Amstetten, ÖsterreichDr. Alexander I. Shapiro Max-Planck-Institut für
Sonnensystemforschung, Göttingen, Germany Dr. Natalie A. Krivova Max-Planck-
Institut für Sonnensystemforschung, Göttingen, Germany Prof. Dr. Stefan
Dreizler Institut für Astrophysik, Georg-August-Universität Göttingen,
Göttingen, Germany Prof. Dr. Stefan Dreizler Institut für Astrophysik, Georg-
August-Universität Göttingen, Göttingen, Germany Dr. Alexander I. Shapiro Max-
Planck-Institut für Sonnensystemforschung, Göttingen, Germany Prof. Dr. Gibor
BasriUniversity of California, Berkeley, United States of America Prof. Dr.
Sami K. Solanki Max-Planck-Institut für Sonnensystemforschung, Göttingen,
Germany Prof. Dr. Ariane Frey II. Physikalisches Institut, Georg-August-
Universität Göttingen, Göttingen, Germany Prof. Dr. W. Kollatschny Institut
für Astrophysik, Georg-August-Universität Göttingen, Göttingen, Germany Prof.
Dr. Hardi Peter Max-Planck-Institut für Sonnensystemforschung, Göttingen,
Germany 2020 2021

# Exploring the solar paradigm to explain stellar variability

Nina-Elisabeth Nèmec

######  ?contentsname?

  1. Abstract
  2. Zusammenfassung
  3. 1 Introduction
    1. 1.1 Solar variability
      1. 1.1.1 Magnetic activity and sunspots
      2. 1.1.2 Solar brightness variability
    2. 1.2 Stellar variability
      1. 1.2.1 Chromospheric and photometric brightness variations
      2. 1.2.2 The Sun in the stellar context
    3. 1.3 Motivation and thesis outline
  4. 2 Power spectra of solar brightness variations at various inclinations
    1. 2.1 Introduction
    2. 2.2 Methods
      1. 2.2.1 Calculating brightness variations
      2. 2.2.2 Surface flux transport model
      3. 2.2.3 From magnetic fluxes to area coverages
      4. 2.2.4 Model parameters
    3. 2.3 Solar brightness variations as seen by an ecliptic bound observer
      1. 2.3.1 TSI variability during activity cycles of different strengths
      2. 2.3.2 Solar variability in different passbands
    4. 2.4 Solar brightness variations as they would be seen from out of ecliptic
      1. 2.4.1 Effect of inclination on brightness variability
      2. 2.4.2 Disentangling evolution and rotation of magnetic features
      3. 2.4.3 The full time series
    5. 2.5 Conclusions and outlook
  5. 3 Connecting measurements of solar and stellar brightness variations
    1. 3.1 Introduction
    2. 3.2 Conversion from solar to stellar passbands
      1. 3.2.1 SATIRE-S
      2. 3.2.2 Filter systems
      3. 3.2.3 Results
    3. 3.3 Correction for the inclination
      1. 3.3.1 Approach
      2. 3.3.2 Results
    4. 3.4 Modelling Kepler light curves using VIRGO/SPM
    5. 3.5 Conclusions
  6. 4 Where have all the faculae gone? Explaining the transition from faculae to spot domination
    1. 4.1 Introduction
    2. 4.2 Model
      1. 4.2.1 Surface flux transport and filling factors
      2. 4.2.2 Fixing the parameters
    3. 4.3 Extending the model to stars more active than the Sun
    4. 4.4 What happens to the flux?
    5. 4.5 Conclusion
  7. 5 Explaining the dependence of stellar brightness variability on the rotation period
    1. 5.1 Introduction
    2. 5.2 Model
      1. 5.2.1 Flux emergence and transport
      2. 5.2.2 Defining the filling factors
      3. 5.2.3 Calculating the brightness variations
    3. 5.3 Light curves
    4. 5.4 Comparison to observations
    5. 5.5 Conclusions
  8. 6 Summary and prospects
  9. A Power spectra of solar brightness variations for TESS and Strömgren b and y
  10. B Obtaining the spot filling factors used in Sect. 5
  11. C Butterfly diagrams
  12. D Light curves of the full 4–years
  13. Publications
  14. Acknowledgements
  15. Curriculum vitae

## Abstract

The unprecedented precision of broadband stellar photometry achieved with the
planet-hunting missions CoRoT and Kepler initiated a new era in examining the
magnetically-driven brightness variations of hundreds of thousands of stars.
Such brightness variations are well studied and understood for the Sun. The
plethora of data allows to accurately compare solar and stellar brightness
variations. An intriguing question is whether the observed trends in the
stellar photometric variability (e.g. the dependence of the variability on the
stellar rotation period) can be explained by utilising the solar paradigm, in
particular the physical concepts of brightness variations learnt from the Sun.
The goal of this work is to find out, through comparison of observational and
simulated data, if any physical concepts of solar brightness variability have
to be altered to reproduce the distribution of Sun-like stars variabilities.

Comparisons between solar and stellar variability suffer from several
observational biases. Stellar brightness variations are routinely measured in
various spectral passbands and direct measurements of solar variability in
these passbands do not exist. Therefore, measurements of stellar variability
are often compared to measurements of the Total Solar Irradiance variability
(i.e. the spectrally integrated solar radiative flux at 1 AU from the Sun),
introducing potential biases. Additionally, observations of solar variability
are made from the equatorial plane, corresponding to a right angle between the
Sun’s rotation axis and the line-of-sight (the 7.25∘ tilt between the solar
rotation axis and the ecliptic plane can be neglected). In this thesis I build
a model based on a surface flux transport model (SFTM) and the Spectral And
Total Irradiance REconstruction (SATIRE) approach to calculate the effect of
the inclination and different passbands on the solar variability on both the
activity cycle (11 years) and the rotational (27 days) timescales. This model
is presented in Sect. 2. We quantify the rotational variability of the Sun as
it would be observed by different space missions and the effect of the
inclination in Sect. 3. In the next step we extend our model to stars that are
more active than the Sun. This extension is based on the observation that the
solar disk coverage by spots increases faster with the activity than that by
faculae. Until now such a behaviour has not been explained. I demonstrate in
Sect. 4 that the cancellation of small magnetic flux concentrations, which are
associated with faculae, is able to explain this behaviour. In Sect. 5 I
present calculations of brightness variations for fast-rotating stars. I
conclude that in order to model the observed dependence of the stellar
variability on the rotation period, the degree of nesting (i.e. the tendency
of active regions to emerge in the vicinity of previous emergences) of active
regions should increase with decreasing rotation periods.

## Zusammenfassung

Die beispiellose Präzision der Breitband-Sternfotometrie, die mit den
Planetensuchmissionen CoRoT und Kepler erreicht wurde, leitete eine neue Ära
bei der Untersuchung der magnetisch bedingten Helligkeitsschwankungen von
Hunderttausenden von Sternen ein. Solche Helligkeitsvariationen sind für die
Sonne gut untersucht und verstanden. Die Fülle der Daten ermöglicht einen
genauen Vergleich der solaren und stellaren Helligkeitsvariationen. Eine
faszinierende Frage ist, ob die beobachteten Trends in der photometrischen
Variabilität der Sterne (z.B. die Abhängigkeit der Variabilität von der
Rotationsperiode) mit Hilfe des Sonnenparadigmas erklärt werden können,
insbesondere der physikalischen Konzepte der Helligkeitsvariationen, die von
der Sonne abgeleitet wurden. Das Ziel dieser Arbeit ist es, durch Vergleich
von Beobachtungs- und Simulationsdaten herauszufinden, welche physikalischen
Konzepte der solaren Helligkeitsvariabilität geändert werden müssen, um die
Verteilung der sonnenähnlichen Sternvariabilitäten zu reproduzieren.

Vergleiche zwischen Sonnen- und Sternvariabilität leiden unter mehreren
Beobachtungsverzerrungen. Stellare Helligkeitsschwankungen werden routinemäßig
in verschiedenen spektralen Filtersystemen gemessen, und direkte Messungen der
Sonnenvariabilität in diesen Filtersystemen gibt es nicht. Daher werden
Messungen der stellaren Variabilität oft mit Messungen der Variabilität der
gesamten Sonneneinstrahlung (d.h. der wellenlängenintegrierten solaren
Strahlungsintensität im Abstand von 1 AE von der Sonne) verglichen, was zu
Verzerrungen führen kann. Zusätzlich werden Beobachtungen der solaren
Variabilität von der Äquatorebene aus gemacht, die einem rechten Winkel
zwischen der Sonnenrotationsachse und der Sichtlinie entspricht (die Neigung
von 7,25∘ zwischen der Sonnenrotationsachse und der Ekliptikebene kann
vernachlässigt werden). In dieser Arbeit baue ich ein Modell auf, das auf
einem Oberflächenfluss-Transportmodell (SFTM) und dem Ansatz der Spektralen
und totalen Strahlungsrekonstruktion (SATIRE) basiert, um die Auswirkung der
Neigung und verschiedener Filtersysteme auf die solare Variabilität sowohl auf
den Aktivitätszyklus (11 Jahre) als auch auf die Rotationszeitskalen (27 Tage)
zu berechnen. Dieses Modell wird in Sect. 2 vorgestellt. Ich quantifiziere die
Rotationsvariabilität der Sonne, wie sie von verschiedenen Raumfahrtmissionen
beobachtet werden würden, und den Einfluss der Inklination in Sect. 3. Im
nächsten Schritt erweitere ich das Modell auf Sterne, die aktiver sind als die
Sonne. Diese Erweiterung beruht auf der Beobachtung, dass die Bedeckung der
Sonnenscheibe durch Flecken mit der Aktivität schneller zunimmt als die durch
Fackeln. Ein solches Verhalten ist bisher nicht erklärt worden. Ich
demonstriere in Sect. 4, dass die gegenseitige Auslöschung kleiner
magnetischer Flusskonzentrationen, die mit Fackeln verbunden sind, dieses
Verhalten erklären kann. In 5 stelle ich Berechnungen von
Helligkeitsschwankungen für Sterne mit kurzen Rotationsperioden vor. Aktive
Regionen tendieren dazu, sich in der Nähe des Entstehungsorts anderer aktiver
Regionen zu bilden. Ich komme zu dem Schluss, dass zur Modellierung der
beobachteten Abhängigkeit der stellaren Variabilität von der Rotationsperiode
diese Tendenz mit sinkender Rotationsperiode verstärkt werden muss.

##  ?chaptername? 1 Introduction

###  1.1 Solar variability

####  1.1.1 Magnetic activity and sunspots

The Sun is only one of the many magnetically active stars known to exist in
the Universe. It holds a special position among those stars for us, as it
makes the Earth habitable. The magnetic activity of the Sun is quite dynamic
and its activity directly influences life on Earth. While phenomena like the
Northern and Southern polar lights are a beautiful manifestation of the
interaction between the solar magnetic activity and the Earth (a connection
normally referred to as space weather), the same interaction occasionally
makes our electrical grids and satellite systems susceptible to power outages.

The magnetic activity of the Sun reveals itself in many shapes and forms. Most
prominent of those are the so-called sunspots that appear on the solar surface
from time to time. Sunspot observations have a long history, with the earliest
observations made in ancient China. With the invention of the telescope in the
early 1700s, the Western world started more detailed studies of sunspots.
While there have been many observers, the first one to publish sunspot
observations was Fabricius (1611). These observations already revealed that
the Sun is rotating, as he tracked the movement of the sunspots across the
solar disc. The top panel in Fig. 1.1 gives the so-called butterfly diagram.
Each wing on this diagram has a time-span of about 11–years. That the
latitudinal position of the sunspot varies throughout the cycle was already
pointed out by Carrington (1858) and Spoerer (1883). They found that spots
emerge at higher latitudes at the beginning of the cycle and then migrating
towards the equator at the end of the cycle. Such a behaviour gives rise to
the characteristic butterfly wing-like shape. Spots also only emerge up to
latitudes of $\pm$30 degree around the equatorial region. The bottom panel in
Fig. 1.1 gives the average daily sunspot number and the 11-year periodicity is
even more prominently visible here. The solar cycle is also sometimes referred
to as Schwabe’s cycle, after its discoverer Heinrich Schwabe (Schwabe, 1844).
When Hale and Nicholson (1938) unraveled that sunspots are magnetic phenomena,
more questions opened up regarding the underlying dynamo action that produces
the observed patters in the sunspots. However, the exact way the solar dynamo
operates is still unclear even after the more than 80 years of active
research.

?figurename? 1.1: Sunspot areas as a function of time. Top panel: Latitudinal
dependence of the rotationally averaged sunspot area. Bottom panel:
rotationally averaged daily sunspot number as a function of time. Courtesy of
David Hathaway. http://solarcyclescience.com/solarcycle.html.

####  1.1.2 Solar brightness variability

Solar variability has various manifestations that are related to the change in
the solar activity. One of the most exciting of them is the variability of the
solar brightness, which is the main focus of this thesis. The solar irradiance
in a broad sense is the power per unit area received from the Sun at a
distance of 1 astronomical unit (AU). The amount of irradiance is wavelength
and time dependent. One common measure is the Total Solar Irradiance (TSI),
which is the measure of the spectrally integrated solar radiative flux at a
distance of 1 AU. The TSI is measured by spaceborne radiometers, which contain
an absorptive blackened cavity, which is maintained in thermal equilibrium.
The incident radiation is absorbed in the cavity and changes its temperature.
By accurate measurements of the cavity temperature the intensity of the
incident radiation can thus be determined.

Already the first spaceborne TSI measurements in the late 1970s revealed that
the amount of irradiance that the Earth receives is not constant, but varies
on multiple time-scales, including variations over the 11-year activity cycle.
We show the Physikalisch- Meterologisches Observatorium Davos (PMOD) composite
(Fröhlich, 2006)111version42_65_1709, ftp://ftp.pmodwrc.ch/pub/datain orange
lines Fig. 1.2, and the daily sunspot area as returned by the Spectral And
Total Irradiance REconstruction (SATIRE, Yeo et al., 2014) model in blue lines
for comparison. Evidently, the rise and fall in the TSI and sunspot area
happen at the same time, counterintuitively, the Sun is the brightest at
sunspot maximum. This is because of the presence of a second manifestation of
magnetic fields on the solar surface: bright faculae (green lines in Fig.
1.2). Faculae are more diffuse in nature than the dark spots and their area is
generally larger. At sunspot maximum, the irradiance change brought about by
faculae over weights that of spots so that TSI increases. The difference in
amplitude between cycle maximum and cycle minimum is around 0.1%, whereas day-
to-day variations can be considerably larger (up to 0.3%).

