# Measuring the magnetic origins of solar flares, CMEs and Space Weather

Philip Judge  High Altitude Observatory, National Center for Atmospheric
Research, Boulder CO 80307-3000, USA  Matthias Rempel  High Altitude
Observatory, National Center for Atmospheric Research, Boulder CO 80307-3000,
USA  Rana Ezzeddine  Department of Astronomy, University of Florida, 211
Bryant Space Sciences Center, Gainesville, Florida, 32611, USA  Lucia Kleint
Université de Genève, Centre Universitaire d’Informatique, 7, route de Drize,
1227 Carouge, Switzerland  Ricky Egeland  High Altitude Observatory, National
Center for Atmospheric Research, Boulder CO 80307-3000, USA  Svetlana V.
Berdyugina  Leibniz-Institut fuer Sonnenphysik (KIS), Schöneckstrasse 6,
79104, Freiburg, Germany  Thomas Berger  Space Weather Technology, Research
and Education Center, 3775 Discovery Drive N433 Boulder, CO 80309  Paul Bryans
High Altitude Observatory, National Center for Atmospheric Research, Boulder
CO 80307-3000, USA  Joan Burkepile  High Altitude Observatory, National Center
for Atmospheric Research, Boulder CO 80307-3000, USA  Rebecca Centeno  High
Altitude Observatory, National Center for Atmospheric Research, Boulder CO
80307-3000, USA  Giuliana de Toma  High Altitude Observatory, National Center
for Atmospheric Research, Boulder CO 80307-3000, USA  Mausumi Dikpati  High
Altitude Observatory, National Center for Atmospheric Research, Boulder CO
80307-3000, USA  Yuhong Fan  High Altitude Observatory, National Center for
Atmospheric Research, Boulder CO 80307-3000, USA  Holly Gilbert  High Altitude
Observatory, National Center for Atmospheric Research, Boulder CO 80307-3000,
USA  Daniela A. Lacatus  High Altitude Observatory, National Center for
Atmospheric Research, Boulder CO 80307-3000, USA

###### Abstract

We take a broad look at the problem of identifying the magnetic solar causes
of space weather. With the lackluster performance of extrapolations based upon
magnetic field measurements in the photosphere, we identify a region in the
near UV part of the spectrum as optimal for studying the development of
magnetic free energy over active regions. Using data from SORCE, Hubble Space
Telescope, and SKYLAB, along with 1D computations of the near-UV (NUV)
spectrum and numerical experiments based on the MURaM radiation-MHD and
HanleRT radiative transfer codes, we address multiple challenges. These
challenges are best met through a combination of near UV lines of bright Mg
II, and lines of Fe II and Fe I (mostly within the $4s-4p$ transition array)
which form in the chromosphere up to $2\times 10^{4}$ K. Both Hanle and Zeeman
effects can in principle be used to derive vector magnetic fields. However,
for any given spectral line the $\tau=1$ surfaces are generally geometrically
corrugated owing to fine structure such as fibrils and spicules. By using
multiple spectral lines spanning different optical depths, magnetic fields
across nearly-horizontal surfaces can be inferred in regions of low plasma
$\beta$, from which free energies, magnetic topology and other quantities can
be derived.

Based upon the recently-reported successful suborbital space measurements of
magnetic fields with the CLASP2 instrument, we argue that a modest space-borne
telescope will be able to make significant advances in the attempts to predict
solar eruptions. Difficulties associated with blended lines are shown to be
minor in an Appendix.

Sun: atmosphere - Sun: magnetic fields

##  1 Introduction

Commenting on the origin of solar flares, in 1960 Gold & Hoyle wrote

> The requirements of the theory can therefore be stated quite definitely.
> Magnetic field configurations must be found that are capable of storing
> energy densities hundreds of times greater than occur in any other form, and
> that are stable most of the time. A situation that occurs only a small
> fraction of the time must be able to lead to instability in which this
> energy can rapidly be dissipated into heat and mass motion…

These requirements have remained unchanged in the intervening six decades. We
now have a “standard model” of flares (Carmichael, 1964; Sturrock, 1966;
Hirayama, 1974; Kopp & Pneuman, 1976), involving a loss of equilibrium of
arcades of closed magnetic flux systems, particle acceleration and,
frequently, ejection of plasmoids. The instability can only arise when the
free magnetic energy stored in the coronal magnetic fields exceeds certain
thresholds. This is, however, a necessary but not sufficient condition for
flaring and plasmoid ejection, because other global and local properties of
the magnetic field, notably the topology, also determine the stabilility of
coronal MHD systems (Low, 1994).

Enormous interest in solar eruptions is driven by society’s need to understand
origins of “space weather” (e.g. Eddy, 2009) as well as its intrinsic
scientific challenges. However, much like weather prediction before the advent
of modern instrumentation and computers, predicting the properties of
eruptions and flares has remained a difficult challenge primarily due to the
lack of quantitative data available for analysis of the pre-flare state and
flare-triggering mechanism(s). In particular, the only high-resolution, high
cadence, data available to date has been the magnetic field state in the
photosphere, which has been used to define “boundary conditions” to the
overlying coronal magnetic field, where most of the free energy is stored
prior to eruption. Herein lies the basic problem: magnetic fields measured in
the photosphere are not ideally suited to the task. We elaborate upon this
below.

The situation described by Gold & Hoyle (1960) draws analogy with trying to
predict the timing and location of lightning strikes (Judge, 2020). By
measuring updrafts and/or observing cloud buildup one can identify the causes
of the buildup of electrostatic energy, but lightning occurs in a predictable
way only in a statistical sense. In-situ measurements of electric fields are
required even then to make probabilistic forecasts of lightning strikes to the
ground, without reference to intra- or inter- cloud discharges. In this sense
the weaknesses found in previous prediction methods is perhaps unsurprising.

The difficulty of the task has been summarized in the write-up of an inter-
agency “All Clear Workshop” (Barnes et al., 2016) in which the authors
conclude with the sobering statement that

> For M-class flares and above, the set of methods tends toward a weakly
> positive skill score (as measured with several distinct metrics), with no
> participating method proving substantially better than climatological
> forecasts.

Steady improvements in the performance of flare forecasts has been documented
in more recent years (e.g. Leka et al., 2019). However, these authors make
explicit the need to augment existing photospheric magnetic data in order to
proceed, with a heterogeneous mix of theory and observation:

> quantitative “modern” forecasts incorporate …physical understanding as they
> often characterize coronal magnetic energy storage by proxy, with the
> parameterizations of photospheric magnetograms.

In other words, measurements of the photospheric magnetic field can be related
only indirectly and through certain ad-hoc parameterizations, to the magnetic
free energy of the overlying corona, which is the ultimate origin of
instabilities leading to flares and CMEs.

With the availability of a decade of high-resolution photospheric magnetic
field data from NASA’s Solar Dynamics Observatory (SDO) mission, recent
efforts have focused on statistical pattern recognition methods for flare
prediction that generally fall under the “machine learning (ML)” sub-field of
artificial intelligence research. For example, a study by Bobra & Couvidat
(2015) used photospheric vector magnetic data of 2071 active regions from the
Helioseismic and Magnetic Imager (HMI) instrument on SDO. They predicted
strong flares using a Support Vector Machine (SVM) algorithm applied to a set
of 25 parameters derived from the vector magnetic field measured in SHARPs
(Space-weather HMI Active Region Patches), which are cut-outs around sunspot
active regions. They achieved a high True Skill Statistic (TSS) of $\approx$
0.8, but at the cost of getting a significant number of false positives, a
common characteristic of all ML flare prediction algorithms optimized for TSS
in the strongly unbalanced training sets which naturally contain many more “no
flare” examples than “flare” examples. Later studies (Park et al., 2018; Liu
et al., 2019; Chen et al., 2019; Li et al., 2020) applied recurrent
architectures such as Long Short-Term Memory (LSTM) systems and deep learning
systems including Convolution Neural Network (CNN) and autoencoder
architectures to explore feature sets that extend beyond the magnetic field-
derived features of earlier ML flare prediction systems. However these studies
do not demonstrate significant increases in skill scores relative to earlier
approaches.

