# Stellar surface information from the Ca II H&K lines
I. Intensity profiles of the solar activity components

M. Cretignier,1 A.G.M. Pietrow,2 and S. Aigrain1
1Department of Physics, University of Oxford, OX13RH Oxford, UK
2Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482
Potsdam, Germany
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The detection of Earth-like planets with the radial-velocity method is
currently limited by the presence of stellar activity signatures. On
rotational timescales, spots and plages (or faculae) are known to introduce
different RV signals, but their corrections require better activity proxies.
The best-known chromospheric activity proxies in the visible are the Ca II H &
K lines, but the physical quantities measured by their profiles need to be
clarified. We first investigate resolved images of the Sun in order to better
understand the spectrum of plages, spots, and the network using the Meudon
spectroheliogram. We show that distinct line profiles are produced by plages,
spots, and by the network component and we also derived the center-to-limb
variations of the three profiles. Some care is required to disentangle their
contributions due to their similarities. By combining disk-integrated spectra
from the ISS high-resolution spectrograph with SDO direct images of the Sun,
we managed to extract a high-resolution emission spectrum of the different
components, which tend to confirm the spectra extracted from the Meudon
spectroheliogram datacubes. Similar results were obtained with the HARPS-N
Sun-as-a-star spectra. We concluded using a three-component model that the
temporal variation of the popular $S_{\text{index}}$ contains, on average for
the $24^{\rm th}$ solar cycle: $70\pm 12\%$ of plage, $26\pm 12\%$ of network
and $4\pm 4\%$ of spots. This preliminary investigation suggests that a
detailed study of the Ca II H & K profiles may provide rich information about
the filling factor and distribution of different types of active regions.

###### keywords:

exoplanets – stars: activity – stars: chromospheres – methods: data analysis –
techniques: spectroscopic

††pubyear: 2023††pagerange: Stellar surface information from the Ca II H&K
lines  I. Intensity profiles of the solar activity components–D

##  1 Introduction

Stellar activity is today the main limitation for the upcoming ultra-stable
spectroscopic era, which targets a radial velocity (RV) precision below 50  cm
s-1. Among the stellar activity components, faculae/plages and spots are known
to dominate in the RV budget of Sun-like stars (Meunier et al., 2022).

Sunspots are the most readily visible, and perhaps best known, manifestations
of inhomogeneities in the solar magnetic field concentration. A recent
detailed review on sunspots were given by Solanki (2003) and Borrero &
Ichimoto (2011). They tend to emerge from a region with enhanced magnetic
activity within the span of several days and have been observed to persist up
to two months (Schrijver & Zwaan, 2000). They are characterized by a dark core
known as the umbra, surrounded by a brighter, radially extending structure
called the penumbra. A spot that lacks the penumbra is called a pore (Verma &
Denker, 2014) and has a shorter lifetime (Keppens, 2000).

The intensity in the visible of umbra and penumbra as compared to the quiet
Sun continuum tends to be 40% and 80% respectively (Lites et al., 1993; Beck &
Chapman, 1993), with a magnetic field orientation vertical in the umbra and
mostly horizontal in the penumbra (Lites et al., 1993; Skumanich et al.,
1994). The photospheric magnetic field in the umbra varies gradually between
1800 and 3700 G, and then lowers to a much lower range of 700 to 1000 G for
the penumbra (Brants & Zwaan, 1982; Beck & Chapman, 1993; Solanki, 2003;
Borrero & Ichimoto, 2011), far exceeding the average quiet Sun magnetic field
of 46 G (Trelles Arjona et al., 2021).

The definition of "plage" has always been, and to some degree still is, a
loose observational term. In solar physics it tends to be used to describe
both the photospheric and chromospheric parts of a region with a quiet-Sun-
like photosphere in the continuum that is covered by a hot and bright
chromospheric canopy that connects regions of opposite polarity. In white
light, plage manifests itself on the disk as abnormal granulation and becomes
bright close to the limb, where it is also known as ’faculae’. Plage can also
be seen as bright inter-granular structures on the disk when using a narrow
band filter. These structures have historically also been called faculae,
which likely contributed to the muddiness of the definition (Stix, 2002;
Buehler et al., 2015; Carlsson et al., 2019; Pietrow et al., 2020).

Plages accompany sunspots and pores, sometimes forming as a precursor to them,
and also tend to outlive the spots, with lifetimes up to two or three times
longer (Foukal, 2013). They are magnetic phenomena with photospheric fields
around 1500 G (Buehler et al., 2015), stronger than the quiet Sun, but weaker
than those of sunspots to the point where convection is hardly affected.
Together with enhanced networks, plage accounts for about half of the total
magnetic flux on the Sun, and are responsible for the majority of the
variations in the UV flux, as well as the total solar irradiance (Morosin et
al., 2020; Pietrow, 2022; da Silva Santos et al., 2022; Chowdhury et al.,
2022). Faculae are often described as the photospheric counterpart of plage
(e.g. Dumusque et al., 2014), however they can also form in the quiet Sun as
part of the enhanced network, or at the boundary of super granules (Muller,
1983; Buehler et al., 2019), and thus the word should be used with caution
(e.g. all plages correspond to faculae, but not all faculae correspond to
plage). In this work, we define plages as large (>8 millionth of solar
hemisphere MSH), bright, continuous regions in chromospheric diagnostics, and
faculae as the more concentrated bright areas underneath, excluding the darker
quiet Sun-like pattern in between.

Network can be found in the quiet Sun, and typically presents itself as
elongated chains of bright points nested within the intergranular lanes.
Network tends to be much more spread out than plage and not connected to
active regions (Buehler et al., 2019). While plage and network share many
characteristics, such as being identified as bright points in the photosphere
and expanding into a bright canopy in the chromosphere, several differences
have been found between the two that can be used to distinguish them. The
primary difference is that network tends to consist of narrower flux tubes
than those found in the plage, resulting on average in temperatures that are
about 200 K higher, a magnetic field that is about 150 G stronger, a 5%
increase in grain contrast, and a more vertically oriented field (Solanki,
1986; Solanki & Brigljevic, 1992; Buehler et al., 2019). Despite the higher
temperature, magnetic field strength and inclination of the individual flux
tubes, the relatively lower fill-factor of said tubes results in a less bright
chromospheric canopy compared to plage (e.g. Worden et al., 1998; Worden et
al., 1999). In this work we define network as bright regions in the
chromosphere that are above the average brightness of the quiet chromosphere,
and below the brightness cutoff for plage.

###  1.1 Standard activity proxies

An activity proxy is an observable that aims to measure a given property of an
active region (AR) or that aims to be at least proportional to it. Usually,
most activity proxies were dedicated to be a tracer of the total filling
factor $f_{\text{tot}}(t)$ of active regions.

In photometry, both spots and faculae have been widely investigated with
results sometimes in contradiction such as for faculae (Chapman & Klabunde,
1982; Unruh et al., 1999; Foukal et al., 1990; Berger et al., 2007; Toriumi et
al., 2020). For starspots, results are more established and their modulation
has been largely investigated thanks to large photometric surveys (Messina &
Guinan, 2002; Mirtorabi et al., 2003; Lanza et al., 2009; Bonomo & Lanza,
2012; Nielsen et al., 2013; Roettenbacher et al., 2013; Lanzafame et al.,
2018; Breton et al., 2021). However, as faculae are dimmer in photometry, yet
dominate in RVs due to their larger filling factor (Meunier et al., 2015),
stable photometry is not sufficient to forecast suitable RV standard stars
(Bastien et al., 2014). Furthermore, bright and dark regions can cancel out in
photometry, whereas their effects add up in RVs.

High resolution spectra have also been used to investigate active regions.
Indeed, reaching an RV precision better than 1  m s-1usually requires high
resolution spectra ($\mathcal{R}>100\,000$), from fiber-feed spectrographs.
Most of them working in the visible due to the richness of atomic transitions
in the blue part of the spectrum for G to K dwarfs stars and the absence of
strong tellurics bands. Because of the conservation of the etendue (Schroeder,
2000), small fibers are needed to avoid the development of large optical
elements such as the grating in high-resolution instruments, which would be
too expensive. Consequently, the stellar point-spread function (PSF) on sky is
larger than the fiber sky-projected diameter and part of the stellar light is
not captured (Pepe et al., 2021), leading to absolute flux variations as large
as 50% induced by seeing variations, guiding errors or change in weather
conditions. As a consequence, high-resolution spectra are usually not
photometrically calibrated, which prevents the photometric signatures of ARs
to be measured. Even if projects of diffraction-limited spectrographs111aiming
to photometric accurate spectra using adaptive optics are being investigated
(Mello et al., 2018; Jovanovic et al., 2016; Bechter et al., 2018, 2019), none
of them are currently in use.

From spectra, the extraction of information on faculae/plages and spots is
more difficult. Due to the color dependency of spots (Desort et al., 2007;
Carleo et al., 2020; Kossakowski et al., 2022; Miyakawa et al., 2021), some
authors tend to use chromatic indices such as differential RVs (Zechmeister et
al., 2020), whereas other have tried to measure the magnetic field via Zeeman
broadening (Robinson, 1980; Berdyugina et al., 2003; Anderson et al., 2010;
Reiners et al., 2013; Lienhard et al., 2023) or molecular bands (Afram &
Berdyugina, 2015; O’Neal & Neff, 1997; O’Neal et al., 1998, 2001, 2004).

In general, the whole spectrum tends to be affected by stellar activity. For
example, other stellar lines distortions such as filling of the line core has
been reported many times (Stenflo & Lindegren, 1977; Basri et al., 1989;
Brandt & Solanki, 1990; Malanushenko et al., 2004; de la Cruz Rodríguez et
al., 2013; Druett et al., 2017; Thompson et al., 2017; Vissers et al., 2021;
Cretignier et al., 2021; Druett et al., 2022; Pietrow et al., 2022). More
generally, using the varying sensitivity of different stellar lines has been
proposed as a way to reduce degeneracies in models. As an example, combining
low and high temperature sensitive lines can recover the effective temperature
of stars (Gray, 1988; Gray & Johanson, 1991; Caccin et al., 2002; Teixeira et
al., 2016; da Silva Santos et al., 2018) and this idea has been naturally
extrapolated for active regions as well (Chapman et al., 2001; Catalano et
al., 2002; Penza et al., 2004; Frasca et al., 2005; O’Neal, 2006). Using lines
formed at different depth in the stellar atmosphere also demonstrated
interesting insights on the effect of stellar activity on the convective
motions (Cretignier et al., 2020a; Al Moulla et al., 2023).

A simple element that would help to disentangle AR contributions would be to
know the spectrum of plages/faculae and spots across the solar disk
$I_{\text{AR}}(\lambda,\mu)$ over a large bandpass and at high-resolution, the
variable $\mu=\cos(\theta)$ representing the heliocentric angle and $\theta$
the angle between the observer and the normal to the surface vector ($\mu=1$
at stellar center and $\mu=0$ at the stellar limb).

While such studies have been conducted for the quiet Sun in recent years
(Ramelli et al., 2017, 2019; Pietrow et al., 2023c; Ellwarth et al., 2023),
there is still a lack of information for ARs center-to-limb variation (CLV).
Such products do not exist or only for peculiar lines (Avrett et al., 2015;
Rajhans et al., 2023) and projects are in preparation, such as a solar
telescope on ESPRESSO222https://poet.iastro.pt. A sunspot atlas spectrum can
be found in Wallace et al. (2005) and is already used to simulate stellar
activity (Dumusque et al., 2014), but such a spectrum was only taken at a
single $\mu$ angle and similar observations do not exist for facula.

