Fields and Flares: Understanding the Complex Magnetic Topologies of Solar Active Regions

Fields and Flares:
Understanding the Complex
Magnetic Topologies of
Solar Active Regions


Sophie A. Murray

Fields and Flares:

Understanding the Complex

Magnetic Topologies of

Solar Active Regions



Sophie A. Murray

School of Physics

University of Dublin, Trinity College

A thesis submitted for the degree of

PhilosophiæDoctor (PhD)

2013 January

Declaration


I, Sophie A. Murray, 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.
Name: Sophie A. Murray

Signature: …………………………………. Date: …………..


The thesis work was conducted from October 2008 to September 2012 under the supervision of Dr. Peter T. Gallagher and Dr. D. Shaun Bloomfield at Trinity College, University of Dublin.
In submitting this thesis to the University of Dublin I agree that the University Library may lend or copy the thesis upon request.
Name: Sophie Murray

Signature: …………………………………. Date: …………..








Summary


Sunspots are regions of decreased brightness on the visible surface of the Sun (photosphere) that are associated with strong magnetic fields. They have been found to be locations associated with solar flares, which occur when energy stored in sunspot magnetic fields is suddenly released. The processes involved in flaring and the link between sunspot magnetic fields and flares is still not fully understood, and this thesis aims to gain a better understanding of these topics. The magnetic field evolution of a number of sunspot regions is examined using high spatial resolution data from the Hinode spacecraft.

Photospheric magnetic field data is first investigated, and significant increases in negative vertical field strength, negative vertical current density, and field inclination angle towards the vertical are observed just hours before a flare occurs, which is on much shorter timescales than previously studied. These parameters then return to their pre-flare ‘quiet’ state after the flare has ended. First observations of spatial changes in field inclination across a magnetic neutral line (generally believed to be a typical source region of flares) are also discovered. The changes in field inclination observed in this thesis confirm field configuration changes due to flares predicted by a number of previous works.
3D magnetic field extrapolation methods are then used to study the coronal magnetic field, using the photospheric magnetic field data as a boundary condition. Significant geometrical differences are found to exist between different field configurations obtained from three types of extrapolation procedure (potential, linear force free, and non-linear force free). Magnetic energy and free magnetic energy are observed to increase significantly a few hours before a flare, and decrease afterwards, which is a similar trend to the photospheric field parameter changes observed. Evidence of partial Taylor relaxation is also detected after a flare, as predicted by several previous studies.
The research presented in this thesis gives insight into photospheric and coronal magnetic field evolution of flaring regions. The magnetic field changes observed only hours before a flare could be useful for flare forecasting. Field changes observed due to the flare itself have confirmed currently proposed magnetic field topology changes due to flares. The results outlined show that this particular field of research is vital in furthering our understanding of the magnetic nature of sunspots and its link to flare processes.

“Were it not for magnetic fields, the Sun would be as uninteresting as most astronomers seem to think it is..”

- R.B Leighton, 1969

Acknowledgements

I must first thank my supervisors, Shaun Bloomfield and Peter Gallagher, for their help and guidance over the past four years. Peter, thank you for agreeing to supervise me a second time, your experience and advice has been vital on so many occasions. Shaun, thank you for all your wise suggestions and advice over those Monday afternoon meetings. Your keen eye for detail has been invaluable!
Thank you also to all in the Astrophysics Research Group, particularly to the postgrads (past and present) that I shared the fourth-floor office with. I also thank the staff in the School of Physics, anyone who has helped me over the years I have spent in TCD.
I must extend my appreciation to colleagues outside of TCD who have given me advice throughout my PhD. Particular thanks must go to those who provided codes I have used during analysis. Thank you also to the AXA Research Fund, who agreed to fund my PhD.
Finally, thanks to my family and friends for putting up with me on this long stressful journey!

List of Publications

Refereed

  1. Murray, S. A., Bloomfield, D. S., & Gallagher, P. T (2012)
    “Variation of Magnetic Field Inclination across a Magnetic Neutral Line during a Solar Flare”,
    Astrophysical Journal Letters, in prep

  2. Bloomfield, D. S., Gallagher, P. T, Carley, E., Higgins, P. A., Long, D. M., Maloney, S. A., Murray, S. A., O’Flannagain, A., Perez-Suarez, D. Ryan, D., & Zucca, P. (2013b)
    “A Comprehensive Overview of the 2011 June 7 Solar Storm”,
    Astronomy & Astrophysics, submitted

  3. Murray, S. A., Bloomfield, D. S., & Gallagher, P. T (2013a)
    “Evidence for Partial Taylor Relaxation from Changes in Magnetic Geometry and Energy during a Solar Flare”,
    Astronomy & Astrophysics, in press (DOI: 10.1051/0004-6361/201219964)

  4. Murray, S. A., Bloomfield, D. S., & Gallagher, P. T (2011)
    “The Evolution of Sunspot Magnetic Fields Associated with a Solar Flare”,
    Solar Physics, 277, 1, 45-57

Contents
List of Figures
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thesis.nls

Chapter 1 Introduction

The Sun is a middle-aged G2V star currently situated on the main sequence of a Herzsprung-Russel (HR) diagram. This is a graphical representation of stars obtained by plotting absolute magnitude against spectral class (see Figure 1.1). The Sun has a luminosity  W, mass  kg, and radius  m (Phillips, 1992). It was formed by the gravitational collapse of an interstellar gas cloud. While on the main sequence, the Sun’s energy is supplied by hydrogen fusion reactions in the core (see Section 1.1). When this ceases (after  years), the Sun will progress to the red giant phase. Red giants are larger, more luminous stars, of spectral type K or M, with lower effective surface temperatures. After another  years, the red giant sheds it outer layers to form a planetary nebula (a ring-shaped nebula formed by an expanding shell of gas around the aging star). The star will leave behind its core, becoming a compact object known as a white dwarf.

The Sun is the only star that we can observe at high angular resolution, and is therefore ideal for improving our understanding of Sun-like stars in the Universe. Events on the Sun also have a direct impact on life and technologies in the near-Earth environment. Solar eruptions can cause large amounts of energetic particles, plasma, and radiation across the entire electromagnetic spectrum, to hurtle towards the Earth. This can damage spacecraft instrumentation and electrical powergrids, disrupt radio and GPS communications, as well as divert polar airline flights and spacewalks (amongst other effects). As a consequence, solar physics is on the cutting edge of current physics research. The need to accurately predict these solar eruptions increases as our society grows more technologically dependent. This thesis examines regions on the Sun that are the source of these eruptions, as improving our understanding of the basic physics behind the eruption processes will aid prediction methods.

In this chapter, the basic theory behind the processes on the Sun relevant to this thesis will be discussed, beginning with a description of its structure. The Sun can be separated into distinct regimes, namely the interior, surface zones, atmosphere, and its extension, the solar wind. See Figure 1.2 for an illustration of the various solar layers, which will be described in more detail in the following sections.

Figure 1.1: A Hertzsprung-Russel colour-magnitude diagram. The colour bar indicates spectral class and temperature range. The Sun is a G2 spectral class star located on the main sequence. HIPPARCOS (black dots) and ÒGliese 1991Ó (dark gray dots) catalogues were used to create the plot, excluding binary systems (Higgins, 2012).

Figure 1.2: Cut of the layers of the Sun, courtesy of NASA. The solar core is the source of energy, where fusion heats plasma up to  K. Radiative diffusion transports the energy from the core in the radiative zone, out to 0.7. The convection zone is heated from the base of the tachocline, allowing convective currents to flow to the photosphere, the visible surface of the Sun. Temperatures rise from  K in the photosphere to  K in the chromosphere, then rise rapidly in the transition region to over 10 K in the solar corona.

1.1 Internal Structure

Nuclear fusion of hydrogen into helium accounts for the energy source in the solar core (out to 0.25 , temperature of 15 MK, and density of  kg m). The dominant process is the proton-proton chain, which provides 99% of the Sun’s energy. The proton-proton chain consists of a series of reactions. First, deuterium (H) is formed from the collision of two protons (p),

(1.1)

where e is a positron, and is an electron neutrino. Alternatively, the chain can be started by a proton-electron-proton reaction,

(1.2)

where e is an electron. The next steps are then,

(1.3)
(1.4)

where is a gamma ray, and and are helium isotopes with one and two neutrons respectively. Note that Equations 1.1 and 1.3, or Equations 1.2 and 1.3, must occur twice for each time Equation 1.4 occurs.

The other 1% of the Sun’s energy comes from the carbon-nitrogen-oxygen (CNO) cycle, more details of which can be found in Phillips (1992). For both the p-p chain and CNO cycle nuclear reaction sequences, the net result can be written,

(1.5)

The neutrinos escape from the Sun and carry away a small amount of energy. The energy, in ray photons, positron annihilation, and particle kinetic energy, are all available to heat the Sun, i.e., to provide its luminosity.

Energy is transported via radiation from the core out to a radius of . The temperature in the radiative zone drops to  MK, with the radiation field closely approximated by a black body, for which the spectral radiance, , can be described by the Planck function (Planck, 1900),

(1.6)

where is the intensity of radiation per unit frequency at a temperature , is Planck’s constant, is the speed of light, and is Boltzmann’s constant.

At the top of the radiative zone, the temperature is low enough that elements such as iron are not fully ionised and radiative processes are less effective, and convective processes become more important. Thus, from out to the solar surface, energy is transported by convection. The Schwarzschild criterion (Schwarzchild, 1906) indicates when convection is likely to occur, when the radial gradient of the temperature is greater than the adiabatic temperature gradient,

(1.7)

Here, the adiabatic temperature describes that of a convecting cell, while the radiative temperature is that exterior to the cell. This criterion basically determines whether a rising (sinking) globule of gas will continue to rise (sink), or if it will return to its original depth. In the convection zone, heat from below can no longer be transmitted towards the surface by radiation alone, and heat is transported by material motion. As the gas rises it cools and begins to sink. When it falls to the top of the radiative zone, it heats up and starts to rise once more. This process repeats, creating convection cells and the visual effect of boiling on the Sun’s surface (i.e., granulation). Temperatures of  K can be found in this zone, and at the top (on the surface)  K. It is worth noting that the tachocline is found at the base of the convection zone, specifically where the convection zone transitions into the stable radiative zone. This is where solar magnetic fields are regenerated via the solar dynamo on solar cycle time scales (Rempel & Dikpati, 2003), which will be discussed in more detail in Section 1.3.1.

