. Other macroscopic quantities are
related to averages weighted by the distribution. In other words, an average quantity,
gav (r̂, t), can be obtained by:
gav (r̂, t) =
1
n(r̂, t)
Z
g(r̂, v̂, t)f (r̂, v̂, t)d3 v
For example, the thermal energy of the plasma,
(2.30)
3
2 kB T
is simply a measure of the
average kinetic energy of the particles h 21 mv 2 i (where angle brackets denote an average).
Therefore, the thermal energy of the plasma and hence the temperature can be obtained
by:
3
kB T =
2
1
mv 2
2
1 m
=
n(r̂, t) 2
Z
v 2 f (r̂, v̂, t)d3 v
(2.31)
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[Ê + v̂ × B̂], where q is the electric charge of the particle, Ê is the electric field,
and B̂ 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, vx , t)dxdvx ] =
∂t
[f (x, vx , t)vx − f (x + dx, vx , t)vx ]dvx
+ [f (x, vx , t)ax (x, vx , t) − f (x, vx + dvx , t)ax (x, vx + dvx , t)]dx +
∂f
∂t
coll
(2.32)
where ax =
∂vx
∂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
Fx ∂f
∂f
+ vx
+
=
∂t
∂x
m ∂vx
∂f
∂t
(2.33)
coll
Extending to three dimensions gives
∂f
F̂
+ v̂ · ∇r f +
· ∇v f =
∂t
m
∂f
∂t
(2.34)
coll
where
∂
∂
∂
+ ĵ
+ k̂
∂x
∂y
∂z
∂
∂
∂
≡ î
+ ĵ
+ k̂
∂vx
∂vy
∂vz
∇r ≡ î
∇v
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
+ v̂ · ∇r f + [Ê + v̂ × B̂] · ∇v f = 0
∂t
m
(2.35)
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 Boltzmann equation by taking its moments. The ith moment involves multiplying the velocity
distribution in the Boltzmann equation by velocity to the ith power, v i , i.e.
F̂
∂v i f
+ v̂ · ∇r v i f +
· ∇v v i f =
∂t
m
∂v i f
∂t
(2.36)
coll
The zeroth moment (v 0 ) leads directly to the conservation of mass (or charge),
otherwise known as the continuity equation.
∂
n(r̂, t) = −∇ · [n(r̂, t)û(r̂, t)]
∂t
(2.37)
where û(r̂, 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 (v 1 ) 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,
mni
dû
= −∇p + ni F̂ + Ŝij
dt
(2.38)
65
2. THEORY
where ∇p is the pressure gradient, and Ŝij 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 (v 2 ) gives the conservation of energy, when both sides are scaled by 12 nm. This is also known as the energy transport
equation and will be used a number of times in this thesis.
∂[ 12 nmhv 2 i]
1
= −∇( nmhv 2 iû) + p∇ · û + ∇ · q̂ + Ŝcoll
∂t
2
(2.39)
where q̂ 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
plasma, it can be re-expressed as 23 kB T , or even
1
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
=−
∇(pû) + p∇ · û + ∇ · q̂ + Ŝcoll
γ − 1 ∂t
γ−1
(2.40)
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 ∂
∂us
∂
1 ∂p
=−
(pus ) − p
− Fc − n2 Λ(T ) + Scoll + h
γ − 1 ∂t
γ − 1 ∂s
∂s
∂s
(2.41)
In this equation, γ is the adiabatic constant, p is pressure, us is the plasma flow velocity (along the axis of the magnetic field, s), Fc is the conductive heat flux, n2 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 = nkB T ,
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
by the divergence of the heat flux, Fc , which in turn is given by Fc = −κ ∂T
∂s , where
κ is the thermal conductivity and
∂T
∂s
is the temperature gradient. In this treatment
Spitzer conductivity is assumed, i.e. κ = κ0 T 5/2 where κ0 ≈ 10−6 is a constant. Meanwhile the radiative loss rate, n2 Λ(T ), can be approximated in the range 106 – 107 K
the parameterisation of Rosner et al. (1978), i.e., n2 Λ(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 ∂nkB T
=−
γ − 1 ∂t
∂s
5/2 ∂T
κ0 T
− χn2 T −1/2 + h
∂s
[ergs cm−3 s−1 ] (2.42)
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
kB ∂T
[κ0 T 5/2
]=
n
∂s
∂s
γ − 1 ∂t
[ergs cm−3 s−1 ]
(2.43)
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
dt =
2kB
nL
ds
κ0 (γ − 1) T 5/2
[s]
(2.44)
Integrating the LHS from 0 to τc , the characteristic conductive timescale, and the RHS
70
2.2 Hydrodynamics
again from −L to L, gives
τc =
nL2
4kB
κ0 (γ − 1) T 5/2
[s]
(2.45)
Finally, assuming that the plasma is monatomic, (γ = 5/3), gives
τc = 4 × 10−10
nL2
T 5/2
[s]
(2.46)
The radiative timescale is similarly derived, i.e. by neglecting conductive losses and
heating in Equation 2.42, giving
χnζ T α =
∂T
1
nkB
γ−1
∂t
[ergs cm−3 s−1 ]
(2.47)
Again this assumes density is constant. Rearranging gives
dt =
kB
T −α dT
(γ − 1)χnζ−1
[s]
(2.48)
Integrating the LHS from 0 to τr , the characteristic radiative cooling timescale, and
the RHS from 0 to T gives
τr =
kB
T 1−α
(γ − 1)(1 − α)χ nζ−1
[s]
(2.49)
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
τr = 3.45 × 103
T 3/2
n
[s]
(2.50)
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 −
τc
[K]
(2.51)
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 −
2 τr
[K]
(2.52)
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
t∗ = τc0 [(
τr0 7/12
)
− 1]
τc0
[s]
(2.53)
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
T∗ = T0 (
τr0 −1/6
)
τc0
[K]
(2.54)
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
"
ttot = τc0
τr0
τc0
7/12
#
2τr0
−1 +
3
τc0
τr0
5/12 "
1−
τc0
τr0
1/6 TL
T0
#
[s] (2.55)
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
ttot
2τr0
=
3
TL
1−
T0
[s]
(2.56)
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 substrands 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 Å); and the short channel (0.5–4 Å). 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 = C i
∞
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 GOES15 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 /(Ci Gi )
(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
∞
Gi (λ)f (T, λ)dλ/Gi
bi (T ) =
(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 Å 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]
81
(3.8)
3. INSTRUMENTATION
Figure 3.5: Relationship between temperature and XRS flux ratio as determined by
Thomas et al. (1985).
and
1055 b8 = −3.86 + 1.17T − 1.31 × 10−2 T 2 + 1.78 × 10−4 T 3
[1055 W m−2 cm3 ] (3.9)
where temperature is in units of MK. Hence they determined the emission measure to
be
EM = 1055 Bi /(1055 bi (T ))
[cm−3 ]
(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
ER = EM × Λ(T )
[ergs s−1 ]
(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
Lrad =
τ
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 Imager (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 differentiate 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) · P R(E) + BG(E)
(3.13)
where P R(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 nonthermal 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 European 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 terminator, 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 Mexico. 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 Å)
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 Å 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 Å, 1700 Å, and 4500 Å. 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 Å), while others are even double-peaked (e.g. 94 Å). 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 Å, 1700 Å, and 4500 Å
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 MEGSPhotometer (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 Monitor (SAM); Multiple Extreme-ultraviolet Grating Spectrograph A (MEGS-A); Multiple
Extreme-ultraviolet Grating Spectrograph B (MEGS-B); Extreme-ultraviolet SpectroPhotometer (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 extensive catalogue of solar flare thermal properties. Using this, we confirm that the peak
emission measure and total radiative losses scale with background-subtracted GOES Xray 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 Å 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 Feldman 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. Previous 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) (Section 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 Xray 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
4.1
http://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 Å; dotted curve and 1–8 Å; 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 Å 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
Flux
4.3 Background Subtraction Method
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
4.2
http://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
by plugging the ratio of the background fluxes, RB = B4B /B8B , 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. However, 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 Subtraction (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
channel (B8B , B4B ), 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
temperature, TB , calculated using the background values, (B8B , B4B ). 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 Figures 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 Figure 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.
4.3
http://www.solarmonitor.org/TEBBS/
125
4. FLARE THERMAL PROPERTIES OVER THREE SOLAR CYCLES
126
Figure 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). Overplotted 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.
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 + βlog10 B8
[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
EM = 10γ B8δ
[cm−3 ]
(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
B8 = ηEM ζ
[W m−2 ]
(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:
Lrad = 10ε B8θ
[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
132
Figure 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.
