11institutetext: Rosseland Centre for Solar Physics, University of Oslo,
Postboks 1029 Blindern, N-0315 Oslo, Norway 22institutetext: Institute of
Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, N-0315
Oslo, Norway
22email<EMAIL_ADDRESS>
# Comparison of chromospheric diagnostics in a 3D model
atmosphere: H$\alpha$ linewidth and mm continua

Sneha Pandit 1122 Sven Wedemeyer 1122 Mats Carlsson 1122 Mikołaj Szydlarski
1122

###### Abstract

Context. The H$\alpha$ line, one of the most studied chromospheric
diagnostics, is a tracer for magnetic field structures, while its line core
intensity provides an estimate of the mass density. The interpretation of
H$\alpha$ observations is complicated by deviations from Local Thermodynamic
Equilibrium (LTE) or instantaneous statistical equilibrium conditions.
Meanwhile, millimetre continuum radiation is formed in LTE, hence the
brightness temperatures from Atacama Large Millimetre-submillimetre Array
(ALMA) observations provide a complementary view of the activity and the
thermal structure of stellar atmospheres. These two diagnostics together can
provide insights into the physical properties like temperature stratification,
magnetic structures, and mass density distribution in stellar atmospheres.

Aims. In this paper, we present a comparative study between synthetic
continuum brightness temperature maps for millimetre wavelengths (0.3 mm to
8.5 mm) and the width of the H$\alpha$ 6565 Å line.

Methods. The 3D radiative transfer codes Multi3D and Advanced Radiative
Transfer (ART) are used to calculate synthetic spectra for the H$\alpha$ line
and the mm continua respectively, from an enhanced network atmosphere model
with non-equilibrium hydrogen ionisation generated with the state-of-the-art
3D radiation magnetohydrodynamics (rMHD) code Bifrost. We use Gaussian Point
Spread Function (PSF) for simulating the effect of ALMA’s limited spatial
resolution and calculate the H$\alpha$ vs. mm continuum correlations and
slopes of scatter plots for the original and degraded resolution of the whole
box, quiet sun and enhanced network patches separately.

Results. The H$\alpha$ linewidth and mm brightness temperatures are highly
correlated and the correlation is highest at a wavelength 0.8 mm i.e. in ALMA
Band 7. The correlation systematically increases with decreased resolution. On
the other hand, the slopes decrease with increasing wavelength. The
degradation of resolution does not have a significant impact on the calculated
slopes.

Conclusions. With decreasing spatial resolution the standard deviations of the
observables, H$\alpha$ linewidth and brightness temperatures decrease and the
correlations between them increase, but the slopes do not change
significantly. Hence, these relations may prove useful to calibrate the mm
continuum maps observed with ALMA.

###### Key Words.:

Line: profiles, Radiative transfer, Methods: numerical, Sun: radio radiations,
Sun: chromosphere, Radio continuum: stars

##  1 Introduction

One of the most commonly used chromospheric diagnostics is the H$\alpha$ line,
i.e. the transition between atomic levels 3 and 2 in the hydrogen atom (Cram &
Mullan, 1985). Due to its high opacity in the line core, the H$\alpha$ line
core forms in the low plasma beta regime, i.e. with more dominant magnetic
pressure than the plasma pressure, which results in magnetic fields being the
main structuring agent in the upper chromosphere (Leenaarts et al., 2012).
Hence, the H$\alpha$ line is a very good diagnostic for the magnetic
structures in the chromosphere. The chromospheric mass density can be traced
with the H$\alpha$ line core intensity (Leenaarts et al., 2012). The low
atomic mass of hydrogen results in high-temperature sensitivity of the
H$\alpha$ line width through significant thermal Doppler broadening (Cauzzi et
al., 2009). Even though the H$\alpha$ line is a useful diagnostic of stellar
chromospheres, since it forms in non-local thermodynamic equilibrium (NLTE),
it is not particularly suitable for the determination of the chromospheric
temperature stratification (Pasquini & Pallavicini, 1991).

Based on radiative transfer calculations for a solar atmospheric radiation-
magnetohydrodynamics simulation (Carlsson et al., 2016), Leenaarts et al.
(2012) show that the H$\alpha$ line width is correlated with the temperature
of the plasma while the line core intensity is a diagnostic of the mass
density in the chromosphere. These relations come from the fact that the
H$\alpha$ line is a strongly scattering line such that the source function is
non-locally determined by the mean intensity. A high local temperature means
that the atomic absorption profile gets wide (due to the low atomic mass, the
width of the profile is dominated by thermal broadening) and since the source
function is rather insensitive to the local conditions, we also get a wide
intensity profile. A high mass density in the chromosphere leads to a line
core formation at a larger height where the source function is lower, hence a
lower line core intensity.

Figure 1: Horizontal cross-section of temperature (left) and the absolute
magnetic field ($|B|=\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$) (right) from a
snapshot of the Bifrost simulation at a constant height of 1.5 Mm from the
surface.

It is well established that the continuum radiation at millimetre (mm)
wavelengths is formed at chromospheric heights (see, e.g., Wedemeyer et al.,
2016, and references therein). The continuum radiation at millimetre
wavelengths (here 0.3–8.5 mm) originates from free-free emission in the
chromosphere, and the two main opacity sources are H and H- free-free
absorption (Dulk, 1985). These processes result in a Local Thermodynamic
Equilibrium (LTE) source function as they are coupled to the local property of
the plasma: the electron temperature. Hence, we can use the Rayleigh-Jeans law
and interpret the emergent intensity (the observed brightness temperature
($T_{\mathrm{b}}$)) in the millimetre wavelength domain as local electron
temperature (Loukitcheva et al., 2015). Thus, the millimetre continuum is a
very convenient diagnostic to understand the temperature stratification in
solar and stellar atmospheres (Wedemeyer-Böhm, 2007; Wedemeyer et al., 2020).

The H$\alpha$ line opacity is proportional to the hydrogen $n=2$ level
population; the H$\alpha$ optical depth scales with the $n=2$ column density
(Leenaarts et al., 2007). The availability of free electrons is dependent on
the n=2 population of hydrogen in the upper atmosphere (Molnar et al., 2019).
The formation of the H$\alpha$ and mm continuum is influenced by the plasma
temperature, the H(n=2) level populations and electron populations.

Traditional H$\alpha$ activity indicators, like the linewidth, Full Width at
Half Maximum (FWHM) or integrated fluxes depend on several wavelengths across
the line profile (Hanuschik, 1989; Gizis et al., 2002; Marsden et al., 2014;
Molnar et al., 2019). The H$\alpha$ line core is formed higher in the
atmosphere than the wings. Therefore, while calculating these activity
indicators from observations, we inherently look at a large volume of plasma
in the chromosphere (Vitas et al., 2009), which makes determining the thermal
stratification difficult. In this regard, the millimetre continuum has an
additional advantage as a complementary diagnostic, namely that — to a first
approximation — the radiation at a given millimetre wavelength originates from
more or less the same height range with the average formation height range
increasing as a function of wavelength. While this assumption is probably
valid on average, we note that the exact formation height ranges might show
variations that are currently investigated (see, e.g., Eklund et al., 2021;
Wedemeyer et al., 2022; Hofmann et al., 2022, and references therein). On the
other hand, the formation height of the H$\alpha$ line is in the range of 0.5
Mm to 3 Mm above the photosphere, which is a large range (Leenaarts et al.,
2012).

