Title: Magnetic field determination of chromospheric spicules using the Weak Field Approximation.
AUTHOR: Matheus Aguiar-Kriginsky Silva
Master’s Thesis
Master’s degree in Advanced Physics and Applied Mathematics (With a speciality/Itinerary Astrophysics and Relativity)
at the
UNIVERSITAT DE LES ILLES BALEARS
Academic year 2018-2019
Date: September 18, 2019
UIB Master’s Thesis Supervisor: Ramón Oliver
Content
1 Introduction 5
1.1 Overview. . . 5
1.2 The Sun’s atmosphere . . . 5
1.2.1 The photosphere . . . 5
1.2.2 The chromosphere . . . 6
1.2.3 The corona . . . 8
1.3 Chromospheric spicules . . . 8
2 Characterising radiation 11 2.1 The polarisation ellipse . . . 11
2.2 Special cases . . . 13
2.2.1 Linear polarization . . . 13
2.2.2 Circular polarization . . . 13
2.3 Complex notation . . . 14
2.4 Non-monochromatic radiation . . . 14
2.5 The Stokes parameters . . . 15
3 The weak field approximation 17 3.1 The Zeeman effect . . . 18
3.2 The Doppler effect and the Doppler width . . . 20
3.3 The WFA equations . . . 21
3.4 Magnetic filling factorf . . . 22
4 The Ca II 8542 Å line 23
5 The magnetic field and the Stokes V parameter 24
6 Observational data 26
7 Data Analysis 28
7.1 Correlation analysis . . . 28
7.2 StokesV asymmetry. . . 29
7.3 Bayesian inference . . . 29
7.3.1 Likelihood distribution . . . 30
7.3.2 Prior Distribution . . . 30
8 Results 31 8.1 Noise characterisation . . . 31
8.2 StokesV asymmetry . . . 34
8.3 BLOS results . . . 36
8.4 General results . . . 38
9 Conclusions 39
10 References 41
1 Introduction
1.1 Overview
The main goal of this study is determine the magnetic field values of chromospheric spicules from high cadence, high spatial resolution data using the weak field approximation. A study of the magnetic field values is performed, with an analysis of the possible implications of the results obtained and their validity.
1.2 The Sun’s atmosphere
We are currently orbiting a giant mass of plasma with a nuclear reactor at its core and an enor- mously complex structure. The Sun is a G-type main-sequence star, just past the middle age of its 10 billion-year life, and it has been the subject of our interest since the beginning of times.
The evolution of life is directly linked to the presence and characteristics of the Sun, since it serves as a source of energy that enables water on Earth to be in liquid form, it is the source of the winds that drive the dynamics of our atmosphere and it provides a gravitational "lock" for our planet to orbit.
Given its significance, the Sun has been observed and studied for a long time. Understanding the inner workings of our host star is one of the most important scientific goals of today.
Just like the Earth, the Sun also has a very dynamic atmosphere. The presence of high tempera- tures and strong magnetic fields is the rudder that guides the apparently chaotic behaviour of the Sun’s atmosphere. The solar atmosphere is usually divided in three main layers: the photosphere, the chromosphere and the corona.
1.2.1 The photosphere
The photosphere is innermost layer of the Sun, with an average thickness of 100 km and temper- atures around 5800 K. It is the layer we see with our eyes, as it produces most of the visible light that reaches us from the Sun (see Figure 1, left).
The photosphere is divided into small bright regions called granules (Figure 1, right), which are formed as a result of the upwelling of hot plasma bubbles from the interior of the Sun. What we are really looking at is the top of the outermost layer of the solar interior, the convection zone.
The granules pop in and out of existence in a few minutes, with a diameter of approximately 1000 km.
Figure 1: Left: Photosphere seen with the Solar And Heliospheric Observatory (SOHO) in 2003.
The dark spots around the equator and in the southern hemisphere are sunspots. Right:
Granulation of the photosphere as seen by the Swedish 1-m Solar Telescope (SST) in 2010.
In regions of strong magnetic activity, if the magnetic field from the Sun breaks through the surface into the atmosphere, an active region is formed. The active regions can be easily located on images from the photosphere as dark regions called sunspots (see Figure 1, left).The earliest plausible records of sunspot classification dates back to 800 BC found in the Book of Changes, a Chinese ancient book [17]. The amount of sunspots present in the solar photosphere is directly related to the magnetic cycle of the Sun.
Our understanding of the physical processes that come into action in the photosphere to produce the phenomena that we observe is much deeper than what we know about the upper solar at- mosphere. The reason is rather simple: many of the approximations that simplify the physical description of the solar atmosphere rendering the problem mathematically simpler are only valid in the photosphere, while in the solar chromosphere and corona they no longer hold.
1.2.2 The chromosphere
Above the photosphere lies the layer known as chromosphere. It is called “chromo” because of its reddish color which can be spotted in solar eclipses (see Figure2). The reddish color is due to the recombination of hydrogen present in the chromosphere, specifically the Hα transition.
The first reports of chromospheric phenomena were seen during solar eclipses, and date back to medieval times [19]. The first documented observation of what is unequivocally chromospheric emission is from the 18th century [44]:
“Captain Stannyan, in a report on the eclipse of 1706, observed by him at Berne, noticed that the emersion of the Sun was preceded by a blood-red streak of light, visible for six or seven seconds on
the western limb... This outer envelope... seems to be made up not of overlying strata of different density, but rather of flames, beams and streamers, as transient as those of our own aurora borealis.
It is divided into two portions... the outer portion... may almost, without exaggeration, be likened to ‘the stuff that dreams are made of ’, since it is chiefly due to the ‘corona’... At its base, and in contact with the photosphere, is what resembles a sheet of scarlet fire... This is the ‘chromosphere’, a name first proposed by Frankland and Lockyer in 1869... in allusion to the vivid redness of the stratum, caused by the predominance of hydrogen in these flames and clouds."
At its base, the chromosphere is slightly cooler than the top of the photosphere, but the temperature increases again with height before reaching a plateau halfway to the top of the layer.
The chromosphere is a very enigmatic and chaotic layer, since the plasma there transitions between two very different situations. At the lower atmosphere, the plasma motions are not constrained to the shape of the magnetic field, and the gas pressure is higher than the pressure caused by the presence of the magnetic field. The opposite happens at the top of the solar atmosphere, since the magnetic pressure is much higher than the gas pressure, and the plasma is forced to move along the magnetic field lines. The chromosphere, being about 2000 km thick, is a transition region between these two regimes, and the dynamics of the plasma present in this layer are very complicated to study and comprehend.
