Investigating Magnetic Fields in the Solar Chromosphere

Stockholm University Department of Astronomy

LICENTIATE THESIS Investigating magnetic ﬁelds in the solar chromosphere

Author: Alexander G.M. Pietrow

Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

Supervisor: Dan Kiselman

Co-Supervisor: Jaime de la Cruz Rodr´ıguez

Thesis Mentor: Arjan Bik

November 25, 2020 Abstract

Solar plage has been the topic of many studies since its initial description in the mid 19th century, but as of yet it has not been understood to the point where we can reproduce all aspects of these active regions in quasi-realistic numerical models. To a large extent, this is caused by an incomplete understanding of the magnetic structure that drives the activity in these areas. Detailed measurements have been done of the magnetic field configuration of plage in the photosphere since the late 20th century, but only a handful of papers have managed to make any measurements at all in the higher situated chromosphere, despite the fact that the magnetic field vector of plage is important in understanding chromospheric magnetic fields in general, as well as the heating processes of the higher atmosphere. In Pietrow et al.(2020) we add to these measurements by introducing what is to our knowledge the first full Stokes inversion of chromospheric plage, which allowed us to estimate the magnetic field vector at an optical depth of log τ = 3.5. The obtained value is B = 440 90 G in the plage with an inclination of − | | ± 10◦ 16◦ with respect to the local vertical. Our reported magnetic field strength matches with a ± recent study by Morosin et al.(2020), but is higher by a factor of two or more compared to previous studies that measured the field using other methods. Additionally we measure an average magnetic field strength of B = 300 50 G in a fibrillar region close to the plage. | | ± In this thesis we explore the difficulties of measuring this magnetic field vector. Since plage exists in a complex environment, we will begin with a general description of the structure and properties of the solar atmosphere and the layers from which it is composed, as well as review the types of active regions that can be found in the solar atmosphere. Our focus then narrows to the chromosphere, the diagnostic properties of spectral lines that are sensitive to this layer (mainly the Ca II 8542 Å line), plage regions, and plage chromospheric magnetic fields. Ad- ditionally, we touch upon the theory of radiative transfer and how physical characteristics of the atmosphere can be inferred from polarised light. We also give attention to the observing process with the Swedish 1-m Solar Telescope (SST) and the workings of the reduction pipeline and post-reduction methods as well as the process spectropolarimetric inversions. Finally, once we have understood why and how this project has been done, we summarize our findings and compare them to current literature.

Sammanfattning

De områden på solen som kallas plage har varit foremål¨ for¨ många studier sedan de forst¨ beskrevs i mitten av 1800-talet. Men annu¨ har vår forståelse¨ inte nått så långt att vi kan re- producera alla aspekter av dessa aktiva regioner i numeriska simuleringar. Detta beror till stor del på en ofullstandig¨ forståelse¨ av magnetfaltets¨ struktur i dessa områden. Detaljerade matningar¨ av magnetfaltskonﬁgurationen¨ i plageområden i fotosfaren¨ har gjorts sedan slutet av 1900-talet, men bara ett fåtal studier har lyckats gora¨ några matningar¨ alls i den hogre¨ lig- gande kromosfaren.¨ Detta trots att kannedom¨ om magnetfaltsvektorn¨ i plage ar¨ viktig for¨ att forstå¨ kromosfariska¨ magnetfalt¨ i allmanhet¨ liksom for¨ uppvarmningsprocesserna¨ i den hogre¨ solatmosfaren.¨ I en artikel av Pietrow et al.(2020) presenteras ett bidrag till dessa m atningar¨ i form av vad som kan vara den forsta¨ inversionen av den fullstandiga¨ Stokes-vektorn for¨ att kartlagga¨ magnetfaltet¨ i en plage-kromosfar.¨ Detta mojliggjorde¨ en uppskattning av den magnetiska faltvektorn¨ vid ett optiskt djup om log τ = 3.5. Det erhållna vardet¨ ar¨ B = 440 90 G − | | ± i plage med en inklination på 10◦ 16◦ från solytans normal. Vardet¨ på den magnetiska falt-¨ ± styrkan stammer¨ val¨ overens¨ med resultatet i en aktuell studie av Morosin et al.(2020), men ar¨ en faktor två eller mer hogre¨ an¨ tidigare arbeten som anvant¨ andra metoder. Samtidigt erhålls ett medelvarde¨ på den magnetiska faltstyrkan¨ på B = 300 50 G i ett narliggande¨ område med | | ± ﬁbriller. I denna avhandling diskuteras svårigheterna med att få fram dessa resultat. Eftersom plage ex- isterar i ett komplicerat sammanhang kommer forst¨ en oversikt¨ av solatmosfaren¨ och dess olika komponenter samt de aktiva regionerna. Fokus laggs¨ sedan på kromosfaren,¨ de spektrallinjer som anvands¨ som diagnostika (framfor¨ allt Ca II 8542 Å ) samt plageområden och deras magnetfalt.¨ Dartill¨ berors¨ strålningstransport och hur polariserat ljus kan anvandas¨ for¨ att harleda¨ fysikaliska tillståndet i solatmosfaren.¨ Det redogors¨ for¨ hur observationerna med det svenska solteleskopet SST på La Palma går till och hur data reduceras och behandlas, liksom hur de spektropolarimetriska inversionerna går till. Slutligen sammanfattas resultaten och jamf¨ ors¨ med vad som hittills rapporterats i litteraturen.

3 List of papers

Paper I A.G.M. Pietrow, D. Kiselman, J. de la Cruz Rodr´ıguez, C. J. D´ıaz Baso, A. Pastor Yabar, R. Yadav, Inference of the chromospheric magnetic ﬁeld conﬁguration of solar plage using the Ca II 8542 Å line, Astronomy & Astrophysics, Submitted 25/06/2020, accepted 21/09/2020

A printed version is available in an appendix at the end of this thesis.

Author’s contribution I performed all of the reduction of the data presented in the paper, except for the application of the deep learning algorithm. This includes the development of several additional new steps. I did two observing campaigns for the project, though in the end we chose to use a dataset acquired by JdlCR and RY. I performed all the analysis presented in the paper and created all ﬁgures apart from Fig. 1. I wrote the ﬁrst draft of the paper, which was later redacted and expanded with help of the co-authors. My main contributions are all sections except Section 2 ’The Ca II 8542 Å line in plage regions’, which was written by JdlCR. Contents

1 Introduction3

2 The solar atmosphere and active regions5 2.1 The solar atmosphere...... 5 2.2 Active Regions...... 8 2.3 Sunspots...... 10 2.4 Plage...... 10 2.5 Fibrils...... 12

3 Chromospheric diagnostics 15 3.1 Chromospheric spectral lines...... 15 3.2 LTE vs Non-LTE...... 17 3.3 Polarization of light...... 18 3.4 Zeeman eﬀect...... 20 3.5 Polarized radiative transfer equation...... 23

4 Chromospheric magnetic fields 25 4.1 Inferring Solar magnetic fields...... 25 4.2 Magnetic fields in plage...... 26

5 Observations and data reductions 29 5.1 The Swedish 1-m Solar Telescope...... 29 5.2 The SSTRED data-reduction pipeline...... 32 5.3 The MOMFBD routine...... 34

6 Post-reduction techniques 37 6.1 Fringe Removal...... 37 6.2 Improving S/N...... 39

7 Inversions 41 7.1 Inversion methods for solar observations...... 41 7.2 The Levenberg-Marquardt algorithm and regularization...... 42 7.3 Initialization of the line-of-sight velocity...... 44

5 7.4 Inversion strategy...... 44

8 Summary of Paper I 47

9 List of acronyms 49

10 Acknowledgements 51

11 Paper I 63

Introduction

This thesis deals with the study of chromospheric magnetic fields in bright active regions on the Sun called plage regions. In Paper I we have observed plage and inferred the magnetic field vector at a specific chromospheric height. In Chapter2 we will give a general overview of the solar atmosphere and its distinct layers. Furthermore, we will discuss the different types of active regions that can be found on the Sun, in order to build an understanding of the solar features discussed in Paper I and the environment in which they exist. Additionally, Chapter2 gives a brief overview of the solar exterior, and prepares us for Chapter3 which describes the most important chromospheric spectral lines and how they are used to obtain information about magnetic fields via polarimetry. Magnetic fields in the chromosphere will be discussed in Chapter4, as well as past and current measurements and the limitations that we currently face. The next three chapters deal with the methods used to acquire the data, reduce them and eventually infer physical parameters from them. Throughout these chapters we will connect the processes and definitions to Paper I and then contextualize them in the final chapter where we discuss the work done in Paper I and how the obtained results fit into our current understanding of the subject. However, before we can do any of that, we need to start by defining the solar atmosphere and the active regions that can be found in it.

The solar atmosphere and active regions

In this chapter we focus on the general structure of the solar atmosphere, the layers from which it is composed, and active regions. This will allow us to understand the environment in which the active region studied in Paper I exists.

2.1 The solar atmosphere

The solar atmosphere is most often defined as the part of the Sun from which photons can directly escape into space. It is customary to divide the solar atmosphere into four separate layers, each with their own physical properties. Going from the ’surface’ outwards, we have the photosphere, the chromosphere, the transition region and the corona. The photosphere is the lowest of the four, and also the thinnest with a span of only a few hundred kilometers. Despite its small height range, most of the Sun’s visible emission originates from this layer. The granulation pattern is a defining feature of the photosphere, consisting of small granules with a typical diameter of 1000 km, which represent the top of convective cells coming up from deeper inside the Sun. Their centers are bright because they consist of hot rising plasma while their edges (or intergranular lanes) are dark and represent falling material. There are also larger scale mesogranules and supergranules. In the photosphere we can find active regions like and pores and sunspots, whose frequency depends on the phase of the 11 year solar cycle. The relatively high densities in this layer make conditions close to local thermodynamic equilibrium (LTE)1. Photospheric magnetic fields are generally concentrated around active regions (Priest 2014). The photosphere is sometimes still referred to as the solar surface, which is a remnant from the now antiquated belief that the Sun was a rigid body (Kaiser 1845). A more modern definition of the solar surface is based on optical depth, where τv = 1. As this varies with wavelength, it is customary in the field to express this value in terms of the optical depth at the continuum wavelength of λ = 500 nm. The photosphere, in a quiet Sun environment, is defined as reaching

1This is discussed in detail in Chapter 2.

5 6

Figure 2.1: Left: A representation of a 1D time-independent semi-empirical model of the solar atmosphere showing the variation of temperature as a function of height and the approximate regions of formation for various spectral lines. We added a colorbar to indicate the location of the photosphere, chromosphere and transition region. Reproduced from Vernazza et al.(1981). Right: The thicknesses of the components of the Sun’s atmosphere.

Figure 2.2: A part of active region AR12713 observed in Ca II 8542 Å showing the upper photosphere in the line wing (left) and the chromosphere in the line core (right). In these images we can make out ﬁbrils, plage and pores. Image taken with the SST. 2. The solar atmosphere and active regions 7

from 100 km below τ500 = 1 to 525 km above τ500 = 1 (Carroll & Ostlie 2017). The temperature of the photosphere ranges from approximately 9400 K at the bottom to 4400 K according to Vernazza et al.(1976) and illustrated in Fig. 2.1. Above this height, the temperature rises again and we enter the chromosphere.

Above the photosphere we find the chromosphere. It is much less bright than the photosphere, emitting only about 0.1% as much light. There is no fixed boundary for the thickness of the chromosphere, as this varies with time, location and solar activity. However, we can define a mean chromosphere according to the VAL model (Vernazza et al. 1976; Avrett 1985) as starting after the photosphere and ending at 2100 km above τ500 = 1 and thus spanning roughly 1600 km in total. Similarly to the produced light, the gas density also drops by more than a factor of 1000. The temperature increases with altitude from 4400 K to roughly 10000 K, as can be seen in Fig. 2.1. The VAL model is a 1D mean model of the solar atmosphere that assumes that thermodynamic properties vary only with height and are time-independent. It was replaced by the updated FAL model (Fontenla et al. 1993). Curves exist for four different situations. FAL-A, for faint Quiet Sun, FAL-C, for average Quiet Sun, FAL-F, for bright Quiet Sun and FAL-P for plage. These days the C model is still widely used as a standard and for creating initial guesses for inversion codes2, but beyond that it fails to realistically represent the dynamical, time-dependent and inhomogeneous nature of the solar atmosphere. In its stead, more sophisticated 3D computational models are being developed, like for example the Bifrost code (Gudiksen et al. 2011). Codes like Bifrost can model the photosphere, especially quiet Sun, accurately but doing so with the chromosphere is more challenging as not all heating processes are yet understood. In Paper I we use the FAL-C model as an initial guess for our input atmosphere. The name of the layer was suggested by Lockyer(1868a).

Because the temperature increases in the higher levels of the chromosphere, we end up with a partially ionised atmosphere with conditions far from LTE. The work done in Paper I will mostly focus on the chromosphere.

Above the chromosphere the model shows a very rapid temperature rise from 104 K to about 105 K over a height of approximately 100 km, after which the temperature rises more slowly up to about 1 million K. The thin region where the temperature increases rapidly is called the transition region, it is mostly observed in the UV and extreme UV parts of the spectrum. Above the transition region we have the corona. This outermost layer of the atmosphere is the warmest and least dense. It extends from about 2200 km above τ500 = 1 upwards without a well-deﬁned upper boundary. It has an energy output of roughly 1 millionth of the intensity produced by the photosphere. The density at this point is so low that it is virtually transparent to most radiation (except radio). Ground-based observations of the corona are done by obscuring the much brighter lower layers of the atmosphere by using a coronograph or a solar eclipse. The plasma inside the corona is very susceptible to magnetic ﬁelds. This is best seen during eclipses that take place during a solar minimum, as the corona takes on the shape of a dipole from an observational viewpoint.

2See chapter7. 8

The corona can be subdivided into three structural components. The lower corona is called the K corona (from Kontinuierlich or continuous) and produces continuous white light emission. This light is the result of free electrons scattering the photospheric radiation, a highly polarizing eﬀect. Photospheric spectral lines in this layer are washed out due to the large Doppler shifts that result from the high thermal velocities of electrons in this region.

The F corona (named after Fraunhofer) is produced by dust that scatters photospheric light. Compared to electrons, these dust particles are much slower and much more massive, which results in a much lower Doppler broadening of the Fraunhofer lines. The F corona merges with the zodiacal light, which is the faint glow along the ecliptic that is caused by the reﬂection of sunlight by interplanetary dust. This light is also visible in the optical.

The E corona (from Emission) is a source of emission lines caused by highly ionized atoms. The low number density prevents recombination and allows the occurrence of forbidden transitions which leads to the production of spectral lines that are generally only found in low density interstellar gas environments (Carroll & Ostlie 2017). Beyond the corona is the heliosphere, which is the cavity formed by the Sun in the surrounding interstellar medium that extends past Pluto.

2.2 Active Regions

The definition of active regions (ARs) has become more refined throughout the field of solar physics over the last several decades as our observational methods allowed for observations in more and more wavelengths and at better resolutions. Starting from areas with sunspots, later including faculae and even later chromospheric features like plage and fibrils (D’Azambuja 1956). The effect of this ’living definition’ can still be felt today by the plethora of slightly different definitions found in current literature. Some sources define them as extended areas on the Sun where spots, plages and filaments occur (Foukal 2008). In other sources we find the definition of an AR to be a region that contains at least one sunspot that is visible in broadband optical light (van Driel-Gesztelyi & Green 2015). Perhaps the most authoritative source on the matter is the SWPC (Space Weather Prediction Center) of the NOAA (National Oceanic and Atmospheric Administration), which is the agency that designates serial numbers to active regions. This numbering system started on the 5th of January 1972 and has been counting consecutively since then. For example, the name of the AR observed in Paper I is AR127123 (NASA 2009). According to their definition, solar regions qualify for assignment of a serial number if they meet one or more of the following conditions (private communication, SWPC Customer Support, 2020):

1. Contain a conspicuous spot group (Class C or larger)4.

3In some databases it will be denoted as AR2713, to keep with the four number format. Any region that is observed after the 14th of June 2002 will be above 10000. 4Modiﬁed Zurich Class (McIntosh 1990) 2. The solar atmosphere and active regions 9

2. Contain a class A or B group confirmed by at least two observers, preferably with observations more than one hour apart. 3. Produce a solar flare with an X-ray burst. 4. Plage has a white-light brightness of at least of 2.5 (on a linear scale 1-5, 5=flare) and has an extent of at least five heliographic degrees. 5. Plage is bright near the west limb and is suspected of growing. A similar and perhaps more complete definition is suggested by van Driel-Gesztelyi & Green (2015) who define active regions as ”the totality of observable phenomena in a 3D volume represented by the extension of the magnetic field from the photosphere to the corona, revealed by emissions over a wide range of wavelengths from radio to X-rays and γ-rays (only during flares) accompanying and following the emergence of strong twisted magnetic flux (kG, 1020Mx)5 ≥ through the photosphere into the chromosphere and corona”. ARs in their most basic form consist of a bipolar magnetic field configuration, but they can grow much more complex with many bipoles. These magnetic fields can manifest different phenomena in different layers of the solar atmosphere. In the photosphere we will see sunspots or pores and faculae, which are caused by concentrated and dispersed magnetic fields respectively. Higher up in the chromosphere we find filaments connecting the opposing polarities and dispersed fields as plages. Loops are seen in the transition region and the corona, connecting the opposing polarities. The effects of ARs do not end there however, as these magnetic fields dominate the interplanetary magnetic field. Currently three empirical laws exist regarding the formation of ARs and the 11-year solar cycle. 1. Sporer’s¨ law, also known as the butterfly diagram, shows the relation between latitude of ARs and solar cycle phase. Active regions tend to emerge at higher latitudes at the start of the cycle and appear progressively closer to the equator as the cycle progresses. 2. Joy’s law states that there is a systematic preference against perfect east-west alignment of bipolar ARs, with the leading spot6 tending to be closer to the equator. 3. Hale’s law states that ARs that are bipolar and aligned roughly east-west will have opposite leading polarities on the other hemisphere. The polarities alternate between solar cycles. We define three different scales of active regions following van Driel-Gesztelyi & Green(2015). The smallest being Ephemeral regions with magnetic flux values between 3 1018 and 1 1020 Mx, · · typically lasting between a few hours and a day. Small ARs, which have pores and not spots, have flux values between 1 1020 and 5 1021 Mx and last between a few days and a week. · · Large ARs, typically with spots, have a flux between 5 1021 and 3 1022 Mx and last in the · · 51 maxwell = 1 gauss cm2 6The westernmost spot 10 order of several weeks to a month. In Fig. 2.2 we see an active region in the photosphere and chromosphere containing fibrils, plage and pores and in the next section we explain what these regions are in more detail.