The Atacama Millimetre/submillimetre Array (ALMA) (Wootten & Thompson, 2009)
provides observations with a high spatial and temporal resolution for the mm
continuum. The H$\alpha$ linewidth and mm continua both depend on the
temperature and the electron density of the volume of plasma where they are
formed. This work is focused on using these two as complementary diagnostics
similar to the case study on co-observational data for ALMA Band 3 and IBIS in
Molnar et al. (2019). In this paper, we compare the mm continuum and H$\alpha$
linewidth synthesised from a 3D realistic Bifrost model atmosphere to
understand how these can be used as complementary diagnostics for the
chromosphere. The model and the spectral synthesis are described in Sect. 2.
In Sect. 3, we present the quantitative comparison of the two diagnostics
which is further qualitatively discussed in Sect. 4. The study is concluded in
Sect. 5.

##  2 Methods

A snapshot of a numerical 3D simulation of the solar atmosphere (see Sect.
2.1) is used as input for radiative transfer codes to compute the H$\alpha$
line intensity (see Sect. 2.2) and the continuum intensity at millimetre
wavelengths (see Sect. 2.5). The resulting maps for different wavelengths are
then degraded to a spatial resolution typically achieved with ALMA as
described in Sect. 2.6 before the H$\alpha$ line width is determined and
compared to the mm brightness temperatures.

###  2.1 3D model atmosphere

Bifrost is a 3D radiation magnetohydrodynamics (rMHD) simulation code which
includes physics relevant to chromospheric conditions (Gudiksen et al., 2011).
The model used for this study is taken from a continuation of the enhanced
network simulation en024048 at 17 min after the publicly released snapshots
from Carlsson et al. (2016). The computational box of this model extends 24 Mm
$\times$ 24 Mm horizontally, and vertically from 2.4 Mm below the visible
surface to 14.4 Mm above it, thus encompassing the upper part of the
convection zone, the photosphere, chromosphere, transition region and corona.
The computational box consists of 504 $\times$ 504 $\times$ 496 grid points
with a horizontal resolution of 48 km, which corresponds to an angular size of
approximately 0.066 arcsec. The grid is non-equidistant in the vertical
direction varying from 19 km in the photosphere and chromosphere up to 5 Mm
height and then increasing to 100 km at the top boundary. Both the top and
bottom boundaries are transparent, whereas lateral boundary conditions are
periodic. At the bottom boundary, the magnetic field is passively advected
with no extra field fed into the computational domain. For further details see
Carlsson et al. (2016, and references therein).

The temperature and absolute magnetic field showing the loops of the enhanced
network with the quiet region at typical chromospheric heights are shown in
Fig. 1. As seen in Figs. 11 and 12 in Carlsson et al. (2016), the simulation
has different regions. The en024048 simulation features an enhanced network
(EN) patch with loops in the middle of the computational domain as seen in
Fig. 2 in Loukitcheva et al. (2015) or (see the black square in Figs. 3 and
4), whereas the outer parts are more representative of magnetically less
active, quiet Sun (QS) conditions. In the absolute magnetic field slice
through 1.74 Mm (Fig. 1b), the footpoints of the magnetic loops are seen. The
gas temperatures at those heights are comparatively lower at locations with
higher magnetic strengths. And, the loops show significantly higher
temperatures in the range of 2 to 3 MK from the footpoint to the apex of the
loop.

Figure 2: Comparison of the linewidth as defined by Molnar et al. (2019)
(upper panel), by Leenaarts et al. (2012) (middle panel) and as defined as the
FWHM (lower panel). The red, black and green lines show normalised spectra for
the VAL C model, Bifrost QS and the observed spectrum for $\alpha$ Cen A
(Porto de Mello et al., 2008), respectively. The solid blue lines in the top
panel mark $\pm 1\AA$, while the dotted lines show the considered maxima and
the dashed lines show the calculated linewidths in all three cases.

###  2.2 H$\alpha$ spectral line synthesis

Multi3D (Leenaarts & Carlsson, 2009) employs the accelerated lambda iteration
method developed by Rybicki & Hummer (1992) with the extension to treat
effects of partial frequency redistribution using the angle averaged
approximation by Uitenbroek (2001). We used a five-level plus continuum
hydrogen model atom. Microturbulence is not introduced, as Leenaarts et al.
show that the synthesised spectra are similar to observational spectra even
without using microturbulence. Further details of synthesising the H$\alpha$
spectra in 3D using Multi3D can be found in Leenaarts et al. (2012).

###  2.3 H$\alpha$ observations of $\alpha$ Cen A

In this study, H $\alpha$ observations of the solar-like star $\alpha$ Cen A
are used for comparison. The data is acquired from Porto de Mello et al.
(2008) and Lyra & Porto de Mello (2005). The H$\alpha$ observations are
collected at Observatório do Pico dos Dias (OPD), operated by the Laboratório
Nacional de Astrofísica (LNA), CNPq, Brazil on the ESPCOUDE 1.60m telescope.
The data are wavelength-calibrated, Doppler-corrected, and flux-normalised to
unity by Porto de Mello et al..

###  2.4 Definition of linewidth

As this study is based on forward modelling, the linewidth definition by
Leenaarts et al. (2012) is used. However, we note that different definitions
of the H$\alpha$ linewidth are used in the literature. The core width
definition by Leenaarts et al. (2012) and the definition used by Cauzzi et al.
(2009) and Molnar et al. (2019) are compared to the Full Width Half Maximum
(FWHM) definition for the linewidth in Fig. 2

The Leenaarts et al. (2012) H$\alpha$ line-core width calculated for the
Bifrost model as the full width at $1/10^{th}$ maximum, i.e. the separation
between the line wings at $I=I_{min}+0.1(I_{max}-I_{min})$. The representative
wavelengths for the red and blue line wing that span the core width are
calculated by first identifying the two spectral sampling points which are
closest to the $1/10^{th}$ of the intensity between the pseudo continuum and
the line core and then using linear interpolation.

Molnar et al. (2019) follow the definition by Cauzzi et al. (2009) for their
observational study. There, the linewidth is defined as the separation of the
line profile wings at half of the line depth, where the maxima are defined at
$\pm 1\AA$ from the line core (referred to as Molnar et al. (2019) definition
of linewidth, here onward). The line core intensity calculated for the Bifrost
model used in this study reveals the structure of the EN loops. The variation
in the activity in the model atmosphere among QS and EN regions is also
visible from the variation of the H$\alpha$ line core intensity. It is to be
noted that throughout the paper, the wavelengths mentioned are vacuum
wavelengths.

The resulting line core width according to the Leenaarts et al. (2012)
definition is shown in Fig. 3a. Please note that the plotted value ranges for
these maps have been limited to the 99th to 1st percentile for better
visibility, in particular, to highlight the imprints of the loop structures in
contrast to the less active surroundings.

###  2.5 Synthesis of millimetre continuum brightness temperatures

The intensity of the continuum radiation at millimetre wavelengths for the 3D
input model (see Sect. 2.1) is calculated with the Advanced Radiative Transfer
(ART) code by de la Cruz Rodríguez et al. (2021). The output intensities
$I_{\lambda}$ are then converted to brightness temperatures $T_{b}$ by using
the Rayleigh-Jeans approximation:

$T_{b}=\frac{\lambda^{4}}{2k_{B}c}I_{\lambda},$ (1)

where, $\lambda,k_{B}$ and $c$ are the wavelength and Boltzmann constant and
the speed of light, respectively. In total, maps for 34 wavelengths are
computed, covering the range that in principle can be observed with ALMA. That
includes both currently available receiver bands and bands that might be
offered for solar observations in the future. The resulting maps for the
wavelengths of 3.0 mm and 0.8 mm are presented in Figs. 3 and 4, respectively,
while further maps for selected wavelengths are shown in Sect. A (Figs.
19-26).

