# Mapping Solar Magnetic Fields from
the Photosphere to the Base of the Corona

Ryohko Ishikawa1∗, Javier Trujillo Bueno2,3,4,
Tanausú del Pino Alemán2,3, Takenori J. Okamoto1, David E. McKenzie5,
Frédéric Auchère6, Ryouhei Kano1, Donguk Song1, Masaki Yoshida1,7,
Laurel A. Rachmeler8, Ken Kobayashi5, Hirohisa Hara1, Masahito Kubo1,
Noriyuki Narukage1, Taro Sakao9, Toshifumi Shimizu9, Yoshinori Suematsu1,
Christian Bethge10, Bart De Pontieu11,12,13, Alberto Sainz Dalda11,14,15,
Genevieve D. Vigil5,16, Amy Winebarger5,
Ernest Alsina Ballester17, Luca Belluzzi17,18, Jiří Štěpán19,
Andrés Asensio Ramos2,3, Mats Carlsson12,13, Jorrit Leenaarts20
1National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588 Japan
2Instituto de Astrofísica de Canarias, 38205 La Laguna, Tenerife, Spain
3Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain
4Consejo Superior de Investigaciones Científicas, Spain
5NASA Marshall Space Flight Center, Huntsville, AL 35812, USA
6Institut d’Astrophysique Spatiale, 91405 Orsay Cedex, France
7Department of Astronomical Science, School of Physical Sciences,
SOKENDAI (The Graduate University for Advanced Studies)
8National Oceanic and Atmospheric Administration,
National Centers for Environmental Information, Boulder, CO 80305, USA
9Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency,
Sagamihara, Kanagawa 252-5210, Japan
10Cooperative Institute for Research in Environmental Sciences,
University of Colorado at Boulder, Boulder, CO 80305, USA
11Lockheed Martin Solar & Astrophysics Laboratory, Palo Alto, CA 94304, USA
12Rosseland Centre for Solar Physics, University of Oslo, NO-0315 Oslo, Norway
13Institute of Theoretical Astrophysics, University of Oslo, NO-0315 Oslo,
Norway
14Bay Area Environmental Research Institute, Moffett Field, CA 94035, USA
15Stanford University, HEPL, Stanford, CA 94305-4085, USA
16Universities Space Research Association, Huntsville, AL 35805, USA
17Istituto Ricerche Solari Locarno, CH - 6605 Locarno Monti, Switzerland
18Leibniz-Institut für Sonnenphysik (KIS), Schöneckstr. 6, D-79104, Freiburg,
Germany
19Astronomical Institute, Academy of Sciences of the Czech Republic,
25165 Ondrejov, Czech Republic
20Institute for Solar Physics, Department of Astronomy, Stockholm University,
AlbaNova University Centre, SE-106 91, Stockholm, Sweden
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> Routine ultraviolet imaging of the Sun’s upper atmosphere shows the
> spectacular manifestation of solar activity; yet we remain blind to its main
> driver, the magnetic field. Here we report unprecedented spectropolarimetric
> observations of an active region plage and its surrounding enhanced network,
> showing circular polarization in ultraviolet (Mg ii $h$ & $k$ and Mn i) and
> visible (Fe i) lines. We infer the longitudinal magnetic field from the
> photosphere to the very upper chromosphere. At the top of the plage
> chromosphere the field strengths reach more than 300 gauss, strongly
> correlated with the Mg ii $k$ line core intensity and the electron pressure.
> This unique mapping shows how the magnetic field couples the different
> atmospheric layers and reveals the magnetic origin of the heating in the
> plage chromosphere.

## 125-character teaser

A novel space experiment achieves an unprecedented map of solar magnetic
fields from the photosphere to the base of the corona

## Main Text

### Introduction

The chromosphere is a very important region of the solar atmosphere, with an
extension of several thousand km, located between the relatively cool surface
layers of the photosphere and the overlying hot corona (?, ?, ?). Although the
temperature of the chromospheric plasma does not exceed $10^{4}$ K, the fact
that its density is much larger than that of the extended and rarified corona
implies that much more mechanical energy is required to sustain the
chromosphere than the million-degree corona. Moreover, from the visible
photospheric surface to the chromosphere-corona transition region (TR) the
plasma density decreases exponentially by several orders of magnitude, more
rapidly than the magnetic field strength. As a result, the $\beta=1$
corrugated surface, where the ratio of gas to magnetic pressure is unity, lies
inside the chromosphere. Above the $\beta=1$ surface the magnetic field
essentially dominates the structuring and dynamics of the plasma. This,
together with the fact that the non-thermal energy needed to heat the corona
must propagate through the chromosphere, explains why it is indeed a crucial
interface region to solve many of the key problems in solar and stellar
physics.

It is impossible to fully understand the solar chromosphere without mapping
its magnetic structure, especially in the relatively hot layers of the upper
chromosphere and TR where $\beta<1$ (?, ?, ?, ?, ?, ?, ?, ?, ?). To this end,
we need to measure and model the polarization of ultraviolet (UV) spectral
lines originating in such atmospheric regions (?). The theoretical
investigations reported in the just quoted review paper led us to a series of
suborbital space experiments called CLASP, which required the development of
novel instrumentation (supplementary material 1A). The present investigation
is based on a unique dataset acquired by the Chromospheric LAyer
Spectropolarimeter (CLASP2), a suborbital space experiment that on 11 April
2019 allowed us to measure the first ever spectrally resolved Stokes profiles
across the Mg ii $h$ & $k$ lines in active and quiet regions of the solar
disk.

### Methods

We focus on the CLASP2 measurements in an active region plage and its
surrounding enhanced network (see Figures 1A and 1B, and note the position of
the spectrograph’s slit). These are poorly understood regions of the solar
disk with large concentrations of magnetic flux, located at the footpoints of
many coronal loops through which the mechanical energy that energizes the
corona propagates (?, ?). Although CLASP2 successfully measured the four
Stokes parameters of several lines around 280 nm (see Figure S2), in this
paper we use only the intensity (Stokes $I$) and circular polarization (Stokes
$V$) profiles of the Mg ii $h$ & $k$ lines and of two nearby Mn i lines. The
observed $V/I$ spectra of these four lines can be seen in panel D of Figure 1,
which also shows the detection of weaker circular polarization signals in
other spectral lines. We point out that in this paper we always deal with the
fractional circular polarization $V(\lambda)/I(\lambda)$ (hereafter, $V/I$),
with $\lambda$ the wavelength.

Interestingly, CLASP2 detected clear circular polarization signals not only
within the bright region of the plage, but also at the enhanced network
elements located at d and e (Figure 1A). The Stokes $V/I$ profiles of the Mg
ii $h$ & $k$ resonance lines have two external lobes and two inner lobes,
which encode information on the longitudinal component of the magnetic field
in the middle chromosphere and at the top of the upper chromosphere,
respectively (supplementary materials 2A$-$2C). The $V/I$ profiles of the two
Mn i lines at 279.91 nm and at 280.19 nm have only two lobes, which provide
information on the longitudinal field component in the lower chromosphere
(supplementary material 2D). In addition, in order to obtain the longitudinal
field component in the underlying photosphere, we have used the Stokes
profiles of two visible Fe i lines measured by the Hinode spacecraft during
the CLASP2 flight (supplementary material 1B).

