The Diagnostic Potential of the He I D3 Spectral Line in the Solar Atmosphere

Tine Libbrecht What if we look at the Sun in a new light? With the Swedish 1-m Solar Telescope providing excellent spatial resolution, we explore the solar The diagnostic potential of chromosphere via the He I D3 spectral line. This line has been largely unused in the last decades and the historical data have poor spatial the He I D3 spectral line in resolution. Therefore, almost uniquely off-limb targets could be observed. Now, due to instrumental progress, we are able to observe The diagnostic potential of He I D the the solar atmosphere the on-disk solar atmosphere via He I D3 with high spatial resolution and high temporal cadence to obtain spectroscopic and spectropolarimetric data. This thesis proposes the He I D3 line as a new way to Tine Libbrecht study reconnection events in the solar atmosphere. 3 spectral in line solar atmospherethe

Tine Libbrecht started her PhD studies in 2014 at the Institute for Solar Physics in the Department of Astronomy at Stockholm University

ISBN 978-91-7797-602-8

Department of Astronomy

Doctoral Thesis in Astronomy at Stockholm University, Sweden 2019 The diagnostic potential of the He I D3 spectral line in the solar atmosphere Tine Libbrecht Academic dissertation for the Degree of Doctor of Philosophy in Astronomy at Stockholm University to be publicly defended on Friday 26 April 2019 at 13.00 in FB52, AlbaNova universitetscentrum, Roslagstullsbacken 21.

Abstract The research question of my PhD is in a way a simple one: what can observations of the He I D3 line teach us about the solar chromosphere? This optical spectral line at 5876 Å is generally formed in the upper chromosphere, and is sensitive to the local magnetic field. The He I D3 line is also indirectly sensitive to heating of the transition region and corona, since it is resulting from a transition that occurs between levels in the triplet system of neutral helium. These levels are generally populated via an ionization-recombination mechanism under the influence of EUV radiation originating in the transition region and corona. The He I D3 line was used as a flare diagnostic in the seventies and in the subsequent decades also to measure magnetic fields in prominences. However, due to the poor spatial resolution and low signal-to-noise of that data, almost exclusively off-limb targets have been studied. The on-disk absorption of He I D3 is very weak and localized. Recent instrumental developments allow for the acquisition of high spatial resolution on-disk spectroscopic and spectro-polarimetric data of He I D3 with different instruments at the SST, opening the possibility of studying all types of targets in the chromosphere in a new light. During my PhD, I have focused on the study of reconnection targets via high-resolution observations of He I D3 with TRIPPEL and CRISP at the SST, in co-observation with space-borne instruments. Subsequently, a theoretical study has aimed at in-depth understanding of He I D3 line formation in small-scale reconnection events. The data which I have obtained and analyzed during my PhD has provided new insights in Ellerman bombs and flares. 4 Our He I D3 observations have suggested that the temperature of Ellerman Bombs is higher than 2×10 K based on the discovery of helium emission signatures in these events. This result is unexpected, since previous modeling in the literature estimates the temperatures of Ellerman Bombs below 104 K. Subsequently, 3D non-LTE radiative transfer calculations have revealed the detailed physical mechanisms to generate He I D3 emission in these events. The calculations also confirmed that temperatures between 2×104 - 106 K are required to populate the helium triplet levels. In the context of flares, we measured strong downflows in the chromosphere via He I D3, revealing detailed dynamics in the deep atmosphere during a flare. Spectro-polarimetry was used to measure the magnetic field during a flare and to propose its magnetic topology. In conclusion, the He I D3 line is an excellent probe for reconnection targets in the solar atmosphere. Detailed dynamics as well as the magnetic field configuration can be derived using the line. Our findings encourage the use of the He I D3 spectral line as a diagnostic for the chromosphere and open up a range of applications that is yet to be exploited.

Keywords: Sun, Line formation, Chromosphere, Spectro-Polarimetry, Flares, Reconnection.

Stockholm 2019 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-166996

ISBN 978-91-7797-602-8 ISBN 978-91-7797-603-5

Department of Astronomy

Stockholm University, 106 91 Stockholm

THE DIAGNOSTIC POTENTIAL OF

THE HE I D3 SPECTRAL LINE IN THE SOLAR ATMOSPHERE

Tine Libbrecht

The diagnostic potential of the He I D3 spectral line in the solar atmosphere

Tine Libbrecht ©Tine Libbrecht, Stockholm University 2019

ISBN print 978-91-7797-602-8 ISBN PDF 978-91-7797-603-5

Cover image: observation of a C3.6-class flare in the He I D3 line core obtained with the CRisp Imaging Spectro- Polarimeter at the Swedish 1-m Solar Telescope. Data observed on 2015-05-05 by Hiva Pazira.

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List of Papers

The following papers, referred to in the text by their Roman numerals, are included in this thesis.

PAPER I: Observations of Ellerman bomb emission features in He I D3 and He I 10830 Å T. Libbrecht, J. Joshi, J. de la Cruz Rodríguez, J. Leenaarts, and A. Asensio Ramos (2016), Research Article, Astronomy and Astrophysics, 2016. DOI: https://doi.org/10.1051/0004-6361/201629266

PAPER II: Chromospheric condensations and magnetic ﬁeld in a C3.6 class ﬂare studied via He I D3 spectro-polarimetry T. Libbrecht, J. de la Cruz Rogríguez, S. Danilovic, J. Leenaarts, H. Pazira (2019), Research Article, Astronomy and Astrophysics, 2019. DOI: https://doi.org/10.1051/0004-6361/201833610

PAPER III: Line formation of He I D3 and He I 10830 Å in a small-scale reconnection event T. Libbrecht, J. Pirés Bjørgen, J. Leenaarts, J. de la Cruz Ro- dríguez, V.Hansteen, J. Joshi (2019), Research Article Manuscript, in preparation.

Reprints were made with permission from the publishers.

Author’s contribution

PAPER I: The data was observed and reduced by J. Joshi and myself. I have conducted the full analysis, made all ﬁgures and written the manuscript, taking into account suggestions from the co- authors.

PAPER II: The data was observed by H. Pazira and reduced by J. de la Cruz Rodríguez and myself. I have conducted the full analysis, made all ﬁgures and written the manuscript, taking into account suggestions from the co-authors.

PAPER III: The MHD simulation was run by V. Hansteen. The 3D radiative transfer calculations were performed mostly by J. Pirés Bjørgen and continued by myself. I have conducted the full analysis, made all ﬁgures and written the manuscript, taking into account suggestions from the co-authors.

The thesis is based on the unpublished licentiate thesis: “Exploring the diagnostic value of He I D3 in the solar chromosphere”, written by myself and defended on November 14th, 2016. I have reused Chapters 1 – 4 from the licentiate thesis for this PhD thesis, but all text has been fully updated with recent work, developments and references. I expanded the text of Section 2.4.6, 3.4.3, 4.1.1, 4.1.2, 4.3.1 and 4.3.3 with some paragraphs on new results and added Section 4.3.4. I moved the section on ﬂares from Chapter 4 to Chapter 6. Chapter 5 has rewritten and expanded with Sections 5.3, 5.4, and extra paragraphs in 5.5, 5.6 and 5.7. Chapter 6 has been added, of which Sections 6.1 and 6.2 are new and 6.3 is paraphrased from Chapter 4 of the licentiate. The abstract and summaries are newly written for this thesis.

Contents

List of Papers i

Author’s contribution iii

1 Introduction to the Solar Atmosphere 7

2 Helium Line Formation 11 2.1 Discovery of Helium ...... 11 2.2 Atomic Structure ...... 12 2.3 Line Formation Mechanisms ...... 14 2.4 Review on Helium Line Formation Research ...... 16

3 Polarization and Inversion Codes 31 3.1 Polarization of Light ...... 31 3.2 Polarizing Mechanisms ...... 32 3.3 The Polarized Radiative Transfer Equation ...... 39 3.4 Inversion of Observed Stokes Parameters of Helium Lines . . 41

4 Observations of He I D3 53 4.1 Telescopes and Instruments ...... 53 4.2 Spectral Proﬁles of He I D3 and He I 10830 Å ...... 59 4.3 Overview of He I D3 Observations and Results ...... 59

5 Ellerman bombs 71 5.1 Definition ...... 71 5.2 Flux Emergence ...... 72 5.3 Magnetic reconnection ...... 74 5.4 Ellerman bomb topologies ...... 77 5.5 Spectral Signatures and Forward Modeling ...... 77 5.6 Connection between EBS and UV bursts ...... 80 5.7 He I D3 and He I 10830 Å observations and MHD simulations 82 6 Flares 85 6.1 Definition and the flare standard model ...... 85 6.2 Magnetic fields and spectro-polarimetry of flares ...... 88 6.3 Helium observation and line formation of flares ...... 90

7 Summary and connection to the papers 93

Sammanfattning xcv

Publications not included in this thesis xcvii

Acknowledgements xcix

References ciii 1. Introduction to the Solar Atmosphere

Throughout history, the Sun has been studied from the Earth by means of radiation originating in the solar atmosphere. The only probe for the core of the Sun are solar neutrinos while accelerated particles escape from the solar outer layers via the solar wind. Apart from neutrinos and solar wind particles, the study of radiation escaping from the solar atmosphere is our only way of observing the Sun. The layer where photons in the optical continuum escape is called the photosphere. When observing the Sun in the optical continuum or in photospheric spectral lines (such as most Fe I lines), the granulation pattern becomes ev- ident. This pattern emerges as a result of convective motions, giving rise to bright granules where plasma rises - surrounded by darker and narrow intergranular lanes in which the plasma flows down. Magnetic features become apparent in the photosphere in the form of magnetic bright points, pores and sunspots. An important parameter when studying the solar atmosphere is the plasma- β parameter, defined as the ratio between the gas pressure Pgas and the magnetic pressure Pmagn: Pgas 2µ0Pgas β = = 2 , (1.1) Pmagn B with µ0 the permeability of the vacuum and B the magnetic field strength. The photosphere consists of high-β plasma. The density is high, meaning that collisions govern line formation in the photosphere and therefore, most photospheric lines can assumed to be formed in local thermodynamic equilibrium (LTE). The chromosphere is the layer situated above the photosphere. In the chromosphere, the pressure drops by several orders of magnitude over a short distance (500-1000 km) which completely changes the physical conditions in the atmosphere. The chromosphere is a low-β plasma in which the magnetic force dominates the dynamics. The chromosphere is usually observed in the core of the strongest lines of the solar spectrum, such as hydrogen Balmer lines, and resonance lines of Ca II and Mg II. These lines have such large opacities that line core photons escape from the chromosphere while line wing photons

7 Figure 1.1: The chromosphere as seen in the Hα line core λ0 (left panel). The right panel shows the Hα red wing at λ0 + 1.05 with contributions from both the chromosphere and upper photosphere. Observed with SST/CRISP by Jaime de la Cruz Rodríguez and Jorrit Leenaarts. escape from the photosphere. Chromospheric features are entirely different from photospheric ones and include canopies of ﬁbrils, loops, jets, etc. (see Fig. 1.1). Due to the small density, collisional rates are low so that the approximation of LTE is typically invalid for chromospheric lines, which presents a challenge to modelling these lines. Additional complications to chromospheric modelling are: effects of partial and non-equilibrium ionization, partial frequency redistribution of scattered photons, and 3D radiative transfer effects. A review of chromospheric diagnostics and their required modelling efforts is given in Section 4 of de la Cruz Rodríguez and van Noort (2017). The transition region is the interface region between the chromosphere and corona in which the temperature suddenly and very steeply rises from O(104) K to O(106) K. The steep temperature rise occurs when all neutral hydrogen (and helium) is ionized away. Lines of highly ionized species become visible such as Si IV and O IV lines. The transition region is a topic of many recent and present studies, due to the launch of the Interface Region Imaging Spectrograph in 2013 (IRIS, De Pontieu et al. 2014). The hot corona is sampled in lines of highly ionized species (e.g. Fe XII 193 Å or Fe XVIII 94 Å) from space with instruments such as the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012) on board of the Solar Dynamics Observatory. Observable features in the corona are for example large scale magnetic loops above active regions that are bright due to heating. Coronal lines are formed in conditions which are called coronal equilibrium. Elec- trons are excited collisionally and decay radiatively, meaning that observed

8 line intensities are directly related to temperature of the corona. This property facilitates modelling of the corona. The reason why the chromosphere and corona are heated so violently is still presently debated, and referred to as the coronal heating problem (for recent reviews see e.g. Parnell and De Moortel 2012; Klimchuk 2015; De Moortel and Browning 2015). It is clear that heating by dissipation of Alfvén and magneto-acoustic waves is contributing to some extent, but not all observed heating can be explained by waves. Additional sources of magnetic heating are necessary, for example by small-scale reconnection events in chromosphere and transition region. In order to make progress on this problem - and on a better understanding of the chromosphere in general - a combination of MHD-modelling, radiative transfer modelling and high-resolution observations in the chromosphere and transition region is necessary. This thesis focuses on rather atypical chromospheric diagnostics: the He I D3 spectral line at 5876 Å and the more commonly used He I 10830 Å line, both originating from transitions in the triplet system of neutral helium (see Sect. 2.2). These lines are chromospheric, not because of their large opacity but due to their exceptional formation mechanism (discussed in Chapter 2). The energy levels of He I 10830 Å and He I D3 are populated proportional to the amount of incident extreme ultraviolet (EUV) radiation originating in the transition region and the corona. Therefore, the He I 10830 Å and He I D3 lines are sensitive to heating in those layers. As opposed to most chromospheric diagnostics, He I D3 and He I 10830 Å are usually formed in a thin layer in the upper chromosphere and have modest opacities, which means that their line proﬁles can be calculated under the assumption of complete frequency redistribution. However, additional modelling efforts are required because of their sensitivity to the radiation ﬁeld.

He I D3 and He I 10830 Å are formed at a height that is not sampled by any other spectral line able to provide magnetic ﬁeld measurements. Fig. 1.2 shows the formation height of He I 10830 Å and He I D3 together with the most commonly used chromospheric diagnostics. Figure 1.2 suggests that the helium formation height largely coincides with Ca II H formation height. However, this is likely only true within the quiet sun regime in which these calculations are performed. Moreover, no spectro-polarimetric observations of Ca II 1 H are currently available . Therefore, He I 10830 Å and He I D3 remain the

1The Ca II H&K lines are routinely observed with CHROMIS, a recently installed instrument at the SST. Ca II H&K are magnetically sensitive but CHROMIS is currently not equipped to conduct polarimetric observations. Moreover, the Ca II H&K lines are expected to be less effective for magnetic ﬁeld measurements due to a low UV photon count and their short wavelength which reduces the strength of the Zeeman splitting (de la Cruz Rodríguez and van Noort, 2017).

9 Figure 1.2: Some common chromospheric diagnostics and their formation height at the line core. The helium line formation height is given in grey-scale and has a continuous contribution along the line of sight because it is optically thin in this atmosphere. Courtesy of Mats Carlsson. only tools available to conduct spectro-polarimetric measurements in the upper chromosphere, providing a unique opportunity to infer the magnetic field at those heights. Because He I D3 and He I 10830 Å are usually optically thin and not typically have a profiles with self-reversal, velocity measurements via these lines are more straightforward than via typical optically thick chromospheric diagnostics such as Ca II H&K or Hα. For example, a blue asymmetry in an Hα profile with self-reversal could mean either upward velocities at the formation height of the blue peak, or downward velocities at the formation height of the line core. This type of ambiguity is absent in the He I D3 and He I 10830 Å lines, providing an opportunity for detailed velocity measurements in the chromosphere. The structure of the thesis is as follows: we discuss helium spectral line formation (Chapter 2), spectro-polarimetry and inversions (Chapter 3), high- resolution observations of the chromosphere via helium (Chapter 4), an example of small-scale magnetic reconnection events called Ellerman bombs (Chap- ter 5) and large-scale reconnection, i.e. flares (Chapter 6). This information should provide the reader with a review so that the papers that are included in this thesis can be read and situated within the research field by astronomers or scientists in other fields.

10 2. Helium Line Formation

2.1 Discovery of Helium

The element helium was discovered in solar spectra before it was detected on Earth. Since the early 19th century, spectral lines in the solar spectrum were known but their origin was not understood at that time. Joseph von Fraunhofer named prominent spectral lines and regions with letters, some of which are still commonly used today, see Fig. 2.1 and Table. 2.1. Gustav Kirchoff and Robert Bunse made progress in the 1860’s by dedicating spectral lines to atomic elements. They were the ﬁrst to realize that solar absorption lines correspond to emission lines produced in laboratory experiments when heating chemical elements. They correctly concluded that solar spectral lines are the result of absorption by atoms and molecules in the solar atmosphere. In 1868, an unknown yellow and bright spectral line was observed in eclipse spectra by the French astronomer Pierre-Jules-César Janssen and his co-workers. This line was designated with the Fraunhofer name D3 because of its proximity to the Na I D1&D2 lines. Independently, Norman Lockyer observed the line as well and concluded that it was not produced by any known element at the time. Therefore he named the new element after “Helios”. The existence of the new element helium was doubted by the scientiﬁc community at the time and it was believed that helium - if it existed - only occurred on the Sun. Then in 1895, the element was detected on Earth in laboratory experiments of Scottish chemist Sir William Ramsay. He treated Cleveite (i.e. UO2 with

Figure 2.1: The visible solar spectrum and its most prominent absorption lines labeled with their Fraunhofer designation. The x-axis indicates wavelength in Å.

11 Table 2.1: An example of some Fraunhofer designations for spectral lines as shown in Fig. 2.1.

Fraunhofer Element Wavelength designation [Å]

C Hα 6562.8

D1 Na I 5895.9

D2 Na I 5889.0

D3 He I 5875.6 H Ca II 3968.5 K Ca II 3933.7

10% of the uranium substituted by rare earth elements1) with acid and he iso- lated helium gasses resulting from the chemical reaction. In the same year, Per Teodor Cleve and Abraham Langlet from Uppsala independently recovered helium from Cleveite and were able to measure the atomic weight of helium.

2.2 Atomic Structure

The Hamiltonian for an atom in the absence of an external magnetic ﬁeld is given by H = H0 + Hee + Hso. (2.1)

H0 describes the kinetic energy and the potential energy of an electron in the electric ﬁeld of an atomic nucleus, with a distance r j between the electron and the nucleus 2 h¯ 2 Ze H0 = ∑ − ∇ − . (2.2) j 2me r j

The term Hee is the Coulomb energy between the electrons 1 e2 Hee = ∑ , (2.3) 2 i6= j |r j − ri| and Hso is the spin-orbit coupling between the angular momentum l j and the spin s j of the electron Hso = ∑ξ(r j)l j · s j, (2.4) j

1The 17 rare-earth elements are cerium (Ce), dysprosium (Dy), erbium (Er), europium (Eu), gadolinium (Gd), holmium (Ho), lanthanum (La), lutetium (Lu), neodymium (Nd), praseodymium (Pr), promethium (Pm), samarium (Sm), scandium (Sc), terbium (Tb), thulium (Tm), ytterbium (Yb), and yttrium (Y).

12 Figure 2.2: Schematic view of some of the energy levels of neutral helium. In the side panel, a zoom on He I D3 and He I 10830 Å and their splitting according to the J quantum number is shown. Figure from Centeno et al. (2008).

with ξ(r j) the spin-orbit coupling constant. In light atoms such as helium, Hee > Hso so that Hso can be treated as a perturbation to H0 + Hee. This is called LS-coupling. In the LS-coupling scheme, energy states are often written as terms

2S+1 (o) LJ , (2.5) where quantum numbers S and L represent the combinations of angular momentum of respectively spin s j and orbit l j. J is related to the total angular momentum which can have discrete values between |L − S| ≤ J ≤ |L + S| and adds a splitting to the 2S+1L-levels. The parity of the state is determined from (−1)(∑ j l j). The superscript o in the term is written when there is odd parity. In LS-coupling, the selection rules for electric dipole transitions require a parity change, ∆J = 0,±1,∆S = 0,∆L = 0,±1 and no L = 0 → L = 0 transitions. The ∆S = 0 selection rule divides neutral helium into para-helium and ortho-helium, behaving almost like two separate atoms, see Fig. 2.2. For para- helium, the two electrons have a total spin S = 0 and the corresponding energy levels are referred to as the singlet states. The singlet terms are therefore 1S, 1L, 1D, etc. Ortho-helium has two electrons with a total spin that adding up to

13 Table 2.2: Components of He I 10830 Å and He I D3. Wavelength information taken from the online NIST Atomic Spectra Database (Kramida et al., 2015) with the helium levels from Morton et al. (2006).

