Licentiate thesis
Constraints on the gas temperature in the solar atmosphere from multiwavelength inversions
João Manuel da Silva Santos
Supervisor:
Jaime de la Cruz Rodríguez Co-supervisor:
Jorrit Leenaarts
October, 2018
In this Licentiate thesis I review the properties of the solar atmosphere and the diagnostic value of different spectral lines in the visible and ultraviolet (UV) along with the millimeter (mm) continua in the electromagnetic spectrum of the Sun.
While the solar atmosphere has been routinely observed in high-resolution from ground-based opti- cal telescopes such as the Swedish Solar Telescope (SST), and more recently in the UV from space tele- scopes such as the Interface Region Imaging Spectrograph (IRIS), radio observations lag behind despite their great usefulness. This is likely to change thanks to the Atacama Large Millimeter Array (ALMA) that only started observing the Sun in 2016 with a few limitations, but the first results are promising.
ALMA observations probe the solar chromosphere at different heights by tuning into slightly different frequencies at potentially milliarcsecond scales if the full array is able to operate with the longest base- lines. This new spectral window onto the Sun is expected to advance various fields of research such as wave propagation and oscillations in the chromosphere, thermal structure of filaments/prominences, triggering of flares and microflares, and more generally chromospheric and coronal heating, because the mm-intensities can be modelled by simply assuming local-thermodynamic equilibrium.
In da Silva Santos et al. (2018) we find that coordinated observations from SST, IRIS and ALMA will
permit us to estimate with greater accuracy the full thermodynamical state of the plasma as a function
of optical depth based on experiments with a snapshot of a three-dimensional magnetohydrodynamic
simulation of the Sun’s atmosphere. Particularly, the mm-continuum improves the accuracy of inferred
temperatures in the chromosphere. Here we expand on the Why and How this can be done. The goal is
to better constrain the temperature stratification in the solar atmosphere in order to understand chromo-
spheric heating.
I den här Licentiatavhandlingen granskar jag egenskaperna hos solens atmosfär och provvärdet av olika spektrallinjer i det synliga och ultravioletta (UV) tillsammans med millimeter (mm) kontinuum i solens elektromagnetiska spektrum.
Medan solens atmosfär har observerats rutinmässigt i högupplösning från markbaserade optiska teleskop som svenska solteleskopet (SST) och nyligen i UV från rymdteleskop som ”Interface Region Imaging Spectrograph” (IRIS), så är radioobservationer inte så vanliga trots deras stora användbarhet.
Detta kommer sannolikt att förändras tack vare ”Atacama Large Millimeter Array” (ALMA) som började följa solen i 2016 med några få begränsningar, men de första resultaten är lovande. ALMA-observationer genomsöker solkromosfären i olika höjder genom att justera lite i olika frekvenser vid potentiellt mil- libågsekundersskala om den fulla uppställningen kan arbeta med de längsta baslinjerna. Detta nya spek- tralfönster till solen förväntas ge framsteg inom flera forskningsområden såsom vågförökning och oscil- lationer i kromosfären, termisk struktur av filament / protuberanser, uppkomsten för ”flares” och ”mi- croflares”, och mer allmänt kromosfärisk och koronal uppvärmning, eftersom mm-intensiteterna kan modelleras genom att helt enkelt anta lokal-termodynamisk jämvikt.
I da Silva Santos et al. (2018) finner vi att samordnade observationer från SST, IRIS och ALMA
tillåter oss att med högre noggrannhet uppskatta det fullständiga termodynamiska tillståndet för plas-
man som en funktion av optiskt djup baserat på experiment med en ögonblicksbild av en tredimen-
sionell magnetohydrodynamisk simulering av solens atmosfär. I synnerhet förbättrar mm-kontinuumet
noggrannheten av antagna temperaturer i kromosfären. Här expanderar vi på Varför och Hur det kan
göras. Målet är att bättre begränsa temperaturstratifieringen i solatmosfären för att förstå kromosfärisk
uppvärmning.
Paper included in this thesis:
da Silva Santos, J. M., de la Cruz Rodríguez, J. & Leenaarts, J. 2018, Temperature constraints from inversions of synthetic solar optical, UV and radio spectra, A&A, 620, A124
The author of this licentiate thesis performed all the calculations (with a code written by de la Cruz
Rodríguez et al.) and analysis, and produced all the text and figures in the included article, herein Paper I,
reproduced with permission of © ESO.
I thank Jaime de la Cruz Rodríguez for his tireless support and for reviewing this document. I thank Jorrit Leenaarts for his useful comments and remarks during the course of this work. I thank Per Calisendorff and Dan Kiselman for the swedish proofreading.
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.
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.
This research has made use of NASA’s Astrophysics Data System (ADS) Bibliographic Services.
Abstract
Sammanfattning Author’s contribution Acknowledgments
1 The solar atmosphere 1
1.1 General properties . . . . 2
1.2 Scale heights . . . . 2
1.3 Radiative transfer . . . . 4
1.4 LTE vs non-LTE . . . . 7
1.5 Introduction to line formation . . . . 8
1.6 Polarized radiative transfer . . . . 10
1.6.1 Measuring magnetic fields in the radio . . . . 12
2 Radiative diagnostics 15 2.1 The visible . . . . 15
2.1.1 The calcium lines . . . . 16
2.2 The ultraviolet . . . . 19
2.2.1 The magnesium lines . . . . 20
2.3 The radio . . . . 21
2.3.1 The formation of the sub-mm/mm continuum . . . . 22
2.3.2 First results of ALMA solar observations . . . . 24
3 Simulations and modeling 27 3.1 Forward modeling . . . . 27
3.2 Inverse modeling . . . . 31
3.2.1 The Levenberg-Marquardt method . . . . 33
3.2.2 Regularization . . . . 35
3.3 Response functions . . . . 35
4 Summary and Outlook 37
5 Future prospects 39
References 41
Paper I 53
1.1 The solar photosphere at 396.37 nm. . . . 1
1.2 A model for the average quiet-Sun. . . . 4
1.3 Definition of specific intensity. . . . 5
1.4 Polarization states. . . . 11
1.5 Iron line in a Milne-Eddington atmosphere. . . . . 11
1.6 The true longitudinal magnetic field versus the restored one using mm-circular polarization. 13 2.1 The solar atmosphere observed at different visible frequencies. . . . . 16
2.2 Term diagram for the singly ionized calcium atom. . . . 17
2.3 The solar spectrum around the calcium lines. . . . 18
2.4 IRIS observation of an active region. . . . 19
2.5 Average quiet-sun spectrum in the IRIS window containing the h and k lines of singly ionized magnesium. . . . 21
2.6 The formation of the millimeter continuum. . . . 23
2.7 ALMA observations of the solar disk. . . . 25
2.8 Plasmoids observed in the ultraviolet, X-rays and radio. . . . 26
3.1 Real vs simulated sunspot. . . . 28
3.2 Snapshot of a 3D radiation-MHD simulation. . . . . 30
3.3 Inversion of the magnesium lines. . . . 32
3.4 Optimisation with the Levenberg-Marquardt algorithm. . . . 34
3.5 Response functions to temperature perturbations. . . . . 36
List of Tables 2.1 The effective Landé factor for a few transitions in the calcium atom. . . . 17
2.2 Average brightness temperatures of different structures on the Sun. . . . 26
AIA Atmospheric Imaging Assembly ALMA Atacama Large Millimeter Array CF contribution function
CRD complete redistribution
DKIST Daniel K. Inouye Solar Telescope EUV extreme-ultraviolet
FOV field-of-view FUV far-ultraviolet
IRIS Interface Region Imaging Spectrograph LM Levenberg-Marquardt
LOS line-of-sight
LTE local thermodynamic equilibrium MHD magnetohydrodynamics NIR near-infrared
NUV near-ultraviolet PRD partial redistribution QS quiet-Sun
RTE radiative transfer equation SDO Solar Dynamics Observatory SSE sum of squared errors SST Swedish Solar Telescope STiC STockholm inversion Code TE thermodynamic equilibrium UV ultraviolet
1D, 2D, 3D one-, two-, three- dimensional
The solar atmosphere
A pproximately 4.6 billion years ago, the gravitational collapse of a rotating molecular cloud gave birth to what seems to be a rather ordinary, yet fascinating G2V-type dwarf star — the Sun — and with it, a group of orbiting planets, moons and other smaller celestial bodies composing the Solar System. 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 10 000 - 170 000 years later (Mitalas and Sills, 1992). Only the surface layers of the Sun — its atmosphere — can be directly observed, but what they reveal is no less intriguing.