All such circular polarization signals are produced by the Zeeman effect
caused by the magnetic field that permeates the solar atmosphere. To determine
the longitudinal field component from the Stokes $I$ and $V$ profiles measured
by CLASP2 in the above-mentioned chromospheric lines we applied the weak field
approximation (WFA; supplementary material 1D). Figure 2 shows an example of
the observed profiles, with the colored curves indicating the corresponding
WFA fits. The generally stronger photospheric magnetic field values reported
here result from the application of a suitable inversion code to the Stokes
profiles observed by the Hinode spacecraft (supplementary material 1E).

### Results and Discussion

Figure 3 gives the variation, along the spatial direction of the
spectrograph’s slit, of the longitudinal component of the magnetic field. The
figure shows this quantity in the photosphere (green curves), in the lower
chromosphere (blue symbols), in the middle chromosphere (black symbols) and at
the top of the upper chromosphere (red symbols). Note that the error bars
($1\sigma$) of the inferred values, due to the noise in $V/I$, are very small,
thanks to the high-precision CLASP2 measurements. The observed bright plage
region extends from $-98$ arcsec to about 28 arcsec, with a small dark region
located around $-62$ arcsec (see Figure 1A).

In the bright plage region, we see clearly in Figure 3 that the magnetic field
always shows a single magnetic polarity, except perhaps at the $-65$ arcsec
and $+12$ arcsec positions, where the retrieved opposite polarity fields may
not be statistically significant because they are comparable to or smaller
than the $1\sigma$ uncertainty. As expected, the strongest magnetic fields are
found in the photosphere of the observed plage, where the magnetic field
appears to be organized into small regions with strong magnetic concentrations
(with longitudinal field components as large as 1250 G) separated by small
regions with longitudinal field values of the order of 10 G (see the green
curve and its spatial fluctuation, which has typical scales of less than 10
arcsec). In the lower chromosphere (blue symbols) the magnetic field also
shows a substantial fluctuation on similar spatial scales, but the amplitude
of the spatial variation is significantly smaller and this variation is not
exactly in phase with the one in the underlying photosphere (e.g., note that
the spatial locations of the blue-symbol peaks do not coincide exactly with
those of the green curve). In the low chromosphere the maximum longitudinal
magnetic field values are about 700 G (blue symbols). In the middle
chromosphere of the plage and at the top of its upper chromosphere the black
and red symbols of Figure 3 indicate that the longitudinal component of the
magnetic field varies from almost zero gauss (slit positions around $+12$ and
$-65$ arcsec) to more than 300 G (locations a, c and $-44$ arcsec), showing a
smoother spatial variation than in the lower chromosphere (blue symbols) and
photosphere (green curve). The longitudinal magnetic field values found near
the TR (red symbols, inferred from the inner lobes of the Stokes $V/I$
profiles of Mg ii $h$ & $k$) are not much weaker than those found in the
middle chromosphere (black symbols, determined from the outer $V/I$ lobes of
the $h$ line), although it must be noted that the latter values are a lower
limit (supplementary material 2C). Finally, note that at almost all of the
spatial positions where the fluctuating green curve reaches its local minima,
the photospheric longitudinal field values are smaller than at the same
locations in the overlying layers (e.g., see position b and the corresponding
Figure S5). However, at the spatial positions where the longitudinal component
of the magnetic field in the lower chromosphere (blue symbols) reaches its
smallest values, we find similar ones in the middle chromosphere and at the
top of the upper chromosphere (black and red symbols, respectively). These and
the previously mentioned fact, namely, that the spatial fluctuations of the
longitudinal field values decrease with height, confirm that the plage
magnetic field is highly structured in the photosphere, with strong and weak
field variations, and that it expands rapidly merging and spreading
horizontally in the overlying chromosphere where the field is weaker and has
smoother spatial variations.

We have found a remarkably high correlation between the longitudinal component
of the magnetic field in the middle and top layers of the plage chromosphere
and the intensity at the center and the emission peaks of the Mg ii $k$ line
(see Figure 3). As shown in Table 1, the coefficient that quantifies this
correlation increases with height in the plage atmosphere, and it is important
to note that the intensity in the $k$ line-core emission peaks correlates with
the temperature at its formation height (?). Moreover, from inversions of the
observed Mg ii $h$ & $k$ intensity profiles (supplementary material 2E), we
have found (see Table 1) that in the middle and the upper chromosphere the
longitudinal component of the magnetic field is also significantly correlated
with the electron pressure (i.e., with the product of the temperature and the
electron density). These pieces of empirical information strongly support that
the heating of the upper chromosphere of active region plages is of magnetic
origin.

Consider now the supergranulation cell outlined by its bright chromospheric
network, which is located above 60 arcsec in Figure 1A. The slit of the
spectrograph crosses network features at 63 arcsec (location d) and at 97
arcsec (location e). As shown by the green curve of Figure 3, the longitudinal
component of the magnetic field in the photosphere of these two network
elements is about 220 G. At these network locations we find longitudinal field
strengths of about 160 G and 80 G in the lower and middle chromosphere,
respectively (see Figure 4 for location d). Figure 3 shows that the magnetic
fields in the atmospheres of these two network elements have opposite
polarities. As seen in the top panels of Figure 4, the inner lobes of the
Stokes $V/I$ profiles of the $h$ & $k$ lines are so weak that their circular
polarization amplitudes are compatible with zero within the error bars, and
this is not due to cancelations of the circular polarization signals during
the 150.4 s integration time (supplementary material 1D). The absence of
significant inner lobes in such $V/I$ profiles leads us to conclude that the
magnetic field in the very upper chromosphere of the observed network features
is weaker than about 10 G. Therefore, in the atmospheres of these network
elements the longitudinal field values vary approximately from 220 G in the
photosphere, to 160 G in the lower chromosphere, to at least 80 G in the
middle chromosphere, and to less than 10 G at the top of the upper
chromosphere.

We foresee three possible scenarios compatible with the above-mentioned
results from the CLASP2 spectropolarimetric observations of the network
features: (a) the magnetic loops of the network do not reach the very top of
the chromosphere, and return to the photosphere after reaching chromospheric
heights, (b) the magnetic fields in the very upper chromosphere of the network
regions are too weak to be detected through the Zeeman effect in the Mg ii
resonance lines, and (c) the magnetic field in the top layers of the network
chromosphere becomes nearly perpendicular to the line of sights of the CLASP2
observation, so that there is hardly any longitudinal field component to
measure. Our CLASP2 circular polarization measurements provide quantitative
evidence that the commonly held picture of a wineglass-shaped magnetic canopy
of network fields that fill the entire quiet solar chromosphere above a
certain height is too simplistic to describe the expansion of the magnetic
field in the observed enhanced network regions (?, ?). An advanced inversion
code is currently under development, and we hope that its future application
to the four Stokes profiles observed by CLASP2 (see Figure S2) will allow us
to disentangle among the three options mentioned above.

Although a suborbital space experiment can only provide a few minutes of
observing time, the unprecedented observations achieved by CLASP2 have
demonstrated the enormous diagnostic potential of spectropolarimetry in the
spectral region of the Mg ii $h$ & $k$ lines. Of particular interest are the
circular polarization signals observed in the Mg ii and Mn i resonance lines.
This is the first time that the longitudinal component of the magnetic field
is simultaneously determined at several heights from the photosphere to the
very top layers of the chromosphere of active region plages and enhanced
network features, at each position along the spatial direction of the
spectrograph’s slit. This opens up the possibility of mapping the magnetic
field throughout the solar atmosphere over large fields of view, an empirical
information that is crucial for obtaining more accurate coronal field
estimates and for deciphering how the magnetic field couples the different
atmospheric layers and channels the mechanical energy that heats the
chromosphere and the corona of our nearest star.

