11institutetext:  Zentrum für Astronomie der Universität Heidelberg,
Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany
11email<EMAIL_ADDRESS>22institutetext: Astronomical
Observatory of Vilnius University, Saulėtekio al. 3, Vilnius LT-10257,
Lithuania 33institutetext: Leibniz-Institut für Astrophysik Potsdam, An der
Sternwarte 16, D-14482 Potsdam, Germany 44institutetext: Theoretical
Astrophysics, Department of Physics and Astronomy, Uppsala University, Box
516, SE-751 20 Uppsala, Sweden 55institutetext: GEPI, Observatoire de Paris,
PSL Research University, CNRS, Place Jules Janssen, 92190 Meudon, France

# The solar photospheric silicon abundance according to CO5BOLD

Investigating line broadening, magnetic fields, and model effects

S.A. Deshmukh 11 H.-G. Ludwig 1155 A. Kučinskas 22 M. Steffen 33 P.S. Barklem
44 E. Caffau 55 V. Dobrovolskas 22 P. Bonifacio 55

(Submitted: ; Received 23 August 2021; accepted 1 August 2022)

###### Abstract

Context. In this work, we present a photospheric solar silicon abundance
derived using CO5BOLD model atmospheres and the LINFOR3D spectral synthesis
code. Previous works have differed in their choice of a spectral line sample
and model atmosphere as well as their treatment of observational material, and
the solar silicon abundance has undergone a downward revision in recent years.
We additionally show the effects of the chosen line sample, broadening due to
velocity fields, collisional broadening, model spatial resolution, and
magnetic fields.

Aims. Our main aim is to derive the photospheric solar silicon abundance using
updated oscillator strengths and to mitigate model shortcomings such as over-
broadening of synthetic spectra. We also aim to investigate the effects of
different line samples, fitting configurations, and magnetic fields on the
fitted abundance and broadening values.

Methods. CO5BOLD model atmospheres for the Sun were used in conjunction with
the LINFOR3D spectral synthesis code to generate model spectra, which were
then fit to observations in the Hamburg solar atlas. We took pixel-to-pixel
signal correlations into account by means of a correlated noise model. The
choice of line sample is crucial to determining abundances, and we present a
sample of 11 carefully selected lines (from an initial choice of 39 lines) in
both the optical and infrared, which has been made possible with newly
determined oscillator strengths for the majority of these lines. Our final
sample includes seven optical Si I lines, three infrared Si I lines, and one
optical Si II line.

Results. We derived a photospheric solar silicon abundance of
$\log\varepsilon_{\text{Si}}$ = $7.57\pm 0.04$, including a $-0.01$ dex
correction from Non-Local Thermodynamic Equilibrium (NLTE) effects. Combining
this with meteoritic abundances and previously determined photospheric
abundances results in a metal mass fraction
$\mathrm{Z}/\mathrm{X}=0.0220\leavevmode\nobreak\ \pm\leavevmode\nobreak\
0.0020$. We found a tendency of obtaining overly broad synthetic lines. We
mitigated the impact of this by devising a de-broadening procedure. The over-
broadening of synthetic lines does not substantially affect the abundance
determined in the end. It is primarily the line selection that affects the
final fitted abundance.

###### Key Words.:

Sun: abundances – Stars: abundances – Sun: photosphere – Hydrodynamics –
magnetohydrodynamics (MHD)

##  1 Introduction

The chemical composition of the solar photosphere serves as a widely applied
yardstick in astronomy, since it is considered largely representative of the
chemical make-up of the present-day Universe. The determination of solar
abundances has a long history, dating back at least to the 1920s (Payne, 1925;
Unsöld, 1928; Russell, 1929). Among the elements which are spectroscopically
accessible in the solar photosphere, silicon plays a somewhat special role.
Besides being relatively abundant, it is used as a reference to relate the
solar photospheric composition to the composition of type-I carbonaceous
chondrites which are believed to constitute fairly pristine samples of
material from the early Solar System (e.g. Anders & Grevesse, 1989; Lodders,
2003; Lodders et al., 2009; Palme et al., 2014). This important aspect of the
silicon abundance led to many investigations trying to establish an ever more
precise and accurate solar reference value. Over the years, atomic and
observational data have improved, and increasingly sophisticated modelling
techniques have been applied, including time-dependent three-dimensional (3D)
model atmospheres (e.g. Caffau et al., 2011; Pereira et al., 2013; Asplund,
2000) and treatment of departures from thermodynamic equilibrium (NLTE) (e.g.
Bergemann et al., 2019; Amarsi et al., 2019; Steffen et al., 2015).

From the above, it is tempting to conclude that the determination of the
abundance of a particular element in the solar photosphere is an entirely
objective process. Unfortunately, this is not the case, due to several
judicious decisions a researcher must make along the way, namely: i) which
observational material to use, ii) where to place the continuum when
normalising the spectrum, iii) which lines have accurate atomic data (meaning
oscillator strengths and broadening constants), and iv) which lines are
largely unaffected by blends. These aspects influence the final outcome of an
analysis, and we list some previous works’ results below to illustrate the
evolution of the fitted solar silicon abundance with time. Indeed, as we shall
later show, statistical uncertainties are of secondary importance here, and it
is primarily the line selection that dominates the final outcome (including
uncertainties). Moreover, in the present analysis process, we found that more
sophisticated model atmospheres do not necessarily give a more coherent
picture, since they can bring to light modelling shortcomings which were not
recognised before.

Holweger (1973) determined an LTE solar silicon abundance of
$\log\varepsilon_{\text{Si}}=7.65\pm 0.07$ based on 19 Si I lines whose
oscillator strengths were measured by Garz (1973). Wedemeyer (2001) derived a
1D NLTE correction of -0.010 dex for silicon. Together with a correction of
the scale of Garz by Becker et al. (1980), they arrived at a silicon abundance
of $\log\varepsilon_{\text{Si}}=7.550\pm 0.056$. Some of the first multi-
dimensional (multi-D) studies were carried out by Asplund (2000) and Holweger
(2001), who arrived at $\log\varepsilon_{\text{Si}}=7.51\pm 0.04$ and
$\log\varepsilon_{\text{Si}}=7.536\pm 0.049$, respectively. The statistical
equilibrium of silicon and collisional processes were investigated by Shi et
al. (2008), and an abundance of $\log\varepsilon_{\text{Si}}=7.52\pm 0.06$ was
found, taking an extended line sample into account. They also specified that
the NLTE effects on optical silicon lines are weak, but it was later found
that near-infrared lines have sizeable NLTE effects (Shi et al., 2012;
Bergemann et al., 2013). Shchukina et al. (2012) conducted an NLTE analysis of
65 Si I lines, and found $\log\varepsilon_{\text{Si}}=7.549\pm 0.016$.
Shaltout et al. (2013) obtained a 3D LTE solar silicon abundance of
$\log\varepsilon_{\text{Si}}=7.53\pm 0.07$ and a 1D NLTE abundance of
$\log\varepsilon_{\text{Si}}=7.52\pm 0.08$, using the aforementioned -0.010
dex correction.

More recently, Scott et al. (2015) conducted a 3D LTE study and found an
abundance as low as $\log\varepsilon_{\text{Si}}=7.52\pm 0.03$. The result was
later corroborated by Amarsi & Asplund (2017) who derived a 3D NLTE correction
of $-0.01$ dex to this abundance. This analysis used nine Si I lines and one
Si II line in the optical wavelength range. The final abundance was calculated
by means of a weighted average. All in all, over the last twenty years, a
slight downward trend of the derived solar abundance of silicon has become
apparent, but, on a level which is on the edge of being statistically
significant.

In the present work, we apply CO5BOLD model atmospheres (Freytag et al., 2012)
to derive the photospheric abundance of silicon in the Sun. Primarily, the
motivation to do so was the availability of new data for the oscillator
strengths of silicon lines (Pehlivan Rhodin, 2018). This enlarged the set of
silicon lines that could be potentially useful in an abundance analysis,
adding oscillator strengths for near-infrared lines. Additionally, we intended
to relate solar abundances so far derived with CO5BOLD models (for a summary,
see Caffau et al., 2011) to the meteoritic abundance scale.

The necessary spectral synthesis calculations were performed with the LINFOR3D
code111https://www.aip.de/Members/msteffen/linfor3d in LTE approximation, with
only a few exceptions. To compare this with observations, we developed a
custom spectral fitting routine that accounts for correlated photometric noise
in the observations. Our analysis stands out by using a sizeable number of
lines with carefully calculated line broadening constants, new oscillator
strengths, and investigating systematic shortcomings of our 3D model
atmosphere. The last point became important since we found that our model was
predicting systematically overly broadened lines, seen also in Caffau et al.
(2015) for oxygen lines.

The rest of the paper is structured as follows: Section 2 discusses the
details of the model atmospheres used in this study, the methodology of line
selection, and spectral synthesis. It also touches on the role of magnetic
field effects pertaining to these topics. Section 3 describes the fitting
routine and the implemented correlated noise model. Our results, and the
differences due to model choice, are shown in Section 4, discussed in Section
5, and summarised in Section 6. Finally, the appendices explain model
atmosphere differences in abundance and broadening (Appendix A), differences
in regard to NLTE (Appendix B), the choice of the broadening kernel (Appendix
C), the centre-to-limb variation of the continuum (Appendix D), and the
partition functions used in LINFOR3D (Appendix E) in further detail.

##  2 Stellar atmospheres and spectral synthesis

###  2.1 Model atmospheres

Systematic errors from spectral synthesis and fitting come from the use of 1D
hydrostatic model atmospheres, and the assumption of LTE when it is not valid
to do so. 1D hydrostatic model atmospheres rely on mixing-length theory (Böhm-
Vitense, 1958; Henyey et al., 1965), and introduce additional free parameters
such as micro- and macro-turbulence (Gray, 2008). Spectral lines generated
from these model atmospheres are too narrow to fit observations, if they do
not take macroscopic broadening into account, necessitating the use of free
parameters to fit observations. 3D model atmospheres, on the other hand,
should in principle be able to reproduce line shapes, shifts and asymmetries,
and spectral lines synthesised with these can be directly fit to observations.

The prominent state-of-the-art radiative-convective equilibrium 1D models of
solar and stellar atmospheres such as ATLAS (Kurucz, 2005), MARCS (Gustafsson,
1975; Gustafsson et al., 2008) and PHOENIX (Allard & Hauschildt, 1995; Allard
et al., 2001) use classical mixing-length theory, and the efficiency of
convective energy transport here is controlled by a free parameter
$\alpha_{\text{MLT}}$. This, along with the requirement of fitting the micro-
and macro-turbulence free parameters during line synthesis, is a major
drawback of 1D models.

3D atmospheres account for the time-dependence and multi-dimensionality of the
flow, and a spectral synthesis using these atmospheres reproduces line shapes
and asymmetries. Prominent examples include STAGGER (Magic et al., 2013),
MuRAM (Vögler et al., 2005), Bifrost (Gudiksen et al., 2011), and Antares
(Leitner et al., 2017). In this work, we use CO5BOLD , a conservative
hydrodynamics solver able to model surface convection, waves, shocks and other
phenomena in stellar objects (Freytag et al., 2012). As 3D atmospheres are
understandably expensive to run, their output can be saved as a sequence where
flow properties are recorded, commonly called a sequence of ‘snapshots’. These
snapshots are then used in spectral synthesis codes, such as LINFOR3D (used in
this study) or MULTI3D (Leenaarts & Carlsson, 2009). The parameters of the
CO5BOLD model atmospheres used in this work are summarised in Table 2. We use
20 model snapshots to compute line syntheses with LINFOR3D. Throughout this
work, the ‘Model ID’ (see Table 2) for each model will be used to refer to it.

The msc600, m595, b000 and b200 models all use a short characteristics scheme
with double Gauss-Radau quadrature and 3 $\mu$-angles. The n59 model uses a
long characteristics Feautrier scheme with Lobatto quadrature and 4+1
$\mu$-angles. In the current version of CO5BOLD the former scheme is given the
name “MSCrad” and the latter is given the name “LCFrad” (Freytag et al., 2012;
Steffen, 2017).

Table 1:  3D model properties

Model ID | Box Size | Resolution | Grid Points | $T_{\text{eff}}$ | ¡$B_{\textrm{z}}$¿ | Rad. Trans.
---|---|---|---|---|---|---
(d3gt57g44) | [Mm3] | [km3] | $N_{\mathrm{x}}\times N_{\mathrm{y}}\times N_{\mathrm{z}}$ | [K] | [G] | Module
msc600 | $8.0\times 8.0\times 2.3$ | $32\times 32\times 10\ldots 15$ | $250\times 250\times 207$ | $5773\pm 9.4$ | – | MSCrad
n59 | $5.6\times 5.6\times 2.3$ | $40\times 40\times 15$ | $140\times 140\times 150$ | $5774\pm 16$ | – | LCFrad
m595 | $5.6\times 5.6\times 2.3$ | $40\times 40\times 15$ | $140\times 140\times 150$ | $5775\pm 15$ | – | MSCrad
b000 | $5.6\times 5.6\times 2.3$ | $40\times 40\times 15$ | $140\times 140\times 150$ | $5750\pm 19$ | 0 | MSCrad
b200 | $5.6\times 5.6\times 2.3$ | $40\times 40\times 15$ | $140\times 140\times 150$ | $5793\pm 17$ | 200 | MSCrad

222 “Model ID” lists the name of a 3D model used in this paper (note all
models share the prefix “d3gt57g44”), “Box Size” the geometrical size of its
computational domain, “Resolution” the applied grid spacing, “Grid Points” the
number of mesh points per dimension, “$T_{\text{eff}}$” the effective
temperature of the model, “¡$B_{\mathrm{z}}$¿” the mean magnetic field
component in vertical direction, and “Rad. Trans. Module” the name of the
radiation transport module used (see Sec. 2.1 for an explanation). Different
from the other models, the msc600 model uses a variable grid spacing in the
vertical direction; for that reason the covered range is provided. We note
that ‘$\pm$’ in “$T_{\text{eff}}$” should not be interpreted as an
uncertainty: it is the natural dispersion of effective temperature in the time
series illustrating fluctuations between snapshots. Each model has
$\log(\mathrm{g})=4.44$ and solar metallicity.

