11institutetext: Lund Observatory, Division of Astrophysics, Department of
Physics, Lund University, SE-22100 Lund, Sweden
11email<EMAIL_ADDRESS>22institutetext: Zentrum für Astronomie der
Universität Heidelberg, Landessternwarte, Königstuhl, DE–69117 Heidelberg,
Germany
22email<EMAIL_ADDRESS>
# Solar photospheric spectrum microvariability

I. Theoretical searches for proxies of radial-velocity jittering

Dainis Dravins and Hans-Günter Ludwig 1122

(Received June 9, 2023; accepted August 18, 2023)

###### Abstract

Context. Extreme precision radial-velocity spectrometers enable extreme
precision stellar spectroscopy. Searches for low-mass exoplanets around solar-
type stars are limited by various types of physical variability in stellar
spectra, such as the short-term jittering of apparent radial velocities on
levels of $\sim$2 m s-1.

Aims. To understand physical origins of radial-velocity jittering, the solar
spectrum is assembled, as far as possible, from basic principles. Solar
surface convection is modeled with time-dependent 3D hydrodynamics, followed
by the computation of high-resolution spectra during numerous instances of the
simulation sequence. The behavior of different classes of photospheric
spectral lines is monitored throughout the simulations to identify
commonalities or differences between different classes of lines: weak or
strong, neutral or ionized, high- or low-excitation, atomic or molecular.

Methods. Synthetic spectra are examined. With a wavelength sampling
$\lambda$/$\Delta\lambda$ $\sim$1,000,000, the changing shapes and wavelength
shifts of unblended and representative Fe i and Fe ii lines are followed
during the simulation sequences. The radial-velocity jittering over the small
simulation area typically amounts to $\pm$ 150 m s-1, scaling to $\sim$2 m s-1
for the full solar disk. Flickering within the G-band region and in
photometric indices of the Strömgren $uvby$ system are also measured, and
synthetic G-band spectra from magnetic regions discussed.

Results. Most photospheric lines vary in phase but with different amplitudes
among different classes of lines. Amplitudes of radial-velocity excursions are
greater for stronger and for ionized lines, decreasing at longer wavelengths.
By matching precisely measured radial velocities to such characteristic
patterns should enable to remove a significant component of the stellar
‘noise’ originating in granulation.

Conclusions. The granulation-induced amplitudes in full-disk sunlight amount
to $\sim$2 m s-1, with the differences between various line-groups an order of
magnitude less. To mitigate this jittering, a matched filter must recognize
dissimilar lineshifts among classes of diverse spectral lines with precisions
$\sim$10 cm s-1 for each line group. To verify the modeling toward such a
filter, predictions of center-to-limb dependences of jittering amplitudes for
different classes of lines are presented, testable with spatially resolving
solar telescopes connected to existing radial-velocity instruments.

###### Key Words.:

Sun: photosphere – Sun: line profiles – stars: solar-type – techniques:
spectroscopic – stars: line profiles – exoplanets

##  1 Introduction

Extreme precision stellar spectroscopy is now enabled at several
observatories, using extreme precision radial-velocity spectrometers designed
for a wavelength stability corresponding to Doppler shifts of $\sim$1 m s-1 or
less. Such are operating at several large and very large telescopes, e.g.,
CARMENES on Calar Alto (Quirrenbach et al., 2014, 2018), ESPRESSO at the ESO
VLT on Paranal (Pepe et al., 2014, 2021); EXPRES at the Lowell Discovery
Telescope in Arizona (Jurgenson et al., 2016; Blackman et al., 2020); HARPS
(Mayor et al., 2003; Coffinet et al., 2019) and NIRPS (Bouchy et al., 2017) at
ESO on La Silla; HARPS-North and the forthcoming HARPS-3 on La Palma
(Cosentino et al., 2012; Thompson et al., 2016), HPF on the Hobby-Ebery
Telescope in Texas (Kanodia et al., 2021; Mahadevan et al., 2014); iLocater on
LBT in Arizona (Crass et al., 2022); KPF, the Keck Planet Finder on Maunakea
(Gibson et al., 2016); MAROON-X at Gemini-North on Maunakea (Seifahrt et al.,
2018, 2022); NEID at WYIN on Kitt Peak (Schwab et al., 2016); SPIRou at CFHT
on Maunakea (Donati et al., 2020; Hobson et al., 2021), Veloce Rosso at AAT in
Australia (Gilbert et al., 2018), and a few others. Such performance
requirements also are in the design criteria of future instruments for
extremely large telescopes (Halverson et al., 2016; Marconi et al., 2022;
Plavchan et al., 2019; Szentgyorgyi et al., 2018).

The primary task for these spectrometers is to search for periodic modulations
of stellar radial velocities induced by orbiting exoplanets. The detection of
planets with successively smaller mass requires to recognize successively
smaller velocity amplitudes: for an Earth-mass planet orbiting a solar-mass
star in a one-year orbit, the expected signal amounts to at most only 0.1 m
s-1 (e.g., Hall et al., 2018) The precision of current and foreseen
instruments now actually begins to approach these levels, soon enabling such
very challenging observations. However, radial-velocity fluctuations of much
greater amplitude are contributed by stellar phenomena such as convective
motions in granulation and supergranulation, p-mode oscillations, modulation
by dark starspots and bright magnetically active regions carried in and out of
sight by stellar rotation; changes in surface structure during long-term
magnetic activity cycles, and other. The requirement of mitigating effects of
stellar activity for radial-velocity measurements was realized already long
ago (e.g., Dravins, 1989) and to segregate such, often irregular, effects from
the more regular modulation induced by planets, comprehensive data analysis
methods have been developed to scrutinize spectra recorded with current
instruments. These typically include the calculation of cross-correlation or
other statistical functions to find the wavelength displacement relative to a
template, with various adjustments applied using the asymmetries or widths of
such functions. The level of magnetic activity can be estimated from
simultaneously measured chromospheric emission, usually in the cores of the Ca
ii H & K-lines but also feasible from statistics of Zeeman broadening
(Lienhard et al., 2023). Extensive comparisons between different methods
advocated by different groups have been carried out, testing software routines
on data with both a synthetic planetary signal injected, and for sequences of
pure stellar observations (Dumusque, 2016; Dumusque et al., 2017; Hara & Ford,
2023; Zhao et al., 2022).

While such routines do succeed in eliminating much of the non-planetary
signals, even the currently best modeling of stellar variability is unable to
extract planetary signals with amplitudes much less than 1 or 2 m s-1. The
limitations are thus no longer instrumental or technical but lie in
understanding the complexities of atmospheric dynamics and spectral line
formation, manifest as a jitter of the apparent radial velocity and flickering
of the photometric irradiance. As concluded in reviews of the field (Fischer
et al., 2016), and in reports from working groups for precision radial
velocities and exoplanet detection (Crass et al., 2021; Rackham et al., 2023),
some significantly different approach will be required in order to reach
precisions of 10 cm s-1 or better, as ultimately required to find an exoEarth.

The aim of this paper is not to further refine methods in analyzing observed
spectra but instead to start to try to understand the physical origins of
radial-velocity jittering in solar and stellar atmospheres. To reach
challenging goals of precisions in the cm s-1 range, it will probably not
suffice to apply statistical analyses on observed data but to somehow exploit
the information content of the many thousands of different spectral lines in
solar-type stars.

We examine time sequences of synthetic solar spectra, as computed from
hydrodynamic simulations of the solar photosphere. In principle, this amounts
to watching movies of high-resolution solar spectral atlases, searching for
systematic patterns of variability, in particular such that correlate with
excursions in apparent radial velocity. The paper is organized as follows:
Section 2 describes the background, the aims, the feasibility constraints, and
places the velocity jittering over the simulation areas in the context of the
full solar disk. Section 3 describes how the overall synthetic spectra are
obtained, and Section 4 how the properties of their individual spectral lines
are measured. Section 5 analyzes the wavelength jittering of groups of
selected Fe i and Fe ii lines. Section 6 examines the behavior of the
molecular G-band and surrounding atomic lines, including a discussion of their
formation physics and effects by magnetic fields. Section 7 evaluates the
fluctuations in color indices from broader-band $uvby$ Strömgren photometry.
Section 8 documents searches for such spectral-line parameters that failed to
yield significant correlations with excursions in radial velocity. The
concluding Section 9 outlines how it should be possible to diminish the
jittering signal arising from surface convection in searches for low-mass
exoplanets. The subsequent Paper II (Dravins & Ludwig, 2024) will evaluate
signatures of microvariabilty in observations of the Sun seen as a star, also
guided by the conclusions from this Paper I.

Figure 1: Synthetic solar spectra from the CO 5BOLD N59 model simulation,
centered on 525 nm; seen over gradually narrower wavelength ranges, decreasing
by a factor of three for each row from top down. Left column: Disk center,
$\mu$ = cos $\theta$ = 1, middle: $\mu$ = 0.41, right: $\mu$ = 0.09. The
intensity scale is common for all frames, i.a., illustrating how the limb
darkening is more pronounced at shorter wavelengths.

##  2 Synthesizing spectra from line formation physics

A copious amount of work has been invested in analyzing and dissecting
observed spectra, using a multitude of statistical and other functions, but
appears to hit a boundary on the m s-1 level. Instead of scrutinizing observed
spectra from ‘top down’, we now try to assemble and construct not-yet-observed
stellar spectra from ‘bottom up’, as far as possible from basic principles. By
numerically simulating stellar atmospheric structure and dynamics ‘ab initio’
for both quiet and magnetically active regions, and knowing the physical
conditions in the model, signatures from dynamic stellar surface
inhomogeneities can be traced forward into the complete synthetic spectrum.
Different classes of stellar surface features (hot or cold, magnetic or not)
affect different classes of spectral lines (weak or strong, neutral or
ionized, high- or low-excitation, atomic or molecular) to varying degrees and
their effects should be possible to identify and calibrate also in the
spectrum of the full stellar disk.

Such an approach is made feasible by two developments. Firstly, on the
theoretical side, convectively driven gas motions near stellar surfaces can
now be modeled in detail with time-dependent 3D [magneto-]hydrodynamics, and
such simulations are now established as quite realistic descriptions of the
spectral-line forming regions in the photospheres of at least solar-type
stars. A more recent development is the computation of complete stellar
spectra from such models; Chiavassa et al. (2018); Dravins et al. (2021a, b);
Ludwig et al. (in prep.). These produce complete and complex stellar spectra,
incorporating a huge number of spectral lines from databases such as
VALD111VALD, The Vienna Atomic Line Database of atomic and molecular
transition parameters of astronomical interest and others (Heiter et al.,
2015; Ryabchikova et al., 2015), and are able to so do so in sampling
wavelengths with a hyper-high spectral resolution ($\lambda$/$\Delta\lambda$
$\sim$1,000,000)222The term hyper-high is used to denote spectral resolutions
$\lambda$/$\Delta\lambda\gtrsim$ 106 since ultra-high is already in common use
for describing spectrometers with the much lower resolutions of ‘only’
$\sim$200,000., spanning all observable spectral ranges. Details in the
resulting spectra depend on a complex interplay between, for example,
atmospheric opacity in different atmospheric inhomogeneities, blending of
multiple spectral lines, and as seen under different viewing angles across the
stellar disk in different wavelength regions. Of course, the modeling entails
various approximations, but since the synthetic spectra are free from random
photometric noise, spectral microvariability such as the radial-velocity
jittering of different spectral features can be followed throughout the
simulation sequences by measuring the relative wavelength displacements of any
feature anywhere in the spectrum and correlations with other spectral
parameters identified.