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## Acknowledgments

CLASP2 is an international partnership between NASA/MSFC, NAOJ, JAXA, IAC, and
IAS; additional partners include ASCR, IRSOL, LMSAL, and the University of
Oslo. The Japanese participation was funded by JAXA as a Small Mission-of-
Opportunity Program, JSPS KAKENHI Grant numbers JP25220703 and JP16H03963,
2015 ISAS Grant for Promoting International Mission Collaboration, and by 2016
NAOJ Grant for Development Collaboration. The USA participation was funded by
NASA Award 16-HTIDS16_2-0027. The Spanish participation was funded by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Advanced Grant agreement No. 742265). The
French hardware participation was funded by CNES funds CLASP2-13616A and
13617A. The CLASP2 team acknowledges Dr. Shin-nosuke Ishikawa, who led the
development of the critical component of the polarization modulation unit
(PMU) at ISAS/JAXA. Hinode is a Japanese mission developed and launched by
ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as
international partners. It is operated by these agencies in cooperation with
ESA and NSC (Norway). IRIS is a NASA Small Explorer Mission developed and
operated by LMSAL with mission operations executed at NASA Ames Research
Center and major contributions to downlink communications funded by ESA and
the Norwegian Space Centre.

## Funding

R.I. acknowledges the NAOJ Overseas Visit Program for Young Researchers
(FY2019) and a Grant-in-Aid for Early-Career Scientists JP19K14771. J.T.B.
acknowledges the funding received from the European Research Council through
Advanced Grant agreement No. 742265. E.A.B. and L.B. acknowledge the funding
received from the Swiss National Science Foundation through grants
200021-175997 and CRSII5-180238. B.D.P. and A.S.D. acknowledge support from
NASA contract NNG09FA40C (IRIS). J. Š. acknowledges financial support by the
Grant Agency of the Czech Republic through grant 19-20632S and project
RVO:67985815. M.C. acknowledges support from the Research Council of Norway
through its Centres of Excellence scheme, project number 262622, and through
grants of computing time from the Programme for Supercomputing.

## Author contributions

R.I. and J.T.B conceived the investigation, critically discussed each
development step and wrote the manuscript. R.I. led the data analysis, while
J.T.B and T.dP.A. led the theoretical investigations applied to this study,
which included discussions with E.A.B., L.B., J.Š. and A.A.R. The comparison
between the CLASP2 and IRIS data was done by T.J.O., who also processed the
Hinode/SOT data. D.E.M., R.I, J.T.B, and F.A. are the CLASP2 Principal
Investigators from USA, Japan, Spain, and France, respectively. T.J.O and
L.A.R., the CLASP2 project scientists, helped to coordinate the observations
with IRIS and Hinode. B.D.P. conducted the IRIS observation and together with
A.S.D. performed the inversions based on the IRIS2 database. C.B., T.J.O. and
D.S. contributed to the data calibration. R.K., D.S., M.Y., K.K., H.H., M.K.,
N.N., T.S., T.S., Y.S., G.D.V., and A.W. contributed to the development of the
instrument. M.C. and J.L. provided information on models of the solar
chromosphere.

## Competing interests

There are no competing interests.

## Data and materials availability

All data needed to evaluate our conclusions are presented in the paper and/or
in the Supplementary Materials. The calibrated CLASP2 data will be made
publicly available soon through the Virtual Solar Observatory. The Level 2
Hinode/SOT-SP data are available at https://csac.hao.ucar. edu/sp_data.php,
the IRIS data at https://iris.lmsal.com/data.html, and the SDO data at
http://jsoc. stanford.edu.

Fig. 1. CLASP2 and Hinode data. (A) Broadband Lyman-$\alpha$ image obtained by
the CLASP2 slit-jaw system. The red line indicates the radially oriented slit
of the CLASP2 spectrograph, which covers 196 arcsec. (B) Longitudinal
component of the photospheric magnetic field inferred from the Stokes profiles
observed by Hinode/SOT-SP in Fe i visible lines. (C) Variation along the slit
of the intensity profile $I(\lambda)$ observed in the spectral region of the
Mg ii $h$ & $k$ lines. The labels at the top of panel C indicate the location
of Mg ii $k$ at 279.64 nm, Mg ii $h$ at 280.35 nm, and the Mn i lines at
279.91 nm and 280.19 nm. (D) Fractional circular polarization
$V(\lambda)/I(\lambda)$ observed by CLASP2 around 280 nm. The $I$ and $V/I$
spectra are the result of temporally averaging the individual Stokes
parameters during 150.4 s. The vertical axes indicate the distance in
arcseconds along the spatial direction of the CLASP2 slit, measured from its
center. The small gap seen in panels C and D results from the lack of data in
a few deteriorated pixels.

Fig. 2. Example of the Stokes $I$ and $V/I$ profiles observed by CLASP2. The
profiles correspond to Mg ii $k$ at 279.64 nm, Mg ii $h$ at 280.35 nm, Mn i at
279.91 nm, and Mn i at 280.19 nm at location c in Fig. 1, where the
longitudinal field retrieved from the Mg ii $h$ & $k$ lines is among the
strongest ones. The gray curves show the corresponding Stokes $I$ profiles,
normalized to the maximum intensity of the Mg ii $k$ line. The $V/I$ error
bars indicate the $\pm{1}\sigma$ uncertainties resulting from the photon
noise. The WFA fits and the inferred longitudinal magnetic field values are
shown in blue for the Mn i lines, in black for the external $V/I$ lobes of Mg
ii $h$, and in red for the inner $V/I$ lobes of Mg ii $h$ & $k$.

Fig. 3. Spatial variation of the longitudinal component of the magnetic field
($B_{\mathrm{L}}$). Blue symbols: $B_{\mathrm{L}}$ in the lower chromosphere
determined from the Mn i lines. Black symbols: $B_{\mathrm{L}}$ in the middle
chromosphere determined from the external lobes of the $V/I$ profiles in the
Mg ii $h$ line. Red symbols: $B_{\mathrm{L}}$ at the top of the upper
chromosphere determined from the inner lobes of the $V/I$ profiles in the Mg
ii $h$ & $k$ lines. Green curve: $B_{\mathrm{L}}$ in the photosphere obtained
from the Hinode/SOT-SP observations, after spatially smearing the data to
mimic the CLASP2 resolution (supplementary material 1E). The error bars of the
CLASP2 data represent 1$\sigma$ errors. Note that the thin black curves give
the normalized intensity observed by CLASP2 at the $k_{\rm 3}$ line center and
at the $k_{\rm 2v}$ emission peak of the Mg ii $k$ line (solid and dashed-
dotted; see the inset of Fig. S4). The lower panel is a zoom of the upper one.

Fig. 4. CLASP2 data corresponding to one of the network features. $V/I$
profiles at location d indicated in Fig. 1. See the caption of Fig. 2 for
further explanations. We find $B_{L}=-5\pm 7$ G at the top of the upper
chromosphere (after averaging the longitudinal field values retrieved from the
inner $V/I$ lobes of the Mg ii $h$ & $k$ lines), $B_{L}=78\pm 7$ G in the
middle chromosphere (from the external $V/I$ lobes of Mg ii $h$), and
$B_{L}=164\pm 15$ G in the lower chromosphere (from the $V/I$ profiles of the
Mn i lines).