Figure 2: August 1999 total solar eclipse [27]. The solar chromosphere can be identified as the red ring. The outer “cloud” surrounding the Sun is the solar corona.
The combined effects of magnetic guidance and small-scale gas thermodynamics [19] lead to a vast amount of fine structures such as jets, spicules(see Figure 3), fibrils, mottles, etc. that make the chromosphere a very unique part of the solar atmosphere. In particular, chromospheric spicules are the subject of study of this work, so they rightfully deserve a section of their own.
Figure 3: Off-limb spicules viewed in a high-resolution image observed by the Solar Optical Telescope (SOT) [16].
1.2.3 The corona
The corona is the outermost layer of the solar atmosphere. It is 10−12 times less dense than the photosphere but it is much warmer, with temperatures reaching 106 K. This rather unusual and counter-intuitive configuration of the solar atmosphere still baffles solar physicists, and it is known as the coronal heating problem.
The corona is much fainter than the photosphere, and it is only visible to the naked eye during solar eclipses, when the solar disk is obscured by the Moon (see Figure 2).
As mentioned above, in the corona the magnetic field drives the motion of the plasma, which means that the images of the solar corona are closely related to the topology of the magnetic field that is present in the region. A representative example of such magnetic structures are coronal loops (see Figure 4). The plasma serves as a tracer to delimit the shape of the magnetic field.
In regions where the Sun’s magnetic field opens to the interplanetary space, a region known as coronal hole appears. These regions are usually seen as dark spots in X-ray and UV images of the Sun (see Figure5), and their density is lower than that of the rest of the corona. The configuration of the magnetic field allows the particles to escape, and coronal holes are thought to be the main source of high speed solar wind streams.
1.3 Chromospheric spicules
High spatial resolution off-limb observations of the solar chromosphere taken during or outside a sloar eclipse show that the chromosphere is populated by many rapidly changing hairlike features, called spicules [5]. They were first observed by Secchi (1877) [35], while Roberts (1945) baptised them as “spicules" [32].
Figure 4: Magnetic loops observed by the Transition Region and Coronal Explorer (TRACE) [18].
Figure 5: Coronal holes seen as dark regions by the Solar Terrestrial Relations Observatory (NASA) [40].
Spicules have been the subject of observations for over a century, but because of the poor spatial resolution of the available instruments (see Figure 6) only qualitative studies of their properties were initially conducted. With the advent of high-resolution observations from modern instruments such as the Hinode Solar Optical Telescope (see Figure 3), a massive leap in the understanding of
spicules and their properties has taken place in recent years.
Figure 6: Observations of spicules through a tunable Zeiss Hα filter. Source: R. B. Dunn [5].
Spicules are seen as thin, elongated jet-like features that are highly dynamic. Since their discovery, spicules have attracted increased interest because of their potential to act as an energy and mass bridge between the dense photosphere and the tenuous but extremely hot corona [5, 41, 45]. A comprehensive study of the properties of chromospheric spicules was conducted by Pereira et. al.
(2012) [31], and it is enlightening to briefly sum up their results here:
• There are apparently two types of spicules. Type I spicules are seen to clearly rise and fall, with lifetimes of 150–400 s and upward velocities of 15–40 km/s. Meanwhile, type II spicules have shorter lifetimes of 50–150 s, faster upward velocities (30–110 km/s) and do not appear to fall down, but they fade around their maximum length instead.
• Type II spicules are by far the most common type, seen in quiet sun and coronal holes. On the other hand, type I spicules are mostly seen in active regions.
• The driver of the motion of type I spicules seems to be linked to magnetoacoustic shock waves, while the driver of type II spicules is still not clear.
• The path followed by type II spicules in a time-distance diagram seems to be rather linear, while the trajectory of type I spicules is best fitted by a parabolic motion.
• The maximum length of chromospheric spicules is around 7000 km, being specially longer in coronal holes. This is probably due to the fact that in such regions they are less inclined because of the magnetic field structure.
Another important feature of chromospheric spicules is their oscillatory behaviour, specially of transverse nature, with periods between 1 and 15 minutes [45, 10,22, 29].
2 Characterising radiation
The primary physical parameters used in this project are the so-called Stokes parameters. This section is specifically designed to define them in such a way that most readers can have an approx- imate idea of what they represent and what is done experimentally to measure their values.
The Stokes parameters offer one of several ways to describe the polarisation properties of a radiation beam. The polarisation phenomena are directly related to the freedom that the magnetic (or electric) field has to span the plane perpendicular to the plane of propagation.
The brief description given in this section is largely based on the book Polarization in Spectral Lines [11], where a much more thorough discussion is presented.
2.1 The polarisation ellipse
The description of polarisation can be done in terms of the polarisation ellipse, either using the magnetic field or the electric field because both quantities are related by the expression
B(~~ r, t) =~n×E(~~ r, t), (1) where ~n is the unit vector in the direction of propagation of the radiation wave. In a right- handed coordinate system in which the direction zˆ corresponds to the direction of propagation, the electromagnetic wave has the expressions
Ex(~r, t) =E1cos(kz−ωt+φ1), (2) Ey(~r, t) = E2cos(kz−ωt+φ1), (3) where E1, E2, φ1, φ2 are positive constants that specify the amplitudes and phases of the electric field components and k, ω are the wave number and angular frequency, respectively. In a fixed plane (for example, z = 0) perpendicular to z, equations (2) and (3) reduce toˆ
Ex(~r, t) =E1cos(ωt+φ1), (4) Ey(~r, t) =E2cos(ωt+φ2). (5)
Figure 7: Polarisation ellipse [11].