λ in Air [Å] Levels 3 3 5875.599 1s2p P2 - 1s3d D1 3 3 5875.614 1s2p P2 - 1s3d D2 3 3 5875.615 1s2p P2 - 1s3d D3 3 3 5875.625 1s2p P1 - 1s3d D1 3 3 5875.640 1s2p P1 - 1s3d D2 3 3 5875.966 1s2p P0 - 1s3d D1

3 3 10829.091 1s2s S1 - 1s2p P0 3 3 10830.250 1s2s S1 - 1s2p P1 3 3 10830.340 1s2s S1 - 1s2p P2

S = 1 and its energy levels are called the triplet states 3S, 3L, 3D, etc. The He I 10830 Å and He I D3 lines are both transitions between energy levels in the triplet system in neutral helium as indicated in Fig. 2.2. The He I 10830 Å line is the transition between the ground level of ortho-helium 3 3 1s2s S1 and the ﬁrst excited state 1s2p P0,1,2. The He I D3 line is the tran- 3 3 sition between 1s2p P0,1,2 and 1s3d D1,2,3 in ortho-helium. Due to the level splitting according to quantum number J, the He I 10830 Å line has 3 components and He I D3 has 6 components, listed in Table 2.2. Since electric dipole radiative transitions do not allow for a change in spin (∆S=0), transitions between singlet and triplet states of neutral helium are forbidden via the electric dipole transition. Moreover, the ground state of ortho- helium 1s2s 3S is also suppressed to decay into the ground state of para-helium 1s2s 1S by the need of a parity change. Therefore, the ground state of ortho- helium has an extremely long lifetime of the order of 104 s before it would decay into the actual ground state of helium.

2.3 Line Formation Mechanisms

Helium is the ﬁrst noble gas in the periodic table. In the ground state, the atom has two electrons in its 1s orbital. Therefore, helium has a very high ﬁrst ionization potential of 24.587 eV. This property, along with the fact that radiative transitions between the singlet and triplet states are forbidden via the electric

14 dipole transition, raises questions on how the triplet states are populated and how triplet lines such as He I D3 and He I 10830 Å obtain their opacity. Af- ter all, a large amount of energy is needed to ionize helium - either through photoionization or collisionally - or to directly populate the triplet levels collisionally from the helium ground state. Moreover, collisional rates are low in the chromosphere due to the limited (electron) density. For clarity of the review in Sec. 2.4, we present here the three possible mechanisms which are able to populate triplet levels of neutral helium:

• Photoionization-Recombination Mechanism (PRM): electrons in the ground 2 1 state of neutral helium (1s S0) are ionized via EUV photons shortward of λ ≤ 504 Å. The photoionization rate Rik from an atom with the electron in state i to the ionization state k is given by

∞ Z Jν αi jνdν Rik = 4π , (2.6) νi j hν

with Jν the incident radiation ﬁeld, αi j(ν) the photoionization cross- section, νi j is the frequency of the transition and ν is the frequency over which we integrate. After photoionization, the electrons recombine into both the singlet system (S = 0) and the meta-stable triplet system (S = 1) of helium. This process is also referred to as electron cascades. The power P per volume emitted by electron cascades between level i and level j is set by P = niAi jhνi j, (2.7)

with ni the population of level i, Ai j is the Einstein coefﬁcient for spontaneous radiative decay, h is the Planck constant and νi j is the frequency associated with the transition. The EUV photons which ionize helium are impinging on the chromosphere from the corona and transition region (a detailed discussion is given in Sec. 2.4). The PRM is schematically presented in Fig. 2.3.

• Collisional Excitation Mechanism (CM): the electrons in the helium ground state are excited through collisions with free electrons. The collisional transition rate from level i to level j is given by Z ∞ niCi j = nine σi j(v)v f (v)dv, (2.8) v0 ≡ nineqi j(T), (2.9)

with ni the population of level i, Ci j the number of collisional excitations from state i to state j, ne is the electron density, v0 is a threshold velocity 1 2 given by 2 mv0 = hν0, σi j(v) is the collisional cross-section, f (v) is the

15 Figure 2.3: Schematic representation of the photoionization-recombination mechanism (PRM). CI stands for coronal illumination. Figure from Centeno et al. (2008).

velocity distribution (Maxwell distribution in LTE) and v is the velocity over which we integrate. In LTE, excitation balances are calculated via the Boltzmann equation.

• Collisional Ionization-Recombination Mechanism (CRM): helium gets collisionally ionized with rates given in Eq. 2.8 after which the electrons recombine spontaneously in electron cascades as described by Eq. 2.7. In literature, CRM and CM are not always distinguished. It is usually safe to assume that if CRM takes place, CM will take place as well and vice versa, since the ionization energy is at 24.587 eV while the 3 excitation energy for the ground state of the triplet system (1s2s S1) is 19.820 eV. In LTE, ionization balances are given by the Saha equation as shown for helium in Fig. 2.4 for different electron densities.

Once the triplet levels are populated via one of the above mechanisms, the He I 10830 Å and He I D3 lines are formed via scattering of continuum photons. This (usually) results in absorption lines on-disk and emission lines off-limb.

2.4 Review on Helium Line Formation Research

2.4.1 Helium Line Intensity

The observed intensities of He I spectral lines have been raising questions since Goldberg (1939) pointed out the problem in 1939. He studied chromospheric 1932 eclipse spectra in many optical He I lines, both from triplet and singlet states. Assuming spontaneous de-excitation (Eq. 2.7) and the Boltzmann dis-

16 Figure 2.4: Ionization balance for neutral helium as given by the Saha equation at different electron densities Ne.

tribution for level populations (LTE), he estimated the temperature of the chromosphere at different heights above the solar limb. Two important results were obtained from this. First, his results showed that the temperature in the chromosphere increases with height, which was unexpected at that time. Second, the observed intensity of the singlet lines was anomalously faint in comparison to the triplet lines at the temperatures that he estimated (between 4000-7000 K). Goldberg (1939) then proposed the idea of the meta-stable helium triplets states being (over)populated by a photoionization-recombination mechanism (PRM) driven by an excess of UV-radiation, even though he did not comment on a possible cause for this excess.

In 1975, Jordan (1975) found a discrepancy between He I and He II observed resonance line strengths (e.g. He I 584 Å and He II 304 Å) and the ones that she calculated assuming of coronal equilibrium and optically thin line formation. She pointed out that the observed line intensities were stronger by a factor ﬁve compared to the calculated ones, which was not the case for lines of other ions formed at similar temperatures. According to her, the mechanism causing this discrepancy should enfold atoms and ions excited by electrons at a higher temperatures than the temperature at which the ionization equilibrium is operating.

In summary, the entire helium spectrum (He I singlet and triplet lines and He II lines) as observed from the Sun could not be reproduced by modelling efforts of Goldberg (1939) and Jordan (1975), and many other authors after them who have attempted this.

17 2.4.2 Coronal Heating and Early Measurements of the Solar UV Flux (50’s-70’s)

Goldberg’s results indicated an increasing temperature with height in the chromosphere. Coronal (and chromospheric) heating was on the verge of being discovered. In the same period, Grotrian (1933), Edlén and Swings (1942) and Edlén (1945) found that observed emission lines from the corona during solar eclipses are originating from iron and other known elements in very high ionization states instead of being related to a new element “coronium”. These findings implied that the solar corona is heated to temperatures of the order of 106 K. Meanwhile, theoretical work by Alfvén (1942) demonstrated the existence of magneto-hydrodynamic waves in the solar atmosphere. Alfvén concluded that the dissipation of the waves at different heights could possibly lead to heating of the corona up to the observed temperatures. At that time, the temperature of the chromosphere was still debated and the observed intensities of neutral helium triplet lines lead Redman (1942) to propose a chromosphere at a temperature of 35 000 K. Miyamoto (1951) con- tradicted this by saying that the observed line intensities of the Fraunhofer lines were consistent with a much cooler chromosphere at 10 000 K. He proposed the PRM as playing an important role in the population of the triplet states of helium in the upper chromosphere, realizing (as opposed to Goldberg) that the required UV-radiation originates in the corona. The main criticism on the first proposals for the PRM was based on the doubt whether the UV-flux of the corona was sufficiently large to explain the observed intensity of helium lines in the chromosphere. Therefore, several authors such as Athay (1963) and Zirin and Howard (1966) proposed a chromosphere having a high-temperature region at around 20 000 K (in later models often referred to as the “Lyman plateau”), which would explain population of helium triplet levels through the collisional excitation mechanism (CM) and collisional ionization-recombination mechanism (CRM). During the fifties and sixties, several rocket missions with an on-board spectrograph aimed at obtaining spectra in (E)UV-wavelength ranges. Also satellite missions contributed such as OSO-IV (Goldberg et al., 1968) obtaining spectroheliograms in EUV lines. In 1971, Hirayama (1971) used a UV-flux provided by rocket data (Hinteregger, 1965) to calculate the population of the helium triplet levels and he concluded that the EUV-flux from the corona is sufficiently intense to populate helium triplet levels via a PRM in both prominences and the chromosphere. An observational leap forward was made with the NASA mission Skylab, a space station which orbited Earth between 1973 and 1979. It was equipped with a versatility of instruments that were able to measure and image the so-

18 lar (E)UV-ﬂux. It provided spectroheliograms in (E)UV and the optical to visualize the chromosphere and corona. Many features were discovered and discussed as for example coronal holes observed in He II 304 (Tousey et al., 1973). These coronal holes were spatially linked with weakening of helium lines.

2.4.3 Observational Constraints In the 70’s and 80’s, observational constraints for helium modelling were pointed out by several authors. Any model that attempts to explain the solar helium spectrum should be able to reproduce not only the observed intensities but also a number of additional observational constraints as listed here.

1. He I and He II lines are weakened in coronal holes while transition region lines from other elements are only slightly affected (Tousey et al., 1973; Zirin, 1975).

2. He I triplet lines such as He I D3 and He I 10830 Å show enormous limb brightening: a factor 20 over 1 arcmin (Zirin, 1975).

3. A dark band is seen between the He I D3 emission region and the solar limb (Zirin, 1975).

4. The line proﬁles of resonance lines such as He I 584 Å, He I 504 Å, and He II 304 Å should be reproduced correctly (Milkey, 1975). However, observations have not been conclusive whether or not the He I 584 Å line shows a central reversal.

5. He I 10830 Å (and He I D3 ) filtergrams show spatial structure on a sub- arcsecond scale (Lites et al., 1985). Constraints 1 – 3 are naturally explained by the PRM as discussed in Zirin (1975). He proposed a model in which neutral helium is excited by coronal EUV-radiation, meanwhile producing He II resonance line photons at 304 Å, which themselves also contribute to ionize neutral helium. He successfully calculated the He I D3 intensity and the emission heights using a photoionization- recombination model. Principal doubts in his model are the uncertainty on the coronal EUV-flux and the accuracy of the fit of the model to the He I D3 line emission. Constraint 4 was put forward by Milkey (1975) who wrote a comment to Zirin (1975). He pointed out that if it is true that the PRM is responsible for nearly all triplet level population, then profiles of resonance lines of He I and He II should have a self-reversal in the line core. The large absorption of photoionizing photons by neutral helium would convert them into resonance

19 line photons at large line center opacities. The resonance lines hence would undergo many scatterings and escape by frequency redistribution to the wings of the line. Constraint 5 presents an observational counter argument for the PRM and was raised by Lites et al. (1985). Studying He I 10830 Å echelle spectra from the Vacuum Tower Telescope at Sacramento Peak Observatory, they noticed spatial structure of less than 1 arcsec. However, if the triplet level population is set by (E)UV-radiation from the corona, the impinging radiation field on the chromosphere would be spatially averaged over the solid angle defined by the distance corona-chromosphere and be rather smooth - unable to cause structure on sub-arcsec scale. Moreover, there is no evidence that the coronal (E)UV radiation field itself possesses fine-structure on those small spatial scales.

2.4.4 Semi-Empirical 1D Model Atmospheres

Meanwhile, Vernazza et al. (1973) had published their famous semi-empirical one-dimensional model of the solar atmosphere, see Fig. 2.5. They solved the radiative transfer equations for statistical equilibrium in complete redistribution for H, H-, Si I and C I atoms. The atmospheric model was retrieved by trial-and-error guesses to ﬁt the observed spectral lines of the atoms in the model. Remarkable is the “Lyman plateau” in the upper chromosphere at around 20 000 K which was needed to account for the observed intensities of hydrogen lines. A similar high-temperature region was proposed previously to argue in favor of collisional excitation and ionization for neutral helium. Almost immediately thereafter, Milkey et al. (1973) used this model to calculate helium line intensities for a 14-level He I atom in statistical equilibrium. They also used UV data from OSO-IV. They found that they needed a thick region of material at 45 000 K in the upper chromosphere to reproduce the strength of resonance lines at 584 Å and at 537 Å. Then they concluded that the triplet levels are populated by the photoionization-recombination mechanism in low-temperature regions (T ≤ 10 000 K) and by collisional excitation in high-temperature regions (T ≥ 20 000 K). The overpopulation of the triplet states compared to the singlet states was attributed to the decay of singlet states producing resonance line photons. With each resonance line photon escaping, the depopulation of the singlet states becomes larger. A similar mechanism is absent for the triplet states. The proposed model of Milkey et al. (1973) did roughly reproduce quiet sun helium spectra but not eclipse spectra. In the 1980’s, the discrepancies between observations and theory were known and pointed out (see Sec. 2.4.3). Progress on the problem was made by more sophisticated theoretical calculations and more advanced observational work in both solar and stellar context (e.g. Livshits et al. 1976; Mango et al.

20 Figure 2.5: VAL-model as ﬁrst published in 1973 by Vernazza et al. (1973).

1978; Kahler et al. 1983; Obrien and Lambert 1986; Zarro and Zirin 1986; Pozhalova 1988; Batalha and de La Reza 1989). However, the dominant mechanism populating neutral helium triplet levels remains unclear. In 1990 and following years, Fontenla et al. (1990) updated the previous VAL-models to current FAL-models including more realistic physics relevant to the chromosphere and transition region, such as ambipolar diffusion (i.e. neutral atoms diffusing in opposite direction as protons and electrons) and conductive heat transport in a partially ionized gas. Diffusion effects allow hydrogen to be present at relatively high temperatures and the new FAL-models therefore do not exhibit the “Lyman"-plateau any longer. The Lyman line forming region is now placed between 40 000 – 70 000 K in the transition region, showing a steep temperature gradient. The authors specifically studied the formation of helium lines within their models (Fontenla et al., 1993) and claim that they can reproduce observed central intensities of helium resonance lines, which are enhanced by diffusion effects. However, the computed spectral profile of He I 584 Å shows a self- reversal which seemed to be in contradiction with observations and the He II resonance lines are still rather weak. Moreover, the models fail to reproduce enhancement of He I 10830 Å absorption in plages as compared to the quiet sun. The influence of coronal irradiation is studied and continued in more de-

21 tail by Avrett et al. (1994) and it is concluded that the spectral line profiles not only depend on coronal UV irradiation but are also greatly affected by the geometrical thickness of the chromosphere. Their plage model had a thinner chromosphere which explains the weakening of He I 10830 Å lines as compared to the quiet sun. Andretta and Jones (1997) conducted similar calculations using modified VAL-C models (the temperature plateau was made smaller) and solving the radiative transfer equations in non-LTE. One of their interesting results con- cerned a model where only the photoionization-recombination mechanism was acting to excite neutral helium (no transition region was present). For fixed coronal illumination it was found that neutral helium triplet lines weaken with increasing electron density. The increased number of collisions allow transitions from the triplet system to the singlet system which leads to a depopulation the triplet levels. This is an opposite density dependence of what was found by Livshits et al. (1976). The calculations also showed that He II resonance lines are less sensitive to coronal illumination than He I triplet lines, suggest- ing a collisional excitation mechanism for He I 304 Å, which was also found by Fontenla et al. (1993). In the end, Andretta and Jones (1997) propose a mixed two-layer line formation model (see Fig. 2.6), in which the photoionization-recombination mechanism is acting in the upper chromosphere under influence of coronal (E)UV radiation. Collisional excitation would dominate in the transition region at temperatures above 20 000 K. In this scenario, the authors dedicate the observed small-scale structure in He I 10830 Å to density perturbations in the transition region where He I is excited collisionally. The large scale correlation between UV radiation and He I 10830 Å intensity (such as in coronal holes) would originate via the photoionization-recombination mechanism in the chromosphere. In the discussion so far, modelling efforts for helium have mostly concentrated on quiescent and plage regions. Mauas et al. (2005) studied the problem in a solar active region in the context of helium abundance determination. The abundance of helium in the solar photosphere is not well known because it is undetectable in the visible photospheric spectrum, hence the helium abundance has been determined by theoretical stellar evolution models with an often used value of [He] = NHe/NH = 0.1. However, there are large uncertain- ties to this result because measurements of solar wind abundance in the fast solar wind yield results of ∼ 0.05. In the fast solar wind, elements with a low first ionization potential (FIP) are usually enhanced compared to their photospheric abundance, while elements with a high FIP usually maintain their photospheric abundance. Helium seems to be an exception to this behavior because its abundance is reduced in the fast solar wind and it is suspected that

22 Figure 2.6: Mixed two-layer line formation model as proposed by Andretta and Jones (1997). the fractionization according to FIP should occur in the chromosphere. There- fore, the physics of helium ionization in active regions and coronal holes in the chromosphere should be understood to explain its abundance in the solar wind (see Chapter 9 of Schrijver and Siscoe (2011) for a review on the solar wind and FIP effects). Mauas et al. (2005) estimated the coronal (E)UV irradiance using observations in the UV from the Solar EUV Monitor (SEM, Hovestadt et al. 1995) and the EUV Imaging Telescope (EIT, Delaboudinière et al. 1995). Those observations were co-temporal with observations of multiple chromospheric lines from the Solar and Heliospheric Observatory (SOHO). Constructing semi-empirical models to ﬁt the data, they concluded that impinging (E)UV radiation on the chromosphere was of crucial importance for He I triplet lines while it has only a limited effect on He UV lines in active regions, similar as to what was found earlier for quiet sun and plages. Their models could ﬁt the observations rather

23 Figure 2.7: Synthetic profiles for He I 10830 Å and He I D3 with different values for coronal irradiance (CI). Figure from Centeno et al. (2008). well for different helium abundances hence more observational constraints on the transition region structure are necessary to estimate the helium abundance via this method. Centeno et al. (2008) synthesized He I 10830 Å and He I D3 lines from VAL and FAL models with a prescribed coronal irradiance and found that the two components of the He I 10830 Å line are getting saturated at different optical depths. The optical depth corresponds to the population of the level which is set by EUV irradiance, as shown in Fig. 2.7. The authors hence proposed that the ratio of the blue and red component in the He I 10830 Å line is a powerful tool to estimate coronal irradiance in observations of the line profiles. No such effects for He I D3 were found because this line does not reach sufficiently high optical depths to saturate in their models. The large sensitivity of He I 10830 Å and He I D3 to coronal irradiance is a strong argument in favour of the PRM for helium triplet line formation.

2.4.5 Diffusion and Burst Models for Helium Line Formation In order to meet the observational criteria described in Sec. 2.4.3, several alternative scenarios have been proposed which are based on the idea that electron

24 temperatures are different from the temperature at which helium ionization is acting (Jordan, 1975). Shine et al. (1975) proposed that the weakening of the line in coronal holes might be due to diffusion effects. The authors investigated a mechanism in which thermal diffusion causes He I to move upwards in the atmosphere towards regions of higher temperature, while not immediately getting ionized because of the high ionization potential. This mechanism explains the population of triplet levels through collisions in high temperature regions while still having sufficient He I present, which would enhance the intensity of neutral helium triplet lines. Moreover, such diffusion effects would be mostly absent in coronal holes because of a more moderate temperature gradient in the transition region. However, they did not calculate He I triplet line intensities. Another - more or less opposite - diffusion mechanism has been studied by Shoub (1983). The transition region electrons in his model have a non- Maxwellian velocity distribution with a high-velocity tail in lower temperature regions. This is the result of diffusion of electrons from higher to lower temperature. This mechanism also provides a situation in which the electrons have a higher temperature than at which He I ionization operates. The author ar- gues that this mechanism will enhance emission of He I and He II resonance lines due to collisional excitation of the triplet levels by non-thermal electrons. Shoub (1983) obtained enhanced values of collisional excitation rates for He I and He II resonance transitions but he did not calculate line profiles. The He I diffusion mechanism as proposed by Shine et al. (1975) got criticized by Wolff and Heasley (1984) because of an unrealistic formulation of the physics. Wolff and Heasley (1984) studied the ratio of He I 10830 Å over He I D3 in dwarf G and K stars and compared the observations with computed non-LTE line intensities. However, they obtained too weak He I 10830 Å and He I D3 line intensities for a realistic (E)UV-flux and therefore suspect that the PRM is not the dominant line formation mechanism in these stars. Laming and Feldman (1992) proposed a burst model to investigate He I and He II intensities in flares. The burst is simulated by raising the electron temperature of small regions of the plasma in times that are short compared to the ionization equilibration time for helium. Electron collisions are then considered as the mechanism to excite electrons into the neutral helium triplet states. The above diffusion and burst models were theoretically developed and unified into “velocity redistribution” models (Andretta et al., 2000; Smith and Jordan, 2002; Judge and Pietarila, 2004; Pietarila and Judge, 2004). The idea is that cold helium atoms diffuse into hotter regions due to turbulent motions, locally increasing the He I abundance. This will enhances helium triplet lines and helium EUV resonance lines due to collisional excitation.