###  2.6 Image degradation

In principle, the synthesised beam for interferometric observations with ALMA,
i.e. the effective point spread function (PSF), depends on several factors
such as the angle of the Sun in the sky with respect to the array baselines
and the configuration of the antennas. The resulting beam would be typically
elliptical, while the Fourier space of the source (here the Sun) would only be
sampled sparsely (see, e.g., Högbom, 1974; Boone, 2013). For this theoretical
study, circular Gaussians are used as PSFs as a first approximation as this
simplified approach suffices to demonstrate the effect of spatial resolution.
The more detailed modelling of the ALMA beam would depend on the exact array
configuration, receiver band (some of which are not used for solar
observations yet), and details of the imaging process, which would go beyond
the scope of this study (Wedemeyer et al., 2022).

As mentioned above, the exact shape and width of the synthetic beam varies
even for observations in the same receiver band. For instance, the data sets
found on the Solar ALMA Archive (SALSA, Henriques et al., 2022) have widths
around 2” including the beam with a width of 1.92” $\times$ 2.30” for the data
set 2016.1.01129.S. That data set was also used by Mohan et al. (2021),
although their choice of imaging parameters results in a beam width of only
1.75” $\times$ 1.91”. However, based on a comparison to H$\alpha$
observations, the authors instead chose a PSF with a width of 1.95” $\times$
2.03”. For simplicity of this study, the width of the beam (i.e. the PSF) is
set to 2” at a wavelength of $\lambda=3.0$ mm and is then linearly scaled as a
function of the observing wavelength $\lambda$. The considered wavelength
range includes the receiver bands that are already available for solar
observations with ALMA (bands 3, 5 and 6, i.e. 2.59 - 3.57, 1.42 - 1.90 and
1.09 - 1.42 mm) but in addition also bands 1,2,4, and 7-10, thus covering the
whole range from 0.3 mm to 8.5 mm that might be offered in the future. The
same wavelength-dependent Gaussian kernels are used for degrading the
millimetre brightness temperature maps and the H$\alpha$ intensity maps.
Please note that all three considered snapshots (with 5 minutes solar time
between snapshots) are synthesised and then degraded with the different beams.
In addition, corresponding time-averaged ALMA maps are calculated so that they
can be compared to the observational data used by Molnar et al. (2019).

###  2.7 Semi-empirical models

For comparison, the 1D semi-empirical reference models VAL C (Vernazza et al.,
1981) and FAL C (Fontenla et al., 1993) are used, which are both
representative of quiet Sun conditions. Corresponding H$\alpha$ line profiles
and mm continuum intensities for these 1D semi-empirical atmosphere models are
calculated with the radiative code RH (Uitenbroek, 2001; Pereira & Uitenbroek,
2015).

##  3 Results

###  3.1 Dependence on spatial resolution

Figure 3 shows the comparison between the EN region and QS region for the
original and the degraded resolution corresponding to the assumed ALMA
resolution at a wavelength of 3 mm. The calculated H$\alpha$ line core width
for the whole box (panel a) is very similar to the results in Fig. 9a in
Leenaarts et al. (2012). The H$\alpha$ line core width from the simulation
(Fig. 9a in Leenaarts et al., 2012) appears to be formed at a lower height in
the chromosphere in comparison to the observation (Fig. 16b in Leenaarts et
al., 2012). As the employed wavelength points are very close to the line core,
one should expect to see chromospheric features in the line core width map. In
contrast, however, the continuum intensity for 3 mm at original resolution as
shown in Fig. 3b differs notably from the line core width shown in Fig. 3a.

The maps appear more similar at reduced resolution after convolution with a
circular Gaussian kernel corresponding to the spatial resolution at the
wavelength of 3 mm (see panels d and e). The black and red boxes on these maps
denote the chosen EN and QS regions for further comparison. The EN region is
chosen to be centred at the EN as seen in the degraded resolution, and the QS
region is placed at the bottom right corner of the simulation box. Both boxes
are of the same size of 9.5 Mm $\times$ 9.5 Mm (200 pixels $\times$ 200
pixels).

The panels c and f of Fig. 3 show the scatter contour plots for the three sets
of pixels: the whole simulation box, the QS box (red square in panels
a,b,d,e), and the EN box (black square in panels a,b,d,e). The distribution
for the whole simulation box is shown in red contours with the colour bar for
the number of pixels falling into the individual bins. The QS and EN
distributions are represented by green and blue contours, respectively. The
distributions at the original resolution are substantially more spread than
the distributions at degraded resolution because of the associated loss of
variations on small spatial scales. This effect is already clear from a visual
comparison of the H$\alpha$ linewidth maps in panels a and d and likewise for
the mm brightness temperature maps in panels b and e, respectively. The red,
green and blue points in panels c and f represent the average values for the
whole box, QS and EN, respectively. The impact of applying the ALMA beam can
be seen from the spatial power spectral density plots in Fig. 5. As seen in
the plots for 3.0 mm and 0.8 mm, the power spectral density (PSD) falls
drastically for small spatial scales that are not resolved with respective PSF
(beam). Another notable observation is that both definitions of linewidths
show similar values of power at larger scales. At smaller scales the mm PSD is
close to the PSD of the line core width definition, implying better
correlations (see Table 1). Also, the observational 3 mm data shows a very
similar trend in PSD to the mm spatially degraded data as expected.

In addition, the H$\alpha$ linewidth and the $T_{\mathrm{b}}$ at the mm
wavelengths for the VAL C and FAL C semi-empirical model atmospheres are
plotted for comparison. The VAL C model has consistently lower brightness
temperature than the FAL C model as seen in Fig. 6. When plotted on the
linewidth vs temperature planes, these quiet sun models lie in between the EN
and QS distributions, demonstrating the agreement between the data from our
study and the semi-empirical models. Applying the beam changes the mean
$T_{\mathrm{b}}$ value for the QS and EN pixel sets as shown in Table 1. The
mean EN $T_{\mathrm{b}}$ decrease and mean QS $T_{\mathrm{b}}$ increase leave
the mean $T_{\mathrm{b}}$ of the whole box unchanged. This change in the mean
and standard deviation of the brightness temperatures is further shown in Fig.
6 and discussed in Sect. 4.1. The $T_{\mathrm{b}}$ at 3 mm for the VAL C and
FAL C models are calculated as 7009 K and 7624 K, respectively, which are thus
similar to the corresponding average brightness temperature of 6968 K across
the whole horizontal extent of the here employed 3D model (see Table 1). The
value for FAL C agrees with the calculations reported by Loukitcheva et al.
(2004). The differences are further discussed in comparison to observations in
Sect. 4.2 It should also be noted that the average QS $T_{\mathrm{b}}=6621$ K
found for the 3D model is lower than the reference value of 7300 K suggested
by White et al. (2017) for Band 3 ($\lambda=3.0$ mm) at the solar disk-centre.
Please refer to Sect. 4.2 for further discussion.

Figure 3: Comparisons of the H$\alpha$ line core width (a,d) and ALMA at 3.0
mm brightness temperature maps (b,e) at original (top row) and degraded
(bottom row) resolution. The black and red boxes on these maps denote the
chosen EN and QS regions respectively. The last column is a correlation
contour plot between the H$\alpha$ linewidth and ALMA brightness temperature
at 3.0 mm, with contours with linear fits (dashed lines) and means (circles)
of the three cases: the whole simulation box in red, the QS box in green and
the EN region box in blue. The red and green star data points are for the FAL
C and VAL C 1D semi-empirical models and the black circle with error bars is
an observational data point for G2V type star $\alpha$ Cen A with linewidth
0.62 $\AA$. Figure 4: Same as Fig. 3 but for a wavelength of 0.8 mm.

