11institutetext: Rosseland Centre for Solar Physics, University of Oslo,
Postboks 1029 Blindern, NO-0315 Oslo, Norway 22institutetext: Institute of
Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, NO-0315
Oslo, Norway
22email<EMAIL_ADDRESS>
# EMISSA: Exploring millimetre indicators of solar-stellar activity

III. Comparison of Ca II indices and millimetre continua in a 3D model
atmosphere

Sneha Pandit 1122 Sven Wedemeyer 1122 Mats Carlsson 1122

###### Abstract

Context. Amongst several spectral lines, some of the strongest chromospheric
diagnostics are offered by the Ca II H & K lines. These lines can be used to
gauge the temperature stratification of the atmosphere since the line core and
wings are formed in different regions of the solar atmosphere. Furthermore,
the Ca II lines act as tracers for the magnetic structure of the solar
atmosphere, as the line cores are formed in the upper chromosphere even though
they are formed in non-local thermodynamic equilibrium (NLTE). In contrast,
the formation of millimetre (mm) continuum radiation occurs under local
thermodynamic equilibrium (LTE) conditions. As a result, the brightness
temperatures obtained from observations with the Atacama Large
Millimetre/Submillimetre Array (ALMA) offer a complementary perspective on the
activity and thermal structure of stellar atmospheres.

Aims. The overall aim is to establish more robust solar/stellar activity
indicators using ALMA observations in comparison with classical diagnostics,
such as the s index and infrared triplet (IRT) index.

Methods. We employed the 1.5D radiative transfer codes RH1.5D and advanced
radiative transfer (ART) to compute the synthetic spectra for the Ca II lines
and the millimetre (mm) continua, respectively. These calculations were
performed using an enhanced network atmosphere model, which incorporates non-
equilibrium hydrogen ionisation generated by the state-of-the-art 3D radiation
magnetohydrodynamics (rMHD) Bifrost code. To account for the limited spatial
resolution of ALMA, we simulated the effect using a Gaussian point spread
function (PSF). Additionally, we analysed the correlations and slopes of
scatter plots between the Ca II indices and mm continuum for the original and
degraded resolutions, focusing on the entire simulation box, quiet Sun
regions, and enhanced network patches separately. The activity indices
generated from these lines could further be used to compare the spectra of
Sun-like stars with the solar spectrum.

Results. We present a comparative study between synthetic continuum brightness
temperature maps at mm wavelengths (0.3 mm to 8.5 mm) and the Ca II activity
indices; namely, the s index and infrared triplet (IRT) index. The Ca II
activity indices and mm brightness temperatures are weakly correlated at the
high resolution, with the highest correlation observed at a wavelength of 0.3
mm, corresponding to ALMA band 10. As the resolution decreases, the
correlation consistently increases. Conversely, the slopes exhibit a
decreasing trend with increasing wavelength, while the degradation of
resolution does not noticeably affect the calculated slopes.

Conclusions. As the spatial resolution decreases, the standard deviations of
the Ca II activity indices and brightness temperatures decrease, while the
correlations between them increase. However, the slopes do not exhibit
significant changes. Consequently, these relationships could be valuable for
calibrating the mm continuum maps obtained through ALMA observations.

###### Key Words.:

Sun: chromosphere, Sun: activity, Methods: numerical, Line: formation, Line:
profiles, Radiative transfer

##  1 Introduction

For the Sun, as the closest star to Earth, spectra can be obtained at high
spatial and temporal resolution. In contrast, the observed spectra of other
stars are not as detailed but are the only basis for computing activity
indices. These indices are generated by collapsing the resolved spectra by
averaging, integrating, or convolving with a filter (Strassmeier et al.,
1989). Activity indicators are calculated using spectral lines sensitive to
the magnetic activity, as with the s index, a proxy based on the Ca II H and K
flux integrated over the filters as defined by Vaughan et al. (1978), or the
H$\alpha$ index as defined in Gizis et al. (2002). The abundance of
spatiotemporally resolved data available for solar spectra allows for the use
of these indices to retrieve information about the atmosphere. Furthermore,
these indices can be used to compare the solar spectra with those of other
stars to gain insights into their atmospheres and stratification.

The chromosphere is responsible for radiating away most of the energy that is
transported to the outer solar atmosphere. The chromosphere is where the
dynamical properties change from gas-pressure-dominated to magnetic force-
dominated. The thermodynamical conditions in the chromosphere go from local
thermodynamic equilibrium (LTE) to non-equilibrium conditions. The
chromosphere is partially ionised, and the ionisation state changes from
almost neutral to full ionisation (Carlsson & Stein, 2002). All these
transitions make chromospheric physics very complex and it remains the most
poorly understood region of the Sun (see Carlsson et al., 2016, and references
therein).

To study this layer and the temperature stratification of the Sun, Ca II H & K
lines are used as diagnostics. The line core for Ca II H & K and the infrared
triplet (IRT) are formed at chromospheric heights. The wavelengths away from
the core are subsequently formed at lower layers and the wings are formed at
deeper photospheric heights following local thermodynamic equilibrium (LTE).
Observations in the Ca II H & K lines imply a reversal of the atmospheric
temperature profile with a chromospheric temperature rise. See Table 1 for the
average formation heights as derived in the empirical study by Bjørgen et al.
(2018).

K line | Formation height range | Average formation height | H line | Average formation height below the K line
---|---|---|---|---
| (Mm) | (Mm) |  | (km)
$K_{3}$ | 0.5-4.0 | 1.9 $\pm$ 0.6 | $H_{3}$ | 150
$K_{2V}$ | 0.3-3.0 | 1.3 $\pm$ 0.5 | $H_{2V}$ | 150
$K_{2R}$ | 0.3-3.0 | 1.0 $\pm$ 0.5 | $H_{2R}$ | 100
Table 1: Formation heights of the features from the photosphere as described
in Bjørgen et al. (2018) based on Bifrost models derived from Carlsson et al.
(2016).

The Ca II H and K lines are of interest due to their high depth and width
resulting from the most probable transition between the highly populated 4s
energy level and the immediate higher shell, 4p (Uitenbroek, 1990b; Wedemeyer-
Böhm & Carlsson, 2011). On the other hand, the infrared triplet of Ca II is
between 3d and 4p. It consists of three strong lines at 849.802, 854.209, and
866.214 nm, whose cores are formed in the chromosphere as well (Bjørgen et
al., 2018; Jorgensen et al., 1992). They have smaller energy gaps and are
brighter than the H & K lines, as seen further in Fig. 3. The line core
intensities for the IRT lines are higher than the H & K line cores. The H and
K lines are formed much higher in the less dense upper chromosphere where 3D
and Partial Redistribution (PRD) effects play a more essential role in line
formation than in the infrared triplet of Ca II (Uitenbroek, 1990a).