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 Imager (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
Feldman et al. (1996b)
Battaglia et al. (2005)
TEBBS
β
0.185
0.33±0.29
0.26+0.04
−0.01
κ
-9
-12
-9+1
−2
Sample Size
868
85
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:
B8 = 3.5 × 10βT +κ
[W m−2 ]
(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,
+17.3
values for α and β are found to be 28+198
−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
Garcia & McIntosh (1992)
TEBBS (Eqn 4.2)
TEBBS (lower edge)
γ
53.04
53±0.1
53.4
δ
0.83
0.86±0.02
0.96
Sample Size
710
50,703
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
Battaglia et al. (2005)
Hannah et al. (2008)
TEBBS
η
3.6×10−50
1.15×10−52
1×10−61±1
ζ
0.92±0.09
0.96
1.15±0.02
Sample Size
85
6,740
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. However, 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 Aschwanden 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 singlewavelength study at 335 Å by Aschwanden (2012). Multi-wavelength studies using all
six coronal filters (94, 131, 171, 193, 221, and 335 Å) 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
Fλ (t) − Fλ (tb ) =
X
dEM (T, t)
Rλ (T ) =
EMk (T, t)Rλ (Tk )∆Tk .
dT
k
146
(5.1)
5.2 Data Analysis
Here, Fλ (t) is the measured flux in channel λ at time, t, tb is the time at which the
background flux is taken,
dEM (T,t)
dT
is the differential emission measure which is a func-
tion 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):
dEM (T )
= EMp exp
dT
−[log10 (T ) − log10 (Tp )]2
2σT2
.
(5.2)
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)
"
σdev
1 X
=
(ff it,λ − fobs,λ )2
nλ
#1/2
(5.3)
λ
where nλ is the number of coronal channels, i.e. 6, and ff it,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 Å 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
10-25
10-26
10
335
94
10-28
10-29
106
A
E
GO
-27
10-30
105
-8
S1
E
GO
S
0.5
107
4A
RHESSI 6 keV
304
171
193
211
131
RHESSI 3 keV
Response [DN cm5 s-1 pix-1]
10-24
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 Å and 0.5-4 Å 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
14033
6310
62
100
4969
54 73
64
124
9
53
125
329
88
836187
85
143
144
74
129112
9236
146
13
89
103
52
141
98
25 114
142
79
123
110
43
24
56
18
145
38
122
99
78
55
137
35
126
30
91
0
72
39
50
86
20
84
58
42
47
19
65
1
5
44
105
116
107
121
106
93 26
51
119
40
22
115
17
46
127
11
8108
70
147
14
128
12
94
21
77
71
109
27 15
102
130
37
90
7 68
148
113
66
48
117
228
118
41104
45
138
139
59
131
6782
9775
133
134
16
95
111
135136
80 604
76101
31
120
32
Emission measure log(EMp [cm-3])
50
49
48
34
81
96
132
6
23
47
46
5.0
5.5
6.0
6.5
7.0
Temperature log[T(MK)]
7.5
8.0
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 Å) to log10 (T ) ∼ 7.3 (94, 131, 193 Å), 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
and standard deviation of log10 (EMp ) = 49.2 ± 0.6, in units of cm−3 . 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
of log10 (σT ) = 0.50 ± 0.13, which corresponds to a temperature factor of 100.5 ≈ 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
20
TGOES/TAIA= 1.4_
1.4+ 0.4
tGOES-tAIA= 27+
27_ 26
GOES temperature T[MK]
Median = 1.3
15
10
10
5
N = 149
1
1
0
-100
10
AIA temperature T[MK]
-50
0
50
tGOES-tAIA [s]
100
GOES flare peak emission measure log(EMp [cm-3])
52
log(EMGOES)-log(EMAIA)=-0.1+
)=-0.1_ 0.4
51
Median =-0.2
50
49
48
47
N = 149
46
46
47
48
49
50
51
AIA flare peak emission measure log(EMp [cm-3])
52
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 Mand 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
lower than the AIA values and imply an average ratio of EMGOES /EMAIA = 10−0.1±0.4 ,
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 February 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
10
TRHESSI/TAIA= 1.9_
1.9+ 1.0
tRHESSI-tAIA= 23+
23_ 25
RHESSI temperature T{MK]
Median = 1.6
8
6
10
4
2
N = 61
1
1
0
-100
10
AIA temperature T(MK)
-50
0
50
tRHESSI-tAIA [s]
100
RHESSI emission measure log(EMp [cm-3])
52
log(EMRHESSI)-log(EMAIA)=-0.9_
)=-0.9+ 1.4
51
Median =-0.8
50
49
48
47
N = 61
46
46
47
48
49
50
51
AIA emission measure log(EMp [cm-3])
52
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
5.3.1
Discussion
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 log10 T-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 (Equation 3.9),
1055 b8 (T ) = −3.86+1.17T −1.31×10−2 T 2 +1.78×10−4 T 3
156
[1055 W m−2 cm3 ]
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 (Equation 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).
EM = B8 /b8 (T )
[cm−3 ]
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 log10 T)
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
can be then directly computed with the GOES response functions ρ8 (T ) = b8 1055 and
ρ4 (T ) = ρ8 (T ) × R(T ),
Z
B4 =
dEM (T )
ρ4 (T )dT
dT
(5.4)
157
5. MULTI-INSTRUMENT TEMPERATURE COMPARISONS
0.7
Isothermal
GOES filter ratio R=B4/B8
0.6
0.5
Multi-thermal
0.4
0.3
0.2
0.1
0.0
1
σT=1.00
σT=0.90
σT=0.80
σT=0.70
σT=0.60
σT=0.50
σT=0.40
σT=0.30
σT=0.20
σT=0.10
17 MK
10
Temperature T[MK]
100
GOES temperature bias qT=TGOES/T(σT)
5
4
σT=1.00
σT=0.90
σT=0.80
σT=0.70
σT=0.60
σT=0.50
3
σT=0.40
2
σT=0.30
1
σT=0.20
σT=0.10
σT=0.00
0
1
qT=1.7
Isothermal
10
Temperature T[MK]
100
Figure 5.5: Top: The filter ratio of the GOES 0.5-4 Å to the 1-8 Å 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
B8 =
dEM (T )
ρ8 (T )dT
dT
(5.5)
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 GOESderived 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
qGOES
TGOES
≈
=
Tp
22
Tp,M K
0.9σT
,
(5.6)
where all variables have the same meaning as above except for Tp,M K 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
= 1.4 ± 0.3
qGOES
(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
F () = F0
exp (−/kB T ) dEM (T )
dT
dT
T 1/2
[keV s−1 cm−2 keV−1 ]
(5.9)
where F0 ≈ 8.1 × 10−39 keV s−1 cm−2 keV−1 . 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 (n2 dV ) in the element of volume dV corresponding to temperature
range dT ,
dEM (T )
dT
dT = n2 (T ) dV
(5.10)
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 multithermal 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
qF = F6 /F12 = 103 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
RHESSI flux F(E) (arbitrary units)
F6
F12
Tp,mt=10 MK, σT=0.5
100
Tiso=53 MK
10-5
10-10
E1= 6
1
E2=12
Tiso=10 MK
10
Energy E[keV]
100
15
15
10
RHESSI temperature bias qT=Tiso/Tp(σT)
RHESSI flux ratio qF=F 6I/F12
iso-thermal
σT=0.30
σT=0.40
5
σT=0.50
σT=0.60
σT=1.00
0
106
qF=3.7
multi-thermal
107
Temperature T[MK]
σ
σTT=1.00
=0.90
σT=0.80
σT=0.70
σT=0.60
σT=0.50
10
σT=0.30
qT=5.3
5
σT=0.20
σT=0.10
Isothermal
0
106
108
σT=0.40
107
Temperature T[MK]
108
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 TM T = 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,
qRHESSI
TRHESSI
≈
=
Tp
60
σ1/2
T
(5.11)
Tp,M K
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 hightemperature part of the DEM in the range of Tp ≈ 10–20 MK is not so well constrained
with AIA, to which only the 193 Å line (with a Fe XXIV line) and the 94 Å 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’ hightemperature 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 Å 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 density would have to be high (&1011 cm−3 ) 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
6.2.1
Observations & Data Analysis
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
GOES/XRS
Ion
Fe XXIV
Fe XXII
Fe XIX
Fe XVIII
Fe XVI
Fe XV
Fe XIV
Wavelength [nm]
0.05–0.4 – Short
0.1–0.8 – Long
Wavelength [nm]
19.20
11.71
10.83
9.39
33.54
28.41
26.47
Temperature [MK]
>4
Temperature [MK]
15.8
12.6
10.0
7.9
6.3
2.5
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 descending 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
T (t) = T0 + θt + µt2
[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
is very narrowly peaked around zero with a full width half max of .10−4 MK s−2 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). Depending 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. Therefore, 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
3 −3
L = (1.4 × 10 )
3
Tmax
p
[cm]
(6.2)
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.
L=
1
T2
kB (1.4 × 103 )3 n
[cm]
(6.3)
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 = LRT V ± 1.8 · LRT V . 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 · LRT V .
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 Equation 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 modelpredicted cooling times with the observations discussed in Section 6.2.2.