Figure 5: Normalised power spectral density $kP_{k}$ plots for time-averaged datasets of mm brightness temperatures, and both definitions of linewidths at 3.0 mm and 0.8 mm. The solid lines represent the original resolution data and the dashed lines, the spatially degraded data, green for (Leenaarts et al., 2012) linewidth, red for Molnar et al. (2019) definition of linewidth and blue for the corresponding mm wavelength. The vertical lines represent the dominant spatial scales (in arcsec) for the original resolution, for (Leenaarts et al., 2012) definition of linewidth and Molnar et al. (2019) definition of linewidth respectively. The black dashed line represents the PSF corresponding to the mm wavelength, showing the resolution limit for the degraded dataset. In the left panel, the spatial power spectral density for the Band 3 ALMA observations (ADS/NRAO.ALMA#2016.1.01129.S) as used by Molnar et al. (2019) and shown in Fig. 11 are plotted as the black solid line. The power spectral density was calculated for all time steps in the time range considered by Molnar et al. (2019) and then averaged in time. Figure 6: Comparison of the mean brightness temperatures for mm wavelengths at the resolution of the original simulations (in black) and the resolution of the observations from the ALMA observatory (in red). In both these cases the whole simulation box (solid lines), the QS box(dotted lines) and the EN box (dashed lines) are taken separately. The pink and blue shaded regions show one standard deviation region around the mean for the original resolution and the degraded resolution case respectively. The green and blue dashed lines show brightness temperature trends for FAL C and VAL C models respectively. The black dashed vertical lines mark 0.8 mm and 3 mm. Figure 7: Comparison of the calculated Pearson correlation coefficients with two different resolutions. Black vertical lines show the ALMA band 3 range and black horizontal lines show the Molnar et. al. correlation coefficient = 0.84. and the red vertical line shows the maximum correlation in the original resolution case which is at 0.8 mm. Figure 8: Comparison of the calculated slopes for the scatter plots, legend same as Fig. 6. The black lines show the Molnar et. al. slope = 0.0612 at 3 mm. Figure 9: Same as Fig. 3, for time-averaged dataset. Figure 10: Same as Fig. 4, for time-averaged dataset. | Original resolution | Reduced resolution
---|---|---
| wavelength | Whole box | QS | EN | Whole box | QS | EN
Single snapshot: | line core width calculated with |  (Leenaarts et al., 2012) formula. |
Temperatures | 0.8 mm | 5.275 | 5.124 | 5.590 | 5.275 | 5.126 | 5.593
(kK) | 3.0 mm | 6.968 | 6.621 | 7.835 | 6.968 | 6.637 | 7.848
Correlation | 0.8 mm | 0.764 | 0.798 | 0.785 | 0.778 | 0.834 | 0.808
coefficients | 3.0 mm | 0.524 | 0.516 | 0.628 | 0.453 | 0.520 | 0.813
Slopes | 0.8 mm | 0.047 | 0.052 | 0.045 | 0.041 | 0.047 | 0.040
| 3.0 mm | 0.026 | 0.030 | 0.029 | 0.012 | 0.022 | 0.020
Time-averaged: | line core width calculated with |  (Leenaarts et al., 2012) formula. |
Temperatures | 0.8 mm | 5.111 | 4.881 | 5.609 | 5.111 | 4.881 | 5.609
(kK) | 3.0 mm | 6.862 | 6.367 | 7.891 | 6.862 | 6.371 | 7.878
Correlation | 0.8 mm | 0.668 | 0.705 | 0.712 | 0.696 | 0.751 | 0.755
coefficients | 3.0 mm | 0.461 | 0.398 | 0.583 | 0.470 | 0.316 | 0.755
Slopes | 0.8 mm | 0.035 | 0.048 | 0.035 | 0.031 | 0.047 | 0.031
| 3.0 mm | 0.017 | 0.018 | 0.023 | 0.009 | 0.008 | 0.016
Single snapshot: | linewidth calculated with | Molnar et al. (2019) formula. |
Correlation | 0.8 mm | 0.733 | 0.746 | 0.797 | 0.770 | 0.813 | 0.854
coefficients | 3.0 mm | 0.443 | 0.443 | 0.545 | 0.384 | 0.411 | 0.819
Slopes | 0.8 mm | 0.042 | 0.043 | 0.040 | 0.044 | 0.048 | 0.041
| 3.0 mm | 0.020 | 0.023 | 0.022 | 0.014 | 0.022 | 0.022
Time-averaged: | linewidth calculated with | Molnar et al. (2019) formula. |
Correlation | 0.8 mm | 0.729 | 0.692 | 0.837 | 0.771 | 0.724 | 0.894
coefficients | 3.0 mm | 0.479 | 0.343 | 0.609 | 0.610 | 0.294 | 0.829
Slopes | 0.8 mm | 0.040 | 0.045 | 0.043 | 0.039 | 0.046 | 0.043
| 3.0 mm | 0.019 | 0.015 | 0.025 | 0.015 | 0.008 | 0.025
Table 1: Brightness temperatures in kK, Correlation coefficients and Slopes
for the three pixel sets (whole box, QS and EN) each for two resolutions
(original and degraded) for the single snapshot and time-averaged dataset,
with both the definitions used for calculating the linewidth.

The wavelength of 0.8 mm is of particular interest for this study, too, as
will be discussed in Sects. 3.2 and 4.2. The results for $\lambda=0.8$ mm are
presented in Fig. 4, which corresponds to the above-discussed Fig. 3 for
$\lambda=3.0$ mm. While the H$\alpha$ line core width map at the original
resolution (panel a) is the same in both figures, the mm brightness
temperature maps (panel b) and the degraded maps (panels d and e) differ. The
H$\alpha$ line core width map (Fig. 4a) and the mm continuum map at 0.8 mm
(Fig. 4b) are already very similar at the original resolution and even more so
at the lower resolution as shown in Fig. 4d-e. The PSF at 0.8 mm is smaller in
width compared to the 3.0 mm PSF by a factor of $0.8/3.0$, which can be seen
from a visual comparison of Fig. 3e and Fig. 4e. The 3.0 mm maps appear more
blurred, mainly as a result of the wider PSF. The same effect can be seen for
the H$\alpha$ line width maps, which were derived from the H$\alpha$ line
intensity maps after these were all degraded accordingly for each wavelength
point.

The slopes in the EN and QS cases are very similar to each other and to that
for the whole box as seen in Table 1. In contrast to Fig. 3c,f, the scatter
plots in Fig. 4c,f show that the slopes for the QS and EN distributions (0.052
and 0.045) are very similar to the values for the maps at degraded resolution.
On the other hand, for 3 mm, the values are 0.03 and 0.029, which show a
higher departure from each other. The mean $T_{b}$ values for the QS region
and the whole box are quite close as seen in Table 1, as the atmosphere is
mainly QS. The VAL C and FAL C data points are closer to the mean $T_{b}$ for
the EN region, possibly because of the reasons discussed before. Treating the
synthetic mm and the H$\alpha$ linewidth map with the generated PSFs for
corresponding wavelengths (see Appendix A for representative plots similar to
Figs. 3, 4 for ALMA Bands), the scatter and the linear fits were plotted and
the slopes and the correlation coefficients were calculated. The trends in
correlations and slopes for all considered wavelengths are presented in Sect.
3.2 and shown in Figs. 7 and 8, respectively.

