New insights from the millimeter continuum

A multiwavelength approach to solar chromospheric heating

New insights from the millimeter continuum

João Manuel da Silva Santos

João Manuel da Silva Santos A m ul tiw av elengt h appr oac h to solar c hr omospheric hea ting

Department of Astronomy

ISBN 978-91-7911-382-7

A multiwavelength approach to solar chromospheric heating

New insights from the millimeter continuum

João Manuel da Silva Santos

Academic dissertation for the Degree of Doctor of Philosophy in Astronomy at Stockholm University to be publicly defended on Thursday 21 January 2021 at 13.00 in sal FR4, AlbaNova universitetscentrum, Roslagstullsbacken 21 and online via Zoom, public link is available at the department website.

Abstract

The chromosphere is an intermediate layer of the Sun's atmosphere where radiative equilibrium breaks down. The standard chromospheric diagnostics such as the Mg II h and k and Ca II H and K spectral lines are formed under nonlocal thermodynamic equilibrium (NLTE) and they are only partially sensitive to the local conditions. Consequently, the interpretation of their profiles is not straightforward. In contrast, millimeter (mm) continuum radiation is produced by thermal free-free collisional interactions in the chromosphere under most solar conditions, and the observed brightness temperatures are better proxies for plasma temperatures. Observations at these long wavelengths have been recently enabled thanks to the Atacama Large Millimeter/submillimeter Array (ALMA), but the Sun remains largely unexplored in this spectral range.

In this thesis I explore the diagnostic potential of the mm continuum to study the solar chromosphere using inversions and radiation-magnetohydrodynamics (r-MHD) simulations. In particular, this work takes an unprecedented look at solar active-regions in the mm using some of the first solar ALMA observations.

In Paper I, we investigated whether the mm continuum helps to constrain temperatures in NLTE inversions of the MgII and CaII resonance lines using synthetic data from a 3D r-MHD simulation. In Paper II, we applied the same inversion technique to observational data in order to constrain temperature and microturbulence in plage, and we detected signatures of wave heating in coordinated observations with the IRIS satellite. In Paper III, we reported the first results of a comprehensive effort to characterize the visibility of small-scale heating events in an active-region using multiwavelength observations from the mm to the extreme-ultraviolet. We detected multiple, dynamic, transient brightenings -- we called them "millimeter bursts", and we investigated magnetic reconnection using a simulation.

This thesis shows that ALMA offers a complementary spectral diagnostic to the existing ones at visible and ultraviolet wavelengths and it underscores the importance of mm continuum observations for constraining models of the solar atmosphere.

Keywords: Sun, chromosphere, radiative transfer, waves, reconnection.

Stockholm 2020

http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-186959

ISBN 978-91-7911-382-7 ISBN 978-91-7911-383-4

Department of Astronomy

Stockholm University, 106 91 Stockholm

A MULTIWAVELENGTH APPROACH TO SOLAR CHROMOSPHERIC HEATING

João Manuel da Silva Santos

A multiwavelength approach to solar chromospheric heating

New insights from the millimeter continuum

João Manuel da Silva Santos

©João Manuel da Silva Santos, Stockholm University 2020 ISBN print 978-91-7911-382-7

ISBN PDF 978-91-7911-383-4

Cover image: (false-color) ultraviolet image of a solar active region in the core of the Mg II k line taken by the IRIS satellite in April 2019. IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre.

Printed in Sweden by Universitetsservice US-AB, Stockholm 2020

List of included papers

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

PAPER I: Temperature constraints from inversions of synthetic solar optical, UV, and radio spectra

da Silva Santos, J. M., de la Cruz Rodríguez, J. & Leenaarts, J., Astronomy & Astrophysics, 620, A124, (2018).

DOI: https://doi.org/10.1051/0004-6361/201833664

PAPER II: The multi-thermal chromosphere. Inversions of ALMA and IRIS data

da Silva Santos, J. M., de la Cruz Rodríguez, J., Leenaarts, J., Chintzoglou, G., De Pontieu, B., Wedemeyer, S. & Szydlarski, M., Astronomy & Astrophysics, 634, A56, (2020).

DOI: https://doi.org/10.1051/0004-6361/201937117

PAPER III: ALMA observations of transient heating in a solar active re- gion

da Silva Santos, J. M., de la Cruz Rodríguez, J., White, S. M., Leenaarts, J., Vissers, G. J. M. & Hansteen, V. H., Astronomy &

Astrophysics, 643, A41, (2020).

DOI: https://doi.org/10.1051/0004-6361/202038755

Reprints are made with permission from Astronomy & Astrophysics, © ESO

Author’s contribution

PAPER I: I performed all the calculations and analysis. I wrote most of the text and made all the figures. JdLC and JL participated in the discussion.

PAPER II: I coaligned the Hinode and IRIS rasters, prepared the data cubes for the inversions and performed all the calculations and analysis.

I wrote most of the text and made all the figures. JdLC and JL provided comments. BDP is the principal investigator of the ALMA proposal. SW and MS reduced the ALMA data. GC coaligned the IRIS and ALMA data. All coauthors participated in the discussion.

PAPER III: I proposed the project and wrote the ALMA proposal with the help of JdLC, JL, GJMV and SMW. I coaligned the ALMA and SDO data and performed all the analysis. I wrote most of the text and made all of the figures. JdLC and JL provided comments.

SMW reduced the ALMA data. GJMV helped to process the AIA

data and run the Ellerman Bomb detection code. VHH provided

the r-MHD simulation snapshot. JdLC calculated the opacity

of the millimeter continuum in the simulation. All coauthors

participated in the discussion.

This thesis

Reuse of material from licentiate thesis of the author

This thesis is partly based on the author’s licentiate thesis (defended on Novem- ber 15, 2018), but the text has been extensively revised, updated, and restruc- tured: chapter 1 has been expanded and split into two (chapters 1 and 2 here);

chapter 2 has been expanded (chapter 3 here), and chapter 3 has been expanded and split into two (chapters 4 and 5 here). The contribution from the licentiate is less than 50% in chapters 1–5, while chapters 6, 7, and 8 are entirely new. Of the papers included in this thesis, only Paper I was part of the licentiate.

Figures

All figures in this thesis are original and the data sources are given in the text.

Formal acknowledgments

The Institute for Solar Physics is a national research infrastructure under the

Swedish Research Council. It is managed as an independent institute associated

with Stockholm University through its Department of Astronomy. The SST

is operated on the island of La Palma by the Institute for Solar Physics in

the Spanish Observatorio del Roque de los Muchachos of the Instituto de

Astrofísica de Canarias. This thesis was possible thanks to the resources

provided by the Swedish National Infrastructure for Computing (SNIC) at the

High Performance Computing Center North (HPC2N) at Umeå University, and

the National Supercomputer Centre (NSC) at Linköping University. I gratefully

acknowledge financial support from grants of the Swedish Research Council

(2015-03994), the Swedish National Space Agency (128/15) and the Knut och

Alice Wallenberg foundation (2016.0019). This thesis has made extensive use

of NASA’s Astrophysics Data System (ADS) Bibliographic Services and the

Python libraries matplotlib and seaborn for figure production and astropy

and sunpy for handling of astronomical data. Further acknowledgments are

given at the end.