6.4
6.4.1
Results & Discussion
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
3kB n0
∂T
= −χnζ0 T α + h
∂t
[ergs s−1 ]
(6.4)
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
H=V
0
ttot
∂T (t)
−19 2
−1/2
3kB n0
+ 1.2 × 10 n0 T (t)
dt
∂t
187
[ergs]
(6.5)
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 overplotted 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,
V =
EM
n20
(6.6)
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 categorically 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 (Section 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 = 3nkB T
[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
1.1±0.1
H = 10−2.7±0.4 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
2010 May 05
2010 Jun 12
2010 Jun 13
2010 Oct 16
2010 Nov 06
2011 Feb 09
2011 Feb 13
2011 Feb 14
2011 Feb 15
2011 Feb 16
2011 Feb 16
2011 Feb 18
2011 Mar 07
2011 Mar 07
2011 Mar 08
2011 Mar 08
2011 Mar 09
2011 Mar 15
2011 Mar 24
2011 Jul 27
2011 Aug 03
2011 Aug 04
2011 Aug 09
2011 Aug 09
2011 Sep 07
2011 Sep 22
2011 Sep 24
2011 Sep 25
2011 Sep 25
2011 Sep 25
2011 Sep 25
2011 Sep 28
2011 Oct 02
2011 Oct 20
Date
GOES
Start Time
17:13:00
00:30:00
05:30:00
19:07:00
15:27:00
01:23:00
17:28:00
17:20:00
01:44:00
01:32:00
14:19:00
12:59:00
07:49:00
09:14:00
03:37:00
10:35:00
23:13:00
00:19:00
12:01:00
15:48:00
04:29:00
03:41:00
03:19:00
07:48:00
22:32:00
10:29:00
17:19:00
02:27:00
04:31:00
15:26:00
16:51:00
13:24:00
00:37:00
03:10:00
GOES Observed
Cargill
T Peak
Class Cooling [s] Cooling [s] [MK]
M1.3
212
241
21.0
M2.0
295
112
20.6
M1.0
267
55
15.4
M3.2
115
122
17.3
M5.5
389
257
17.4
M1.9
112
154
16.9
M6.6
422
113
20.4
M2.3
128
50
17.9
X2.3
501
213
24.5
M1.0
540
187
17.5
M1.7
118
66
15.9
M1.5
215
112
20.1
M1.6
196
88
21.4
M1.8
157
77
17.6
M1.5
2025
209
13.0
M5.4
386
85
20.2
X1.5
542
181
23.8
M1.1
140
82
19.6
M1.0
341
107
16.7
M1.1
959
48
14.0
M1.7
136
60
19.7
M9.3
451
194
18.0
M2.5
1430
310
17.7
X7.3
233
225
32.5
X1.8
281
145
21.8
X1.4
2934
378
20.2
M3.1
498
10
19.5
M4.6
228
125
20.1
M7.4
1152
209
18.2
M3.8
325
108
15.7
M2.2
202
272
18.5
M1.3
240
76
17.9
M3.9
681
215
21.0
M1.6
1360
1825
32.4
T Peak T Min
Time [s] [MK]
17:17:59 2.5
00:56:13 1.6
05:37:12 7.9
19:11:44 2.5
15:35:06 2.0
01:30:18 2.5
17:34:30 2.5
17:24:44 2.5
01:53:20 2.5
01:37:52 7.9
14:24:44 6.3
13:02:28 2.5
07:52:21 7.9
09:19:01 7.9
03:46:13 2.5
10:41:02 2.5
23:21:36 2.5
00:22:20 2.5
12:06:01 2.5
15:59:24 7.9
04:31:39 6.3
03:54:14 2.0
03:49:06 2.5
08:03:23 1.6
22:37:42 2.5
10:44:13 7.9
17:22:10 7.9
02:31:54 2.5
04:39:56 2.5
15:32:44 2.5
16:55:36 2.5
13:27:29 2.5
00:45:59 2.0
03:15:25 7.9
T Min
Density
Loop Half
XRT Half
XRT Filter
Decay Ph.
Time [s] [1012 cm−3 ] Length [cm] Length [cm]
Energy [ergs]
17:21:31
0.9
1.0
0.5
Al thick
(...)
01:01:09
1.8
0.7
(...)
(...)
1.2
05:41:39
1.2
0.7
(...)
(...)
(...)
(...)
19:13:40
1.2
1.0
1.2
Al mesh
15:41:35
0.6
1.8
(...)
(...)
3.4
01:32:11
0.9
0.8
(...)
(...)
(...)
4.2
17:41:32
1.6
1.0
1.7
Ti poly
17:26:52
3.0
0.5
0.4
Be thick
0.5
02:01:42
1.2
1.7
0.3
Be thin
(...)
01:46:53
0.6
1.3
2.1
Ti poly
(...)
14:26:43
1.3
0.7
0.9
Be thin
0.3
13:06:03
1.6
2.2
0.6
Be thick
0.5
07:55:38
1.5
0.3
(...)
(...)
(...)
09:21:38
1.6
0.4
(...)
(...)
(...)
04:19:58
0.4
1.7
(...)
(...)
8.2
10:47:28
2.1
0.7
0.5
Be thick
2.9
23:30:38
1.3
1.6
(...)
(...)
13.4
00:24:40
2.1
0.8
0.5
Be thick
(...)
12:11:43
1.2
0.7
0.8
Al med
0.6
16:15:23
1.1
1.0
(...)
(...)
(...)
04:33:55
2.3
0.9
(...)
(...)
0.3
04:01:45
0.8
1.3
(...)
(...)
7.2
04:12:56
0.6
1.1
1.2
Al thick
(...)
08:07:16
1.8
1.9
(...)
(...)
(...)
22:42:24
1.4
1.0
(...)
(...)
6.4
11:33:08
0.4
2.7
1.7
Al thick
(...)
17:30:28
10.9
0.3
(...)
(...)
(...)
02:35:43
1.4
1.0
1.9
Al med + Al
0.9
04:59:09
0.7
2.0
(...)
(...)
13.5
15:38:09
1.1
0.7
0.3
Be thick
2.3
16:58:59
0.6
1.5
(...)
(...)
(...)
13:31:30
2.0
0.6
0.9
Al med
0.6
00:57:20
0.9
2.0
(...)
(...)
5.2
03:38:06
0.2
10.9
(...)
(...)
(...)
Table 6.2: Events used in this study with observed and model-predicted cooling times and other thermodynamic properties
2011 Oct 21
2011 Oct 31
2011 Dec 25
2011 Dec 26
2011 Dec 29
2012 Jan 17
2012 Jan 18
2012 Jan 19
2012 Jan 23
2012 Feb 06
2012 Mar 05
2012 Mar 06
2012 Mar 06
2012 Mar 06
2012 Mar 14
2012 Mar 17
2012 Mar 23
2012 Apr 27
2012 May 07
2012 May 08
2012 May 10
2012 May 10
2012 May 17
2012 Jun 03
2012 Jun 09
2012 Jun 10
2012 Jun 30
2012 Jun 30
2012 Jul 02
2012 Jul 02
2012 Jul 04
2012 Jul 05
2012 Jul 05
2012 Jul 05
2012 Jul 06
2012 Jul 06
2012 Jul 06
2012 Jul 07
Date
GOES
Start Time
12:53:00
14:55:00
18:11:00
02:13:00
21:43:00
04:41:00
19:04:00
13:44:00
03:38:00
19:31:00
02:30:00
12:23:00
21:04:00
22:49:00
15:08:00
20:32:00
19:34:00
08:15:00
14:03:00
13:02:00
04:11:00
20:20:00
01:25:00
17:48:00
16:45:00
06:39:00
12:48:00
18:26:00
00:26:00
19:59:00
14:35:00
01:05:00
10:44:00
13:05:00
01:37:00
08:17:00
23:01:00
03:10:00
GOES Observed
Cargill
T Peak
Class Cooling [s] Cooling [s] [MK]
M1.3
426
260
16.3
M1.1
2033
92
29.0
M4.1
197
130
18.9
M1.5
787
151
17.2
M2.0
393
124
18.2
M1.0
942
88
15.1
M1.7
461
122
15.9
M3.2
5318
50
14.3
M8.7
1427
171
18.6
M1.0
2783
340
12.9
X1.1
2499
142
19.1
M2.2
1369
128
18.3
M1.4
365
126
18.0
M1.0
321
62
19.0
M2.8
397
113
16.0
M1.4
264
184
19.0
M1.0
172
138
18.5
M1.1
616
103
14.8
M1.9
1410
312
14.6
M1.4
150
57
18.4
M5.9
258
43
18.5
M1.8
328
45
16.3
M5.1
1256
139
15.8
M3.4
231
97
14.9
M1.9
200
193
18.5
M1.3
317
107
17.5
M1.1
130
64
17.3
M1.6
185
80
17.7
M1.1
351
174
17.2
M3.8
694
186
18.9
M1.3
138
84
18.6
M2.5
208
206
19.2
M1.8
150
565
19.9
M1.2
1010
119
14.4
M3.0
150
52
22.0
M1.6
187
421
17.8
X1.1
178
239
26.7
M1.2
472
77
20.0
T Peak T Min
Time [s] [MK]
12:57:27 2.5
15:00:45 7.9
18:15:15 2.5
02:22:06 2.5
21:48:00 7.9
04:46:06 6.3
19:09:08 2.5
15:14:51 7.9
03:49:43 1.6
19:42:31 2.0
03:52:33 6.3
12:36:54 7.9
21:06:18 7.9
22:52:31 2.5
15:17:38 2.0
20:37:41 2.0
19:39:35 7.9
08:22:32 2.5
14:18:50 2.5
13:06:30 6.3
04:16:52 6.3
20:23:23 2.5
01:38:06 6.3
17:54:26 2.0
16:50:17 2.5
06:43:21 6.3
12:51:33 6.3
18:31:08 6.3
00:33:41 2.0
20:04:29 2.0
14:39:24 7.9
01:09:34 7.9
10:47:23 7.9
13:11:03 7.9
01:39:23 2.0
08:23:06 7.9
23:07:05 2.5
03:13:11 7.9
T Min
Density
Loop Half
XRT Half XRT Filter Decay Ph.