###  2.2 Line sample

Silicon is an interesting element as it is an important electron donor in
late-type stars. Though it seems that Si I’s ionisation potential of 8.15 eV
would mean a significant amount of silicon to be present in the form of Si II,
the higher ionisation potential of Si II is unfavourable for the appearance of
strong lines in the solar spectrum at wavelengths longer than 3000 Å (Moore,
1970; Russell, 1929). We therefore primarily consider Si I lines and two
carefully chosen Si II lines.

The Si I and II line list below (Table 3) was compiled from the line lists
used in the solar abundance determination by Holweger (2001), Wedemeyer
(2001), Shi et al. (2008), and Amarsi & Asplund (2017). The initial line
selection was performed by synthesising line profiles and comparing to
observations, thus checking these for blends.

We solely used the observational data by Neckel & Labs (1984) (hereafter the
’Hamburg spectrum’) and have worked with the disc-centre and disc-integrated
spectrum. Doerr et al. (2016) show the resolution of the Hamburg spectrum to
be up to $520,000$ compared to the often-used Liege spectrum (Delbouille et
al., 1973), which practically was shown to have a resolution of $\sim
216,000$. The higher spectral resolution allows for a more continuous
representation of the pixel-to-pixel variations in the spectrum, and we
utilise a noise model that represents the covariance between pixels.
Additionally, the Liege spectrum covers a range of $3000-10000$ Å, while the
Hamburg spectrum covers a range of $3290-12510$ Å, affording us access to very
clean near-infrared lines. Furthermore, it remains unclear whether the Liege
spectrum is a true disk-centre spectrum or an integral over a narrow range of
angles around disk-centre, and the available documentation does not precisely
establish this.

From the original 39 lines, we chose a subset of 11 based on comparisons
between disk-centre and disk-integrated spectra, the line shape and the
precision of oscillator strength. We focus on lines that do not have strong
line blends that can interfere with abundance determination. Each line was
weighted on a scale of $1-3$ based on these criteria, and the resulting
weights were used to compute the mean abundance. This idea of a weighted mean
was inspired by the work of Amarsi & Asplund (2017).

There are various sources of error that can enter during the spectroscopic
fitting, but the choice of oscillator strength for each spectral line is often
the largest one. We use semi-empirical $gf$-values from Pehlivan Rhodin (2018)
(hereafter also PR18) for every line in our chosen sub-sample, which have been
updated with respect to previous experimental values measured by Garz (1973)
with accurate lifetime renormalisations by O’Brian & Lawler (1991a, b)
(hereafter also G73+OBL91). The $gf$-values from PR18 were obtained by
combininb calculated level lifetimes with experimental branching ratios.
Moreover, the calculations were validated against existing measurements of
level lifetimes and oscillator strengths. An advantage of the PR18 data is the
homogeneity of the extensive set of line transitions, ranging from UV to
infrared wavelengths. On average across all lines in Table 3, the new values
give a $\Delta\log{gf}=0.024$ decrease with respect to the previous
measurements, and lead to a respective increase of the mean abundance by the
same value when using all lines. Using the new values of the oscillator
strengths lowers the overall scatter in fitted abundance values by $0.07$ dex
and the formal uncertainty by $0.01$ dex.

Figure 1: Individual fitted Si abundances for a set of eight Si I lines and
one Si II line as a function of equivalent width (left) and excitation
potential (right) for eight Si I lines and one Si II line. Only the Si I lines
are used to determine the mean and RMSE (see definition in Eq. 8), given in
the legend in the left panel. G73+OBL91 indicates oscillator strengths from
Garz (1973) normalised according to the results of O’Brian & Lawler (1991a);
the oscillator strength for the Si II line for comparison was taken from
Kurucz (2014); PR18 indicates oscillator strengths from Pehlivan Rhodin
(2018).

We investigated the overall performance of the PR18 $gf$-values on the set of
lines used in Scott et al. (2015) and Amarsi & Asplund (2017). Fig. 1 depicts
the resulting Si abundances. The use of the PR18 data reduces the overall
scatter in the Si abundances from $0.036$ dex (applying the data of G73+OBL91)
to $0.018$ dex. In all cases, there is no discernible dependence on line
strength or excitation potential. The only Si II line at
$6371\leavevmode\nobreak\ \mathrm{\AA}$ that we consider sufficiently reliable
for abundance analysis may indicate a slight ionisation imbalance. However,
this imbalance is of similar magnitude for both sets of oscillator strengths.
Moreover, the line has a high excitation potential and is partially blended
(see Fig. 4) so that the apparent imbalance could be the result of systematics
in the analysis.

In previous studies, strong infrared Si I lines above $10000$ Å often lacked
reliable experimental oscillator strengths (Borrero et al., 2003; Shi et al.,
2008; Shchukina et al., 2012). Shi et al. (2012) investigate near-infrared
lines in nearby stars with both LTE and NLTE, and find that there is a larger
departure from LTE for the infrared lines than optical ones. They point out
that weak lines are insensitive to NLTE effects, whereas stronger lines show
visible effects. The new oscillator strengths also afford us the use of near-
infrared lines in our sample. These lines are not as affected by blends as the
optical lines (Shi et al., 2008), but some show strong NLTE cores. We do not
include these lines to determine a final abundance. During fitting, we clip
points that are more than $0.5\%$ of relative (normalised) flux further from
the observations after an initial fit, removing weak line blends and deviating
line cores from the abundance determination.

Table 2:  Atomic data for spectral lines of Si i and Si ii.

$\lambda$ | Transition | $\log gf$ | $\log gf$ | Weight | E${}_{\text{low}}$ \- E${}_{\text{upp}}$ | E${}_{\text{lim, low}}$ \- E${}_{\text{lim, upp}}$ | n*${}_{\text{low}}$ | n*${}_{\text{upp}}$ | $\sigma_{0}$ | $\alpha$
---|---|---|---|---|---|---|---|---|---|---
[Å] |  | old | new |  | [cm-1] – | [cm-1] | [cm-1] – | [cm-1] |  |  | [a.u.] |
Si I | – |  |  |  |  |  | – |  | – |  |  |  |  |
5645.61* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{3}{\rm P}_{\rm 2}$ | – 2.04(3) | G73+OBL91 | – 2.067 | 1 | 39760 – | 57468 | 65748 – | 65748 | 2.054 | 3.640 | 1791 | 0.223
5665.55* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 0}$ – | $5\rm p\,^{3}{\rm P}_{\rm 1}$ | – 1.94(3) | G73+OBL91 | – 2.025 | 2 | 39683 – | 57329 | 65748 – | 65748 | 2.051 | 3.609 | 1772 | 0.222
5684.48* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 2}$ – | $5\rm p\,^{3}{\rm S}_{\rm 1}$ | – 1.55(3) | G73+OBL91 | – 1.606 | 2 | 39955 – | 57542 | 65748 – | 65748 | 2.062 | 3.656 | 1798 | 0.221
5690.43* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{3}{\rm P}_{\rm 1}$ | – 1.77(3) | G73+OBL91 | – 1.802 | 3 | 39760 – | 57328 | 65748 – | 65748 | 2.054 | 3.609 | 1772 | 0.222
5701.11* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{3}{\rm P}_{\rm 0}$ | – 1.95(3) | G73+OBL91 | – 1.981 | 3 | 39760 – | 57296 | 65748 – | 65748 | 2.054 | 3.602 | 1769 | 0.222
5708.40 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 2}$ – | $5\rm p\,^{3}{\rm P}_{\rm 2}$ | – 1.37(3) | G73+OBL91 | – 1.388 | 0 | 39955 – | 57468 | 65748 – | 65748 | 2.062 | 3.640 | 1788 | 0.223
5772.15 |  $4{\rm s}\,^{1}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{1}{\rm S}_{\rm 0}$ | – 1.65(3) | G73+OBL91 | – 1.643 | 0 | 40992 – | 58312 | 65748 – | 65748 | 2.105 | 3.841 | 2036 | 0.208
5780.38 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 0}$ – | $5{\rm p}\,^{3}{\rm D}_{\rm 1}$ | – 2.25(3) | G73+OBL91 | – 2.156 | 0 | 39683 – | 56978 | 65748 – | 65748 | 2.051 | 3.536 | 1691 | 0.228
5793.07* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{3}{\rm D}_{\rm 2}$ | – 1.96(3) | G73+OBL91 | – 1.893 | 2 | 39760 – | 57017 | 65748 – | 65748 | 2.054 | 3.544 | 1704 | 0.228
5797.86 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 2}$ – | $5{\rm p}\,^{3}{\rm D}_{\rm 3}$ | – 1.95(3) | G73+OBL91 | – 1.830 | 0 | 39955 – | 57198 | 65748 – | 65748 | 2.062 | 3.582 | 1755 | 0.223
5948.54 |  $4{\rm s}\,^{1}{\rm P}^{{\rm o}}_{\rm 1}$ – | $5\rm p\,^{1}{\rm D}_{\rm 2}$ | – 1.13(3) | G73+OBL91 | – 1.179 | 0 | 40992 – | 57798 | 65748 – | 65748 | 2.105 | 3.714 | 1845 | 0.222
6125.02 |  $3{\rm p^{3}}\,^{3}{\rm D}^{{\rm o}}_{\rm 1}$ – | $5\rm f\,^{3}{\rm D}_{\rm 2}$ | – 1.465 | K07 | – | 0 | 45276 – | 61598 | 114716 – | 65748 | 1.257 | 5.141 | 3354 | 0.348
6142.49 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 3}$ – | $5\rm f\,^{3}{\rm D}_{\rm 3}$ | – 1.296 | K07 | – | 0 | 45321 – | 61597 | 114716 – | 65748 | 1.257 | 5.141 | 3354 | 0.348
6145.02 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 2}$ – | $5\rm f\,^{3}{\rm G}_{\rm 3}$ | – 1.311 | K07 | – | 0 | 45294 – | 61562 | 114716 – | 65748 | 1.257 | 5.119 | 3295 | 0.341
6237.32 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 1}$ – | $5\rm f\,^{3}{\rm F}_{\rm 2}$ | – 0.975 | K07 | – | 0 | 45294 – | 61304 | 114716 – | 65748 | 1.257 | 4.968 | 3081 | 0.351
6243.82 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 2}$ – | $5\rm f\,^{3}{\rm F}_{\rm 3}$ | – 1.244 | K07 | – | 0 | 45294 – | 61305 | 114716 – | 65748 | 1.257 | 4.968 | 3081 | 0.351
6244.47 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 2}$ – | $5\rm f\,^{1}{\rm D}_{\rm 2}$ | – 1.091 | K07 | – | 0 | 45294 – | 61305 | 114716 – | 65748 | 1.257 | 4.968 | 3081 | 0.351
6976.51 |  $4{\rm p}\,^{3}{\rm D}_{\rm 1}$ – | $6{\rm d}\,^{3}{\rm F}^{{\rm o}}_{\rm 2}$ | – 1.07(3) | G73+OBL91 | – | 0 | 48020 – | 62350 | 65748 – | 65748 | 2.487 | 5.681 | 4600 | 0.530
7003.57 |  $4{\rm p}\,^{3}{\rm D}_{\rm 2}$ – | $6{\rm d}\,^{3}{\rm F}^{{\rm o}}_{\rm 3}$ | – 0.793 | G73+OBL91 | – | 0 | 48102 – | 62377 | 65748 – | 65748 | 2.493 | 5.704 | 4700 | 0.531
7034.91 |  $3\rm d\,^{1}{\rm D}^{{\rm o}}_{\rm 2}$ – | $5\rm f\,^{3}{\rm F}_{\rm 3}$ | – 0.78(3) | G73+OBL91 | – | 0 | 47352 – | 61562 | 65748 – | 65748 | 2.442 | 5.119 | 3232 | 0.338
7226.21 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 1}$ – | $4\rm f\,^{1}{\rm D}_{\rm 2}$ | – 1.41(3) | G73+OBL91 | – | 0 | 45276 – | 59111 | 114716 – | 65748 | 1.257 | 4.065 | 1745 | 0.307
7405.79 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 1}$ – | $4\rm f\,^{3}{\rm F}_{\rm 2}$ | – 0.72(3) | G73+OBL91 | – | 0 | 45276 – | 58775 | 114716 – | 65748 | 1.257 | 3.966 | 1585 | 0.304
7415.36 |  $3{\rm p}\,^{3}{\rm D}^{{\rm o}}_{\rm 2}$ – | $4\rm f\,^{3}{\rm F}_{\rm 2}$ | – 0.65(3) | G73+OBL91 | – | 0 | 45294 – | 58774 | 114716 – | 65748 | 1.257 | 3.966 | 1585 | 0.304
7680.27* |  $4{\rm p}\,^{1}{\rm P}_{\rm 1}$ – | $5\rm d\,^{1}{\rm D}^{{\rm o}}_{\rm 2}$ | – 0.59(3) | G73+OBL91 | – 0.678 | 2 | 47284 – | 60301 | 65748 – | 65748 | 2.437 | 4.487 | 2107 | 0.494
7918.38 |  $4{\rm p}\,^{3}{\rm D}_{\rm 1}$ – | $5\rm d\,^{3}{\rm F}^{{\rm o}}_{\rm 2}$ | – 0.51(3) | G73+OBL91 | – 0.666 | 0 | 48020 – | 60645 | 65748 – | 65748 | 2.487 | 4.636 | 2934 | 0.234
7932.35 |  $4{\rm p}\,^{3}{\rm D}_{\rm 2}$ – | $5\rm d\,^{3}{\rm F}^{{\rm o}}_{\rm 3}$ | – 0.37(3) | G73+OBL91 | – 0.472 | 0 | 48102 – | 60705 | 65748 – | 65748 | 2.493 | 4.664 | 2985 | 0.234
10288.94* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 0}$ – | $4\rm p\,^{3}{\rm S}_{\rm 1}$ | – | – | – 1.622 | 2 | 39683 – | 49400 | 65748 – | 65748 | 2.493 | 4.664 | 739 | 0.230
10371.26 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $4\rm p\,^{3}{\rm S}_{\rm 1}$ | – | – | – 0.789 | 0 | 39760 – | 49400 | 65748 – | 65748 | 2.493 | 4.664 | 739 | 0.230
10603.43 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $4\rm p\,^{3}{\rm P}_{\rm 2}$ | – | – | – 0.394 | 0 | 39760 – | 49188 | 65748 – | 65748 | 2.493 | 4.664 | 727 | 0.231
10689.72 |  $4{\rm p}\,^{3}{\rm D}_{\rm 1}$ – | $4\rm d\,^{3}{\rm F}^{{\rm o}}_{\rm 2}$ | – | – | – 0.017 | 0 | 48020 – | 57372 | 65748 – | 65748 | 2.493 | 4.664 | 1418 | 0.234
10694.25 |  $4{\rm p}\,^{3}{\rm D}_{\rm 2}$ – | $4\rm d\,^{3}{\rm F}^{{\rm o}}_{\rm 3}$ | – | – | \+ 0.155 | 0 | 48102 – | 57450 | 65748 – | 65748 | 2.493 | 4.664 | 1445 | 0.750
10749.37 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $4\rm p\,^{3}{\rm P}_{\rm 1}$ | – | – | – 0.268 | 0 | 39760 – | 49061 | 65748 – | 65748 | 2.493 | 4.664 | 721 | 0.231
10784.56 |  $4{\rm p}\,^{3}{\rm D}_{\rm 2}$ – | $4\rm d\,^{3}{\rm F}^{{\rm o}}_{\rm 2}$ | – | – | – 0.746 | 0 | 48102 – | 57372 | 65748 – | 65748 | 2.493 | 4.664 | 1417 | 0.296
10786.85 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 1}$ – | $4\rm p\,^{3}{\rm P}_{\rm 0}$ | – | – | – 0.380 | 0 | 39760 – | 49028 | 65748 – | 65748 | 2.493 | 4.664 | 719 | 0.231
10827.09 |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 2}$ – | $4\rm p\,^{3}{\rm P}_{\rm 2}$ | – | – | \+ 0.227 | 0 | 39955 – | 49188 | 65748 – | 65748 | 2.493 | 4.664 | 728 | 0.231
12390.15* |  $4{\rm s}\,^{1}{\rm P}^{{\rm o}}_{\rm 1}$ – | $4\rm p\,^{3}{\rm P}_{\rm 1}$ | – | – | – 1.805 | 2 | 40992 – | 49061 | 65748 – | 65748 | 2.493 | 4.664 | 730 | 0.234
12395.83* |  $4{\rm s}\,^{3}{\rm P}^{{\rm o}}_{\rm 2}$ – | $4\rm p\,^{3}{\rm D}_{\rm 1}$ | – | – | – 1.723 | 2 | 39955 – | 48020 | 65748 – | 65748 | 2.493 | 4.664 | 675 | 0.231
Si II | – |  |  |  |  |  | – |  | – |  |  |  |  |
6347.10 |  $4{\rm s}\,^{2}{\rm S}_{\rm 1/2}$ – | $4{\rm p}\,^{2}{\rm P}^{{\rm o}}_{\rm 3/2}$ | \+ 0.170 | K14 | \+ 0.182 | 0 | 65500 – | 81299 | 131838 – | 131838 | 2.572 | 2.943 | 390 | 0.190
6371.36* |  $4{\rm s}\,^{2}{\rm S}_{\rm 1/2}$ – | $4{\rm p}\,^{2}{\rm P}^{{\rm o}}_{\rm 1/2}$ | – 0.040 | K14 | – 0.120 | 1 | 65500 – | 81299 | 131838 – | 131838 | 2.572 | 2.943 | 390 | 0.190

