Imaging spectropolarimetry of solar active regions Gautam Narayan

Transcription

1 Imaging spectropolarimetry of solar active regions Gautam Narayan Department of Astronomy Stockholm University

2 Cover image: Small-scale solar convection in strong magnetic field. The image shows a line-of-sight velocity map obtained from Milne-Eddington inversions of data recorded on 26 May 2008 with CRISP at the Swedish 1-m Solar Telescope. c Gautam Narayan, Stockholm 2011 ISBN Universitetsservice, US-AB, Stockholm 2011 Department of Astronomy, Stockholm University

3 Doctoral Dissertation 2011 Department of Astronomy Stockholm University SE Stockholm Abstract Solar magnetic fields span a wide range of spatial scales from sunspots and plages to magnetic bright points. A clear understanding of the physical processes underlying the evolution of these magnetic features requires high-resolution spectropolarimetric observations of solar active regions and comparisons with synthetic data from simulations. This thesis is based on observations with the Swedish 1-m Solar Telescope (SST) and the CRISP imaging spectropolarimeter which, processed with a sophisticated image restoration technique, produce data of unsurpassed quality. The Fe I nm line is used for all the spectropolarimetric observations. It appears likely that present telescopes resolve the fundamental scales of penumbral filaments. However, the penumbrae of sunspots are still not fully understood, with various theoretical models competing to explain their fine structure and flows. We analyze spectropolarimetric observations with a resolution close to the SST diffraction limit of 0.16 arcsecond. Using inversion techniques, we map the line-of-sight velocities and the magnetic-field configuration of dark-cored penumbral filaments. Over the past decade, sunspots and quiet sun magnetic fields have received considerable attention, with intermediate plage regions being somewhat neglected. We perform a detailed analysis of a plage region and present the first observational evidence of a small-scale granular magneto-convection pattern associated with a plage region. Magnetic bright points are believed to be formed due to magnetic field intensification caused by flux-tube collapse involving strong downflows. Although magneto-hydrodynamic (MHD) simulations agree with this view, only a few observations with adequate spatial resolution exist in support of the simulations. We present several cases of bright-point formation associated with strong downflows, which qualitatively agree with simulations and past observations. However, we find the field intensification to be transient rather than permanent.

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5 Inmemoryofmygrandmother,Amma

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7 Preface This thesis presents a study of solar active regions. It is based on the following scientific publications: I CRISP spectropolarimetric imaging of penumbral fine structure Scharmer, G. B., Narayan, G., Hillberg, T., de la Cruz Rodriguez, J., Löfdahl, M. G., Kiselman, D., Sütterlin, P., van Noort, M., Lagg, A., 2008, Astrophysical Journal, 689, L69 II Small-scale convection signatures associated with a strong plage solar magnetic field Narayan, G., & Scharmer, G. B. 2010, Astronomy & Astrophysics, 524, A3 III Transient downflows associated with the intensification of small-scale magnetic features and bright point formation Narayan, G., 2011, accepted for publication in Astronomy & Astrophysics. The articles will be referred to in the text by their Roman numerals. For the papers of which I am not the sole author, my contributions are as follows: In Paper I, I carried out the inversions and participated in the analysis. In Paper II, I reduced the data, carried out the inversions, made all the figures, and wrote the initial version of the paper which was extended by my co-author.

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9 Contents 1 Introduction The Sun Photospheric magnetic fields Sunspots Penumbral models Flux-tube-based penumbral models Convective penumbral models Comparison of the penumbral models Flux emergence Small scale solar magnetic fields Faculae Photospheric magnetic field diagnostics Spectral lines Zeeman effect Stokes parameters Polarized radiative transfer Polarimetric Analysis Inversion techniques Helix Additional techniques used Straylight correction Polarimetry with SST Instrumentation SST and CRISP Data Reduction Polarimetric calibration Flat-fielding and dark subtraction MOMFBD Cavity map correction Summary of papers 41 Acknowledgements 43

10 Bibliography 47

11 List of Figures 1.1 Solar structure A light bridge Dark-cored penumbral filaments Evershed effect The uncombed penumbral model Gappy model and striations A flux emergence event Emerging flux in simulations Small scale features An active region with faculae The physics of faculae Reference geometry for the radiative transfer equation Steps involved in a Stokes inversion technique Essence of genetic algorithms Observed vs fitted profiles without stray-light correction Observed vs fitted profiles with stray-light correction The CRISP Cavity map correction

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13 1 1 Introduction 1.1 The Sun In elementary terms, the Sun is the star around which the planets in our solar system revolve. It is a uniform spherical object, with a mass of about kilograms and a radius of around kilometers. It is composed mainly of hydrogen, helium and minute quantities of heavier elements like carbon, oxygen, iron, and others. Magnetic fields on the Sun are ubiquitous. Regions in the solar atmosphere, where the magnetic field is strong, are known as solar active regions. The Sun can be divided into several layers defined by the dominant physical processes as illustrated in Fig Figure 1.1: Solar structure. Image source: NASA Core: This is the innermost layer, where the Sun produces its energy by nuclear fusion.

14 2 Introduction Radiative zone: In this zone heat is transported from the core to the outer layers by radiation. Convective zone: In this zone heat is transported to the solar surface by convection. The convective processes overshoot and leave imprints on the solar surface in the form of granulation, mesogranulation and super-granulation. Photosphere: This is the visible layer of the Sun, also known as the solar surface. The photosphere offers us the opportunity to observe manifestations of magnetic fields, convection and the interaction between them. The photosphere is approximately 500 kilometers thick. Chromosphere: This is a thin layer of the Sun s atmosphere just above the photosphere, roughly 2,000 kilometers thick. Corona: This is separated from the chromosphere by a thin layer called the transition region. The corona extends all the way from the transition region and far out into the solar system where it eventually turns into the solar wind. 1.2 Photospheric magnetic fields The photosphere is the visible surface of the Sun. In the absence of strong magnetic fields, granulation is the dominant feature in the photosphere. Granulation is a pattern consisting of up-flowing hot matter, making up the bright center parts of granules, and cool downflowing matter surrounding the granules, visible as dark intergranular lanes. Each granule is roughly 1000 kilometers in diameter. Magnetic flux in the solar photosphere is highly fragmented, occurring in a large range of spatial scales from km to 100 km and below. On the top of the scale are large sunspots, while small-scale magnetic features like micropores and magnetic bright points are at the bottom of the scale. Features like pores lie in the intermediary scale. The interaction of magnetic fields and convection lead to several of interesting and important processes in the Sun: 1. Formation of strong field structures, 2. Modification of convection by strong magnetic field, 3. Global luminosity variations due to local heat flux deficits (sunspots) or heat flux excesses (plages, faculae, network), 4. Transport of energy to the chromosphere and corona.

