# Universal Scaling Laws for Solar and Stellar Atmospheric Heating: Catalog of
Power-law Index between Solar Activity Proxies and Various Spectral
Irradiances

Shin Toriumi Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210,
Japan  Vladimir S. Airapetian Sellers Exoplanetary Environments Collaboration,
NASA Goddard Space Flight Center, Greenbelt, MD, USA  Department of Physics,
American University, Washington, DC, USA  Kosuke Namekata National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Yuta Notsu Laboratory for Atmospheric and Space Physics, University of
Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA  National Solar
Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA  Department of Earth
and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama,
Meguro-ku, Tokyo 152-8551, Japan

###### Abstract

The formation of extremely hot outer atmospheres is one of the most prominent
manifestations of magnetic activity common to the late-type dwarf stars,
including the Sun. It is widely believed that these atmospheric layers, the
corona, transition region, and chromosphere, are heated by the dissipation of
energy transported upwards from the stellar surface by the magnetic field.
This is signified by the spectral line fluxes at various wavelengths, scaled
with power-law relationships against the surface magnetic flux over a wide
range of formation temperatures, which are universal to the Sun and Sun-like
stars of different ages and activity levels. This study describes a catalog of
power-law indices between solar activity proxies and various spectral line
fluxes. Compared to previous studies, we expanded the number of proxies, which
now includes the total magnetic flux, total sunspot number, total sunspot
area, and the F10.7 cm radio flux, and further enhances the number of spectral
lines by a factor of two. This provides the data to study in detail the flux-
flux scaling laws from the regions specified by the temperatures of the corona
($\log{(T/{\rm K})}=6$–7) to those of the chromosphere ($\log{(T/{\rm K})}\sim
4$), as well as the reconstruction of various spectral line fluxes of the Sun
in the past, F-, G-, and K-type dwarfs, and the modeled stars.

G dwarf stars (556); Solar analogs (1941); Solar magnetic fields (1503); Solar
chromosphere (1479); Solar transition region (1532); Solar corona (1483);
Solar spectral irradiance (1501); Stellar chromospheres (230); Stellar coronae
(305)

††journal: ApJS

##  1 Introduction

Late-type dwarf stars, including the Sun, commonly exhibit magnetic activity
in a variety of forms. In their turbulent outermost envelope, the convection
zone, magnetic flux is generated and enhanced by the dynamo mechanism (Brun &
Browning, 2017; Charbonneau, 2020; Fan, 2021). The produced magnetic flux
emerges to the photosphere and builds up active regions, including
sunspots/starspots (Solanki, 2003; Berdyugina, 2005; Strassmeier, 2009; Cheung
& Isobe, 2014; Toriumi, 2014). Active regions contain highly concentrated
magnetic flux that drives eruptive processes such as flares and coronal mass
ejections via magnetic reconnection (Benz & Güdel, 2010; Fletcher et al.,
2011; Shibata & Magara, 2011; Maehara et al., 2012; Davenport, 2016; Benz,
2017; Toriumi et al., 2017; Toriumi & Wang, 2019), and coronal mass ejections
accompanying flares expand into interplanetary space (Collier Cameron &
Robinson, 1989; Chen, 2011; Harra et al., 2016; Veronig et al., 2021; Namekata
et al., 2021). It is widely believed that the magnetic flux covering the
entire stellar surface transports the energy from the surface upwards and
heats the outer stellar atmospheres, known as the chromosphere, transition
region, and corona (Güdel, 2004). However, the exact mechanism of atmospheric
heating is still unclear (Klimchuk, 2006).

The comparison between the full-disk magnetograms and the associated X-ray and
extreme ultraviolet (EUV) images of the Sun clearly shows that the rate of
atmospheric heating strongly depends on the surface magnetic flux. Empirical
relationships between the surface magnetic flux and quasi-steady X-ray
emission flux of the Sun and Sun-like stars have been well characterized as a
function of the rotation period and average magnetic field strength. For
example, by measuring the total unsigned magnetic flux ($\Phi$) and X-ray flux
($F_{\rm X}$) of various structures such as the quiet Sun, X-ray bright
points, active regions, entire solar disk, G, K, M dwarfs, and T Tauri stars,
Pevtsov et al. (2003) found that the two parameters showed a power-law scaling
with a power-law index in excess of unity, $F_{\rm X}\propto\Phi^{\alpha}$,
where $\alpha=1.15$. Similar values ($\alpha\gtrsim 1$) were obtained by other
studies (Fisher et al., 1998; Vidotto et al., 2014; Reiners et al., 2022). It
was found that the X-ray flux of late-type dwarf stars decreases with the
rotation period or the Rossby number $Ro$ (defined as the rotation period
divided by the convective turnover time: Noyes et al. 1984) in the regime of
$Ro\gtrsim 0.1$, while it is saturated for $Ro\lesssim 0.1$ (Pizzolato et al.,
2003; Wright et al., 2011; Vidotto et al., 2014; Reiners et al., 2014; Takasao
et al., 2020). Recently, studies also investigated the dependence of magnetic
field strength on $Ro$ (Reiners et al., 2022).

These relationships can be attributed to the stellar evolution (Skumanich,
1972). The rotation speed, which is fastest immediately after starbirth,
determines the efficiency of stellar dynamo, and hence, the average magnetic
field strength and X-ray luminosity driven by the magnetic heating of the
corona. As the stellar evolution progresses, the stellar wind driven by the
magnetic field carries away the angular momentum, which decreases the rotation
speed. As a result, the dynamo action, average field strength, and X-ray
luminosity weaken. For detailed discussions on the stellar evolution and
activity, see reviews by Güdel (2007), Testa et al. (2015), Brun & Browning
(2017), and Vidotto (2021).

High-energy radiation can create detrimental conditions for habitability of
exoplanets around active host stars. Specifically, the X-ray and EUV
radiations, collectively referred to as XUV, emitted from active regions and
stellar flares can evaporate the planetary atmospheres by the photoionization-
driven heating that expands the exosphere, thereby igniting ionospheric and
hydrodynamic escape. Therefore, investigating the dependence of spectral line
irradiances on the stellar magnetic activity is important for elucidating the
stellar atmospheric heating as well as understanding their effects on
exoplanets (Linsky, 2019; Airapetian et al., 2020).

Despite its importance for the exoplanetary atmospheric evolution and
habitability, it is difficult to observe stellar EUV flux, especially of
wavelengths longer than 360 Å owing to the strong absorption by the
interstellar medium (see, e.g., Cruddace et al. 1974 for absorption cross-
section). Therefore, EUV spectrum is estimated and reproduced by using the
scaling laws between EUV and other observable wavelengths, such as X-ray and
Ca II K, or by obtaining the differential emission measure distributions
(Sanz-Forcada et al., 2011; Linsky et al., 2014; Chadney et al., 2015;
Youngblood et al., 2016, 2017; Ayres, 2020, 2021; Johnstone et al., 2021).
However, considering the stellar atmospheric heating is moderated by the
surface magnetic field, physical correspondence can be obtained directly by
measuring the scaling relations between irradiances and the surface magnetic
flux.

Accordingly, Toriumi & Airapetian (2022, hereafter TA22) derived the power-law
correlations between the total unsigned magnetic flux of the Sun over 10 years
and the irradiances of emission lines of various wavelengths, i.e., various
temperature domains. As a result, it was found that the acquired correlations
strikingly replicated in Sun-like G-type stars at five spectral lines: X-rays,
Fe XV 284 Å, C II 1335 Å, Ly$\alpha$, and Mg II h & k. This indicated that the
extremely hot outer atmospheres of the Sun and Sun-like stars are heated by a
common mechanism, which is independent of the stellar age or activity level.

The obtained power-law index for the soft X-ray band in TA22, $\alpha=1.16\pm
0.03$, is highly consistent with the preceding studies by, e.g., Pevtsov et
al. (2003), wherein the exponent was $\alpha=1.15$. Furthermore, we found that
the other coronal line fluxes can be consistently scaled with the above-unity
exponents. Such values have been explained using theoretical models based on
RTV scaling laws (Fisher et al., 1998; Zhuleku et al., 2020; Takasao et al.,
2020) and numerical simulations, wherein Alfvén waves propagating in the
corona loop heat the atmosphere via turbulent dissipation (Shoda & Takasao,
2021). For chromospheric lines, the $\alpha$ values lie below unity in TA22,
which is in agreement with previous studies (Skumanich et al., 1975; Schrijver
et al., 1989; Harvey & White, 1999; Rezaei et al., 2007; Barczynski et al.,
2018).

In TA22, we examined the correlations of multiple lines to the total unsigned
magnetic flux of the Sun and compared them with the stellar observations.
However, by expanding the activity proxy to the historical records of sunspot
number, sunspot area, and the F10.7cm radio flux, and by further enhancing the
number of lines to be investigated, we can provide the means to synthesize the
spectral irradiances over a wide range of wavelength, based on the combination
of the obtained power-law indices and proxies of the Sun in the past, other
Sun-like stars, and numerical models. Therefore, in this study, we create a
catalog of power-law scaling factors for various lines and activity proxies by
analyzing solar synoptic observations. Considering the number of lines was
particularly small for the transition region temperatures in TA22, this study
also leads to a better understanding of how $\alpha$ changes as the
temperature changes from the chromosphere to the corona.

In Section 2, we provide detailed descriptions of the data that are analyzed,
while Section 3 explains how we measure the power-law correlations. Section 4
provides the catalog of the power-law index. We show the temperature and
wavelength dependence of the power-law index in Section 5 based on the
obtained scalings and demonstrate how to synthesize the line and band
irradiances in Section 6. Finally, Section 7 summarizes and discusses the
implications of the obtained results.

##  2 Data

In this study, we investigated the thermal responses of upper atmospheres to
the magnetic flux on the surface by comparing the light curves of spectral
lines and bands of various wavelengths, or various formation temperatures,
with the multiple proxy data representing the solar magnetic activity. As
proxies, we adopted two kinds of the total unsigned magnetic flux, both of
which were derived from the line-of-sight (LOS) magnetic field, total sunspot
number, total sunspot area, and the F10.7cm radio flux between May 2010 and
February 2020.

Table 1 summarizes the key information of the proxies and spectral
lines/bands, such as the formation temperature, central wavelength and
spectral window for calculating the irradiance, and the data source. All
spectral irradiances were converted to values at 1 au. The F10.7cm flux was
used both as a proxy of solar activity and as a light curve data representing
the solar atmospheres.