The appearance of the solar atmosphere is markedly different when observed at different wave- lengths, therefore, multiwavelength observations are often needed to understand it. That is because the range of physical conditions changes drastically from the bottom to the top of the atmosphere, which brings together different domains of physics, each producing different observables. In Paper I we argue that only from a combination of as many as possible of those observables we can accurately infer the full thermodynamical state of the plasma as function of height based on simulations. In particular, we focused on the additional diagnostic potential of the millimeter (mm) radiation since a new window onto the Sun has been recently opened at these wavelengths with new instrumentation.
Understanding the diverse phenomena in solar observations requires knowledge of the physics of radiation transport through the atmosphere and how it interacts with matter, which we briefly review here.
This first chapter presents essential concepts of radiative transfer and borrows from several textbooks such as Böhm-Vitense (1989), del Toro Iniesta (2003), Rutten (2003) and Priest (2014) to where we refer for a more comprehensive explanation of the physics of the solar atmosphere.
Figure 1.1: The photosphere of the Sun observed using a wide band filter around 396.37 nm at the Swedish Solar Telescope in
the island of La Palma (Credit: V. Henriques, 2010).
1.1 General properties
The solar atmosphere is defined as the part of the Sun from where photons are able to escape di- rectly into space, and it is essentially divided into four layers: the photosphere, the chromosphere, the transition region and the corona. We now provide a short overview of their general properties (see e.g., Priest, 2014, for more details).
The photosphere is a few hundred-km thin shell that emits most of the Sun’s visible light. Its appear- ance is characterized by the ubiquitous granulation pattern made of not only small convection cells (or granules, see Fig. 1.1) with a typical diameter of 1 Mm, but also larger structures such as mesogranules and supergranules with sizes up to 70 Mm. It also harbours pores and sunspots that increase in number during the solar maximum 1 .
The chromosphere is a more rarefied and warped layer that is known for its dynamic fibrils at the disk 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 10 Mm but lasting only 10-180 s (for the type-II), and up to 40 Mm for the macrospicules often seen in polar coronal holes.
The transition region is where the plasma is heated to coronal temperatures over very short heights ( . 100 km), and it is visible at UV wavelengths.
The corona can be identified as the million-kelvin-hot plasma forming loops of different sizes (up to 700 Mm) at the top of the atmosphere but also composing the solar-wind that streams through space.
Its visibility is optimal from space at X-rays and extreme-UV (EUV) wavelengths. Coronal holes are indistinguishable from their surroundings in the photosphere and chromosphere, but they appear as voids in the coronal emission since they contain cooler and less dense gas.
From the bottom to the top of the atmosphere the particle density decreases exponentially having typical values of 10 23 m −3 in the photosphere, 10 19 m −3 in the chromosphere, 10 15 m −3 in the transition region and 10 12 m −3 at a height of one solar radius (R ).
Most of the solar radiation is emitted from the quiet-Sun (i.e., areas of low activity), whereas the active-Sun, despite being capable of flaring up in the EUV and X-rays, is a much more irregular and unpredictable component of the total energy output. For example, the energy released during a flare 2 can vary between ∼ 10 29 − 6 × 10 32 erg with an X-class flare having a peak luminosity that is as much as 0.01% of the luminosity of the Sun.
1.2 Scale heights
A static star is said to be in hydrostatic equilibrium when the gravitational force is balanced by the pressure gradient. The balance of these radial forces can be written as:
dP
dr = −ρg (1.1)
where g = GM /r 2 is the gravitational acceleration on the material of density ρ at a given radius r , with G being the gravitational constant. The Ideal Gas Law can be used as an equation of state (i.e. to relate density and pressure) as follows:
P = ρ kT
µ (1.2)
1
The period of greatest activity during the 11-year cycle.
2
A sudden increase in brightness across all the electromagnetic spectrum typically occurring around sunspots.
where k is the Boltzmann constant, T is the temperature and µ is the mean mass of all particles in the gas, which in the case of the Sun can be shown to be approximately µ ≈ 0.6 m p (for a fully ionized gas) with m p being the proton mass. Dividing Eq. (1.2) by Eq. (1.1):
P
|dP /dr| = kT
µg = kT r 2
µGM ≡ H (1.3)
defines a characteristic pressure scale height H . We can then estimate the characteristic temperature for the stellar interior by taking the pressure scale height to be the stellar radius R .
T int ∼ GM µ
kR ∼ 13 × 10 6 M M
R R
[K] (1.4)
This means that temperatures in stellar interiors are of the order of ten million kelvin. The temperatures at the visible surface are much lower, though. We call the amount of energy emitted by the stars per unit area and time the flux F . This quantity can be compared to the flux emitted by a black body 3 which, according to the Stefan-Boltzmann’s law, is given by:
F = σT 4 (1.5)
where σ is the Boltzmann constant. The effective temperature T eff is the temperature a black body would need to have to radiate the same amount of energy. For the Sun, F = 6.3 × 10 10 erg cm − 2 s − 1 — the so-called solar constant — therefore, T eff ≈ 5 800 K.
The large difference in temperature between the interior and the surface implies a large difference in H in these regions. In fact, the characteristic temperature in the surface layers is near the stellar effective temperature. For the Sun, setting T = T eff in Eq. (1.3) implies that H ≈ 300 km.
If we assume the temperature to be constant in the atmosphere, substituting Eq. (1.2) and Eq. (1.3) in Eq. (1.1), now in a 1D Cartesian frame, the pressure gradient in the z direction can be written as:
dP (z)
dz − P (z)
H = 0 (1.6)
whose solution is of the form P (z) ∝ e −z/H . An analogous expression can be obtained for the density.