?figurename? 1.2: Disk area coverages of different solar magnetic features and
activity proxies as a function of time. The spot ($A_{s}$) and facular
($A_{f}$) disk area coverages are from Yeo et al. (2014). The TSI values are
from the PMOD (Fröhlich, 2006) composite and the S-index values are from
Bertello et al. (2016).

While sunspots and faculae are photospheric phenomena, the magnetic fields
also expand throughout the whole solar atmosphere. The layer directly above
the photosphere is called chromosphere. Temperatures rise from about 5770K to
50k - 100k K, and because of this high temperatures chromospheric spectral
lines appear in emission. One of the most famous chromospheric spectral
features are the Ca II H&K lines at 3933.66 Å and 3968.47 Å, respectively.
Monitoring of the Ca II H&K lines started in the late 1960s when Olin Wilson
founded the Mount-Wilson HK program, establishing the Mount-Wilson Observatory
(MWO) S-index (S-index for short, Vaughan and Preston, 1980) of chromospheric
activity. Measurements of the solar Ca II H&K lines were not made directly,
but by using the reflected light of the Moon. However, solar data from the
Mount-Wilson observatory are sparse, hence other observations from the NSO
Sacramento Peak, the Kodaikanal Observatory and the Lowell Observatory Solar-
Stellar Spectrograph have to be combined to obtain a composite that stretches
from the 1960s to now (Egeland et al., 2017). The solar S-index variation is
shown in panel d of Fig. 1.2. What is clear from Fig. 1.2 is that the Sun is
the brightest, when its chromospheric emission is the highest.

###  1.2 Stellar variability

The umbrella-term Sun-like covers cool, main-sequence dwarf stars with
convective envelopes and of spectral classes F, G, and M. The term solar-
analogues however narrows it down to cool stars with similar fundamental
parameters (effective temperature and metallicity as the Sun.

####  1.2.1 Chromospheric and photometric brightness variations

The MWO program not only observed the chromosperic variability of the Sun, but
is most-well known for its monitoring of roughly 2,000 Sun-like stars. These
long-term observations revealed that stars exhibit variations in their
chromospheric emissions as well, with some stars even displaying cyclic
variations (Baliunas et al., 1995; Saar and Brandenburg, 1999). This
variations can be related to the rotation period (Skumanich, 1972; Noyes et
al., 1984), with young stars rotating faster and having a tendency for
irregular or shorter cycles, whereas their older peers rotate slower and
exhibit more pronounced, longer cycles. Vaughan and Preston (1980) reported
that there is an apparent gap (now called the Vaughan-Preston Gap), between
highly active stars and stars with lower activity levels, later confirmed by
Brandenburg et al. (1998), who split the stars further into an active and
inactive branch. While the existence of the active branch has come under
scrutiny in more recent studies, the clustering of more inactive stars has
shown to be robust even with longer data-sets and different methods (see e.g.
Distefano et al., 2017; Boro Saikia et al., 2018; Olspert et al., 2018).

Complimentary ground-based photometric studies of the long-term variability of
solar-type stars have confirmed the existence of stellar activity cycles as
observed from the Mount-Wilson Survey. Lockwood et al. (1997) found, that with
increasing chromospheric variability,the photometric variability increases.
Additionally, as noted by Radick et al. (1998), younger, fast rotating stars
with higher chromosperic emission are the dimmest in terms of photometry at
high chromospheric emission levels, whereas older, slower rotating stars have
their photometric brightness maximum at the time of maximum chromospheric
activity, as observed from the Sun. Observational studies of the Sun (e.g.
Foukal, 1993; Chapman et al., 1997) have shown that with increasing spot area,
the ratio between facular and spot areas decreases. Shapiro et al. (2014) have
used the dependence of spot and facular areas on the solar S-index to
extrapolate from the solar case to more active stars and found that the anti-
correlation between photometric and chromospheric variability can be explained
by the transition from faculae to spot dominated on the activity timescale.
Reinhold et al. (2019) later found that this transition coincides with the
Vaughan-Preston gap.

These studies of the photometric activity were mostly long-term studies
conducted on the activity cycle timescale and data was only available for some
thousands of stars. Additionally, ground-based observations are also
challenging for various reasons (seeing, day-night-cycle, weather, only to
name a few). The field of stellar variability studies in terms of photometric
observations was truly revolutionized with the introduction of space born
observatories, such as the Convection, Rotation and planetary Transits (CoRoT
Bordé et al., 2003; Baglin et al., 2006), the Kepler telescope (Borucki et
al., 2010) and the Transiting Exoplanet Survey Satellite (TESS, Ricker et al.,
2014). While the primary goal of these missions was to detect exoplanets with
the transit method, their high precision and high cadence also opened up
possibilities to study stellar variability on the rotation period and below
(e.g. Basri et al., 2013) and retrieve the rotation periods of stars to
further explore the rotation period–stellar age relation (Walkowicz and Basri,
2013). The rotational variability of a sample of stars with detected rotation
periods and effective temperatures between 5500 and 6000 K from McQuillan et
al. (2014) is given as a function of rotation period in Fig. 1.3. This figure
reveals how sophisticated the picture of stellar variability is. While there
is a dependence of the variability on the rotation periods for stars with
periods lager than ten days, this not the case for the faster rotators.

####  1.2.2 The Sun in the stellar context

With some thousands of stars observed by ground-based, long-term photometric
and chromospheric activity monitoring programs and hundreds of thousands of
stars observed by space missions, we finally address an important question:
How does the Sun compare to Sun-like stars in terms of its variability?

We distinguish between the long-term variability on the cycle time-scale and
the short-term variability on the rotational timescale. When Vaughan and
Preston (1980) and Noyes et al. (1984) compared the solar chromosperic
emission to the chromosperic emission of Sun-like stars from the MWO survey,
they found that the Sun lies somewhat between the active and inactive branch,
suggesting some transition phase between those branches. Radick et al. (2018)
narrowed the analysis down with roughly 70 Sun-like stars, of which almost 80%
are actually even solar-analogues, they found that the solar-chromospheric
variability is even higher than the linear dependence of chromospheric
emission of the S-index as a function of the latter suggests. In terms of the
long-term photometric variability, stars with similar chromospheric activity
as the Sun showed considerably larger variations in brightness amplitude than
0.1%, by factors of 10 to 30. Importantly, Radick et al. (2018) acknowledged,
that the chosen solar cycle for this type of comparisons is of importance and
that also the inclination (that is the angle between a star’s rotation axis
and the line-of-sight of the observer) plays a role.

In order to compare the solar rotational variability to stars observed by
Kepler, we calculated the rotational variability of the Sun using the TSI of
the SATIRE-S model from solar cycles 21–24 (1976–2019). The rotational
variability of the Sun is represented by the red star in Fig. 1.3. Two things
are uncanny. Firstly, there is large spread of variabilities on the timescales
of the solar rotation and secondly the solar variability is unambiguously low
compared to its peers. Amazo-Gómez et al. (2020) have argued, that algorithms
like the auto-correlation function used by e.g. McQuillan et al. (2014) have
difficulties detecting the rotation period even of the Sun, as the light-
curves of those stars are rather irregular. Other factors complicate the
comparison in this picture as well. The inclination might be different for the
stars in the sample and in this instance also the passband of the instruments
used introduces hurdles that need to be overcome. Detection biases are one
side of the story, however Reinhold et al. (2020) found that the light curves
of stars for which they could detect rotation periods are vastly different
from the solar paradigm. These stars show more regular patterns in their
variability, which raises the question, if we can model the observed
distribution of brightness variations of Sun-like stars by simply
extrapolating from the solar paradigm, or if any physical concepts, such as
the distribution of the magnetic field on the surfaces of stars have to be
altered to reproduce the observed photometric trends.

?figurename? 1.3: Rotational variability as a function of the rotation rate of
Kepler stars with effective temperatures between 5500 and 6000 K with known
rotation periods. Grey points are the data from McQuillan et al. (2014), and
the red star represents the median value of the solar rotational variability
calculated using the SATIRE-S TSI for cycles 21–24.

###  1.3 Motivation and thesis outline

The aforementioned observational findings have lead to several questions, one
of the most intriguing ones being if the lessons learnt from the solar
paradigm can be applied to explain stellar variabilities. One of the main
goals of this thesis is to find out through comparison of observational and
simulated data, which, if any physical concepts of solar brightness
variability have to be altered to reproduce the distribution of variabilities
of Sun-like stars. Central to this approach is the model I present in Sect. 2.
In Sect. 3 I directly apply this model to quantify the effect of the
inclination and spectral passbands used by past and current space missions to
facilitate the solar-stellar comparison. I present a physics-based explanation
for the transition between faculae and spot domination of more active stars
using the model in Sect. 4. In Sect. 5 I aim to aim to model the observed
distribution of variability across rotation rates from 4 days down to 30 days
by employing the SFTM and changing the characteristics of the distribution of
the magnetic features. In Sect. 6 I present a summary of the most important
findings of this thesis and provide examples of further applications of our
model.

##  ?chaptername? 2 Power spectra of solar brightness variations at various
inclinations

The contents of this section are identical to the printed version of Nèmec,
N.-E. et al., A&A, Vol. 636, A42, 2020, reproduced with permission ©ESO. DOI:
10.1051/0004-6361/202037588
Contributions to the paper: I developed the model, produced the results and
provided the main scientific interpretation.

###  2.1 Introduction

Recent planet-hunting missions such as CNES’ Convection, Rotation and
planetary Transit (CoRoT, Baglin et al., 2006; Bordé et al., 2003), NASA’s
Kepler (Borucki et al., 2010), and the Transiting Exoplanet Survey Satellite
(TESS, Ricker et al., 2014) have opened up new possibilities for studying
stellar variability up to timescales of the rotational period and, in some
cases, even beyond. (Reinhold et al., 2017; Montet et al., 2017). The plethora
of data obtained by these missions underlines the need for a better
understanding and modelling of stellar brightness variations. One of the
possible approaches for such an approach to modelling is to rely on the solar
paradigm; that is, to take a model which reproduces the observed variability
of solar brightness and extend it to other stars. For example, such an
approach has been used by Witzke et al. (2018) , who extended the Spectral And
Total Irradiance REconstruction (SATIRE, Fligge et al., 2000; Krivova et al.,
2003) model of solar brightness variability to calculate brightness variations
over the timescale of the activity cycle in stars with different metallicities
and effective temperatures. Later, Witzke et al. (2020a) utilised a similar
model to investigate how the amplitude of the rotational stellar brightness
variability as well as the detectability of stellar rotation periods depend on
the metallicity. Here, we perform one more extension of the SATIRE model to
study how the amplitude of solar brightness variability depends on the angle
between solar rotation axis and directions to the observer (hereafter, the
inclination).

The brightness variability of the Sun is brought about by magnetic features
(such as dark spots and bright faculae) on its surface (see, e.g. reviews by
Solanki et al., 2013; Ermolli et al., 2013). The visibility of the magnetic
features and their brightness contrasts depend on the position of the observer
relative to the solar rotation axis. This causes the solar brightness
variability to depend on the inclination. A quantitative assessment of such a
dependence is of particular importance in attempting to answer the question of
how solar photometric variability compares to that of other stars. To properly
address this question, we need to take into account that the Sun is observed
from its near-equatorial plane (i.e. at inclinations close to $90^{\circ}$),
while stars are observed at random, mostly unknown, inclinations.

The effect of the inclination on solar variability can only be assessed with
models since solar brightness has never been measured out of the ecliptic. For
example, to account for possible long-term climate response to the change of
the Earth’s orbital inclination in relation to the solar equator, Vieira et
al. (2012) developed a model based on combining synoptic maps and disk images
obtained from the Helioseismic and Magnetic Imager (HMI, Schou et al., 2012)
data. They found that on timescales of several thousands of years, the total
solar irradiance (TSI) variability due to the change in the Earth’s orbital
inclination is negligibly small.

A number of studies have modelled the dependence of solar brightness
variability on the inclination over the timescale of the 11-year activity
cycle. These studies have been motivated by ground-based observations of Sun-
like stars that have revealed the Sun exhibits lower photometric variability
on the activity cycle timescale than most Sun-like stars with near-solar
levels of magnetic activity (Lockwood and Skiff, 1990; Lockwood et al., 2007;
Radick et al., 2018). Schatten (1993) proposed that this enigmatic behaviour
of the Sun is due to its equator-on view from the Earth. He found that the
amplitude of the activity cycle in solar brightness significantly increases
with decreasing inclination. Later, Knaack et al. (2001) and Shapiro et al.
(2014) employed a more accurate model and also found an increase of the
variability for the out-of-ecliptic observer, but the effect of the
inclination appeared to be considerably weaker than that reported by Schatten
(1993). All in all, the current consensus is that the effect of inclination
cannot explain the low variability of the Sun on the activity cycle timescale
and, consequently, other explanations for this have been proposed (Shapiro et
al., 2016; Witzke et al., 2018; Karoff et al., 2018).

Schatten (1993), Knaack et al. (2001), and Shapiro et al. (2014) assumed an
axisymmetric band-like distribution of faculae and spots. Such an assumption
is justifiable for modelling solar brightness variations on the activity cycle
timescale but it does not allow modelling brightness variability on the solar
rotational timescale. Indeed, the activity cycle variability is caused by the
overall modulation with regard to the solar surface coverage by magnetic
features from activity minimum to maximum and it depends only on the time-
averaged surface distribution of magnetic features (which can be approximated
by the axisymmetric band-like structure rather well). In contrast, rotational
variability is caused by the evolution of individual magnetic features and
their transits across the visible solar disc as the Sun rotates. Consequently,
it depends on the exact distribution of magnetic features.

An attempt to model the effect of the inclination on the rotational solar
brightness variability was recently carried out by Shapiro et al. (2016). They
used distribution of magnetic features on the visible solar disk provided by
Yeo et al. (2014) and obtained the distribution of magnetic features on the
far-side of the Sun (part of which would become visible for the observer not
bound to the Earth) assuming that the near and far sides of the Sun are point-
symmetric with respect to each other through the centre of the Sun. They found
that an observer bound to the ecliptic plane witnesses the Sun to be spot-
dominated on the rotational timescale, but with decreasing inclination the
amplitude of the rotational variability decreases (in contrast to the
brightness variability on the activity cycle timescale, which increases with
decreasing inclination) and the facular contribution becomes dominant. Despite
being more advanced relative to previous studies, the assumption of the point-
symmetric distribution of solar magnetic features employed in Shapiro et al.
(2016) does not account for the appearance and disappearance of magnetic
features which rotate in and out of the visible solar disc. This has led to
the contribution of a number of artefacts which did not allow for a study of
the effects of the inclination on the detectability of stellar rotation
periods. These effects might play, however, an important role in understanding
the observed distribution of rotation periods in Kepler stars (Reinhold et
al., 2019; van Saders et al., 2019). Also, these artefacts hindered the
accurate assessment of the inclination effect on the timescale of solar
rotation. Such an assessment is, in turn, needed for the interpretation of the
data from the planet-hunting missions. For example, the Kepler data indicated
that also solar brightness variability on the timescale of solar rotation
appears to be lower than that of most of the stars with known near-solar
fundamental parameters and rotation periods Reinhold et al. (2020).