15 1.2 Photospheric magnetic fields Sunspots Sunspots are the most prominent of the large scale magnetic features observed in the photosphere. They form in active regions, which typically have a bipolar magnetic field configuration. A typical sunspot diameter is that of 30 granules (30000 km) and has a lifetime of around a few weeks (Schlichenmaier 2009). Sunspots usually comprise a dark part called the umbra and a surrounding filamentary part called the penumbra. Sunspots without any penumbra are referred to as pores. The magnetic field strength in sunspots varies between G in the darkest umbral regions and between G in the outermost penumbral regions (Solanki 2003). The umbra appears dark because it is around K cooler than the quiet Sun. This is because the strong magnetic field suppresses convection. Umbral fine-structure consists of bright umbral dots and light bridges. Light bridges are bright lanes which divide the umbra into two or more parts. Dark lanes have been observed both in umbral dots (Bharti et al. 2007) and lightbridges (Lites et al. 2004; Scharmer et al. 2007). Figure 1.2 shows a light bridge with a dark core. Penumbral fine-structure mostly consists of penumbral filaments with or without dark cores and penumbral grains. While it is well known that sunspots are manifestations of magnetic fields, not much is known about the exact nature of the sub-surface field. Two models describing the structure of sunspot magnetic fields are the monolithic model and the jellyfish model. The monolithic model describes a sunspot as a single flux tube, extending from the upper photosphere to the deeper layers of the photosphere. The jellyfish model is described by Parker (1979) as a model in which the photospheric field is monolithic but the sub-photospheric field is fragmented into smaller flux tubes due to the interchange instability. A consequence of this model is the presence of field free gaps. Intrusion of field free plasma from the convection zone below can occur within these gaps. Umbral dots are thought to be signatures of such a process. The field free gaps are narrow and the convection is thus confined by a strong surrounding field leading to narrowing of convection upwards. This explains the tiny brightenings in the form of umbral dots which are seen in observations. Recent simulations of Schüssler & Vögler (2006) demonstrate the formation of umbral dots in the form of narrow upflow plumes that become almost field-free near the surface layers. These simulated umbral dots also have central dark lanes. The upflow plumes have a cusp-like shape in their top parts. The upflow plumes and umbral dots in these simulations form naturally in a strong vertical magnetic field, this is radiatively driven magnetoconvection and not due to intrusion of field-free plasma from the convection zone as in the jellyfish model. Lightbridges are also manifestations of field-free convection according to simulations of Nordlund (2006). Convective heat transport pushes open and

16 4 Introduction Figure 1.2: The figure shows a sunspot recorded at nm with the SOUP filter on 15 Aug It is a Stokes I image at 50 pm from the line center. The axes are in units of arcseconds. The light bridge is the feature within the rectangle. maintains a gap with a strongly reduced field strength. Dark cores associated with the lightbridges also formed in the simulations Penumbral models The most crucial problem associated with sunspot penumbrae (and umbrae) is that of the heat flux. The brightness of penumbrae is around 75% of the quiet sun solar granulation and that of the umbra is 20% (Spruit & Scharmer 2006). This brightness of sunspots can neither be explained by radiative transport nor by heat conduction (Schlichenmaier 2009). That leaves us with convection, which has not been fully suppressed by the strong magnetic fields, to account for the heat transportation in sunspots.

17 1.2 Photospheric magnetic fields 5 The key to understanding penumbral fine structure lies in measuring their spectro-polarimetric signatures in order to derive their physical properties like magnetic fields and velocities. This requires high resolution spectro-polarimetric observations combined with sophisticated techniques like adaptive optics and image restoration. An important step forward was taken with the commissioning of SST in the year 2002, which turned out to be a watershed year for sunspot physics. During this year dark-cored penumbral filaments (Scharmer et al. 2002) were discovered using the SST. Figure 1.3 shows dark cored penumbral filaments in a sunspot. Knowledge of the characteristics of the velocity field and the magnetic field is essential for understanding the nature of the penumbral fine structure. Flows related to sunspots were first discovered by Evershed (1909) using the Kodaikanal solar telescope in India. Evershed found a radial, predominantly horizontal, flow of plasma that increased in strength towards the outward penumbra. This is referred to as the Evershed flow. An observational signature of the Evershed effect is the presence of redshift in the limb facing side of the penumbra and a blue shift in the disk center facing side of the penumbra. This is demonstrated in Fig An important quantity associated with the magnetic field topology is the net circular polarization (NCP) that has been observed in sunspots. NCP is related to the asymmetry of the Stokes V profile. See Sect. 3.3 for details about Stokes V asymmetries. The NCP links the flow field with the magnetic field (Schlichenmaier 2009) because the asymmetry of the observed Stokes V profile can be explained by the presence of co-spatial gradients of the magnetic vector and the line-of-sight (LOS) velocity. Penumbral models try to explain not only the penumbral fine structure observed in temperature encoded as intensity, but also the associated physical properties like the magnetic field and flow field. Models trying to explain the penumbra are either based on flux tubes or convection Flux-tube-based penumbral models Uncombed penumbra model: This model was put forward by Solanki & Montavon (1993) to explain the observed NCP in sunspots. Sanchez Almeida & Lites (1992) showed the presence of strongly asymmetric Stokes V profiles observed on the limb side penumbra of sunspots far from disk center and interpreted the asymmetries as strong gradients in penumbral magnetic field inclination along the LOS. Solanki et al. (1993) explained that such large magnitude of gradients implied presence of strong curvature forces that would destroy the sunspot. Solanki & Montavon (1993) came up with a two component interlocked or uncombed model to overcome this problem. The fundamental idea behind this model is that penumbral filaments are manifestations of nearly horizontal flux tubes embedded in a vertical magnetic

18 6 Introduction Figure 1.3: The top panel shows a nm image of sunspot recorded with the SOUP filter on 04 Jul It is a Stokes image at +50 pm from the line center. The bottom panel is a zoom in of the rectangular region marked in the top panel and shows dark-cored penumbral filaments in more detail. The axes are in units of arcseconds.

19 1.2 Photospheric magnetic fields 7 N DC (b) Figure 1.4: A Dopplergram showing the Evershed effect: blue shift in the disk center side and red shift in the limb facing side implies radial outflow in the penumbra. DC disk center, N solar north. Figure reproduced from Langhans et al. (2005) Figure 1.5: An illustration of the uncombed penumbral model showing a horizontal field (shown as dark flux tubes) embedded in a more vertical magnetic field (shown as gray flux tubes). Figure kindly provided by Nic Brummell. field. This is illustrated in Fig Thus this model avoids strong curvature due to the smooth gradients in the inclinations and at the same time explains

20 8 Introduction the observed NCP due to a strong flow parallel to the horizontal component of the magnetic field. Siphon flow model: This model was first proposed by Meyer & Schmidt (1968) to explain the Evershed effect in sunspot penumbrae and was further explored by Montesinos & Thomas (1997). The model explains the Evershed effect as a quasi-stationary flow along a flux tube due to the pressure gradient between the foot points of a flux tube. The presence of the pressure gradient is attributed to a difference in the magnetic field strengths of the foot points, which is an inherent assumption of the model. Dynamic flux tube model: This dynamic model was developed by Schlichenmaier et al. (1998a,b) inspired by a static model of Jahn & Schmidt (1994). In the Jahn & Schmidt (1994) model, the excess brightness of the penumbra is explained by heat exchange between the penumbra and the neighboring quiet sun, combined with energy transport through convection by an interchange of magnetic flux tubes. Schlichenmaier et al. (1998a,b) simulated the evolution of a penumbral filament as a single thin flux tube, initially located along the magnetopause. However, instead of interchange convection they found an alternate method of heat transfer where the flux tube is radiatively heated by the adjacent fieldfree atmosphere and rises through the sub-photospheric layers. A hot upflow is formed within the tube, leading to a decrease in its magnetic field strength. The upflow cools radiatively and starts to move radially outwards. This radial flow is interpreted as the Evershed flow. In later, more refined simulations of the moving flux tube, Schlichenmaier (2002, 2003) found new effects associated with the overshooting of the upflow formed inside the flux tube due to the magnetic field decreasing during the evolution. The overshoot leads to oscillations, which give a serpentine nature to the flows. Downward pumping model: This model is also based on the siphon flow model (Meyer & Schmidt 1968; Montesinos & Thomas 1997). A major limitation of the siphon flow model was the inability of the flux tubes to be submerged due to the magnetic buoyancy forces and the magnetic curvature forces. Thomas et al. (2002) and Weiss et al. (2004) proposed a mechanism, in which downward pumping of magnetic flux by the small-scale, turbulent granular convection that surrounds a sunspot, keeps the flux tube submerged. They explain the filamentary structure of sunspot penumbrae to be the result of this downward pumping.