Table 1. Correlation coefficients (CI for intensity and CP for electron
pressure). The table gives the corresponding correlation coefficient between
the longitudinal component of the magnetic field, $B_{\mathrm{L}}$, inferred
at different heights in the atmosphere, and (a) the observed Mg ii $k$-line
intensity ($k_{\rm 3}$ or $k_{\rm 2v}$) and (b) the inferred electron pressure
near the upper chromosphere (see supplementary material 2E). The linear
Pearson correlation coefficients have been calculated considering only the
bright plage region pixels located between $-98$ and $+28$ arcsec (see Figure
1C).

Wavelength window for $B_{\mathrm{L}}$ | Atmospheric layer | CI ($k_{\rm 3}$) | CI ($k_{\rm 2v}$) | CP
---|---|---|---|---
Inner lobes of Mg ii $h$ & $k$ | Top of upper chromosphere | 0.87 | 0.81 | 0.74
External lobes of Mg ii $h$ | Middle chromosphere | 0.80 | 0.75 | 0.67
Mn i lines around 280 nm | Lower chromosphere | 0.65 | 0.63 | 0.55
Fe i lines around 630.2 nm | Photosphere | 0.48 | 0.44 |

## Supplementary Materials

Supplementary Material and Methods
Supplementary Text
Figs. S1 to S7
References (15-58)

##  1 Supplementary Material and Methods

1A. The CLASP2 instrument and the observation

The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) (?, ?, ?, ?, ?, ?)
was designed to measure the variation with wavelength of the Stokes $I$, $Q$
and $U$ parameters of the hydrogen Lyman-$\alpha$ line at 121.6 nm, where $I$
is the intensity and $Q$ and $U$ the linear polarization. This suborbital
rocket experiment was carried out in September 2015, and provided the first
measurement of the linear polarization signals produced by scattering
processes and the Hanle effect in the Lyman-$\alpha$ line  (?, ?, ?, ?).
CLASP2 is the result of a non-negligible modification of CLASP, but
maintaining its novel design based on the inverse Wadsworth configuration
(?). The aim was to measure the Stokes $I$, $Q$, $U$ and $V$ (circular
polarization) parameters across the Mg ii $h$ & $k$ lines near 280 nm in quiet
and active regions of the solar disk with a spectral resolution of 0.01 nm, a
temporal resolution of 3.2 s and a spatial resolution of at least 2 arcsec (?,
?). The resolutions estimated from the pre-flight measurements were $\sim$1.1
arcsec and $\sim$0.01 nm. A comparison of the spectroscopic data taken by
CLASP2 and IRIS during the observation of the plage target suggests that these
spatial and spectral resolutions are representative for the CLASP2 flight
measurements.

Previous space instruments developed for measuring polarization signals in
solar ultraviolet lines failed due to in-flight degradation problems (?), or
lacked the polarimetric sensitivity and resolution (spectral, spatial and
temporal) needed for accurately determining magnetic fields in the upper solar
chromosphere and TR of quiet and active regions (?, ?, ?, ?, ?).

The CLASP2 instrument consists of a Cassegrain telescope with an aperture of
27 cm, a slit-jaw (SJ) monitor system, and a spectropolarimeter to measure the
Stokes $I$, $Q$, $U$ and $V$ profiles in the wavelength window of the Mg ii
$h$ and $k$ lines over the 196 arcsec covered by the spectrograph’s slit. The
width of the slit corresponds to 0.53 arcsec. The SJ system is the same one
used in CLASP (?); it takes fast cadence (0.6 s) images of the chromosphere
using a Lyman-$\alpha$ broad-band filter (FWHM=7 nm). The plate scale of the
SJ monitor system is 1.03 arcsec per pixel (hereafter, arcsec/pix).

The spectropolarimeter has two optically symmetric channels, which provide
simultaneous measurements of two orthogonal polarization states. The
continuously rotating waveplate of the Polarization Modulation Unit (PMU) (?)
is located in front of the slit and modulates the incident radiation (one
rotation takes 3.2 s). At the downstream of the slit, a diffraction grating
mounted in the inverse Wadsworth configuration serves both as a spectral
dispersion element and as beam splitter. Each resulting beam finally goes
through a linear polarization analyzer, mounted 90∘ from each other, and is
measured by its corresponding CCD camera. The two cameras record the modulated
intensity every 0.2 s in synchronization with the waveplate rotation. After
the dark and gain corrections, we perform the demodulation and derive the
wavelength variation of the four Stokes parameters for each channel beam and
for each rotation of the waveplate. Then, we correct for pointing and
wavelength drifts and co-register the Stokes spectra. Finally we obtain $I$,
$Q/I$, $U/I$ and $V/I$ by combining the information from the two channels. The
plate scales of the spectropolarimeter are 0.527 arcsec/pix along the slit and
0.00498 nm/pix in the dispersion direction.

CLASP2 conducted the observations at three locations on the solar disk during
its ballistic flight. From 16:52:53 to 16:53:07 UT on April 11 2019, it
observed the center of the solar disk to verify that the instrumental
polarization is negligibly small compared with the required polarization
accuracy of 0.1%. From 16:53:40 to 16:56:16 UT, it observed an active region
plage which was located at the East side of active region NOAA 12738 (Fig.
S1A). About 130 arcsec of the slit covered part of the bright region of the
plage, while the remaining 70 arcsec crossed a surrounding quieter region
including two network elements of a supergranulation cell. The data obtained
during this 150.4 s had a very stable pointing. For the analysis of this
paper, in order to increase the signal-to-noise ratio we have used the
temporally averaged spectra from the plage target. From 16:56:25 to 16:58:45
UT, CLASP2 observed a quiet region near the solar limb, with the slit also
radially oriented with respect to the center of the solar disk.

Figure S2 shows the variation along the slit of the intensity (Stokes $I$) and
fractional polarization ($Q/I$, $U/I$ and $V/I$) observed by CLASP2 in the
plage target. The polarization signals theoretically expected for the Mg ii
$h$ & $k$ lines are clearly seen, both the circular polarization produced by
the Zeeman effect and the linear polarization caused by scattering processes
and its possible modification by the Hanle and magneto-optical effects (?, ?,
?, ?). In addition, CLASP2 detected clear circular polarization signals
produced by the Zeeman effect in Mn i lines, as well as in other spectral
lines which will be the subject of other publications. In this paper we focus
on the longitudinal component of the magnetic field that we can directly infer
from the Stokes $I$ and $V/I$ profiles of the Mg ii and Mn i resonance lines,
while in future investigations we will apply a new (presently under
development) non-LTE inversion technique to the Stokes profiles observed by
CLASP2 across the Mg ii resonance lines.

1B. The Hinode/SOT observation

The Hinode satellite (?) performed coordinated observations of the CLASP2
plage target. In particular, the Spectro-Polarimeter (SP) (?, ?) of the Solar
Optical Telescope (SOT) (?, ?, ?) measured the four Stokes profiles of the Fe
i photospheric lines at 630.15 nm and 630.25 nm with a wavelength sampling of
2.16 pm.