These studies have used only photospheric measurements of magnetic fields.
Investigations have also been carried out that include information from higher
atmospheric layers, e.g., by incorporating SDO Atmospheric Imaging Array (AIA)
data into existing ML flare prediction systems (Nishizuka et al., 2018; Jonas
et al., 2018), or by using extrapolated coronal magnetic field models (Korsós
et al., 2020). Chromospheric UV spectral lines from IRIS have also been
scrutinized for additional flare prediction utility (Panos & Kleint, 2020).
These studies all show some degree of improvement in flare prediction metrics
when including chromospheric or derived coronal information. More recently,
Deshmukh et al. (2020) showed that combining Topological Data Analysis (TDA)
of the radial magnetic field structure in active regions with the SHARPs
vector magnetic field metadata results in similar or slightly higher skill
scores compared to using selected SHARPs metadata alone. This result implies
that inclusion of high-resolution imaging of active region and flows may be an
important complement to spectroscopic measurements when deducing atmospheric
conditions that evolve to a flaring state.

Physically, however, solar flare prediction based upon photospheric data is
ultimately confronted with basic difficulties, beyond the well-known
disambiguation problem associated with native symmetries in the Zeeman effect.
The photosphere is a dense radiating layer of high density ($10^{-7}$ g cm-3)
plasma. Outside of sunspot umbrae the photospheric magnetic field is unable to
suppress convective motions which therefore control the magnetic fields
threading the fluid, a case where the plasma-$\beta=$ gas pressure / magnetic
pressure, is generally $>1$. In traversing the chromosphere, emerging magnetic
fields drop in strength algebraically but the plasma densities and energy
densities drop exponentially (by 7 and 5 orders of magnitude, respectively),
so that the tenuous plasma at the coronal base is in a low-$\beta$ state. One
immediate practical problem is that tiny dynamical changes which may be
unobservable in the dense photosphere can have enormous consequences for the
tenuous plasma and magnetic fields above: the problem is observationally ill-
posed. As a result of these different photospheric and coronal plasma regimes,
the time scales for significant photospheric magnetic field evolution are
typically measured in minutes to hours whereas the dynamic evolution and
reconnection of the coronal magnetic field configurations leading to flares,
after a quasi-static buildup phase, is measured in seconds. This fundamental
mismatch in dynamical time scales evidently cannot be overcome through any
type of analysis of photospheric magnetic field data and leads to the
conclusion that quantitative analysis of the magnetic field and flows in the
chromosphere and corona are necessary to catalyze significant breakthroughs.

Figure 1: Contribution functions are shown as a function of temperature from
the model C of Fontenla et al. (1993). The emergent intensities are over-
plotted as solid lines for a ray from close to disk center
($\cos\theta=0.887$, $\theta$ the angular distance from disk center as
measured from Sun center). The Ca II lines at 393 and 854 nm span from 3.8 to
4.0 in log$T$ (the latter forming in deeper cooler layers than the former),
and the 1083 nm line of He I forms within the last pressure scale height of
the chromosphere. Notice the extension of contributions for the Mg II
resonance lines beyond all others, to $\log T_{e}\geq 4.3$, and the multiple
lines of Fe II between 259 and 261 nm whose cores form close to 3.9-4.0 in
log$T$, and weaker lines in the 275 nm region with stronger contributions from
lower temperatures. A modest atomic model for Fe II was adopted to examine the
line cores formed in non LTE in the upper chromosphere. The calculations used
the RH computer program Uitenbroek (2001). Far more complete LTE calculations
are made in the Appendix where spectral line blends are discussed.

In order to address this and other questions, our purpose here is to explore
the entire solar spectrum to attempt to identify more direct ways to identify
the free energy and topology of magnetic fields above active regions. In this
way we hope to put the prediction of solar magnetic eruptions, the source of
all major space weather events, on a firm observational basis. The specific
problem we address is reviewed in section 2. It will involve spectro-
polarimetry.

Before proceeding, we point out that our goals may on first reading appear
similar to efforts measuring only intensities of EUV and UV emission lines
(e.g. De Pontieu et al., 2020), an endeavor with a history of 6+ decades yet
able to study only the effects and not causes of magnetic energy storage and
release. Instead our work is closer to an important study by Trujillo Bueno et
al. (2017), but the latter focuses more on the physical processes behind the
magnetic imprints across the solar UV spectrum, including the near-UV spectral
region of the Mg II $h$ & $k$ lines. We focus instead on finding an optimal
set of observations which target the problem of the build-up and release of
magnetic energy. We argue that spectro-polarimetry of the near-UV solar
spectrum is indeed a profitable avenue for research.

##  2 Observational signatures of energy build-up and release

###  2.1 Measuring magnetic fields in and above chromospheric plasma

Two approaches are in principle possible: in-situ measurements and remote
sensing. Clearly in-situ methods are far beyond the capabilities of known
technology. The state-of-the-art mission Parker Solar Probe (Fox et al., 2016)
has a perihelion distance of just below ten solar radii, far above active
regions where most of the free magnetic energy exists. Therefore we limit our
discussion to measuring magnetic fields remotely, an area which has a firm
physical basis (Landi degl’Innocenti & Landolfi, 2004).

Briefly, two types of remote-sensing measurements have been made: direct
measurements of light emitted from within coronal plasma, and measurements
close to the corona’s lower boundary, some 9 pressure scale heights above the
photosphere. Coronal measurements include (see, for example, reviews by Judge
et al., 2001; Casini et al., 2017; Kleint & Gandorfer, 2017, for more complete
references):

  * •

Measurements of line-of-sight (LOS) field strengths using differences between
ordinary and extraordinary rays of magneto-ionic theory at GHz frequencies
Gelfreikh (1994).

  * •

Measurements of field strengths across surfaces determined typically by a
harmonic of gyro-resonant motions of electrons at frequencies of several GHz,
frequencies uncontaminated by bremsstrahlung opacity only when the field
strength exceeds $\approx$200 G (White, 2004).

  * •

Serendipitous level-crossings in atomic ions produces changes in EUV spectra
through level mixing by the Zeeman effect (Li et al., 2017), yielding magnetic
field strengths.

  * •

A long history of measurements of magnetic dipole lines during eclipses or
with coronagraphs hold great promise for measuring vector magnetic fields in
the corona. The infrared, coronagraphic DKIST facility offers significant new
opportunities for advancements using such lines (e.g. Judge et al., 2013).

  * •

Occultations of background radio sources by the corona have revealed
integrated properties of coronal magnetic fields, weighted by electron density
through the Faraday effect (Bird & Edenhofer, 1990), a technique also applied
to signals from spacecraft away from the Sun-Earth line (Bird, 1982).

Of the second type, we can list:

  * •

Several strong lines measured using spectro-polarimetric measurements of the
Zeeman effect from the ground can yield LOS components of chromospheric
magnetic fields. Very high photometric precision is however required for
vector fields, beyond most current capabilities (Uitenbroek, 2010), owing to
the second-order $\epsilon^{2}$ signature of the Zeeman effect in the small
ratio $\epsilon$ of magnetic splitting to the line Doppler width (defined
below). These include Na I D lines, Ca II H, K and infrared triplet lines.
Synthetic Ca II line profiles are shown in the non-LTE calculations using the
RH non-LTE program including partial redistribution (Uitenbroek, 2001) in
Figure 1.

  * •

The He I 1083 nm triplet has a proven record of magnetic field determinations
in a variety of cool structures such as spicules (Centeno et al., 2010),
filaments (Kuckein et al., 2012; Díaz Baso et al., 2019) and cool loops
(Solanki et al., 2003), in the vicinity of the base of the corona before,
during and after flares (Judge et al., 2014) even extending far into the
corona (Judge et al., 2012; Schad et al., 2016). Both the Zeeman and Hanle
effects contribute to the formation of this line, owing to the large
contribution of anisotropic radiation incident on the helium atoms to the
statistical equilibrium in the triplet states (e.g. Trujillo Bueno & Asensio
Ramos, 2007).