###  2.1 Photometric irradiance flickering

Corresponding challenges in microvariability exist for the photometric
flickering, a limiting parameter in identifying photometric transits of small
planets across stellar disks and also, more generally, a diagnostic tool for
studying the presence of granulation on stellar surfaces, as predicted from
hydrodynamic models for different classes of stars (e.g. Ludwig, 2006;
Lundkvist et al., 2021). With $\sim$106 granules across the solar disk,
typical brightness contrasts of $\sim$20 % in each granule reduces to $\sim$2
10-4 after averaging over such a million random locations (e.g. Nèmec et al.,
2020). Precision photometry from space reveals the temporal power spectra of
such flickering, and can be used to infer stellar properties (Cranmer et al.,
2014; Kallinger et al., 2014; Samadi et al., 2013a, b; Sulis et al., 2020).

Table 1: Identifiers for the CO 5BOLD models used to compute synthetic solar spectra. The identifiers are unique and are also used in other work. The spectra were sampled in 47 $\times$ 47 grids across the simulation areas of 140 $\times$ 140 horizontal spatial points extending over $hsize$ in $x$ and $y$, at 19 or 20 temporal snapshots, and for up to 21 angular directions. T${}_{\textrm{eff}}$ [K] | log $\varg$ [cgs] | Temp. RMS [K] | Model | Spectral points | boxsize:$hsize$ [cm] | boxsize:$vsize$ [cm]
---|---|---|---|---|---|---
5782 | 4.44 | 12 | N58 - d3gt57g44n58 | 47 $\times$ 47 $\times$ 19 $\times$ 13 | 5.60 108 | 2.25 108
5774 | 4.44 | 16 | N59 - d3gt57g44n59 | 47 $\times$ 47 $\times$ 20 $\times$ 21 | 5.60 108 | 2.25 108
Figure 2: Examples of selected lines, here some Fe i lines around $\lambda$
525 nm from the N59 model. Top: synthetic spectra at solar disk center, $\mu$
= 1, bottom: $\mu$ = 0.41. Clean lines selected for detailed analysis are
marked in red (strong-medium-weak marked as A-B-C). Lines become noticeably
broader toward the limb. Figure 3: Example of temporal variability of line
profiles during the N59 simulation sequence, sampled at 20 instances of time.
Left columns: the strong Fe i $\lambda$ 436.59 nm, $\chi$ = 2.99 eV line in
the blue; Right columns: the weak Fe i $\lambda$ 675.27 nm, $\chi$ = 4.64 eV
line in the red. Upper row shows profiles for disk center $\mu$ = 1; bottom
row shows integrated sunlight, corresponding to a full solar disk, synthesized
using center-to-limb profiles from the small simulation area. Zoomed-up frames
to the right show line centers on a uniform ‘wavelength’ scale in units of
radial velocity (at top). In the zoomed-up frames, tick marks on the vertical
axes correspond to those for the full profiles.

###  2.2 Limitations of theory

Ideally, one could envision detailed magnetohydrodynamic simulations embracing
entire stellar surfaces, producing time-variable spectral atlases for varying
mixtures of different types of surface structures on different classes of
stars. This is not yet practical and the scope has to be limited. We start
with the solar spectrum, both because it is the most studied and modeled one,
and because theoretical predictions can be observationally tested in quite
some detail. Realistic 3D simulation volumes can cover groups of multiple
photospheric granules and permit to follow how their appearance changes when
seen under different angles, from disk center out toward the limb. Still,
simulated solar surface areas can cover only a small fraction of the full
surface.

As described in Sect. 5 below, the equivalent of a full solar disk is obtained
from this simulation area of 5.6 $\times$ 5.6 Mm2 by summing appropriately
weighted spectra from various computed center-to-limb angles, producing a
spectrum that retains the fluctuations from the small simulation area. The
actual solar hemisphere area is larger by a factor of $\sim$100,000 and the
projected solar disk by $\sim$50,000. In case of statistically independent
random fluctuations (neglecting details of geometric projection effects and
statistical weighting by limb darkening), the averaged full-disk amplitudes
should then decrease by a factor equal to the square root of this number.
However, away from disk center, spectra are also computed for up to eight
different azimuthal directions, thus providing some further averaging.
Nevertheless, even if such spectra are dissimilar as seen from various oblique
directions, such samples are from the same instance in time and cannot be seen
as statistically fully independent. While it is awkward to precisely quantify
their level of randomness, the amplitudes for the actual full solar disk can
be expected to average out to $\sim$150-200 times smaller than those from our
modest simulation volume. As will be seen below, the jittering of spectral
lines averaged over the simulation area typically extend over $\sim$300 m s-1,
which would then reduce to $\sim$1-2 m s-1 in actual integrated sunlight. Such
are the magnitudes of microvariability to be studied here.

The modeling discussed in this first paper concerns mainly the non-magnetic
granulation. This covers most of the solar surface, and provides most of the
solar flux. Analyses of the time-variable spectra from the simulation volume
enable to identify relations between various parameters and to extrapolate
what amounts should remain in actual full-disk spectra. However, to properly
reproduce also larger-scale features such as supergranular flows or p-mode
oscillations will require (much) larger simulation volumes. Also, a
significant part of the spectrum variability originates in regions of magnetic
granulation and spots. Magnetohydrodynamic 3D simulations for such are
available and their synthetic spectra can be computed, foreseen to be
discussed in future papers.

Irrespective of the computational sophistication, the accuracy of synthetic
spectra is ultimately limited by the quality of atomic and molecular data.
Current synthetic spectra include more than half a million spectral lines,
with their wavelengths and oscillator strengths originating from laboratory
measurements or theoretical predictions. The accuracy of those data,
originating from a mixed variety of sources, is somewhat variable. While the
overall spectrum does reproduce the observed one to a rather high degree,
detailed examination often reveals certain differences in relative line
strengths, of blending lines, or other. For some studies, these limitations
may be circumvented: e.g., to gain a knowledge of the absolute amount of
convective blueshifts, an identical spectral synthesis may be repeated with
the same input data but for a static model atmosphere, where convective
lineshifts cannot exist, and the difference to the hydrodynamic case then
evaluated (Dravins et al., 2021a). In a somewhat analogous vein, we do not
need to precisely reproduce the exact strength of each individual spectral
feature but we rather examine the temporal jittering in wavelength and changes
in shapes and strengths of representative types of lines, even if their
absolute wavelengths or exact oscillator strengths would not be precisely
known.

Figure 4: Temporal variability of idealized line profiles during the N58
simulation sequence, sampled at 19 instances of time. Left columns:
Differently strong Fe i lines at $\lambda$ 620 nm, $\chi$ = 3 eV; Right
columns: Lines of similar strength but with different lower excitation
potentials, $\chi$ = 1 eV (black) and 5 eV (red). Top row: Profiles at disk
center $\mu$ = 1; bottom row: $\mu$ = 0.59. To limit cluttering, profiles are
shown for only two (orthogonal) azimuth angles. The right-hand frames show the
line centers zoomed up, with ‘wavelength’ scales equal to those in the
previous Figure 3. In these zoomed-up frames, tick marks on the vertical axes
correspond to those for the full profiles. Figure 5: Example of 5-parameter
Gaussian-type fit to a typical Fe i line profile in integrated sunlight.
Profiles from each temporal snapshot are thin black lines; their average is
bold-dashed red, and the fitted profile is yellow circles. In the zoomed-up
frame at right, tick marks of the vertical scale are carried over from the
left frame.

###  2.3 Realistic project aims

If microvariability signatures generated in stellar atmospheres can be
adequately understood, one could then determine what observational efforts
eventually would be required to detect and calibrate those. Not less
motivating for both the theoretical and observational aspects, is that both of
them are expected to improve in the near future. With enhanced computational
capacity, numerical simulations are being extended to greater solar and
stellar volumes, also incorporating an increased variety of magnetic features,
sunspot umbrae, penumbrae, and other. Observationally, several light feeds
from solar telescopes are now connected to extreme-precision radial velocity
spectrometers and are expected to continue collecting data over at least a
solar activity cycle or longer. Of additional interest will be data from
across a spatially somewhat resolved solar disk measured with comparable
precision, e.g., monitoring the characteristics in wavelength jittering in
quiet areas versus magnetic ones. Such behavior can be theoretically predicted
but corresponding observations are not yet available, although projects are in
progress (Leite et al., 2022).

The variability of the optical spectrum of the Sun seen as a star has
previously been studied mainly in features such as Ca ii H & K or H$\alpha$
but signals in such chromospheric lines are unlikely to precisely relate to
radial velocities, where the signal comes from photospheric lines only.
However, the number of photospheric features that potentially can be examined,
combined or correlated with one another is enormous and here only a limited
number of selected samples can be surveyed. The aim will be to outline
possible paths toward the identification of spectral signatures to serve as
proxies for calibrating (at least part of) the radial-velocity jittering in
sunlight, and eventually also for other stars. Such proxies would need to be
measurable at ground-based observatories independently of the overall radial
velocity. These might include, e.g., velocity differences between various
line-groups, changes in line-depths or widths, or perhaps even broad-band
photometric colors. A typical photospheric absorption line subtends a
wavelength width of $\sim$10 km s-1, and a radial-velocity signal of 1 m s-1
thus amounts to only 1 part in 10,000. Quite possibly, proxies for such
signals are comparably minute and probably may require extensive modeling to
be securely identified.

The value of understanding photospheric spectrum microvariability extends to
also areas outside monitoring stellar radial motion, such as searches for
varying fundamental constants in nature, using precise differences of
spectral-line wavelengths in solar-type stars (Berke et al., 2023a, b; Murphy
et al., 2022).

Figure 6: Radial-velocity shifts of synthetic isolated Fe i lines at $\lambda$
620 nm in the N58 model. Relations between shifts at solar disk center, $\mu$
= 1, for weak, medium and strong lines, are shown for different excitation
potentials. Each point represents one temporal snapshot during the simulation.
Dotted reference lines mark the identity relations, the other fitted lines
indicate how the dependence of convective blueshift on line-strength and
excitation potential generally remains valid also when the spatially averaged
radial velocity fluctuates.

##  3 CO 5BOLD modeling of the solar spectrum

Building upon previous work (Dravins et al., 2017a, b, 2018, 2021a, 2021b),
profiles of synthetic spectral lines are computed from 3D atmospheric
structures produced in hydrodynamic simulations, using those as grids of
space- and time-varying model atmospheres. Two different non-magnetic solar
models are used here: N58 and N59 (Table 1). Both these have solar
temperature, surface gravity and metallicity but differ in some computational
aspects, and the details of their granular evolution geometries are somewhat
different (although, of course, statistically similar). Computationally, the
N59 model is a successor to N58, which was examined in Dravins et al. (2021a,
b). For example, while the radiation balance in N58 was calculated using five
wavelength bins, N59 uses 12 and hence, the photospheric energy budget should
be better represented. For each model, the horizontal simulation area is 5.6
$\times$ 5.6 Mm2, with each modeling pixel in the 140 $\times$ 140 grid of
hydrodynamic simulation equal to 40 $\times$ 40 km2 (Table 1). In the
simulation sequences, the gradual evolution of granular structures can be
followed. Their duration of about two solar hours permitted some 20 temporal
snapshots to be selected sufficiently far apart in time to ensure largely
uncorrelated granulation flow patterns between successive samples. For such
snapshots, spectra for all different elevation and azimuth angles are computed
on a grid of 47 $\times$ 47 points (each horizontally 120 $\times$ 120 km2),
downsampled by merging 3 $\times$ 3 pixels in the hydrodynamic simulation.
Since successive snapshots are chosen well apart in time, their later
averaging should have a largely random character but, for a given snapshot,
the spectra at different emerging angles originate from the same granular
structures (viewed from different directions), making their level of mutual
randomness more awkward to estimate.

Details about the computation of these types of models are in Tremblay et al.
(2013), where also representative synthetic granulation images are shown, both
for models at solar temperature, and others. Further details are in Ludwig et
al. (in prep).