333 An asterisk next to the wavelength signifies the line was in the chosen
subsample. References to the “old” $gf$-values are G73+OBL91: Garz (1973),
renormalised by $+0.097$ dex according to O’Brian & Lawler (1991a), K07:
Kurucz (2007), and K14: Kurucz (2014). “new” $\log gf$ values come from
Pehlivan Rhodin (2018). The “Weight” column specifies our weighting for each
line. The right portion of the table shows ABO theory parameters.
“E${}_{\text{low}}$” & “E${}_{\text{upp}}$” show the lower and upper energy
levels of the transitions in cm-1, and “Elim” shows the series limit energy
for that level (see e.g. Heiter et al. (2021) for details).
“n*${}_{\text{low}}$” & “n*${}_{\text{upp}}$”, are the effective quantum
numbers associated with the states of the transitions. “$\sigma_{0}$” is the
line broadening cross section in atomic units and “$\alpha$” describes the
power law velocity dependence.

###  2.3 Spectral synthesis

In this study, a 3D hydrodynamical solar model atmosphere with an initial
silicon abundance of $\log\varepsilon_{\text{Si}}=7.55$ was used for the line
synthesis. Each line is synthesised with the LINFOR3D code over 20 atmospheric
snapshots. The lines are synthesised in full 3D, including Doppler shifts, and
the profile is averaged in time and space (horizontally). We generate a set of
syntheses for each spectral line, each with a different $\log
gf\varepsilon_{\odot}$ value. We alter $\log{gf}$ in this case, with a
distinct stepping around a central value for each line depending on the
sensitivity of the lines to oscillator strength changes.

Our investigation of the solar silicon abundance is mostly 3D LTE, but the
$-0.01$ dex correction due to NLTE effects determined by Amarsi & Asplund
(2017) was used globally for our calculated abundance. We investigate NLTE
effects on broadening and abundance in appendix B.

###  2.4 Line broadening

Aside from the abundance, we also aim to investigate the effects of broadening
required to fit synthetic spectra to observations. The broadening we see in
the observed lines can be attributed to broadening by macroscopic velocity
fields, thermal broadening, and atomic line broadening due to collisions with
neutral particles and electrons. For broadening due to macroscopic velocity
fields, though a tendency towards less turbulent flow is expected in numerical
models due to limited spatial resolution, our 3D syntheses are broader than
the observations, even prior to applying instrumental profile broadening. The
result is not unique to our work (see, e.g. Caffau et al., 2015), nor is it
unique to CO5BOLD models (see Appendix A for details).

####  2.4.1 Collisional broadening

Solar spectral lines are subject to broadening from atomic effects. These are
pressure broadening effects arising from van der Waals and other interatomic
forces. It turns out that the $7680$ Å line shows sizeable Stark broadening,
meaning the Stark effect is non-negligible for some solar silicon lines and
illustrating the importance of accurately modelling these effects. We use the
theory by Anstee, Barklem and O’Mara (Anstee & O’Mara, 1991, 1995; Barklem et
al., 1998) to describe the line broadening effects of neutral particles,
predominantly neutral hydrogen (hereafter ABO theory). Note that the
broadening constants were specifically calculated using existing tables from
the above papers or through extended calculations, and are shown in Table 3.
LINFOR3D additionally takes the Stark broadening of lines into account,
employing line widths from from the Vienna Atomic Line Database (VALD) (Kupka
et al., 1999, 2000; Ryabchikova et al., 2015) and assuming temperature
dependence from the Unsöld theory (Unsöld, 1955).

The theory of collisional line broadening for neutral species (Anstee &
O’Mara, 1991, 1995) was extended to singly ionised atoms in Barklem & O’Mara
(1998). For the lines corresponding to ionised Si in this study, we performed
specific calculations assuming $E_{p}=-4/9$ atomic units (the average value of
the energy denominator of the second-order contribution to the interaction
energy), which is expected to give reasonable estimates, though this
assumption is less secure for ions than for neutrals (see Roederer & Barklem,
2018; Barklem & Aspelund-Johansson, 2005). Equation 1 of Anstee & O’Mara
(1995) describes the line broadening cross section $\sigma$ (in atomic units)
as a function of the relative collision velocity $v$, with respect to a
reference value of $v_{0}=10^{4}$ m s-1, with the parameter $\alpha$ giving
the power law velocity dependence:

$\sigma(v)=\sigma_{0}\left(\frac{v}{v_{0}}\right)^{-\alpha}.\,$ (1)

The line width at a given temperature can then be obtained from this relation
by analytic integration over the Maxwellian velocity distribution (see
Equation 3 of Anstee & O’Mara (1995)).

####  2.4.2 Rotational broadening

For disk-integrated spectra, we consider fixed solid body rotation assuming a
synodic $v\sin{i}=1.8$ km s-1 is present in the observations from the Hamburg
spectrum (Bartels, 1934; Gray, 2008). In total, $1$ vertical $16$ inclined
rays were used ($4$ $\mu$-angles and $4$ $\phi$-angles) with Lobatto
quadrature (Abramowitz & Stegun, 1965). Rotational broadening is not included
for disk-centre spectra.

###  2.5 Magnetic fields

####  2.5.1 Effects on model atmospheres

It is known that weak magnetic fields are present in the solar surface layers
and there is evidence that the mean magnetic flux density for these is of the
order of $10^{2}$ G (Bueno et al., 2004; Nordlund et al., 2009). Numerical 3D
convection models predict that the granulation pattern is profoundly affected
in regions with high flux density (Cattaneo et al., 2003; Vögler et al., 2005;
Cheung et al., 2008), in agreement with observations. Additionally,
photospheric material becomes more transparent in magnetic concentrations due
to their lower density, allowing one to see into deeper, hence hotter, layers
of the solar atmosphere (Stein & Nordlund, 2000). In these regions, where flux
tubes are also heated through heat influx from the surrounding material
(Spruit, 1976), weak spectral lines will experience a brightening of their
core, meaning magnetic fields can act on temperature-sensitive lines not only
directly through the Zeeman effect, but also indirectly due to temperature
stratification in line-forming regions (Fabbian et al., 2012).

####  2.5.2 Effects on spectral synthesis

In 1D model atmospheres, Borrero (2008) showed that magnetic fields have non-
negligible effects on spectral line synthesis, and Fabbian et al. (2010)
expanded this to 3D atmospheres, finding an abundance correction for Fe of the
order of $\sim+0.01$ dex when using magneto-convection models. Fabbian et al.
(2012) used 28 iron lines and found the average solar iron abundance to be
$\sim 0.03-0.11$ dex higher when using a magneto-convection model with
$\left\langle B_{\mathrm{vert}}\right\rangle=200$ G and investigated models
with average magnetic flux densities of $\left\langle
B_{\mathrm{vert}}\right\rangle=0,50,100,200$ G. They also showed that Zeeman
broadening gains importance in the infrared, and that the largest contribution
to higher abundance is the indirect effect of line-weakening caused by a
warmer stratification as seen on an optical depth scale.

Following this, Fabbian & Moreno-Insertis (2015) were able to reproduce the
observations of 2 O I spectral lines with blends with the use of a 3D MHD
photospheric model with a uniform vertical magnetic field of $200$ G. They
required an oxygen abundance several centidexes higher than those suggested by
Asplund et al. (2009) to fit observations and again showed the need to
consider magneto-convection processes when considering problems sensitive to
the shape of spectral features, such as spectral synthesis and fitting.

As shown in Table 2, we use two magnetic model atmospheres to investigate the
effects of the magnetic field strength on the fitted abundance and broadening
values. Zeeman broadening is not taken into account in the spectral synthesis.
Note that Shchukina et al. (2015) and Shchukina et al. (2016) also show that
such vertical fields as in Fabbian et al. (2012) overestimate the effects when
compared with a 3D MHD model with a self-consistent small-scale dynamo and
more realistic magnetic fields. With this in mind, and with the fact that the
magnetic models used here are not of a high enough resolution to
quantitatively investigate the effects of magnetic fields, we consider only
differential effects. Section 4 shows these differences and concludes that the
extra broadening present in the msc600 model syntheses could be partly
attributed to the lack of magnetic fields.