  * •

Recently, sub-orbital rockets have carried sensitive spectropolarimeters to
observe the resonance lines of hydrogen (Kano et al., 2017) and singly-ionized
magnesium (Trujillo Bueno et al., 2018; Ishikawa et al., 2021). Both Zeeman
and Hanle effects have revealed information on magnetic fields near the very
top of the solar chromosphere.

  * •

Measurements at 30 GHz and above made with the ALMA interferometer facility
can in principle yield line-of-sight magnetic field strengths using
Gelfreikh’s (1994) theory of continuum polarization (Loukitcheva, 2014, 2020),
but we have been unable to find reports of solar polarization measurements
with ALMA. It can observe from 0.4 to 8.8 mm wavelengths (750 and 34 Ghz
respectively), thereby sampling heights from 600 km to 2000 km in the
chromosphere. Transition region plasma under non-flaring conditions is tenuous
and thin. It contributes very little to the far brighter chromospheric
emission at the high frequencies measured by ALMA.

Next we assess the ability of these techniques to address the specific problem
of interest.

Figure 2: Images are shown to highlight the corrugated surfaces at which the
centers of UV lines are formed (close to $\tau=1$) in MURaM calculations of
the upper solar chromosphere. The calculations are from a numerical experiment
where a magnetic flux system is emerging from beneath the solar atmosphere.
The emergent fields carry with them plasma, which is revealed by the extended
heights of the $\tau=1$ surfaces, which can exceed 10 Mm. Typically the line
shifts in the lower transition region are smaller than the line widths (Athay
& Dere, 1991). Therefore the $\tau=1$ surfaces were computed without taking
into account Doppler shifts, at the centers of lines of Mg II $k$, and several
lines of Fe II. The $x-y$ images of height of formation are plotted in order
of decreasing opacity from Mg II to Fe II 2769.75 Å. In the bottom right
panel, slices of these $\tau=1$ surfaces are shown, taken along the black line
in the five other panels. But the black lines in this final panel are magnetic
lines of force in the $y-z$ plane. Evidently, a combination of these and other
lines of iron with a wide range of opacities spans the ranges of excursions of
the surfaces away from a horizontal plane.

###  2.2 Establishing methods to measure free magnetic energy and topology

While information on magnetism above the photosphere is contained in all the
above strategies, few offer ways to assess the necessary quantities concerning
magnetic free energy as magnetic fields emerge, build and change topology as
it evolves and is suddenly released. If we consider measurements within the
corona, we would have to probe the 3D magnetic structure. However, the corona
is optically thin (with the exception of gyro-resonance emission which yields
just the magnetic field strength within resonant surfaces, whose geometry is
not known from one line of sight), which means that recovery of the 3D coronal
field would require measurements from at least two different lines of sight
(e.g. Kramar et al., 2006). Until suitable instruments are flown on spacecraft
away from the Sun-Earth line, the required stereoscopic views will be
unavailable. Therefore we must seek alternatives.

When chromospheric, transition region and coronal plasmas are connected by
magnetic lines of force (Judge, 2021), the plasma pressure near the top of the
chromosphere is close to that of the overlying corona. Accurate measurements
of magnetic fields using spectral lines formed in plasma up to a few times
$10^{4}$ K can therefore be used to inform us on the evolving magnetic state
of the overlying corona (Trujillo Bueno et al., 2017). Selected spectral line
profiles formed across the upper chromosphere are shown in Figure 1, intended
to illustrate near UV lines in comparison to familiar lines commonly observed
from the ground. The solar spectrum between 250 and 281 nm is dominated by
line transitions of Mg II and many lines belonging to the $4s-4p$ transition
array in neutral and singly ionized iron. In the figure, contributions to the
intensities of Mg II and typical Fe II lines are shown as a function of
electron temperature. This abscissa was chosen instead of height and continuum
optical depth because in the particular 1-dimensional model adopted, the
temperature gradient in the transition region is very large, making lines
appear to originate from the same heights.

The regime of plasma $\beta=8\pi p/B^{2}<1$ in upper chromospheric, transition
region and coronal lines is the same, if the photospheric magnetic field has
expanded to fill the volume. In active region plasma this appears to be the
case (Ishikawa et al., 2021). When measurements are made close to the force-
free state ($\beta\ll 1$), then theory can be reliably invoked in at least two
ways: application of the magnetic virial theorem (Chandrasekhar, 1961), and
extrapolation of fields with observational boundary conditions compatible with
the magnetostatic conditions (Wiegelmann & Sakurai, 2021). Any waves and/or
tangential discontinuities measured will also be of importance to assessing
the free magnetic energy and its evolution. Fleishman et al. (2017) have
confirmed these theoretical concepts quantitatively using model simulations.

Skeptics of our statements might rightfully recall that the chromosphere is
finely structured (see, e.g., the remarkable structure in narrow-band images
of Cauzzi et al., 2008; Wang et al., 2016; Robustini et al., 2018). It is also
known, for example from limb observations, that transition region plasma at
intermediate temperatures has contributions from structures far from a level
surface which might otherwise be amenable to mathematical techniques to
estimate free energies and extrapolated magnetic fields. A non-level,
“corrugated” surface of unit optical depth in a given spectral line can lead
to spurious inferences of magnetic structure even within low-$\beta$ plasmas
(see, e.g., section 4.3 of the paper by Trujillo Bueno et al., 2017, and our
Figure 2 below). The question then arises, is the interface between
chromosphere and corona just too finely structured to make measurements of
magnetic fields there of little use?

Hints from two pieces of work suggest that this is not the case. Firstly, the
CLASP2 measurements reported by Ishikawa et al. (2021) have shown that, at an
angular resolution of 1” which is far smaller than the scales of large coronal
structures leading to flares, the fields measured in the core of the Mg II $h$
& $k$ lines across active regions are unipolar and spatially smooth over large
areas. Any mixed polarity fields undetected by CLASP2 must lie below 1” scales
and will therefore not penetrate more than a few hundred km above the
photosphere. But the fields measured by CLASP2 may well contain unseen
tangential discontinuities arising from small-scale angular displacements
across flux surfaces (Judge et al., 2011). Such features would constitute an
additional source of free magnetic energy whose consequences might then be
explored using the very techniques advocated for below. Secondly, Judge (2021)
has shown that the attention to the dynamics and fine structure of transition
region plasma has been over-emphasized, statistically. Most of the
chromosphere-corona transition region on the Sun is indeed in a thermal
interface connected by magnetic lines of force, and not isolated, as was
claimed by Hansteen et al. (2014) and references therein, all of them based on
circumstantial evidence accumulated using spectral line intensities alone.

One example is shown in Figure 2. This figure is based on a MURaM flux
emergence simulation in a 98 Mm wide domain, reaching from -8 to 75 Mm in the
vertical direction. The snapshot corresponds to a highly energized corona, in
which a strong bipolar field has emerged into a previously quiet region. Ten
minutes after this, a flare and a mass ejection occurred. The simulation is
based on the Rempel (2017) version of MURaM, which uses a simplified
description of the chromosphere. The $\tau=1$ surfaces were computed using
parameterized li¡ne opacities at line center and integrating downwards from
the corona. The opacities were derived using a function, which for the Mg II
$k$-line for example, are of the form

$\kappa_{\nu=\nu_{0}}=\frac{\pi
e^{2}f_{ij}}{m_{e}c}\frac{n_{\mathrm{i}}}{n_{\mathrm{Mg^{+}}}}\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}\frac{n_{\mathrm{Mg}}}{n_{\mathrm{H}}}n_{\mathrm{H}}\phi_{\nu_{0}},$
(1)

in which the lower level population $n_{\mathrm{i}}$ is expanded as a product
of ionization fraction $\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}$ (a
function of electron temperature $T_{e}$), the abundance
$\frac{n_{\mathrm{Mg}}}{n_{\mathrm{H}}}$, and the hydrogen density
$n_{\mathrm{H}}$ from the model. The first factor is the frequency-integrated
cross section of the $k$ line from fundamental constants and the absorption
oscillator strength $f_{ij}$ between levels $i$ and $j$, and $\phi_{\nu}$ the
line profile at frequency $\nu$, normalized so that
$\int_{0}^{\infty}\phi_{\nu}d\nu=1$. The profile is dominated for these ions
by an assumed microturbulence of 10 km s-1. The term
$\frac{n_{\mathrm{Mg}^{+}}}{n_{\mathrm{Mg}}}$ is computed assuming that
neutral Mg is negligible, but allowing for electron impact ionization to Mg2+
and higher ions and radiative recombination using data from Allen (1973).