The Atacama Millimetre/submillimetre Array (ALMA) (Wootten & Thompson, 2009)
provides observations with a high spatial and temporal resolution for the
millimetre (mm) continuum. The Ca II lines and mm continua both probe
chromospheric heights. It is established that the continuum radiation observed
at millimetre (mm) wavelengths is formed in the chromosphere (see, e.g.,
Wedemeyer et al., 2016, , and references therein). This continuum radiation in
the millimetre wavelengths (here 0.3–8.5 mm) arises from free-free emission
within the chromosphere, where the primary sources of opacity are H and H-
free-free absorption (Dulk, 1985). These processes lead to an LTE source
function that depends on the electron temperature of the plasma. Therefore, we
can interpret the emergent intensity (the observed brightness temperature
($T_{\mathrm{b}}$) in the millimetre wavelength domain as local electron
temperature (Loukitcheva et al., 2015). The opacity at mm wavelengths is
directly dependent on the electron density, which deviates from equilibrium
conditions in the chromosphere (see, e.g., Dulk, 1985; Loukitcheva et al.,
2004; Wedemeyer et al., 2016, and references therein). While the formation
process operates under LTE conditions, the non-equilibrium electron densities
impact the diagnostic information derived from ALMA observations (see, e.g.,
Molnar et al., 2019; da Silva Santos et al., 2020). The millimetre continuum
is nevertheless a useful diagnostic tool for probing the thermal properties of
the atmospheres of the Sun and other stars as long as the non-equilibrium
nature of the electron density is carefully taken into account.

Activity is a range of phenomena occurring in the outer layers of a star.
Variations in the magnetic fields can be observed through variations of the
observed intensity, which are used to calculate activity indices (Skumanich et
al., 1975; Harvey & White, 1999). The most commonly used indicator of
chromospheric activity is the s index, the ratio of the flux in the core of
the Ca II H and K lines to the nearby continuum (Wilson, 1968; Vaughan et al.,
1978). The mm continua observations of the Sun can reveal the 3D structure of
the gas and even the magnetic field in the outer layers as shown by da Silva
Santos et al. (2018); De la Luz (2016). The continuum wavelengths probe
slightly different heights (Loukitcheva et al., 2015), giving us information
of electron temperatures at different heights based on the mm wavelengths.
This study aims to use ALMA’s novel capabilities for a re-evaluation of the
Calcium II activity indicators using a comparative solar-stellar study, with
the solar model serving as a fundamental reference.

Solar observations from ALMA have been compared with the H$\alpha$ line by
Molnar et al. (2019); Tarr et al. (2023); Pandit et al. (2023), but there are
fewer studies comparing mm solar data with Ca II lines. Harvey & Solis Team
(2020) have compared nearly co-observed Ca II 8542 data from SOLIS with ALMA
data and reported striking similarities. Molnar et al. (2021) compared co-
observed IBIS Ca II 8542 and H$\alpha$ data with ALMA Bands 3 and 6 data
statistically to model the propagation of waves based on the velocities found
with the Ca II 8542 line. In da Silva Santos et al. (2020) the ALMA data was
compared with IRIS Mg II lines, and the authors recommended a closer
comparison of ALMA data with NLTE lines like the Ca II lines. None of the
studies mentioned have quantified the similarities between Ca II and mm data
for the Sun and this study is trying to fill that gap.

There has been a continuous effort undertaken to understand the activity based
on the solar cycle. Pietarila & Livingston (2011) have found that the line
core intensity and the line bisector at the line wings of Ca II 854.2 nm line
are sensitive to the solar activity over 25 years covered in their study.
Brajsa et al. (2018) carried out a study to compare ALMA Band 6 data with
theoretical models of different active phenomena. Deland & Cebula (1993)
studied the Mg II index for the Sun over two solar cycles to measure long-term
UV solar activity. The H Balmer lines variability based on the solar long-term
variability has been addressed by Criscuoli et al. (2023). There are numerous
studies that have employed the Sun-as-a-star approach. Our work takes this
approach to understand the Ca II indices in detail at different activity
levels to add to the effort.

This work is focused on using and comparing Ca II intensities and millimetre
brightness temperatures as complementary diagnostics. In this paper, we
compare the mm continuum and Ca II activity indicators 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 Section 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) was used as input for radiative transfer codes for the synthesis of Ca II
lines (see Sect. 2.2) and the continuum intensity at millimetre wavelengths
(see Sect. 2.4). The resulting maps for different wavelengths were then
degraded to a spatial resolution typically achieved with ALMA, as described in
Sect. 2.5 before the activity indicators were determined (see Sect. 2.7) and
compared to the mm brightness temperatures.

###  2.1 Bifrost

Bifrost is a 3D simulation code that provides an unprecedented view of the
solar atmosphere with high resolution (Gudiksen et al., 2011). The
computational box measures 24 Mm x 24 Mm horizontally and extends 2.4 Mm below
the visible surface and 14.4 Mm above, covering the upper regions of the
convection zone, photosphere, chromosphere, transition region, and corona
(Carlsson et al., 2016). It consists of 504 x 504 x 496 grid points with a
horizontal resolution of 48 km and a variable grid separation in the vertical
direction ranging from 19 km in the photosphere and chromosphere up to 5 Mm
height and then increasing to 100 km at the top boundary. Both top and bottom
boundaries are open, and the magnetic field is passively advected at the
bottom boundary, with no additional field fed into the computational domain
(Gudiksen et al., 2011; Carlsson et al., 2016). The name of the model used in
this project is en024048, with ’en’ standing for enhanced network, the
horizontal size of the computational domain 24 Mm, and the horizontal grid
size 48 km (more relevant details are discussed in Pandit et al. (2023) and
references therein).

As seen in Wedemeyer et al. (2020), the simulation has different regions. At
the lateral boundaries, it is more quiet and hence, less magnetically active.
The en024048 simulation features an enhanced network (EN) patch with loops in
the middle of the computational domain, whereas the outer parts are more
representative of magnetically less active, quiet Sun (QS) conditions (see
Fig. 1 in Pandit et al., 2023, for more details about the temperature and
absolute magnetic field). This diverse behavior exhibited by the model
atmosphere is reflected in the varying intensity of the Ca II lines (as seen
in Fig. 4).

###  2.2 Radiative Transfer (RH) code

The Ca II lines are synthesised from the Bifrost model atmosphere using the
radiative transfer code RH1.5D. The multilevel atoms and molecules are solved
for statistical equilibrium and radiative transfer equations in a self-
consistent manner using the RH code, as detailed in Pereira & Uitenbroek
(2015); Uitenbroek (2001). The formation of the Ca II H and K wings occurs in
the photosphere under LTE, while the cores form in the chromosphere, where the
low densities cause significant departures from LTE. If the LTE approximation
is used to synthesise line cores, it leads to unrealistic emission profiles
since the temperature increases in the layers above the temperature minimum.
Uitenbroek (1990b) has emphasised that PRD effects are crucial for the
formation of these lines. Thus, the RH’s non-LTE PRD capability is employed to
synthesise spectra for the Ca II H & K lines, similarly to what is seen in the
studies from Bergemann et al. (2019); Bjørgen & Leenaarts (2017).

Figure 1 shows the agreement between the averaged Ca II data synthesised using
snapshot 420 of the same Bifrost model ’en024048’, which is — minutes more
evolved than the snapshot used in the study. This data is used to save
computational time as it was produced for another work. As can be seen in the
figure, the average Ca II line profiles follow the ATLAS profiles quite
neatly, especially in the (photospheric) line wings. The average Bifrost line
profiles were produced without including line blanketing to speed up the
calculations when calculating the spectra for all 504x504 columns. When
calculating the indices in the paper, we do include line blanketing in a
sparse grid (as explained in Figs. 2 and 3).