Contents

List of included papers i

Author’s contribution iii

This thesis v

Abbreviations ix

List of Figures x

1 Introduction 11

1.1 The solar atmosphere . . . . 12

1.1.1 General properties . . . . 12

1.1.2 Scale heights . . . . 14

1.2 The chromospheric heating paradigms . . . . 15

1.3 The case for the chromosphere . . . . 17

1.4 Outline . . . . 19

2 Radiative transfer 21 2.1 Absorption and emission . . . 21

2.2 The radiative transfer equation . . . . 23

2.3 To LTE or to NLTE . . . . 24

2.4 Complete and partial frequency redistribution . . . . 26

2.5 Spectral line formation . . . . 27

2.6 Polarization in spectral lines . . . . 29

3 Radiative diagnostics 33 3.1 The visible and infrared Ca

lines . . . . 34

3.2 The ultraviolet Mg

lines . . . . 38

3.3 Optically thin lines . . . . 42

3.4 The millimeter/submillimeter continuum . . . . 45

3.4.1 Formation properties . . . . 45

3.4.2 Observations and results . . . . 48

3.5 Observatories . . . . 50

3.5.1 The Solar Dynamics Observatory (SDO) . . . . 50

3.5.2 The Interface Region Imaging Spectrograph (IRIS) . . . . 54

3.5.3 The Solar Optical Telescope (SOT) on board Hinode . . . . . 58

3.5.4 The Atacama Large Millimeter/submillimeter Array (ALMA) 59

4 Inversions 63

4.1 Depth-stratified inversions . . . . 65

4.1.1 The Levenberg-Marquardt algorithm . . . . 68

4.1.2 Response functions . . . . 70

4.1.3 Regularization . . . 71

4.2 Differential Emission Measure analysis . . . . 74

5 Simulations 77 6 Summary of papers 83 6.1 Paper I . . . . 83

6.2 Paper II . . . . 84

6.3 Paper III . . . . 85

7 Future perspectives 87

8 Sammanfattning 89

9 Acknowledgments 91

References xciii

Abbreviations

AD Ambipolar diffusion

AIA Atmospheric Imaging Assembly

ALMA Atacama Large Millimeter/submillimeter Array

AR Active-region

CF Contribution function

DEM Differential emission measure

EB Ellerman bomb

EM Emission measure

EUV Extreme-ultraviolet

FUV Far-ultraviolet

HMI Helioseismic and Magnetic Imager IRIS Interface Region Imaging Spectrograph

LM Levenberg-Marquardt

LOS Line-of-sight

LTE Local thermodynamic equilibrium

ME Milne-Eddington

MF Microflare

MHD Magnetohydrodynamics

NEQ Nonequilibrium

NF Nanoflare

NLTE Nonlocal thermodynamic equilibrium

NUV Near-ultraviolet

PWV Precipitable water vapor

QS Quiet-Sun

RF Response function

RTE Radiative transfer equation SDO Solar Dynamics Observatory

SJI Slit-jaw imager

SOT Solar Optical Telescope

SST Swedish Solar Telescope

STiC STockholm inversion Code

TP Total power

UV Ultraviolet

UVB Ultraviolet burst

List of Figures

1.1 Full-disk images of the Sun taken by SDO and ALMA. . . 13

1.2 Temperature and mass density in the ALC7 model. . . 15

1.3 SST/CRISP Hα filtergram of an active region in the Sun. . . . 18

2.1 Spectral synthesis in LTE vs NLTE. . . 25

2.2 Near-UV spectrum of the QS at the disk center. . . 28

2.3 Synthetic stokes vector for the Fe

6173 Å line. . . . 31

3.1 The visible quiet solar spectrum around the Ca

lines. . . 35

3.2 SST/CHROMIS observations of an active region. . . . 37

3.3 Average quiet-Sun NUV spectrum observed with IRIS. . . 39

3.4 The effect of velocity gradients in the Mg

k line. . . 40

3.5 IRIS observations of AR 12738 in April 2019. . . . 41

3.6 Synthetic solar EUV spectrum computed using ChiantiPy. . . 43

3.7 IRIS and ALMA observations of AR 12470 in December 2015. 48 3.8 Response functions of the SDO/AIA channels. . . . 51

3.9 Wavelength response functions of the SDO/AIA UV channels. 51 3.10 SDO/AIA and SDO/HMI diagnostic images of the Sun. . . 53

3.11 IRIS spectra and slit-jaw images of AR 11974. . . 55

3.12 IRIS raster scans of AR 11974. . . . 57

3.13 Hinode raster scan of a plage region. . . 58

3.14 Transmission curves of the atmosphere at the ALMA site. . . . 60

3.15 ALMA Band 6 and Band 3 brightness temperature maps of solar active regions. . . 62

4.1 Step-by-step diagram of a typical inversion scheme. . . 64

4.2 Example of an inversion of IRIS Mg

h and k lines. . . . 67

4.3 Response functions to temperature perturbations in the ALC7. 71 4.4 Influence of regularization weights on the inversions. . . 73

4.5 Differential emission measure templates. . . 75

5.1 Formation of the mm continuum in a 3D Bifrost atmosphere. 79

5.2 Synthetic 3 mm continuum from a 3D flux emergence simulation. 81

1. Introduction

A pproximately 4.6 billion years ago, the gravitational collapse of a molec- ular cloud gave birth to a fairly ordinary, yet fascinating star – the Sun.

Its energy source lies hidden under thick layers of plasma, so opaque that a photon generated in the core will collide and be re-emitted countless times before escaping some tens of thousands of years later. Only the surface layers of the Sun – the solar atmosphere – can be directly observed, and one of them – the chromosphere – is the main focus of this thesis.

The second half of the twentieth century witnessed tremendous progress in our understanding of the Sun’s atmosphere owing to advances in instrumentation and observing techniques. The popularization of adaptive optics for ground- based observations in the optical and infrared and the launch of space telescopes that continuously monitor the Sun for long periods and offer a seeing-free view at shorter wavelengths in the ultraviolet (UV) and X-rays have played a pivotal role. The ever-increasing computing power has allowed to process larger volumes of data and simulate the physical mechanisms that are believed to take place in the Sun’s atmosphere in soaring detail and complexity.

Nonetheless, we lack a definitive explanation for many aspects of the structure, dynamics, and coupling of the different atmospheric layers at a fundamental level. In this thesis I use the largely unexplored millimeter (mm) wavelength range to shed light on some open problems in solar physics.