##  3 Spectral fitting routine

The synthesised lines are fitted to the observations from the Hamburg spectrum
via $\chi^{2}$ minimisation, where we define $\chi^{2}$ as

$\chi^{2}=\Delta x^{\mathrm{T}}\mathbf{C}^{-1}\Delta x,\,$ (2)

where $\Delta x$ is the vector of residuals and $\mathbf{C}$ is the covariance
matrix of the data. We use maximum likelihood estimation using $\chi^{2}$
rather than the conventional least-squares minimisation in order to
incorporate the errors due to covariance in the spectrum.

In this analysis, we fit disk-centre intensities for abundance determination,
but disk-integrated spectra are also used when choosing the line sample. To
mask weak blends, a window is chosen around each line for the fitting
procedure. An example is shown in Fig. 2.

Figure 2: Synthesised fitted line profile (black line) against the
observations from the Hamburg atlas (points). Grey points were cut from the
initial fit; blue points show areas where the deviation between initial fit
and data is too high, and so the subsequent fits do not use the offending
points; red points are used for computing all fitted quantities. Removing the
points on right hand side removes the strong blend, and sigma-clipping handles
weaker ones during fitting. $w_{\mathrm{K}}$ is the width of the broadening
kernel used. Note that this line is not used in the final determination of the
silicon abundance because of the large line blends.

The routine involves first loading the observations and determining the
covariance between pixels for a given line (see Section 3.2 for details).
Then, during the fitting, instrumental profile broadening and rotational
broadening (for disk-integrated spectra) are applied to the syntheses. A
monotonic cubic interpolation across abundance is used to find the closest
synthesis matching the observations, now also fitting the wavelength shift, to
create an array of syntheses. Finally, for the nine abundances in steps of
$0.05$ dex that were synthesised, the abundance is fit by interpolating
through the array of syntheses and constructing a spectrum from the overall
best fitting points. Sigma-clipping is employed to improve subsequent fits.
The tight window is necessary with our particular combination of syntheses and
observations to minimise the effects that weak line blends have on fitting.

Some infrared lines exhibited strong NLTE cores, and we do not include these
lines in our final sample used to determine abundance. However, we did attempt
to fit these lines by masking the core and fitting the line up to $70\%$
residual intensity, inspired by the work of Shi et al. (2008). Their work
revolves around accurately treating NLTE effects, and we noticed that infrared
lines shown in this work could be fit in LTE up to this $70\%$ intensity. In
the end, clipping points at the $0.5\%$ level was more versatile and useful
than simply masking the line cores. This choice of $0.5\%$ eliminates stronger
offending blends while still retaining important characteristics in the line
profile. We fitted some infrared lines with strong NLTE cores with this
method, but ultimately chose to leave them out of the subsample used to derive
an abundance since they require excess negative broadening (see Sec. 3.1) to
fit, which could be indicative of smaller NLTE effects present in the fitted
parts of the line profile.

###  3.1 Profile (de)broadening

In the present investigation, we found that in many cases the synthetic line
profiles were already broader than the observations, even prior to the
application of instrumental broadening. Such a problem is specific to spectral
synthesis calculations based on 3D model atmospheres (Caffau et al., 2015), as
broadening effects of the stellar velocity field in 1D hydrostatic models are
added in an ad-hoc fashion via fitting micro- and macro-turbulent broadening
to the observations. Hence, mismatches of the overall broadening between model
and observations cannot occur.

In our case, we could either broaden the observations or de-broaden the
syntheses. Unsurprisingly, broadening the observations a priori by $2.5$ km
s-1 (the value that allowed our syntheses to fit the observations well)
resulted in better statistical fits, as the lines appeared more Gaussian-like.
However, this removes information about the line, such as weak blends and the
overall shape. Instead, we chose to implement the capability to de-broaden our
syntheses, rendering them narrower to better fit the observations as opposed
to broadening observations instead. This is stated as equivalent to a negative
broadening, or broadening with a kernel that has a negative full width at
half-maximum (FWHM). We hoped to maintain the ability of 3D line profiles that
allows one to identify weak blends that can degrade fits without being
immediately recognised. This provides us a fully invertible method when it
comes to fitting the broadening value of syntheses. Note that the broadening
value we fit is still much smaller than the values of micro- and macro-
turbulence fitted in 1D models, and our 3D model still reproduces line shapes
and asymmetries.

We formally associate broadening with convolution, and de-broadening (negative
broadening) as deconvolution and use a kernel inspired by Dorfi & Drury
(1987). This kernel is composed of a double exponential decay centred on zero
(see Appendix C for details of the implementation), and is referred to
throughout this work as the $G_{n}$ kernel, where $n$ is an integer
representing the ’order’ of the kernel. We can convolve the $G_{1}$ kernel
(Equation 3) with itself to produce more Gaussian-like kernels, but note that
this brings back the noise amplification that we aimed to mitigate with the
use of the $G_{1}$ kernel:

$g_{1}(x,x^{\prime})=G_{1}(x-x^{\prime})=\frac{\alpha}{2}\exp(-\alpha\left\lvert
x-x^{\prime}\right\rvert).$ (3)

Here, $x$ is position (in velocity space) and $\alpha$ controls the width of
the kernel.

Gaussian and sinc functions were both considered as instrumental profile
broadening kernels, but these do not allow for efficient and optimal
computation of a de-broadened profile, since as the Fourier transform of these
kernel functions rapidly goes to zero precluding a deconvolution, small
disturbances cause large spikes in the wings of debroadened spectra. Across
all lines (and across the downsample), the choice between the $G_{1}$ and
$G_{3}$ kernels does not affect the fitted abundance. The $G_{1}$ kernel does
result in slightly better statistical fits, and so we favour it in this study.

###  3.2 Photometric noise model

Often, the pixel-to-pixel correlation of the signal in the spectrum is
ignored, despite being rather commonly present due to instrumental
imperfections during detection as well as steps during data reduction. The
Hamburg atlas was rebinned at steps of $3.8$ mÅ. This rebinning introduced a
correlation between pixels. We implement a photometric noise model that
considers the relation between neighbouring pixels in the observations, whose
correlated signal introduces correlation in the noise values in each pixel
(given by the root mean-squared error (RMSE) of a window). In order to model
this, we require a representation of the covariance matrix of the noise. This
covariance matrix, or correlated noise matrix, can then be applied to the set
of observations in the fitting routine to describe the pixel-to-pixel
correlation of the noise in the spectrum. Due to the nature of the covariance
matrix, it must be positive semi-definite. To accurately estimate the matrix
of correlated noise, we fit the autocorrelation function of continuum regions
of the spectrum with an exponential decay.

##  4 Results

###  4.1 Line syntheses

The spectral line syntheses for the chosen $11$ lines are shown in Fig. 4. The
lines are synthesised with the non-magnetic msc600 model atmosphere and use
the $G_{1}$ broadening kernel during fitting. We also use a covariance matrix
for pixel-correlated noise, which results in an increase in mean abundance of
$0.001$ dex compared to assuming uncorrelated noise. Our final derived
photospheric solar silicon abundance is $\log\varepsilon_{\text{Si}}$ =
$7.57\pm 0.04$, including the $-0.01$ dex correction from NLTE effects (Amarsi
& Asplund, 2017).

We find that simultaneously fitting the continuum for the entire selection of
spectral lines systematically lowers the abundance by $0.01$ dex. A local fit
of the continuum is therefore not representative of the spectrum on a larger
scale, and we rely on the normalisation provided by the Hamburg atlas. This
comparison is shown for two lines in Fig. 3. Broadening the observations a
priori by $2.5$ km s-1(to counter the overly broadened syntheses) increases
the abundance across all configurations by $0.02$ dex as this convolves weak
line blends into the primary line shape, rendering the observations more
Gaussian-like and removing information (such as the observational line shape)
in the process.

Figure 3:  Comparison of two silicon line profiles while locally fitting the
continuum. Observations are shown in black and the fitted syntheses at
$5701.11$ and $5708.40\leavevmode\nobreak\ \mathrm{\AA}$ are shown in red and
blue, respectively, along with the fitted continuum value. Across a wider
wavelength range, the various fitted continua are not consistent with one
another.  Figure 4: Synthesised fitted line shapes (black solid line) against
observations from the Hamburg spectrum (points) for 11 silicon lines for the
msc600 model. Grey points were removed before fitting, blue points show areas
where sigma-clipping was employed to remove poorly fitting points, and red
points were used for determining final fitted quantities. Green lines show the
residuals increased by a factor of 10 for readability. In each panel, the name
of the line, the reduced $\chi^{2}$ value, the width of the broadening kernel
$w_{\mathrm{K}}$, and the LTE abundance $\log\varepsilon_{\text{Si}}$ are
shown. The final weighted mean abundance of $7.57\pm 0.04$ includes the
$-0.01$ dex NLTE correction, a fixed continuum, no pre-broadening, and de-
broadening.

###  4.2 Magnetic field effects

Figure 5 shows the difference in abundance and broadening fitted when
comparing different magnetic model atmospheres: one without a magnetic field
and one with a magnetic field of 200 G (b200).

The fitted results of the magnetic models should not be used on an absolute
scale, as the model atmosphere grids are too coarse to resolve the detailed
structure of small magnetic flux concentrations. Additionally, as the
effective temperatures of these models are further from the nominal $T_{\rm
eff}=5772$ K (Prša et al., 2016), we apply a correction to each line’s fitted
abundance to account for this change in temperature. The correction was
derived from the snapshot-to-snapshot variation of $T_{\rm eff}$ and
equivalent width. The highest correction, for the b000 model, was $+0.015$ dex
for the Si II lines, while the Si I lines averaged a $-0.005$ dex correction.
Si I and Si II lines show opposite trends in each model. The models are used
for differential comparison, noting that increasing the magnetic field
strength increases both the fitted abundance and broadening values. A possible
implication is that the over-broadening of synthesised lines is caused by a
lack of magnetic fields in the model atmosphere, as magnetic field lines
constrain the flow of material in the solar atmosphere, reducing turbulence
and thereby the line broadening. Fig. 6 illustrates this point for all $4$
models alongside $2$ models with much higher and lower effective temperatures.
Though the b200 model has a higher effective temperature than the other solar-
type models, its still has a lower vertical RMS velocity. Comparing to the
t63g45mm00 model at $6233$ K, the increase in effective temperature in the
b200 model to overcome the magnetic field effects would need to be much
greater than the current model’s value.

Additionally, a magnetic field strength of $200$ G still does not give the
full amount of de-broadening required to fit the observations and is higher
than the value of up to $75$ G expected in the majority of the quiet
photosphere (Ramírez Vélez et al., 2008). Again, as shown by Shchukina et al.
(2015) & Shchukina et al. (2016), a vertical field of $200$ G would
overestimate the effects when compared with a self-consistent 3D MHD model
with a small-scale dynamo. The syntheses do not include the $1$ km s-1
instrumental broadening, meaning the nominal $w_{\mathrm{K}}\approx 1$ km s-1
– hence even further de-broadening from magnetic fields would be required.
Therefore, our results are suggestive but not definitive that the lack of
magnetic fields contributes to the over-broadening of spectral lines, and the
high magnetic field strengths required to produce the required broadening
would not be consistent with 3D MHD models with small-scale dynamos.

Figure 5: Comparison of abundance and broadening between the b000 (red
points), and b200 (blue points) models. The red and blue horizontal dashed
lines follow the same colour scheme and show the weighted average values.
Triangles indicate the lines used in the subsample. Increasing the magnetic
field strength increases fitted abundance and decreases negative broadening.
Figure 6:  Vertical RMS velocity profiles in and around line-forming regions
for the b000, b200 models (solid lines) with the msc600, n59, and 2 other
models at much lower and higher effective temperatures shown for comparison
(dotted lines). The b200 model clearly has lower RMS velocity than the other
solar-type models, even though its effective temperature is higher.

###  4.3 Effects of model differences

The models presented in this work have different spatial resolutions and
utilise different radiation transport (RT) schemes. As a comparison, Fig. 7
shows the difference between the fitted abundance and broadening values in the
msc600, m595 and n59 models. We find that using the coarser models predicts a
slightly higher abundance, and the lines are not synthesised as broad as when
using the finer msc600 model, which is shown in Fig. 8. This is in line with
the reduced level of turbulence in the n59 model due to its lower resolution.
The n59 model also has a significantly higher extra viscosity relative to the
msc600 model, and utilises a long characteristic RT scheme while the msc600
model uses a newer, multiple short characteristic scheme. The m595 model has
the same parameters as msc600, except the spatial resolution, which is that of
n59. Its fitted broadening lies between n59 and msc600, but is closer to the
latter model. This suggests that, rather than the spatial resolution, it is
the RT scheme and viscosity parameters that primarily affect the line
broadening; though the spatial resolution does have a small effect. All models
use the same opacity table and equation of state. With the chosen line sample,
generally positive broadening is required for the n59 model; however, when
considering all lines, the majority still require negative broadening to fit
the observations.

Figure 7: Comparison of abundance and broadening between the msc600 (red
points), m595 (green points) and n59 (blue points) models. Triangles indicate
the lines used in the subsample. The red, green and blue horizontal dashed
lines follow the same colour scheme and show the weighted average values.
Figure 8:  Line profiles from msc600 (black) and n59 (red) models with the
residuals magnified by a factor of $10$ in green. The difference in the line
profiles is greatest in the line core.