Even if the full spectrum is sometimes studied, researchers tend rather to
investigate the chromospheric signatures of the ARs. The hot plage canopy is
visible in most chromospheric diagnostics such as the Ca ii (Shapiro et al.,
2014; Milbourne et al., 2019; Pietrow et al., 2020; Morosin et al., 2020),
Magnesium (Sasso et al., 2017; Li et al., 2023), and H $\alpha$ (Socas-Navarro
& Uitenbroek, 2004; Frasca et al., 2008; Flores et al., 2016, 2018). Note
that, in the context of RV, the question of good activity proxies is still an
open question for which the answer may depends on the stellar spectral type
(Meunier & Delfosse, 2009; Robertson et al., 2016; Maldonado et al., 2019;
Schöfer et al., 2019; Lafarga et al., 2021) and recent studies tend to show
that the best activity proxy for solar-type star is the unsigned magnetic
field (Milbourne et al., 2019; Haywood et al., 2022).

Figure 1: Illustration of the Ca II K line profile obtained by the ISS
spectrograph ($\mathcal{R}\sim 300^{\prime}000$). The peculiar K1 minima, K2
maxima and K3 minimum are indicated. The Mount Wilson triangular bandpass,
from which the Mount-Wilson $S_{\text{index}}$ ($S_{\text{MW}})$ is derived,
is also indicated at the bottom superposed on the bandpass used in this work
to derive $S_{\text{index}}$ .

###  1.2 The physics of the Ca II H&K lines

The most widely used chromospheric lines are the Ca II H & K lines. They are
amongst the opaquest lines in the visual spectrum and, like the Mg II h & k
lines, they typically have wide damping wings and a central reversal in their
cores. The wings of the lines sample the higher photosphere up to the
temperature minimum, and can be modelled in LTE (Rutten et al., 2004). The
line cores have a characteristic reversed shape (see Fig.1) that is caused by
the rise in temperature in the chromosphere above the temperature minimum.
Afterwards the line decouples from the Planck function and an absorbing core
forms (Vernazza et al., 1981; Leenaarts et al., 2013; Bjørgen et al., 2018a).
The H1/K1 minima correlate with the temperature at the higher photosphere,
also known as the temperature minimum. However, it has been shown that this
value is underestimated due to the line being already partially decoupled from
the Planck function (Shine et al., 1975; Bjørgen et al., 2018a). The H2/K2
peaks form in the lower chromosphere with the R peaks forming lower than the V
peaks. Their brightness temperature correlates with the gas temperature in the
lower chromosphere, but again should not be taken as an absolute value. The
peaks, along with the H1/K1 minima, move apart when observed further away from
disk center, and tend to disappear altogether when averaged over large areas
(Wilson & Vainu Bappu, 1957; White & Livingston, 1978; Druzhinin et al., 1987;
Pietrow et al., 2023c). The H3/K3 minimum forms in the higher chromosphere and
can be used to measure the vertical velocity at this height.

The strong variation in line shape of Ca II H & K as a result of chromospheric
activity (e.g. de la Cruz Rodríguez et al., 2013; Varun et al., 2018) combined
with broad sensitivity across various heights in the solar atmosphere, makes
these lines an excellent activity probe. Other peculiar distortions of the
lines were reported such as the broadening of the line as measured by distance
of the K2 maxima (Wilson & Vainu Bappu, 1957) or line asymmetry (White &
Livingston, 1978). Some authors have tried constructing different metrics from
the lines, and attempting to interpret them based on physical solar quantities
(Pasquini, 1992; Livingston et al., 2007; Bertello et al., 2012; Dineva et
al., 2022), but no clear relations have been found.

Usually, the study of the Ca II H & K lines is mainly performed via a bandpass
integration in the core of the lines, that people called $S_{\text{index}}$
metrics. The most widely used convention is the Mount Wilson
$S_{\text{index}}$ ($S_{MW}$) that is characterized by a triangular band pass
integration (Duncan et al., 1991).

In this work, we performed a wide investigation on the solar Ca II H & K
lines, to better understand how ARs modify their line profiles. We used the
Sun as a starting point, since we can test it against resolved observations.
We then used the knowledge obtained on the Sun to better characterize the
stellar activity on $\alpha$ Cen B in a subsequent paper II (Cretignier et
al., prep).

The paper analyses the data coming from four solar instruments: the Meudon
spectroheliograph, the SDO AIA1700 filtergrams, the disk-integrated Ca II H&K
high resolution spectra from ISS and HARPS-N solar telescope. The structure of
the paper is divided in two main parts. The first part (Sect. 2) focuses on
the analysis of low resolution spectra obtained from spatially resolved
observations of the meudon spectroheliographs. The second part of the paper
(Sect. 3) focuses on the combination of disk-integrated high resolution
spectra with spatially resolved observation. Finally, we conclude in Sect. 4.

##  2 Spatially resolved spectra from the Meudon spectroheliograph

The Meudon specroheliograph is a low-resolution ($R\sim 40\,000$) slit
spectrograph located at the Meudon observatory in Paris that takes daily
observations of the Sun with three different filters (Ca II K, Ca II H, H
$\alpha$ ). The instrument was upgraded in 2018 (Malherbe & Dalmasse, 2019)
and can be used to study a vast diversity of solar active events (Malherbe et
al., 2023). Contrary to several other Sun-as-a-star instruments that work with
disk-integrated spectra or restricted field of view, the full solar surface is
scanned by the slit spectrograph and a spectrum is extracted producing
therefore a datacube $I(\lambda,x,y)$. Such datasets are crucial to better
constrain stellar activity since they directly lead to the intensity profile
$I(\lambda,\mu)$. We recall333see e.g Thompson (2006) for a paper about change
of units in solar images that if $x$ and $y$ are the sky coordinates with
their origin at the solar disk centre and given in fractions of the total
radius, the center-to-limb angle $\mu$ is given by:
$\mu=\sqrt{1-x^{2}-y^{2}}$.

The section is divided as follows: the processing of the Meudon
spectroheliogram datacubes is described in Sect. 2.1. The analysis of peculiar
events is discussed in Sect. 2.2. We then extracted the emission profile per
unit area of plages, spots and of the network from Meudon datacubes in Sect.
2.3, before to use numerical simulations to recreate the disk-integrated
spectra of plages and spots in Sect. 2.4.

###  2.1 Datacubes processing

The images have sizes of 2048 $\times$ 2048 pixels, whereas the wavelength
axis is sampled by 98 and 94 data points for the Ca II K and Ca II H
respectively. The wavelength step is 0.093 Å, equivalent to a spectral
resolution of $R\sim 40\,000$. The stellar radius is about 880 pixels on
average444Due to the eccentricity of the Earth’s orbit. leading to an angular
resolution of 1.1", providing slightly more than $\pi\times 880^{2}\simeq$ 2
million spectra per observation. One pixel covering roughly 0.41 MSH.

Table 1: Table of the dataset observations downloaded from Meudon spectroheliograph. Six events of active regions crossing the solar disk were selected. A spot dominated event in September 2017, a plage event in May 2018 and 4 events in 2022, the most active solar activity period. A quiet period in 2020 was also selected for sanity check of segmentation algorithms. The first day ($T_{1}$) and the last day ($T_{2}$) of each period as well as the number of datacubes ($\\#D_{3}$) extracted for each line is displayed. The total is displayed in the last row. Each datacube contains roughly 2 000k of spectra. The number of spectra extracted after the segmentation of the datacubes are also indicated. Both the number of spectra for spots ($\\#_{s}$) and plages ($\\#_{p}$) are given in thousands of units (k). $T_{1}$ [date] |  $T_{2}$ [date] | #$D_{3}$(K/H) |  $\\#_{s}$(K/H) |  $\\#_{p}$(K/H)
---|---|---|---|---
01-09-2017 | 07-09-2017 | 6/6 | 37/37 | 122/96
24-05-2018 | 31-05-2018 | 9/10 | 0/0 | 63/61
21-02-2020 | 29-02-2020 | 6/6 | 0/0 | 10/21
18-04-2022 | 27-04-2022 | 10/10 | 48/51 | 709/665
15-05-2022 | 24-05-2022 | 12/15 | 30/48 | 1176/1295
18-06-2022 | 24-06-2022 | 6/6 | 12/15 | 289/259
06-07-2022 | 18-07-2022 | 24/24 | 93/116 | 1255/820
- | - | 73/77 | 221/267 | 3624/3216

All the observations are publicly available and can be obtained via a web
page555https://bass2000.obspm.fr/home.php query. Due to the size of a single
datacube ($\sim$ 2048 $\times$ 2048 $\times$ 100), only 6 events of ARs
crossing the stellar disk were analysed. In total, 73 and 77 epochs were hence
obtained for the Ca II K and Ca II H lines respectively. Among the 6 events,
we selected an event dominated by a spot in September 2017, a single plage
event in May 2018 and four periods with large number of ARs in 2022. The
datasets in 2017 and 2018 were taken at the minimum of the solar cycle, but
represent ARs that are typical in size. The 2022 probes a mid-activity solar
level. We also extracted datacubes for a period when no clear AR was visible
in 2020. This dataset will be used as a quality control for the segmentation
algorithm. A summary of the observations can be found in Table.1. Even if the
number of epochs analysed may seem low, it already represents a total of 51
million of spectra to analyse.

####  2.1.1 Quiet Sun CLV correction

All datacubes were first normalised by the center-to-limb variation of the
quiet Sun. Such normalisation, also known as the third solar spectrum (SS3,
Stenflo, 2015; Ramelli et al., 2019), are common way to analyse the residual
intensities (Harvey & White, 1999). The CLV law was fit independently for each
wavelength element, producing as well CLV datacubes. The solar surface was
first binned into a 15 $\times$ 15 grid where the median intensity inside each
cell was computed in order to remove the signatures of active regions that
will never dominate in area compared to the quiet Sun.

The CLV was approximated by a 5th order polynomial following Neckel & Labs
(1984) and Neckel & Labs (1994). In order to obtain the coefficients of the
CLV relation, we simultaneously fit a basis of 2D polynomial function of
degree 10 in $x$ and $y$ directions of the images:

$\begin{array}[]{rccccl}I(\lambda,x,y)&=&P_{10}(\lambda,x,y)&+&\text{CLV}_{5}(\lambda,x,y)\\\
&=&\displaystyle\sum_{i,j=0}^{i+j\leq 10}C_{i,j}(\lambda)\cdot x^{i}\cdot
y^{j}&+&\displaystyle\sum_{k=0}^{5}c_{k}(\lambda)\cdot\mu^{k}\end{array}$ (1)

The first term was necessary to correct for instrumental systematics666see
also Shen et al. (2018) for similar systematics and correction using Zernike
polynomials, while the the second is the CLV relation of the quiet Sun. Note
that since CLV is described by radial polynomials, we removed the components
of the 2D polynomial basis that were already present in the CLV
relation777which is the case when $i=0$ and $j$ is a even number, or $j=0$ and
$i$ is even..

Even if dividing out by CLV profiles of the quiet Sun allows to precisely
study the residuals, a first issue was related to the absolute flux level of
the CLV profiles. These are caused by variable observational conditions that
affect the absolute flux level, and the difficulty of setting the continuum
level of the lines due to the restricted bandpass of the instrument. An
inaccurate flux level will prevent the comparison of the line profiles
obtained by different instruments at later stage due to different flux units.
To fix it, we used as a reference profile the HARPS-N master quiet spectrum
(see Sect. 3.3), since HARPS-N has a much wider bandpass that allows to fit a
more reliable and accurate continuum.

The HARPS-N master quiet spectrum was degraded to the Meudon spectral
resolution in order to produce a reference spectrum $I_{\text{ref}}(\lambda)$,
while each CLV law was disk-integrated to produce a master quiet disk-averaged
stellar line $I_{\Sigma}(\lambda)$. In order to determine the proper scaling,
we initially tried to fit a simple linear or quadratic relation with a free
offset between the two profiles, but doing so often led to non-physical units,
such as negative flux values close to the limb. In order to avoid this issue,
we removed the free offset in the model and imposed a boundary condition on
the fit, namely that the minimum of the CLV datacubes888which is expected to
be at the limb of the core of the line:
$I_{\text{min}}\simeq\text{CLV}_{5}(\lambda_{c},\mu=0)$ $I_{\text{min}}$ goes
to 0. In practice, such conditioned change of units is obtained by fitting the
quadratic relation between the master quiet spectrum of both instruments:
$I_{\text{ref}}(\lambda)=a\cdot(I_{\Sigma}(\lambda)-I_{\text{min}})+b\cdot(I_{\Sigma}(\lambda)-I_{\text{min}})^{2}$.
The parameters $a$ and $b$, and the constant $I_{\text{min}}$ can then be used
to scale the Meudon flux units toward HARPS-N flux units.