Time [s] [1012 cm−3 ] Length [cm] Length [cm]
Energy [ergs]
13:04:33
0.6
0.9
1.3
Al med
(...)
15:34:39
2.8
1.2
(...)
(...)
(...)
18:18:33
1.3
1.2
(...)
(...)
1.2
02:35:13
0.9
1.3
(...)
(...)
2.2
21:54:34
0.8
2.1
(...)
(...)
(...)
05:01:49
0.9
1.2
0.8
Al med
1.8
1.5
19:16:50
1.0
1.2
1.1
Al med
16:43:30
1.1
0.9
1.0
Be thick
(...)
04:13:31
1.0
1.4
(...)
(...)
23.4
20:28:55
0.2
0.7
(...)
(...)
(...)
04:34:12
0.9
1.4
(...)
(...)
46.9
12:59:43
0.8
1.4
(...)
(...)
(...)
21:12:23
1.0
0.7
(...)
(...)
(...)
22:57:53
2.6
0.5
(...)
(...)
0.5
15:24:15
1.1
0.9
(...)
(...)
1.9
20:42:06
1.0
1.1
(...)
(...)
(...)
19:42:28
1.0
0.7
(...)
(...)
(...)
08:32:48
1.1
1.3
(...)
(...)
1.4
14:42:20
0.3
2.0
(...)
(...)
5.3
13:09:00
2.4
0.2
(...)
(...)
(...)
04:21:11
2.8
0.4
(...)
(...)
2.6
1.1
20:28:51
2.8
0.3
0.9
Ti poly
01:59:03
0.6
1.7
(...)
(...)
12.4
17:58:17
1.2
0.7
(...)
(...)
1.9
16:53:38
1.0
0.8
(...)
(...)
(...)
06:48:38
1.0
1.1
(...)
(...)
0.7
12:53:43
1.6
1.0
(...)
(...)
0.2
18:34:13
1.4
1.2
(...)
(...)
0.4
00:39:33
0.8
1.7
(...)
(...)
0.7
20:16:03
0.9
2.4
1.0
Al thick
5.3
14:41:43
1.2
1.9
(...)
(...)
(...)
01:13:03
0.6
2.0
(...)
(...)
(...)
10:49:53
0.1
1.5
(...)
(...)
(...)
13:27:53
0.6
1.0
(...)
(...)
(...)
01:41:53
4.1
0.5
(...)
(...)
0.8
08:26:14
0.3
1.8
(...)
(...)
(...)
23:10:04
1.2
2.6
(...)
(...)
(...)
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 temperature, 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 studies. However, the improved background subtraction revealed that flares of given
GOES classes have higher temperatures and lower emission measures than previously 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 log10 T-space) may not be the most suitable parameterisation in this regime. This is supported by Graham et al. (2013) who found
asymmetric DEMs (in log10 T-space) with much more rapid fall-offs on the hightemperature tail. It was concluded that accurate determinations of flare temperature 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 predominantly 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 versatile 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 Å) 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 (Dudı́k 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 wellapproximate 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
GOES-6
GOES-6
GOES-6
GOES-6
GOES-10
GOES-12
06-Nov-1980 – 17-Dec-1982
24-Apr-1984
20-May-1984 – 24-Jun-1988
06-Mar-1989 – 02-Nov-1992
02-Apr-2001 – 15-Apr-2001
28-Oct-2003 – 7-Sep-2005
Long Channel
(10−4 W m−2 )
(...)
13
(...)
12
18
17
Short Channel
(10−4 W m−2 )
1.8
1.2
1.2
1.2
4.7
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:
T (t) = T0 (1 + t/τc0 )−2/7
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 Equation 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 ∂
∂us
∂
=−
(pus ) − p
− Fc − n2 Λ(T ) + Scoll + h
γ − 1 ∂t
γ − 1 ∂s
∂s
∂s
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 ∂(pus )
∂us ∂Fc
=−
−p
−
γ − 1 ∂t
γ − 1 ∂s
∂s
∂s
(B.1)
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
γ−1
∂p
∂us
∂p
+p
+ us
∂t
∂s
∂s
+p
∂us
∂Fc
=−
∂s
∂s
(B.2)
By assuming that the plasma flow velocity is small compared to the sound speed,
∂us
∂s
→ 0 and
∂p
∂s
→ 0. Using the first of these consequences and also substituting in the
ideal gas law, p = nkB T , Equation B.2 becomes:
1 p
γ−1n
∂n
∂n
+ us
∂t
∂s
=−
∂Fc
∂s
(B.3)
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
∂(nus )
=−
∂t
∂s
(B.4)
Expanding the RHS and rearranging gives
us
∂n
∂n
∂us
=−
−n
∂s
∂t
∂s
(B.5)
214
B.1 Cooling due to Conduction
This form of the equation assumes that the cross-sectional area, A, does not vary
appreciably along the loop. Substituting this into Equation B.3, recalling that
∂p
∂s
∼ 0,
and rearranging gives
∂
∂s
1
pus − Fc
γ−1
=0
(B.6)
1
pus − Fc = fenth (t)
γ−1
(B.7)
This implies that
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.
Fc =
1
pus
γ−1
(B.8)
or equivalently that the flow velocity is the ratio of the conductive flux and the thermal
energy density
us =
(γ − 1)Fc
p
(B.9)
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
area, A, does not change appreciably along the loop. This gives Fc = κ0 T 5/2 ∂T
∂s , where
κ0 = 10−6 . By using this relation, Equation B.9, and the ideal gas law (n = p/kB T ),
we can rewrite Equation B.3 like so:
1 p ∂T
∂
∂T
∂T 2
= − (κ0 T 5/2
) − κ0 T 3/2 (
)
γ − 1 T ∂t
∂s
∂s
∂s
215
(B.10)
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, −k 2 .
We can therefore employ the separation of variables technique where we rewrite T as
T0 θ(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/τc
dt
(B.11)
where τc is the conductive cooling timescale given τc =
p
1
γ−1 κ T 7/2 k2 ,
0 0
or alternatively,
by Equation 2.46.
Integrating both sides and then multiplying both sides by T0 gives our final result
of Equation 2.51:
T (t) = T0 (1 + t/τc0 )−2/7
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):
T (t) = T0
(1 + α)t
1−
τr0
1/(1+α)
(B.12)
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
= −n2 Λ(T )
γ − 1 ∂t
(B.13)
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
= −n2 λT −α
γ − 1 ∂t
(B.14)
Antiochos (1980) found that Equation B.12 was a solution of Equation B.14. Therefore 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 = nkB T . 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:
τr =
p/(γ − 1)
n2 Λ(T )
(B.15)
Substituting in the radiative loss function parameterisation and the ideal gas law gives
τr =
1 kB T 1+α
γ−1 λ n
(B.16)
or alternatively
n=
1 kB T 1+α
γ − 1 λ τr
(B.17)
We now begin the proof by substituting Equation B.16 into Equation B.12 and
rearranging to give:
T 1+α = T01+α − (1 + α)t
(γ − 1)λn
2kB
(B.18)
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 −2 d
dτr0
dn−2
n
=
(pT α ) + (pT α )
− (1 + α)
γ−1λ
dt
dt
dt
(B.20)
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
γ−1
p dT
dp
α
+
T dt
dt
= −(1 + α)n2 λT −α
(B.21)
Finally substituting the ideal gas law into the LHS and rearranging gives Equation B.14:
1 ∂p
= −n2 λT −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
220
References
Antiochos, S.K. (1980). Radiative-dominated cooling of the flare corona and transition region.
Astrophysical Journal , 241, 385–393. 68, 72, 172, 216, 217, 218
Antiochos, S.K. & Sturrock, P.A. (1978). Evaporative cooling of flare plasma. Astrophysical Journal , 220, 1137–1143. 68, 72, 172, 213
Aschwanden, M.J. (2004). Physics of the Solar Corona. An Introduction. Praxis Publishing
Ltd. xv, xvii, xviii, xxvii, 11, 13, 32, 33, 45, 46, 53, 59, 208
Aschwanden, M.J. (2007). RHESSI Timing Studies: Multithermal Delays. Astrophysical
Journal , 661, 1242–1259. 162
Aschwanden, M.J. (2011). The State of Self-organized Criticality of the Sun During the Last
Three Solar Cycles. I. Observations. Solar Physics, 274, 99–117. 191, 203, 204
Aschwanden, M.J. (2012). The Spatio-temporal Evolution of Solar Flares Observed with
AIA/SDO: Fractal Diffusion, Sub-diffusion, or Logistic Growth?