This figure illustrates approximate line formation heights, sufficient for the
present paper. The excursions in height corresponding to the emerging flux
system vary from a maximum near 10 Mm for Mg II $k$, to a few Mm for a line of
Fe II at 276.9 nm. As can be seen, these corrugations can be probed with
multiple spectral features with different opacities. Thus measurements at the
top of the chromosphere, properly sampling the potential excursions of the
isotherms and isobars in the transition between chromosphere and corona,
appear to be a promising approach to the problem. At the very least, such new
magnetic measurements could be compared in detail with advanced simulations of
the Sun’s atmosphere (Gudiksen et al., 2011; Rempel, 2017). Below we will
suggest one approach to this problem based upon machine learning techniques.

Figure 3: A low-resolution UV irradiance spectrum from the Whole Heliospheric
Interval in 2008 is shown (from
https://lasp.colorado.edu/lisird/data/whi_ref_spectra). The near UV region has
a photon flux density far higher than the majority of vacuum UV lines below
200 nm. The exception is H L$\alpha$ which roughly 10x smaller than the flux
density at 260 nm and longer wavelengths.

###  2.3 The near UV solar spectrum and the chromosphere-corona interface

A standard, low resolution solar UV spectrum is shown in Figure 3. After an
extensive search, we have identified a spectral range of 256-281 nm as a
promising part of the solar spectrum for measurement of vector magnetic fields
at the base of the corona. Our search, driven by the need to probe in detail
magnetic structure in height above the solar surface and/or depth along the
LOS of prominences,  led to the choice of important transitions in the 4s-4p
transition array of Fe II. While there are strong Fe II 4s-4p transitions in
the region near 234 nm, the solar fluxes are weaker than in the 256-281 nm
range. Our wavelength selection is significantly broader than the 279.2-280.7
nm spectral region of CLASP2, which includes the Mg II $h$ & $k$ line cores
and magnetically-sensitive wings (Auer et al., 1980; Belluzzi & Trujillo
Bueno, 2012; Alsina Ballester et al., 2016; del Pino Alemán et al., 2016) the
3p-3d lines (del Pino Alemán et al., 2020), and photospheric lines of Mn I
(Ishikawa et al., 2021).  Both Hanle and Zeeman effects can be brought to bear
on the problem depending on the strength of the magnetic fields found across
the solar atmosphere. The Hanle effect can be important to remove the 180o
ambiguity present in Zeeman effect measurements. The rationale for this
spectral region is based upon requirements demanded by the desire to measure
vector magnetic fields within force-free ($\approx$ low $\beta$) plasma at the
coronal base, over an area covering a typical active region. An angular
resolution sufficient to resolve the thermodynamic fine structure is also a
constraint. Such features are most clearly seen only in images of features
within the narrow spectral lines chromospheric images. This constraint poses
difficulties for current observations at radio and sub-mm wavelengths, and in
hard X-rays. Soft X-rays ($\leq 1$ keV) have no known sensitivity to magnetic
fields of solar magnitude, and along with EUV photons cannot escape from or
penetrate deep into the chromosphere. We are then left to consider vacuum UV
to infrared wavelengths.

We further eliminate wavelengths above the atmospheric cutoff at 310 nm
because of the lack of spectral lines that can span many decades in optical
depths above the chromosphere. The Balmer and higher line series of hydrogen
are poor choices for Zeeman diagnostic work, the lightness of hydrogen makes
lines broad and patterns overlapping. For Ba-$\alpha$, seven anomalous Zeeman
patterns will lie within half a wave number for typical solar field strengths
even in the photosphere, thereby making the observation of anomalous Zeeman
patterns extremely difficult. The Hanle effect has been explored by Štěpán &
Trujillo Bueno (2010) in their comparison of 1D radiative transfer
calculations with limb observations of the quiet Sun by Gandorfer (2000). The
value of H$\alpha$ as a diagnostic of magnetic field remains a subject for
further research.

The strongest line in the chromospheric visible spectrum is Ca II $K$ has just
$1/20$ the opacity of the Mg II $k$ line (see Figures 1 and 5). Also shown in
Figure 1 is the well-known line of He I at 1083 nm which is present in active
region plasma. However, the line, belonging to the triplet system, populated
largely by recombination following EUV photoionization, tends to form where
coronal ionizing photons (and perhaps even hot particles) cannot penetrate the
chromosphere. Our non-LTE calculations (Figure 1) reveal that significant
opacity in the line is narrowly confined between surfaces of column mass well
below the height of the NUV Mg II lines, because of the depth of penetration
of vacuum UV radiation from the overlying corona in the continua of H and He.
When included along with the Ca II resonance and infrared triplet lines, the
optical- near UV regions accessible from the ground extend neither high
enough, nor do they sample opacity with the fineness of UV lines (Figures 1, 2
and 5. Lines of Mn I are sufficiently close to the strong Mg II lines for the
wings of the latter to contribute to opacity, but this was not accounted for
in the plot. Therefore heights of formation of Mn I lines are lower limits. ).

The solar irradiance spectrum at vacuum UV wavelengths (below 200 nm, and
above 91 nm below which the Lyman continuum absorbs line radiation), including
H L$\alpha$ and many chromospheric and transition region lines, is shown in
Figure 3. All are substantially weaker than the Mg II lines. The polarimetric
measurements necessary to infer magnetic fields require large photon fluxes.
The VUV region is a factor of at least 10-100 times dimmer than the NUV. Also,
the Zeeman signatures vary in proportion to the wavelengths observed. Further,
most of the transition region lines are optically thin, so that there is no
single “depth of formation” of a given line. The exceptions are the Lyman
lines of hydrogen and 58.4 nm resonance line of He I. But using only this
region of the spectrum would also fail to sample surfaces of multiple optical
depths, and thus fail to address the need to span the anticipated corrugated
isothermal and isobaric surfaces (Figure 2), particularly in active regions,
the sources of most solar eruptions.

Figure 4: A smoothed UV spectrum of $\alpha$ Cen A, obtained by the Hubble
Space Telescope, is displayed as a proxy for the mean solar intensity
spectrum. The large number of lines of Fe I and Fe II are indicated and
annotated with their multiplet terms. The Mg II spectrum is also annotated
including the $3s-3p$ $h$ and $k$ lines, and the $3p-3d$ lines. Notice that
the permitted Fe II lines (shown with solid bars with downward ticks) consist
of the sextet-D resonance lines $(4s-4p)$ near 2600 Å, and quartet transitions
at other wavelengths that occur between metastable (even parity) quartet lower
levels and odd-parity quartets of the same transition array ($4s-4p)$.

In contrast, the NUV region (from say 200 to 310 nm, the atmospheric cutoff)
is comparatively brighter, containing lines (Mg II) whose cores form within
lower transition region. Compared with lines accessible from the ground, lines
have smaller Zeeman signatures, but have increased linear polarization
generated by the higher radiation anisotropies at UV wavelengths (much
stronger limb-darkening). Thus, the Hanle signals will statistically be
stronger than at visible wavelengths. Further, the region contains a plethora
of lines of neutral and singly ionized iron within the $4s-4p$ transition
array, including resonance lines near 260 nm (see Figure 4). Their complex
spectra span and sample multiple depths across the chromosphere, extending
into lower transition region plasma (see in particular the contours of unit
optical depth surfaces for several lines in the lower right panel of Figure 2,
and the plot of relative optical depths in Figure 5).