1.2 Solar Surface and Atmosphere

The photosphere is the visible surface of the Sun, with a total density of  cm (neutral hydrogen, electron and helium densities), and a thickness of less than 500 km. It is usually defined as where the optical depth, , equals 2/3 for radiation at a wavelength of 5000 Å (visible light). Note that the ‘2/3’ value can be found from the Eddington-Barbier expression, telling us that the solar surface flux is equal to times the source function (defined in Section 2.3.1) at an optical depth of 2/3 (Rutten, 2003). Optical depth expresses the quantity of light removed from a beam by scattering or absorption during its path through a medium, and can be defined as,

(1.8)

where is the intensity of radiation at the source, and is the observed intensity after a given path. Optical depth will be discussed further in Section 2.3.1. The photosphere has a radiation spectrum similar to that of a blackbody at 5778 K. The strongest lines observed in the spectrum are Fraunhofer absorption lines due to the tenuous layers of the atmosphere above the solar surface.

The main features of interest to this thesis on the solar surface are sunspots, which are discussed in detail in Section 1.3. As mentioned previously, the solar surface is covered by granulation, representing the tops of convective cells rising from the solar interior. There are two main cell sizes: granules of the order of  km across and with lifetimes of  minutes, and supergranules that are typically  km in diameter and have lifetimes of  days (Simon & Leighton, 1964; Rieutord & Rincon, 2010). The boundaries of supergranules contain magnetic field concentrations that give rise to the magnetic network in the layer above the photosphere, known as the chromosphere.

The temperature of the chromosphere first falls with height to 4400 K (see Figure 1.3), at the temperature minimum. From here, the temperature rises with height to  K at the chromosphere, with a total density of  cm, and a thickness of  km. The brightness of the photosphere tends to overwhelm the chromosphere in the optical continuum, however the hotter chromospheric temperatures mean strong H emission is present. Plage regions are a typical example of a feature in the chromosphere. These are bright regions above the photosphere that are typically found near sunspots, of opposite polarity to the main spot. Other interesting features of note in the chromosphere are filaments, which are long, dark structures on the solar disk. If found on the solar limb, they are referred to as prominences. Hair-like structures, known as fibrils, and dynamic jets, known as spicules (plasma columns), are also observed in the chromosphere. Spicules can be found on the solar limb, and typically reach heights of  3,000 – 10,000 km above the solar surface. They are very short-lived, rising and falling over  minutes.

A so-called ‘transition region’ exists above the chromosphere, where, across a height of  km, temperatures increase from 0.01 MK to 1 MK, and densities decrease from  cm to  cm. It marks a point where magnetic forces dominate over gravity, gas pressure and fluid motion (compared to the photosphere). This concept will be discussed in more detail in Section 2.1.4. Most observations of the transition region are obtained from UV and EUV wavelengths. This is because the extreme temperatures in this region result in emission of UV and EUV from carbon, oxygen, and silicon ions (Mariska, 1993).

Figure 1.3: 1D static model of solar electron temperature (; K), electron density (; cm), and neutral hydrogen density (; cm) as a function of height. Variation of , , and are shown using logarithmic scales. Modified Figure by Phillips et al. (2008), based on one-dimensional model calculations of Vernazza et al. (1981), Fontenla et al. (1988), and Gabriel (1976).

Figure 1.3 presents an illustration that summarises the changing temperature and density in the solar atmosphere. This is a 1D static model of the solar atmosphere, showing layers stratified into photosphere, chromosphere, transition region and corona. However, it must be noted that this stratification is a simplified view of the atmosphere; in reality the solar atmosphere is an inhomogeneous mix of these zones due to complex dynamic processes such as heated upflows, cooling downflows, field line motions and reconnections, hot plasma emission, acoustic waves, and shocks (Aschwanden, 2005). Figure 1.4 shows typical examples of observations of these solar atmospheric layers, using data from the Solar Dynamics Observatory (SDO) spacecraft. The 4500 Å image shows the photospheric layer, with a number of dark sunspot regions near disk centre. The 304 Å image is indicative of the chromosphere, 171 Å image the quiet corona, and 2111 Å image the active corona. All of these three wavelengths show brighter intensity in the sunspot regions, as well as some loops along the limb (particularly evident in the solar northwest).

Figure 1.4: Full-disk images of the Sun at various wavelengths on 2012 July 12, observed mainly in the extreme ultraviolet by the Atmospheric Imaging Assembly (AIA) instrument onboard SDO. Upper left panel is a white light 4500 Å image ( K, photosphere), upper right panel at 304 Å ( K, chromosphere and transition region), lower left panel at 171 Å ( K, quiet corona and upper transition region), and lower right panel at 211 Å ( K, active region corona).

Above the transition region is the corona, the uppermost part of the solar atmosphere, and an extremely hot and tenuous region. Total densities range from  cm at the base ( 2,500 km above the photosphere), to  c‡m for heights  R above the photosphere (Aschwanden, 2005). The total density here represents electron, proton, and helium densities, as plasma is fully ionised in the corona due to the high temperatures (compared to the presence of neutral hydrogen in the photosphere and chromosphere). The temperature is at  K in the corona. Temperatures can be hotter in regions associated with sunspots222In regions of increased magnetic field density, such as sunspots, temperatures of 2 – 6 MK are observed, while temperatures of 1 – 2 MK are observed across the quiet Sun. (see Section 1.3), where large complex loop structures exist. The amount of complexity of the magnetic field structures has been found to be related to the occurrence of solar flares (a topic of investigation in this thesis, see Section 1.4.1 for a description). Note that there are two distinct magnetic zones in the corona with different magnetic topologies, closed and open magnetic field regions. In closed-field regions, closed-field lines connect back to the solar surface, and these regions are the source location of the slow solar wind333Stream of charged particles, consisting mainly of electrons and protons. ( km s). Open magnetic field regions pervade the solar north and south poles. Regions of open field known as coronal holes can generally be found at the poles, and can also be found intermittently closer to the equator. Open field regions connect the solar surface to interplanetary space, and are the source of the fast solar wind ( km s).

The solar wind is a constant out-stream of charged particles of plasma from the solar atmosphere, consisting mainly of electrons and protons at energies of keV. It was predicted by Parker (1958), who assumed the wind is steady, isothermal, and spherically symmetric. He derived an equation of motion that reveals the existence of the solar wind,

(1.9)

where is the distance from the centre of the star, is the solar wind speed, is Boltzmann’s constant, and is the gravitational constant. The right hand side (RHS) of the equation was found to be negative for temperatures, T, observed in the lower corona. However, the gravitational force term decreases as , so it will eventually become smaller than the first term on the RHS, which decreases only as . Thus, there is a critical radius, , beyond which the RHS changes sign and the left hand side (LHS) goes through zero,

(1.10)

Analysis of these conditions for Parker’s solar wind model shows that there exists five classes of solutions to Equation 1.9, which are shown in Figure 1.5. Solutions I and II are excluded as they are double-valued (two values of velocity at the same height), and confined to small and large respectively. Solution II is also disconnected from the surface. Solution III is supersonic close to the Sun, which does not satisfy observations. Solution IV, known as the ‘solar breeze’ solution, remains subsonic. Solution V is the considered the standard solar wind solution, starting subsonically near the Sun, and reaching supersonic speeds. This solution passes through the critical point, where is undefined at and . At this point the coefficient of and the RHS of Equation 1.9 vanish simultaneously. Although Solution V is generally considered the correct solution, it must be noted that it is only an approximation, as the assumptions of radial expansion and isothermality are not entirely correct in reality.

Figure 1.5: The five classes of solutions to the Parker solar wind model for a steady, isothermal, spherically symmetric outflow.

Gas pressure dominates over magnetic pressure in the solar wind, and thus the solar wind drags out solar magnetic field lines. These field lines become wound up as a result of solar rotation, to form a Parker spiral. This is an Archimedean spiral described by the equation,

(1.11)

where can be taken at the solar surface,  rad s is the solar angular rotation rate, and is a longitude angle (Zirin, 1998). The resulting Archimedean spirals leave the Sun near-vertically to the surface at an angle of , and cross the Earth orbit at an angle of . The solar wind eventually terminates when it reaches the edge of the heliosphere. The termination shock is the location where the solar wind slows from supersonic to subsonic speeds, which was observed by the Voyager II mission in August 2007 (Burlaga et al., 2008). In this thesis, the extension of the solar magnetic field into interplanetary space is not of concern, but the structure of the magnetic field on the solar surface and in the solar atmosphere is. Active region magnetic fields are the main focus of investigation, which will be discussed in the following section.

1.3 Sunspots

Active regions (ARs) are localised volumes of the Sun’s outer atmosphere where powerful and complex magnetic fields, emerging from subsurface layers, form loops that extend into the corona. They give rise to features such as sunspots, plage, fibrils and filaments in the photosphere and chromosphere (see review by Solanki, 2003). Sunspots appear as dark zones in the photosphere, mainly located between latitudes of 60 N and 60 S. Figure 1.6 shows a typical AR (NOAA 10030444The National Oceanic and Atmospheric Administration numbers ARs consecutively as they are observed on the Sun. An AR must be observed by two observatories before it is given a number, or before this if a flare is observed to occur in it. The consecutive numbering began on 1972 January 5.) on the solar disk (left, SOHO/MDI continuum image), with a zoom-in (right, Swedish Solar Telescope image) showing sunspots in this AR. The dark interior of a sunspot is known as the umbra and the lighter area surrounding this is known as the penumbra555Note that an umbra existing without a penumbra is known as a pore.. This Section will discuss sunspots in detail, as they are the main focus of study in this thesis, beginning with their formation and the solar cycle, flux emergence, sunspot structure, and finally sunspot evolution.

Figure 1.6: Left shows a full-disk SOHO/MDI continuum image of NOAA AR 10030 on 2002 July 15. Right shows a zoom-in of this region using observations from the Swedish Solar Telescope. Images are courtesy of SOHO (NASA & ESA) and the Royal Swedish Academy of Sciences.

1.3.1 Solar Cycle

The dynamo (Parker, 1955a) describes the process by which the Sun’s magnetic field is generated, and thus is a starting point for describing the formation of sunspots. The effect describes the sub-surface magnetic field lines becoming distorted, twisted, and more complex in shape under the effect of the rotation of solar material. The effect describes the way in which magnetic fields are stretched out and wound around the Sun by solar differential rotation. The differential rotation rate has the general form,

(1.12)

where is the angular velocity in degrees per day, is the latitude, and A, B, and C are constants. The values of these constants differ depending on the techniques used to make the measurement, and the time period selected. Snodgrass & Ulrich (1990) quote current accepted values as A  degrees/day, B  degrees/day, and A  degrees/day, obtained using cross-correlation measurements from magnetograph observations.