###  3.2 Correlations and slopes

The correlation of the H$\alpha$ linewidth maps and the mm brightness
temperature maps and the dependence of this correlation on millimetre
wavelength and thus spatial resolution are quantified through the Pearson
correlation coefficient between the H$\alpha$ linewidth maps and mm brightness
temperature maps and linear fits the distributions as shown, for instance, in
Figs. 3, 4. The slopes, as well as the intercept of the linear fits, are
calculated by linear regression for the whole box, and the EN and QS regions
separately.

The resulting correlation coefficients as a function of wavelength are shown
in Fig. 7 at original resolution (black lines) and after degradation with the
synthetic ALMA PSFs (red lines) corresponding to the wavelength. It should be
noted that the mm brightness temperature maps exhibit differences as a
function of wavelength due to the different formation height ranges at a given
wavelength. Although the formation heights can vary notably across the model,
they nonetheless increase on average with increasing wavelength. Consequently,
the mm brightness temperature maps at original resolution show a maximum
correlation with the H$\alpha$ linewidth maps at a given wavelength as can be
seen most clearly from the peak for the QS correlation (black dotted line) for
wavelengths in the range from 0.5 mm to 1.8 mm. Peaks are also present for the
whole box and the EN region with a decrease towards shorter and longer
wavelengths in all cases.

Reducing the spatial resolution has a notable impact on the correlation
coefficients (see Fig. 7), resulting in generally higher values as compared to
the correlation at the original resolution. The QS region correlations are
lower than for the whole box in both cases and the EN region correlations are
slightly higher in the degraded resolution case and for most wavelengths, for
the original resolution. While also the correlation for the degraded
resolution maps exhibits peaks, there is a systematic increase in the
correlation between the degraded resolution maps as a function of wavelength
for longer wavelengths. This effect can be attributed primarily to the spatial
resolution, i.e., the width of the PSF which increases with wavelength.
Consequently, the structure below a corresponding (small) spatial scale is
blurred, which affects the resulting correlation coefficient. With increasing
wavelength more and more of this atmospheric fine structure is blurred as
already discussed in Sect. 3.1 and seen from a comparison of Figs. 3 and 4
(see Table 1 for the correlation coefficients for 3.0 mm and 0.8 mm). At the
longest wavelengths, however, even structure on relatively large spatial
scales is lost. Consequently, the data ranges for these very blurred maps
become very narrow, which then results in a high correlation coefficient. The
impact of a lower spatial resolution on the data ranges is also visible in
Fig. 3c,f and 4c,f: The shown distributions become notably narrower as a
function of spatial resolution for the whole box, QS region and EN region,
being broadest at the original resolution and narrowest when degraded with the
3.0 mm PSF. Most importantly, the highest correlation for the degraded maps
with a value of 0.844 is found for the EN region at 1.1 mm, whereas the QS
region has a maximum of 0.839 at 1.0 mm. The correlation for the whole box
peaks with 0.799 at a wavelength of 1.0 mm, too. These results suggest that
(i) the highest correlation between the H$\alpha$ line core width and mm
continuum brightness temperature maps is found for wavelengths at 1.0 mm,
which corresponds to ALMA Band 7, and that (ii) the correlation between
observed H$\alpha$ line core width maps and co-simultaneously obtained ALMA
Band 7 brightness temperature maps is expected to be higher than what was
found by Molnar et al. (2019) for Band 3 observations (at 3.0 mm). See Sect.
4.2 for further discussion.

The blue contours in Fig. 4c,f, which represent the EN region, show higher
H$\alpha$ line core widths and mm brightness temperatures. On the other hand,
the green contours, for QS regions, show lower intensities in mm and lower
H$\alpha$ linewidths, in both the resolutions, more distinct in the degraded
resolution case. As the smaller network-internetwork features get blurred, the
higher (lower) intensities in the QS (EN) region get blurred with neighbouring
lower (higher) intensities and hence the green (blue) contours shift downwards
(upwards). The slopes of the fitted lines for the three distributions change
with resolution. The slope decreases for the EN case but increases for the QS
case with the degraded resolution as can be seen in Fig. 8 for all the cases
in the dataset.

The slopes show a general decreasing trend with wavelength. As the wavelength
increases, the radiation originates from a slightly higher layer of the
atmosphere which is on average slightly hotter. As seen in Fig. 6, the
temperatures at different wavelengths are also dependent on the formation
heights, in addition to the resolution. This is evident from the decreasing
slopes shown in Fig. 8. For the degraded resolution case, the difference in
the slope for QS and EN regions is more than the original resolution case. In
particular, for the EN case, as seen in Figs. 3 and 4, in the degraded
resolution case (panels d and e), the brightness temperatures decrease,
decreasing slopes significantly. For the QS case, on the other hand, the
slopes remain comparable, or approximately the same in both resolutions. This
is explained by the blurring effect due to the application of the ALMA PSFs,
but in the QS region, the features are more equally distributed. Hence the
slopes do not significantly change with the resolution in the millimetre range
of interest (Bands 3-6). The line widths derived from the time-averaged
H$\alpha$ intensity maps following the definition by Molnar et al. (2019)
produce very similar trends and also indicate the best match to occur at
wavelengths corresponding to ALMA Band 7. Please refer to Sects. 3.3, 4.2 for
further discussion.

###  3.3 Dependence on temporal resolution

Figures 9 and 10 show the original and ALMA resolution maps for the dataset
averaged over 10 minutes. The comparison of features in Figs. 9a,b, 3a,b and
Figs. 10a,b, 4a,b respectively, shows that the time averaging causes the
features to smooth out, as expected. The QS, EN and whole box contours show
less spread in the time-averaged case than in the single snapshot case as
observed in Figs. 9c,f, 3c,f and Figs. 10c,f, 4c,f respectively. When
calculating the H$\alpha$ line core width from the time-averaged maps and
comparing it to the time-averaged mm brightness temperature maps, the
correlation between the H$\alpha$ line core width maps and the mm brightness
temperature maps at original resolution decreases, whereas the correlation
increases significantly for the corresponding maps at reduced (i.e. ALMA)
resolution. The correlation is highest for the EN case and the lowest for the
QS case at both resolutions. This can be attributed to the differences in the
small-scale features in the QS region, which get averaged over time and hence
do not match in the two data sets at degraded resolution. Similarly, the
slopes for the QS region are also lower than for the whole box, whereas the
slopes for the EN region are higher than for the whole box. The slopes do not
show very drastic changes from the single snapshot case, but they are
monotonically decreasing for all pixel sets.

##  4 Discussion

The H$\alpha$ line core widths correlate best with the mm continuum brightness
temperatures at a wavelength of 0.8 mm. The Pearson correlation coefficients
for the degraded maps are determined as 0.778 for the whole box, 0.808 for the
QS region, and 0.834 for the EN region (see Table 1). These results imply
that, despite the similarities of the H$\alpha$ linewidth with the
observational ALMA Band 3 data found by Molnar et al. (2019), a closer match
with Band 7 is to be expected. The immediate next step therefore would be to
compare observational ALMA Band 7 data with co-observed H$\alpha$ data but, to
the best knowledge of the authors, no such data is currently available for the
study presented here.

###  4.1 Effect of spatial resolution

One of the motivations of this study is to use high-resolution synthetic data
at mm wavelengths to investigate how they can complement other diagnostics,
like H$\alpha$, for determining the properties of small-scale features in the
solar atmosphere. Even though the resolution of ALMA observational data is
outstanding compared to previous observations at millimetre wavelengths, the
resolution is very low compared to what is typically achieved at optical
wavelengths (see, e.g., White et al., 2006; Wedemeyer et al., 2016, and
references therein).