25 2.4.6 Non-Equilibrium Helium Ionization and 3D Modelling

Velocity redistribution models incorporate electrons at temperatures which are higher than ionization temperatures. However, non-equilibrium ionization also provides a situation in which ionization is decoupled from electron temperatures. Golding et al. (2014) studied non-equilibrium (non-E) helium ionization using 1D modelling carrying out a full non-E time-dependent simulation and compared their results to a run under the assumption of statistical equilibrium (SE). They found that in the case of shocks, there are considerable non-E effects which influence the line formation. During a shock and with the assumption of SE, all He II ionizes instantaneously consuming much of the heat released in the shock. In non-E, the He II ionization rate is lower meaning more heat is produced in the shock and more 304 Å photons are produced collisionally. Therefore, the triplet levels are more populated in the non-E case with respect to the SE case and the He I 10830 Å and He I D3 opacities are enhanced. The ionization balance between the He I and He II ground levels is set by photoionization but can be influenced by collisions during shocks due to the additional heat present. The work by Golding et al. (2014) has shown that the helium ionization balance has a substantial effect on the energy balance and hence the thermal structure in the chromosphere and transition region. The limit on modelling helium ionization and line formation is that it has been always conducted in one dimension. Progress in 2D and 3D radiative magnetohydrodynamics (RMHD) has been made since Leenaarts et al. (2007) have conducted 2D RMHD simulations of non-equilibrium hydrogen ionization. The method has been implemented in the stellar atmosphere code Bifrost (see Gudiksen et al. (2011) for a description of Bifrost) and later, also non-equilibrium ionization of helium has been implemented (Golding et al., 2016). Golding et al. (2016) have performed 2D RHMD simulations including non-equilibrium ionization of both hydrogen and helium and taking into account the effect of Lyman-α radiation and EUV radiation from the corona, which is calculated self-consistently within the simulation. Similarly as to Golding et al. (2014), the results of these simulations show that there are large variations in chromospheric temperature because the possible long ionization time scales do not immediately balance perturbations in temperature. Leenaarts et al. (2016) have used the 3D extension of the 2D simulations by Golding et al. (2016) to study line formation of He I 10830 Å and the cause of observed spatial sub-arcsecond structure in this line. The EUV field has been taken into account self-consistently instead of using prescribed values derived from observations, as has been done in all line formation studies before. More-

26 over, a 3D evaluation of the radiation field allows to study geometrical effects on line formation. This has never been done before because all previous modelling has been done in 1D. The simulations have confirmed that the main population of the triplet levels is caused by the PRM in the quiet sun. Contributions of direct collisional excitation are negligible in the model. The main source of photoionizing radiation originates in the corona from plasma at temperatures between 0.5 − 2 · 106 K. Interestingly, there is a second source of ionizing photons in the transition region of gas between 8 − 20 · 104 K. The contribution of ionizing photons from the transition region is smaller than from the corona, but the transition region gas at temperatures sufficient to produce the photons is highly spatially structured and the photons originate spatially close to the chromosphere. This means that the photoionizing radiation impinging on the chromosphere from the transition region possesses spatial structure and causes the observed sub-arcsecond structure in He I 10830 Å and He I D3 lines. This effect is demonstrated by comparing observed He I 10830 Å intensities to transition region and coronal images, see Fig. 2.8. It is clear that the spatial structure as observed in He I 10830 Å equivalent width measurements correlates with transition region emission of Si IV, a line having its equilibrium formation temperature at around 8 · 104 K. Leenaarts et al. (2016) also studied electron density dependence and found - similar as Livshits et al. (1976) - that the triplet level population is an increasing function of electron density. Density perturbations in the chromosphere thus can also cause spatial variations in observations of He I 10830 Å and He I D3. Substantial progress has been made also for EUV resonance lines of He I and He II by Golding et al. (2017). Effects of 3D, non-equilibrium ionization, radiative transfer, and coronal illumination were investigated. It turns out that non-equilibrium ionization and electron cascades give rise to enhanced intensities of He I 504 Å, He I 584 Å and He II 304 Å. The line profile of He I 584 Å shows a small self-reversal. The authors include a discussion on observations of this line and argue that a small self-reversal in the modeled profile is not in contradiction with the observations. Spectra with high resolving power would shed light on this issue. It now seems that intensities of He I triplet lines and He I and He II resonance lines are largely reproduced in the latest models, but there are many aspects left for further investigation. The simulations conducted by Leenaarts et al. (2016) have included the non-equilibrium helium ionization in the MHD part, but not for the calculation of the background coronal EUV irradiance. They have only studied He I 10830 Å intensities and no line profiles. Also, it is unclear to what extend the results are applicable to active regions and plages, and they are most likely not valid for flares or similar high-energetic phenom-

27 ena. The question still remains which line formation mechanism dominates under different solar conditions such as: prominences, ﬂares, ﬂux emergence regions, plages, etc.

Figure 2.8: Comparison between observations of chromospheric He I 10830 Å with the SST, transition region Si IV 1400 Å emission observed with IRIS and coronal line emission in several lines observed with SDO/AIA. Figure from Leenaarts et al. (2016). The blue box in panel a indicates the area in which the results are applicable (quiet-sun-like regime). The red contours in panel j outline areas with high He I 10830 Å equivalent width. The same contours are plotted in panel k and l, indicating that they coincide quite well with Si IV 1400 Å emission.

In fact, Paper I and Paper III in this thesis concentrate on helium lines in Ellerman bombs (EBs, see Chapt. 5), which is an example of a target in which the inﬂuence of the corona and transition region photons on helium line forma-

28 tion is negligible. EBs are small-scale reconnection events that take place in the deep chromosphere. Paper I discusses the discovery of emission signatures in He I D3 and He I 10830 Å in EBs, something that was unexpected since this would either mean that EUV photons are either locally generated in the EB or that the helium triplet levels would get collisionally excited. Both scenarios suggest higher temperatures for EBs than what was previously estimated before in the literature. In Paper I we propose a temperature range between 2·104 −105 for our observed EBs. The lower temperature limit was estimated assuming LTE, while the higher temperature limit was obtained from assuming that all observed broadening was collisional via thermal Doppler broadening – a quite unlikely case, but it does provide an upper limit. However, Paper III aims at really understanding helium line formation in EBs, wherefore we made use of an MHD simulation in which a synthetic EB was present. Using this model as an input atmosphere, we calculated the helium spectrum using the 3D non-LTE code Multi3D, taking into account coronal illumination self-consistently. We ﬁnd that the photoionization- recombination mechanism is still dominating over any collisional populating mechanisms. However, the EUV photons are generated locally at the bottom boundary of the EB at a height of 0.8 − 1 Mm, ionizing He I after which it recombines radiatively into the triplet states. In this case, the emission in He I D3 and He I 10830 Å is due to high electron density in a shell right outside the bottom boundary of the EB, causing a strong coupling to the local conditions. The temperature in this shell is between 7 · 103 − 104 K, which is higher than the temperature at which the continuum is created, so that He I D3 and He I 10830 Å are in emission. The photoionizing radiation is originating thermally inside the EB, at temperatures between 2 · 104 − 106 K and electron densities between 1011 −1013 cm−3. This result suggests - similarly as in Paper I - that helium emission signatures in EBs indicate temperatures of at least 2 · 104 K.

29 30 3. Polarization and Inversion Codes

3.1 Polarization of Light

Observed sunlight consists of electromagnetic waves which can be described by the time-dependent electric ﬁeld vector E(t). In a Cartesian coordinate system, we pose that the electromagnetic wave is traveling in the z-direction without loss of generality. E(t) is then deﬁned by the values of Ex(t) and Ey(t) equal to

Ex(t) = Ex(0)cos(ωt − φ1), (3.1) Ey(t) = Ey(0)cos(ωt − φ2), with ω the circular frequency of the electromagnetic wave and φ1,2 the phase. Polarization depends on the values of Ex(0), Ey(0) and the phase difference δ = φ1 − φ2. The most general case of polarization is elliptical polarization. The time evolution of the electric ﬁeld vector then describes an ellipse when projected on the xy-plane. Special cases of polarization are: • Circular polarization:

– Ex(0) = Ey(0) π – φ1 = φ2 ± 2 • Linear polarization:

– φ1 = φ2 Astronomers generally measure the intensity of light instead of the electric ﬁeld amplitude. Polarization is hence parameterized according to the four Stokes parameters which are deﬁned as 2 2 I = hEx i + hEy i, Q = hE2i − hE2i, x y (3.2) U = 2hExEy cosδi,

V = 2hExEy sinδi.

31 For polarization on a microscopical scale corresponding to individual waves, the Stokes parameters obey

I2 = Q2 +U2 +V 2. (3.3)

3.2 Polarizing Mechanisms

Polarization of light is produced when there is an anisotropy or a preferred direction present in the medium, for example due to magnetic ﬁelds. We discuss the mechanisms which cause and affect spectral line polarization of He I 10830 Å and He I D3.

3.2.1 Zeeman effect In Sec. 2.2, we have discussed the Hamiltonian of an atom. When a magnetic field is present, the Hamiltonian has an extra term HB given by e e2 H = − B · (L + 2S) + B2r2sin2 θ. (3.4) B 2mc 8mc In the presence of a magnetic field, the energy levels J are splitted into 2J + 1 sublevels, each with a different value for quantum number M. Hence, M indicates the level splitting with M = −J,J + 1,...,J − 1,J. If HB Hso, then the linear Zeeman effect is acting which causes a splitting of the energy levels. The splitting is linearly proportional to the magnetic field strength. When Hso and HB are comparable, the incomplete Paschen-Back regime starts having its effect. The level splitting increases non-linearly with magnetic field. If Hso HB, the complete Paschen-Back effect is acting. The influence of the Paschen-Back effect is illustrated in Fig. 3.1 for He I 10830 Å and He I D3. For He I 10830 Å, the incomplete Paschen-Back effect kicks in around 100 G and for He I D3 around 10 G as can be read from the upper panels. When the splitting is becoming large, level crossings can occur. These are shown in the lower panels of Fig. 3.1 and occur at 350 G for He I 10830 Å and at 10 G for He I D3. For the electric dipole transition between two J levels with splitting according to M, the selection rules ∆M = Mu − Ml = 0,±1 apply (index u refers to the upper level and index l to the lower level). A transition occurring between two J levels of which at least one of the J levels is different from zero will hence be split into three or more components, which can be grouped according to:

• ∆M = −1 → σr , shifted to the red with respect to λ0

32 Figure 3.1: Energy splitting as a function of magnetic ﬁeld strength. Top left: magnetic splitting for He I 10830 Å. Top right: magnetic splitting for He I D3. Bottom left: zoom on the magnetic splitting to show level crossings for He I 10830 Å. Top right: zoom on the magnetic splitting to show level crossings for He I D3. Figure from Asensio Ramos et al. (2008).

• ∆M = 0 → π , symmetrical around λ0

• ∆M = 1 → σb , shifted to the blue with respect to λ0

λ0 is the wavelength of the transition in absence of magnetic splitting. σr and σb can be mirrored on each other with respect to λ0. The wavelength of the transitions in the linear Zeeman regime can be calculated via

λ 2eB = − 0 (g M − g M ), λJuJl MuMl λ0 2 u u l l (3.5) 4πmec or expressed in frequency

νJuJl MuMl = ν0 − νL(guMu − glMl), (3.6)

e0B with ν0 the central frequency of the line unaffected by splitting, and νL = 4πmc the Larmor frequency. The splitting is dependent on the Landé factor g of each level, which equals

3 S(S + 1) − L(L − 1) g = + , (3.7) 2 J(J + 1)

33 Table 3.1: Relative strength of the transitions between the splitted levels in the different Zeeman components. In the Table, we omitted the subscript l hence all J and M are given for the lower level of the transition, unless subscript u is speciﬁed.

Ju = J + 1 Ju = JJu = J − 1

3(J+M+1)(J+M+2) 3(J−M)(J+M+1) 3(J−M)(J−M−1) σb Mu = M + 1 2(J+1)(2J+1)(2J+3) 2J(J+1)(2J+1) 2J(2J−1)(2J+1) 3(J−M+1)(J+M+1) 3M2 3(J−M)(J+M) π Mu = M (J+1)(2J+1)(2J+3) J(J+1)(2J+1) J(2J−1)(2J+1) 3(J−M+1)(J−M+2)) 3J(J+M)(J−M+1) 3(J+M)(J+M−1) σr Mu = M − 1 2(J+1)(2J+1)(2J+3) 2(J(J+1)(2J+1) 2J(2J−1)(2J+1) in the LS-coupling scheme, valid for light atoms (see Sec. 2.2). The relative strength of the transitions between the splitted levels in the different Zeeman components is given by Table 3.1. The way in which the Zeeman effect produces polarization can be understood by considering classical Lorentz theory of the electron. This describes an atom as an oscillating electron around a nucleus. The electron obeys the following equation of motion in the presence of a magnetic ﬁeld:

d2x e dx dx = −4π2ν x − 0 × B − γ . (3.8) dt2 0 mc dt dt

The first term describes the oscillation of the electron in the atom, the second term is the Larmor precession of a charge in a magnetic field and the last term is the damping term caused by emission of radiation. Detailed calculations (as given in e.g. Chapter 3 of Landi Degl’Innocenti and Landolfi 2004) show that this equation of motion gives rise to three oscillation modes at three frequencies corresponding to σr, σb and π, schematically presented in Fig. 3.2. Two special cases are defined:

• Transverse Zeeman effect: B and the observers line of sight are perpendicular. Only linear polarization is observed from the three components. The π component is polarized parallel to B, the σ components are polarized perpendicular to B.

• Longitudinal Zeeman effect: B and the observers line of sight are parallel. Only circular polarization is observed from σr and σb (the sign depends on the pointing of B). The π component is not observed in this geometrical conﬁguration.

34 Figure 3.2: Different Zeeman components generating polarization depending on the line of sight and the direction of the magnetic ﬁeld. Note that this is valid only for emission lines. The sign of the circular polarization is reversed for absorption lines. Figure from Landi Degl’Innocenti and Landolﬁ (2004).

The circular polarization produced through the longitudinal Zeeman effect is stronger than the linear polarization originating from the transverse Zeeman effect. For not too strong ﬁelds (weak ﬁeld approximation), the Stokes V signals are an order of magnitude larger than the Stokes Q and U signals.

3.2.2 Atomic level polarization In the absence of a magnetic field, the magnetic sublevels with different quantum number M are degenerate in energy, but they can still have different population levels. This happens when the atom is embedded in an anisotropic or polarized radiation field, or when anisotropic particle distributions are present (for example non-thermal electron beams in flares, see e.g. Judge (2015) and Sec. 6.1). Atomic level polarization is important for helium atoms in the upper chromosphere, since the incident solar radiation field is anisotropic. The anisotropy of the radiation field increases with height in the solar atmosphere. Two situations can be distinguished: • Atomic alignment: caused by e.g. anisotropy in the incident radiation field. The magnetic sublevels of different |M| are populated differently, giving rise to linear polarization.

• Atomic orientation: caused by e.g. polarization of the incident radiation

35 ﬁeld. The magnetic sublevels of equal |M| are populated differently, giving rise to circular polarization.

In the presence of a magnetic ﬁeld, the population of the magnetic sublevels M is altered by the Hanle effect see Par. 3.2.3.

3.2.3 Hanle effect

In the same way as the classical theory of the electron can provide an intuitive grasp on the Zeeman effect, it can also be useful to understand the Hanle effect at least qualitatively. A general property of scattering is that it polarizes light (except forward scattering), as illustrated in panel (a) of Fig. 3.3. (In the quantum mechanical description, scattering polarization is equivalent to atomic level alignment). When unpolarized light is scattered, the electric field excites the oscillators of the scattering atom. The oscillators are damped by emitting radiation. Depend- ing on the line of sight of the observer, which in Fig. 3.3 is along the x-axis, the light is linearly polarized along the y-axis. In generally terms, the light is linearly polarized perpendicularly to the scattering plane. The same scattering event is shown in panel (b) of Fig. 3.3, but the oscillators are now parameterized as one linear oscillator along the x-axis and two circular oscillators in opposite direction. The z-component of the electric field of the incident radiation excites the circular oscillators with frequency ν0 in such a way that they have a well defined phase relation (or coherence) between them and the scattered light is linearly polarized in the z-direction. In Fig. 3.3 panel (c), a magnetic field along the x-axis is introduced. The circular oscillators now have different frequencies: ν0 + νL and ν0 − νL with νL the Larmor frequency. Therefore, the well defined phase relation is lost and the polarization of the scattered light describes a distinct pattern called a “rosette” during the radiative decay of the circular oscillators. Part of the light is depolarized, part of it is linearly polarized along the main axis of the “rosette”. 2πνL The shape of the “rosette” is determined by the parameter H = γ , with γ given in Eq. 3.8 equal to the the Einstein coefficient Aul of the spontaneous radiative decay. When the magnetic field is strong (H 1), the pattern of the polarization becomes symmetrical (called a “daisy”), shown in panel (d) of Fig. 3.3. Therefore, the light is now completely unpolarized in the scattering direction. The situation is different for forward scattering. No polarization is present for zero magnetic field and forward scattering. However, when there is a magnetic field present (inclined to the scattering direction), the Hanle effect will

36 Figure 3.3: An illustration of scattering and the Hanle effect. Figure from Landi Degl’Innocenti and Landolﬁ (2004).

create linear polarization signals in the magnetic ﬁeld direction, see e.g. Fig. 5.17 of Landi Degl’Innocenti and Landolﬁ (2004).

In summary, the Hanle effect will act when the magnetic field is inclined with respect to the symmetry axis of the incident radiation field. For off limb observations for example, the Hanle effect will not be affecting the polarization if the direction of the magnetic field is radial. The Hanle effect introduces a (de)polarization and rotation on the scattered radiation. In the quantum mechanical description, the Hanle effect alters the atomic level alignment of the magnetic sublevels of the atom.

37 3.2.4 Regimes

We have discussed the mechanisms which produce and affect polarization of light in spectral lines of helium. Depending on the magnetic ﬁeld strength, different mechanisms dominate:

• B = 0: Very weak ﬁeld regime. Only scattering polarization/atomic level alignment can inﬂuence the polarization of helium lines.

• νL ∼ A: Hanle regime. The splitting of the magnetic sublevels is of the same order of the natural line width. The polarization of helium lines is affected by the Hanle effect but not yet by the Zeeman effect.

• νL A, νL < ∆νD: Saturated Hanle regime or weak Zeeman regime. The Hanle effect has depolarized the linear polarization as introduced from scattering/atomic level polarization. Splitting of the levels due to the Zeeman effect starts to be measurable but is still small compared to the Doppler width ∆νD of the line. The Doppler width equals

r ν 2kT ∆ν = 0 · , (3.9) D c m

with k the Boltzman constant, c the speed of light, T the gas temperature, and m the mass of the atom.

• νL ∼ ∆νD: strong Zeeman regime. The splitting is of the same order of magnitude as the Doppler width of the line and strong polarization signals are produced.

• νL > ∆νD: intense ﬁeld regime. The splitting of the levels is measurable in the Stokes I component. Polarimetry is not strictly necessary to measure the magnetic ﬁeld strength, which can now just be derived from the amount of splitting of the levels.

For He I D3 and He I 10830 Å, linear polarization is mostly dominated by atomic level polarization and the Hanle effect for magnetic fields below 100 Gauss. For stronger magnetic fields, linear polarization of helium lines is affected by the transverse Zeeman effect. However, Trujillo Bueno and Asensio Ramos (2007) showed that atomic level polarization can substantially affect linear polarization signals of the He I 10830 Å line up till magnetic field values of 1000 Gauss.