Table 1: Summary of the Observables

Feature | $\log{(T/{\rm K})}$ | Wavelength (Å) | Basal | Minimum | Maximum | Unit | Source
---|---|---|---|---|---|---|---
Radial magnetic flux | 3.8 | 6173.3 | $1.18\times 10^{23}$ | $1.16\times 10^{23}$ | $3.35\times 10^{23}$ | Mx | SDO/HMI
LOS magnetic flux | 3.8 | 6173.3 | $7.02\times 10^{22}$ | $6.85\times 10^{22}$ | $2.52\times 10^{23}$ | Mx | SDO/HMI
Sunspot number | 3.8 | WL | 0 | 0 | $220$ | – | WDC-SILSO (ver 2.0)
Sunspot area | 3.8 | WL | 0 | 0 | $3120$ | MSH | USAF/NOAA
F10.7cm radio |  $\sim$6 | $10.7\times 10^{8}$ | $68.83$ | $63.67$ | $466.57$ | sfu | DRAO
Total solar irradiance | 3.8 | WL | – | $1358.5$ | $1362.3$ | W m-2 | SORCE/TIM
X-rays 1–8 Å | 6–7 | 1–8 | 0 | $1.00\times 10^{-9}$ | $4.81\times 10^{-5}$ | W m-2 | GOES/XRS
X-rays 5.2–124 Å | 6–7 | 5.2–124 | $2.11\times 10^{-4}$ | $1.85\times 10^{-4}$ | $1.01\times 10^{-3}$ | W m-2 | SORCE/XPS
Fe XV 284 Å | 6.4 | $284.15\pm 1.50$ | $9.36\times 10^{-6}$ | $5.68\times 10^{-6}$ | $1.27\times 10^{-4}$ | W m-2 | SORCE/XPS
Fe XIV 211 Å | 6.3 | $211.32\pm 1.50$ | $1.20\times 10^{-5}$ | $9.88\times 10^{-6}$ | $6.75\times 10^{-5}$ | W m-2 | SORCE/XPS
X-rays (XRT) | $6.2\pm 0.1$ | 5–60 | $5.00\times 10^{-5}$ | $4.71\times 10^{-5}$ | $1.01\times 10^{-3}$ | W m-2 | Hinode/XRT
Fe XII 193$+$195 Å | 6.2 | $193.50\pm 2.50$ | $6.16\times 10^{-5}$ | $5.66\times 10^{-5}$ | $1.72\times 10^{-4}$ | W m-2 | SORCE/XPS
Fe XII 1349 Å | 6.2 | $1349.40\pm 1.00$ | $3.64\times 10^{-6}$ | $3.23\times 10^{-6}$ | $5.66\times 10^{-6}$ | W m-2 | SORCE/SOLSTICE
Fe X 174 Å | 6.1 | $174.53\pm 1.50$ | $5.64\times 10^{-5}$ | $5.40\times 10^{-5}$ | $0.90\times 10^{-4}$ | W m-2 | SORCE/XPS
Fe XI 180 Å | 6.1 | $180.41\pm 1.50$ | $4.57\times 10^{-5}$ | $4.31\times 10^{-5}$ | $0.95\times 10^{-4}$ | W m-2 | SORCE/XPS
F10.7cm radio |  $\sim$6 | $10.7\times 10^{8}$ | $68.83$ | $63.67$ | $466.57$ | sfu | DRAO
Fe IX 171 Å | 5.9 | $171.07\pm 1.50$ | $5.50\times 10^{-5}$ | $5.32\times 10^{-5}$ | $0.73\times 10^{-4}$ | W m-2 | SORCE/XPS
N V 1238 Å | 5.3 | $1238.90\pm 1.15$ | $1.62\times 10^{-5}$ | $1.55\times 10^{-5}$ | $2.39\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
N V 1242 Å | 5.3 | $1242.95\pm 1.00$ | $1.04\times 10^{-5}$ | $9.89\times 10^{-6}$ | $1.54\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
C IV 1548 Å | 5.1 | $1548.25\pm 1.20$ | $1.11\times 10^{-4}$ | $1.07\times 10^{-4}$ | $1.53\times 10^{-4}$ | W m-2 | SORCE/SOLSTICE
C IV 1551 Å | 5.1 | $1550.73\pm 0.95$ | $6.58\times 10^{-5}$ | $6.38\times 10^{-5}$ | $9.02\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
C III 1175 Å | 5.0 | $1175.70\pm 1.75$ | $5.52\times 10^{-5}$ | $5.35\times 10^{-5}$ | $8.24\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
He II 256 Å$+$blends | 4.9 | $256.30\pm 3.00$ | $5.53\times 10^{-5}$ | $5.20\times 10^{-5}$ | $1.21\times 10^{-4}$ | W m-2 | SORCE/XPS
He II 304 Å | 4.9 | $304.00\pm 1.00$ | $4.25\times 10^{-4}$ | $4.09\times 10^{-4}$ | $6.19\times 10^{-4}$ | W m-2 | SORCE/XPS
Si IV 1393 Å | 4.9 | $1393.85\pm 1.30$ | $4.45\times 10^{-5}$ | $4.27\times 10^{-5}$ | $7.66\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
Si IV 1402 Å | 4.9 | $1402.85\pm 0.85$ | $2.32\times 10^{-5}$ | $2.25\times 10^{-5}$ | $3.91\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
Si III 1206 Å | 4.8 | $1206.60\pm 1.25$ | $8.59\times 10^{-5}$ | $8.32\times 10^{-5}$ | $1.66\times 10^{-4}$ | W m-2 | SORCE/SOLSTICE
He I 10830 Å | 4.5 | $10830.40\pm 0.25$ | $0.0292$ | $0.0270$ | $0.0308$ | W m-2 | SORCE/SIM & SOLIS/ISS
C II 1335 Å | 4.3 | $1335.25\pm 1.90$ | $1.57\times 10^{-4}$ | $1.52\times 10^{-4}$ | $2.46\times 10^{-4}$ | W m-2 | SORCE/SOLSTICE
H I 1216 Å (Ly$\alpha$) | 4.3 | $1215.70\pm 2.00$ | $5.73\times 10^{-3}$ | $5.60\times 10^{-3}$ | $8.94\times 10^{-3}$ | W m-2 | SORCE/SOLSTICE
O I 1302 Å | 4.2 | $1302.20\pm 0.85$ | $4.16\times 10^{-5}$ | $3.93\times 10^{-5}$ | $5.40\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
O I 1305 Å | 4.2 | $1305.50\pm 1.75$ | $9.14\times 10^{-5}$ | $8.77\times 10^{-5}$ | $1.17\times 10^{-4}$ | W m-2 | SORCE/SOLSTICE
Mg II k 2796 Å | (3.9) | $2796.38\pm 0.78$ | $0.0136$ | $0.0135$ | $0.0180$ | W m-2 | SORCE/SOLSTICE
Mg II h 2803 Å | (3.9) | $2803.48\pm 0.65$ | $0.0097$ | $0.0096$ | $0.0126$ | W m-2 | SORCE/SOLSTICE
Cl I 1351 Å | (3.8) | $1305.50\pm 1.75$ | $9.06\times 10^{-6}$ | $8.57\times 10^{-6}$ | $1.17\times 10^{-5}$ | W m-2 | SORCE/SOLSTICE
Ca II K 3934 Å | (3.8) | $3933.66\pm 0.50$ | $0.0114$ | $0.0111$ | $0.0130$ | W m-2 | SORCE/SIM & SOLIS/ISS
Ca II H 3968 Å | (3.8) | $3968.47\pm 0.50$ | $0.0139$ | $0.0139$ | $0.0155$ | W m-2 | SORCE/SIM & SOLIS/ISS
H I 6563 Å (H$\alpha$) | (3.8) | $6562.80\pm 0.50$ | $0.0369$ | $0.0360$ | $0.0448$ | W m-2 | SORCE/SIM & SOLIS/ISS
Ca II 8542 Å | (3.8) | $8542.10\pm 0.50$ | $0.0347$ | $0.0346$ | $0.0392$ | W m-2 | SORCE/SIM & SOLIS/ISS

Note. — Listed above the horizontal line are the solar activity proxies, while
the rest are the spectral lines and bands whose irradiances are compared with
the proxies. F10.7cm radio flux is registered as both proxy and spectral band.
The temperatures of optically thick chromospheric lines are given in
parentheses. All irradiances were converted to the values at the distance of 1
au from the Sun.

###  2.1 SDO/HMI

To calculate the total unsigned magnetic flux in the visible hemisphere of the
Sun, we used the full-disk magnetograms obtained by the Helioseismic and
Magnetic Imager (HMI; Scherrer et al. 2012; Schou et al. 2012) aboard the
Solar Dynamics Observatory (SDO), which was launched in February 2010 and
began observations in May 2010. This determines the beginning of the target
period in this study.

HMI obtains full-disk continuum images, magnetograms, and Dopplergrams with
cadences of 45 s and 720 s by acquiring the spectropolarimetric signals of the
Fe I 6173.3 Å line. In this study, we analyzed four LOS magnetograms of 720 s
cadence at 0, 6, 12, and 18 UT for each day, which were reduced from the
original $4096\times 4096$ pixels to $1024\times 1024$ pixels by averaging
over a $4\times 4$ pixel tile.111http://jsoc.stanford.edu/data/hmi/fits By
integrating the LOS field strength $B_{\rm LOS}$ over the entire solar disk,
two types of total magnetic flux were obtained: One is the radial unsigned
magnetic flux, wherein the radial field strength at each pixel, which is
estimated by correcting the viewing angle from the disk center ($\theta$),
$B_{\rm LOS}/\cos{\theta}$, is integrated over the disk, $\Phi_{\rm
rad}=\int|B_{\rm LOS}/\cos{\theta}|\,dS$; The other is the LOS unsigned
magnetic flux, where the LOS field strength is simply integrated over the
disk, $\Phi_{\rm LOS}=\int|B_{\rm LOS}|\,dS$. In both cases, the noise levels
were estimated by fitting a Gaussian function to the distribution of the field
strength in each magnetogram, as in Hagenaar (2001). The reductions of
magnetic flux due to binning the magnetograms from the original $4096\times
4096$ pixels to $1024\times 1024$ pixels were 18.9% and 23.9% for the solar
maximum (2014 October 23) and minimum (2019 March 1), respectively. Therefore,
a typical reduction of 20% is expected to occur.

###  2.2 WDC-SILSO

In 2015, the daily sunspot number was recalibrated and released as a new
dataset (version 2). We obtained this dataset from the WDC-SILSO
webpage.222https://www.sidc.be/silso/datafiles Refer to Clette et al. (2014)
for a general account on the sunspot number and recalibrated record.

###  2.3 USAF/NOAA

Since 1977, the areas of sunspot groups were measured and recorded by the US
Air Force (USAF) and the National Oceanic and Atmospheric Administration
(NOAA),333http://solarcyclescience.com/activeregions.html following the record
by the Royal Greenwich Observatory. Using this dataset, we calculated the
daily total sunspot area on the visible hemisphere of the Sun. The sunspot
areas are measured in units of millionths of the solar hemisphere (MSH), which
is equivalent to $3\times 10^{6}\ {\rm km}^{2}$.

###  2.4 SORCE/TIM

The daily total solar irradiance (TSI) data (level 3, version 19) obtained by
the Total Irradiance Monitor (TIM; Kopp et al. 2005) on board the Solar
Radiation and Climate Experiment (SORCE; Rottman 2005) were downloaded from
the data archive.444https://lasp.colorado.edu/home/sorce/data/ SORCE operated
from February 2003 to February 2020, which determines the end of the analysis
period in this study. However, there are some gaps in observation owing to the
degradation of the battery capacity (longest one from August 2013 to February
2014; Woods et al. 2021).

Whereas the TSI increases as the solar activity increases, it is occasionally
reduced owing to individual sunspot transit and does not correlate well with
other proxies such as the total magnetic flux and total sunspot number.
Therefore, the TSI was used for reference purposes only.

###  2.5 GOES/XRS

As one of the X-ray datasets, we analyzed the soft X-ray flux over 1–8 Å,
measured by the X-Ray Sensor (XRS) onboard the GOES satellite. In this study,
we used the daily-averaged “science quality” level 2 data, acquired by the
GOES-15 satellite from May 2010 to February
2020.555https://www.ngdc.noaa.gov/stp/satellite/goes-r.html To determine the
noise level, we referred to the value of $\lesssim 3\times 10^{-9}\ {\rm W\
m}^{-2}$ at $10^{-5}\ {\rm W\ m}^{-2}$ or less provided by Simões et al.
(2015).

###  2.6 SORCE/XPS and SOLSTICE

The irradiances of emission lines and bands from X-rays to near UV were
derived using the XUV Photometer System (XPS; Woods & Rottman 2005) and the
Solar Stellar Irradiance Comparison Experiment (SOLSTICE; McClintock et al.
2005) on board the SORCE satellite. The data were obtained from the SORCE data
archive.

From the XPS daily spectral data (level 4, version 12), which spans over 1 to
400 Å with a spectral resolution of 1 Å, we measured the irradiances of X-rays
5.2–124 Å (ROSAT heritage band), Fe IX 171 Å, Fe X 174 Å, Fe XI 180 Å, Fe XII
193$+$195 Å (combined), Fe XIV 211 Å, He II 256 Å$+$blends, Fe XV 284 Å, and
He II 304 Å. From the SOLSTICE daily spectral data (level 3, version 18),
which covers from 1150 to 3100 Å with a resolution of 1 Å, we estimated the
irradiances of C III 1175 Å, Si III 1206 Å, H I 1216 Å (Ly$\alpha$), N V 1238
Å, N V 1242 Å, O I 1302 Å, O I 1305 Å, C II 1335 Å, Fe XII 1349 Å, Cl I 1351
Å, Si IV 1393 Å, Si IV 1402 Å, C IV 1548 Å, C IV 1551 Å, Mg II k 2796 Å, and
Mg II h 2803 Å.

In this dataset (i.e., SORCE/SOLSTICE daily spectral data: level 3, version
18), the geocoronal effects were removed from Ly$\alpha$. For each line, we
referred to Ayres (2021) for the central wavelength and spectral window to
calculate the irradiance and the CHIANTI database for the corresponding
formation temperature. All irradiances have been corrected to their respective
value at 1 au. The noise levels were estimated by referring to the irradiance
uncertainty shown in the dataset.