From UT 16:46:57 to 17:51:31 of 2019 April 11 the plage region was scanned
using the fast mapping mode, with an exposure time of 3.2 s at each slit
position and a spatial plate scale of 0.32 arcsec/pix. The resulting field of
view (FOV) was 299 arcsec (solar E-W) by 161 arcsec (solar N-S), which
included the spatial positions of the slit of the CLASP2 spectrograph (Fig.
1B). The spatial resolution that can be achieved with the Hinode/SOT-SP is
about 0.32 arcsec, with a spatial sampling of 0.16 arcsec (?, ?), but with the
chosen fast mapping mode the spatial resolution was $\sim 0.64$ arcsec. Thanks
to these nearly simultaneous coordinated observations with Hinode, we obtained
a map of the photospheric longitudinal magnetic field strength through the
application of a Milne-Eddington inversion code (supplementary material 1E).

The co-alignment of the CLASP2 and Hinode/SOT-SP data was done using images
from the Atmospheric Imaging Assembly (AIA) (?) and the Helioseismic and
Magnetic Imager (HMI) (?) on board the Solar Dynamics Observatory (SDO) (?).
AIA takes full disk images of the solar atmosphere in multiple wavelengths
with a temporal cadence of 12 s and HMI provides full disk measurements of
photospheric longitudinal magnetic fields (magnetograms) every 45 s, both of
them with a moderate spatial resolution of $\sim$1.5 arcsec. The AIA and HMI
images are calibrated and co-aligned using the data reduction procedure
aia_prep.pro in SolarSoft (see https://www.lmsal.com/solarsoft/). The AIA 30.4
nm channel provides images very similar to those given by the CLASP2 SJ
monitor system (?). Firstly, the co-alignment of the CLASP2 and SDO data was
done by cross-correlating the AIA 30.4 nm (Fig. S1A) and CLASP2/SJ images
(Fig. S1B). Secondly, the co-alignment of the SDO and Hinode/SOT images was
done by cross-correlating the photospheric longitudinal magnetic field maps
obtained by SOT-SP and HMI. The HMI magnetograms were temporally averaged over
the duration of CLASP2 observations of the plage target.

1C. The IRIS observation

Coordinated observations of the CLASP2 plage target were also carried out with
the Interface Region Imaging Spectrograph (IRIS) (?), from 16:36:45 to
17:51:27 UT of 2019 April 11. The roll angle of the IRIS spacecraft was set to
$71^{\circ}$ and the slit of its spectrograph was almost parallel to the
CLASP2 slit (Fig. S1).

The IRIS raster scans, each with 16 steps and a 1 arcsec gap between adjacent
slit positions, cover a FOV of 15 arcsec (perpendicular to the slit) $\times$
175 arcsec (parallel to the slit), with a cadence of 28 s (see the green
dashed boxes in Fig. S1). Five raster scans occurred during the CLASP2
observations. With its spectrograph (SG), IRIS measured the Stokes $I$ spectra
in two wavelength ranges, in the near ultraviolet (NUV) and in the far
ultraviolet (FUV), but it also provided slit-jaw images (SJIs). The co-spatial
and co-temporal Mg ii $h$ and $k$ NUV spectra, and the SJIs with a broad-band
passband of 0.4 nm centered at 279.6 nm (i.e., around the center of the Mg ii
$k$ line, panel C of Fig. S1) were used for comparison with the CLASP2 data.
Binnings along the spatial and dispersion directions were carried out on-
board, yielding spatial and wavelength plate scales of 0.33 arcsec/pix and 5.1
pm/pix for the NUV SG data, and a spatial plate scale of 0.33 arcsec/pix for
the SJIs, which implies a spatial resolution of 0.66 arcsec because of the
Nyquist theorem. Without binning the IRIS spatial resolution in the NUV would
be 0.4 arcsec, both for the SG and SJIs (?).

The spatial resolution of the CLASP2 spectra can be mimicked using the IRIS
SJI data. The pseudo SJI at 279.6 nm is made by applying the IRIS passband to
the CLASP2 intensity spectra, while the IRIS SJI is spatially convolved with
Gaussians of several widths. The best cross-correlation between the pseudo SJI
and the real IRIS SJI was found for a Gaussian FWHM of $\sim$1.2 arcsec (panel
D of Fig. S1).

In order to get a better context of the plage target, with IRIS we also
carried out dense raster scans before and after the CLASP2 observation. The
scan step size was set to 0.35 arcsec and the FOV covered by the SG was 112
arcsec (perpendicular to the slit) $\times$ 175 arcsec (parallel to the slit),
taking about 30 min to complete one raster scan. Figure S3 shows the resulting
raster intensity image at the fixed wavelength position of the Mg ii $k$ line
center, taken at the closest observing time to the CLASP2 observation. These
high spatial resolution IRIS data are helpful to understand the structure of
the observed plage in the upper chromosphere.

1D. Derivation of the longitudinal magnetic field component with the WFA

The weak-field approximation (WFA) is the simplest and most direct technique
to estimate the longitudinal component of the magnetic field from the observed
intensity and polarization profiles (?, ?). The WFA is applicable in the weak-
field regime, namely when the Zeeman splitting
$g_{\mathrm{eff}}\Delta\lambda_{B}$ (with $g_{\mathrm{eff}}$ the effective
Landé factor) is significantly smaller than the line’s Doppler width
$\Delta\lambda_{D}$. These quantities are given by

$\Delta\lambda_{B}=\frac{e\lambda^{2}_{0}}{4\pi m_{e}c}B,$ (1)

and

$\Delta\lambda_{D}=\frac{\lambda_{0}}{c}\sqrt{\frac{2k_{B}T}{m}+\xi^{2}},$ (2)

where $e$ is the absolute value of the electron charge, $m_{e}$ the mass of
the electron, $\lambda_{0}$ the wavelength of the spectral line, $c$ the speed
of light, $T$ the plasma temperature, $k_{B}$ the Boltzmann’s constant, $m$
the mass of the atom, and $\xi$ the non-thermal velocity. Therefore,

${\color[rgb]{0,0,0}{\cal
R}}\,=\,g_{\mathrm{eff}}\frac{\Delta\lambda_{B}}{\Delta\lambda_{D}}\propto
g_{\mathrm{eff}}\frac{\lambda_{0}B}{\sqrt{2k_{B}T/m+\xi^{2}}}.$ (3)

In the chromosphere, where the UV spectral lines here considered originate,
the temperature and the non-thermal velocity are larger and the magnetic field
strength is weaker than in the underlying photosphere. Typically, ${\cal
R}{\approx}0.06$ for the Mg ii lines (assuming $T=8000$ K and $B=300$ G) and
${\cal R}{\approx}0.3$ for the Mn i lines (assuming $T=4000$ K and $B=500$ G)
when the non-thermal velocity $\xi$ is not included in the estimation (see
supplementary materials 2C and 2D for the effective Landé factors). Given that
in the chromosphere $\xi$ is very significant, in reality such $\cal R$ values
are even smaller (i.e., the weak field regime holds for both spectral lines).