From the N58 model, profiles of single, idealized and isolated spectral lines
for Fe i at $\lambda$ 620 nm were computed for five different line strengths
and three different lower excitation potentials, $\chi$ = 1, 3, and 5 eV. Four
different center-to-limb positions $\mu$ = cos $\theta$ = 1.00, 0.87, 0.59,
0.21, and four azimuth angles $\psi$ = 0, $90^{\circ}$, $180^{\circ}$ and
$270^{\circ}$ were treated in spectral line synthesis carried out in local
thermodynamic equilibrium (LTE). Such calculations were made for 19 temporally
sampled snapshots during the extent of the simulation. Each profile was
computed for 201 wavelength points covering $\pm$15 km s-1. The sampling
stepsize thus corresponds to a hyper-high spectral resolution
$\lambda$/$\Delta\lambda$ = 2,000,000. Hyper-high resolution is required to
fully resolve intrinsic line asymmetries and to resolve wavelength shifts on
levels of cm s-1. Spectra from this series have been discussed in Dravins et
al. (2017a, b, 2018), and also in Dravins et al. (2021b).

For the N59 model, complete and complex spectra were computed, incorporating
practically all relevant spectral lines from databases such as VALD.
Calculations were made for the spectral range of 120-1000 nm, and for 20
largely uncorrelated instances in time during the simulation. The wavelength
sampling corresponds to the hyper-high value $\lambda$/$\Delta\lambda$
$\sim$900,000. For each instance in time, four center-to-limb angles $\mu$ =
1.0, 0.79, 0.41, and 0.09 are provided. The spectra were computed for a total
of 21 emergent directions: eight azimuth angles in steps of $45^{\circ}$ for
$\mu$ = 0.79 and 0.41, and four angles in steps of $90^{\circ}$ for $\mu$ =
0.09. With intensities in absolute units, it becomes possible to obtain also
synthetic broad-band colors. Time-averaged spectra from analogous types of
calculations were presented previously in Dravins et al. (2021a, b) for
different classes of stars, but now the spectra are time-resolved.

Despite the spatial averaging over the model area, the total volume of these
data is significant: with some 3 million data points in each spectral
realization, and for each of the 21 position angles, at each of the 20
instances in time, the total amounts to more than a billion spectral intensity
values, requiring some selectivity in the handling of spectra and in choosing
and limiting lines for analysis. In the figures, plots for specific center-to-
limb positions refer to averages over all computed azimuth angles, unless
noted otherwise. Integrated sunlight is computed as for a full disk (and
labeled thus in the figures) by appropriately interpolating and summing
spectra from different $\mu$-values (Fig.1). The numerical values, however,
refer to those obtained from the small simulation area while – as discussed
above – those for the actual Sun are estimated to be smaller by a factor
$\sim$150-200. Further degrees of freedom are offered by stellar rotational
broadening; such variants were treated in Dravins et al. (2017a, b) but here
we limit ourselves to rotationally unbroadened lines only.

Each of these N58 and N59 model spectra offers its advantages and limitations.
The advantage with idealized single lines in N58 is that these can highlight
possible subtle contrasts between lines of only slightly differing strengths
or excitation potentials. However, such lines might not represent
observational realities, where not all combinations of line parameters are
present and with often multiple blending lines in crowded spectra. Such
complete spectra are instead produced in N59 but require some effort in
selecting representative and distinct spectral features. With more than
600,000 lines included, any selected line will unavoidably incorporate also
some weak blends which might have a different temporal behavior than the main
line, thus adding some astrophysical noise. The complete spectra in N59 should
illustrate what could be practically measurable while the idealized lines in
N58 should help to understand the physical origins of possibly identified
relations.

A limitation comes from the circumstance that reliable modeling is for the
lower photospheric levels in the atmosphere while higher chromospheric
structures are not treated. Consequently, lines that are (partially) formed in
the uppermost photosphere or lower chromosphere cannot be expected to be
precisely reproduced. On one hand, such chromospherically influenced lines –
perhaps Na i D, the Mg i triplet or Balmer lines from hydrogen – are not
likely to be used for measuring precision radial velocities. However, we will
still examine their behavior since their modeling may confirm trends when
going to stronger lines formed in higher atmospheric layers and they should
follow photospheric events more closely than chromospheric Ca ii H & K-line
emission.

Table1 gives the unique identifiers of the models, which enables comparisons
with other work based on the CO 5BOLD grid (Freytag et al., 2012). The number
of sampling points for each spectral line profile is the product of horizontal
spatial resolution points in x and y, the number of temporal snapshots
extracted during the simulation sequence, and the number of different
directions (combined elevation and azimuth angles) for the emerging radiation.
The x-y horizontal hsize and z vertical vsize extents of the hydrodynamic
simulation volume are there given in centimeters.

Figure 7: Differences in radial-velocity shifts between lines with different
strengths (but similar excitation potentials). These data at $\mu$ = 1 are
from synthetic isolated Fe i lines at $\lambda$ 620 nm in the N58 model. Each
point represents the difference between instantaneous radial velocities (e.g.,
strong line minus weak) at one temporal snapshot during the simulation.
Straight lines are fits to the data points. Different scales between the
frames are adapted to reflect how convective blueshifts deviate for lines with
increasingly different strengths.

##  4 Synthetic spectral lines

Figure 1 shows complete synthetic spectra for three center-to-limb angles from
the N59 model: disk center $\mu$ = 1, $\mu$ = 0.41, and $\mu$ = 0.09. These
data are from one particular instance in time; however the differences between
successive temporal snapshots or between different azimuth angles are much too
small to be visible on this scale.

Figure 2 gives an example of the selection of differently strong Fe i lines in
one particular wavelength region while Figures 3 and 4 illustrate the temporal
variability in lines from the N59 and N58 models.

###  4.1 Fitting line parameters

To enable searches for dependences among various line parameters, the
information resident in the line profiles must somehow be parameterized. We
stress that the line profiles from each instant in time are not the spatially
resolved ones. Such local profiles, originating from specific granular
inhomogeneities, display grossly varying shapes (e.g., Asplund, 2005;
Bergemann et al., 2019; Dravins & Nordlund, 1990), and (even disregarding the
data quantity) would not lend themselves to easily understandable
quantification. Each of the current temporally resolved line profiles was
spatially averaged over more than 2000 spatial points across the
hydrodynamically modeled stellar surface, embracing numerous granulation
structures, thereafter resulting in profiles that are closely reminiscent of
the temporally averaged ones, and that can be described with commonly used
parameters. However, these still retain noticeable variability over time since
the simulation area is tiny compared to the full solar disk.

To quantify these profiles, various tests of line-fitting parameters were
made, finding a suitable approximation by fitting with five-parameter
Gaussian-type functions of the type
$y_{0}-a\cdot\exp[-0.5\cdot(|x-x_{0}|/W)^{c}]$, yielding $y_{0}$ as the
continuum level; $a$ for the line’s absorption depth; $W$ as a measure of the
line width, $c$ indicating the degree of the line’s boxiness or pointedness,
and $x_{0}$ for the central wavelength.

Although the profiles of course are not precisely Gaussian, these functions
appear reliable to use. Different varieties were tested (Lorentzian, etc.),
but it was found that such Gaussian-type functions gave robust estimates for
all basic line parameters. For the wavelength positions, this fitting thus
provides one value that refers to the displacement of the line as a whole. For
asymmetric profiles, a single number for the line wavelength is not a uniquely
defined measure, but depends on whether the fit is made to the central or more
extended parts of the profile, and with what type of weighting. However, the
physical variability among our line profiles is much greater than possible
ambiguities stemming from the exact shape of the fitting functions, and – in
any case – here we do not aim at determining absolute line quantities as in
Dravins et al. (2021a), but only their relative changes in time. Fig.5 gives a
representative example of such a fit.

With a significantly greater sample (producing more stable averages), one
could parameterize with respect to also line asymmetries or correlations
between different portions of the line profile (e.g., Ludwig & Steffen, 2008).
However, studies of more subtle effects are currently constrained by the
limited number of temporal samples. Data from ab-initio simulations, where the
character of the output cannot be precisely predicted, have a character
similar to that of noisy observational data in that they need to be arranged
and averaged in order to discern trends. The parameters of wavelength shift
and line width appear to be the most readily measurable quantities (also
observationally), likewise for the center-to-limb dependences in spatially
resolved spectra.

Figure 8: Differences in radial-velocity shifts between Fe i lines of
different strength in different spectral regions of integrated sunlight,
versus shifts in medium-strong Fe i lines at $\lambda$ 520 nm. Each point
represents one temporal snapshot during the N59 simulation. The spectra
correspond to those of a full solar disk, as synthesized from a sequence of
center-to-limb spectra, although the numerical values [m s-1] refer to the
small simulation area (for the actual Sun, divide by a factor $\sim$150-200).
Note that the vertical scales differ among the frames. The fitted straight
lines suggest the trends but also indicate the limitation from finite
simulation sequences.  Figure 9: Radial-velocity shifts between Fe i and Fe ii
lines of relatively similar strengths (strong minus medium) versus those with
greater differences (strong minus weak), in different spectral regions of
integrated sunlight. The numerical values refer to the small simulation area
in the N59 model (for the actual Sun, divide by a factor $\sim$150-200). All
frames have identical scales.

###  4.2 Variability in radial velocity and irradiance

Considerable efforts have been made in studying the Sun as a star and in
searching for statistical relations between various parameters of solar and
stellar variability, including work by Cegla et al. (2019), Collier Cameron et
al. (2019, 2021), de Beurs et al. (2022), Dumusque et al. (2015, 2021),
Haywood et al. (2016), Lanza et al. (2016, 2019), Maldonado et al. (2019),
Marchwinski et al. (2015), Thompson et al. (2020); references therein, and
also as reviewed by Hara & Ford (2023). These studies use various statistical
approaches to search for possible correlations and empirical proxies for
radial-velocity fluctuations. Considerations of other spectral types include
Dumusque et al. (2011a) and Meunier et al. (2017a, b).

An estimate of the trends in our models indicates that excursions around
$\pm$100 m s-1 correspond to about $\pm$1% in irradiance. As discussed above,
actual full-disk amplitudes will be reduced from the simulation area by a
factor of $\sim$ 200 or 150, leading to about 0.5 or 0.7 m s-1 per 50 ppm in
irradiance. These values seem consistent with those deduced by Cegla et al.
(2019), Meunier et al. (2015), and Meunier & Lagrange (2020) for the
granulation-induced signal in apparent solar full-disk radial velocity
although further contributions may come from larger-scale structures such as
supergranulation (Meunier & Lagrange, 2019), not yet accessible to detailed 3D
modeling. A further modeling caution is that brightness variations are also
driven by p-mode oscillations which are incompletely treated in the small
simulation volumes.

We also note recent related work by Kitiashvili et al. (2023), where synthetic
radial-velocity fluctuations are computed from a 3D model of the G0 V star
HD209458, revealing amplitudes of $\sim\pm$200 m s-1 over their simulation
area of 12.8 Mm width.

##  5 Photospheric Fe i and Fe ii lines

An examination of the simulation sequences shows that different absorption
lines generally fluctuate in phase although with subtle (but systematic)
differences between lines of different strength or located in different
spectral regions. The absolute amounts of convective blueshift in time-
averaged spectra of course show their familiar dependence on line-strength
(e.g., Dravins et al., 2021a) but there are also line-strength dependences of
the amplitudes in the temporal variation of these blueshifts.

To identify systematics in the spectral-line jittering, various correlations
were examined between different groups of lines, using the parameters
determined from the five-parameter Gaussian-type fitting as described above.
From the N58 model, idealized single Fe i lines at $\lambda$ 620 nm were
computed with a nominal zero rest wavelength shift, and thus the instantaneous
radial velocity value equals the displacement relative to this reference point
of zero. Five different line-strengths were computed in N58, each for the
three different lower excitation potentials of 1, 3, and 5 eV. In all cases,
the trend of fitted parameters with changing line-strengths or excitation
potentials was found to be monotonic and in the following figures, only a few
representative cases are selected to be plotted.