This thesis demonstrates the diagnostic potential of the mm continuum for the solar chromosphere. This had been investigated using radiation-magnetohy- drodynamics (r-MHD) simulations of quiet solar conditions (Wedemeyer-Böhm et al. 2007; Loukitcheva et al. 2015, 2017a) and only recently in more active conditions (Martínez-Sykora et al. 2020a). Here I discuss the diagnostic value of the mm continuum in the framework of inversions and through the use of some of the first interferometric maps of the Sun provided by the Atacama Large Millimeter/submillimeter Array (ALMA). The focus is on active-regions, which have been scarcely investigated at mm wavelengths both from the observational and modeling standpoints. The synergies with other observatories operating at higher frequencies are emphasized.

In this chapter, I provide an overview of the solar atmosphere and build the

case for the chromosphere. Section 1.4 outlines the structure of this thesis and

summarizes the topics that we have addressed in the included papers.

1.1 The solar atmosphere

The solar atmosphere is defined as the part of the Sun from where photons escape into space and it is essentially divided into four layers: the photosphere, the chromosphere, the transition region and the corona. In the next section I provide a short overview of their general properties (see e.g., Aschwanden 2005; Priest 2014, for a more comprehensive review).

1.1.1 General properties

The photosphere is a few hundred-km thin "shell" that emits most of the Sun’s visible light. Its appearance is characterized by the ubiquitous granulation pattern made not only of small convection cells (or granules) with a typical diameter of 1 Mm, but also of larger structures such as mesogranules and super- granules with sizes up to 70 Mm. It also harbours magnetic field concentrations such as faculae, pores, and sunspots that appear in greater numbers at the maximum of the eleven-year solar cycle.

The chromosphere is a more rarefied and warped layer that is known for its fibrils at the disk and prominences and spicules at the limb. The latter are jet-like structures that can reach up to 5 Mm in height and last 3-10 min (for the type-I) or up to 10 Mm but lasting only 10-180 s (for the type-II), and up to 40 Mm for the macrospicules. Plage regions are bright chromospheric patches associated with concentrations of mostly unipolar and nearly vertical magnetic fields where coronal loops are rooted. The chromosphere is partially ionized.

The transition region is where the plasma is heated (or cooled) to (or from) coronal temperatures over a very short height range (. 100km), and it is visible at UV and centimeter (cm) wavelengths.

The corona can be identified as the million-kelvin-hot plasma forming loops up to 700 Mm long and also composing the solar wind that streams through the solar system. Its visibility is optimal from space in the X-ray and extreme-UV (EUV) wavelength range. Coronal holes appear as voids in the coronal emission since they contain cooler and less dense gas. The corona is fully ionized.

The particle density decreases exponentially from the bottom to the top of

the atmosphere; typical values are 10

m

in the photosphere, 10

m

in

the chromosphere, 10

m

in the transition region, and 10

m

at a height

of one solar radius. The magnetic field strength also decreases with height,

but much more slowly. Particularly in active-regions, radio observations have

shown that magnetic field strengths can be of the order of 1−2 kG in the corona

(e.g., White & Kundu 1997; Brosius & White 2006), which is of the order

of photospheric values. The plasma motions become increasingly dominated

by the magnetic pressure with height both in the quiet-Sun (QS) and in the

active-Sun. The ratio of gas to magnetic pressures is called plasma β .

Figure 1.1: Full-disk images of the Sun taken by SDO and ALMA on April 13, 2019.

Left: SDO/HMI 6173 Å continuum; middle: ALMA Band 6 brightness temperature (single-dish, range: 5200-7400 K); right: SDO/AIA Fe

171 Å.

Everything we have learned about the Sun’s atmosphere was through the study of the properties of the light that it emits. Comparison of images taken at different wavelengths provides interesting insight into the structuring of its layers. Figure 1.1 shows a collage of three images of the solar disk that show the photosphere, chromosphere and corona at the same instant. The image

taken at mm wavelengths shows an inhomogeneous temperature structure across the solar disk; higher temperatures coincide with active-regions (ARs) where the hot coronal loops are rooted, whereas lower temperatures are found in the internetwork. The EUV image shows emission in the Fe

line at logT ∼ 5.8 K (see Section 3.5.1). Because we observe higher layers of the atmosphere towards the solar limb, the figure shows that there is a positive temperature gradient in the chromosphere as evidenced by the limb-brightening effect (see Section 3.4) in contrast with the limb-darkened photosphere.

The existence of the chromosphere and corona in the Sun and other cool stars is a fundamental problem as it implies that large amounts of nonthermal energy are needed to break the radiative equilibrium (Hall 2008). The source of this extra heating is unknown but a few different candidates have been proposed (see Section 1.2). Most of this energy is absorbed by neutral hydrogen atoms in the chromosphere because of their large first ionization energy and converted into hydrogen ionization like a "thermal buffer" (Ayres et al. 2009).

The freed electrons will contribute to collisional excitation and subsequent radiative cooling in spectral lines of H, Mg

, Ca

, and Fe

on short time scales (e.g., Anderson & Athay 1989; Carlsson & Leenaarts 2012). For this reason, temperature rises slowly over several pressure scale heights in the

"chromospheric plateau" in the QS (e.g., Avrett & Loeser 2008). When the plasma becomes fully ionized, the temperature skyrockets to a million degrees in the corona that cools off mainly at EUV wavelengths.

1ALMA project 2018.1.01518.S at https://almascience.nrao.edu/asax/

1.1.2 Scale heights

In order to understand the structure of the solar atmosphere it is instructive to introduce the concept of hydrostatic equilibrium – the gravitational force is solely balanced by the pressure gradient. Assuming spherical symmetry, the hydrostatic equilibrium equation for a static star can be written as

dP

dr = −ρg, (1.1)

where P is the gas pressure, g = GM/r

is the gravitational acceleration on the material of mass density ρ at a given radius r, and G is the gravitational constant. The Ideal Gas Law can be used as an equation of state, that is, to relate density and pressure,

P = ρ kT

¯µ , (1.2)

where k is the Boltzmann constant, T is the temperature and ¯µ is the mean mass of all particles in the gas. Dividing Eq. 1.2 by Eq. 1.1,

P

|dP/dr| = kT

¯µg = kTr

¯µGM ≡ H, (1.3)

defines a characteristic pressure scale height H. For the Sun, setting ¯µ ∼ m

, where m

is the proton mass, and T = T

≈ 5780 K in Eq. 1.3 implies that H ≈ 180 km. The effective temperature T

is defined as the temperature a black body

would need to have to radiate the same amount of energy.

If we assume the temperature to be constant and substitute Eq. 1.2 and Eq. 1.3 in Eq. 1.1, the solution of the differential equation in a 1D Cartesian system for the height z is P(z) ∝ e

. An analogous expression can be obtained for the density. Pressure and density drop very rapidly with height in such a way that the temperature scale height is much larger than the pressure scale height (see Fig. 1.2). Because of the relative thinness of the atmospheric scale height, the emergent spectrum of the Sun can, in general, be modeled in terms of a one-dimensional (1D) plane atmosphere to good approximation.