###  2.7 Hinode/XRT

The X-Ray Telescope (XRT; Golub et al. 2007) aboard the Hinode satellite
captures the full-disk synoptic soft X-ray images roughly twice a day at 6 UT
and 18 UT except for the interruption periods owing to, e.g., CCD bakeout and
other engineering operations (Takeda et al., 2016). Montana State University
provides the daily averaged electron temperature ($T_{\rm e}$), emission
measure ($EM$), and soft X-ray irradiance (5–60 Å)
data,666http://solar.physics.montana.edu/takeda/XRT_outgoing/irrad/ which are
derived with the filter ratio method based on the isothermal spectrum (5–60 Å)
under the assumption of coronal elemental abundance in the CHIANTI atomic
database (version 10: Del Zanna et al. 2021).

The filter pairs used for this method are Ti_poly/Al_mesh from February 2008
to May 2015 and Al_poly/Al_mesh from June 2015 to June 2021. The correction
factors for the stray light and filter contamination are selected to ensure
that the $T_{\rm e}$ and $EM$ values in the Cycle 24/25 minimum (around 2019)
are nearly the same as those in the Cycle 23/24 minimum (around 2008).

Considering the filter-ratio method diagnoses the plasmas over a wide
temperature range (Narukage et al., 2011), we determined the XRT temperature
to be $\log{(T/{\rm K})}=6.2\pm 0.1$ by taking the average and standard
deviation of the $T_{\rm e}$ values between May 2010 and February 2020.

###  2.8 F10.7cm Radio Flux

The 10.7 cm (2.8 GHz) band radio flux, F10.7cm, is an excellent proxy of solar
activity and has been measured consistently in Canada since 1947 (Tapping,
2013). The transparency of the Earth’s atmosphere to this microwave signal
makes it possible to monitor solar activity with a high duty cycle.

In this study, we used the daily F10.7cm flux data obtained by the Dominion
Radio Astrophysical Observatory (DRAO), specifically, the “adjusted” data,
which are corrected to the values at 1
au.777https://www.spaceweather.gc.ca/forecast-prevision/solar-
solaire/solarflux/sx-en.php The data are shown in solar flux units (sfu),
which corresponds to $10^{-22}\ {\rm W}\ {\rm m}^{-2}\ {\rm Hz}^{-1}$.

When the Sun is quiet with no flaring activity, the formation of F10.7cm flux
can be described as a combination of thermal radiation from the transition
region to the upper chromosphere (temperatures of 20,000–30,000 K),
gyroresonance radiation from active regions, and thermal radiation from the
active region corona ($>1$ MK) (Gary & Hurford, 1994). The variation
component, which is used in this study, is mostly due to the active region
corona, and hence, we assumed the corresponding temperature to be
$\log{(T/{\rm K})}\sim 6$. For the data uncertainties, we assumed that the
average error was no more than 0.5% by referring to Tapping & Charrois (1994).

###  2.9 SORCE/SIM and SOLIS/ISS

For the chromospheric lines from the visible to near infrared range, we
analyzed the daily spectral data of Ca II K 3934 Å, Ca II H 3968 Å, H I 6563 Å
(H$\alpha$), Ca II 8542 Å, He I 10830 Å measured by the Integrated Sunlight
Spectrometer (ISS; Bertello et al. 2011) of the Synoptic Optical Long-term
Investigations of the Sun (SOLIS). These spectra are provided as relative
intensities with respect to the nearby continuum levels. Therefore, the daily
spectral irradiance data of the SORCE’s Spectral Irradiance Monitor (SIM;
Harder et al. 2005) (level 3, version 27), spanning from 2400 to 24200 Å with
a resolution of 10–340 Å, were incorporated to obtain the absolute
intensities. Note that the SOLIS/ISS observation was terminated in October
2017.

##  3 Derivation of Power-law Index

###  3.1 Light Curve

Figure 1: Time series of all solar activity proxies (left column) and light
curves of all spectral lines and bands (second to fourth columns) analyzed in
this study. In each panel, the typical noise level is shown as an error bar on
the right. The basal flux is shown as a horizontal dashed line with the value
provided at the bottom right.

All the daily data used in this study, i.e., the activity proxies and
line/band light curves, are shown in Figure 1, whereas the minimum and maximum
values of these observables are shown in Table 1. As shown in Figure 1, the
irradiance of each line/band varies as the solar activity waxes and wanes. The
spikes in the curves indicate that when active regions and other magnetic
elements transit across the solar disk, the surface magnetic flux, spot
number, and spot area increase (dimming in case of TSI), whereas the $EM$ and
irradiances in the Sun’s upper atmospheres are enhanced.

In contrast, some lines present weak correlations with the solar activity. In
particular, H$\alpha$ and Ca II 8542 Å increase brightness only during the
declining phase of the cycle, and the long-term trend of He I 10830 Å shows an
almost inverse correlation with the activity. This may be because these
chromospheric lines usually appear in absorption on the disk (Avrett et al.,
1994; Brajša et al., 1996).

###  3.2 Power-law Index

To obtain the scaling relationships between the activity proxies ($P$) and
irradiances ($F$), we first obtained the basal fluxes ($P_{0}$ and $F_{0}$)
and daily variations (residuals: $\Delta P=P-P_{0}$ and $\Delta F=F-F_{0}$).
Then, we created a scatter plot of the residuals for each pair of the proxies
and irradiances ($\Delta F$ vs. $\Delta P$). The basal fluxes can be
considered as the surface magnetic flux and the resultant magnetically-driven
high-temperature emissions that are always present as background components.
Therefore, they can be measured during the deepest solar minimum. The
residuals indicate the appearance of magnetic fields, such as active regions
and plages, and the associated heating of the upper atmospheres. Additionally,
it is possible to set wide dynamic ranges for scatter plots by taking
residuals.

The basal flux was defined as, of the total of 3,592 days, from May 2010 to
February 2020, the median of the values on the days that met the following
conditions:

  * •

The final one year, i.e., the deepest solar minimum from March 2019 to
February 2020;

  * •

When the total sunspot number is 0; and

  * •

When the radial total unsigned magnetic flux is less than the 10th percentile
for the entire period.

As a result, the number of unspotted days that satisfy these conditions was
86. However, depending on the observables, the actual number of unspotted days
that was used for taking the medians may differ.

As the observation of SOLIS/ISS was terminated in 2017, for the chromospheric
lines observed by this telescope, we considered the median of the 268 days
that met the following conditions:

  * •

One year centered on December 2008; and

  * •

When the total sunspot number is 0.

The basal fluxes for all observables are summarized in Table 1 and denoted by
horizontal dashed lines in Figure 1. We set the basal fluxes for the total
sunspot number, total sunspot area, and the GOES soft x-ray flux (1–8 Å) as 0,
0 MSH, and 0 W m-2, respectively.

As described above, the basal flux for each time series was calculated as the
median of spotless day data. This is because it is not known whether the
minimum value in a time series is truly the lowest value due to data gaps.
Therefore, the minimum values in Table 1 are smaller than the basal fluxes in
most cases.

Figure 2: Double logarithmic scatter plots of irradiances vs. the total radial
unsigned magnetic flux. In each panel, the straight line indicates the result
of a linear fitting to the double logarithmic plot. The power-law index
$\alpha$, correlation coefficient CC, and data number $N$ are provided at the
top left. Figure 3: Same as Figure 2 but for the total LOS unsigned magnetic
flux. Figure 4: Same as Figure 2 but for the total sunspot number. Figure 5:
Same as Figure 2 but for the total sunspot area Figure 6: Same as Figure 2 but
for the F10.7cm radio flux.

Figures 2 to 6 show the scatter plots of irradiances (residual: $\Delta F$)
vs. the solar activity proxies (residual: $\Delta P$) of radial total unsigned
magnetic flux (Figure 2), LOS total unsigned magnetic flux (Figure 3), total
sunspot number (Figure 4), total sunspot area (Figure 5), and the F10.7cm flux
(Figure 6). Here, only the data points where both $\Delta P$ and $\Delta F$
were positive are plotted. The fractions of data points that were not used due
to negative values of $\Delta P$ or $\Delta F$ are typically 13–17% for the
SORCE data and 51–60% for the SOLIS/ISS data.

Each figure shows the result of a linear fit to a double logarithmic plot: The
linear fit was applied to the $(\log{\Delta P},\log{\Delta F})$ data, where
both $\Delta P$ and $\Delta F$ were positive, to obtain $\alpha$ and $\beta$
as in the following equation:

$\displaystyle\Delta P=10^{\beta}\Delta F^{\alpha},$ (1)

or equivalently,

$\displaystyle\log{\Delta P}=\beta+\alpha\log{\Delta F}.$ (2)

We assumed that both $\log{\Delta P}$ and $\log{\Delta F}$ have errors. Also,
we applied a uniform weight for each observable because giving weights to
smaller data points allows for wider dynamic ranges over which the linear fit
is performed.888We also tested the differential weighting method, which puts
more weight on larger data. However, the fitting results were not much
different from the uniform weighting cases, especially for $\log{\Delta P}$
with broad dynamic ranges such as the total radial unsigned magnetic flux.
Therefore, we adopted the uniform weighting method in favor of the effective
dynamic ranges. The degree of dispersion of the data points was also examined
by measuring the linear Pearson correlation coefficient, ${\rm CC}(\log{\Delta
P},\log{\Delta F})$.

It should be noted here that the observation data for which the power-law
scalings were calculated are not evenly distributed between May 2010 and
February 2020: There exist observational gaps for each observable as they
appear as gaps in the light curves in Figure 1.

Table 2: Power-law Indices and Correlations between Irradiance and Total
Radial Magnetic Flux

Feature | $\log{(T/{\rm K})}$ | Power-law Index $\alpha$ | Offset $\beta$ | Correlation Coefficient CC | Data Points $N$ | LS Deviation
---|---|---|---|---|---|---
X-rays 1–8 Å | 6–7 | $1.42\pm 0.04$ | $-38.6\pm 0.8$ | 0.893 | 3243 | 0.431
X-rays 5.2–124 Å | 6–7 | $1.16\pm 0.03$ | $-29.9\pm 0.7$ | 0.926 | 2994 | 0.247
Fe XV 284 Å | 6.4 | $1.15\pm 0.03$ | $-30.6\pm 0.7$ | 0.919 | 3009 | 0.258
Fe XIV 211 Å | 6.3 | $1.15\pm 0.03$ | $-30.9\pm 0.7$ | 0.924 | 2998 | 0.248
X-rays (XRT) | $6.2\pm 0.1$ | $0.95\pm 0.03$ | $-25.2\pm 0.6$ | 0.932 | 2966 | 0.222
Fe XII 193$+$195 Å | 6.2 | $1.14\pm 0.03$ | $-30.5\pm 0.7$ | 0.925 | 2998 | 0.246
Fe XII 1349 Å | 6.2 | $0.71\pm 0.02$ | $-22.3\pm 0.5$ | 0.836 | 2978 | 0.236
Fe X 174 Å | 6.1 | $1.15\pm 0.03$ | $-31.1\pm 0.7$ | 0.924 | 2998 | 0.248
Fe XI 180 Å | 6.1 | $1.15\pm 0.03$ | $-30.9\pm 0.7$ | 0.925 | 2998 | 0.247
F10.7cm radio |  $\sim$6 | $1.24\pm 0.03$ | $-26.8\pm 0.7$ | 0.939 | 3200 | 0.225
Fe IX 171 Å | 5.9 | $1.15\pm 0.03$ | $-31.4\pm 0.7$ | 0.925 | 2998 | 0.247
N V 1238 Å | 5.3 | $0.82\pm 0.02$ | $-24.3\pm 0.5$ | 0.888 | 3014 | 0.233
N V 1242 Å | 5.3 | $0.85\pm 0.02$ | $-25.1\pm 0.5$ | 0.882 | 2989 | 0.239
C IV 1548 Å | 5.1 | $0.85\pm 0.02$ | $-24.2\pm 0.5$ | 0.898 | 3080 | 0.233
C IV 1551 Å | 5.1 | $0.81\pm 0.02$ | $-23.6\pm 0.5$ | 0.875 | 3072 | 0.248
C III 1175 Å | 5.0 | $0.82\pm 0.02$ | $-23.6\pm 0.5$ | 0.903 | 3055 | 0.218
He II 256 Å | 4.9 | $1.14\pm 0.03$ | $-30.8\pm 0.7$ | 0.923 | 3001 | 0.249
He II 304 Å | 4.9 | $1.15\pm 0.03$ | $-30.4\pm 0.7$ | 0.923 | 2998 | 0.250
Si IV 1393 Å | 4.9 | $0.90\pm 0.02$ | $-25.3\pm 0.5$ | 0.923 | 3089 | 0.215
Si IV 1402 Å | 4.9 | $0.83\pm 0.02$ | $-24.1\pm 0.5$ | 0.914 | 3096 | 0.214
Si III 1206 Å | 4.8 | $0.89\pm 0.02$ | $-24.7\pm 0.5$ | 0.924 | 3126 | 0.214
He I 10830 Å | 4.5 | $1.09\pm 0.06$ | $-28.2\pm 1.4$ | 0.453 | 1419 | 0.381
C II 1335 Å | 4.3 | $0.79\pm 0.02$ | $-22.5\pm 0.5$ | 0.924 | 3102 | 0.193
H I 1216 Å (Ly$\alpha$) | 4.3 | $0.89\pm 0.02$ | $-23.3\pm 0.5$ | 0.939 | 3105 | 0.193
O I 1302 Å | 4.2 | $0.84\pm 0.02$ | $-24.6\pm 0.5$ | 0.822 | 2971 | 0.300
O I 1305 Å | 4.2 | $0.83\pm 0.02$ | $-24.1\pm 0.5$ | 0.815 | 3011 | 0.307
Mg II k 2796 Å | (3.9) | $0.95\pm 0.02$ | $-24.4\pm 0.5$ | 0.949 | 3120 | 0.187
Mg II h 2803 Å | (3.9) | $0.97\pm 0.03$ | $-25.2\pm 0.6$ | 0.944 | 3097 | 0.200
Mg II k$+$h | (3.9) | $0.96\pm 0.02$ | $-24.5\pm 0.6$ | 0.951 | 3120 | 0.187
Cl I 1351 Å | (3.8) | $0.83\pm 0.02$ | $-24.9\pm 0.5$ | 0.783 | 2928 | 0.312
Ca II K 3934 Å | (3.8) | $0.87\pm 0.03$ | $-23.1\pm 0.8$ | 0.723 | 1755 | 0.214
Ca II H 3968 Å | (3.8) | $0.86\pm 0.04$ | $-22.7\pm 0.9$ | 0.539 | 1624 | 0.273
H I 6563 Å (H$\alpha$) | (3.8) | $-1.46\pm 0.14$ | $29.9\pm 3.1$ | $-0.152$ | 1487 | 0.643
Ca II 8542 Å | (3.8) | $-1.52\pm 0.45$ | $31.3\pm 10.1$ | $-0.014$ | 1678 | 0.714