Astrophysical Journal ,
757, 94. 146
Aschwanden, M.J. & Alexander, D. (2001). Flare Plasma Cooling from 30 MK down to 1
MK modeled from Yohkoh, GOES, and TRACE observations during the Bastille Day Event
(14 July 2000). Solar Physics, 204, 91–120. 150, 167
Aschwanden, M.J. & Boerner, P. (2011). Solar Corona Loop Studies with the Atmospheric
Imaging Assembly. I. Cross-sectional Temperature Structure. Astrophysical Journal , 732,
81. 74
221
REFERENCES
Aschwanden, M.J. & Shimizu, T. (2013). Multi-Wavelength Observations of the SpatioTemporal Evolution of Solar Flares with AIA/SDO: II. Hydrodynamic Scaling Laws and
Thermal Energies. Astrophysical JournalSubmitted . 145, 146, 147, 150, 153, 184
Aschwanden, M.J. & Tsiklauri, D. (2009). The Hydrodynamic Evolution of Impulsively
Heated Coronal Loops: Explicit Analytical Approximations. Astrophysical Journal Supplemental Series, 185, 171–185. 38
Aschwanden, M.J., Stern, R.A. & Güdel, M. (2008). Scaling Laws of Solar and Stellar
Flares. Astrophysical Journal , 672, 659–673. 140, 144
Aschwanden, M.J., Zhang, J. & Liu, K. (2013). Multi-Wavelength Observations of the
Spatio-Temporal Evolution of Solar Flares with AIA/SDO: I. Universal Scaling Laws of
Space and Time Parameters. Astrophysical JournalSubmitted . 145, 146
Babcock, H.W. (1961). The Topology of the Sun’s Magnetic Field and the 22-YEAR Cycle.
Astrophysical Journal , 133, 572. xvi, 21, 24
Bahcall, J.N., Huebner, W.F., Lubow, S.H., Parker, P.D. & Ulrich, R.K. (1982).
Standard solar models and the uncertainties in predicted capture rates of solar neutrinos.
Reviews of Modern Physics, 54, 767–799. 6
Battaglia, M., Grigis, P.C. & Benz, A.O. (2005). Size dependence of solar X-ray flare
properties. Astronomy & Astrophysics, 439, 737–747. xxiii, 105, 106, 126, 128, 129, 134,
135, 136, 137, 139, 145, 155
Battaglia, M., Fletcher, L. & Benz, A.O. (2009). Observations of conduction driven
evaporation in the early rise phase of solar flares. Astronomy & Astrophysics, 498, 891–900.
38
Bhattacharya, P.K., Chernoff, H. & Yang, S.S. (1983). Nonparametric Estimation of
the Slope of a Truncated Regression. Annals of Statistics, 11, 505–514. 127
Bornmann, P.L. (1990). Limits to derived flare properties using estimates for the background
fluxes - Examples from GOES. Astrophysical Journal , 356, 733–742. xxi, 106, 107, 111,
114, 115, 116, 120
222
REFERENCES
Bradshaw, S.J. & Cargill, P.J. (2005). The cooling of coronal plasmas. Astronomy &
Astrophysics, 437, 311–317. 68, 172, 208
Bradshaw, S.J. & Cargill, P.J. (2010). A New Enthalpy-Based Approach to the Transition
Region in an Impulsively Heated Corona. Astrophysical Journal Letters, 710, L39–L43. 208
Brown, J.C. (1971). The Deduction of Energy Spectra of Non-Thermal Electrons in Flares
from the Observed Dynamic Spectra of Hard X-Ray Bursts. Solar Physics, 18, 489–502. 35
Brown, J.C. (1974). On the Thermal Interpretation of Hard X-Ray Bursts from Solar Flares.
In G.A. Newkirk, ed., Coronal Disturbances, vol. 57 of IAU Symposium, 395. xxv, 162, 164
Brown, J.C., Turkmani, R., Kontar, E.P., MacKinnon, A.L. & Vlahos, L. (2009).
Local re-acceleration and a modified thick target model of solar flare electrons. Astronomy
& Astrophysics, 508, 993–1000. 35
Brueckner, G.E., Howard, R.A., Koomen, M.J., Korendyke, C.M., Michels, D.J.,
Moses, J.D., Socker, D.G., Dere, K.P., Lamy, P.L., Llebaria, A., Bout, M.V.,
Schwenn, R., Simnett, G.M., Bedford, D.K. & Eyles, C.J. (1995). The Large Angle
Spectroscopic Coronagraph (LASCO). Solar Physics, 162, 357–402. 17
Cargill, P.J. (1993). The Fine Structure of a Nanoflare-Heated Corona. Solar Physics, 147,
263–268. 68, 172
Cargill, P.J., Mariska, J.T. & Antiochos, S.K. (1995). Cooling of solar flares plasmas.
1: Theoretical considerations. Astrophysical Journal , 439, 1034–1043. 40, 68, 69, 174, 179,
192, 201, 213, 217
Carmichael, H. (1964). A Process for Flares. NASA Special Publication, 50, 451–+. 18, 33
Carrington, R.C. (1859). Description of a Singular Appearance seen in the Sun on September
1, 1859. Monthly Notices of the Royal Astronomical Society, 20, 13–15. 2
Carroll, B.W. & Ostlie, D.A. (1996). An Introduction to Modern Astrophysics. xv, 5, 8,
9, 11
223
REFERENCES
Carroll, B.W. & Ostlie, D.A. (2006). An introduction to modern astrophysics and cosmology. xvi, 23
Christe, S., Hannah, I.G., Krucker, S., McTiernan, J. & Lin, R.P. (2008). RHESSI
Microflare Statistics. I. Flare-Finding and Frequency Distributions. Astrophysical Journal ,
677, 1385–1394. 105
Cox, D.P. & Tucker, W.H. (1969). Ionization Equilibrium and Radiative Cooling of a
Low-Density Plasma. Astrophysical Journal , 157, 1157–+. 84
Culhane, J.L., Vesecky, J.F. & Phillips, K.J.H. (1970). The Cooling of Flare Produced
Plasmas in the Solar Corona. Solar Physics, 15, 394–413. 173, 185, 186
Davis, R. (1994). A review of the homestake solar neutrino experiment. Progress in Particle
and Nuclear Physics, 32, 13–32. 6
Dennis, B.R. & Schwartz, R.A. (1989). Solar flares - The impulsive phase. Solar Physics,
121, 75–94. xvii, 34
Dere, K.P., Landi, E., Mason, H.E., Monsignori Fossi, B.C. & Young, P.R. (1997).
CHIANTI - an atomic database for emission lines. Astronomy & Astrophysics Supplemental ,
125, 149–173. 58, 181
Doschek, G.A., Cheng, C.C., Oran, E.S., Boris, J.P. & Mariska, J.T. (1983). Numerical Simulations of Loops Heated to Solar Flare Temperatures - Part Two - X-Ray and
Ultraviolet Spectroscopy. Astrophysical Journal , 265, 1103. 68, 172
Dudı́k, J., Kašparová, J., Dzifčáková, E., Karlický, M. & Mackovjak, Š. (2012). The
non-Maxwellian continuum in the X-ray, UV, and radio range. Astronomy & Astrophysics,
539, A107. 207
Dulk, G.A. & Dennis, B.R. (1982). Microwaves and hard X-rays from solar flares - Multithermal and nonthermal interpretations. Astrophysical Journal , 260, 875–884. xxv, 162,
164
224
REFERENCES
Dwivedi, B.N., Curdt, W. & Wilhelm, K. (1999). Analysis of Extreme-Ultraviolet OffLimb Spectra Obtained with SUMER/SOHO: Ne VI-Mg VI Emission Lines. Astrophysical
Journal , 517, 516–525. 18
Emslie, A.G., Dennis, B.R., Shih, A.Y., Chamberlin, P.C., Mewaldt, R.A., Moore,
C.S., Share, G.H., Vourlidas, A. & Welsch, B.T. (2012). Global Energetics of Thirtyeight Large Solar Eruptive Events. Astrophysical Journal , 759, 71. 30, 107, 144, 192
Feldman, U. (1992). Elemental abundances in the upper solar atmosphere. Phys. Scr , 46,
202–220. 18
Feldman, U., Mandelbaum, P., Seely, J.F., Doschek, G.A. & Gursky, H. (1992). The
potential for plasma diagnostics from stellar extreme-ultraviolet observations. Astrophysical
Journal Supplemental Series, 81, 387–408. 58, 133
Feldman, U., Doschek, G.A., Mariska, J.T. & Brown, C.M. (1995). Relationships
between Temperature and Emission Measure in Solar Flares Determined from Highly Ionized
Iron Spectra and from Broadband X-Ray Detectors. Astrophysical Journal , 450, 441–+. 104,
140
Feldman, U., Doschek, G.A. & Behring, W.E. (1996a). Electron Temperature and Emission Measure Determinations of Very Faint Solar Flares. Astrophysical Journal , 461, 465–+.
104
Feldman, U., Doschek, G.A., Behring, W.E. & Phillips, K.J.H. (1996b). Electron
Temperature, Emission Measure, and X-Ray Flux in A2 to X2 X-Ray Class Solar Flares.