38 3.3 The Polarized Radiative Transfer Equation

To calculate the emerging Stokes vector I = (I,Q,U,V)T , one must solve the radiative transfer equation for polarized light

d I = −KKII + j. (3.10) dz

The z-axis is deﬁned towards the observer and K is the total absorption matrix

K = κc1 + κ0Φ, (3.11) with κc the opacity of the continuum, κ0 the opacity at line center and Φ de- ﬁned in Eq. 3.13. The emission vector j equals

j = κcSce0 + κ0Sle0, (3.12)

T with e0 = (1,0,0,0) , Sc the continuum source function and Sl the line source function. The absorption matrix or propagation matrix Φ equals

  ηI ηQ ηU ηV ηQ ηI ρV −ρU  Φ =  . (3.13) ηU −ρV ηI ρQ  ηV ρU −ρQ ηI

The ηI profiles describe absorption phenomena which are the same for all Stokes components, the profiles ηQ,U,V describe dichroic absorption phenomena and the dispersion profiles ρQ,U,V describe magneto-optical effects (Fara- day rotation and pulsation). If we define χ and γ respectively as the azimuthal and inclination angle between the observed line of sight and the magnetic field vector, then the absorption profiles are given by

1 η + η η = r b (1 + cos2 γ) + η sin2 γ , (3.14) I 2 2 p 1 η + η η = η − r b sin2 γ cos2χ, (3.15) Q 2 p 2 1 η + η η = η − r b sin2 γ sin2χ, (3.16) U 2 p 2 η − η η = r b cosγ, (3.17) V 2

39 and the dispersion proﬁles by 1 ρ + ρ ρ = ρ − r b sin2 γ cos2χ, (3.18) Q 2 p 2 1 ρ + ρ ρ = ρ − r b sin2 γ sin2χ, (3.19) U 2 p 2 ρ − ρ ρ = r b cosγ, (3.20) V 2 which are composed of the red, blue and parallel absorption and dispersion proﬁles:

ηb,r = ∑ Sb,r · H(a,v − vLOS ± vB), (3.21) MuMl ηp = ∑ Sp · H(a,v − vLOS), (3.22) MuMl ρb,r = ∑ Sb,r · 2F(a,v − vLOS ± vB), (3.23) MuMl ρp = ∑ Sp · 2F(a,v − vLOS), (3.24) MuMl where the summation adds the contributions from relevant transitions according to ∆M = Mu −Md. Sb,r,p are the Zeeman strengths of the proﬁles, calculated according to Table 3.6, H and F are the Voigt and Faraday functions respectively, and the wavelength separations are given by ν − ν v = 0 , (3.25) ∆νD ∆νLOS vLOS = , (3.26) ∆νD

νJuJl MuMl vB = , (3.27) ∆νD with νJuJl MuMl as given in Eq. 3.6. The damping parameter given by Γ + Γ + Γ a = stark vdW rad , (3.28) ∆νD accounting for Stark and van der Waals broadening governed by collisions, while radiative broadening is given by the lifetime of the energy level. ∆νD is the Doppler broadening given by Eq. 3.9. Note that the equations given from Eq. 3.14 onward are only valid for the Zeeman effect. If one wants to include scattering and the Hanle effect, one needs to use the expressions given in Par. 7.6b of Landi Degl’Innocenti and Landolﬁ (2004). A quantitative treatment of the Hanle effect lies outside the scope of this thesis.

40 3.4 Inversion of Observed Stokes Parameters of Helium Lines

Inversion codes use observations of the Stokes parameters to retrieve corresponding atmospheric conditions and structure. We give a short schematic view of how inversion codes work:

Deﬁne an initial guess for the atmosphere in terms of the used parameters e.g. the magnetic ﬁeld vector B, the Doppler width ∆νD, etc.

Synthesize the Stokes parameters. Take into account the relevant physics (for helium): Zeeman effect, anisotropic radiation pumping, Hanle effect.

Compare the synthetic Stokes proﬁles with the observed Stokes pro- ﬁles. Iteratively minimize χ2 via updating the parameters of the guess atmosphere.

Obtain the final synthetic profiles (fits) and the parameters of the final atmosphere, inferred from observations.

Hence the two main parts of an inversion code are (1) solving the radiative transfer equation and (2) minimizing the merit function χ2. The polarized radiative transfer equation (Eq. 3.10) can be solved formally by deﬁning an evolution operator O(τ0,τ). This operator allows to calculate the Stokes parameter at optical depth point τ0 using the Stokes parameter at optical depth point τ. Both numerical approximations and analytical expressions exist for O, but a full discussion of this topic lies outside the scope of this thesis. We will discuss only the solutions of the polarized radiative transfer equation which are used by inversion codes HELIX+ (see Par. 3.4.2) and HAZEL (see Par. 3.4.3). These two inversion codes are currently the most commonly used codes which can invert the He I D3 and He I 10830 Å lines. An algorithm which is used in many inversion codes to minimize χ2 is the Levenberg-Marquardt algorithm, which we discuss in Sec. 3.4.1.

41 3.4.1 Levenberg-Marquardt algorithm

We define our observed Stokes profiles as Oi with index i the total number of wavelength points in all Stokes parameters. These observations have a noise level given by σi. Si(β ) are the synthetic profiles which depend on a set of parameters β describing the atmosphere, e.g. the magnetic field vector B, Doppler width ∆νD, the line of sight velocity vLOS, etc. The merit function is then defined as 2 2 [Oi − Si(β )] χ (β ) = ∑ 2 . (3.29) i σi The algorithm works iteratively, hence the initial value of χ2 is usually large. The initial parameters β are then perturbed by an amount δ which reduces the value of χ2. This is done using the assumption of linearity for the synthesis:

Si(β + δ ) ≈ Si(β ) + Jiδ , (3.30) where the Jacobian Ji equals ∂S (β ) J = i , (3.31) i ∂ββ also known as response functions. Using this expression, χ2(β + δ ) can be written as

2 2 [Oi − Si(β ) + Jiδ ] χ (β + δ ) ≈ ∑ 2 , (3.32) i σi w w2 wO − S(β ) − Jδ w ≈ w w , (3.33) w σ w (O − S(β ))T (O − S(β )) − 2(O − S(β ))T Jδ + (δ J)T Jδ = (3.34), σ T σ with Eqs. 3.33 and 3.34 in vector notation. An optimal choice for the parameter δ should minimize the value of χ2(β +δ ). Therefore, the derivative of χ2(β + δ ) with respect to δ is set to zero, which yields

(JT J)δ = JT [O − S(β )]. (3.35)

This expression calculates δ using the result for β from the previous iteration. However, the algorithm as presented in Eq. 3.35 turns out to be unstable and does not usually lead to convergence. Therefore, Marquardt introduced a damping parameter λ, to be used as

(JT J + λ1)δ = JT [O − S(β )], (3.36)

42 with 1 the identity matrix. When the value of λ is large, the algorithm approximately equals the steepest descent method which guarantees to lead to a decrease in the value of χ2. In each successful iteration step, the value of λ is then decreased (by e.g. a factor 10), where the algorithm aims for the minimum in a more direct way. This means that the Levenberg-Marquardt algorithm uses less iterations to reach the minimum than the steepest descent method. How- ever, the more direct aim of the Levenberg-Marquardt can lead to an increase in the value of χ2 and when that happens, the value of λ is increased again. The danger the Levenberg-Marquardt algorithm is to get stuck in a local minimum and missing the global minimum of the χ2 function in the parameter space. Therefore, most inversion codes combine the Levenberg-Marquardt algorithm with other χ2 minimizing algorithms to first find the approximate location of the global minimum and afterwards refine the convergence with the Levenberg-Marquardt algorithm.

3.4.2 HELIX+

The HELIX+ inversion code developed by Andreas Lagg (Lagg et al., 2004; Lagg, 2007) makes use of a Milne-Eddington atmosphere to ﬁt helium lines. This is an atmosphere for which all quantities are constant with optical depth, except the source function which is a linear function of the optical depth: S(τ) = S0 + S1τ. In a Milne-Eddington atmosphere, the Eddington-Barbier relationship is not an approximation but is valid exactly:

S0 + µS1 = Ic. (3.37)

Furthermore, the Polarized Radiative Transfer Equation (Eq. 3.10) has an analytical solution, also known as the Unno-Rachkowsky solution (Unno, 1956):

S I = S + µ 1 (1 + η )(1 + η )2 + ρ2 + ρ2 + ρ2 , (3.38) 0 ∆ I I Q U V S Q = −µ 1 (1 + η )2η + (1 + η )(η ρ − η ρ ) + ρ R, (3.39) ∆ I Q I Q U U V Q S U = −µ 1 (1 + η )2η + (1 + η )(η ρ − η ρ ) + ρ R, (3.40) ∆ I Q I Q V V Q U S V = −µ 1 (1 + η )2η + (1 + η )(η ρ − η ρ ) + ρ R, (3.41) ∆ I V I U Q Q U V with ∆ and R deﬁned as

2 2 2 2 2 2 2 2 2 ∆ = (1 + ηI) (1 + ηI) − ηQ − ηU − ηV + ρQ + ρU + ρV − R (3.42), R = ηQρQ + ηU ρU + ηV ρV . (3.43)

43 The Zeeman effect and Paschen-Back effect are included in the code. In a later version of the code, the possibility has been included to also take into account the Hanle effect using slab geometry, based on how this is implemented in the HAZEL code (Asensio Ramos et al., 2008). The HELIX+ code fits 8 parameters: magnetic field strength B, magnetic field inclination θ, magnetic field azimuth χ, Doppler width ∆νD, line-of-sight velocity vLOS, damping a, the gradient of the source function S1 and the ratio between line opacity and the continuum opacity at the central wavelength η = κ0 (in the notation used by HELIX+, Eqn. 3.21-3.24 have to still be mul- 0 κc tiplied by η0). The source function intercept S0 is determined from Eq. 3.37 and the observing angle µ is known from observations. In case of multiple components, the filling factor α is also a fitting parameter. The fitting algorithms available in HELIX+ are Levenberg-Marquardt (as discussed in Sec. 3.4.1) and PIKAIA. PIKAIA is a genetic algorithm suitable to locate the global minimum in a multi-dimensional parameter space and known for its robustness. The robustness of PIKAIA has also been demonstrated in Lagg et al. (2004) via a comparison with a fitting algorithm called UOBYQA (Unconstrained Optimization BY Quadratic Approximation). The advantage of the PIKAIA algorithm is that it does not require the calculation of the response functions, which is usually the most computationally expensive part of an inversion code. In the HELIX+ code, one can use as many inversion cycles as one wishes, defining the algorithm for each cycle. It is customary to alternate the PIKAIA algorithm with the Levenberg-Marquardt algorithm to first approach the location of the global minimum and then refine it. We conclude the paragraph by summing up some advantages and disadvantages of HELIX+. Advantages: • Can easily handle multi-component fits with for example 5 components to fit the He I 10830 Å line, as was done by Sasso et al. (2011) • Other lines in the same spectral region can be fitted simultaneously, such as the Si I line at 10827 Å, close to He I 10830 Å • In each inversion cycle, the maximum relative change for each parameter compared to the previous result can be specified • Provides software to examine the inversion results Disadvantages: • The Milne-Eddington approximation is not a realistic physical description of how helium is formed in the upper chromosphere. Inversions of

44 off-limb proﬁles is not possible with Milne-Eddington. Therefore, when the Hanle effect is included, a slab geometry has to be used which is the default in the HAZEL code.

• The multiple components are combined using a ﬁlling factor. It is not possible to concatenate the components in order to model the effect of one component on another one.

3.4.3 HAZEL

The HAZEL inversion code has been developed by Andrés Asensio Ramos (Asensio Ramos et al., 2008) to invert the He I 10830 Å and He I D3 lines. The code uses slab geometry with constant property slabs. The slab is hanging at a certain height h above the solar surface and is illuminated from below by continuum radiation. This geometry allows to properly take into account scattering and is hence especially useful to model off-limb observations but can just as well be applied on-disk. For the following equations, we assume that continuum radiation radiation is unpolarized and that opacities κ0 and κc are taken into account in Eqs. 3.21- 3.24 instead of in Eqs. 3.11 and 3.12. The total absorption matrix K in the polarized radiative transfer equation d I = −KKII + j, (3.44) dz then actually equals Φ given by Eq. 3.13. Dividing this expression by ηI we obtain dI = K ∗I − S, (3.45) dτ ∗ with S = j/ηI and K = K/ηI. An evolution operator O(τ1,τ0) is deﬁned which allows to calculate the Stokes parameter at optical depth point τ1 from the Stokes parameter at optical depth point τ0. The formal solution of Eq. 3.45 then equals

Z τ1 I(τ1) = O(τ1,τ0)I(τ0) − O(τ1,τc)K(τc)S(τc)dτc, (3.46) τ0 where τ0, τ1 and τc apply to the optical depth at the bottom, top and center of the grid cell respectively. The derivation of this equation is given in Chapter 9 of del Toro Iniesta (2007). Assuming a constant K matrix within one grid cell, the expression for the evolution operator equals

R τ1 τ −Kdτ O(τ0,τ1) = e 0 . (3.47)

45 For a constant property slab with optical depth τ, this simplifies to O(τ) = e−Kτ . (3.48) Returning to Eq. 3.45, an analytical solution has been obtained by Asensio Ramos et al. (2008) using formal solutions proposed by Trujillo Bueno (2003) employing the evolution operator of Eq. 3.48: −K ∗τ ∗ −1 −K ∗τ I = e I sun + [K ] (1 − e )S, (3.49) with I sun the continuum radiation illuminating the slab from below, acting as a boundary condition. This radiation is also referred to as pumping radiation. The evolution operator is then calculated using ∞ (−1)n eK = ∑ K n. (3.50) n=0 n! HAZEL takes into account all relevant physics such as the Zeeman effect, atomic level polarization and quantum coherences between levels, the Hanle effect, the Paschen-Back effect and level crossings. The slab geometry is a natural geometry to describe especially scattering polarization and the Hanle effect. For the one slab case, eight parameters are fitted with HAZEL: magnetic field strength B, magnetic field inclination θ, magnetic field azimuth χ, Doppler width ∆νD, line-of-sight velocity vLOS, damping a, opacity τ and the height of the slab h. When two slabs are combined, the additional fitting parameter β is defined equal to the ratio of the source functions of the two slabs. A nice feature of HAZEL is that it offers two possibilities to combine slabs: (1) fit a filling factor α which corresponds to a weighted average of both components, (2) stack the slabs on top of one another and use the emergent radiation from the lower slab as a boundary condition for the upper slab. The latter approach does not require an extra fitting parameter. A detailed discussion of both ap- proaches is given in Paper I. The fitting algorithms available in HAZEL are Levenberg-Marquardt (as discussed in Sec. 3.4.1) and a Direct algorithm which divides the parameter space in hypervolumes to find the approximate location of the global minimum. The location of the minimum is then refined with the Levenberg- Marquardt algorithm. A sequence of several inversion cycles can be defined with the possibility of choosing the inversion algorithm for each cycle. Some advantages and disadvantages of the HAZEL code are listed below. Advantages: • The slab geometry corresponds better to the physical situation in which the helium lines are formed as compared to a Milne-Eddington atmosphere

46 • Slabs can be combined with either a ﬁlling factor, or they can be stacked on top of one another (a detailed description of this is given in Paper I)

• HAZEL comes with a GUI to synthesize or invert manually and examine the ﬁtting result graphically Disadvantages:

• HAZEL uses tabulated values for the continuum radiation which illumi- nates the slab. These values correspond to the center to limb variation of the continuum. However, local variations in intensity (such as sunspots) are not taken into account. This was also pointed out by Schad et al. (2015).

• A maximum of (only) 2 slabs can be deﬁned in HAZEL

• The DIRECT algorithm can introduce in certain cases a discrete spread in the variables obtained via inversion

However, recently a new version of HAZEL has been developed by A. Asen- sio Ramos, and is called HAZEL 2.0. The beta version is available on github and has many new features implemented:

• HAZEL 2.0 uses a python environment to easily run small as well as big synthesis and inversion jobs from the command line.

• The photosphere is modeled using LTE code SIR. The outgoing radiation is used as an input on the chromospheric component.

• An unlimited number of photospheric, chromospheric, telluric and stray- light models can be added in the order that the user chooses.

• The convergence properties of the Levenberg-Marquardt algorithm has been improved and extra features such as randomization of the input variables have been implemented.

3.4.4 Ambiguities Inversion codes have their main application in inferring the full magnetic ﬁeld vector from observations, i.e. magnetic ﬁeld strength |B|, inclination γ and azimuth χ. However, the azimuth χ cannot usually be unambiguously derived from observations. The polarized radiative transfer equation for the Zeeman effect depends only on cos2χ and sin2χ (see Eqs. 3.15, 3.16, 3.18 and 3.19) and is therefore invariant under the following transformation

χ → π + χ. (3.51)

47 Figure 3.4: Hanle diagram for the blue component of He I D3 (this component R Qdν consists of 5 blended transitions, see Table 2.2). On the y-axis isp ˜Q = R Idν and R Udν on the x-axis isp ˜U = R Idν . The solid lines correspond to constant azimuth χ and the dashed lines correspond to constant magnetic field strength |B| given in units ◦ of Gauss. The entire diagram is given for a magnetic field inclination of θB = 90 with respect to the radial direction on the Sun. Figure from Landi Degl’Innocenti and Landolfi (2004).

The Hanle effect deals with its own ambiguities. To understand these better, one can consider Hanle diagrams as calculated from theory, shown in ◦ Fig. 3.4. This diagram is given for a perfectly horizontal ﬁeld (θB = 90 with θB the angle between the solar vertical and the magnetic ﬁeld vector). The degree of linear polarization in Q and U is given on the y and x-axis respectively. The Hanle diagram indicates that the Hanle saturation regime for the red component of the He I D3 line is reached at around 100 Gauss. The loops are caused by level crossing interferences. It is worthwhile to note that the Hanle diagram for the blue component of He I D3 is quantitatively different from the one in Fig. 3.4 and does not exhibit loops. Hanle diagrams are invariant under the transformation

θB → π − θB, χ → −χ, (3.52) for scattering angles of 90◦ (off-limb) or 0◦ (forward scattering). Another am-

48 Figure 3.5: Hanle diagram for red component of the He I 10830 Å line, which consists of 2 blended components listed in Table 2.2. The diagram is given for the saturation regime and for a fixed magnetic field strength of 25 Gauss. Solid lines represent fixed inclination angles θB. Dashed lines represent fixed azimuth angles χ. Figure from Merenda et al. (2006). biguity arises from the transformation

χ → π + χ, (3.53) similar to the 180◦ Zeeman ambiguity. However, this transformation yields an opposite sign for the Stokes V parameter, implying that the ambiguity can be solved when circular polarization signals are observed. Additional ambiguities can arise in the Hanle saturation regime, which are called the Van Vleck ambiguities. Linear polarization in this regime is no longer sensitive to magnetic ﬁeld strength but only to its direction. The ambiguities can be determined from theory, as was done by e.g. Merenda et al. (2006). They have constructed an overview of the Van Vleck ambiguities for the red component of the He I 10830 Å line for a magnetic ﬁeld strength of |B| = 25 Gauss, shown in Fig. 3.5. It is clear that the ambiguities arise in the ◦ region located in the middle between the Van Vleck angles θB = 54.74 and ◦ θB = 125.2 , where solid lines cross over one another. For a transformation of π χ → χ ± , (3.54) 2 49 Figure 3.6: Calculation of all global minima of the merit function χ2 with HAZEL for a certain set of prominence observations in He I 10830 Å. Black regions indicate the global minima, black dashed lines correspond to the Van Vleck angles, contours of constant Q/I are given by red dashed lines and constant U/I by green solid lines. Figure from Orozco Suárez et al. (2014).

certain values for θB and γ exist which will leave the linear polarization signals in Stokes Q and U unchanged. More detailed explanations of the Van Vleck ambiguity are given in e.g. Casini et al. (2005); López Ariste and Casini (2005); Martínez González et al. (2015). Asensio Ramos et al. (2008) have proposed to find the possible ambiguities numerically instead of theoretically, using the DIRECT algorithm of the HAZEL code. With sufficient iterations, the algorithm succeeds in finding all global minima in the parameter space of the merit function χ2. This method was applied by e.g. Orozco Suárez et al. (2014) with the results presented in Fig. 3.6. There are 8 global minima, corresponding to 8 ambiguous solutions. When Stokes V signal is observed, only 4 ambiguities remain, connected by the 180◦ and the 90◦ ambiguity, as indicated in Fig. 3.6. Several methods have been proposed in the literature to get rid of the 180◦ ambiguity of the Hanle effect, given by the transformation in Eq. 3.52. The first

50 one is presented in the theoretical work presented by Landi Degl’Innocenti et al. (1987), which shows that simultaneous observations of Balmer lines (which are optically thick) and the He I D3 line (which is optically thin) can remove the 180◦ ambiguity. A second method is to use solar rotation and observe the target at several scattering angles (which will all be close to but not exactly equal to 90◦). The 180◦ ambiguity is dependent on the scattering geometry and only occurs for scattering degrees of 0◦ (forward scattering) and 90◦ (off-limb scattering). A third method is explained in Landi Degl’Innocenti and Landolfi (2004), and is based on the fact that the He I D3 line consists of two (resolved) components with each their different Hanle diagram. Therefore, observations of the He I D3 line provide as much information as the observation of two separate spectral lines with different optical depths, with the potential of re- moving ambiguity. The same is not true for the He I 10830 Å line, because the blue component of He I 10830 Å has an upper level with quantum number J = 0, therefore this level is no subject to atomic level alignment or atomic level polarization (see Par. 3.2.2). In addition to ambiguities arising from intrinsic properties of the Zeeman and Hanle effect, it has been shown that degeneracies in inversions results can arise from noise in the observed Stokes profiles (Asensio Ramos et al., 2008) or inversion noise (Casini et al., 2005). For example, Casini et al. (2009a) have performed inversions of synthetic data of simultaneous He I 10830 Å and He I D3. They found significantly lower errors in the case of multi-line inversions, arguing in favor of building instruments that perform simultaneous observations of multiple diagnostics.