In the xy-plane, the tip of the electric field describes an ellipse, as shown in Figure 7. This is the so-called polarisation ellipse. The (x0, y0) axes represent the proper axes of the polarization ellipse, the long axis being directed alongx0. Defininga and bas the semi-major and semi-minor axes, we obtain
Ex0(t) =acos(ωt−φ0), (6)
Ey0(t) =bsin(ωt−φ1). (7)
From the rotation matrix that defines the relation between the old axes (x, y) and the new ones (x0, y0) we can obtain a relation between the parameters a, b, φ and the constants E1, E2, φ1, φ2
a2+b2 =E12+E22, (8)
ab=−E1E2sinφ1 −φ2, (9)
a2−b2 = (E12−E22) cos(2α) + 2E1E2cos(φ1−φ2) sin(2α), (10)
(E12 −E22) sin(2α) = 2E1E2cos(φ1−φ2) cos(2α). (11)
The above relations show that the properties of the polarisation ellipse depend on bilinear combi- nations of the electric field components. Now we introduce the following notations
PI =E12+E22, (12)
PQ =E12−E22, (13)
PU = 2E1E2cos(φ1−φ2), (14)
PV = 2E1E2sin(φ1−φ2), (15)
from which the quantities a, b, α can be easily obtained, except in the case PQ = PU = 0 which leaves α undefined. These four quantities are not independent, as they satisfy the relation
PI2 =PQ2 +PU2 +PV2. (16)
2.2 Special cases
There are a few special values ofE1, E2 and (φ1−φ2) for which the polarisation ellipse degenerates into either a circle or a segment.
2.2.1 Linear polarization
If PV = 0, which entails that either E1, E2 vanish or, alternatively, that φ1 =φ2 or φ1 = φ2 +π, we have
a=√
PI, b= 0,
which means that the electric field describes a segment in the plane perpendicular to the direction of propagation, and the monochromatic wave is said to be linearly polarised.
2.2.2 Circular polarization
If E1 =E2 and (φ1−φ2)=±π/2, the polarisation ellipse degenerates into a circle, with
PV =±PI, a= qPI
2 , b=±q
PI
2 .
If PV =±PI, the electric field vector rotates clockwise (+) or counter-clockwise (−) as viewed by an observer facing the radiation source.
2.3 Complex notation
The description of the electric vibration presented above can be done more easily using complex numbers, with the convention that a physical quantity is represented by the real part of such complex number. Equation (2), for example, can be rewritten in the form
Ex(~r, t) = Re
E1ei(kz−ωt)
, (17)
where E1 =E1eiφ1. The quantitiesPI, PQ, PU, PV are then given by the expressions
PI =E1∗E1+E2∗E2, (18)
PQ =E1∗E1− E2∗E2, (19)
PU =E1∗E2 +E2∗E1, (20)
PV =i(E1∗E2 − E2∗E1). (21)
2.4 Non-monochromatic radiation
So far we have only considered a pure monochromatic wave. The electric fields of most light sources, however, have a non-zero angular spread and a finite frequency bandwidth. In this case, it is impossible to define an instantaneous polarisation ellipse. It is however possible to define appropriate average quantities in order to define PI, PQ, PU, PV.
Considering a surface elementΣperpendicular to the direction of the wavevector~kand a coordinate system (xyz) as shown in Figure 8, it is possible to show [11] that at any given point P onΣ we can conduct appropriate averages, denoted by <...>, over time intervals much longer than the average wave period and over the surfaceΣ so that
PI =<E1∗(P, t)E1(P, t)>+>E2∗(P, t)E2(P, t)>, (22)
PQ =<E1(P, t)∗E1(P, t)>−<E2∗(P, t)E2(P, t)>, (23)
PU =<E1∗(P, t)E2(P, t)>+<E2∗(P, t)E1(P, t)>, (24)
PV =i[<E1(P, t)∗E2(P, t)>−<E2∗(P, t)E1(P, t)>]. (25)
Figure 8: Coordinate system defined to describe the propagation of a non-monochromatic wave [11].
2.5 The Stokes parameters
Now that we now that the quantities PI, PQ, PU, PV fully describe the polarization properties of a radiation beam even in the non-monochromatic case, we find ourselves in a position to relate their values to magnitudes that can be measured experimentally. In order to give these definitions, we need to imagine 3 types of detectors:
1. A detector capable of measuring, in absolute units, the electromagnetic energy falling on its acceptance area.
2. A linear polariser (also called analyser), a device that is transparent to the electric field vibration along a given axis (known as the transmission axis) and totally opaque to the electric vibration along the axis perpendicular to the former
3. A filter transparent to positive (negative) circular polarization and opaque to negative (pos- itive) circular radiation.
Finally, in order to give a proper definition of the Stokes parameters, a particular direction in the plane perpendicular to the direction of propagation needs to be fixed. This direction is defined as the reference direction or reference axis.
2.5.0.1 Stokes I
The Stokes parameterI is defined as the energy measured by the detector 1 defined above per unit time and per unit cross-sectional area. Given its definition, the first Stokes parameter is simply called the intensity of the radiation beam.
2.5.0.2 Stokes Q and U
Suppose now that we alternatively place linear polarisers with their respective transmission axes directed at 0◦, 45◦, 90◦ and 135◦, all angles reckoned counter-clockwise for an observer looking at the beam from the detector.
We can then define the Stokes parameter Qas the difference between the energy measured by the detector per unit time and per unit cross-sectional area when a linear polariser with 90◦ is placed between the beam and the detector and when a linear polariser with 0◦ is placed instead.
Similarly, the StokesU parameter is defined as the difference in the energies measured when placing a 135◦ linear polariser between the beam and a 45◦ linear polariser.
2.5.0.3 Stokes V
If we now place alternatively a filter transparent only to positive circular radiation and a filter transparent only to negative circular radiation, the last Stokes parameter (V) is defined as the difference between the energies measured by the detector. A schematic view of the definition of the Stokes parameters Q,U and V is pictured in Figure 9.
Figure 9: Representation of the definition of the Stokes Q,U and V parameters [11].
Making use of the definitions of the Stokes parameters, it is possible to show that they satisfy the relations
I =κPI, (26)
Q=κPQ, (27)
U =κPU, (28)
V =κPV, (29)
where κ is a constant. This means that the Stokes parameters are measurable quantities that unequivocally describe the polarization of the radiation beam coming from a source, which in the case of this study is the solar atmosphere.
3 The weak field approximation
Explaining the phenomenon of line formation in a magnetised plasma is not a simple task. The problem is to relate the polarization aspects of the emerging radiation, described by its Stokes
parameters, to the physical aspects and parameters that describe the medium it comes from and the magnetic field vector present there.