The semi-empirical ALC7 model of the average QS EUV spectra (Avrett &

Loeser 2008) is valid for a static, non-magnetic atmosphere where the thermo- dynamic parameters vary only with height as described in the above. Figure 1.2 shows the stratification of temperature and mass density in ALC7. Note that given the nonlinear mapping between intensities and plasma parameters, models of average spectra do not necessarily correspond to "mean atmospheres", and the Sun shows more complicated, time-dependent stratifications (e.g., Carlsson

& Stein 2002a). Alternatively, r-MHD simulations can provide synthetic spectra that can be compared to observations (see Chapter 5).

2A black body is an idealized source of radiation that does not reflect any light and whose properties only depend on a characteristic temperature set by the thermodynamic equilibrium.

Figure 1.2: Temperature (solid line) and mass density (dashed line) in the ALC7 model (Avrett & Loeser 2008) of the quiet-Sun.

1.2 The chromospheric heating paradigms

While the photosphere has been scrutinized in detail and it is now fairly well characterized (see review by Nordlund et al. 2009), the fine thermal structure and dynamics of the chromosphere – and its heating problem, have been a matter of debate for a long time (e.g., Athay 1976). The total energy losses in the QS chromosphere have been estimated to be of the order of 4 × 10

ergcm

s

(Withbroe & Noyes 1977) or 1.4 × 10

ergcm

s

(Anderson & Athay 1989), whereas the coronal losses are only a few percent of those; AR requirements are even larger (e.g., Priest 2014). Open questions remain as to where the energy deposited in the chromosphere comes from, what impact it has on the observed dynamics, and how it relates to coronal heating at all. This is a complex problem since both layers are not isolated from each other but continuously exchange mass and energy (see review by Parnell & De Moortel 2012). In this thesis I keep the discussion mostly at the chromospheric level.

Most of the solar radiation is emitted from the QS, whereas the active-Sun is a much more irregular and unpredictable component of the total energy output.

Therefore, explaining why the entire solar disk and beyond is covered with EUV emission from highly ionized elements (see Fig. 1.1) ultimately requires that we understand the QS.

Early studies proposed that acoustic shocks could be the main heating

source (Biermann 1946; Schwarzschild 1948), but over the years this has been

questioned (Kalkofen 2007, and references therein). This was investigated in

detail in a series of papers using 1D radiation-hydrodynamic simulations with

the RADYN code (Carlsson & Stein 1992, 1994, 1995, 1997, 2002a) that showed

that the Ca

H and K lines could emerge from a generally cool chromosphere bumped by shock waves in contrast with the warm-chromosphere paradigm.

However, these models were not able to explain the UV continuum, which hinted for a missing heating mechanism possibly related to magnetic fields (Carlsson & Stein 2002b). Later observations have shown that acoustic waves are insufficient on their own (e.g., Fossum & Carlsson 2006; Vecchio et al.

2009; Bello González et al. 2009, 2010; Sobotka et al. 2016).

It can be shown that MHD waves are solutions of the MHD equations for a uniform plasma (e.g., Goossens 2003). Photospheric granular motions in weakly magnetized regions generate acoustic waves that propagate upwards and become MHD waves in the chromosphere, while in strong magnetic concentrations such waves can be generated directly (e.g., Priest 2014); eventually, "slow- mode" and "fast-mode" magnetoacoustic waves dissipate their energy in shocks in the chromosphere, whereas the (transverse) "intermediate-mode" (Alfvén waves) propagates upward to the corona. Edwin & Roberts (1983) showed that cylindrical flux tubes with different internal and external conditions give rise to kink and sausage modes. Both compressible and incompressible waves are ubiquitous at chromospheric temperatures (see review by Jess et al. 2015).

Magnetoacoustic shocks have been observed in plage regions (e.g., Hansteen et al. 2006; De Pontieu et al. 2007) and may account for a significant fraction of the radiative losses there (Sobotka et al. 2016; Abbasvand et al. 2020a). The recent study by Abbasvand et al. (2020b) showed that the acoustic energy flux balances the radiative losses in the mid chromosphere but it is insufficient in the upper chromosphere, so different energy sources may be important in different layers of the atmosphere. We discuss wave heating in plage in Paper II.

Another class of heating mechanisms includes magnetic reconnection phe- nomena. This can be intuitively understood from the buildup of magnetic tensions that lead to breaking and reconnection of field lines converting mag- netic energy into heat, turbulence, and plasma kinetic energy. This can occur in a variety of ways in the solar atmosphere; for example, photospheric granular motions drag magnetic field lines around and cause twists and braids that lead to Ohmic dissipation in current sheets; it can also occur from the emergence of new magnetic flux tubes into the preexisting ambient field, or as the result of instabilities (e.g., Priest 2014).

Bi-directional jets in small-scale explosive events (Innes et al. 1997), mi- croflares (Hong et al. 2016), Ellerman bombs (Watanabe et al. 2011), and ultraviolet bursts (Peter et al. 2014) are perhaps some of the clearest evidence for magnetic reconnection in ARs. The recent work by Joshi et al. (2020) suggests that the QS is likewise teeming with small-scale reconnection events.

Ellerman bombs (EBs) are brightenings in the wings of the Balmer lines

and UV continuum (e.g., Rutten et al. 2013), whereas ultraviolet bursts (UVBs)

are bright in transition region diagnostics (e.g., Young et al. 2018). Both have

been interpreted as the result of magnetic reconnection but they seldom appear together (e.g., Chen et al. 2019). We discuss EBs and UVBs in Paper III.

1.3 The case for the chromosphere

The reasons why there are still so many open questions about the chromosphere since its discovery by Lockyer (1868) lie in the complicated physics that it entails and the limitations associated to the spectral diagnostics available until recently (Judge 2006; Rutten 2007; de la Cruz Rodríguez & van Noort 2017).

However, its study is important because the chromosphere is not only testing ground for plasma physics, but it is also the source of all nonthermal energy and mass that reaches the corona and is eventually carried away by the solar wind that interacts with Earth’s magnetosphere (Schwenn 2006).

On the theoretical side, the chromosphere marks a transition in several domains of physics: from LTE to non-LTE, from high to low plasma β , from single- to multi-fluid MHD, and from complete to partial frequency redistribu- tion, which complicates the interpretation of observations and makes modeling computationally intensive (see review by Carlsson et al. 2019). Usually some simplifications have to be made for tractability.

On the observational side, there had been a paucity of continuous, high- quality spectroscopic data in the near-UV (NUV), where two important Mg

lines can be found (see Section 3.2), until the launch of the Interface Region Imaging Spectrograph (IRIS, De Pontieu et al. 2014) in 2013 (see Section 3.5.2).