51 52 4. Observations of He I D3

4.1 Telescopes and Instruments

4.1.1 Fabry-Pérot Interferometer versus Slit-Spectrograph Solar observations are multi-dimensional having at maximum all of the following dimensions or a subset of those: two spatial directions x and y, spectral information λ, time evolution t and the Stokes vector I. Depending on the science goal, one wants to achieve for example optimal spatial resolution and therefore sacriﬁce spectral sampling or obtain high signal-to-noise polarimetric measurements and sacriﬁce temporal cadence. The science goal will hence determine the type of instrument to use and set the observational parameters such as exposure times, etc. The SST is currently host to two Fabry-Pérot interferometers and a slit- spectrograph, capable of providing observations in all dimensions (x,y,λ,t,I). The CRisp Imaging Spectro-Polarimeter (CRISP, Scharmer 2006; Scharmer et al. 2008) is operational in the visible to the near-infrared (∼ 500 − 900 nm) while CHROMIS operates in the blue part of the visible spectrum to observe mainly the Ca II H&K lines and Hβ (e.g. Bjørgen et al. 2018). The TRI-Port Polarimetric Echelle-Littrow spectrograph (TRIPPEL, Kiselman et al. 2011) operates mainly in the visible spectrum, although infrared observations have been conducted in 2015 with a rental infrared CCD. We discuss CRISP and TRIPPEL in more detail since we have used data of those two instruments in Paper I, Paper II and Paper III.

CRISP The CRISP instrument is a dual Fabry-Pérot interferometer (FPI). It consists of a set of high-resolution etalons (HRE) and a set of low-resolution etalons (LRE), as shown Fig. 4.1. A cavity is present between the etalons for multiple reflections to take place. The transmitted light hence interferes with the multiple reflected rays, each having a different path length introducing a phase difference. Therefore, interference patterns arise from each set of etalons as shown in Fig. 4.2. A prefilter is needed to select the wavelength range of the spectral line. When combining the transmission profiles of the HRE, LRE and the prefilter, an extremely narrow wavelength region can be selected (Fig. 4.2).

53 Figure 4.1: Schematic setup of the CRISP instrument. The red line is the light beam incident from the right upper corner from the figure. WB is the wide band camera, HRE are the high-resolution etalons, LRE are the low-resolution etalons, FPI stands for Fabry-Pérot interferometer, LCs are the liquid cristals with polarizing properties, PBS is the polarizing beam splitter, NBT is the narrow-band transmitted beam camera and NBR is the narrow-band reflected camera. Adapted figure from de la Cruz Rodríguez et al. (2015).

Figure 4.2: The different transmission curves as a function of wavelength for the high-resolution etalons (HRE), the low-resolution etalons (LRE) and the preﬁlter. Figure from de la Cruz Rodríguez et al. (2015).

54 CRISP can hence take high spatial resolution images with an extremely narrow spectral range. To obtain spectral information, the etalons are tilted to obtain images at several wavelength points, sampling a spectral line. The inclusion of more wavelength points will increase the quality of the spectra, at the cost of losing temporal cadence. One has to keep in mind that the spectra are not internally consistent since the wavelength points are not observed co- temporary. If too many wavelength points are included, the sun will have changed within the time of sampling one spectral line. The image quality of CRISP is improved during the observations via the adaptive optics system (AO) and after observations during the reduction by applying corrections for seeing using the Multi-Object Milti-Frame Blind De- convolution (MOMFBD,van Noort et al. 2005) code. Polarimetric measurements are possible with CRISP using the polarizing retarders1. If we return now to the dimensions of the observations, CRISP provides information in all dimensions (x,y,λ,t,I), with very high spatial resolution x and y, good temporal cadence t, with the possibility of measuring of the full Stokes vector I, usually at the cost of limited spectral sampling λ.

TRIPPEL The TRI-Port Polarimetric Echelle-Littrow spectrograph (TRIP- PEL, Kiselman et al. 2011) is a slit-spectrograph. A schematic view of the instrument is given in Fig. 4.3. An echelle spectrograph makes use of a blazed grating at a large blaze angle. The reflected light contains all the rays with different path length creating an interference pattern. However, the selected wavelengths are all superposed on one another, split into different orders. Therefore, a prefilter is needed to select the right order. The slit has a reflecting surface on the incident beam side. The light that is not introduced in the spectrograph can be reflected and recorded to obtain slit-jaw images. These are useful to provide context to the spectra. Currently, there are no standard polarizing units installed at TRIPPEL to conduct polarimetric measurements. However it is possible to build a set-up to do so. With TRIPPEL, three different spectral regions can be observed simultaneously, although one needs to be aware of atmospheric refraction. If spectral regions are observed with large wavelength separation (e.g. optical and infrared), the difference between the projected slit positions can be of the order of several arcsecond, depending on airmass and Earth-atmospheric properties. In 2015, a scanning method has been implemented for TRIPPEL by Guus Sliepen. The correlation tracker is used to move the beam over the slit. Raster scans of variable size can be taken with a step size of minimum 0.1 arcsec.

1Fig. 4.1 shows polarizing liquid crystals instead of retarders. The liquid crystals have been replaced with retarders in 2016.

55 Figure 4.3: Schematic view of the TRIPPEL instrument as seen from above. The beam enters from the left and ﬁrst passes through the slit, then hits the ﬁeld lens, the Littrow doublet lens, the grating and follows back the same path to get directed towards the different ports A, B and C with pick-off mirrors. Figure form Kiselman et al. (2011)

The observations taken with TRIPPEL have the dimensions (x,y,λ,t,(I)). The information in the x and λ (corresponding to a single spectrum) has high spatial and spectral resolution. The y dimension is obtained via scanning and will suffer from seeing effects. The quality of the observations is improved by the AO during the observations. A disadvantage of slit-spectrographs as compared to imaging instruments is that restoration techniques to correct for seeing are not implemented in standard pipelines. (A restoration method has been proposed by Keller and Johannesson (1995) and this is a topic of ongoing research.) The temporal cadence is low when taking large rasters. In case one sacriﬁces spatial information, good temporal cadence can be achieved (i.e. sit-and-stare spectra or small raster scans).

4.1.2 Overview of Facilities We give an overview of telescopes and instruments which are currently providing most of the high-resolution observations of He I 10830 Å or He I D3 used in scientiﬁc publications. The overview is limited to instruments and telescopes which are operational at the moment (with the exception of TIP-II at the VTT and HESP at the SST). We focus on ground-based facilities providing high-resolution data. There are several telescopes observing low resolution or full-disk He I 10830 Å images and spectra which are not listed here.

DST The Richard B. Dunn Solar Telecope (DST) is a 76 cm telescope located at Sacramento Peak in New Mexico. Several instruments are installed at the DST which are able to observe the He I 10830 Å and He I D3 lines. The Facility IR Spectro-polarimeter (FIRS, Jaeggli et al. 2010) has an optical and

56 an infrared beam so it can conduct measurements of Stokes parameters of an optical line and an infrared line simultaneously. It is equipped with 4 slits im- proving the cadence of the raster scans with a factor 4. Another beam splitter allows to operate FIRS together with the Interferometric Bi-dimensional Spec- trometer (IBIS, Cavallini 2006). IBIS is a Fabry-Pérot instrument which needs preﬁlters to select the right wavelength range. At the DST, a He I D3 preﬁlter is present hence it is possible to take He I D3 images with spectral tuning. The DST operates a higher-order adaptive optics system that is used in combination with all instruments.

GREGOR GREGOR (Schmidt et al., 2012) is the German 1.5 m telescope located on Tenerife. It is equipped with the Gregor Infrared Spectrograph (GRIS, Collados et al. 2012). GRIS is used in combination with TIP-II (Col- lados et al., 2007) to conduct spectro-polarimetric measurements of Stokes parameters in the infrared (1.0 − 1.8µm) hence it can observe the He I 10830 Å line. The GREGOR telescope is equipped with a high order adaptive optics (HOAO). GREGOR also operates the GREGOR Fabry-Pérot Interferometer (GFPI, Puschmann et al. 2012) in the optical. In the future, it will be possible to simultaneously operate GRIS in the infrared and GFPI in the optical.

NST Big Bear Solar Observatory is operating the 1.6m New Solar Telescope (NST, Goode and Cao 2012). The telescope provides high-resolution He I 10830 Å images using a narrow-band ﬁlter. No spectral or polarimetric information is available for the line. An higher order adaptive optic system is installed to improve the image quality.

NVST The New Vacuum Solar Telescope (NVST, Liu et al. 2014) is a 1m Chinese telescope, operational since 2012. The telescope is equipped with an imaging system making use of Lyot ﬁlters. There are also two spectrometers available: the multi-bands spectrometer (MBS) designed for the optical and a high dispersion spectrometer (HDS) designed for the infrared. Liu et al. (2014) mention explicitly that HDS can conduct spectral measurements of He I 10830 Å but so far, no data has been used in scientiﬁc publications. An AO system is available to improve the image quality.

57 Table 4.1: Current telescopes and instruments providing high-resolution observations of He I 10830 Å or He I D3 . It should be noted that all of this information applies to observing He I 10830 Å and He I D3 . For example the NST has an instrument capable of polarimetry, but no polarimetric He I 10830 Å data has been recorded.

2 Telescope/ Type Spectral Telescope FOV Stokes AO 10830 Å He I D3 Instrument resolution1 [cm] [arcsec]

DST/FIRS Spectrograph 200,000 (IR) 76 25 or 75 yes yes yes no DST/IBIS Fabry-Pérot 200,000 (VIS) 76 80 yes yes no yes GREGOR/GRIS Spectrograph 200,000 (IR) 150 60 yes yes yes no NST Imaging / 160 70×70 no yes yes no SST/CRISP Fabry-Pérot 100,000 (VIS) 100 60×60 yes yes no yes SST/TRIPPEL Spectrograph 200,000 (VIS) 100 40 no yes yes yes THEMIS/MTR Spectrograph 300,000 (VIS) 90 120 yes no no yes 1 Spectral resolution is dependent on wavelength hence the values indicated are approximately valid for either the visible (VIS) or the infrared (IR) 2 The FOV is either given by the projected slit length for spectrographs (one dimension) or by the image size for imaging instruments (two dimensions). The FOV of DST/IBIS is circular.

SST The Swedish 1m Solar Telescope (SST, Scharmer et al. 2003) is located on the island of La Palma. It operates TRIPPEL, see 4.1.1 in the optical and near-infrared. TRIPPEL can observe three spectral regions simultaneously, meaning He I 10830 Å and He I D3 can be observed co-temporarily (as what has been done in Paper I). Spectro-polarimetric measurements are possible with CRISP (see 4.1.1) a Fabry-Pérot instrument. A He I D3 preﬁlter is available for CRISP hence high resolution images with spectral tuning can be obtained. Higher order adaptive optics have been operational since 2003, improv- ing the image quality for all instruments. For the fall of 2019, the installation of a new instrument called Helium spectro-polarimeter (HESP) is planned. This instrument is a microlense array designed by Michiel van Noort and capable of observing polarized spectra of all pixels in the FOV (the FOV will be of the order of 10 × 10 arcsec). These polarized spectra are tilted and partially superposed before they are recorded by a CCD, so that advanced algorithms are necessary to extract the single spectra from the CCD.

THEMIS The Télescope Héliographique pour l’Etude du Magnétisme et des Instabilités Solaires (THEMIS, López Ariste et al. 2000) is a 90 cm telescope located on Tenerife. It can be operated in the multiline spectroscopy mode (MTR) allowing it to measure Stokes proﬁles in spectral lines such as the He I D3 line. An adaptive optics system is currently being installed at the telescope.

58 VTT The German 70 cm Vacuum Tower Telescope (VTT) on Tenerife has until recently operated the TIP-II (Collados et al., 2007) but it is now placed at Gregor under the name GRIS. TIP-II was used in combination with an echelle spectrograph to measure Stokes parameters in near infrared spectral lines (1.0- 1.8 µm) as for example the He I 10830 Å line.

4.2 Spectral Proﬁles of He I D3 and He I 10830 Å

In Fig. 4.4 we show raster scans and spectral profiles of He I D3 and He I 10830 Å in a flux emergence region. The lines have multiple components, splitted according to quantum number J (see Table 2.2), but only two components are resolved in solar conditions due to Doppler broadening. The He I D3 is typically very weak on the solar disk and therefore, a continuum correction is usually necessary to be able to see the line in images or raster scans. The He I D3 line suffers from multiple telluric blends and some solar blends as shown in panel c of Fig. 4.4. Paper I discusses a method based on Principle Component Analysis (PCA) to correct for telluric blends. He I 10830 Å has a telluric blend in its red wing and a Si I blend in its blue wing. The Si I 10827Å line is magnetically sensitive and is formed in the upper photosphere to lower chromosphere. When observing this spectral window, the combination of Si I 10827 Å and He I 10830 Å is able to provide information on the magnetic field in both the photosphere and the chromosphere. The lines have often been used together, see e.g. Joshi et al. (2016) for a recent example.

4.3 Overview of He I D3 Observations and Results 4.3.1 Prominences and Filaments Prominences are bright arcade-like structures which appear above the solar limb as observed in certain chromospheric spectral lines such as e.g. Hα, He I 10830 Å, He I D3 and He II 304 Å. Fig. 4.5 shows an example of a prominence as observed in the Hα and He I D3 line with SST/CRISP. Prominences consist of relatively cool material of temperatures between T ∼ 7·103 −2·104 K embedded in a hot corona of T ∼ 1 · 106 K. Prominences correspond to on- disk features called ﬁlaments which appear dark (in the relevant spectral lines) against the continuum background. Observational aspects of solar prominences are reviewed by Parenti (2014). The interpretation of emergent intensities from prominences is compli- cated by the fact that some lines get optically thick in the prominence plasma (e.g. continuum and resonance lines of H I, He I and He II). Moreover, the

59 Figure 4.4: The He I D3 and He I 10830 Å lines in a flux emergence region, observed with SST/TRIPPEL (same dataset as used in Paper I). a. Continuum- corrected line core raster scan in He I D3. b. Spectrum of He I 10830 Å corresponding to the vertical line in panel a. c. The average spectrum as shown in panel b. Spectral lines are identified. Telluric lines are indicated with “Atm”. d. Line core raster scan in He I 10830 Å. e. Spectrum of He I 10830 Å corresponding to the vertical line in panel d. f. The average spectrum as shown in panel e. Spectral lines are identified. Telluric lines are indicated wit “Atm”.

Figure 4.5: Prominence observed with SST/CRISP in the Hα line core (left) and the He I D3 line core (right). The prominence in He I D3 is very weak so that it is invisible in the He I D3 line core without proper scaling (middle).

60 prominence emergent spectral lines are subject to non-LTE effects because of the strong influence of incident radiation in the prominence (Hirayama, 1963). Labrosse and Gouttebroze (2004) have showed that a Prominence-to- Corona Transition Region (PCTR) has to be taken into account to realistically model He I triplet line formation in prominences. Their results indicate that the the photo-ionization recombination mechanism is dominating He I triplet level populations in prominence models including a PCTR. Their models also indicate that He I D3 and He I 10830 Å are almost always optically thin in prominences. The magnetic field configuration for prominences is a topic of ongoing debate. It is known that prominences occur above polarity inversion lines, i.e. the line of zero polarity between positive and negative polarities in the photospheric magnetic field. The main question considers stability of prominences: which magnetic field configurations can support and isolate cool plasma in its hot coronal environment. Observations could possibly play a key role to (in)validate magnetic field configurations proposed by theory or MHD simulations. Therefore, measurements of the full magnetic field vector B are required, not only its strength but also inclination and azimuth. Using observations of the full Stokes vector of He I 10830 Å or He I D3 and fitting the profiles with an inversion code can provide such measurements of B. In fact, Par. 13.4 of Landi Degl’Innocenti and Landolfi (2004) provides a list of reasons which argue why the He I D3 line is the best diagnostic to conduct magnetic field measurements in prominences. The main reason to prefer He I D3 over Hα is because He I D3 is optically thin. Hanle theory for optically thick lines requires more com- plicated descriptions, in which polarization is dependent on thermodynamic properties of the plasma (such as velocity fields) and on the geometry of the observed structure. Full 3D non-LTE radiative transfer models are needed to accurately model the Hα line. Helium lines in prominences are formed by scattering of continuum radiation. Since the magnetic field strengths are weak (order of ∼ 10 Gauss) and scattering is an important process, the magnetic field has to be measured via the Hanle effect. The first wave of attempts at measuring the magnetic field using the Hanle effect was in the 70’s and 80’s. Theoretical descriptions of the Hanle effect were developed and improved by Sahal-Brechot et al. (1977); Sahal-Brechot and Bommier (1977); Bommier (1980); Landi Degl’Innocenti (1982). Applying Hanle theory allowed to measure the magnetic field vector in prominences using the He I D3 line (Bommier and Sahal-Brechot, 1978; Bom- mier et al., 1981; House and Smartt, 1982; Landi Degl’Innocenti, 1982; Athay et al., 1983; Querfeld et al., 1985). Some results of these early inversions are given here: • Magnetic field strengths between 2−20 Gauss were found for large sam-

61 ples of prominences.

• For a majority of the prominences the magnetic ﬁeld vector points in opposite sense as the photospheric magnetic ﬁeld polarity (these are called inverse prominences).

• The inclination angle was usually assumed to be horizontal or close to horizontal, i.e. the horizontal field hypothesis (Leroy et al., 1983). In several cases, the magnetic field vector was found to be inclined with respect to the horizontal plane of up till 30◦. These inclinations were interpreted as dips in the magnetic field lines by e.g. Leroy et al. (1983); Bommier et al. (1994).

• Azimuthal measurements yield results in all possible ranges. Angles of around ∼ 25◦ between the long axis of the prominence and the magnetic field vector appear most often (Leroy et al., 1983). Aforementioned results were obtained by authors who (to variable extent) took into account the 180◦ ambiguity which arises from the transformation given in Eq. 3.52. They either assumed the horizontal field solution or applied techniques to solve the ambiguities as summarized in Par. 3.4.4. However, the Van Vleck ambiguity or ambiguities introduced by noise were not considered and the horizontal field hypothesis was criticized by e.g. Merenda et al. (2006), especially for polar crown prominences which exhibit clear vertical structure. Moreover, the magnetic topology in (footpoints of) prominences is a still-standing problem. From the late 90’s and 2000 onward, observations with increased spatial resolution from slit-spectrographs were provided by DST and THEMIS (Pale- tou et al., 2001; Casini et al., 2003) for He I D3 and by the VTT for He I 10830 Å (Trujillo Bueno et al., 2002b). Maps in 2D of the magnetic field vector in prominences were calculated by Casini et al. (2003, 2005). There were also advances in theoretical work (Trujillo Bueno et al., 2002b,a; Trujillo Bueno and Manso Sainz, 2002) and inversion techniques (López Ariste and Casini, 2002, 2003). The latter constructed a database of 200 000 synthetic He I D3 profiles under the assumption of the optically thin limit and collisionless plasma, and used the technique of principle component analysis (PCA) to invert observed He I D3 profiles. The aforementioned authors mostly confirmed that magnetic fields are pre- dominantly horizontal and that the average field strength is around ∼ 20 Gauss. However, structures in prominences were found with magnetic fields that are significantly larger (> 50 Gauss) than the average magnetic field (Paletou et al., 2001; Trujillo Bueno et al., 2002b; López Ariste and Casini, 2002, 2003; Wiehr and Bianda, 2003).