In the solar chromosphere, the plasma is not in Local Thermodynamic Equilibrium (LTE), because the density is so low that particle collisions are much less frequent than in the photosphere. The LTE approximation is very useful since it serves as a simplification that aids in solving the radiation transfer (RT) problem. Properly synthesising a spectral line requires much more work, as it entails solving, simultaneously and self-consistently, the RT and the statistical equilibrium equations.
This system is a non-local, non-linear problem that requires the numerical integration of the RT equation.
Because of the expansion of the magnetic field with height in the solar atmosphere, the value of the magnetic field intensity decreases, reaching lower values in the chromosphere than those found in the photosphere. This makes the process of experimentally measuring the chromospheric magnetic fields a task for the most sensitive instruments available. We are then faced with two challenges:
obtaining significant experimental values and later being able to interpret what they could possibly mean.
The most realistic way to interpret the experimental data would be full Non-LTE spectral line inversions, solving the coupled system of the RT and the statistical equilibrium equations. However, computational costs limit the capability to solve such problem, and efforts so far have had limited success, only being applied under specific simplifications and to a few spectral lines.
There are also peculiar lines, such as the HeI 10830 Å and D3 multiplets, that have a specific formation mechanism that only allows them to be formed in a thin layer at the top of the chromo- sphere. There are methods, such as Milne-Eddington and slab inversions, that try to profit from these type of lines in order to infer information about the chromosphere.
Here we present a different alternative, called the Weak Field Approximation (WFA), which serves as a mechanism to infer information about the magnetic field present in the medium and its prop- erties relying on a few assumptions that can be valid under some circumstances for chromospheric material. This method allows us to directly infer physical information about the astrophysical plasma without having to deal with the RT problem.
3.1 The Zeeman effect
First discovered in the laboratory in 1896 and used for measuring the magnetic field of the solar atmosphere since 1908, the Zeeman effect is the most reliable tool to solar physicists for the investigation of the magnetic phenomena taking place in the Sun [25].
According to the theory of atomic spectra, the presence of a magnetic field B~ affects the energy levels of the atoms inside the medium. The total Hamiltonian of the atomic system H now corresponds to the original Ho with the addition of the magnetic HamiltonianHB, defined by
HB= e0h
4πmc(J~+S)~ ·B~ + e20
8mc2(B~ ×~r)2 (30)
where e0, m are the electron’s charge and mass, h is Planck’s constant, c is the speed of light and J,~ S~ are the total angular momentum and total spin of the electronic cloud.
The second term in the equation above is known as the diamagnetic term, and for most astro- physical and laboratory plasmas it is negligible in comparison to the first term. It needs to be considered only in the presence of extremely large magnetic fields, such as those found in magnetic white dwarfs [2]. We can safely write then
HB = e0h
4πmc(J~+S)~ ·B.~ (31)
Assuming that the magnetic field is so weak that the energy associated with it is much smaller than the original energy intervals introduced by Ho, perturbation theory [28] can be applied to quantify the effect ofHB on the original energy levels. If the original HamiltonianHo is invariant under rotations, the total angular momentumJ and its projectionM along an arbitrary z axis are good quantum numbers, so that Ho is diagonal in the basis|αJ M >.
According to perturbation theory, the correction to the original energy levels is given by the matrix
< αJ M|HB|αJ M0 >=µoB < αJ M|(J~+S)~ ·~b|αJ M0 >, (32)
where µo is Bohr’s magneton (µo = 9.27×10−21 erg G−1), and~b is a unit vector in the direction of the magnetic field. If we choose to align the z direction with the direction of B, the matrix~ elements defined above are given by the expression
< αJ M|HB|αJ M0 >=µoBgM δM M0, (33) where g is the Landé factor:
g = 1 + < αJ||S||αJ >~
pJ(J+ 1) , J 6= 0. (34)
The wavelengths of the transitions between two atomic energy levels can be computed in a simple way in the case of infrared and visible lines:
λJ JM M0 0 =λ0−∆λB(g0M0 −gM) (35) where
∆λB = λ0e0B
4πmc2, (36)
λ0 being the wavelength of the transition when the energy levels are unperturbed.
Having defined the Landé factor, it is useful for the purposes of this study to introduce the so- called effective Landé factorg¯[24]. Given two atomic levels labelled 1 and 2, ordered by increasing energy, the effective Landé factor is defined as
¯ g = 1
2(g1+g2) + 1
4(g1−g2)[J1(J1+ 1)−J2(J2+ 1)). (37)
3.2 The Doppler effect and the Doppler width
The discussion in the previous section was aimed to demonstrate how the magnetic field of a medium can be estimated based on the spectral lines of elements. However, the atoms are in constant movement inside any medium and the spectral profiles are sensitive to these motions, in a phenomenon called the Doppler effect.
Assuming that the velocities are not relativistic, the motion of an atom along the line-of-sight (LOS) of the detector will cause a shift in the measurement of the frequency of the atomic transition, given by the expression
νo0 =νo
1−ω c
, (38)
where νo is the original transition frequency,νo0 is the observed frequency and ω is the component of the velocity of the atom along the line-of-sight.
The velocity ω is usually decomposed into two contributions: the bulk velocity ωA, which cor- responds to the velocity of the ambient medium, and an additional random velocity caused by thermal or micro-turbulent motions ωT. Assuming a Maxwellian distribution of velocities for ωT, it is given by the expression
ωT = s
2kBT
µM +ξ2, (39)
whereT is the temperature of the medium, kB is the Boltzmann constant,µis the atomic weight, M is the mass of the atomic species and ξ is the micro-turbulent velocity. Having defined ωT, different reduced variables are introduced in the analysis of dispersion profiles. Based on our purpose to analyse the effect of the magnetic field upon the atomic energy transitions, we will make use of one of these variables: the normalized Zeeman splitting vB:
vB= ∆λB
∆λD, (40)
where ∆λD is called the Doppler width:
∆λD =λ0ωT
c . (41)
3.3 The WFA equations
Having defined all the useful quantities necessary to understand the regime of applicability of the WFA, in this section we proceed to enumerate the conditions that need to be satisfied for the WFA equations to be satisfied.