In the weak-field regime, the following equation is strictly valid when the
line-of-sight component of the magnetic field is constant in the atmospheric
region where the Stokes $V$ profile of the spectral line under consideration
is formed:

$V(\lambda)=-\frac{e\lambda^{2}_{0}}{4\pi
m_{e}c}g_{\mathrm{eff}}B_{\mathrm{L}}\biggl{(}\frac{\partial
I}{\partial\lambda}\biggr{)}=-4.67\times
10^{-12}g_{\mathrm{eff}}\lambda^{2}_{0}B_{\mathrm{L}}\biggl{(}\frac{\partial
I}{\partial\lambda}\biggr{)},$ (4)

where $\lambda_{0}$ is in nm and $B_{\mathrm{L}}=B{\rm cos}{\theta_{B}}$ is
the longitudinal component of the magnetic field, with $B$ the magnetic
strength in gauss and $\theta_{B}$ the angle between the magnetic field vector
and the line of sight (LOS). After calculating the derivative of the Stokes
$I$ profile with respect to wavelength, the scaling factor that provides the
best fit to the circular polarization profile is obtained by means of the
least-square minimization method. The 1-$\sigma$ uncertainties of the
retrieved $B_{\mathrm{L}}$ values were estimated from the $V(\lambda)$
measurement errors, the main source of which is the photon noise.

As is well-known, the magnetic field in the solar atmosphere is not constant
along the LOS. Nevertheless, as discussed in supplementary material 2C, the
WFA is still applicable to estimate the longitudinal field component in the
line formation region, provided that the magnetic field is in the weak field
regime and that the spatial gradients of the LOS component of the magnetic
field are not too extreme in the region of formation of the Stokes $I$ and $V$
spectral features.

As mentioned in supplementary material 1A, in order to increase the signal-to-
noise ratio we have averaged the Stokes profiles over the 150.4 s duration of
the plage-target observation. Although the solar chromosphere is very dynamic,
with significant variations of the macroscopic velocity and of the kinetic
temperature and density of the plasma over shorter time scales, its magnetic
structure is expected to be much simpler than its thermodynamics, especially
in the $\beta{<}1$ regions where it is almost force-free, and to show a
relatively slow time evolution (?). The important question is whether the
longitudinal field component inferred from the temporally-averaged Stokes
profiles is significantly affected by their temporal evolution, mainly
dominated by the dynamics and thermodynamics of the chromospheric plasma. We
have checked that the wavelength derivative of the observed intensity profiles
of the Mg ii $h$ & $k$ lines at each temporal step of 3.2 s shows similar
patterns and does not lead to any significant cancellation or reduction of the
temporally-averaged signals.

We point out that what the WFA provides is the average longitudinal field
strength over the resolution element (i.e., the net magnetic flux divided by
the area of the resolution element). This quantity is equal to the
longitudinal magnetic field strength if the magnetic filling factor ($f$) is
unity in the atmospheric region of formation of the considered spectral line
interval (in a two-component model $f$ is the fraction of the resolution
element covered by the magnetic component). Because of the rapid expansion of
the magnetic field in active region plages the magnetic filling factor rapidly
approaches unity as one goes to higher atmospheric layers. We assume $f=1$ in
our inferences from the CLASP2 data.

1E. Determination of the photospheric magnetic field from the Hinode/SOT-SP
data

The photospheric magnetic field in plage regions is generally strong, of the
order of kG, and thus the WFA is not generally applicable to infer the
magnetic field from the Fe i lines around 630 nm. Therefore, in order to
obtain the LOS component of the magnetic field in the photosphere of the
observed plage region,  we have used the SOT-SP Level 2 information given by
the Community Spectro-polarimetric Analysis Center (CSAC) of the High Altitude
Observatory (DOI:10.5065/D6JH3J8D. See also https://www2.hao.ucar.edu/csac).
This information stems from the application of the MERLIN Milne-Eddington (ME)
inversion code to the SOT-SP Level 1 data (DOI:10.5065/D6T151QF), which
provides the magnetic field vector (field strength $B$, inclination
$\theta_{B}$, and azimuth $\chi_{B}$), the line-of-sight velocity, the source
function, the Doppler broadening as well as the macroturbulence and a stray
light factor. To this end, MERLIN assumes a single atmospheric magnetic
component and fits the observed intensity profile with
$fI_{mag}+(1-f)I_{stray}$, where $I_{mag}$ is the intensity from the magnetic
component and $I_{stray}$ the stray light profile. The fraction $(1-f)$ of
stray light needed to fit the observed intensity spectrum varies between 0 and
1, and the filling fraction $f$ can in principle be interpreted as the
fraction of the pixel occupied by the assumed single atmospheric magnetic
component. The $f$ values provided by MERLIN at the spatial points of the
plage crossed by the CLASP2 slit vary between 0.2 and 0.95 (see the brightest
region in Fig. 1A). The quantity $B\cos{\theta_{B}}$ is a good estimation of
the longitudinal field strength $B_{\mathrm{L}}$ if the assumption of a single
atmospheric magnetic component within each pixel is reasonable and if there is
no significant degeneracy between the values inferred for $f$ and $B$. In this
paper, we have assumed this to be the case, reason why Figure 1B and the green
curve of Figure 3 show $B\cos{\theta_{B}}$.

The Hinode/SOT-SP data has better spatial resolution than that of the CLASP2
data. In order to properly compare the spatial variation of the photospheric
longitudinal magnetic field, $B_{\mathrm{L}}$, with the ones we have obtained
from the Stokes $I$ and $V/I$ profiles observed by CLASP2, we have smeared the
$B_{\mathrm{L}}$ map obtained from the SOT-SP observations to mimic the CLASP2
spatial resolution. The convolution of the IRIS data with a Gaussian with a
FWHM of 1.2 arcsec provides the best cross-correlation with the CLASP2 data
(supplementary material 1C). Because the spatial resolution of the SOT-SP and
IRIS instruments are comparable, we have applied the same gaussian smoothing
to the SOT-SP map (see the green curve of Fig. 3).

##  2 Supplementary Text

2A. Intensity spectra

Figure S4 shows the temporally and spatially averaged intensity spectra
observed by CLASP2 outside the bright part of the plage target, which covers
60 arcsec (see Fig. 1C). The investigation of this paper is based on the
Stokes $I(\lambda)$ and $V(\lambda)/I(\lambda)$ profiles of the Mn i and Mg ii
lines indicated in Figure S4 (hereafter, Stokes $I$ and $V/I$).

The Stokes $I$ profiles of the Mg ii $h$ and $k$ lines show the well-known two
peak structure (see the inset of Fig. S4), where we distinguish three
features: $k_{3}$ (line center depression), $k_{2\mathrm{v}}$ and
$k_{2\mathrm{r}}$ (the violet and red emission peaks), and $k_{1\mathrm{v}}$
and $k_{1\mathrm{r}}$ (the violet and red minima) of the Mg ii $k$ line (for
the Mg ii $h$ line the same features are dubbed $h_{3}$, $h_{2\mathrm{v}}$,
$h_{2\mathrm{r}}$, $h_{1\mathrm{v}}$, and $h_{1\mathrm{r}}$).

In Figure 3 and Table 1, we provide information at the $k_{3}$ and
$k_{2\mathrm{v}}$ wavelengths. At most of the pixels along the slit, the
$k_{2\mathrm{v}}$ intensity peaks were successfully identified. However, in
some pixels of the bright region of the plage target, the Mg ii $h$ & $k$
intensity spectra showed blue-shifted single-peak profiles, without any clear
$k_{3}$ or $h_{3}$ line-center depression. For such pixels we have used the
$k_{3}$ wavelength of the spatially averaged intensity profile.