Astrophysical Journal , 460, 1034–+. xxiii, 104, 105, 106, 112, 126, 128, 134, 135, 137, 140,
144
Feldman, U., Dammasch, I., Landi, E. & Doschek, G.A. (2004). Observations Indicating
That 1 × 107 K Solar Flare Plasmas May Be Produced in Situ from 1 × 106 K Coronal
Plasma. Astrophysical Journal , 609, 439–451. 18, 133
Fisher, G.H., Canfield, R.C. & McClymont, A.N. (1985a). Flare loop radiative hydrodynamics. V - Response to thick-target heating. VI - Chromospheric evaporation due to
225
REFERENCES
heating by nonthermal electrons. VII - Dynamics of the thick-target heated chromosphere.
Astrophysical Journal , 289, 414–441. 38
Fisher, G.H., Canfield, R.C. & McClymont, A.N. (1985b). Flare loop radiative hydrodynamics. V - Response to thick-target heating. VI - Chromospheric evaporation due to
heating by nonthermal electrons. VII - Dynamics of the thick-target heated chromosphere.
Astrophysical Journal , 289, 414–441. 68, 172
Fludra, A., Bentley, R.D., Culhane, J.L., Jakimiec, J., Lemen, J.R., Sylwester, J.
& Moorthy, S.T. (1990). The Decay Phase of Three Large Solar Flares. In L. Dezso, ed.,
The Dynamic Sun, 266–+. 18
Fludra, A., Doyle, J.G., Metcalf, T., Lemen, J.R., Phillips, K.J.H., Culhane, J.L.
& Kosugi, T. (1995). Evolution of two small solar flares. Astronomy & Astrophysics, 303,
914–+. 38
Garcia, H.A. (1988). The empirical relationship of peak emission measure and temperature
to peak flare X-ray flux during solar cycle 21. Advances in Space Research, 8, 157–160. 128,
136, 139
Garcia, H.A. (1994). Temperature and emission measure from GOES soft X-ray measurements. Solar Physics, 154, 275–308. 179
Garcia, H.A. & McIntosh, P.S. (1992). High-temperature flares observed in broadband
soft X-rays. Solar Physics, 141, 109–126. xxiii, 104, 126, 128, 129, 136, 139
Ghezzi, I. & Ruggles, C. (2007). Chankillo: A 2300-Year-Old Solar Observatory in Coastal
Peru. Science, 315, 1239–. 2
Golub, L., Deluca, E., Austin, G., Bookbinder, J., Caldwell, D., Cheimets, P.,
Cirtain, J., Cosmo, M., Reid, P., Sette, A., Weber, M., Sakao, T., Kano, R.,
Shibasaki, K., Hara, H., Tsuneta, S., Kumagai, K., Tamura, T., Shimojo, M.,
McCracken, J., Carpenter, J., Haight, H., Siler, R., Wright, E., Tucker, J.,
Rutledge, H., Barbera, M., Peres, G. & Varisco, S. (2007). The X-Ray Telescope
(XRT) for the Hinode Mission. Solar Physics, 243, 63–86. 91
226
REFERENCES
Graham, D.R., Fletcher, L. & Hannah, I.G. (2011). Hinode/EIS plasma diagnostics in
the flaring solar chromosphere. Astronomy & Astrophysics, 532, A27. 139
Graham, D.R., Hannah, I.G., Fletcher, L. & Milligan, R.O. (2013). The Emission
Measure Distribution of Impulsive Phase Flare Footpoints. Astrophysical Journal , 767, 83.
xxvii, 166, 201, 206, 207
Hale, G.E. & Nicholson, S.B. (1925). The Law of Sun-Spot Polarity. Astrophysical Journal ,
62, 270. 21
Handy, B.N., Acton, L.W., Kankelborg, C.C., Wolfson, C.J., Akin, D.J., Bruner,
M.E., Caravalho, R., Catura, R.C., Chevalier, R., Duncan, D.W., Edwards,
C.G., Feinstein, C.N., Freeland, S.L., Friedlaender, F.M., Hoffmann, C.H.,
Hurlburt, N.E., Jurcevich, B.K., Katz, N.L., Kelly, G.A., Lemen, J.R., Levay,
M., Lindgren, R.W., Mathur, D.P., Meyer, S.B., Morrison, S.J., Morrison,
M.D., Nightingale, R.W., Pope, T.P., Rehse, R.A., Schrijver, C.J., Shine, R.A.,
Shing, L., Strong, K.T., Tarbell, T.D., Title, A.M., Torgerson, D.D., Golub,
L., Bookbinder, J.A., Caldwell, D., Cheimets, P.N., Davis, W.N., Deluca,
E.E., McMullen, R.A., Warren, H.P., Amato, D., Fisher, R., Maldonado, H.
& Parkinson, C. (1999). The transition region and coronal explorer. Solar Physics, 187,
229–260. 28
Hannah, I.G. & Kontar, E.P. (2012). Differential emission measures from the regularized
inversion of Hinode and SDO data. Astronomy & Astrophysics, 539, A146. xxvii, 206, 207
Hannah, I.G., Christe, S., Krucker, S., Hurford, G.J., Hudson, H.S. & Lin, R.P.
(2008). RHESSI Microflare Statistics. II. X-Ray Imaging, Spectroscopy, and Energy Distributions. Astrophysical Journal , 677, 704–718. xxiii, 105, 126, 129, 137, 138
Hanser, F.A. & Sellers, F.B. (1996). Design and calibration of the GOES-8 solar x-ray
sensor: the XRS. In E. R. Washwell, ed., Society of Photo-Optical Instrumentation Engineers
(SPIE) Conference Series, vol. 2812 of Society of Photo-Optical Instrumentation Engineers
(SPIE) Conference, 344–352. xviii, 77
227
REFERENCES
Heyvaerts, J., Priest, E.R. & Rust, D.M. (1977). An emerging flux model for the solar
flare phenomenon. Astrophysical Journal , 216, 123–137. 33
Hirayama, T. (1974). Theoretical Model of Flares and Prominences. I: Evaporating Flare
Model. Solar Physics, 34, 323–338. 33
Hock, R.A., Chamberlin, P.C., Woods, T.N., Crotser, D., Eparvier, F.G.,
Woodraska, D.L. & Woods, E.C. (2012a). Extreme Ultraviolet Variability Experiment
(EVE) Multiple EUV Grating Spectrographs (MEGS): Radiometric Calibrations and Results. Solar Physics, 275, 145–178. 99
Hock, R.A., Woods, T.N., Klimchuk, J.A., Eparvier, F.G. & Jones, A.R. (2012b).
The Origin of the EUV Late Phase: A Case Study of the C8.8 Flare on 2010 May 5. ArXiv
e-prints. 187
Hurford, G.J., Schmahl, E.J., Schwartz, R.A., Conway, A.J., Aschwanden, M.J.,
Csillaghy, A., Dennis, B.R., Johns-Krull, C., Krucker, S., Lin, R.P., McTiernan, J., Metcalf, T.R., Sato, J. & Smith, D.M. (2002). The RHESSI Imaging Concept.
Solar Physics, 210, 61–86. xix, 86, 87
Inan, U.S. & Golkowski, M. (2011). Principles of Plasma Physics for Engineers and Scientists. Cambridge University Press, 2011. xviii, 61, 63, 65, 66
Jakimiec, J., Sylwester, B., Sylwester, J., Serio, S., Peres, G. & Reale, F. (1992).
Dynamics of flaring loops. II - Flare evolution in the density-temperature diagram. Astronomy & Astrophysics, 253, 269–276. 217
Jiang, Y.W., Liu, S., Liu, W. & Petrosian, V. (2006). Evolution of the Loop-Top Source
of Solar Flares: Heating and Cooling Processes. Astrophysical Journal , 638, 1140–1153. 74,
173, 185, 187, 189, 192
Kano, R., Sakao, T., Hara, H., Tsuneta, S., Matsuzaki, K., Kumagai, K., Shimojo,
M., Minesugi, K., Shibasaki, K., Deluca, E.E., Golub, L., Bookbinder, J., Caldwell, D., Cheimets, P., Cirtain, J., Dennis, E., Kent, T. & Weber, M. (2008).
The Hinode X-Ray Telescope (XRT): Camera Design, Performance and Operations. Solar
Physics, 249, 263–279. 91
228
REFERENCES
Kliem, B. (1990). Fragmentary energy release due to tearing and coalescence in coronal current
sheets. Astronomische Nachrichten, 311, 399. 33
Kliem, B. (1995). Coupled Magnetohydrodynamic and Kinetic Development of Current Sheets
in the Solar Corona. In A.O. Benz & A. Krüger, eds., Coronal Magnetic Energy Releases,
vol. 444 of Lecture Notes in Physics, Berlin Springer Verlag, 93. 32
Klimchuk, J.A. (2006). On Solving the Coronal Heating Problem. Solar Physics, 234, 41–77.