Note. — The first and second columns show the spectral lines and their
formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the
power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the
number of data points $N$ of each double logarithmic scatter plot of
irradiance versus total radial magnetic flux. Column 7 presents the least-
square deviation of the linear fit to the double logarithmic plot.

Table 3: Power-law Indices and Correlations between Irradiance and Total LOS
Magnetic Flux

Feature | $\log{(T/{\rm K})}$ | Power-law Index $\alpha$ | Offset $\beta$ | Correlation Coefficient CC | Data Points $N$ | LS Deviation
---|---|---|---|---|---|---
X-rays 1–8 Å | 6–7 | $1.43\pm 0.04$ | $-38.7\pm 0.9$ | 0.887 | 3230 | 0.443
X-rays 5.2–124 Å | 6–7 | $1.16\pm 0.03$ | $-29.8\pm 0.7$ | 0.910 | 2982 | 0.271
Fe XV 284 Å | 6.4 | $1.15\pm 0.03$ | $-30.5\pm 0.7$ | 0.905 | 2997 | 0.279
Fe XIV 211 Å | 6.3 | $1.15\pm 0.03$ | $-30.8\pm 0.7$ | 0.911 | 2986 | 0.269
X-rays (XRT) | $6.2\pm 0.1$ | $0.96\pm 0.03$ | $-25.3\pm 0.6$ | 0.932 | 2953 | 0.221
Fe XII 193$+$195 Å | 6.2 | $1.14\pm 0.03$ | $-30.4\pm 0.7$ | 0.911 | 2986 | 0.267
Fe XII 1349 Å | 6.2 | $0.71\pm 0.02$ | $-22.3\pm 0.5$ | 0.829 | 2966 | 0.238
Fe X 174 Å | 6.1 | $1.15\pm 0.03$ | $-31.0\pm 0.7$ | 0.911 | 2986 | 0.269
Fe XI 180 Å | 6.1 | $1.15\pm 0.03$ | $-30.8\pm 0.7$ | 0.911 | 2986 | 0.268
F10.7cm radio |  $\sim$6 | $1.23\pm 0.03$ | $-26.3\pm 0.7$ | 0.928 | 3193 | 0.243
Fe IX 171 Å | 5.9 | $1.15\pm 0.03$ | $-31.2\pm 0.7$ | 0.911 | 2986 | 0.268
N V 1238 Å | 5.3 | $0.83\pm 0.02$ | $-24.3\pm 0.5$ | 0.881 | 3001 | 0.238
N V 1242 Å | 5.3 | $0.85\pm 0.02$ | $-24.9\pm 0.5$ | 0.871 | 2980 | 0.250
C IV 1548 Å | 5.1 | $0.84\pm 0.02$ | $-23.9\pm 0.5$ | 0.896 | 3070 | 0.232
C IV 1551 Å | 5.1 | $0.82\pm 0.02$ | $-23.6\pm 0.5$ | 0.866 | 3059 | 0.257
C III 1175 Å | 5.0 | $0.82\pm 0.02$ | $-23.6\pm 0.5$ | 0.893 | 3041 | 0.228
He II 256 Å | 4.9 | $1.15\pm 0.03$ | $-30.6\pm 0.7$ | 0.909 | 2989 | 0.271
He II 304 Å | 4.9 | $1.15\pm 0.03$ | $-30.4\pm 0.7$ | 0.909 | 2986 | 0.272
Si IV 1393 Å | 4.9 | $0.90\pm 0.02$ | $-25.2\pm 0.5$ | 0.912 | 3078 | 0.229
Si IV 1402 Å | 4.9 | $0.84\pm 0.02$ | $-24.2\pm 0.5$ | 0.908 | 3083 | 0.220
Si III 1206 Å | 4.8 | $0.89\pm 0.02$ | $-24.6\pm 0.5$ | 0.928 | 3111 | 0.207
He I 10830 Å | 4.5 | $1.07\pm 0.06$ | $-27.6\pm 1.3$ | 0.472 | 1419 | 0.374
C II 1335 Å | 4.3 | $0.80\pm 0.02$ | $-22.6\pm 0.5$ | 0.927 | 3092 | 0.189
H I 1216 Å (Ly$\alpha$) | 4.3 | $0.90\pm 0.02$ | $-23.2\pm 0.5$ | 0.939 | 3095 | 0.191
O I 1302 Å | 4.2 | $0.85\pm 0.02$ | $-24.6\pm 0.5$ | 0.837 | 2962 | 0.288
O I 1305 Å | 4.2 | $0.83\pm 0.02$ | $-24.0\pm 0.5$ | 0.817 | 2998 | 0.303
Mg II k 2796 Å | (3.9) | $0.95\pm 0.02$ | $-24.3\pm 0.6$ | 0.945 | 3110 | 0.194
Mg II h 2803 Å | (3.9) | $0.98\pm 0.03$ | $-25.2\pm 0.6$ | 0.943 | 3087 | 0.200
Mg II k$+$h | (3.9) | $0.96\pm 0.02$ | $-24.4\pm 0.6$ | 0.947 | 3109 | 0.193
Cl I 1351 Å | (3.8) | $0.83\pm 0.02$ | $-24.8\pm 0.5$ | 0.781 | 2919 | 0.312
Ca II K 3934 Å | (3.8) | $0.85\pm 0.03$ | $-22.5\pm 0.8$ | 0.737 | 1755 | 0.209
Ca II H 3968 Å | (3.8) | $0.84\pm 0.04$ | $-22.1\pm 0.9$ | 0.570 | 1624 | 0.264
H I 6563 Å (H$\alpha$) | (3.8) | $-1.44\pm 0.15$ | $29.2\pm 3.3$ | $-0.126$ | 1487 | 0.653
Ca II 8542 Å | (3.8) | $1.50\pm 0.44$ | $-36.9\pm 9.9$ | $0.014$ | 1678 | 0.714

Note. — The first and second columns show the spectral lines and their
formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the
power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the
number of data points $N$ of each double logarithmic scatter plot of
irradiance versus total LOS magnetic flux. Column 7 presents the least-square
deviation of the linear fit to the double logarithmic plot.

Table 4: Power-law Indices and Correlations between Irradiance and Total
Sunspot Number

Feature | $\log{(T/{\rm K})}$ | Power-law Index $\alpha$ | Offset $\beta$ | Correlation Coefficient CC | Data Points $N$ | LS Deviation
---|---|---|---|---|---|---
X-rays 1–8 Å | 6–7 | $1.98\pm 0.06$ | $-9.8\pm 0.1$ | 0.819 | 2682 | 0.419
X-rays 5.2–124 Å | 6–7 | $1.41\pm 0.04$ | $-6.1\pm 0.1$ | 0.815 | 2578 | 0.293
Fe XV 284 Å | 6.4 | $1.44\pm 0.04$ | $-7.0\pm 0.1$ | 0.809 | 2585 | 0.306
Fe XIV 211 Å | 6.3 | $1.42\pm 0.04$ | $-7.3\pm 0.1$ | 0.813 | 2581 | 0.298
X-rays (XRT) | $6.2\pm 0.1$ | $1.11\pm 0.03$ | $-5.5\pm 0.1$ | 0.833 | 2479 | 0.228
Fe XII 193$+$195 Å | 6.2 | $1.42\pm 0.04$ | $-7.0\pm 0.1$ | 0.814 | 2581 | 0.297
Fe XII 1349 Å | 6.2 | $0.94\pm 0.03$ | $-7.9\pm 0.1$ | 0.738 | 2538 | 0.236
Fe X 174 Å | 6.1 | $1.42\pm 0.04$ | $-7.5\pm 0.1$ | 0.813 | 2581 | 0.298
Fe XI 180 Å | 6.1 | $1.42\pm 0.04$ | $-7.4\pm 0.1$ | 0.813 | 2581 | 0.297
F10.7cm radio |  $\sim$6 | $1.46\pm 0.04$ | $-1.1\pm 0.1$ | 0.867 | 2861 | 0.258
Fe IX 171 Å | 5.9 | $1.42\pm 0.04$ | $-7.8\pm 0.1$ | 0.813 | 2581 | 0.297
N V 1238 Å | 5.3 | $1.10\pm 0.03$ | $-7.5\pm 0.1$ | 0.816 | 2557 | 0.232
N V 1242 Å | 5.3 | $1.11\pm 0.04$ | $-7.7\pm 0.1$ | 0.792 | 2547 | 0.249
C IV 1548 Å | 5.1 | $1.14\pm 0.04$ | $-6.9\pm 0.1$ | 0.814 | 2578 | 0.243
C IV 1551 Å | 5.1 | $1.08\pm 0.03$ | $-7.0\pm 0.1$ | 0.812 | 2575 | 0.232
C III 1175 Å | 5.0 | $1.13\pm 0.03$ | $-7.0\pm 0.1$ | 0.801 | 2580 | 0.250
He II 256 Å | 4.9 | $1.42\pm 0.04$ | $-7.2\pm 0.1$ | 0.814 | 2582 | 0.297
He II 304 Å | 4.9 | $1.42\pm 0.04$ | $-6.8\pm 0.1$ | 0.812 | 2581 | 0.299
Si IV 1393 Å | 4.9 | $1.15\pm 0.04$ | $-7.0\pm 0.1$ | 0.815 | 2583 | 0.247
Si IV 1402 Å | 4.9 | $1.05\pm 0.03$ | $-7.1\pm 0.1$ | 0.826 | 2579 | 0.217
Si III 1206 Å | 4.8 | $1.08\pm 0.03$ | $-6.5\pm 0.1$ | 0.831 | 2591 | 0.222
He I 10830 Å | 4.5 | $1.17\pm 0.08$ | $-5.5\pm 0.1$ | 0.352 | 1344 | 0.399
C II 1335 Å | 4.3 | $1.02\pm 0.03$ | $-6.3\pm 0.1$ | 0.837 | 2574 | 0.205
H I 1216 Å (Ly$\alpha$) | 4.3 | $1.14\pm 0.04$ | $-4.9\pm 0.1$ | 0.805 | 2585 | 0.250
O I 1302 Å | 4.2 | $1.17\pm 0.04$ | $-7.6\pm 0.1$ | 0.754 | 2525 | 0.287
O I 1305 Å | 4.2 | $1.18\pm 0.04$ | $-7.2\pm 0.1$ | 0.722 | 2539 | 0.310
Mg II k 2796 Å | (3.9) | $1.21\pm 0.04$ | $-5.0\pm 0.1$ | 0.828 | 2588 | 0.250
Mg II h 2803 Å | (3.9) | $1.23\pm 0.04$ | $-5.2\pm 0.1$ | 0.822 | 2584 | 0.259
Mg II k$+$h | (3.9) | $1.23\pm 0.04$ | $-4.8\pm 0.1$ | 0.828 | 2590 | 0.254
Cl I 1351 Å | (3.8) | $1.12\pm 0.04$ | $-8.1\pm 0.1$ | 0.738 | 2517 | 0.277
Ca II K 3934 Å | (3.8) | $0.85\pm 0.04$ | $-4.7\pm 0.1$ | 0.656 | 1659 | 0.218
Ca II H 3968 Å | (3.8) | $0.85\pm 0.04$ | $-4.7\pm 0.1$ | 0.531 | 1529 | 0.254
H I 6563 Å (H$\alpha$) | (3.8) | $-1.60\pm 0.19$ | $-0.5\pm 0.3$ | $-0.097$ | 1399 | 0.667
Ca II 8542 Å | (3.8) | $1.61\pm 0.38$ | $-6.0\pm 0.7$ | $0.023$ | 1586 | 0.696

Note. — The first and second columns show the spectral lines and their
formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the
power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the
number of data points $N$ of each double logarithmic scatter plot of
irradiance versus total sunspot number. Column 7 presents the least-square
deviation of the linear fit to the double logarithmic plot.