2.3 Sunspots

Possibly the earliest observation of a sunspot dates back to 467 BC. Anaxagoras of Clazomenae potentially saw one by eye and postulated that part of the Sun had broken off. He thought that the spot was a gash left in the Sun from where a chunk broke off. He then predicted that this part would fall onto the Earth as a meteor in the near future. By complete coincidence a large meteor ’the size of a wagon’ fell near the town of Aegospotami. This was seen as a confirmation for Anaxagoras’ theory that was cited for centuries after that (Bicknell 1968). Around this time several other observations were made of sunspots by Greek and Chinese astronomers (e.g. Vaquero 2007), all by eye. Sunspots were observed by telescope almost immediately after it was invented. Fabricius was the first to publish about them in 1611, closely followed by Galileo in 1613 (Fabricius 1611; Galilei 1613). However Thomas Harriot and Christoph Scheiner are both recorded to have observed them before that (Solanki 2003). Following the discovery of the telescope, astronomy and the study of the Sun slowly became more systematic and reliable over the course of many decades, leading to many rapid revisions in our understanding of the Sun and its features. The solar cycle and the empirical laws described in the prior section were made possible due to well over two centuries of tracking the location and number of sunspots. The regions themselves were better understood with the advent of higher-resolution telescopes and later polarimetry, which made clear that sunspots are magnetic structures that appear as a dark core surrounded by a lighter radially extending structure, called umbra and penumbra respectively. Sunspots are active regions with the strongest magnetic field, where convection is reduced or suppressed. In the photosphere this is between 1000 and 1500 G when averaged over the entire sunspot, but varies gradually from values between 1800 and 3700 G in the umbra and 700 to 1000 G in the penumbra, with the average field strength in the quiet Sun being 100 G. Both parts of the sunspot are colder and darker than the quiet Sun, with temperatures in the umbra ranging between 3900 and 4800 K and 5400-5500 K in the penumbra. The umbra radiates only 20-30% of the bolometric flux of the quiet Sun, while the penumbra radiates 75-85% as much. A sunspot without a penumbra is called a pore (Solanki(2003), see Fig. 2.3).

2.4 Plage

Plage is a chromospheric bright region, typically seen in the vicinity of pores and sunspots. Plage was ﬁrst observed in the chromosphere by Lockyer(1869) using a spectrograph and later named by Deslandres(1893). It is commonly believed that the regions are poetically named after the French word for ’beach’ due to their resemblance to sand in white-light observations. However, this does not seem to be the case from Deslandres’ original paper, where he writes 2. The solar atmosphere and active regions 11

Figure 2.3: Maps of the solar atmosphere in diﬀerent wavelength bands showing AR 12713 in diﬀerent atmospheric heights ranging from the photosphere to the corona. From left to right: HMI continuum, AIA 1700 Å, HMI photospheric magnetogram, AIA 304 Å and AIA 171 Å taken by the Solar Dynamics Observatory (SDO, Pesnell et al. 2011). The black square represents the observed area for Paper I. 12

’Les facules sont, par definition,´ les plages brillantes de la surface solaire...’ In this context ’plage’ is interpreted as ’region’, ’area’ or ’zone’, alluding to the Latin term ”Plagae” (singular ”plaga”) meaning ”regions”, ”areas”, ”spots.” Nowhere in the paper does Deslandres describe it as a beach or make a comparison, in fact he suggests calling them ’facular flames’. It seems that myth comes from the fact that the French word changed meaning since then. Before this term caught on these regions went under the name of ’flocculi’ (Latin for ’tufts of wool’) as defined by Hale & Ellerman(1904). Plage is classically defined as a bright region observed in Hα and other chromospheric lines (Carroll & Ostlie 2017). According to Dunn & Zirker(1973) there is a smooth transition between quiet Sun and plage, with the appearance of chromospheric structures having the form of ’bright and dark mottles’ in the quiet Sun, ’bright mottles’ packed together to form smaller ’plagettes’ in enhanced network and plage with extensive bright areas and dark fibrils in active regions. Nowadays, many authors identify plage regions by only looking at the photospheric magnetic field concentrations, regardless of the Hα intensity that is associated with those regions. A comprehensive definition describes plage as areas where the magnetic field in the photosphere is confined in the intergranular lanes and forms a magnetic canopy in the chromosphere that is hot and bright in most chromospheric diagnostics such as the Ca II H & K lines, the Ca II infrared triplet lines, Mg II h & k and Hα (Carlsson et al. 2019; Chintzoglou et al. 2020). This excludes superpenumbra, pores, or elongated fibrillar structures. In Paper I we do not have any detailed information on photospheric magnetic fields. Therefore we delineate the plage regions by first selecting an area where the Ca II 8542 Å line core is bright. This is then manually adjusted to exclude fibrils7. Plage regions are important structures that act as the footpoints of coronal loops and the origins of fibrils, making them an important interface for coronal heating (e.g. Reardon et al. 2009; Carlsson et al. 2019; Chitta et al. 2018; Yadav et al. 2020). The photospheric magnetic field in plage is around 1500 G (Buehler et al. 2015).

2.5 Fibrils

Fibrils form a dense canopy at chromospheric heights that connect regions of opposite polarity and appear like the quiet Sun in the photosphere. They are also known as mottles or straws if found on the disk and spicules if seen on the limb (Rutten 2006; Kianfar et al. 2020). Spicules were first observed by Father Angelo Secchi who described them as vertical flames in the solar atmosphere (Secchi 1877) and named by Roberts(1945). Both Lippincott(1955) and Loughhead(1968) credit the naming of fibrils to D’Azambuja. The first report of fibrils in the Ca II K line was done by (Zirin 1974). Fibrils have a lifetime of between 3 to 5 minutes and are therefore highly dynamic (Gafeira et al. 2017; Kianfar et al. 2020). In Ca II K and Hα fibrils are more opaque than in Ca II 8542 Å , where we can see further down than in the prior two lines. The lower and upper fibrillar structures tend to align with the magnetic field most of the time, but not always (de la Cruz Rodr´ıguez & Socas-Navarro 2011), This misalignment could

7See Section 2.5. 2. The solar atmosphere and active regions 13 be due to a decoupling between ions and neutral atoms (Leenaarts et al. 2015; Mart´ınez-Sykora et al. 2016; Carlsson et al. 2019), see Fig. 2.2.

Chromospheric diagnostics

In this chapter we describe the most important spectral lines that are used to study the chromosphere and discuss their merits. We also discuss the basic theory behind radiative transfer, polarization and how this is used to infer physical parameters from spectro-polarimetric observations.

3.1 Chromospheric spectral lines

Physical conditions in the chromosphere can be inferred through the analysis of the spectral features that form there. In the following section we will discuss the most common chromospheric diagnostics as described in Carlsson et al.(2019). We illustrate several of these lines in Fig. 3.1, showing their relative heights in a slice of a Bifrost model (Gudiksen et al. 2011). In this image we see how the activity inside the solar atmosphere can aﬀect the line formation height. For this reason we describe height in the atmosphere in terms of log τ500, rather than a given amount of km above the solar surface. In the optical we have several chromospheric lines that have been the focus of solar research for a long time due to their accessibility for ground-based observations. The most important amongst these are the Ca II H & K and Ca II 8542 Å lines. The former are a pair of lines that can be found at 3968 and 3934 Å respectively. These lines both have the ground state of Ca II as a lower level, which is the dominant ionization stage up to about 13000 K. Their opacity is strongly correlated with the column mass and the source function is partially coupled to the local Planck function, making these lines a good diagnostic for the local temperature. Additionally, the short wavelength of the lines translates to a higher diﬀraction limited spatial resolution. The Ca II IR triplet lines are found at 8498, 8542 and 8662 Å respectively. All three have a metastable level as their lower levels, which means that their opacity is more temperature dependent than the H and K lines. The most used of the three is Ca II 8542 Å , as it has the strongest magnetic sensitivity of the three and is least troubled by blends. As a result, the Ca II 8542 Å line has become one of the foremost chromospheric diagnostics

15 16

Figure 3.1: Several diagnostics shown in a slice of a Bifrost model. Optically thick lines are displayed at τ = 1 and for the optically thin He II 10830 Å line we show the contribution function to intensity. Additionally the plasma β = 1† line is given. Image reproduced from Carlsson et al.(2019). S ee Chapter 4.1. † for temperatures, line-of-sight velocities, and the magnetic ﬁeld vector (see, e.g., Socas-Navarro et al. 2000a;L opez´ Ariste et al. 2001; Pietarila et al. 2007; de la Cruz Rodr´ıguez et al. 2012, 2015a; Quintero Noda et al. 2017; Henriques et al. 2017; Centeno 2018;D ´ıaz Baso et al. 2019). It is also the line that we use for our observations in Paper I.

Next we have the Hα line and He II 10830 Å . The former is very temperature sensitive due to the lower level having a relatively high excitation energy. Coupled with the low atomic weight of hydrogen, this results in large amounts of thermal broadening. Because of this, the width of the line is used as a diagnostic for temperature, but compared to the other lines it is not a good diagnostic for non-thermal motion. The He II 10830 Å line is found on the opposite side of the optical spectrum from the Ca II H and K lines, resulting in a potential diffraction limited resolution that is 3x lower. Despite this, the line is an important diagnostic for the upper chromosphere due to its high excitation energy that is primarily sensitive to recombination after photoionization by the coronal radiation field. Hydrogen and helium absorption prevent this process from happening lower in the chromosphere. This peculiar line formation mechanism results in an optically thin line formed in a thin layer, see Fig. 3.1. The line is a diagnostic to magnetic fields but only clearly visible in ARs.

In the ultraviolet we have the Mg II h & k lines at 2803 and 2796 Å respectively. These lines are comparable to the Ca II H & K lines, but have a higher opacity due to the larger abundance of magnesium. The self-reversed core is mostly decoupled from the Planck function, making them poor diagnostics for temperature, but in turn they are sensitive to Doppler shifts which makes an excellent diagnostic for the line-of-sight velocity at τ = 1. These lines are observed with the IRIS satellite. 3. Chromospheric diagnostics 17

3.2 LTE vs Non-LTE

Considering the simple case of a gas in an ’ideal box’, we can get to a state where the gas and black body radiation will reach a steady-state equilibrium condition, where there is no net flow of energy inside the box and every process occurs at the same rate as its inverse process, like for example the absorption and emission of photons. We call this condition thermodynamic equilibrium. Here the distribution of atoms over different excitations is given by the Boltzmann distribution, and the distribution of atoms between different stages of ionization is given by the Saha equation. The former being

nr,s gr,s (χ , χ , )/kT = e− r s− r t , (3.1) n g " r,t #LT E r,t where nr,s is the number of atoms per unit volume in level s of ionization stage r, gr,s the statistical weight of level s in stage r, χr,s the excitation energy of level s in stage r, measured from the ground level (r, 1) of stage r and χ χ = hν for a radiative transition between levels r,x − r,t (r, s) and (r, t), where level s is “higher” (more internal energy) than level t. The Saha equation is

3/2 Nr+1 1 2Ur+1 2πmekT χ /kT = e− r , (3.2) n N U h2 " r #LT E e r ! where Ne is the electron density, me the electron mass, with Nr and Nr+1 the total population densities in the two successive ionization stages r and r +1, χr = hνthreshold the ionization energy of stage r (the minimum energy needed to free an electron from the ground state of stage r). Ur is the partition function given by χr,s/kT Ur = gr,se . (3.3) Xs This model works well for an ’ideal box’, but not for a star which has a net outwards flow of energy and a varying temperature. This means that gas particles or photons can move from their initial position to a new hotter or cooler location. These particles then interact with the environment by means of collisions and by absorbing and emitting photons. However, if the distance over which temperature changes is large when compared to the distance travelled by gas particles and photons (the mean free path), we create a de facto ’ideal box’ from which the particles and photons cannot escape. This local environment will behave in much the same way as our idealized example where the Boltzmann and Saha equations are valid. This is typically called the local thermodynamic equilibrium (LTE, Carroll & Ostlie 2017). Another property that comes with an ’ideal box’ is that the source function of the gas equals the blackbody Planck function (S ν = Bν) as long as the radiation is negligible and the level populations can be described by the Saha-Boltzmann distribution (Rutten 2003). However, things get more complicated when LTE does not hold and collisional processes no longer dominate over radiative processes. This situation, called non-LTE (or NLTE), thus no 18 longer can be described as an ’ideal-box’. We can no longer make any assumptions about the source function, and have to consistently calculate it from the opacities and emissivities. Typ- ically one assumes statistical equilibrium implicitly, which implies time invariant populations and radiation fields and is usually done by using the Maxwell distribution and complete redistribution in frequency and angle. Therefore in order to integrate the radiative transfer equation one has to compute the source function explicitly by solving the statistical equilibrium equation. As shown above, this depends on the rates and the radiation field. Typically this non-linear and non-local problem is solved in an iterative way, which requires a much larger computational effort than with LTE calculations (Rutten 2003; Kubat´ 2014). See Chapter7 for more information on inversions.

3.3 Polarization of light

George G. Stokes first described a method of quantifying the polarization of light in Stokes (1852), introducing four quantities that are functions of observables of the electromagnetic wave. These quantities are now known as the Stokes parameters. With these parameters it is possible to express the polarization state of light. Perhaps the easiest way to derive the Stokes parameters is when they are likened to a set of filters that transmit 50% of the fully polarized incident light. The first filter is isotropic, not discriminating between the states. The second and third filters transmit linearly polarized light and have a respectively horizontal and diagonal transmission axis. The fourth filter is opaque to linear states, only letting through circularly polarized light. If each of these filters is then positioned in an unpolarized beam of light and we measure the intensity at the end of the setup. Then we can express the Stokes parameters in terms of the four measured intensities I0, I1, I2, I3 and get the following equations:

S 0 = 2I0, (3.4) S = 2I 2I , (3.5) 1 1 − 0 S = 2I 2I , (3.6) 2 2 − 0 S = 2I 2I . (3.7) 3 3 − 0

In this representation S 0 is simply the incident irradiance and the remaining three parameters specify specific polarization states. S 1 represents a horizontal state when S 1 > 0 or a vertical state when S 1 < 0. If S 1 = 0 then the light has no preferential orientation along these axes. In this case it can still have a preferential orientation along the diagonals and have a circular polarization or be unpolarized. S 2 represents the diagonals in a similar way to S 1 and S 3 reveals whether the beam is right handed (S 3 > 0), left handed (S 3 < 0) or neither if the value is zero. These solutions are not unique, and other equivalent definitions exist. However, this one gained popularity over the others thanks to Stokes’ work and is now the standard in most scientific fields that work with polarization 3. Chromospheric diagnostics 19

Figure 3.2: a) A schematic representation of the theoretical measurement setup of the Stokes parameters. b) A cartoon of the polarization with the arrows pointing in the direction of the light if the wave is travelling towards the reader in a visual representation of Eq. 3.15- 3.18.

Following the derivation from Hecht(1970) we can then deﬁne the Stokes parameters from the general expression for a quasi-monochromatic wave propagating in the z direction.