Here we take a different approach than Shapiro et al. (2016) and we utilise a
surface flux transport model (SFTM, Cameron et al., 2010) to obtain the
distribution of solar magnetic features over the entire solar surface (i.e. on
both near and far sides of the Sun). This distribution is then fed into the
SATIRE model to calculate the solar brightness variability for different solar
activity levels, various photometric filter system used in stellar
observations, and at different inclinations. In particular, we show how the
change of the inclination affects the power spectrum of solar brightness
variations. This allows for a study of the impact of the inclination on
brightness variability depending on the timescale of the variability. In Sect.
2.2, we describe how we compute the solar disc area coverages by magnetic
features from the SFTM and then calculate the brightness variations following
the SATIRE model. We also list the main parameters of the model and explore
their impact on the brightness variations. In Sect. 2.3, we show how the
strength of an individual cycle affects the solar photometric variability in
different passbands before we move to different inclinations in Sect. 2.4. In
Sect. 2.4, we also decompose the solar brightness variability into components
arising from the evolution of magnetic features and from the solar rotation.
We present our main conclusions in Sect. 2.5.

###  2.2 Methods

####  2.2.1 Calculating brightness variations

We built our method based on the SATIRE model, in which brightness variations
on timescales longer than a day are attributed to the emergence and evolution
of magnetic field on the surface of the Sun, as well as on solar rotation
(Fligge et al., 2000; Krivova et al., 2003). The photospheric magnetic
features are divided into three main classes: sunspot umbra ($u$), sunspot
penumbra ($p$), and faculae ($f$). The intensities of these features and that
of the quiet Sun ($q$) depend on the wavelength and the cosine of heliocentric
angle $\theta$ ($\mu=\mathrm{cos}\theta$), but they are also time-independent.
The intensities were computed by Unruh et al. (1999) (following Castelli and
Kurucz, 1994) with the use of the spectral synthesis code ATLAS9 (Kurucz,
1992). The 1D atmospheric structures of umbra, penumbra, and quiet Sun were
calculated using radiative equilibrium models, while the facular model is a
modified version of FAL-P by Fontenla et al. (1993).

The spectral solar irradiance $S(t,\lambda_{w})$ (i.e. spectral radiative flux
from the Sun, normalised to one AU), where $t$ is the time and $\lambda_{w}$
the wavelength (which should not be confused with $\lambda$ used for the
latitude later in this paper), is calculated by summing the intensities
weighted by the corresponding fractional disc area coverages of the magnetic
features (designated with the index $k$) as given by

$S(t,\lambda_{w})=S^{q}(\lambda_{w})+\sum_{mn}\sum_{k}(I_{mn}^{k}(\lambda_{w})-I_{mn}^{q}(\lambda_{w}))\,\alpha_{mn}^{k}(t)\Delta\Omega_{mn}.$
(2.1)

Here the summation is done over the pixels of the magnetograms and the $m$ and
$n$ indexes are the pixel coordinates (longitude and latitude, respectively),
$\alpha_{mn}^{k}$ is the fraction of pixel ($m$,$n$) covered by the magnetic
feature $k$, $\Delta\Omega_{mn}$ is the solid angle of the area on the solar
disc corresponding to one pixel, as seen from the distance of 1 AU, and Sq is
the quiet Sun irradiance, defined as

$S^{q}(\lambda_{w})=\sum_{mn}I_{mn}^{q}(\lambda_{w})\Delta\Omega_{mn}.$ (2.2)

The solid angles of pixels as well as corresponding intensity values depend on
the vantage point of the observer. Consequently, the solar irradiance values
$S(t,\lambda_{w})$ given by Eq. (2.1) also depend on the vantage point of the
observer and, in particular, on the inclination.

####  2.2.2 Surface flux transport model

To simulate the full surface distribution of magnetic features, we use the
SFTM in the form presented in Cameron et al. (2010). The SFTM describes the
passive transport of the radial component of the magnetic field B, considering
the effects of differential rotation $\Omega(\lambda)$ (with $\lambda$ being
the latitude), meridional flow $\nu(\lambda)$ at the solar surface, and a
horizontal surface diffusion thanks to a non-zero diffusivity $\eta_{H}$. The
emerged active regions gradually disperse due to the radial diffusion
$\eta_{r}$, with the flux finally decaying after cancellation between opposite
polarities, where they overlap. The governing equation is

$\begin{split}\frac{\partial B}{\partial t}&=-\Omega(\lambda)\frac{\partial
B}{\partial\phi}-\frac{1}{R_{\odot}\cos\lambda}\frac{\partial}{\partial\lambda}(\nu(\lambda)B\cos(\lambda))\\\
&+\eta_{H}\left(\frac{1}{R_{\odot}^{2}\cos\lambda}\frac{\partial}{\partial\lambda}\left(\cos(\lambda)\frac{\partial
B}{\partial\lambda}\right)+\frac{1}{R_{\odot}^{2}\cos^{2}\lambda}\frac{\partial^{2}B}{\partial\phi^{2}}\right)\\\
&+D(\eta_{r})+S(\lambda,\phi,t),\end{split}$ (2.3)

where $R_{\odot}$ is the solar radius, $\phi$ is the longitude of the active
region, and $D$ is a linear operator that describes the decay due to radial
diffusion with the radial surface diffusivity $\eta_{r}$. For the linear
operator $D$ the form of Baumann et al. (2006) was used. The horizontal
diffusivity $\eta_{H}$ was taken to be 250 km2s-1 as in Cameron et al. (2010)
and the radial surface diffusivity $\eta_{r}$ was set to 25 km2s-1 according
to Jiang et al. (2011b). The time average (synodic) differential rotation
profile was taken from Snodgrass (1983) and is given as (in degree per day):

$\Omega(\lambda)=13.38-2.3\cdot\sin^{2}\lambda-1.62\cdot\sin^{4}\lambda.$
(2.4)

The time-averaged meridional flow is expressed following van Ballegooijen et
al. (1998), namely,

$\nu(\lambda)=\begin{cases}11\cdot\sin(2.4\lambda)\quad\text{m/s,}&\text{where}\quad\lambda\leq
75^{\circ}\\\ 0,&\text{otherwise}.\end{cases}$ (2.5)

The source term $S(\lambda,\Phi,t)$ in Eq. (2.3) describes the magnetic flux,
which is prescribed to be in the form of two patches with opposite polarities
(van Ballegooijen et al., 1998; Baumann et al., 2004). The patches are centred
at $\lambda_{+}$ and $\phi_{+}$ for the positive polarity patch and
$\lambda_{-}$ and $\phi_{-}$ for the negative polarity patch. The field of
each patch is given by

$B^{\pm}(\lambda,\phi)=B_{\mathrm{max}}\left(\frac{0.4\Delta\beta}{\delta}\right)^{2}e^{-2[1-\cos(\beta_{\pm}(\lambda,\phi))]/\delta^{2}},$
(2.6)

where $B^{\pm}$ is the flux density of the positive and negative polarity,
$\beta_{\pm}$($\lambda$,$\phi$) are the heliocentric angles between point
($\lambda$, $\phi$) and the centres of the polarity patches, $\Delta\beta$ is
the separation between the two polarities and $\delta$ is the size of the
individual polarity patches, taken to be 4∘. $B_{\mathrm{max}}$ is a scaling
factor introduced by Cameron et al. (2010) and Jiang et al. (2011b) and was
fixed to 374 G. This value was found by forcing the total unsigned flux to
match the measurements from the Mount Wilson and Wilcox Solar Observatories.

Jiang et al. (2011a) constructed a semi-empirical source term
$S(\lambda,\Phi,t)$ for the 1700–2010 period so that its statistical
properties reflect those of the Royal Greenwich Observatory sunspot record.
Here we adopt the $S(\lambda,\Phi,t)$ term from Jiang et al. (2011a) but with
one important modification. As an observer stationed at a vantage point
outside the ecliptic sees both the near- and far-sides of the Sun (as defined
by the Earth-bound observer), it is crucial to avoid any systematic
differences between the active region distributions on the two sides. To this
purpose we have modified $S(\lambda,\Phi,t)$ so that the emergence of active
regions happens at random longitudes, whereas the butterfly-like shape of
their latitudinal emergence, as well as the number of emergences and the tilt-
angle distributions, over the course of the cycle is preserved.

All in all, the adapted source term describes the emergence of active regions
on the solar surface in a statistical way. We stress that the goal of this
study is not to reproduce the exact solar light curve as it would be seen from
outside the ecliptic, but to study the effect of the inclination on the power
spectrum of solar brightness variations at different levels of solar activity.
The statistical representation of the source term is fully sufficient for this
purpose.

####  2.2.3 From magnetic fluxes to area coverages

The SFTM returns simulated magnetograms, with a pixel-size of
1${}^{\circ}\rm\times$ 1∘. We follow the approach of Dasi-Espuig et al. (2014)
and divide each pixel ($m$,$n$) into 100 sub-pixels, with a size of
0.1${}^{\circ}\rm\times$ 0.1∘ each.

To calculate the brightness variations, we need to distinguish between spots
and faculae. The spot areas and positions at the day of emergence have been
provided by Jiang et al. (2011a) together with the source term
$S(\lambda,\Phi,t)$. After spots emerge, their positions on the solar surface
are affected by the differential rotation described by Eq. (2.4) and the
meridional flow described by Eq. (2.5). The spot sizes are calculated by
following a decay law during their evolution. We have found studies in the
literature that support linear and parabolic decay laws and different values
for the decay rate (Moreno-Insertis and Vazquez, 1988; Martinez Pillet et al.,
1993; Petrovay and van Driel-Gesztelyi, 1997; Baumann and Solanki, 2005;
Hathaway and Choudhary, 2008). As Baumann and Solanki (2005) found, it is not
possible to distinguish between a linear and parabolic decay law from, for
example, the area distribution of sunspots. For simplicity, we chose a linear
decay law of:

$A(t)=A_{0}-R_{d}\cdot(t-t_{0}),$ (2.7)

where $A(t)$ is the area on a given day $t$ and $t_{0}$ is the day on which
the spot has its maximum area $A_{0}$ (provided in the input). The decay rate
$R_{d}$ is measured in microsemi-hemispheres (MSH) per day and is a semi-free
parameter of the model, which will be discussed in more detail in Sect. 2.2.4.
The decay rate $R_{d}$ has been studied extensively before. In particular,
Martinez Pillet et al. (1993) have reported several values of the decay rate,
ranging from 25 to 47 MSH day-1. The value we found to be most optimal for our
model is 80 MSH day-1 (see a detailed description of the procedure used to
determine $R_{d}$ in Sect. 2.2.4). The slightly higher value, compared with
observational estimates, which we obtained for our modelling can be explained
by the low spatial resolution of the source term in Eq. (2.6). A group of
spots might be represented by one large spot (due to the resolution of the
source term), which then decay with a rate that is equal to the sum of the
decay rates of the individual spots.

Having established the spatial and temporal spot distribution, we can then
correct the simulated magnetograms for the spot magnetic flux, which is
important for the masking of the faculae. The correction is done on the
original 1${}^{\circ}\rm\times$ 1∘ grid corresponding to the SFTM output
since, in contrast to the spot distribution which is calculated on the
0.1${}^{\circ}\rm\times$ 0.1∘ grid, we calculate more diffuse facular
distribution on the original grid. If a 1${}^{\circ}\rm\times$ 1∘ pixel is
found to be free of spots, the correction is equal to 0 and the magnetic field
in the pixel is directly taken from the SFTM. If a given pixel is found to be
partially covered by spots the magnetic field in the pixel is corrected as,

$B^{\prime}_{\mathrm{(m,n)}}=B_{\mathrm{m,n}}-B_{\mathrm{spot}}\cdot
a^{s}_{\mathrm{m,n}},$ (2.8)

where $B_{\mathrm{m,n}}$ is the pixel field returned by the SFTM,
$B_{\mathrm{spot}}$ is the mean magnetic field of a spot, and $a^{s}_{\rm mn}$
is the fractional coverage of the pixel (m,n) by spots. The value of $B_{\rm
spot}$ is taken from observations. Keppens and Martinez Pillet (1996) have
measured the umbral and penumbral field strength of solar sunspots. We do not
distinguish between umbral and penumbral regions and we use an area-weighted
average of the values of 800 G reported in Keppens and Martinez Pillet (1996).

The remaining magnetic field $B^{\prime}_{\mathrm{(m,n)}}$ (with
$B^{\prime}_{\mathrm{(m,n)}}=B_{\mathrm{(m,n)}}$ for pixels free of spots) is
then attributed to faculae and is calculated following the SATIRE approach:

$\alpha^{f}_{\rm m,n}=\begin{cases}\frac{B^{\prime}_{\rm m,n}}{B_{\rm
sat}}&\text{if B${}_{\mathrm{mn}}$ $<$ B${}_{\mathrm{sat}}$}\\\ 1&\text{if
B${}_{\mathrm{mn}}$ $\geq$ B${}_{\mathrm{sat}}$},\end{cases}$ (2.9)

where Bsat is the saturation threshold, in accordance to the SATIRE-S model
(Krivova et al., 2003; Wenzler et al., 2004; Ball et al., 2012). In this
model, the facular filling factor increases linearly with the magnetic field
strength, eventually reaching unity at a saturation. Given that the SFTM
provides information only at time of the maximum area and during the
subsequent decay of the active regions, we need to additionally consider the
growth phase of the spots (i.e. take into account that they do not emerge
instantaneously). We employ a linear growth law with a constant rate $R_{g}$
similar to the decay law given by Eq. (2.7). For $R_{g}$ we have not found any
appropriate studies so that it is treated as a free parameter (see the next
section).