62 Merenda et al. (2006) have used the He I 10830 Å line to measure the magnetic field vector in a polar crown prominence (i.e. prominences consisting of obvious vertical strands of plasma). Since the Hanle saturation regime for the He I 10830 Å line kicks in at around 10 Gauss (as opposed to 100 Gauss for the He I D3 line), the Van Vleck ambiguities have to be considered carefully. However, the authors showed that their observations were situated out of the domain where the Van Vleck ambiguities occur. Merenda et al. (2006) found a magnetic field vector which was substantially more vertical than what was ◦ previously measured (θB ∼ 30 ). Moreover, for pixels located along the direction of the slit at one slit position, a rotation of B was found around a fixed direction in space. These results are particularly interesting in the light of the current controversy considering magnetic field direction in prominence footpoints. The discussion essentially boils down to vertical vs. horizontal magnetic field (with respect to the local vertical). Observations of the He I D3 with THEMIS and inversions with the PCA code of López Ariste and Casini (2002) continue to yield pre-dominantly horizontal magnetic fields (Schmieder et al., 2013, 2014b; Levens et al., 2016a). However, the inverted maps appear usually noisy and it is not entirely clear how noise and ambiguities are affecting the inversion results. Then, there are observational hints pointing toward a vertical magnetic field in prominences, e.g. vertical plasma structuring and inclined velocity fields (θ ∼ 45◦ as found by Schmieder et al. 2010). Moreover, quasi-vertical bubbles and tornado-like structures with apparent rotations have been observed in prominences as well (Dudík et al., 2012; Orozco Suárez et al., 2012; Wede- meyer et al., 2013; Su et al., 2014; Schmieder et al., 2017). Orozco Suárez et al. (2014) have used the He I 10830 Å line (raster scans from the VTT/TIP) to construct 2D magnetic field maps, using HAZEL. Since the observations are situated within the Hanle saturation regime, ambiguities are considered carefully (see Fig. 3.6). The authors end up with two ambiguous solutions for the magnetic field vector: a semi-horizontal and a semi-vertical solution. Martínez González et al. (2015) used He I 10830 Å observations of a prominence with helical structure in its legs. All pixels in the observations are situated within the Van Vleck ambiguity region. The authors used HAZEL to invert the magnetic field vector and consider a quasi-vertical and a quasi- horizontal solution. They find that the magnetic field exhibits helicity in both cases and the authors show that the vertical solution is the stable one. More- over, the helicity of the field lines is consistent with the vertical solution. Levens et al. (2016b, 2017) have studied the magnetic field of several atypical prominences using He I D3 observations and found a pre-dominantly horizontal field for a tornado-like prominence and a more vertical field for an

63 eruptive prominence. A recent study by Hanaoka and Sakurai (2017) did a statistical study using full-disk He I 10830 Å data of active region filaments, to study chirality depending on hemisphere: in the northern hemisphere deviates the magnetic field usually clockwise from the average orientation of the filament formed at the polarity inversion line while opposite on the southern hemisphere. In summary, the magnetic field toplogy of prominences is an ongoing topic of debate. From the observational point of view, the largest uncertainty is found in inversion results, due to noise and ambiguities. A paper by Milic´ et al. (2017) has also pointed out the importance of 2D radiative transfer effects in scattering polarization for “typical lines" such as He I 10830 Å. Their inferred magnetic field using a 1D slab model was almost always more horizontal and weaker than the input values generated from a 2D prominence model. The situation is different for measurements of magnetic fields in on-disk filaments as observed in He I 10830 Å by Kuckein et al. (2009, 2010); Sasso et al. (2011); Kuckein et al. (2012a,b); Xu et al. (2012); Sasso et al. (2014). These authors studied (flaring) active region filaments and they measured higher magnetic field strengths than in prominences. For example Kuckein et al. (2009) has measured magnetic fields in a filament up till ∼800 Gauss while Xu et al. (2012) has measured strengths of ∼150 Gauss in the filament above granulation and ∼750 Gauss above an active region. The observed spectropolarimetric profiles are dominated by the Zeeman effect so that the Hanle effect and ambiguities were not taken into account. The topologies that were found are consistent with twisted flux rope models (Kuckein et al., 2012a; Xu et al., 2012; Sasso et al., 2014). Casini et al. (2009b) wrote a reply to Kuckein et al. (2009) that scattering polarization effects should be present for active region filaments, even in highly magnetized regions. The authors suggest that a depolarization could occur due to the presence of unresolved highly entangled magnetic field. In the same line of thought, Díaz Baso et al. (2016) argued that the active region filaments might be quite transparent so that inverting the filament with one Zeeman component only yields wrong results: the observed Stokes V signal might be dominated by the signal from the photosphere below the filament, harboring strong magnetic fields. Díaz Baso et al. (2016) propose a model in which a lower atmosphere component has a magnetic field of around 600 Gauss while a filament in a higher layer has a magnetic field of around 10 Gauss with an azimuth difference of 90 degrees between the two components. The authors show that the resulting polarized profiles have Zeeman-only signatures since the scattering polarization signatures of the two components have opposite signs. In such a model, active region filaments could have similar magnetic field strengths as (quiescent) prominences.

64 Figure 4.6: Spicules as observed with SST/TRIPPEL in He I D3 (left spectrum), slit-jaw images in Ca II H continuum (middle) and core (right). The white dashed line indicates the true position of the slit in Ca II H images. The true slit position differs from the projected one (as seen by a dark line on the image) because of atmospheric refraction.

4.3.2 Spicules

Spicules are thin tube-like chromospheric plasma structures which occur almost everywhere on the Sun. Spicules are believed to be capable of transport- ing mass and energy between photosphere and corona and hence possibly play an important role in coronal mass supply and heating (e.g. De Pontieu et al. 2011). Spicules are observed with highest contrast at the limb in chromospheric lines such as Hα, Ca II H&K, Ca II 8542 Å, Mg II h&k, He I 10830 Å and He I D3. An example of spicule observations with SST/TRIPPEL in He I D3 and IRIS is given in Fig. 4.6. Similar as to prominences, helium triplet lines are used to measure the magnetic field for off-limb observations, in this case spicules. One can ask why other spectral lines are not used in this case, since spicules are visible in many diagnostics (as opposed to prominences). However, weak magnetic fields and the off-limb geometry imply again that measurements have to be conducted via the Hanle effect. Hanle theory is well understood for the optically thin helium triplet lines and implemented in ready-to-use inversion codes such as HAZEL. The Hanle effect in for example the Ca II 8542 Å line, has been investigated only recently (Manso Sainz and Trujillo Bueno, 2010; Carlin and Asensio Ramos, 2015; Štepánˇ and Trujillo Bueno, 2016) and is not implemented in e.g. the inversion code NICOLE, which is a non-LTE code capable of inverting the Ca II 8542 Å line based on Zeeman polarization (Socas-Navarro et al., 2015). The reason that Ca II 8542 Å is not used for magnetic field measurements

65 in spicules/prominences is the same as for Hα (see Sec. 4.3.1): Hanle polarization for optically thick lines depends on thermodynamic and geometrical properties of the plasma. However, people have attempted to investigate the magnetic ﬁeld using Hα lines in prominences (e.g. Bommier et al. 1994; López Ariste and Casini 2005) and (at least theoretically) in spicules (de Ker- tanguy 1998). The same might be true for the Ca II 8542 Å line in the future. The Hanle saturation regime for this line kicks in at magnetic ﬁelds of around 10 Gauss, similar as for the He I 10830 Å line.

Observations of spicules in the He I D3 line have been conducted by López Ariste and Casini (2005); Ramelli et al. (2005); Socas-Navarro and Elmore (2005); López Ariste and Casini (2006); Ramelli et al. (2006, 2011) and for the He I 10830 Å line by Trujillo Bueno et al. (2005, 2007); Centeno et al. (2010a,b); Martínez González et al. (2012); Orozco Suárez et al. (2015); Beck et al. (2016). Until recently, most of these observations of spicules were carried out with spectrographs in the “sit-and-stare" mode, integrating for long times (10-60 min) to detect polarization signal in spicules. An average field strength of ∼ 10 Gauss was found in spicules but with local outliers between 30 − 60 Gauss (López Ariste and Casini, 2005; Trujillo Bueno et al., 2005; Ramelli et al., 2006; Centeno et al., 2010b). The inclination of the magnetic field is found to be aligned with the spicule direction, even though Van Vleck ambiguities cannot rule out the (unlikely) case of a field perpendicular to the spicules. Stokes V signal caused by Zeeman polarization was more or less at the noise level for López Ariste and Casini (2005); Trujillo Bueno et al. (2005) while Ramelli et al. (2006) have observed very weak symmetrical Stokes V signal, possibly generated by atomic level polarization. Centeno et al. (2010b) observed clear Stokes V signals in a local patch along the slit with larger magnetic field caused by the Zeeman effect. They propose a scenario in which two magnetic components along the LOS can either add up or cancel out. Raster scan observations in He I 10830 Å were obtained by Martínez González et al. (2012); Orozco Suárez et al. (2015). Martínez González et al. (2012) have focussed on observations of asymmetrical Stokes V profiles in spicules with net circular polarization. The cause of net circular polarization is likely due to the presence of velocity gradients correlating with magnetic fields along the line of sight, with maybe a small contribution from atomic level polarization. Orozco Suárez et al. (2015) investigated the variation of the magnetic field with height above the solar surface and found that the magnetic field is stronger and more vertical at the bottom of the spicules (with strengths up till 80 Gauss). The field got weaker and more horizontal with height (|B| ∼ 30 ◦ Gauss and θB ∼ 50 at a height of 3 Mm above the solar surface). The magnetic field strength was measured via circular polarization signal caused by

66 Figure 4.7: Flux emergence region as observed with SST/CRISP. The same target is shown in Fig. 1.1 in the Hα line. The He I D3 line core is shown (left) and the He I D3 line core continuum corrected (right). Due to seeing effects, some granulation is visible in the continuum corrected image. the Zeeman effect while the ﬁeld inclination and azimuth were determined via linear polarization signal dominated by scattering polarization and the Hanle effect.

4.3.3 On-disk Targets: Active Regions and Flux Emergence

Published on-disk images of He I D3 are rare and usually targeting flares (see Sec. 6.1). It is possible to observe He I D3 with DST/IBIS to obtain images with spectral tuning, but no such data has been published, even though it has been investigated by Reardon (2012). The reason for the scarcity of He I D3 images is that He I D3 on-disk is usually very weak or entirely absent. However, observations with SST/CRISP (e.g. Fig. 4.7) and with SST/TRIPPEL (e.g. Fig. 4.4) have shown that on-disk He I D3 signal is present in active regions and flux emergence regions. To see the signal against the continuum background, a continuum correction is usually necessary. On-disk He I 10830 Å observations are much more common and can give an indication of what we can expect for the He I D3 line (a review on results up till 2006 is given in Solanki et al. 2006). The He I 10830 Å line has been used for studies of waves and oscillations in sunspot atmospheres (Lites, 1986; Chuprakov et al., 2004; Centeno et al., 2006; Bloomfield et al., 2007; Felipe et al., 2010; Kobanov et al., 2015; Su et al., 2016) or magnetic field measurements in sunspots and superpenumbrae (Schad et al., 2012, 2013; Joshi, 2014;

67 Schad et al., 2015; Joshi et al., 2016) and in umbral flashes (Houston et al., 2018) where inversions in He I 10830 Å have been combined for the first time with inversions in the Ca II 8542 Å line. One of the questions addressed is whether the chromospheric magnetic field possesses structure on the same scales as the features that are observed in the chromospheric lines used to measure the field (such as Ca II 8542 Å and He I 10830 Å). Due to the changing physical regime (high-β to low-β plasma), the small-scale structure of the magnetic field is lost when moving from the photosphere to the chromosphere. The gas pressure drops and the magnetic field expands and fills the space available in a smooth way (e.g. Schad et al. (2015) for He I 10830 Å). However, Joshi et al. (2016) has found that it is true that the magnetic field strength above a sunspot is smooth, but he observed fine structure of the inclination of the magnetic field above a sunspot penumbra in the upper chromosphere. In addition to sunspots, multiple studies of flux emergence regions via He I 10830 Å are available. Magnetic field strength, velocity fields, and height of chromospheric loops in flux emergence regions have been investigated via inversions of He I 10830 Å (Solanki et al., 2003; Lagg et al., 2004; Sasso et al., 2006; Lagg et al., 2007; Xu et al., 2009, 2010; Asensio Ramos and Trujillo Bueno, 2010; Merenda et al., 2011; Balthasar et al., 2016; González Manrique et al., 2017, 2018) or using He I 10830 Å filtergrams (Vargas Domínguez et al., 2014). The physics of flux emergence regions is described in Par. 5.2. Solanki et al. (2003) used the He I 10830 Å line to infer the magnetic field strength in rising chromospheric loops in a flux emergence region. They found a location in which strong magnetic fields with opposite polarity were sepa- rated by a narrow region with low values for the magnetic field. The authors have interpreted this as evidence for the existence of current sheets in the upper chromosphere. Such observations support coronal heating theories based on magnetic dissipation at current sheets (i.e. reconnection). The same data set has been used by Lagg et al. (2007) to study supersonic downflows located in the chromosphere over a growing pore in the photosphere. The authors also observed He I 10830 Å emission associated with the downflows. They propose that the supersonic velocities are related to draining of the rising chromospheric loop. The downflows are still considerably slower than free-fall speed but are likely to be slowed down by underlying plasma in the loop. The emission is possibly generated at the transition from supersonic to subsonic flow speeds. Similar scenarios have also been proposed by Balthasar et al. (2016); González Manrique et al. (2017) and González Manrique et al. (2018) to explain supersonic downflows at footpoints of filaments observed in He I D3. The topic of He I D3 and He I 10830 Å in other on-disk targets such as

68 plages, quiet sun or coronal holes remains largely unexplored. However, He I 10830 Å signal is omnipresent on the solar disk and He I D3 has been studied in plages (e.g. Landman 1981). Studies of plages and quiet sun in He I 10830 Å and to some extent in He I D3 should hence be possible.

4.3.4 Coronal targets

The He I D3 and He I 10830 Å lines can also be used to study targets that are situated very high in the atmosphere at coronal heights, for example the observation of a coronal loop at an estimated height of ∼ 70 Mm (Schad et al., 2016), or coronal rain observed outside the limb at ∼ 15 Mm (Schad, 2018). Even these targets are situated at coronal heights, some plasma at chromospheric temperatures should be left for He I te be present at all. This connects also to Paper II, in which the observed flare loops in He I D3 absorption are estimated to be at a height of ∼ 14 Mm based on geometrical arguments. The observations of Schad et al. (2016); Schad (2018) and Paper II encourage the use of He I 10830 Å and He I D3 for (off-limb) coronal targets. Especially since HAZEL is equipped to fit limb observations, as opposed to for example the NICOLE (Socas-Navarro et al., 2015) or STIC (de la Cruz Ro- dríguez et al., 2016, 2019) inversion codes, commonly used to invert the Ca II 8542 Å line. Therefore, magnetic field estimates via Ca II 8542 Å in limb targets have to be conducted via the weak-field approximation (Robustini et al., 2019; Kuridze et al., 2019).

4.3.5 Reconnection targets My PhD research has mostly concentrated on reconnection targets: Ellerman bombs and ﬂares. Since both of these topics have been the subject of my papers included in this thesis, I discuss these in the following two chapters separately: see Chapter 5 for Ellerman bombs and Chapter 6 for ﬂares.

69 70 5. Ellerman bombs

5.1 Deﬁnition

Ellerman bombs (EBs) were first observed by Ellerman (1917) who called the events “hydrogen bombs” at the time. The spectra of hydrogen Balmer lines exhibit very clear signatures: spatially narrow and bright emission in the wings of the line and an unaffected line core with strong absorption, see Fig. 5.1. EBs have sometimes been referred to as “moustaches” in the literature, due to the shape of the line profile (Severny, 1956). Ellerman bombs are transient events with life times of the order of several minutes up to 1 hour. Measurements of the life time of EBs are dependent on temporal cadence of the observations since EBs tend to reappear at the same location. The size of EBs is of the order of ∼ 1 arcsec. Ellerman bombs have been observed for 100 years now (Severny, 1956; Bruzek, 1972), but substantial progress on the topic has been made in the last two decades by high-resolution observations with the Flare Genesis balloon telescope (Georgoulis et al., 2002; Pariat et al., 2004, 2006) the Hinode space- craft (Matsumoto et al., 2008; Hashimoto et al., 2010) and with SST/CRISP (Watanabe et al., 2011; Vissers et al., 2013; Nelson et al., 2013; Vissers et al., 2015; Reid et al., 2015; Rutten et al., 2015; Reid et al., 2016). These high- resolution observations have revealed detailed properties of EBs as listed here. • EBs occur in active regions and are usually associated with regions of enhanced magnetic activity and/or flux emergence. Observations of mixed polarities, flux emergence, flux cancellation and bi-directional jets at EB locations established magnetic reconnection as the physical driver for EBs (Georgoulis et al., 2002; Pariat et al., 2004; Socas- Navarro et al., 2006; Watanabe et al., 2008, 2011; Nelson et al., 2013; Reid et al., 2016; Nelson et al., 2016; Yang et al., 2016; Toriumi et al., 2017; Centeno et al., 2017). • EBs are situated under the canopy of fibrils that is observed in optically thick Balmer lines. When sampled in the line core of Hα (or Hβ, etc.), no EB signal is observed. A high-resolution example of EBs as visible in the wings of Hα and shielded in the core, is given in Fig. 1 of Watan- abe et al. (2011). When the fibrils are velocity shifted, the observed

71 Figure 5.1: Left: Hα spectrum of an EB as ﬁrst observed by Ellerman (1917). Right: Hβ spectrum of an EB observed with SST/TRIPPEL (same data set as used in Paper I).

EBs show asymmetry in their wings with blue asymmetries prevailing (Bruzek, 1972; Kitai, 1983; Dara et al., 1997). • EBs exhibit small-scale structure with a ﬂame-like morphology, sometimes with an upside-down “Y” shape. They are rooted in intergranular lanes in the deep photosphere and pointing upwards, giving them an elongated shape depending on viewing angle (Kurokawa et al., 1982; Georgoulis et al., 2002; Pariat et al., 2007; Hashimoto et al., 2010; Watan- abe et al., 2011). • A review by Rutten et al. (2013) has stressed that EBs are not to be con- fused with magnetic bright points. The latter also occur in intergranular lanes and show wing brightening in Hα, just as EBs. However they are smaller and shorter lived than EBs and unlike EBs they show continuum brightenings - not only Balmer line wing brightening. Magnetic bright points are essentially locations of small scale vertical ﬂux tubes pinch- ing through the photosphere. They are bright because one observes the (deeper located) walls of the Wilson depression.

5.2 Flux Emergence

EBs occur most commonly in flux emergence regions (FERs). Magnetic flux emergence from the bottom of the solar convection zone to the corona is a complex process, evolving through layers with huge differences in terms of pressure, temperature and physical regimes (high β to low β plasma). The study of flux emergence is relevant to many areas of research such as dynamo models, active region and sunspot formation, Ellerman bombs and reconnection,

72 Figure 5.2: A schematic view of flux emergence in the convection zone, photosphere, chromosphere and corona. The green drop indicates the location of Ellerman bombs in field line dips. Reconnection will remove the dense material and allow the field lines to fully emerge into the chromosphere and corona. Figure from Toriumi (2014). and flare and CME launching. There are standard models of how flux emergence is proceeding from the convection zone to the corona, as summarized in Fig. 5.2 and reviewed by e.g. Archontis (2010); Toriumi (2014); Schmieder et al. (2014a). However, many aspects of the physics remain topics of active research in the community, using high-resolution observations of all layers of the solar atmosphere, helioseismology, and also MHD-modeling. The commonly accepted idea is that flux emergence starts with a rising twisted flux tube from the bottom of the convection zone (i.e. the tachocline). The flux tube is generated by the solar dynamo under influence of differen- tial rotation. Density perturbations render the flux tube buoyant which will result in a rise of the Ω-shaped tube through the convection zone. This process acts via the Parker instability in which material is drained down from the top of the Ω-shaped loop via the field lines (Parker, 1955; Spruit, 1981). The rise of the flux tube may be partially driven by convection (Parker, 1988) but at the same time it has been found that convection can destroy untwisted

73 flux tubes (Emonet and Moreno-Insertis, 1998; Jouve and Brun, 2009). Con- vection breaks apart the flux tube in small magnetic loops so that emergent flux at the photosphere appears in small-scale magnetic elements of different polarities (Cheung et al., 2010; Toriumi and Yokoyama, 2012; Rempel and Cheung, 2014) as depicted by dark and light gray dots in the photosphere on Fig. 5.2. Observationally, granulation in emerging flux regions becomes elongated. Eventually the magnetic elements of equal polarity will group together to form pores and can develop into full-size sunspots. Once the flux tube is entering the photosphere, its emergence is halted for several reasons: (1) the photosphere does not support convection, (2) the Parker instability is removed because the magnetic tension balances the buoy- ancy (Magara, 2001) and (3) parts of the flux tube will never emerge because they are trapped by dense material (Magara and Longcope, 2001; Fan, 2001). Therefore, the magnetic field lines are found to have an undulatory shape with dips where the dense material is located (Strous and Zwaan, 1999). Such configurations are also referred to as sea-serpent field lines or serpentine field lines, as shown by the blue line in Fig. 5.2. The upper parts of the loops that have emerged through the photosphere are allowed to rise further via the magnetized Rayleigh-Taylor instability (Parker, 1966), this is an instability that occurs when material with higher density floats on top of material with lower density. Dense material accumulates in the dips of the field lines, called bald patches. These are locations in which the magnetic field satisfies Bz = 0 and B ·∇Bz > 0 (Pariat et al., 2004). The dense material in the bald patches will prevent the loops from fully emerging - as shown by the green drop in Fig. 5.2 - until magnetic reconnection takes place. It has been shown that the location of bald patches where magnetic reconnection occurs correspond to the locations of Ellerman bombs in many cases (∼ 70%) (Bernasconi et al., 2002; Georgoulis et al., 2002; Pariat et al., 2004). Reconnection provides a mechanism that can remove dense material from magnetic field line dips, allowing the field lines to fully emerge into the chromosphere and the corona. These large magnetic loops in the corona are called arch filaments, as shown by the red field line in Fig. 5.2. The next section describes the process of magnetic reconnection in more detail.