From the N59 model with complete spectra, groups of clean Fe i lines were
selected in representative wavelength regions: around $\lambda$ 430 nm,
$\lambda$ 520 nm (Fig. 2), and $\lambda$ 670 nm, and an Fe ii sample around
$\lambda$ 435 nm. These regions were selected to span different wavelength
regions, where also lines of different relative strength were present (labeled
as strong-medium-weak). However, the absolute line strengths differ among the
regions: Fe i lines in the red are rather weaker than lines in the blue (Fig.
3), and at longer wavelengths there are too few Fe ii lines for a meaningful
sample. Also, a further subdivision in N59 with respect to excitation
potential was not seen as practical. The instantaneous radial velocity value
for each of those lines is obtained as the displacement relative to each
line’s average over the simulation sequence.

The number of temporally separated snapshots during the simulation sequences
is 19 for the N58 model, and 20 for N59. Each snapshot represents the spatial
average over the simulation area, and these are the number of points plotted
in the figures below. The values shown are averages for the selected lines
from each region; however it was found that, within each wavelength region,
the radial-velocity shifts are almost identical for different lines with the
same strength; differences appear only between different line-strengths,
different wavelength regions, and different excitation potentials. This also
implies that the noise in these lineshift plots originates from temporal
fluctuations in the granulation structure during the 3D-simulation, not
photometric or other uncertainties in the computed spectra. To decrease such
noise requires spatially and/or temporally more extended simulations.

Figure 10: Spatially resolved jittering across the solar disk. RMS values for
radial velocities of idealized Fe i lines at $\lambda$ 620 nm are shown for
differently strong lines at excitation potentials $\chi$ = 1 and 5 eV. Bold
curves show averages over the small simulation area: solid for strong lines;
dashed for weak. Curves for $\chi$ = 1 eV are black, 5 eV are red. Multiple
thin lines illustrate the differences among different azimuth angles at each
center-limb position.

Figure 6 shows fluctuations in the radial velocities as obtained from fits to
isolated, differently strong Fe i lines at $\lambda$ 620 nm in the N58 model;
each point marks a different time during the simulation. For reference, the
horizontal axis shows the instantaneous radial velocity for a weak line with
the excitation potential 1 eV, while the plot is populated with data points
for medium and strong lines, as in Fig. 4. The fitted straight lines confirm
that the convective blueshifts systematically decrease with increasing line-
strength (left panel). There is also a certain dependence on excitation
potential (right panel), with higher-excitation lines exhibiting slightly
greater blueshifts. However, such trends seen for these idealized lines, are
less obvious to identify in the real solar spectrum because not many really
high-excitation lines of significant strength exist. These plots are for solar
disk center, $\mu$ = 1.

Figure 7 compares differences in instantaneous radial-velocity shifts between
pairs of idealized Fe i lines at $\lambda$ 620 nm with modest differences in
strength (strong minus medium; horizontal axis) versus simultaneous
differences between more dissimilar lines (strong minus weak; vertical axis).
The plots show systematic trends, such that a given velocity shift between
lines of rather similar strengths most often corresponds to a simultaneous but
greater shift between lines with greater differences in strength. There,
amplitudes between weak and strong lines have some further dependence on
excitation potential, although there is scatter among the limited number of
data points.

While Figs. 6 and 7 are for idealized lines at solar disk center, Figure 8
shows these types of relations for the simulated full solar disk, now measured
for essentially unblended lines selected in the complete solar spectrum from
the N59 model. The spectrum for the integrated full solar disk is obtained by
integrating over a disk, where the spectra from the four computed center-to-
limb angles ($\mu$ = 1.0, 0.79, 0.41, 0.09) are first averaged over their
respective azimuth angles, then interpolated to intermediate center-to-disk
positions, and finally summed with appropriate wavelength-dependent limb-
darkening values. This is equivalent to the spectrum of a tiny solar disk,
where the fluctuation amplitudes of radial velocity, irradiance, and other
still refer to the small simulation area. As discussed above, the amplitudes
for the actual full solar disk, will be $\sim$150-200 times smaller, although
the exact number is awkward to precisely determine since it depends on
understanding the degree to which temporal and azimuthal samples from the
simulations can be considered statistically independent.

As reference for the radial-velocity, the values for medium-strength lines at
$\lambda$ 520 nm (Fig. 2) make up the horizontal axis in Fig.8. The vertical
axes show simultaneous lineshift differences between pairs of line-groups with
different strengths and/or in various wavelength regions. Despite a scatter of
points, all fitted lines indicate qualitatively similar trends, with the
greatest amplitudes between strong lines around $\lambda$ 520 nm and weak ones
in the far red at $\lambda$ 670 nm.

Figure 9 summarizes the main trends: velocity jittering in different lines is
almost always in phase, but the amplitudes depend on line-strength, ionization
level, and wavelength region. The greatest amplitudes are noted for Fe ii,
while for Fe i, both the jittering amplitudes and their differences between
differently strong lines decrease for longer wavelengths in the red. These
data are from essentially unblended lines selected in the complete solar
spectrum from the N59 model. Straight lines at bottom right show the various
fitted trends.

As seen from the horizontal axis of Fig. 8, the total jittering amplitude over
the simulation area is on the order of $\pm$100-200 m s-1, reducing to
$\sim$1-2 m s-1 in integrated sunlight. The difference between weak and strong
lines is an order of magnitude smaller (Fig. 9), perhaps $\sim$0.1-0.2 m s-1 ,
requiring highly precise measurements for detection. However, once such shifts
between groups of different spectral lines can be reliably identified, that
will constitute a specific signature (for the current discussion, at least of
the non-magnetic) granulation, and suitably matched filtering should be able
to identify and remove much of that particular atmospheric signature.

Figure 11: Radial-velocity fluctuations among different line-groups, as
function of absorption-line depth. Standard deviations and jittering ranges in
the simulation of integrated sunlight are shown, with numbers from the small
simulation area (for the actual Sun, divide by a factor of about 150-200).
Stronger and deeper lines display greater amplitudes. The trends extend to the
even stronger Na i D1 line which, at 136 and 404 m s-1, is off scale and not
included for the fitted lines.

###  5.1 Mechanism of radial-velocity differences

For solar-type stars, the greatest convective blueshifts are found in weak
lines formed in the deeper photosphere. However, here we find that the
variability of that quantity is smallest in those weak lines and instead
increases for stronger lines predominantly formed in the upper photosphere.
The mechanism can be tentatively understood by examining a typical 3D
atmospheric structure, e.g., Fig. 14 in Nordlund et al. (2009). That figure
shows temperatures as a function of optical depth at several horizontal
locations across a simulation area. When plotted on an optical-depth scale
(not versus geometric height), the temperature dependence with optical depth
is rather similar in both hot granules, and in cool downflows. The reason is
the steep temperature dependence of the opacity, and any given optical depth
is reached at about the same temperature, irrespective of whether the
temperature gradient is high (as in rising granules) or more modest (as in
sinking downflows). Local line strengths, however, depend also on the vertical
gradients.

As illustrated in that figure, the smallest temperature fluctuations occur at
optical depths around $\tau$ = 0.3, reaching greater but comparable values
both deeper down ($\tau$ = 1) and higher up, $\tau$ = 0.1. Weaker spectral
lines form at optical depths around about $\tau$ = 0.5, where the fluctuations
between various horizontal locations are smaller than those on the more
elevated formation heights of the stronger lines (perhaps $\tau$ = 0.1). At
such levels, the lines thus experience greater jittering amplitudes. A
velocity disturbance coming from below will propagate upward into layers of
reduced optical depth and lower density, where the fluctuation amplitudes may
roam more freely. However, what matters is not only the excursion of the
temperature as such, but rather how spectral-line parameters respond to
temperature disturbances. As seen in observed spectra of stars slightly hotter
and slightly cooler than the Sun, temperature divergences from the solar value
cause greater relative line-strength changes if the excursion is toward cooler
temperatures (i.e., toward higher regions of stronger-line formation) than if
it gets hotter; e.g., Fig. 4 in Biazzo et al. (2007). This holds for both Fe i
and Fe ii lines, although their signs of change are opposite. Such
differential behavior is an established indicator of stellar temperature
changes (Gray, 1994).

###  5.2 Jittering amplitudes

Due to their smallness, some model predictions may not be easy to verify in
current observations of the Sun seen-as-a-star. However, the modeling could
instead be tested in some detail with spatially resolved observations across
the solar disk. In Fig. 10. the root-mean-square values for the velocity
fluctuations are shown for idealized Fe i lines at $\lambda$ 620 nm, for two
line-strengths and two excitation potentials. At the various center-to-limb
positions ($\mu$ = cos $\theta$ = 1.00, 0.87, 0.59, 0.21), also the values for
different azimuth angles are shown, together with their averages. The multiple
thin curves indicate the physical spread of 3D dynamics in the model and also
serve to remind that averaging over rather extensive simulation sequences will
be required to reliably identify also more subtle dependences or higher-order
effects.

The jittering is clearly greatest at disk center, gradually decreasing toward
the limb. Over much of the disk (50% of the disk area is at $\mu$ $>$ 0.71),
the stronger line displays the greatest jittering amplitude but closer to
limb, the weak line starts to dominate. Near disk center, the low-excitation
lines jitter more but that reverses near the limb.

Figure 11 instead shows jittering amplitudes for the full simulated solar
disk, for different selected lines, now including the quite strong Mg i
$\lambda$ 457 nm. Also, the even stronger Na i D lines were measured. Those
will likely not be used for radial-velocity observations but their data serve
to confirm trends seen already in weaker lines. Figure 11 uses data from the
full spectrum in the N59 model, and the velocity standard deviations refer to
excursions around the mean velocity, as averaged over the simulation sequence.

Figures 10 and 11 provide specific model predictions which should be possible
to test observationally. For measurements of integrated sunlight, the
amplitudes from the simulation area have to be reduced by a large factor but
only by a small amount (if any) in spatially resolved data. Such observations
should become practical with forthcoming solar telescopes, where light from
sections of a spatially somewhat resolved solar image is fed into a high-
precision instrument, e.g., the PoET project at the ESPRESSO spectrometer of
ESO in Chile (Leite et al., 2022).

Figure 12: Spectral region surrounding the G-band, indicating measured
sections: G-band core (red), full G-band (red + brown), nearby Fe i lines
(blue), and pseudocontinuum reference points (red). This plot from the CO
5BOLD N59 model is for disk center, $\mu$=1.  Figure 13: G-band region in
synthetic non-magnetic CO 5BOLD N59 solar spectra. Top frame: Intensity
spectra at disk center $\mu$ = 1 (black), at $\mu$ = 0.41 (green), and close
to limb, $\mu$ = 0.09 (red), separately normalized to a very local
’continuum‘. Middle frame: spectral ratios relative to disk center for $\mu$ =
0.79 (blue) and $\mu$ = 0.41 (green). Bottom frame: Range of temporal
variability is illustrated by spectral ratios between snapshot instances of
highest (5806 K) and lowest (5735 K) model temperatures; same vertical scale
as middle frame.

##  6 Specific spectral regions

There might exist spectral features that are both sensitive to fluctuations in
the photospheric structure and also practical to observe as possible proxies
for excursions in radial velocity. Parameters of strong chromospheric lines
have often been used to quantify solar activity, in particular the Ca ii H & K
emission, Balmer lines from hydrogen and high-excitation lines from helium.

Nevertheless, all those have shortcomings in that they do not precisely
reflect local conditions in the photosphere. The strong chromospheric lines
are formed mainly in high atmospheric layers with their greater amplitudes in
gas motions, driven by magnetic structures, while the weaker He lines are
partially excited by X-rays radiating downward from the transition region. On
the other hand, precision radial velocities are measured from photospheric
spectra only, and one needs to find features that are particularly sensitive
to photospheric changes and that also appear promising to examine in observed
spectra.