As mentioned above, the RT equation is a set of relations between the Stokes parameters and the polarization features of the emerging radiation. This equation is usually complex and difficult to solve, although under a set of conditions it can be simplified. Those conditions are:
1. The magnetic field is uniform along the line of sight.
2. The magnetic field is weak or, more precisely, the Zeeman splitting is much smaller than the Doppler width. This condition essentially means
¯
gvB = ¯g∆λB
∆λD 1. (42)
When these two conditions are satisfied, a perturbation scheme can be applied to the RT equations, reducing them to simpler relations between the Stokes parameters. The relation we are going to explore in this study is
V(λ) = −∆λBcos(θ) ∂I
∂λ
, (43)
where θ is the angle between the magnetic field vector and the direction of propagation of the radiation beam (see Figure 10). Equation (3.3) appears at first order of the perturbation scheme, and at this order Qand U must vanish.
The first condition detailed above seems restrictive, but it can be satisfied in some situations. For example, off-limb observations of the solar atmosphere provide little inhomogeneity along the line of sight under some circumstances. There are ways to identify possible signs of gradients along the line-of-sight in observed Stokes profiles, providing useful mechanisms to reassure that the magnetic field is indeed constant along the LOS direction.
The second condition can be rewritten as
¯ gλ0eoB
4πmc s
2kBT
µM +ξ2 1 ⇒ ¯gB
q
1.663×10−2Tµ +ξ2 1.4×10−7λ0
, (44)
Figure 10: Coordinate system defined to describe the magnetic field vector [11]. ~ea and ~eb are the vectors that define the plane perpendicular to the direction of propagation of the radiation beam.
Ω~ represents the direction of propagation of the electromagnetic wave.
where B is expressed in gauss G, T in K, µ in atomic units [u], λ0 in Å and ξ in km/s. This means that ultimately condition 2 is a restriction on the value of the magnetic field that can be considered “weak" given a line (through the values of ¯g, λ0 and µ) and a medium (through the values of T and ξ).
It appears then that given the right choice of spectral line and instrumentations, the WFA can present itself as a very powerful experimental tool to infer the physical conditions of the solar atmosphere, as the current project intends to verify.
3.4 Magnetic filling factor f
The relation between the Stokes parameter V and the derivative of the Stokes I parameter pre- sented in the previous section assumes one important thing about the magnetic field: it is uniform across the resolution element [11,3] (pixel). This condition can be satisfied when numerically mod- elling the plasma, but it should be addressed carefully when conducting studies over experimental data.
The magnetic field can have small-scale variations that lead to the assumption to be invalid even for telescopes with high-resolution imaging capabilities. One approach that is widely used to address this issue is the introduction of the concept of the magnetic filling factor f. This quantity represents the idea that the magnetic field covers only a fraction of the pixel, with the remaining fraction (1-f) corresponding to a field-free or non-magnetic solar area. The relation presented must be modified, resulting in the expression
V(λ) = −f∆λBcos(θ) ∂I
∂λ
. (45)
There are more sophisticated methods that try to allow for more complicated structuring of the magnetic field, but it is not the aim of this study to analyse them in detail. Furthermore, since we are dealing with chromospheric magnetic fields which are assumed to fill the whole of the plasma in that layer, we will assume that the actual value of the filling factorf is very near to 1, allowing us to safely choosef = 1for the remaining of this work. This section is aimed merely to inform the
reader about the possible limitations and further analysis is needed should the WFA be applied to other magnetic structures.
4 The Ca II 8542 Å line
In this study we make use of spectropolarimetric observations in the Ca II 8542 Å line, which is the central line in the Ca II infrared triplet. It is formed from a transition between the 42P3/2 and 32D5/2 levels (Figure 11).
Figure 11: Grotrian diagram of the Ca II ion, with the infrared triplet highlited in red. Source:
Esmu Igors [12].
This is one of the most frequently used lines to study the orientation and strength of the chromo- spheric magnetic field [7]. It is easily accessible to ground-based telescopes due to its location in the near-IR side of the solar spectrum.
The formation height of this line ranges from theτ5000 = 1level, the classical photospheric surface, to around 1500 km above such surface[15,6]. The line core is formed in the central part of the chro- mosphere. It is then a broad line which is magnetically sensitive, allowing for spectropolarimetric studies of the magnetic structures inside the chromosphere.
As noted in Section 3.3, the choice of line and magnetic region effectively limits the validity of the WFA by means of the maximum value of the magnetic field that can be present in the plasma.
Choosing an average chromospheric temperature of 12000 K and a micro-turbulent velocity of 1 km/s coupled with the properties of the line yields the condition
B 2400 G. (46)
The magnetic field intensity in the chromosphere is expected to be much lower than such value due to the expansion of the magnetic field lines with height. It would be reasonable to say that the WFA is safely applicable to the chromosphere when the Ca II 8542 Å line is used. A thorough study of the applicability of the WFA to the chromosphere with this line conducted by Centeno et. al. [8] shows that the WFA should not be applied to regions with magnetic fields higher than 1500 G. This is reassuring since we are not expecting to infer such values, and the only limitation to the applicability of the WFA is the need for the magnetic field to be constant along the line of sight.
5 The magnetic field and the Stokes V parameter
As seen in Section3.3, the magnetic field needs to be independent of the optical depth (∂B/∂τ = 0) for the WFA equations to hold. In this work, the magnetic field is itself the object of study, with the WFA serving as the source of information about its value. This means that apparently there is no way to check if the magnetic field is indeed independent of the optical depth.
However, the presence of a magnetic field gradient along the line of sight can leave footprints in the profiles of the circularly polarized radiation (Stokes V) [37]. In a static atmosphere in LTE, the usual profile of the Stokes V parameter as a function of wavelength is characterised by two lobes of equal area and amplitude, with a zero crossing at the line center λ0 [4]. A typical asymmetric Stokes V profile is shown in Figure12.
There can be three main types of asymmetry in the circular polarization profile:
• Amplitude asymmetry. The amplitudes of the red and blue lobes (ar and ab in Figure 12) are different. We can define a measure of such asymmetry as
δa= |ab−ar|
max(|ab|,|ar|). (47)
δa= 1 corresponds to the completely asymmetric case (one lobe), while δa= 0 means that the lobes are identical in amplitude and there is no asymmetry.
• Area asymmetry. The areas of the red and blue lobes (Ar andAb in Figure12) are different.
We can define a measure of such asymmetry as δA= |Ab−Ar|
max(|Ab|,|Ar|) (48)
Figure 12: Typical asymmetric Stokes V profile. Source: Sheminova (2005) [36]
δA = 1 corresponds to the complete area asymmetry case (one lobe), while δA = 0 means that there is no area asymmetry.