####  2.1.2 Correction of residual systematics

After such processing, several artefacts in the form of diagonal lines can be
observed on the images (already visible on the raw datacubes as displayed in
the top row of Fig.3). Those artefacts are likely created by scattered light
in the spectrograph or changes in weather conditions during the scanning by
the slit spectrograph (a scan being typically 1 min long). To complicate
matters, the angles of those rays change from one day to the next. However,
the angle remains identical for all the wavelengths. For that reason, the mean
along the wavelength axis was taken in order to produce a high signal-to-noise
ratio master image of the Sun $I_{\text{master}}(x,y)$ used to search for the
angle of the rays.

Figure 2: Representation of a subpart of the Meudon spectroheliogram datacubes
(5 of the 96 wavelengths) for an observation taken on 4 September 2017. Top:
Raw datacubes. The images of the Sun at five different wavelength around the
line center $\lambda_{c}$ of Ca II K are plotted (see Fig.3). From left to
right: $\lambda_{c}-0.6$, $\lambda_{c}-0.3$, $\lambda_{c}$, $\lambda_{c}+0.3$
and $\lambda_{c}+0.6$ Å. A zoom on the central part of the Sun, containing
most of the active regions, is displayed in Fig.3. Bottom: Datacubes residual
intensities after fitting out the quiet CLV and correcting from instrumental
systematics.

Figure 3: Extracted spectra from the Meudon spectroheliogram for 4 September
2017. Top panel: Segmentation maps for spots (blue), plages (red) and network
(purple) is represented for the central part of the Sun (see dashed square in
Fig.3). The wavelengths displayed are the same as in Fig.3. Bottom panel:
Average spectrum $I_{\text{AR}}(\lambda,\mu\simeq 1)$ obtained from the
segmented regions. The quiet spectra from Meudon ($R\sim 40\,000$), HARPS-N
($R\sim 120\,000$) and ISS ($R\sim 300\,000$) are displayed as comparison
respectively in black, dark grey and light grey. HARPS-N and ISS spectra are
shown with arbitrary offsets.

The algorithm used to detect the rays is a "rotating-stacking algorithm"
similar to the one developed in Cretignier et al. (2021) to detect the ghosts
on the HARPS detector. The algorithm consists in computing the median of the
pixel intensity for a given angle direction $\theta$, before computing the
standard deviation of the distribution. The variance of the distribution will
be the smallest once pixels are summed along the best angle, while a flat
distribution is expected when summing along a random angle. The optimal angle
$\theta_{\text{max}}$ is found with a grid search between -$90^{\circ}$ and
$90^{\circ}$ with steps of $0.25^{\circ}$. Once the optimal angle
$\theta_{\text{max}}$ is found, the median along the angle is used to correct
the image. The correction is performed for each wavelength independently. We
will refer hereafter to these images as the flat corrected images.

An example of the datacubes, after that the CLV of the quiet Sun has been fit
out and straight rays corrected, is shown in the bottom row of Fig.3. Some
minor residuals remains, but they are not a matter of concern, as they will be
diluted when spectra from a given $\mu$ angle will be stacked at later stage
(see Sect. 2.2), taking advantage of the significant number of epochs
processed and the number of ARs.

####  2.1.3 Identification of ARs

Having corrected for the CLV of the quiet Sun $I_{\text{quiet}}(\lambda,\mu)$,
we segmented the flat corrected images, before extracting the relative
intensity profiles of the AR compared to the quiet chromosphere, namely
$I_{\text{AR}}(\lambda,\mu)/I_{\text{quiet}}(\lambda,\mu)$. The results of the
segmentation are displayed in Fig.3. The cutoff values were set by eye as good
trade-off values. The thresholds were defined as follows: we considered a spot
as any pixel with a flux intensity on the first wavelength element (located on
the wing of the Ca II H&K lines at $\lambda=3964.097$Å and
$\lambda=3929.292$Å) lower than 75% the quiet photosphere, which usually
selects the umbrae and penumbrae of the spots. Note that the spatial
resolution of the Meudon spectroheliograph does not allow an easy
identification of both components and disentangling umbrae from penumbrae on
stellar spectra is unlikely at the moment. On the other hand, we considered as
a plage any pixel brighter than 225% the solar quiet chromosphere in the core
of the Ca II lines and brighter than 95% the quiet photosphere using the first
wavelength element. The last condition was mandatory since, as showed in Sect.
2.2, spots also present a clear emission in the core of the line which is also
visible in Fig.3 for the most southern spot in the image. The 30% of pixels
closest to the poles were excluded since ARs are not expected above latitudes
$l\simeq 45^{\circ}$ according to the butterfly diagram (Pastor Yabar et al.,
2015). No active region at limb angle smaller than $\mu<0.30$ were attempted
to be detected due to the large noise close to the limb enhanced by the border
instability in the CLV fit.

We tested the segmentation on the 2020 quiet dataset to assess the level of
contamination expected. No region was flagged as spotted; however, we observe
that some pixels were classified as plages likely due to stochastic noise in
the core of the Calcium lines or network pixels. The average contamination per
frame is however lower than 0.001% of the stellar surface and are therefore
negligible validating our choice of the threshold for both types of ARs.

We also used the 2020 dataset to extract the network intensity spectra, since
no other activity components were expected during this period of time. The
network was detected by a brighter pixel condition using the average of the 5
wavelength elements the closest to the line center. The lower boundary
segmentation criterion was however slightly more difficult to set, compared to
plage for instance, since this component is closer to the noise precision of
the datacubes. Indeed, this threshold helps to disentangle real signals from
noise, which can not be set with a fixed threshold since the value will depend
on the S/N of the observations.

We used the property that no relative intensity smaller than 1 is expected
from any activity components during this period of time and such values are
only produced by the stochastic noise in the images. We therefore compute the
half $\sigma$-width of the distribution computed as the median $(I_{50})$
minus the $16^{\rm th}$ percentile ($I_{16}$) and used the same value
projected on the positive side multiplied by 3 in order to use an equivalence
to a $3\sigma$ definition. Any pixel brighter than
$I/I_{\text{quiet}}>I_{\text{min}}=I_{50}+3\cdot(I_{50}-I_{16})$ and not
already classified as a plage was categorised as network.

Once images were segmented, the spectra from the 1) quiet Sun 2) plages 3)
spots and 4) the network were extracted. The segmentation resulted in a large
dataset of more than $\sim$250 000 spectra for spots, $\sim$350 000 spectra
for the network and $\sim$3 000 000 spectra for plages. Ultimately, all the
spectra were corrected for the solar rotation using a solid body rotation
model $v_{\text{eq}}\cdot\sin i$ with an equatorial velocity $v_{\text{eq}}=2$
km s-1. The inclination of the Sun observed from the Earth was obtained from
the JPL Horizons Ephemerids (Giorgini et al., 1996). Deviations from the solid
body model (such as differential rotation) were neglected given the spectral
resolution of the instrument.

###  2.2 Analysis of the intensity profiles $I_{\text{AR}}(\lambda,\mu)$ for
peculiar events

We first investigated the spectra coming from Meudon spectroheliograph in
order to see if substantial variations are observed as a function of $\mu$
angle along half a rotational phase ($\sim$ 13 days) of the Sun. We analysed
two isolated AR crossing events: a spot-dominated and a plage-dominated event
that took place in 2017 and 2018 respectively. We show the results obtained
with the Ca II H line as there were more observations available of these
events when compared the Ca II K line, however the results of the latter are
very similar. The 2020 dataset was used to study the spectra of the network.

####  2.2.1 The 2020 quiet Sun

We used the 2020 solar dataset to extract the signal from the network, because
no other ARs were visible at that time. Since it is difficult to follow a
single network region during a rotational phase, we displayed in Fig.4 the
superposition of all the frames used in the 2020 dataset. The filling factor
for the network was in average around 2.7% which is slightly to large compared
to the SDO value around 1.0% (see Sect. 3.1) and may highlight that the
segmentation also picked up spurious pixels. The implications of this will be
discussed later.

We chose to highlight the CLV by binning the network in different $\mu$ angle
rings, which shows how the $\mu$ angle changes across the surface. This figure
gives a better appreciation of how close to the solar limb are $\mu<0.30$
spectra and hence why they are so difficult to extract.

The network has a weak core emission and is roughly 20% brighter than the
quiet solar chromosphere. We can also observe that the profiles becomes
slightly broader closer to the limb, but such effect will be confirmed later
(see Sect.2.3). Interestingly enough, we also observe a deep at 3970Å, which
is the location of the $H_{\varepsilon}$ Balmer line.

Figure 4: Extraction of the profiles on the Ca II H line for the quiet phase
in 2020. The color highlights different $\mu$ bin angle and values are
reported in the legend. Note that all the observations of the 2020 dataset
have been agglomerated together. The profile of the network does not
significantly change from center-to-limb and is described by a core 20%
brighter than the quiet solar chromosphere. The small deep on the right side
of the core emission is the Balmer $H_{\epsilon}$ line.

####  2.2.2 The 2018 plage event

We analysed the series of datacubes taken during the May 2018 plage-dominated
event, which first appeared on the eastern limb on 27 May 2018. This region
was given the NOAA designation of 12712 and had a filling factor of $f\simeq
1\%$ at disk center. In Fig.5, we show the extracted intensity profiles of the
plage at 10 different locations on the solar disk. Instead of displaying the
average location of the AR in the ($x$, $y$) sky-coordinates, we used the
($\nu$, $\mu$) coordinates system, with $\nu$ being the local radial velocity
due to the stellar rotation ranging from $-2$ to $2$ km s-1from left to right,
and $\mu$ the cosine of the heliocentric angle $\theta$. To avoid
contamination from other ARs, only spectra around the expected longitude and
latitude of the AR were extracted.

We can observe that the relative intensity of the plage spectra with the quiet
chromosphere does not change significantly from the limb to the solar center.
The classical double peak signature of the lines is not visible, except for
$\mu<0.3$. The reason for this is the relatively low spectral resolution of
the instrument. Because the separation of the double H2 peaks increases
towards the limb (Smith, 1960; Bjørgen et al., 2018b; Pietrow2023a), this
effect can be detected only for the smallest $\mu$ values.

Figure 5: Similar to Fig.4 for the $30$ May 2018 plage-dominated crossing
event. The solar rotation vector is oriented to the top. Only spectra at the
AR expected latitude and longitude were extracted. Bottom: Segmentation map
for the plages, each color represents an observation taken on a different day.
Top: Mean relative intensity profiles extracted from the segmentation maps
presented below, using ther same color codes as the bottom panel. An arbitrary
offset value was applied for each curve, but the quiet chromospheric level is
highlighted for comparison (dotted lines). The average location of the active
region ($v,\mu$) is written in the legend. The velocity $v$ is in  km s-1.

The present example shows that, at least for that specific AR, no strong
variation is observed from one exposure to the next day and the profile
remains coherent across the rotational phase. Small variations are still
observed, but those are small compared to the main line profile. The intensity
profile of a plage is slightly brighter in the wings, whereas the core is made
of a single core emission twice as bright as the quiet chromosphere. This
"universality" of the intensity Ca II H & K plage profile will be confirmed
later when a larger dataset of ARs will be analysed (see Sect. 2.3).