Pressure and density drop very rapidly with height (Section 1.1) such that the temperature scale height is usually much larger than the pressure scale height. Because of the relative thinness of the atmospheric scale height, the emergent spectrum of the Sun can, in general, be modelled in terms of a one-dimensional (1D) plane atmosphere in which local conditions vary only with height z to good approximation.
The average of the quiet-Sun (QS) spectra has been used in such modelling attempts such as the VAL model (Vernazza-Avrett-Loeser, Vernazza et al., 1981) that is valid for a static atmosphere where the thermodynamic properties vary only with height as we have described. However, creating a model for the average spectra is not the same as averaging the models for different resolved spectra, which can alter the way observations are interpreted (Carlsson and Stein, 2002a).
Figure 1.2 depicts the variation of temperature and mass density as function of height in the ALC7 model (Avrett and Loeser, 2008) of the average QS, which is a more recent update on the old VAL model. The temperature profile is characterized by a decline in temperature from 6 600 K at the bottom of the photosphere to a minimum value of approximately 4 400 K at a height of 500 km from where the temperature rises through the lower chromosphere and up to the chromospheric plateau, and then more
3
A black body is an idealized source of radiation that does reflect any light and whose properties do not depend on the constituent material
but only on a characteristic temperature set by the thermodynamic equilibrium.
0 1000 2000 3000 Height [km]
10
310
410
510
6T em p er a tu re [K ]
photosphere chromosphere corona
10
−510
−410
−310
−210
−110
010
110
2M a ss d en sit y [k g m
−2]
Figure 1.2: The mean variation of temperature (solid line) and mass density (dashed line) with height in the solar atmosphere according to the ALC7 model.
abruptly through the transition region to a few million degrees in the corona. In contrast, the mass density decreases steadily through the atmosphere.
One-dimensional, static, semi-empirical models such as the aforementioned one cannot accurately represent the variety of complicated time-dependent structures of the solar atmosphere, although they have the virtue of producing reasonable spectra which can be readily compared to observations. More realistic calculations should be performed in 3D in both the magnetohydrodynamics (MHD) and radiation transport, at least for a few important spectral lines as we further discuss in Chapters 2 and 3.
1.3 Radiative transfer
The radiation field within a star can be described in terms of the (monochromatic) specific intensity 4 (or surface brightness) I ν 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 ν ( r , s , t) = dE ν
dSdtdνdΩ = dE ν
dA cos θdtdνdΩ (1.7)
where the angles θ is defined with respect to the normal n to the area dA as illustrated in Figure 1.3 adapted from Maciel (2016).
The specific 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 5 . This is cer- tainly true in the ”vacuum” of space, but in the interior and atmosphere of the Sun I ν also depends on the interactions with matter.
There is an infinitesimal probability dp ν of a photon with a certain frequency being absorbed in a thin slab of thickness ds along the direction s. This is expressed by the linear absorption coefficient α ν :
α ν ≡ dp ν
ds = κ ν ρ [ cm −1 ] (1.8)
4
Or alternatively: I
λ= I
νc/λ
2[W m
−2sr
−1Hz
−1].
5
The 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 cos θ/d
2[sr].
Figure 1.3: Definition of specific intensity (adapted from Maciel, 2016).
where κ ν is the monochromatic mass extinction coefficient (in cm 2 g − 1 ). The fraction of specific inten- sity lost due to absorption (or extinction) in the element ds along the ray is:
dI ν I ν
= −dp ν = −α ν ds (1.9)
Integrating this equation along the path with total thickness D yields:
Z D 0
dI ν I ν
= − Z D
0
α ν (s) ds
I ν (D) = I ν (0) × exp
− Z D
0
α ν (s) ds
, (1.10)
with I ν (0) being the intensity of radiation at the source. Therefore, the transparency of the slab can be quantified by the quantity in the exponent which we call monochromatic optical depth ( τ ν ):
τ ν ≡ Z D
0
α ν ds (1.11)
A path is said to be optically thick at a given frequency if τ ν > 1 and optically thin if τ ν < 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 former are the result of increased opacity due to specific elements in the overlying atmosphere.
The sources of opacity (absorption and scattering) depend on how photons interact with matter, namely atoms, ions and free electrons. The four primary sources of opacity are:
• Bound-bound transitions: when an electron in a atom or ion makes a transition between two or- bitals; high opacity at discrete transition wavelengths
• 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 scattered by a nearby ion absorbing a photon; is is also
a source of continuum opacity. It can also happen that the electron is slowed down by the ion
emitting a photon (bremsstrahlung).
• Elastic processes: a photon is scattered by a free electron (Thomson scattering); important when most of the gas is completely ionized. A photon may also be scattered by a loosely-bound electron to an atom (Compton or Rayleigh scattering depending on the wavelength).
We will revisit free-free processes in Section 2.3.1 because they are important for understanding millimeter (mm) radiation.
More 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 j ν . This is described by the radiative transfer equation (RTE) given as follows:
dI ν
ds = j ν − α ν I ν
dI ν
α ν ds = S ν − I ν dI ν
dτ ν
= S ν − I ν (1.12)
where S ν ≡ j ν /α ν is the source function. Equation (1.12) describes the change of intensity over the path length ds . To integrate it along the s direction, we multiply both sides by the integrating factor e τ
νand we derive the formal solution 6 of the RTE:
I ν (τ ν ) = I ν (0) e −τ
ν+ Z τ
ν0
S ν (t ν )e −(τ
ν−t) dt ν (1.13)
where t ν is just a dummy integration variable in optical depth. In an atmosphere where α ν and j ν are constant, S ν does not also depend on the location, therefore we can take it out of the integral in Eq. 1.13 and we simply obtain:
I ν (τ ν ) = I ν (0) e −τ
ν+ S ν (1 − e −τ
ν) (1.14) This result can be used to intuitively understand the basics of line formation as we discuss in Section 1.5 in greater detail.
In general, I ν 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 (as in Eq. (1.6)). We define the viewing angle µ ≡ n · z = cos θ between the line of sight and the z -axis such that I ν = I ν (µ, z) . In these terms τ ν can then be redefined as the vertical optical depth τ ν ≡
Z ∞ z
0α ν dz for a distant observer located at z = ∞ . Then Eq. (1.12) can be rewritten as follows:
µ dI ν
dτ ν = I ν − S ν (1.15)
The emergent intensity from the horizontal slab as seen from an external observer at τ ν = 0 is given by:
I ν (µ, τ ν = 0) = Z ∞
0
S ν (t ν )e −t
ν/µ dt ν /µ; (µ > 0) (1.16)
If, for simplicity, we assume S ν to be a linear function of optical depth: S(t ν ) = a + b t ν where a and b are constants, then, the integral in Eq. (1.16) can be shown to be equal to a + bµ , which implies that:
I ν (µ, 0) = S ν (τ ν = µ) (1.17)
6
The one that can be written although S
νis not necessarily a known explicit function of τ
ν.