• Zero-crossing wavelength. The wavelength of the zero-crossing of the circular polarization profile is different from the line centerλ0.
Extensive studies of these three types of asymmetries have been conducted since the late 1980’s [14,13,21,20,38], providing a thorough insight about the reasons for the existence of such profiles.
The shift in the zero-crossing wavelength of the Stokes V profile is associated with bulk flows, and when the associated velocity calculated from such shift differs from the Doppler velocity of the intensity profile, it is due to a local motion inside the fluid element at that specific position [14]. This means that the source of this type of asymmetry is simply a non-zero plasma motion with respect to the observer. Were we to use the full WFA equations up to second order in the perturbation scheme, we would need to filter out profiles of this type since the assumption of constant velocity along the line of sight needs to be met in addition to the previously presented conditions. However, since we are only using the first order approximation, which does not rely on such assumption, we will not need to discard them.
Amplitude and area asymmetries are strongly linked to variations in the magnetic field configura- tion combined with velocity shifts along the line of sight [36, 39]. This means that although we are not concerned with velocity shifts, we need to control these asymmetries since they are caused by a gradient in the magnetic field.
The ideal way to proceed would be to filter out these asymmetries separately, but we will only control the amplitude asymmetries. This because of the evidence [38] that the amplitude asym- metries are usually much larger than the area asymmetries. This means that we can control both
asymmetries by only focusing on measuring the amplitude asymmetry, without having to deal with noise-related issues.
6 Observational data
The observations used in this work were obtained with the CRISP imaging spectropolarimeter at the Swedish 1-m Solar Telescope [33] in 3 June 2016. They consist of spectropolarimetric data (a total of 15 wavelengths1) in the Ca II 8542 Å line. The cadence of the data is 36.33 s. Figure 13 shows examples of the four Stokes parameters at the line centre.
Figure 13: Images of the four Stokes profiles at the Ca II 8542 Å line centre at 16:42:30 UT on 3 June 2016. The red cross corresponds to the position of the Stokes profiles shown in Figure 14,
and the blue rectangles correspond to the area under study shown in Figure 22.
1The precise spectral positions (in Å) are: 8540.50, 8540.75, 8541.00, 8541.25, 8541.50, 8541.75, 8542.00, 8542.05, 8542.15, 8542.25, 8542.50, 8542.75, 8543.00, 8543.25, 8543.50.
Figure 14: Stokes profiles at the red cross in Figure 13.
The target of the observations was an active region, since the primary goal of the observational campaign was the study of coronal rain. This fact justifies the low cadence of the data, which makes it rather difficult to completely characterise the properties of the spicules present in the data. The only study conducted in this work is the analysis of the values of the magnetic field as inferred with the WFA.
The data were reconstructed with the Multi-Object Multi-Frame Blind Deconvolution (MOMFBD) [26, 43], and the CRISP data reduction pipeline was applied for standard data processing [9].
7 Data Analysis
Before performing the inference of the magnetic field values we need to have a plan of attack. We are looking for data points where the conditions that lead to the applicability of the WFA are satisfied, and with that in mind we need to design an algorithm that can find them.
7.1 Correlation analysis
Given the fact that the expression
V(λ) =−∆λBcos(θ) ∂I
∂λ
(49)
is essentially a linear relation between V(λ) and ∂I/∂λ, a good starting point is to estimate the linearity of such relation when treating observational data.
The approach we take is to calculate the Pearson correlation coefficient [30] R between V(λ) and
∂I/∂λ. The coefficientR between two variablesx and y is defined as
R=
Pn
i=1(xi−x)(y¯ i−y)¯ pPn
i=1(xi−x)¯ 2pPn
i=1(yi−y)¯ 2 (50)
|R|=1 means total linear correlation, and R=0 means that there is no linear correlation. With this definition we have set the first step in our algorithm, in which we choose to discard every data point where |R| < 0.9. On the left panel of Figure 15, a case where |R| < 0.9 is shown, meaning that such point is filtered out from the analysis. A case where |R| > 0.9 is shown on the right panel for comparison.
Figure 15: Plot of StokesV against ∂I/∂λfor a point where |R| <0.9 (left) and a point where
|R| >0.9. On top of each panel are shown the corresponding values of R.
7.2 Stokes V asymmetry.
Following the discussion presented in Section5, we need to be careful not to include in our analysis points that possess a detectable gradient of the magnetic field along the line of sight. The next step in our algorithm will then be to analyse the data points that pass the correlation analysis filter, and we will set an upper bound to the value of the amplitude asymmetry that we will accept, δa. Since it is almost impossible to have δa= 0 due to numerical accuracy and the noise present in the data, we will need to set an upper limit to δa based on the noise level compared to the Stokes V profiles depending on the temporal properties of the noise. We deal with these issues in Sections8.1 (noise characterisation) and 8.2 (StokesV asymmetry). An example of V profile that does not meet the asymmetry criterion is shown in the left panel of Figure 16 alongside one that does meet such criterion on the right panel.
Figure 16: Examples of StokesV profiles in which the correlation criterion |R|>0.9 is met. Left:
the asymmetry criterion is not satisfied. Right: The asymmetry criterion is satisfied.
7.3 Bayesian inference
Once the appropriate data points have been chosen, a Bayesian inference method is used to estimate the mean value of the line-of-sight magnetic field component (BLOS) in every pixel, as well as its Highest Posterior Density (HPD), that serves as a measure of how well constrained the magnetic field value is given the observational data.
The philosophy behind the Bayesian analysis is to use measurements of observed quantities in order to infer a probability distribution about the parameters that determine the value of such quantities. This procedure is based on Bayes’ theorem:
P(params|data) α P(data|params)·P(params), (51)
where:
• P(params|data) is called the posterior probability, the resulting probability distribution of the unknown parameters given the experimental measurements made.
• P(data|params)is called the likelihood distribution. It represents the probability of measur- ing the experimental values of the known quantities given a particular value of the unknown parameters.
• P(params)is the prior distribution, a probability distribution which reflects the belief one has of the values that the unknown parameters can take before the measurements are performed.
In order to arrive at the posterior distribution which will ultimately give information about the unknown parameters, it is insightful to separately describe the likelihood and prior distributions on detail.