We have also lacked sufficient spectral/temporal resolution in the optical in some of the strongest Fraunhofer lines such as the Ca

H, K lines formed in the chromosphere (see Section3.1). This has improved with the SST/CHROMIS instrument that observes the H and K lines at approximately λ /D ∼ 0.08"

resolution (Scharmer 2017). But there is still a problem of signal-to-noise ratio in the polarimetric signals, principally in linear polarization, for instance, in Ca

8542 Å (e.g., Pietrow et al. 2020); they are often too weak outside of strongly magnetized regions and cannot be as widely used to infer the magnetic field vector in the chromosphere as it is usually done for the photosphere. This is relevant because magnetic fields are believed to play a role in enhancing atmospheric heating (e.g., Narain & Ulmschneider 1996; Carlsson et al. 2019) as shown by the correlation between magnetic field strength and chromospheric emission in the Mg

and Ca

lines (e.g., Schmit et al. 2015; Leenaarts et al.

2018) and mm continuum (e.g., Loukitcheva et al. 2009).

The continuum at mm wavelengths is conveniently formed in the chro- mosphere with an LTE source function (see Section 3.4), and the intensities are nearly linear functions of electron temperature. However, observations at these long-wavelength (e.g., White et al. 1992, 2006; Bastian et al. 1993a;

Loukitcheva et al. 2014; Iwai & Shimojo 2015; Iwai et al. 2016) have had insuf-

Figure 1.3: SST/CRISP Hα filtergram of an active region in the Sun in May 2017.

The FOV is 52"×48" and the pixel scale is 0.057". Credit: J. M. da Silva Santos, G. J.

M. Vissers, and H. Pazira.

ficient resolution (worse than ∼ 5", or 3500 km) for probing the chromosphere for which there is evidence of sub-structures at scales of 100 km and possibly shorter, for example, in spicules (de Pontieu et al. 2007), ultraviolet-bursts (Pe- ter et al. 2014), vortexes (Wedemeyer-Böhm et al. 2012), penumbral microjets (Katsukawa et al. 2007), and canopy fibrils (Kianfar et al. 2020), among others (see Fig. 1.3).

Naturally, mm observations have been largely ignored for decades until the

advent the ALMA interferometer that started to observe the Sun in 2016, albeit

not at the fullest of its capabilities (see Section 3.5.4). However, the applicability

of such data is vast: from waves, to magnetic reconnection, molecular line

observations, relativistic particles, etc., so the potential is large (see review by

Wedemeyer et al. 2016). This thesis makes a strong case for the need to observe

the Sun at radio wavelengths in high-resolution and paired with other spectral

diagnostics in the ultraviolet and visible range.

1.4 Outline

The main goal of this thesis is to better understand the physical processes that take place in the solar chromosphere using ALMA observations. In that process we further improve our understanding of the formation of the mm continuum itself, especially in ARs which are not as well-researched as the QS at these wavelengths.

The first question that this thesis undertakes is how well the plasma pa- rameters can be inferred (or inverted) from observations of spectral lines and continua at different wavelengths (Paper I). We investigated whether the mm continuum improves the accuracy of inverted temperatures using simulated data.

We then applied the same inversion techniques to real NUV and mm obser- vations (2016.1.00050.S, PI: B. De Pontieu) in Paper II. The main goal was to better constrain the atmosphere in a plage region, which are places known for enhanced temperatures and microturbulence. We also discuss the evidence for magnetoacoustic shocks and nonequilibrium effects shown by the observations.

Paper III makes use of some of the ALMA data that was taken for the project 2018.1.01518.S (PI: J. M. da Silva Santos). The goal was to investi- gate Ellerman bombs, ultraviolet bursts and other AR transient brightenings using mm observations in order to constrain their formation heights and de- termine their role in the energy balance in the chromosphere. We investigate the visibility of small-scale reconnection in the chromosphere through mm continuum enhancements using a 3D r-MHD simulation snapshot.

The remainder of this thesis is organized as follows: in Chapter 2, I provide

a general overview of radiative transfer concepts that are needed to interpret

observations, while a more detailed description of the formation mechanisms

of selected chromospheric diagnostics is given in Chapter 3; a brief overview

of the instrumental details of the observatories that we used is also given in

Chapter 3; in Chapter 4, I present the inversion methods that we used in order

to interpret the observations; in Chapter 5, I discuss MHD modeling of the

chromosphere and present the simulations that we used for comparisons with

observations; I provide a summary of Papers I–III in Chapter 6; I draw some

future perspectives in Chapter 7, and I provide a plain-language summary in

Swedish in Chapter 8.

2. Radiative transfer

Understanding the observed phenomena in the Sun requires knowledge of the physics involved in the transport of radiation through the atmosphere and how it interacts with matter. In this chapter I provide a brief overview of the basics of radiative transfer based on several textbooks such as Rybicki & Lightman (1986), Böhm-Vitense (1989), del Toro Iniesta (2003), Rutten (2003), and Landi Degl’Innocenti & Landolfi (2004) to which I refer for a more comprehensive explanation of the physics of the solar atmosphere.

2.1 Absorption and emission

The radiation field within a star can be described in terms of the (monochro- matic) specific intensity

I

that represents the energy flow at a given location r along the direction s, per-unit area dA, time dt, frequency dν and solid angle dΩ, which can be mathematically expressed by

I

(r, s, t) = dE

dSdtdνdΩ = dE

dAcosθdtdνdΩ , (2.1)

where the angle θ is defined with respect to the normal n to the area dA, and I

is given in units of [ergs

cm

sr

Hz

]. The intensity is conserved along the ray, i.e. it is independent of the distance to the source because it is defined per solid-angle which itself decreases with inverse squared distance

. This is certainly true in the "vacuum" of space, but in the interior and atmosphere of the Sun I

also depends on the interactions with matter.

There is an infinitesimal probability d p

of a photon with a certain fre- quency being absorbed in a thin slab of thickness ds along the direction s. This is expressed by the linear absorption coefficient α

(in [cm

]),

α

= d p

ds = κ

ρ , (2.2)

where κ

is the (monochromatic) mass extinction coefficient (in [cm

g

]) . The fraction of specific intensity lost due to absorption (or extinction) in the

1Or alternatively: I_λ=Iνc/λ².

2The solid angle Ω of an object whose area A makes an angle θ with the line of sight seen from a distance d is: Ω = A cosθ/d²[sr].

element ds along the ray is:

dI

I

= −d p

= −α

ds, (2.3)

Integrating this equation along the path with total thickness D yields I

(D) = I

(0) exp



− Z

0

α

(s)ds



, (2.4)

with I

(0) being the intensity at the source. The transparency of the slab can be quantified by the quantity inside the exponent, which we call monochromatic optical thickness τ

(D),

τ

(D) = ^Z

0

α

(s)ds. (2.5)

A medium is said to be optically thick at a given frequency if τ

(D) > 1 and optically thin (or transparent) if τ

(D) < 1. For example, the layers immediately above the bottom of the solar photosphere (few hundred kilometers) are optically thin in the near-UV (NUV), visible and near-infrared (NIR), but optically thick in many spectral lines that appear as dark absorption patches superimposed on a continuous spectrum.