Figure 6: Same as Fig.5 for the spot-dominated crossing event, on $4$
September 2017. Only the stable sunspots from the northern hemisphere were
extracted. Segmentation maps of plage were also indicated (grey area). The
profiles are characterized, as expected, by dark wings, but also a clear core
emission. Depending on the configuration of plage and spots (see main text),
an enhancement of the core emission can be seen, as displayed for the south
hemisphere in Fig.7. Inner panels represent the SDO AI131 filtergrams probing
the corona.

####  2.2.3 The 2017 spot event

After investigating the behaviour of a plage and of the network, we
investigated the spot event that started on 30 August 2017 when the AR
appeared and was assigned the NOAA designation of 12674. This region comprised
out of two large spots that were interconnected by a plage region.
Additionally, a smaller spot (NOAA 12673) was visible on the southern
hemisphere during this crossing. The spots on the northern hemisphere remained
largely unchanged during their crossing, while the southern spots evolved
rapidly once they crossed the meridian.

Figure 7: Same as Fig.6 but extracting only the emerging sunspots from the
southern hemisphere. A rapid evolution of the sunspots (and surrounding plage)
is observable as soon as the solar meridian is crossed. The spectra from the
pores before the sudden emergence are similar to those in Fig.6. However, once
the meridian crossed, a huge flux excess becomes visible in the core. Its
origin seems related to a peculiar magnetic field configuration as visible
from the AI131 filtergrams.

The emission spectra as a function of the solar rotational phase are drawn in
Fig.6 and Fig.7 for the northern and southern spot regions respectively. The
stable sunspots regions from the North show a time-independent profile with
darker photospheric wings and a core emission, similarly to the results
obtained for plage, although less bright. The main feature to notice is the
darker photospheric profile, which is expected because of the segmentation
criterion. It can be seen by eye that the recovered spot spectra are narrower
than those recovered for the plage, a behavior which we will discuss further
in Sect. 2.3.

Figure 8: Stacked profiles of the quiet spectrum
$I_{\text{quiet}}(\lambda,\mu)$ (left panel) and relative active spectra
$I_{\text{AR}}(\lambda,\mu)/I_{\text{quiet}}(\lambda,\mu)$ for seven bins of
$\mu$ angle between $\mu=1.0$ and $\mu=0.3$ (color code). For the quiet Sun,
profiles up to $\mu=0.1$ are shown. Both the spatial average (first row),
spatial dispersion (second row) and their ratio (third row) are displayed. A
null spatial dispersion implies universal profiles (see main text). Different
regions are displayed such as spots (second column), plage (third columns) and
the network (fourth columns). Spots are darker in the wings but present an
emission in their core. The plage emission profiles are always brighter, but
present a similar core emission as spots. The network exhibits a weak emission
profile slightly brighter in the core of the line. The result of the Gaussian
fit on the intensity line profiles for the different active regions is
displayed in Fig.9.

While the northern regions show a time-independent behaviour, things are
radically different for southern hemispheric regions. Indeed, before the AR
crossed the meridian, the intensity profiles of the pores were almost
identical to those of the northern hemisphere AR. However, once it crossed the
meridian, we observed a rapid evolution of the AR. Note that this AR will
produce one of the largest, and best studied flare recorded during the cycle
on the 6th of September 2017 (e.g. Yang et al., 2017; Hou et al., 2018; Inoue
et al., 2018; Romano et al., 2018; Wang et al., 2018; Verma, 2018; Inoue &
Bamba, 2021; Zou et al., 2019, 2020; Vissers et al., 2021; Pietrow et al.,
2023a). That is however not included in our Meudon dataset.

A strong emission in the core of the lines is visible on the evolving spot
(also visible in the middle panel of Fig.3). It is unclear what is causing
such different spectra. Is that spectrum a specific behaviour of spots in
evolution or could it be contamination from adjacent plage that "covers" the
sunspot?

Unlike the AR seen on the northern hemisphere, (classified as a dipole $\beta$
configuration999See Hale et al. (1919) and Nikbakhsh et al. (2019) for a
detail description of the classification and the modified Zürich/McIntosh
(McIntosh, 1990) systems. We refer the reader to the SDIC page for an overview
of the classification systems.
https://www.sidc.be/educational/classification.php), we can see that the spots
in the southern region are surrounded by plages (classified as a complexe
$\beta\gamma\delta$ configuration), which means that the resulting magnetic
field strength and vector will be very different. This was shown by Anan et
al. (2021), who found that the plage magnetic field in the high chromosphere
tends to be horizontal, while Pietrow et al. (2020) and Morosin et al. (2020)
found a mostly vertical field for a plage in between two pores. Note that the
latter studies were done using the Ca ii 8542 Å line, which forms lower down
than the He i 10830 Å line studied by the former (e.g. de la Cruz Rodríguez et
al., 2019). As the wings are darker than the quiet chromosphere, we are still
observing a sunspot, but the 3D magnetic field geometry is likely different.

This was confirmed by looking at other AIA filtergrams that probe higher
atmospheric layers such as the corona. We displayed the 131 Å filter in the
inner panels of the Fig.6 and Fig.7. Looking those filters, the southern
region also shows a clear anomalous bright excess, that may indicate that the
magnetic field loops were contained at a different height compared to the
northern hemisphere.

###  2.3 Analysis of the intensity spectrum CLV of active regions

We extended the sample size by merging all the datasets coming from different
epochs, and by binning the spectra in 7 bins from $\mu=0.3$ to $\mu=1$. Doing
so assumes that no time-dependency exists in the spectrum itself, or to say it
differently that the spectrum of the network, spots and plages are
"universal". This hypothesis is therefore the same as assuming similar
physical conditions (in temperature, velocity flows, magnetic field strength
and direction) for all ARs of that type, which is clearly not true, but we
believe that this is still a valid way of representing the average behavior of
these regions.

We represent in the top row of Fig.8 the spatial average of the spectra for
different $\mu$ angles101010The spatial average profiles are provided in
machine readable format in the online supplementary material on CDS, as well
as the spatial dispersion in the bottom row. The spatial dispersion is given
by the standard deviation among the different spectra averaged. The ratio
between the spatial average and spatial dispersion can be understood as a sort
of "universality" measurement. Absolute ratio larger than 1 giving a rough
delimitation for the "universality" region.

At first glance, there are already clear differences between the spots and
plage, with plage (third column) being broader and brighter at disk center,
which is in line with the observations of (Smith, 1960). The profile for the
spots (second column) is thinner and similar to the one published for the MgII
lines in Avrett et al. (2015). The network (fourth column) as a limited
intensity roughly ten times smaller than plages.

Figure 9: Parameters of the Gaussian fit on the active spectra of a spot
(blue), plage (red) and network (green) as a function of the $1-\mu$ angle
(see Fig.8). The parameter of the Gaussian function is indicated in the title
of each subplot. For each parameter, a linear relation as a function of
$1-\mu$ angle (star center on the left, limb on the right) was fit and the
relation displayed in the legend. The CLV relation for plage show rather a fix
intensity emission profile in agreement with the universal emission plage
profile (Oranje, 1983). A notable broadening of 14.6  km s-1from center-to-
limb for plages can be observed already reported in (Engvold, 1966, 1967). For
spots, no trend with $\mu$ angle can be clearly extracted even if the mean of
the parameters strongly differ from the plage emission profile.

The spatial dispersion of the quiet Sun follows a constant level of 0.5% of
noise that is likely the superposition of stellar signals, such as granulation
and p-modes oscillations (Pietrow et al., 2023c), with the limited algorithmic
precision of the CLV fit. This is confirmed by the two largest dispersions
($\sim$$1.5\%$) observed for $\mu<0.3$, which shows the difficulty of
obtaining stable fits close to the edges of the solar disk. A small excess is
visible in the core of the line, which may be explained by other structures
too small to be spatially resolved ($<0.40$ MSH).

A weak spatial dispersion is observed for plages, meaning that the plage
profile seems "universal" as already suggested in Oranje (1983). The standard
deviation of the profile across the surface (0.8 at the line core) being
typically 4.5 times smaller than the mean profile (that peaks at 3.5).

In contrary to the plage, spots present more diverse behaviour. The
"universality" of spots is limited to their darker photospheric wings, whereas
more diverse behaviours are observed in the core of the lines (1.0 of standard
deviation against a mean profile at 2.5). This was already visible in Fig.3,
where the bottom spot presented a clear emission, whereas the one at the top
was still partially dark. We showed in Sect. 2.2.3 that this behaviour was
likely related to a geometrical effect due to the peculiar configuration with
adjacent plages.

Further possible explanations include that spots with different umbra versus
penumbra ratio exhibit different chromospheric signatures, since both
components have rather different profiles (see Fig.2 of Yoon et al. (1995)).
Umbrae typically represents between 15-30% of the sunspot size (Tlatov &
Pevtsov, 2014). Also, because the cores of the Ca ii lines probe the upper
part of the chromosphere (Vernazza et al., 1981; Bjørgen et al., 2018a), we
should consider that spots are partially covered by the surrounding fibrils
and other structures (e.g. Fig. 1 of Cauzzi & Reardon (2012), Fig. 1 of Yadav
et al. (2022), Fig. 1 of Pietrow et al. (2022), and Fig. 3 of Morosin et al.
(2022)).

In order to extract the $\mu$ dependency of the AR profiles, we fit a Gaussian
to each emission profile111111Note that we also tried to extract other
parameters of the profiles (such as distribution moments) in order to confirm
that the trends observed were real and not induced by the choice of the
Gaussian parametrisation.. The parameters for the Gaussian fit, namely the
offset, amplitude, full-width at half maximum (FWHM) and center are displayed
for the Ca II K (Fig.9) and Ca II H (Fig.21) lines respectively. The offset
parameter can be understood as the photospheric brightness, while the
amplitude is associated with the chromospheric brightness. A similar behaviour
is observed for both lines and we will only describe the behaviour of the Ca
II K line. A linear relation as a function of $\mu$ was fit for each Gaussian
parameter to highlight any linear CLV of the active spectra.

The main conclusions drawn from these figures are that plage profiles as a
function of $\mu$ see 1) their photospheric component becoming dimmer, 2)
their chromopsheric component becoming brighter, 3) their profile becoming
significantly broader. The broadening of the line is similar to the one
observed on the quiet Sun by Pietrow et al. (2023c), meaning that this
behavior might be intrinsic to the line itself and how it changes from the
center to the limb. The clear broadening of the lines detected towards the
limb is also in agreement with old solar physics studies (Engvold, 1966,
1967). Note that similar broadening toward the limb has also been detected for
photospheric lines (Bruls & Solanki, 2004). However, the darkening of the
photospheric component (the offset parameter) towards the limb contradicts the
idea of faculae becoming brighter towards the limb (Unruh et al., 1999), but
it is hard to compare the photospheric level in the wings of the Ca II H & K
to the one observed lower in the solar atmosphere by G-band continuum images.

For spots, there seems to be some trends also, but the dispersion is much
larger than for plage. This is notably visible on the amplitude parameter of
the Gaussian that shows strong jitter, that is seemingly uncorrelated with the
$\mu$ angle. Both activity profiles are red shifted by $\sim$ 1.5  km s-1 but
we do not know if the Meudon spectroheliograph is stabilised and accurate in
RVs. This is particularly true given that the network seems redshifted by 3
km s-1, while such shift is not confirmed with the Ca II H line (see appendix
Fig.21). We do however note that the trends are very similar in Ca II H and Ca
II K, suggesting that the spread that we see is tied to physical processes
rather than instrumental ones.