Figure 1: FTS solar ATLAS data in both Ca II H and K band and IRT band is
shown (in red) in comparison with the average Ca II data (in black) generated
from the Bifrost model without line blanketing included.

The Kurucz line
list111https://lweb.cfa.harvard.edu/amp/ampdata/kurucz23/sekur.html has an
extensive list of background lines which contribute to the opacity. However,
lines with lower transition probability have an insignificant impact on the
intensity. To reduce the required computation time, which is substantial for a
model with 504x504 columns, the lines affecting the calculated s index the
most were selected for the test run. The criterion for selection for this list
was that the resulting s index is within 1% of the calculated s index with the
whole line list for each 50th column (121 columns in total).

Figure 2: Resolved spectra simulated with all Kurucz lines in the range
389.1067 nm to 401.1067 nm and 847.33 nm to 870.74 nm shown in black fine
lines in comparison with the run considered in the study with comparatively
fewer Kurucz lines shown in red dots for column (0,0), the filters for the
different bands required for calculating the indices are shown in thick black
lines. Figure 3: Same legend as in Fig. 2, for the bands of interest for the
calculation of the s index and IRT index. The panels show these bands: V band
(389.1067 nm to 391.1067 nm), K line (393.3664 nm), H line (396.847 nm), R
band (399.1067 nm to 401.1067 nm), V band for IRT (847.33 nm to 847.83 nm),
1st triplet line (849.8 nm), 2nd triplet line (854.2 nm), 3rd triplet line
(8662 nm), and R band for IRT (870.24 nm to 870.74 nm).

In Figure 2, the intensity is calculated with all the background lines in the
Kurucz line list. The calculation of the s index requires the lines within the
predefined triangular filters and continuum bands from Vaughan et al. (1978)
on either side (red and blue) of the lines shown in the figure by solid black
lines. The same is also shown for the IRT index. An alternative method of
saving computational power is to alter the computational wavelength grid and
calculate only the required intensities within the filters. The red dots show
the spectra generated with the finalised grid for the final run. This
wavelength grid table was also chosen for the final run as it was observed
that the calculated s index is within 1% of the calculated s index with the
whole grid.

The outcome of both the runs for the wavelength ranges used for calculating
the s index and IRT index in the first column (0,0 column) is shown in Fig. 3.
The panels show the following bands: V band, K, H, R bands, V band for the
infrared triplet, first triplet line (849.8 nm), second triplet line (854.2
nm), third triplet line (866.2 nm), and R band for IRT. The grey lines show
the runs with the whole Kurucz line list and correspondingly saturated
wavelength grid. The almost constant continuum intensity is due to the lack of
strong lines so this constant value at 35.6 nW/(Hz m2 sr) is nearly the true
continuum. Red dots show the run with the finalised grid and the Kurucz line
lists truncated at the 99th percentile. The lines are in good agreement, and
the blue and red bands that indicate the denominators of the respective
indices are reaching the same heights in intensity at the continuum. The
averaged synthetic line profiles generated from the Bifrost model are observed
to follow the FTS ALTAS spectra.

Figure 4: Maps showing the integrated intensities, within the 1st and 99th
percentile, for the wavelength ranges of interest for the calculation of the s
index and the IRT index, namely, the continuum, triangular bandpasses of
$2\AA$ centred at lines K and H, and rectangular bandpasses of $2\AA$ around
lines in the infrared triplet.

Figure 4 shows the intensity maps for different wavelengths formed in
different layers from the Bifrost simulation box. The continuum is formed in
the photosphere which is seen from the synthetic observations. The H and K
line cores on the other hand show the footpoints of the enhanced network which
is in the chromosphere. Similarly in the infrared triplet lines, the inter-
network and network regions are seen prominently.

###  2.3 Ca II IRT observations of $\alpha$ Cen A

In this study, Ca II IRT observations of the solar-like star $\alpha$ Cen A
are used for comparison. The data is acquired using MUSICOS (MUlti SIte
COntinuous Spectroscopy) high-resolution echelle spectrograph (R=35000,
$\delta\lambda$ 0.024 nm at $\lambda$ 850 nm) mounted at the 1.6 m telescope
of the Observatório do Pico dos Dias (OPD), operated by the Laboratório
Nacional de Astrofísica (LNA/MCTI), Brazil from Lorenzo-Oliveira et al.
(2016). The wavelength range covers approximately 550-880 nm divided into 50
spectral orders. The $\alpha$ Cen A echelle spectrum is reduced following the
standard Image Reduction and Analysis Facility (IRAF) procedures: bias
subtraction, flat field, scattered light corrections, 1D extraction, and
Doppler and blaze function corrections. The continuum in the processed spectra
around the Ca IRT lines (834.0 nm and 877.1 nm) is normalised to unity using
low-order Legendre polynomials by Lorenzo-Oliveira et al. (2016).

###  2.4 Synthesis of millimetre continuum brightness temperatures

Using the advanced radiative transfer (ART) code developed by de la Cruz
Rodríguez et al. (2021), the intensity of the continuum radiation at
millimetre wavelengths is computed for the 3D input model described in Sect.
2.1. The output intensities ($I_{\lambda}$) are then converted into brightness
temperatures ($T_{b}$) by applying the Rayleigh-Jeans approximation, as shown
in the following equation:

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

Here, $\lambda$, $k_{B}$, and $c$ represent the wavelength, Boltzmann
constant, and speed of light, respectively. This process is repeated for 34
wavelengths, covering a range that can be observed using the available and
potential receiver bands of ALMA, as demonstrated in Pandit et al. (2023).

###  2.5 Image degradation

We adopted the same procedure for image degradation described in Section 2.6
of Pandit et al. (2023) to degrade the Ca II data before computing the
integrated intensities and indices. The synthetic beam’s exact shape and width
can vary even for observations within the same receiver band depending on the
array configuration, and thus the diameter of the synthesised aperture and the
position of the source in the sky with respect to the array baseline. For
instance, the band 3 data sets available in the Solar ALMA Archive (SALSA,
Henriques et al., 2022) have widths of approximately 2”, including the beam
with a width of 1.92” $\times$ 2.30” for the 2016.1.01129.S data set. To
simplify this study, the beam’s width (i.e., the point spread function, PSF)
is set to 2” at a wavelength of $\lambda=3.0$ mm and then linearly scaled as a
function of the observing wavelength $\lambda$. The wavelength range we
considered includes the receiver bands that are currently available for solar
observations with ALMA (bands 3, 5, and 6, i.e., 2.59–3.57 mm, 1.42–1.90 mm,
and 1.09–1.42 mm, respectively), as well as bands 1, 2, 4, and 7-10, covering
the entire range from 0.3 mm to 8.5 mm that may be available in the future.
This assumes that the interferometric array is a large circular dish with a
full width at half maximum (FWHM) in the x-y directions of
$2"\times(\text{wavelength in mm})/3\text{ mm}$ for the synthetic Gaussian
kernel. A circular wavelength-dependent Gaussian point spread function (PSF)
is then applied to both the millimetre brightness temperature maps and the Ca
II intensity maps.