21 1.2 Photospheric magnetic fields Convective penumbral models Convective roll model: In this model, proposed by Danielson (1961), a penumbral filament is assumed to be made up of two radially aligned rolls rotating in opposite directions. This leads to downflows in the sides and an upflow in the top of the filament. The rolls are assumed to be elongated convection cells, aligned to the horizontal component of the sunspot magnetic field. The Gappy penumbra model: In this model, proposed by Spruit & Scharmer (2006); Scharmer & Spruit (2006), the origin of penumbral filaments is interpreted as due to convection in field-free, radially aligned gaps just below the visible surface of the penumbra, intruding into a nearly potential field above. This is illustrated in Fig Overturning convection within the gaps is responsible for energy transportation to the surface. A radially outward flow equivalent to the horizontal component of the convection is interpreted as the Evershed flow in the gappy penumbra model (Scharmer et al. 2008b). Figure 1.6: An illustration depicting the gappy model as a convecting gap in the penumbral magnetic field, with the umbra assumed to be towards the right. Colors represent temperature, with red corresponding to a lower temperature and yellow corresponding to a higher temperature. The dark core is shown as the top part of gap. The blue lines represent the magnetic field lines wrapping around the gap. Striations associated with the field lines are also shown. Figure reproduced from Spruit et al. (2010). The model explains the dark cores in penumbral filaments as an opacity effect, corresponding to the top of a nearly field-free cusp, which outlines a region of higher gas pressure together with an overall drop in temperature with

22 10 Introduction height. The folding of field lines over the gap explains the observed strong inclinations and related asymmetries Comparison of the penumbral models Uncombed penumbra model: Support for this model mainly comes from two-component inversions performed on spectropolarimetric data at low spatial resolution close to 1 arcsec (Bellot Rubio et al. 2003, 2004; Borrero et al. 2004, 2005, 2006), which reproduced better the observed Stokes profiles than one-component inversions. However, the issue here is the spatial resolution of such observations, that prevents a detailed and accurate inference of relevant properties from the Stokes profiles. This is supported by conflicting interpretations from observations of a similar nature, for example Westendorp Plaza et al. (2001b,a) who found that magnetic field strength increases with height in the atmosphere and Martínez Pillet (2000) who demonstrated that the opposite interpretation was possible. Spruit & Scharmer (2006); Scharmer & Spruit (2006) point out that the model proposed by Solanki & Montavon (1993) involves flux tubes with circular cross-sections in which the internal field lines are aligned with the flux tube and the external field lines are wrapped around it and such a configuration is magnetostatically unstable. In such a model the surrounding magnetic field should vanish at the top and bottom and the magnetic field in the sides will increase. This would create compressional forces on the flux-tube which will destroy it within 10 seconds. Scharmer (2009) has pointed out a list of other shortcomings concerning flux tube models: 1. The models assume the presence of a flux tube. 2. They are 1-dimensional. 3. The curvature forces of the surrounding magnetic field are ignored. 4. Only a single flux tube is simulated and influence of neighboring flux tubes is ignored. Dynamic flux tube model: This model explains certain characteristics of penumbral observations, namely that of inward and outward moving penumbral grains (Schlichenmaier 2002), strong upflows in bright grains (Rimmele & Marino 2006), the behavior of moving magnetic features (Sainz Dalda & Bellot Rubio 2008). It is consistent with the two-component model and produces realistic NCP maps (Schlichenmaier 2009). However, since this model is based on a single thin flux tube, the problems associated with the uncombed penumbra model listed above applies equally to this model. Another problem with this model is its inability to fully address the penumbral heat transport problem.

23 1.2 Photospheric magnetic fields 11 Downward pumping model: Observations show that in the outermost parts of the penumbra the field is almost horizontal. Borrero & Solanki (2008) found evidence of field lines in the outer penumbra dipping down into the photosphere. This gives support to the downward pumping model. However, this model fails to explain the large azimuths variations in inclinations seen in the inner penumbra. Convective roll model: Zakharov et al. (2008) found evidence for apparent lateral motions of bright and dark structures inside bright filaments in penumbral filaments in SST observations of a sunspot close to the solar limb. They interpreted this as observation of convective rolls. Similar observations with the Hinode spacecraft had earlier been presented by Ichimoto et al. (2007). Zakharov et al. (2008) demonstrated that the observed motion in penumbral filaments could not be related to motion of magnetic fields but was due to convective processes and concluded that they had found evidence for the convective roll model of Danielson (1961) to be the source of heat flux of the penumbra. Scharmer (2008, 2009) pointed out that closed rolls can sustain the energy content of bright filaments for only a few minutes, but the observed life time of bright filaments is of the order of one hour or more as shown in the observations of Langhans et al. (2007). The gappy penumbra model: Scharmer (2008, 2009) interpreted the apparent twisting motions of penumbral filaments as seen in observations (Zakharov et al. 2008; Ichimoto et al. 2007) as evidence for overturning convection in field free gaps. Spruit et al. (2010) demonstrated that the apparent twisting motions are connected to striations observed in penumbral filaments. They explained the origin of striations as a corrugation of the boundary between the overturning convection and the surrounding field lines (See Fig. 1.6). This corrugation effect is also seen in the small scale solar magnetic features (faculae) (Carlsson et al. 2004). Based on 3-D radiative magneto-hydrodynamic simulations, Scharmer et al. (2008b) concluded that the Evershed flow infact represents the horizontal component of penumbral convection. This has been supported with separate simulations of a full sunspot (Rempel et al. 2009; Rempel 2011). In the gappy model, the dark cores are also locations of strongly inclined fields due the folding of the near-potential magnetic field over the gap. In Paper I we investigate the variations of field-strength and inclination angles over rudimentary dark-cored penumbral filaments in a large pore. The gappy model predicts the presence of upflows as well as downflows in the interior penumbra. So far, no clear detections of such convective flows in observations have been published.

24 12 Introduction Flux emergence The emergence of magnetic flux into the solar atmosphere from regions beneath the solar surface is an important process in the production of structures and dynamics observed in the solar photosphere. Understanding of the flux emergence process is crucial in understanding the connection between processes occurring in the convection zone like the dynamo process and the magnetic features in the photosphere. Such a process is driven by buoyancy and advection. Magnetic flux concentrations have lower densities in order to maintain the pressure equilibrium with the surroundings and hence are buoyant (Nordlund et al. 2009). They are also advected by convective upflows. Figure 1.7: A time sequence of snapshots of the line-of-sight magnetic field. The time is increasing from left to right and top to bottom. The flux emergence event is characterized by an expanding cloud of opposite polarity signals. The observations were recorded on 27 May 2008 with CRISP. The field of view is approximately

25 1.2 Photospheric magnetic fields 13 The rising flux also interacts with convective downflows leading to formation of Ω loops. Downflows being stronger than upflows leads to an asymmetry in the flux transportation with downward transportation dominating. This is referred to as magnetic flux pumping. Fig. 1.7 depicts a flux emergence event in which a cloud of opposite polarity magnetic fields emerges and slowly expands. A striking feature is that there are many smaller mixed polarity fields within the emerging flux region which form later. Similar trends have been seen in simulations of Cheung et al. (2008), who explain this as the action of convective motions on field lines, distorting them into a serpentine shape which explains the multitude of mixed polarity signals. Fig. 1.8 shows an emerging flux event from the simulations of Cheung et al. (2008). Our observations appear quite similar to the simulations. Figure 1.8: Snapshots of the vertical magnetic field from two time steps (increasing left to right) from the simulations of Cheung et al. (2008). Reproduced by permission of the AAS and Mark Cheung Small scale solar magnetic fields Small-scale solar magnetic fields produce features of similar spatial scales as normal granulation or smaller. These features are most often found in intergranular lanes. Some small-scale magnetic features appear dark in continuum images, like micropores and protopores, while some appear bright. Bright magnetic features are referred to by a wide variety of names such as network bright points, magnetic bright points, facular points, internetwork bright points, G-band bright points, filigree, etc. There are subtle differences between the various bright magnetic features, depending on in which wavelength and where on the solar surface they are observed. While facular points refer to photospheric bright points in active regions, network bright points are bright points found in boundaries of supergranular cells. Bright points often form interconnected linear structures filling the intergranular lanes. Such structures are known as filigree. High spatial