The monochromatic optical path is defined as dτ

(s) = α

(s)ds.

The sources of opacity (absorption and scattering) depend on how pho- tons interact with matter, namely atoms, ions, and free electrons. In stellar atmospheres, the four primary sources of opacity are:

• Bound-bound transitions: when an electron in a atom or ion makes a transition between two orbitals; high opacity at discrete transition wavelengths (or line opacity).

• Bound-free absorption: a photon is capable of removing an electron from the atom; it is a source of continuum opacity.

• Free-free absorption: a free electron is accelerated by the Coulomb field of nearby ion absorbing a photon; this is referred to as (inverse) bremsstrahlung and it is a source of continuum opacity.

• Elastic processes: a photon is scattered by a free electron (Thomson scattering) or by bound electrons in an atom or molecule (Rayleigh scattering); these are sources of continuum opacity.

Free-free processes are further discussed in Section 3.4 because they are important for understanding mm/sub-mm continuum radiation.

There may also be sources of radiation within the path ds that add to the in-

tensity (the inverse of the aforementioned absorption processes). These are quan-

tified by the spontaneous emission coefficient j

in [ergcm

s

Hz

sr

]

such that dI

(s) = j

(s)ds.

2.2 The radiative transfer equation

Generally one is usually interested in computing the balance between absorption and emission taking into account not only the absorption along the s direction in the atmosphere but also the radiative emissivity. This is described by the radiative transfer equation (RTE) given as follows

dI

ds = j

− α

I

, dI

dτ

= S

− I

, (2.6)

where S

= j

/α

is the source function. Equation 2.6 describes the change of intensity over the optical path dτ. To integrate it along the s direction, we multiply both sides by the integrating factor e

and we derive the formal solution

of the RTE (e.g., Rutten 2003),

I

(D) = I

(0)e

+ Z

0

S

(s)e

dt

(s). (2.7) In an atmosphere where α

and j

are constant, S

does not also depend on the location, therefore we can take it out of the integral in Eq. 2.7 and we simply obtain

I

(D) = I

(0)e

+ S

h 1 − e

i

. (2.8)

In general, I

can be a complicated function of r, ν and n, but given the relative thinness of the atmosphere we can employ the plane-parallel approximation considering only a directional dependence along the z-axis. We define the viewing angle µ = n · z = cosθ between the line of sight and the z-axis such that I

= I

(µ, z). In these terms τ

can then be redefined as the vertical optical depth τ

= − ^R

α

dz for a distant observer located at z = ∞. Then, since we are looking from the outside in, Eq. 2.6 can be rewritten as follows

µ dI

dτ

= I

− S

. (2.9)

The emergent intensity from the horizontal slab as seen from an external ob- server at τ

(∞) = 0 is given by

I

(µ, τ

= 0) = ^Z

0

S

(τ

)e

dτ

/µ, (µ > 0). (2.10) Another useful quantity to define is the mean intensity J

,

J

= 1 4π

Z I

dΩ. (2.11)

1The one that can be written although Sνis not necessarily a known explicit function of τν.

2.3 To LTE or to NLTE

In Section 1.1.2, I described the black body radiation as the energy emitted by an idealized body in the thermodynamic equilibrium (TE) with a characteristic temperature. The energies of particles in a system in the TE follow Maxwell- Boltzmann statistics with a well-defined temperature. Gas and light particles will then come into equilibrium described by a single temperature. In such system, every absorption of a photon is balanced by an emission process. A star cannot ever be in perfect TE because temperature varies with location and there is a net outward flow of energy and matter. However, if we consider that the distance over which temperature significantly changes is large compared to the mean-free-paths, then we can think of the particles as effectively confined to a volume of approximately constant temperature – local thermodynamic equilibrium (LTE).

This is not a bad approximation at least for the matter because the mean- free-path between collisions of hydrogen atoms is several billion times smaller

than the temperature scale height (Section 1.1.2). In LTE, the distribution of populations in different excitation states i and j with energies E

and E

in an atom is given by the Boltzmann equation,

n

n

= g

g

exp



− hν

kT



, (2.12)

where g

and g

are the statistical weights and ν

= (E

−E

)/h is the frequency corresponding to the energy difference. The number density of atoms in the ionization stage I + 1 relative to stage I is given by the Saha equation,

n

n

= 1 n

2U

U

 2πm

kT h



exp 

− χ

kT

 (2.13)

where n

is the electron density, m

is the electron mass, U

and U

are the partition functions of the ionization stages (sum over all excitation states), χ

is the ionization energy, and h is the Planck constant.

For photons, LTE means that the source function equals the Planck function (S

= B

) which is defined by

B

= 2hν

c

1

exp(hν/kT ) − 1 , (2.14)

where c is the speed of light. In the limit of very high temperatures or low frequencies e

≈ 1 + hν/kT , therefore Eq. 2.14 reduces to the Rayleigh- Jeans law,

B

≈ 2ν

kT

c

. (2.15)

1For a photosphere consisting of mainly of neutral hydrogen with density of ρ = 2.1 × 10⁻⁴kgm⁻³the particle mean free path is l ∼ 2 × 10⁻⁴m (Carroll & Ostlie 2007).

Figure 2.1: Spectral synthesis in LTE vs NLTE. Profiles of the Mg

k and triplet lines from the ALC7 model (Fig. 1.2) for a LTE and NLTE level populations.

Equation 2.14 (or 2.15) can be solved for T , which is then called brightness (or radiation) temperature T

(or T

) by equating S

= I

. In more general cases, S

is not necessarily given by the Planck function. For a bound-bound transition between two energy levels with quantum numbers i and j, the line source function is given by

S

= 2hν

c

 n

g

n

g

− 1



, (2.16)

which is generally valid without imposing any equilibrium condition but com- plete redistribution is assumed (see Section 2.4). In LTE, the line source function simplifies to S

= B

if the occupation numbers n

, n

follow Saha- Boltzmann statistics. The problem is then how to compute S

when LTE does not hold – this is referred to as non-LTE (or NLTE).

When the radiative processes cannot be neglected compared to the colli- sional processes which, for example, can occur in a low density gas such as the chromosphere, then the population ratio can be computed explicitly by assuming statistical equilibrium: the radiation field and level populations do not vary with time. The statistical equilibrium equation is given by

N

∑

n

P

− n

∑

P

= 0, (2.17)

where P

≡ R

+C

are the rates at which particles change from level i to j, which include both radiative (R

) and collisional (C

) processes. In a bound- bound or bound-free transition R

depends on the mean intensity. Solving the radiative transfer problem under these circumstances is much more complicated because S

has to be computed explicitly by solving Eq. 2.17, which involves computing the rates, which in turn depend on the radiation field itself. This is a nonlocal problem that has to be solved self-consistently in an iterative way.