###  2.4 Simulating disk-integrated spectra from active intensity profiles

The last sections focused on disk resolved observations where ARs could be
picked out by simple thresholding techniques, leading to
$I_{\text{AR}}(\lambda,\mu)$. However, this is of course not possible with
disk-integrated stellar spectra and a natural question that raises is how
those spectra would look in disk-integrated observations? We therefore
simulated the behaviour of the Ca II K line using the activity profiles
obtained in the previous Sect. 2.3. Several codes exist today to recreate
light curves photometry, stellar spectra, CCF or RV time series based on an AR
configuration. As examples, we can cite SOAP (Boisse et al., 2012; Dumusque et
al., 2014) that has recently be upgraded to produce spectra using a GPU (Zhao
& Dumusque, 2023), but also StarSim (Herrero et al., 2016), and Starry (Luger
et al., 2019). Creating a spectrum is really straightforward once
$I_{\text{quiet}}(\lambda,\mu)$ and $I_{\text{AR}}(\lambda,\mu)$ are known.
The main issue is to precisely obtain such spectra across the stellar disk, on
a long band pass and at high-resolution. Such products have only been obtained
for the quiet Sun recently (Löhner-Böttcher et al., 2019).

We created a solar disk image in a 880 $\times$ 880 grid. The $\mu_{i}$ angle
of each cell are known and each cell contains a spectrum
$I_{\text{quiet}}(\lambda,\mu_{i})$ or $I_{\text{AR}}(\lambda,\mu_{i})$. For
the ARs spectra, we used the spatial averaged profiles (top middle and right
panel in Fig.9) rather than the analytical Gaussian fit approximation in
Fig.9. To extract an analytical approximation, we fit a polynomial function of
degree 4 on each wavelength independently
$P_{\text{AR}}(\lambda,\mu)=\sum_{i=0}^{4}A_{i}(\lambda)\cdot\mu^{i}$. Note
that those parameters are in fact the ratio profiles of the AR with the quiet
Sun: $I_{\text{AR}}(\lambda,\mu)/I_{\text{quiet}}(\lambda,\mu)$ and therefore
need to be rescaled afterwards. To derive $I_{\text{quiet}}(\lambda,\mu)$, a
similar polynomial function
$I_{\text{quiet}}(\lambda,\mu)=\sum_{i=0}^{4}Q_{i}(\lambda)\cdot\mu^{i}$ was
used on the profiles of the quiet Sun (top left panel in Fig.8). All the
profiles are then shifted to reproduce the $\pm 2$ km s-1solar rotation across
the stellar disk. Differential rotation was neglected for simplicity in the
toy model. Once all the spectra are known across the stellar disk, an AR
covering $\mu_{j}$ locations can be placed and the integral over the surface
performed to simulate a disk-integrated Sun-as-a-star observation:

$I_{\text{obs}}(\lambda)=\sum_{i}I_{\text{quiet}}(\lambda,\mu_{i})+\sum_{j\neq
i}I_{\text{quiet}}(\lambda,\mu_{j})\cdot P_{\text{AR}}(\lambda,\mu_{j})$ (2)

Even if this $I_{\text{obs}}(\lambda)$ is the natural product obtained by
several codes, this intensity spectrum is not the one as measured by high-
resolution spectra. Indeed, because high-resolution spectra are not accurate
in flux, researchers rather study their ratio or difference with a reference
spectrum after the continuum of both spectra have been adjusted to match. We
reproduced that behaviour, using as reference the spectrum obtained by
integrating the full quiet Sun: $I_{\text{quiet}}(\lambda)$. The continuum of
the observed spectrum $I_{\text{obs}}(\lambda)$ was adjusted to the continuum
level of $I_{\text{quiet}}(\lambda)$ by computing the scaling factor $C$
between both spectra outside the core of the line, where the core was defined
as $\pm 1$ Å around the line center. Such renormalisation tends to match the
photospheric flux level between the reference spectrum and the observations.
Once the observed spectra scaled, we analysed the difference of both:

$\Delta I(\lambda)=C\times I_{\text{obs}}(\lambda)-I_{\text{quiet}}(\lambda)$
(3)

We displayed in Fig.12 the series of profiles obtained for an AR going from
the solar center to the limb with a filling factor of $f=1\%$. We did not
study the network component, since this component has a weak rotational
modulation. We note that spots and plage both have near identical profiles now
that the photospheric flux has been renormalised. The main difference between
the profile is a slight difference of amplitude and of width, already visible
from the Gaussian parameters in Fig.9, the spot profile being $\sim$25%
thinner than the plage profile and 40% less bright. We notice that both ARs
are less bright on the limb compared to the center, because of the $\mu(t)$
projection effect of the AR. Such effect likely shows that integrating the
flux in the core of the line is a good proxy for their filling factor
$f_{\text{AR}}(t)$. Such relation between the core integration and the plage
filling factor have already been established (Skumanich et al., 1975; Ortiz &
Rast, 2005), but we highlight that the authors never tried to split the
contribution from plage and spots in the Ca II H & K lines, and that good
correlation were found mainly because the plage filling factor dominates the
spot filling factor by one order of magnitude (Chapman et al., 2001). Such a
splitting of both contributions will be done later (see Sect. 3.2.2 and Sect.
3.3.2).

Figure 10: Recreated disk-integrated spectra of the Ca II K line (bottom
panels) from the polynomial CLV relation fit on profiles in Fig.8. We
simulated the effect of a 1% filling factor AR at the equator at five
different rotational phases (top panels) equidistant by $20^{\circ}$
($\sim$1.5 days). The spectra are normalised by the photospheric flux level,
as would high-resolution spectra observations be (see plain text), explaining
why the spot does not look darker and only exhibits core emission. A clear
brightening of plages (red curve) and spot (blue curve) occurs from limb to
star center, mainly due to the increased apparent size. The amplitude of the
peak ($A$) as well as the FWHM ($F$) in  km s-1 are indicated in the legend of
each subplot. Note that the last phase ($\mu=0.16$) is an extrapolation of the
polynomial CLV, since outside the fitting domain ($\mu<0.30$).

Figure 11: Same as Fig.12 for the Ca II H line. A similar behaviour is
observed, a different behaviour of the Balmer $H_{\epsilon}$ at $\lambda=3970$
Å is also visible. The line becoming either deeper or shallower depending if
the AR is a spot or a faculae respectively. A behaviour already noticeable in
Martinez Pillet et al. (1990).

Figure 12: Same as Fig.12 for different $\mu$-distributions of ARs. By
construction, the filling factor is fixed to $f=1\%$ in all the cases. The
stellar rotation was removed to focus on the CLV contribution. If spots do not
show clear effect, the plages show on the $<\mu>=0.66$ spread configuration a
weakening and broadening of 10% compared to the $<\mu>=1$ packed
configuration.

The intensity at center being 0.15% and 0.09%, for a $f=1$% plage and spot AR
respectively. Detecting such effect on stellar observation is not
straightforward since the signal-to-noise ratio in the extreme blue is often
limited. For a plage, the signal is easier to detect since plages can easily
cover up to 10% of the solar surface but spots rarely exceed 1% in size (see
Sect. 3.1).

We performed the same reconstruction for the Ca II H line in Fig.12. A similar
behaviour is observed that highlights once again the similarity of both Ca II
H & K lines. However, an interesting extra feature is visible on the right
side of the line, which is the behaviour of the Balmer line $H_{\epsilon}$ at
$\lambda$=3970 Å. As visible, the line is becoming deeper for a plage, but
shallower for a spot; a behaviour already visible in Martinez Pillet et al.
(1990). This difference theoretically opens a way to lift the degeneracy
between the spot and plage contributions. However, its amplitude does not
exceed 0.01% for a filling factor $f=1\%$ which is extremely challenging to
measure in stellar observations. The further information brought by the
$H_{\epsilon}$ Balmer line was also raised for solar activity studies recently
(Krikova et al., 2023).

We pushed further the investigations by simulating the behaviour of the Ca II
K line for different configurations of ARs. In the previous simulation, the
decrease of the line intensity was dominated by the size projection of the ARs
that decrease linearly from center-to-limb. In order to remove the projection
dependency and focus on the CLV effect, we simulated five different
distributions of ARs, that are all configured with a filling factor of $f=1$%.

The simulations are created using seven small circular121212we used circular
shape on the projected surface for simplicity and since the intrinsic shape is
irrelevant for the final integrated profile. ARs, for which the six outermost
ARs are gradually moved toward the limb. We kept an AR at center since it is
likely to have an AR at $\mu>0.90$ when randomly observing a star due to
geometrical consideration. The result of this simulation is showed in Fig.12.
For this peculiar simulation, we also removed the rotational velocity to avoid
any broadening from that component. As visible, once the filling factor is
fixed, the peak intensity is now far less affected compared to Fig.12. The
profile of spots is almost constant, indicating a low CLV effect. For plage,
when the six ARs are moved to $\mu=0.60$ (a disk-averaged value of
$<\mu>\,=0.66$) the profile becomes broader by 11% and the amplitude smaller
by 15%.

The present example shows that the CLV relation has a weak, but measurable
effect, in disk-integrated spectra in particular for the plages. Therefore,
different configurations lead to slightly different integrated profiles.
However, it is unclear if disk-integrated spectra could be inverted to infer
the distribution of ARs due to degenaracies between spots or plages ratio and
their locations.

##  3 Combining spatially resolved observations with high-resolution disk-
integrated spectra

A strong advantage of the Meudon spectroheliograms is that the individual
spectra coming from different active regions can directly be obtained from the
spatially resolved surface elements of the datacubes. However, such advantage
is at the expense of the spectral resolution which is therefore limited. In
this section, we solved for the issue by combining disk-integrated high-
resolution spectra from ISS and HARPS-N spectrograph, with spatially resolved
information from SDO. We used the SDO public database to segment the solar
surface and extract the filling factor of the different active regions. The
choice of using SDO filtergrams, rather than the Meudon spectroheliograms is
justified by the more extended and better temporal sampling of the SDO space
mission compared to Meudon spectroheliograms.

The section is divided as follows: we first extracted the filling factor time
series of the different activity components using SDO filtergrams in Sect.
3.1. We combined these filling factors time series with the ISS spectra in
Sect. 3.2 in order to derive the emission profile of the ARs per unit of
filling factor. The same analysis is conducted in Sect.3.3 for HARPS-N. The
profiles obtained between the instruments are commented in Sect.3.4. Finally,
an estimation of the contribution of the different types of ARs in an
$S_{\text{index}}$ -like metric is investigated in Sect.3.5.

Figure 13: Extraction of the filling factor time series for plages, spots and
network by segmentation of the AIA1700 SDO filtergrams. The dataset begins
from August 2010 to March 2023 and probes the full $24^{\rm th}$ magnetic
cycle as well as the beginning of the $25^{\rm th}$ solar cycle. Top: Plage
filling factor time series $f_{\text{plage}}(t)$. The maximum and minimum
values of the network filling factor are indicated with the blue dashed lines.
Middle: Spot filling factor time series $f_{\text{spot}}(t)$. Bottom: Network
filling factor time series $f_{\text{network}}(t)$. Unlike the two other
components, the network time series has been cleaned from discontinuities and
outliers.

###  3.1 Fillling factor of plages, spots and network from SDO images

The Solar Dynamics Observatory (SDO, Pesnell et al., 2012) is a solar observer
launched in February 2010. The satellite is equipped with the Atmospheric
Imaging Assembly (AIA, Lemen et al., 2012) producing filtergrams in several
bandpasses (Title & AIA Team, 2006), probing different layers of the solar
atmosphere. Among the SDO/AIA filters, AIA1700 measures the ultraviolet flux
continuum at 1700Å, which probes the temperature minimum, located between the
photosphere and the lower chromosphere from $4500$K to $5000$K (Simões et al.,
2019). More information on the filters can be found on the official NASA web
site131313https://www.nasa.gov/content/goddard/sdo-aia-1700-angstrom/. An
advantage of this filter is that both bright plages and dark spots are visible
and can therefore be segmented from the same image (see Fig.18). The same can
be done with the wavelengths close the K1/H1 minima in the Meudon
spectroheliogram datacubes (see the second and fourth panel in Fig.3).