7.3.1 Likelihood distribution
As mentioned above, the likelihood distribution represents the probability distribution of the mea- sured quantities given a particular value of the unknown parameters. On the case of this study, the measured quantities are the Stokes I andV. The quantity that determines their specific value, according to the WFA, is BLOS:
V(λ) = kBLOS∂I
∂λ, k =−f λ0e0
4πmc2. (52)
Given this information, one can ask what would the distribution of the values of V look like for a given value ofBLOS. If the WFA is to be believed, one would expect the mean of such distribution to be kBLOS∂I
∂λ. The presence of noise and any other random processes that occur at the time of the observations is what gives the standard deviation (σ) of the distribution a value different from zero. Should the WFA be an exact relation and the measurements be ideal, the distribution would be a Dirac delta and an observer would always measure the correct values of V and I that are related by the actual value of BLOS.
The simplest way then to define the likelihood distribution is to say that the measured value of V is given by a Gaussian distribution centred at kBLOS∂I∂λ with standard deviation σ. The choice of a Gaussian distribution is obviously arbitrary, but there is no particular reason to choose a more elaborate probability distribution.
7.3.2 Prior Distribution
Having determined the likelihood distribution, the next step is to choose appropriate prior distri- butions. As defined earlier, the prior distribution of the unknown parameters is the distribution that one expects them to have prior to the measurements. It needs then additional information provided by the knowledge that the physicist has about them.
There are two unknown parameters; BLOS and σ. Assuming that they are independent of each other, the prior distribution factorises into:
P(params) =P(BLOS, σ) = PB(BLOS)×Pσ(σ), (53) where PB(BLOS) is the prior distribution ofBLOS and Pσ(σ) is the prior distribution of σ
All that is known about σ is that it is positive, since it is a standard deviation. One can choose then a distribution that only allows positive values, for example a half-Cauchy distribution.
Regarding BLOS, one could argue that there is no way of knowing what values it could possibly have prior to its determination. This ignorance must be reflected on the choice of a distribution for its prior. The easiest way to reflect it is to choose a uniform distribution. This choice gives the same probability to all the values inside its range of definition. One thing that can be done in order to give a reasonable width for PB(BLOS) is to have a first guess of where the true value of BLOS must be. This is done by performing a least-squares fit between V and ∂λ∂I. BLOS can then be firstly approximated from the slope (see Equation52), and this value is where the uniform distribution is centred, with a total width of 500 G.
In order to carry out the product between the likelihood and the prior distributions and obtain the posterior distribution, a numerical sampler is needed. This is done with PyMC3 [1], a Python package devoted mainly to Bayesian inference problems. PyMC3 makes use of Markov Chain Monte Carlo methods to sample the posterior, and it also includes a wide functionality for summarising model statistics and the output. One particular statistic of interest in this study is the Highest Posterior Density (HPD) interval. A 100 (1 -α)% HPD is a region that satisfies the following two conditions:
• The posterior probability of that region is 100(1 -α)%.
• The minimum density of any value within that region is equal or larger than the density of any point outside that region.
No error bars will be given in this study, but a 95% HPD interval is given as a measure of the most probable values that BLOS can take at each spatial point and how localised such values are.
8 Results
8.1 Noise characterisation
Before performing any analysis, a study of the properties of the noise present in the data is necessary. The first step is to select an area of the field of view where the signal consists mostly of noise. An example is shown in Figure17. The time evolution of the images makes the selection of the area troublesome: because of the telescope rotation, the number of points inside the area diminishes from 70000 pixels on the first image to 11200 on the last frame. This means that there are less points to use in a statistical analysis and the properties of the noise could be affected.
Once the area is selected, a Gaussian distribution is fitted to the V data in order to obtain the standard deviation and characterise the noise. This procedure is done for each of the 15 spectral positions, for each time. Three examples can be found in Figure18.
An overview of some of the results is shown in Figure19. The noise standard deviation is plotted normalised by the continuum intensity Ic, which is determined by averaging the Stokes I at the line wings over an area on the solar disk [23]. In order to view the possible effects of the decrease
Figure 17: Areas selected for the analysis of the noise. Top: the area selected is plotted on the line center image of the Stokes I (left) and Stokes V (right) for the initial time frame. Bottom:
Same plots as the top row but for the last time frame of the dataset.
in data points during the observational time, the average noise profile for the five initial times is plotted together with the noise average of the last five times. There is a noticeable similarity between both averages, and while the discrepancies on the line wings could be caused by the lack of points in the last times, it could just as easily be an effect of the temporal evolution of the noise, since the only assumption made here is that it is of Gaussian nature, not stationary.
Figure 18: Statistical analysis results of the noise for a given time frame at three different spectral positions.
Figure 19: Comparison between the normalised average standard deviation of the noise present in the StokesV signal for the first and last five times.
8.2 Stokes V asymmetry
Having determined the approximate standard deviation of the noise present in the data as a function of time and wavelength, a study of the experimentally measured asymmetry of the Stokes V profile can be done.
The first thing to be pointed out is that there are only 15 wavelengths for whichV has measured values. The amplitude asymmetry relies on having the values of the extrema of V, and this can only happen if they are located precisely in one of these 15 spectral positions. This is virtually im- possible, and the extrema have to be approximated from the measured values. This approximation is not only compromised by the lack of spectral points, but also by the presence of noise.
The approach used in this study consists of fitting two kinds of curves to the Stokes V spectral profile, and choosing the "true" profile as the one which best fits the data.
The first model is very intuitive: the sum of two Gaussian curves. By choosing two Gaussians, both lobes of V can have different shapes and different absolute values of their extrema, which allows for amplitude asymmetries.
The second model is more specific of the problem at hand: the derivative of an exponentially modified Gaussian. The derivative of a normal Gaussian has no amplitude or area asymmetry, and fitting such a derivative toV would leave no place for the asymmetry analysis. The exponentially modified Gaussian is not symmetrical, i.e., it is skewed. The derivative of such function will have both area and amplitude asymmetries, and its form is already similar to the StokesV profile. This model is also used because fitting two Gaussians is not always reliable.
After fitting both models to the data, the Bayesian information criterion (BIC) [34] is used to choose the model that best fits the data. This criterion favours the model with the least number of free parameters while also favouring how likely it is to have the data if the ideal model represented the true behaviour of the physical parameter (similar to the likelihood distribution). An example of the results is shown in Figure20. Both models provide a good fit to the data, and in this particular example the modified Gaussian model has a lower BIC than the sum of Gaussians model (−216 vs −213).