###  4.4 Disk-centre and disk-integrated spectrum differences

Comparisons between disk-centre and disk-integrated spectra for the msc600
model are shown in Fig. 9. Disk-integrated spectra show a $0.01$ dex higher
abundance on average, and a broadening $0.03$ km s-1 more negative than the
disk-centre spectra across the full sample of $39$ lines. The correspondence
between disk-centre and disk-integrated fitted abundance and broadening is
hence quite satisfactory. The infrared lines not chosen in the subsample show
higher deviation than most optical lines, perhaps due to NLTE core effects
that are more prominent in the disk-integrated spectrum. Additionally, the
largest deviation is given by the Si II $6347.10\leavevmode\nobreak\
\mathrm{\AA}$ line, which is not used in the subsample because of this large
deviation. The other Si II at $6371.36\leavevmode\nobreak\ \mathrm{\AA}$ line
is used in the subsample and shows a large uncertainty in the fitted
abundance.

Figure 9: Differences between disk-centre and disk-integrated fitted
abundances for the msc600 model. Red triangles indicate the lines in the
subsample used for the final abundance calculation.

The fitted abundance and broadening values for the disk-centre and disk-
integrated syntheses, as well as the old and new oscillator strengths used in
this study are given in Table 3. Only the disk-centre spectra and new
$gf$-values are used for determining the final abundance.

Table 3: Fitted 3D LTE abundance and broadening values for disk-centre (DC)
and disk-integrated (DI) spectra with their formal statistical uncertainties.

Wavelength | Wλ (DC) | Wλ (DI) |  $\log\varepsilon_{\text{Si}}$ (DC) |  $\log\varepsilon_{\text{Si}}$ (DI) |  $w_{\mathrm{K}}$ (DC) |  $w_{\mathrm{K}}$ (DI)
---|---|---|---|---|---|---
[$\mathrm{\AA}$] | [m$\mathrm{\AA}$] | [m$\mathrm{\AA}$] |  |  | [km s-1] | [km s-1]
Si I |  |  |  |  |  |
5645.6128* | $37.285\pm 0.004$ | $38.465\pm 0.004$ | $7.583\pm 0.005$ | $7.590\pm 0.006$ | $-0.62\pm 0.01$ | $-0.60\pm 0.02$
5665.5545* | $40.850\pm 0.003$ | $42.100\pm 0.003$ | $7.589\pm 0.004$ | $7.600\pm 0.005$ | $-0.36\pm 0.02$ | $-0.40\pm 0.04$
5684.4840* | $66.447\pm 0.003$ | $65.793\pm 0.003$ | $7.560\pm 0.006$ | $7.566\pm 0.007$ | $-0.70\pm 0.01$ | $-0.89\pm 0.01$
5690.4250* | $54.123\pm 0.003$ | $54.875\pm 0.003$ | $7.566\pm 0.006$ | $7.581\pm 0.005$ | $-0.83\pm 0.01$ | $-1.12\pm 0.01$
5701.1040* | $40.941\pm 0.004$ | $41.240\pm 0.005$ | $7.554\pm 0.006$ | $7.554\pm 0.007$ | $-0.67\pm 0.01$ | $-0.81\pm 0.02$
5708.3995 | $83.932\pm 0.001$ | $81.971\pm 0.002$ | $7.547\pm 0.003$ | $7.557\pm 0.004$ | $-0.61\pm 0.01$ | $-0.79\pm 0.01$
5772.1460 | $58.461\pm 0.003$ | $58.254\pm 0.003$ | $7.578\pm 0.005$ | $7.590\pm 0.004$ | $-0.31\pm 0.02$ | $-0.26\pm 0.25$
5780.3838 | $34.914\pm 0.004$ | $38.252\pm 0.005$ | $7.618\pm 0.007$ | $7.664\pm 0.007$ | $+1.94\pm 0.02$ | $+2.49\pm 0.00$
5793.0726* | $47.079\pm 0.003$ | $47.932\pm 0.004$ | $7.553\pm 0.005$ | $7.565\pm 0.006$ | $+0.27\pm 0.02$ | $+0.49\pm 0.02$
5797.8559 | $44.799\pm 0.003$ | $46.259\pm 0.002$ | $7.476\pm 0.005$ | $7.497\pm 0.003$ | $-0.67\pm 0.01$ | $-0.48\pm 0.03$
5948.5410 | $97.658\pm 0.002$ | $94.248\pm 0.003$ | $7.581\pm 0.005$ | $7.598\pm 0.006$ | $-0.60\pm 0.01$ | $-0.82\pm 0.01$
6125.0209 | $34.496\pm 0.005$ | $34.737\pm 0.006$ | $7.517\pm 0.006$ | $7.529\pm 0.007$ | $-0.58\pm 0.02$ | $-0.53\pm 0.03$
6142.4832 | $39.323\pm 0.004$ | $38.765\pm 0.006$ | $7.408\pm 0.006$ | $7.413\pm 0.007$ | $-1.10\pm 0.01$ | $-1.12\pm 0.02$
6145.0159 | $44.040\pm 0.004$ | $42.187\pm 0.005$ | $7.487\pm 0.006$ | $7.479\pm 0.006$ | $-1.17\pm 0.01$ | $-1.36\pm 0.01$
6237.3191 | $81.782\pm 0.002$ | $76.779\pm 0.003$ | $7.468\pm 0.003$ | $7.462\pm 0.003$ | $-1.23\pm 0.01$ | $-1.39\pm 0.01$
6243.8146 | $57.363\pm 0.003$ | $54.134\pm 0.004$ | $7.574\pm 0.004$ | $7.567\pm 0.005$ | $-0.96\pm 0.01$ | $-1.13\pm 0.01$
6244.4655 | $54.674\pm 0.003$ | $52.042\pm 0.004$ | $7.390\pm 0.005$ | $7.388\pm 0.005$ | $-0.98\pm 0.01$ | $-1.15\pm 0.01$
6976.5129 | $56.438\pm 0.003$ | $52.806\pm 0.003$ | $7.654\pm 0.003$ | $7.648\pm 0.004$ | $-1.22\pm 0.01$ | $-1.22\pm 0.01$
7003.5690 | $75.507\pm 0.002$ | $68.469\pm 0.003$ | $7.583\pm 0.003$ | $7.569\pm 0.005$ | $-1.43\pm 0.01$ | $-1.39\pm 0.01$
7034.9006 | $83.976\pm 0.002$ | $76.443\pm 0.003$ | $7.575\pm 0.003$ | $7.565\pm 0.004$ | $-1.13\pm 0.01$ | $-1.28\pm 0.01$
7226.2079 | $43.345\pm 0.004$ | $42.227\pm 0.004$ | $7.587\pm 0.005$ | $7.597\pm 0.006$ | $-0.24\pm 0.02$ | $-1.47\pm 0.05$
7405.7718 | $110.020\pm 0.002$ | $101.650\pm 0.002$ | $7.634\pm 0.004$ | $7.655\pm 0.004$ | $-1.31\pm 0.00$ | $-1.27\pm 0.01$
7415.9480 | $105.090\pm 0.002$ | $99.125\pm 0.002$ | $7.558\pm 0.004$ | $7.590\pm 0.005$ | $-0.83\pm 0.01$ | $-0.87\pm 0.01$
7680.2660* | $102.790\pm 0.002$ | $95.068\pm 0.002$ | $7.669\pm 0.004$ | $7.698\pm 0.005$ | $-0.93\pm 0.01$ | $-0.92\pm 0.01$
7918.3835 | $103.440\pm 0.002$ | $97.816\pm 0.002$ | $7.690\pm 0.003$ | $7.723\pm 0.004$ | $-0.82\pm 0.01$ | $-0.85\pm 0.01$
7932.3479 | $127.030\pm 0.002$ | $117.070\pm 0.002$ | $7.689\pm 0.004$ | $7.706\pm 0.004$ | $-0.61\pm 0.01$ | $-0.81\pm 0.01$
10288.9440* | $88.692\pm 0.001$ | $85.213\pm 0.001$ | $7.522\pm 0.002$ | $7.538\pm 0.003$ | $-0.37\pm 0.01$ | $-0.39\pm 0.01$
10371.2630 | $199.010\pm 0.001$ | $189.420\pm 0.001$ | $7.621\pm 0.002$ | $7.668\pm 0.003$ | $-0.63\pm 0.00$ | $-1.08\pm 0.00$
10603.4250 | $295.370\pm 0.001$ | $281.120\pm 0.001$ | $7.686\pm 0.002$ | $7.742\pm 0.002$ | $-1.24\pm 0.00$ | $-1.88\pm 0.00$
10689.7160 | $228.360\pm 0.001$ | $205.700\pm 0.001$ | $7.712\pm 0.003$ | $7.741\pm 0.002$ | $-0.91\pm 0.00$ | $-1.28\pm 0.00$
10694.2510 | $251.900\pm 0.001$ | $230.870\pm 0.001$ | $7.679\pm 0.002$ | $7.727\pm 0.002$ | $-1.15\pm 0.00$ | $-1.51\pm 0.00$
10749.3780 | $330.400\pm 0.000$ | $308.620\pm 0.001$ | $7.677\pm 0.001$ | $7.713\pm 0.002$ | $-1.44\pm 0.00$ | $-2.08\pm 0.03$
10784.5620 | $106.910\pm 0.001$ | $98.584\pm 0.001$ | $7.645\pm 0.002$ | $7.664\pm 0.002$ | $-0.54\pm 0.01$ | $-0.72\pm 0.01$
10786.8490 | $296.440\pm 0.001$ | $278.200\pm 0.001$ | $7.651\pm 0.001$ | $7.687\pm 0.002$ | $-1.29\pm 0.00$ | $-1.98\pm 0.03$
10827.0880 | $473.350\pm 0.000$ | $433.190\pm 0.001$ | $7.662\pm 0.001$ | $7.678\pm 0.002$ | $-2.32\pm 0.01$ | $-1.97\pm 0.02$
12390.1540* | $89.770\pm 0.001$ | $87.503\pm 0.001$ | $7.615\pm 0.002$ | $7.613\pm 0.003$ | $-0.11\pm 0.03$ | $-0.29\pm 0.83$
12395.8320* | $111.000\pm 0.001$ | $106.810\pm 0.001$ | $7.613\pm 0.002$ | $7.609\pm 0.003$ | $+0.08\pm 0.03$ | $-0.38\pm 0.02$
Si II |  |  |  |  |  |
6347.1087 | $55.354\pm 0.002$ | $48.835\pm 0.002$ | $7.675\pm 0.005$ | $7.547\pm 0.005$ | $+0.91\pm 0.01$ | $+1.43\pm 0.01$
6371.3714* | $37.936\pm 0.004$ | $32.226\pm 0.006$ | $7.610\pm 0.010$ | $7.469\pm 0.011$ | $+0.59\pm 0.02$ | $+1.25\pm 0.03$

444 $w_{\mathrm{K}}$ is the width of the broadening kernel. The old and new
$\log{gf}$ values are also provided. An asterisk next to the wavelength
signifies the line was in the chosen subsample.

###  4.5 Comparisons with meteoritic abundances

Type-I carbonaceous chondrites (CI chondrites) constitute a special class of
meteorites. Their chemical composition of refractory elements is believed to
reflect the composition of the early solar system (e.g., Lodders et al.,
2009). Conventionally, meteoritic abundances are given relative to silicon on
the so-called cosmochemical scale, here for an element X written as

$\mu_{\mathrm{X}}\equiv 10^{6}\times\frac{n_{\mathrm{X}}}{n_{\mathrm{Si}}},$
(4)

where $n_{\mathrm{X}}$ denotes the number density per volume of element X. In
this paper, we gave abundances on the astronomical scale defined by

$\varepsilon_{\mathrm{X}}\equiv
10^{12}\times\frac{n_{\mathrm{X}}}{n_{\mathrm{H}}}\,.$ (5)

With these definitions we have555$\lg\equiv\log_{10}$
$\lg\mu_{\mathrm{Si}}=6$, and $\lg\varepsilon_{\mathrm{H}}=12$. Conversion
from the abundance of an element X from the astronomical to the cosmochemical
scale reads

$\lg\mu_{\mathrm{X}}=\lg\varepsilon_{\mathrm{X}}-\lg\varepsilon_{\mathrm{Si}}+6$
(6)

which necessitates the knowledge of the silicon abundance on the astronomical
scale. Equation (6) shows that an increase of the silicon abundance – as found
in this work in comparison to earlier results – leads to a corresponding
decrease of an abundance given on the cosmochemical scale. Since the abundance
on the cosmochemical scale involves two abundances on the astronomical scale
the conversion generally leads to an increase of the resulting uncertainty for
an abundance – assuming that there are no significant correlations among the
individual input uncertainties. Dealing with a ratio of two metal abundances
here has the advantage that settling effects over the lifetime of the Sun
should cancel out to first order (Lodders et al., 2009), and we can directly
compare to the corresponding meteoritic ratios. In Fig. 10 we compare these
abundance ratios on the cosmochemical scale in ascending order of the $50\%$
condensation temperatures of the elements. All condensation temperatures are
from Palme et al. (2014) and meteoritic abundances are taken from Lodders
(2021, her Tab. 3, present solar system). We use the solar photospheric
abundances of non-volatile elements based on CO5BOLD models presented in
Caffau et al. (2011). The uncertainties indicated in Fig. 10 are dominated by
the uncertainties of the spectroscopically determined photospheric abundances.
The uncertainty of the silicon abundance noticeably contributes here. The
error bars are perhaps over-estimated since some compensatory effects due to
error correlations might be present. With the exception of hafnium (a well-
known problem case) meteoritic and photospheric abundances are consistent with
each other on the $1\,\sigma$ level. However, for being able to identify
possible differences a reduction of the uncertainties appears desirable.