26 14 Introduction resolution observations recorded with SST have resolved the small bright features that make up filigree in plage regions and they have been identified and named as ribbons, flowers and strings (Berger et al. 2004; Rouppe van der Voort et al. 2005). Figure 1.9 shows some examples of these features. The granules in the vicinity of small scale magnetic features have a lower contrast and are smaller. This type of granulation is called abnormal granulation (Dunn & Zirker 1973; Title et al. 1992). Such granulation is clearly seen in the continuum map of Fig A review of small-scale magnetic features is given by Solanki (1993). Larger structures like sunspots have been observed to decay into smallscale magnetic features and features like pores have been observed to form by coagulation of several small-scale magnetic features. This suggests that there is perhaps a fundamental building block of magnetic flux and since 90% of the flux outside of sunspots was found to be in a strong field state this led to the development of the thin flux tube model of photospheric magnetic fields. In this model a magnetic feature is assumed to be an isolated, vertically oriented flux tube which is in magneto-hydrostatic equilibrium with the surrounding non-magnetic atmosphere. The gas pressure and the magnetic pressure within the tube together balance the gas pressure of the surroundings. The formation of small scale magnetic features can be understood as a two step process. In the first step the magnetic flux is swept into intergranular lanes and is concentrated due to horizontally diverging granular flows. This process is referred to as flux expulsion. There is a limit up to which the magnetic flux can be compressed by the flux expulsion process. This limit is set by the equipartition value of 500 G (Takeuchi 1999). The limit exists because the Lorentz force prevents further increase of the flux concentration by opposing the surrounding convective motions. This causes a reduction in the horizontal heat transfer into the flux tube from surrounding granules. The energy imbalance caused due to the lowering of horizontal heat flux (from the surroundings) and radiative losses at the surface leads to the flux tube becoming cooler compared to its surroundings. The cooling leads to a reduced gas pressure within the flux tube causing the downflow in the flux tube to accelerate, leading to the superadiabatic effect (Parker 1978). This leads to an increase of the magnetic field strength of the flux tube due to its partial evacuation. Partial evacuation causes a depression of the τ = 1 surface (Wilson depression), which means one observes deeper down in the flux tube and since it is hotter at the flux tube bottom it appears brighter The super adiabatic effect induces a convective instability in the flux tube. The convective instability can be in the form of a monotonically increasing upflow or downflow (Schüssler 1990). In the case of a downflow instability, the already existing downflow is enhanced leading to further intensification of the magnetic field and a stable final state. This is known as convective collapse. The observational signature of a convective collapse process is the presence of strong downflows within the fluxtube, accompanied by the intensification

27 O1 B0 25 O0 O2 B1 20 S0 F3 R3 S1 O3 R2 15 B3 S2 10 F2 P S3 R0 F1 B2 F0 5 M R O1 B0 25 O0 O2 B1 20 S0 F3 R3 S1 O3 R2 15 B3 S2 10 F2 P S3 R0 F1 B2 F0 5 M R Figure 1.9: Small scale features. Ribbons (marked R1, R2, etc.), flowers (F1, F2, etc.), strings (S1, S2, etc.), Micropore (M), and protopore (P). Left:restored CRISP continuum image at Fe I nm; Right: Fe I nm line minimum map. The observations were recorded on 26 May 2008 with CRISP. The axes are in units of arcseconds. Reproduced from Paper II. 1.2 Photospheric magnetic fields 15

28 16 Introduction of its magnetic field strength. In the case of an upflow instability, the flux tube is dispersed due to the field strength getting weaker and the instability getting stronger. This is sometimes referred to as reverse convective collapse. Convective collapse produces kilogauss (kg) strength flux tubes. Observations also show the presence of sub-kg features. These are thought to be smaller features. It has been shown that the convective collapse of smaller flux tubes is less effective due to efficient radiative energy exchange of smaller flux tubes with the surroundings (Venkatakrishnan 1986). Simulations have confirmed both these scenarios (Grossmann-Doerth et al. 1998; Danilovic et al. 2010). The destruction of small-scale magnetic features is not yet fully understood. Three processes which could possibly lead to their dissipation are: 1. Interchange instability (Steiner 1999), 2. Reconnection with magnetic fields of opposite polarity, 3. Reverse convective collapse. In Paper II we analyze the properties of the magnetic field and LOS velocity associated with a solar plage region. We also identify the various small scale magnetic features which make up the extended strong field region and study their magnetic field and LOS velocity signatures. In Paper III we present eight cases of formation of small bright magnetic features accompanied by strong transient downflows and a simultaneous transient field intensification. We also investigate possible reconnection events in three of the cases Faculae In observations away from the disk center towards the solar limb, small-scale magnetic flux concentrations show an increase in contrast against the quiet sun background and become apparent in white light as solar faculae. Figure 1.10 shows an example of such an observation. The standard facular model also consists of a flux tube in a magnetohydrostatic equilibrium with a field-free surrounding atmosphere. Such a flux-tube has a lower density and thus a lower opacity. Towards the limb, where the surface is viewed at an angle, one sees the hot granule wall behind the tube. This is known as the hot-wall effect, explained in Fig Looking at Fig 1.10, it is obvious that faculae are actually partially bright granules. There is also a dark lane centerward of the faculae, explained by the cool material inside the flux tube. The observations also show striations in the faculae. These can be explained by the variation of magnetic field strength of flux tubes which cause changes in the density and opacity leading to observed striations. High-resolution observations of solar faculae show that they have an asymmetric continuum intensity profile with a steep increase in contrast on the disk center side and a gradual decrease towards the limb. The wide contrast profiles cannot be explained with only the hot wall effect. Simulations of Keller et al. (2004), Carlsson et al. (2004), and Steiner (2005)

29 1.2 Photospheric magnetic fields 17 Figure 1.10: An SST wide band image at nm close to limb showing an active region with faculae (bright regions). The data was recorded on 26 June 2006 with the SOUP filter. Axes are in units of pixels with a pixel scale of /pixel. Figure 1.11: A cartoon explaining the limb brightening of faculae (Keller et al. 2004). Reproduced by permission of the AAS and Christoph Keller.

30 18 Introduction reproduce qualitatively the contrast profiles of observations. The conclusion from the simulations is that limbward extension of brightness occurs from seeing the granule behind the facular magnetic flux concentration. Berger et al. (2004) have provided observational evidence supporting this.

31 19 2 Photospheric magnetic field diagnostics 2.1 Spectral lines Spectral lines are formed as a result of perturbation of an atom resulting in transitions of electrons in the atom from one energy level to another. Spectral lines are used to deduce the physical properties of the atmosphere which is emitting the radiation. This is done by studying the strength and shape of the spectral lines originating from known transitions in known atoms. Diagnosing magnetic field with some precision also requires measurements of polarization. 2.2 Zeeman effect When atoms are in the presence of a magnetic field, the resulting spectral lines are split into differently polarized components. This effect was discovered in 1896 by the Dutch physicist Pieter Zeeman and is known as the Zeeman effect. In 1908, George Ellery Hale found sunspots to be associated with strong magnetic fields by measuring the widening and splitting in the observed sunspot spectra. This was the first detection of magnetic fields outside of the Earth. The theory behind the Zeeman effect has a quantum mechanical origin. In quantum mechanics, the state of the atom is defined by four quantum numbers L,S,J, and M J. L represents the orbital angular momentum of the electrons, S represents the spin, J represents the total angular momentum, and M J represents the magnetic quantum number, which determines the total angular momentum in any one direction. This direction corresponds to the direction of the external magnetic field. M J can take values between J and J. J can take values between L S and L+S. When there is no external magnetic field (B = 0) then all M J states are identical in energy. If B 0 then there is a shift in energy and a consequent Zeeman splitting of the spectral line. This splitting ( λ B ) is given by λ B = λ λ 0 = e 4πc 2 m e g λ 2 0 B (2.1) where λ 0 is the original position or the central wavelength, λ is the new position, c is the speed of light, B = B is the magnetic field, m e is the mass of the