We now introduce the concept of monochromatic flux F ν which represents the net flow of energy per second through an area placed at location r perpendicular to n. This is defined by:
F ν ( r , n , t) = Z
I ν cos θ dΩ = Z 2π
0
Z π 0
I ν cos θ sin θ dθ dφ (1.18)
where φ is the azimuthal angle on the plane dA in Fig. 1.3. For a plane-parallel atmosphere we can assume axial symmetry, therefore the previous equation reduces to:
F ν (z) = 2π Z π
0
I ν cos θ sin θ dθ = 2π Z 1
−1
I ν cos θ d( cos θ)
= 2π Z 1
0
µ I ν dµ
| {z }
F
ν+(z)
− 2π Z −1
0
µ I ν dµ
| {z }
F
ν−(z)
(1.19)
where we separated the flux into the in-going ( F ν − ) and the out-going ( F ν + ) part. From the relation (1.17) follows that I ν (µ, 0) = a + bµ . Inserting this into Eq. (1.19), we obtain the flux emitted ( µ > 0 ) from the surface:
F ν + = 2π Z 1
0
µ (a + bµ)dµ = π
a + 2b
3
= π S ν
τ ν = 2
3
(1.20) which is the so-called Eddington relation. This means that the flux coming from the stellar surface ( F = F /π ) at any angle equals the source function at the optical depth τ ν = 2/3 .
1.4 LTE vs non-LTE
In Section 1.2 we 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 7 than the temperature scale height (Section 1.2).
In LTE, the number of atoms with energy in different states of excitation is given by the Boltzmann distribution, while the number of atoms in different stages of ionization is given by the Saha equation.
These two can be combined in the Saha-Boltzmann distribution given as follows:
n c n i
= 1 N e
2g c g i
2πm e kT h 2
3/2
e −χ
ci/kT (1.21)
7
For a photosphere consisting of mainly of neutral hydrogen with density of ρ = 2.1 × 10
−4kg m
−3the particle mean free path is
l ∼ 2 × 10
−4m (Carroll and Ostlie, 2007).
where n i is the total population density of the level i , n c is the number of ions in ionization level c , N e is the electron density, m e is the electron mass, g c and g i are the statistical weights and χ ci is the ionization energy from level i to state c .
The photons do not ”sense” a constant temperature in the atmosphere because their mean-free-path is comparable to the temperature scale height, so LTE is not strictly valid. In this case the assumption of LTE means that the source function equals the Planck function S ν = B ν which is defined by:
B ν = 2hν 3 c 2
1
exp (hν/kT ) − 1 [ Wsr − 1 m − 2 Hz − 1 ] (1.22) In the limit of very high temperatures or low frequencies e hν/kT ≈ 1 + hν/kT , therefore Eq. (1.22) reduces to:
B ν (T ) ≈ 2ν 2 kT
c 2 (1.23)
Equation 1.22 (or 1.23) can be solved for T which is then labelled brightness (or radiation) tempera- ture T b (or T rad ) by equating S ν = I ν . In more general terms, S ν is not necessarily given by the Planck function. It can be shown (e.g. Rutten, 2003) that for a bound-bound transition between two energy levels with quantum numbers u and l the (frequency-independent) 8 S ν is given by:
S ν = 2hν 3 c 2
1
n
lg
un
ug
l− 1 (1.24)
which is generally valid without imposing any equilibrium condition. In LTE, the line source function simplifies to S ν = B ν if the occupation numbers n l , n u in the denominator follow the Boltzmann distribution. The problem is then how to compute it when LTE does not hold — non-LTE.
When the radiative processes cannot be neglected compared to the collisional processes which, for example, can occur in a low density gas such as the chromosphere, then the population ratio n l /n u
has to be computed explicitly usually by assuming statistical equilibrium: the radiation field and level populations do not vary with time. The statistical equilibrium equation is given by:
X
j6=i
n j P ji − n i X
j6=i
P ij = 0 (1.25)
where P ij ≡ R ij + C ij are the rates at which the particles change from level i to j which include both radiative ( R ij ) and collisional ( C ij ) processes. In a bound-bound or bound-free transition R ij depends on the mean intensity itself.
To compute realistic electron densities in the chromosphere we cannot compute the populations of all species using Saha-Boltzmann distributions because the degree of ionization is decoupled from the local temperature, that is, S ν is not given by the Planck function. Integrating the RTE (Eq. (1.15)) under these circumstances is a much more complicated problem because S ν has to be computed explicitly by solving Eq. (1.25), which involves computing the rates which in turn depend on the radiation field itself.
This is a non-linear and non-local problem that has to be solved self-consistently in an iterative way.
1.5 Introduction to 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
8
That is assuming complete frequency redistribution.
gives an absorption line spectrum, just like what we observe on the Sun and on most stars (Kirchhoff’s Laws). We are now able to understand how absorption and emission lines are formed in the terms discussed in the previous sections.
We can interpret the first term on the right-hand side of Eq. (1.14) as the amount of light that is left after being absorbed through a path with optical depth τ ν , while the second term gives the contribution of radiation absorbed and re-emitted along the path.
Then it automatically follows that when the gas is optically thick ( τ ν 1 ): I ν ≈ S ν , and no lines emerge.
In the optically thin case ( τ ν 1 ): e −τ
ν≈ 1 − τ ν , therefore we derive:
I ν = I ν (0) − [I ν (0) − S ν ]τ ν (1.26) If I ν (0) = 0 , which means the gas is not back-lit, then I ν = τ ν S ν . Because the extinction coefficient ( α ν ) 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 ν (0) 6= 0 and: (1) I ν (0) > S ν then the second term on Eq. (1.26) is positive so there is an amount of radiation proportional to τ ν that is subtracted from the background originating an absorption line; (2) I ν (0) < S ν then there is a positive contribution to the background radiation, so one would see an emission line on top of the I ν (0) intensity level.
The solar (visible, near-IR and near-UV) spectrum is populated with numerous lines that appear in absorption because the decreasing temperature implies that the source function also decreases with height (assuming LTE), that is, the intensity coming from deeper layers is larger than the source function for the top layers. However, at wavelengths shorter than ∼ 1600 Å we only see a number of strong emission lines (such as Lyman- α 1216 Å) coming from the chromosphere, transition region and beyond, since there is essentially no background light at these wavelengths because of the exceedingly high continuous absorption coefficient, while the source function is increasing outwards (Böhm-Vitense, 1989).
Line formation can be, in fact, more complicated than what we have described because the source function does not necessarily equal the Planck function, therefore it can, for example, decrease while the temperature increases outwards, or it can have a complicated variation in height producing dips and reversals in the line shapes.
For the sake of simplicity one assumes complete frequency redistribution (CRD) to obtain Eq. (1.24) for the source function of a bound-bound transition. This means that S ν is independent of the angle and frequency of the light beam considered. 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 (e.g. Linsky, 1985). In partially-coherent scattering, more commonly known as partial frequency redistribution (PRD), the frequency and direction of the incoming and outgoing photon during scattering is not random but can be correlated. PRD effects are important in Ly α , Ly β , Mg II h and k, Ca II H and K, among a few others. (e.g. Hubeny and Mihalas, 2014; Sukhorukov and Leenaarts, 2017, and references therein). We discuss the properties of the Ca II lines in Section 2.1.1 and the Mg II lines in Section 2.2.1.