###  6.1 The G-band

One interesting region is the G-band. This molecular bandhead around 430 nm
mainly consists of transitions between rotational and vibrational sublevels of
the CH molecule, blended with numerous atomic lines. The high line density
causes an absence of any real continuum across the whole bandhead. As
discussed below, the detailed physics of its spectrum formation is not fully
understood, although its formation must largely reflect photospheric
structures. Line identifications in that region are shown in the Sacramento
Peak solar atlas (Beckers et al., 1976), in the IRSOL Second Solar Spectrum
atlas (Gandorfer, 2002), and in the graphic version (Mrotzek et al., 2016) of
the Göttingen solar flux atlas (Reiners et al., 2016). Here, we examine the
theoretically expected types of variability in advance of actual measurements
in Paper II.

###  6.2 Non-magnetic G-band spectra

Inside the G-band, we do not attempt to follow radial velocities since the
massive blending makes it awkward to reliably segregate wavelength
displacements from changes in line profiles induced by varying blends driven
by, e.g., temperature fluctuations. Instead, the total spectral flux within
different regions is examined. While the exact boundaries of the G-band are
not uniquely defined, we adopt the full G-band as 429.84 – 431.33 nm, with its
central core 430.71 – 430.88 nm (Fig. 12). In its immediately surrounding
region, ten reasonably clean Fe i lines were selected to provide also radial-
velocity measures: five lines shortward, and five longward of the G-band. As a
near-continuum reference, 32 segments were selected throughout the region (the
same ones later used for observational studies in Paper II). This enables to
search for differing behavior among spectral features in this region.

Synthetic G-band spectra at different center-to-limb locations are shown in
Fig. 13, together with their ratios. There is a striking dependence such that
the relative brightness of the G-band core greatly increases when moving from
disk center toward the limb. By contrast, the changes induced by temperature
fluctuations in the model are quite modest. Spectral variability is thus more
pronounced with respect to the spatial disk position, not with temporal
changes. Figure 14 shows the types of correlations present, here selected for
$\mu$ = 0.79 as representative for much of the emerging flux from the full
disk. The data show integrated fluxes of each spectral feature inside its
respective wavelength interval (the flux, not the absorption). Naturally,
fluxes increase during hotter instants of the simulation but the responses
differ among different spectral features, with the G-band center showing
greater spread than Fe i lines. However, the radial velocities (being the
averages from the ten surrounding Fe i lines) fail to show any obvious
correlation with the G-band fluxes.

Figure 14: Temporal flickering of the flux in the G-band region of synthetic
non-magnetic CO 5BOLD N59 solar spectra, measured at $\mu$ = 0.79. The lower
row shows how the effective temperature of the model area at each selected
time of the simulation affects features in the region. The G-band flux is
plotted in the three leftmost upper frames; the rightmost one shows radial
velocities for the full disk measured in the nearby Fe i lines surrounding the
G-band.

###  6.3 Magnetic G-band spectra

The complex molecular structure of the G-band implies that its formation may
be influenced by numerous effects, as already suggested in the examples of
Fig. 13. A primary observation is that magnetic field concentrations in the
quiet Sun outside sunspots are spatially largely coincident with areas that
are bright in the G-band. Such regions of magnetically affected granulation
may cover a non-negligible fraction of the Sun and have a different structure
and generally exhibit smaller convective wavelength shifts than non-magnetic
areas. Due to their area coverage, these facular and plage regions are the
second most significant contributors to the emerging solar radiation (after
the extensive regions of non-magnetic granulation). Much of the solar-cycle
variation in apparent solar radial velocity appears to be due to the varying
area coverage of such magnetic plage regions (Meunier et al., 2010a, b) and it
would be desirable to identify some independent measure of their coverage.

####  6.3.1 Solar filigree and their magnetic fields

A significant fraction of small-scale magnetic fields in the photosphere are
traced by the elongated, sinuous and crinkly solar filigree, also referred to
as bright points (although they seldom are point-like). These small brightness
enhancements on subarcsecond scales are now understood to form inside
intergranular lanes, near the outer edges of granules, where a local minimum
of gas pressure (contributing to the horizontal pressure gradient that drives
horizontal gas motions inside and outside granules) is compensated by the
internal magnetic pressure in the flux tubes to achieve a balance with their
surroundings.

Solar filigree were first observed (and named) by Dunn & Zirker (1973). They
could be resolved once subarcsecond angular resolution was achieved for
imaging the photosphere in narrow-band monochromatic images, initially in the
far wings of H$\alpha$ (Dravins, 1974; Dunn & Zirker, 1973), later at also
other wavelengths and soon understood to be largely cospatial with magnetic
field concentrations (Mehltretter, 1974). The appearance of solar filigree in
various spectral regions on the quiet Sun is described by Berger & Title
(2001); Berger et al. (1995, 2004); Rouppe van der Voort et al. (2005) and
Sánchez Almeida et al. (2004); a review is by Bellot Rubio & Orozco Suárez
(2019). Among these, observations in the G-band are particularly extensive.

Figure 15: Region around the central G-band in CO 5BOLD simulations for zero
magnetic field (dashed black curves) and for B = 1600 G (160 mT); solid
colored curves; for three different center-to-limb positions. The spectra are
individually normalized to their local pseudo-continua.

####  6.3.2 Modeling G-band spectral brightness

Specifics about the G-band formation attracted attention already a long time
ago when observations of its center-to-limb variation suggested it to be
formed on a rough and corrugated solar surface (Pecker, 1949). More recent
work has mapped also the center-to-limb change in convective wavelength shift
(Langhans & Schmidt, 2002). Solar filigree are well visible in the G-band and
numerous attempts have been made to model their brightness. There is general
agreement that the magnetic pressure inside flux tubes implies a reduced
density relative to their surroundings. At a given geometric height, a lower
gas pressure causes a lower opacity than in adjacent non-magnetic regions,
enabling to see deeper and hotter layers. In these deep photospheric layers,
the CH abundance is reduced by dissociation and its weaker absorption lines
permit more continuum flux to shine through (Carlsson et al., 2004; Kiselman
et al., 2001; Shelyag et al., 2004; Schüssler et al., 2003; Stein, 2012;
Steiner et al., 2001, 2003).

The detailed physics may be more complicated because of effects from non-LTE
and three-dimensional radiative line transfer. The hot walls of the flux tube
perimeter emit enhanced ultraviolet radiation into the flux tube from its
sides, driving photo-dissociation of molecules (Steiner et al., 2001). Since
the G-band largely contains saturated absorption lines, the amount of its
weakening is sensitive to specific line-broadening parameters (Sánchez Almeida
et al., 2001). The appearance of the solar surface is further affected by its
geometric corrugation: granules have their continuum optical depth unity level
some 80 km above the photospheric average, with the magnetic flux tubes some
200–300 km below (Carlsson et al., 2004). For those flux tubes that are
inclined relative to the line of sight, the view toward deeper layers may be
obstructed.

It should also be noted that not all magnetic elements necessarily are bright:
In the quiet Sun, G-band brightenings almost always correspond to elements
with strong magnetic field but in developed plages, the correlation becomes
weaker (Riethmüller & Solanki, 2017). Also, non-magnetic bright points may
appear, at least in simulations, caused by vertically extending vortices
(vortex tubes) in the photosphere (Calvo et al., 2016).

####  6.3.3 Other molecular bands

The low gas densities and high transparency of fluxtubes produce enhanced
contrast in also other spectral features, in particular molecular lines. The
violet CN band around 388 nm gives a contrast about 1.4 times greater than the
G-band (Kiselman et al., 2001; Rutten et al., 2001; Zakharov et al., 2005),
although observationally that short-wavelength region is more demanding than
the G-band. Responses for further molecular species are discussed by
Berdyugina et al. (2003); Ji et al. (2016); Steiner et al. (2001), and
Tritschler & Uitenbroek (2006), that of extended spectra by Norris et al.
(2017, 2023).

####  6.3.4 Magnetic CO 5BOLD models

After injecting a certain magnetic flux in 3D atmospheric structures, its
development can be followed into magnetic flux concentrations around
intergranular lanes, the magnetic forces being balanced by gas pressure and
dynamic effects. The granular motions and their development is disturbed,
resulting in a more disrupted granulation structure with a generally smaller
convective blueshift. Such surface magnetoconvection has been modeled, for
example, by Beeck et al. (2015a, b), Khomenko et al. (2018), Norris et al.
(2023), and Salhab et al. (2018). Spectral synthesis illustrates how the
magnetic properties are reflected in the shapes of spectral lines, depending
on their magnetic sensitivity (Holzreuter & Solanki, 2012, 2015).

Here we use a series of magnetic CO 5BOLD solar models (Ludwig & Steffen,
2022; Ludwig et al., 2023). The total net magnetic flux through each
horizontal plane is kept constant with the field passing orthogonally through
the lower and upper model boundaries, with flux densities of 800-1600-2400 G
(80-160-240 mT). This results in a somewhat deformed granulation structure,
from which spectra of the G-band region were computed. Values on the order of
160 mT seem characteristic for actual intergranular field strengths in solar
filigree (Bellot Rubio & Orozco Suárez, 2019; de Wijn et al., 2009; Salhab et
al., 2018; Stein, 2012). Figure 15 compares the resulting spectra between the
zero-field case and that of this representative intergranular flux density of
160 mT. The overall spectral contrast varies with center-to-limb distance and
changes sign when approaching the limb.

This is further illustrated in Fig. 16, which shows how the modeled brightness
varies across the solar disk for different magnetic field strengths. The
G-band flux is here given by the pseudo-continuum levels near the edges of the
G-band region: 428.8 – 428.9, and 431.9 – 432.0 nm. For moderate flux
densities of 800 and 1600 G (80 and 160 mT), this continuum brightness
overtakes the zero-field case in positions toward the limb, thus illustrating
the crossover to bright facular regions, as distinctly observed in the G-band.
For the strongest fields treated here, the continuum intensity becomes faint
throughout and apparently something akin to umbral dots begins to form. With
even stronger magnetic flux concentrations, surface convection is inhibited
and starspots develop, forming umbrae and penumbrae structures, as modeled for
different stellar types by Panja et al. (2020). While outside the scope of the
present paper, corresponding synthetic spectra from also such magnetic umbrae
and penumbrae may be combined into a study of fluctuations of spatially
resolved features. Pairs of spectral lines with disparate Landé
$\varg$${}_{\textrm{eff}}$-factors and thus different magnetic sensitivities
should be able to diagnose field properties (Smitha & Solanki, 2017), although
it might be challenging to perform a detailed synthesis of a full spectrum
that precisely incorporates all Zeeman components from all relevant lines.
Although the G-band is a very crowded region, there still exist instances of
somewhat isolated groups of CH lines that are predicted to produce a
measurable circular polarization signal through the Zeeman effect (Uitenbroek
et al., 2004).

##  7 Flickering in photometric colors

We now examine what information might be extracted from fluctuations over even
broader wavelength regions. Precision measurements of stellar irradiance, made
from space, reveal the brightness flickering due to time-variable granulation.
This has been theoretically modeled for different spectral types (Lundkvist et
al., 2021; Rodríguez Díaz et al., 2022; Samadi et al., 2013a, b), and its
observation permits various stellar parameters to be inferred (Bastien et al.,
2016; Ness et al., 2018; Van Kooten et al., 2021). Such photometric flicker is
a concern for exoplanet transit photometry (Sulis et al., 2020).

The connection between simultaneous fluctuations in radial velocity and
brightness has been studied. Clear correlations are seen for hotter and more
active stars, but are not identified for the smaller amplitudes in cooler
stars of solar temperature (Giguere et al., 2016; Oshagh et al., 2017; Sulis
et al., 2023). In general, stars with greater photometric variability
(probably caused by spots) also have increased levels of jitter in radial
velocity. Thus, photometric stability may be used as a criterion to select
stars that remain particularly quiet in radial velocity (Hojjatpanah et al.,
2020; Tayar et al., 2019).