Figure 20: Estimation of the StokesV asymmetry: comparison between the model fitting of a sum of two Gaussians and the modified Gaussian derivative.
Once the fitting is finished and the model that best fits the data is chosen, it is time to introduce a criterion based on the properties of the noise that accepts or rejects the spectral profile based on the asymmetry of the Stokes V parameter. The extrema of the fitted function are determined, and the difference between their absolute values is computed. If such difference is smaller than the standard deviation of the noise at that spectral position, i.e.,
|ab−ar|< εσ (54)
this spatial point asymmetry can be attributed to the presence of noise. εhas been chosen to be0.5, based in a comparison between this type of analysis and the one conducted by Kuridze et. al (2019) [23] on off-limb CRISP SST data. Their analysis of the asymmetry is based on approximating the extrema of the StokesV parameter directly from the experimental values, without fitting a model to the profiles. Their asymmetry condition is to limitδato be smaller than 0.4. If their asymmetry analysis is performed on the data points that pass the filter presented here,δais never higher than 0.3, which is ultimately a more restrictive condition than the one proposed by Kuridze et. al.
It is important to remark that in order for a point under study to be considered for the Bayesian inversion, its I and V parameters must first pass the correlation coefficient filter and then the asymmetry filter.
8.3 B
LOSresults
For all the data points that have “survived” the analysis detailed above, the Bayesian inference of BLOS detailed in Section 7.3 is performed to obtain the posterior distribution of BLOS and the 95% HPD interval. An example of the results for a particular pixel is shown in Figure 21.
Figure 21: Results from the Bayesian inference for the point marked with the red cross in Figure 13. The mean value and the 95% HPD interval of σ (in normalised counts units) and BLOS (in
G) are shown.
Figure 22 shows the points inside the blue rectangle of Figure 13 that satisfy all the imposed conditions, plotted on the intensity image of a particular time instant. The red rectangle represents the location of a particular spicule which is shown in more detail in Figure 23.
Figure 22: Results from the Bayesian Inference for a particular time instant. The blue line represents the position of the solar limb.
The spicule shown in Figure 23 is seen to last for at least five time intervals, or roughly 2.5 min. This particular spicule is special because there is information about its magnetic field over five consecutive frames, while the immense majority of spicules in our data set only has such information for one to two frames. A histogram of the inferred BLOS values for the five temporal images is shown in figure 24. The average BLOS value is 246 G, with an average 95% HDI of [186−300] G.
Figure 23: Detailed inference results for the rectangular box shown in Figure22.
Figure 24: Histogram of the inferred BLOS values for the spicule shown in Figure 23.
8.4 General results
The results for one particular spicule are interesting but of limited statistical significance. Mea- suring theBLOS of the overall field of spicules is then the next step in the analysis. A histogram of the inferredBLOS values over all the temporal frames inside the rectangle from Figure 13is shown in Figure 25. The average BLOS is 213 G and has a 95% HDP interval of [160−285] G.
Figure 25: Histogram of the inferred BLOS values for the whole data points inside the rectangle from Figure 13.
9 Conclusions
This is the first time the WFA is used to study the properties of the magnetic field of spicules.
There have been a few other studies that used different techniques to infer the magnetic field intensity of these structures, and their results differ significantly from the ones reported here.
Trujillo Bueno et al. (2005) ([42]) used a theoretical modelling of the Hanle and Zeeman effects together with spectropolarimetric observations in the He I 10830 line to infer the magnetic field vector in a field of quiet-sun spicules. They used observations carried out in 2001 with the Tenerife Infrared Polarimeter (TIP) mounted on the German Vacuum Tower Telescope (VTT) at the Ob- servatorio del Teide. The height above the solar limb of their observations was of 2000 km, similar to the heights of the analysis conducted in this study. Their data consist of several slit images, the equivalent of drawing a “straight line” over the data presented here at each time frame at a particular height above the visible limb. Furthermore, a temporal sum was applied to increase the Signal to Noise Ratio (SNR), resulting in an effective cadence of almost 5 min. Their inference provided the full magnetic vector, resulting in magnetic field strength values of the order of 10 G.
They did not discard, however, the possibility of the presence of much stronger magnetic fields in future observations, as is reported here.
If a temporal sum is conducted on the data presented here to match the effective cadence of the data analysed by Trujillo Bueno et al., the average BLOS values inferred decrease from 210 G to 180 G. This means that the temporal sum is not enough to explain the discrepancies between both studies. The difference must therefore lie on the type of spicules being analysed, since on this study an active region was targeted, and the magnetic field is much higher on active regions than on quiet Sun areas.
The temporal averaging with an effective cadence that can be higher than the mean lifetime of a spicule can have a big impact on the final BLOS value. The spicule may be present for a few frames of the observational data, but for the rest of them the space it occupies may be empty and therefore the average BLOS is lowered.
Another study conducted by Centeno et al. [8] using data from the TIP instrument at the VTT set a new lower bound value of 50 G for the magnetic field values on spicule regions, a study also conducted over a quiet Sun region. This result serves as further evidence that even in quiet Sun regions the magnetic fields can be high, and that the data analysed here represents a significant step forward on the study of the chromospheric magnetic field over active regions.
The values of BLOS reported here (213 G on average) are not only much higher than the ones reported in previous studies. Another interesting thing to note is the fact that they are all positive.
This is strange, but it is not difficult to explain from a geometrical point of view. The active region under study had just started to disappear beyond the West limb due to solar rotation, and due to that fact there was a significant inclination of the spicules toward the observer. Even if the spicules are oscillating transversally along the line -of-sight, the magnetic field inclination is forcing them to stay pointing predominantly towards the observer.
As noted earlier, the WFA is essentially a perturbation theory. To first order, the proportionality between ∂I/∂λ and V holds, while Q and U are zero. The careful reader may notice that on the observations (Figure 13) there appears to be a small but noticeable signal of Q and U. The signal is very weak and buried under the noise levels, but it appears to be present. If Q and U
are not zero, this means that the following orders of the perturbation scheme are not negligible, and further calculations should be performed. This signal of Q and U is mostly noticeable over the solar disk and not inside the region that was studied here. This means that the second order effects introduced on the data analysed are small in our region of interest.
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