###  2.6 Semi-empirical models

The 1D semi-empirical reference models representative of quiet Sun conditions
VAL C (Vernazza et al., 1981) and FAL C (Fontenla et al., 1993), the plage
model by Fontenla et al. (2009), and the sunspot model by Severino et al.
(1994) as provided by Loukitcheva (priv. comm.) are used to compare with the
3D results. Corresponding Ca II line profiles and mm continuum intensities for
these 1D semi-empirical atmosphere models are calculated with the radiative
code RH 1.5D (as described in Sect. 2.2).

###  2.7 Calculation of activity indices

From the intensities calculated in the run set for the whole simulation box,
namely, 504 $\times$ 504 columns (see Sect. 2.2), the calcium activity indices
were calculated. The definition from Vaughan et al. (1978) was used. Figure 4
shows the individual integrated intensities over the defined filters as per
Vaughan et al. (1978) for the components of the calcium s index. The continuum
intensity image shows photospheric granulation, where the V band has slightly
lower continuum intensity than the R band as expected from the wavelength-
dependence of the solar flux spectrum. The H and K line-integrated intensities
over the triangular filter centred at the line core show the internetwork
regions and the enhanced network loop footpoints with significantly higher
intensities. By definition of the s index, the triangular filter for the H
line has half the height of that of the K line, namely, equal to the
continuum. The s index is calculated by using the standard definition from
Vaughan et al. (1978) as:

$S={\alpha_{c}}\times 8\times\frac{F_{H}+F_{K}}{F_{V}+F_{R}},$ (2)

where $F_{H}$ is the integrated intensity over a triangular filter between
396.738 nm and 396.956 nm, centred at 396.847 nm and $F_{K}$ is the integrated
intensity over a triangular filter between 393.2574 nm and 393.4754 nm,
centred at 393.3664 nm, and $F_{V}$ (389.1067 nm to 391.1067 nm) and $F_{R}$
(399.1067 nm to 401.1067 nm) are the averaged intensities integrated over the
rectangular filter over the whole box (Sowmya et al., 2021). The factor of 8
in Eq. (2) was introduced to account for the differences in integration time
in the different filters centred at the line cores as compared to the
continuum observations as described in Hall et al. (2007). The factor is kept
in the definition to allow for comparison with other published values. The
instrumental calibration factor $\alpha_{c}$ is set to a value of 2.4 for the
Mount Wilson Observatory (Vaughan et al., 1978) but it is set to 1 in this
study as the synthetic data is not subject to instrumental properties.

The Ca infrared triplet index is defined based on the definition of the s
index (Marsden et al., 2014):

$\text{CaIRT
index}=\frac{F_{8498}+F_{8542}+F_{8662}}{F_{V_{IRT}}+F_{R_{IRT}}},$ (3)

where $F_{8498}$, $F_{8542}$ and $F_{8662}$ are the fluxes measured in 2 Å
rectangular bandpasses centred on the respective Ca II IRT lines, while
$F_{V_{IRT}}$ and $F_{R_{IRT}}$ are the fluxes in 5 Å rectangular bandpasses
centred on two continuum points, 8475.8 Å and 8704.9 Å, at either side of the
IRT lines (Petit et al., 2013, Eq. 1).

##  3 Results

###  3.1 The s index

As seen in Fig. 4a, the photospheric granulation is visible in the integrated
intensities in the continuum. In panels b and c, the integrated intensities
for K and H lines using the triangular filters defined by Vaughan et al.
(1978) are shown respectively. The s index calculated from these integrated
intensities is shown in Fig. 5a. Figure 5 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. Panel a
shows the s index calculated as per Eq. 2 based on the procedure described in
Sect. 2.7. The shock pattern from the lower chromosphere is observed from this
map similar to the K and H lines in Fig. 4 b,c. On the other hand, the 3 mm
map (panel b) shows a layer at heights higher than the height represented by
Ca II H&K integrated intensity shown in Fig. 4 b,c. The maps appear more
similar at reduced resolution after convolution with a circular Gaussian
kernel corresponding to the spatial resolution at a wavelength of 3 mm,
namely, with a PSF of 2” (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 enhanced network 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) (similar to the analysis in Pandit et al., 2023).

Following the same procedure as described in Pandit et al. (2023), Fig. 5
presents scatter contour plots for three different sets of pixels: the entire
simulation box, the QS box (indicated by a red square in panels a, b, d, e),
and the EN box (indicated by a black square in panels a, b, d, and e). The
distribution for the entire simulation box is depicted by red contours,
accompanied by a colour bar indicating the number of pixels within each bin.
The QS and EN distributions are represented by green and blue contours,
respectively. The distributions at the original resolution are substantially
more spread out than the distributions at degraded resolution because of the
associated loss of variations on small spatial scales. This effect is evident
from the visual comparison of the s index maps in panels a and d, as well as
the mm brightness temperature maps in panels b and e. The average values for
the whole box, QS, and EN are denoted by red, green, and blue points,
respectively, in panels c and f. The mean s index values for the whole box, QS
and EN regions, 0.257, 0.237, and 0.307 are shown by a circle corresponding to
the colours mentioned before. The s index value depends on the activity level
as there are notable differences in these means.

Furthermore, for comparison, the s index and the $T_{b}$ at the mm wavelengths
are plotted for the VAL C and FAL C semi-empirical model atmospheres. The VAL
C model has consistently lower brightness temperature than the FAL C model as
seen in Fig. 6 in Pandit et al. (2023). Also, both these semiempirical models
exhibit lower s indices than the calculated averages for the whole box, EN and
in the case of the VAL C model, even lower than QS. This issue is further
addressed in Sect. 4. When plotted on the s index 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
similar to (see Sect.3.1 in Pandit et al., 2023).

In Figure 6 the results for 0.3 mm are presented. The wavelength 0.8 mm was of
particular interest in the study by Pandit et al. (2023), and thus the
corresponding results are presented in Fig. 7. In addition to these short
wavelengths, the results for a longer wavelength of 4.5 mm in ALMA band 2 are
shown in Fig. 8. The panel a in all the four Figs. 5, 6, 7, and 8 is the same,
but the panels b show the brightness temperatures at respective wavelengths
and panels d and e are the corresponding degraded maps, using the PSF
corresponding to the wavelength. The PSF at 0.3 mm is smaller in width
compared to the 3.0 mm PSF by a factor of 0.3/3.0, which can be seen from a
visual comparison of Fig. 5e and Fig. 6e. The 3.0 mm maps appear more blurred,
mainly as a result of the wider PSF. The same effect can be seen for the s
index maps, which were calculated from the Ca II lines intensity maps after
these were all degraded accordingly for each wavelength point.

Figure 5: Comparisons of the s index (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 s index 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. The 1D solar semiempirical data with respective
filling factors using Severino et al. (1994) sunspot model and Fontenla et al.
(2009) Plage model H along with VAL C data are represented by upwards facing
triangles and downwards facing triangles. Figure 6: Same as Fig. 5 but for a
wavelength of 0.3 mm. Figure 7: Same as Fig. 5 but for a wavelength of 0.8 mm.
Figure 8: Same as Fig. 5 but for a wavelength of 4.5 mm. Figure 9: Comparison
of the calculated Pearson correlation coefficients between s index and mm
brightness temperature with two different resolutions Figure 10: Comparison of
calculated slopes for scatter plots of s index mm brightness temperatures.