The magnetic polarity of sunspot pairs reverses and then returns to its original state in a  22-year Hale cycle (Hale & Nicholson, 1925), with the Sun’s magnetic poles reversing every 11 years. A number of models have been put forward to explain the 22-year cycle, perhaps most notably the Babcock-Leighton model, which was first proposed by Babcock (1961) and further elaborated by Leighton (1964, 1969). This is illustrated in Figure 1.7, and consists of multiple stages (van Driel-Gesztelyi, 2008):

Figure 1.7: Illustration of the - effect that is the source of the solar magnetic field, beginning with a poloidal field on the left, winding up towards a toroidal field in the middle, and the emergence of complex active regions on the right (Carroll & Ostlie, 2006).
  • About three years before the onset of the new sunspot cycle, the new field to be involved is approximated by a dipole field symmetric about the rotation axis- a pure poloidal field (see left panel in Figure 1.7).

  • The originally poloidal field is pulled into a helical spiral in the activity belts, with the resulting field becoming toroidally amplified ( effect, see middle panel in Figure 1.7). This is due to solar differential rotation, i.e., the rate of solar rotation is observed to be fastest at the equator, decreasing as latitude increases. Further amplification is reached by the -effect, i.e., twisting by the effect of the radial differential rotation shear (see right panel in Figure 1.7).

  • Each -shaped loop erupting through the surface (see Section 1.3.2) produces a bipolar active region with preceding and following magnetic polarities. The sense of positive and negative polarities is equivalent to Hale’s polarity law, which states that active regions on opposite hemispheres have opposite leading magnetic polarities, alternating between successive sunspot cycles. With the term in solar differential rotation (Equation 1.12) SpörerÕs Law is followed, i.e., the first active regions of the cycle are expected at higher latitudes , and tend to emerge at progressively lower latitudes as a cycle progresses.

  • The general poloidal field is neutralised and reversed due to flux cancellation, with flux diffused out of following polarity spots which are closer to the poles (known as Joy’s Law).

  • After 11 years, there is renewed winding up by differential rotation with the polar field reversed; the 22 year cycle restores it to the original polarity.

This model has its limitations; being heuristic, semi-quantative and kinematical, amongst others. Recent models have aimed to find fully dynamical solutions of the induction equation (defined in Chapter 2) together with the coupled mass, momentum and energy relations for the plasma (see the review by Charbonneau (2010)). For example, a large focus is placed on the meridional circulation of the poloidal field in the advective dynamo model of Choudhuri et al. (1995). However, the Babcock-Leighton model is sufficient to give a general overview of the source of the solar magnetic field.

Schwabe (1843) first discovered this 11-year solar cycle, with the magnetic polarities of sunspots reversing from one cycle to the next. The behaviour of the entire solar magnetic field is governed by this reversal. At the peak of the sunspot cycle (solar maximum), the greatest number of sunspots is observed, with increased solar activity. The opposite is the case at solar minimum, with very few visible spots. As the cycle progresses, more spots form closer to the equator, in a butterfly wing-like development (Spörers Law). Figure 1.8 illustrates features of the solar magnetic cycle. A number of periods of prolonged sunspot number minimum/maximum are marked in the top of the Figure, e.g, the Maunder Minimum from 1645 to 1715. The bottom of Figure 1.8 shows the butterfly-like development during the solar cycle. It is worth noting that the polarity of the foremost spots in one of the solar hemispheres is opposite to that in the other hemisphere (Hale’s polarity law).

Figure 1.8: Upper panel: 400 years of regular sunspot number observations, based on an average of measurements from multiple observatories worldwide. The sunspot records are shown in blue, with a polynomial fit in black. Prior to 1749 (shown in red) only sporadic observations of sunspots are available. Lower panel: Sunspot area in equal area latitude strips (colours indicate % of strip area as per legend) over 140 years of observations. Both images courtesy of NASA.

1.3.2 Flux Emergence

The formation of sunspots originates in solar sub-surface magnetic field lines within the convection zone, which are distorted and become more complex in shape under the effect of the Sun’s rotation. This is the effect mentioned previously, describing a twisted geometry of a rising flux tube666Magnetic flux tubes can be thought of as bundles of magnetic field lines.. Since magnetic pressure adds to the gas pressure inside a magnetic flux tube (Parker, 1955b), local hydrostatic equilibrium requires,

(1.13)

The pressure within the flux tube is the sum of the gas pressure and the magnetic pressure ; this is balanced by the external gas pressure , while the magnetic field outside of a flux tube is assumed to be negligible. The continuous shearing of the magnetic field eventually causes a build-up of magnetic field in the azimuthal direction. The magnetic pressure () associated with these field lines forces out the infused plasma in order to maintain a pressure balance with the surrounding plasma. Thus, and buoyancy occurs.

An -shaped loop of magnetic flux rises buoyantly from the convection zone, breaking through the photosphere. The emerging magnetic flux tube is described by Thomas & Weiss (2008) as a closed-packed bundle of nearly vertical magnetic flux. The flux bundles are fragmented and twisted into separate strands as they reach the surface. The sunspot magnetic field is thus organised into a flux rope, a collection of twisted flux tubes. Upon emergence, the field lines become very dense and the small flux elements often coalesce to form pores on the surface. As more pores emerge, they grow and move towards each other, coalescing and thus forming a larger sunspot.

Although the structure of flux ropes near the photosphere has been well-studied, the deeper structure of sunspot regions and the processes involved in flux emergence remain poorly understood. There has been much previous work using simulations to try to accurately model an emerging sunspot. For example, much progress has been made by the Solar Multidisciplinary University Research Initiative (MURI) consortium; an example of their work can be found in Figure 1.9. This is a reproduction of Figure 1 of Abbett & Fisher (2003), who present a combined subsurface-atmospheric model of an emerging sunspot in 3D.

Figure 1.9: Snapshot of a simulation of the model corona being driven by an emerging loop. A buoyant flux rope is shown in the lower rectangular box. The vertical component of the magnetic field along the photospheric lower boundary is shown as a grey-scale image (white areas represent positive polarity and black areas represent negative polarity). Magnetic field lines are traced in blue in the atmosphere above (Abbett & Fisher, 2003).

It is worth noting that, as a flux-tube reaches the photosphere, the magnetic pressure is greater than the gas pressure in the tube reaching into the atmosphere above. This causes the magnetic fields to fan apart, forming a magnetic field structure that will be discussed more thoroughly in the next section.

1.3.3 Field Structure

Sunspot magnetic fields are from 100 to 5000 times more intense than the surrounding quiet field, i.e., thousands of gauss (G) versus tens to hundreds of G, where 1 G Tesla. This inhibits convection and hence heat transport. This lowers the temperature to  K, and the sunspots appear darker than the rest of the surface. The opacity also decreases with decreased temperature and density in the sunspot, and deeper geometrical levels in the spot can be seen than in the surrounding photosphere. A typical sunspot structure is shown in Figure 1.10, including typical temperatures for the various features, from observations by the Hinode spacecraft (see Chapter 3). It is worth noting that as a sunspot approaches the limb, the width of the penumbrae on the diskward side decreases at a greater rate than the side of the spot near the limb (known as the Wilson effect; Loughhead & Bray, 1958).

Figure 1.10: Sunspot observations taken with the Solar Optical Telescope (SOT) onboard the Hinode spacecraft. The left panel shows a white light image, with dark umbra and pores ( K), surrounding penumbra ( K), and granules in the quiet outlying area ( K). The right panel shows a vector magnetogram of the same region, black indicating negative polarity (into surface) and white positive polarity (out of surface). The direction of the magnetic field is shown by the red arrows, with its strength indicated by the length of the arrow (Benz, 2008).

The magnetic field of a sunspot is almost vertical (i.e., radial) at the centre of the spot, with umbral field strength usually between  G. The inclination of the field from vertical increases with increasing radius, while field strength decreases, with the field at inclination at the edge of the sunspot and field strength dropping below  G. The field strength also decreases with height (Solanki, 2003). See Figure 1.11 for a sketch of the typical magnetic field configuration of a sunspot. This shows the fan-like structure of field lines mentioned Section 1.3.2. Beckers & Schröter (1969) found that the radial variation of could be best represented by , where is the central umbral field strength and the fractional radius, so that the field falls to half its central value at the edge of the spot.

Figure 1.11: Sketch of the Parker sunspot model, showing the conventional idea of the magnetic field configuration of a sunspot. The heavy line represents the visible surface of the Sun (Parker, 1979).

Beckers & Schröter also noted that the inclination of the magnetic field in the penumbra varies azimuthally. Figure 1.12 reproduces Figure 1 of Thomas et al. (2002), showing a more detailed sketch of the ‘interlocking-comb’ structure777This term was coined in a study by Thomas & Weiss (1992), however this form of structure has also been called by various other names: ‘spines’ (Lites et al., 1993), ‘fluted’ (Title et al., 1993), and ‘uncombed’ (Bellot Rubio et al., 2003) are some examples. of the magnetic field in a sunspot penumbra. Bright radial filaments, where the magnetic field is more inclined towards the vertical (known as ‘spines’), alternate with dark filaments in which the field is nearly horizontal. Langhans et al. (2005) found the inclination of the bright components with respect to the vertical to be in the inner penumbra, increasing to towards the outer boundary. The inclination of the dark component increases outwards from approximately 40 in the inner penumbra, and are nearly horizontal in the middle penumbra. Within the dark filaments, some magnetic flux tubes extend radially outward beyond the penumbra along an elevated magnetic canopy, while other ‘returning’ flux tubes dive back below the surface (known as ‘sea serpents’). The sunspot is surrounded by a layer of small-scale granular convection (squiggly arrows in Figure 1.12) embedded in the radial outflow. The outflow is associated with a long-lived supergranule (large curved arrow). The submerged parts of the returning flux tubes are held down by the granular convection (vertical arrows).

Figure 1.12: Sketch of an ‘interlocking-comb’ structure, in which granular convection plays a key role in submerging the returning penumbral flux tubes and establishing the structure of the penumbral magnetic field (Thomas et al., 2002).

Even smaller features exist in sunspot regions, and moving magnetic features (MMFs) are of particular interest in this thesis. They were first detected by Sheeley (1969) as small bright features moving radially outwards from sunspots. Sheeley postulated that these features could be the manifestation of magnetic field erupting through the surface. Magnetic field measurements by Harvey & Harvey (1973) later confirmed that these bright features represented bipolar magnetic field concentrations. More recent observations have showed that MMFs are extensions of the sunspot penumbral field (Ravindra, 2006; Sainz Dalda & Bellot Rubio, 2008), their orientation correlating well with the sign and amount of twist in the sunspot field (having a U-shaped configuration according to Lim et al., 2012). Figure 1.13 reproduces Figure 3 from Sainz Dalda & Bellot Rubio (2008) showing a sketch of a penumbra with MMFs. As mentioned previously, the inclined fields of the penumbra resemble ‘sea serpents’, flanked by more vertical field lines representing penumbral ‘spines’. The ‘sea serpents’ propagate across the penumbra and reach the moat, where they become bipolar MMFs. Note a sunspot moat is an annular region of typical radius 10 – 30 Mm (Sheeley, 1969), cleared of stationary magnetic flux, that surrounds sunspots as a moat surrounds a castle. MMFs emerge and move outward at approximately the moat flow speed of  kms (surface outflows beyond the penumbra) to form the outer boundary of the moat. However, the small spatial scale () of MMFs means much of the details of their structure remains unknown. These features are investigated in more detail in Chapter 4 , where high-resolution photospheric magnetic field observations are used to study the evolution of small-scale magnetic field structures near a sunspot.