####  2.2.4 Model parameters

?figurename? 2.1: Comparison of the power spectra of solar brightness variations produced by our model to those given by the PMOD-composite as well as SATIRE-S and SATIRE-T2 models for cycle 22 (panel a) and cycle 23 (panel b) and the combined timeseries (panel c). The vertical dashed black line indicates the synodic solar rotation period of 27.3 days. ?tablename? 2.1: List of the parameters used in our model Parameters | Description | Best value
---|---|---
Rd | decay rate spots | 80 MSH day-1
Rg | growth rate spots | 600 MSH day-1
Bsat | saturation threshold faculae | 500 G

To find the best set of model parameters, we compare the power spectra of the
computed TSI time series to the power spectra of TSI from other sources. We
use the Physikalisch-Meterologisches Observatorium Davos (PMOD) composite
(Fröhlich, 2006, version 42_65_1709,ftp://ftp.pmodwrc.ch/pub/data), which
provides TSI measurements over several decades. We also use the TSI output
from the SATIRE-S (Yeo et al., 2014) and SATIRE-T2 (Dasi-Espuig et al., 2016)
solar irradiance variability reconstruction models. In SATIRE-S the
distribution of magnetic features on the solar surface is derived from full
disk images and magnetograms of the Sun, whereas in SATIRE-T2 it is derived
from a SFTM but with a different source term than employed in this study.

In cycle 21, both the PMOD composite and SATIRE-S contain a significant amount
of data gaps that would affect the power spectra. We therefore restrict
ourselves to use cycles 22 and 23 for the determination of the best parameter
set. We show the power spectra of the solar brightness variations as presented
by PMOD, SATIRE-S and SATIRE-T2 in Fig. 2.1. One striking difference between
the datasets is that SATIRE-S and SATIRE-T2 show higher power values compared
to the PMOD-composite at periods below five days for both considered cycles.
We attribute this to aliasing effects being present in the two SATIRE-models.
Both, SATIRE-S and SATIRE-T2, give one instantaneous value of the TSI per day,
whereas the PMOD-composite gives daily averages. Consequently, the difference
between the power spectra appears because of the comparison between
instantaneous values (affected by aliasing) and daily averages. To avoid
aliasing issues in our model output, we calculate solar brightness with a six-
hour cadence. We found that this leads to similar values of spectral power
starting from timescales of about two days as the PMOD-composite.

We found our best set of parameters (see Table LABEL:paper_1_best_params) by
comparing the power spectra obtained with the output of our model to those
obtained with the PMOD composite. Namely, we calculated the $\chi^{2}$ values
using the parts of the power spectra below the solar rotation period (i.e. we
only considered periods shorter than 27.3 days). Despite having used only low-
period parts of the power spectra for the fit, we find that we are still able
to maintain a reasonable agreement on longer timescales as well. Our
calculations seem to slightly overestimate the variability on the activity
timescale, which can be attributed to the absence of ephemeral regions in our
model (see discussion in Dasi-Espuig et al., 2016).

?figurename? 2.2: Effect of the different parameters of the model on the
brightness variations. Panel a) shows the effect of the decay rate Rd on the
spot component only, panel b) the effect of Bsat on the total power spectrum,
and panel c) the effect of different growth rates Rg on the total power
spectrum compared to not having the spot growth included as depicted by the
red curve. Rd and Rg are in units of MSH day-1. The vertical dashed black line
indicates the synodic solar rotation period at 27.3 days.

Let us also check how the different free parameters of our model affect the
power spectrum of solar brightness variations returned by the model. The
effects of the spot decay rate $R_{d}$ (panel a), Bsat value (panel b), and
spot growth rate $R_{g}$ (panel c) are illustrated in Fig. 2.2. With
decreasing spot decay rate, $R_{d}$, the overall area coverage of the spots is
increasing, which affects timescales longer than about 10 days (Shapiro et
al., 2020). The prominent peak at the rotation period for the $R_{d}=26.5$ MSH
day-1 is a result of the long lifetime of the spots. The longer the spot
lives, the higher the probability it reoccurs at the next rotation which leads
to the formation of the rotation harmonic in the power spectrum.

The effect of the saturation threshold, $B_{sat}$, is shown in Fig. 2.2 b. We
note that the facular filling factors are primarily regulated via this
parameter. On the activity cycle timescale, faculae are the dominant source of
variability, whereas on timescales, below 100 days, the spot component is the
main driver of the variability. A value of 500 G for $B_{\rm sat}$ leads to
the best fit compared to the PMOD-composite. In contrast to the effect of the
decay rate, $R_{d}$, the growth rate , $R_{g}$, shows the highest impact on
timescales below 10 days (see right panel of Fig. 2.2). The value of 600 MSH
day-1 gives the best agreement with the PMOD composite on those timescales.

###  2.3 Solar brightness variations as seen by an ecliptic bound observer

####  2.3.1 TSI variability during activity cycles of different strengths

?figurename? 2.3: Power spectrum of the TSI for different cycles as seen by an
ecliptic bound observer. The vertical dashed black line indicates the synodic
solar rotation period at 27.3 days.

Until now, we considered the TSI variability during cycles 22 and 23. To
understand the solar brightness variations in the context of stellar
variability, it is important to explore different activity levels. With our
source term we can calculate solar brightness variations back to 1700. In Fig.
2.3, we compare power spectra of the TSI variability as returned by our model
for cycles 16 (one of the weakest cycle over the last 300 years), 19 (the
strongest cycle observed so far), 22, and 23. For cycle 16 and 23, a small
peak at the rotation period of about 27 days can be seen. The profile of the
power spectrum for cycle 19 is rather surprising, with two peaks on periods
slightly below (25 days) and above (32 days) the rotation period (see also
Fig. 2.5 where the double peak structure is more easily visible). Shapiro et
al. (2020) explained such a double-peak structure by the cancellation of spot
and facular contribution to the rotation signal. Witzke et al. (2020a) further
analysed the connection between the power spectrum profile and detectability
of the rotation period.

Recently a lot of effort has been put into determining stellar rotation
periods from photometric observations by the Kepler telescope (see e.g.
Reinhold et al., 2013; McQuillan et al., 2014; Angus et al., 2018). In what
appears to be an intriguing result, the detection of the rotation period of
old stars with near-solar level of magnetic activity seems to be challenging
due to the low amplitude of the irradiance variability, short lifetime of
spots, and the cancellation of the rotational signal from spots and faculae
(Aigrain et al., 2015; Shapiro et al., 2017; Reinhold et al., 2019). In
agreement with previous studies (e.g. Lanza and Shkolnik, 2014; Aigrain et
al., 2015), our analysis indicates that the same star can be deemed as
periodic or non-periodic (according to the definition of McQuillan et al.,
2014), depending on whether it is observed at high or low activity.

####  2.3.2 Solar variability in different passbands

In this section, we explore solar brightness variations as they would be
observed in different passbands. We multiply the computed spectral irradiance
given by Eq. (2.1) with the response functions of different filter systems and
then integrate over the corresponding wavelength ranges. We consider the
Strömgren filters b and y which have been widely used in ground-based
observations to study long-term stellar photometric variability (Radick et
al., 2018), as well as the Kepler and TESS passbands. The transmission curves
and the quiet-Sun spectrum (according to the SATIRE model) are shown in Fig.
2.4. The Strömgren b and y filters are centred at 476 and 547 nm,
respectively, so that Strömgren b is located around the maximum of the solar
spectrum, while Strömgren y is shifted to the red. The primary goal of Kepler
was to find planets around solar-type stars and its filter profile covers
almost the whole visual wavelength range. TESS is aimed at observing a large
number of M dwarfs and is, consequently, more sensitive to the red part of the
spectrum.

?figurename? 2.4: Response functions of the different filter systems used in
this work. The quiet-Sun irradiance as used by SATIRE is shown in grey.
?figurename? 2.5: Power spectra of solar brightness variations in different
filter systems for different cycles as observed from the ecliptic. Panel a)
shows cycle 19, panel b) cycle 22. The vertical dashed black line indicates
the synodic solar rotation period at 27.3 days.

We compare the different filter systems and their effect on the measured
variability for different cycles as observed by a solar equator-bound observer
in Fig. 2.5. Interestingly, the shapes of the power spectra are very similar
on timescales below about a year. On timescales below 1 year, the variability
in the two narrow-band Strömgren filters shows the highest power, followed by
Kepler, whereas the brightness variations as they would be observed by TESS
show the lowest amplitude.

On timescales above one year the variability in the Kepler, TESS and Strömgren
y passband have similar strength, whereas the signal in Strömgren b is
considerably lower. For the Strömgren b filter, Shapiro et al. (2016) have
found that the facular and spot contributions to the variability almost cancel
each other out, hence, the variability is low. The compensation is less
pronounced in the other passbands.

###  2.4 Solar brightness variations as they would be seen from out of
ecliptic

In the following, we refer to the inclination as the viewing angle of the
observer with respect to the solar rotation axis. An inclination of 90∘
corresponds to an observer in the solar equatorial plane, while inclinations
of <90∘ refer to a displacement of the observer from the equatorial plane
towards the North pole.

####  2.4.1 Effect of inclination on brightness variability

We now consider the variability during cycles 19 and 22 as it would be
observed by Kepler. The power spectra of brightness variations as they would
be seen at 90∘ (i.e. from the equatorial plane), at 57∘ (which is the mean
value of the inclination for a random distribution of orientations of rotation
axes), and at 0∘ (i.e. the view at the solar North pole) are plotted in Fig.
2.6.

?figurename? 2.6: Power spectra of solar brightness variations in the Kepler
passband with at different inclinations and two different cycles. Panel a)
shows cycle 19 and b) cycle 22. The vertical dashed black lines indicate the
synodic solar rotation period at 27.3 days. ?figurename? 2.7: Ratios of the
power spectra of solar brightness variations for Kepler as shown in Fig. 2.6.
Blue lines represent the ratios of 57∘ to 90∘ and orange lines between 0∘ and
90∘. Panel a) shows cycle 19 and b) cycle 22. The vertical dashed black line
indicates the synodic solar rotation period at 27.3 days. The horizontal solid
black line indicates a ratio of 1.

The power at the rotational timescale drops with decreasing inclination, but
the variability on the activity timescale increases. This effect is not strong
between 90 and 57∘ inclination, but significant between 90 and 0∘.
Interestingly, the double-peak structure of cycle 19 that has been described
before for the ecliptic-bound observer, is also present for the inclination of
57∘, although the peaks are less pronounced. For the observer at 0∘, the power
in the signal below 100 days is significantly lower than for the 90 and 57∘
vantage point. However, on timescales longer than 100 days, the power becomes
higher compared to the other vantage points. We discuss this result in more
detail in Sect. 2.4.2. We also show the power spectra of brightness variations
as observed by TESS and in the two Strömgren filters in the Appendix (Fig.
A.1–A.3) for cycle 19 only.

The impact of the inclination on the power spectrum becomes more evident in
Fig. 2.7, where we show the ratios between the power as it would be measured
at inclinations of 57∘ and 0∘ relative to that obtained by an ecliptic-bound
observer. In agreement with Fig. 2.6 the power on timescales below 200 days
decreases with decreasing inclinations, whereas longward of 200 days the power
increases with decreasing inclination. The reason for the increase of the
variability is due to several effects. Most noteworthy are the effects of
foreshortening and centre-to-limb variations (CLV). In the wavelength regime
where Kepler operates, the facular contrast (compared to the quiet Sun) is
higher at the limb due to limb-darkening, whereas the spot contrast is the
strongest at disc centre, as seen by an ecliptic bound observer. With
decreasing inclination, the effect of CLV on the facular component is less
pronounced and the facular contribution to the brightness variations is
increasing (conversely, the effect of the spots is decreasing). While the
effect of foreshortening is decreasing with decreasing inclination, it is not
enough to compensate for the stronger contrast of the faculae. For a more
detailed discussion see Shapiro et al. (2016). The distribution of the
magnetic features (in particular the spot distribution) is also important, as
we discuss in the next section.

####  2.4.2 Disentangling evolution and rotation of magnetic features

?figurename? 2.8: Comparison of the power spectra of the solar brightness
variations in the Kepler passband, with and without taking the solar rotation
into account (orange and blue, respectively). Panel a) shows 90∘ inclination,
b) 57∘ and c) 0∘. The vertical dashed black lines indicates the synodic solar
rotation period at 27.3 days.

The solar brightness variability is caused by changes in the solar disc
coverage by magnetic features. These changes are in turn due to (1) emergence
and evolution of magnetic features and (2) the solar rotation, which causes
transits of individual magnetic features across the visible solar disk (see,
e.g. Solanki and Stenflo, 1984, and references therein). Our model allows us
to pinpoint the contribution of the solar rotation to the solar brightness
variability. This can be done by disregarding the free term in Eq. (2.4), i.e.
by looking at the non-rotating Sun from a fixed direction. We note that by
doing this we still preserve the differential rotation term.

In Fig. 2.8 we compare the power spectra of solar brightness variations over
cycle 22 calculated with and without taking solar rotation into account
(orange and blue lines, respectively). Figure 2.8 a shows power spectra as
recorded by an ecliptic-bound observer. The solar rotation does not play a
substantial role at timescales below about four to five days (the orange and
blue lines in Fig. 2.8 a are very close to each other). The variability at
such timescales is apparently due to the evolution of individual magnetic
features. The variability at timescales between five days and the solar
rotation period is mainly due to the solar rotation. Interestingly, while the
rotation itself becomes unimportant at timescales above the rotation period
the two power spectra are still different up to the timescale of about four to
five years. This is because the variability of the rotating Sun is determined
by the longitudinal-averaged distribution of magnetic features. The
variability of the non-rotating Sun is given by the distribution seen from a
fixed vantage point. Since the emergence of magnetic features is random over
longitude, the two described distributions are the same if averaged over a
sufficiently long time interval (so that blue and orange lines almost coincide
at timescales larger than four to five years). At the same time, at timescales
shorter than four to five years, the distributions might still be different
since they depend on the specific realisation of emergences of magnetic
features. Consequently, this part of the power spectrum depends on the
specific longitudinal location of the vantage point.

Figure 2.8 b illustrates the case of a 57∘ inclination, which looks very
similar to the case of the ecliptic-bound observer. Fig. 2.8 c represents the
view from the observer located over the solar North pole. Naturally, the solar
rotation does not contribute to the brightness variability as it is determined
solely by the evolution of the magnetic features and the modulation of their
emergence rate over the solar activity cycle. Therefore, the blue and orange
curves in Fig. 2.8 c coincide at all timescales.

Fig. 2.8 allows us to better understand the origin of the decrease of short-
timescale variability with decreasing inclination as seen in Figs. 2.6–2.7.
The emergence of active regions is confined to about $\pm$ 30–40∘ centred
around the equator. Consequently, even though the variability at timescales
shorter than four to five days is not affected by the solar rotation, it is
strongly decreased due to the effect of foreshortening.