The calculated Pearson correlation coefficients between the s index and the mm
brightness temperatures show that these indices correlate best with the lowest
wavelength (see Table 2 for the correlation coefficients for 0.3 mm and 4.5
mm). The black lines which show the correlation at the original resolution,
are lower than the correlations for the degraded resolution as expected. Also,
the correlation coefficients are highest for EN and lowest for QS cases in
both the resolutions, respectively, similar to the study by Pandit et al.
(2023).

The increase in the correlation coefficients in the case of the whole box and
the EN region at the degraded resolution is primarily because of the spatial
resolution, namely, the width of the PSF which increases with wavelength.
Hence, the structure at spatial scales below the corresponding angular
resolution is blurred, which affects the resulting correlation coefficient.
With increasing wavelength (and thus decreasing angular resolution) more of
this atmospheric fine structure becomes blurred (as seen from a comparison of
Figs. 6 and 8). However, at the longest wavelengths 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 as seen in Fig. 9. Most of the original resolution
data is very weakly correlated with the Pearson coefficient ¡ 0.3. With
degraded resolution, the differences in the nearby pixels get smoothed out,
which effectively reduces the corresponding data ranges.

The slopes calculated for the scatter plots show sudden decay with the
wavelengths increasing beyond 1 mm. The slopes are highest at the lowest
wavelength and for the wavelengths higher than 1 mm, the slopes are stably
decreasing. This is also evident from Figs. 5-8 (panels c,f). For the QS
region at degraded resolution at wavelengths longer than 3 mm, the slopes are
very close to zero (also observed from Figs. 5c,f,8c,f). As the wavelength
increases, the QS part in the mm maps and s index map becomes progressively
different in terms of structure and the network-internetwork regions become
more and more misaligned; this is evident from the green contours being more
concentrated in shorter ranges of s index and temperatures. At higher mm
wavelengths, the radiation is formed significantly higher up in the
atmosphere, whereas the Ca II H&K fluxes, (the basic constituents of the s
index) are formed primarily in the lower atmosphere. Consequently, the scatter
plots exhibit correlation coefficients and slopes that approach zero or become
negative, indicating a lack of correspondence between the two diagnostics.

| Original resolution | Reduced resolution
---|---|---
| wavelength | Whole box | QS | EN | Whole box | QS | EN
Temperatures | 0.3 mm | 4.722 | 4.691 | 4.85 | 4.722 | 4.691 | 4.850
(kK) | 0.8 mm | 5.275 | 5.124 | 5.590 | 5.275 | 5.126 | 5.593
| 3.0 mm | 6.968 | 6.621 | 7.835 | 6.968 | 6.637 | 7.848
| 4.5 mm | 7.661 | 7.250 | 8.715 | 7.661 | 7.268 | 8.699
s index |  |  |  |
Correlation | 0.3 mm | 0.704 $\pm$ 1.29E-03 | 0.645 $\pm$ 1.27E-03 | 0.744 $\pm$ 1.28E-03 | 0.729 $\pm$ 1.17E-03 | 0.666 $\pm$ 3.22E-03 | 0.767 $\pm$ 2.32E-03
| 0.8 mm | 0.431 $\pm$ 1.62E-03 | 0.348 $\pm$ 1.64E-03 | 0.476 $\pm$ 1.62E-03 | 0.510 $\pm$ 1.49E-03 | 0.396 $\pm$ 4.49E-03 | 0.566 $\pm$ 2.66E-03
| 3.0 mm | 0.249 $\pm$ 1.85E-03 | 0.101 $\pm$ 1.86E-03 | 0.339 $\pm$ 1.85E-03 | 0.534 $\pm$ 1.57E-03 | 0.095 $\pm$ 4.85E-03 | 0.701 $\pm$ 2.19E-03
| 4.5 mm | 0.223 $\pm$ 2.01E-03 | 0.034 $\pm$ 1.97E-03 | 0.350 $\pm$ 1.99E-03 | 0.591 $\pm$ 1.59E-03 | -0.099 $\pm$ 6.38E-03 | 0.811 $\pm$ 1.49E-03
Slopes | 0.3 mm | 0.165 $\pm$ 3.31E-04 | 0.114 $\pm$ 6.75E-04 | 0.214 $\pm$ 9.61E-04 | 0.175 $\pm$ 3.26E-04 | 0.120 $\pm$ 6.74E-04 | 0.224 $\pm$ 9.34E-04
| 0.8 mm | 0.063 $\pm$ 2.63E-04 | 0.039 $\pm$ 5.32E-04 | 0.087 $\pm$ 8.04E-04 | 0.077 $\pm$ 2.56E-04 | 0.046 $\pm$ 5.36E-04 | 0.104 $\pm$ 7.58E-04
| 3.0 mm | 0.029 $\pm$ 2.28E-04 | 0.010 $\pm$ 5.11E-04 | 0.051 $\pm$ 7.05E-04 | 0.049 $\pm$ 1.56E-04 | 0.009 $\pm$ 5.11E-04 | 0.082 $\pm$ 4.17E-04
| 4.5 mm | 0.027 $\pm$ 2.28E-04 | 0.004 $\pm$ 5.50E-04 | 0.051 $\pm$ 6.87E-04 | 0.043 $\pm$ 1.16E-04 | -0.009 $\pm$ 4.59E-04 | 0.082 $\pm$ 2.95E-04
IRT index |  |  |  |
Correlation | 0.3 mm | 0.535 $\pm$ 1.77E-03 | 0.428 $\pm$ 1.72E-03 | 0.621 $\pm$ 1.74E-03 | 0.559 $\pm$ 1.64E-03 | 0.446 $\pm$ 4.20E-03 | 0.645 $\pm$ 3.29E-03
| 0.8 mm | 0.307 $\pm$ 1.91E-03 | 0.212 $\pm$ 1.92E-03 | 0.383 $\pm$ 1.92E-03 | 0.382 $\pm$ 1.83E-03 | 0.253 $\pm$ 4.87E-03 | 0.472 $\pm$ 3.45E-03
| 3.0 mm | 0.176 $\pm$ 1.96E-03 | 0.068 $\pm$ 1.97E-03 | 0.289 $\pm$ 1.93E-03 | 0.467 $\pm$ 1.69E-03 | 0.090 $\pm$ 4.40E-03 | 0.660 $\pm$ 2.16E-03
| 4.5 mm | 0.164 $\pm$ 2.02E-03 | 0.019 $\pm$ 1.99E-03 | 0.296 $\pm$ 2.04E-03 | 0.514 $\pm$ 1.81E-03 | -0.139 $\pm$ 5.54E-03 | 0.763 $\pm$ 1.53E-03
Slopes | 0.3 mm | 0.033 $\pm$ 1.04E-04 | 0.022 $\pm$ 2.36E-04 | 0.043 $\pm$ 2.76E-04 | 0.035 $\pm$ 1.03E-04 | 0.024 $\pm$ 2.38E-04 | 0.046 $\pm$ 2.72E-04
| 0.8 mm | 0.012 $\pm$ 7.29E-05 | 0.007 $\pm$ 1.64E-04 | 0.017 $\pm$ 2.07E-04 | 0.015 $\pm$ 7.09E-05 | 0.008 $\pm$ 1.63E-04 | 0.021 $\pm$ 1.96E-04
| 3.0 mm | 0.005 $\pm$ 6.09E-05 | 0.002 $\pm$ 1.52E-04 | 0.011 $\pm$ 1.75E-04 | 0.010 $\pm$ 3.79E-05 | 0.002 $\pm$ 1.34E-04 | 0.017 $\pm$ 9.65E-05
| 4.5 mm | 0.005 $\pm$ 6.07E-05 | 0.001 $\pm$ 1.63E-04 | 0.011 $\pm$ 1.72E-04 | 0.008 $\pm$ 2.75E-05 | -0.003 $\pm$ 1.15E-04 | 0.016 $\pm$ 6.90E-05