As discussed above, 3D simulations suggest relations where a 1 m s-1
fluctuation in velocity caused by granular evolution would correspond to a
brightness fluctuation on the order of 100 ppm in the light from the full
solar disk. However, the longer-term photometric variability of the Sun is
dominated by sunspots and active regions, even if searches for photometry as a
proxy for short-term variations have been made (Kosiarek & Crossfield, 2020).

The required micromagnitude accuracies in absolute stellar fluxes cannot
realistically be achieved from the ground, but differential measures of
photometric colors might perhaps be feasible from exceptional observing
locations (e.g., Schmider et al., 2022), despite requiring a good
understanding of atmospheric extinction, especially at the shorter
wavelengths. Color changes reflect stellar temperature variations and should
carry information analogous to the flickering in brightness, possibly being a
candidate for a proxy of excursions in radial velocity. Since our simulations
provide absolute fluxes, synthetic broad-band photometric colors can be
computed. For CO 5BOLD models, certain synthetic colors have previously been
computed by Bonifacio et al. (2018) and Kučinskas et al. (2018), and for the
STAGGER grid by Chiavassa et al. (2018).

Figure 16: Emission levels in pseudo-continua around the G-band for different
magnetic field strengths, relative to the zero-field case at solar disk
center, and for different center-to-limb positions. For 800 and 1600 G (80 and
160 mT), the crossover to bright regions (faculae) is seen when moving toward
the limb. For the strongest magnetic fields, the fluxes become faint;
apparently dark pores begin to form.

A photometric system developed to give maximum sensitivity to physical
parameters in specifically solar-type stars is the Strömgren one with its four
passbands of $u-v-b-y$, originally proposed by Strömgren (1956, 1966), with
later calibrations by Crawford & Barnes (1970), Önehag et al. (2009), and
others. In particular, it is being used to study properties of solar near-
twins, e.g., Meléndez et al. (2010) and Dalle Mese et al. (2020).

These filters are defined such that the $y$ and $b$ passbands (centered around
547 and 467 nm) contain no strong spectral features for solar-type type stars,
and thus the slope of the spectrum, measured as the color index $(b-y)$, is
free of significant effects from line blanketing, becoming an indicator of
stellar temperature. The $v$ filter, centered at 411 nm, is placed longward
from the region where the Balmer lines from hydrogen begin to crowd, as those
start to approach the Balmer limit (although the filter transmits H$\delta$).
The near-ultraviolet $u$ filter is centered at 350 nm with its longward part
remaining below the Balmer jump while its shortward part extends to the
telluric atmospheric absorption cutoff (Fig.17). Following standard usage, the
fluxes $F$ inside each photometric $u-v-b-y$ band are converted to magnitudes
according to $m=-2.5\,{\log}{{}_{10}}(F/F{{}_{0}})$, where $F{{}_{0}}$ is some
reference flux. However, here we are concerned only with the temporal
fluctuations in color indices, not their absolute calibration. For an overview
of photometric systems, see Bessell (2005).

The exact realization of observational $uvby$ photometry involves nuances such
as the construction of the transmission filter optics, properties of the
detector (e.g., whether integrating the energy flux inside the filter passband
or the number of photons; here the former is applied). We approximate the four
filter passbands with Gaussians centered on 350, 411, 467, and 547 nm, with
respective standard deviations 12.74, 8.07, 7.64, and 9.77 nm. At the lowest
transmission, they are truncated at 0.001 where adjacent filters would start
to overlap at filter edges. These profiles closely approximate those used in
other work, and are shown in Figure 17 together with the transmitted flux from
the N59 solar model.

From measurements in these filter passbands, three indices are widely used for
stellar classification: the color index $(b-y)$, and the two color-index
differences $m{{}_{1}}\equiv(v-b)-(b-y)$, measuring the amount of absorption-
line blanketing in the 410 nm region, and $c{{}_{1}}\equiv(u-v)-(v-b)$,
quantifying the pressure-dependent Balmer discontinuity. The color index
$(b-y)$ reflects stellar temperature and the metal-line index $m{{}_{1}}$
indicates the amount of metallic-line absorption. The Balmer discontinuity
index $c{{}_{1}}$ provides a measure of the atmospheric pressure, which, being
a function of the surface gravity, gives the stellar luminosity. Greater
values of $(b-y)$ imply cooler temperatures; increased metallicity is
reflected in a larger $m_{1}$, and greater luminosity (lower pressure) as a
greater value of $c_{1}$. During the solar activity cycle, spectral brightness
changes are driven by magnetic activity, influencing the signals in especially
the Strömgren $v$ and $b$ filters (Witzke et al., 2018).

Figures 18 and 19 show relations between model fluctuations in the various
$uvby$ indices, also relating to the simultaneous jittering in apparent radial
velocities for representative Fe i lines of different strength. Their
parameters of continuum level, depth, width, shape and wavelength position are
those obtained from the five-parameter Gaussian-type fit described above.
Clear trends are that instances of enhanced continuum brightening correlate
with a bluer $(b-y)$ color, weaker line absorption, and an apparently lower
metallicity. However, relations among other measures of line-depth, line-width
and line-shape are less obvious or absent, consistent with observational
findings by Oshagh et al. (2017) and Sulis et al. (2023).

Irrespective of whether correlations with specific spectral signatures can be
identified, the data illustrate what amplitudes might be expected in the
flickering of full-disk sunlight in different color indices. For the data
shown in Fig. 18, the RMS fluctuations in $(b-y)$, $m{{}_{1}}$, and
$c{{}_{1}}$ are, respectively, 0.0023, 0.0033, and 0.0017. The greatest
changes occur in $m{{}_{1}}$, due to changing absorption-line strengths in the
blue while the modulation of the luminosity index is only half of that. In
actual full-disk sunlight, these amplitudes should be divided by a factor
$\sim$150 or 200, then shrinking from a few millimagnitudes to only some 10 or
20 $\mu$mag. The corresponding RMS fluctuations in the continuum level of 1.4%
at 525 nm over the model area correspond to some 100 ppm in full integrated
sunlight, in line with what was deduced by Cegla et al. (2019). Plots for
separate center-to-limb positions show the relative fluctuation amplitudes
increasing from disk center toward the limb but are not qualitatively
different. The expected performance of space missions such as PLATO readily
permits to detect such levels of photometric variability (Heller et al., 2022;
Morris et al., 2020), although their search for exoplanets will depend also on
how to segregate other types of stellar variability. Observed day-to-day solar
irradiance fluctuations, driven by the presence of faculae and appearance of
spots, are typically greater by an order of magnitude, and those during a
solar activity cycle by another order of magnitude (Lean et al., 2020;
Marchenko et al., 2019).

Figure 17: Top row: Spectral flux transmitted thorough the four Strömgren
$u-v-b-y$ filters, whose transmission functions are in the bottom row.  Figure
18: Relations between parameters of a representative Fe i line of medium
strength at 520 nm (Fig.2) in integrated sunlight from the N59 model, and
flickering in photometric Strömgren indices. Quantitites refer to relative
variations over the small simulation area.

##  8 Lack of correlation in other line parameters

Relations were sought between the velocity jittering and other spectral-line
parameters which could have been suspected to vary in phase during the
simulation sequences. However, as seen in Fig. 20, there appears to be no
sensible correlation with either continuum brightness, line-width or line
shape (the quantities in the five-parameter Gaussian-type fit described in
Sect. 4.1). An exception is that a weak relation appears to exist with line-
depth, at least for the strongest lines. Such a correlation actually is
expected and was seen in clean disk-center data by Dravins et al. (2021b).
Locations of increased irradiance display enhanced convective blueshift
combined with a stronger line absorption, caused by enhanced vertical
gradients in regions of line formation. However, while this is seen when
granules are viewed directly from above, the effect rapidly diminishes away
from disk center and largely gets washed out in the integrated sunlight of
Fig. 20.

Figure 19: Search for relations between radial velocities of Fe i lines of
different strengths at 520 nm (Fig.2), and photometric Strömgren indices in
integrated sunlight, as computed over the small simulation area of the N59
model (for the actual Sun, divide the numerical values by a factor
$\sim$150-200). Rows from top down: Weak, medium, and strong lines.

##  9 Search for Earth-like exoplanets

Under realistic conditions, the spectral-line parameter that can be most
precisely measured is its wavelength (rather than the depth, width or shape,
which are more sensitive to the exact adjustments in the spectroscopic
instrumentation). Given the results in this paper, that is satisfactory since
significant mutual correlations between line-groups could be identified for
precisely such wavelength jittering. Various other parameters, such as the
fluctuating widths or depths of absorption lines, or flickering in broad-band
photometric indices fail to show convincing correlations with excursions in
radial velocity.

Using the results from hydrodynamic simulations, it is thus possible to
identify patterns of velocity jittering. Characteristics of simultaneous
jittering in radial velocity between different groups of lines, as shown in
Figs. 8 and 9, discern the radial-velocity contributions originating from
fluctuations in photospheric granulation, the signature of which should be
possible to deduce and deduct.

In practice, this will require to obtain radial-velocity measures for groups
of individually selected spectral lines within distinct batches of stronger
and weaker lines, and in different wavelength regions. Jittering amplitudes
from each such homogeneous line-group are expected to show systematic
differences, e.g., such that the excursions of strong short-wavelength Fe ii
lines would be greater by a certain factor than, e.g., those of weak Fe i
lines at long-wavelengths. A ‘matched filter’ fitted to such a variability
pattern should identify that component of velocity jittering which is specific
to granular convection. While the present work permits to estimate the rough
parameters for such a filter, its more precise numbers will require more
extensive simulations, the validity of which could be confirmed in such
spatially resolved observations that are feasible already with current
instrumentation, as suggested in Figs. 10 and 11. Of course, other sources of
jittering originate from oscillations and various magnetic phenomena over
other timescales, requiring separate attention.

The observational requirements are stringent but probably not impossible. The
total jittering amplitude in full-disk sunlight amounts to $\sim$1-2 m s-1,
with differences between various line-groups perhaps one tenth of that. To
mitigate these fluctuations thus requires to recognize the characteristic
patterns of dissimilar lineshifts among classes of various spectral lines with
precisions $\sim$10 cm s-1 for each absorption-line group. Although certainly
challenging, such requirements correspond to what in any case is required in
searches for truly Earth-like exoplanets; e.g., HARPS-3 (Farret Jentink et
al., 2022; Thompson et al., 2016) or ANDES (Marconi et al., 2022).

From other considerations, it has also been recognized that there is
additional information content in individual spectral lines, as compared to
statistical correlation functions (Al Moulla et al., 2022; Cretignier et al.,
2020; Dumusque, 2018; Meunier et al., 2017b; Wise et al., 2022). Recent
efforts for the Sun seen as a star include the extraction of radial-velocity
data separately from lines with different depths (Miklos et al., 2020), and
the study of how spectral portions differ during epochs of magnetically more
or less active periods (Thompson et al., 2020).

Most of our present modeling is limited to the non-magnetic photosphere and to
such short-term fluctuations that are comparable to granular evolution
timescales. This quiet photosphere covers the largest fraction of the solar
surface and contributes the greatest part of the total flux (and presumably
that of other solar-type stars) but magnetic features seem responsible for
much of the variability over the longer timescales of stellar rotation and
activity cycles.

The following Paper II (Dravins & Ludwig, 2024) will explore observed solar
microvariability on also such somewhat longer timescales, aiming to identify
further measurable photospheric parameters that might be used as proxies for
radial-velocity fluctuations.

Figure 20: Radial velocities show lack of correlation with several plausibly
anticipated fluctuations in various line parameters. No sensible correlations
were found here, except a hint for a line-depth dependence in the strongest
lines. The measures for line depth, width, shape are normalized to 100% for
the averages over the simulation sequence. The calculations are for integrated
sunlight from the small simulation area of the N59 model (for the actual Sun,
divide the line-shift values by a factor $\sim$150-200).