68, 172
Klimchuk, J.A. & Cargill, P.J. (2001). Spectroscopic Diagnostics of Nanoflare-heated
Loops. Astrophysical Journal , 553, 440–448. 68, 172
Klimchuk, J.A., Patsourakos, S. & Cargill, P.J. (2008). Highly Efficient Modeling of
Dynamic Coronal Loops. Astrophysical Journal , 682, 1351–1362. 68, 74, 172, 208
Kopp, R.A. & Pneuman, G.W. (1976). Magnetic reconnection in the corona and the loop
prominence phenomenon. Solar Physics, 50, 85–98. 33
Kosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., Tachikawa, S.,
Hashimoto, T., Minesugi, K., Ohnishi, A., Yamada, T., Tsuneta, S., Hara, H.,
Ichimoto, K., Suematsu, Y., Shimojo, M., Watanabe, T., Shimada, S., Davis, J.M.,
Hill, L.D., Owens, J.K., Title, A.M., Culhane, J.L., Harra, L.K., Doschek, G.A.
& Golub, L. (2007). The Hinode (Solar-B) Mission: An Overview. Solar Physics, 243, 3–17.
91
Landi, E., Landini, M., Dere, K.P., Young, P.R. & Mason, H.E. (1999). CHIANTI
– an atomic database for emission lines. III. Continuum radiation and extension of the ion
database. Astronomy & Astrophysics Supplemental , 135, 339–346. 82, 106
Landi, E., Feldman, U. & Dere, K.P. (2002). CHIANTI – An Atomic Database for Emission Lines. V. Comparison with an Isothermal Spectrum Observed with SUMER. Astrophysical Journal Supplemental Series, 139, 281–296. 82, 106
229
REFERENCES
Landi, E., Del Zanna, G., Young, P.R., Dere, K.P. & Mason, H.E. (2012). CHIANTI
– An Atomic Database for Emission Lines. XII. Version 7 of the Database. Astrophysical
Journal , 744, 99. xvii, 43, 58, 180
Leboef, J.N., Tajima, T. & Dawson, J.M. (1982). Dynamic Magnetic X Points. Physics
of Fluids, 25, 784–800. 32
Lee, L.C. & Fu, Z.F. (1986). Multiple X line reconnection. I - A criterion for the transition
from a single X line to a multiple X line reconnection. Journal of Geophysical Research, 91,
6807–6815. 32
Lemen, J.R., Title, A.M., Akin, D.J., Boerner, P.F., Chou, C., Drake, J.F.,
Duncan, D.W., Edwards, C.G., Friedlaender, F.M., Heyman, G.F., Hurlburt,
N.E., Katz, N.L., Kushner, G.D., Levay, M., Lindgren, R.W., Mathur, D.P.,
McFeaters, E.L., Mitchell, S., Rehse, R.A., Schrijver, C.J., Springer, L.A.,
Stern, R.A., Tarbell, T.D., Wuelser, J.P., Wolfson, C.J., Yanari, C., Bookbinder, J.A., Cheimets, P.N., Caldwell, D., Deluca, E.E., Gates, R., Golub, L.,
Park, S., Podgorski, W.A., Bush, R.I., Scherrer, P.H., Gummin, M.A., Smith, P.,
Auker, G., Jerram, P., Pool, P., Soufli, R., Windt, D.L., Beardsley, S., Clapp,
M., Lang, J. & Waltham, N. (2012). The Atmospheric Imaging Assembly (AIA) on the
Solar Dynamics Observatory (SDO). Solar Physics, 275, 17–40. xix, xx, 95, 97, 98
Lin, R.P., Dennis, B.R., Hurford, G.J., Smith, D.M., Zehnder, A., Harvey, P.R.,
Curtis, D.W., Pankow, D., Turin, P., Bester, M., Csillaghy, A., Lewis, M.,
Madden, N., van Beek, H.F., Appleby, M., Raudorf, T., McTiernan, J., Ramaty,
R., Schmahl, E., Schwartz, R., Krucker, S., Abiad, R., Quinn, T., Berg, P.,
Hashii, M., Sterling, R., Jackson, R., Pratt, R., Campbell, R.D., Malone, D.,
Landis, D., Barrington-Leigh, C.P., Slassi-Sennou, S., Cork, C., Clark, D.,
Amato, D., Orwig, L., Boyle, R., Banks, I.S., Shirey, K., Tolbert, A.K., Zarro,
D., Snow, F., Thomsen, K., Henneck, R., McHedlishvili, A., Ming, P., Fivian,
M., Jordan, J., Wanner, R., Crubb, J., Preble, J., Matranga, M., Benz, A.,
Hudson, H., Canfield, R.C., Holman, G.D., Crannell, C., Kosugi, T., Emslie,
A.G., Vilmer, N., Brown, J.C., Johns-Krull, C., Aschwanden, M., Metcalf,
230
REFERENCES
T. & Conway, A. (2002). The Reuven Ramaty High-Energy Solar Spectroscopic Imager
(RHESSI). Solar Physics, 210, 3–32. 85, 86
López Fuentes, M.C., Klimchuk, J.A. & Mandrini, C.H. (2007). The Temporal Evolution of Coronal Loops Observed by GOES SXI. Astrophysical Journal , 657, 1127–1136.
185
Mariska, J.T. (1992). The solar transition region. 55, 56
Mazzotta, P., Mazzitelli, G., Colafrancesco, S. & Vittorio, N. (1998). Ionization
balance for optically thin plasmas: Rate coefficients for all atoms and ions of the elements H
to NI. Astronomy & Astrophysics Supplemental , 133, 403–409. xviii, 57, 58, 82, 133
McKenzie, D.L. & Feldman, U. (1992). Variations in the relative elemental abundances of
oxygen, neon, magnesium, and iron in high-temperature solar active-region and flare plasmas.
Astrophysical Journal , 389, 764–776. 18
McTiernan, J.M. (2009). RHESSI/GOES Observations of the Nonflaring Sun from 2002 to
2006. Astrophysical Journal , 697, 94–99. 145, 155
McTiernan, J.M., Kane, S.R., Loran, J.M., Lemen, J.R., Acton, L.W., Hara, H.,
Tsuneta, S. & Kosugi, T. (1993). Temperature and Density Structure of the 1991 November 2 Flare Observed by the YOHKOH Soft X-Ray Telescope and Hard X-Ray Telescope.
Astrophysical Journal Letters, 416, L91. 185
Milligan, R.O., Gallagher, P.T., Mathioudakis, M., Bloomfield, D.S., Keenan,
F.P. & Schwartz, R.A. (2006a). RHESSI and SOHO CDS Observations of Explosive
Chromospheric Evaporation. Astrophysical Journal Letters, 638, L117–L120. 35
Milligan, R.O., Gallagher, P.T., Mathioudakis, M. & Keenan, F.P. (2006b). Observational Evidence of Gentle Chromospheric Evaporation during the Impulsive Phase of a
Solar Flare. Astrophysical Journal Letters, 642, L169–L171. 35
Milligan, R.O., Kennedy, M.B., Mathioudakis, M. & Keenan, F.P. (2012). Timedependent Density Diagnostics of Solar Flare Plasmas Using SDO/EVE. Astrophysical Journal Letters, 755, L16. xxvi, 180, 181, 207
231
REFERENCES
Mitalas, R. & Sills, K.R. (1992). On the photon diffusion time scale for the sun. Astrophysical Journal , 401, 759. 10
Moore, R.L. & Datlowe, D.W. (1975). Heating and cooling of the thermal X-ray plasma
in solar flares. Solar Physics, 43, 189–209. 68, 172, 185
Newton, H.W. & Nunn, M.L. (1951). The Sun’s rotation derived from sunspots 1934-1944
and additional results. Monthly Notices of the Royal Astronomical Society, 111, 413. 23
Parker, E.N. (1955). Hydromagnetic Dynamo Models. Astrophysical Journal , 122, 293. xvii,
25, 28
Parker, E.N. (1963). The Solar-Flare Phenomenon and the Theory of Reconnection and
Annihiliation of Magnetic Fields. Astrophysical Journal Supplemental Series, 8, 177. 32
Pesnell, W.D., Thompson, B.J. & Chamberlin, P.C. (2012). The Solar Dynamics Observatory (SDO). Solar Physics, 275, 3–15. 94
Peter, H., Bingert, S., Klimchuk, J.A., de Forest, C., Cirtain, J.W., Golub, L.,
Winebarger, A.R., Kobayashi, K. & Korreck, K.E. (2013). Structure of solar coronal
loops: from miniature to large-scale. Astronomy & Astrophysics, 556, A104. 74, 208
Petschek, H.E. (1964). Magnetic Field Annihilation. NASA Special Publication, 50, 425. 32
Phillips, K.J.H. (1992). Guide to the sun. 4
Phillips, K.J.H. (2004). The Solar Flare 3.8-10 keV X-Ray Spectrum. Astrophysical Journal ,
605, 921–930. 91
Phillips, K.J.H. & Feldman, U. (1995). Properties of cool flare with GOES class B5 to C2.