Table 2: 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 s index and IRT index.

###  3.2 The IRT index

An analysis similar to the one presented for the s index was also done for the
Ca infrared triplet (IRT) lines. The integrated intensities for the Ca IRT
lines over the rectangular filters described in Fig. 3 are shown in Fig. 4
(d,e and f). In comparison to the K and H lines, the IRT lines are formed at
slightly lower heights in the lower chromosphere. Structures showing reverse
granulation features are seen and the intensities are slightly higher at the
footpoints of the loops, in comparison to the internetwork regions where shock
fronts are observed. As shown by Moe et al. (2024), the synthetic Ca II 854.2
nm data from Bifrost shows narrower profiles than the observed solar data
derived with the CRISP instrument at the Swedish 1-m Solar Telescope (SST).
This means that the Bifrost model in effect underestimates the IRT index. The
IRT index at the original resolution calculated using the fluxes and the
continuum (as per Eq. 3) are shown in Fig. 11 a, which shows lower layer in
the atmosphere in comparison to Fig. 5a. Similar to the s index case, the
original resolution maps for the IRT index and mm brightness temperature are
shown in panels a and b and the degraded resolution maps are shown in panels d
and e, respectively. The scatter contour plots are plotted for both
resolutions in panels c and f, respectively, similar to the treatment in the
previous section. Also, similar plots for a wavelength of 0.3 mm showing
highest correlation coefficient, 0.8 mm, being of particular interest in
Pandit et al. (2023), and 4.5 mm for illustrating wavelength formed at higher
heights are also shown respectively in Figs. 12, 13, and 14. The mean values
of the IRT index for the whole box, QS and EN regions are 0.262, 0.259 and
0.273, shown by filled circles of the red, green and blue colour,
respectively.

Figure 11: Same as Fig. 5 but for IRT index, and the black circle with error
bars is an observational data point for G2V type star $\alpha$ Cen A with
calculated IRT index of 0.315. Figure 12: Same as Fig. 11 but for a wavelength
of 0.3 mm. Figure 13: Same as Fig. 11 but for a wavelength of 0.8 mm. Figure
14: Same as Fig. 11 but for a wavelength of 4.5 mm. Figure 15: Comparison of
the calculated Pearson correlation coefficients between IRT index and mm
brightness temperature with two different resolutions Figure 16: Comparison of
the calculated slopes for the scatter plots for IRT index and mm brightness
temperatures.

Figures 15 and 16 reveal that the correlation coefficients and slopes peak at
the lowest wavelengths, similar to the correlation coefficients and slopes
derived for the s index. However, the slopes for the IRT index are an order of
magnitude lower, evident from the fact that the range of the calculated IRT
index is an order of magnitude lower than the calculated s index. The
correlations are slightly lower as well, which can be attributed to the IRT
index plane representing a slightly lower atmospheric region compared to the s
index plane. This distinction is evident in Fig.11a and Fig.5a. Moreover,
Fig.6b and Fig.12b demonstrate that the 0.3 mm map bears more similarity to
the s index than to the IRT index. Thus, the correlation coefficient for the
IRT index is lower than that for the s index.

It is important to note that the general trends in the correlation
coefficients and slopes are very similar for both the s index and the IRT
index. In the original resolution case, the correlation coefficients decrease
from 0.3 mm to 1.0 mm, but the decrease is not significant for longer
wavelengths. In the degraded resolution case, however, the correlation
coefficients notably increase at longer wavelengths due to the larger
corresponding PSFs. The slopes also exhibit a drastic decrease between 0.3 mm
and 1.0 mm and this trend is observed in both the original resolution and
degraded resolution cases. With further increases in wavelength, the decrease
in slopes becomes less significant. Additionally, Table 2 demonstrates that
for degraded resolution, the slopes are consistently higher compared to the
corresponding original resolution case, but follow the same pattern of a
sudden decrease at smaller wavelengths and a gradual decrease at longer
wavelengths.

Figures 6c&f and 12c&f demonstrate larger slopes at lower wavelengths,
indicated by the strong correspondence between the linear fits for the whole
box, QS, and EN regions. Conversely, in Figs. 8c&f and 14c&f, the lines appear
almost horizontal due to the lack of corresponding distributions at longer
wavelengths. For the QS, the slopes become more negative as the wavelength
increases. This trend is also observed in the correlation coefficients for QS
at the degraded resolution, as shown in Table 2. In Figs. 9 and 15, the
correlations decrease and become negative for wavelengths longer than 3 mm.
This is explained by the respective patterns of shock fronts in Figs. 5, 8,
11, and 14 (panels d and e). The QS region shows shock patterns that do not
correspond well to each other, resulting in negative slopes in panel f, as
observed in Table 2. Conversely, the EN region exhibits strong correspondence,
reflected in the contours with highly positive slopes and higher correlation
coefficients compared to the original resolution case. The calculated
uncertainties, using bootstrapping in the case of correlation coefficients and
by using the residues in the case of slopes, are 2 to 3 orders of magnitude
smaller than the values of correlation coefficients and slopes as seen in
Table 2, making these results reliable.

##  4 Discussion

###  4.1 Formation heights

In Table 1, the formation heights of the line feature from the solar
photosphere ($\tau_{500}=1$ surface) are listed. Bjørgen et al. (2018),
constrained the formation height range using SST/CHROMIS observations and
different models for Ca II absorption-emission peaks generated from Bifrost
atmospheres. The average formation heights for the Ca II K line are provided
with respect to the height where the average optical depth is unity, whereas
the Ca II H formation height range is provided with respect to the K line. Our
results also show very similar formation heights. The average formation height
for the line cores of K and H are found to be 1.73 Mm and 1.62 Mm,
respectively, which are within the range described in Bjørgen et al. (2018).
As seen in Fig. 4b,c, the K line core map shows fewer shock fronts and
internetwork features, which are formed at slightly lower heights, than the H
line. Further, the formation heights of the IRT line cores are lower than
those of the H&K lines, namely 0.95 Mm, 1.17 Mm, and 1.11 Mm for 849.8 nm,
854.2 nm, and 866.2 nm, respectively. The infrared triplet line cores are
formed at lower heights than the H&K lines (as seen in Fig. 15 in Bjørgen et
al., 2018), which is very evident from the integrated intensity maps in Fig.
4.