Figure 1.13: Sketch of a penumbra showing ‘sea-serpent’ field lines in between more vertical ‘spine’ fields, with a MMF at the moat. The ovals indicate the polarity of the different structures when observed at disk center (Sainz Dalda & Bellot Rubio, 2008).

Thomas et al. (2002) and Schlichenmaier (2002) (among others) proposed that MMFs are the continuation of the penumbral fields that harbour the Evershed flow, later confirmed by the observations of Sainz Dalda & Bellot Rubio (2008). The Evershed flow is a horizontal outward flow of plasma across the photospheric surface of the penumbra, from the inner border with the umbra towards the outer edge. It was first discovered by Evershed (1909). The speed of the flow tends to vary from  km at the border between the umbra and the penumbra to a maximum of approximately double this value in the middle of the penumbra. The flow speed then falls off to zero at the outer edge of the penumbra. In the chromosphere and transition region the flow reverses to an inflow (with generally higher velocities than the outflow), termed the inverse Evershed flow.

Although much work has been done to try to understand the structure of sunspots, the intricacies of sunspot fine structure is still not fully resolved (Thomas & Weiss, 2004). Studying the 3D magnetic field topology of ARs is important in improving our understanding of sunspot structure. Early work focused on simple extrapolation methods to study the full solar global field. For example, Figure 1.14 shows a typical global potential field extrapolation (explained in Chapter 2) obtained using the Potential Field Source Surface model (PFSS, Schrijver & De Rosa, 2003) with observations from . The observations correspond to the same time as those used for Figure 1.4. Recently, more advanced techniques have been used to study smaller FOVs, investigating the topology of sunspot magnetic fields in greater detail (Régnier & Priest, 2007b; De Rosa et al., 2009; Conlon et al., 2010). 3D extrapolations will be used in Chapter 6 in order to investigate the magnetic topology of small flux elements in a sunspot region.

Figure 1.14: Example of extrapolations obtained using observations from the spacecraft. Multiple coronal wavelengths observations are shown on the left from AIA (211 Å, 193 Å, and 171 Å), and a magnetogram from HMI on the right. The white lines show the field line traces obtained by the PFSS model. The enormous complex NOAA region 11520 at disk centre (Hale classification , or McIntosh classification) was the source of an X1.4-class flare (see Table 1.2) at 15:30 UT on 2012 July 12.

1.3.4 Classification Schemes

It is worth mentioning how sunspots are traditionally classified. Hale et al. (1919) determined a classification scheme for sunspots, with unipolar spots designated , bipolar spots , and multipolar spots . Künzel (1960) added a classification for more complex regions that are generally associated with flaring (as will be discussed in Section 1.4). Mount Wilson Observatory developed this classification further after examining regular measurements of sunspot polarities, as detailed in Table 1.1, and this classification scheme will be referred to throughout this thesis. Examples of , , , and regions are shown in Figure 1.15. Also, the sunspot shown in Figure 1.10 is a class region.

Class Description
Unipolar sunspot group.
Sunspot group having both positive and negative magnetic polarities
(bipolar), with a simple and distinct division between the polarities.
Complex sunspot group in which the positive and negative polarities are
so irregularly distributed as to prevent classification as a bipolar group.
Sunspot group that is bipolar but sufficiently complex that no single,
continuous line can be drawn between spots of opposite polarities.
Sunspot group in which the umbrae of the positive and negative polarities
are within 2 degrees of one another and within the same penumbra.
Sunspot group of class but containing one (or more) spots.
Sunspot group of class but containing one (or more) spots.
Sunspot group of class but containing one (or more) spots.
Table 1.1: The Modified Mount Wilson sunspot classification scheme (Bray & Loughhead, 1964).

Figure 1.15: Examples of (upper left panel), (upper right), (lower left), and (lower right) regions, as defined by the Modified Mount Wilson Classification Scheme. Images are courtesty of SolarMonitor.org.

McIntosh (1990) developed a new classification of sunspots based on their likelihood of flaring, which is shown in more detail in Figure 1.16. The general form is Zpc, where Z is a modified Zürich class888The original Zürich classification scheme was developed by Waldmeier (1947). (left column of Figure 1.16, classifying sunspot group), p is the type of principal sunspot (middle column, primarily describing the penumbra), and c is the degree of compactness in the interior of the group. McIntosh found that Fkc class spots are much more likely to flare: F being defined as a ‘bipolar group with penumbra on spots at both ends of the group’, k meaning a large and asymmetric penumbra, and c suggesting a compact sunspot distribution. Sunspots will generally change classification at various stages of a their life cycle, often beginning with a simple structure and becoming more complex. Sunspot evolution will be described in more detail in the next section.

Figure 1.16: The three-component McIntosh classification, with examples from each category (as described in McIntosh (1990)). The left column shows the modified Zürich class, the middle column primarily describes the penumbra, and the right column indicates the degree of compactness in the interior of the group.

1.3.5 Field Evolution

Flux emergence was described in detail in Section 1.3.2, however there are multiple stages of sunspot evolution that can be examined as an AR crosses the solar disk. Considering that ARs emerge in generally days, and spend % of their life decaying (Harvey, 1993), studying the evolution of ARs is a necessary step in improving the understanding of AR magnetic field structure.

After an AR emerges, the region will generally continue to grow, and often an increase in complexity of the region becomes apparent. Flare activity may increase as complexity increases, a concept that will be discussed in more detail in Section 1.4.1. Van Driel-Gesztelyi (2000) studied the long-term evolution of an AR with photospheric magnetic field and X-ray observations, and found solar flares occurring in the first three rotations only (the evolution lasted several months). However, coronal mass ejections (CMEs) were found to occur from emergence phase through to the decay phase of the AR (relating to removal of excess shear, which will be defined in Section 1.4.1).

Once ARs reach maximum maturity, they begin to decay; sunspots diminish in size and flux, then pores and small spots disappear (van Driel-Gesztelyi, 1998). Flaring activity is strongly diminished. The decay-phase time scale of an AR differs depending on the type of region, but if there are multiple sunspots the leading spot can generally last for a number of months, with trailing spots splitting first. Schrijver & Harvey (1994) note an average time scale of disappearance to be  4 months for solar cycle maximum, and months at solar minimum. Plage areas generally expand and dissolve into the chromospheric network. As the magnetic structure simplifies, a bipolar nature generally becomes more obvious. Following the loss of bipolar structures, remnants of the two main polarities may be tracked as monopolar structures, which often merge with other AR remnants. Larger composite unipolar regions often drift towards the poles, and can give rise to coronal holes (van Driel-Gesztelyi, 2008).

Figure 1.17: The evolution of NOAA AR 11476 from 5–17 May 2012 in 4500 Å images (upper row, SDO/AIA observations), LOS magnetic field (middle row, SDO/HMI observations), and 171 Å images (lower, SDO/AIA observations). Note that the magnetogram on May 13 (fourth column, middle row) was taken from the SOLIS instrument (chromospheric magnetic field), as no HMI data was available that day. Left to right columns show observations from 2012 May 6 (at Solar X , Solar Y ), May 8 (, ), May 10 (, ), May 13 (, ), and May 15 (, ) respectively. Images courtesy of SolarMonitor.org.

Figure 1.17 shows the evolution of an AR as observed by the SDO spacecraft. The evolution is shown with LOS photospheric magnetograms from the HMI instrument, and 4500 Å (temperature minimum/photosphere) and 171 Å (quiet corona, upper transition region) wavelength images taken by the AIA instrument. The AR moved across the solar disk over a period of 13 days. It first emerged on the eastern limb on 2012 May 5, however was not given a classification until May 6. It was designated NOAA 11476, and determined to be a class region (see first column of Figure 1.17). By May 7 it had evolved into a region, and was the source of multiple medium-magnitude flares, which became more frequent on May 8 (second column of Figure 1.17). The AR was designated on May 9, i.e., the most complex classification likely to flare. The region was a source of several flares larger in magnitude than previously observed. More larger flares occurred on May 10 as the AR neared disk centre (third column of Figure 1.17), with the largest magnitude flare (GOES M5.7, which will be defined in Section 1.4) beginning at 04:11 UT on May 11. Only medium-magnitude flares occurred from the May 13 onwards, and the AR showed clear signs of decay, with trailing elements disappearing (fourth column of Figure 1.17). As the AR neared the western limb on May 15 it became a classification AR (fifth column of Figure 1.17), still with some flaring, but of smaller magnitude and less frequent. The AR was last observed on the western limb on May 17, when it was designated a -class region.

Deng et al. (1999) also tracked an AR over 5 days as it traversed the solar disk. Deng et al. found that increasing AR area and complexity, as well as emergence of new flux, led to a large solar flare occurring. With such large changes observed in the magnetic field structure over the course of an AR’s life, it is clear that studying sunspot magnetic field evolution is a useful tool for improving our understanding of sunspot structure. Also, clear trends in flare activity over the course of the evolution period indicate a worthy area of investigation. This thesis will investigate the link between sunspot field evolution and flaring processes, but over much shorter timescales than months of a full sunspot life cycle. Magnetic field evolution will be examined over hours leading up to and after a solar flare.

1.4 Solar Flares

Sunspots decay and fragment by the motion of supergranulation and solar rotation. This, along with the emergence of new magnetic field, can lead to very complex and stressed magnetic fields. Solar flares occur when the energy stored in the sunspot magnetic fields is suddenly released, converting magnetic energy to kinetic energy of near-relativistic particles, mass motions, and radiation emitted across the entire electromagnetic spectrum. Large solar energetic eruptive events can include CMEs as well as flares (Green et al., 2001; Švestka, 2001; Zhang et al., 2001), but flares are the particular phenomena of interest in this thesis999It is worth noting that whenever ‘eruptions’ are mentioned in the text, CMEs are also being referred to.. In the soft X-ray range, flares are classified as A-, B-, C-, M- or X- class according to the peak flux measured near Earth by the GOES spacecraft over  Å (in Watts m). Each class has a peak flux ten times greater than the preceding one, with X-class flares having a peak flux of order 10 Wm. Table 1.2 outlines this in more detail. The radiated energy of a large flare may be as high as 10 erg (10 J).