Astronomy & Astrophysics, 304, 563–+. 104
Phillips, K.J.H., Sylwester, J., Sylwester, B. & Kuznetsov, V.D. (2010). The Solar
X-ray Continuum Measured by RESIK. Astrophysical Journal , 711, 179–184. 133
Pottasch, S.R. (1964a). On the chemical composition of the solar corona. Monthly Notices
of the Royal Astronomical Society, 128, 73. 18
232
REFERENCES
Pottasch, S.R. (1964b). On the Interpretation of the Solar Ultraviolet Emission Line Spectrum. Space Science Reviews, 3, 816–855. 18
Priest, E. & Forbes, T. (2000). Magnetic Reconnection. 32, 33
Raftery, C.L. (2012). EUV and X-ray Spectroscopy of the Active Sun. Ph.D. thesis, PhD
Thesis, 2012. xvii, xviii, 44, 51, 57
Raftery, C.L., Gallagher, P.T., Milligan, R.O. & Klimchuk, J.A. (2009). Multiwavelength observations and modelling of a canonical solar flare. Astronomy & Astrophysics,
494, 1127–1136. xix, 90, 144, 145, 155, 173, 179, 185
Ray, T.P. (1989). The winter solstice phenomenon at Newgrange, Ireland: accident or design?
Nature, 337, 343–345. 2
Reeves, K.K. & Warren, H.P. (2002). Modeling the Cooling of Postflare Loops. Astrophysical Journal , 578, 590–597. 68, 172
Rosner, R., Tucker, W.H. & Vaiana, G.S. (1978). Dynamics of the quiescent solar corona.
Astrophysical Journal , 220, 643–645. 59, 70, 71, 107, 140, 144, 181, 187, 193, 217
Rousseeuw, P.J. (1984). Least Median of Squares Regression. Journal of the American Statistical Association, 79, 871–880. 127
Ryan, D.F., Milligan, R.O., Gallagher, P.T., Dennis, B.R., Tolbert, A.K.,
Schwartz, R.A. & Young, C.A. (2012). The Thermal Properties of Solar Flares over
Three Solar Cycles Using GOES X-Ray Observations. Astrophysical Journal Supplemental
Series, 202, 11. 103, 141, 144, 179, 189, 200
Ryan, D.F., Chamberlin, P.C., Milligan, R.O. & Gallagher, P.T. (2013). Decay-phase
Cooling and Inferred Heating of M- and X-class Solar Flares. Astrophysical Journal , 778,
68. 133, 144, 171, 194, 201, 202
Ryan, D.F., O’Flannagain, A.M., Aschwanden, M.J. & Gallagher, P.T. (2014).
The Compatibility of Flare Temperatures Observed with AIA, GOES, and RHESSI. Solar
Physics, 289, 2547–2563. 143, 169, 201
233
REFERENCES
Sabine, E. (1852). On Periodical Laws Discoverable in the Mean Effects of the Larger Magnetic
Disturbances. No. II. Philosophical Transactions of the Royal Society of London, 142, 516–
526. 2
Sarkar, A. & Walsh, R.W. (2008). Hydrodynamic Simulation of a Nanoflare-heated Multistrand Solar Atmospheric Loop. Astrophysical Journal , 683, 516–526. 68, 172
Schwabe, M. (1843). Die Sonne. Von Herrn Hofrath Schwabe. Astronomische Nachrichten,
20, 283. 21
Schwarzchild, K. (1906). Göttinger Nachrichten, 1, 41. 10
Serio, S., Reale, F., Jakimiec, J., Sylwester, B. & Sylwester, J. (1991). Dynamics
of flaring loops. I - Thermodynamic decay scaling laws. Astronomy & Astrophysics, 241,
197–202. 217
Smith, D.M., Lin, R.P., Turin, P., Curtis, D.W., Primbsch, J.H., Campbell, R.D.,
Abiad, R., Schroeder, P., Cork, C.P., Hull, E.L., Landis, D.A., Madden, N.W.,
Malone, D., Pehl, R.H., Raudorf, T., Sangsingkeow, P., Boyle, R., Banks, I.S.,
Shirey, K. & Schwartz, R. (2002). The RHESSI Spectrometer. Solar Physics, 210, 33–60.
86, 153
Snodgrass, H.B. & Ulrich, R.K. (1990). Rotation of Doppler features in the solar photosphere. Astrophysical Journal , 351, 309–316. 23
Sterling, A.C., Doschek, G.A. & Feldman, U. (1993). On the absolute abundance of
calcium in solar flares. Astrophysical Journal , 404, 394–402. 18
Stoiser, S., Brown, J.C. & Veronig, A.M. (2008). RHESSI Microflares: II. Implications
for Loop Structure and Evolution. Solar Physics, 250, 315–328. 140, 141, 202
Sturrock, P.A. (1966). Model of the High-Energy Phase of Solar Flares. Nature, 211, 695–
697. 33
Suchocki, J.A. (2004). Conceptual Chemistry. Benjamin Cummings, 2nd edn. xvii, 49
234
REFERENCES
Sweet, P.A. (1958). The Neutral Point Theory of Solar Flares. In B. Lehnert, ed., Electromagnetic Phenomena in Cosmical Physics, vol. 6 of IAU Symposium, 123. 32
Sylwester, J. (1988). Coronal loops, X-ray diagnostics. Advances in Space Research, 8, 55–66.
18
Thomas, R.J., Crannell, C.J. & Starr, R. (1985). Expressions to determine temperatures
and emission measures for solar X-ray events from GOES measurements. Solar Physics, 95,
323–329. xix, 78, 80, 81, 82, 83, 84, 106, 114, 115, 136, 156
Thompson, W.T., Davila, J.M., Fisher, R.R., Orwig, L.E., Mentzell, J.E., Hetherington, S.E., Derro, R.J., Federline, R.E., Clark, D.C., Chen, P.T.C.,
Tveekrem, J.L., Martino, A.J., Novello, J., Wesenberg, R.P., StCyr, O.C.,
Reginald, N.L., Howard, R.A., Mehalick, K.I., Hersh, M.J., Newman, M.D.,
Thomas, D.L., Card, G.L. & Elmore, D.F. (2003). COR1 inner coronagraph for
STEREO-SECCHI. In S.L. Keil & S.V. Avakyan, eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 4853 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1–11. 17
Tsuneta, S., Ichimoto, K., Katsukawa, Y., Nagata, S., Otsubo, M., Shimizu, T.,
Suematsu, Y., Nakagiri, M., Noguchi, M., Tarbell, T., Title, A., Shine, R.,
Rosenberg, W., Hoffmann, C., Jurcevich, B., Kushner, G., Levay, M., Lites, B.,
Elmore, D., Matsushita, T., Kawaguchi, N., Saito, H., Mikami, I., Hill, L.D. &
Owens, J.K. (2008). The Solar Optical Telescope for the Hinode Mission: An Overview.
Solar Physics, 249, 167–196. 26
Wall, J.V. & Jenkins, C.R. (2003). Practical Statistics for Astronomers. Cambridge University Press, Cambridge. 191
Warren, H.P. (2006). Multithread Hydrodynamic Modeling of a Solar Flare. Astrophysical
Journal , 637, 522–530. 68, 74, 172, 208
Warren, H.P. & Doschek, G.A. (2005). Reconciling Hydrodynamic Simulations with Spectroscopic Observations of Solar Flares. Astrophysical Journal Letters, 618, L157–L160. xxvii,
208, 209
235
REFERENCES
Warren, H.P. & Winebarger, A.R. (2007). Static and Dynamic Modeling of a Solar Active
Region. Astrophysical Journal , 666, 1245–1255. 68, 172
White, S.M., Thomas, R.J. & Schwartz, R.A. (2005). Updated Expressions for Determining Temperatures and Emission Measures from Goes Soft X-Ray Measurements. Solar
Physics, 227, 231–248. xviii, 78, 82, 84, 106, 107, 115, 116, 136, 137, 156, 167, 189
Widing, K.G. & Feldman, U. (1992). Elemental abundances and their variations in the
upper solar atmosphere. In E. Marsch & R. Schwenn, ed., Solar Wind Seven Colloquium,
405–410. 18
Withbroe, G.L. (1978). The thermal phase of a large solar flare. Astrophysical Journal , 225,
641–649. 74, 173, 187, 189
Woods, T.N., Eparvier, F.G., Hock, R., Jones, A.R., Woodraska, D., Judge, D.,
Didkovsky, L., Lean, J., Mariska, J., Warren, H., McMullin, D., Chamberlin,
P., Berthiaume, G., Bailey, S., Fuller-Rowell, T., Sojka, J., Tobiska, W.K.
& Viereck, R. (2012). Extreme Ultraviolet Variability Experiment (EVE) on the Solar
Dynamics Observatory (SDO): Overview of Science Objectives, Instrument Design, Data
Products, and Model Developments. Solar Physics, 275, 115–143. 98
Yokoyama, T. & Shibata, K. (1998). A Two-dimensional Magnetohydrodynamic Simulation
of Chromospheric Evaporation in a Solar Flare Based on a Magnetic Reconnection Model.
Astrophysical Journal Letters, 494, L113. 35
Young, P.R. & Mason, H.E. (1997). The Mg/Ne abundance ratio in a recently emerged flux
region observed by CDS. Solar Physics, 175, 523–539. 18
236