###### Acknowledgements.

The work by DD is supported by grants from The Royal Physiographic Society of
Lund. HGL gratefully acknowledges financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID
138713538–SFB 881 (‘The Milky Way System’, subproject A04). Extensive use was
made of NASA’s ADS Bibliographic Services and the arXiv® distribution service.
We thank the referee for very detailed and insightful comments.

## References

  * Al Moulla et al. (2022) Al Moulla, K., Dumusque, X., Cretignier, M., et al. 2022, A&A, 664, A34
  * Asplund (2005) Asplund, M. 2005, ARA&A, 43, 481
  * Bastien et al. (2016) Bastien, F. A., Stassun, K. G., Basri, G., et al. 2016, ApJ, 818, 43
  * Beckers et al. (1976) Beckers, J. M., Bridges, C. A., & Gilliam, L. B. High resolution spectral atlas of the solar irradiance from 380 to 700 nanometers, Sacramento Peak Observatory, Air Force Geophysics Laboratory, 1976
  * Beeck et al. (2015a) Beeck, B., Schüssler, M., Cameron, R. H., et al. 2015a, A&A, 581, A42
  * Beeck et al. (2015b) Beeck, B., Schüssler, M., Cameron, R. H., et al. 2015b, A&A, 581, A43
  * Bellot Rubio & Orozco Suárez (2019) Bellot Rubio, L. & Orozco Suárez, D. 2019, Liv. Rev. Solar Phys., 16, 1
  * Berdyugina et al. (2003) Berdyugina, S. V., Solanki, S. K., & Frutiger, C. 2003, A&A, 412, 513
  * Bergemann et al. (2019) Bergemann, M., Gallagher, A. J., Eitner, P., et al. 2019, A&A, 631, A80
  * Berger & Title (2001) Berger, T. E. & Title, A. M. 2001, ApJ, 553, 449
  * Berger et al. (1995) Berger, T. E., Schrijver, C. J., Shine, R. A., et al. 1995, ApJ, 454, 531
  * Berger et al. (2004) Berger, T. E., Rouppe van der Voort, L. H. M., Löfdahl, M. G., et al. 2004, A&A, 428, 613
  * Berke et al. (2023a) Berke, D. A., Murphy, M. T., Flynn, C., et al. 2023a, MNRAS, 519, 1221
  * Berke et al. (2023b) Berke, D. A., Murphy, M. T., Flynn, C., et al. 2023b, MNRAS, 519, 1238
  * Bessell (2005) Bessell, M. S. 2005, ARA&A, 43, 293
  * Biazzo et al. (2007) Biazzo, K., Frasca, A., Catalano, S., et al. 2007, Astron. Nachr., 328, 938
  * Blackman et al. (2020) Blackman, R. T., Fischer, D. A., Jurgenson, C. A., et al. 2020, AJ, 159, 238
  * Bonifacio et al. (2018) Bonifacio, P., Caffau, E., Ludwig, H.-G., et al. 2018, A&A, 611, A68
  * Bouchy et al. (2017) Bouchy, F., Doyon, R., Artigau, É., et al. 2017, The Messenger, 169, 21
  * Calvo et al. (2016) Calvo, F., Steiner, O., & Freytag, B. 2016, A&A, 596, A43
  * Carlsson et al. (2004) Carlsson, M., Stein, R. F., Nordlund, Å., et al. 2004, ApJ, 610, L137
  * Cegla et al. (2019) Cegla, H. M., Watson, C. A., Shelyag, S., et al. 2019, ApJ, 879, 55
  * Chiavassa et al. (2018) Chiavassa, A., Casagrande, L., Collet, R., et al. 2018, A&A, 611, A11
  * Coffinet et al. (2019) Coffinet, A., Lovis, C., Dumusque, X., et al. 2019, A&A, 629, A27
  * Collier Cameron et al. (2019) Collier Cameron, A., Mortier, A., Phillips, D., et al. 2019, MNRAS, 487, 1082
  * Collier Cameron et al. (2021) Collier Cameron, A., Ford, E. B., Shahaf, S., et al. 2021, MNRAS, 505, 1699
  * Cosentino et al. (2012) Cosentino, R., Lovis, C., Pepe, F., et al. 2012, Proc. SPIE, 8446, 84461V
  * Cranmer et al. (2014) Cranmer, S. R., Bastien, F. A., Stassun, K. G., et al. 2014, ApJ, 781, 124
  * Crass et al. (2021) Crass, J., Gaudi, B. S., Leifer, S., et al. 2021, arXiv:2107.14291
  * Crass et al. (2022) Crass, J., Aikens, D., Mason, J., et al. 2022, Proc. SPIE, 12184, 121841P
  * Crawford & Barnes (1970) Crawford, D. L. & Barnes, J. V. 1970, AJ, 75, 978
  * Cretignier et al. (2020) Cretignier, M., Dumusque, X., Allart, R., et al. 2020, A&A, 633, A76
  * Dalle Mese et al. (2020) Dalle Mese, G., López-Cruz, O., Schuster, W. J., et al. 2020, MNRAS, 494, 2995
  * de Beurs et al. (2022) de Beurs, Z. L., Vanderburg, A., Shallue, C. J., et al. 2022, AJ, 164, 49
  * de Wijn et al. (2009) de Wijn, A. G., Stenflo, J. O., Solanki, S. K., et al. 2009, Space Sci. Rev., 144, 275
  * Donati et al. (2020) Donati, J.-F., Kouach, D., Moutou, C., et al. 2020, MNRAS, 498, 5684
  * Dravins (1974) Dravins, D. 1974, Chromospheric Fine Structure, IAU Symp. 56, 257
  * Dravins (1989) Dravins, D., 1989, in R.J.Rutten, G.Severino, eds. Solar and Stellar Granulation, Proc. NATO ASI Ser. C 263, 153
  * Dravins & Ludwig (2024) Dravins, D., Ludwig, H.-G. 2024, A&A, to be submitted (Paper II)
  * Dravins & Nordlund (1990) Dravins, D., & Nordlund, Å. 1990, A&A, 228, 184
  * Dravins et al. (2017a) Dravins, D., Ludwig, H.-G., Dahlén, E., et al. 2017a, A&A, 605, A90
  * Dravins et al. (2017b) Dravins, D., Ludwig, H.-G., Dahlén, E., et al. 2017b, A&A, 605, A91
  * Dravins et al. (2018) Dravins, D., Gustavsson, M., & Ludwig, H.-G. 2018, A&A, 616, A144
  * Dravins et al. (2021a) Dravins, D., Ludwig, H.-G., & Freytag, B. 2021a, A&A, 649, A16
  * Dravins et al. (2021b) Dravins, D., Ludwig, H.-G., & Freytag, B. 2021b, A&A, 649, A17
  * Dumusque (2016) Dumusque, X. 2016, A&A, 593, A5
  * Dumusque et al. (2015) Dumusque, X., Glenday, A., Phillips, D. F., et al. 2015, ApJ, 814, L21
  * Dumusque et al. (2017) Dumusque, X., Borsa, F., Damasso, M., et al. 2017, A&A, 598, A133
  * Dumusque (2018) Dumusque, X. 2018, A&A, 620, A47
  * Dumusque et al. (2011a) Dumusque, X., Udry, S., Lovis, C., et al. 2011a, A&A, 525, A140
  * Dumusque et al. (2021) Dumusque, X., Cretignier, M., Sosnowska, D., et al. 2021, A&A, 648, A103
  * Dunn & Zirker (1973) Dunn, R. B. & Zirker, J. B. 1973, Sol. Phys., 33, 281
  * Farret Jentink et al. (2022) Farret Jentink, C., Mortier, A., Snik, F., et al. 2022, Proc. SPIE, 12182, 1218231
  * Fischer et al. (2016) Fischer, D. A., Anglada-Escude, G., Arriagada, P., et al. 2016, PASP, 128, 066001
  * Freytag et al. (2012) Freytag, B., Steffen, M., Ludwig, H.-G., et al. 2012, J. Comp. Phys., 231, 919
  * Gandorfer (2002) Gandorfer, A. 2002, The Second Solar Spectrum: A high spectral resolution polarimetric survey of scattering polarization at the solar limb in graphical representation. Volume II: 3910 Å to 4630 Å, Zurich: vdf Hochschulverlag, 2002
  * Giguere et al. (2016) Giguere, M. J., Fischer, D. A., Zhang, C. X. Y., et al. 2016, ApJ, 824, 150
  * Gibson et al. (2016) Gibson, S. R., Howard, A. W., Marcy, G. W., et al. 2016, Proc. SPIE, 9908, 990870
  * Gilbert et al. (2018) Gilbert, J., Bergmann, C., Bloxham, G., et al. 2018, Proc. SPIE, 10702, 107020Y
  * Gray (1994) Gray, D. F. 1994, PASP, 106, 1248
  * Hall et al. (2018) Hall, R. D., Thompson, S. J., Handley, W., et al. 2018, MNRAS, 479, 2968
  * Halverson et al. (2016) Halverson, S., Terrien, R., Mahadevan, S., et al. 2016, Proc. SPIE, 9908, 99086P
  * Hara & Ford (2023) Hara, N. C. & Ford, E. B. 2023, Ann. Rev. Statistics and Its Application, 10, 623
  * Haywood et al. (2016) Haywood, R. D., Collier Cameron, A., Unruh, Y. C., et al. 2016, MNRAS, 457, 3637
  * Heiter et al. (2015) Heiter, U., Lind, K., Asplund, M., et al. 2015, Phys. Scr, 90, 054010
  * Heller et al. (2022) Heller, R., Harre, J.-V., & Samadi, R. 2022, A&A, 665, A11
  * Hobson et al. (2021) Hobson, M. J., Bouchy, F., Cook, N. J., et al. 2021, A&A, 648, A48
  * Hojjatpanah et al. (2020) Hojjatpanah, S., Oshagh, M., Figueira, P., et al. 2020, A&A, 639, A35
  * Holzreuter & Solanki (2012) Holzreuter, R. & Solanki, S. K. 2012, A&A, 547, A4
  * Holzreuter & Solanki (2015) Holzreuter, R. & Solanki, S. K. 2015, A&A, 582, A101
  * Ji et al. (2016) Ji, K.-F., Xiong, J.-P., Xiang, Y.-Y., et al. 2016, Research in Astronomy and Astrophysics, 16, 78
  * Jurgenson et al. (2016) Jurgenson, C., Fischer, D., McCracken, T., et al. 2016, Proc. SPIE, 9908, 99086T
  * Kallinger et al. (2014) Kallinger, T., De Ridder, J., Hekker, S., et al. 2014, A&A, 570, A41
  * Kanodia et al. (2021) Kanodia, S., Halverson, S., Ninan, J. P., et al. 2021, ApJ, 912, 15
  * Khomenko et al. (2018) Khomenko, E., Vitas, N., Collados, M., et al. 2018, A&A, 618, A87
  * Kiselman et al. (2001) Kiselman, D., Rutten, R. J., & Plez, B. 2001, Recent Insights into the Physics of the Sun and Heliosphere: Highlights from SOHO and Other Space Missions, IAU Symp. 203, 287
  * Kitiashvili et al. (2023) Kitiashvili, I., Granovsky, S., & Wray, A. 2023, Predictive Power of Computational Astrophysics as a Discovery Tool, IAU Symp. 362, 324
  * Kosiarek & Crossfield (2020) Kosiarek, M. R. & Crossfield, I. J. M. 2020, AJ, 159, 271
  * Kučinskas et al. (2018) Kučinskas, A., Klevas, J., Ludwig, H.-G., et al. 2018, A&A, 613, A24