1.4.1 Standard Model

The basic physics that govern flares is still not well understood despite many years of observations, but the ‘standard model’ is widely accepted. The model suggests three main phases of the flare: pre-flare, impulsive, and decay phase. In the pre-flare phase there is a build up of stored magnetic energy. This phase will be discussed in detail throughout the thesis, however the energy can be built up via, e.g., twisting or shearing101010Magnetic shear angle is defined as the angle between the measured transverse field and calculated potential field. of the field lines (see Section 1.4.2).

GOES class Peak flux
[
A 10
B 10
C 10
M 10
X 10
Table 1.2: GOES classification scheme for solar flares.

Plasma is heated to high temperatures in the impulsive phase, with strong particle acceleration and a rapid upflow of heated material. The impulsive phase is clearly seen in hard X-rays (HXR) and radio, but intense emission is also observed in optical, UV and EUV. It is generally believed that the impulsive phase is driven by magnetic reconnection, where the stored energy that has built up is released, which accelerates coronal particles.

The so-called CSHKP model is a standard model of large-scale magnetic reconnection to explain solar flares and CMEs, named after pioneers in this line of research (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974; Kopp & Pneuman, 1976). Reconnection is believed to occur when the shearing and twisting of magnetic field lines pushes the coronal magnetic field to an unstable state, one which wishes to reach a preferred lower energy state (Aschwanden, 2005). The driving force behind all reconnection models is a scenario where two regions of oppositely directed magnetic flux converge (see Figure 1.18). Usually solar plasma is assumed to be perfectly conducting and so the magnetic fields are said to be ‘frozen-in’ to the plasma; see Section 2.1.3 for a more detailed description. However, here the magnetic field across the boundary (diffusion zone) is zero and the frozen-in condition breaks down. This allows plasma to diffuse and reconnect, resulting in a lower energy topology.

Figure 1.18: Basic 2D model of the reconnection process, where two oppositely directed regions of magnetic flux converge. A diffusion region forms at the boundary where magnetic field is zero, creating horizontal outflows.
(\(\mathrm{http://www.aldebaran.cz/astrofyzika/plazma/reconnection\_en.html}\)).

Note that the impulsive phase of a flare is often described in more detail by the ‘thick-target model’, originally developed by Brown (1971). In this model, once the eruption has begun a reconnection ‘jet’ of fast moving material collides with the soft X-ray (SXR) loop below, producing an MHD fast shock. This shock produces a HXR loop-top source and further acceleration. Electrons and ions stream down the legs of the loop, and produce HXRs by Bremsstrahlung radiation when they meet the chromosphere. Sometimes chromospheric material is heated so rapidly that energy cannot be radiated away, and pressure gradients can build up in the plasma. The plasma thus expands to fill the SXR loops. This process is known as chromospheric evaporation. As the eruption progresses, more and more field lines reconnect, producing an arcade of loops seen in SXR. This occurs over a timescale of  tens of minutes (Phillips, 2008). It is worth noting that in recent times a number of issues have been raised regarding the way the energetics and plasma physics are described in the the thick target model (see, e.g., Brown et al. (1990)). The model faced problems with high electron beam densities, leading to a reworking of the theory to include local re-accelerations of fast electrons throughout the HXR source itself (rather than just streaming down to the chromosphere) (Brown et al., 2009).

The decay phase of the standard model can be simply described as when the plasma cools down. A summary of the process of flaring outlined by the standard model is shown via an illustration in Figure 1.19.

Figure 1.19: Standard solar flare model, as illustrated by Shibata (1999).

Figure 1.20: Figure 2 of Sammis et al. (2000), showing peak flare intensities (Wm) for the largest flare of each of the 2,789 sunspot groups studied as a function of peak area in disk fraction. Each magnetic Hale class is plotted separately (colours indicated in the legend). Note that regions producing no flares are omitted.

This thesis will explore the connection between sunspot magnetic field structure and solar flares. Early studies of solar flare activity found a close relationship between flares and sunspots, and was the reason for the extra classification scheme developed by Künzel (1960) for the Hale clasification scheme. Zirin & Liggett (1987) examined eighteen years of observations from Big Bear Solar Observatory, and found a link between the existence of configuration sunspot groups and very large flares. Later statistical studies confirmed this early research, for example a statistical study by Sammis et al. (2000) described relationships between active region size, peak flare flux, and magnetic classification. Their Figure 2 is reproduced in Figure 1.20, showing that bigger flare events (in terms of peak flux) occur in more complex Hale magnetic classification sunspot groups ( producing the biggest). It must be noted that the values plotted are largest quantities at any time in the AR’s lifetimes (not necessarily at flare peak). The tendency of larger flares occurring in more complex regions could also be seen in the AR evolution illustrated in Figure 1.17, as the AR produced its largest-magnitude flare at the height of its complexity. Thus the degree of complexity of active regions has been found to be linked to the likelihood of flaring and maximum flare magnitude: larger, more numerous flares occur near larger and more complex sunspot groups. The following subsections will explore the relationships between sunspots and flares further, reviewing previous work in the field.

1.4.2 Trigger Mechanisms

Much previous work has expanded upon the link between sunspot complexity and flaring, to focus on possible flare trigger mechanisms. Hagyard et al. (1984) found that points of flare onset are where both magnetic shear and field strength are the strongest, suggesting that a flare is triggered when the magnetic shear stress exceeds some critical amount. Shearing is taken to mean that the field is aligned almost parallel to the magnetic neutral line (NL, where field falls to zero) rather than perpendicular to it, as would be observed in a potential field configuration (Schmieder et al., 1996). Many early theoretical studies suggested a link between both the emergence of new flux and the shearing and twisting of field lines with the flare trigger mechanism (see Rust et al., 1994, for a review).

Flux emergence has been found to play a significant role, with Feynman & Martin (1995) showing that eruptions of filaments are most likely to occur when new flux emerges nearby in an orientation favourable for reconnection. More recently, Wallace et al. (2010) found the emergence of small-scale flux near a magnetic NL to be a trigger to a B-class flare, with pre-flare flows observed along two loop systems in the corona  minutes before the flare began. The earliest indication of activity in the event studied by Wallace et al. was a rise in non-thermal velocity 1 hour before flare start. An increase in non-thermal velocity was suggested by Harra et al. (2001) as an indicator of turbulent changes in an AR prior to a flare that are related to the flare trigger mechanism. Computational simulations have also been used to investigate flux emergence as a flare trigger, for example Kusano et al. (2012) used 3D magnetohydrodynamic simulations to investigate small-scale flux emergence of opposite polarity as possible flare triggers.

Aside from non-thermal velocity, a number of other pre-flare indicators have been discovered across multiple wavelengths (see review by Benz (2008)). Des Jardins & Canfield (2003) found changes in H observations up to  hours before the start of 11 eruptive flares, a phenomenon they called ‘moving blueshift events’. Des Jardins & Canfield concluded that reconnection in the chromosphere or low corona plays an important role in establishing the conditions that lead to solar flare eruptions. Nonthermal emission was observed in microwaves and hard X-Rays by Asai et al. (2006) during the pre-flare phase of an X-class flare, from  minutes before flare start. A faint ejection associated with the flare was also observed in EUV images. Joshi et al. (2011) also observed significant pre-flare activities for minutes before the onset of a flare impulsive phase, in the form of an initiation phase observed at EUV/UV wavelengths, followed by an X-ray pre-cursor phase. Although multi-wavelength pre-flare observations in the chromopshere and corona have been well-studied, currently no observations of magnetic field changes on such short timescales before flaring have been found. In this thesis, the magnitude and orientation of the photospheric magnetic field will be investigated over a number hours before flare start.

As analytical methods have improved, and more high-resolution magnetic field data has become available, studies have begun to focus more on increasing twist and magnetic helicity as important flare triggers (e.g., Harra et al., 2009; Georgoulis, 2011a; Li, 2011). Note that magnetic helicity is a measure of magnetic topological complexity, e.g., twists and kinks of field lines (see Canfield & Pevtsov, 1998). However, early theoretical work hinted at the importance of these factors, with Hood & Priest (1979) estimating a value of twist of 2.5 radians as a critical threshold after which a flux tube will become linearly unstable to kinking. Pevtsov et al. (1996) confirmed this theory with X-ray and vector magnetic field observations of NOAA AR 7154, finding twist in the AR to exceed this threshold before an M-class flare occurred. The use of sophisticated 3D magnetic field extrapolations can now improve upon earlier studies, giving a more accurate depiction of field line topology (Chandra et al., 2009; Jing et al., 2012). 3D extrapolations will be used in Chapter 6 to investigate possible flare triggers for the event studied there.

1.4.3 Flaring Process

Figure 1.21: Depiction of a magnetic field configuration during flaring by Tanaka (1986). The configuration is shown at pre-flare (left column), flare (middle column), and post-flare (right column) states. Upper row shows a view from above the configuration, and the lower row shows a side-on view.

Analysing the magnetic field configuration of ARs is an important step in understanding the flaring process. Tanaka (1986) depicts a possible evolution of large-scale fields in a flare, shown in Figure 1.21. The depiction includes an ensemble of sheared fields containing large currents and a filament located above the NL in the pre-flare state. Tanaka suggests a flare may be triggered by a filament eruption at this NL location. More sophisticated flare models were later developed, e.g., Antiochos (1998) described a ‘breakout’ model for large eruptive flares, beginning with newly-emerged, highly-sheared field held down by an overlying un-sheared field. Reconnection takes place between the un-sheared overlying flux and flux in additional neighboring systems. This reconnection transfers un-sheared flux to the neighboring flux systems, thereby removing the overlying field and the restraining pull. Hence, reconnection allows the innermost core field to ‘break out’ to infinity, without opening the overlying field itself and thus violating an upper limit of free energy known as the Aly-Sturrock limit (Aly, 1991; Sturrock, 1991). This limit states that an open state of magnetic field contains the largest energy of any possible field configuration. Antiochos note that, while a bipolar AR does not have the necessary complexity, a sunspot has the correct topology for this model.

Figure 1.22: Flare model of Moore et al. (2001) showing the magnetic field configuration before, during, and after the onset of an explosion that is unleashed by internal tether-cutting reconnection. The dashed curve indicates the photospheric NL; the arc in the background of the panels defines the chromospheric limb; the dashed area in the upper left panel shows a filament of chromospheric temperature plasma; the dark gray areas in the other panels are ribbons of chromospheric flare emission (at the feet of reconnected field lines).