2B. Detailed description of the circular polarization observed in Mg ii $h$
and $k$

Figure S5 shows the intensity and circular polarization profiles of the Mg ii
and Mn i lines at the location b indicated in Figure 1. The Mg ii $h$ line
shows larger $V/I$ amplitudes than the Mg ii $k$ line ($g_{\mathrm{eff}}=4/3$
and $7/6$, respectively), as expected from their effective Landé factors. The
$V/I$ profiles of the Mg ii resonance lines show four lobes, both inside and
outside the bright region of the observed plage, with the inner lobes (located
just around the line centers, $\Delta{\lambda}<0.015$ nm) presenting
significantly smaller amplitudes than the external ones. A comparison between
the intensity (gray line in Fig. S5) and the $V/I$ profiles indicates that the
inner lobes extend from the $k_{3}$ ($h_{3}$) center till the
$k_{2\mathrm{v}}$ and $k_{2\mathrm{r}}$ ($h_{2\mathrm{v}}$ and
$h_{2\mathrm{r}}$) peaks of the intensity profiles, while the external lobes
appear in the outer wings of the $k_{2\mathrm{v}}$ and $k_{2\mathrm{r}}$
($h_{2\mathrm{v}}$ and $h_{2\mathrm{r}}$) peaks. Figure S5 also shows the best
fits produced by the wavelength derivative of the intensity profile, which is
also normalized to $I(\lambda)$ (i.e., to the intensity at each spectral
pixel). We point out that the relatively small amplitudes of the inner lobes
of the plage $V/I$ profiles are caused by the corresponding flattened
intensity profiles in the spectral region between $k_{2\mathrm{v}}$
($h_{2\mathrm{v}}$) and $k_{2\mathrm{r}}$ ($h_{2\mathrm{r}}$) peaks.

2C. Application of the WFA to the Mg ii $h$ and $k$ lines

Radiative transfer calculations in semi-empirical models of the solar
atmosphere with constant magnetic fields weaker than 1000 G show that, while
the WFA provides an excellent fit to the inner lobes of the $V/I$ profiles of
the Mg ii $k$ line ($g_{\mathrm{eff}}=7/6$), it overestimates the amplitudes
of the external lobes due to the impact of partial frequency redistribution
(PRD) on the fractional circular polarization (?, ?). This implies that if the
conditions under which the WFA hold are satisfied, then its application to the
inner lobes alone provides a reliable estimation of $B_{\mathrm{L}}$ in the
region of the solar atmosphere where these lobes originate, while its
application only to the external lobes would underestimate $B_{\mathrm{L}}$ in
the corresponding atmospheric region of formation. The same is valid for the
Mg ii $h$ line ($g_{\mathrm{eff}}=4/3$), but the underestimation of the
$B_{\mathrm{L}}$ values that result from the application of the WFA to the
external lobes of this line is less significant (i.e., the use of the $V/I$
external lobes of the $h$ line leads to an error of about 10%, while it is a
bit larger for the $k$ line).

There is also a blend in the external blue wing of the $k$ line. Thus, in this
paper we have applied the WFA separately to the inner lobes of the $V/I$
profiles of $h$ and $k$ and to the external lobes of the $V/I$ profiles of
only the $h$ line.

In the solar atmosphere the magnetic field changes along the LOS, but if this
variation is not too strong the WFA still provides reliable estimations of the
longitudinal field component. We illustrate this in Figure S6, the right panel
of which shows the results of PRD radiative transfer calculations in a semi-
empirical model of the quiet solar atmosphere (?) permeated by a vertical
magnetic field with a strength varying exponentially with height as indicated
in the left panel. The black filled circles show the calculated $V/I$ profile,
while the solid curves show the fits obtained when applying the WFA only to
the external $V/I$ lobes (gray curves) and only to the inner $V/I$ lobes
(green curve). The longitudinal magnetic field strength inferred from the
inner lobes is $B_{\rm c}=15$ G (this value corresponds to the top of the
model’s upper chromosphere; see the green circle in the left panel). The
application of the WFA to the external lobes gives instead $B_{\rm w}=158$ G,
which corresponds to the middle chromosphere (see the gray circle in the left
panel). These spatial locations are in agreement with the information provided
by the contribution function of the inner and external lobes of the $V/I$
profile, as shown by the colored regions of Figure S6. The same conclusion,
namely that the inner lobes of the $V/I$ profiles of the Mg ii $h$ & $k$ lines
provide information on the magnetic field at the top of the upper chromosphere
and the external lobes at middle chromospheric heights, is confirmed by the
response function of $V/I$ to magnetic field perturbations in semi-empirical
models of the solar atmosphere (?). The just quoted paper, which explains how
the calculations of Figure S6 were done, shows also that in a plage model the
response function of the external lobes peaks closer to the upper chromosphere
than in a quiet Sun model.

On average, the Mg ii $k$ line forms only about 50 km higher than the Mg ii
$h$ line (?, ?), a distance significantly smaller than the pressure scale
height in the solar chromosphere. The red symbols in Figure 3 show the
$B_{\mathrm{L}}$ values that result from averaging those obtained
independently from the inner $V/I$ lobes of the Mg ii $h$ & $k$ lines. From
the information provided above, the magnetic field strengths given by the red
symbols in Figure 3 can be associated to the top of the upper chromosphere
(i.e., near the TR), while the black symbols to the middle chromosphere.

2D. Application of the WFA to the Mn i lines

The CLASP2 spectral window includes three resonance lines of Mn i resulting
from transitions between the ${\rm a}^{6}{\rm S}_{5/2}$ level of the ground
term and the following upper levels of increasing energy: ${\rm y}^{6}{\rm
P}_{3/2}$, ${\rm y}^{6}{\rm P}_{5/2}$ and ${\rm y}^{6}{\rm P}_{7/2}$. One of
these lines lies in the blue near wing of Mg ii $k$, very close to the
$k_{1\mathrm{v}}$ wavelength, while the other two lines lie in the far wing
between the Mg ii $k$ & $h$ line cores (see Fig. S4). These two lines at
279.91 nm and 280.19 nm are the Mn i lines we have used in this paper, because
they show very clear circular polarization signals produced by the Zeeman
effect (see Fig. 1), each line showing two $V/I$ lobes.

A non-LTE radiative transfer investigation that will be published elsewhere
shows that the $V/I$ signals of such Mn i lines originate in the lower
chromosphere, near the temperature minimum region in standard solar semi-
empirical models. Accordingly, the lower chromosphere is the approximate
region of formation we assign to the blue-symbol values in Figure 3. Manganese
has nuclear spin $I=5/2$ and the same theoretical investigation reveals that
the ensuing hyperfine structure (HFS) with the crossings and repulsions
between the hyperfine F-levels have a significant impact on the Stokes $I$ and
$V$ profiles of the Mn i lines observed by CLASP2. The inclusion of HFS widens
the Stokes $I$ profiles and lowers the amplitude of Stokes $V/I$ profiles
while also separating their peaks. Moreover, that investigation shows that the
WFA is applicable to the two Mn i lines we have selected, when using
$g_{\mathrm{eff}}=1.94$ for the 279.91 nm line and $g_{\mathrm{eff}}=1.7$ for
the 280.19 nm line (these are the fine-structure effective Landé factors).