32 20 Photospheric magnetic field diagnostics electron, and e is the electric charge. g is the g factor defined as g = g l M l g u M u (2.2) where g l, g u are the Landé factors and M l, M u are the magnetic quantum numbers of the lower and upper transition states, respectively. The Landé factor g k is defined in the LS (Russell-Saunders) coupling scheme with quantum numbers J k, L k, S k as g k = S k(s k + 1) L k (L k + 1), (2.3) 2J k (J k + 1) where k = (u,l) denotes the upper and lower levels. The splitting of an atomic spectral line into three components is called normal Zeeman splitting. This situation occurs rarely, for example when S = 0 and the Landé factor is unity. If the condition S = 0 is true for both upper and lower transitions, then from Eq. (2.2) we have g = 1,0,1. The three components are referred to as a triplet. The Zeeman triplet consists of two shifted σ components and an unshifted π component. When one observes such that the magnetic field B is oriented towards the LOS then the observer sees only the σ components which are circularly polarized in opposite directions with respect to each other. This is referred to as the longitudinal Zeeman effect. When observing in a direction perpendicular to the field B, the observer sees all the three components. The π component is linearly polarized parallel to B and the two σ components are linearly polarized perpendicular to B. This is known as the transverse Zeeman effect. More commonly, we have the anomalous Zeeman splitting with multiplets instead of triplets. However, due to the broad nature of solar lines, multiplets are unresolved. It is common to treat a multiplet as a triplet in the weak field regime (Stix 2004). For anomalous splitting an effective Landé factor can be computed as g eff = 1 2 (g l + g u )+ 1 4 (g l g u )(J l (J l + 1) J u (J u + 1)). (2.4) The Fe I nm line used for observations in this thesis has an effective Landé factor, g eff = Stokes parameters An unpolarized monochromatic beam can be described by just an intensity I. If the beam of radiation is polarized, then we need three more quantities to describe it completely. Q and U denote the amounts of light linearly polarized along the x- and y- axes, respectively, while V denotes the amount of circularly polarized light.

33 2.4 Polarized radiative transfer 21 These are known as the Stokes parameters. For a mathematical definition we describe a single monochromatic light wave propagating along the z direction with the following equations: E x = ξ x cos(ωt φ x ) E y = ξ y cos(ωt φ y ) (2.5) where ξ x and ξ y are the amplitudes and φ x φ y are the phase differences and ω is the angular frequency. For a fully polarized wave, the four Stokes parameters are then defined as I = ξ 2 x + ξ 2 y Q = ξ 2 x ξ 2 y U = 2ξ x ξ y cos(φ x φ y ) V = 2ξ x ξ y sin(φ x φ y ), (2.6) with I 2 = Q 2 +U 2 +V 2. Natural light is never perfectly monochromatic and tends to consist of many superimposed independent wave trains, each wave train having its own amplitude and phase. For such radiation the Stokes parameters are temporal averages, I = ξx 2 + ξy 2 Q = ξx 2 ξy 2 U = 2 ξ x ξ y cos(φ x φ y ) V = 2 ξ x ξ y sin(φ x φ y ). (2.7) 2.4 Polarized radiative transfer The radiative transfer equation (RTE) for polarized light in a plane-parallel atmosphere is: di = K(I S) (2.8) dτ where I = (I,Q,U,V), K is 4 4 matrix called the total absorption matrix (or propagation matrix), S is the source function vector and means transpose. In local thermodynamic equilibrium (LTE) conditions, S = (B ν (T),0,0,0), where B ν (T) is the Planck function at temperature T. K can be expressed as K = κ c 1+κ 0 Φ (2.9) where κ c is the continuum opacity and κ 0 is the line center opacity. 1 is a unit 4 4 matrix. Eq. (2.9) can be written as K = 1+η 0 Φ, (2.10)

34 22 Photospheric magnetic field diagnostics Z B γ χ Y X Figure 2.1: Reference geometry for RTE. The z axis points towards the observer. where η 0 is the ratio of the line center to continuum opacity. The propagation matrix was defined by Unno (1956) as η I η Q η U η V K = η Q η I 0 0 η U 0 η I 0, η V 0 0 η I containing the absorption profiles defined as η I = 1+ η 0 2 η Q = η 0 2 η U = η 0 2 [ φ p sin 2 γ + 1 ] 2 [φ b + φ r ](1+cos 2 γ), (2.11) [ φ p 1 ] 2 [φ b + φ r ] sin 2 γ cos2χ, (2.12) [ φ p 1 ] 2 [φ b + φ r ] sin 2 γ sin2χ, (2.13) η V = η 0 2 [φ r φ b ]cosγ, (2.14) where, φ b,p,r are the generalized absorption coefficients, and the angles γ and χ are the azimuthal and inclination angles of the magnetic field vector with respect to the LOS (Fig. 2.1). Rachkovsky (1962) introduced the dispersion

35 2.4 Polarized radiative transfer 23 profiles in the propagation matrix, η I η Q η U η V K = η Q η I ρ V ρ U η U ρ V η I ρ Q, η V ρ U ρ Q η I where ρ Q = η 0 2 ρ U = η 0 2 [ ψ p 1 ] 2 [ψ b + ψ r ] sin 2 γ cos2χ, (2.15) [ ψ p 1 ] 2 [ψ b + ψ r ] sin 2 γ sin2χ, (2.16) ρ V = η 0 2 [ψ r ψ b ]cos γ. (2.17) ψ b,p,r are the anomalous dispersion coefficients which introduce the magnetooptical effects. The generalized absorption coefficients and the anomalous dispersion coefficients are defined as φ b,r = H(a,v ± v B ),φ p = H(a,v), (2.18) ψ b,r = 2F(a,v ± v B ),ψ p = 2F(a,v), (2.19) where a is the damping parameter, v is the wavelength shift, and v B is the Zeeman splitting. These parameters, expressed in units of Doppler width λ D are defined as a = Γ λ 2 0 4π λ d, v = λ λ 0 λ vlos λ D, (2.20) v B = λ B λ D, where Γ is the damping factor, λ B is defined in Eq. (2.1), and λ vlos is the Doppler shift due to the motion of the atom along the LOS. H and F are the Voigt and Faraday-Voigt functions respectively. They are defined as H = a π e y2 (v y) 2 + a 2 dy, F = 1 (v y)e y2 2π (v y) 2 + a 2 dy. (2.21)

36 24 Photospheric magnetic field diagnostics A Milne-Eddington (ME) atmosphere is defined as one in which all the atmospheric parameters are constant with optical depth except the source function, which varies linearly with the continuum optical depth as S = S 0 + S 1 τ. (2.22) In such an atmosphere, where the propagation matrix is a constant the RTE (Eq. 2.8) has an analytical solution given by the four Stokes profiles, Q = µs 1 U = µs 1 V = µs 1 I = S 0 + µs 1 (1+η I) [ (1+η I ) 2 + ρq 2 + ρ2 U + ] ρ2 V, (2.23) [ (1+η 2 I )η Q +(1+η I )(η V ρ U η U ρ V )+ρ Q W ], (2.24) [ (1+η 2 I )η U +(1+η I )(η Q ρ V η V ρ Q )+ρ U W ], (2.25) [ (1+η 2 I )η V +(1+η I )(η U ρ Q η Q ρ U )+ρ V W ], (2.26) where = (1+η I ) [(1+η 2 I ) 2 ηq 2 η2 U η2 V + ρ2 Q + ρ2 U + ρ2 V and W = η Q ρ Q + η U ρ U + η V ρ V s. ] W 2 Eq. ( ) represent the analytical solution of the RTE (Eq. 2.8) in a ME atmosphere and are known as the Unno-Rachkovsky solutions. The analytical solutions in a ME atmosphere depend on the following nine parameters: 1. B, the magnetic field strength, 2. γ, the inclination angle of the magnetic field, 3. χ, the azimuth angle of the magnetic field, 4. v LOS, the line-of-sight velocity, 5. η 0, the ratio of the line-to-continuum absorption coefficient, 6. λ d the Doppler width, 7. a, the damping parameter, 8. S 0, the source function parameter, 9. S 1, the slope of the source function.