The richness of the Fe II spectrum itself has been known for decades in
astronomy (e.g. Viotti et al., 1988). Figure 4 identifies lines and multiplets
of Fe plotted above a smoothed Hubble Space Telescope spectrum of $\alpha$ Cen
A, which is a good proxy for disk-averaged intensity spectrum of the Sun.
(There is no disk-averaged solar spectrum of comparable quality). All of the
strong transitions of iron lie near or below the Mg II lines near 280 nm. NUV
spectra obtained at the solar limb during the SKYLAB era revealed a
spectacular array of Fe II lines in emission against the darkness of space
(Doschek et al., 1977). Particularly strong in the limb spectra (Figure 10)
are multiplets in the sextet system near 259-263 nm, and quartet transitions
in the 273 nm region. These Fe II lines were readily detected up to 8” above
the limb in the SKYLAB data, the limb itself is already far higher than the
photosphere. Such emission is very likely associated with the kind of
structures modeled in Figure 2. A downside of the richness of the spectra of
iron is the possibility of blended lines. The important issues of blended
spectra lines at NUV wavelengths and effects of missing opacity are addressed
in the Appendix. They are shown to be a minor issue, affecting only a few
lines in the regions between 256 and 281 nm, owing to the overwhelming
strength of the Fe II lines formed high enough in the chromosphere to aid in
the diagnosis of magnetism there. Between 281 and 310 nm, there are just 10
“wide” lines, i.e. strong enough to exhibit wing absorption profiles, compared
with 38 listed by Moore et al. (1982) between 256 and 281 nm. None of the
lines above 281 nm form high enough in the Sun’s atmosphere to contribute
significantly beyond measurements of the chromospheric magnetic field at
shorter wavelengths.

Figure 5: Properties of spectral lines calculated using line center opacities
as a function of wavelength are shown. Circles show lines of singly-charged
ions, squares show lines of neutral species. Symbol sizes reflect critical
Hanle field strengths. Relative optical depths of the lines are shown on the
ordinate. The Mn I lines should include wing opacity from the Mg II line, but
this was omitted here. The Mn I optical depths are thus a lower limit. The “?”
for the He I 1083 nm line refers to the unknown relative optical depth of this
line because of peculiarities in the formation of this multiplet in the Sun
(e.g. Judge et al., 2015, who discuss this multiplet in the context of a well-
observed flare).

Relative optical depths at the centers of the lines in this region, and “Hanle
field strengths” are shown in Figure 5, including lines of He, Mg, Ca, Mn and
Fe. These Hanle field strengths (see equations 2 and 3 below) are those near
which the emergent polarized spectra are strongly sensitive to magnetic field
strength and direction, (Landi degl’Innocenti & Landolfi, 2004, see below).

###  2.4 Spectral signatures of magnetic fields

The Zeeman effect is well-known in laboratory and astrophysics as a
characteristic splitting in intensity ($I$) spectra. But in the Sun and stars,
the Stokes parameters $I,Q,U$ and $V$ are used to measure magnetic fields in
commonly encountered situations where the widths of the lines exceed the
Zeeman splitting (this is assumed in the following discussion). As noted
above, Zeeman signals depend upon the quantity
$\varepsilon=\omega_{B}/\Delta\omega_{0}$, where $\omega_{B}$ is the Larmor
frequency of the gyrating ion:

$\hbar\omega_{B}=\mu_{0}B.$ (2)

Here $\mu_{0}$ is the Bohr magneton, and $B$ the magnetic field strength in G.
$\Delta\omega_{0}$ is the line width (Doppler width for chromospheric and
coronal lines). The amplitude of circular polarization measured by Stokes $V$
varies linearly with $\epsilon$, the linear polarization profiles $Q$ and $U$
have amplitudes varying as $\varepsilon^{2}$. While $B$ is measured in Gauss,
it is important to remember that unless contributions from non-magnetic
regions to $I,Q,U$ and $V$ can be determined, the polarized components can be
used to determine only the average flux density per unit area in Mx cm-2, not
field strength.

Table 1: Prominent lines and blends in the solar chromospheric spectrum

$\lambda$ nm | Landé g |  log $\omega_{L}$ |  log $\sum_{i}A_{ji}$ | $B_{HANLE}$ |  log $\tau_{0}$ | Ion | blend
---|---|---|---|---|---|---|---
lab. (air) | factor |  rad sec-1 |  sec-1 | G |  |  |  severity∗ Ion $\lambda$
256.254 | 1.21 | 7.03 | 8.49 | 28.8 | -2.54 | Fe II |
256.348 | 1.10 | 6.99 | 8.49 | 31.9 | -2.87 | Fe II | 1 Fe I .341
256.691 | 0.83 | 6.86 | 8.49 | 42.2 | -3.30 | Fe II |
257.437 | 1.30 | 7.06 | 8.51 | 28.4 | -4.33 | Fe II | 1 Co I .351
257.792 | 1.33 | 7.07 | 8.49 | 26.4 | -3.32 | Fe II | 1 Mg I .888
258.258 | 1.47 | 7.11 | 8.49 | 23.9 | -3.19 | Fe II |
258.588 | 1.50 | 7.12 | 8.46 | 21.7 | -2.03 | Fe II | 2 Fe I .588
259.055 | 1.50 | 7.12 | 8.48 | 22.8 | -4.47 | Fe II | 1 Co I .059
259.154 | 1.49 | 7.12 | 8.49 | 23.5 | -3.24 | Fe II |
259.373 | 2.17 | 7.28 | 8.49 | 16.2 | -4.02 | Fe II | 1 Mn I .372
259.837 | 1.50 | 7.12 | 8.46 | 21.7 | -1.93 | Fe II |
259.940 | 1.56 | 7.14 | 8.46 | 21.1 | -1.39 | Fe II | 1 Fe I .957
260.709 | 1.50 | 7.12 | 8.66 | 34.8 | -1.99 | Fe II |
261.107 | 1.90 | 7.22 | 8.49 | 18.4 | -4.23 | Fe II | 1 Cr II? .104
261.187 | 1.59 | 7.15 | 8.46 | 20.5 | -1.79 | Fe II |
261.382 | 1.50 | 7.12 | 8.46 | 21.7 | -2.20 | Fe II |
261.762 | 1.66 | 7.16 | 8.46 | 19.6 | -2.36 | Fe II |
262.041 | 1.87 | 7.22 | 8.66 | 27.9 | -3.75 | Fe II |
262.167 | 3.34 | 7.47 | 8.46 | 9.8 | -2.76 | Fe II |
262.567 | 1.50 | 7.12 | 8.46 | 21.9 | -2.08 | Fe II |
262.829 | 1.50 | 7.12 | 8.66 | 34.8 | -2.21 | Fe II |
263.105 | 1.50 | 7.12 | 8.46 | 21.7 | -2.02 | Fe II |  1 Fe II .045†
263.132 | 1.50 | 7.12 | 8.46 | 21.7 | -1.97 | Fe II |
264.112 | 1.87 | 7.22 | 8.51 | 19.8 | -4.62 | Fe II | 1 Ti I .089
268.351 | 1.84 | 7.21 | 8.51 | 20.1 | -4.99 | Fe II | 2 Cr II .345
269.283 | 1.93 | 7.23 | 8.47 | 17.2 | -4.40 | Fe II | 1 Fe I .265, Fe II .260
270.938 | 2.10 | 7.27 | 8.47 | 15.8 | -4.30 | Fe II | 1 Cr II .931
271.441 | 1.50 | 7.12 | 8.60 | 30.3 | -3.09 | Fe II |
271.670 | 1.33 | 7.07 | 8.46 | 24.8 | -3.11 | Fe II | 1 Mn II .680
271.903 | 1.25 | 7.04 | 8.25 | 16.3 | -4.45 | Fe I | 2 Fe I .903
272.090 | 1.17 | 7.01 | 8.28 | 18.4 | -4.76 | Fe I |
272.358 | 1.00 | 6.94 | 8.27 | 21.1 | -5.22 | Fe I |
272.488 | 1.20 | 7.02 | 8.47 | 27.7 | -3.05 | Fe II | 1 Fe I .495
272.754 | 1.50 | 7.12 | 8.60 | 30.4 | -3.05 | Fe II |
273.073 | 0.80 | 6.85 | 8.47 | 41.6 | -3.19 | Fe II |
273.245 | 1.45 | 7.10 | 8.46 | 22.8 | -4.92 | Fe II | 1 Fe II .233
273.697 | 1.50 | 7.12 | 8.60 | 30.3 | -3.22 | Fe II | 1 Fe I .696
273.731 | 2.00 | 7.25 | 8.27 | 10.5 | -5.12 | Fe I |
273.955 | 1.43 | 7.10 | 8.61 | 32.1 | -2.31 | Fe II |
274.241 | 1.67 | 7.17 | 8.28 | 12.9 | -5.01 | Fe I | 1 Ti I .230, V II .243
274.320 | 0.50 | 6.64 | 8.47 | 66.6 | -2.81 | Fe II |
274.407 | 2.50 | 7.34 | 8.27 | 8.4 | -5.51 | Fe I |
274.648 | 0.90 | 6.90 | 8.47 | 37.0 | -2.59 | Fe II |
274.698 | 1.37 | 7.08 | 8.60 | 33.2 | -2.66 | Fe II | 1 Fe I .698
274.918 | 1.20 | 7.02 | 8.60 | 38.0 | -3.02 | Fe II |
274.932 | 1.07 | 6.97 | 8.46 | 30.9 | -2.38 | Fe II |
274.949 | 0.00 | 4.31 | 8.60 | … | -3.24 | Fe II |
275.014 | 1.58 | 7.14 | 8.25 | 12.8 | -5.04 | Fe I |
275.574 | 1.17 | 7.01 | 8.46 | 28.1 | -2.17 | Fe II |
275.633 | 2.00 | 7.25 | 8.28 | 10.7 | -5.52 | Fe I | 2 Fe I .633, Cr II .630
276.181 | 1.50 | 7.12 | 8.60 | 30.4 | -3.25 | Fe II | 1 Fe I .178, Cr I .174
276.893 | 1.50 | 7.12 | 8.60 | 30.3 | -3.08 | Fe II |
277.273 | 1.50 | 7.12 | 8.61 | 30.6 | -3.14 | Fe II | 1 Fe I .283
277.534 | 1.34 | 7.07 | 8.46 | 24.2 | -6.06 | Fe II | 1 ?
279.565 | 1.17 | 7.01 | 8.43 | 25.3 | 0.00 | Mg II |
279.482 | 1.98 | 7.24 | 8.57 | 21.3 | -5.65 | Mn I |
279.827 | 1.70 | 7.17 | 8.57 | 24.8 | -5.79 | Mn I |
280.108 | 0.84 | 6.87 | 8.57 | 50.3 | -5.96 | Mn I |
280.270 | 1.33 | 7.07 | 8.43 | 21.9 | -0.30 | Mg II |
285.213 | 1.00 | 6.94 | 8.69 | 55.8 | -2.20 | Mg I |
393.366 | 1.17 | 7.01 | 8.17 | 14.3 | -1.55 | Ca II |
854.200 | 1.10 | 6.99 | 7.00 | 1.0 | -2.50 | Ca II |