E = E (t) cos(ω ¯ t δ (t))x ˆ, (3.8) x 0x − x

E = E (t) cos(ω ¯ t δ (t))y ˆ. (3.9) y 0y − x

Here, E(t) = Ex(t)+Ey(t), ω is the angular frequency and δ the phase. We know that the incident irradiance is given by 2 2 I = 0c Ex(t) + Ey (t) , (3.10) where the angled brackets indicate the time-averagedD E D value,E0 is the vacuum permittivity and c the speed of light. When averaged over a relatively long time interval compared to the coherence time of the wave 9 (typically of the order of 10− s for natural light), we get a time averaged irradiance of the shape of c I = 0 E2 + E2 . (3.11) 2 0x 0y D E D E 20

This is equivalent to 2I0 and thus S 0 by the definition in Eq. 3.4. In order to get S 2 we first need an expression for the intensity transmitted by a horizontal linear polarizer. This is equal to the 0c 2 x component of the total field, which can be written down as I1 = 2 E0x . Plugging this into Eq. 3.5 then gives us c D E S = 0 E2 E2 . (3.12) 1 2 0x − 0y (x ˆ+yˆ) D E D E We get S by defining E (t) = E(t), meaning that I = c E2 and therefore 2 45 √2 2 0 45 c D E S = 0 2E2 E2 cos δ , (3.13) 2 2 0x 0y D E where δ = δ δ . The last parameter requires us to first evaluate I , which can be constructed y − x 3 from the E45 linear polarizer defined above and a quarter wave plate. If we place the fast 1 π axis alongy ˆ, we will then find that the y-component is ahead of the x-component by 2 .A right handed beam, which means that the proposed filter I3 has to be opaque to left handed polarization, which is achieved by placing the retarder before the polarizer. This retardation can π be introduced in the equation by adding a 2 term to the phase component. The field is the same π as for S 2 but with the added phase. (δ0 = δ + 2 ) This gives us the final parameter in the shape of c S = 0 2E2 E2 sin δ . (3.14) 3 2 0x 0y D E In solar physics we denote S 0, S 1, S 2 , S 3 as I, Q, U, V. Additionally it has become common 0c practice to drop the constant 2 from the equations, giving us the following four equations in terms of amplitudes and relative phase:

2 2 I = E0x + E0y , (3.15) Q =D E2 E D E2 E , (3.16) 0x − 0y D 2 E 2 D E U = 2E0xE0y cos δ , (3.17) D 2 2 E V = 2E0xE0y sin δ . (3.18) D E We note that the Stokes parameters obey I2 Q2 + U2 + V2. ≥

3.4 Zeeman eﬀect

Spectral lines are the result of electrons transitioning between specific energy levels in atoms and molecules. These energy levels are defined by the quantum numbers L, S, J and M in the Hamiltonian for an atom in a uniform external magnetic field Bext. When the magnetic field associated to the spin-orbit coupling (Bint) dominates over Bext, we are in the regime where we

1Light polarized along the fast axis encounters a lower index of refraction and travels faster through wave plates than light polarized along the slow axis. 3. Chromospheric diagnostics 21

Figure 3.3: Components of oscillation along the magnetic field. The π component keeps its unperturbed frequency while the σ components have their frequency shifted by ν . Reproduced from Sanchez et al. ± L (1992). can use perturbation theory on the Hamiltonian, where the magnetic part can be used as the perturbation for the rest of the Hamiltonian. In first order, this gives us the energy splitting that is produced by Bext on the unperturbed energy level, which splits a given level with total angular momentum J, into 2J+1 sublevels. We call this effect the Zeeman effect (Zeeman 1896). An intuitive way to describe this is by looking at the classical description of a radiating atom2. In this case we can decompose the oscillation of an electron into three separate components, each with their own frequency (see Fig. 3.3). From this we can see that two different situations arise, depending on the viewing direction of the observer. If we observe along the the direction of the magnetic field, then we will find both clockwise and counter-clockwise circular polarization at a respective frequency of ν + g¯ν and ν g¯ν , where ν is the unperturbed frequency and ν the 0 l 0 − l 0 l Larmor frequency which scales linearly with Bext. However, if we observe perpendicularly to the magnetic field, then we will find linear polarization parallel to the magnetic field and linear polarization perpendicular to the field at a respective frequency of ν = ν and ν = ν g¯ν . 0 0 ± l Hereg ¯ is the effective Lande´ factor, which acts as a scalar that represents the sensitivity of the line to a magnetic field. In the photosphere there are many lines available, which has given us the chance to select the ones with the highest Lande´ factor for observations. These lines have Lande´ factors that are typically around 2. Far fewer lines are available in the chromosphere, especially unblended ones. Therefore, chromospheric lines like Ca II 8542 typically have a Lande´ factor closer to 1. The shifted components are commonly known as the σ components and the stationary component as the π component. These two special cases of the Zeeman effect are known as the Longitudinal Zeeman effect (only circular polarization) and Transverse Zeeman effect (only linear polarization) respectively. Of course, usually we are not aligned perfectly in either way and see a combination of the two. In quantum theory we can generalize this example to allow all values for J. Depending on the transition, this is described by the normal Zeeman effect

2Which holds only for a spectral line that has J=0 as the lower level and J’=1 as the upper level. 22 which is equivalent to the classical case and a number of other transitions. However, unlike in the classical case, some transitions produce more than three frequencies, which is called the anomalous Zeeman effect. The split introduced by either Zeeman effect is typically less than other line broadening effects, and therefore not measurable directly in the intensity profile. In- stead we measure the polarization effects that it produces. For this reason the Zeeman splitting is also known as Zeeman broadening (Sanchez et al. 1992; Stix 2002; Landi Degl’Innocenti 2004). The longitudinal Zeeman effect produces a stronger signal than the transverse Zeeman effect. The circular Stokes V component is typically asymmetrical to the line center and more sensitive to weaker fields. The linear Stokes Q and U components are typically symmetrical around line center and are much harder to measure. In fact, measuring these components is the main challenge in Paper I. Doppler broadening can be a further complication to measuring the Zeeman broadening. This effect is caused by the distribution of atomic and molecular velocities and their projection with respect to our line-of-sight. These different velocities result in different Doppler shifts, which cumulatively result in a broadening of the line when averaged. The width of a spectral line due to Doppler broadening is given by

λ0 2kBT 2 ∆λD = + vturb. (3.19) c r m 3 With vturb, being the microturbulent velocity (or microturbulence ) and λ0 the lines wavelength at rest. On the other hand, the Zeeman broadening is given by

λ2e B ∆λ = 0 0 ext . Z 2 (3.20) 4πmec

Some intuitive understanding into the Stokes parameters and their sensitivity can be gained by looking at the equations used to derive the so called weak field approximation (WFA) (see Eq. 3.21 and Eq. 3.22). These are only valid in the regime where the Zeeman broadening is less than the Doppler broadening of the line,g ¯∆λ /∆λ 1. For the Fe I 6302 Å line, which is Z D commonly use to measure photospheric magnetic fields, this would be B 1500 G, and for ext Ca II 8542 Å it would be B 1300 G. Both of which are values that we typically only find in ext flares and sunspots. As long as this limit is valid, we find in Landi Degl’Innocenti(2004) that the polarization of light scales with the first and second derivative of the intensity and are given by ∂I(λ) V(λ) B (3.21) ∝ || ∂λ and

3A parameter that accounts for unresolved motion along the LOS, but generally is used as a fudge parameter to compensate for model deﬁciencies. 3. Chromospheric diagnostics 23

∂2I(λ) Q(λ) U(λ) B2 . (3.22) ∝ ∝ ⊥ ∂λ2

Here, we see that Stokes V is proportional to the longitudinal part of Bext and Stokes Q and U are proportional to the square of the transverse part of Bext.

3.5 Polarized radiative transfer equation

Light produced in the lower layers of the Sun travels through the solar atmosphere to eventually fall onto our detector. While crossing the semi-transparent atmosphere, the light is altered by means of absorption or scattering and new light can be added by the medium itself. The amount of absorption depends on the absorption coeﬃcient αν of the material and the amount of material that the light passes through. When the light is unpolarized, this process is described by the transfer equation (Rybicki & Lightman 1979; Rutten 2003) dI ν = α I + j . (3.23) ds − ν ν ν

Here Iν is the intensity at a given wavelength, s the geometrical path length and j the monochromatic emissivity. This equation can be re-written to include the source function (S ν = jν/αν), after which it has the shape of dI ν = α (I + S ) . (3.24) ds − ν ν ν When including polarization, the radiative transfer equation (RTE) stays mostly the same, except that the absorption is now handled by the absorption matrix K and I and S become vectors. Following the notation from Landi Degl’Innocenti(2004), if we neglect continuum scattering, we can express the RTE as the sum of the continuum term and the line term

I 1 0 0 0 I S c hI hQ hU hV I S L  − −  d Q  0 1 0 0 Q hQ hI rV rU Q    = kc     + κL −    . (3.25) ds U −  0 0 1 0 U hU rV hI rQ U         −     V   0 0 0 1   V   h r r h   V         V U Q I           −       Continuum    Line                | {z } | {z } Here κL(= kl/kc) is the ratio between the frequency-integrated line absorption coe fficient (kl) and the continuum absorption coefficient (kc) at the given wavelength. In the absorption matrix for the spectral line we find three different contributing terms, each of them sensitive to the magnetic field in a certain geometry. On the diagonal we find the terms hI, which can amplify or attenuate the incoming radiation regardless of its polarization. The dichroism factors hQ,U,V , account for the same process, but are selective to only a certain polarization state. Finally, 24

there are the dispersion terms rQ,U,V , which describe the dephasing between the different states. These terms together can be used to infer the magnetic field of the medium that the light is travelling through, as both the h and r depend on the wavelength in the line, the Doppler shift and the Zeeman components. Numerically solving the radiative polarized transfer equation is an important part of doing spectropolarimetric inversions. We refer to Landi Degl’Innocenti (2004) for a more complete description of the mathematics behind this representation. However, even without exactly defining the h and r terms, we can see that the continuum term 4 of the RTE is not affected by magnetic fields and kc remains the same as in a non-magnetized medium. This means that all information on the magnetic field is contained in the propagation matrix of the spectral line.

4This is true for the ﬁelds typically found on the Sun, but no longer the case for extremely strong ﬁelds like those found on magnetic white dwarfs (B 107). ≥ 4

Chromospheric magnetic ﬁelds

In this chapter we will discuss how magnetic ﬁelds are measured on the Sun and cover current literature on the topic, focusing on plage.

4.1 Inferring Solar magnetic ﬁelds

The inference of magnetic fields has been a part of solar physics ever since Hale first measured the Zeeman split introduced by the magnetic field of a sunspot in several lines (Hale 1908), although first reported by Lockyer(1868b) as found by del Toro Iniesta(1996). The importance of the subject grew over the years since, but only truly took off in the 1960’s, when vector magnetometry (the practice of looking at both the linear and transverse magnetic field) became a popular tool for interpreting MHD structures of the solar atmosphere. The field then slowly advanced with the advent of higher resolution observations and more powerful computers with larger storage capacity (Stenflo 2017).

Hale remarked that the shape of the corona suggests a global dipole field along the Sun and that the radial structure of Hα fibrils around sunspots suggests the presence of a strong magnetic field. The former claim was not so easy to measure, with the earliest attempts only happening decades later (e.g. Harvey 1969; Lin et al. 2000; Yang et al. 2020).

The bulk of magnetic field estimates to date are done in the photosphere, due to the fact that we can model the spectral lines with relatively simple models and are often able to assume LTE. Additionally, magnetic-field strengths here are stronger than in the layers above. Additionally, the Lande´ factors are typically larger. In the lower photosphere magnetic features cover at most 1% of the quiet solar surface, but expand dramatically with height to the point where the entire ’surface’ is covered in the middle chromosphere, with much more complex fields (Solanki 1998). It was less computationally heavy to model these observed photospheric lines as they could be considered to be under LTE and Zeeman sensitive lines in the chromosphere are broad, optically thick and have a complex line formation.

25 26

The combination of the factors that make chromospheric magnetic field measurements harder than photospheric measurements is perhaps best illustrated by comparing the timelines. Early chromospheric magnetic field measurements began in the later part of the 20th century, with notable observations being made by Rust(1967) who used the H α line to measure a 5 G field in a prominence and Leroy(1977) who also observed prominences via the He D 3 line and got a similar result. The chromosphere is largely dominated by magnetic fields due to the low density of material in this layer. The density of the gas decreases exponentially with height, while the magnetic field decreases at a slower pace. In the photosphere only strong fields of 1.2 kG and above can com- pete with the gas pressure, which means that the gas generally dominates outside regions with strong magnetic field concentrations like sunspots and pores. In the chromosphere the pressure decreases and with that the required field strength required for domination. In plasma physics the term β is used to denote the ratio between the gas pressure and the magnetic pressure. β = 1 is typically reached around half way up the chromosphere in quiet sun (average height 0.9 Mm, see Fig 3.1), but close to 0 Mm in strong magnetic regions. The chromosphere functions as a sort of transition between the layer where plasma motions dominate, and the layer where magnetic fields dominate. This decreasing pressure is also why fields spread out horizontally, going from concentrated regions to volume filling regions (D´ıaz Baso et al. 2019). It is important to understand the topology of the magnetic fields to understand the chromosphere because of this. The heating of the chromosphere requires a large amount of energy, roughly an order of magnitude more than that required to heat the corona and heliosphere combined. The temperature increase with height cannot be explained with radiative heating and needs to be aided by other processes that are in part driven by magnetic fields (Carlsson et al. 2019). Understanding chromospheric heating is an important step towards understanding coronal heating and a better understanding of these systems may make space weather predictions possible.

4.2 Magnetic ﬁelds in plage

Plage has been extensively studied in the photosphere, which has led to the view that the magnetic field in the lower layers is concentrated in the intergranular spaces (e.g. Topka et al. 1992). This field has been measured in the photosphere and has been found to be mostly vertical, with average values slightly above 10◦ from the local vertical, but only when far enough from larger active regions like sunspots. The field strength is measured around 1500 G in the photosphere (at log τ = 0.9, Buehler et al. 2015). Plage close to larger sunspots has been observed to 500 − have field inclinations as large as 48◦ (e.g. Bernasconi et al. 1994; Sanchez Almeida & Martinez Pillet 1994; Pillet et al. 1997). Similar to the quiet sun magnetic fields, the plage fields spread out with height due to the decreasing gas pressure, going from very localized fields to a volume filling field which decreases strongly with height (Solanki et al. 1992, and references therein). This was confirmed by Buehler et al.(2015), who detected the expansion of magnetic elements as a function of height using inversions. 4. Chromospheric magnetic fields 27

The fact that magnetic fields in plage are mostly vertical (in regions far enough from sun spots) further complicates efforts to measure complete field vectors at larger heights. Studies of plage in the photosphere suggest that the field might become more inclined further up (e.g., Solanki et al. 1992; Buehler et al. 2015), however, this has not been proven and with only a handful of chromospheric plage magnetic field measurements have been made to date, leaving the strength and topology of magnetic fields in plage a subject of ongoing discussion. Current measurements are limited and values vary quite widely, between 60 and 450 G in the three papers that we summarize below. The lower value was reported by Asensio Ramos et al. (2017) for the transverse component1 of the magnetic field. This value is the median value of a Bayesian hierarchical model of the linear polarization of the data. The distribution itself is lopsided towards the higher values, with a tail that goes past 200 G. In the case of this measurement it is important to remark that the plage that was used is a young emerging-flux region and that the measurements were performed over elongated fibrillar structures. Therefore these measurements do not fit our definition of plage, which should be kept in mind when comparing the value to the rest. The high end of the range was reported by Morosin et al.(2020), who studied a plage region in three different spectral lines with a spatially-coupled weak-field approximation in order to infer the stratification of the field in different opacity windows. Their reported value is for the longitudinal magnetic field. However, if the field is still mostly vertical at these heights, the value should be close to B . Finally, a value of B = 200 G is mentioned | | | | in the review paper by Carlsson et al.(2019) as the ’canonical value’. The values that were presented in Paper I match closely to the values of Morosin et al.(2020).

1 We define the transverse magnetic field with respect to the line of sight as B0 and the longitudinal magnetic ⊥ field in the same reference frame as B0. Their unprimed counterparts represent the same quantities with respect to the solar surface. ||

Observations and data reductions

In this chapter we take an in-depth look at the telescope and instruments with which our data were taken as well as the reduction process that we applied to the data.