2E. Thermodynamical parameters inferred from CLASP2 intensity spectra

As shown in Table 1, there appears to be a good correlation between the
magnetic field measured with CLASP2 and the Mg ii $k_{\mathrm{3}}$ and
$k_{\mathrm{2v}}$ intensities and, as pointed out in the main text, the
intensity in the $k$ line-core spectral features correlates with the
temperature at its formation height (?). To further investigate the
relationship between the magnetic field and the local thermodynamics
properties of the chromosphere, we exploit the IRIS2 database (?). The latest
version of this database contains 150,000 spectral profiles around the Mg ii
$h$ & $k$ lines, and the associated model atmospheres (i.e., temperature,
velocity, turbulence, electron density, as a function of optical depth),
calculated using the STiC inversion code which includes the effects of
radiative transfer with PRD (?).

For each CLASP2 Stokes $I$ spectral line profile of the Mg ii $h$ & $k$ lines,
we determine the profile in the IRIS2 database that has the smallest weighted
Euclidean distance to the CLASP2 profile. Since we are interested mostly in
the chromospheric properties, we place enhanced weighting on the wavelength
regions between Mg ii $k_{\mathrm{1v}}$ and Mg $k_{\mathrm{1r}}$ and between
Mg ii $h_{\mathrm{1v}}$ and Mg $h_{\mathrm{1r}}$, as well as the Mg ii triplet
line formed around 279.89 nm, which are all formed in the chromosphere. The
associated model atmosphere from IRIS2 is then utilized to investigate its
correlation with the CLASP2 magnetic field measurements. We find that in the
middle and in the upper chromosphere the electron pressure (i.e., the product
of temperature and electron density) shows a high cross-correlation with the
magnetic field (see Table 1). Figure S7 shows the product of temperature and
electron density along the CLASP2 slit, as derived from the IRIS2 inversions,
in comparison with the CLASP2 magnetic field measurements. The cross
correlation values are similar to those of the Mg ii $k_{\mathrm{3}}$ and
$k_{\mathrm{2v}}$ intensities shown in Figure 3. The high correlation with the
electron pressure provides support for a close relationship between magnetic
field strength in the chromosphere and local heating. This is because heating
leads to an increase in local temperature, density, and ionization.

We note that the IRIS2 database was created for spectral line profiles at the
spatial resolution (0.4 arcsec) and temporal resolution ($2-30$ s) of IRIS.
The CLASP2 intensity measurements are averaged over 150.4 s with a lower
spatial resolution. Such averaging generally leads to smoother spectral line
profiles since the spatio-temporal averaging tends to smear out subtle
spectral features. To find the best matches with the CLASP2 observations we
thus first perform spectral smoothing on the IRIS2 profiles so that the
profiles show a similar level of detail as those of the CLASP2 measurements.

Inversions of complex chromospheric spectral lines formed under non-LTE
conditions have become possible thanks to advances in numerical radiative
transfer and inversion techniques (?). The derived thermodynamic parameters
are approximations and not necessarily unique solutions to the underlying
atmospheric conditions. In addition, the inherent properties of the radiative
transfer in spectral lines imply that they are sensitive to local conditions
at varying levels, depending on the height range considered (?, ?). Despite
these inherent limitations (which are not always easy to quantify), tests in
which inversion results are compared with “ground truth” from numerical
simulations suggest that they can provide valuable information about the
atmosphere responsible for the observed emissions (?, ?).

With these caveats in mind, our results suggest that there exists a
correlation between two independent “measurements” in the atmosphere: the
line-of-sight magnetic field (as derived from the Stokes $V$ CLASP2
measurements), and the product of temperature and electron density (as derived
from comparison with STiC inversions in the IRIS2 database).

Fig. S1. CLASP2 plage target. (A) SDO/AIA 30.4 nm image taken at 16:54:29 UT
on 2019 April 11. (B) Simultaneous CLASP2/SJ image. (C) Simultaneous IRIS SJI
at 279.6 nm. (D) Comparison between the pseudo SJI made from the CLASP2
spectra (black) and the IRIS SJI at 279.6 nm (red). The IRIS SJI is smoothed
with a Gaussian of FWHM = 1.2 arcsec, which provides the highest correlation
with the pseudo SJI. The yellow solid box represents the CLASP2/SJ FOV of the
the plage target, while the yellow solid line indicates the CLASP2 slit. The
green dashed and solid boxes represent the FOVs observed by IRIS SJ and SG.

Fig. S2. Stokes spectra observed by CLASP2 in the plage target. Each Stokes
parameter was temporally averaged over the 150.4 s observing time. We point
out that while the $V/I$ signals are due to the Zeeman effect in several lines
contained in the CLASP2 spectral window, the $Q/I$ and $U/I$ signals are
dominated by the scattering of anisotropic radiation and the Hanle and
magneto-optical effects across the Mg ii $h$ & $k$ lines (?, ?, ?, ?).

Fig. S3. Synoptic view of a dense IRIS raster scan of the plage target.
Intensity image at the fixed wavelength position of the Mgii $k$ line center,
taken on 2019 April 11 from 15:57:54 to 16:25:26 UT. The yellow line shows the
approximate position of the CLASP2 slit.

Fig. S4. Temporally and spatially averaged intensity observed by CLASP2
outside the plage. The spectral lines used in this paper are indicated. The
inset shows the well-known features of the Mg ii $k$ line. The vertical axis
shows the averaged number of electrons per 3.2 s (one PMU rotation).

Fig. S5. The $I$ and $V/I$ profiles observed by CLASP2 at location b of Fig.
1. $V/I$ profiles of Mg ii $k$ at 279.64 nm, Mg ii $h$ at 280.35 nm, Mn i at
279.91 nm, and Mn i at 280.19 nm at the location b indicated in Figure 1,
where the $B_{\mathrm{L}}$ values in the photosphere are smaller than in the
chromosphere. The circular polarization $V(\lambda)$ is normalized to
$I(\lambda)$ (i.e., to the intensity at each spectral pixel). The gray curves
show the corresponding Stokes $I$ profiles, normalized to the maximum
intensity of Mg ii $k$. The $V/I$ error bars indicate the $\pm{1}\sigma$
uncertainties resulting from the photon noise. The WFA fits and the inferred
longitudinal magnetic field values are shown in blue for the Mn i lines, in
black for the external $V/I$ lobes of Mg ii $h$, and in red for the inner
$V/I$ lobes of Mg ii $h$ & $k$.

Fig. S6. Performance of the WFA with height-dependent magnetic fields. Left
panel: variation with height of the strength of a vertical magnetic field in a
semi-empirical model of the quiet solar atmosphere (?). Right panel: the black
filled circles show the $V/I$ profile of the Mg ii $h$ line calculated at the
model’s disk center, while the solid curves show the fits obtained when
applying the WFA only to the external $V/I$ lobes (gray curves, which give
$B_{\rm w}=158$ G) and only to the inner $V/I$ lobes (green curve, which gives
$B_{\rm c}=15$ G). The colored areas show the $V/I$ contribution function,
which indicate that the inner lobes originate at the top of the model’s upper
chromosphere (green open circle in the left panel) and the external lobes in
the middle chromosphere (gray open circle in the left panel).

Fig. S7. Spatial variation of the longitudinal magnetic field and of the
product of temperature and electron density. The purple curve shows the
product of temperature and electron density (i.e., the electron pressure) near
the upper chromosphere (i.e., at $\log\tau_{500}=-5.4$, with $\tau_{500}$ the
continuum optical depth at 500 nm). The blue and red symbols show the LOS
component of the magnetic field in the lower chromosphere and at the top of
the upper chromosphere, respectively, as in Fig. 3.