Several broadening mechanisms are usually convoluted and play a role in explaining the line profiles, namely the natural broadening, collisional broadening and Doppler broadening producing the classical Voigt profile 9 , apart of course, from the macroscopic flows of plasma that often originate asymmetrical
9
The Voigt profile is the result of a convolution of a Lorentzian function describing the radiation damping and a Gaussian function describing
the Doppler broadening.
line shapes. In addition, one often has to include what is called microturbulence ξ micro as a fudge param- eter, i.e. an ad hoc term to compensate model deficiencies, but also to account for potential (random) unresolved motions along the line-of-sight. The total Doppler broadening ∆ν D is then given by:
∆ν D = ν 0 c
q
ξ 2 thermal + ξ 2 micro (1.27)
where ξ thermal ≡ ν 0 /c p
2kT /m is the thermal Doppler broadening.
Magnetic fields also impact in line formation. In fact, this will make us redefine the radiative transfer problem (Section 1.6). For now, we mention an additional broadening mechanism: the so-called Zeeman effect, which occurs when a spectral line is split in a presence of a magnetic field. The effective Landé factor g eff is a dimensionless proportionality constant between the magnetic moment and the angular quantum number; the higher its value the larger the splitting in the energy levels due to magnetic fields.
The separation between the Zeeman components is directly proportional to the magnetic field strength and the square of the wavelength, which means that it becomes more noticeable towards the infrared as we further discuss in Section 2.1.
1.6 Polarized radiative transfer
The radiative transfer equation for the monochromatic specific intensity as we have described so far is valid for unpolarized 10 light travelling through isotropic media. However, if the radiation is partially or fully polarized, namely by the presence of a vector magnetic field B, we need more quantities to specify the polarization state.
If we think of light as a transverse electromagnetic wave, we can decompose the electric field vector E of a monochromatic light wave into its x and y components:
E x = A x cos (wt − φ x ) E y = A y cos (wt − φ y )
(1.28)
where A x and A y are the amplitudes, φ x and φ y are the phases and w = 2πν is the angular frequency.
The Stokes parameters are given by:
I ν ≡ hA 2 x i + hA 2 y i Q ν ≡ hA 2 x i − hA 2 y i
U ν ≡ h2A x A y cos (φ x − φ y )i V ν ≡ h2A x A y sin (φ x − φ y )i
(1.29)
where the brackets denote time averages. Q and U describe linear polarization and V describes circular polarization as illustrated in Figure 1.4. We define a four-vector I≡ (I ν , Q ν , U ν , V ν ) T , where the index T stands for transpose, and simply call it Stokes vector.
By measuring the orientation of the electric field of the incoming light waves we are thus able to infer the magnetic field that induced such polarization on the radiation.
Equation (1.15) can be re-written as a vector differential equation in the following form:
d I
dτ = K ( I − S ) (1.30)
10
The electric field of the waves vibrates at random angles with the same amplitude.
Figure 1.4: Polarization states.
6301 6302 6303 6304 λ (˚ A) 0.4
0.6 0.8 1.0
I/I
cB=100 B=1000 B=2000
6301 6302 6303 6304 λ (˚ A)
−0.01 0.00 0.01 0.02
Q/I
c6301 6302 6303 6304 λ (˚ A)
−0.006
−0.004
−0.002 0.000
U/I
c6301 6302 6303 6304 λ (˚ A)
−0.10
−0.05 0.00 0.05 0.10
V /I
cFigure 1.5: Stokes profiles of the Zeeman sensitive line Fe I 6302 Å in a Milne-Eddington atmosphere for different magnetic field strengths (100, 1000 and 2000 G). The profiles are normalized to the continuum intensity.
where K is the propagation matrix defined by:
K ≡
η I η Q η U η V η Q η I ρ V −ρ U η U −ρ V η I ρ Q
η V ρ U −ρ Q η V
(1.31)
containing all the absorption ( η ) and dispersion ( ρ ) profiles whose expressions can be found in ra- diative transfer textbooks such as del Toro Iniesta (2003). In LTE and complete redistribution, S = (B ν (T ), 0, 0, 0) T with B ν (T ) given by Eq. (1.22).
One of the simplest solutions of the polarized RTE is obtained by assuming that S ν is a linear function of τ ν and that the thermodynamic and magnetic variables are constant throughout the atmosphere; this is called the Milne-Eddington approximation (see review by del Toro Iniesta and Ruiz Cobo, 2016). In this case the RTE has an analytic solution (Unno, 1956; Rachkovsky, 1962, 1967) that can be quickly computed and compared with observations. Although it represents a crude approximation, it is applicable to a wide range of conditions on the Sun whenever the physical parameters do not vary significantly over thin slabs of plasma where the lines are formed like in the photosphere (Section 3.2).
Figure 1.5 illustrates how a typical observation of the four Stokes profiles of the magnetically sensi-
tive Fe I 6302 Å line would look like synthesized under the ME approximation for arbitrarily chosen line-
to-continuum absorption coefficient ratio η 0 = 10 , constant line-of-sight velocity v LOS = 10 km s − 1 ,
damping parameter a = 1 , microturbulence ξ micro = 1 km s − 1 , thermal broadening ξ thermal = 0.035 Å
and different magnetic field strengths of | B | = [100, 1000, 2000] G with field inclination θ = 45 ◦ and
azimuth angle φ = 0 ◦ .
At the moment of writing full-Stokes spectropolarimetric observations are being carried out using instrumentation at ground-based 1 m-class optical telescopes (Section 2.1) such as the CRISP instrument (Scharmer, 2006; Scharmer et al., 2008) at the Swedish Solar Telescope (SST, Scharmer et al., 2003), the NIRIS instrument (Cao et al., 2012) at the New Solar Telescope (NST, Goode and Cao, 2012) and the GRIS instrument (Collados et al., 2012) at GREGOR (Schmidt et al., 2012), and from space with the Solar Optical Telescope (Tsuneta et al., 2008) onboard Hinode (Kosugi et al., 2007) and the HMI instrument (Scherrer et al., 2012) onboard the Solar Dynamics Observatory (SDO, Pesnell et al., 2012), while the community eagerly awaits for the next generation of 4 m-class telescopes, namely the upcoming Daniel K. Inouye Solar Telescope (DKIST).