We downloaded 4518 images from the public
archive141414https://sdo.gsfc.nasa.gov/data/aiahmi/, aiming for a cadence of 1
image per day on a baseline of 13 years, between 2010 and 2023. In this work,
we focus on detecting plages and spots that constitutes large ARs. Because of
this, we restricted ourselves to the 512 $\times$ 512 pixel filtegrams that
are faster to process than the raw 4048 $\times$ 4048 images. The stellar
radius is on average about 198 pixels, hence the spatial resolution is around
5” arcsec (or 8 MSH per pixel), which prevents the detection of small
structures (such as pores) but which is sufficient for plages and spots that
are larger than 20 MSH (Tlatov & Pevtsov, 2014). All the selected images
passed a visual inspection to reject anomalous filtergrams, which occurs when
the exposure began with improper setting of the alignment of the filter or
when the filtergrams were not properly centered by the SDO data extraction
pipelines.

Each SDO image is processed in a similar way as the Meudon spectroheliogram
datacubes to remove the CLV signature of the quiet Sun (see Sect. 2). A
difficulty consists in selecting a good brightness threshold for the flat
corrected images to disentangle plages from the enhanced network (Dineva et
al., 2022). We selected a conservative value that was performing a
satisfactory result visually. The thresholds brightness for plage and spots
were set to brighter and darker than 25% the quiet Sun values respectively.
Again, we did not attempt to detect any ARs at limb angle smaller than
$\mu<0.3$ due to the large amount of noise close to the limb. The network time
series suffered from discontinuities and clear outliers, mainly induced by
issues during the CLV fit that bias the $3\sigma$ threshold to large value
(therefore producing small filling factor values). Such time-domain signatures
are unexpected, since a smooth component is visible everywhere else on the
time series. We corrected the time series from the discontinuities by
"sticking" back the different part of the signal and cleared the outliers
using the average of the next and previous day.

The time series for the filling factor of plages, spots and network are shown
in Fig.13. Our result is similar to the one obtained by Denker & Verma (2019)
and Milbourne et al. (2021). The $24^{\rm th}$ magnetic cycle is clearly
identified as well as the start of the $25^{\rm th}$ cycle. Cycle $24^{\rm
th}$ was the quietest cycle in the past century (Love, 2021) with a 100-day
smoothed filling factor of 4% at maximum for plages. The rotational modulation
due to plages is clearly visible on top of this. The threshold used for the
plage segmentation seems adequate since their filling factor value converges
toward 0 at the minimum of the solar cycle, whereas the network saturates to
its basal minimal intensity around 1%, as in Milbourne et al. (2021).

On average, spots are roughly ten times smaller than plages, in agreement with
Chapman et al. (2001). They were almost absent from the $24^{\rm th}$ cycle,
except for a large sunspot event in 2015. Due to the dataset used, the spatial
resolution is unable to detect pores or spots < 8 MSH. However, most of the
sunspots are larger than 100 MSH (Tlatov & Pevtsov, 2014; Nagovitsyn &
Pevtsov, 2016) and are therefore confidently detected. The magnetic cycle is
far less clear with spots, which is in contradiction with several studies
presenting sunspot number (Love, 2021). It may indicate that the number of
spots shows a clear magnetic cycle, but the total area does not, which seems
to be a peculiarity of the 24th solar cycle151515See the sunspot area in
https://lasp.colorado.edu/lisird/data/sfo_sunspot_indices/.

Figure 14: Core integration of the Ca II K lines obtained from the uncorrected
and corrected ISS spectra. The $S_{\text{index}}$ metrics have been rescaled
in unit to provide the closest 1:1 relation with the SDO plages filling factor
of Fig.13. Top panel: Core integration with a rectangular 1Å band pass in
order to obtain a $S_{\text{index}}$ metric from the uncorrected ISS spectra.
Middle panel: Same as top panel, but from the ISS corrected spectra. Bottom
panel: SDO plage filling factor $f_{\text{plage}}(t)$ from Fig.13. The Pearson
correlation coefficient $\mathcal{R}$ between plage filling factor and the
core integrated emission was boosted from $\mathcal{R}=0.69$ to
$\mathcal{R}=0.81$. The effect was even more impressive for the Ca II H line
slightly more affected, with a correlation increased from $\mathcal{R}=0.51$
to $\mathcal{R}=0.80$.

While the modulation of the signals is similar, we noted a difference in the
absolute filling factor values for spots in comparison with Milbourne et al.
(2021). For instance, looking their Fig.2, we can see that their spot filling
factor rises to $\sim$ 0.75% for the 2015 event, while ours remains at $0.40$
%. A difference of filling factor could be due to different threshold
brightness (which are always arbitrary to some extent), but the reason for the
discrepancy here is likely physical. The difference in filling factor mainly
arises because we observe a different solar atmospheric layer. Indeed, while
we used the AIA1700 filter that probes the chromosphere, Milbourne et al.
(2021) used HMI that mainly probes the photosphere. In order to confirm this
hypothesis, we compared for a given date the spot filling factor obtained
between HMI and AIA1700. Defining sunspots as any region darker than 90% the
quiet photosphere for HMI, we found that spot filling factor was roughly twice
as large in HMI as in AIA1700. Furthermore, 80% of the regions missing from
our segmentation of AIA1700 are brighter than 75% the quiet solar photosphere
in HMI, which indicates that our SDO segmentation likely misses the penumbrae,
but catches the umbrae of sunspots.

Note that because the present paper aims to study the behaviour of the Ca II H
& K lines that are formed in the chromosphere, we are not particularly
interested in physical quantities measured in the photosphere, justifying the
use of AIA1700 rather than HMI for this work.

###  3.2 Combining the spatially resolved SDO filtergrams with ISS disk-
integrated spectra

The Integrated Sunlight Spectrometer (ISS) of the Synoptic Optical Long-term
Investigations of the Sun (SOLIS, Keller et al., 2003) that was installed at
the National Solar Observatory has taken daily disk-integrated images of the
Sun at extremely high spectral resolution ($\mathcal{R}$ $\sim$ 300 000) for
almost a decade, from 2007 to 2017. Public spectra are available and can be
downloaded from an archive161616https://solis.nso.edu/pubkeep/ . Several lines
are available with ISS, but in this paper, we focus on the Ca II H & K lines.
Standard products are the daily stacked observations with a S/N in the
continuum of the CaII lines of S/N$\sim 800$.

####  3.2.1 Processing of the disk-integrated ISS spectra

Despite the impressive baseline and spectral resolution of the dataset, the
spectrograph suffered from clear scattered light or background correction
issues, which can be highlighted when computing the integral of the core
emission. We show in Fig.14 (top panel), the time series of the rectangular
core integration of the Ca II K lines. Strong peak excursions are visible at
different times (see e.g BJD=$2\,455\,450$) and a clear degradation of the
instrument is visible toward the end of its life.

A simple way to correct for background scattering consists in fitting a
parabolic relation using the wings of the Ca II lines that probe the
photosphere. Doing so will absorb any real signature coming from that solar
layer, but these are assumed to be small compared to the chromopheric
signatures. Moreover, such photospheric normalisation are standard when
studying spectra time series where residuals spectra are compared to a
reference spectrum. We defined the reference spectrum
$I_{\text{ref}}(\lambda)$ as the median of the 10% quietest spectra. An
example of the line profile correction is displayed in Appendix.C (Fig.20).
The parabolic fit is performed on the wings selecting the wavelength range
where $I_{\text{ref}}(\lambda)>0.10$, which excludes the core of the lines.

Similarly to the Meudon spectra (see Sect. 2), another issue was related to
the lack of reliable continuum level due to the restrictive bandpass. We
therefore also calibrated the ISS flux units to match the HARPS-N flux units
by fitting a linear relation between both master spectra. The calibration
leads to a relation: $f_{\text{HARPS-N}}=0.741\times f_{\text{ISS}}-0.005$ and
was used to convert the flux units of all the ISS spectra.

We show the previous rectangular core integration obtained with the clean
spectra in middle panel of Fig.14. By comparing the two time series with the
plage filling factor of SDO (see Sect. 3.1), we observed that the
$S_{\text{index}}$ obtained with the clean spectra was now strongly correlated
with the SDO $f_{\text{plage}}(t)$ time series ($\mathcal{R}\simeq 0.80$ in
average for both lines), while a moderate correlation ($\mathcal{R}\simeq
0.60$) was obtained with the uncorrected spectra.

Figure 15: Extraction of the intensity emission profiles per unit of area from
spots $I_{\text{spot}}(\lambda)$, plages $I_{\text{plage}}(\lambda)$ and the
network $I_{\text{ntw}}(\lambda)$ using the ISS spectra from 2010-09-01 to
2017-10-23. In order to extract the profiles, the ISS spectra time series
(colormap) was fit with a multi-linear model of SDO spot (green), plage
(orange) and network (blue) filling factor according to Eq.4, plus a free
offset parameter. The SDO time series are the same as the one presented in
Fig.13 except that only intersection of SDO and ISS observations were kept.
The residuals spectra are displayed on the right. Some rotational behaviour
are still visible in residuals and are either due to non-universality or CLV
such as the broadening of the plage emission spectrum mentioned in Sect. 2.2.

####  3.2.2 Extraction of the ISS activity emission profiles per unit of area

The Meudon spectroheliogram showed that spots and plages present different
emission profiles. Unfortunately, the moderated spectral resolution of the
instrument prevents the extraction of intensity profile for high-resolution
spectrographs. Since the goal is to obtain the emission intensity profile per
unit of area, we can wonder if SDO filtergrams and ISS disk-integrated spectra
could not be combined in a joint analysis. A solution for it consists in
fitting the filling factors of plage and spots obtained from SDO onto the ISS
spectra time series to recover the emission profile per unit of area.

We first selected all the days where simultaneous SDO and ISS spectra were
available, that corresponds to 1617 days between September 2010 and October
2017. Because we are interested in rotational timescale modulation, a precise
match of the observations is not required, and we considered 1-day as the
maximum time difference acceptable. We then modeled the spectra time series
$\delta(\lambda,t)=I(\lambda,t)-I_{\text{ref}}(\lambda)$ of ISS, with
$I_{\text{ref}}(\lambda)$ the median of the 1% spectra with the smallest
integrated flux emission171717Because of the reference choice, these filling
factor values have to be understood as relative values compared to the mean
reference epochs level $\Delta f_{X}(t)=f_{X}(t)-f_{X}(t_{\text{ref}})$. The
residual spectra time series was fit by a three-component model described as
the superposition of the spots $f_{\text{spot}}(t)$, plage
$f_{\text{plage}}(t)$ and network $f_{\text{ntwk}}(t)$ SDO filling factors:

$\delta(\lambda,t)=I_{\text{spot}}(\lambda)\cdot
f_{\text{spot}}(t)+I_{\text{plage}}(\lambda)\cdot
f_{\text{plage}}(t)+I_{\text{ntwk}}(\lambda)\cdot f_{\text{ntwk}}(t)$ (4)

Such a model assumes only three components, which seems to be relatively true
for the Ca II H & K lines based on Meudon spectroheliogram, but which could be
wrong for other lines. For example, the $H_{\alpha}$ line would require
another component for the filaments that are darker compared to the quiet
chromosphere (e.g. Kuckein et al., 2016; Zhu et al., 2019). Furthermore, this
model assumes that the AR profiles do not change over time, which is only
partially true as seen in Sect. 2.2 since ARs evolve over time and due to the
CLV. However, these are secondary effects and we are more interested to derive
the average behaviour of the ARs.

On top of that, it is important to note that, for a target with an inclination
of 90 degrees (equator-on), any AR will spend $75\%$ of its time at $\mu>0.5$,
and $90\%$ at $\mu>0.30$ due to the spherical geometry of stars (e.g. see Fig.
2 of Pietrow et al., 2023b). This means that for most of the time AR are
visible, the Gaussian parameters of the ARs change by less than 15%.