In the ‘tether-cutting’ scenario of Moore et al. (2001), as shown in Figure 1.22, there is strong shear at low altitudes along the NL before the flare occurs (upper left panel). The upper right panel of Figure 1.22 shows reconnection via ‘tether-cutting’ occurring in the region, and lower left panel shows the completion of the early reconnection. Finally, the lower right panel of Figure 1.22 shows a rising plasmoid distending the outer field lines, which continue the process of reconnection to form the expanding flare. In the Moore et al. model the plasmoid eventually escapes as a CME. Note the main difference between this and the ‘breakout model’ is that reconnection occurs above the unstable structure in the ‘breakout’ model, whereas it occurs below it in the ‘tether-cutting’ scenario.

Field topology studies have been used to place constraints on theoretical models, for example Mandrini (2006) reviewed a number of flaring active-region topologies, finding that magnetic reconnection can occur in a greater variety of magnetic configurations than traditionally thought. The reader is referred to the reviews of Priest & Forbes (2002a) and Schrijver (2009), and references therein, for more recent developments in eruptive event models.

1.4.4 Flaring Locations

Although sunspot regions are closely associated with solar flares, it is rare that flares occur in the sunspot umbra itself (Moore & Rabin, 1985). The magnetic neutral line (NL, where the field falls to zero) is often found to be a region of flare activity, a locus across which the line-of-sight (LOS) field component changes sign. The review of Moore & Rabin (1985) noted that the most common flare-productive field configuration is characterised by strong shear across the NL. For example, Hagyard et al. (1984) found flare onset to occur across a NL that exhibited strong field strength and magnetic shear, as mentioned in Section 1.4.2. This location will be investigated further in Chapter 5 for its importance to flaring.

It has been found that in the lead up to a flare occuring, field lines rooted to the NL run nearly parallel rather than perpendicular to it (see, e.g, the tether-cutting scenario of Moore et al., 2001). The field near such a NL is far from potential and holds ample free magnetic energy for flares. Magnetic energy changes over the course of a flare observation period will be investigated in Chapter 6. It is worth noting that there is some debate over what overlays a NL; sheared magnetic arcades or helical magnetic flux ropes. However, Georgoulis (2011b) examined the magnetic pre-flare configuration near the NL and suggested instead that numerous small-scale magnetic reconnections, constantly triggered in the NL area, can lead to effective transformation of mutual helicity (i.e, crossing of field lines) to self magnetic helicity (i.e., twist and writhe) that, ultimately, may force the magnetic structure above NLs to erupt to be relieved from its excess helicity.

1.4.5 Observed Magnetic Field Changes

Numerous observational studies have confirmed the importance of emergence and shearing to flare phenomena. Zirin & Wang (1993) investigated flux emergence and sunspot group motions, which resulted in complicated flow patterns leading to flaring. Wang et al. (1994) used vector magnetograms to observe magnetic shear in five X-class solar flares; in all cases increasing along a substantial portion of the magnetic NL. They suggested flux emergence being key to eruption, but the increase in shear persisted much longer after the flare rather than decreasing as per model predictions, and no theoretical explanation was given. Recent evidence has furthered the idea that emerging-flux regions and magnetic helicity are crucial to the pre-flare state (e.g., Liu & Zhang, 2001; Wang et al., 2002; Chandra et al., 2009).

However, these parameters are not the only ones of interest when studying the links between AR magnetic fields and flaring. Wang et al. (2002) studied LOS magnetic field observations during three X-class flares, and found changes in magnetic flux and transverse field due to the flare. Sudol & Harvey (2005) also used LOS magnetgrams to observe abrupt and permanent changes in the LOS field after 15 X-class flares. Sunspot magnetic field inclination has also been found to change dramatically due to solar flares (Liu et al., 2005; Li et al., 2009; Gosain, 2012), as well as current density (de La Beaujardiere et al., 1993; Su et al., 2009; Canou & Amari, 2010). These parameters will be investigated further in Chapters 4 and 5. It is clear that significant changes in various magnetic field parameters occur due to solar flares, but most previous work has focused on investigating changes due to the flare itself. The research in this thesis will examine the full evolution of ARs associated with flare events (over shorter timescales than previously studied) looking for changes leading up to the flare as well as afterwards.

Early investigations were limited by the lack of high-resolution solar vector magnetic field data, mainly using only LOS observations of the magnetic field. Wang (2006) listed predictions of results from future flare observations, when high-resolution vector magnetograms from Hinode and Solar Dynamics Observatory would become available,

  • Transverse magnetic fields at a flaring NL will increase rapidly following flares.

  • The unbalanced flux change will be more prominent when the regions are closer to the limb due to the enhanced projection effect there.

  • Evershed flow will decrease in the outer boundary of a -configuration, as outward-inclined fields will become more vertical.

  • In the initial phase of the flare, two flare footpoints may move closer before they start the usual separation motion.

  • As a consequence of the reconnection, some current will be able to be measured near the photosphere, and therefore, an increase of the magnetic shear near the flaring NLs may be detected.

This list is not exhaustive (other indicators will be discussed throughout this thesis in detail), but it summarises a selection of current theories and observations well. The research described in this thesis uses this newly available high-resolution data to test currently proposed theories, by examining magnetic field parameters in active regions for any significant changes before and after a flare. Photospheric magnetic field evolution is examined in Chapters 4 and 5, while 3D magnetic field extrapolation methods are used in Chapter 6 to study the coronal magnetic field.

1.5 Outline of Thesis

The research contained in this thesis examines the evolution of a number of ARs undergoing flaring using various analysis techniques. Examination of the magnetic field conditions in an AR before a flare occurs could produce useful indicators for flare forecasting. Studying differences in AR topology between pre- and post- flare states helps to test the validity of currently proposed changes in magnetic topology during solar flares. Through examination of the evolution of a sunspot, changes in the topology observed, if any, can give an insight into how and when a flare might occur from this kind of region.

In this chapter, the introductory theory behind the research project has been presented, starting with some background to the Sun, and then focusing on sunspots and solar flares in greater detail. In Chapter 2, some basic theory needed to understand the analysis techniques throughout this thesis are introduced. Fundamentals of magnetohydrodynamics are described to introduce 3D magnetic field extrapolations, and an introduction to radiative transfer is also presented. The instruments used to obtain data for analysis in this thesis are described in Chapter 3. Specifically instruments onboard the Hinode spacecraft are described, including the software methods used to process the raw data obtained. The techniques used for analysis in Chapters 4,  5, and  6 are also introduced in Chapter 3, including some background theory not previously described in Chapter 2.

In Chapter 4, the results of examining photospheric magnetic field information are presented, from small flux elements of a sunspot region undergoing flaring. Data from the Hinode spacecraft are used to observe the AR magnetic field before and after a B-class flare. Distinctive magnetic field changes are observed leading up to and after the flare in an area of chromospheric flare brightening. In Chapter 5, a magnetic neutral line location is examined with Hinode observations during a period of observation in which a C-class flare occurred. Both temporal and spatial changes are observed across the magnetic NL during the observation period.

While 2D photospheric magnetic field observations are studied in Chapters 4 and  5, the 3D coronal field is examined in the third research chapter, using magnetic field extrapolation methods. In Chapter 6, the results of the 3D extrapolations are described, using the observations of the B-class flare event previously examined in Chapter 4. Magnetic geometry and energy changes are observed both before and after the flare in the region of flare brightening. Finally, the main conclusions of the research presented in this thesis are described in Chapter 7, with some directions for future work.

Chapter 2 Theory

   
In this chapter, the theory needed to understand the various analysis techniques used in this thesis is presented, beginning with a background to magnetohydrodynamics. Fundamental magnetohydrodynamic equations are outlined, and the basic theory behind 3D magnetic field extrapolations is described. Radiative transfer is then discussed, including the various assumptions that must be made in order to obtain a model of the solar atmosphere.
   

2.1 Magnetohydrodynamics

The Sun is composed of material called plasma, which is defined as a state similar to gas in which a certain fraction of atoms are ionised. In the Sun, this ionisation is caused by the extreme temperatures and pressures that exist, and matter in and around the Sun can be treated as electrically charged fluid. Magnetohydrodynamics (MHD) describes the flow of electrically conducting fluid in the presence of electromagnetic (EM) fields. Its equations govern the magnetised plasma of the Sun. It is useful to include a description of basic MHD in studies of AR magnetic fields in order to understand the dynamics of solar plasma, thus this section introduces some fundamental theory needed to understand this complex topic.

2.1.1 Maxwell’s Equations

Maxwell’s equations (Maxwell, 1861) are a set of four fundamental equations governing electromagnetism (i.e., the behavior of electric and magnetic fields). For time varying fields they are defined (in differential microscopic form111There are numerous forms of these equations, including integral and macroscopic form, but the form defined here is what will be referred to throughout the rest of this thesis.) as,

(2.1)
(2.2)
(2.3)
(2.4)

where B is the magnetic field, is the magnetic permeability of free space, j is the current density, is the speed of light in a vacuum, E is the electric field, is the permittivity of free space, and is the charge density.

Equation 2.1 is generally referred to as Ampere’s Law, which means that either currents or time-varying electric fields may produce magnetic fields. If it is assumed that the typical plasma velocity, v, is much less than , the second term on the RHS of Equation 2.1 can be ignored (MHD approximation). Thus Ampere’s Law becomes,

(2.5)

Equation 2.2 is often described as Gauss’s Law for magnetic fields (no magnetic monopoles). Faraday’s Law is defined in Equation 2.3, meaning that time-varying magnetic fields can give rise to electric fields. Finally, Equation 2.4 is known as Poisson’s Law (or Gauss’s Law for electric fields), and means that electric charges may give rise to electric fields.

Ohm’s Law couples the plasma velocity to Maxwell’s equations, and is written as,

(2.6)

where is the plasma conductivity. It expresses that the electric field in the frame moving with the plasma (E) is proportional to the current, or that the moving plasma in the presence of magnetic field B is subject to an electric field (in addition to E).

2.1.2 Induction Equation

Ampere’s Law (Equation 2.5) and Ohm’s Law (Equation 2.6) can be combined to give,

(2.7)

where is the magnetic diffusivity. Taking the curl of both sides, and substituting for Faraday’s Law (Equation 2.3) gives,

(2.8)

Since , Equation 2.8 can be rewritten using Equation 2.2 to derive the induction equation for the solar magnetic field,

(2.9)

The first term of the RHS of the equation describes advection, and the second term diffusion.

The magnetic Reynold’s number defines the ratio of the advection and diffusion terms of the induction equation, such that,

(2.10)

Replacing vector quantities by their magnitudes, and considering , where is a typical length scale, the equation can be written as,

(2.11)

where is a typical velocity. For example in a sunspot  m (typical radius),  m s (super granular motion at surface), and  m s (medium-sized sunspot at the photosphere; (Aschwanden, 2005)). This gives a value of . For the solar corona, (since  m and  m s).