Table 5: Power-law Indices and Correlations between Irradiance and Total
Sunspot Area

Feature | $\log{(T/{\rm K})}$ | Power-law Index $\alpha$ | Offset $\beta$ | Correlation Coefficient CC | Data Points $N$ | LS Deviation
---|---|---|---|---|---|---
X-rays 1–8 Å | 6–7 | $1.18\pm 0.04$ | $-9.2\pm 0.1$ | 0.796 | 2621 | 0.423
X-rays 5.2–124 Å | 6–7 | $0.83\pm 0.03$ | $-5.7\pm 0.1$ | 0.767 | 2532 | 0.317
Fe XV 284 Å | 6.4 | $0.83\pm 0.03$ | $-6.5\pm 0.1$ | 0.760 | 2534 | 0.322
Fe XIV 211 Å | 6.3 | $0.82\pm 0.03$ | $-6.8\pm 0.1$ | 0.770 | 2532 | 0.311
X-rays (XRT) | $6.2\pm 0.1$ | $0.65\pm 0.02$ | $-5.1\pm 0.1$ | 0.770 | 2415 | 0.251
Fe XII 193$+$195 Å | 6.2 | $0.82\pm 0.03$ | $-6.5\pm 0.1$ | 0.770 | 2532 | 0.311
Fe XII 1349 Å | 6.2 | $0.56\pm 0.02$ | $-7.6\pm 0.0$ | 0.650 | 2484 | 0.263
Fe X 174 Å | 6.1 | $0.82\pm 0.03$ | $-7.1\pm 0.1$ | 0.770 | 2532 | 0.311
Fe XI 180 Å | 6.1 | $0.82\pm 0.03$ | $-6.9\pm 0.1$ | 0.770 | 2532 | 0.311
F10.7cm radio |  $\sim$6 | $0.88\pm 0.03$ | $-0.7\pm 0.1$ | 0.807 | 2804 | 0.297
Fe IX 171 Å | 5.9 | $0.82\pm 0.03$ | $-7.3\pm 0.1$ | 0.770 | 2532 | 0.311
N V 1238 Å | 5.3 | $0.66\pm 0.02$ | $-7.2\pm 0.1$ | 0.725 | 2504 | 0.276
N V 1242 Å | 5.3 | $0.66\pm 0.02$ | $-7.4\pm 0.1$ | 0.700 | 2492 | 0.285
C IV 1548 Å | 5.1 | $0.68\pm 0.02$ | $-6.5\pm 0.1$ | 0.694 | 2515 | 0.302
C IV 1551 Å | 5.1 | $0.63\pm 0.02$ | $-6.7\pm 0.1$ | 0.718 | 2514 | 0.270
C III 1175 Å | 5.0 | $0.66\pm 0.02$ | $-6.6\pm 0.1$ | 0.712 | 2515 | 0.282
He II 256 Å | 4.9 | $0.81\pm 0.03$ | $-6.8\pm 0.1$ | 0.771 | 2532 | 0.308
He II 304 Å | 4.9 | $0.82\pm 0.03$ | $-6.3\pm 0.1$ | 0.769 | 2532 | 0.312
Si IV 1393 Å | 4.9 | $0.66\pm 0.02$ | $-6.6\pm 0.1$ | 0.744 | 2524 | 0.270
Si IV 1402 Å | 4.9 | $0.63\pm 0.02$ | $-6.8\pm 0.0$ | 0.737 | 2521 | 0.258
Si III 1206 Å | 4.8 | $0.65\pm 0.02$ | $-6.2\pm 0.1$ | 0.728 | 2529 | 0.273
He I 10830 Å | 4.5 | $0.74\pm 0.05$ | $-5.3\pm 0.1$ | 0.311 | 1332 | 0.415
C II 1335 Å | 4.3 | $0.59\pm 0.02$ | $-6.0\pm 0.0$ | 0.733 | 2507 | 0.245
H I 1216 Å (Ly$\alpha$) | 4.3 | $0.67\pm 0.02$ | $-4.6\pm 0.1$ | 0.708 | 2521 | 0.291
O I 1302 Å | 4.2 | $0.70\pm 0.02$ | $-7.2\pm 0.1$ | 0.645 | 2458 | 0.328
O I 1305 Å | 4.2 | $0.71\pm 0.03$ | $-6.9\pm 0.1$ | 0.627 | 2475 | 0.346
Mg II k 2796 Å | (3.9) | $0.72\pm 0.02$ | $-4.6\pm 0.1$ | 0.745 | 2528 | 0.293
Mg II h 2803 Å | (3.9) | $0.74\pm 0.02$ | $-4.9\pm 0.1$ | 0.744 | 2522 | 0.300
Mg II k$+$h | (3.9) | $0.73\pm 0.02$ | $-4.4\pm 0.1$ | 0.748 | 2528 | 0.294
Cl I 1351 Å | (3.8) | $0.66\pm 0.02$ | $-7.8\pm 0.1$ | 0.666 | 2463 | 0.300
Ca II K 3934 Å | (3.8) | $0.53\pm 0.03$ | $-4.5\pm 0.1$ | 0.492 | 1643 | 0.262
Ca II H 3968 Å | (3.8) | $0.53\pm 0.03$ | $-4.5\pm 0.1$ | 0.410 | 1514 | 0.282
H I 6563 Å (H$\alpha$) | (3.8) | $-1.00\pm 0.13$ | $-0.8\pm 0.3$ | $-0.093$ | 1388 | 0.667
Ca II 8542 Å | (3.8) | $1.01\pm 0.35$ | $-5.7\pm 0.9$ | $0.011$ | 1567 | 0.698

Note. — The first and second columns show the spectral lines and their
formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the
power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the
number of data points $N$ of each double logarithmic scatter plot of
irradiance versus total sunspot area. Column 7 presents the least-square
deviation of the linear fit to the double logarithmic plot.

Table 6: Power-law Indices and Correlations between Irradiance and F10.7cm
Radio Flux

Feature | $\log{(T/{\rm K})}$ | Power-law Index $\alpha$ | Offset $\beta$ | Correlation Coefficient CC | Data Points $N$ | LS Deviation
---|---|---|---|---|---|---
X-rays 1–8 Å | 6–7 | $1.21\pm 0.03$ | $-8.1\pm 0.0$ | 0.918 | 2978 | 0.334
X-rays 5.2–124 Å | 6–7 | $0.89\pm 0.02$ | $-5.0\pm 0.0$ | 0.932 | 2853 | 0.207
Fe XV 284 Å | 6.4 | $0.86\pm 0.02$ | $-5.8\pm 0.0$ | 0.938 | 2856 | 0.192
Fe XIV 211 Å | 6.3 | $0.87\pm 0.02$ | $-6.1\pm 0.0$ | 0.935 | 2852 | 0.199
X-rays (XRT) | $6.2\pm 0.1$ | $0.78\pm 0.02$ | $-4.7\pm 0.0$ | 0.912 | 2741 | 0.218
Fe XII 193$+$195 Å | 6.2 | $0.87\pm 0.02$ | $-5.8\pm 0.0$ | 0.935 | 2852 | 0.198
Fe XII 1349 Å | 6.2 | $0.59\pm 0.02$ | $-7.1\pm 0.0$ | 0.811 | 2771 | 0.227
Fe X 174 Å | 6.1 | $0.87\pm 0.02$ | $-6.3\pm 0.0$ | 0.935 | 2852 | 0.199
Fe XI 180 Å | 6.1 | $0.87\pm 0.02$ | $-6.2\pm 0.0$ | 0.935 | 2852 | 0.198
Fe IX 171 Å | 5.9 | $0.87\pm 0.02$ | $-6.6\pm 0.0$ | 0.935 | 2852 | 0.198
N V 1238 Å | 5.3 | $0.70\pm 0.02$ | $-6.6\pm 0.0$ | 0.881 | 2812 | 0.216
N V 1242 Å | 5.3 | $0.72\pm 0.02$ | $-6.8\pm 0.0$ | 0.865 | 2797 | 0.232
C IV 1548 Å | 5.1 | $0.72\pm 0.02$ | $-5.9\pm 0.0$ | 0.877 | 2852 | 0.233
C IV 1551 Å | 5.1 | $0.67\pm 0.02$ | $-6.1\pm 0.0$ | 0.881 | 2845 | 0.212
C III 1175 Å | 5.0 | $0.69\pm 0.02$ | $-6.0\pm 0.0$ | 0.886 | 2826 | 0.207
He II 256 Å | 4.9 | $0.88\pm 0.02$ | $-6.0\pm 0.0$ | 0.932 | 2854 | 0.204
He II 304 Å | 4.9 | $0.88\pm 0.02$ | $-5.6\pm 0.0$ | 0.934 | 2852 | 0.200
Si IV 1393 Å | 4.9 | $0.73\pm 0.02$ | $-6.0\pm 0.0$ | 0.901 | 2847 | 0.209
Si IV 1402 Å | 4.9 | $0.67\pm 0.02$ | $-6.2\pm 0.0$ | 0.904 | 2855 | 0.190
Si III 1206 Å | 4.8 | $0.72\pm 0.02$ | $-5.6\pm 0.0$ | 0.896 | 2871 | 0.215
He I 10830 Å | 4.5 | $0.81\pm 0.05$ | $-4.6\pm 0.1$ | 0.438 | 1416 | 0.386
C II 1335 Å | 4.3 | $0.65\pm 0.02$ | $-5.5\pm 0.0$ | 0.907 | 2845 | 0.180
H I 1216 Å (Ly$\alpha$) | 4.3 | $0.73\pm 0.02$ | $-4.0\pm 0.0$ | 0.892 | 2863 | 0.222
O I 1302 Å | 4.2 | $0.70\pm 0.02$ | $-6.5\pm 0.0$ | 0.802 | 2773 | 0.283
O I 1305 Å | 4.2 | $0.72\pm 0.02$ | $-6.2\pm 0.0$ | 0.760 | 2788 | 0.321
Mg II k 2796 Å | (3.9) | $0.78\pm 0.02$ | $-4.0\pm 0.0$ | 0.910 | 2872 | 0.214
Mg II h 2803 Å | (3.9) | $0.79\pm 0.02$ | $-4.2\pm 0.0$ | 0.908 | 2856 | 0.218
Mg II k$+$h | (3.9) | $0.78\pm 0.02$ | $-3.8\pm 0.0$ | 0.914 | 2868 | 0.209
Cl I 1351 Å | (3.8) | $0.72\pm 0.02$ | $-7.2\pm 0.0$ | 0.773 | 2744 | 0.299
Ca II K 3934 Å | (3.8) | $0.65\pm 0.03$ | $-4.2\pm 0.0$ | 0.707 | 1752 | 0.220
Ca II H 3968 Å | (3.8) | $0.64\pm 0.03$ | $-4.1\pm 0.0$ | 0.549 | 1620 | 0.271
H I 6563 Å (H$\alpha$) | (3.8) | $-1.09\pm 0.11$ | $-1.7\pm 0.2$ | $-0.133$ | 1483 | 0.651
Ca II 8542 Å | (3.8) |  $\cdots$ |  $\cdots$ | $0.000$ | 1674 |  $\cdots$

Note. — The first and second columns show the spectral lines and their
formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the
power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the
number of data points $N$ of each double logarithmic scatter plot of
irradiance versus radial F10.7cm radio flux. Column 7 presents the least-
square deviation of the linear fit to the double logarithmic plot.