Taking scattering into account, the source function (in a two-level atom) can be written as (e.g., Hubeny & Mihalas 2014)

S = (1 − ε) ¯J+ εB, (2.18)

where ¯ J is the frequency-averaged mean intensity and ε is the destruction prob- ability, which depends on the ratio of the collision rates C

to the spontaneous emission A

: if C

> A

, ε → 1 and S

→ B

(LTE); for chromospheric lines ε  1 and S departs significantly from the Planck function (NLTE).

NLTE effects have profound impact in the profiles of the Mg

lines as depicted in Fig. 2.1. They were computed from the ALC7 QS model (see Sec- tion 1.1.2) using the STiC code, and they are plotted in brightness temperature scale. In LTE the line source function has no choice but to follow the monotonic temperature rise through the ALC7 chromosphere, therefore the line appears in emission with a peak T

≈ 6700 K that corresponds to the temperature at an altitude of 2 Mm (see Fig. 1.2). In NLTE, S

is set by ¯ J

and it decouples from temperature at lower heights (see also the appendix of Rutten 2017b); the peak T

is lessened and is lower than the gas temperature (see also Leenaarts et al. 2013a). Observations have shown that profiles with central reversal are the most common ones on disk as a result of NLTE and opacity effects. The formation of the Mg

NUV lines is further discussed in Section 3.2.

2.4 Complete and partial frequency redistribution

For simplicity one assumes complete frequency redistribution (CRD) to obtain Eq. 2.16 for the source function of a bound-bound transition. This means that S

is independent of the angle and frequency. This approximation is valid for almost all lines in the solar spectrum including the ones forming in the photosphere and chromosphere, except for the strongest chromospheric lines.

In partially-coherent scattering, more commonly known as partial frequency

redistribution (PRD), the frequency and direction of the incoming and outgoing

photon during scattering are not random but can be correlated. PRD effects

are important in lines of abundant elements such as Lyα, Lyβ , Mg

h and k,

Ca

H and K, among a few others formed in low density environment where

collisions do not occur frequently enough to destroy coherency (e.g., Linsky

1985; Hubeny & Mihalas 2014).

2.5 Spectral line formation

We known from laboratory experiments that when a gas in a container is heated it produces emission line spectra, whereas a cold gas illuminated from behind from a light source with a continuous spectrum gives an absorption line pattern, just like what we observe on the Sun (Kirchhoff’s Laws).

The formulas given in Section 2.2 can be used to intuitively understand the basics of line formation. We can interpret the first term on the right-hand side of Eq. 2.8 as the amount of light that is left after being absorbed through a path with optical thickness τ

(D), while the second term gives the contribution of radiation absorbed and re-emitted along the path. It follows that when the gas is optically thick (τ

( D)  1): I

≈ S

, and no lines emerge. In the optically thin case (τ

( D)  1): e

≈ 1 − τ

(D), therefore we derive

I

= I

( 0) − [I

( 0) − S

]τ

(D). (2.19) If I

( 0) = 0, which means the gas is not back-lit, then I

= τ

( D)S

. Because the extinction coefficient α

is larger at line frequencies, the intensity will increase accordingly and one expects to see emission lines. This is, for example, the case of the coronal features when observed at the Sun’s limb. If I

( 0) 6= 0 and: (1) I

(0) > S

then the second term on Eq. 2.19 is positive so there is an amount of radiation proportional to τ

(D) that is subtracted from the background originating an absorption line; (2) I

( 0) < S

then there is a positive contribution to the background radiation, so one would see an emission line on top of the I

(0) intensity level.

The solar (visible, NIR and NUV) spectrum is populated with numerous lines that appear in absorption because the source function follows the decreas- ing temperature with height, that is, the intensity coming from deeper layers is larger than the source function for the top layers. However, at wavelengths shorter than ∼1700 Å in the FUV we see a number of strong emission lines coming from the chromosphere and transition region (such as Lyα 1216 Å) because of the high continuous absorption coefficient, while the temperature is increasing outwards (Böhm-Vitense 1989).

Line formation can be more complicated than what I have described here, particularly for the chromospheric lines because the source function does not follow the Planck function, therefore it can decrease while the temperature increases outwards, or it can have a complicated variation in height producing dips and reversals in the line shapes. It may also happen that different compo- nents contribute to the emission/absorption along the line-of-sight (LOS). This is further discussed in Chapter 3.

Apart from the permitted spectral lines by the electric dipole rules, there

are also forbidden lines. Some lines formed in the solar corona such as Fe

5303 Å, Fe

6375 Å , which occur via collisional de-excitation, are forbidden lines because the transitions are very unlikely (Aschwanden 2005).

Several broadening mechanisms are usually convoluted and play a role in explaining the line profiles. The main mechanisms are:

• Natural broadening. The energy of the excited state has some spread associated to the lifetime of the level as stated by the uncertainty principle;

This produces a Lorentz-type profile.

• Doppler broadening. Atoms (or molecules) have a distribution of ve- locities that will result in different frequency shifts in the frame of the observer. If the velocity field is Maxwellian, the Doppler width is given by

∆ν

= ν

c r 2kT

m , (2.20)

where ν

is the rest frequency and m is the mass of an atom.

• Microturbulence. There could also be turbulent velocity fields. They are usually assumed to have a Gaussian distribution with root-mean-square v

, which adds in quadrature to the Doppler broadening (Eq. 2.20) and produces a gaussian profile.

• Collisional. Also known as pressure broadening; it is caused by collisions between gas particles. This produces a Lorentz-type profile.

The combination of these mechanisms produces a Voigt profile with a Gaussian- like core and damped wings (at least in the QS). Other broadening mechanisms include the splitting of energy levels in the presence of an electric field (Stark effect) or a magnetic field (Zeeman effect, see Section 2.6), isotopic broadening, for example, in the Ca

8542 Å line (e.g., Leenaarts et al. 2014), and opacity broadening that lead to complex line profiles (see Sections 3.1, 3.2).

Figure 2.2: Near-UV spectrum of the QS at the disk center recorded by the IRIS slit

shows many absorption lines and two prominent emission lines of Mg

.

2.6 Polarization in spectral lines

The RTE for the monochromatic specific intensity as I have described so far is valid for "unpolarized"

light traveling through isotropic media. If the radiation is partially or fully polarized, namely by the presence of a vector magnetic field, we need to specify the polarization state.

If we think of light as a transverse electromagnetic wave, we can decompose the electric field vector EEE into two orthogonal components E

and E

,

E

= A

cos(wt − φ

),

E

= A

cos(wt − φ

), (2.21) where A

and A

are the amplitudes, φ

and φ

are the phases and w = 2πν is the angular frequency. The Stokes parameters are given by

I

= κ ( hA

i + hA

i) Q

= κ ( hA

i − hA

i) U

= κ h2A

A

cos(φ

− φ

) i

V

= κ h2A

A

sin(φ

− φ

) i,

(2.22)

where the brackets denote time averages and κ is some dimensional quantity.