5.1 The Swedish 1-m Solar Telescope

The observations used in Paper I were acquired with the Swedish 1-m Solar Telescope (SST, Scharmer et al. 2003a), located at an altitude of 2360 m at the Observatorio del Roque de los Muchachos (ORM) on the Canary island of La Palma. This is a popular location for professional telescopes and has been a preferred location for European night-time telescopes since early site testing in the 1970’s (Beer et al. 2016). A similar conclusion was made for day-time telescopes, where contemporary site testing marked the future location of the SST as the best in the world for such telescopes (Scharmer et al. 1999). This opinion has not changed since, as a location next to the SST has recently been pointed out as a potential site for the next generation European large aperture solar telescope (EST, Sosa Mendez´ et al. 2011). Many variables are considered when ranking sites for a telescope, however a measure for atmospheric turbulence that we call the Fried parameter (r0, Fried 1966) is the most important for solar telescopes due to the atmospheric heating caused by the observed target. Light beams are disturbed when propagating through the atmosphere, with the amplitude of disturbance depending on the level of turbulence and the associated refractive-index inhomogeneities. Each turbulence cell deflects the light at a slightly different velocity and direction. When such a beam is later brought into focus onto a detector, the quality of the image will be affected by the distortion of the shape and intensity of the wavefront. The sum of these distortions is called seeing, and the Fried parameter denotes the diameter of a circular area over which the root mean square wavefront error caused by the atmosphere is equal to 1 radian. An unaided telescope can only reach the diffraction limit when r0 is larger than the telescope aperture (D, Fried 1966). At the SST we typically want an r0 of at least 10. The SST is a ø 97 cm vacuum tower telescope with an alt-azimuth turret design. The turret

29 30 of this domeless refractor is made as compact as possible to minimize the disturbance of the wind. The large gears and bearings inside the turret allow for a very high stiffness and precision, resulting in a high stability against wind shaking (Mitchell 1912; Dunn 1964; Wyller & Scharmer 1985; Scharmer et al. 1999). The fused-silica entrance window doubles as its first optical element, and is designed as a singlet lens with a low thermal expansion coefficient. The light is directed downwards from the turret into the main tube by means of two folding mirrors under a 45◦ angle. The light then travels to the bottom of the tower, where it is reflected at an angle into the Schupmann corrector (Schupmann 1913). This system re-images the 1-m singlet objective onto a 25 cm corrector that consists of a negative lens and a mirror. The chro- matic aberrations caused by the objective are then effectively canceled out by the Schupmann corrector after the light makes a double pass through the corrector lens. An achromatic image is formed at the secondary focus. After this the beam reaches a tip-tilt mirror that compensates for image motions from seeing and tracking errors. Remaining aberrations are countered by an 85-element hexagonal adaptive optics system that works with a Shack-Hartmann wavefront sensor with a matching geometry (Scharmer et al. 2003b, 2019). The turret design does not remove field rotation, which has to be taken care of during the data-reduction phase. Because effective exposure times in solar observations of this kind are by definition short, this will not introduce any smearing as would be the case during longer exposures. After being re-imaged, the beam is split into a red and blue beam by a dichroic beamsplitter. The correlation tracker camera is located on the blue beam. Images taken with this camera are used for a ’live view’ of the blue beam, as well as fed into the correlation tracker computer, which calculates the shift between two consecutive images and translates this into a tip and a tilt offset for the tip-tilt mirror. The rest of the light in the blue beam is used for scientific acquisition by the CHROMIS system. CHROMIS, which stands for ’CHROMospheric Imaging Spectrometer’ is a spectrometer based on a dual Fabry–Perot´ interferometer interferometer (FPI, Fabry & Perot 1901), designed to function at the wavelength range of 3800 to 5000 Å. The system is optimised for observing the upper chromosphere via the Ca II H & K lines (Scharmer et al. 2019). The WFS camera is located on the red beam and controls the AO to compensate for the aberrations that are left after the tip-tilt mirror counters the image shift. The rest of the light in the red beam is used for scientific acquisition with the CRISP instrument. The CRisp Imaging Spec- troPolarimeter (CRISP, Scharmer et al. 2008), is an FPI spectropolarimeter operating from 5100 to 8600 Å. Apart from the difference in spectral operating range, CRISP is capable of measuring polarization by using ferro-electric modulation combined with a polarizing beamsplitter. CRISP uses three cameras in its system, one for wide band-imaging and two for simultaneous narrow-band imaging. CHROMIS has one narrow-band and two wide-band cameras. The heart of an FPI interferometer is the etalon. This is a pair of partially reflective optical plates separated by a distance d. When light enters the etalon, it will reflect multiple times where certain wavelengths will experience constructive interference, while others experience destructive interference. By controlling the incidence angle, and d, it is possible to tune which wavelengths are transmitted and which are not. This makes it an excellent tool for single line spectroscopy. 5. Observations and data reductions 31

Figure 5.1: Schematic drawing of the SST reproduced from Scharmer et al.(2003a). Field lens and mirror are seen in A, the Schupmann corrector system is seen in B and the tip-tilt, AO setup and re-imaging lens are shown in C. 32

However, a fundamental problem with the FPI interferometer is that the transmitted wavelength shifts with the angle of incidence, which can result in a variable central wavelength across the field of view, which has to be corrected during the reduction phase (Keller et al. 2015). In the case of both CRISP and CHROMIS, quasi-monochromatic images are produced by using two FPI etalons in tandem. The first high spectral resolution etalon (HRE) defines the observed wavelength and the second lower spectral resolution etalon (LRE) suppresses the first few higher order peaks. All other peaks are removed by a prefilter that limits the spectral range of the incoming light. A theoretical representation of the approximate CRISP transmission profile at 630 nm is shown in Fig. 5.2a. Both instruments use the combination of high reflectivity for the HRE and low reflectivity for the LRE, as initially proposed in Scharmer(2006). This setup ensures a wide enough spectral passband for the LRE to accommodate the wavelength shifts introduced by the cavity errors of the HRE. This ensures a uniform throughput and, apart from wavelength shifts, a largely stable transmission profile across the FOV (Scharmer et al. 2019). The image scales for CRISP and CHROMIS are 0.06”/pixel and 0.04”/pixel respectively. Usually exposure times for CRISP and CHROMIS are 17.5 and 12.5 ms respectively. These are the respective timescales required to ”freeze” the seeing while also exposing long enough to avoid domination by the read noise, allowing for image reconstruction.

5.2 The SSTRED data-reduction pipeline

After the data is saved and transferred to Stockholm, a series of reduction procedures are required in order to create a science ready data-cube1. During these steps instrumental and atmospheric effects are corrected, images are de-rotated and fringes and other non-source signals are removed. We will provide an outline of this process of the section, but for more details the reader is referred to de la Cruz Rodr´ıguez et al.(2015b) andL ofdahl¨ et al.(2018). In order to process SST data, the first step is to run a setup script that analyzes the directory tree for that day’s observed data. Here both science and calibration data will be identified and saved into a configuration file and an IDL script containing the recommended processing steps. Additionally a work directory is made for each of the two instruments. Like with night-time astronomy, data corrections and calibrations should be applied in reverse order, thus starting at the detector and moving towards the source (de la Cruz Rodr´ıguez et al. 2015b). This implies that atmospheric corrections have to be applied at the very last step. How- ever, this is not possible due to the fact that we cannot do polarimetric calibration before all narrow-band states are co-aligned and de-stretched. Additionally image restoration is required before FPI wavelength shifts can be corrected, as otherwise the spectra would become spatially incoherent between scans. Because of these necessities, the data processing path becomes highly complicated. First the darks, which are taken shortly after the science observations, are summed and made

1A FITS cube with the following parameters: [x,y,time,λ,Stokes(I,Q,U,V)]. 5. Observations and data reductions 33

Figure 5.2: a) CRISP and CHROMIS can achieve quasi-monochromatic images by combining a prefilter, a high resolution FPI etalon to select the wavelength and a low resolution FPI etalon to remove higher order peaks. Image reproduced from de la Cruz Rodr´ıguezet al.(2015b). b) Current optical setup at the SST including the AO, WFS and the CRISP and CHROMIS re-imaging systems. Image reproduced from Scharmer et al.(2019). 34 ready to subtract from all other data frames. This thermal noise or dark current is created in the CCD detector by the thermal movement of electrons on the chip. Cooling down the detector lowers the dark current. This is not the case for the CRISP/CHROMIS cameras as the integration time is very short (at most 16 ms) and the dark current can be ignored compared to the read-out-noise of the detector. Next, the flats are summed and likewise prepared to be applied to the frames. These maps remove inhomogeneities in the sensitivity of individual pixels on the CCD. In order to do this an exposure is needed of a flat and smooth surface in order to assure that all inhomogeneities in the final flat are from differences in pixel sensitivities. The Sun does typically not have a smooth surface, but in a quiet sun region the average area can be assumed as such. Therefore, in solar observations it is customary to point to a quiet region around the disk center and move in circles around it while taking exposures with the AO randomized. This ensures that all solar surface structure is averaged out after the frames are summed. The statistics for both dark and flat frames are checked before summing, and outliers are removed to ensure that the final dark and flat frames are not ruined by shutter glitches, birds crossing the FOV, etc. Bad pixels are found and set to zero in the gain tables. Later steps interpolate these pixels with the average value of surrounding pixels. Pinhole observations allow for inter-camera alignment. To achieve this a pinhole array is placed in the Schupmann focus, which is assumed to be the same in all camera states and can therefore be used to find the relative orientation (rotation and mirroring) of each camera. The light beam entering the SST passes four lenses and four mirrors before the optical bench. They modify the polarization of the incoming light as a function of the pointing angle. Addi- tionally, the optical table containing the CRISP and CHROMIS instrument also hosts a plethora of optical elements that can affect polarization. The polarimetric calibration is performed based on van Noort & Rouppe van der Voort(2008) where calibration data is obtained by using a linear polarizer and a quarter-wave plate, both installed just after the telescope exit window. The quarter-wave plate makes a full circle with 10◦ steps for two (typically) linear-polarization angles. This is enough to accurately infer the modulation matrix.

5.3 The MOMFBD routine

After the darks, flats and camera alignment are taken care of, we can now focus on the image restoration, more specifically the process called Multi-Object Multi-Frame Blind Deconvolution (MOMFBD, van Noort et al. 2005). This is used to remove aberrations that were not correctable by the AO (e.g. areas outside the isoplanatic patch surrounding the FOV of the wave front sensor) or ones that were introduced by the instrumentation. A post-acquisition image-restoration technique is capable of correcting these higher-order aberrations because it does not suffer from time constraints. In the case of MOMFBD this is done by dividing the FOV up into several subfields smaller than the isoplanic patch (typically 5”), which allows us to assume that the point ≈ spread function (PSF) inside each of these subfields is spatially invariant. It is then assumed that any distortion inside such a subfield is a convolution of the real image that has a constant 5. Observations and data reductions 35

Figure 5.3: Dataﬂow in the SSTRED pipeline. Input is given by grey boxes with rounded corners and consists of the raw data and polarization model. Diamond boxes represent ﬁtting and other processing. Output products are represented by white boxes with rounded edges. MOMFBD is indicated separately in a black circle. Reproduced from de la Cruz Rodr´ıguezet al.(2015b). 36

PSF and the aberrations (Lofdahl¨ 2002). The default way to parameterize these aberrations is by use of Karhunen–Loeve` (KL) modes (Roggemann & Welsh 1995). The applied KL modes are based on an expansion of Zernike modes (Zernike 1934) that have uncorrelated coefficients. Because they come from Zernike modes, KL modes are indexed in the same dominating order as was proposed by Noll(1976). However, when using these modes in this order, the expected variance from the atmosphere does not decrease monotonically. For this reason the modes are reordered to ensure this and truncated after the desired precision is reached. The used order is: 2–6, 9, 10, 7, 8, 14, 15, 11–13, 20, 21. With these modes, an unknown wavefront aberration can be expanded into a linear combination of the set. Due to the high cadence required to take full spectro(polarimetric) measurements with CRISP and CHROMIS, only a few frames are collected per wavelength position (and polarization state). This is typically too few to apply MOMFBD. This is solved by simultaneous collection of wide- band (WB) images together with the narrow-band observations for either instrument. On top of that, a second defocused WB camera is included for both instruments to allow for phase diversity (PD) wavefront sensing. This allows for additional constraints for the MOMFBD process by introducing a known parabolic difference in phase over the pupil (Gonsalves 1982; van Noort et al. 2005). A physical chopper is used for CRISP to synchronise its three cameras in order to avoid unwanted variations due to digital readout. Additional constraints are given by the fact that seeing effects change with wavelength position as the images are not exactly co-temporal. Once the aberrations are minimized, a mosaic is created out of the separate subimages. Finally the resulting cubes are corrected for the polarization introduced by the telescope and optical table. This is done in reverse order, starting with the demodulation matrix for the table and moving, after which the same is done for the telescope. Finally the individual scans are derotated using the turret log pointing position and alignment of granulation patterns in the WB images. 6

Post-reduction techniques

After the SSTRED reduction process we found that the polarimetric signal in the data was not strong enough to work with straight away. In order to get a signal that can be inverted, we need to remove fringes and beat down the noise in the individual Stokes images. In this chapter we will go through the steps that were applied in Paper I and delve into the theory behind each of them.

6.1 Fringe Removal

The CRISP frames have two fringe patterns on them that can be seen already in the raw data. These consist of a stable high-frequency pattern that appears to move in the ﬁnal derotated cubes. The origin of these fringes is unknown1, and until recently no centralised tools existed to remove them. For this reason, most people using SST data developed their own tools to remove them (e.g. Vissers et al. 2020 and of course Paper I). In this section we will cover the methods used in Paper I to remove these fringes. The high-frequency fringes are currently removed in the newest release of the SSTRED pipeline, and the method for removing the low frequency fringes presented in Paper I is currently being considered for inclusion. The high-frequency fringes vary in amplitude depending on which Stokes parameter we ob- 3 serve. The largest amplitude is found in Stokes U, at 1.5 10− I , where I is the continuum · c c intensity. Stokes V has the second largest amplitude at half the value of Stokes U, Stokes Q only has 13% of the amplitude and Stokes I 5%. We can neglect the fringes in Stokes I due to their low amplitude and the strong signal in this parameter, but have to remove them in the remaining three before any inferences can be made from this data. We found that the easiest way to remove them is by using a two-dimensional Fourier transform of the image (Lim 1990). This lets us see the frequency domain of the data, which in the simplest terms gives us a map of the periodicity of patterns in the data. The higher the frequency, the further away things are from the center, which represents 0 frequency. Since these fringes are clean, periodic and have

1Some guesses are made at the end of this section.

37 38

Figure 6.1: Fringe removal from reduced data. a) A cartoon of the low frequency fringe extraction process. Each wavelength point (blue dot) contains a linear combination of the fringes found in the left and right wing points (yellow boxes), which are denoted by several lines (black boxes). These fringes on either side can be extracted from the wings applying PCA to the outer most non-continuum wing points. A linear combination of these two fringes is then subtracted from the full spectrum. b) A Stokes U line wing image, showing both the high- and low-frequency fringes. c) The same image after the high-frequency fringes have been removed with FFT filtering. d) The same image after the low frequency fringe was removed using our PCA fringe removal technique. Reproduced from Paper I. a high frequency, they will show up in the Fourier image as a pair of bright dots relatively far from the center. This is good because most of the data will be concentrated in the center, which makes it easy to distinguish between the two. If we then remove the peaks from the Fourier map and transform back the image, we will effectively remove the fringes. Of course it is important to make sure that the mask always covers the peaks, but does not remove any additional data. We found that this is hard to achieve in the final data product, as the images have been derotated, and thus the peaks move in the Fourier plane over time. We solve this by applying this step before generating the final wide- and narrow-band cubes. This method is similar to the procedure used for removing fringes in observations made by the Sunrise telescope (Pillet et al. 2010). The current SSTRED pipeline uses the images from the polarimetric calibration to get the fringe patterns instead of the data images. In Fig. 6.1 b and c we can see the effects of this method on our data. The low-frequency fringes cannot be removed in the same way, as their peaks are too close to the center and masking that frequency would lead to significant loss of data. Instead we remove these fringes by applying Principal Component Analysis (PCA, Pearson 1901). This is a technique that allows for a reduction of the dimensionality of datasets, making the result easier to interpret while minimizing the information loss. This is done by restructuring the data into orthogonal variables that are sorted by variance. These variables, commonly known as principal components, are defined by the dataset alone and free of a priori bias, which makes PCA an adaptive data analysis technique with many different uses (e.g. Jolliffe & Cadima 2016; Skumanich & Lopez´ Ariste 2002; Eydenberg et al. 2005). An intuitive way of visualizing this is by seeing PCA as a fit of a n-dimensional ellipsoid to the data. Each of the n axis of the ellipsoid represents a principal component, with their length representing the variance in that component. If an axis is very short compared to the rest, excluding it will hardly affect the goodness of the 6. Post-reduction techniques 39

ﬁt to the data as only a small amount of information is discarded. In Paper I we apply PCA in two ways, once in the most classical way to reduce noise (See noise reduction section below) and once in a new way to remove the low frequency fringes.

4 The low-frequency fringes in our data had an intensity of roughly 10− Ic and are variable over wavelength. This variability combined with the fact that they have a low frequency means that we cannot remove them with Fourier ﬁltering, which is why we looked for other methods. After studying the structure and behavior of the fringes in our data, we found that the pattern seems to transition smoothly between each wavelength step, changing completely when the left and right wing are compared. We then assumed that the fringes can be modelled as a linear combination of the pattern found in the wings (Buehler 2018, private communication) as shown in the equation below,

f = f (1 λ/n) + f (λ/n). (6.1) λ red · − blue · Here, fλ is the fringe intensity at wavelength position λ, fred and fblue are the extracted fringe intensities from the red and blue wing respectively, and n is the number of data points in the observation (see Fig. 6.1a). This method requires us to have the fringes on both wings, and for them not to be contaminated with signal. Fortunately, the wings of Stokes Q and U typically do not have signal when it comes to plage and pores, and we can find the fringes and noise in the wings. In order to reduce the noise as much as possible, we apply PCA on the two outermost non-continuum wing points on both sides of the line. The first component will mostly consist of the fringe, while the second will mostly be noise. We then use these obtained fringes as fred and fblue in Eq. 6.1. We do not know what causes these fringes, as they can be produced by a wide range of optical elements such as detector windows, mirrors, filters, (polarizing) beam-splitters, etc, (Casini & Li 2019). The appearance of fringes on top of the signal is one of the challenges that we are faced with in astronomical observations. Sometimes they are unavoidable in a system due to its design. For a more in-depth look on the topic of polarized fringes we refer to dedicated studies thereof (e.g. Lites 1991; Semel 2003; Clarke 2004).