##  4 Catalog of Power-law index

Tables 2 to 6 summarize the power-law index $\alpha$, offset $\beta$,
correlation coefficient CC, number of data points $N$, and least-square
deviation of the linear fit for all scatter plots in Figures 2 to 6. The
overall trend is that the higher temperature lines and bands show higher CCs.
For each line, among different proxies, the total magnetic fluxes and F10.7cm
flux tend to show higher CCs compared to the sunspot number and the area.
Because He I 10830 Å often falls below its basal flux level (i.e., $\Delta F$
often becomes negative), we created scatter plots by taking the absolute value
of $\Delta F$. For F10.7cm vs. Ca II 8542 Å (Figure 6), the scaling factors
$\alpha$ and $\beta$ are not provided in Table 6 owing to the failure of the
linear fit. These chromospheric lines and H$\alpha$, Ca II K 3934 Å, and Ca II
H 3968 Å generally had poorer CCs and least-square deviations.

##  5 Dependence of Power-law Index

###  5.1 Temperature Dependence

Figure 7 shows the exponent of irradiances with respect to the total radial
unsigned magnetic flux, plotted as a function of temperature. Note that
H$\alpha$ and Ca II 8542 Å are omitted because they exhibited negative
proportionalities with the magnetic flux (i.e., $\alpha<0$). As He I 10830 Å
showed an anti-phased variation with the activity proxies (Figure 1) and a
weak anti-correlation (Table 2), we plotted $\alpha$ calculated by taking the
absolute value of $\Delta F$.999In this study, we measured the irradiance at
the line core of He I 10830 Å and found a weak anti-correlation, while
Livingston et al. (2007) showed a strong correlation between its equivalent
width and the solar activity.

Compared to the previous study (Figure 3 in TA22), the increase in the number
of observables, especially for the transition region temperatures, allows for
scrutinizing the change of $\alpha$ from the chromosphere to the corona.

For the coronal temperatures, $\alpha>1$ for most observables, which is in
agreement with many previous studies (see Section 1). However, for Hinode/XRT,
$\alpha$ was slightly below unity owing to several possible reasons. For
example, the field of view of XRT was only about $2048\arcsec\times
2048\arcsec$, and hence, if there is a bright coronal structure outside the
limb, XRT may miss its contribution and underestimate irradiance, especially
during the solar maximum. The exclusion of images that contain saturated
pixels due to flares may also lead to the underestimation of irradiance.
Furthermore, the combination of filters used to create the XRT light curve was
changed, making it difficult to compare the long-term evolution. Fe XII 1349 Å
also had a coronal formation temperature at $\log{(T/{\rm K})}=6.2$, but
$\alpha$ was well below unity, even smaller than the chromospheric line in the
same wavelength range. This may be attributed to the fact that this line is
much weaker than the other lines, owing to which the irradiance cannot be
easily determined.

The result that the $\alpha$ values for most chromospheric lines take less
than unity also supports the previous analyses (see Section 1). However, it is
newly found that most of the transition region lines also take $\alpha<1$ as
in the chromosphere.

Herein the formation temperature of He I 10830 Å was set to $\log{(T/{\rm
K})}=4.2$, however, it should be noted that this line was formed by the
combination of multiple mechanisms (e.g., Andretta & Jones, 1997): (1) EUV
photons in the corona invade the upper chromosphere and photoionize the
neutral He atoms. When the generated He ions are recombined, they form a group
of He I lines; (2) When electrons with temperatures of 20,000 K or higher
collide with the He atoms between the chromosphere and corona, collisional
excitation occurs, and as the electrons return to the ground state, He I lines
are produced. Therefore, the fact that $\alpha$ of He I 10830 Å is close to
the coronal values (i.e., $\alpha>1$) indicates that the mechanism (1) is more
effective. This may also be related to that the other He lines (He II 256 Å
and 304 Å) show $\alpha$ values that are above unity.

Figure 7: Power-law indices $\alpha$ for the scatter plots of the total radial
unsigned magnetic flux and irradiances of various spectral lines/bands (Figure
2 and Table 2), plotted as a function of temperature. Errors on estimating
$\alpha$ are indicated by vertical bars, whereas the horizontal bars show the
temperature ranges for the three X-ray data, GOES/XRS 1–8 Å, SORCE/XPS 5.2–124
Å, and Hinode/XRT 5–60 Å. He I 10830 Å, which shows an inverse proportionality
against activity proxies, is indicated by an open circle. The $\alpha=1$ level
is indicated by a skyblue line.

###  5.2 Wavelength Dependence

Figure 8: Same as Figure 7 but plotted as a function of wavelength. He I 10830
Å, which shows an inverse proportionality against activity proxies, is
indicated by an open circle. F10.7cm radio flux, which corresponds to the
wavelength of $10.7\times 10^{8}$ Å, is indicated by the open rightfacing
triangle at the rightmost end.

Figure 8 shows the dependence of the power-law index $\alpha$ on the spectral
line wavelength. As shown in Figure 7, He I 10830 Å was plotted despite its
inverse proportionality against the solar activity proxies, while F10.7cm
($=10.7\times 10^{8}$ Å) radio flux is shown in the infrared range for
visualization purposes only.

As seen in the figure, $\alpha$ displays a V-shaped profile with the apex
located at the near UV range around 1000–2000 Å. The value increases from
below unity to above unity as the wavelength shifts from near UV both towards
the EUV and X-rays and the infrared and radio waves. This is because the
corresponding spectral lines and bands are sensitive to increasingly higher
temperature plasmas.

##  6 Applications: Reconstruction of Solar XUV Irradiances

We determined the scaling laws between the solar activity proxies and
irradiances of various lines and bands. That is, using the obtained $\alpha$
and $\beta$ values, it is possible to calculate the irradiance of these
lines/bands from any of these proxies, expressed as:

$\displaystyle F=10^{\beta}(P-P_{0})^{\alpha}+F_{0}.$ (3)

We can even estimate the irradiances from proxies for targets having no
observation of the upper atmospheres. For example, irradiances can be
estimated from surface magnetic field distributions calculated by the solar
dynamo models or surface flux transport models, the surface magnetic field
distribution acquired by the stellar Zeeman-Doppler Imaging, or the starspot
sizes estimated from the visible light curve of the Sun-like stars.

XUV irradiance estimates are often based on scaling relationships with other
spectral lines or bands (e.g., Chamberlin et al., 2007, 2020; Linsky et al.,
2014, see also Section 1). However, since the model in this work uses the
daily solar activity proxies, although it cannot be used for short time scales
like solar and stellar flares, longer-term variations such as rotational
modulations and solar cycle variations can be estimated based on more physical
relationships, i.e., atmospheric heating owing to surface magnetic field.

Figure 9: Reconstruction of irradiances over the past centuries using the
historical records of activity proxy observations. The top three rows are the
monthly-averaged total sunspot number, total sunspot area, and the F10.7cm
radio flux. The four remaining rows are the synthesized X-rays 5.2–124 Å, Fe
XV 284 Å, Ly$\alpha$, and Mg II k 2796 Å. The black, blue, and orange curves
indicate the monthly data synthesized based on the sunspot number (top row),
spot area (second row), and the F10.7cm flux (third row), respectively. The
basal fluxes are shown as horizontal dashed lines. The gray hatch indicates
the period during which the scalings between the proxies and irradiances are
measured in this study (2010 May to 2020 February).

To demonstrate this approach, Figure 9 shows the “backcasting” of solar
irradiances in the past centuries based on the long-term solar observations.
In fact, the reconstruction of spectral radiations using the historical
records has been one of the key scientific targets for understanding the
atmospheric/chemical interactions of the Earth and planets (see, e.g., Kopp &
Shapiro, 2021). Here we used the total spot number (Section 2.2) since January
1749, total spot area (Section 2.3) since May 1874, and the F10.7cm radio flux
(Section 2.8) since February 1947. The irradiances that were reconstructed are
those whose scaling laws were verified by a comparison with stellar data in
TA22, i.e., X-rays 5.2–124 Å, Fe XV 284 Å, Ly$\alpha$, and Mg II k 2796 Å.
Although it is possible to reconstruct daily irradiances by using the daily
proxy data, for a better visualization, we synthesized monthly light curves
based on the monthly-averaged proxies.

The relative difference between two of the synthesized irradiances is
expressed as:

$\displaystyle d_{\rm TSN,TSA}=\frac{|F^{\rm TSN}-F^{\rm TSA}|}{(F^{\rm
TSN}+F^{\rm TSA})/2},$ (4) $\displaystyle d_{\rm TSN,F10.7}=\frac{|F^{\rm
TSN}-F^{\rm F10.7}|}{(F^{\rm TSN}+F^{\rm F10.7})/2},$ (5)

where $F^{\rm TSN}$, $F^{\rm TSA}$, and $F^{\rm F10.7}$ are the irradiances
based on the total sunspot number, total sunspot area, and the F10.7cm radio
flux, respectively. For the period during which the irradiances are derived
from multiple proxies, the median values of the relative differences are
$d_{\rm TSN,TSA}=14.5$% and $d_{\rm TSN,F10.7}=52.9$% for X-rays 5.2–124 Å,
$d_{\rm TSN,TSA}=22.2$% and $d_{\rm TSN,F10.7}=63.7$% for Fe XV 284 Å, $d_{\rm
TSN,TSA}=4.3$% and $d_{\rm TSN,F10.7}=18.6$% for Ly$\alpha$, and $d_{\rm
TSN,TSA}=2.7$% and $d_{\rm TSN,F10.7}=12.3$% for Mg II k 2796 Å. These values,
up to approximately 20% for the transition-region and chromospheric lines and
up to approximately 50% for the coronal lines, can be referred to as typical
errors when reconstructing irradiances using this method.

Possible sources of errors for this irradiance reconstruction method include
the errors in the proxy data (see, Clette et al., 2014, for errors in the
sunspot number data) and those in the power-law indices (i.e. $\alpha$ and
$\beta$ in Tables 2 to 6). Also, the fact that the power laws were derived
only for the Cycle 24, which showed a very weak activity, may cause additional
errors (see Section 7 for further discussion).

##  7 Summary and Discussion

In this study, we used the methodology described in TA22 to derive the scaling
laws between the solar activity proxies (not only the radial magnetic flux but
also the LOS flux, total sunspot number, total sunspot area, and the F10.7cm
flux) and the irradiances of various spectral lines and bands. By further
increasing the number of lines, especially of the transition region
temperatures, we investigated the variation of power-law index $\alpha$ from
the chromospheric to coronal temperatures, as shown in Figure 7.

Our results provide the framework for estimating spectral irradiances from the
proxy data. If one of the five proxies is given, one can estimate the
line/band irradiances by using the power-law indices $\alpha$ and offsets
$\beta$ provided in Tables 2 to 6. For instance, we can estimate the
irradiances from the total magnetic flux or total sunspot area of the Sun-like
stars obtained from modeling and observations. To demonstrate the usefulness
of this method, we reconstructed selected irradiances over the past centuries
based on the historical records of solar observations (Figure 9). The relative
differences between the synthesized irradiances was up to 20% for the
chromospheric and transition-region lines and up to 50% for the coronal lines,
which can be considered as the typical errors of the method.

It is also necessary to specify the limitations of this method. The scaling
laws were obtained from daily solar synoptic data over the last decade.
Therefore, this method can only be applied for reconstructing irradiance
variations of time scales longer than a day (i.e., quasi-stationary component)
and not for synthesizing transient brightenings, such as solar and stellar
flares (time scales of tens of minutes to hours). Additionally, because the
last 10 years was one of the weakest solar activity cycles in the last few
hundred years (e.g., Pesnell, 2020), one has to extrapolate the scalings to
obtain the irradiances of stronger cycles, as shown in Figure 9.

In addition, irradiances can only be reproduced for stars with almost the same
parameters as the current Sun. For example, the chemical abundance is fixed to
that of the current Sun, and hence, reproducing irradiances of stars with
significantly different abundances can be challenging. Nonetheless, it has
been verified by TA22 that the scalings are universal among G-type stars,
regardless of age or activity level. Therefore, the method discussed here can
be used as far as the irradiance synthesis is conducted for the main-sequence
G-dwarfs.