1.6.1 Measuring magnetic fields in the radio
Observations with the Atacama Large Millimetre Array (ALMA) may be an alternative way of mea- suring the longitudinal (or vertical) component of chromospheric magnetic fields because the tempera- ture gradient in the chromosphere produces an observable net circular polarization in the mm-continua (see review by Wedemeyer et al., 2016). This aspect had already been pointed out by, for example, Kundu and McCullough (1972), who measured the polarization degree at 9.5 mm of the full solar disk at very low spatial resolution (1.6’), but did not estimate the magnetic field itself. Later, Bogod and Gel- freikh (1980) and Grebinskij et al. (2000) proposed a method for determining magnetic fields via thermal bremsstrahlung (Section 2.3) based on the magneto-ionic theory of radiation that treats separately the or- dinary mode (o-mode) and the extraordinary mode (x-mode) of light (Ratcliffe, 1959). In a magnetized medium, the two modes have different refractive indices and the opacity for free-free emission is higher in the x-mode than in the o-mode, which means that they become optically thick at different heights in the atmosphere. In an atmosphere with a positive temperature gradient, the x-mode becomes optically thick in higher, hotter layers than the o-mode, which leads to a net polarization. Grebinskij et al. (2000) proposed determining the slope ζ (or spectral index) of the brightness temperature spectrum at a given frequency ν that is given by:
ζ = d log I ν
d log ν (1.32)
with I ν ∝ T b (ν) according to what was discussed in Section 1.4. The longitudinal magnetic field B l at the height where the emission at a given mm-wavelength is formed can then be estimated as follows:
B l = P ζ
ν[ Hz ] 2.8 × 10 −6
( G ) (1.33)
where P ≡ V ν /I ν is the degree of circular polarization. Therefore, all it takes to measure B l with ALMA is to measure P and calculate the slope of T b around a given frequency.
ALMA receivers use linearly polarized feeds (labelled X and Y) allowing to compute the four cross- correlations of the electric field (XX, YY, XY, and YX) for each antenna baseline (e.g. Shimojo et al., 2017a). Ignoring the instrumental effects, the correlation products for a pair of antennas are given by:
XX = [I ν + Q ν cos (2χ) + U ν sin (2χ)]
Y Y = [I ν − Q ν cos (2χ) − U ν sin (2χ)]
XY = [−Q ν sin (2χ) + U ν cos (2χ) + iV ν ] Y X = [−Q ν sin (2χ) + U ν cos (2χ) − iV ν ]
(1.34)
Figure 1.6: Longitudinal component of the magnetic field in a snapshot of a 3D MHD simulation at the formation height of the mm-radiation versus the restored values using the circular polarization method (from Loukitcheva et al., 2017a).
where χ is the parallactic angle.
Linear polarization signals are expected to be null due to the differential Faraday rotation which oc- curs when linearly polarized waves have their plane of polarization rotated by an angle ∆φ in the pres- ence of a magnetic field by an amount proportional to the integrated electron density and magnetic field strenght along the line-of-sight: ∆φ ∝ ν −2
Z
n e B l ds (e.g. Hatanaka, 1956; Grognard and McLean, 1973). Any linear polarization of the mm-radiation is washed out by the different ∆φ from different parts of the source. This effect is very strong at low frequencies meaning that within the observing band- width the emission will cancel out, so XX = YY = I ν . In practice this equality is not true due to noise, so by comparing XX with YY images it is possible estimate the signal-to-noise level in the observations (Shimojo et al., 2017a).
Equation (1.34) also implies that measuring XY and YX is a proxy for Stokes V ν . Unfortunately, at
the moment of writing ALMA does not yet provide full-Stokes observations of the Sun because it would
require calibration of the leakage terms in XY and YX that we omitted. The full expressions can be found in Shimojo et al. (2017a).
Eventually, when full-polarimetry becomes available, ALMA is going to be capable of measuring a few percent in P in the solar chromosphere according to model predictions (Wedemeyer et al., 2016, and references therein). The method described above should work well as demonstrated by Loukitcheva et al. (2017a) from where we borrow the plots shown in Fig. 1.6. The authors find that there is a tight correlation between the LOS magnetic field in the simulation snapshot and the inferred magnetic field at the height where the radiation is formed, despite the latter being usually underestimated by some
∼ 10 % in regions with |P| > 0.01 %. The method seems to work better at longer mm-wavelengths.
It is emphasized that spatial resolution better than 1 00 is needed for a robust application of this method
since lower resolution significantly blurs polarization signals. This should be possible with ALMA.
Radiative diagnostics
”Look here, I have succeeded at last in fetching some gold from the sun.”
Gustav Kirchhoff, 1890 In 1814, Fraunhofer realized that the visible solar spectrum is populated with numerous dark bands at specific frequencies, but only a few decades later Kirchhoff and Bunsen established the link between the absorption lines and the chemical elements. The Fraunhofer lines were then recognized as some of the visible Balmer lines, namely H α (6563 Å), H β (4861 Å), H γ (4340 Å) and H δ (4102 Å), the magnesium triplet lines Mg I b 1 (5184 Å), b 2 (5173 Å), b 4 (5167 Å), and the resonance doublets of sodium Na I D 1 (5896 Å), D 2 (5890 Å) and calcium Ca II H (3969 Å), K (3934 Å), among many others. It became clear that the electromagnetic spectrum of the Sun encodes information of its very nature.
Ground based observations are constrained to relatively longer wavelengths in the visible, NIR, and a few radio windows, whereas observations from space are not limited by the transmittance of the Earth’s atmosphere and thus enable us to probe more energetic phenomena on the Sun in the UV and X-rays.
Ideally, different diagnostics should be combined to benefit from their complementarity. This was the case of some of the most recent studies by, for example, Libbrecht et al. (2017, 2018); Leenaarts et al.
(2018); Shetye et al. (2018); Robustini et al. (2018), who presented state-of-the-art observations with the SST with different combinations of lines, and Nóbrega-Siverio et al. (2017); Rouppe van der Voort et al.
(2017); Gošić et al. (2018), who reported on coordinated observations with SST and IRIS. We refer to de la Cruz Rodríguez and van Noort (2017) for a review on radiative diagnostics for the chromosphere.
In Section 1.5 we presented the basics of line formation based on key aspects of radiative transfer. In this chapter we present in greater detail the formation mechanisms of some of the most commonly used radiative diagnostics of the solar atmosphere from high to low frequencies, although with an emphasis on the chromosphere.
2.1 The visible
The visible wavelength range features many of the most frequently used diagnostics for the lower atmosphere. The photosphere is commonly observed, for example, at neutral iron lines, some of which are Fe I 5247, 5250, 6173, 6301, 6302 Å etc., (e.g Beckers and Schröter, 1969; Skumanich and Lites, 1985; Lites et al., 1996; Domínguez Cerdeña et al., 2003; Centeno et al., 2007), and at the TiO 7058 Å band and CH (G-band) at 4305 Å (e.g. Makita, 1968; Steiner et al., 2001; Rutten et al., 2001; Berdyugina et al., 2003; Wang et al., 2018). The interface between the top of the photosphere and low chromosphere can be observed, for example, in the Na I and Mg I lines (e.g. Leenaarts et al., 2010; Rutten et al., 2011;
Quintero Noda et al., 2018). The chromospheric diagnostic par excellence is H α (see reviews by Rutten,
Figure 2.1: The same region of solar atmosphere observed at the core of Fe I 6302 Å (left), Ca II 8542 Å (middle) and H α (right) on the disk. Credit: G. Vissers and J. da Silva Santos, SST 2017.