####  2.4.3 The full time series

In the previous sections, we limit our analysis to selected individual solar
activity cycles. The source term used in the SFTM provides information from
1700 to 2009. We now consider the solar brightness variations for this whole
interval, with respect to different inclinations, limiting ourselves to
calculating solar brightness variation in the Kepler passband, which we
present in Fig. 2.9. The differences in the power spectra between the 90∘ and
57∘ degree vantage point are small (shown earlier in the paper), whereas the
difference between 90 and 0∘ is pronounced. On the timescale above one year,
the variability as observed from an inclination of of 0∘ becomes stronger, due
to the stronger facular contribution to the solar brightness variations.

For all inclinations, a pronounced peak at around 10.8 years is visible, which
corresponds to the average length of a cycle in our considered sample. On the
rotational timescale, however, no peak is seen and no peaks above or below the
rotation period appear.

?figurename? 2.9: Power spectra of the solar brightness variations for the
full time series of over 300 years as it would be observed by Kepler for
different inclinations. The vertical dashed black lines indicates the synodic
solar rotation period at 27.3 days.

###  2.5 Conclusions and outlook

We employed the Surface Flux Transport model (SFTM, in the form of Cameron et
al., 2010) with the source term from Jiang et al. (2011a) to obtain the
distribution of magnetic flux on the entire solar surface. This distribution
was then converted into surface area coverages of solar magnetic features and
the SATIRE approach was utilised for calculating brightness variations. This
allowed us to model the brightness variability of the Sun at different
activity levels as it would be seen from any arbitrary vantage point and in
different filter systems.

We analysed the dependence of the power spectrum of solar brightness
variations on the inclination. While the decrease of the inclination leads to
an increase of the variability on the timescale of the solar activity cycle,
the variability decreases at shorter timescales. In particular, it decreases
on the timescale of solar rotation. Since the Sun is always seen equator-on,
its variability is higher than of another star with the same activity level,
but seen from a higher latitude. Consequently the higher variability of solar-
like stars cannot be due to the inclinations of their rotation axis alone. The
effect of the inclination strengthen the conclusions of Reinhold et al. (2020)
that stars with near-solar fundamental parameters and rotation periods have on
average significantly higher variability on the solar rotation timescale than
the Sun.

Our calculations also indicate that the power spectrum of solar brightness
variations does not have a clear peak at the rotation period, not only for the
ecliptic-bound observer (see Shapiro et al., 2017; Witzke et al., 2020a), but
also for the out-of-ecliptic observer. This factor might play an important
role in explaining the deficiency of stars with detected near-solar rotation
periods (see van Saders et al., 2019; Witzke et al., 2020a).

Our model also allowed us to decompose the contributions of solar rotation and
evolution of magnetic features into solar brightness variability. In
particular, we have shown that the variability on timescales below five days
is mainly due to the evolution of magnetic features and not due to the solar
rotation.

The SFTM model is also capable of simulating stars more active than the Sun
(Isık et al., 2018), so we plan to extend the present study to model
brightness variations of stars that are more active than the Sun. In combining
it with the results of Witzke et al. (2018, 2020a), we also plan to extend the
model to stars with different fundamental parameters.

##  ?chaptername? 3 Connecting measurements of solar and stellar brightness
variations

The contents of this section are identical to the printed version of Nèmec,
N.-E., et al., A& A, 638, A56,2020, reproduced with permission ©ESO.
DOI:10.1051/0004-6361/202038054
Contributions to the paper: I produced the results and provided the main
scientific interpretation.

###  3.1 Introduction

Dedicated planet-hunting photometric missions such as CoRoT (Convection,
Rotation and planetary Transit, see Baglin et al., 2006; Bordé et al., 2003),
Kepler (Borucki et al., 2010), and TESS (Transiting Exoplanet Survey
Satellite, see Ricker et al., 2014), and also the Gaia space observatory (Gaia
Collaboration, 2016) have made it possible to measure stellar brightness
variability with unprecedented precision. In particular, they allow studying
stellar brightness variations caused by transits (as the star rotates) and the
evolution of magnetic features, that is, bright faculae and dark spots. Such
variations are often referred to as rotational stellar variability. The
plethora of stellar observational data rekindled an interest in the questions
of how typical our Sun is as an active star, and more specifically, how the
solar rotational variability compares to that of solar-like stars.
Furthermore, these data allow probing whether the solar activity paradigm is
also valid for other stars. This requires comparing the stellar properties and
behaviour with those of the Sun. While the solar variability has been measured
for more than four decades now by various dedicated space missions (see e.g.
Fröhlich, 2012; Ermolli et al., 2013; Solanki et al., 2013; Kopp, 2016, for
reviews), a comparison between solar and stellar brightness measurements is
far from straightforward (see e.g. Basri et al., 2010; Reinhold et al., 2020;
Witzke et al., 2020b). Firstly, solar and stellar brightness variations have
been measured in different spectral passbands. Because the amplitude of the
solar rotational variability strongly depends on the wavelength (Solanki et
al., 2013; Ermolli et al., 2013), the solar and stellar brightness records can
be reliably compared only after conversion from one passband to another.
Secondly, the solar brightness variations have (so far) only been measured
from the ecliptic plane, which is very close to the solar equatorial plane
(the angle between the solar equator and ecliptic plane is about 7.25∘). The
values of the angle between the line of sight of the observer and the rotation
axes of the observed stars (hereinafter referred to as the inclination) are
mostly unknown.

Studies comparing solar and stellar rotational brightness variations have used
different types of solar brightness measurements. Reinhold et al. (2020), for
instance, used the total solar irradiance (TSI), that is, the solar radiative
flux at 1 AU integrated over all wavelengths. More commonly, however, the
solar variability was characterised (see e.g. Basri et al., 2010; Gilliland et
al., 2011; Harrison et al., 2012) using measurements by the Variability of
solar IRradiance and Gravity Oscillations / Sun PhotoMeters (VIRGO/SPM)
(Fröhlich et al., 1995, 1997) instrument on board the Solar and Heliospheric
Observatory (SoHO). VIRGO/SPM measures solar brightness in three filters with
a bandwidth of 5 nm each. Neither VIRGO/SPM nor TSI measurements can be
directly compared to records of stellar brightness variability, which
typically cover wavelength ranges broader than the VIRGO/SPM filters, but much
narrower than the TSI. Accurate estimations of solar variability in passbands
used for stellar measurements have therefore so far been missing. Some effort
has previously been made to model the solar rotational variability as it would
be observed out of ecliptic (e.g. Vieira et al., 2012; Shapiro et al., 2016;
Nèmec et al., 2020b). In particular, Shapiro et al. (2016) and Nèmec et al.
(2020b) (hereinafter N20) have shown that the amplitude of the solar
brightness variations on the rotational timescale decreases with decreasing
inclination. Because of its almost equator-on view, the Sun would therefore
appear on average more variable than stars with the same activity level that
are observed at random inclinations. At the same time, an easy-to-use receipt
for correcting the variability for the inclination effect is lacking so far,
and consequently, the inclination has not yet been quantitatively accounted
for in solar-stellar comparison studies.

In this paper we seek to overcome these two hurdles and quantify solar
variability in passbands that are used by different stellar space missions and
at different inclinations. In Sect. 3.2 we employ the spectral and total
irradiance reconstruction (SATIRE; Fligge et al., 2000; Krivova et al., 2003)
model of solar brightness variations to show how the actual solar brightness
variations are related to solar brightness variations as they would be
observed in spectral passbands used by stellar missions. We also establish the
connection between the TSI and VIRGO/SPM measurements. In Sect. 3.3 we follow
the approach developed by N20 to quantify the effect of the inclination on the
brightness variations. We discuss how the Sun as observed by Kepler can be
modelled using light curves obtained by VIRGO/SPM in Sect. 3.4 before we
summarise our results and draw conclusions in Sect. 3.5.

###  3.2 Conversion from solar to stellar passbands

####  3.2.1 SATIRE-S

The SATIRE model Fligge et al. (2000); Krivova et al. (2003) attributes the
brightness variations of the Sun on timescales longer than a day to the
presence of magnetic features on its surface, such as bright faculae and dark
spots. The two main building blocks of SATIRE are the areas and the positions
of the magnetic features on the solar disc as well as contrasts of these
features relative to the quiet Sun (i.e. regions on the solar surface free
from any apparent manifestations of magnetic activity). The contrasts of the
magnetic features as a function of disc position and wavelength were computed
by Unruh et al. (1999) with the spectral synthesis block of the ATLAS9 code
(Kurucz, 1992; Castelli and Kurucz, 1994). The 1D atmospheric structure of the
two spot components (umbra and penumbra) and of the quiet Sun were calculated
using radiative equilibrium models produced with the ATLAS9 code, while the
facular model is a modified version of FAL-P by Fontenla et al. (1993).

Various versions of the SATIRE model exist. In this section we employ the most
precise version, which is SATIRE-S, where the suffix “S” stands for the
satellite era Ball et al. (2014); Yeo et al. (2014). SATIRE-S uses the
distribution of magnetic features on the solar disc obtained from observed
magnetograms and continuum disc images and spans from 1974 to today, covering
four solar cycles. As especially the early ground-based observations contain
gaps in the data, we used the SATIRE-S model as presented by Yeo et al. (2014)
(version 20190621), where the gaps in spectral solar irradiance (SSI) and TSI
have been filled using the information provided by solar activity indices.
SATIRE-S was shown to reproduce the apparent variability of the Sun as
observed, in both the SSI and in the TSI (see Ball et al., 2012, 2014; Yeo et
al., 2014; Danilovic et al., 2016, and references therein). The spectral
resolution of the SATIRE output is 1 nm below 290 nm, 2 nm between 290 nm and
999 nm, and 5 nm above 1000 nm. This is fully sufficient for the calculations
presented in this study.

####  3.2.2 Filter systems

In this section we multiply the SATIRE-S SSI output with the response function
of a given filter and integrate it over the entire filter passband to obtain
the solar light curve in the corresponding filter. It is important to take the
nature of the detectors used in different instruments into account (see e.g.
Maxted, 2018). In particular, while solar instruments (e.g. VIRGO/SPM and all
TSI instruments) measure the energy of the incoming radiation, charge-coupled
devices (CCDs) used in Kepler, Gaia, and TESS count the number of photons and
not their energy. In order to obtain the solar light curve, $LC$, as it would
be measured by the instrument counting photons, we therefore follow

$LC=\int\limits_{\lambda_{1}}^{\lambda_{2}}R(\lambda)\cdot
I(\lambda)\frac{\lambda}{h\cdot c}\,d\lambda,$ (3.1)

where $\lambda_{1}$ and $\lambda_{2}$ are the blue and red threshold
wavelengths of the filter passband, $R(\lambda)$ is the response function of
the filter, and $I(\lambda)$ is the spectral irradiance at a given wavelength,
$h$ is the Planck constant, and $c$ the speed of light.

?figurename? 3.1: Response functions for the various filter systems used in
this study. For comparison, the quiet-Sun spectrum used by SATIRE-S is plotted
in grey in each panel. Top panel: Kepler, TESS, and CoRoT. Middle panel: Three
Gaia passbands. Bottom panel: Three VIRGO/SPM channels.

First we consider several broad-band filters used by the planet-hunting
missions: CoRoT, Kepler, and TESS. The spectral passbands employed in these
missions are shown in the top panel of Fig. 3.1, along with the quiet-Sun
spectrum calculated by Unruh et al. (1999) and used in SATIRE-S. Clearly, the
CoRoT and Kepler response functions are very similar to each other because
both missions focused on G stars. TESS is designed to observe cooler stars
than Kepler, hence the response function is shifted towards the red part of
the spectrum.

Gaia measures stellar brightness in three different channels (Gaia
Collaboration, 2016). Gaia G is sensitive to photons between 350 and 1000 nm.
Additionally, two prisms disperse the incoming light between 330 and 680 nm
for the Blue Photometer (hereafter referred to as Gaia GBP) and between
640–1050 nm for the Red Photometer (hereafter, referred to as Gaia GRP). The
response functions are shown in the middle panel in Fig. 3.1. We employ the
revised passbands used for the second data release of Gaia (Gaia DR2, Evans et
al., 2018) for the calculations.

Solar-stellar comparison studies have often used the solar variability as
measured by the VIRGO/SPM instrument. SPM comprises three photometers, with a
bandwidth of 5 nm operating at 402 nm (blue), 500 nm (green), and 862 nm
(red). The response functions are shown in Fig. 3.1 in the bottom panel. We
refer to these filters from now on as VIRGO-blue, -green, and -red.

####  3.2.3 Results

?figurename? 3.2: Normalised fluxes for the different filter systems compared
to the TSI (black solid line). Top panels: Kepler, CoRoT, and TESS. Middle
panels: Three Gaia passbands. Bottom panels: Three VIRGO/SPM channels.
?figurename? 3.3: $R_{30}$ in different filter systems compared to the TSI
(black solid line) for a time span of 1000 days over solar cycle 22. Top
panels: Kepler, CoRoT, and TESS. Middle panels: Three Gaia passbands. Bottom
panels: Three VIRGO/SPM channels. See the main text for the definition of
$R_{30}$. ?figurename? 3.4: Linear regression between the $R_{30}$ values
calculated with the TSI and with the solar light curves as they would be
recorded in different filter systems. The values of the slope, $s$, are given
in the legend. The black lines have a slope equal to 1.

Figure 3.2 shows the solar light curve for the period of 2456700 – 2456850 JD
(24 February 2014 – 11 July 2014) as it would be observed in different
passbands. This corresponds to a 150-day interval during the maximum of cycle
24. This interval was chosen arbitrarily to display the effect of the filter
systems on the solar variability. For this, we first divided each light curve
in 90-day segments. This time span corresponds to Kepler quarters. This is
motivated by the way Kepler observations are gathered and reduced. We note
that the detrending by the Kepler operational mode is applied here for purely
illustrative purposes and was not used for the calculations presented below.
Within each segment, we subtracted the mean value from the fluxes before
dividing the corresponding values by the mean flux in each segment. In all
stellar broad-band filters, the light curve is remarkably similar in shape to
the TSI (solid black curve), although the amplitude can differ. As might be
expected, the difference in the amplitude of the variability is somewhat more
conspicuous in the blue filters. The $Gaia\leavevmode\nobreak\ G_{RP}$ light
curve is basically identical to the TSI light curve, whereas the variability
in $Gaia\leavevmode\nobreak\ G_{BP}$ and in the narrow VIRGO-blue filter show
far stronger variability than the TSI.