Another limitation is that H$\alpha$ and Ca II 8542 Å cannot be reproduced as
they brighten only in the declining phase of the solar cycle (Figure 1) and
show weak CCs against activity proxies. Based on the Sun-as-a-star monitoring,
Maldonado et al. (2019) reported that H$\alpha$ and other Balmer lines
(H$\beta$ and H$\gamma$) are inversely correlated with the sunspot number and
Ca II K intensity. However, these authors only used data over three years.
Meunier & Delfosse (2009) analyzed the data for several cycles and showed that
although H$\alpha$ and Ca II indices were positively correlated with the
activity cycle in the long term, their CCs varied with the phase of the
activity cycle. Therefore, the negative or no correlations for H$\alpha$ and
Ca II 8542 Å found in this study may be attributed to the timescale or the
activity phase of our sampling. It is also important to analyze spatially
resolved data of the Sun to investigate how individual structures such as
plages, filaments, and sunspots affect the chromospheric lines and spectra of
the Sun as a whole (e.g., Diercke et al., 2022).

For the active G-type main-sequence stars that emit superflares, Notsu et al.
(2019) found a strong positive correlation between the brightness variation
amplitude of visible light curves, which is an indicator of the starspot size,
and the Ca II 8542 Å and H & K intensities, as opposed to the expectation from
this study. It is possible that the solar Ca II 8542 Å line fluxes are in the
saturated regime in the atmospheres of solar-like stars, where they only show
a weak dependence on the Ca II K intensity (see Figure 5 of Takeda et al.,
2012). Cincunegui et al. (2007), who studied various stars ranging from F to
M, showed that although H$\alpha$ and Ca II H & K were strongly correlated as
a whole, this general trend was lost for individual stars. Reiners et al.
(2022) showed that H$\alpha$ in M-dwarfs had a positive correlation with the
magnetic flux with an exponent of $\alpha=1.43$. However, these authors noted
that H$\alpha$ requires a minimum average magnetic field strength of several
hundred G to ensure a detectable emission. Therefore, the chromospheric lines
that appear in absorption on the Sun may have different formation mechanisms
compared to the chromospheres of active stars. One possible explanation of
this difference can be attributed to the frequency of occurrence of coronal
flare events in active stars, which can heat the chromosphere via electron
beams and excite hydrogen line emissions. In contrast, frequent solar
microflares can mostly heat the transition region and do not contribute much
to the chromospheric heating.

This points to the importance of estimating spectral irradiances using the
scaling laws as well as examining the relationships between starspots and the
upper atmospheric variations by actually conducting long-term monitoring of
stars at multiple wavelengths. To this end, Toriumi et al. (2020) proposed the
methodology of estimating the size of stellar active regions by acquiring the
light curves for many different rotational phases, not only in the visible
band but also in the XUV band. Recently, it has become possible to track the
growth of starspots based on the long-term changes of dips in stellar visible
light curves and the starspot mapping technique (Namekata et al., 2019, 2020);
however, if there is contemporaneous XUV observation, we can also obtain clues
to understand how active region atmospheres evolve. For instance, whether the
rotational modulations of visible light and H$\alpha$ are correlated,
uncorrelated, or anti-correlated is a key to probe the chromospheric activity
of starspots (Maehara et al., 2021; Namekata et al., 2022; Schöfer et al.,
2022), which should be expanded to the XUV range.

In this study, we derived the scaling laws between the solar activity proxies
and the irradiances. However, the mutual relations between irradiances of
different spectral lines may also be utilized to investigate the physical
processes of the solar and stellar atmospheres (e.g., Linsky et al., 2014,
2020; Youngblood et al., 2017; France et al., 2018). In TA22, the power-law
indices were also obtained by dividing the total 10-year period into four
phases according to the solar activity, and it was found that $\alpha$ was
smallest during the cycle maximum and largest during the minimum. Although the
$\alpha$ values were derived only for the entire 10-year period in this study,
it is possible that $\alpha$ depends on activity phase, and this may cause
differences between the Sun and other stars. Future studies on such mutual
relations and cycle dependence are expected.

Another possible direction is to reconstruct the XUV irradiances using radio
fluxes. Currently, observations of the radio photosphere are performed for a
limited sample of G-type stars (Villadsen et al., 2014). However, the next
generation Very Large Array (ngVLA) can supposedly detect radio photospheres
of many more main-sequence stars (Carilli et al., 2019). In this study, strong
correlations were found between the F10.7cm (2.8 GHz) radio flux and the XUV
irradiances, which indicates that the radio fluxes can be useful proxies for
reconstructing stellar XUV line fluxes.

Understanding of the basal fluxes requires further investigations. For the
late-type stars, Schrijver (1987) found the power-law scalings of the X-ray
emission with the Ca II and Mg II emissions by subtracting the basal fluxes
for the chromospheric lines. Schrijver (1987) interpreted the basal fluxes as
a component due to pure acoustic heating of unmagnetized atmosphere. In this
study, however, we defined the basal fluxes as the medians of unspotted values
in the minimum of the solar activity cycle and, by subtracting the basal
fluxes from the light curves, we derived the power-law scalings. Magnetic
fluxes are ubiquitously distributed even in the quiet Sun during the cycle
minimum, causing the atmospheric heating above. Therefore, in this study, the
basal fluxes can be represented by the minimum magnetic flux and associated
heating (Schröder et al., 2012). This view may be supported by the fact that
non-thermal broadening is detected for the chromospheric and transition-region
lines during the cycle minimum (e.g., Ayres et al., 2021) (for further
discussions, see Testa et al., 2015; Linsky, 2019).

In TA22 and this study, the scalings were examined only for selected lines and
bands of the chromospheric to coronal temperatures. However, it is important
to extend these relations for the continuum components by evaluating the
scaling relationships between the entire XUV spectrum and the activity
proxies, such as the total magnetic flux, for every single wavelength bin,
including the continuum, rather than extracting the emission lines only. This
would make it possible to reconstruct the whole XUV spectra for the F-, G-,
and K-type stars. Radiative energy distributions over the wavelength for
planet-hosting stars can provide critical information for assessing the
efficiency of atmospheric escape from the (exo)planets orbiting them.
Derivation of such scalings requires further analysis that we defer to forth-
coming publications.

The authors acknowledge the comments suggested by the referee, which helped to
improve the quality of the paper. The authors would like to thank Dr. Aki
Takeda for useful comments on the XRT synoptic data. Data are courtesy of the
science teams of SDO, WDC-SILSO, USAF/NOAA, SORCE, GOES, Hinode, DRAO, and
SOLIS. HMI is an instrument on board SDO, a mission for NASA’s Living With a
Star program. Hinode is a Japanese mission developed and launched by
ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK)
as international partners. Scientific operation of the Hinode mission is
conducted by the Hinode science team organized at ISAS/JAXA. This team mainly
consists of scientists from institutes in the partner countries. Support for
the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.),
NASA, ESA, and NSC (Norway). ISS data were acquired by SOLIS instruments
operated by NISP/NSO/AURA/NSF. This work was supported by JSPS KAKENHI Grant
Nos. JP20KK0072 (PI: S. Toriumi), JP21H01124 (PI: T. Yokoyama), JP21H04492
(PI: K. Kusano), JP21J00316 (PI: K. Namekata), and JP21J00106 (PI: Y. Notsu).
V.S.A. was supported by the GSFC Sellers Exoplanet Environments Collaboration
(SEEC), which is funded by the NASA Planetary Science Division’s Internal
Scientist Funding Model (ISFM), NASA’s TESS Cycle 1, HST Cycle 27 and NICER
Cycle 2 project funds.