5.3 Magnetic reconnection

Magnetic topologies are exactly conserved in a plasma with zero resistivity, meaning that no magnetic diffusion takes place. Under such conditions, the magnetic ﬁeld lines are “frozen in”: they advect together with the plasma ﬂows without changing their topology. This is the case in ideal MHD and the induc-

74 Figure 5.3: Sweet-Parker model for reconnection, figure from Zweibel and Ya- mada (2016). Magnetic field lines are given in blue, the current sheet is depicted in orange, gray arrows indicate flows, 2L is the length and 2δ is the width of the boundary layer. tion equation equals ∂B = ∇ × (v × B). (5.1) ∂t In non-ideal MHD, magnetic diffusion has to be taken into account and the induction equation becomes

∂B = ∇ × (v × B) + η∇2B, (5.2) ∂t

c2 where η = 4πσ and σ is the conductivity. Magnetic reconnection takes place in a plasma in which the field lines are almost completely frozen in, but some small yet finite diffusion is possible. The magnetic Reynold number RM is essentially the ratio of advection over diffusion in a plasma. If we scale the quantities in Eq. 5.2 as follows: ∇ ∼ L−1, v ∼ V, B ∼ B then we obtain for the magnetic Reynolds number LV R = . (5.3) M η In astrophysical systems such as the Sun, the magnetic Reynold number is 8 14 typically very large, of the order of RM ' 10 − 10 (Zweibel and Yamada, 2016). The simplest geometry in which reconnection takes place is “X”-point reconnection, see Fig. 5.3. The first model to describe reconnection in 2D was

75 proposed in 1957 and is called the Sweet-Parker model (Parker, 1957; Sweet, 1958). A current sheet is located where magnetic fields of opposite direction are brought together. Plasma flows in from the top and bottom of the topology depicted in Fig. 5.3 and flows out from the left and right side of the topology. It is expected that mass conservation approximately holds locally so that the inflow velocity vi relates to the outflow velocity vo as

viL ≈ voδ, (5.4) where 2L is the length and 2δ the width of the boundary layer. The outﬂow kinetic energy is of the order of the magnetic energy around the current sheet 1 B ρv2 ≈ , (5.5) 2 o 8π which shows that the outﬂow velocity equals the Alfvén speed vA B vo ≈ vA = √ . (5.6) 4πρ The Sweet-Parker model assumes a steady state so the induction equation reduces to ∇ × (v × B) + η∇2B = 0. (5.7) Using this equation, an order of magnitude estimate yields then the relationship v B ηB i ≈ . (5.8) δ δ 2 If we combine this result with Eqs. 5.4, 5.6 and 5.3, we obtain

vA vi ≈ √ . (5.9) RM This equation shows that within the Sweet-Parker model, the reconnection rate is only a fraction of the Alfvén speed. Therefore, this model is also known as the model for slow 2D magnetic reconnection. Observations of flares seem to require much larger reconnection rates, therefore models for fast reconnection in 2D have been proposed by Petschek (1964), in which the fast reconnection rate equals vA vi ≈ . (5.10) log(RM) Both models for slow and fast reconnection have been unified by Priest and Forbes (1986). A detailed discussion of the Petschek and unified reconnection models - as well as the complex world of 3D reconnection - lies outside the scope of this thesis.

76 5.4 Ellerman bomb topologies

We have now discussed that the simplest reconnection geometry requires a current sheet that exists when opposite magnetic field lines are pushed closely together (see Sect. 5.2). We also reviewed that Ellerman bombs occur in flux emergence regions in places with serpentine field lines (see Sect. 5.3). So what are the exact magnetic topologies of Ellerman bombs? Multiple scenarios are possible, indicated as (a), (b) and (c) in Fig. 5.4 and described in detail by Georgoulis et al. (2002). Scenario (a) depicts a situation where dips in the serpentine field lines and converging velocity fields in the photosphere cause large magnetic field gradients to form a current sheet. “X”- shape reconnection takes place along the current sheet and outflows in the form of bi-directional jets are present, one jet going down towards the photosphere, another one going up in the atmosphere. Scenario (b) is very similar to (a), except that the flows in the photosphere are not opposite. One large flow is present compressing the serpentine field lines together, giving rise to multiple locations in which possible current sheets can form. In this case, bi-directional flows will also be present. The third scenario (c) is different from (a) and (b) in the sense that it does not have the “X”-shape reconnection. It is a situation in which newly emerging flux interacts with pre-existing horizontal magnetic field of a canopy in an active region. The newly emerged flux might have the same polarity but a different direction compared to the pre-existing field. This topology leads to the presence of “quasi separatrix layers” (QSLs) which are part of more complex 3D-reconnection descriptions. In these cases, no null-points or “X”-point separatrices are present. Observational evidence for bi-directional jets associated with EBs has been found by e.g. Watanabe et al. (2008); Matsumoto et al. (2008); Watanabe et al. (2011); Vissers et al. (2015). MHD simulations of Ellerman bombs essentially confirm the resistive serpentine flux emergence and reconnection scenario (Isobe et al., 2007; Archontis and Hood, 2009; Nelson et al., 2013; Danilovic, 2017; Danilovic et al., 2017; Hansteen et al., 2017). In 3D simulations, EB topologies are generally more complex, meaning that the examples in Fig. 5.4 should be seen as idealized and schematic cases.

5.5 Spectral Signatures and Forward Modeling

Spectral signatures of Ellerman bombs are summarized in e.g. Rutten et al. (2015); Rutten (2016) and in the Introduction of Paper I. We have summarized EB spectral signatures in Table 5.1. We note that EBs are not visible in EUV lines such as He II 304 Å, nor in the photospheric continuum.

77 Figure 5.4: Schematic view of different Ellerman bomb magnetic topologies as proposed by Georgoulis et al. (2002). MDF stands for moving dipolic feature. QSL stands for quasi-separatrix layer.

78 Table 5.1: EB spectral signatures.

Spectral Line Spectral Proﬁle Citations (e.g.)

Hα,Hβ, etc Moustache Ellerman (1917); Rutten et al. (2013)

Ca II H&K Moustache Kitai (1983); Pariat et al. (2007) Hashimoto et al. (2010)

Ca II 8542 Å Moustache Fang et al. (2006); Socas-Navarro et al. (2006) Pariat et al. (2007); Vissers et al. (2013) Hong et al. (2014); Li et al. (2015) Rezaei and Beck (2015) 1600 Å, 1700 Å continua Bright Qiu et al. (2000); Vissers et al. (2013)

Mg II h&k, C II 1330Å Moustache Vissers et al. (2015); Tian et al. (2016) Kim et al. (2015); Grubecka et al. (2016) Paper I

Mg II triplet 2799 Å Enhanced Vissers et al. (2015); Hong et al. (2017b) Paper I

Si IV 1400Å Enhanced, broad Vissers et al. (2015); Kim et al. (2015) Sometimes self-reversal Tian et al. (2016); Paper I

He I D3, He I 10830 Å Moustache Paper I

EB signatures at various wavelengths are important because they provide constraints on modeling of EB atmospheres. Multiple studies have attempted to synthesize EB spectra in Hα, Ca II H, Ca II 8542 Å or Mg II h&k. Those studies start from a model atmosphere, typically a FAL model, and insert an ad-hoc temperature and/or density increase at a certain height in the upper photosphere or lower chromosphere. When comparing synthetic line proﬁles with observations, temperature, density and formation height of EBs can be estimated. Temperature enhancements are found of 1500 K by Kitai (1983), 600-1300 K by Fang et al. (2006), 5000 K by Bello González et al. (2013), 4000 K by Berlicki and Heinzel (2014), 2700-3000 K by Li et al. (2015), 3350 K by Grubecka et al. (2016) and 600-3000 K by Fang et al. (2017). They all located EBs close to or slightly higher than the height of the temperature minimum in the (quiet) solar atmosphere. Some of these forward modeling efforts have been criticized by Rutten et al. (2015) for several reasons:

• The formation height of EBs is estimated to be in the lower chromosphere, slightly above the temperature minimum by most forward modeling attempts. This is the result of the constraint that EBs are invisible in the optical continuum. However, high-resolution images of EBs at large viewing angle have indicated that EBs are rooted into intergranular

79 lanes in the (deep) photosphere. Also, EBs are shielded by the canopy of optical thick ﬁbrils in the core of Hα (and other Balmer lines) and are therefore not chromospheric. However, part of this discussion might be in nomenclature, since it also depends on how the chromosphere is deﬁned (see Rutten (2016) for a discussion).

• Not all spectral signatures have been taken into account in forward modeling efforts. EBs should be invisible in Mg I b and Na I D lines while this property has not been checked in any modeling effort.

Moreover, most of the forward modeling efforts have been conducted before it was discovered that (some) EBs have signatures in IRIS lines such as Mg II h&k, Si IV 1400 Å, C II 1330 Å (Vissers et al. 2015; Tian et al. 2016 and Paper I). Due to these observations, a new type of event has been described in the literature: UV bursts.

5.6 Connection between EBS and UV bursts

With the launch of the Interface Region Imaging Spectrograph in 2014 (IRIS, De Pontieu et al. 2014), observations of the chromosphere and transition region in the UV have been routinely accessible. Shortly after, the first observation of UV bursts was reported (Peter et al., 2014). UV bursts were observed as strong emission in the Si IV 1400 Å doublet, with a distinct spectral charac- teristic: very broad emission profiles with a central reversal. The ratio of the two emission profiles of the Si IV doublet equals two, indicating that the doublet is formed under optically thin conditions. Therefore, the double peak is interpreted as due to the presence of a strong bidirectional jet with velocities of up to 75 km s-1. Another spectral property of UV bursts is that the broad emission profiles are blended with narrow absorption lines originating from singly ionized species situated in cooler material in chromospheric layers on top of the UV burst location. The entire spectrum indicates hence strong reconnection events in the deep atmosphere. Based on the assumption of coronal equilibrium, the authors estimate the temperatures and density of these events to be of the order of T ' 105 K and located in the solar photosphere. Peter et al. (2014) proposed a link between EBs and UV bursts but since they did not have optical spectra available, this connection was not directly studied in their paper. Soon after, several instances were reported of EBs having UV burst type Si IV spectra (Vissers et al., 2015; Kim et al., 2015; Tian et al., 2016; Hong et al., 2017b) and Paper I, for a review on UV bursts see Young et al. (2018). However, the link between UV bursts and EBs has been confusing to many.

80 This is due to their naming being based on observationally different charac- teristics: a UV burst requires strongly enhanced Si IV 1400 Å profiles, while an EB requires a moustache shaped Hα profile with a line core that is unaltered compared to surrounding Hα fibril spectra. The observational evidence so far indicates that some EBs are also UV bursts - though not all of them, and some UV bursts are also EBs - though again not all of them. Often it is not known whether the observed EBs are also UV bursts and vice versa because simultaneous observations in the UV and the optical are not readily available. The assumptions of Peter et al. (2014) have been criticized by Judge (2015) placing the events not in the photosphere but in the mid- to high chromosphere. The events discussed in Peter et al. (2014) are likely UV bursts without an EB counterpart, although one can never be certain since no corresponding optical spectra are available. From the physical point of view, the mechanism is the same for EBs and UV bursts: small-scale magnetic reconnection (see Sect. 5.2) and even the range of magnetic topologies leading to EBs and UV bursts is likely the same. The difference between both events is their formation height and energy. EBs are generally estimated to be located in the upper photosphere to lower chromosphere with energy estimates of around 1027 erg (Shimizu, 2015) while UV bursts are located in the chromosphere or transition region and with similar to slightly higher energy of the order of 1028 erg (Young et al., 2018). How- ever, as stated before, there is an overlap between EBs and UV bursts. As both energy and formation height are continuous properties, EBs gradually evolve into UV bursts with increasing energy and formation height. There is a range of energies and formation heights where the reconnection events have observational properties of both EBs and UV bursts. This is the case for the events discussed in Paper I. Due to the discovery of UV bursts, the fact that their properties partially overlap with EBs, and that they are estimated to have temperatures of the order of T ∼ 105 K has given rise to a controversy in the field. Using forward modeling techniques, all EBs have been estimated to have temperatures cooler than or of the order of T ≤ 104 K (see Sect. 5.5). However, none of the forward modeling efforts have taken into account EB/UV burst signals in Si IV 1400 Å which is the diagnostics that indicates temperatures T > 104 K for EBs and UV bursts. Following the observations reported of Peter et al. (2014), some studies have been considering different types of modeling. For example Rutten (2016) has combined almost all EB diagnostics in a model assuming LTE for the event onset, and calculated that EB temperatures should be ranging between T ∼ 1.5−2×104 K. Another approach was modeling using radiative hydrodynamics (RHD, Reid et al. 2017; Hong et al. 2017a). None of the RHD studies

81 have found hints that EBs could be hotter than 104 K. Only Hong et al. (2017a) has synthesized the Si IV 1400 Å line but has not found any enhancement in the line in their models. The events that they synthesized were hence pure EBs without a UV burst counterpart. So far, no study has been succeeded at simultaneously reproducing UV bursts in Si IV 1400 Å with an EB counterpart, such as the events that are observed in Vissers et al. (2015) and Paper I. This issue has been addressed extensively in Vissers et al. (2019). The observations consist of EBs with a UV burst counterpart observed simultaneously by the SST and IRIS. The events were analyzed using the Stockholm Inversion Code (STIC, de la Cruz Rodríguez et al. 2016, 2019). Any attempt to obtain a strong signal in Si IV 1400 Å resulted into unrealistic enhancement of the Mg II h&k lines. Pos- sible causes of this inconsistency might be that the code works in 1D while perhaps 3D is needed, or the difference in timing between the observations of sometimes almost 30 s.

5.7 He I D3 and He I 10830 Å observations and MHD simulations

The spectral signatures and temperatures of EBs and UV bursts have been extensively discussed in the literature. However, Paper I has added a new EB signature to the list: emission signatures in the form of a moustache line proﬁle for He I D3 and to some extend also for He I 10830 Å. The events that were discussed in Paper I have both properties of EBs and UV bursts, with enhanced Mg II h&k and Si IV 1400 Å spectra. We have interpreted the fact that EBs are visible in He I D3 as a suggestion for high temperature of these events, due to the line formation mechanism of He I D3 and He I 10830 Å (see Chapter 2). We suspect that the He I D3 emission is likely formed by either thermal collisions or by ionization-recombination due to the local presence of EUV radiation. In either case, high EB temperatures seem to be needed. A rough estimate is given with a range of temperatures between T ' 2 × 104 − 105 K. In order to study helium line formation during these events in more detail, an synthetic EB resulting from a radiative MHD simulation is used to synthesize helium spectra, see Paper III . Radiative MHD simulations have been used before to study spectral diagnostics of EBs and UV bursts Nelson et al. (2013); Danilovic (2017); Hansteen et al. (2017); Danilovic et al. (2017). Those studies demonstrated that the spectral diagnostics of EBs and UV bursts are reasonably well-reproduced during the reconnection events found in the MHD simulations. The synthetic EB/UV burst used in Paper III reaches tem-

82 peratures of > 106 K in the upper photosphere. However, the synthetic Si IV 1400 Å doublet and the Hα spectra calculated by Hansteen et al. (2019) using the same rMHD snapshot, have similar properties as the observed ones. This indicates that we are observationally blind to hot temperatures in the low atmosphere: the nominal chromospheric diagnostics are not sensitive to coronal temperatures, while any coronal radiation gets absorbed in the overlying cooler layers. Paper III demonstrates the presence of EUV radiation generated locally in the EB at a height of ∼ 0.8 − 1 Mm. The ionizing radiation is generated at temperatures between 2 · 104 − 106 K and at electron densities between 1011 − 1013 cm−3. Subsequently this EUV radiation propagates from the hot EB region and is absorbed in a thin shell around the EB. In this shell, neutral helium is ionized and recombines into the triplet system. Due to high electron density in this location (1012 − 1013 cm−3), there is a coupling to local conditions with temperatures of 7 · 103 − 104 K. These temperatures are higher than the temperature at which the photospheric background is formed, so that emission in He I D3 and He I 10830 Å is present. This result conﬁrms that helium emission signatures are an indicator for EB temperatures ≥ 2 · 104 K.

83 84 6. Flares

6.1 Deﬁnition and the ﬂare standard model

Flares happen as the result of sudden magnetic energy release by reconnection. Magnetic energy is converted into particle energy, heat, waves, and plasma motion. Observationally, there is the presence of emission across the spectrum. Flares are observable in many wavelength bands, but with rather different char- acteristics and response, see Fig. 6.1. Flares usually have a pre-flare phase with a gradual energy build-up, the impulsive phase of a couple of minutes during and right after the reconnection happened, and a decay phase of a couple of hours. Some recent reviews on observations and modelling aspects of flares are written by Benz (2008); Milligan (2015); Fletcher et al. (2011); Holman et al. (2011); Janvier et al. (2015); Schmieder et al. (2015). Flares occur preferably in active regions with a complex magnetic field topology, often at sites where δ-sunspots are present: these are sunspots with opposite magnetic field polarity in their umbrae enclosed by one and the same penumbra. Often there is also new magnetic field emerging in the active region (see Sect. 5.2). The standard flare scenario shown in Fig. 6.2 is able to explain a range of different observational properties of flares. Flares are the result of magnetic reconnection (see Sect. 5.3) situated in the corona. Due to reconnection, electrons - and possibly ions - get accelerated towards the chromosphere, although the precise acceleration mechanism is still a topic of ongoing research. The observational evidence for particle acceleration is the presence of hard X-rays (HXRs) caused by Bremsstrahlung, i.e. X-rays emitted while free electrons scatter off ions (free-free radiation). The HXRs have a non-thermal energy distribution. The accelerated electrons are often referred to as an electron beam and travel through the corona and transition region without considerably losing energy (thin-target approximation) until they reach the cold and dense chromosphere and deposit their energy (thick-target approximation, Brown 1971). The locations of observed HXRs are called the flare footpoints. The energy deposition of the accelerated electrons in the chromosphere heats narrow horizontal elongated regions in the chromosphere: the flare ribbons, which are radiating soft thermal X-rays. The flare ribbons are also readily observed in Hα and

85 Figure 6.1: Schematic overview of flare variation with time in different wavelength bands. The pre-flare takes a couple of minutes, the impulsive phase can take around 3-10 min, the flash phase 5-10 min and the decay a couple of hours. Figure from Benz (2002). other chromospheric optical lines. The Neupert effect (Neupert, 1968) is an empiric relation observed in almost all flares (except the most energetic X-flares) stating that the flux of the soft X-rays (SXRs) observed in flares equals the accumulated HXR flux over time: Z t 0 0 FSXR(t) ∝ FHXR(t )dt . (6.1) 0 This relationship strongly suggests a causal relationship between accelerated electrons and thermal SXRs. As a result of the energy deposition in the chromosphere, both chromospheric “evaporation” and chromospheric “condensation” occur. These are misleading terms for the presence of strong upflows and downflows respectively, originating from a high-pressure plug in the chromosphere where the accelerated electrons deposit their energy (Milligan, 2015). Chromospheric evaporation is the process of chromospheric material being heated and rising

86 Figure 6.2: The standard flare scenario. Figure from Christe et al. (2017) up along the flare loops, which in return also emit SXRs and EUV radiation. The upflow velocities reach up to −300 km s-1 and are most easily observed in spectral lines of highly ionized species, e.g. Fe XXIV. Chromospheric condensation exhibits downflow velocities of the order of 50 km s-1 which are measured in cooler chromospheric species such as the Mg II triplet at 2799 Å (Graham and Cauzzi, 2015). Chromospheric condensations were also measured via He I D3 in Paper II, showing that this line is excellent to capture flare dynamics in the chromosphere. The He I D3 observations reveal shocks and a “bounce-back” effect of the deep compressed chromospheric layers - a process that has not been observed before. The validity of the standard model for flares is still heavily debated, especially with regard to the presence of electron beams. Theoretical concerns have been raised with regard to the stability of the beam, the rate at which the corona would be depleted of free electrons, and the deposition depth of the energy. An alternative scenario has been proposed by Fletcher and Hudson (2008) in which electrons are accelerated locally by Alfvén waves.

87 Flares are often classified according to their maximum X-ray flux as measured by the GOES satellite and corresponds to the order of magnitude of X-ray flux in the 1−8 Å channel. The scale is given by a letter A, B, C, M and X which means an X-ray flux larger than 10−8, 10−7, 10−6, 10−5 and 10−4 W m-2, respectively.