If (as in a sunspot) then the advection term dominates and the induction equation becomes . This is for ideal MHD, i.e., in the limit of perfect electrical conductivity (, ). The field is said to be ‘frozen-in’ (see Section 2.1.3), with the field lines in this perfectly conducting plasma behaving as if they move with the plasma. If the field is very strong, plasma is constrained to flow along it, and if the field is weak, it is advected with the plasma.

If the diffusion term dominates and the induction equation becomes . Magnetic field irregularites will then diffuse away over a time scale of . For a sunspot, this yields a diffusion times of  seconds, thus it is not the main method of particle motion here (unless there are short length scales or large gradients in the field). Note for flares, the diffusion time is  seconds, which gives a lengthscale of 10 m. However, the finest instrument resolution is currently a few hundred km, leading to the need for better resolution instruments to fully analyse the solar flare process.

2.1.3 Magnetic Flux Freezing

Conservation of magnetic flux in a perfectly conducting fluid is one of the most fundamental conservation laws of MHD. Also known as Alfvn’s Theorem, it implies that magnetic field lines are ÔfrozenÕ into the fluid, so that the field lines and the plasma move together. The idea of MHD is that magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. In ideal MHD, the fluid is a perfect conductor, and applies to partially ionised plasmas which are strongly collisional and have little or no resistivity.

A more detailed description of this theorem begins by considering a gas parcel threaded with magnetic field. The flux anchored to the gas parcel, will change in two ways:

(2.12)

where . One part of the change of flux anchored to a surface comes from a temporal change in the magnetic flux density over the surface (first term on RHS), while the other part comes from a change of surface boundary due to gas motion (second term on RHS; ).

Stoke’s theorem, , can be used to reduce the last part of Equation 2.12 to . This gives

(2.13)

which means the magnetic flux anchored to a gas parcel does not change in the ideal MHD regime, it remains constant in time for any arbitrary contour. Thus magnetic field lines must move with the plasma, i.e., they are ‘frozen’ into the perfectly conducting fluid.

Note that in Equation 2.13, can be written in terms of Faraday’s Law (i.e, Equation 2.3). Alfvn’s theorem is thus a direct result of Faraday’s law applied to a medium of infinite electrical conductivity. Motions along the field lines do not change the field but motions transverse to the field carry the field with them, i.e., the field is dragged with the plasma or vice-versa. One important consequence of this theorem is that the topology of the field lines is preserved, and, in particular, that crossing field lines cannot reconnect. This presents a puzzle in many situations where ideal MHD is supposed to hold to a very good first-order approximation, such as in many astrophysical systems, when the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field.

2.1.4 Equation of Motion

The equation of motion (momentum equation), , is described by,

(2.14)

with the external forces F on the RHS indicating the gradient of the gas pressure, Lorentz force, and gravitational force onto the plasma, respectively. Note on the LHS of the equation of motion that is the convective time derivative,

(2.15)

The Lorentz term of the equation of motion can be expanded upon. Using Ampere’s law (Equation 2.1) with the j B term of Equation 2.14, and the vector identity , gives

(2.16)

where the first term on the RHS is the magnetic pressure force, and the second term on the RHS is the magnetic tension force. The magnitude of the plasma pressure, p222p=2, where n is the number of hydrogen atoms, k is Boltzmann’s constant and is the temperature. The factor of 2 is included as both electrons and protons contribute., and magnetic pressure, are compared by the plasma parameter, which is defined as,

(2.17)

If , the gas pressure dominates (e.g., in the solar photosphere) and the influence of the magnetic field is negligible (with plasma motions dominating over the magnetic field forces). If , the magnetic pressure dominates (e.g., in the solar corona) and the magnetic field is often assumed to be force free for the purposes of extrapolation procedures (see below). Figure 2.1333Note that the units for in the plot are CGS units (rather than SI), with magnetic pressure defined as and gas pressure as . The CGS version of magnetic pressure will be used for calculations in Chapter 6. illustrates the change in value throughout the solar atmosphere as depicted by Gary (2001). After a high plasma in the photosphere, the value decreases to low values through the chromosphere and corona, where the magnetic field structures are observed to suspend plasma in loops and filaments. In the extended upper atmosphere rises again, and the magnetic field is advected out with the solar wind plasma flow to ultimately form the Parker spiral (as described in Section 1.2).

2.2 3D Magnetic Field Extrapolations

In Chapter 6 the evolution of the 3D coronal magnetic field in an active region is investigated using three types of extrapolation procedure: potential, linear force free (LFF), and non-linear force free (NLFF). This section aims to discuss the theory and techniques behind these procedures.

Figure 2.1: Plasma as a function of height over an AR (magnetic field strengths between 150 G for a plage region, and 2500 G for a sunspot). The dotted lines divide the photosphere (), chromosphere and corona (), and the solar wind () (Gary, 2001).

First the special condition of magnetohydrostatic equilibrium is applied to the equation of motion (Equation 2.14). Flows are neglected, so that , and it is assumed there is no time variation, so that . Hence, the equation of motion becomes,

(2.18)

The corona is considered to be generally force free (Gold & Hoyle, 1960), dominated by the relatively stable magnetic field in a low- plasma. The gas pressure term in Equation 2.18 is negligible compared to the Lorentz term, and gravity can also be considered negligible high in the upper solar atmosphere. Equation 2.18 thus reduces to,

(2.19)

This is known as the force-free approximation, which all three types of 3D extrapolation mentioned above assume (Gary, 2001). The approximation results in the current being parallel vectorially to the magnetic field, with a proportionality factor termed the force-free field parameter, and is a scalar function of position (i.e., is a spatially varying function to be determined). There are three general forms of the force-free relation,

j (2.20)
j (2.21)
j (2.22)

A potential field configuration is defined as one containing no currents, resulting in the case of Eqn. 2.20 where . When is non-zero but constant throughout a given volume the field configuration is referred to as LFF (Eqn. 2.21). Finally, when is allowed to vary spatially (i.e., differing from field line to field line, but constant along one field line) the field configuration is referred to as NLFF (Eqn. 2.22). This specific case allows for the existence of both potential and non-potential fields within the given volume. The following subsections will examine these three forms in more detail.

2.2.1 Potential and Linear Force-Free Fields

For a current-free potential field, assuming the force-free approximation, Ampere’s Law reduces to . The most general solution to this is

(2.23)

where is the scalar magnetic potential. Substituting this into Equation 2.2 gives , showing that potential magnetic fields satisfy Laplace’s equations. Green’s functions are often used to solve a potential magnetic field, first proposed by Schmidt (1964), and further developed by Sakurai (1982). An example of a typical global potential field extrapolation is shown in Figure 2.2, as previously discussed in Section 2.2.3.

Figure 2.2: Low-resolution equivalent of a full-disk magnetogram, with computed field lines from a potential field extrapolation overlayed: black indicates closed field lines, and green and pink indicate open field lines (for field pointing away from or towards the Sun, respectively). The computed field lines start from a set of uniformly distributed points on the solar surface (Schrijver & De Rosa, 2003).

If the magnetic field is not potential, then using Ampere’s Law we obtain,

(2.24)

Using the vector identity with Equation 2.24 gives,

(2.25)

However, Equation 2.2 shows the first term on the RHS is zero, thus

(2.26)

so that is constant along each field line, although it may vary from field line to field line in the case of a NLFF field. Note that if , the magnetic field is potential. Figure 2.3 shows a typical example of a LFF extrapolation. Gary (1989) outlines typical methods for LFF extrapolations, discussing their limitations and usefulness (see Section 2.2.3 for a comparison with potential and NLFF cases).

Figure 2.3: LFF extrapolated field lines with a value of  Mm, overlayed on STEREO A EUVI images of NOAA 10956. The field lines start from a user-defined footpoint, comparing the ability of three cost functions to recover the geometry of the active region loop (Conlon et al., 2010).

2.2.2 Non-Linear Force-Free Fields

If is constant along a field line, then , where . Taking the curl of this gives,

(2.27)

Using the vector identity with Equation 2.2 and the above equation gives,

(2.28)
(2.29)

Thus there are two coupled equations for B and for the case of a NLFF field, which need to be solved together. It is worth noting that for the LFF case, Equation 2.28 reduces to , known as the Helmholtz equation. There are several different approaches to a NLFF solution currently available, e.g., Grad-Rubin style current-field iteration procedures, boundary integral methods, magnetofrictional methods, and optimization approaches (see the review by Schrijver et al., 2006). Figure 2.4 shows examples of the various types of NLFF extrapolation procedures, featured in the review of De Rosa et al. (2009).

Figure 2.4: Representative field lines in the central portion of the active region for each NLFFF model listed in Table 1 of De Rosa et al. (2009). Black field lines indicate (closed) lines that intersect the lower boundary twice, and red and green field lines represent field lines that leave the box through either the sides or top, with colour indicative of polarity.

2.2.3 Comparison

Potential field extrapolations can be thought of as the first-order approximation to the coronal magnetic topology. They are often used to represent solar global magnetic fields, which are dominated by simple dipolar configurations with few currents (Liu & Lin, 2008, see Figure 2.2). In contrast, LFF field extrapolations have been considered sufficient to represent the large-scale, current-carrying coronal magnetic fields present on whole active region size scales (Gary, 1989; Wheatland, 1999, see Figure 2.3). However, NLFF field extrapolations intrinsically contain more information on the complex nature of solar magnetic fields and more accurately represent coronal magnetic structures on size scales smaller than whole active regions. For example, Wiegelmann et al. (2005) compared field extrapolations from photospheric measurements to observed chromospheric magnetic fields, finding the potential solution provided no agreement with any observed field lines, the LFF solution provided 35% agreement, while the NLFF solution provided 64% agreement. Figure 2.5 shows Figure 1 of Wiegelmann et al. (2005), which illustrates these differences.

It should be noted that there are still inconsistencies between existing NLFF extrapolation methods, particularly regarding the treatment of boundary conditions (see, e.g., the comparison paper of De Rosa et al., 2009, as in Figure 2.4). However, for the purposes of this thesis the NLFF extrapolation is considered the most accurate representation of sunspot magnetic field lines.

Figure 2.5: Observed and extrapolated fields compared by Wiegelmann et al. (2005). The upper row shows the original observed chromospheric loops, second row the potential field extrapolation, third row the LFF field extrapolation, and fourth row the NLFF field extrapolation. The bottom row shows one loop for comparison of all extrapolation types (black line: original, white: potential field, orange: LFF field, yellow: NLFF field.)

2.3 Radiative Transfer

Although analysis of the 3D structure of ARs is the ultimate goal of this thesis, first magnetic field data must be obtained to be used as inputs to the extrapolation procedures described above. The input vector magnetic field information is often obtained using radiative transfer techniques. Radiative transfer is described as the physical phenomenon of energy transfer in the form of EM radiation. The convection found below the solar surface rarely carries a significant fraction of energy flux into the photosphere, and the dominant mechanism of energy transport through surface layers of a Sun-like star is radiation (Gray, 2005). Radiative transfer theory is used to create model atmospheres that can be compared to observations. This section serves to introduce the reader to the basic theory behind radiative transfer, which will be built upon to describe specific analysis techniques in Section 3.2.