Landé g-factors are computed using LS coupling. The final column lists the
severity of the blend (as assessed by Moore et al., 1982), and the wavelength
of the blended line (decimal nm). ∗The severity is: 0=unblended; 1=moderate
blend; 2=severe blend. The data in Figure 5 are taken directly from this
table. †The strong line of Fe II multiplet 1 is blended with a line of
multiplet 171.

The Hanle effect has a quite different and complementary behavior independent
of the Doppler width $\Delta\omega_{0}$. It requires a pre-existing state of
polarization induced by symmetry breaking effects such as anisotropic and/or
polarized incident radiation. When the radiation is anisotropic, such as
radiation emerging into the upper chromosphere from the photosphere below, the
atom develops atomic polarization (a particular imbalance of populations of
magnetic substates called the atomic “alignment”). In the presence of a
magnetic field, these populations and associated atomic coherences
(essentially representing the entanglement of neighboring atomic states) are
altered by the magnetic field in the regime where

$g_{ji}\omega_{B}\mathcal{T}_{j}\sim 1,$ (3)

where $g_{ji}$ is the effective Landé $g-$factor for the transition from upper
level $j$ to lower $i$, and $\mathcal{T}_{j}=1/\sum_{i}A_{ji}$ is the inverse
lifetime of level $j$, the sum taken over all lower levels $i$. Figure 5 shows
spectral lines of Fe and Mg, their wavelengths, optical depths, with symbol
sizes indicating the magnetic field for which equality holds in equation (3),
using data compiled in the NIST spectroscopy database. These Hanle field
strengths vary between roughly 1 G and 70 G for the different lines, field
strengths that are anticipated in quiet and active regions of the solar
chromosphere. The data shown in Figure 5 are listed in Table 1, the final
column of which which also addresses the potentially serious problem of
spectral line blends. This problem is shown in the Appendix to be easily dealt
with.

In a classical picture, when $g_{ji}\omega_{B}\mathcal{T}_{j}\approx 1$, the
ion on average gyrates through 1 radian before it emits photons sharing the
common upper level. When $\omega_{B}\mathcal{T}_{j}\gg 1$, the “strong-field
limit” of the Hanle effect, the radiating ions exhibit many gyrations before
emitting a photon and all “memory” of the particular orientation of the
radiating ions is statistically averaged out. This condition is the same as
that found for coronal forbidden lines where $\mathcal{T}_{j}$ is typically
between 1 and 0.01 seconds, not $10^{-8}$ seconds for the permitted lines
studied here (Casini et al., 2017). The only signature of the magnetic field
is in its direction projected on to the plane-of-the-sky, modulo 90o. If
$\omega_{B}\mathcal{T}_{i}\ll 1$ (the weak field limit) the magnetic field
enters as a small perturbation in expressions for the emergent radiation and
the magnetic effects cannot easily be observed. But near the Hanle regime
(equation 3) the magnetic field is encoded as a rotation and de-polarization
of the emergent linearly polarized radiation. The symmetry broken by the mis-
alignment between the directed radiation vector and magnetic field vector
resolves ambiguities associated with Zeeman measurements. This effect may be
useful for identifying tangential discontinuities (changes of magnetic field
direction not strength, across a flux surface).

In practice, both the Zeeman and Hanle effects can be used in a complementary
fashion to find signatures of magnetism in the Sun’s atmosphere using such
spectra (Trujillo Bueno et al., 2018; Ishikawa et al., 2021).

Finally, we have investigated whether molecular transitions are significant in
this spectral region, using the line list complied by R. Kurucz111
http://kurucz.harvard.edu/linelists.html and the synthesis codes by Berdyugina
et al. (2003). In particular, we have found that OH transitions from the
electronic A–X system are abundant in this spectral region, with stronger
lines toward the red. They form below most chromospheric lines owing to
chromospheric stratification and the rapid drop of molecular number density
with gas pressure, but remarkably, can also form above deeper photospheric
lines. A few moderately strong OH features may be visible at wavelengths
between strong atomic lines in the quiet sun spectrum. They become especially
important in models of sunspot umbrae. Numerous molecular CO lines in this
spectral region are quite weak, but they are so many that the continuum level
is effectively reduced by a few percent, with stronger lines toward the blue.
If observed, both OH and CO lines are potentially interesting for sensing
magnetic fields using both the Zeeman and Hanle effects  (Berdyugina &
Solanki, 2002; Berdyugina et al., 2003; Berdyugina & Fluri, 2004), as well as
temperature and pressure near the interface between the photosphere and
chromosphere. (The reader can refer to the paper by Landi Degl’Innocenti, 2007
for elementary theory of molecules in magnetic fields applied to solar
conditions).

We conclude that, based on known atomic and molecular data, there are many
lines of Fe II whose cores are not strongly blended. The case of blended lines
from sunspot umbrae requires further work outside of the scope of this paper.

###  2.5 Diagnostic Sensitivity of Mg II k

Figure 6: Observing geometry for uniform field computations with HanleRT used
in the panels of Figure 7. Note the convention for negative LOS $\theta$.