  * Langhans & Schmidt (2002) Langhans, K. & Schmidt, W. 2002, A&A, 382, 312
  * Langhans et al. (2002) Langhans, K., Schmidt, W., & Tritschler, A. 2002, A&A, 394, 1069
  * Lanza et al. (2016) Lanza, A. F., Molaro, P., Monaco, L., et al. 2016, A&A, 587, A103
  * Lanza et al. (2019) Lanza, A. F., Collier Cameron, A., & Haywood, R. D. 2019, MNRAS, 486, 3459
  * Lean et al. (2020) Lean, J. L., Coddington, O., Marchenko, S. V., et al. 2020, Earth and Space Science, 7, 00645
  * Leite et al. (2022) Leite, I., Cabral, A., Abreu, M., Santos, N. 2022, J. Phys.: Conf. Ser. 2407, 012020
  * Lienhard et al. (2023) Lienhard, F., Mortier, A., Cegla, H. M., et al. 2023, MNRAS, 522, 5862
  * Ludwig (2006) Ludwig, H.-G. 2006, A&A, 445, 661
  * Ludwig & Steffen (2008) Ludwig, H.-G. & Steffen, M. 2008, Precision Spectroscopy in Astrophysics, ESO Astrophysics Symposia, Springer, 133
  * Ludwig & Steffen (2022) Ludwig, H.-G. & Steffen, M. 2022, Cool Stars 21, Toulouse, France. Zenodo. https://doi.org/10.5281/zenodo.7447826
  * Ludwig et al. (2023) Ludwig, H.-G., Steffen, M. & Freytag, B. 2023, A&A, submitted
  * Lundkvist et al. (2021) Lundkvist, M. S., Ludwig, H.-G., Collet, R., et al. 2021, MNRAS, 501, 2512
  * Mahadevan et al. (2014) Mahadevan, S., Ramsey, L. W., Terrien, R., et al. 2014, Proc. SPIE, 9147, 91471G
  * Maldonado et al. (2019) Maldonado, J., Phillips, D. F., Dumusque, X., et al. 2019, A&A, 627, A118
  * Marchenko et al. (2019) Marchenko, S. V., Woods, T. N., DeLand, M. T., et al. 2019, Earth and Space Science, 6, 2379
  * Marchwinski et al. (2015) Marchwinski, R. C., Mahadevan, S., Robertson, P., et al. 2015, ApJ, 798, 63
  * Marconi et al. (2022) Marconi, A., Abreu, M., Adibekyan, V., et al. 2022, Proc. SPIE, 12184, 1218424
  * Mayor et al. (2003) Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20
  * Mehltretter (1974) Mehltretter, J. P. 1974, Sol. Phys., 38, 43
  * Meléndez et al. (2010) Meléndez, J., Schuster, W. J., Silva, J. S., et al. 2010, A&A, 522, A98
  * Meunier & Lagrange (2019) Meunier, N. & Lagrange, A.-M. 2019, A&A, 625, L6
  * Meunier & Lagrange (2020) Meunier, N. & Lagrange, A. M. 2020, A&A, 642, A157
  * Meunier et al. (2010a) Meunier, N., Desort, M., & Lagrange, A.-M. 2010a, A&A, 512, A39
  * Meunier et al. (2010b) Meunier, N., Lagrange, A.-M., & Desort, M. 2010b, A&A, 519, A66
  * Meunier et al. (2015) Meunier, N., Lagrange, A.-M., Borgniet, S., et al. 2015, A&A, 583, A118
  * Meunier et al. (2017a) Meunier, N., Lagrange, A.-M., Mbemba Kabuiku, L., et al. 2017a, A&A, 597, A52
  * Meunier et al. (2017b) Meunier, N., Lagrange, A.-M., & Borgniet, S. 2017b, A&A, 607, A6
  * Miklos et al. (2020) Miklos, M., Milbourne, T. W., Haywood, R. D., et al. 2020, ApJ, 888, 117
  * Mrotzek et al. (2016) Mrotzek, N., Lemke, U., Hinrichs, J., et al. 2016, Spectral Atlas of the Sun from 405 to 2300 nm, Institut für Astrophysik, Göttingen, http://www.astro.physik.uni-goettingen.de/research/flux_atlas/atlas_print_version_4000-23000.pdf
  * Morris et al. (2020) Morris, B. M., Bobra, M. G., Agol, E., et al. 2020, MNRAS, 493, 5489
  * Murphy et al. (2022) Murphy, M. T., Berke, D. A., Liu, F., et al. 2022, Science, 378, 634
  * Nèmec et al. (2020) Nèmec, N.-E., Işık, E., Shapiro, A. I., et al. 2020, A&A, 638, A56
  * Ness et al. (2018) Ness, M. K., Silva Aguirre, V., Lund, M. N., et al. 2018, ApJ, 866, 15
  * Nordlund et al. (2009) Nordlund, Å., Stein, R. F., & Asplund, M. 2009, Living Rev. Solar Phys., 6, 2
  * Norris et al. (2017) Norris, C. M., Beeck, B., Unruh, Y. C., et al. 2017, A&A, 605, A45
  * Norris et al. (2023) Norris, C. M., Unruh, Y. C., Witzke, V., et al. 2023, MNRAS, in press, arXiv:2306.04669
  * Önehag et al. (2009) Önehag, A., Gustafsson, B., Eriksson, K., et al. 2009, A&A, 498, 527
  * Oshagh et al. (2017) Oshagh, M., Santos, N. C., Figueira, P., et al. 2017, A&A, 606, A107
  * Panja et al. (2020) Panja, M., Cameron, R., & Solanki, S. K. 2020, ApJ, 893, 113
  * Pecker (1949) Pecker, J.-C. 1949, Annales d’Astrophysique, 12, 9
  * Pepe et al. (2014) Pepe, F., Molaro, P., Cristiani, S., et al. 2014, Astron. Nachr., 335, 8
  * Pepe et al. (2021) Pepe, F., Cristiani, S., Rebolo, R., et al. 2021, A&A, 645, A96
  * Plavchan et al. (2019) Plavchan, P., Vasisht, G., Beichman, C., et al. EarthFinder Probe Mission Concept Study, NASA-JPL-George Mason University-Caltech, 66 pp. 2019, arxiv:2006.13428
  * Rackham et al. (2023) Rackham, B. V., Espinoza, N., Berdyugina, S. V., et al. 2023, RAS Techniques and Instruments, 2, 148
  * Reiners et al. (2016) Reiners, A., Mrotzek, N., Lemke, U., et al. 2016, A&A, 587, A65
  * Riethmüller & Solanki (2017) Riethmüller, T. L. & Solanki, S. K. 2017, A&A, 598, A123
  * Rodríguez Díaz et al. (2022) Rodríguez Díaz, L. F., Bigot, L., Aguirre Børsen-Koch, V., et al. 2022, MNRAS, 514, 1741
  * Rouppe van der Voort et al. (2005) Rouppe van der Voort, L. H. M., Hansteen, V. H., Carlsson, M., et al. 2005, A&A, 435, 327
  * Rutten et al. (2001) Rutten, R. J., Kiselman, D., Rouppe van der Voort, L., et al. 2001, Advanced Solar Polarimetry – Theory, Observation, and Instrumentation, ASP Conf. Ser. 236, 445
  * Ryabchikova et al. (2015) Ryabchikova, T., Piskunov, N., Kurucz, R. L., et al. 2015, Phys. Scr, 90, 054005
  * Quirrenbach et al. (2014) Quirrenbach, A., Amado, P. J., Caballero, J. A., et al. 2014, Proc. SPIE, 9147, 91471F
  * Quirrenbach et al. (2018) Quirrenbach, A., Amado, P. J., Ribas, I., et al. 2018, Proc. SPIE, 10702, 107020W
  * Salhab et al. (2018) Salhab, R. G., Steiner, O., Berdyugina, S. V., et al. 2018, A&A, 614, A78
  * Samadi et al. (2013a) Samadi, R., Belkacem, K., & Ludwig, H.-G. 2013a, A&A, 559, A39
  * Samadi et al. (2013b) Samadi, R., Belkacem, K., Ludwig, H.-G., et al. 2013b, A&A, 559, A40
  * Sánchez Almeida et al. (2001) Sánchez Almeida, J., Asensio Ramos, A., Trujillo Bueno, J., et al. 2001, ApJ, 555, 978
  * Sánchez Almeida et al. (2004) Sánchez Almeida, J., Márquez, I., Bonet, J. A., et al. 2004, ApJ, 609, L91
  * Schmider et al. (2022) Schmider, F.-X., Abe, L., Agabi, A., et al. 2022, Proc. SPIE, 12182, 121822O
  * Schüssler et al. (2003) Schüssler, M., Shelyag, S., Berdyugina, S., et al. 2003, ApJ, 597, L173
  * Schwab et al. (2016) Schwab, C., Rakich, A., Gong, Q., et al. 2016, Proc. SPIE, 9908, 99087H
  * Seifahrt et al. (2018) Seifahrt, A., Stürmer, J., Bean, J. L., et al. 2018, Proc. SPIE, 10702, 107026D
  * Seifahrt et al. (2022) Seifahrt, A., Bean, J. L., Kasper, D., et al. 2022, Proc. SPIE, 12184, 121841G
  * Shelyag et al. (2004) Shelyag, S., Schüssler, M., Solanki, S. K., et al. 2004, A&A, 427, 335
  * Smitha & Solanki (2017) Smitha, H. N. & Solanki, S. K. 2017, A&A, 608, A111
  * Stein (2012) Stein, R. F. 2012, Living Rev. Solar Phys., 9, 4
  * Steiner et al. (2001) Steiner, O., Hauschildt, P. H., & Bruls, J. 2001, A&A, 372, L13
  * Steiner et al. (2003) Steiner, O., Hauschildt, P. H., & Bruls, J. 2003, Astr. Nachr. 324, 398
  * Strömgren (1956) Strömgren, B. 1956, Vistas in Astronomy, 2, 1336
  * Strömgren (1966) Strömgren, B. 1966, ARA&A, 4, 433
  * Sulis et al. (2020) Sulis, S., Lendl, M., Hofmeister, S., et al. 2020, A&A, 636, A70
  * Sulis et al. (2023) Sulis, S., Lendl, M., Cegla, H. M., et al. 2023, A&A, 670, A24
  * Szentgyorgyi et al. (2018) Szentgyorgyi, A., Baldwin, D., Barnes, S., et al. 2018, Proc. SPIE, 10702, 107021R
  * Tayar et al. (2019) Tayar, J., Stassun, K. G., & Corsaro, E. 2019, ApJ, 883, 195
  * Thompson et al. (2016) Thompson, S. J., Queloz, D., Baraffe, I., et al. 2016, Proc. SPIE, 9908, 99086F
  * Thompson et al. (2020) Thompson, A. P. G., Watson, C. A., Haywood, R. D., et al. 2020, MNRAS, 494, 4279
  * Tremblay et al. (2013) Tremblay, P.-E., Ludwig, H.-G., Freytag, B., et al. 2013, A&A, 557, A7
  * Tritschler & Uitenbroek (2006) Tritschler, A. & Uitenbroek, H. 2006, ApJ, 648, 741
  * Uitenbroek et al. (2004) Uitenbroek, H., Miller-Ricci, E., Asensio Ramos, A., et al. 2004, ApJ, 604, 960
  * Van Kooten et al. (2021) Van Kooten, S. J., Anders, E. H., & Cranmer, S. R. 2021, ApJ, 913, 69
  * Wise et al. (2022) Wise, A., Plavchan, P., Dumusque, X., et al. 2022, ApJ, 930, 121
  * Witzke et al. (2018) Witzke, V., Shapiro, A. I., Solanki, S. K., et al. 2018, A&A, 619, A146
  * Zakharov et al. (2005) Zakharov, V., Gandorfer, A., Solanki, S. K., et al. 2005, A&A, 437, L43
  * Zhao et al. (2022) Zhao, L. L., Fischer, D. A., Ford, E. B., et al. 2022, AJ, 163, 171