## References

  * Airapetian et al. (2020) Airapetian, V. S., Barnes, R., Cohen, O., et al. 2020, International Journal of Astrobiology, 19, 136, doi: 10.1017/S1473550419000132
  * Andretta & Jones (1997) Andretta, V., & Jones, H. P. 1997, ApJ, 489, 375, doi: 10.1086/304760
  * Avrett et al. (1994) Avrett, E. H., Fontenla, J. M., & Loeser, R. 1994, in Infrared Solar Physics, ed. D. M. Rabin, J. T. Jefferies, & C. Lindsey, Vol. 154, 35
  * Ayres et al. (2021) Ayres, T., De Pontieu, B., & Testa, P. 2021, ApJ, 916, 36, doi: 10.3847/1538-4357/abfa92
  * Ayres (2020) Ayres, T. R. 2020, ApJS, 250, 16, doi: 10.3847/1538-4365/aba3c6
  * Ayres (2021) —. 2021, ApJ, 908, 205, doi: 10.3847/1538-4357/abd095
  * Barczynski et al. (2018) Barczynski, K., Peter, H., Chitta, L. P., & Solanki, S. K. 2018, A&A, 619, A5, doi: 10.1051/0004-6361/201731650
  * Benz (2017) Benz, A. O. 2017, Living Reviews in Solar Physics, 14, 2, doi: 10.1007/s41116-016-0004-3
  * Benz & Güdel (2010) Benz, A. O., & Güdel, M. 2010, ARA&A, 48, 241, doi: 10.1146/annurev-astro-082708-101757
  * Berdyugina (2005) Berdyugina, S. V. 2005, Living Reviews in Solar Physics, 2, 8, doi: 10.12942/lrsp-2005-8
  * Bertello et al. (2011) Bertello, L., Pevtsov, A. A., Harvey, J. W., & Toussaint, R. M. 2011, Sol. Phys., 272, 229, doi: 10.1007/s11207-011-9820-8
  * Brajša et al. (1996) Brajša, R., Pohjolainen, S., Ruždjak, V., et al. 1996, Sol. Phys., 163, 79, doi: 10.1007/BF00165457
  * Brun & Browning (2017) Brun, A. S., & Browning, M. K. 2017, Living Reviews in Solar Physics, 14, 4, doi: 10.1007/s41116-017-0007-8
  * Carilli et al. (2019) Carilli, C., Butler, B., Golap, K., Carilli, M. T., & White, S. M. 2019, BAAS, 51, 243
  * Chadney et al. (2015) Chadney, J. M., Galand, M., Unruh, Y. C., Koskinen, T. T., & Sanz-Forcada, J. 2015, Icarus, 250, 357, doi: 10.1016/j.icarus.2014.12.012
  * Chamberlin et al. (2007) Chamberlin, P. C., Woods, T. N., & Eparvier, F. G. 2007, Space Weather, 5, S07005, doi: 10.1029/2007SW000316
  * Chamberlin et al. (2020) Chamberlin, P. C., Eparvier, F. G., Knoer, V., et al. 2020, Space Weather, 18, e02588, doi: 10.1029/2020SW002588
  * Charbonneau (2020) Charbonneau, P. 2020, Living Reviews in Solar Physics, 17, 4, doi: 10.1007/s41116-020-00025-6
  * Chen (2011) Chen, P. F. 2011, Living Reviews in Solar Physics, 8, 1, doi: 10.12942/lrsp-2011-1
  * Cheung & Isobe (2014) Cheung, M. C. M., & Isobe, H. 2014, Living Reviews in Solar Physics, 11, 3, doi: 10.12942/lrsp-2014-3
  * Cincunegui et al. (2007) Cincunegui, C., Díaz, R. F., & Mauas, P. J. D. 2007, A&A, 469, 309, doi: 10.1051/0004-6361:20066503
  * Clette et al. (2014) Clette, F., Svalgaard, L., Vaquero, J. M., & Cliver, E. W. 2014, Space Sci. Rev., 186, 35, doi: 10.1007/s11214-014-0074-2
  * Collier Cameron & Robinson (1989) Collier Cameron, A., & Robinson, R. D. 1989, MNRAS, 238, 657, doi: 10.1093/mnras/238.2.657
  * Cruddace et al. (1974) Cruddace, R., Paresce, F., Bowyer, S., & Lampton, M. 1974, ApJ, 187, 497, doi: 10.1086/152659
  * Davenport (2016) Davenport, J. R. A. 2016, ApJ, 829, 23, doi: 10.3847/0004-637X/829/1/23
  * Del Zanna et al. (2021) Del Zanna, G., Dere, K. P., Young, P. R., & Landi, E. 2021, ApJ, 909, 38, doi: 10.3847/1538-4357/abd8ce
  * Diercke et al. (2022) Diercke, A., Kuckein, C., Cauley, P. W., et al. 2022, A&A, 661, A107, doi: 10.1051/0004-6361/202040091
  * Fan (2021) Fan, Y. 2021, Living Reviews in Solar Physics, 18, 5, doi: 10.1007/s41116-021-00031-2
  * Fisher et al. (1998) Fisher, G. H., Longcope, D. W., Metcalf, T. R., & Pevtsov, A. A. 1998, ApJ, 508, 885, doi: 10.1086/306435
  * Fletcher et al. (2011) Fletcher, L., Dennis, B. R., Hudson, H. S., et al. 2011, Space Sci. Rev., 159, 19, doi: 10.1007/s11214-010-9701-8
  * France et al. (2018) France, K., Arulanantham, N., Fossati, L., et al. 2018, ApJS, 239, 16, doi: 10.3847/1538-4365/aae1a3
  * Gary & Hurford (1994) Gary, D. E., & Hurford, G. J. 1994, ApJ, 420, 903, doi: 10.1086/173614
  * Golub et al. (2007) Golub, L., Deluca, E., Austin, G., et al. 2007, Sol. Phys., 243, 63, doi: 10.1007/s11207-007-0182-1
  * Güdel (2004) Güdel, M. 2004, A&A Rev., 12, 71, doi: 10.1007/s00159-004-0023-2
  * Güdel (2007) —. 2007, Living Reviews in Solar Physics, 4, 3, doi: 10.12942/lrsp-2007-3
  * Hagenaar (2001) Hagenaar, H. J. 2001, ApJ, 555, 448, doi: 10.1086/321448
  * Harder et al. (2005) Harder, J., Lawrence, G., Fontenla, J., Rottman, G., & Woods, T. 2005, Sol. Phys., 230, 141, doi: 10.1007/s11207-005-5007-5
  * Harra et al. (2016) Harra, L. K., Schrijver, C. J., Janvier, M., et al. 2016, Sol. Phys., 291, 1761, doi: 10.1007/s11207-016-0923-0
  * Harvey & White (1999) Harvey, K. L., & White, O. R. 1999, ApJ, 515, 812, doi: 10.1086/307035
  * Johnstone et al. (2021) Johnstone, C. P., Bartel, M., & Güdel, M. 2021, A&A, 649, A96, doi: 10.1051/0004-6361/202038407
  * Klimchuk (2006) Klimchuk, J. A. 2006, Sol. Phys., 234, 41, doi: 10.1007/s11207-006-0055-z
  * Kopp et al. (2005) Kopp, G., Lawrence, G., & Rottman, G. 2005, Sol. Phys., 230, 129, doi: 10.1007/s11207-005-7433-9
  * Kopp & Shapiro (2021) Kopp, G., & Shapiro, A. 2021, Sol. Phys., 296, 60, doi: 10.1007/s11207-021-01802-8
  * Linsky (2019) Linsky, J. 2019, Host Stars and their Effects on Exoplanet Atmospheres, Vol. 955, doi: 10.1007/978-3-030-11452-7
  * Linsky et al. (2014) Linsky, J. L., Fontenla, J., & France, K. 2014, ApJ, 780, 61, doi: 10.1088/0004-637X/780/1/61
  * Linsky et al. (2020) Linsky, J. L., Wood, B. E., Youngblood, A., et al. 2020, ApJ, 902, 3, doi: 10.3847/1538-4357/abb36f
  * Livingston et al. (2007) Livingston, W., Wallace, L., White, O. R., & Giampapa, M. S. 2007, ApJ, 657, 1137, doi: 10.1086/511127
  * Maehara et al. (2012) Maehara, H., Shibayama, T., Notsu, S., et al. 2012, Nature, 485, 478, doi: 10.1038/nature11063
  * Maehara et al. (2021) Maehara, H., Notsu, Y., Namekata, K., et al. 2021, PASJ, 73, 44, doi: 10.1093/pasj/psaa098
  * Maldonado et al. (2019) Maldonado, J., Phillips, D. F., Dumusque, X., et al. 2019, A&A, 627, A118, doi: 10.1051/0004-6361/201935233
  * McClintock et al. (2005) McClintock, W. E., Rottman, G. J., & Woods, T. N. 2005, Sol. Phys., 230, 225, doi: 10.1007/s11207-005-7432-x
  * Meunier & Delfosse (2009) Meunier, N., & Delfosse, X. 2009, A&A, 501, 1103, doi: 10.1051/0004-6361/200911823
  * Namekata et al. (2019) Namekata, K., Maehara, H., Notsu, Y., et al. 2019, ApJ, 871, 187, doi: 10.3847/1538-4357/aaf471
  * Namekata et al. (2020) Namekata, K., Davenport, J. R. A., Morris, B. M., et al. 2020, ApJ, 891, 103, doi: 10.3847/1538-4357/ab7384
  * Namekata et al. (2021) Namekata, K., Maehara, H., Honda, S., et al. 2021, Nature Astronomy, 6, 241, doi: 10.1038/s41550-021-01532-8
  * Namekata et al. (2022) —. 2022, ApJ, 926, L5, doi: 10.3847/2041-8213/ac4df0
  * Narukage et al. (2011) Narukage, N., Sakao, T., Kano, R., et al. 2011, Sol. Phys., 269, 169, doi: 10.1007/s11207-010-9685-2
  * Notsu et al. (2019) Notsu, Y., Maehara, H., Honda, S., et al. 2019, ApJ, 876, 58, doi: 10.3847/1538-4357/ab14e6
  * Noyes et al. (1984) Noyes, R. W., Weiss, N. O., & Vaughan, A. H. 1984, ApJ, 287, 769, doi: 10.1086/162735
  * Pesnell (2020) Pesnell, W. D. 2020, Journal of Space Weather and Space Climate, 10, 60, doi: 10.1051/swsc/2020060
  * Pevtsov et al. (2003) Pevtsov, A. A., Fisher, G. H., Acton, L. W., et al. 2003, ApJ, 598, 1387, doi: 10.1086/378944
  * Pizzolato et al. (2003) Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., & Ventura, P. 2003, A&A, 397, 147, doi: 10.1051/0004-6361:20021560
  * Reiners et al. (2014) Reiners, A., Schüssler, M., & Passegger, V. M. 2014, ApJ, 794, 144, doi: 10.1088/0004-637X/794/2/144
  * Reiners et al. (2022) Reiners, A., Shulyak, D., Käpylä, P. J., et al. 2022, A&A, 662, A41, doi: 10.1051/0004-6361/202243251
  * Rezaei et al. (2007) Rezaei, R., Schlichenmaier, R., Beck, C. A. R., Bruls, J. H. M. J., & Schmidt, W. 2007, A&A, 466, 1131, doi: 10.1051/0004-6361:20067017
  * Rottman (2005) Rottman, G. 2005, Sol. Phys., 230, 7, doi: 10.1007/s11207-005-8112-6
  * Sanz-Forcada et al. (2011) Sanz-Forcada, J., Micela, G., Ribas, I., et al. 2011, A&A, 532, A6, doi: 10.1051/0004-6361/201116594
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207, doi: 10.1007/s11207-011-9834-2
  * Schöfer et al. (2022) Schöfer, P., Jeffers, S. V., Reiners, A., et al. 2022, A&A, 663, A68, doi: 10.1051/0004-6361/201936102
  * Schou et al. (2012) Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229, doi: 10.1007/s11207-011-9842-2
  * Schrijver (1987) Schrijver, C. J. 1987, A&A, 172, 111
  * Schrijver et al. (1989) Schrijver, C. J., Cote, J., Zwaan, C., & Saar, S. H. 1989, ApJ, 337, 964, doi: 10.1086/167168
  * Schröder et al. (2012) Schröder, K. P., Mittag, M., Pérez Martínez, M. I., Cuntz, M., & Schmitt, J. H. M. M. 2012, A&A, 540, A130, doi: 10.1051/0004-6361/201118363
  * Shibata & Magara (2011) Shibata, K., & Magara, T. 2011, Living Reviews in Solar Physics, 8, 6, doi: 10.12942/lrsp-2011-6
  * Shoda & Takasao (2021) Shoda, M., & Takasao, S. 2021, A&A, 656, A111, doi: 10.1051/0004-6361/202141563
  * Simões et al. (2015) Simões, P. J. A., Hudson, H. S., & Fletcher, L. 2015, Sol. Phys., 290, 3625, doi: 10.1007/s11207-015-0691-2
  * Skumanich (1972) Skumanich, A. 1972, ApJ, 171, 565, doi: 10.1086/151310
  * Skumanich et al. (1975) Skumanich, A., Smythe, C., & Frazier, E. N. 1975, ApJ, 200, 747, doi: 10.1086/153846
  * Solanki (2003) Solanki, S. K. 2003, A&A Rev., 11, 153, doi: 10.1007/s00159-003-0018-4
  * Strassmeier (2009) Strassmeier, K. G. 2009, A&A Rev., 17, 251, doi: 10.1007/s00159-009-0020-6
  * Takasao et al. (2020) Takasao, S., Mitsuishi, I., Shimura, T., et al. 2020, ApJ, 901, 70, doi: 10.3847/1538-4357/abad34
  * Takeda et al. (2016) Takeda, A., Yoshimura, K., & Saar, S. H. 2016, Sol. Phys., 291, 317, doi: 10.1007/s11207-015-0823-8
  * Takeda et al. (2012) Takeda, Y., Tajitsu, A., Honda, S., et al. 2012, PASJ, 64, 130, doi: 10.1093/pasj/64.6.130
  * Tapping (2013) Tapping, K. F. 2013, Space Weather, 11, 394, doi: 10.1002/swe.20064
  * Tapping & Charrois (1994) Tapping, K. F., & Charrois, D. P. 1994, Sol. Phys., 150, 305, doi: 10.1007/BF00712892
  * Testa et al. (2015) Testa, P., Saar, S. H., & Drake, J. J. 2015, Philosophical Transactions of the Royal Society of London Series A, 373, 20140259, doi: 10.1098/rsta.2014.0259
  * Toriumi (2014) Toriumi, S. 2014, PASJ, 66, S6, doi: 10.1093/pasj/psu100
  * Toriumi & Airapetian (2022) Toriumi, S., & Airapetian, V. S. 2022, ApJ, 927, 179, doi: 10.3847/1538-4357/ac5179
  * Toriumi et al. (2020) Toriumi, S., Airapetian, V. S., Hudson, H. S., et al. 2020, ApJ, 902, 36, doi: 10.3847/1538-4357/abadf9
  * Toriumi et al. (2017) Toriumi, S., Schrijver, C. J., Harra, L. K., Hudson, H., & Nagashima, K. 2017, ApJ, 834, 56, doi: 10.3847/1538-4357/834/1/56
  * Toriumi & Wang (2019) Toriumi, S., & Wang, H. 2019, Living Reviews in Solar Physics, 16, 3, doi: 10.1007/s41116-019-0019-7
  * Veronig et al. (2021) Veronig, A. M., Odert, P., Leitzinger, M., et al. 2021, Nature Astronomy, 5, 697, doi: 10.1038/s41550-021-01345-9
  * Vidotto (2021) Vidotto, A. A. 2021, Living Reviews in Solar Physics, 18, 3, doi: 10.1007/s41116-021-00029-w
  * Vidotto et al. (2014) Vidotto, A. A., Gregory, S. G., Jardine, M., et al. 2014, MNRAS, 441, 2361, doi: 10.1093/mnras/stu728
  * Villadsen et al. (2014) Villadsen, J., Hallinan, G., Bourke, S., Güdel, M., & Rupen, M. 2014, ApJ, 788, 112, doi: 10.1088/0004-637X/788/2/112
  * Woods et al. (2021) Woods, T. N., Harder, J. W., Kopp, G., et al. 2021, Sol. Phys., 296, 127, doi: 10.1007/s11207-021-01869-3
  * Woods & Rottman (2005) Woods, T. N., & Rottman, G. 2005, Sol. Phys., 230, 375, doi: 10.1007/s11207-005-2555-7
  * Wright et al. (2011) Wright, N. J., Drake, J. J., Mamajek, E. E., & Henry, G. W. 2011, ApJ, 743, 48, doi: 10.1088/0004-637X/743/1/48
  * Youngblood et al. (2016) Youngblood, A., France, K., Loyd, R. O. P., et al. 2016, ApJ, 824, 101, doi: 10.3847/0004-637X/824/2/101
  * Youngblood et al. (2017) —. 2017, ApJ, 843, 31, doi: 10.3847/1538-4357/aa76dd
  * Zhuleku et al. (2020) Zhuleku, J., Warnecke, J., & Peter, H. 2020, A&A, 640, A119, doi: 10.1051/0004-6361/202038022