Figure 7:  _Top:_ “Heat Map” of the Mg II k core polarization amplitude for a
uniform magnetic field inclined at $\theta_{B}=-30^{\circ}$ from vertical.
Gray regions are where the amplitude falls below 0.1% of the intensity, here
taken as a typical instrumental SNR. _Bottom:_ Derivative of polarization
amplitude data from the top panel with respect to magnetic field to produce a
“heat map” of the response. The color scale gives the relative change in
amplitude of the spectral feature in response to a 20% change in $|B|$.
Regions of the map brighter than the 0.1% value on the color scale
(highlighted by black box) indicate an observing geometry that could infer
$|B|$ to within 20% assuming an instrumental polarization SNR of 0.1% of the
intensity. See text for further details.

In Figures 1 and 2 we saw how the Mg II $k$ line has contributions from plasma
that is highest in the Sun’s atmosphere. The $h$ line is a factor of two less
opaque, but it has a higher Landé g-factor than $k$ and so is more sensitive
to the Zeeman effect (Ishikawa et al., 2021). However, its upper $J=1/2$ level
is unpolarizable and so it can have no Hanle effect in the line core.

Both the Zeeman and Hanle effects can and should be used to probe
chromospheric magnetism. The Zeeman effect produces the largest polarimetric
signals when fields are relatively strong (average flux densities of a few
hundred Mx cm-2 were measured using the weak field limit applied to the Mg
lines by Ishikawa et al., 2021). However, for weaker fields, the Hanle effect
becomes the dominant source of polarized light variations in response to the
solar magnetic fields. Following early theoretical work by Belluzzi & Trujillo
Bueno (2012), the scattering polarization, Hanle and Zeeman effects across the
$h$ and $k$ lines was investigated in semi-empirical 1D models (Alsina
Ballester et al., 2016; del Pino Alemán et al., 2016, 2020). The Hanle field
of the $k$ line is 23 G (see Figure 5). These authors demonstrated that, for
field strengths of the order of 10 G, measurable variations of the order of
$\sim$1% in the fractional linear polarization of Mg II $k$ are produced by
the Hanle effect. Here, we extend this work with a larger grid of LOS angles
and magnetic field strengths (see Figure 6 for the geometry) and show
polarization amplitude and magnetic response “heat maps” in Figure 7. As in
del Pino Alemán et al. (2020) we used HanleRT to compute the intensity and
polarization from a Mg II 3-term atom including PRD effects in atmosphere C of
Fontenla et al. (1993). We computed profiles for a grid of uniform magnetic
field vectors inclined at $\theta_{B}=-30^{\circ}$ from vertical, with fixed
azimuth, and with $|B|$ ranging from 1 to 200 G at 1 G intervals. These values
of $|B|$ are about 20 and 10 times smaller and larger, respectively, than the
Hanle field. We computed the emergent Stokes profiles for 13 LOS viewing
angles $\theta$ ranging from $+75^{\circ}{}$ to $-75^{\circ}{}$ at 15∘
intervals, as well as $\mu=\cos(\theta)=0.1$, which corresponds to
$\theta\sim\pm 84.5$∘ or a limb distance of about 5′′. We examine the relative
variation in the amplitude of the fractional polarization (hereafter simply
“response“), $|B|\cdot dA_{i}/dB$, where $i$ indicates the polarization $Q$,
$U$, or $V$, and $A_{i}$ is the amplitude, $A_{i}=({\rm max}(i)-{\rm
min}(i))/I\cdot 100\%$, within the Mg II k core region. The top panel of 7
shows the amplitude $A_{i}$, while the bottom panel shows the response with
respect to magnetic field over the full range of $|B|$ and LOS $\theta$. The
units of the response are $\%I/\%B$, and we have scaled the data such that the
denominator represents a 20% change in $|B|$. Under this normalization, a
response of 1 corresponds to a 1% amplitude change for a 20% change in $|B|$.

In order to assess the practical application of these calculations, we
consider noise at the level of $\sim 0.1\%$ of intensity ($10^{-3}I$). Those
regions in the response heat map brighter than that value on the color scale
are where the instrument would be able to infer components of the magnetic
field to within 20%. We conclude that we would be able to reasonably estimate
this vector orientation for magnetic field strengths of $\sim$5 to 50 G at
observation angles from about $\theta=\pm 45^{\circ}$ to the limb due to the
Hanle effect in $Q$ and/or $U$, and stronger line-of-sight fields at nearly
any observation angle from the Zeeman effect in Stokes $V$. The largest
sensitivity in $Q$ and $U$ occurs for observations near the limb and field
strengths below the critical Hanle field. (Dark vertical strips in the
response of Stokes $U$ are due to local maxima in the amplitude curve near the
critical Hanle field for Mg II $k$; compare to the top panel of Figure 7). The
$V$ amplitude is constant and zero at $\theta=60^{\circ}$ as this is the angle
where the magnetic field vector is orthogonal to the line-of-sight. Measurable
linear polarization is produced at that geometry, however its variation with
magnetic field is too low to be used as a diagnostic. “Heat map” diagrams for
other field inclination angles are qualitatively similar, with the regions of
maximum response due to the Hanle and Zeeman effects shifting in relation to
the field & LOS geometry.

We conclude that diagnosing magnetic field from the Hanle effect in Mg II $k$
is feasible for a large range of geometries and with achievable signal-to-
noise ratios. The range of field strengths to which the Hanle effect is
sensitive complement those higher field strengths ($|B|\geq 50$G) to which the
Zeeman effect is sensitive, for the case of the Mg II $k$ line.

Additional work on the Hanle effect in the lines of Fe II is underway and will
be reported elsewhere. Importantly, there is at least one line (at 2744.01 Å)
with a Hanle field above 60 G, and several where the critical field exceeds
that of the Mg II $k$ line.

###  2.6 Addressing the “corrugation problem”

Figure 2 shows how state-of-the-art numerical models lead to contours in
optical depth which can be far from horizontal. The free energy in these MHD
simulations generated by convection and emerging flux leads to highly dynamic
evolution in the lower density regions of most interest to us at the top of
the chromosphere. The geometric distortions of the $\tau=1$ surfaces in these
tenuous regions naturally reflect the low inertia of the plasma, subjected to
Lorentz, pressure gradient and gravitational forces, evolving in response to
the forcing from below. But, as noted earlier, the extreme corrugations of
order 10 Mm are not negligible compared with the scale of the emerging flux
system itself. The question then arises, how can we infer magnetic field at a
constant geometric height, if we are to apply mathematical techniques to
estimate the free energy and topology within the corona? Progress can be
anticipated without such techniques, by direct comparison of numerical
prediction and measurements. But this question must be addressed if we are to
measure the development of free energy in the overlying corona in time, in
general.

Let us assume that the calculations are a reasonable, “first order”
approximation to the kind of geometric distortions of the $\tau=1$ surfaces of
sets of lines. The advantage of observing the 259-281 nm region is that we
know the order of the opacities of each line in a sequence: the Mg II lines
are the most opaque, next come the sextet system transitions of Fe II and then
the quartet systems, each with its spread of oscillator strengths. Thus, given
a calculation, we can (as in Figure 2) find the $\tau=1$ surfaces. We could
compute Stokes profiles and associate a magnetic field from such synthetic
data, knowing the heights of each line’s formation.

But this is not the situation we face with the remotely-sensed Stokes data, we
have no knowledge other than the Stokes profiles of many lines, on a two
dimensional geometric image of the Sun at each wavelength. Fortunately, this
problem fits well into the picture of machine learning algorithms (e.g. Bobra
& Mason, 2021). Synthetic data can be used to train an algorithm to find level
geometric surfaces (Asensio Ramos & Díaz Baso, 2019) of direct use for
mathematics to derive quantities of interest. The large coverage of optical
depths of the 256-281 nm region and the variety of Hanle field strengths
strongly suggest that, if the models are sufficiently complete to represent
the atmospheric dynamics and temperature structure, then such an algorithm
should have a non-zero probability of yielding the desired results. The
combined use of Hanle and Zeeman effects enhances this probability, at least
in principle. We are exploring ways in which this program of research might be
implemented.