37 2.4 Polarized radiative transfer 25 While the ME atmosphere allows for an analytical solution, such an atmosphere is not always preferable. When atmospheric parameters vary with optical depth, then analytical solutions of the RTE are not possible and one has to resort to numerical solutions of the RTE. Quite a few numerical techniques have been developed to solve the RTE. These techniques are based on the formal integration of the RTE. One of the earliest techniques was implemented by Auer et al. (1977). In this technique the differential equations of transfer are solved with a second-order finite difference (Feautrier) scheme. A second technique, which exploits the fact that the diagonal elements of the propagation matrix are all the same, thereby transforming the differential equations to integral equations was implemented by Rees et al. (1989). This is the diagonal element lambda operator (DELO) method. Another technique based on Taylor expansion of the Stokes vector is the Hermitian method (Bellot Rubio et al. 1998).

38

39 27 3 Polarimetric Analysis 3.1 Inversion techniques The polarized radiative transfer equation (RTE) described in Chapter 2 is a powerful tool for analysis of solar active regions. The RTE can be used as a mathematical tool to model solar spectra. The technique of using the analytical or numerical solutions of the RTE, assuming a theoretical model atmosphere to generate synthetic Stokes profiles, is known as forward modeling. The synthesized Stokes profiles can be fitted with the observed profiles. This fitting returns model atmosphere parameters, such as magnetic field strength, inclination and azimuth angles, and the line-of-sight velocity. The process of fitting a synthetic profile to an observed profile iteratively, until a best possible fit is achieved, is called an inversion technique. This process is summarized in the form of a flowchart shown in Fig Solve RTE Theoretical model atmosphere Synthetic Stokes profile Observed Stokes profile Fitting procedure New model atmosphere Poor fit Good fit Final model atmosphere Figure 3.1: Steps involved in a Stokes inversion technique

40 28 Polarimetric Analysis A standard fitting scheme involves a set of N discrete data points [x i,y i ] with an associated measurement error given by σ i σ(y i ). If we want to fit a linear function to this data defined by y i = ax i + b, then the fitness, which describes how good the linear function for a given [a,b] fits the data, is defined by the χ 2 merit function, χ 2 N [ ] y(xi ;a,b) y 2 i (a,b) =. (3.1) i=1 In polarimetric analysis the merit function is defined as (del Toro Iniesta 2003) χ 2 = 1 v 3 s=o λ σ i [ I obs s (λ) Is syn (λ) ] 2 w 2 s,λ, (3.2) where Is obs represents the four observed Stokes parameters, Is syn represents the four synthesized Stokes parameters and λ represents the scanned wavelength points. The merit function is scaled with the degrees of freedom, which is the difference between the number of observed parameters and the number of free parameters used in the model, denoted by v. w s,λ is a weighting function. Thus inversion techniques correspond to procedures used to minimize the χ 2 described in Eq. (3.2) to physically meaningful values. The procedures are usually some type of optimization algorithm. Optimization is a class of computational techniques which involves finding the extrema of a mathematical function. Optimization schemes can be iterative or heuristic. Iterative procedures are based on computing of gradients which involves calculation of the derivatives of χ 2. Such procedures are based on finding the minimum and so have the drawback that the final solution may be a local minimum. Heuristic algorithms are based on finding the global minimum by an educated guess approach in which a candidate solution is iteratively worked upon to find a better solution. 3.2 Helix Helix (Lagg et al. 2004) is an ME inversion code which uses a genetic algorithm (PIKAIA; Charbonneau 1995) based optimization scheme. Genetic algorithms are a class of heuristic algorithms which are inspired by the biological evolutionary process of natural selection (survival of the fittest). They employ mathematical implementation of biological processes like genotype, phenotype, reproduction, crossover, and mutation. Genotype is the genetic content of an individual and phenotype is the actual characteristics of the individual. Phenotype is essentially the decoded version of genotype. Reproduction involves transferring of the genotype from the parents to the offspring by recombination of genetic material which involves chromosomal crossover. Changes are introduced by mutations. The flowchart in Fig. 3.2 shows the very basic steps involved in a genetic algorithm.

41 3.2 Helix 29 Select random initial population Evaluate fitness of every individual create new population by breeding best fit individuals Evaluate fitness of every indvidual of new population Test for convergence. NO Does fittest phenotype match target phenotype? YES STOP Figure 3.2: Essence of genetic algorithms In PIKAIA, a population is made up of strings which are defined in terms of two randomly generated real numbers each consisting of 8 decimal digits, which encode phenotypes (candidate solutions). Genotypes are then just the digits of the two strings put together by the encoding process. Breeding involves taking two individuals from the population based on fitness and using a random crossover and mutation scheme to derive offspring populations. Crossover operations, acting constantly, produce jumps in values and thus ensure a wide exploration of the parameter space. Thus a PIKAIA based optimization is extremely robust and finds a global minimum independent of the initial parameter values. PIKAIA allows giving a range for all the parameters to be fitted along with initial values. This range ensures that the results stay within a physically useful regime of solutions. The trade off is that PIKAIA and other genetic algorithms are slow in converging. However, Helix assumes a ME atmosphere and the computation of synthetic profiles are not computationally intensive. Therefore one can afford to go for the robustness of fits provided by PIKAIA. Helix also has the option for a hybrid optimization scheme in which PIKAIA is used only for one pixel and the neighboring pixels use the

42 30 Polarimetric Analysis PIKAIA result as a starting guess, but use faster gradient based iterative schemes for converging. The function to minimize in Helix is defined as δ 1, where δ = [I obs (λ) I syn (λ) I c σ I ] 2 w 2 I + [Q obs (λ) Q syn (λ) σ Q ] 2 w 2 Q + [U obs (λ) U syn (λ) σ U ] 2 w 2 U + [V obs (λ) V syn (λ) (3.3) σ V ] 2 w 2 V. Here, [I,Q,U,V] obs represents the four observed Stokes parameters, [I,Q,U,V] syn represents the four synthesized Stokes parameters, λ represents the scanned wavelength points, w I, w Q, w U, and w V are wavelength dependent weighting function, I c is the continuum intensity and σ I,Q,U,V are parameters defined as: σ I = 1 n I obs I λ c 1 (3.4) σ Q,U,V = 1 n (Q,U,V) obs (3.5) λ Note that the function defined by Eq. (3.3) is not strictly a χ 2 as in Eq. (3.2). Helix works with an input file which primarily gives the model atmospheric parameters (described in chap. 2) with ranges. We can select various parameters to be free or fixed parameters depending on the observations. It is possible to use multi-component atmospheres via a filling-factor. For higher accuracy it is possible to convolve the synthesized Stokes profiles with the profile of the filter used in the observations before fitting with the observed Stokes profiles. Straylight correction is also possible in Helix. Helix was used for analysis of the data in Paper I, Paper II, and Paper III. 3.3 Additional techniques used Some other techniques used to analyse polarimetric data are based solely on the properties of the observed Stokes profiles. Some of the quantities measured in this thesis are: Stokes V asymmetries: Milne-Eddington inversions model the magnetic field and velocity without gradients and cannot produce asymmetric Stokes profiles. We therefore also characterize the asymmetries of the Stokes V profile. Two kinds of asymmetries used in the literature are the area asymmetry (δa) and the amplitude asymmetry (δa).