Stokes Q and U describe linear polarization and V describes circular polariza- tion. The Stokes vector is defined as III = (I

,Q

,U

,V

)

, where the superscript

|

indicates transposition. The different components of the Stokes vector can be measured using spectropolarimetry. Stokes V can be used to measure longitudi- nal fields, while Stokes Q and U are proxies for transverse fields (parallel to the solar surface).

Using the definition of the Stokes vector and continuum optical depth τ

, τ

= −

Z

z₀

α

dz, (2.23)

where α

, is the absorption coefficient of the continuum and the zero point z

is at the observer, so that the integration is performed in the opposite direction of the ray path, Eq. 2.6 can be rewritten as a vector differential equation as given below (e.g., del Toro Iniesta 2003),

dIII

dτ

= K ( III −SSS), (2.24)

where K is the propagation matrix,

K =









η

η

η

η

η

η

ρ

−ρ

η

−ρ

η

ρ

η

ρ

−ρ

η









, (2.25)

1The electric field of the waves vibrates at random angles with the same amplitude.

that contains all the absorption (η) and dispersion (ρ) profiles whose expres- sions can be found in radiative transfer textbooks such as del Toro Iniesta (2003, pag. 116). In LTE and complete redistribution, SSS = (B

(T ),0,0,0)

.

The formal solution of the RTE for polarized light can be written as III(0) = ^Z

0

O( 0,τ

) K(τ

) SSS(τ

) dτ

, (2.26) where O is the evolution operator, which is a linear operator that describes the transformation of the Stokes vector between two optical depths τ

and τ

such that III(τ

) = O(τ

, τ

) III(τ

) . In general, the solution can only be obtained numerically since O does not have a simple analytical form.

There are two main mechanisms through which magnetic fields can induce polarization signals in spectral lines: the Zeeman effect and the Hanle effect.

The latter is beyond the scope of this thesis but I refer to Landi Degl’Innocenti

& Landolfi (2004) for the full theory.

Our knowledge of photospheric and chromospheric magnetic fields is mainly based on observations of the Zeeman effect that consists in the splitting of the energy levels of the atom by an external magnetic field. An energy level with angular momentum J is divided into 2J + 1 sublevels with magnetic quantum number M = −J,...,+J and the different subtransitions (or Zeeman components) are elliptically polarized. Let J and J

be the lower and upper levels of an absorption line and M and M

their corresponding magnetic sub- levels. Transitions with ∆M = 0 give rise to π components, while ∆M = −1 and ∆M = +1 originate σ

and σ

components. The wavelength shifts of each component are given by

λ

= λ

− eλ

4πm

c

B(gM − g

M

), (2.27) where e is the electron charge, λ

is the wavelength of the unperturbed transition and B is the magnetic field strength. The Landé factor g is a dimensionless proportionality constant between the magnetic moment and the angular quantum number (expression given in, e.g., Landi Degl’Innocenti & Landolfi 2004). So there are two ways of measuring magnetic fields with the Zeeman effect: (1) at high field strengths (or at the long wavelengths) the Stokes I profiles will appear split; (2) depending on the orientation of the magnetic field relative to the observer, different Zeeman components will be either circularly or linearly polarized, thus giving rise to Q, U, and V signals.

For example, the Fe

6173 Å Zeeman triplet line that is observed from space by SDO/HMI (see Section 3.5.1), has J = 1 and J

= 0, therefore M = (1,0,−1) and M

= (0,0,0). This produces one unshifted π component and two shifted σ components by an amount (in [Å])

∆λ

= 4.6686 × 10

λ

¯gB, (2.28)

Figure 2.3: Synthetic stokes vector for the Fe

6173 Å line from a Milne-Eddington atmosphere. The solid and dashed profiles correspond to B = 1000 G and B = 500 G.

The profiles are normalized to the continuum intensity.

where ¯g is the effective Landé factor. Figure 2.3 shows the Stokes parameters of

the iron line from a Milne-Eddington (ME) atmospheric model (see Chapter 4)

in which the physical parameters are constant with height. In this example,

for B = 1000, θ = 45

(inclination angle), and ϕ = 0 (azimuth angle). For

T = 6000 K, the thermal broadening is larger than the Zeeman broadening

and no splitting is visible in intensity. Note how much weaker Q and U are

compared to I. Measuring high signal-to-noise ratio linear polarization signals

can be challenging, especially in the QS. This is expected to greatly improve

with the new generation of ground-based optical telescopes such as the 4-m

Daniel K. Inouye Solar Telescope (DKIST, Rimmele et al. 2020) in Maui,

Hawaii or the European Solar Telescope (EST, Collados et al. 2013) in the

future.

3. Radiative diagnostics

In the early eighteenth century, Fraunhofer realized that the visible solar spec- trum consisted of numerous dark bands at specific frequencies (Fraunhofer 1817), but only a few decades later Kirchhoff and Bunsen established the link between absorption lines and chemical elements (Kirchhoff 1860; Kirchhoff

& Bunsen 1860). The Fraunhofer lines were then recognized as some of the visible Balmer lines, namely Hα (6563 Å), Hβ (4861 Å), Hγ (4340 Å) and Hδ (4102 Å), the magnesium triplet lines Mg

b

(5184 Å), b

(5173 Å), b

(5167 Å), and the resonance doublets of sodium Na

D

(5896 Å), D

(5890 Å) and calcium Ca

H (3969 Å), K (3934 Å), among many others. It became clear that the spectrum of the Sun encodes information of its very nature.

Ground-based observatories are constrained to the visible, NIR, and a few radio windows, whereas observations from space are not limited by the transmittance of the Earth’s atmosphere, enabling us to probe more energetic phenomena on the Sun in the UV and X-rays. The first observations of the Sun in the UV were reported by Durand et al. (1949) after several rocket launches with a spectrograph onboard. The spectra showed very broad emission features coincident with the Mg

h (2804 Å) and k (2796 Å) lines. Observations in the EUV were important for the development of early semi-empirical models such as the widely used VAL models (Vernazza et al. 1981). Brightness temperature measurements at mm and cm wavelengths were also crucial for those models.

The continuum radiation between 400 Å and 1600 Å and between 0.1 mm and 20 mm originate in the chromosphere. Solar radio radiation was first reported by Hey (1946) after the second World War.

In Chapter 2, I provide an introduction to the formation of spectral lines and continua from textbook radiative transfer principles. In this chapter I present in more detail the formation mechanisms of some of the most commonly used radiative diagnostics for the solar chromosphere and corona, in particular the ones within the scope of Papers I–III. Other chromospheric diagnostics such as the Lyman α and Hα lines (e.g., Rutten 2007), the He

D

and 10830 Å lines (e.g., Libbrecht 2019, PhD thesis), and various other lines in the FUV (e.g., Doschek et al. 1977) are not discussed in this thesis. I refer to de la Cruz Rodríguez & van Noort (2017) for a more complete overview of photospheric and chromospheric diagnostics and modeling, and I point to Del Zanna &

Mason (2018) for a comprehensive review of EUV and X-ray diagnostics and

observational techniques.