The decomposition to recover $I_{\text{spot}}(\lambda)$,
$I_{\text{plage}}(\lambda)$ and $I_{\text{ntwk}}(\lambda)$ is displayed for
the Ca II K line in Fig.15 and for the Ca II H line in the appendix (Fig.22).
Similar intensity spectra are obtained for both lines. The emission profiles
are compatible with those published in literature. For example, the double-
peak of the plage emission profile is almost identical to the one published by
Oranje (1983) with K2R higher than K2V, whereas the spot emission profile is
compatible with the one published by Yoon et al. (1995). A thinner emission
profile was also observed in penumbrae (see Fig.1 of Sheminova et al. (2005)).
Finally, the network spectrum shows a broader peak separation of the K2 maxima
compared to plage with core depletion. So far as we know, we are not aware of
any studies focusing on the profiles of the network in the Ca II H & K lines.

The similarity of the spot and plage profiles is a huge concern since it means
that most of the band pass to disentangle both contributions come from the
wavelength range cover by the wings of their profiles, which has strong
implication for the minimum S/N needed to extract both components. The same
conclusion holds for the plage and network components. The peak separation
between the K2 maxima is gradually increasing from spots, plage to network
with separation of 17.5  km s-1, 21.0  km s-1 and 30.0  km s-1 respectively.

Looking at those three intensity profiles, we can easily understand that the
different parametrization of the lines profiles (such as distance between the
K2 maxima, K1 minima or similar handmade metrics) had poor chance to
disentangle the three signatures and their investigations has lead so far to
none meaningful result (Keil et al., 1998; Livingston et al., 2007; Bertello
et al., 2011; Dineva et al., 2022). The time series extracted from them were
certainly a complex mixture of spot, plage and network signals strongly
affected by their respective ratio area.

In a similar fashion, integration with bandpasses (Gomes da Silva et al.,
2018) will extract with difficulty the three contributions that are strongly
similar. The $S_{\text{index}}$ and its triangular integration was likely the
weighted average of all the contributions which is a meaningful metric if the
chromospheric flux excess, independently on the nature of the ARs, want to be
measured (for instance to measure the global stellar activity level). However,
such metric has a limited interest for RV correction for which each activity
component is known to contribute in a different way (Milbourne et al., 2021).

Figure 16: Same as Fig.15 for the HARPS-N solar spectra. The double peak broad
plage emission profile is recovered as well as its broader equivalence for the
network. A noisier spot emission profile is recovered, where the larger noise
can be explained both by the lower S/N of HARPS-N in the extreme blue
(S/N${}_{4000}\sim$ 450) and by larger degeneracy in the Eq.4, since HARPS-N
has mainly probed the minimum of the solar magnetic cycle from 20 July 2015
(equivalent to time index 1123 in Fig.15) to 28 June 2018, and only a few
spots events have been measured so far.

###  3.3 Combining the spatially resolved SDO filtergrams with HARPS-N disk-
integrated spectra

HARPS-N is a high-resolution spectrograph ($\mathcal{R}\sim$ 110 000)
installed on the TNG at La Palma. A solar telescope has been installed in 2015
(Dumusque et al., 2015) and is observing continuously the Sun with 5-min
exposure time to mitigate the p-mode. A public data release was presented in
Dumusque et al. (2021) that has been used in several occasions for different
stellar activity studies (Milbourne et al., 2021; Collier Cameron et al.,
2021; Langellier et al., 2021; de Beurs et al., 2022; Haywood et al., 2022;
Camacho et al., 2022; Hara & Delisle, 2023; Lienhard et al., 2023; Al Moulla
et al., 2022, 2023).

####  3.3.1 Processing of the disk-integrated HARPS-N spectra

We processed 535 daily stacked publicly available HARPS-N spectra from 20 July
2015 to 28 June 2018 (Dumusque et al., 2021). We analysed the 1D order-merged
spectra produced by the official DRS. As usual, with high-resolution
spectrographs, spectra are not photometrically calibrated due to differential
extinction, filters in the instrument, change in weather condition but also
since the fiber sky-projected size is usually smaller than the on-site seeing
value (Pepe et al., 2021). As a consequence, part of the light is not captured
inducing large variations in the continuum from one exposure to the next. The
spectra were continuum normalised with RASSINE (Cretignier et al., 2020b). The
code181818A Python code publicly available on GitHub:
https://github.com/MichaelCretignier/Rassine_public uses the concept of alpha
shapes to perform a filtering between the stellar lines and the upper envelop
of the continuum. As part of the standard RASSINE process, all the spectra
were evenly resampled on a 0.01 Å wavelength grid by a linear interpolation.

Unlike the ISS, the background scattered light is highly limited in HARPS-N
due to the intrinsic RV precision that the instrument is aiming for.
Unfortunately, the Ca II H & K lines are contaminated by two ghosts (Dumusque
et al., 2021) which are spurious reflections of the physical echelle-order
inside the spectrograph. In the official DRS, the $S_{\text{index}}$ is
corrected for the ghost effect, but such correction is not performed at the
spectrum level and the 1D spectrum are therefore still contaminated. We post-
processed the spectra using YARARA (Cretignier et al., 2021) since the
pipeline possesses a recipe to mitigate the ghost contamination using
Principal Component Analysis (PCA). Since YARARA also corrects for stellar
activity, we disabled the stellar activity correction. As part of the standard
post-processing, spectra are daily stacked in order to mitigate short
timescale phenomena and in order to boost the S/N. Its value in the continuum
of the CaII lines is in average around S/N$\sim 450$.

####  3.3.2 Extraction of the HARPS-N activity emission profiles per unit of
area

We performed the same analysis as in the previous Sect. 3.2.2, but combining
SDO filling factors with HARPS-N solar observations. Because of the ghosts
falling on both Ca ii lines (Dumusque et al., 2021), only the Ca II K line,
slightly less contaminated, was analysed and the Ca II H line was dropped.
Compared to the ISS dataset, less observations are available and most of them
probe the minimum of the solar cycle. Hence, we only fit for a linear trend
($N=1$) for the polynomial detrending.

The solution for the Eq.4 is plotted in Fig.16. It leads to emission profiles
similar to the one obtained by ISS in the previous section (see Sect. 3.2.2).
The plage profile once again shows the characteristic double peak emission
with a brighter K2R than K2V maximum. The network profile is also similar to
the ISS one. The amplitude of the spot profile is smaller compared to ISS and
may be induced by the non-universality of the spot chromospheric emission
profiles, already detected with Meudon (see Sect. 2.2).

We also observed that the extracted emission profile of spots is noisier than
the one obtained with ISS, even if a similar thin emission profile with double
peaks is visible. The noisier profile can be due to the lower S/N observations
(S/N$\sim 450$) or to the number of observations four times less numerous, but
also to larger degeneracies in the model. Indeed, contrary to the ISS
observations that were probing the full $24^{\rm th}$ solar cycle, HARPS-N has
so far mainly probed the minimum part of the solar cycle for which less spot
events are visible. Therefore, the current dataset is more subject to
degeneracy and spurious correlations.

###  3.4 Comparison between the instruments

We now discuss and compare the different profiles obtained by the different
instruments.

A good agreement was found between both ISS and HARPS-N spectrograph, ISS
producing smoother profiles as a result of the larger S/N. Globally the
profiles are in agreement with intensity profile per unit of filling factor
$I_{X}$ around 0.15% for plage and network. A small disagreement is detected
for the intensity of spots with intensity of 0.25% for HARPS-N against 0.40%
for ISS. Such a discrepancy may be explained by the results of Meudon that the
spot signature likely depends on the magnetic field geometry and is therefore
non-universal.

When comparing Meudon with the two other instruments, results are more
difficult to interpret. It should be reminded that the scaling of the flux
units, on the HARPS-N reference, was more difficult to obtain on the Meudon
datacubes compared to the ISS disk-integrated spectra. Hence, the flux units
of Meudon are less reliable in absolute.

Quantitatively, all the previous activity profiles are recovered. For
instance, a thinner/larger intensity profile of spot/network are obtained
compared to plage. In Sect.2.4, we recovered similarly $0.15\pm 0.05$% for the
plage intensity profile per unit of filling factor. However, the spot profile
intensity was about $0.09\pm 0.06$% which is far weaker than the value
obtained with the previous instruments. However, we note than only a few spots
events have been measured with Meudon, and therefore the statistical size of
the sample may be too small to produce reliable average values.

Finally, whereas the intensity profile for the network was found of a similar
level than for the plage on ISS and HARPS-N, results obtained on Meudon were
rather providing a contribution 5 times smaller compared to plages (see the
amplitude in Fig.9). Since the same profiles were obtained with ISS and
HARPS-N, the discrepancy with Meudon either occurs because of the SDO
segmentation values and/or Meudon segmentations ones.

Figure 17: Normalised distribution to the $S_{\text{index}}$ contribution of
the three activity components: network (blue), plage (orange) and spot
(green). Vertical lines highlight the median values of the distributions that
are also specified in the legend of the figure with the standard deviation
values. The different contributions are obtained by the integration of the
different terms in Eq.4 fitted on the ISS data ($24^{\rm\text{th}}$ solar
cycle) with a 1Å band pass rectangular kernel. The compositions are separately
investigated for the full time series (left), and three different activity
levels: high (top right), mid (middle right) and low (bottom right). Between
low and high activity level, the contribution gradually decreases for the
network and increase for plages.

We suspect that the Meudon precision was not good enough to perform an
accurate segmentation of the network or that an extra activity component
contaminates the network segmentation maps. For instance, two datacubes taken
a few tens of hours apart were showing filling factor variation larger than 2%
which is too large compared to the temporal variation of the network (see
bottom panel of SDO in Fig.13). This indicates that the Meudon datacubes
either contain another bright activity component or noisy pixels. This could
also be identified in Fig.3 where the segmentation seems to pick up the quiet
solar chromosphere as well.

On the 2020 dataset, we remarked that the average filling factor was roughly
2.7 times larger than the SDO value in 2020. Hence, it is likely that our
network signal is diluted by the same amount due to imperfect segmentation.
Therefore, the difference between the plage and network intensity could be a
factor of $5/2.7\simeq 1.9$, rather than 5.

Another observation to raise is the difficulty to disentangle the network
component from plage using a single filter for the segmentation. Trivially,
since we defined in the present work plages as region brighter than a given
threshold $I>I_{\text{min}}$, increasing or decreasing the threshold value
$I_{\text{min}}$ will naturally convert some network regions as plages or
reversely. If plage can easily be identified as concentrated or packed ARs,
the network is made of small-scale structures more diffuse.

Another major difference between the Meudon and SDO dataset is also naturally
the dates of the observations. For Meudon, we only used the 2020 dataset to
form the network sample because this latter was free of any ARs. However, it
could be possible that the network component itself is reduced in intensity at
the solar minima compared to its average magnetic cycle value. To check this
eventuality, we compared the network spectra obtained from the April 2022
datacubes with those of the 2020 dataset. We found that, on this dataset, the
intensity of the network was on average twice larger compared to its intensity
in 2020. Using again that the signal is diluted by 2.7 would produce a profile
$2\times 2.7=5.4$ larger, reducing the tension with the extracted network
amplitude from HARPS-N and ISS data.

Note that since the network component seems to evolve with time (or has
similarly a large variance spatially), it violates the hypothesis of the model
used in Eq.4 and therefore the result on ISS/HARPS-N may also be subject to
bias. As a consequence it is likely that the network intensity is likely
between the lower value of the Meudon instrument
($I_{\text{plage}}/I_{\text{ntwk}}\sim 1.9$) and ISS values
($I_{\text{plage}}/I_{\text{ntwk}}\sim 1$). We note that Worden et al. (1998)
derived a factor 1.5 for the contrast ratio between plage and enhanced network
in the core of the CaII K line during the $22^{\text{th}}$ solar cycle.