6.2 Improving S/N

Once the fringes are removed from our data, we still have relatively noisy profiles in Stokes Q and U. This is partially due to our short integration times and partially because plage is a difficult target for current generation instruments. For this reason we tested and eventually combined several methods of noise reduction. In this section we will cover the methods used in Paper I to reduce the noise. The first thing that we did was to apply a convolutional neural network (CNN, LeCun et al. 1995) fromD ´ıaz Baso et al.(2019) to our data. This network was trained on CRISP spectro- 40 polarimetric data and therefore could be directly applied to our observations in Stokes Q, U and V, which lowered our noise floor by a factor of four. In recent years neural networks have been applied in order to reduce the noise in data in several studies (e.g.D ´ıaz Baso & Asensio Ramos 2018; Fu et al. 2018; Xu et al. 2020, and references therein). On top of that, their performance seems to be consistently above that of classical noise reduction methods due to the highly non-linear properties of their components. It is relatively easy to train a network when you have a training set with noisy and clean pairs of images. Unfortunately, we do not have the possibility of obtaining noise-free observations or an accurate model of the noise to train the network with. Instead,D ´ıaz Baso et al.(2019) chose a di fferent approach known as Noise2Noise (Lehtinen et al. 2018), which allows for the training of a neural network with only noisy data. This is achieved by taking advantage of the temporal redundancy in our observations, where we can assume that the time between two consecutive scans is short enough that the only difference between two measurements of a certain wavelength is the noise. This allows us to teach the network to remove these differences that are the noise. The network is trained on a large number of input frames containing observations and zero-mean noise (e.g. no fringes), and does not require a clean data set to work towards. For additional details on the implementation of the network, we direct the reader to (D´ıaz Baso et al. 2019) and the project GitHub2. We then decided to apply a second noise-reduction technique on top of the CNN which is a more traditional version of applying PCA then in the previous section (e.g. Gonzalez´ et al. 2008; Casini et al. 2012; Ng 2017; Casini & Li 2018). Here we apply PCA over the entire wavelength range of a single scan of an individual Stokes parameter. In our case this resulted in a set of 23 principle components with most of the information contained in the first 15 components and the rest being noise. We then simply reconstructed the data without the last 8 components into a cleaner image with minimal loss of data. This method was only applied to Stokes Q and U. We did not apply this method to Stokes I and V as the signal was strong enough already. We then applied the two most traditional methods for improving the signal to noise, binning and stacking. This is a very strong method as the signal improves with the square root of the amount of pixels that are binned or frames that are stacked. However, one should be careful with doing this as binning too many pixels might alter the shape of the profiles and so does stacking if the area evolves in between frames. In order to find this limit for stacking we compared the profiles of several pixels in consecutive frames and the stacked profile to the individual ones. The limit turned out to be two frames, as significant changes were seen when three or more were stacked. A similar process was used for the binning, where individual pixels were compared to their neighbors and the binned profile, here the limit was also 2x2 binning. This means that together these two methods helped us gain a factor of two in signal. The final mean noise after 3 the application of these combined techniques was estimated to be 1 10− I in Stokes U. · c

2https://github.com/cdiazbas/denoiser 7

Inversions

After the reduction and post reduction steps of the previous chapters, one can use the resulting polarimetric dataset to infer physical properties of the observed regions of the solar atmosphere. In this chapter we will explain how inversion codes ﬁt these parameters as well as the methods used to prevent non-physical solutions.

7.1 Inversion methods for solar observations

The thermodynamic and magnetic properties of the observed plasma are not directly observable quantities, instead we can infer the influence of these properties at a certain depth from the observed polarized light of a specific spectral line. The collective term for inferring this information out of such observations is ’spectropolarimetric inversions’ or ’inversions’ for short. In the most general terms, inversion codes iterate over a range of thermodynamical and magnetic parameters which are then used to synthesize spectropolarimetric profiles which in turn are compared to our data to see how well they match. This inverted way of fitting the data has so far been one of the most successful approaches to this problem (de la Cruz Rodr´ıguez & van Noort 2017; de la Cruz Rodr´ıguez et al. 2019). Various approaches have been taken when it comes to inversion codes, with differing assumptions depending on the atmospheric layer (LTE vs NLTE) and variables of interest. It is often difficult to estimate the constrainable information in the spectra which can lead to different methods being applied for differing datasets. How- ever, it is often possible to apply some a-priori restrictions to the inversions based on physical arguments. Finding the right recipe for obtaining a plausible fit is therefore more often than not compared to an art that has to be mastered than a set process that one can follow. An important part of inversions is the data that the models are fitted to, with the sophistica- tion of the fitted atmosphere largely depending on the number of spectral lines that are given to the code. A single line contains a large amount of information, but only for the part of the atmosphere from which it originated. Adding lines that originate at different heights in the atmosphere or even have different sensitivities to a certain variable can greatly reduce the degeneracy

41 42 of potential solutions. In Paper I we only inverted a single line along with a measurement in the continuum, which led to diﬃculties in getting the inversions to converge to a smooth physical result. To combat this, we have applied several new and traditional methods to help constrain these results. We will discuss these methods later in this chapter.

The first inversions of this kind were done by Harvey et al.(1972) and Auer et al.(1977) and were based on the Milne-Eddington (ME) model, which describes a line by assuming a constant opacity, a linear source function and several other non-physical fit parameters. However, because the solution can be integrated analytically it is possible to very efficiently calculate and fit the line profiles. Many codes exist that utilize ME to make sophisticated inversions, differing mostly in their minimization strategy. A detailed overview of these codes and a comparison of their performance can be found in Borrero et al.(2014).

These codes were followed by more sophisticated ones that employed a fully stratified physical model of the atmosphere in LTE like SIR (Ruiz Cobo & del Toro Iniesta 1992) and SPINOR (Frutiger et al. 2000). The first NLTE inversions were performed by Mein et al.(1987) which could derive temperature and velocity fluctuations based on disturbances in the Fourier coefficients of the observed lines. Late attempts were made to invert the Ca II IR triplet lines with a code that later grew into NICOLE (Socas-Navarro et al. 1998, 2000b, 2015). The HAZEL code was developed specifically for inverting the He D3 lines (Asensio Ramos et al. 2008). The code that we used for Paper I is The STockholm inversion Code (STiC, de la Cruz Rodr´ıguez et al. 2016; de la Cruz Rodr´ıguez et al. 2019) which is a parallel code capable of full Stokes observations of multiple LTE and NLTE lines by using the Levenberg-Marquardt (LM) algorithm and regularization (both of which will be explained below). The code assumes statistical equilibrium and plane-parallel geometry and makes the calculations on a pixel-by-pixel basis, which makes it a 1.5D code. In the simplest terms, STiC synthesizes an atmosphere based on the input parameters and then fits it to the data, where certain spectral points can have more weight than others. It then uses the LM algorithm to minimize the differences between the fitted data and the synthesized spectra and regularization to weed out nonphysical solutions.

In Paper I we only use the Ca II 8542 Å line for our inversions, a line that loses sensitivity to velocity and microturbulence in the plage. Because of this limitation we had to come up with some additional methods to help STiC converge to a solution. We could not use the Ca II K data, as we have time averaged (stacked) the observations.

7.2 The Levenberg-Marquardt algorithm and regularization

The Levenberg-Marquardt (LM) algorithm (Levenberg 1944; Marquardt 1963) is a widely used alternative to the Gauss-Newton method for minimizing a nonlinear least squares model in an iterative manner. LM is widely used for minimizing the differences between a guessed model and observed profile in inversion codes due to its efficiency compared to other algorithms. This is because the method is a combination of the Gauss-Newton and gradient descent methods, using both of relative strengths. 7. Inversions 43

In the Gauss-Newton method, the assumption is that the merit function is locally quadratic, and a minimum can be found locally. With gradient descent a minimum is found by choosing parameters that reduce the merit function in the direction of steepest descent. LM uses a dampening parameter (λ) to diﬀerentiate between the two methods, where a low value makes it behave like Gauss-Newton and a large value results in behavior closer to gradient descent (Weisstein 2000; Gavin 2011; de la Cruz Rodr´ıguez et al. 2019). The merit function is commonly deﬁned as

n 2 1 Ok S k(p, xk) χ2(p, x) = − . (7.1) n σk Xk=1 " # Here, p is a vector containing the n parameters of the model, Ok is the k-th measured data point, S k is the matching prediction of our model computed at the point x and σk is the error (or noise) of the k-th measurement. The LM algorithm can be written down as

(JT J + λ diag(JT J))δp = JT [O S(p)], (7.2) − where J (= ∂S/∂p) is the Jacobian matrix of S, λ the dampening parameter and δp a small perturbation over the parameters of p. The hybrid nature of the LM algorithm ensures that it will take large steps according to gradient descent when far from the solution, and then converge more accurately with Gauss-Newton once closer to a minimum. This allows us to converge to a new model p by comparing the χ2 of the former and current step.In STiC this is repeated for a pre-defined amount of iterations, until the desired value of χ2 is reached. The input model is then randomized and the process repeated a set amount of times in order to try to avoid the code from getting stuck in a local minimum. It is therefore important to start with an initial guess that has as much information as possible. Regardless of how good the initial guess is, we will get a degeneracy of solutions for a given data set. For example, line broadening can be fitted by both temperature and microturbulence, but a solution with very low temperatures that are compensated by high microturbulence values would be unphysical. We also might prefer a solution with a smooth atmosphere over one that has jumps in temperature. These degeneracies can be overcome by setting regularization requirements, which encourage a certain family of solutions over others. We can apply different regularization types to each atmospheric parameter as well as decide how much we weigh them. STiC uses Tikhonov (or ridge regression) regularization (Tikhonov & Arsenin 1977), which can be used to penalize certain families of solutions by adding limitations to the derivatives of specific parameters. This can be a limit on the first derivative to favor parameters that are smoother along the optical depth, a limitation in the second derivative to prevent solutions that oscillate or solutions that minimize a certain parameter can be favored over those who don’t. For example if two models with similar χ2 are found, we can favor the one with a lower microturbulence. 44

7.3 Initialization of the line-of-sight velocity

Typically a value of 0 km/s is used as an initial guess for the line-of-sight velocity of the input model. However, when working with strangely shaped profiles or low signal to noise, a better initial estimate can improve the result. We developed a linear approximation for the line of sight velocity based on findings by Skumanich & Lopez´ Ariste(2002). In this paper the authors show that when applying PCA to a large amount of similar profiles, the leading Eigenprofiles of the new basis will represent a Taylor series. From this an approximation for the velocity can be obtained, as well as the magnetic splitting parameter. However, for this to work, it is required that the first Eigenprofile represents the median profile of the input and the second represents the derivative thereof. For spectroscopic data with many points along the wavelength axis this is the case, but results are varied for CRISP/CHROMIS data due to the low amount of wavelength points. Instead we choose to skip the PCA step and simply construct a basis out of the median profile and its derivative. Using this basis, a first order approximation of the line of sight velocity can be made. This is done by reconstructing a red- and a blue-shifted profile with our basis and fitting a linear relation to the resulting factors. This basis will not be fully orthonormal, but still gives a usable estimation of the velocity where the PCA basis breaks down completely. In Paper I we use a median Ca II K plage profile and its derivative for our basis. This will give the best results in plage, but poorer approximations in regions with profiles that are drastically different from the average. Due to the raised core Ca II 8542 Å profiles in plage, these regions have the biggest degeneracy for possible solutions, while it is relatively straight forward in the rest of the FOV. As an additional step, we used the Ca II K line core to get a velocity estimate for the chromosphere and the line wings to get an independent estimate of the photosphere. We then combined these two maps into a simple model atmosphere where the chromospheric velocities were used for log(τ ) < 3.5 and the photospheric velocities for the lower atmosphere. (Hereafter we 500 − drop the 500 from log(τ500) to simplify the notation.) An arctangent function was used to connect the two in a smooth way. The resulting input model can be seen in Fig. 4 of Paper I.

7.4 Inversion strategy

Inverting observations of plage with only Ca II 8542 Å is challenging due to the raised core profiles, as the relatively flat line cores make it very hard for an inversion code to constrain the Doppler width of the line and derive accurate values of microturbulence. Even the line- of-sight velocity itself is difficult to constrain. Additionally the amplitude of Stokes Q,U,and V is not only modulated by the magnetic field strength (in the weak-field regime), but also by the gradient of the source function. This additional dependence scales the amplitude of the Stokes parameters, scaling them up when the gradient is large and scaling them down when the gradient becomes very shallow. An example of this effect is visible in the lower panels of Fig. 1 of Paper I around x 14”. ≈ These limitations mean that we are subject to degeneracies that would ordinarily be compen- 7. Inversions 45

Figure 7.1: Example of node interpolation using quadratic Bezier (solid gray), cubic Bezier (solid orange), straight segments (blue), and discontinuous with slope delimiter (green dots). The node values are indicated with black crosses. Reproduced from de la Cruz Rodr´ıguezet al.(2019). sated by other lines. In our inversions we overcame this by taking an iterative approach to our inversions, initially only fitting the temperature to a FAL-C atmosphere with a line-of-sight velocity estimated by the method above and with regularization set up to favor smooth temperature stratification and a low microturbulence and zero weight on Stokes Q, U and V. Once a satisfactory fit of the temperature was reached, we repeated the inversion but now with nodes in vLOS and vturb. From this atmosphere we could then focus on the magnetic field by adding weight to the last three Stokes parameters and including a guess for the field vector to the model. In order to ensure that the values for B are reliable, we set the regularization to favor values ⊥ with B = 0, to ensure that if a non-zero value was returned for a given pixel we were certain ⊥ that it was necessary to explain the observed profiles. In Fig. 7.1 we can see an example of a 7 node atmosphere that has been interpolated using three different methods. The amount of nodes and inversion steps can be seen in Table 1 of Paper I and a schematic representation of the inversion process can be found in Fig. 7.2. The full inversion with all steps took roughly a week to complete on the local cluster when using 1000 CPU cores. 46

Input model

Interpolate atmosphere based on node positions and Perturb nodes values

Input data and Synthesize spectrum weights

Compare ﬁt Keep new model and decrease λ

Yes

No Is the maximum No Is the ﬁt good Is it better than the amount of iterations enough? last model? reached?

No Yes Yes Keep old model and increase λ

Save model

Did we use the desired amount of No Modify amount nodes for each of nodes. parameter?

Yes

Done!

Figure 7.2: Flowchart of the inversion process. Input is given by grey boxes with rounded corners and consists of the post-reduced data and atmospheric model. Square boxes represent ﬁtting and other processing. Round boxes represent checks. The output product is represented by a diamond. 8

Summary of Paper I

So far we have introduced concepts used in Paper I, covering the structure of the solar atmosphere, active regions, the polarization of light, chromospheric diagnostics, observation methods, data reduction and data inversions. Throughout these chapters we have touched upon their connection to Paper I to help contextualize the material. In this chapter we place everything in the context of Paper I. The goal of the project was to measure the magnetic field vector of a plage region. Constrain- ing the values for magnetic fields in plage is an important step in understanding chromospheric magnetic fields and potentially the heating processes of the higher atmosphere. Additionally, these values could be of interest for numerical modellers, as it is currently impossible to reproduce all aspects of solar plage in current quasi-realistic numerical models. For example, plage regions in Bifrost simulations look like coronal holes in higher layers (Carlsson et al. 2019). The magnetic field configuration in the lower atmosphere is one of the few free parameters in such simulations, and current literature only has a very limited amount of proxy-based estimates of the field strength (see Chapter4). Taking such data is at the limit of current capabilities due to the requirement of spectro-polarimetric measurements with a high signal-to-noise ratio in all four Stokes parameters to be able to invert the field, a high spectral resolution in order to sufficiently sample the lines and a high enough cadence stay below the evolution time of plage. Specific observations were required for this project, where the observing angle had to be as close as possible to 45◦ to maximize the signal in Stokes Q and U of the mostly vertical field while not losing too much resolution to the projection effects. Due to these requirements, it took several observing runs to obtain data of the desired situation with a high enough r0. The observations used had an observing angle of 37◦ and were taken with a high cadence program that only observed one line for CRISP and CHROMIS: Ca II 8542 Å and Ca II K respectively. From this data we selected the best frames based on seeing stability and applied the reduction and post reduction methods described in Chapter5. This was a largely iterative process where different methods were evaluated to maximize the signal in Stokes Q and U. Once a high enough

47 48 signal-to-noise ratio was reached we inverted our data as described in Chapter7, which allowed us to infer the inverted parameters inside different regions of our FOV. In the paper we defined four regions of interest: two manually defined plage regions based on an intensity threshold of blurred data where we made sure not to contaminate the selected area with fibrils. Besides that two fibrilar regions were selected for their strong signal in B and divided based on their ⊥ azimuth. One of the regions was a more typical fibrilar region that is further away from the plage and therefore less affected by it, while the other region originates from between the plage regions and seems to be affected by the plage in terms of azimuth, inclination and magnetic field strength. We have marked these regions in Fig. 6a of Paper I.