###  4.6 Mass fractions of hydrogen, helium, and metals

In this section we calculate the mass fractions of hydrogen $\mathrm{X}$,
helium $\mathrm{Y}$, and metals $\mathrm{Z}$ which are of particular interest
for stellar structure. Our intention is not so much to provide the absolute
numbers but rather to demonstrate the involved uncertainties. Serenelli et al.
(2016) and Vinyoles et al. (2017) both advocate the use of meteoritic
abundances for elements heavier than C, N, O in the Sun. We follow this idea,
and augment the 12 photospheric abundances from Caffau et al. (2011) (Li, C,
N, O, P, S, K, Fe, Eu, Os, Hf, Th) and the newly derived silicon abundance by
the meteoritic abundances given by Lodders (2021). To this end, we need to
bring photospheric and meteoritic abundances onto the same scale. For relating
the abundances we use the abundance of silicon only. Using Eq. (5) we obtain

$\lg\varepsilon^{\diamond}_{\mathrm{X}}=\lg\mu^{\diamond}_{\mathrm{X}}+\lg\varepsilon^{\odot}_{\mathrm{Si}}-6\hskip
20.00003pt\mbox{or}\hskip
20.00003pt\varepsilon^{\diamond}_{\mathrm{X}}=\frac{n^{\diamond}_{\mathrm{X}}}{n^{\diamond}_{\mathrm{Si}}}\,\varepsilon^{\odot}_{\mathrm{Si}},$
(7)

where diamonds ($\diamond$) indicate meteoritic values, the Sun symbol
($\odot$) solar photospheric values. The basic assumption underlying Eq. (7)
is that
$\varepsilon^{\diamond}_{\mathrm{Si}}=\varepsilon^{\odot}_{\mathrm{Si}}$, that
is to say, silicon is not subject to differentiation effects, neither in the
solar photosphere, nor in the meteorites.

Equation (7) shows that in the conversion of the meteoritic abundances we have
to take into account the uncertainties of the individual species including the
silicon abundance as measured in meteorites. The normalisation puts the
meteoritic silicon abundance on a value of $10^{6}$, however, with an
uncertainty of 3.4 %. The uncertainty of the solar silicon abundance
contributes to the meteoritic uncertainties in the conversion to the
astronomical scale. An input neither directly obtained from meteorites nor
from photospheric spectroscopy is the helium abundance. Here, we assume a
ratio $n^{\odot}_{\mathrm{He}}/n^{\odot}_{\mathrm{H}}=(8.38\pm 0.39)\times
10^{-2}$ as given by Lodders (2021) for the present Sun which is motivated
from helioseismic measurements. With these ingredients and atomic weights
assumed to be precisely known we ran Monte-Carlo error propagations and
obtained $\mathrm{X}=0.7382\pm 0.0084$, $\mathrm{Y}=0.2456\pm 0.0086$,
$\mathrm{Z}=0.0162\pm 0.0015$, and $\mathrm{Z}/\mathrm{X}=0.0220\pm 0.0020$
(independent of the abundance of helium). The uncertainties of $\mathrm{X}$
and $\mathrm{Y}$ are dominated by the uncertainty of the helium abundance. In
order to further quantify the uncertainties of $\mathrm{Z}$ we present the
build-up of the overall uncertainty of $\mathrm{Z}$ by sequentially adding
sources of uncertainty. We obtain the sequence
$\sigma_{\mathrm{Z}}=(65,77,79,148)\times 10^{-5}$ when adding the
uncertainties of the meteoritic abundances, of the photospheric abundance of
silicon, of all photospheric abundances except CNO, and of all contributions
including from CNO, respectively. This perhaps odd-appearing procedure is
motivated by the fact that the individual sources of uncertainty are not
simply additive (rather additive in quadrature), and there is some coupling
emergent from the presence of the photospheric silicon abundance in the
conversion given by Eq. (7). Keeping this limitation in mind, we see that the
contribution to the overall uncertainty by the meteorites is not
insignificant, and noticeably increased by the uncertainty of the photospheric
silicon abundance. We emphasise that the “meteoritic” abundances include the
noble gases (particularly neon). Their assumed values are coming from
measurements other than those in meteorites. The photospheric abundances other
than of CNO contribute little to the overall uncertainty, the most important
contribution stems – perhaps unsurprisingly – from CNO. We conclude that the
procedure of using the average of several refractory elements (e.g., Lodders
et al., 2009) is a good way to reduce the overall uncertainty on the mass
fraction of metals. The precision to which the CNO elements are measured in
the solar photosphere is most important. Beyond that, any reduction of the
uncertainty of photospheric or meteoritic abundances is helpful.

Figure 10: Elemental abundance differences between CI chondrites
($\mu_{X}^{\diamond}$) and solar photospheric composition ($\mu_{X}^{\odot}$)
according to CO5BOLD in ascending order of the condensation temperatures of
the elements. The silicon abundance difference is zero by definition, but
shown to illustrate the scale of the error.

##  5 Discussion

###  5.1 Comparisons between models and line samples

Both the choice of model atmosphere and the line sample affect the final
abundance (and the level of broadening or debroadening required). Across all
configurations, prior broadening of the observations increases the fitted
abundance. Moreover, there is a spread of fitted abundances for even a single
model based on the chosen line sample and fitting method. Overall, it is the
combination of more or less judicious decisions taken during the fitting
process that ultimately determines the final fitted abundance value.

###  5.2 Comparisons with previous works

Our derived abundance of $7.57\pm 0.04$ is $0.06$ dex higher than the
abundance recently presented in Asplund et al. (2021). As a comparison to
their work, we calculated a mean abundance using the line sample they
presented (leaving out the $6741.61$ Å line) alongside their weights, and find
an abundance of $7.55\pm 0.02$, which is $0.04$ dex higher than that presented
in their work. Part of the difference in fitted abundance stems from the new
oscillator strength data used; when using the same oscillator strength data,
we find an abundance of $7.54\pm 0.03$. This is consistent with the average
difference of the oscillator strength data used for these lines. The
equivalent widths obtained by our line profile fitting for all models are
higher than those given in Amarsi & Asplund (2017), and are shown in Table 6.
The corresponding differences in abundance are shown in Fig. 11. Altogether,
the dominant difference comes from the new oscillator strength data ( $0.04$
dex) and equivalent widths ( $0.02$ dex), while line selection, weighting and
3D model are of secondary importance ( $0.01$ dex).

Table 4:  Equivalent widths for the four models presented in this study
alongside those given in Amarsi & Asplund (2017) (AA17).

Wavelength | msc600 | b000 | b200 | n59 | m595 | AA17
---|---|---|---|---|---|---
[$\AA$] | [m$\AA$] | [m$\AA$] | [m$\AA$] | [m$\AA$] | [m$\AA$] |
Si I |  |  |  |  |  |
5645.6128 | 37.4 | 37.4 | 37.5 | 37.4 | 37.4 | 35.0
5684.4840 | 66.5 | 67.0 | 67.0 | 66.5 | 66.5 | 63.7
5690.4250 | 54.2 | 54.2 | 54.6 | 54.2 | 54.1 | 52.6
5701.1040 | 40.9 | 40.9 | 41.1 | 40.9 | 40.9 | 39.5
5772.1460 | 58.8 | 58.8 | 59.4 | 58.8 | 58.8 | 56.0
5793.0726 | 47.4 | 47.4 | 47.9 | 47.4 | 47.4 | 45.8
7034.9006 | 83.4 | 83.4 | 84.6 | 83.5 | 83.2 | 74.0
7226.2079 | 43.4 | 43.4 | 43.7 | 43.4 | 43.4 | 38.7
Si II |  |  |  |  |  |
6371.3714 | 38.1 | 37.6 | 37.6 | 37.6 | 38.3 | 36.6

666 Our higher equivalent widths lead to a $+0.02$ dex increase in abundance
across all cases (using the weighting of AA17).

Figure 11:  Differences in equivalent width (top) and fitted abundance
(bottom) in comparison to the values in Amarsi & Asplund (2017). The numbers
in the bottom panel give the mean abundance difference using the weighting
scheme of Amarsi & Asplund (2017). The difference in abundance includes all
sources, not just the difference in equivalent width.

###  5.3 Uncertainties

We use the root-mean-squared error (RMSE) of the selected sample to capture
the final uncertainty on the fitted abundance. This uncertainty represents the
uncertainties through the entire fitting procedure, including uncertainties in
oscillator strengths as well as statistical uncertainties. The RMSE is given
by

$\mathrm{RMSE}=\sqrt{\frac{1}{N}\sum_{i}^{L}{(y_{i}-\hat{y})^{2}}},$ (8)

where $\hat{y}$ is the weighted mean abundance, $y_{i}$ is the weighted
abundance calculated for a single line, $L$ is the number of lines used to
determine the weighted mean and $N$ is the sum of the weights. We chose this
estimator since each line yields a different fitted abundance, and the RMSE
naturally incorporates this scatter. Additionally, since we compute the RMSE
on the final fitted abundances, the uncertainties in fitting and in the
oscillator strengths are already represented in the scatter. On average, the
oscillator strength uncertainty is $8.57\%$, which is compatible with our RMSE
uncertainty in abundance. For other fitted quantities, we use the statistical
uncertainties from the fitting procedure and other relevant sources of error,
such as uncertainties in ABO theory parameters for broadening.

Despite careful line selection, new oscillator strength values and an improved
broadening theory, there is still a substantial scatter in individual fitted
abundance values. The scatter is included in the uncertainties by means of the
RMSE. The (maximum - minimum) scatter in abundance values is $0.15$ dex for
our main configuration, with an RMSE of $0.04$. Using the line list of Amarsi
& Asplund (2017) decreases the scatter to $0.06$ dex and the RMSE to $0.02$.
Again, this shows that the choice of lines to use has a non-negligible impact
on the final derived abundance.

We make use of the ‘mpfitcovar’ routine (Markwardt, 2009) which takes in the
spectrum and correlated noise model and fits the free parameters of our
routine (normally abundance, broadening and wavelength shift). We use
$\chi^{2}$ statistics to find the best fitting parameter as well as the errors
on those parameters. There is some correlation between the quantities
themselves, shown in Fig. 12 for the msc600 model.

Figure 12: Correlation coefficient between fitted parameters for the msc600
model. $\Delta\lambda$ is the wavelength shift, $\log\varepsilon_{\text{Si}}$
is the abundance and $w_{\mathrm{K}}$ is the width of the broadening kernel.

##  6 Conclusions

We have presented a 3D LTE analysis of 39 silicon lines using CO5BOLD model
atmospheres and the LINFOR3D spectral synthesis code. Of these, a total of
$11$ were selected for the abundance analysis, comprising of 7 optical Si I
lines, 3 near-infrared Si I lines and 1 Si II line. New oscillator strengths
from Pehlivan Rhodin (2018) were used, enabling the use of infrared lines
alongside optical ones and providing smaller uncertainties for oscillator
strengths. Compared to the previous experimental strengths from Garz (1973),
the new $\log(gf)$ values and weighting scheme decrease the formal statistical
uncertainty across the relevant lines from $0.07$ dex to $0.04$ dex. An
improved broadening theory also helped to constrain statistical uncertainties
further.

Our main conclusions are as follows:

  * •

We find a photospheric solar silicon abundance of
$\log\varepsilon_{\text{Si}}$ = $7.57\pm 0.04$, including the $-0.01$ dex
correction from NLTE effects investigated in Amarsi & Asplund (2017). The
$0.06$ dex increase with respect to the recent studies by Asplund et al.
(2021), Amarsi & Asplund (2017) suggests that the determination of the solar
silicon abundance is not yet a firmly solved problem. Our advocated
configuration uses the $G_{1}$ broadening kernel, the higher resolution msc600
model and our chosen subsample of lines.

  * •

Several factors affect the fitted abundance and broadening, but the line
selection plays the primary role. We focus on lines that are devoid of major
blends, have updated oscillator strengths, and also where syntheses match well
with observed line shapes. Notably, the near-infrared lines give higher
abundances than optical lines on average.

  * •

The over-broadened line syntheses we see in this work are not specific to the
CO5BOLD atmospheres. Comparisons were made with STAGGER \+ BALDER (see
Appendix A for details), and we are able to refit their abundances and
broadening values using both the msc600 and n59 models. Broadening-wise, their
syntheses lie between the msc600 and n59 models described here. Additionally,
the overly broadened line syntheses are caused by the combination of various
effects, including velocity fields, atomic broadening and neglect of magnetic
field effects – we did not find a single definite cause for over-broadening.

  * •

Using a magnetic model with a magnetic field strength of $200$ G increases the
fitted abundance and reduces the Doppler broadening when compared to the same
model with a magnetic field strength of $0$ G. These differential results
could point towards over-broadened syntheses resulting from a lack of
consideration of magnetic field effects, particularly that strong magnetic
fields impede turbulent flow in the atmosphere (Cattaneo et al., 2003),
resulting in narrower lines. However, the vertical magnetic fields that would
be required to offset the negative broadening are much too large to be
consistent with 3D MHD models with small-scale dynamos (Shchukina et al.,
2015, 2016), meaning magnetic field effects likely do not play a major role in
the overbroadening of the line syntheses.

  * •

The atomic broadening cross-sections for the silicon lines presented in this
work are remarkably large, which subsequently increases the effect of
collisional broadening. Alongside the effects of magnetic fields, the
collisional broadening may then also contribute to the over-broadening in the
line syntheses.