43 3.3 Additional techniques used 31 The area asymmetry is defined as δa = (A b A r )/(A b + A r ), (3.6) where A b is the area of the blue lobe of the Stokes V profile and A r is the area of the red lobe of the Stokes V profile. Area asymmetries are produced by velocity gradients along the line-of-sight (Solanki 1993). The amplitude asymmetry is defined as δa = (a b a r )/(a b + a r ), (3.7) where a b is the peak amplitude of the blue lobe of the Stokes V profile and a r is the peak amplitude of the red lobe of the Stokes V profile. Amplitude asymmetries can be produced by vertical, horizontal or temporal velocity gradients in the solar atmosphere (Solanki 1993). A measure related to the Stokes V asymmetries is the broad band circular polarization (BBCP) or the net circular polarization (NCP). When the circular polarization V(λ) is integrated over a wide spectral band, then the presence of asymmetric Stokes V profiles leads to a non-zero net polarization. This is referred to as the BBCP (Illing et al. 1975). When V(λ) is integrated over a spectral line then the measured non-zero net polarization due to asymmetries is known as NCP. Mathematically NCP is defined as λr λ b V(λ)dλ/I c, (3.8) where λ b and λ r are the wavelength limits over which the integration is carried out and I c is the continuum intensity. Stokes V zero-crossing shifts: This refers to the wavelength at which V(λ)=0 between the two wings of the Stokes V profile. This can be interpreted in terms of the Doppler effect, to obtain velocities which then correspond to flows within magnetic flux elements. Total linear polarization: This gives a measure of the transverse magnetic field and is expressed as Q 2 +U 2 dλ/i c, (3.9) where I c is the continuum intensity. Total circular polarization: This gives a measure of the longitudinal magnetic field and is expressed as V dλ/i c. (3.10)

44 32 Polarimetric Analysis 3.4 Straylight correction In solar astronomy, stray-light is the problem of contamination, wherein the light recorded in a pixel has components of light from surrounding pixels mixed in it. This contamination occurs either due to atmospheric causes or due to instrumental issues. The importance of spatial stray-light when interpreting observed data from any existing solar telescope is hardly contested today, since independently developed 3D simulation codes agree in producing much higher granulation RMS contrast than observed. The highest RMS continuum contrast in the present SST/CRISP data is a little over 8% whereas the true contrast (obtained from 3D simulations) at 630 nm is about 14% (Wedemeyer-Böhm & Rouppe van der Voort 2009). The difference in RMS granulation contrast between the simulations and observations suggests a stray-light level of about 40%. Stray-light has a strong effect on the polarimetric measurements of small-scale solar features with sizes close to the resolution of the SST. In general, stray-light is stronger for Stokes I than for polarized light, because only a small fraction of the solar surface is covered with magnetic fields that can contribute to polarized stray-light. We found that, for small bright points associated with strong downflows, the fitted Stokes I profiles were red-shifted relative to the observed Stokes I profiles and that the opposite was true for the Stokes V profiles as shown in Fig We suspected stray-light to be the reason for the poor quality of Figure 3.3: Observed (solid) and fitted (dashed) Stokes I and V for a small bright point. the fits. In support of this argument found that the velocities obtained from the Stokes V zero-crossing shifts were almost twice stronger than those from the inversions. We reasoned that Stokes V profiles indicate locally stronger downflows than the I profiles since the effect of stray-light is larger for Stokes I than for Stokes V. This explains why the observed localized downflows appear stronger in Stokes V than inversions, which compute velocities using both

45 3.4 Straylight correction 33 Stokes I and Stokes V profiles giving a final result which is a kind of average. The stronger effect of stray-light on Stokes I than on Stokes Q, U or V for small-scale magnetic structure has prompted counter-measures in Stokes inversion codes used today. These counter-measures consist in, e.g., introducing free stray-light parameters, magnetic filling factors or local stray-light profiles (e.g., HELIX and MILOS (Orozco Suárez & Del Toro Iniesta 2007)) to improve the estimates of physical quantities. To establish that stray-light definitely explains the poorly fitted Stokes I and V profiles of Fig. 3.3, we ran several test inversions with Helix using a fixed local stray-light compensation. The local stray-light profile is implemented in Helix as I obs = αi mag +(1 α)i obs P, (3.11) where I obs is the observed Stokes I, (1 α) is the stray-light fraction, I mag is Stokes I for the (unknown) magnetic component, and denotes convolution with the stray-light point spread function P (assumed to be a gaussian). The plots in Fig. 3.4 show the observed and fitted Stokes I and V profiles, (a) (b) (c) (d) Figure 3.4: Observed (solid) and fitted (dashed) Stokes I and V for different straylight parameters. Wavelengths along the x-axis are in units of må. Obtained values of LOS magnetic field (B LOS ), LOS velocity (V LOS ), and the magnetic field strength (B) are given at the top of the plots. for α = 0.99 (nearly no stray-light), 0.8, 0.7 and 0.6 (40% stray-light). For these plots, P was assigned an assumed FWHM of 40 pixels (2.8 arcsec). The

46 34 Polarimetric Analysis fits with α = 0.99 correspond to no stray-light correction (Fig. 3.3) and show the relative wavelength shifts between the Stokes I and V profiles, discussed above. Increasing the stray-light (decreasing α) leads to dramatic improvement of the fits. The improvement is obvious already with 20% stray-light, with 40% stray-light the fits look nearly perfect. In order to eliminate the possibility that the improved fits shown in these plots is just a fortunate coincidence, we repeated the inversions for all pixels in a pixel map using different values for the stray-light parameter α (0.99, 0.8, 0.7, 0.65, 0.6, 0.5, 0.4 and 0.3) and calculated the RMS difference between the observed and fitted Stokes I (di rms ) and V (dv rms ) profiles (with the profiles normalized to the continuum intensity). The results are shown in Table 3.1. Table 3.1: RMS difference for different stray-light parameters α Straylight di rms dv rms % % % % % % % % The reason for the dv rms values being much lower than di rms is that only a small fraction of the FOV is magnetic. Thus, increasing the assumed straylight leads to systematically better fits for both Stokes I and V. As expected from the plots shown, the improvement is relatively large when decreasing α from 0.99 to 0.7 and smaller when decreasing α from 0.7 to 0.6. Based on the Stokes V fits, there is a shallow optimum around α = 0.6. The V fits are good in an absolute sense: dv rms approaches 1.8 times the noise level ( ). Based on both the Stokes I and V fits, the optimum α would be about 0.5. Setting α to 0.3 gives significantly poorer fits for Stokes I. We concluded that the explanation for the poor fits was primarily due to stray-light at a level roughly consistent with what is needed to explain the observed granulation RMS contrast. Reducing α below 0.6 is tempting but could correspond to a corrected granulation RMS contrast that is unrealistically high. We settled for a somewhat conservative value of α = 0.6. The stray-light correction described above was used for data presented in Paper III.

47 35 4 Polarimetry with SST 4.1 Instrumentation A polarimeter is an instrument that can measure polarization properties of light. In solar astronomy, a polarimeter generally consists of a telescope followed by a chain of optics, a modulator and finally a detector. The chain of optics can be reflecting and transmitting optical surfaces like mirrors, beamsplitters, and filters. The modulator is a device used to modulate the input Stokes vector into four different intensity states. Modulators are usually a combination of a retarder and an analyzer. Retarders used in solar astronomy include piezo-elastic retarders and simple rotating wave retarder in combination with a half-wave plate or ferro-electric liquid crystals. Analyzers can be Glan polarizers or polarizing beam-splitters. In our case the modulator consists of nematic Liquid Crystal Variable Retarders (LCVR) together with a polarizing beam-splitter used as an analyzer. The challenge of solar polarimetry is to obtain high resolution, high signalto-noise ratio (SNR), and high cadence observations. High spatial resolution demands short exposure times, since we want the effects of atmospheric seeing to be as small as possible. Atmospheric seeing refers to the distortions caused to the signal (solar rays) due to the turbulence in the atmospheric layers. High SNR demands longer integration times, since polarimetric signals are weak. A high cadence is needed to keep up with the evolution time scales of solar features. Thus a compromise between the mutually exclusive requirements of sensitivity, cadence, and resolution is needed (van Noort & Rouppe van der Voort 2008) SST and CRISP The Swedish 1-m Solar Telescope (Scharmer et al. 2003) is situated on the island of La Palma in the Spanish Canary archipelago. It is located at an altitude of about 2400 m above sea level. This is one of the best sites in the world for astronomical observations, offering excellent seeing conditions. Together with an adaptive optics system (Scharmer et al. 2003), the SST has produced some of the sharpest ever diffraction limited images of the Sun. The SST optics consist of a single high quality lens which doubles up as a vacuum window. This minimizes the number of optical surfaces which is advantageous for narrow band imaging and polarimetry. It has a secondary optical system consisting of a field mirror and a 24 cm corrector comprising a lens and mir-