2007, 2008, 2010, and references therein), so much so that it conferred the name chromosphere to the reddish glow beyond the solar disk observed during eclipses. Other common chromospheric diagnostics are the Ca II H and K lines along with the infrared triplet at 8498, 8542, and 8662 Å (e.g. Shine and Linsky, 1972; Martinez Pillet et al., 1990; Cauzzi et al., 2008; Reardon et al., 2009; de la Cruz Rodríguez et al., 2013a; Quintero Noda et al., 2016, 2017) and the He I D 3 at 5876 Å and He I 10830 Å (e.g. Harvey and Hall, 1971; Zirin, 1975; Bommier et al., 1994; Lagg et al., 2004; Socas-Navarro and Elmore, 2005;
Asensio Ramos et al., 2008; Leenaarts et al., 2016; Xu et al., 2016; Libbrecht et al., 2017, 2018).
Figure 2.1 shows how markedly different the appearance of the solar atmosphere is when observed at some of these wavelengths. Narrow band images taken at core of Fe I 6302 Å shows the structure of the sunspot and pores in the photosphere as well as the top of the granules and bright plage. The core of H α reveals the chromospheric canopy with dense dynamic elongated fibrils, which are generally analogous to the structures seen at the core of Ca II 8542 Å but with less opacity in the latter.
2.1.1 The calcium lines
In Paper I we focused on the aforementioned Ca II lines as the primary diagnostics of the chromo- sphere in the visible range. Figure 2.2 shows the term diagram for the relevant transitions in a model of the Ca II atom from Bjørgen et al. (2018). Both the H and K lines and the infrared triplet lines share the same upper orbital (4p) but have different lower terms 1 with the H and K lines having a sharp term (3s 2 S e ) and the triplet lines having a diffuse (or metastable) term (3d 2 D e ).
The interest in the H an K lines is not a novelty; they have been observed for a long time because they are the only resonance doublet in the visible spectrum from an abundant element in its dominant stage of ionization (most calcium is ionized and in the ground state) in the photosphere and low chromosphere, hence their large opacity (e.g. Linsky and Avrett, 1970). Consequently, they are more controlled by the collisional rates in the lower chromosphere than H α is, thus more sensitive to local temperature (Cauzzi et al., 2008). The cores of these lines are formed in non-LTE somewhere in the chromosphere, but their wings are formed in the photosphere with their opacities following LTE (e.g. Sheminova, 2012), therefore they can be used to probe the temperature stratification in the lower atmosphere (Rouppe van der Voort, 2002; Henriques, 2012).
1
The electron states in an atom are described by the electron orbital configuration followed by term symbols of the form
2S+1L
Pwhere S
is the total spin quantum number, P is the parity (odd or even) and L is the total orbital quantum number in spectroscopic notation (L=0→ S,
L=1→ P and L=2→ D).
Table 2.1: The effective Landé factor for a few transitions of the Ca II atom.
line K (λ3934) H (λ3969) (λ8498) (λ8542) (λ8662)
g
eff1.17 1.33 1.07 1.1 0.87
Figure 2.2: Term diagram for the Ca II atom showing some of the relevant atomic bound-bound permitted (solid line), forbidden (dashed line) and bound-free (dotted) transitions and their respective wavelengths in ångstroms; PRD transitions are indicated in orange; the total angular momentum quantum number is written above the horizontal lines (from Bjørgen et al., 2018).
As shown on the left panel of Fig. 2.3, the H and K lines have very broad wings which are blended with a few telluric 2 lines and lines of other species in the solar photosphere. They are usually seen in absorption except above the limb where they appear in emission and they are narrower than on the disk. Emission in the cores of H and K on the disk occurs essentially in towards plage regions and on an irregular pattern in the network outlining super-granulation boundaries (Priest, 2014, and references therein).
These lines have also been used to probe the chromospheric structure above sunspots and, in par- ticular, a phenomenon called umbral flashes first described by Beckers and Tallant (1969). They are repetitive brightenings in the low umbral chromosphere with a period of three minutes (e.g. Rouppe van der Voort et al., 2003, and references therein).
Furthermore, it has long been known, at least since Babcock and Babcock (1955); Leighton (1959), that regions of high magnetic field strengths are correlated with enhanced emission in H and K. Likewise, the infrared triplet lines are well-know for being good diagnostics of magnetic fields (e.g. Socas-Navarro et al., 2000), with the 8542 Å line being arguably the best because of its larger opacity and higher Landé factor (Pietarila et al., 2007; de la Cruz Rodríguez et al., 2012). Despite the H and K lines having slightly larger effective Landé factors than the infrared triplet (see Table 2.1), the squared-wavelength dependence in the Zeeman effect implies that the infrared triplet lines produce stronger polarization signals (Section 1.5). Moreover, they can be modelled in 1D assuming CRD to good approximation (e.g. Uitenbroek, 1989), which is surely more convenient than the H and K lines which require 3D PRD computations (e.g. Socas-Navarro et al., 2000; Bjørgen et al., 2018).
Even so, the triplet lines have not been as widely used as their H and K siblings until a decade ago.
Some of the first high-spatial-resolution ( λ/D ∼ 0.23 00 ) observations in a large field-of-view of the 8542 Å line with the Interferometric BIdimensional Spectrometer (IBIS, Cavallini, 2006) at the Dunn
2
Absorption features due to gases in the Earth’s atmosphere such as H
2O, O
2and CO
2.
3930 3932 3934 3936 λ (˚ A )
K1V K2V K2R
K1R K3
8475 8500 8525 8550 8575 8600 8625 8650 8675
λ (˚ A )
8662 8542
8498
Figure 2.3: Left: the solar spectrum around the K line of singly ionized calcium; the absorption and emission features close to the core are labelled according to the designations established by Hale and Ellerman (1904) and are analogously defined for the H line. Right: the calcium infrared triplet lines (black thick line) and zoom-in close the core of the 8542 Å line.
Solar Telescope revealed a wealth of features from the reversed granulation, to bright points and dark fibrils (Rutten, 2007; Cauzzi et al., 2008), while observations at the SST with the CRISP instrument (Scharmer, 2006; Scharmer et al., 2008) at a resolution of λ/D ∼ 0.18 00 demonstrated the disk counter- parts of type II spicules (Rouppe van der Voort et al., 2009), and umbral flashes (de la Cruz Rodríguez et al., 2010, 2013b).
The right panel in Fig. 2.3 shows the solar spectrum in the window containing the infrared triplet of which we highlight the 8542 Å line. We can see that its core is characterized by a transition (marked by vertical dashed lines) between the shallower wings and the steeper flanks that confer large Doppler sensitivity. It also marks the transition between photospheric LTE formation, with the line S ν decreasing with height with the electron temperature (Section 1.5), and chromospheric non-LTE formation with S ν decreasing steeply with height with increasing electron temperatures because the radiative rates start to dominate over the collisional ones in the less-dense chromosphere. This quick transition between the two regimes over such a short wavelength range originates a gap in the line opacity around the temperature minimum (Cauzzi et al., 2008).
For a while, the comparison between K-filtergrams and the data taken at 8542 Å in higher spectral resolution had been somewhat disappointing because the H and K images did not show the same fine de- tails as the infrared line. The latter is, in fact, easier to observe than H or K because the seeing conditions are better towards the red where the photon flux is higher allowing better spectral resolution (Reardon et al., 2009). More importantly, the large passband ( ∼ 0.3-0.6 Å) of the filters at blue wavelengths that had been used caused significant blend of photospheric and chromospheric signal, so the fine structures as seen in H α and in the infrared lines could not be resolved, though they were predicted to exist (e.g.