In the plage we find a median inclination of γ = 10◦ 16◦ with respect to the local vertical. ± The total magnetic field that we found in these regions is B = 440 90 G for. The inclination | | ± values align well with earlier estimates made in the photosphere (e.g. Bernasconi et al. 1994, Sanchez et al. 1992, Pillet et al. 1997). The field strength is more than twice what was suggested by Carlsson et al.(2019), but comparable to the value found in a parallel study done by Morosin et al.(2020), where a spatially coupled weak field approximation was used to infer the magnetic field.

In the fibrils far from the plage we find a median inclination of γ = 50◦ 13◦ and a field of ± B = 296 50 G. The field is above the upper limit of 200 G suggested by Mooroogen et al. | | ± (2017) that was based on the analysis of magnetohydrodynamic kink waves in chromospheric fibrils. The most significant limitation on our data was the time evolution of the plage and fibrils, which required a high cadence and limited our observations to a single line per instrument. The inversions could be better constrained by co-observing with multiple current generation telescopes with chromospheric spectro-polarimetric capabilities. However, we believe that in order to make significant improvements to our measurements, an integral-field spectrometer should be employed to simultaneously observe multiple lines while keeping the same cadence. Alternatively a telescope with more light-gathering power and appropriate instruments could be used, like the new 4-m DKIST telescope in Hawaii or the planned similarly sized EST. 9

List of acronyms

AO Adaptive optics AR Active region CCD Charge-coupled device CHROMIS CHROMospheric ImagingSpectrometer CNN Convolutional neural network CRISP CRisp Imaging SpectroPolarimeter DKST Daniel K. Inouye Solar Telescope EST European Solar Telecope FAL Fontenla, Avrett, Loeser FITS Flexible Image Transport System FFT Fast Fourier transform FOV Field of view FPI Fabry-Perot´ interferometer HAZEL Hanle and Zeeman Light HMI Helioseismic and Magnetic Imager HRE High spectral resolution etalon IRIS Interface Region Imaging Spectrograph KL Karhunen–Loeve`

49 50

LM Levenberg-Marquardt LRE Lower spectral resolution etalon LTE Local thermodynamic equilibrium MHD Magnetohydrodynamic MOMFBD Multi-Object Multi-Frame Blind Deconvolution MR Milne-Eddington NICOLE Non-LTE Inversion COde usingthe Lorien Engine nLTE Non-local thermodynamic equilibrium NOAA National Oceanic and Atmospheric Administration ORM Observatorio del Roque de losMuchachos PCA Principle component analasys PD Phase diversity PSF Point spread function RTE Radiative transfer equation SDO Solar Dynamics Observatory SIR Stokes Inversion based on Response functions SST Swedish 1-m Solar Telescope STiC STockholm inversion Code SWPC Space Weather Prediction Center VAL Vernazza, Avrett, Loeser WB Wide band WFA Weak ﬁeld approximation WFS Wavefront sensor 10

Acknowledgements

I thank Dan Kiselman for his ongoing support and tireless reviewing of this document, and Jaime de la cruz Rodriguez for his useful comments and remarks during the course of this work. Additionally, I thank Arjan Bik for being my mentor throughout this process, Flavio Calvo for our mathematical discussions and Anders Nyholm for the discussions on historical content featured in the thesis and for proof reading it. Finally, I thank Dominique Petit for proof reading the work and Dan Kiselman for help with the Swedish translation of my abstract. The Swedish 1- m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof´ısica de Canarias. The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2017-00625). Computations were performed on resources provided by the Swedish Infrastructure for Computing (SNIC) at the PDC Centre for High Performance Computing (Beskow, PDC-HPC), at the Royal Institute of Technology in Stockholm as well as the National Supercomputer Centre (Tetralith, NSC) at Linkoping¨ University. This work was supported by the Knut and Alice Wallenberg Foundation. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. We acknowledge the community eﬀort devoted to the development of the following open-source packages that were used in this work: numpy (numpy.org), matplotlib (matplotlib.org), astropy (astropy.org).

List of Figures

2.1 Left: Reproduced from Vernazza et al.(1981). Right: Own work, based on Fig. 11.12 of (Carroll & Ostlie 2017)...... 6 2.2 Own work...... 6 2.3 Own work, taken from AIA and HMI observations...... 11

3.1 Reproduced from Carlsson et al.(2019)...... 16 3.2 Own work...... 19 3.3 Reproduced from Sanchez et al.(1992)...... 21

5.1 Reproduced from Scharmer et al.(2003a)...... 31 5.2 Reproduced from Scharmer et al.(2019)...... 33 5.3 Reproduced from de la Cruz Rodr´ıguez et al.(2015b)...... 35

6.1 Reproduced from Paper I/ (Pietrow et al. 2020)...... 38

7.1 Reproduced from de la Cruz Rodr´ıguez et al.(2019)...... 45 7.2 Own work...... 46

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Paper I

Inference of the chromospheric magnetic ﬁeld conﬁguration of solar plage using the Ca ii 8542 Å line A.G.M. Pietrow1, D. Kiselman1, J. de la Cruz Rodríguez1, C. J. Díaz Baso1, A. Pastor Yabar1, and R. Yadav1

Institute for Solar Physics, Dept. of Astronomy, Stockholm University, Albanova University Centre, SE-106 91 Stockholm, Sweden e-mail: [email protected]

Received June 26, 2020; accepted September 22, 2020

ABSTRACT

Context. It has so far proven impossible to reproduce all aspects of the solar plage chromosphere in quasi-realistic numerical models. The magnetic field configuration in the lower atmosphere is one of the few free parameters in such simulations. The literature only offers proxy-based estimates of the field strength, as it is difficult to obtain observational constraints in this region. Sufficiently sensitive spectro-polarimetric measurements require a high signal-to-noise ratio, spectral resolution, and cadence, which are at the limit of current capabilities. Aims. We use critically sampled spectro-polarimetric observations of the Ca ii 8542 Å line obtained with the CRISP instrument of the Swedish 1-m Solar Telescope to study the strength and inclination of the chromospheric magnetic field of a plage region. This will provide direct physics-based estimates of these values, which could aid modelers to put constraints on plage models. Methods. We increased the signal-to-noise ratio of the data by applying several methods including deep learning and PCA. We 3 estimated the noise level to be 1 10− Ic. We then used STiC, a non-local thermodynamic equilibrium (NLTE) inversion code to infer the atmospheric structure and magnetic· field pixel by pixel. Results. We are able to infer the magnetic field strength and inclination for a plage region and for fibrils in the surrounding canopy. In the plage we report an absolute field strength of B = 440 90 G, with an inclination of 10◦ 16◦ with respect to the local vertical. This value for B is roughly double of what was| | reported± previously, while the inclination matches± previous studies done in | | the photosphere. In the fibrillar region we found B = 300 50 G, with an inclination of 50◦ 13◦. | | ± ± Key words. plages, magnetic fields, Sun:atmosphere Sun:chromosphere, Methods:observational

1. Introduction an important interface for coronal heating (Reardon et al. 2009; Carlsson et al. 2019; Chitta et al. 2018; Yadav et al. 2020). They Observationally derived chromospheric magnetic field vectors have been extensively studied in the photosphere, leading to the are a vital part of our attempt to understand solar magneto- view that the magnetic field in the lower layers is concentrated in hydrodynamical processes, as is shown by the continuous effort the intergranular spaces (e.g., Buente et al. 1993). The inclina- that has been made in this direction over the last three decades tion of the field in the photosphere is found to be mostly vertical, (e.g., Bernasconi et al. 1994; Solanki et al. 1996; Socas-Navarro with average values slightly above 10◦ from the local vertical et al. 2000a; López Ariste & Casini 2005; Schad et al. 2015; (e.g., Bernasconi et al. 1994; Sanchez Almeida & Martinez Pil- Martínez González et al. 2016; Esteban Pozuelo et al. 2019; let 1994; Pillet et al. 1997). The flux tubes spread rapidly with Yadav et al. 2019). Understanding these processes is of the ut- height due to the exponentially decreasing gas pressure, eventu- most importance in also understanding the energy balance, atmo- ally merging and filling almost the entire atmosphere. This leads spheric stratification, and dynamics of the solar chromosphere. to a decrease of the magnetic field strength with respect to height Plage regions, first observed in the chromosphere by Lock- (Solanki 1992, and references therein). Perhaps the most clear yer(1869) and named by Deslandres(1893), are classically de- detection of the magnetic field canopy effect from photospheric fined as bright regions observed in Hα and other chromospheric observations are the studies by Sanchez Almeida & Martinez Pil- lines. Nowadays, many authors identify plage regions by only let(1994) and Buehler et al.(2015), who detected the expansion looking at the photospheric magnetic field concentrations, re- of magnetic elements as a function of height using inversions. arXiv:2006.14486v3 [astro-ph.SR] 16 Oct 2020 gardless of the Hα intensity that is associated with those regions. However, due to the relatively low Landé factors and broad In this paper we restrict the plage designation to regions where profiles of lines that are useful as chromospheric diagnostics, it is the magnetic field in the photosphere is confined in the intergran- much more challenging to recover magnetic fields at this height ular lanes and forms a magnetic canopy in the chromosphere that when compared to the photosphere. Therefore, the field strength is hot and bright in most chromospheric diagnostics such as the and the magnetic topology in plage are the subjects of an on- Ca ii H&K lines, the Ca ii infrared triplet lines, Mg ii h&k, and going discussion. Carlsson et al.(2019) cite a canonical value Hα. This leaves out superpenumbra, pores, or elongated fibrillar of B = 200 G. Asensio Ramos et al.(2017) report transverse | | 1 structures. Our usage is in line with that of Chintzoglou et al. magnetic fields with a median value of B0 60 G. This result (2020). ⊥ ≈ 1 Plage regions are important structures that act as the foot- We define the transverse magnetic field in our line of sight as B0 ⊥ points of coronal loops and the origins of fibrils, making them and the vertical magnetic field in the same reference frame as B0 . Their || Article number, page 1 of 12 A&A proofs: manuscript no. ms is based on a Bayesian hierarchical model of the linear polar- λ λ =-880 mA˚ λ λ = 0 mA˚ ization of the data whose transverse magnetic field distribution − 0 − 0 has a tail going past 200 G. However, it is important to remark that the plage used in this paper is a young, flux-emerging region 40 where the measurements were performed over elongated fibrillar structures, and therefore they do not fit our definition of plage. 35 However, despite this seeming agreement for the magnetic field strength, it has so far proven impossible to reproduce a plage 30 chromosphere in quasi-realistic numerical models like Bifrost (Gudiksen et al. 2011), which instead create an atmosphere with 25 a calm chromosphere and cold corona, more resembling a coronal hole (Carlsson et al. 2019). 20

In recent years, significant progress has been made in the de- y [arcsec] velopment of codes capable of performing non-local thermody- 15 namic equilibrium (non-LTE) inversions of spectral lines in the chromosphere (Asensio Ramos et al. 2008; Socas-Navarro et al. 2015; de la Cruz Rodríguez et al. 2016; Milic´ & van Noort 2018). 10 These inversions aim to provide a direct physics-based estimate of key parameters by producing a spatially resolved model atmo- 5 sphere consistent with the observations, rather than proxy-based estimates that were the norm before. In this paper we perform 0 non-LTE inversions to get a direct physics-based estimate of the 0 10 20 0 10 20 magnetic field in plage, as well as other key parameters like x [arcsec] temperature, line-of-sight velocity, and microturbulence. This is 1.0 achieved by using polarimetric data with a high spatial, spectral, and temporal resolution in Ca ii 8542 Å as our input. In order 0.8 to fully reconstruct the magnetic field vector, we use a combination of several novel and standard post-reduction techniques to improve the otherwise low signal-to-noise ratio of our Stokes Q 0.6 and U measurements. In Sect.2 we discuss the diagnostic potential of the 0.4 Normalized intensity Ca ii 8542 Å line and the limitations that come with it. In Sect.3 we focus on our observations, the reduction, and post-reduction 0.9 0.6 0.3 0.0 0.3 0.6 0.9 − − − steps. In Sect.4 we discuss the STockholm Inversion Code λ λ [A]˚ (STiC) and the additional steps that we applied to aid the in- − 0 0.5 versions. Response functions and uncertainties are discussed in − ˚ A] Sect.5. The results from our inversions are shown and discussed [ 0 in Sect.6 and our final conclusions can be found in Sect.7. λ 0.0 − λ 0.5 2. The Ca ii 8542 Å line in plage regions 0.5 ii −

The Ca 8542 Å line has become one of the foremost chromo- ˚ A] [ spheric diagnostics for temperatures, line-of-sight velocities, and 0

λ 0.0

the magnetic ﬁeld vector (see, e.g., Socas-Navarro et al. 2000b; −

López Ariste et al. 2001; Pietarila et al. 2007; de la Cruz Ro- λ dríguez et al. 2012, 2015a; Quintero Noda et al. 2017; Henriques 0.5 et al. 2017; Centeno 2018; Díaz Baso et al. 2019). Although the 0 5 10 15 20 atomic level populations must be modeled under the assumption x [arcsec] of statistical equilibrium (Wedemeyer-Böhm & Carlsson 2011), Fig. 1. Plage spectral signatures in the Ca ii 8542 Å line. Top: a relatively simple six-level atom can be used to model the in- ∆λ = ii Monochromatic images acquired in the wing ( 880) mÅ and tensities of the Ca H&K lines as well as the infrared triplet in the core of the line. Middle: Example plage spectra− in Stokes I ex- lines. The profile of this line can be calculated by assuming a tracted from the locations indicated in the upper panels with the same complete redistribution of scattered photons (Uitenbroek 1989; color coding. For comparison, we have also included a quiet-Sun pro- Sukhorukov & Leenaarts 2017). In active regions far from the file colored in gray. Bottom: Stokes Iλ (upper) and Vλ/Iλ spectra (lower) limb, the imprint of atomic polarization can be assumed to be along the slit indicated in the top panels of the figure. much lower than Zeeman-induced polarization (Manso Sainz & Trujillo Bueno 2010; Štepánˇ & Trujillo Bueno 2016). In plage targets, the Ca ii 8542 Å line shows peculiar line 2013). The main feature of those profiles is the apparent absence profiles that originate in a hot magnetic canopy that extends of an absorption Gaussian core. Instead, the core of these pro- over a photosphere where the magnetic fields are tightly con- files can appear completely flat, or it can show weak absorption centrated in the intergranular lanes (de la Cruz Rodríguez et al. and emission features that are modulated by the velocity field unprimed counterparts represent the same quantities with respect to the and a very shallow stratification of the source function (de la solar surface. Cruz Rodríguez et al. 2013; Carlsson et al. 2015). In the top pan-

Article number, page 2 of 12 A.G.M. Pietrow et al.: Inference of the chromospheric magnetic ﬁeld conﬁguration of solar plage using the Ca ii 8542 Å line