To quantify the rotational variability, we computed the $R_{30}$ values (see
e.g. Basri et al., 2013). To do this, the obtained light curves were split
into 30-day segments, and within each segment, we calculated the difference
between the extrema and divided this value by the mean flux in the segment to
derive the relative variability. For the SATIRE-S time series, we directly
considered the difference between the extrema instead of the differences
between the 95th and 5th percentiles of sorted flux values, as is usually done
in the literature with the more noisy Kepler measurements. We calculated
$R_{30}$ values for the period 1974–2019 (i.e. cycles 22–24). This allowed us
to quantify the mean level of solar variability in $R_{30}$ that represents
the full four decades of TSI measurements.

| slope
---|---
Kepler | 1.123 ($\pm$0.007)
CoRoT | 1.110 ($\pm$0.006)
TESS | 0.939 ($\pm$0.004)
Gaia GBP | 1.304 ($\pm$0.005)
Gaia G | 1.131 ($\pm$0.005)
Gaia GRP | 0.944 ($\pm$0.004)
VIRGO/blue | 1.689 ($\pm$0.003)
VIRGO/green | 1.370 ($\pm$0.007)
VIRGO/red | 0.912 ($\pm$0.003)
?tablename? 3.1: Slopes of the linear regressions in Fig. 3.4. | 21 | 22 | 23 | 24 | mean
---|---|---|---|---|---
TSI | 0.743 | 0.806 | 0.682 | 0.492 | 0.681
Kepler | 0.808 | 0.872 | 0.731 | 0.530 | 0.735
CoRoT | 0.801 | 0.866 | 0.726 | 0.526 | 0.730
TESS | 0.680 | 0.739 | 0.615 | 0.445 | 0.620
Gaia GBP | 0.944 | 1.020 | 0.861 | 0.625 | 0.862
Gaia G | 0.817 | 0.883 | 0.741 | 0.537 | 0.744
Gaia GRP | 0.684 | 0.742 | 0.617 | 0.447 | 0.623
VIRGO/blue | 1.252 | 1.352 | 1.167 | 0.846 | 1.154
VIRGO/green | 0.983 | 1.056 | 0.894 | 0.653 | 0.897
VIRGO/red | 0.665 | 0.722 | 0.600 | 0.435 | 0.606
?tablename? 3.2: Cycle-averaged $R_{30}$ in mmag.

In Fig. 3.3 we compare the $R_{30}$ values for all the filter systems
introduced in Sect. 3.2.2 to $R_{30}$ of the TSI for a 1000-day interval
starting 26 July 2013. This interval therefore includes the maximum of solar
24 as well. To better quantify the dependence of $R_{30}$ on the passband, we
show linear regressions between the variability $R_{30}$ in each filter system
and the TSI in Fig. 3.4. The slopes of the linear regressions are listed in
Table 3.1. The Pearson correlation coefficient is above 0.98 for all of the
filter systems. The slope of the linear regressions depends on the filter
system that is considered. For example, TESS and Gaia G regressions have a
slope close to 1, but $G_{BP}$ displays a slope >1, whereas VIRGO-red exhibits
a slope <1\. As expected, the slope is highest for the blue VIRGO filter,
where the amplitude of the variability is highest. We note that the good
agreement of the TSI with the red filters is expected to be valid only for the
rotational variability, which is dominated by spots. In contrast, the solar
irradiance variability on the activity cycle timescale is given by the
delicate balance between facular and spot components, and consequently has a
very sophisticated spectral profile (Shapiro et al., 2016; Witzke et al.,
2018). Thus, values of slopes from Table 3.1 cannot be extrapolated from
rotational to activity cycle timescales (see Shapiro et al., 2016, for the
detailed discussion).

Table 3.2 lists the cycle-averaged values of $R_{30}$ for all passbands in
mmag. Together, Fig. 3.4, and Tables 3.1 and 3.2 show that the TSI is a
passable representative for the variability on the solar rotation timescale as
it would be observed in the TESS, Kepler, CoRoT, Gaia G, Gaia GRP , and VIRGO-
red filters, but it noticeably underestimates the variability in $G_{BP}$,
VIRGO-green, and VIRGO-blue.

?figurename? 3.5: $R_{30}$ over cycle 22 for VIRGO/g+r and Kepler.
?figurename? 3.6: Linear regression between the $R_{30}$ calculated for
VIRGO/g+r and for Kepler light curves. The black solid line represents a
linear regression with slope $=$ 1.

Several studies (see e.g. Basri et al., 2010; Harrison et al., 2012) have
assumed that the amplitude of the rotational solar variability as it would be
measured by Kepler is very close to the amplitude calculated for the combined
green and red VIRGO/SPM light curves (in the following VIRGO/g+r). Here we
test this hypothesis. The variability $R_{30}$ for Kepler compared to
VIRGO/g+r is shown in Fig. 3.5, which is limited to the same time interval as
Fig. 3.3. The two curves are remarkably similar to one another. To test the
similarity quantitatively, we show the linear regression of $R_{30}$ between
Kepler and VIRGO/g+r for four solar cycles (21-24) in Fig. 3.6. The Pearson
correlation coefficient is very high (0.999) and the slope deviates by only
+0.8% of unity averaged over four solar cycles. While these calculations are
related to the amplitude of the rotational variability, $R_{30}$, we
additionally calculate regressions between Kepler and Virgo light curves in
Sect. 3.4. We also directly connect the TSI and VIRGO/g+r rotational
variability. The linear regression between $R_{30}$ in TSI and VIRGO/g+r
results in a slope of 0.88 ($\pm$ 0.002) and a Pearson correlation coefficient
of 0.995.

###  3.3 Correction for the inclination

####  3.3.1 Approach

The results presented in Sect. 3.2.3 are for the Sun viewed from the ecliptic
plane and apply to stars that are viewed approximately equator-on. However,
this is not always the case, and the inclination of a star is often unknown.
Calculations of the solar variability as it would be measured by an out-of-
ecliptic observer demand information about the distribution of magnetic
features on the far side (for the Earth-bound observer) of the Sun. N20 have
used a surface flux transport model (SFTM) to obtain the distribution of
magnetic features of the solar surface, which was then fed into the SATIRE
model to calculate solar brightness variations as they would be seen at
different inclinations.

The SFTM is an advective-diffusive model for the passive transport of the
radial magnetic field on the surface of a star, under the effects of large-
scale surface flows. In this model, magnetic flux emerges on the stellar
surface in the form of bipolar magnetic regions (BMRs). We employed the SFTM
in the form given by Cameron et al. (2010) and followed the approach of N20 to
simulate light curves of the Sun at different inclinations and with various
filter systems. The emergence times, positions, and sizes of active regions in
our calculations were determined using the semi-empirical sunspot-group record
produced by Jiang et al. (2011a). This synthetic record was constructed to
represent statistical properties of the Royal Greenwich Observatory sunspot
record. We additionally randomised the longitudes of the active-region
emergences in the Jiang et al. (2011a) records. Such a randomisation is needed
to ensure that the near and far side of the Sun have on average equal
activity, which is a necessary condition for reliable calculations of the
inclination effect. As a result, our calculations reproduce the statistical
properties of a given solar cycle, but they do not represent the actual
observed BMR emergences for that specific cycle. We stress that in N20 we
developed the model outlined above to study the effect of the inclination on
the power spectra of solar brightness variations. Here we use this model to
explicitly study the dependence of the variability amplitude on the rotational
timescale and its dependence on the inclination in different filters.

####  3.3.2 Results

In the following, we place the observer out of the solar equator towards the
solar north pole. This corresponds to inclinations below 90∘ . We quantify the
rotational variability using the $R_{30}$ metric introduced in the previous
section. To represent an average level of solar activity, we limit the
analysis to cycle 23, which was a cycle of moderate strength.

?figurename? 3.7: Synthetic solar light curves covering solar cycle 23 in the
Kepler passband as it would appear at different inclinations. ?figurename?
3.8: Dependence of the mean variability in $R_{30}$ on the inclination (termed
$\langle R_{30}\rangle_{\rm i}$, where $i$ stands for the inclination). All
values have been normalised to the respective equator-on ($i=90^{\circ}$)
value, here called $\langle R_{30}\rangle_{\rm 90}$. Individual curves
represent different filter systems. Left panel: Kepler, TESS, and CoRoT.
Middle panel: Three Gaia filters. Right panel: Three VIRGO filters and the
TSI. The vertical dashed black line indicates an inclination of 57∘.

We show the calculated solar light curve as it would be observed by Kepler at
various inclinations in Fig. 3.7. For this, we divided the time series for
cycle 23 into 90-day segments, which correspond to Kepler quarters. Within
each quarter, we de-trended the light curves. 90∘ corresponds to an ecliptic-
bound observer, 57∘ represents a weighted mean value of the inclination with
weights equal to the probability of observing a given inclination
($sin(\mathrm{i})$) for the inclination i), and 0∘ corresponds to an observer
facing the north pole. We additionally show 30∘ as an intermediate point
between 57∘ and 0∘. For the inclination values of 30∘, 57∘, and 90∘ , the
variability is brought about by the solar rotation, as well as the emergence
and evolution of magnetic features. A polar-bound observer does not observe
the rotational modulation because there is no transit of magnetic features and
the variability is merely generated by their emergence and evolution (see
Nèmec et al., 2020b, for further details). We emphasise just the reduction in
the amplitude of the variability with decreasing inclination, but also the
change in the shape of the light curve. This is particularly visible in a
comparison of the top and bottom panels of Fig. 3.7. In Fig. 3.8 we show the
change in $R_{30}$ averaged over cycle 23 when we place the observer out of
the ecliptic plane. To facilitate comparison, we normalised each value of the
cycle-averaged variability, $\langle R_{30}\rangle_{\rm i}$, to the
corresponding value for the ecliptic view, $\langle R_{30}\rangle_{\rm 90}$.
Figure 3.8 shows that the rotational variability decreases monotonically with
decreasing inclination. This trend is seen across all considered filter
systems. The differences in the inclination effect among the filter systems
are due to the different dependencies of the facular and spot contrasts (as
well as to their centre-to-limb variations) on the wavelengths.

To evaluate the averaged effect of the inclination, we introduce a new measure
that we call $\langle R_{30}\rangle$ and define as

$\langle R_{30}\rangle=\frac{\sum_{i}\langle
R_{30}\rangle_{i}\cdot\mathrm{sin}(i)}{\sum_{i}\mathrm{sin}(i)},$ (3.2)

where $i$ is the inclination. The factor $\mathrm{sin}(i)$ ensures that the
corresponding values of $\langle R_{30}\rangle_{i}$ are weighted according to
the probability that a star is observed at inclination $i$. The $\langle
R_{30}\rangle$ value represents the variability of the Sun averaged over all
possible inclinations. In other words, if we observed many stars analogous to
the Sun with random orientations of the rotation axes, their mean variability
would be given by the $\langle R_{30}\rangle$ value. Therefore $\langle
R_{30}\rangle$ should be used for the solar-stellar comparison rather than the
$\langle R_{30}\rangle_{90}$ value.

We present $\langle R_{30}\rangle$ normalised to $\langle R_{30}\rangle_{90}$
for all considered filter systems as well as the $\langle R_{30}\rangle$
values themselves in Table 3.3. In the second column of Table 3.3 we give the
inclination-corrected value of the mean rotational variability from Table 3.2
for easier application of our results. On average, all filter systems show a
15% lower variability than the equatorial case. This implies that the slopes
of the linear regressions between the $R_{30}$ values in different passbands
and the TSI have to be corrected for the inclination. When stellar
measurements in the Kepler passband are compared with the TSI records, this
means the following: when stars are observed from their equatorial planes, the
variability in Kepler is about 12% higher than in the TSI (see the slope given
in Table 3.1). However, when the Sun is compared to a group of stars with
random orientations of rotation axes, the inclination effect must be taken
into account as well. It will reduce the stellar variability observed in the
Kepler passband by approximately 15%. Coincidentally, these two effects almost
exactly cancel each other, and the observed TSI variability appears to be a
very good metric for the solar-stellar comparison of Kepler stars in a
statistical sense.

|  $\langle R_{30}\rangle$/$\langle R_{30}\rangle_{90}$ [%] |  $\langle R_{30}\rangle$ [mmag]
---|---|---
TSI | 85.1 | 0.58
Kepler | 84.8 | 0.62
CoRoT | 83.8 | 0.61
TESS | 84.0 | 0.56
Gaia GBP | 83.7 | 0.72
Gaia G | 84.1 | 0.62
Gaia GRP | 84.9 | 0.52
VIRGO/blue | 83.9 | 0.98
VIRGO/green | 85.3 | 0.76
VIRGO/red | 83.4 | 0.50
?tablename? 3.3: $\langle R_{30}\rangle$/$\langle R_{30}\rangle_{90}$ and
$\langle R_{30}\rangle$ values for different filter systems. For $\langle
R_{30}\rangle$ we multiplied $\langle R_{30}\rangle$/$\langle
R_{30}\rangle_{90}$ by the corresponding value for $\langle
R_{30}\rangle_{90}$ from Table 3.2. Time-averaging is performed over solar
cycle 23.

We have shown in Sect. 3.2 that the amplitude of solar rotational variability
as it would be measured by Kepler can be very accurately approximated by
calculating the amplitude of the VIRGO/g+r light curve. However, when
brightness variations of the Sun are compared to those of a large group of
stars with unknown inclinations, the use solar variability averaged over all
possible inclinations should be used rather than the solar variability
observed from the ecliptic plane. We have established that the effect of a
random inclination decreases the variability in the Kepler passband by 15%.
Taking this into account, the relative difference between the variability in
VIRGO/g+r and the solar variability in Kepler averaged over inclinations is
$-14$%. Unlike for the TSI, corrections for the passband and the inclination
only partly compensate for each other. We therefore suggest that the TSI is a
better representative of the Sun as it would be observed by Kepler than
VIRGO/g+r if the inclination of a star is unknown.

###  3.4 Modelling Kepler light curves using VIRGO/SPM

The the previous sections, we have quantitatively validated the argument of
Basri et al. (2010) that the VIRGO/g+r light curve corresponds to the same
variability as the Kepler light curve if both light curves are recorded from
the solar equatorial plane. In this section we perform complementary
calculations: we test if the Kepler light curve can be modelled as a linear
combination of solar light curves in the different VIRGO/SPM channels. We
restrict our calculations to solar cycle 23. All light curves are computed
with the N20 model.