##  3 Discussion

In this paper we have examined the general problem of identifying the sources
of magnetic energy within coronal plasma, which leads to flaring, plasmoid and
CME ejections. By a process of elimination, we arrived at a wavelength range
between 2560 and 2810 Å, which appears to be the most promising range in which
to measure conditions at the coronal base. Our conclusions build upon the
important study of Trujillo Bueno et al. (2017), not only by seeking answers
to questions concerning the magnetic free energy and topology, aimed at
understanding the origins of phenomena responsible for Space Weather. Our
conclusions also differ by proposing a potentially ideal, relatively narrow
spectral region between 256 and 281 nm, expanding significantly the
279.2-280.7 region selected by the CLASP2 mission, to serve as a basis for
future, more targeted research.

To achieve measurements in this region clearly will require a space-based
platform for spectropolarimetry. These conclusions are based upon several
criteria: The spectrum must be bright enough; spectral features must extend as
high as possible; the spectral features must contain information on magnetic
fields through both the Zeeman and Hanle effects; the spectra must be
demonstrably unblended; lines with multiple opacities are needed to span
across the corrugated surfaces that are present at the coronal base.

Thus we suggest that a novel spectropolarimeter, building upon a heritage from
SKYLAB, SMM (UVSP), SoHO (SUMER), IRIS, CLASP and CLASP2, be considered for
flight to address the problems of interest to science and society. Such a
mission would complement the unique capabilties of the Daniel K. Inouye Solar
Telescope which can observe important chromospheric lines (include all those
shown in the top row of Figure 1) from the ground, at a sub-arcsecond spatial
resolution and/or high cadence.

Finally, it is important to note that the CLASP2 measurements of Ishikawa et
al. (2021) achieved signal-to-noise ratios of over 1000 in the Mg II line
cores. This demonstrates that a modest, D=25 cm telescope operating at NUV
wavelengths is sufficient for delivering the weak signals required for
diagnosis of magnetic fields, over active regions at least, where the magnetic
flux density exceeds 100 Mx cm-2.

Thanks to Roberto Casini and Alfred de Wijn for the many fruitful discussions
regarding the Hanle effect and requirements to achieve new results using
diverse spectral lines. NCAR is funded by the National Science Foundation. PGJ
is grateful to the NSF for funding his research at a national center without
any specific goal. The authors thank an anonymous referee for their helpful
comments.

## Line blends and missing opacity

The NUV region is well-known as a crowded region of the solar spectrum, and
also a region where some opacity appears to be “missing”. The effects of this
missing opacity are examined below using comparisons of detailed simulations
and observations. It can be significant at certain wavelengths, but does it
not affect the cores of strong chromospheric lines. (See also discussions by
Fontenla et al., 2011; Peterson et al., 2017). Line blending is however a
concern.

We have addressed blending using data from the Hubble Space Telescope, from
the SO82B spectrograph on SKYLAB, and using sophisticated radiative transfer
models. The models are based on Local Thermodynamic Equilibrium (LTE)
radiative transfer models computed using TURBOSPECTRUM (Plez, 2012), and
adopting a custom calculated Solar 1D, spherical MARCS model atmosphere
(Gustafsson et al., 2008). A complete linelist in the corresponding spectral
region was adopted in the computations from the VALD
database222http://vald.astro.uu.se/ (Ryabchikova et al., 2011). The linelist
includes up-to-date atomic data, including line-broadening parameters from
collisions with neutral hydrogen atoms, calculated ab-initio following Barklem
et al. (1998) for all neutral atoms including dominant Mg II lines and iron
group elements.

A comparison of high resolution Hubble spectra of $\alpha$ Cen A with the
models is informative. A typical example in an interesting region of the NUV
spectrum is shown in Figure 8. The blue line represents the flux spectrum
computed for the Sun, the red line the spectrum of $\alpha$ Cen A. The flux
densities shown are computed at the respective stellar surfaces, with no ad-
hoc adjustments of either scale.

Figure 8: Flux spectra computed for a Solar atmospheric model (blue) using 1D,
LTE radiative transfer calculations (see Section Line blends and missing
opacity) are shown along with spectra of the Sun-like star $\alpha$ Cen A
(red) obtained with the STIS instrument on the HST. The data are a composite
of several exposures from the E230H grating, assembled and processed by Ayres
(2014). The STIS data were shifted according to the redshift of 22.4 km s-1,
and converted to flux at the stellar surface using the angular diameter of
9.09 milli-arcsec. The spectra are compared with no adjustment of flux or
wavelength scale. The qualitative agreement is remarkable. The units are Å
(wavelength) and erg cm-2 s-1 Å-1 (flux, $F=\pi\overline{I}$, where
$\overline{I}$ is the mean disk intensity).

The comparison is remarkable, particularly in the cores of some of the strong
lines, most of which belong to Fe II. Wavelengths are evident (near 2623 Å for
example) where opacity appears to be missing (blue line above red line), but
in the Fe II lines identified in the figure there is no evidence for missing
opacity, if anything, the observed line cores in red lie above the blue line,
indicating emission from the star’s chromosphere.

The model calculations also permit us to identify possible blends (Figure 9).
This figure shows two computations, black is the full model calculation (also
shown in blue in Figure 8), and in this figure blue is a calculation using
opacity only from the Fe II lines. When significant differences between black
and blue lines exist, these are wavelengths where opacities are present in
lines other than Fe II. Conversely, when they agree, the Fe II spectrum
dominates and these are essentially unblended wavelengths. As noted in the
main text, the wavelength region above 281 and below 310 nm does not contain
strong lines formed in the middle-high chromosphere, therefore these are not
shown here. Above 281 nm the lack of such lines makes this a less desirable
region because blending becomes a bigger problem.

Figure 9:  Two calculations are shown for the solar NUV spectrum: one (black)
is made with the full set of lines, and the other (red) is made where lines of
Fe II are the only sources of opacity. Strong lines from the quartet system of
Fe II are annotated, along with lines of Fe I whose opacity is much smaller.
The cores of the strong lines of Fe II are dominated by Fe II itself, the
region above $\approx$ 265 nm is, aside from narrow lines, dominated by wings
of the Mg II $h$ and $k$ lines.

Lastly we can, to some extent, check for blends independently of the model
computations. Rutten & Stencel (1980) were able to find weak lines in the
wings of the Ca II $H$ and $K$ lines looking to data obtained near the solar
limb. In Figure 10 we show (with an arbitrary flux calibration), spectra of
$\alpha$ Cen A with spectra from the SO82B spectrograph on SKYLAB. The latter
are taken from the scans across a quiet region of solar limb obtained on
August 27 1973, the slit being tangential to the limb and stepped by 2”
between pointings. These are the photographic data analyzed by Doschek et al.
(1977). The “intensities” plotted here are photographic densities. But these
densities suffice for us to find significant blends, which will potentially be
revealed by changes in the spectra as the spectrograph slit was repointed from
2” inside the limb to 6” outside, in 2” steps.

Figure 10: The left panel shows the spectral region close to the sextet
resonance lines of Fe II is shown from $\alpha$ Cen A (top), and 5 spectra
representative of the spectra as a function of position relative to the solar
limb. The right panel shows the 2765-2780 Å region. Blue lines mark
wavelengths of Fe II transitions. The three other lines seen in emission in
the right panel are from Cr II at 2766.52 Å, and questionable identifications
of Fe II 2767.50, a transition between high lying states, and of Mg I at
2779.8 Å (Doschek et al., 1977).

Different behaviors are seen depending on the line of interest. In the
wavelength range from 2595 to 2615 Å (left panel, showing resonance lines of
the sextet system) there is no evidence of blended features. But the right
panel of Figure 10 shows a spectral region containing lines of the quartet
system of Fe II. In contrast to the resonance sextet transitions, here we find
that essentially no Fe II lines are strong enough to be unblended. However, a
plot of similar transitions (not shown) between 2750 and 2765 Å has stronger
lines at 2756 and 2762 Å which are not strongly blended. These conclusions
have been verified using NRL report 8653 (Moore et al., 1982), from which the
last column of Table 1 has been constructed. There are only four lines which
are classified as severely blended of all the quartets, sextets of Fe II and
quintets of Fe I.

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