Fig. 2. Fringe removal from reduced data. a) A cartoon of the low-frequency fringe extraction process. Each wavelength point contains a linear combination of the fringes found in the left and right wing points, which are denoted by several lines. These fringes on either side could be extracted from the wings applying principle component analysis (PCA) to the outer most non-continuum wing points. A linear combination of these two fringes are then subtracted from the full spectrum. b) A Stokes U line wing image showing both the high- and low-frequency fringes. c) The same image after the high-frequency fringes have been removed with FFT ( Fast Fourier transform) filtering. d) The same image after the low-frequency fringe was removed using our PCA fringe removal technique. els of Fig.1 we show a subfield from our observations that in- they should be kept in mind in order to properly understand the cludes plage and some quiet-Sun next to it. We have hand-picked inversion results that we present in Sect.6. some locations in the plage region and plotted the correspond- ing profiles in the middle panel. Compared to the quiet-Sun profile (gray), the plage profiles show a relatively raised core with 3. Observations and data processing a diversity of shapes and asymmetries. While the green, blue, and red profiles originate from a non-magnetic photosphere and Region AR12713 was observed on June 15 2018, between 14:23 a magnetic chromospheric canopy, the orange profile is located and 14:48 UT with the Swedish Solar 1-m Telescope (SST, over a magnetic element that is also brighter in the photospheric Scharmer et al. 2003), using both the CRisp Imaging Spec- wings. This line-core effect is therefore not present, but instead troPolarimeter (CRISP, Scharmer et al. 2008) and the CHRO- the whole profile is brighter than that of the quiet-Sun. Mospheric Imaging Spectrometer (CHROMIS; Scharmer 2017) The peculiar shape of the line sets certain limitations on the instruments simultaneously. The region was centred around the diagnostic potential of the Ca ii 8542 Å line in plage targets: heliocentric coordinates (x,y) = ( 55700, 8000), which translates to an observing angle of 37 (µ −= 0.80). This region was se- 1. The relatively flat core of the line makes it very hard for an ◦ lected for this reason, as an observing angle close to 45 of a inversion code to constrain the Doppler width of the line and, ◦ mostly vertical field ensured a signal in both the horizontal and therefore, to derive accurate values of microturbulent mo- vertical line-of-sight components of the magnetic field. The mea- tions. The explanation for the relatively large values of mi- surements were acquired using a high-cadence program for both croturbulent velocity that are required to explain some chro- instruments. We optimized the cadence, given the limitations of mospheric lines (Shine & Linsky 1974; Carlsson et al. 2015; a scanning instrument, to reach a compromise between signal-to- De Pontieu et al. 2015) remains an open question that can noise ratio (S/N) and temporal resolution. This has allowed us to hardly be tackled with observations of this line in plage. apply several of the methods below, as well as to bin consecutive 2. The amplitude of Stokes Q, U, and V is modulated by the scans for a higher S/N. This high-cadence program only allowed magnetic field strength (in the weak-field regime) and by the ii gradient of the source function. When the latter is large, the for one line to be observed. In our case this was Ca 8542 Å for ii amplitude of the Stokes parameters becomes large. This de- CRISP and Ca K for CHROMIS. pendence is explicit even in simple analytical solutions of the For CRISP the cadence was 9.6 s and the observing sequence radiative transfer equation such as the Milne-Eddington one consisted of 23 wavelength positions in the Ca ii 8542 Å line, (see, e.g., Orozco Suárez & Del Toro Iniesta 2007). In those ranging from -0.88 to +0.88 Å relative to line center, with an locations where the source-function gradient becomes very equidistant spacing of 0.055 Å for the inner 21 points and one shallow, the line core becomes particularly flat and therefore doubly spaced point on each end. The CRISP pixel scale is the amplitudes of the Stokes parameters are small. An ex- 0.05800. For CHROMIS the cadence of the observing sequence ample of this effect is visible in the lower panels of Fig.1 was 9.0 s with 28 wavelength positions in the Ca ii K line, rang- around x 14 . However, whenever the source function ≈ 00 ing from -1.51 to +1.51 Å relative to line center. These obser- is not strictly zero, the amplitudes of the Stokes parame- vations have an equidistant spacing of 0.065 Å for the inner ffi ters seem to be su ciently high even within plage patches 21 points, as well as two 0.262 Å steps on each side, a final (x 1600 along the slit in Fig.1). 3. Due≥ to the relatively flat line-core shape, these profiles do not wing point 0.328 Å after that, and one single continuum point at always provide complete information about line-of-sight ve- 4000.08 Å. The CHROMIS pixel scale is 0.037500. Both data locities or about line-of-sight velocity gradients in the chro- sets were reduced using the CRISPRED and CHROMISRED mospheric part of the line. pipelines (Now combined into SSTRED) as described by de la Cruz Rodríguez et al.(2015b) and Löfdahl et al.(2018), which These limitations can affect the sensitivity of the line to a make use of Multi-Object Multi-Frame Blind Deconvolution given physical parameter across the field-of-view (FOV) and (MOMFBD; van Noort et al. 2005; Löfdahl 2002).

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Fig. 3. Overview of AR12713 taken on June 15th 2018 at 14:23 UT. Overlap between CHROMIS and CRISP is marked with a white rectangle. The plage areas P1 and P2 are marked with yellow contours. a) Continuum intensity 4000 Å. b) Ca II K core intensity. c) Ca II K at ∆λ = 0.26 Å. d) Ca ii 8542 Å line core intensity. e) Total linear polarization based on the wavelength average of Stokes Q and U. f) Total circular polarization based on the wavelength average of Stokes V.

Article number, page 4 of 12 A.G.M. Pietrow et al.: Inference of the chromospheric magnetic ﬁeld conﬁguration of solar plage using the Ca ii 8542 Å line

In this paper we mainly focus on the Ca ii 8542 Å data, only 3.3. Binning and stacking using the Ca ii K profile for an initial line-of-sight velocity estimate. Because of this, the CHROMIS data were binned down to After removing the high-amplitude, high-frequency fringes from the CRISP pixel scale and aligned to these observations, as op- the data and denoising it with the neural network, we binned and posed to up-sampling CRISP as is usually the case. In order to stacked the resulting hypercube in order to improve the S/N. De- improve the S/N of the resulting hypercube (t, x, y, λ, s), we ap- spite the high cadence of the observations, we could not stack plied five separate post-processing techniques to the data, which more than two scans in the temporal domain without ending we will cover in order of application. up with non-physical mixed profiles. This maximum was established by comparing the Stokes I profiles of the resulting average to those of the individual scans. As long as these were the same, 3.1. Fourier filtering we could stack images. The same could be done with spatial binning, where the maximum was found to be 2x2 binning. We In our observations we could distinguish a high-frequency fringe stacked two frames and binned the image down two times. diagonally across the image, and a low-frequency fringe perpendicular to it. The high-frequency fringes had the strongest ampli- 3 3.4. Fringe removal and S/N improvement using principle tude in Stokes U, at 1.5 10− Ic. Stokes V had half that, Stokes component analysis Q had 13%, and Stokes·I only 5%. While negligible in Stokes I and Q, these fringes needed to be removed from the Stokes U Principal component analysis (PCA, Pearson 1901) is a linear and V maps before any inferences could be made from them. The orthogonal transformation that converts a set of possibly corre- fringes are stationary on the detector but seem to move in the fi- lated variables into a set of principal components (PC), which nal data cube due to image derotation being applied to the time are linearly uncorrelated variables, sorted by means of variance. series. Such fringes can be removed in the frequency domain by In other words, one could imagine the method as fitting a p- using a two-dimensional Fourier transform of the image (Lim dimensional ellipsoid to the data, with each axis of the ellipsoid 1990). In this work the removal was done by applying a simple representing a principle component and the length of each ra- mask to the frequencies that corresponded to the fringe pattern, dius being defined by its variance. A small radius corresponds to after which the image was transformed back into the spatial do- a small variance, and the general fit will hardly be affected if this main. We then constructed the wide- and narrow-band cubes out axis is omitted, meaning that we only lose a small amount of in- of these filtered images with standard routines. This method is formation. In this work we used PCA in two ways: to remove the similar to the procedure used for removing fringes in observa- low-frequency fringes from our data and to improve the S/N. In tions made by the Sunrise telescope (Pillet et al. 2010); a similar the former case we used the method together with the assump- approach is now part of the CRISPEX pipeline. In Fig.2 b and c tion that the fringe pattern at each wavelength point is a linear we can see the effects of this method. combination of the pattern found in the wings of the line. On top The low-frequency fringes could not be removed in the same of that we assumed that there was no signal in the outer wing way, since masking at lower frequency would lead to significant points of Stokes Q and U, so that the intensities in these points loss of data. Instead we removed these fringes by applying PCA consist of the fringe and background (see Fig.2a). In order to at a later step. separate the fringes from the background, we applied PCA to the two outermost non-continuum wing points on both sides of the line. From each wavelength position we then subtracted an 3.2. Improving S/N with a convolutional neural network approximation of the fringe pattern obtained by taking the linear combination of the two outermost fringe patterns with Recently, several studies have shown how convolutional neural networks (LeCun et al. 1995) could help to reduce the noise in fλ = fred (1 λ/n) + fblue (λ/n), observations, showing a better performance than classical meth- · − · ods. These neural networks exploit the idea of using the pres- where fλ is the fringe intensity at wavelength position λ, fred and ence of spatial correlation to predict the value of a pixel from fblue are the extracted fringe intensities from the red and blue the value of other pixels. In ideal examples, artificial noise can wing respectively, and n is the number of data points in the ob- be used to create a sample where the neural network can learn servation (see Fig.2d,e). from it. However, the characterization of noise in solar applica- After applying the PCA fringe extraction, we applied another tions is not always trivial and the noise is spatially correlated PCA routine to further improve the S/N of the data. Following due to the application of MOMFBD. Using high-cadence obser- the methods described, for example, in Casini et al.(2012), Ng vations, a neural network can be trained to infer the contribution (2017), Casini & Li(2018), and González et al.(2008), we ap- of the noise from the temporal redundancy of the signals them- plied PCA along the wavelength axis of the 3D cubes of the indi- selves. This technique (also known as Noise2Noise; Lehtinen vidual Stokes parameters. The resulting set of 23 principal com- et al. 2018) has specifically been explored on CRISP spectro- ponents contained most of the information in the first 15 com- polarimetric data by Díaz Baso et al.(2019) and therefore could ponents, with the background noise composing the most signif- Q U, V be directly applied to our observations in Stokes , and . icant part of the final eight. Therefore removing these principal For details on the implementation of the network, we direct the 2 components and reconstructing the data with the remaining set reader to the original paper and the project GitHub . This step will improve our S/N ratio, while only losing a minimal amount significantly lowered the noise floor of our observations by a fac- of data. We applied this method to Stokes Q and U. It was not tor of four. necessary to apply it to Stokes I and V due to their relatively higher S/N. After applying the methods discussed above, we es- 3 timated that the background noise value is 1 10− Ic for calculat- 2 https://github.com/cdiazbas/denoiser. ing weights in the inversion. ·

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3.5. Wavelength and intensity calibration Therefore there is a relatively narrow range in optical depth where we have enough sensitivity to do inversions. As discussed Finally we performed an absolute wavelength and intensity cal- in Sect.2, it is challenging to reconstruct a good velocity and ibration using the solar atlas by Neckel & Labs(1984). During microturbulence estimate based on Ca ii 8542 Å alone in plage. these post reduction steps, we extensively used the CRISPEX We attempted to create a better constraint on the velocity by analysis tool (Vissers & Rouppe van der Voort 2012), the introducing a line-of-sight velocity estimate from the available CRISpy python package (Pietrow 2019), and SOAImage DS9 Ca ii K data, which usually has a regular emission core in plage (Joye & Mandel 2003) for data visualization. with a large amplitude.

3.6. Resulting field of view 4.2. Initialization of the line-of-sight velocity In Fig.3 we give an overview of the field of view (FoV) that To get a better line-of-sight velocity estimate we used a method was obtained after the reduction and post-reduction. We focus similar to that suggested by Skumanich & López Ariste(2002). on a single temporally averaged scan taken at 14:23 UT, which Here the authors show that a first order approximation of the was chosen for its good seeing. After binning in space and time velocity map can be made by reconstructing a red- and a blue- / the cadence is 19.2 s and the spatial scale is 0.11600 px. Solar shifted profile with a basis composed of the first and second north and the limb are marked in panel a. The dashed rectangle eigenprofiles of the intensity and fitting a linear relation to the indicates the FoV common between CRISP and CHROMIS and resulting factors. This method works as long as the second eigen- will be subject to data inversions. profile is similar to the derivative of the first. Unfortunately, this In the full CRISP FoV we see two pores, one on the upper was not the case for our data, due to the relatively small number left side and one on the lower right side, best visible in panel b. of wavelength points. We instead chose to directly define a basis In panel d we can see that both of these pores are surrounded that is composed of the average intensity profile in the plage and by unipolar plage regions that are connected with a carpet of its derivative. This basis was not fully orthonormal, but still gave fibrils. These fibrils originate from the lower plage region and a usable estimation of the velocity where the PCA basis breaks are moving up to, and around, the upper plage region. We have down completely. The chromospheric velocity estimate was ob- marked the plage within the white rectangle with a yellow contained by using the line core and the photospheric estimate was tour by selecting the bright areas in panel c. We designate these made by using the outer wing points (see Fig.4 b and c). These two plage regions as P1 and P2. The regions were defined by two maps were then combined into a velocity estimate for the selecting bright regions from the line-core image and then ad- atmosphere where the chromospheric velocities were used for justing them by hand to exclude any fibrillar structure. The size log(τ ) < 3.5 and the photospheric velocities for the lower of these regions is 5409 and 2561 pixels respectively. The last 500 − atmosphere. (Hereafter we drop the 500 from log(τ500) to sim- two panels show the polarization maps for the total linear (e) plify the notation.) An arctangent function was used to connect and circular (f) polarization after post-reduction. the two in a smooth way. The resulting input model can be seen The magnetic field in the canopy is expected to be mostly in Fig.4a. horizontal with respect to the solar surface, while in the plage it is expected to be mostly vertical. However, due to the fact that we observe at a viewing angle of 37◦ (µ = 0.8), we expect to see 4.3. Inversion approach imprints of the plage and the canopy in both polarization maps. We initialized the first cycle of STiC by generating a FAL-C This largely seems to be the case. (Avrett 1985; Fontenla et al. 1993) atmosphere interpolated over 54 depth points in the range of log(τ) = [ 7, 1], which we combined with our initial velocity estimate from− Sect. 4.2. Moreover, 4. Data inversions we used the first three inversion cycles to obtain a model that ap- 4.1. Stockholm Inversion Code proximates the atmosphere in Stokes I and V. We then set an initial guess for the magnetic field in the form of the weak field We performed an inversion of the post-processed Stokes profiles approximation (Landi Degl’Innocenti 2004) for B , 800 G for B 3 0 0 for our observations with the STockholm inversion Code (STiC; , and 1.2 for χ. The last three inversion cycles were|| then used to⊥ de la Cruz Rodríguez et al. 2016; de la Cruz Rodríguez et al. fit Stokes Q and U. An overview of the cycles can be found in Ta- 2019), a parallel non-LTE inversion code that utilises a modi- ble1. We only inverted the part of the FoV that had overlap with fied version of the radiative-transfer code RH (Uitenbroek 2001) the CHROMIS data, as this is the only part where we can get to solve the atomic population densities by assuming statistical an estimate for the velocity and have information on the 4000 Å equilibrium and plane-parallel geometry. STiC uses a regular- continuum. The latter helps to constrain the temperature recon- ized version of the Levenberg-Marquardt algorithm (Levenberg struction in the photosphere. As a final step we subtracted the 1944; Marquardt 1963) to iteratively minimize the χ2 function CRISP cavity error map from the resulting velocity. We raised between observed input data and synthetic spectra of one or more the weights of Stokes Q, U, and V to ensure that the code took lines simultaneously. The code treats each pixel separately as these into account. The initial weight of 1 10 3I was divided a plane parallel atmosphere, fitting them independently of each − c by four for Stokes V and ten for Stokes Q an· U. other. We inverted all four Stokes parameters of the Ca ii 8542 Å line simultaneously, also including a 4000 Å 5. Response functions and uncertainties continuum point from the Ca ii K observations. We could not include the Ca ii K line data due to the temporal averaging that The Ca ii 8542 Å line is sensitive to a large range of log(τ), has been used to increase the S/N of the linear polarization data. with the line core being formed in the chromosphere and its wings reaching down to the photosphere. Additionally, several 3 https://github.com/jaimedelacruz/stic of the aforementioned effects will affect the quality of our in-