We divided all light curves into 90-day segments and calculated the relative
flux within these segments (i.e. we considered the same normalisation of light
curves as shown in Figs. 3.2 and 3.7). Next, we applied multiple linear
regression to fit the Kepler light curve with the VIRGO/SPM green+red light
curves to determine the best set of coefficients for the linear fit. unity
averaged over four solar cycles.

?figurename? 3.9: Comparison between the Kepler and the regressed
VIRGO/green+red light curve. Left:i = 90∘, and right:i=57∘. For the
coefficients of the fit, see the main text.

For an ecliptic-bound observer, we write the multiple linear regression in the
form

$K_{i,\,SPM}=a\cdot V_{g}+b\cdot V_{r},$ (3.3)

where $i$ is the inclination, $V_{g}$ and $V_{r}$ are solar light curves in
VIRGO/SPM blue, green, and red filters (corresponding to the equatorial
plane). The best fit for $i=90^{\circ}$ yields a$=$0.275 ($\pm$ 0.001) and
b$=$0.619 ($\pm$ 0.002). The $r^{2}$ value is 0.999. Such a high correlation
is expected because the VIRGO/SPM and Kepler rotational variability are
similar, as discussed in Sect. 3.2. Next, we applied the multiple-regression
model to simulate the out-of-ecliptic Kepler-like light curve using the light
curves in the SPM channels as input. Because an inclination of 57∘ is often
used to represent the statistical mean of all possible inclination values, we
fit the Kepler light curve observed at $i=57^{\circ}$ with a linear
combination of VIRGO light curves (observed at $i=90^{\circ}$). The best fit
results in the following coefficients: a$=-$0.421 ($\pm$ 0.018) and b$=$1.418
($\pm$ 0.027), with $r^{2}=0.83$.

Figure 3.9 compares the Kepler-like light curve with the regression model
using light curves in all three VIRGO filters for $i=90^{\circ}$ and
$i=57^{\circ}$. For the 90∘ inclination case (left panel in Fig. 3.9), the
differences between the two light curves are basically invisible, but for the
57∘ case, the differences are quite pronounced. These differences have various
origins. In particular, the transits of magnetic features would take different
times for the ecliptic and out-of-ecliptic observer. Furthermore, with
decreasing inclination, the facular contribution becomes stronger, while the
spot contribution weakens (see e.g. Shapiro et al., 2016), which changes the
shape of the light curve. It is therefore necessary to take the actual
distribution of magnetic features into account when light curves are modelled
that were observed at different inclination angles.

###  3.5 Conclusions

We presented recipes for treating two problems that hamper the comparison
between solar and stellar rotational brightness variations: the difference
between the spectral passbands that are used for solar and stellar
observations, and the effect of inclination. To quantify the effect of
different spectral passbands on the rotational variability represented through
the $R_{30}$ metric, we employed the SATIRE-S model. We found that the
rotational variability observed through the filter systems used by the Kepler
and CoRoT missions is about 12% higher than the TSI, whereas the variability
in the TESS passband is about 7% lower. For Gaia G, we find +15% and for Gaia
GBP +30% difference in the amplitude of the rotational variability compared to
the TSI, whereas Gaia GRP shows a difference of -7%. These numbers are valid
for equator-on observations on rotational timescales.

Previous studies have used combinations of the red and green VIRGO light
curves for solar-stellar comparisons (see e.g. Basri et al., 2010; Gilliland
et al., 2011; Harrison et al., 2012). We used linear regressions of the
rotational variability between the two combined VIRGO/SPM passbands and the
solar variability as Kepler would observe it, to test the goodness of this
comparison. We find that the variability in Kepler is 7% higher than that of
VIRGO/green+red. Moreover, we found that the sum of the VIRGO green and red
light curves very accurately represents the solar light curve in the Kepler
passband. This is only valid for the Sun observed from the ecliptic, however.
We showed that a linear combination of VIRGO/SPM passbands cannot accurately
reproduce the solar Kepler light curve observed out of ecliptic.

We have calculated the dependence of the rotational variability on inclination
by following the approach in Nèmec et al. (2020b). In this approach, an SFTM
was used to simulate the distribution of magnetic features on the surface of
the Sun, which was then used to compute the brightness variations with SATIRE.
We find that across all filter systems discussed in this study, the rotational
variability drops by about 15% when it is averaged over all possible
directions of the rotation axis. Because the Kepler rotational variability as
observed from the ecliptic plane is 12% higher than the TSI rotational
variability, we conclude that the TSI is the best proxy for the solar
rotational brightness variations if they were observed by Kepler when the
inclination effect is considered.

##  ?chaptername? 4 Where have all the faculae gone? Explaining the transition
from faculae to spot domination

This section is the draft of a paper by N.-E. Nèmec, A. I. Shapiro, E. Işık,
S. K. Solanki, and N. A. Krivova
Contributions to the paper: I produced the results and provided the main
scientific interpretation.

###  4.1 Introduction

The magnetic fields emerging on solar and stellar surfaces lead to the
formation of surface magnetic features, such as dark spots and bright faculae
(see, e.g., Solanki et al., 2006, for the review of the solar case). One of
the exciting manifestations of these magnetic features are the brightness
variations of the Sun and solar-like stars (Solanki et al., 2013; Ermolli et
al., 2013).

Solar brightness variations have been routinely monitored since the 1970s by
dedicated spaceborne instrumentation (for review see Kopp, 2014). Using
radiometers, these missions measured the total solar irradiance (TSI), which
is defined as the wavelength integrated time–depended solar radiative flux at
a distance of 1 AU. The TSI varies with an 11-year periodicity. Compared to
the minimum, the TSI is 0.1% higher at solar maximum. This implies that the
TSI variability on the cycle timescale is faculae dominated (see, e.g.
detailed discussion and references in Shapiro et al., 2016). This means that
the decrease of the TSI due to the spots is overcompensated by the increase
due to the faculae. However, there is observational evidence, that with
increasing spot coverage, the fractional coverage of faculae to spots
decreases, i.e. the relative importance of the faculae decreases (Foukal,
1993; Chapman et al., 1997).

Long-term monitoring programs of stellar variability started in the 1960’s, as
Olin Wilson founded the Mount-Wilson observations of the Ca II H and K lines
to search for activity cycles in cool main sequence stars (Wilson, 1978). The
emission in the cores of the Ca II H&K lines is formed in stellar
chromospheres and is a good proxy of the magnetic heating in this atmospheric
layer. Thus, it is routinely used to characterise stellar magnetic activity.

The next big step in observations of stellar variability became the
establishment of the Lowell observatory monitoring program (Lockwood et al.,
1992). The program observed stars in the visual spectral domain in the
Strömgren b and y filters, which are centered at wavelengths of 467 and 547
nm, respectively. Simultaneous measurements of stellar photometric brightness
and chromospheric activity revealed a distinction between active younger and
less active older stars (e.g. Soderblom et al., 1991). The Ca II emission and
brightness appear to be correlated for less active stars (i.e. the increase of
the Ca II emission is accompanied by an increase in the photometric
brightness) and anti-correlated in active stars. Interestingly, this
transition from faculae to the spot-domination happens at an activity level
just a bit higher than the Sun has demonstrated over the last 400 years.
Shapiro et al. (2014) (hereinafter S14) showed that this transition can be
explained by extrapolating the solar dependencies of spot and facular disk
coverages on the chromospheric activity. This indicates that the spot disk
coverages increase faster with the activity level than the facular coverages,
so that at certain activity, the stellar variability becomes spot-dominated.
The S14 results imply that the dependencies of solar spot and facular disk
coverages on the activity level are valid not only for the Sun, but also for
other solar-like stars, and, in particular, for stars more active than the
Sun.

We seek to explain the physical mechanism behind this dependence of facular
and spot area coverages with increasing activity level by following the
approach presented in Nèmec et al. 2020b (in the following N20b). N20b have
used a surface flux transport model (SFTM, Cameron et al., 2010) together with
an irradiance model to calculate the brightness variations of the Sun as
represented via power spectra. In the present work, we do not calculate the
brightness per se, but only the area coverages of faculae and spots, following
the N20b approach. We summarize this approach in Sect. 4.2, before we extend
the model to more active stars utilizing the SFTM in the form presented by
Isık et al. (2018), hereafter I18 in Sect. 4.3. We discuss our findings and
the underlying mechanism in Sect. 4.4, before concluding our work in Sect.
4.5.

###  4.2 Model

####  4.2.1 Surface flux transport and filling factors

We obtain the surface distribution and coverage of magnetic features by
following the approach presented in N20b. This method relies on two steps.
Firstly, the surface distribution of magnetic flux is simulated by employing a
surface flux transport model (SFTM), which produces full surface magnetograms.
Secondly, the flux from these magnetograms is attributed to spots and faculae.
We describe the steps needed briefly in the following, but refer to the full
discussion in N20b for more details.

In the SFTM the magnetic flux on the surface of stars is passively transported
due to advection, diffusion and large scale flows, in this instance the
differential rotation and the meridional flow (see Cameron et al., 2010, and
references therein). The magnetic flux in the model is injected as bipolar
magnetic regions (BMRs, e.g. one BMR consists of two flux patches with
opposite polarities). The emergence of the BMRs is introduced via the source
term that determines the position and time of the emerging flux, as well as
the flux of the polarity patches, namely

$B^{\pm}(\lambda,\phi)=B_{\mathrm{max}}\left(\frac{0.4\Delta\beta}{\delta}\right)^{2}e^{-2[1-\cos(\beta_{\pm}(\lambda,\phi))]/\delta^{2}},$
(4.1)

where $\beta_{\pm}$($\lambda$,$\phi$) is the heliocentric angle between a
given point ($\lambda$, $\phi$) and the centres of the polarity patches,
$\Delta\beta$ is the separation between the two polarities and $\delta$ is the
size of the individual polarity patch, taken to be 4∘. $B_{\mathrm{max}}$ is a
scaling factor set to 374 G (Cameron et al., 2010; Jiang et al., 2011b).

The source term was compiled following the approach of Jiang et al. (2011a) by
constructing a semi-empirical input record. This input record contains
statistical properties of the latitude, the size distribution and the tilt-
angles of emergences, reflecting the properties of sunspot observations from
the Royal Greenwich Observatory for the time period 1986–1996, corresponding
to solar cycle 22. Additionally, the longitudes of the emergences are
randomised. This allowed N20b to study the impact of different vantage points
with respect to the ecliptic plane on the solar brightness variations
represented through power spectra. We follow the approach of randomising the
longitudes in this work as well. We remark that this is fully sufficient for
our purposes in the present work, as we are concerned with modelling general
trends of stellar variability. An advantage of employing the SFTM is that the
source term can be changed depending on the objective of research.

To obtain the area coverages of the spots, N20b have used the following
approach: (i) calculate the spot area at the day of emergence from the input
record following the relation between the separation $\Delta\beta$ and the
spot area from Cameron et al. (2010), (ii) follow a linear decay law with a
single decay rate $R_{d}$ for all spots independent of size and (iii) add a
linear growth law with a single growth rate $R_{g}$. Next, the facular
coverages were obtained by first, subtracting the magnetic flux of the spots
from the magnetograms and then using the remaining flux to calculate the
faculae areas following the approach of Krivova et al. (2003) of setting a
saturation threshold $B_{sat}$. While we used the SFTM simulations from
Cameron et al. (2010) in N20b and Nèmec et al. (2020a), the results in this
study are based on the model in I18 in the present work. We discuss the
differences in these two models in more detail in the next section.

####  4.2.2 Fixing the parameters

In Nèmec et al. (2020b) we have fixed the free parameters ($R_{g}$, $R_{d}$,
and $B_{sat}$) by comparing the brightness variations returned by our model to
observations and the SATIRE-S model (Yeo et al., 2014). Since in this study we
are interested in explaining general trends in the behaviour of facular and
spot coverages, we fix the parameters by comparing the coverages returned by
our model to those returned by the SATIRE-S model. The area coverage of the
magnetic features in SATIRE-S are obtained from observed magnetograms and full
disk images and are available starting from 1979. The output of I18 SFTM runs
is available for solar cycle 22.

The statistical source term employed in this work prohibits direct day-to-day
comparison between the area coverages from SATIRE-S and our model. We
therefore sort the area coverages according to ascending facular areas, before
we split the resulting monotonous area coverages into bins containing 58 days.
Within each bin we then calculate the mean value.

First, we fit the output of SATIRE-S following,

$A_{f}=a\cdot\sqrt{A_{s}-c}+d.$ (4.2)

Then we find a combination of the free parameters of our model ($R_{d}$,
$R_{g}$ and $B_{sat}$) which leads to values for $a$, $c$, and $d$ as close to
those obtained with fitting SATIRE-S as possible.

We display the data-sets and their corresponding fits in Fig. 4.1 and give the
coefficients of the fits according to Eq. 4.2 in Table 4.1. Additionally, the
cycle averaged spot coverages (0.00108 for SATIRE-S and 0.00104 for our model)
and the cycle averaged facular coverages (0.0144 for SATIRE-S and 0.0139 for
this work) are consistent. We compare the parameters obtained in this work to
the ones obtained by N20b in Table 4.2. The values of $R_{d}$ and $B_{sat}$
are slightly different in the two models. The reason for this is that for the
compilation of the input record, the sunspot group number is used, which needs
to be converted into individual spot pairs. In N20b we followed the approach
by Jiang et al. (2011a), who used a conversion factor of 2.1 from sunspot
group number to sunspot number, whereas the I18 approach used a conversion
factor of 2.75. The larger factor in I18 resulted in fewer region emergences.
To compensate this, the spot lifetime must be longer and the faculae threshold
has to be lowered to attribute more flux to the faculae in order to preserve
the area coverages.

| a | c | d
---|---|---|---
SATIRE-S | 0.67 | -0.0003 | -0.009
| ($\pm$ 0.057) | ($\pm$ 0.00008) | ($\pm$ 0.003)
Model | 0.57 | -0.0002 | -0.002
| ($\pm$ 0.05) | ($\pm$ 0.00002) | ($\pm$ 0.0008)
?tablename? 4.1: Coefficients of the fit according to Eq. 4.2 of the curves displayed in Fig. 4.1. Parameter | This work | N20b | Description
---|---|---|---
$R_{d}$ [MSH/day] | 47 | 80 | decay rate of spots
$R_{g}$ [MSH/day] | 600 | 600 | growth rate of spots
$B_{sat}$ [G] | 470 | 500 | saturation threshold for faculae
?tablename? 4.2: List of the parameters of our model compared to the N20b
values. ?figurename? 4.1: Comparison between the binned filling factors
returned by the SATIRE-S model (black) and the model presented in this work
(blue). Solid lines represent corresponding fits (see text for details) and
the shaded areas are the 1-$\sigma$ intervals. Vertical bars display the
standard deviation within each bin. Each bin contains 58 data points and is
represented by the mean value.