The comparison between maps at the original resolution and maps at the ALMA
resolution using Gaussian kernels as simplified PSFs for the respective
wavelengths (see Figs. 3 and 4) shows that there are two effects in action:
the change in the formation layer for the mm data, and the decreasing
resolution due to the increasing wavelength. These two effects cannot be
disentangled in observational data, but with a simulation like the one
employed here the physical and instrumental effects can be separated. In Fig.
6, the effect of the change in formation height for the continuum radiation at
mm wavelengths and the change in resolution for the given wavelength are
demonstrated. Although the average brightness temperature of the whole
simulation box remains the same when degrading the resolution at a given
wavelength, the width of the brightness temperature distribution (as
represented by the shaded areas in the figure) reduces with degrading
resolution. The increase in correlations with the degradation of the
resolution can be attributed to this phenomenon. Also, the slopes are affected
as the standard deviations in the data with the reduced resolution, which is
evident from Fig. 6.

To understand this impact of the spatial resolution more quantitatively, the
values of the correlations for all six cases for $\lambda=0.8$ mm and
$\lambda=3.0$ mm are listed in Table 1. The correlations significantly
increase with the degradation of the resolution as observed in Fig. 7. On the
other hand, as seen in Fig. 8 and Table 1, the slopes do not change
significantly. Hence, it can be safely speculated that the H$\alpha$ linewidth
and mm brightness temperatures are correlated and follow the linear fit fairly
well.

In Fig. 16, the formation heights for the continuum at wavelengths of 3.2 mm,
1.8 mm and 0.8 mm, along with the vertical temperature slice through the model
atmosphere are presented. Two more formation height profiles are plotted for
the linewidth definition by Molnar et al. (2019) (cf. Cauzzi et al., 2009) and
for the line core width as defined by Leenaarts et al. (2012). In both cases,
the plotted line is the average formation height of the wavelength points in
the blue and red line wing that result from the calculation of the linewidth.
It is observed that the Leenaarts et al. (2012) definition agrees better with
the 0.8 mm formation height range. The Molnar et al. (2019) linewidth is
rather a diagnostic of the lower atmosphere.

###  4.2 Comparison with observations

Figure 11:  Time-averaged ALMA Band 3 brightness temperature for the same
measurement set as used by Molnar et al. (2019), i.e.
ADS/NRAO.ALMA#2016.1.01129.S from 23 April 2017. The data were averaged over
the period from 17:31 to 17:41 UTC. (a) The whole Field of View (FOV). (b) The
central 24 Mm $\times$ 24 Mm part, also indicated by the red box in panel a
for comparison with the simulated data.

To compare these results with the observational study by Molnar et al. (2019),
we used the same definition of the linewidth as used by them which follows the
approach by (Cauzzi et al., 2009, see Sect. 2.4). The corresponding H$\alpha$
and mm maps at the original and degraded resolution with the scatter contour
plots are shown in Figs. 12 and 13. In Figs. 14c,f and 15c,f, the black dashed
lines show the respective correlation and slope calculated by Molnar et al.
(2019) for the observational data obtained with ALMA in Band 3 (3 mm) on 23
April 2017 (see their Fig. 3). A brightness temperature map similar to the one
shown in Fig. 3 by Molnar et al. (2019) was produced by time-averaging the
corresponding data that is publicly available on the Solar ALMA Science
Archive (SALSA, Henriques et al., 2022) (see Fig. 11). The differences seen in
the data from Molnar et al. and the data taken from SALSA are due to the
difference in the applied imaging including the choice of CLEAN parameters and
the combination of the interferometric and Total Power (TP) data as Molnar et
al. follow the standard approach while the SALSA data is produced with the
Solar ALMA Pipeline (Wedemeyer et al., 2020). Please refer to White et al.
(2017) and Shimojo et al. (2017) for a detailed discussion of the
uncertainties of the interferometric and TP data, which both affect the
resulting absolute brightness temperatures. A cutout from the central part of
the observed field of view with the same extent as the here presented
synthetic observations (i.e., 24 Mm $\times$ 24 Mm) is shown in Fig. 11b to
allow for a direct comparison. Within the 1 and 99 percentile, the brightness
temperature range is observed to be, between 5 kK and 11 kK, in the
observational data is very similar to the range found for the simulated data
(see Fig. 3e), in particular given the aforementioned uncertainties in
absolute brightness temperatures derived from ALMA observations. This
qualitative agreement with observations was already demonstrated for a
different snapshot of the same Bifrost simulation by Loukitcheva et al.
(2015). See also Loukitcheva et al. (2004). As mentioned in Sect. 3.1, the
average brightness temperature at 3.0 mm for the whole computational box of
the 3D model is only $\sim 100$ K higher than the corresponding value for the
VAL C model and only 520 K lower than the FAL C value, which thus all agree
with the reference value of 7300 K derived by White et al. (2017) from ALMA
observations within the aforementioned uncertainties.

Figure 12: Same as Fig. 9, for the Molnar et al. (2019) definition of
linewidth. The black dashed line denotes the linear fit by Molnar et al.
(2019).  Figure 13: Same as Fig. 12, for 0.8 mm. Figure 14: Same as 7, for the
time-averaged data set, for the Molnar et al. (2019) definition of linewidth.
Legend same as Fig. 7 Figure 15: Same as 8, for the time-averaged data set,
for the Molnar et al. (2019) definition of linewidth. Figure 16:  Simulated
logarithmic gas temperature (grey-scale) in a vertical slice of 3D rMHD
Bifrost simulations. The solid lines represent the heights at which the
optical depth is unity at a wavelength of 3.2 mm (red), and 0.8 mm (green),
which correspond to ALMA bands 3, 5 and 7, respectively. The orange, white and
yellow lines show the average heights at which the red and blue sides of the
H$\alpha$ line width according to the Leenaarts et al. (2012), Molnar et al.
(2019) and FWHM definitions of linewidth are formed. Figure 17: Maps of
linewidths for original resolution case for the single snapshot: Left:
Leenaarts et al. (2012) definition, Middle: Molnar et al. (2019) definition
and Right: FWHM.

Figure 18: Left: Distribution of H$\alpha$ line core widths (following the
definition by Leenaarts et al., 2012) for the whole time series. Right:
Corresponding line widths according to the definition by Molnar et al. (2019).
The whole box, QS and EN regions are separately plotted in blue, black and
red, and mean and median values for the three sets of pixels are indicated
with a dot-dashed and dashed lines in the respective colours. Please note the
difference in ranges for linewidths axes in the two plots.

The resolution of the observed ALMA data shown in Fig. 11 appears to be lower
than for the degraded maps presented in Fig. 4e, which can be attributed to
several factors. Firstly, as detailed in Sect. 2.6, the synthetic brightness
temperatures maps are degraded to ALMA’s angular resolution by convolution
with ideal PSFs, each of which is constructed as a circular Gaussian with a
width appropriate for the respective wavelength. This simplified approach
corresponds in principle to a telescope with a filled aperture of the size of
the longest baseline. In reality, however, ALMA consists of a limited number
of antennas in a given configuration, which results in a sparse sampling of
the spatial Fourier space of the observed source. The resulting PSF (or rather
referred to as synthesised beam) of the interferometric array is typically
elongated and tilted. Properly accounting for these detailed instrumental
effects, the additional degradation due to Earth’s atmosphere and the
necessary imaging of the ALMA data (see, e.g., Henriques et al., 2022)
requires a much more detailed and computationally more expensive modelling
approach (Wedemeyer et al., 2022), which goes beyond the scope of the study
presented here. Overall, however, it is safe to assume that the employed
degradation of the synthetic brightness temperature maps is still too
optimistic. The same is true for the synthetic H$\alpha$ data for which
further instrumental effects and the influence of the Earth’s atmosphere are
not accounted for. The statement that the employed instrumental degradation
for ALMA is still too optimistic is confirmed by the power spectral density
shown for the ALMA Band 3 in Fig. 5. The curve for the observational data
starts to drop at a slightly larger spatial scale as compared to the simulated
data and also shows a noise component at smaller spatial scales that is not
included in the simulations here.