###  3.5 Disentangling the S-index composition

To conclude the analysis, we determined the contribution of the different
components to the final $S_{\text{index}}$ value. To do so, we extracted the
intensity map of each contribution ($I_{X}\cdot f_{X}$) based on the Eq.4 for
the ISS decomposition presented in Sect.3.2.2. Each map was then integrated
with a rectangular bandpass of $1\text{\AA}{}$ to mimic an $S_{\text{index}}$
like metric. We display in Fig.17 the contribution of all the components on
the full cycle as well as at three different activity level obtained by
splitting the $S_{\text{index}}$ distribution at $33^{\rm\text{th}}$ and
$66^{\rm\text{th}}$ percentiles. We found out that along the
$24^{\rm\text{th}}$ magnetic cycle, the $S_{\text{index}}$ is composed at
$70\pm 12$ % by plage intensity, $26\pm 12$ % by the network and $4\pm 4$ % by
spots. Because those values are the average value along the cycle and since
the mixture ratio of these three components evolves with time, it is possible
to find some time epochs where the network contribution to the
$S_{\text{index}}$ raises up to 50%. For instance, at the minimum of the solar
magnetic cycle, we found a contribution of $61\pm 13$ % by plage intensity,
$36\pm 15$ % by the network and $4\pm 4$ % by spots. In comparison, at the
maximum of the magnetic cycle, the distributions are $77\pm 7$ % of plage
intensity, $18\pm 7$ % of network and $5\pm 3$ % of spots. Such a change in
the composition of the $S_{\text{index}}$ between high and low activity level
would drastically impact its ability to correct the RVs.

Because the $S_{\text{index}}$ is likely a weighted mixture coming both from
plage and network, and since only191919This is in contradiction with solar
physics studies that also showed the peculiar convective velocities observed
in the network component (Buehler et al., 2019) plage have been found to have
a significant effect on RVs (Milbourne et al., 2021), it could explain why the
$S_{\text{index}}$ has been previously classified as an imperfect activity
proxy for RV correction (Haywood et al., 2016). Furthermore, other studies
also demonstrated that a departure from a strict linear relation may exist
between chromospheric emission and RVs (Meunier et al., 2019). It should be
reminded that RVs are mainly photospheric by nature, and no 1:1 relation exist
between the photosphere and the chromosphere. For this reason, photospheric
activity proxies may be more powerful (Cretignier et al., 2022).

However, more complex extraction of the plage signal from the Ca II H & K
lines could already lead to a better RV correction. This will be shown in our
next paper II (Cretignier et al., prep). Theoretically, it is possible to
elaborate a complex bandpass to extract the different contributions, but none
of them will outperforms the extraction by the real profiles as the one
presented in Fig.15.

##  4 Conclusions

We showed in this paper the different profiles expected for the activity
components given by the Ca II H & K lines. Those results will be used as
Rosetta stones for our next paper II (Cretignier et al., prep).

We first processed the $I(\lambda,x,y)$ Meudon datacubes to show that mainly
three distinct profiles exists for spots, plages and the network. An universal
profile is found for plage that shows a minor CLV such as a broadening toward
the limb. The network intensity was found lower by one order of magnitude, but
that is due partially to a bias that diluted its signal. For spots, an
emission profile $\sim$25% thinner than for plages is measured. This result
was detected on three independent solar instruments. Furthermore, the spot
emission profile seems non-universal, which may be since umbrae and penumbrae
both exhibit different intensity profiles and so different ratio area may lead
to different profiles on the Meudon spectra. Also, investigating a sunspot
along a rotational phase, a peculiar brightness excess was detected when the
AR emerged, that may be related to a peculiar configuration of spots
surrounded by plages or by the peculiar height where the magnetic fields were
confined.

By disk integrating the Meudon spectra and normalising by the photospheric
flux, we recovered a broad plage emission profile with a narrower emission
profile for spots. The plage emission profile also shows a clear and
measurable CLV effect that is visible by the broadening of the line.

By combining SDO filtergrams with ISS disk-integrated spectra of Ca II H & K
lines, we extracted the intensity profile per unit area based on a three-
component model and showed that, once again, a broad emission profile for
plage is detected and a thinner emission profiles for spots, while a broader
profile is obtained for the network compared to plage. Similar results were
obtained with HARPS-N even though noisier profiles were recovered. An
intensity profile for plage $I_{\text{plage}}=0.15$% per percent of filling
factor was detected independently for the three instruments. However,
different values were obtained for spots and for the network. For this latter,
a relative intensity $I_{\text{plage}}/I_{\text{ntwk}}$ between 1 and 2 was
found depending on the instrument used, in agreement with previous estimation
(Worden et al., 1998).

Finally, we estimated that the popular $S_{\text{index}}$ contained in average
for the $24^{\rm th}$ solar cycle: $70\pm 12\%$ of the plage intensity, $26\pm
12\%$ of the network intensity and $4\pm 4\%$ of the spots intensity. If the
spot contribution is almost negligible, the network contribution is not. This
result may explain why this activity proxy is only partially good for RV
correction. Our results for the sunspot contribution are in line with Sowmya
et al. (2023) who showed that sunspots do not contribute in the
$S_{\text{index}}$ , except for very active stars.

The present work could be improved by investigating other strong chromospheric
lines in the visible such as $H_{\alpha}$ and $H_{\beta}$ that could help to
lift the degeneracies between the activity profiles components. Also, the
Meudon datacubes appears to contain a wealth of information that we only
partially explored. A better categorisation of the ARs along the full Meudon
database could be interesting to conduct to better understand the deep
relations between Ca II H & K lines and ARs properties (that we assumed time-
independent during this work). Finally, CLV of the profiles could also be
extracted using the disk-integrated spectra, if filling factor time series in
different $\mu$ bin angle were fit.

In our next paper II (Cretignier et al., prep), we will apply and extrapolate
the information acquired on the Sun to better constrain the stellar activity
on $\alpha$ Cen B. In particular, we will show that, in the context of RVs,
better activity proxies can be extracted.

## Acknowledgements

We acknowledge Paris Observatory for the use of spectroheliograms. Meudon
spectroheliograph data are courtesy of the solar and BASS2000 teams, as part
of operational services of Paris Observatory. M.C. acknowledges the SNSF
support under grant P500PT_211024. AP was supported by the 824135 (SOLARNET –
Integrating High Resolution Solar Physics) grant. SA acknowledges funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 865624)

This research has made use of NASA’s Astrophysics Data System (ADS)
bibliographic services. We acknowledge the community efforts devoted to the
development of the following open-source packages that were used in this work:
numpy (numpy.org), matplotlib (matplotlib.org), and astropy (astropy.org).

## Data Availability

The data underlying this article are publicly available. The datasets were
derived from sources in the public domain: Meudon spectroheliograms
(https://bass2000.obspm.fr/home.php), SDO images
(https://sdo.gsfc.nasa.gov/data/aiahmi/), ISS spectra
(https://solis.nso.edu/pubkeep/) and HARPS-N spectra
(https://dace.unige.ch/pythonAPI/?tutorialId=22).

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##  Appendix A SDO segmentation maps example

We displayed in Fig.18 an example of the SDO AIA 1700 filtergrams
segmentation. The image is a 515 $\times$ 512 resolution image for which the
Sun radius covers in average 220 pixels. The zones flagged as plages, spots
and network based on the threshold values (see Sect. 3.1) are displayed in
dark red, dark blue color and dark green colors.

Figure 18: Example of the segmentation of SDO AIA1700 filtergrams for the same
event as displayed with the Meudon spectroheliograph (see Fig.3): the 4
September 2017 event (exact time in the title). The CLV of the quiet Sun has
already been removed. Plages (red regions) are considered as regions brighter
than 25% the quiet chromosphere, spots (blue regions) as 25% darker than the
quiet chromosphere. The network is defined as regions brighter than
$I>I_{50}+3\cdot(I_{50}-I_{16})$, with $I_{p}$ the intensity value
corresponding to the $p$ percentile of the intensity distribution. Filling
factor of the three components are indicated in the legend. A large plage
($x=300,y=220$) connecting two sunspots and another at ($x=350,y=320$) are the
main AR at that time.

##  Appendix B Meudon segmentation maps example

We displayed in Fig.19 an example of the Meudon Ca II K segmentation map for a
similar epoch as in Fig.18. The image is a 2048 $\times$ 2048 resolution image
for which the Sun radius covers in average 836 pixels. The zones flagged as
plages, spots and network based on the threshold values (see Sect. 2) are
displayed in dark red, dark blue color and dark green colors. By comparing the
SDO and Meudon segmentation maps, a close agreement can be found. However, the
network filling factor of Meudon appears as slightly overestimated compared to
SDO that can be due to a threshold plage value too high. This is notably
visible by a larger amount of network surrounding the plages compared to SDO.
Also, it is unclear if part of the difference is due to the better S/N of SDO
or to its lower spatial resolution. Note that a perfect match between both
frames are not necessarily expected since the segmentation criterion are
performed on filters probing different solar atmospheric layers.

Figure 19: Same as Fig.18 but for the Meudon segmentation. The closest date
matching the SDO observation was selected to compare the segmentation map.
Both segmentation maps are closely in agreement for spots and plages
components, but tend to differ for the diffuse network component.

##  Appendix C Correction of the ISS spectra

ISS high resolution spectra are contaminated by background scattering light
and imperfect correction. Such effects are visible by clear offset flux values
with weird curvature of the photospheric wings and anomalous core emission. In
order to correct for the effect, a reference spectrum made of the 10 percent
of the quietest observations was constructed (mainly made of 2006 observations
as visible in Fig.14). The background correction was obtained by fitting a
polynomial relation between the spectra and the reference spectra:

$I(\lambda,t)-I_{\text{ref}}(\lambda)=A(t)+B(t)\cdot
I_{\text{ref}}(\lambda)+C(t)\cdot(I_{\text{ref}}(\lambda))^{2}$ (5)

Such correction is equivalent to scale the photospheric flux level of the
observations to a reference value. The quadratic fit was performed for
$I_{\text{ref}}(\lambda)>0.10$ in order to avoid the core emission of the
line. The coefficients of the fit were then used to correct the spectra. An
example for the correction is showed in Fig.20. In that example the emission
is corrected to half its raw value.

Figure 20: Example of light scattering correction on an ISS spectrum of the Ca
II K line taken the 2015-12-02. The correction is obtained by adjusting the
photospheric flux ($I_{\text{ref}}(\lambda)>0.1$) between the observed
spectrum $I_{\text{obs}}(\lambda)$ and a reference spectrum at solar minima
$I_{\text{ref}}(\lambda)$. Top panel: a quadratic relation (red curve) is
assumed and fit on the photospheric component (right side of the vertical
dotted line). The relation is then extrapolated on the chromospheric part
corresponding to the line core ($I_{\text{ref}}(\lambda)<0.1$). Uncertainties
from the quadratic fit are visualised by 100 hundreds random realisations in
the $1\sigma$ of the parameters distributions. Bottom panel: Corrected
spectra. The core emission is reduced from 0.0070, to half its value at
0.0035.

##  Appendix D Extension of the analyses for the Ca II H line

The paper mainly focused the analyses on the Ca II K line for simplicity.
However, we noted that qualitatively a similar behaviour was observed for the
Ca II H. Previous authors have already showed the almost perfect linear
relation between both lines (Kneer et al., 1981; Lemaire et al., 1984). We
show the result of the Gaussian fit on the binned $\mu$ profiles of the Meudon
spectroheliograph in Fig.9. Similar behaviour are observed for spots and
plages compared to the Ca II K lines in Fig.9. A notable difference with the
Ca II K line concerns the CLV of the network which is not visible in any
Gaussian parameter.

Figure 21: Same as Fig.9 for the Ca II H line. Even if parameters are slightly
different, similar trend are observed.  Figure 22: Same as Fig.15 for the Ca
II H line. The emission profiles are almost identical to the Ca II K line. A
thin profile for the spot, a broad asymmetric for the plage and a broader
profile with core center depletion for the network.

In a similar way, we performed the ISS decomposition of the Ca II H line with
SDO filling factors (see Fig.22). The emission profiles are almost identical
to the Ca II K line (see Fig.15).