6.2 Magnetic ﬁelds and spectro-polarimetry of ﬂares

Since flares are the result of magnetic reconnection, research has been aim- ing at understanding and estimating the magnetic field topologies and changes during flares. A common way of estimating the large-scale topology of flares is magnetic field extrapolation (Longcope, 2005). This refers to a range of techniques in which a coronal magnetic field is estimated based on the input of a photospheric magnetogram used as a boundary condition. However, the result of extrapolations are subject to errors depending on the boundary condition and on the assumptions that are made, see e.g. (Wiegelmann et al., 2008, 2010). Magnetic fields can also be measured by applying inversion codes to spectropolarimetric data (see Chapt. 3). Spectro-polarimetric inversions of the photosphere are of interest to capture magnetic field changes as a consequence of the reconnection during flares. However, the chromosphere is the layer of the solar atmosphere that gets greatly influenced during flares in terms of heating, accelerated particles, pressure, dynamics, and excitation/ionization of chromospheric species. Chromospheric spectro-polarimetric inversions with magnetic field estimation in high resolution are very rare. This is due to the fact that it is hard to capture flares with ground-based instruments providing chromospheric spectro-polarimetric data. Most studies that have attempted chromospheric estimates of the magnetic field during flares have used the Ca II 8542 Å line (Kleint, 2017; Kuridze et al., 2018) or the He I 10830 Å line (Sasso et al., 2011, 2014; Anan et al., 2018). In the works of Sasso et al. (2011, 2014), the authors have focused on the magnetic field topology of a erupting filament during the flare, by careful inversions of the He I 10830 Å line. Kleint (2017) has mapped photospheric and chromospheric magnetic field changes during flares using the weak-field approximation for the 8542 Å line in the chromosphere and HMI magnetograms in the photosphere. Kuridze et al. (2018) have showed that the Ca II 8542 Å line is formed at higher optical depths during a flare compared to the same location before or after the flare. One possible interpretation of this is that the line is formed deeper in the atmosphere during the flare which would explain the high values of the magnetic field between 1−1.5 kG. This result is similar to Anan et al. (2018), who found an increase of 650 G of the magnetic field

88 Figure 6.3: Fan-spine topology for a circular ribbon ﬂare. (a) depicts a possible initial topology, (b) shows how a current sheet can be formed and (c) displays the ﬁnal topology after reconnection. Figure from Fletcher et al. (2001).

during a flare measured via the He I 10830 Å line. The total measured field is 1.4 kG and the authors also suggest a decreased formation height of the He I 10830 Å line during the flare as explanation for the increased magnetic fields.

For Paper II, we have for the first time used spectro-polarimetric He I D3 data to estimate the magnetic field during a flare. The retrieval of the full magnetic field vector was challenged by low signal-to-noise in the Stokes parameters. Therefore, the orientation of the flare loops was measured using the intensity image and given as an input in the inversion to determine the magnetic field azimuth. In this way, maps of the full magnetic field of a C3.6-class flare are given in the paper. The authors have found very high magnetic field values of 2.5 kG at the flare footpoints where the He I D3 line is in full emission. These footpoints are located above photospheric pores. Similarly to Kuridze et al. (2018); Anan et al. (2018), the authors suggested that the He I D3 line is formed deeper in the atmosphere than usual during the flare at its footpoints. The inversion results of Paper II are compatible with a fan-spine topology of the observed flare, see Fig. 6.3. A fan-spine topology is a common topology for flares (Fletcher et al., 2001; Deng et al., 2013; Zeng et al., 2016) and also for chromospheric jets and surges (Shibata et al., 1994, 2007). This topology occurs in locations were a certain polarity patch is surrounded in a circular fashion by opposite polarity. Small loops will connect from the circular opposite polarity patch into the central polarity. A dome of field lines from the surrounding circular polarity will rise over the lower lying loops. Fig. 6.3 shows how this topology can give rise to the presence of a current sheet and hence magnetic reconnection. When a flare happens in this topology, the observed ribbons are circular. The flare discussed in Paper II only displays half of this topology, hence a half dome and half-circular ribbons.

89 Figure 6.4: He I D3 observation with SST/CRISP of a C3.6 class ﬂare. The left panel shows the line core at λ0 and the right panel shows the red wing of the line at λ0 + 0.693 Å.

6.3 Helium observation and line formation of ﬂares

Flares have been observed via imaging in He I D3 with BBSO (Zirin, 1980; Zirin and Neidig, 1981; Feldman et al., 1983; Zirin and Tang, 1990; Falchi et al., 1992; Cauzzi et al., 1996; Liu et al., 2013). Some of these observations have lead to believe that He I D3 is in absorption for lower energy events such as surges and weak flares, and goes into strong emission in energetic flares only (Zirin, 1980; Zirin and Neidig, 1981; Feldman et al., 1983; Liu et al., 2013). However, the observations presented in Paper II shown also in Fig. 6.4, provide clear proof that He I D3 emission can also be present in small flares, in this case a C3.6 class flare. Emission signals in He I D3 have even been observed in EBs (Paper I) which are many orders of magnitude less energetic than solar flares. Flare observations in the He I 10830 Å line are more common than in the He I D3 line. A variety topics has been studied via He I 10830 Å flare observations such as spectral broadening of He I 10830 Å (You and Oertel, 1992), X- ray threshold values for He I 10830 Å emission (Du and Li, 2008), flare waves (Vršnak et al., 2002), flare triggering by presence of sunspots (Louis et al., 2014), He I 10830 Å polarization in flares (Judge, 2015; Kuckein et al., 2015), and observational studies of flares (e.g. Penn and Kuhn 1995; Malanushenko 1999; Teriaca et al. 2003; Akimov et al. 2014; Zeng et al. 2014; Xu et al. 2016). In Sec. 2.3, we discussed line formation in helium triplet lines, mostly concentrated on quiet sun. The situation is entirely different in flares because

90 of the influence of non-thermal electrons and enhanced EUV radiation. Line formation of He I D3 and/or He I 10830 Å has been investigated by several authors such as Laming and Feldman (1992); Ding et al. (2005); Andretta et al. (2008); Zeng et al. (2014). Laming and Feldman (1992) proposed a burst model to investigate He I and He II intensities in flares. Their model belongs in the class of models that considers electrons at higher temperature than the ionization balance of the atom, as was first suggested by Jordan (1975), see Sec. 2.3. The burst is simulated by raising the electron temperature of small regions of the plasma in times that are short compared to the ionization equilibration time for helium. Electron collisions are then considered as the mechanism to excite electrons into the neutral helium triplet states. Ding et al. (2005) have investigated the influence of electron beams on the helium triplet level population. The results have indicated that the photoionization- recombination mechanism (PRM, Sect. 2.3) dominates when no non-thermal electrons are present and the collisional ionization-recombination mechanism (CRM, Sect. 2.3) dominates when an electron beam is introduced. Based on their calculated line profiles, Ding et al. (2005) propose that strong absorption in He I 10830 Å in the early flare phase and strong He I 10830 Å emission in the maximum of the flare would be an indication of non-thermal electrons in- fluencing the line formation. The latter has been observed by Xu et al. (2016) who found a flare front in absorption in the He I 10830 Å line, calling it a “negative” flare front. Zeng et al. (2014) have considered both the PRM and the burst scenario of Laming and Feldman (1992) and compared it with the observation of a C-class flare in coronal and transition region lines and He I 10830 Å emission. Several patches of He I 10830 Å emission were observed and the estimated radiation energy of those patches was found to be compatible with the photon energy required to generate He I 10830 Å emission via photoionization-recombination. The authors point out some difficulties of the burst scenario: according to their estimates, He I would be ionized quickly when collisional excitation is taking place during the burst. Judge et al. (2015) have focused on the origin of linear polarization of the He I 10830 Å line. Using a slab model, they investigated the anisotropy in the radiation field needed to reproduce the observed Q and U profiles which exhibit a change of sign between the blue and the red component of the He I 10830 Å line. In the flare ribbons, the red component of the He I 10830 Å line is formed at much higher optical depths (∼ 10) than the blue component of the He I 10830 Å line (∼ 1). The slab has produced substantial thermal heating itself and therefore, a change of sign with optical depth is present in the anisotropy of the radiation field which can explain the observed Q and U

91 profiles in the flare. Interestingly, no non-thermal electrons were required to explain the observed linear polarization signatures. There is no consensus yet about the He I D3 and He I 10830 Å line formation mechanism during flares and likely, different mechanisms are acting at different times and targets in the flare. Non-thermal electrons could play a role in the flare footpoints, while PRM is likely acting in the flare loops observed in absorption in Paper II. The results that we have obtained in Paper III in the context of EBs could perhaps be applicable to He I D3 and He I 10830 Å emission signatures in flare footpoints as well, although the role of non-thermal electrons should be investigated. A lot of progress on this topic could be made by studying the He I 10830 Å and He I D3 lines in RHD flare models including electron beams.

92 7. Summary and connection to the papers

The research during my PhD was focused on the diagnostic potential of He I D3 in the solar atmosphere. This thesis aims at providing the reader with an introduction to helium line formation (Chapt. 2), spectro-polarimetry and inversions (Chapt. 3) and helium observations (Chapt. 4). Helium line formation has been poorly understood for more than 75 years and a review of the progress on this topic is given in Sect. 2.4. Quiet sun MHD modelling combined with 3D non-LTE radiative synthesis has confirmed that triplet levels of neutral helium are populated by photoionization-recombination in the quiet sun. The photons are originating in both the corona and in the transition region, the latter causing structure on sub-arcsecond scales in helium images of the upper chromosphere. However, the role of thermal and non-thermal electrons collisions is still unknown in active targets and reconnection events such as flares. Paper I discusses observations of a reconnection event located in the deep chromosphere or upper photosphere: Ellerman bombs (EBs), see Chapt. 5. We discovered that EBs exhibit emission in the wing of He I D3 and to some extend also in He I 10830 Å while the line core of both lines is in absorption. EBs have attracted a lot of attention in recent years because of divergent estimates of their temperatures. On the one hand, the temperature of EBs are estimated via semi-empirical modelling of spectra to be T ≤ 104 K because both the continuum and the line core of Hα are unaltered in the EB compared to its surroundings. On the other hand, broad and intense profiles of the Si IV 1400 Å line coinciding with EB locations suggests formation temperatures of T ∼ 8 · 104 K. In Paper I, we estimated the temperature range of the EBs in our observation to be between T ∼ 2 · 104 − 105 K, based on the presence of emission in He I D3. However, the assumptions made to reach this temperature interval were crude, and detailed 3D modelling is necessary to really understand the line formation of helium in EBs. This topic is addressed in Paper III, where we have made use of an MHD simulated EB and synthesized the helium spectra using the 3D non-LTE code MULTI3D. We find that the triplet-levels of helium are populated by locally generated ionizing radiation in a shell at the boundary

93 between the EB and the underlying chromosphere. Emission in He I D3 and He I 10830 Å is generated with spectral properties that are qualitatively similar to the observed profiles. The synthetic EB is located deep in the chromosphere, where coronal temperatures are reached. In addition, the presence of very high electron densities allows to produce ionizing EUV radiation in the vicinity of where He I 10830 Å and He I D3 emission features are observed. This mechanism of populating the line via locally generated EUV radiation and generating emission in He I 10830 Å and He I D3 could be relevant to other emission events observed in helium such as shocks and flares. It has already been known since the seventies that helium triplet lines reach large opacities in flares and can exhibit strong emission signals. Flares are described in Chapt. 6 with Sect. 6.3 discussing observations and line formation of helium in flares. For Paper II, we have obtained high-resolution polarimetric data of a flare in He I D3 with SST/CRISP. The data set is unique because it shows both strong emission and absorption signals in the flare and because it is the first data available showing on-disk spectro-polarimetric signal in He I D3. Using the HAZEL inversion code, we were able to measure detailed dynamics of the flare in the chromosphere, and to propose the magnetic field topology of the event. The main conclusion of the thesis and papers is that the He I D3 line is a powerful diagnostic tool for reconnection events. Both dynamics and magnetic field of the events can be derived. The helium lines offer a unique view on reconnection events in the chromosphere because their line formation mechanism is different to commonly used chromospheric diagnostics such as Ca II 8542 Å or Hα. Our findings encourage the use of the He I D3 spectral line as a diagnostic for the chromosphere and open up a range of applications that is yet to be exploited.

94 Sammanfattning

Denna avhandling handlar om en på sitt sätt enkel fråga. Vad kan observationer av spektrallinjen He I D3 lära oss om solkromosfären? Denna märkliga spek- trallinje är belägen i den gula delen av spektrum och har en våglängd på 5876 Å. Den anses bildas i den övre kromosfären och vara känslig för det lokala magnetfältet där. He I D3-linjen är också indirekt känslig för uppvärmning i övergångsregionen och koronan, eftersom den härrör från en övergång som inträffar mellan energinivåerna i triplettsystemet för neutralt helium. Dessa nivåer är nämligen vanligtvis befolkade via en joniserings/rekombinations- mekanism under på verkan av extremt kortvågig ultraviolett strålning som här- rör från övergångsregionen och koronan. He I D3-linjen användes för flare-diagnostik på sjuttiotalet och under de följande decennierna också för att mäta magnetfält i protuberanser. På grund av den dåliga spatiala upplösningen och lågt signal/brus-förhållande i observa- tionsdata från den tiden har emellertid nästan uteslutande objekt utanför solski- van studerats. Detta eftersom absorptionslinjen He I D3 är mycket svag på sol- skivan och endast synlig lokalt i småavgränsade områden. Den senaste instru- mentutvecklingen vid det svenska solteleskopet SST på La Palma möjliggör nu spektroskopiska och spektropolarimetriska observationer av He I D3-linjen på solskivans centrum. Detta gör det möjligt att studera alla typer av fenomen i kromosfären i ljuset av denna linje. Avhandlingen fokuserar påstudier av magnetisk rekonnektion i kromosfären via högupplösta observationer av He I D3 med instrumenten TRIPPEL och CRISP vid SST, i samverkan med rymd- burna instrument. Därtill tillkommer en teoretisk studie inriktad på djupare förståelse av linjebildningsprocesserna för He I D3 i småskaliga rekonnektion- shändelser. Avhandlingsarbetet har gett nya insikter i de kortvariga utbrott påsolen som kallas Ellerman-bomber och flares. He I D3-observationerna har satt gränser på den möjliga temperaturen i Ellerman-bomber genom upptäckten av ljusa He I D3-signaturer i dessa händelser. Vidare har beräkningar av strålningstransport i tre dimensioner och med tillåtelse av avvikelser från lokal termodynamisk jämvikt (LTE) ökat vår förståelse för hur He I D3-linjen bildas under sådana omständigheter, vilket har visat att kopplingen till den lokala temperaturen är komplicerad. I samband med ett flare-utbrott mättes starka nedflöden i kromosfären med hjälp av He I D3, vilket avslöjade detaljer i dynamiken i atmosfären under en flare. Spektropolarimetri användes för mätningar av magnetfältet under förloppet och därmed fåinsikter i dess magnetiska topologi. Sammanfattningsvis är He I D3-linjen ett utmärkt diagnostikum för rekon- nektionshändelser i solatmosfären. Detaljerad dynamik såväl som magnetfält- skonfiguration kan härledas med hjälp av linjen. Våra resultat uppmuntrar användningen av He I D3-spektrallinjen för diagnostik av kromosfären och öppnar för en rad nya tillämpningar vilka ännu inte utnyttjas. Publications not included in this thesis

• The high-redshift gamma-ray burst GRB 140515A. A comprehen- sive X-ray and optical study Melandri, A.; Bernardini, M. G.; D’Avanzo, P.; Sánchez-Ramírez, R.; Nappo, F.; Nava, L.; Japelj, J.; de Ugarte Postigo, A.; Oates, S.; Cam- pana, S.; Covino, S.; D’Elia, V.; Ghirlanda, G.; Gafton, E.; Ghisellini, G.; Gnedin, N.; Goldoni, P.; Gorosabel, J.; Libbrecht, T.; Malesani, D.; Salvaterra, R.; Thöne, C. C.; Vergani, S. D.; Xu, D.; Tagliaferri, G. (2015), Research Article, Astronomy and Astrophysics, 2015. DOI: https://doi.org/10.1051/0004-6361/201526660

• The cause of spatial structure in solar He I 1083 nm multiplet images Leenaarts, J.; Golding, T.; Carlsson, M.; Libbrecht, T.; Joshi, J. (2016), Research Article, Astronomy and Astrophysics, 2016. DOI: https://doi.org/10.1051/0004-6361/201628490

• Dissecting bombs and bursts: non-LTE inversions of low-atmosphere reconnection in SST and IRIS observations Vissers, G. J. M.; de la Cruz Rodrguez,´ J.; Libbrecht, T.; Rouppe van der Voort, L. H. M.; Scharmer, G.; Carlsson, M. (2019), Research Arti- cle, submitted to Astronomy and Astrophysics, 2019.

List of non-refereed items:

• Ellerman bomb emission features in He I D3 and He I 10830: observations and modelling Libbrecht, T. (2017), Conference Presentation, 2017. Bibcode: 2017psio.confE..51L

• GCN #16253,#16278,#16290,#16310

Acknowledgements

It seems that it is about to happen, I will be actually defending this PhD! What an adventurous, exhilarating, challenging, at times frustrating, but all-in-all great ride it has been. The path of my PhD has certainly not been straightforward, with the usual obstacles along the way and a rather messy start. One person in our institute that has always been there for me is Jaime. Thank you for your supervision, inspiration, and friendship. I have truly learned a lot from you and I have always been able to count on your help. I am proud to be your first student and perhaps your loudest and most obnoxious one ;-) I remember the daily “fights” about the IRIS-target in La Palma for example. Let’s continue the great collaboration and the joyful criticism of color-choices. Many thanks also to Jorrit who offered a lot of helium expertise, but from the o-so-clean theoretical perspective. Who knows?! Will I continue to cheat on observations with simulations even more? Thanks also for organizing sausage and poker events and it has been fun to hang out together with your kids as well. Let’s see if I continue to like you when you become my postdoc boss ;-) I also like to mention the others in my thesis committee. Matthew, thanks for being a very supportive mentor, it was really good knowing that you were there when needed! Thanks to Jesper as director of grad studies, and Göran for proposing this PhD topic in the first place. In the department I want to thank Rocio for offering administrative support and for being such a wonderful person who makes everyone feel welcome and at home! I’d also like to thank my co-authors and collaborators: Jayant, Jaime, Jorrit, Andrés, Sanja, Hiva, Johan and Viggo. You each had your own indispensable contribution to the papers and to my PhD. Thanks also to Phil Judge for inviting me to Boulder where we had many interesting and deep discussions about helium and also great food and beers in the city. The institute for Solar Physics has been a great working environment because of the wonderful people. I had two awesome office mates Hiva (good that people couldn’t usually hear or see us) and Sanja (the same actually ;-) ). Thank you Hiva for being a great friend and support through some of the more difficult times in the PhD, and for all the great trips and fun stuff that we did together. I miss you! Sanja, thanks for coping with me always chit-chatting, for your great advice on motherhood and kids and for being such a funny and sarcastic friend. Johan, why did you leave us? I am still upset! We have had the greatest times all along our PhDs drinking bears, traveling, cooking dinners, and having so much fun. Now that you sold your soul and are becoming rich, you can certainly hop on a train/plane to visit us very often, beers (or Jäger!) on you! ;-) (Six-packs long forgotten.) Thanks Sara for the great long chats, coffee dates and the shopping spreads, thanks Jayant for inviting us to your wedding in Udaipur - a truly unique experience - and for the fun times in Stockholm, thanks to Carolina and João: immediately some great food and desserts come to mind but also thanks for your companionship, thanks to Dan for the Swedish summary and for enlight- ening us with Swedish culture, thanks to Mats for “anal”izing Paper III in the thesis and thanks to the entire Solar group (past and present) for making working days enjoyable. Thanks also to the PhD students in the Astronomy Department: Ruben, Matteo, Emir, Katia, Maria, Johannes, Saghar, and many more: thanks for all the fun activities such as camping and kayaking (that was certainly Rubens favorite ;-) ), winter BBQ, ice-bathing, board games, pub crawls, and so on... During my PhD, I have been in parental leave: a wonderful time in which I have seen life from a different perspective. The greatest people have accom- panied me in this experience continue to be great friends, thank you Helena, Emir, Leo, Laure, Vincent, Anton, Susanne, Matthias, Helena, Nathalie, Gus and many more moms, dads and babies that I have met and hung out with. Thanks also to Katia for throwing me an awesome baby-shower. I would also like to thank some Belgian friends: “Nerdvana” and “Blond- girls”, you know who you are! Bedankt voor de vele bezoekjes aan Stockholm (Glenn en Enimie strijden om de hoofdprijs: wie kwam het meest op bezoek?) en om nog steeds zo’n goede vrienden te zijn ook al wonen wij niet meer in België. Family is very important to me and I miss mine so dearly: mama, papa en Robbe, bedankt om mij altijd te steunen, voor alle attenties, om zo vaak op bezoek te komen, en voor alle gezellige feestdagen wanneer wij in België zijn. Zoals jullie weten woon ik graag in Stockholm maar hoop ik toch over enkele jaren terug te keren naar België zodat we weer wat meer tijd samen kunnen doorbrengen. Oma, ik wou ook dat ik wat dichter woonde zodat ik vaker op bezoek kon komen, ik mis jou. Bedankt ook Sonja, Peter en Bart om mij helemaal op te nemen in de familie en omdat jullie zoveel voor ons doen. Last but most important of all: Koen, zonder jou was ik zelfs al helemaal nooit in Stockholm beland, en had ik mijn PhD nooit kunnen waarmaken. Jouw overweldigende steun, rust en vertrouwen zijn onmisbaar voor mij! Be- dankt voor de zalige jaren die we hier in Stockholm al gehad hebben en ik kijk uit naar wat er ons nog te wachten staat! June you are undoubtly the best that I produced during my PhD. Ik wou dat ik je de inzichten en levenslessen kon meegeven die ik in de voorbije jaren zelf verzameld heb, maar zoals iedereen zal je zelf je eigen pad zoeken en je eigen keuzes maken. Ik ben nu al heel trots op jou!

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