Secondly, the ALMA image shown in Fig. 11 represents the time-average over an
observational period of 10 min that is produced in the same way as described
by Molnar et al. (2019). Also, the H$\alpha$ observations obtained with IBIS
by Molnar et al. were time-averaged for a comparison with the aforementioned
ALMA data. A period of 10 min is substantial given the dynamic time scales on
which the chromospheric fine structure evolves, the time-averaging further
blurs the resulting images. This effect can be seen by comparing, e.g., Fig.
3e and Fig. 12e but it is also apparent in Fig. 3 by Molnar et al., which
compares snapshots to time-averaged images. In Table 1, the correlation
coefficients and the slopes for both cases are listed.

Thirdly, limited spectral sampling points for the IBIS observations
(Cavallini, 2006) and also the time it takes to scan through the spectral line
profile would contribute to uncertainties. For our simulated data, we have
used more than three times more spectral sampling points for the H$\alpha$
line, namely 101, as compared to the 29 points for the IBIS data used by
Molnar et al. (2019).

As a result of the limited resolution discussed above, emission from different
heights gets mixed within the respective synthesised ALMA beam. This effect is
particularly important to be aware of in view of the highly dynamic and
intermittent nature of the solar chromosphere, which can result in substantial
changes in formation height across small (possibly unresolved) spatial
distances. In addition, it should be noted that the region observed by Molnar
et al. (2019) comprises a mix of plage, network and magnetically quiet
regions, whereas the simulated data contains only EN and QS regions and can
thus only be compared within the respective limitations.

As a result of these effects, differences in the density plot for H$\alpha$
linewidth vs. ALMA brightness temperature for the simulated data in Fig. 12f
and the corresponding density plot for observed data in Molnar et al. (2019,
see their Fig. 4) are expected. It should be noted though that, despite these
effects, the distributions in observed and simulated data are very similar.
The simulated QS distribution coincides with the high-density scatter
distribution at low observed brightness temperatures and low H$\alpha$ line
widths in the plot shown by Molnar et al.. The blue EN contour, which has a
slightly lower slope than the slopes for the QS part and the slope for the
whole box in the simulated data, has its counterpart in the scatter plot for
the observed data. The calculated slopes for the whole box at the degraded
resolution are slightly different, which can be attributed to the
aforementioned differences between the observations and the simulations
reported here. For the observed data, Molnar et al. (2019) report the slope to
be 0.0612 as compared to 0.015 for the whole box at degraded spatial and
temporal resolution for the simulated data (see Table 1 for details).

As shown in Sect. 3.2, changing the spatial resolution affects the
correlations between H$\alpha$ linewidth and mm brightness temperatures. The
correlation for the data observed at 3 mm reported by Molnar et al. (2019) is
0.84, which is close to the correlation of 0.61 found here for the simulated
data at degraded spatial and temporal resolution. The correlation is 0.5 for
the data at the original resolution. The difference in the correlation
coefficients is because the spatial arrangement of the individual pixels
relative to each other does not matter when calculating them. Consequently,
the correlation coefficient increases with decreasing spatial resolution as
the resulting image degradation gradually removes the largest variations on
the smallest scales in both the ALMA and the H$\alpha$ maps. For both data
sets, lowering the spatial resolution results in narrower intensity
distributions towards the mean value.

The height at which the core is formed is higher and the linewidth lower for
the quiet region and the formation height is lower and linewidth is higher for
the EN region (Leenaarts et al., 2012).

###  4.3 Effect of temporal resolution

When time-averaging the spatially degraded H$\alpha$ and mm brightness
temperature maps, the resulting correlation coefficients change with respect
to the results based on single snapshots. As summarised in Table 1, for the
line core width as defined by Leenaarts et al. (2012), the correlation
coefficients for all three pixel sets become lower when time-averaging both at
original and reduced resolution except for a small increase for the
correlation with the 3 mm data across the whole box at reduced resolution (cf.
Fig. 14). The same behaviour is found when using the linewidth definition by
Molnar et al. (2019), maybe except for a small increase of the EN correlations
both at original and reduced spatial resolution.

In general, temporal averaging results in blurring of the maps and lower
correlation coefficients. For the EN region, however, time-averaging can tend
to highlight persistent features, resulting in a slightly higher correlation
coefficient. In contrast, for the QS region, the more uniformly distributed
and short-lived features get averaged over time and space and lose the
similarities. Similarly, the values of slopes for original and degraded
resolutions are very close for individual wavelengths (see Fig. 15). In
conclusion, time averaging causes a decrease in the correlation in the QS case
and a small increase in the EN case for the Molnar definition while the slopes
decrease marginally. This effect becomes slightly more notable for the maps at
reduced spatial resolution as compared to the maps at original resolution. We
conclude that the exact values and behaviour of correlation coefficients as
shown here may depend critically on the spatial and temporal resolution of the
employed data.

###  4.4 Comparison of linewidth definitions

As seen in Fig. 2, the value and the atmospheric layer connected to the
linewidth change significantly based on the definition used. In the case of
the Molnar et al. (2019) definition, as described in Sect. 2.4, the continuum
and thus the line depth are derived from the $\pm 1$ Å region around the line
core. On the other hand for both the Leenaarts et al. (2012) definition and
the FWHM, the continuum is defined based on the true continuum intensity. The
wavelength grid points used for the calculation of the line core width, which
are close to the line core, are formed significantly higher in the atmosphere
as compared to the Molnar et al. approach as illustrated in Fig. 16. The
radiation at the wavelength points in line wings as used for the FWHM
definition are accordingly formed lower in the atmosphere, even lower than for
the Molnar et al. definition. In Fig. 17, the line widths for all three
definitions are compared, implying the above-discussed differences in
connected atmospheric heights, being highest for the line core width
definition by Leenaarts et al. (2012). The FWHM linewidth map much resembles a
granulation pattern as formed in the low photosphere.

The distribution of line core widths as seen in Fig. 18a is in the range of
line core widths calculated by Leenaarts et al. (2012). Using the Molnar et
al. (2019) definition instead produces larger linewidths (see Fig. 18b) which
nonetheless remain smaller than the observationally determined values reported
by Molnar et al. (2019) . The simulated linewidths according to the Molnar et
al. (2019) definition cover a range of 0.75 to 1.0. Å, whereas the values
derived from observations by Molnar et al. (2019) are in the range of 0.9 to
1.3 Å, which is also in line with the observed linewidth for $\alpha$ Cen A
(see Sect. 4.5). The reason behind this discrepancy could be that the
H$\alpha$ opacity used in the simulations including the radiative transfer
calculations is lower than the real Sun (Leenaarts et al., 2012). Similarly,
as seen in Fig. 16, the formation heights of the radiation at the wavelength
positions entering the line core width calculation with the Leenaarts et al.
(2012) definition are closer to the formation heights of the mm continua as
compared to the corresponding heights for the Molnar et al. (2019) definition.
The conclusion that the Leenaarts et al. (2012) definition produces a better
match of the H$\alpha$ line (core) width with the mm continua is also implied
by a closer match in terms of spatial power spectral density as compared to
the Molnar et al. (2019) definition (see Fig. 5). This seems to be true at the
original and at the reduced resolution.