###  4.2 Comparison with semi-empirical models

In this study, the Ca II indices and the brightness temperatures $T_{b}$ at mm
wavelengths are compared for the semi-empirical model atmospheres VAL C and
FAL C. Figure 6 in Pandit et al. (2023) illustrates that the VAL C model
consistently exhibits lower brightness temperature compared to the FAL C
model. The publicly available 3D simulations of the solar atmosphere by
Carlsson et al. (2016) reveal that the Mg II h & k lines are narrow in
comparison to the observations made by IRIS (Carlsson et al., 2019). The Ca II
line profiles for the VAL C and FAL C models are seen to be broader than the
line profiles calculated with the Bifrost model, suggesting that the models
lack sufficient mass at higher temperatures in the upper chromosphere (Pandit
et al., 2023). Consequently, these semi-empirical models of the quiet Sun
exhibit slightly broader profiles, lower integrated intensities, and
consequently lower Ca II indices. However, the Ca II index values for the
semi-empirical models lie within the range of the EN and QS distributions on
the Ca II indices vs temperature planes, indicating an agreement between the
data from the 3D close-to-realistic model and semi-empirical models. On the
other hand, the estimated average s index for the Sun is comparable, near 0.17
as found in Egeland et al. (2017), which is close to the observational value
of s index for the sun-like star $\alpha$ Cen A. The calculated s index from
the 3D model and the 1D semiempirical models slightly overestimate the s
index.

###  4.3 Other Ca II indices

In this study only the s index and IRT index are considered for comparison
with the mm brightness temperatures and not the
$R_{HK},R^{\prime}_{HK},R_{IRT},R^{\prime}_{IRT}$ indicators generated from Ca
II lines. The latter indices are defined in Noyes et al. (1984) which are
derived from Middelkoop (1982), using colour-dependent conversion factor
$C_{cf}(B-V)$ which converts the s index to $R_{HK}$ index as
$R_{HK}=C_{cf}(B-V)\times s.$ The $R_{HK}$ index is fundamentally the Ca II H&
K flux corrected for photospheric radiation, to make it a better chromospheric
indicator. Further $R_{HK}$ index is normalised to bolometric luminosity to
generate $R^{\prime}_{HK}$ index. When comparing the calcium activity
indicators with mm continuum intensities generated from a 3D solar model
atmosphere, $B-V$ and the bolometric luminosity, both would be constants.
Hence, while looking at the relations, it does not change the correlation
coefficients and the slopes discussed in the paper. Furthermore, based on the
stellar sample studied, the exact polynomial used to calculate the colour-
dependent conversion factor $C_{cf}(B-V)$ is different (for example, Mohan et
al., 2022; Boro Saikia et al., 2018; Marsden et al., 2014; Noyes et al., 1984,
and references therein.).

###  4.4 Bandpasses for H&K lines integrated intensities

Sowmya et al. (2021) use rectangular filters instead of triangular filters to
calculate the integrated intensities for calculating the s index. When
comparing the correlations and slopes for the comparison of s index and mm
wavelength brightness temperatures, correlations are highest at the lowest
wavelengths for both the rectangular and the triangular filter integrated
intensities. The only noticeable difference is that the rectangular filter
gives slightly higher integrated intensity values in the integrated signal as
the parts in the line which are slightly away from the core are given more
weight in comparison to the triangular filter. Hence, using the rectangular
filter would contaminate the H & K integrated intensities with information
from the lower chromosphere than the height at which the line cores are
formed. For the purposes of our study, given that this would be beneficial for
stellar astrophysical studies in the future, we used the traditional
definition for s index with triangular filters, as defined by Vaughan et al.
(1978).

###  4.5 Implication for solar-like stars

A direct application to stellar observations is to estimate the stellar
activity level, in particular for Sun-like stars for instance used in the
studies by Liseau et al. (2016) and Mohan et al. (2021, 2022). The data point
for a sun-like star in the Ca II indices–mm brightness temperature plane as
derived from adequate observations then provides constraints on 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 Ca II indices–mm
brightness temperature plane in Figs. 5 (c and f) and 11 (c and f).

For the disk-integrated signal, the centre-to-limb variation for the Ca lines
and ALMA full disk data should be taken into account (see e.g., Alissandrakis
et al., 2022a, b; Nindos et al., 2018, and references therein). As the Ca II
line cores form in the chromosphere and the wings in the photosphere, the limb
effects would vary (brightening or darkening depending on the wavelength) when
going from the core to the wing. In a recent study based on SST observations,
Pietrow et al. (2023) concluded that chromospheric lines, like H$\alpha$, Ca
II H & K, and Ca II 8542, might exhibit a blueshift towards the limb due to
the chromospheric canopies, but the effect on the indices is non-trivial, as
the line core is more affected by the limb effects than the nearby continuum.
Further, the authors show that the line core widths would decrease when going
from the centre to the limb. This is in line with the previous works, where
Engvold (1966) stated that the centre-to-limb variation is a function of the
turbulent velocities in the plasma. Further systematic studies need to be
conducted to understand the limb effects on the correlations between the
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, listed in Table 2. In contrast, the calculated Ca II s index
for $\alpha$ Cen A taken from Henry et al. (1996), is 0.165, although varying
in time between 0.157 and 0.167, and thus in line with the values calculated
for the FAL C and VAL C models (see Figs. 5 c,f and 7 c,f). In comparison, the
corresponding average value derived from the synthetic observables for the
Bifrost simulation is 0.256 and thus substantially higher than the value for
FAL C and VAL C and the value for $\alpha$ Cen A. The IRT index for $\alpha$
Cen A is calculated to be 0.315 using the observational data described in
Sect. 2.3 and thus significantly higher than the average IRT index of 0.262
for the simulated data from the Bifrost model atmosphere, which causes the
respective $\alpha$ Cen A data point to be placed away from the linear fit for
the solar simulation shown in Figs. 12 (c and f) and 13 (c and f). The black
circles show the error bars in these figures.

The wider Ca II line profiles observed for $\alpha$ Cen A compared to the Sun
may be attributed to differences in atmospheric structure and activity levels,
but a thorough investigation is required to draw any firm conclusions.
Additionally, the solar observations are limited to a small, spatially
resolved region on the Sun, while the $\alpha$ Cen A data are integrated
across the entire (spatially unresolved) stellar disk. The used model has a
large filling factor of bright internetwork, which might make the average of
the distributions of limited use for comparison with full-disc integrated
observations of the Sun-like star. Nonetheless, the semi-empirical model shows
similar values of indices with the $\alpha$ Cen A data. In Bjørgen et al.
(2018) the formation of the Ca II lines from the same Bifrost model atmosphere
along with modified model atmospheres including more physical effects such as
non-equilibrium ionisation of He, and instantaneous LTE treatment, are
presented and the Ca II line profiles are narrower than the compared solar
observational line profiles. They conclude that there is a missing component
in all models that explains the wider observed profiles. Two potential
explanations for this discrepancy are the absence of motions at scales smaller
than the photon mean free path, commonly referred to as microturbulence and
insufficiently strong heating processes in the lower chromosphere.