48 36 Polarimetry with SST ror. The secondary system is called as the Schupmann system. This system perfectly compensates the enormous focus variation with wavelength of the 1-m singlet lens (Scharmer et al. 1999). The adaptive optics system consists of a deformable mirror and a wavefront sensor. The basic idea behind adaptive optics is that it tries to correct for atmospheric seeing by evaluating the shape of the distortion of the wavefront and adapting the deformable mirror in way to the counter distortions. The wavefront sensor used in the SST adaptive optics system is a Schack-Hartmann type sensor, which looks at a solar structure through many small parts of the telescope aperture and measures the position of the structure as seen through each part. When the atmosphere distorts the image, it causes these images to move differently depending on their position in the aperture. The positions are measured and translated to commands to the deformable mirror, so that it takes the shape that compensates for the distortions. The CRisp Imaging SpectroPolarimeter (CRISP; Scharmer et al. 2008a) is a dual Fabry-Perot (FP) tunable filter system usable from 510 to 860 nm. It consists of nm wide pre-filters, LCVRs, a high-resolution highreflectivity etalon, followed by a low-reflectivity low-resolution etalon, and a polarizing beam splitter located close to the final focal plane and feeding two 1k 1k-pixel Sarnoff charge-coupled device (CCD) cameras (Scharmer 2006). This constitutes a dual beam polarimetry setup which helps in reducing crosstalk originating from atmospheric seeing. The FPI system has a compact telecentric optical design with a minimum number of optical surfaces and high overall transmission. A third CCD records broad-band images through the pre-filter. All three cameras are synchronized by use of an external chopper. Figure 4.1 shows a block diagram of CRISP. FPI filter systems allow rapid tuning, which makes the overhead of wavelength tuning small compared to the overall integration time. The LCVR is a device whose retardance can be varied by applying different voltages and thus it can be used to modulate the beam with an appropriate modulation scheme based on voltages. The LCVRs are tuned during read-out of the CCDs. Together with the polarizing beamsplitter analyzer four linear combinations of the four Stokes parameters are recorded with this scheme. The CRISP replaced the older Solar Optical Univeral Polarimeter (SOUP) filter as the primary imaging filter at the SST in The SOUP is a Lyot type filter. 4.2 Data Reduction Polarimetric calibration The conversion coefficients from Stokes parameters to the recorded modulated intensity is given by the elements of a modulation matrix M. The modulation matrix for each pixel is determined by a calibration procedure for the

49 4.2 Data Reduction 37 BROAD BAND NARROW BAND CCD CCD Dual FPI CCD PRE FILTER 10/90 LCVRs LRE HRE 50/50 POLARIZING CHOPPER Figure 4.1: A block diagram of CRISP without imaging optics. polarimeter. Polarimeter here refers to all optics starting with the tip-tilt and deformable mirrors of the AO till the CCD. In this procedure we use a calibration optics consisting of a Linear Polarizer (LP) together with a Quarter Wave Plate (QWP). The LP is positioned at several fixed angles, after which the QWP is rotated through 360 degrees. This process generates a large number of input Stokes vectors and determines M. The inverse M is the demodulation matrix that will convert the four images acquired at the different LC states to the Stokes I Q U V representation. The telescope optics are polarizing and this polarization is dependent on the pointing direction. This is compensated by using a telescope model which predicts the polarizing properties of the telescope for a given time by using the azimuth and elevation information from the telescope turret log file. The SST telescope model was implemented by Selbing (2010) Flat-fielding and dark subtraction Accurate flat-fielding of the FPI images is very important for further image restoration and analysis of final polarimetric data. We acquire flat-field data by selecting a quiet sun region, reasonably free from any polarization signals due to magnetic fields, scanning the telescope over that region and averaging a large number of images. This corresponds to recording an image of an object that has constant intensity with no variations over the CCD s field of view, and no polarimetric signals. While flat-fielding broad-band images is straightforward, it is not the same with narrow band images. We record flats for each wavelength position and each LC state used for the science data. Flat-fielding of polarization images is quite complicated because the telescope induces a strong polarization, thereby converting unpolarized light

50 38 Polarimetry with SST (which is assumed while collecting data in flat-field mode) into partially polarized light. This polarizing property of the telescope varies strongly with the azimuth and elevation of the telescope which vary with time. As a consequence of the varying telescope polarization, polarization flats recorded with different LC states will show weak polarization structures that vary strongly with time. In addition, the subsequent optics introduces polarization properties that vary over the field-of-view. If we apply these polarization flats to our data, it is likely that the polarization structures that were relevant only at the time of recording flats are transferred into the data. To overcome this issue we demodulate the flats with the polarimeter modulation matrix individually for each pixel. This demodulation results in four Stokes flats. We then set the Stokes Q, U and V flats to zero, keeping only the Stokes I flats. This constitutes the flat-field image to be used for all four LC states for that wavelength. An in-depth description of the problem and a recent improvement of this procedure is given by Schnerr et al. (2010). A CCD produces a signal even when it is not exposed to any light. This dark level varies from pixel to pixel. To compensate for this, a dark frame is made by averaging images recorded with the beam blocked. This dark frame is then subtracted from the image MOMFBD The Multi-Object Multi-Frame Blind Deconvolution (Löfdahl 2002; van Noort et al. 2005) is a post-processing method for image restoration from residual seeing, not corrected by the AO system. MOMFBD combines a large number of short-exposure images by using a process that separates the useful image information from the seeing-induced image blurring and distortions. Data from each camera, at each wavelength tuning and each LC state, is treated as a separate object and multiple realizations of multiple objects are restored jointly. A near-perfect alignment can be achieved between the different objects, by use of a pinhole calibration process. Images of a pinhole array are recorded simultaneously with all the cameras, and the sub-pixel alignment errors between different cameras, as a function of position within the FOV, is computed. The pinhole array acts as an object that is the same in all the different wavelength and polarization state channels and thus ensures extreme precision. Such precise alignment strongly reduces false signals in the determination of derived quantities, such as magnetograms and Dopplergrams. The exposures have to be short compared to the time scale of the seeing, usually 10 ms. This leads to low signal to noise ratio in each individual data frame, particularly in the core of a dark line. This is compensated by combining many exposures. However, an assumption made by MOMFBD is that the unknown object is the same in all realizations. This limits the time available for a scan to

51 4.2 Data Reduction 39 the solar evolution time scale, or sec at the SST spatial resolution, which can lead to low signal to noise if we scan many positions in the line. The joint processing of the narrow-band channels with the wide-band also helps overcoming this issue Cavity map correction Even with a perfect flat-field, compensated for wavelength shifts of the average quiet sun, there will remain strong and small-scale intensity gradients in the corrected image. This is due to inhomogeneities in the coatings that cause small-scale variations in cavity separation measuring about 0.6 nm RMS. The unwanted intensity gradients are due to wavelength shifts of the high-resolution etalon transmission profile. To compensate for the wavelength shifts caused by cavity errors dominated by the high-resolution etalon, we used the flat-field images. At each pixel, we performed a second-order polynomial fit to the center part of the nm line profile and measured the wavelength shifts at each pixel. These wavelength shifts were subtracted from the velocity maps obtained from the Helix inversions (See Fig. 4.2). Paper I, Paper II, and Paper III are based on Fe I nm spectral line data recorded with the SST/CRISP

52 40 Polarimetry with SST Figure 4.2: The differences in a velocity map before (top) and after (bottom) the cavity map correction. Note gradient from upper right to bottom left in the uncorrected map.