###  4.5 Implications for solar-like stars

The correlation between H$\alpha$ linewidth and mm brightness temperatures
described in Sect. 3.2 can be transferred to the study of sun-like stars. It
should be noted that the study presented here is based on simulated EN and QS
regions, while the corresponding trends for a whole active region could not be
investigated due to the lack of an equivalent consistent simulation. Any
comparison with unresolved observations of other solar-like stars, therefore,
needs to account for the potential contributions of active regions via
corresponding filling factors.

A direct application to stellar observations is to estimate the stellar
activity level, in particular for sun-like stars as for instance used in the
studies by Liseau et al. (2016) and Mohan et al. (2021, 2022). The position in
the H$\alpha$ linewidth - mm brightness temperature plane as derived from
adequate observations then provides constraints for the properties of the
observed stellar atmosphere. This approach is illustrated by comparing the
data points for the semi-empirical 1D models FAL C and VAL C to the averages
for the whole box, EN and QS data points on the H$\alpha$ linewidth - mm
brightness temperature plane in Figs. 3 c,f and 4 c,f. For the disk-integrated
signal, the centre-to-limb variation for the H$\alpha$ line and ALMA full disk
data should be taken into account (see, e.g., Alissandrakis et al., 2022a, b;
Otsu et al., 2022; Nindos et al., 2018, and references therein). As the
H$\alpha$ core forms in the chromosphere and the wings in the photosphere, the
limb effects would be varying (brightening or darkening depending on the
wavelength) when going from the core to the wing. In a recent study based on
SST observations by Pietrow et al. (2022), they concluded that chromospheric
lines might exhibit a blue shift towards the limb due to the chromospheric
canopies, but the resulting effect on the equivalent width or linewidths is
nontrivial. Further, they show that the line core widths would decrease when
going from the centre to limb. More systematic studies need to be conducted in
order to understand the limb effects on the correlations between the two
diagnostics and the implications for disk-integrated stellar observations.

The brightness temperature values at the wavelengths of 0.8 mm and 3 mm for
$\alpha$ Cen A, which are obtained from ALMA observations (see Mohan et al.,
2021, and references therein), are in line with the values derived from ALMA
observations of the Sun and also with those based on the simulations presented
in this study. In contrast, the H$\alpha$ line profile for $\alpha$ Cen A (see
Fig. 2) is with a line core width of $\sim 0.6$ Å significantly broader than
the simulated data based on the Bifrost simulation and also for the FAL C and
VAL C models, which causes the respective data point to be placed off the
linear fit for the solar simulation shown in Figs. 3c,f and 4c,f (see the
black circle with error bars). The $\alpha$ Cen A datapoint is out of the
bounds of the plane in Figs. 12 c,f and 13 c,f as the calculated Molnar et al.
(2019) linewidth is $\leavevmode\nobreak\ 1.1\AA$. The comparatively broad
H$\alpha$ line profile for $\alpha$ Cen A is likely due to differences in
atmospheric structure and the activity level of this star as compared to the
Sun although an in-depth study would be needed to answer this in detail. We
also note that the solar observations are spatially resolved and cover a small
region on the Sun only, whereas the $\alpha$ Cen A data intrinsically
integrated across the whole (spatially unresolved) stellar disk.

##  5 Conclusions

We look at the synthetic H$\alpha$ linewidth and the brightness temperatures
corresponding to the mm wavelengths generated from a 3D close to realistic
Bifrost model. Degrading the spatial resolution of these synthetic
observations to ALMA resolution results in decreased standard deviations of
the synthetic observables: H$\alpha$ line width and mm brightness
temperatures, and increased correlations between them. With increasing
wavelength, the beam size increases and the small structures get blurred out,
resulting in a further increased correlation between the two diagnostics. But
this degradation in terms of spatial resolution does not have much effect on
the mean slopes for the linear fits on the H$\alpha$ linewidth vs mm
brightness temperature distributions. This implies that the instrumental
resolution of ALMA is sufficient to capture the similarities between the two
diagnostics.

The solar results in comparison with the observations of $\alpha$ Cen A
strongly imply that the H$\alpha$ linewidth and mm continuum brightness
temperatures can be considered equivalent indicators, as seen in Fig. 16 that
they form close to each other and that the thermal stratification of a star
can be constrained using the correlations and slopes found in this study in
the absence of mm data, by just using the H$\alpha$ data and vice versa. As
discussed in Wedemeyer et al. (2016, see also references therein), using
multi-wavelength observations along with mm observations can provide better
insights into the thermal structure of stellar atmospheres.

The H$\alpha$ linewidth depends on the temperature in the line-forming region
which is observed using mm continua (Leenaarts et al., 2012). The brightness
temperatures can be directly calculated from the mm continuum intensity maps
which correlate with the H$\alpha$ linewidth very well. The best match in
terms of highest correlation is found for a wavelength of 0.8 mm, which
corresponds to ALMA Band 7. Simultaneous H$\alpha$–ALMA Band 7 observations
have therefore the potential advantage of better constraining the imaging
process and thus resulting in more reliable temperature measurements for the
solar chromosphere. Published simulations of the solar chromosphere (Carlsson
et al., 2016) show too narrow MgII h & k lines, compared with IRIS
observations (Carlsson et al., 2019), indicating that the models have too
little mass at upper chromospheric temperatures. Newer models in higher
resolution including ambipolar diffusion and non-equilibrium ionisation (e.g.,
Martínez-Sykora et al., 2023), flux emergence (Hansteen et al., 2023) or run
with the MURaM code with chromospheric extensions (Przybylski et al., 2022)
show significantly broader magnesium lines. It will be important to check the
reported ALMA-H$\alpha$ relationships with these new models. As the
correlation between the two diagnostics is high even in the case of lower
resolution, this study may prove useful in mitigating the challenging task of
calibration of mm continuum solar data obtained from ALMA.

###### Acknowledgements.

We thank the referee for very valuable comments and suggestions, which
improved considerably the quality of this paper. This work is supported by the
Research Council of Norway through the EMISSA project (project number 286853),
the Centres of Excellence scheme, project number 262622 (“Rosseland Centre for
Solar Physics”) and through grants of computing time from the Programme for
Supercomputing. This paper makes use of the following ALMA data:
ADS/NRAO.ALMA#2016.1.01129.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The authors thank Porto de Mello et al. for their prompt correspondence
regarding the H$\alpha$ observational data of $\alpha$ Cen A. This research
utilised the Python libraries matplotlib (Hunter, 2007), seaborn (Waskom,
2021) and the NumPy computational environment (Oliphant, 2006). The authors
would like to thank Thore Espedal Moe, Mats Ola Sand, Atul Mohan, Juan Camilo
Guevara Gomez, Vasco Henriquez, Shahin Jafarzadeh for their support.

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##  Appendix A Additional brightness temperature maps for single snapshot with
Leenaarts et al. (2012) linewidth

Figure 19: Same as Fig. 3, for 0.4 mm corresponding to ALMA Band 9-10. Figure
20: Same as Fig. 3, for 0.7 mm corresponding to ALMA Band 8. Figure 21: Same
as Fig. 3, for 1.0 mm corresponding to ALMA Band 7.

Figure 22: Same as Fig. 3, for 2.0 mm corresponding to ALMA Band 4

.

Figure 23: Same as Fig. 3, for 3.6 mm corresponding to ALMA Band 3. Figure 24:
Same as Fig. 3, for 4.5 mm corresponding to ALMA Band 2. Figure 25: Same as
Fig. 3, for 6.5 mm corresponding to ALMA Band 1. Figure 26: Same as Fig. 3,
for 8.5 mm corresponding to ALMA Band 1.