  * •

Meteoritic abundances when transformed onto the astronomical scale are
increased with respect to the previous study by Palme et al. (2014) due to the
increase in our photospheric silicon abundance. The differences of the non-
volatile elements available from CO5BOLD-based analyses are – except for
hafnium – consistent with zero, however, with significant uncertainties.
Serenelli et al. (2016) and Vinyoles et al. (2017) both advocate the use of
meteoritic abundances for elements heavier than C, N, O in the Sun, but using
only the silicon abundance for the conversion between cosmochemical and
astronomical scale (e.g., Asplund et al., 2005) would give a $0.05$ dex
increase for non-volatile metals. The sizeable uncertainty of the silicon
abundance found in this study lets the use of multiple elements for
referencing meteoritic and photospheric abundances appear attractive, such as
done in Lodders et al. (2009).

  * •

A local fit of the continuum level is clearly at odds when looking at the
spectrum on a large scale. Performing such a fit results in an artificial
lowering of the equivalent width, and systematically lowers the abundance by
$0.01$ dex.

  * •

We find no strong evidence of NLTE effects severely affecting abundance
calculations in optical lines and the chosen infrared lines beyond the $-0.01$
dex correction included, though the negative broadening required could be
indicative of minor NLTE effects being present. Appendix B explains the
observed NLTE effects in 1D in more detail.

  * •

An NLTE solar silicon abundance of $7.57\pm 0.04$ could improve the
differences for solar neutrino fluxes, sound speed profiles and the surface
helium fraction. Vinyoles et al. (2017) show that a solar model with the
composition proposed in Grevesse & Sauval (1998) statistically performs better
in regard to the solar sound speed profile than a model with a solar
composition proposed in (Asplund et al., 2009). The former composition uses
$\log\varepsilon_{\text{Si}}=7.56\pm 0.01$, while the latter uses
$\log\varepsilon_{\text{Si}}=7.51\pm 0.01$. Serenelli et al. (2016) show that
the silicon abundance of $\log\varepsilon_{\text{Si}}=7.82$ from von Steiger &
Zurbuchen (2016) gives worse fits overall for solar neutrino fluxes, sound
speed profiles, and the surface helium fraction, so a silicon abundance of
$7.57\pm 0.04$, closer to the abundance derived from the Grevesse & Sauval
(1998) composition, clearly results in an improvement relative to (Asplund et
al., 2009).

All in all, our analysis suggests that the photospheric solar Si abundance is
not yet a definitively solved problem. The use of state-of-the-art 3D model
atmospheres and an improved broadening theory is essentially a requirement to
accurately reproduce line shapes. Even with these improvements, our synthetic
line profiles were generally overbroadened with respect to the observations,
and it is unlikely that this feature is unique to CO5BOLD model atmospheres or
to Si lines.

###### Acknowledgements.

S.A.D. and H.G.L. acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
138713538 – SFB 881 (“The Milky Way System”, subproject A04). A.K.
acknowledges support of the ”ChETEC” COST Action (CA16117) and from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 101008324 (ChETEC-INFRA). P.S.B acknowledges support from the
Swedish Research Council through individual project grants with contract Nos.
2016-03765 and 2020-03404. This work has made use of the VALD database,
operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and
the University of Vienna. The authors are grateful for the assistance of A. M.
Amarsi in this work, especially for comparing differences in model
atmospheres, to L. Mashonkina for a discussion on NLTE effects, and to H.
Hartman for providing assistance related to the new oscillator strengths.

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##  Appendix A Comparisons between LINFOR3D/CO5BOLD and BALDER/STAGGER

There was motivation to investigate whether the extra broadening present in
the syntheses produced by LINFOR3D using the CO5BOLD model atmospheres was
unique to these codes. To test this, we compared six LTE Si I line syntheses
using our models against those produced by BALDER (Amarsi et al. 2018), a
custom version of Multi3D (Leenaarts & Carlsson 2009) (data provided by A.
Amarsi, priv. comm.). These data were calculated on a 3D hydrodynamic STAGGER
model solar atmosphere (Collet et al. 2011). These lines were: $5690$, $5780$,
$5793$, $6244$, $6976$ and $7680$ Å.

We use the same $\log{gf}$ values, ABO parameters and central wavelengths for
the lines in both sets of syntheses. After fitting for the abundance
($\log\varepsilon_{\text{Si}}=7.51$ was assumed in the BALDER syntheses), the
results are near-identical, and the fits shown in Figs. 13 & 15 using the n59
and 600 models are excellent. We are able to refit the abundance and
broadening of the BALDER syntheses. The black points represent the BALDER
synthesis, and the red lines are the LINFOR3D syntheses after fitting the
lines synthesised with BALDER.

Table 5:  Fitted broadening values for six lines chosen as a comparison to
STAGGER.

| n59 – | msc600 – | n59 – | msc600 –
---|---|---|---|---
| Stagger | Stagger | Hamburg | Hamburg
$\lambda$ | $w_{\mathrm{K}}$ | $w_{\mathrm{K}}$ | $w_{\mathrm{K}}$ | $w_{\mathrm{K}}$
[$\AA$] | [km s-1] | [km s-1] | [km s-1] | [km s-1]
5690.4250 | $0.49$ | $-0.33$ | $-0.54$ | $-0.83$
5780.3838 | $0.45$ | $-0.40$ | $1.47$ | $1.40$
5793.0726 | $0.47$ | $-0.36$ | $0.72$ | $0.27$
6244.4655 | $0.34$ | $-0.47$ | $-0.79$ | $-0.98$
6976.5129 | $0.43$ | $-0.38$ | $-1.05$ | $-1.23$
7680.2660 | $0.57$ | $-0.08$ | $-0.38$ | $-0.69$
777 The values come from fitting the n59 and msc600 models fitting to the
STAGGER models (first two columns) and to the Hamburg atlas observations (last
two columns). In all cases, the n59 model is less broad than the STAGGER model
syntheses while the msc600 model is broader. However, since the n59 model
generally also requires significant negative broadening to fit the
observations, the STAGGER models must also yield too broad lines compared to
the observations.

Two CO5BOLD model atmospheres were used for the analysis. The msc600 solar
model is a newer model that has a higher spatial resolution and larger box
size than the older n59 model (see Table 2 for details). The n59 model
requires positive broadening to fit the BALDER syntheses, while the msc600
requires negative broadening. Broadening-wise, then, the BALDER syntheses lie
somewhere between our two chosen models - they are broader than the n59 model,
but not as broad as the msc600 model. Table 7 shows the broadening values
required by our models to reproduce the STAGGER syntheses and the observations
from the Hamburg atlas. While the n59 model is less broad than the Stagger
model in all cases and the msc600 model is broader, both of these models still
generally require negative broadening to fit a majority of the lines. Again,
no instrumental broadening (typically $\sim 1$ km s-1 FWHM) was applied to the
synthetic spectra, as should be possible without the presence of over-
broadening. This shows the issue of the syntheses being over-broadened is not
unique to CO5BOLD \+ LINFOR3D but is also present in the syntheses produced by
BALDER \+ STAGGER.

Figure 13:  LINFOR3D \+ CO5BOLD n59 model syntheses (red lines) fit to BALDER
\+ STAGGER (black points). Green lines show residuals increased by a factor of
10. The mean abundance as derived by the CO5BOLD fit is $7.52\pm 0.01$. Figure
14:  LINFOR3D \+ CO5BOLD m595 model model syntheses (red lines) fit to BALDER
\+ STAGGER (black points). Green lines show residuals increased by a factor of
10. The mean abundance as derived by the CO5BOLD fit is $7.52\pm 0.01$. Figure
15:  LINFOR3D \+ CO5BOLD msc600 model model syntheses (red lines) fit to
BALDER \+ STAGGER (black points). Green lines show residuals increased by a
factor of 10. The mean abundance as derived by the CO5BOLD fit is $7.51\pm
0.01$.

##  Appendix B 1D LTE versus NLTE comparisons in LINFOR3D

Following the comparison between model atmospheres and spectral synthesis
routines, four lines were synthesised in both 1D LTE and 1D NLTE as an
additional investigation into the potential extra broadening seen in LINFOR3D.
These lines were $5772.46$, $5948.54$, $7680.27$ and
$12288.15\leavevmode\nobreak\ \mathrm{\AA}$. We fit the 1D NLTE line profiles
with the 1D LTE syntheses for nine different abundance values to investigate
whether the consequent fitted broadening is strongly affected. The abundance
corrections generally remain within the prescribed $-0.01$ dex; the
$12288.28\leavevmode\nobreak\ \mathrm{\AA}$ requires slightly higher
corrections at higher abundances. The employed model atom is the same as that
in Wedemeyer (2001) with $115$ energy levels for Si I and Si II with $84$
transitions. Moreover, the collisional cross-sections for neutral particles
follow Drawin (1967) and Steenbock & Holweger (1984) using a correction factor
of $S_{\mathrm{H}}=0.1$. Fig. 16 shows the fitted broadening at each abundance
point for each line, and Fig. 17 shows the LTE versus NLTE line profiles. In
this small sample, NLTE effects become stronger with increasing abundance
resulting in stronger negative broadening for the $5772.46\leavevmode\nobreak\
\mathrm{\AA}$, $5948.54\leavevmode\nobreak\ \mathrm{\AA}$ and
$12288.15\leavevmode\nobreak\ \mathrm{\AA}$ lines, and less positive
broadening for the $7680.27\leavevmode\nobreak\ \mathrm{\AA}$ line. It should
be noted that the trends are reversed if we fit NLTE to LTE profiles, i.e.
lines that showed negative broadening would instead show positive broadening.
The three lines with a negative NLTE correction also require negative
broadening, suggesting that part of the overly broadened line profiles could
be attributed to NLTE effects. We also found that removing the core of the
line (within $\pm 3$ km s-1) removes the necessity for negative broadening,
illustrating that the NLTE effects are concentrated in the cores of these
lines.

Figure 16: Fitted broadening values for four lines, fitting 1D LTE to 1D NLTE
profiles. Larger NLTE effects are present in $5948.65\leavevmode\nobreak\
\mathrm{\AA}$ and $12228.15\leavevmode\nobreak\ \mathrm{\AA}$ lines than the
$5772.46\leavevmode\nobreak\ \mathrm{\AA}$ and $7680.27\leavevmode\nobreak\
\mathrm{\AA}$ lines. Figure 17:  LTE (points) versus NLTE (solid lines)
profiles for three representative abundance values. The
$5948.54\leavevmode\nobreak\ \mathrm{\AA}$ and $12288.15\leavevmode\nobreak\
\mathrm{\AA}$ show visible NLTE effects in the core that become stronger with
increasing abundance.

##  Appendix C De-broadening of spectral lines

Mathematically, the effect of broadening is described as convolution and de-
broadening as deconvolution. It is well-known that deconvolution is an ill-
posed problem. A robust solution to a deconvolution problem can only be
obtained by suitable regularisation. The problem becomes immediately apparent
when considering a Gaussian (the “kernel”) with which a spectrum is to be de-
convolved. In Fourier space, deconvolution corresponds to a division with the
Fourier transform of the kernel, which is again a Gaussian in this case. The
rapid decline of the wings of a Gaussian leads to a division by almost zero at
distances of a few widths of the Gaussian away from its centre. Any numerical
or physical imprecision leads to large disturbance under such circumstances.
This means even noise-free synthetic line profiles cannot be de-convolved by a
naive division of the raw spectrum by the Fourier transform of a Gaussian
kernel.

The key question is now whether one can suitably regularise the problem or
reduce the level of noise amplification. We followed the latter approach by
seeking a different kernel with less steeply falling wings than a Gaussian. As
starting point we chose the kernel function

$G_{1}(x)=\frac{\alpha}{2}e^{-\alpha\,|x|},$ (9)

normalised to one according to

$\int_{-\infty}^{\infty}dx\,\,G_{1}(x)=1\,.$ (10)

The parameter $\alpha$ controls the width of the kernel. The kernel is
symmetric, and has a peaked shape at the origin. Later, we shall discuss
higher powers in terms of convolutions of the kernel with itself. We index the
resulting kernels by the number of convolutions, which in the present case is
one. Convolving a function $f$ with kernel $G_{1}$ results in a function $g$
according to

$g(x)=\int_{-\infty}^{\infty}dx^{\prime}\,G_{1}(x^{\prime}-x)\,f(x^{\prime})\,.$
(11)

Our choice of the kernel function was inspired by work of Dorfi & Drury
(1987). The authors pointed out that the above kernel is the Green’s function
associated with the differential operator

$\mathbf{1}-\frac{1}{\alpha^{2}}\frac{d^{\,2}}{dx^{2}}\,,$ (12)

where $\mathbf{1}$ indicates the identity operator. This allows one to
formulate the convolution expressed by Eq. 11 as solution of a ordinary
differential equation of second order. Discretising the differential operator,
as well as the functions $f$ and $g$ (simplest on an equidistant $x$-grid),
results in a set of linear equations of the form

$\mathbf{A}\,g=f$ (13)

where $\mathbf{A}$ is a tri-diagonal matrix which can be inverted efficiently.
Remarkably, in this formulation a deconvolution appears even simpler than
convolution: if $g$ is given, a simple matrix multiplication with matrix
$\mathbf{A}$ yields the de-convolved function $f$. It was this feature that
made us choose $G_{1}$ as kernel for deconvolution, and for consistency, also
for convolution. Using the same kernel when convolving a line profile has the
advantage that during fitting there is a continuous transition from
convolution to deconvolution and vice versa. This improves the stability of
the fitting operation when having to deal with a situation where the
broadening or de-broadening is around zero. It should be noted that the $G$
kernels are in fact the Matérn functions with half-numbered indices (Genton
2002).