Reardon et al., 2009; Pietarila et al., 2009). In addition, the low filter transmissions and the lower de- tector efficiencies had to be counter-balanced by long exposure times, thus not permitting to track rapid motions on the chromosphere. These limitations have hindered chromospheric science.
Being able to efficiently scan the doublet lines is important because they sample a greater range of heights than the triplet lines and have larger opacity (e.g. Cauzzi et al., 2008). Also, they provide higher spatial resolution at shorter wavelengths. On the modeling side, the more transitions of the same atom are observed at the same time, the more reliable are the inversion results (Section 3.2).
The observational issues were overcome very recently with the development of the CHROMIS
instrument: a new Fabry-Pérot interferometer installed at the SST (Scharmer, 2017). It was designed
to observe the H and K lines not only at unprecedented spectral resolution ( ∼ 0.13 Å) but also spatial
Figure 2.4: Observation of an active region at three wavelengths near the Mg II k line with IRIS (from Carlsson et al., 2015).
resolution ( λ/D ∼ 0.08 00 or ∼ 60 km) provided by the 1 m-mirror of the SST.
Rouppe van der Voort et al. (2017) reported on the link between intermittent magnetic reconnection and plasmoid 3 ejection associated to UV-bursts using the first data acquired by CHROMIS. UV-bursts are believed to be pockets of hot plasma in the cooler photosphere that is momentarily heated to approxi- mately 8 × 10 4 K (assuming coronal equilibrium) making them visible in UV images (Peter et al., 2014).
Their fast-cadence (15-25 s) observations in the K line was crucial to track the rapid changes during the event and to test the scenario of plasmoid-mediated reconnection.
On the modeling side, Bjørgen et al. (2018) solved the RTE in full 3D non-LTE for the first time using the MULTI3D code developed by Leenaarts and Carlsson (2009). However, comparisons be- tween CHROMIS data and the synthetic spectra from several 3D radiation-MHD simulations of the chromosphere computed with the Bifrost code (Gudiksen et al., 2011) do not seem to agree. Bjørgen et al. (2018) conclude that something is missing in all models that would be responsible for the broad observed profiles. This is further discussed in Section 3.1.
2.2 The ultraviolet
The Lyman- α and - β lines, the far-UV (FUV) continuum, Si IV 1394 Å and 1403 Å, and He II 304 Å probe the upper chromosphere and transition region (e.g. Vernazza et al., 1981; Rutten, 2017; Lemen et al., 2012; de la Cruz Rodríguez and van Noort, 2017, and references therein). Observing these layers in the UV offers a seeing-free view into the largely unknown processes that trigger the steep temperature rise at the base of the transition region.
The very first observations of the Sun in the UV were reported by Durand et al. (1949) after sev- eral rocket launches with a spectrograph onboard. The spectra showed very broad emission coincident with the h and k lines of Mg II. Since then, the resonance UV lines have been observed with improved technology resorting to either rockets or balloons to reach beyond Earth’s atmosphere (e.g. Lemaire and Skumanich, 1973; Doschek and Feldman, 1977; Allen and McAllister, 1978; Kneer et al., 1981; Staath and Lemaire, 1995; Morrill et al., 2001; Morrill and Korendyke, 2008).
A considerable leap forward was achieved with the launch in 2013 of the Interface Region Imaging
3
Observationally, a plasmoid (or ”magnetic islands”) is identified as a moving bright blob generically associated with magnetic reconnection
in surges and flaring activity.
Spectrograph (IRIS, De Pontieu et al., 2014) which became the first telescope observing the chromo- sphere and transition region at a spatial resolution of λ/D ∼ 0.33 − 0.4 00 in the FUV and NUV, respec- tively. IRIS provides high-cadence spectra and slit-jaw images of the solar atmosphere uninterruptedly for at least eight months per year as it orbits around the Earth. The main IRIS diagnostics are the lines of Si IV, Mg II h, k and UV triplet and C II.
Coordinated observations with SST and IRIS have been important, for example, to shed light on chromospheric bright grains and their relation to propagating shock waves through the atmosphere (Martínez-Sykora et al., 2015b), to investigate heating in plage regions (Carlsson et al., 2015), to find disk-counterparts of type II spicules (Rouppe van der Voort et al., 2015) and to constrain the properties of Ellerman bombs 4 (Vissers et al., 2015)
2.2.1 The magnesium lines
In Paper I we studied the Mg II lines as our main temperature diagnostic in the UV, so we now focus on their properties. The physics of line formation for the Mg II atom is well understood since the early works of Dumont (1967); Milkey and Mihalas (1974); Milkey et al. (1975); Ayres and Linsky (1976) among others. Nonetheless, the interpretation of their profiles is not necessarily trivial.
Being a resonance doublet of an element in its dominant ionization stage, the formation of the h and k lines is in a way analogous to the H and K lines of Ca II that we presented in Section 2.1. They are formed from the transition between the upper term 3p 2 P o to the lower (ground) term 3s 2 S e , but since the energy of the excited state is comparatively higher the h and k lines occur in the UV. Similarly, the subordinate triplet lines of Mg II correspond to transitions between the terms 3d 2 D e and 3p 2 P o , which represents a larger energy jump (as large as the ones that originate the h and k lines) compared to their homologous infrared triplet, hence their visibility in the UV range at 2798, 2790.8 and 2797.9 Å.
Because magnesium is roughly 18 times more abundant than calcium in the solar atmosphere (As- plund et al., 2009), the cores of the h and k lines are formed relatively higher in the upper-chromosphere where the conditions are significantly different (Section 1.2). This explains why the h and k lines sys- tematically show double emission reversals, except above sunspots (Morrill et al., 2001) and often in plage (Carlsson et al., 2015) where they are single-peaked, whereas the H and K lines usually do not, but are in absorption or with a single emission peak (Rezaei et al., 2008; Beck and Rezaei, 2011).
Figure 2.5 shows the spectrum in the Mg II h and k window observed by IRIS averaged over a raster scan on the QS (Kayshap et al., 2018). We see that this portion of the UV spectrum includes several (photospheric) absorption lines besides the (chromospheric) emission peaks of the doublet lines with a central reversal in between. The line features are labelled in analogy to the Ca II lines as described in Section 2.1. Two of the UV triplet lines can be seen between the h and k cores, while the other one (not displayed) lies on the blue wing of the k line.
Modeling these lines also requires 3D non-LTE radiative transfer (more importantly between the emission peaks) with PRD of scattered photons in the wings and emission peaks, but the central minimum can be modelled in 3D CRD to good approximation (Leenaarts et al., 2013a). The k line is stronger than the h (Fig. 2.5) meaning that it forms higher in the atmosphere according to Leenaarts et al. (2013a). The brightness temperature of the peaks seem to be well-correlated with the gas temperature at τ = 1 , at least for T b > 6000 K, and the line shapes provide valuable velocity information in the upper chromosphere (Leenaarts et al., 2013b).
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