Article number, page 6 of 12 A.G.M. Pietrow et al.: Inference of the chromospheric magnetic field configuration of solar plage using the Ca ii 8542 Å line versions more at certain heights than others. For this reason it Table 1. Overview of the number of nodes used per inversion cycle. is beneficial to compute response functions (RF) of our param- ``` eters (Beckers & Milkey 1975; Landi Degl’Innocenti & Landi ``` Run # ``` 1 2 3 4 5 6 Degl’Innocenti 1977; Sanchez Almeida 1992; Ruiz Cobo & del Parameter `` Toro Iniesta 1994; Milic´ & van Noort 2017). A RF is the deriva- T 5 7 7 7 0 7 tive of the emerging intensity as a function of a given physical vlos 0 0 2 3 0 3 parameter per δ log(τ). It also allows us to estimate how strongly vturb 0 0 1 1 0 1 the Stokes parameters react to a perturbation at a certain atmo- B0 0 0 0 1 0 3 || spheric height. The stronger the RF, the more sensitive the line B0 0 0 0 1 1 1 ⊥ is at a given log(τ) for the perturbed quantity. The shape of the χ0 0 0 0 1 1 1 RF depends on the magnitude of the perturbation, which should be just above the numerical noise (Milic´ & van Noort 2017). We Note: Columns specify the amount of nodes used per cycle per param- used the built-in RF mode in STiC to compute our response func- eter. The resulting atmosphere was smoothed between each run, except tions and used the same perturbation size as has been used for the for the fifth cycle where we only varied B0 in order to facilitate a better ⊥ inversions. After generating our RFs for our models, we found fit. The initial model was a standard FAL-C atmosphere with 54 points that the maximum response for both magnetic field parameters in the optical depth ranging from -7 to 1, together with our initial veloc- lies between log(τ) = [ 5.5, 3.5]. Outside this optical-depth ity estimate. range, the response is negligible.− − The goodness of the fit can be quantified by computing the between log(τ) = [ 5.5, 3.5], these results are displayed for a χ2 value between the observed and synthetic data. In our case we height of log (τ) = −4.5. − defined the function as − Our chromospheric temperature map is largely consistent 3 q obs syn 2 with the Ca ii 8542 Å line-core-intensity map in Fig.3d. The 2 1 Is (λi) Is (λi; M) χ = − , (1) median temperature is around 4800 K in the fibrillar region, and 4q ws,i Xs=0 Xi=1 ! around 5500 K, with peaks up to 6000 K, in plage. Our values where q is the number of wavelength points of the Stokes pro- are consistent with earlier reported plage temperatures at this files, Isyn are the Stokes profiles from a model M while Iobs are height when taking into account the uncertainty of 430 K, ob- the observed Stokes profiles. The ratio of the weights factor and tained from Eq.2 as shown in Fig.7. Díaz Baso et al.(2019) report 5000 K and da Silva Santos et al.(2020) report values be- noise of the observed Stokes profiles are denoted by ws,i . Additionally we could use the calculated RFs to estimate the tween 6000 and 6500K. The temperature of the fibrillar region is uncertainty in our resulting physical quantities. The latter was also comparable to the values reported by Kianfar et al.(2020). done following the method described in del Toro Iniesta(2003), The reconstruction of the line-of-sight velocity is not very with the exception that we only introduced perturbations at the accurate due to the raised core profiles of the Ca ii 8542 Å line locations of our inversion nodes directly. The uncertainty in a in plage as discussed in Sect.2. When we compare the estimate given physical quantity p can be written as based on Ca ii K (as seen in Fig.4b) to the velocity from the Ca ii 8542 Å line, (see Fig.5b.) we can clearly see the limitations. The velocity in the canopy areas with fibrils is much smoother 3 q 2 obs syn 2 and better constrained than in the plage, where we retrieve a ve- Is (λi) Is (λi; M) ws,i 2 s=0 i=1 − locity field with a more patchy look. σ2 = , (2) p n P P h 3 q i Since the Doppler width constrains the turbulent velocity 2 λ 2 Rp,s( i)ws,i parameter in the inversions, there is low sensitivity to this pa- s=0 i=1 rameter. Having at least two lines from species with a well- P P where n is the number of nodes used in the inversion in model differentiated mass would certainly help to constrain the micro- M and q the amount of wavelength points in the observation and turbulence, but was not the case in our observations. Therefore R is the response function of a Stokes parameter to the physical we have used a low-norm regularization being set to favor lower quantity p. microturbulence values in the reconstruction. The latter resulted 1 We picked one typical pixel in P2, F1, and F2 for which in microturbulence values systematically below 2 km s− in the we calculated the uncertainties in all six fitted parameters us- plage region. As expected, these values were significantly lower ing Eq.2. We displayed these profiles and their fit along with than those reported by de la Cruz Rodríguez et al.(2016), De the retrieved atmospheric parameters and their uncertainties in Pontieu et al.(2015), Carlsson et al.(2015) and da Silva Santos Fig.7. The three profiles have a respective χ2 value of 6, 9, and et al.(2020), where high microturbulence values are required in 5, which correspond to yellow and light green in the map. A order to fit Mg ii h&k profiles in plage. Outside the plage we re- good fit would have a χ2 of roughly 1 if the data is used without gained some sensitivity to microturbulence, and reach values up 1 weights, but when we use weights to be more sensitive to weak to 6 km s− , which is comparable to values derived from similar amplitudes in the polarization, larger values will be obtained for data inversions to those of Kianfar et al.(2020). a similar fit (Milic & Gafeira 2020). The location of these rep- From Fig.5d we see that the magnetic field mostly has pos- resentative pixels is shown in Fig.6. In Fig.8a we displayed the itive polarity in the field of view. We were able to reconstruct a 2 χ values of the entire FoV. longitudinal magnetic field (B0) throughout this FoV, but could || only obtain a nonzero transverse magnetic field (B0 ) in cer- ⊥ 6. Results and discussion tain regions. This could be due to our low-norm regularization, which favors lower values for B0 . This means that higher val- ⊥ In Fig.5, we present the inversion results of the 37 00 2700 field of ues would only be returned if lower values did not fit the data. view that was marked with a white dotted line in Fig.×3. Since the This was done because the transverse magnetic field is strongly maximum sensitivity for B0 and B0 in our response functions lies affected by the noise, and if the regularization was not used we || ⊥ Article number, page 7 of 12 A&A proofs: manuscript no. ms

vlos [km/s] vlos [km/s] 8 4 0 4 8 4 0 4 8 40 1.0

35 0.8 30

0.6 25

weight 20 0.4 y [arcsec] 15 0.2 10

0.0 a) b) c) 5 5.0 2.5 0.0 30 40 30 40 log( ) x [arcsec] x [arcsec] Fig. 4. Simple two-component velocity model created by applying our linear velocity estimates based on the line core and line wings of Ca ii K. We transition from the photospheric velocity estimate to the chromospheric velocity estimate at log(τ) = 3.5 by means of an arctangent function. This velocity model was used together with a FAL-C atmosphere as the initial guess for the inversion code.− From left to right the scaling functions used to compose the velocity model (a), the chromospheric velocity estimate (b) and the photospheric velocity estimate (c).

T [K] VLOS [km/s] Vturb [km/s] B || [G] B [G] 4500 5000 5500 8 4 0 4 80 1 2 3 4 5 500 250 0 250 5000 100 200 300 400 500 40

25 F1 20

y [arcsec] F2 15 P2 10 P1

5 a) b) c) d) e) 30 40 30 40 30 40 30 40 30 40 x [arcsec] x [arcsec] x [arcsec] x [arcsec] x [arcsec]

Fig. 5. Inversion results after six reduction steps in STiC as shown in Table.1. All panels are plotted at log( τ) = 4.5. Two plage regions P1 and P2 are marked in yellow. Inside the region covered with fibrils we mark two regions as F1 and F2. From left to right:− Temperature (a), line-of-sight velocity (b), microturbulence (c), longitudinal magnetic field (d), and the transverse magnetic field (e).

would obtain upper limits for the transverse field in all the pix- We use B0 and B0 to calculate the absolute magnetic field els (Martínez González et al. 2012; Díaz Baso et al. 2019). With ( B ) and its inclination|| ⊥ (γ ). We show these two quantities to- | | 0 the regularization we can be sure that the obtained values are gether with the azimuthal direction of the field (χ0) in Fig.6. Like not dominated by the noise contribution. In Fig.7 (left column) with B, the primed quantities denote the azimuth and inclination the Stokes Q and U fits are not always good. This is common in the line-of-sight frame, whereas their unprimed counterparts when the signal is weak and the field mostly longitudinal (as ex- represent the same quantities with respect to the solar surface. pected in the plage region). Still, there is information in these The map of B0 shows one relatively homogeneous large region ⊥ parameters and we argue that the overall results are meaningful around (x, y) = (3500, 2500) in the area covered with fibrils. Our as witnessed by the appearance of the parameter maps. Notably interpretation is that the inversions have been largely successful the value of the total magnetic field is similar in the several sub- here. This region can be split in two based on the value of χ0 in regions studied.

Article number, page 8 of 12 A.G.M. Pietrow et al.: Inference of the chromospheric magnetic ﬁeld conﬁguration of solar plage using the Ca ii 8542 Å line

Fig. 6. Magnetic field configuration. We mark the same regions as defined in Fig.5. Additionally we mark one pixel in each of the regions that will be used for the calculation of the uncertainties (see Fig.7). a) Ca ii 8542 Å line core image with the azimuth over-plotted in our regions of interest. White arrow: Direction to disk center. Red arrow: Angle between the two pores. b) Azimuth, c) Inclination, and d) Absolute magnetic field.

6 9 5 Temperature [K] Vlos [km/s] Vturb [km/s] 4.0 12500 4 3.5 2 10000 0 3.0 7500 2 2.5 5000 4 2.0 4 2 0 4 2 0 4 2 0 log( ) log( ) log( )

B|0| [G] B0 [G] [deg] 1000 500 150 500 400 100

0 300 50

200 500 0 4 2 0 4 2 0 4 2 0 log( ) log( ) log( )

Fig. 7. Left: Typical observed stokes profiles in Ca ii 8542 Å (black) and the fitted atmosphere (colored). Using Eq.2 we get a χ2 value of 6, 9, and 5 for the three profiles respectively. The pixel locations of these profiles are marked in Fig.6 with the same color coding. Right: Inferred parameters displayed between log(τ) = [ 5.5, 0]. The colors match the profiles on the left. The uncertainties for the same pixels based on Eq.2 are indicated by vertical bars at their respective− node.

Fig.6a and b. We designated these two sub-regions F1 and F2, near F1 and F2. The rest of the FoV shows no preferred direction. consisting of 1334 and 1418 pixels respectively. This is expected inside the plage, but in most of the fibrillar areas In the two plage regions P1 and P2, we see that the map is it is most probably due to a lack of signal. In F1 we find a median value of χ 70 and in F2 χ seem to gradually turn from 70 dotted with pixels indicating 0 G. These pixels suggest that there 0 ≈ ◦ 0 ◦ is a cut-off point where low values for B get forced to 0 G. to about 130◦ as we move downward in the figure. ⊥ From Fig.6a and b, we note that χ0 correlates with the signal It is not possible to disambiguate the entire field because B ⊥ in B0 , with most of the coherent structure being found inside and could only be recovered in certain regions. However, we can see ⊥ Article number, page 9 of 12 A&A proofs: manuscript no. ms

Plage The median inclination for regions P1 and P2 is γ0 = 2 43 18 and γ = 45 16 , respectively. With respect to the ◦ ± ◦ 0 ◦ ± ◦ 0 5 10 15 local vertical we get γ = 7◦ 18◦ for P1 and γ = 8◦ 16◦ for P2. These values match earlier± estimates of the magnetic± field 40 inclination in plage made in the photosphere (e.g., Bernasconi et al. 1994; Sanchez Almeida & Martinez Pillet 1994; Pillet et al. 1997). 35 When looking at the total magnetic field, we find B = 440 90 G for P1 and B = 450 90 G for P2, which is roughly| | twice± what was reported| | by Carlsson± et al.(2019). We note that changes in the detailed selection of the borders of the plage re- 30 gions, even when including the tail ends of fibrils, do not affect these median values significantly.

25 Fibrils In region F1 we find that γ = 50 13 and for region ◦ ± ◦ F2 we find γ = 9◦ 13◦. The inclination of F2 is comparable to that found in plage± regions P1 and P2 and mostly vertical with 20 respect to the surface, while the inclination for F1 is more hori- y [arcsec] zontal. In F2 we find more similarities to the plage, as this region has 15 a median field strength of B = 410 80 G. Region F1 has a lower field strength with B = 296| | 50 G.± This is higher than the upper limit of B 25-200| | G suggested± by Mooroogen et al.(2017), | | ∼ 10 based on the analysis of magnetohydrodynamic kink waves in chromospheric fibrils.

5 7. Conclusions 25 30 35 40 45 We present high spatial, temporal, and spectral resolution spec- x [arcsec] tropolarimetric observations of a plage region in Ca ii 8542 Å . This SST data set allowed us to constrain the thermodynamical Fig. 8. A χ2 map for goodness of fit based on Eq.1. The three selected and magnetic properties of this region and its surroundings. By pixels were chosen to be typical for the region that they represent. Their combining high-cadence observations with the addition of novel profiles are displayed in Fig.7. and established post-processing techniques, we were able to reconstruct the Stokes Q and U profiles sufficiently well to infer B0 in several parts of the field of view. The high cadence re- ⊥ that F1 and F2 are magnetically connected and that the magnetic quired meant that CRISP could only observe Ca ii 8542 Å and field in F2 veers off into either P2 or the area between P1 and P2. the temporal binning prevented us from including Ca ii K in our Also the field in F1 is mostly aligned with the line between the inversions. Our focus on high accuracy in the reconstruction of two pores. The pores themselves lie along a line that is offset by the magnetic field came at the cost of accuracy in the other pa- only 8◦ from the radius vector. We thus conclude that field lines rameters, especially in the plage region where it proved difficult originate from the left part of P2, flow through F2 and into F1, to reconstruct an accurate velocity and microturbulence due to and reach the pore of opposite polarity outside the ROI. the raised core profiles. The inclination map in Fig.6c reflects the B0 map in Fig.5 These maps allowed us to retrieve a magnetic field map in since any pixel without a transverse component⊥ automatically the entirety for B0 and in selected regions for B0 . These two || ⊥ gets interpreted as having γ0 = 0 or 180◦. We excluded these maps allowed us to study the median inclination of the magnetic pixels from our analysis by selecting only those that had a field in plage for the pixels where B0 was above 100 G. For both ⊥ B0 > 100G in order to avoid the bias created by the regular- regions combined, this gave us a value of γ = 10◦ 16◦ with ization.⊥ After applying this mask, we obtained median values of respect to the local vertical. This is a value that matches± previous the magnetic field properties for our regions of interest. Because measurements made in the photosphere. we know that the field in P1 and P2 is close to being vertical and Investigation of our χ0 map showed that it could be disam- that the χ of F1 and F2 is close to being aligned with the radial biguated in one specific location of the canopy. We concluded direction, we could subtract the viewing angle to get values with that the magnetic field flowed from region P2 into region F2. respect to the local vertical, γ = γ‘ 37◦. Both of these regions have an absolute field strength close to − Now that we have shown that the magnetic field in F2 orig- 450 G (450 90 G in P2 and 410 80 G in F2) and similar in- ± ± inates from the left side of P2, we would expect to see a similar clinations of γ = 8◦ 16◦ for P2 and γ = 9◦ 13◦ for F2. The ± ± median value of B for the two regions. By eye we can see a fibrils in F1 have a lower field strength of B = 296 50 G and | | | | ± smooth-looking transition between P2 and F2 in Fig.6d. In the an inclination of γ = 50◦ 13◦. ± next paragraphs we discuss our measurements for the different In a parallel study, Morosin et al.(2020) have studied the plage and fibrillar regions. The uncertainties for B and γ were stratification of canopy fields using three spectral lines that pro- | | 0 propagated from the uncertainties for B0 and B0 , which were vide different opacity windows in the atmosphere (Mg i 5173 Å, obtained by using Eq.2 for the representative⊥ pixels.|| Na i 5896 Å and Ca ii 8542 Å) in combination with a spatially

Article number, page 10 of 12 A.G.M. Pietrow et al.: Inference of the chromospheric magnetic field configuration of solar plage using the Ca ii 8542 Å line coupled weak-field approximation method. Although their ob- de la Cruz Rodríguez, J., Socas-Navarro, H., Carlsson, M., & Leenaarts, J. 2012, servations did not provide a sufficiently high S/N ratio to derive A&A, 543, A34 De Pontieu, B., McIntosh, S., Martinez-Sykora, J., Peter, H., & Pereira, T. M. D. the horizontal component of the field from the 8542 Å line, their 2015, ApJ, 799, L12 inferred B values in the magnetic canopy are close to 450 G in a del Toro Iniesta, J. C. 2003, Introduction to Spectropolarimetry (Cambridge Uni- similar target,k and in agreement with our values. In our study we versity Press) could not study the stratification of the field, but we could de- Deslandres, H. 1893, Knowledge: An Illustrated Magazine of Science, 16, 230 Díaz Baso, C. J., de la Cruz Rodríguez, J., & Danilovic, S. 2019, A&A, 629, A99 rive the full magnetic vector and also the orientation of the field Díaz Baso, C. J., Martínez González, M. J., Asensio Ramos, A., & de la Cruz along fibrils that are anchored outside of the plage target. Rodríguez, J. 2019, A&A, 623, A178 Our results are fundamentally limited by the evolution of Esteban Pozuelo, S., de la Cruz Rodríguez, J., Drews, A., et al. 2019, ApJ, 870, the solar scene, which constrains the cadence. Therefore we 88 Fontenla, J. M., Avrett, E. H., & Loeser, R. 1993, ApJ, 406, 319 believe that significant improvement would require an integral- González, M. J. M., Ramos, A. A., Carroll, T. A., et al. 2008, Astronomy & field spectrometer allowing additional lines to be observed while Astrophysics, 486, 637 keeping the cadence the same, and/or a telescope with more Gudiksen, B. V., Carlsson, M., Hansteen, V. H., et al. 2011, Astronomy & Astro- light-gathering power. physics, 531, A154 Henriques, V. M. J., Mathioudakis, M., Socas-Navarro, H., & de la Cruz Ro- Acknowledgements. JdlCR is supported by grants from the Swedish Research dríguez, J. 2017, ApJ, 845, 102 Council (2015-03994), the Swedish National Space Agency (128/15) and the Joye, W. A. & Mandel, E. 2003, in Astronomical Society of the Pacific Con- Swedish Civil Contingencies Agency (MSB). This project has received fund- ference Series, Vol. 295, Astronomical Data Analysis Software and Systems ing from the European Research Council (ERC) under the European Union’s XII, ed. H. E. Payne, R. I. Jedrzejewski, & R. N. Hook, 489 Horizon 2020 research and innovation program (SUNMAG, grant agreement Kianfar, S., Leenaarts, J., Danilovic, S., de la Cruz Rodríguez, J., & José Díaz 759548). The Swedish 1- m Solar Telescope is operated on the island of La Baso, C. 2020, A&A, 637, A1 Palma by the Institute for Solar Physics of Stockholm University in the Spanish Landi Degl’Innocenti, E. & Landi Degl’Innocenti, M. 1977, A&A, 56, 111 Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Ca- Landi Degl’Innocenti, E. 2004, Astrophysics and Space Science Library, 307 narias. The Institute for Solar Physics is supported by a grant for research infras- LeCun, Y., Bengio, Y., et al. 1995, The handbook of brain theory and neural tructures of national importance from the Swedish Research Council (registra- networks, 3361, 1995 tion number 2017-00625). 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