The fundamental quantity which describes a field of radiation is the specific intensity, . Consider light in a particular frequency range passing through a small area, , into a solid angle, , over a particular time, with the light beam inclined at an angle to the direction perpendicular to the area (normal axis in Figure 2.6). can be defined as the amount of energy passing through unit area (perpendicular to the beam of radiation) per unit time per unit frequency interval into unit solid angle. In other words,

(2.30)

where is the amount of energy (in units of erg) crossing the area . See Figure 2.6 for an illustrative example. The units of the spectral intensity are thus W m sr Hz. Note that if the source emits radiation equally in all directions, the radiation is said to be isotropic (as in the solar interior). In the solar atmosphere there is much more radiation coming from the direction of the Sun’s centre than from the outside direction, i.e., the radiation is anisotropic (Phillips et al., 2008).

Figure 2.6: An illustration of the geometrical portion of the definition of specific intensity. Note here that indicates the inclination angle, which is denoted by in this thesis (Gray, 2005).

2.3.1 Radiative Transfer Equation

The propagation of radiation through a medium can be affected by emission, absorption and scattering processes, and specific intensity plays a part in the definition of these processes. As a beam of radiation travels, it loses energy to the atmosphere by absorption, gains energy by atmospheric emission, and redistributes energy by scattering.

Consider a beam of radiation passing through a medium of thickness (it enters the medium at point , and leaves at ), which may add to or subtract from the radiation, as shown in Figure 2.7. The intensity of the beam, , is increased by processes of emission, and decreased by processes of absorption (including scattering). Let be the emission coefficient at frequency , such that the amount of energy added to the beam is given by,

(2.31)

Similarly, the absorption coefficient is defined as the amount of energy subtracted from the beam,

(2.32)

The change in intensity over length of the material is thus,

(2.33)

Rearranging this equation,

(2.34)

This is known as the radiative transfer equation (RTE), where is the direction of propagation (Rutten, 2003; Phillips et al., 2008). In general, the intensity decreases as the beam travels, due to the absorption of photons being greater than the emission of photons.

Figure 2.7: Radiative transfer in a medium (gray rectangular slab). Radiation enters the medium at height , and leaves it at height . Modified figure originally by Bannister (2007).

There are two physical processes contributing to : absorption where a photon is destroyed and energy ‘thermalised’, and scattering where a photon is deviated in direction and removed from the solid angle. Similarly with , there is emission with creation of photons, and scattering of photons into the direction being considered. The source function, , describes the ratio of the emission coefficient to the absorption coefficient, such that . It is a measure of how photons in a light beam are removed and replaced by new photons by the material it passes through. Equation 2.34 can thus be written,

(2.35)

The source function here can be thought of as the specific intensity ‘emitted’ at some ‘point’ in a hot gas, and has the same units as .

The RTE is commonly written in terms of the optical depth, , which is a measure of transparency, expressing the quantity of light removed from a beam by scattering or absorption during its passage through the medium. For , the plasma is optically thin (transparent), and if it is optically thick (opaque). The optical depth can be defined as , and Equation 2.35 can then be re-written as,

(2.36)

Also note that integrating gives the optical depth as .

2.3.2 Local Thermodynamic Equilibrium

A particularly useful simplification of the RTE occurs under the conditions of local thermodynamic equilibrium (LTE). In LTE, intensive parameters are varying in space and time, but are varying so slowly that, for any point, one can assume thermodynamic equilibrium (i.e., constant temperature) in some neighborhood about that point. Radiation passing through a blackbody volume of gas suffers no change with time, i.e., there is as much radiation absorbed as emitted. So at thermal equilibrium, the ratio of emissivity to absorptivity is a universal function only of radiative wavelength and temperature, i.e, a perfect blackbody emissive power (this is known as Kirchoff’s Law). Therefore, , where is the Planck function, i.e., a blackbody emits radiation in thermal equilibrium according to Planck’s Law (defined in Equation 1.6). Thus, . The RTE in Equation 2.35 can then be written as,

(2.37)

or in terms of optical depth from Equation 2.36,

(2.38)

Generally, LTE is often only applied to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas need not be in thermodynamic equilibrium with each other or with the massive particles of the gas in order for LTE to exist. LTE enables calculations of thermodynamic properties in terms of temperature, density and composition as they change from the solar centre to the surface. In the photosphere the radiation varies from place to place, but sufficiently slowly that the emitting material and radiation are close to thermodynamic equilibrium.

A number of inversion techniques use model atmospheres that assume LTE to make calculations simpler (using photospheric observations for comparison), including the Milne-Eddington approximation, which is described in Section 3.2.1.1. However first the photospheric measurements must be obtained from instruments observing the Sun. The instruments used in this thesis will be described in the next chapter, as well as describing the analysis techniques used on the observations obtained.

Chapter 3 Instrumentation and Analysis Techniques

   
In this chapter, details of the instruments and data analysis techniques used in this thesis are presented, including preparation of the data obtained from them. First some background to the physics used in the measurements is given, as well as a discussion of early observations. Instruments onboard the Hinode spacecraft are the main source of data for this thesis. The Spectropolarimeter onboard Hinode’s Solar Optical Telescope is described in detail, followed by a brief outline of the Broadband Filter Instrument. Data analysis techniques are then described, including details of the atmospheric inversion code, field azimuth disambiguation, coordinate transformations, and 3D magnetic field extrapolation methods.
   

Before the dawn of satellite instrumentation, decades of ground-based longitudinal magnetic measurements could not provide information on the true complexity of active-region magnetic topology, as these did not provide information on the 3D field. Even with good seeing and image stabilisation methods through adaptive optics, atmospheric distortions cause instruments to fall short of resolving sunspot fine structure in the photosphere (angular resolution of  km or better is needed; Lites, 2002). The polarimetric signals from ground-based instruments must also be accumulated over at least a number of seconds in order to achieve the required precision. High spatial resolution observations of the solar magnetic field from spacecraft can now provide this information.

3.1 Solar Magnetic Field Observations

3.1.1 Spectropolarimetry

Spectropolarimetry is used for the investigation of the magnetic field structure of sunspots in this thesis. Its basis comes from spectroscopy and polarimetry. Spectroscopy aims at measuring the flux distribution of a light source as a function of the radiation wavelength (spectra). Emission and absorption lines result from radiative transitions through absorption, emission, recombination and de-excitation of photons, with different elements producing different spectra according to their differing atomic structure. The Sun radiates light across a broad range of the EM spectrum. In the infra-red and visible wavelengths, the solar spectrum is a continuous background crossed by absorption lines, with the greatest intensity found in the visible range. At wavelengths  Å (extreme ultraviolet), the solar spectrum is in the form of emission lines with a relatively weak continuum (broadband EM radiation).

Polarimetry measures the degree to which radiation from a light source is polarised, as well as the polarisation state of the corresponding light. Polarisation is a property of an EM wave describing the orientation of the electric field oscillations. Unpolarised light is observed when segments of light waves occur along the LOS with randomly oriented planes. Linearly polarised light is a confinement of the EM field vector to a given plane along the direction of propagation. Circularly polarised light occurs, for example, when two linearly polarised rays are a quarter of a wavelength out of phase, so that the vector sum of the wave motions rotates as the wave propagates. Figure 3.1 depicts the various types of polarised light, with polarisation being a property of waves describing the orientation of their oscillations.

Figure 3.1: Depiction of the various types of polarisation of light (Phillips, 1992). (a) Linearly polarised light, a confinement of the electric magnetic field vector to a given plane along the direction of propagation. (b) Unpolarised light, segments of light waves occur along line of sight with randomly oriented planes. (c) Circularly polarised light, when two linearly polarised rays are a quarter of a wavelength out of phase, so that the vector sum of the wave motions rotates as the wave propagates.

The Stokes parameters are quantities used to describe the polarisation state of light (Stokes, 1852). The four parameters I, Q, U, and V completely describe polarisation: I is the total unpolarised intensity; Q and U are the components of linear polarisation (where the frames of Q and U are rotated by 45); V is the degree of circular polarisation. In other words,

(3.1)
(3.2)
(3.3)
(3.4)

where and represent right and left circularly polarised light, and represent linearly polarised light in the and directions, and and represent polarisation in the and directions. In a spectral line, the state of polarisation varies as a function of wavelength, so the Stokes parameters are a function of wavelength, i.e., I(), Q(), U(), V().

Figure 3.2 shows examples of Stokes I, Q, U, and V profiles under various circumstances. In the upper left panel, magnetic field strength, inclination angle, and azimuthal angle are all zero. This leads to only a Stokes I profile observed. When the magnetic field strength, , is increased along the LOS only (i.e. inclination, , and azimuthal, , angles are still both zero), the V profile is also observed. The upper right panel shows this for a field strength of 1500 G, with a typical two-lobe structure for Stokes V and the I profile beginning to split due to the presence of the magnetic field. When field inclination is at (and still a 1500 G field), as in the lower left panel, Stokes V disappears, however a typical Stokes Q profile is now observed. Finally, when the azimuthal angle is at (with field strength at 1500 G and inclination at ), the Stokes Q profile disappears and only U profile is observed along with the total intensity I. The two-lobe structure of the V profile, and three-lobe structures of the Q and U profiles can be explained by the Zeeman effect, which will be discussed in the next section.

Figure 3.2: The change of the synthetic emergent Stokes I, Q, U, and V profiles when the magnetic field present in the solar plasma varies. This is depicted in terms of changes in the magnetic field strength (), inclination of the magnetic field vector with respect to the observer’s LOS (), and azimuth of the magnetic field vector in the plane perpendicular to the observer’s LOS (). Modified from a movie originally by Borrero & Ichimoto (2011).

3.1.2 Zeeman Effect

The strong magnetic fields of sunspots can result in large splittings of atomic levels. The Zeeman effect can be defined as the splitting of a spectral line into several components in the presence of a static magnetic field. Atomic and molecular energy levels split into sublevels, which are characterised by their magnetic quantum number, (Condon & Shortley, 1935). More specifically, the presence of the magnetic field breaks degeneracy and an atomic level of total angular momentum J is split into (2J + 1) magnetic sublevels, where ( is orbital angular momentum and is spin angular momentum). For the normal Zeeman effect111The anomalous Zeeman effect appears for transitions where the net spin of the electrons is non-zero, but is not of concern here. (zero spin), the magnetic quantum number selection rule simplifies the available transitions into three allowed states: (