One may question the suitability of the function $G_{1}$ for describing
broadening or de-broadening effects, and might prefer a more Gaussian-shaped
kernel. The Central Limit Theorem states that repeated convolution of a
function with itself (subject to certain regularity conditions) approaches a
Gaussian. We used this property to construct further kernel functions by
convolving $G_{1}$ with itself. We obtained the following sequence of
functions

$G_{2}(x)\equiv G_{1}^{2}(x)=\frac{\alpha}{4}e^{-\alpha\,|x|}(1+\alpha\,|x|),$
(14) $G_{3}(x)\equiv
G_{1}^{3}(x)=\frac{\alpha}{16}e^{-\alpha\,|x|}\left[3(1+\alpha\,|x|)+\alpha^{2}\,|x|^{2}\right],$
(15) $G_{4}(x)\equiv
G_{1}^{4}(x)=\frac{\alpha}{96}e^{-\alpha\,|x|}\left[15(1+\alpha\,|x|)+\alpha^{2}\,|x|^{2}(6+\alpha\,|x|)\right].$
(16)

All functions are normalised to one according to Eq. 10. The possibility of
formulating the convolution operation in Fourier space let it appear desirable
to have the Fourier transforms of the $G_{n}$ function at hand. Defining the
Fourier transform via

$\widehat{G}(k)\equiv\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}G(x)\,\exp(ikx)\,dx$
(17)

we obtained for the (purely real) transforms of the kernel functions

$\widehat{G}_{n}(k)=\frac{\alpha^{2n}}{\sqrt{2\pi}\,(\alpha^{2}+k^{2})^{n}}\hskip
20.00003ptn\in\\{1,2,3,4\\}\,.$ (18)

Figure 18 illustrates the shapes of kernels $G_{1}$ and $G_{3}$. $G_{3}$
already resembles a Gaussian quite well, having a smooth maximum at the origin
and wings that are moderately wider than in the case of a Gaussian. One should
keep in mind that approaching a Gaussian brings back in the problems that we
intended to mitigate, namely significant amplification of noise. Hence, one
should limit the number of repeated convolutions.

Figure 18: Shapes of the kernels $G_{1}$ and $G_{3}$ in comparison with a
Gaussian. For display purposes all functions were normalised to $G_{i}(0)=1$.
All share the same FWHM.

A last point concerns the choice of the broadening parameter $\alpha$. While
the formulae take a simple form by using $\alpha$, physically one likes to
specify the width of the kernel more directly. Fortunately, except for the
pre-factor, all kernels are functions of the product $\alpha\,|x|$ alone,
resulting in a simple (inverse) relation between the full width at half
maximum (FWHM) of a kernel and $\alpha$. For kernel $G_{1}$, the relation
reads $\alpha=2\sqrt{\ln 2}/\mathrm{FWHM}$. The other kernels can be obtained
from $G_{1}$ by two-, three-, and four-fold application of $G_{1}$ with a FWHM
of 0.42243, 0.29746, and 0.24325 times the targeted FWHM, respectively. The
numbers were obtained by solving numerically the transcendent equations for
the widths of the three kernels in question, and the approach was applied when
creating Figure 18.

Figure 19: Broadening and de-broadening effects of kernels G1 and G3: a
Gaussian-shaped line with FWHM of 5 km s-1 was convolved and de-convolved with
kernels having an absolute width of 1 km s-1. Negative broadening values
indicate a deconvolution.

Figure 19 illustrates the impact of convolving and de-convolving a Gaussian-
shaped line profile with kernels $G_{1}$ and $G_{3}$. The FWHM of the line and
the levels of broadening or de-broadening were roughly tailored after the
silicon lines we observe in solar disk-centre spectra. $G_{3}$ has a milder
effect on the line shape than $G_{1}$ at the same FWHM. While not clear from
the plot, it turns out that one can largely mimic the effect of $G_{1}$ by
using $G_{3}$ with a greater width. For the given parameters the “peaky” shape
of $G_{1}$ leaves no imprint in the line shape. So far, the kernels described
above worked in practice, but deconvolution can be handled only for kernels
significantly narrower than the spectral line to be de-broadened.

##  Appendix D Centre-to-limb variation of the continuum and line shapes
predicted by the 3D models

On request by the referee here we report on the performance of our 3D models
in combination with our spectral synthesis codes (Linfor3D for line syntheses,
NLTE3D for spectral energy distributions) when representing the solar centre-
to-limb variation (CLV) of its continuum radiation and the shape of lines. We
start with the CLV, and later point to investigations in the literature which
use features of line shapes that were calculated with the help of CO5BOLD
models.

Figure 22 shows a comparison between prediction of our 3D models with
observations by Neckel & Labs (1994, hereafter NL). NL provide 30 wavelengths
from the near-UV to the near-IR for which they measured relative intensities
over narrow bandwidths (in equivalent Doppler velocity between 1.4 to 1.9 km
s-1). The wavelength points were chosen to be largely free of absorption
lines. We ran pure continuum syntheses for the NL wavelengths for our four 3D
models, and for further comparison for three 1D model atmospheres. The lower
panels (left panels in rotated view) depict the result. For the non-magnetic
models msc600 and n59 the correspondence between observation and model
prediction is very good for $\lambda>0.6\,\mu\mathrm{m}$. We note that the
results for models msc600 and n59 are almost identical so that they are
difficult to distinguish on the scale of the plot. 1D models are added for
setting the scale for a “very good” correspondence: a standard ATLAS9 (Kurucz
1979; Sbordone et al. 2004; Kurucz 2005) model shows a significantly steeper
CLV than found in the observations. Similarly, an LHD model (LHD is a home-
grown 1D stellar atmosphere code that uses the same microphysics as applied in
our 3D models) shows a similar behaviour. Remarkably, our msc600 model
performs even slightly better than the well-known Holweger-Müller solar model
(Holweger 1967; Holweger & Mueller 1974) despite the fact that this semi-
empirical model was constructed to match the CLV.

At $\lambda<0.6\,\mu\mathrm{m}$ the good correspondence deteriorates. However,
as NL point out themselves the notion of observing pure continuum over the
bandwidth used in the measurements becomes questionable. At wavelengths
shorter than $0.6\,\mu\mathrm{m}$ the previously negligible contribution of
line absorption to the observed intensity becomes non-zero but otherwise
undefined. To address this issue we took an extreme stand and calculated the
intensity over $20\,\mbox{\AA}$ wide intervals including all line absorption
with the help of opacity distribution functions (ODFs). We combined line ODFs
from Kurucz’ ATLAS suite (Kurucz 2017) with continuous opacities from our own
opacity package. In each ODF interval the distribution of the line opacity was
represented by a step-function of 12 steps. For each step, the emergent
intensity was calculated and finally integrated over the whole ODF interval so
that we obtained the intensity emerging in the $20\,\mbox{\AA}$ wide ODF
interval. From the construction is is clear that one obtains the average
effect on the intensity of the absorbers present in the ODF interval. The
upper panels (right panels in rotated view) of Fig. 22 illustrate that the
correspondence between model predictions and observations clearly improves for
wavelengths $\lambda<0.6\,\mu\mathrm{m}$, strongly suggesting that indeed
missing line absorption is the reason for the mismatch at these shorter
wavelength in the pure continuum calculations. In fact, the good
correspondence in the near-UV is striking. One has to admit that this is
certainly in part fortuitous since the true contribution of lines to the
observations is unclear, and also whether the line lists going into the
construction of the ODFs are sufficiently complete in the near-UV. To
illuminate the influence of line absorption a bit further we write the
intensity ratio between a location $\mu$ and and disk centre $\mu=1$ as

$\frac{I_{\mu}}{I_{1}}=\frac{c_{\mu}-l_{\mu}}{c_{1}-l_{1}}\approx\frac{c_{\mu}}{c_{1}}\left(1-\frac{l_{\mu}}{c_{\mu}}+\frac{l_{1}}{c_{1}}\right)$
(19)

where $c_{\mu}$ is the continuum intensity at location $\mu$ and $l_{\mu}$ the
intensity reduction by line absorption at this point. The approximate equality
holds for weak line absorption. Equation (19) shows that there is a
compensatory effect by considering intensity ratios. Moreover, since there is
typically a reduction of the intensity ratio with respect to pure continuum
calculations the contribution of line absorption must become larger towards
the stellar limb. It is not straight forward to see why this is: the silicon
lines synthesised for this investigation generally show a mild decrease in
strength towards the limb – particularly the medium to strong lines. However,
we conjecture that in fact molecular lines are a major player here which
significantly increase in strength due to the lower temperatures at which line
formation takes place towards the limb.

Since the contribution of lines is observationally not well-defined we have
also plotted intensity ratios considering ODF sub-intervals only. We expected
that especially the sub-interval with the smallest line contribution would
result in a closer match to the observations which was, however, not
immediately apparent (not shown). We finally remark that the findings
discussed here coincide with results on the CLV based on a CO5BOLD model of an
earlier generation (Ludwig et al. 2010).

For completeness we also investigated the CLV of the magnetic models b000 and
b200. Figure 22 illustrates that the field-free model provides a reasonable
match to the observations while the 200 G model is clearly off. For the given
(somewhat artificial) field configuration one may conclude that the observed
CLV does not permit a mean field strength $\gtrsim 50\,\mathrm{G}$ on the Sun.
Pereira et al. (2013) already arrived at a similar conclusion by comparing a
100 G model with a field-free case.

All spectral synthesis calculations for the 3D models underlying Fig. 22
approximate scattering in the continuum as well as lines as true absorption.
We investigated the effect of this approximation on the CLV in the continuum
by comparing the cases of isotropic coherent scattering and true absorption in
a 1D stratification obtained when averaging (over optical depth surfaces and
time) the msc600 model. As a first step, Fig. 20 illustrates that scattering
generally leads to an increase of the emergent intensity for wavelength
$\lesssim 0.65\,\mu\mbox{m}$, and that effects becomes more pronounced towards
the solar limb. However, quantitatively the changes of the intensity are
modest ($\lesssim 4.5\,\%$). Figure 21 shows that this translates into small
differences ($\lesssim 0.01$) in the CLV, again, mostly at short wavelengths
and close to the limb. The overall limb darkening reduces the effect apparent
in the intensity ratios directly. To see this, we denote by $S$ the intensity
when scattering is treated exactly, by $A$ the intensity when scattering is
treated as true absorption. The difference in the CLV can then be written as

$\frac{S(\mu)}{S(1)}-\frac{A(\mu)}{A(1)}\approx\frac{S(\mu)-A(\mu)}{A(1)}=\frac{A(\mu)}{A(1)}\,\left(\frac{S(\mu)}{A(\mu)}-1\right)\,.$
(20)

The approximate equality comes from the observation that $S(1)\approx A(1)$.
All this means that the very good correspondence between models and
observations shown in Fig. 22 gets only very slightly worse when scattering is
treated correctly but remains very satisfactory.

Figure 20: Ratio of the emergent intensities as function of wavelength and
limb angle cosine $\mu$ between cases when continuum scattering is treated
exactly or as true absorption. The average vertical profile of model msc600
was used as structure in the spectral synthesis calculations.

Figure 21: Difference of the centre-to-limb variation when continuum
scattering is treated exactly or as true absorption. Note the reduction of the
impact of scattering in comparison to Fig. 20.

We now briefly turn to line shapes, particularly line shifts, as predicted by
model n59. As a first example we point to González Hernández et al. (2020,
Fig. 6) where absolute core shifts of 144 Fe I lines are compared to
observations. In their work a very accurate wavelength calibration of the
observed spectra was achieved by utilising a laser frequency comb. Lines with
an equivalent width less than $60\,\mbox{m\AA}$ show a good correspondence
with their observed shifts. One has to keep in mind here that a comparison on
an absolute scale also relies on accurately known laboratory wavelengths. The
reason for the mismatch of lines with equivalent widths greater than
$60\,\mbox{m\AA}$ is not clear but not necessarily related to shortcomings in
the model structure. Moreover, it was also seen in LTE line syntheses based on
STAGGER models (see González Hernández et al. 2020, for further discussion).
As second example, in Löhner-Böttcher et al. (2019, Fig. 19) the observed
shape of the Fe I 6173 Å line is compared to synthetic line profiles computed
with a CO5BOLD model as a function of limb angle. While not all details are
matched the overall correspondence is satisfactory.

Figure 22: Centre-to-limb variation as predicted by our 3D models in
comparison to observations and 1D results. Syntheses excluding (bottom) and
including (top) line absorption are shown. Left panels: models msc600 and n59.
Right panels: models b000 and b200.

##  Appendix E The partition functions of silicon as implemented LINFOR3D

We compared the partition functions of the first three ionisation stages of
silicon as implemented in our spectral synthesis code to data given in the
recent compilation of Barklem & Collet (2016). We find a close to perfect
agreement in the temperature range relevant for the formation of silicon lines
in the solar photosphere, as seen in Fig. 23.

Figure 23: Partition functions implemented in LINFOR3D compared to data of
Barklem & Collet (2016).