###  4.3 Extending the model to stars more active than the Sun

In Sect. 4.2.2 we have shown that our model is able to reproduce the observed
drop in solar facular area with increasing spot area. In the following, we
investigate how $A_{f}$ as a function of $A_{s}$ changes for stars more active
than the Sun. In particular, we address the physical mechanism behind this
behaviour. We can define the time-dependent emergence rate of BMRs on a star
$S_{\star}$ as $S_{\star}(t)=S_{\odot}(t)\cdot\tilde{s}$, where $S_{\odot}(t)$
is the-time dependent solar BMR emergence rate and $\tilde{s}$ is a scaling
factor. This means, that a star with $\tilde{s}=2$ exhibits two times more BMR
emergences compared to the solar case, $\tilde{s}=1$. We note that in contrast
to Isık et al. (2018) we consider stars with solar rotation rate so that while
we change the emergence rate of BMRs, we preserve the solar distribution of
latitudes of emergences (see, Jiang et al., 2011a). We also preserve the solar
distribution of areas of active regions (see Isık et al., 2018, Appendix A.3)
and the histograms in Fig. 4.2. We extend the model in this work to
$\tilde{s}=2,4,8$ and $16$.

?figurename? 4.2: Distribution of the separation of the bipolar magnetic
regions (BMRs) of stars with different activity levels in terms of their
number of emergences. Left: absolute numbers of BMRs per $\Delta\beta$ bin,
right: normalised numbers of BMRs per $\Delta\beta$. The integral for each
distribution in this panel is 1.

The calculated dependencies of $A_{f}$ on $A_{s}$ are shown in Fig. 4.3 top
panel. The binning was done in the same way as described in Sect. 4.2.2. While
the dependencies calculated for different $\tilde{s}$ generally follow the
form of Eq. 4.2, Fig. 4.3 shows that the cases $\tilde{s}>=1$ are not simply a
continuation of the solar case ($\tilde{s}=1$). In general, for a given spot
coverage, calculations with larger $\tilde{s}$ return higher facular coverage.
The exact reason for such a tendency is out of the scope of the present study
but we speculate that it can be attributed to our binning procedure and the
phase shift between maximum contribution of a given BMR to spot and facular
coverages (see discussion in Foukal, 1998).

In Fig. 4.3 bottom panel we compare the fit to the data (blue) with various
other results: the fit to the SATIRE-S values (orange), the relationship
between $A_{f}$ and $A_{s}$ published by S14 and the fit to the $s=1$ case of
our model. While the results agree with each other within error-bars and the
shown 1$\sigma$ intervals (shaded areas in Fig. 4.3 bottom panel), this figure
also indicates, that there might not be an universal dependence of $A_{f}$ on
$A_{s}$. As discussed above the activity level used for calculating the
dependence of $A_{f}$ on $A_{s}$ is an important factor.

?figurename? 4.3: Binned facular disk areas coverages $A_{f}$ as a function of
the spot disk area coverages $A_{s}$ for different activity levels
$\tilde{s}$. Top: binned values for the different simulated stellar activity
levels. Bottom: comparison between the area coverages of the full set of
simulations in this work (blue) to the fit to the observed area coverages
returned by SATIRE-S (orange), empirical relationships (Shapiro et al., 2014,
green), and the fit to the solar activity level $\tilde{s}=1$ (red).

Fig. 4.4 we additionally show the mean values of the disc area coverages by
spots ($\langle A_{s}\rangle$, left panel), faculae ($\langle A_{f}\rangle$
,middle panel) and the ratio between these two values (right panel) as a
function of activity scaling factor, $\tilde{s}$. Clearly, $\langle
A_{s}\rangle$ increases linearly, while $\langle A_{f}\rangle$ deviates from a
linear increase. With higher activity level, the ratio $\langle
A_{f}\rangle$/$\langle A_{s}\rangle$ is dropping, which is inline with the
above discussed behavior of $\langle A_{f}\rangle$ with increasing $\langle
A_{s}\rangle$.

?figurename? 4.4: Mean cycle averaged facular disk areas coverages $\langle
A_{f}\rangle$, mean cycle averaged spot disk areas coverages $\langle
A_{s}\rangle$ and their ratios as a function of the activity scaling factor,
$\tilde{s}$, (see text for details).. The dashed black lines are linear fits
to the data, the colored solid lines connect the points to guide the eye.

###  4.4 What happens to the flux?

It is not surprising that $\langle A_{s}\rangle$ is increasing linearly with
increasing activity level. The spot area coverages are determined by the
separation $\Delta\beta$ of the BMRs given by the source term. As Fig. 4.2
shows, with increasing activity level $\tilde{s}$, the distribution of the
regions with respect to $\Delta\beta$ stays the same, therefore the spot area
coverages are linearly increasing as well.

The same should in principle also apply to the faculae. With the increasing
BMR emergence rate for more active stars, more flux (that is attributed to the
faculae) is emerged on the surface of a star. However, as seen in Fig. 4.4
middle panel, something is happening to the flux once it is emerged, that
decreases $\langle A_{f}\rangle$ with higher activity levels. In the SFTM,
flux is present in both positive and negative polarity. As the model does not
directly include convection, flux is only distributed across the surface by
the surface flows and the surface diffusivities. All in all, there is only one
mechanism to remove flux from the surface and that is through the cancellation
of fluxes with opposite polarities. As activity increases the opposite
polarities come closer to each other so that the cancellation gets more
effective and the ratio between facular and spot coverages drops. We note that
our model also allows to take into account the cancellation of spots with
different polarities. However, since spot areas are much smaller than those of
faculae and also spot lifetimes are much shorter, such cancellations are
barely occurring in our model.

This explanation in reasonably, given that cancellation events have been
observed on the solar surface in active regions (e.g. Green et al., 2011;
Tiwari et al., 2019), in active region plage (Chitta et al., 2019) and even in
quiet Sun regions (Wang Jingxiu et al., 1988). Cancellation events and the
consequent release of magnetic energy have also propose as possible
explanations of both chromospheric and coronal heating through nanoflares (see
e.g. Priest et al., 2018) and an possible trigger mechanism of coronal jets
(McGlasson et al., 2019).

###  4.5 Conclusion

We have used a surface flux transport model (SFTM, as described in Isık et
al., 2018) to simulate the emergence and evolution of magnetic flux on the
surfaces of stars, together with the algorithm of obtaining the area coverages
of magnetic features from Nèmec et al. (2020b) to investigate the curious case
of the decrease of the ratio between facular and spot coverages with
increasing stellar activity.

This allowed us to explain the empirical dependence between facular and spot
coverages reported by Shapiro et al. (2014) as well as to confirm the
extrapolation of this dependence beyond the level of solar activity. At higher
activity levels, the facular area coverage gradually saturates, whereas the
spot area coverage is increasing. This results in a transition from faculae to
spot domination, as the ratio between facular to spot area drops. Our results
strongly point towards the cancellation of magnetic flux in the form of
opposite polarity encounters as the reason for such a behaviour of the faculae
towards higher activity levels.

##  ?chaptername? 5 Explaining the dependence of stellar brightness
variability on the rotation period

This section is the draft of a paper by N.-E. Nèmec, A. I. Shapiro, E. Işık,
T. Reinhold, S. K. Solanki.
Contributions to the paper: I produced the results and provided the main
scientific interpretation.

###  5.1 Introduction

Planetary hunting missions such as the Convection, Rotation and planetary
Transits (CoRoT Baglin et al., 2006; Bordé et al., 2003), Kepler (Borucki et
al., 2010) and the Transiting Exoplanet Survey Satellite (TESS, Ricker et al.,
2014) allowed studying the day-to-day brightness variations of stars with high
precision and cadence. Such brightness variations are well understood for the
Sun and are brought about by the transit of dark spots and bright faculae
across the visible stellar disc (see Solanki et al., 2013; Ermolli et al.,
2013, for review).

Models based on the solar paradigm are a useful tool-set in providing
explanations for the plethora of stellar photometric data. For example, Nèmec
et al. (2020b) have combined the Spectral And Total Irradiance Reconstruction
(SATIRE, Fligge et al., 2000; Krivova et al., 2003) model together with a
surface flux transport model (SFTM, Cameron et al., 2010) to model the power
spectra of solar brightness variations at various inclinations down to several
days. This approach allowed to remove a number of important observational
biases when comparing the solar variability to that of other stars (Nèmec et
al., 2020a; Reinhold et al., 2020). By employing the Nèmec et al. (2020b)
model, Reinhold et al. (2021) have found that rotation periods of most of the
G-dwarfs of near-solar age go undetected. This results provide an explanation
of the discrepancy between the predictions of the number of Sun-like rotators
in the Kepler field (van Saders et al., 2019) and the actual number of
detected ones.

Modelling brightness variations of stars rotating faster than the Sun poses
several challenges. The most prominent one being that the latitudinal
distribution of magnetic features of fast rotators is vastly different from
that of the Sun (Schuessler and Solanki, 1992; Schrijver and Title, 2001) For
instance, for stars with rotation periods below about 3 day various, studies
have reported the presence of polar starspots (Jeffers et al., 2002; Marsden
et al., 2004; Järvinen et al., 2006; Waite et al., 2015). Isık et al. (2018)
have developed a model, that is able to explain the formation of high latitude
spots for those faster rotating stars, by combining thin-flux tube simulations
with the SFTM.

In the present work, we extend the model of I18 to calculate the brightness
variations of stars with various rotation rates, degree of nesting of magnetic
features on their surfaces (i.e. the tendency of magnetic features to emerge
in the vicinity of previous emergences), and observed at various inclinations
(i.e. angles between rotation axis and direction to the observer). We compare
our results to the observational trends deduced by McQuillan et al. (2014)
from the Kepler data and propose a possible explanation of these trends.

###  5.2 Model

Our model consists of two building blocks: calculations of the surface
distribution of magnetic features and the subsequent calculations of the
stellar radiative fluxes. The first block is based on the model developed by
Isık et al. (2018), while the second block builds on the model presented in
Nèmec et al. (2020b).

####  5.2.1 Flux emergence and transport

At its core, the flux emergence and transport (FEAT) model extends the pattern
of emergence and evolution of the magnetic fields observed on the surface of
the Sun to more active stars. At first, an 11–year semi-synthetic active
region record based on the statistics of solar cycle 22 (Jiang et al., 2011a)
is generated. This record contains a number of properties of active regions.
Most importantly, how many regions are emerged in total throughout the cycle
(we call this the emergence rate), the latitude of emergence and the tilt
angles.

The faster rotation rate and the resulting stronger Coriolis force implies
that rising flux tubes are pushed towards higher latitudes. This is taken into
account by thin flux tube simulations (Schuessler et al., 1996; Isık et al.,
2018). For this, the time-latitude distribution of the active regions given by
the input record is mapped to the base of the convection zone, for the case of
$\Omega_{\odot}$. Then the flux-tube simulations were run to follow the
evolution of flux tubes throughout the convection zone until the surface of
the star (for $\Omega_{\star}$), where they emerge in the form of a loop with
two magnetic field patches (we will refer to those as bipolar magnetic
regions, BMRs for short) of opposite polarity as footpoints. Using the lookup
table from flux-tube simulations for a given stellar rotation rate,
($\Omega_{\star}$), the latitudes and tilt angles of emerging flux loops are
then fed into the surface flux transport model (SFTM). Additionally, in
agreement with the observed activity-rotation relation, the BMR emergence rate
can be scaled with the rotation rate. Namely, we define the time-dependent
emergence rate of BMRs on a star as $S_{\star}(t)=S_{\odot}(t)\cdot\tilde{s}$,
where $\tilde{s}$ is a scaling factor. In most of the runs we put it to
$\tilde{s}=\tilde{\omega}\equiv\Omega_{\star}/\Omega_{\odot}$, where
$\Omega_{\star}$ is the rotation rate of a given star and $\Omega_{\odot}$ is
the solar rotation rate.

The SFTM adopted here (Cameron et al., 2010; Isık et al., 2018) describes the
passive transport of the radial component of the magnetic field under the
effects of differential rotation, meridional flow, and supergranular
diffusion. In the SFTM, the flux emerges instantaneously in the form of
bipolar-magnetic regions (BMRs). Once the flux is emerged, it is subject to
differential rotation and the meridional flow, while diffusing at timescales
corresponding to supergranular motions on the Sun. Additionally, the emergence
pattern of BMRs is altered by introducing BMR nesting, which is a feature
observed on the Sun. In our study, we follow the approach of "free-nesting",
where a probability, $p$, is set for a given BMR to be part of a nest (for
further discussion see Isık et al., 2018, Appendix C).

####  5.2.2 Defining the filling factors

The output of the FEAT model are full solar surface maps (360$\degree$ by
180$\degree$) of the distribution of magnetic fields, with a resolution of
1$\degree$ x 1$\degree$ per pixel. As our brightness calculations rely on the
filling factors of spots and faculae, we need to convert the magnetic fields
into filling factors.

The approach presented in N20b follows the evolution of sunspots after their
emergence to calculate the coverage of the solar disk by spots. However, this
approach neglects the spontaneous formation of spots from magnetic flux which
without superposition would emerge as faculae (Kitiashvili et al., 2010). This
is not an issue for calculating solar variability since such spontaneous spot
formations have not been observed on the Sun. However, it might pose problem
for more active stars. Therefore we utilise another approach in this study. In
contrast to N20b, we calculate the disc coverages by spots directly from the
value of the magnetic field returned by the FEAT model. We only take the spot
component of the variability into account, as spots become the dominant source
of the variability for stars more active than the Sun (see Shapiro et al.,
2014). To calculate the coverage of a given pixel of the synthetic magnetogram
returned by FEAT we define two thresholds: a lower cut-off, $B_{min}$ for the
fields and an upper, saturation level, $B_{max}$. The spot coverage of a given
pixel is related to the field in the pixel as

$\alpha^{m,n}=\begin{cases}0\quad\text{if}\quad|B_{mn}|<B_{min}\\\
\frac{|B_{mn}|}{(B_{max}-B_{min})}+1-\frac{B_{max}}{(B_{max}-B_{min})}\quad\text{if}\quad
B_{min}<=|B_{mn}|<B_{max}\\\
1\quad\text{if}\quad|B_{mn}|>=B_{max},\end{cases}$ (5.1)

where $\alpha^{m,n}$ is the spot filling factor in a given pixel with
coordinates m and n on the map, and $|B_{mn}|$ is the absolute value of the
field in said pixel. We find that with this simplified approach and parameters