###  4.6 Comparison with different activity levels

Figure 17: s index (panels a and d), IRT index (panels b and e), and the
brightness temperatures at the mm wavelengths shown in the legend (panels c
and f) as a function of filling factors. The filling factors are considered to
be 0.0001, 0.005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, and 1.0. The upper box
shows plots for the sunspot model by Severino et al. (1994) and the lower box,
for the plage model by Fontenla et al. (2009).

The correlation between the Ca activity indices and the mm brightness
temperature can be used for the study of solar-like stars as demonstrated in
Sect. 4.5. As active region simulations are not immediately available, this
study focused on simulated EN and QS regions. The impact of other types of
regions on (synthetic) observables and resulting activity indicators as they
would be derived from unresolved stellar observations is here explored by
combining the radiative transfer results from 1D models with different filling
factors, alternatively referred to as the fractional area (see, e.g., Alasta
et al., 2019; Balmaceda et al., 2009). For this purpose, the quiet Sun model
VAL C (Vernazza et al., 1981), the plage model H by Fontenla et al. (2009),
and the sunspot umbra model by Severino et al. (1994) as provided by
Loukitcheva (priv. comm.) are used. The umbra model was chosen according to
the detailed study by Loukitcheva & Reardon (2022) that compares various
models to recent (interferometric) ALMA sunspot observations (see also
Loukitcheva et al., 2014). It should be noted that these observations exhibit
brightness temperatures in sunspot umbrae in band 6 (1.3 mm) down to $\sim
5000$ K, which are significantly lower than in quiet Sun regions ($\sim 5900$
K, White et al., 2017), while temperatures of up to $\sim 8000$ K are found in
the penumbra. Even at the lower resolution of ALMA Total Power (TP) single-
dish observations, brightness temperatures of 5700 K are observed in the umbra
(Loukitcheva et al., 2017). In band 3 (3 mm), the brightness temperatures in
the penumbra also reach values of $\sim 8000$ K while the quiet Sun has on
average a brightness temperature of 7300 K (White et al., 2017). In the umbra,
however, the brightness temperature first drops slightly but then increases in
the innermost parts to about $\sim 8500$ K and, thus, beyond the penumbral
value. This finding has been made possible by the unprecedented spatial
resolution achieved with ALMA and is yet to be explained (Loukitcheva &
Reardon, 2022). In conclusion, in comparison to the average Sun, sunspot
umbrae appear as cooler features in band 6 but as slight temperature
enhancements in Band 3 (Loukitcheva et al., 2017).

The resulting brightness temperatures and activity indicators for combinations
of these components for varying the filling factors are shown in Fig. 17. As
the sunspot coverage increases, the calcium indices increase, and so do the
brightness temperatures for mm wavelengths greater than 1.5 mm. For
wavelengths shorter than 1.5 mm, the combined brightness temperatures decrease
because sunspot umbrae appear darker at those wavelengths. With increasing
plage filling factors, all the calcium indices and the brightness temperatures
increase as plage regions are brighter at all the heights from where the mm
continuum emerges at the considered wavelength range. Only values for the
plage filling factor of more than a few percent result in a notable increase
in the continuum brightness temperature for all wavelengths considered and
also for the IRT index and the s index. For the Sun, realistic values for the
plage filling factor vary from 0 at solar minimum to 0.04 - 0.07 at solar
maximum, whereas the filling factor of sunspots ranges from 0 at solar minimum
to 0.002 - 0.005 during solar maximum (Penza et al., 2022). Consequently, as
illustrated in Fig. 17, the resulting change in average continuum brightness
temperature for all considered wavelengths and the activity indices remains
moderate for the Sun. The data points for the combined 1D atmospheres with
different filling factors are plotted in the Ca II index-mm brightness
temperature planes in Figs. 6, 7,12, and 13 (panels c and f) as empty
triangles. The data points are near the linear fit and they are very close to
the respective VAL C data point (the green star) in all four cases.

More active stars can exhibit much larger fractional areas of starspots and
accordingly larger variations of the S-index, even to an extent where the
impact of starspots grows relative to the impact of plage regions (Sowmya et
al., 2023). A corresponding increase of the continuum brightness temperature
for higher sunspot coverage is seen for wavelengths longer than 1.5 mm.
Stellar millimetre observations can therefore provide complementary
constraints for the activity level of a star and its variation with a stellar
activity cycle.

##  5 Conclusions

We investigated synthetic Ca II indices and mm brightness temperatures from a
3D Bifrost model that resembles an enhanced network region surrounded by quiet
Sun. Reducing the spatial resolution of these synthetic observables to ALMA
resolution decreases the standard deviations of the synthetic observables,
leading to increased correlations between them. As the wavelength increases,
the beam size also increases, causing small structures to become blurred out
and further increasing the correlation between the two diagnostics. However,
this decrease in spatial resolution has no significant effect on the mean
slopes of the linear fits on the Ca II indices versus mm brightness
temperature distributions. This suggests that the instrumental resolution of
ALMA is sufficient to capture the similarities between the two diagnostics.
The Ca II indices and mm brightness temperatures are closely linked, forming
in similar layers in the atmosphere for a wavelength range of 0.3-1.0 mm and
providing constraints on the thermal stratification of a star. Previous
studies have shown that using multi-wavelength observations alongside mm
observations can provide better insights into the (average) thermal structure
of stellar atmospheres (Pandit et al., 2023; Mohan et al., 2021; Wedemeyer et
al., 2016, see also references therein).

Although the Ca II line cores are formed in the chromosphere, the s index, and
IRT index are not solely indicative of chromospheric activity. In our model,
the s index, for example, reveals mid-to-upper photospheric and lower
chromospheric features, such as reverse granulation and internetwork
structures. ALMA continuum brightness temperatures are on average too low in
these models according to previous studies (see, e.g., Martínez-Sykora et al.,
2020). The Ca-based activity indices calculated from this model exhibit a
notable similarity and exhibit the highest correlation with the ALMA Band 10
data. Therefore, these indicators of stellar activity are heavily influenced
by photospheric emission, even though the line cores are formed in the
chromosphere. Notably, these indices show clear differences based on activity
level, as the values for the QS and EN regions are significantly different.

The superior sensitivity of millimetre continuum observations towards
chromospheric structures gives a 3D perspective of the chromosphere in a
better manner. Hence, mm brightness temperatures can be used as an alternative
and complementary diagnostic to other chromospheric activity indicators with
different sources for the opacities.

###### 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”). Computational resources have been provided by Sigma2 – the
National Infrastructure for High-Performance Computing and Data Storage in
Norway. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2011.0.00020.SV. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC
and ASIAA (Taiwan), and KAS (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. (2008) and Lorenzo-Oliveira et
al. (2016) for their prompt correspondence regarding the Ca II IRT
observational data of $\alpha$ Cen A. The 1D semiempirical Severino et al.
(1994) sunspot model was kindly provided by Loukitcheva et al. (2014). 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 Mikolaj Szydlarski, Kjell Andresen, Thore Espedal
Moe, and Henrik Eklund for their support.

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