# Solar oxygen abundance

Maria Bergemann,1,2,3 Richard Hoppe,1,2,4 Ekaterina Semenova, 2 Mats Carlsson,
5,6 Svetlana A. Yakovleva,7 Yaroslav V. Voronov,7 Manuel Bautista,8 Ahmad
Nemer,9 Andrey K. Belyaev,7 Jorrit Leenaarts,10 Lyudmila Mashonkina,11 Ansgar
Reiners,12 Monika Ellwarth12

1 The first two authors have contributed equally to the work presented in this
paper
2 Max Planck Institute for Astronomy, 69117, Heidelberg, Germany
3 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17,
DK-2100 Copenhagen Ø, Denmark
4 Ruprecht Karls University, Grabengasse 1, 69117 Heidelberg, Germany
5 Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
6 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
7 Department of Theoretical Physics and Astronomy, Herzen University, St.
Petersburg, 191186, Russia
8 Department of Physics, Western Michigan University, Kalamazoo, Michigan,
49008, USA
9 Princeton University, Department of astrophysical sciences, Princeton, New
Jersey, USA
10 Institute for Solar Physics, Department of Astronomy, Stockholm University,
AlbaNova University Centre, SE-106 91 Stockholm, Sweden
11 Institute of Astronomy of the Russian Academy of Sciences, Pyatnitskaya st.
48, 119017, Moscow, Russia
12 Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-
Hund-Platz 1, D-37077 Göttingen, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Motivated by the controversy over the surface metallicity of the Sun, we
present a re-analysis of the solar photospheric oxygen (O) abundance. New
atomic models of O and Ni are used to perform Non-Local Thermodynamic
Equilibrium (NLTE) calculations with 1D hydrostatic (MARCS) and 3D
hydrodynamical (Stagger and Bifrost) models. The Bifrost 3D MHD simulations
are used to quantify the influence of the chromosphere. We compare the 3D NLTE
line profiles with new high-resolution, R $\approx 700\,000$, spatially-
resolved spectra of the Sun obtained using the IAG FTS instrument. We find
that the O I lines at 777 nm yield the abundance of $\log{\rm A(O)}=8.74\pm
0.03$ dex, which depends on the choice of the H-impact collisional data and
oscillator strengths. The forbidden [O I] line at 630 nm is less model-
dependent, as it forms nearly in LTE and is only weakly sensitive to
convection. However, the oscillator strength for this transition is more
uncertain than for the 777 nm lines. Modelled in 3D NLTE with the Ni I blend,
the 630 nm line yields an abundance of $\log{\rm A(O)}=8.77\pm 0.05$ dex. We
compare our results with previous estimates in the literature and draw a
conclusion on the most likely value of the solar photospheric O abundance,
which we estimate at $\log{\rm A(O)}=8.75\pm 0.03$ dex.

###### keywords:

Atomic data – Radiative transfer – Techniques: spectroscopic – Sun:
photosphere – Sun: chromosphere – Sun: abundances

††pubyear: 2015††pagerange: Solar oxygen abundance–8

##  1 Introduction

Oxygen is (behind H and He) the most abundant chemical element in the Universe
and it is of major relevance in modern astrophysics, across difference fields,
including precision stellar physics, extragalactic astronomy, planet
formation, and galaxy evolution. Most importantly, oxygen determines much of
the opacity in the solar interior (e.g. Bahcall et al., 2005; Serenelli et
al., 2009; Pinsonneault & Delahaye, 2009), therefore its abundance is critical
to the calculation of Standard Solar Models (SSM), which describe the
evolution of the Sun from the pre-main-sequence to the present age of 4.5 Gyr.
Oxygen is also the key element in gas-phase spectroscopic diagnostics on
extragalactic scales, in particular to infer the metallicities from the H II
regions (e.g. Kewley & Ellison, 2008; Moustakas et al., 2010) \- a broadly-
used technique that has been applied to many star-forming galaxies to
establish their metal-content and to quantify the mass-metallicity
relationship across the entire mass range of galaxies in the local universe.
Oxygen is also one of the two - in addition to Mg - most common tracers of
nucleosynthesis in massive stars, which explode as core-collapse supernovae.
Therefore, O abundance is traditionally used in combination with Fe to map the
chemical enrichment and star formation history of stellar populations in the
Milky Way and its satellite galaxies (e.g. Tolstoy et al., 2009; Barbuy et
al., 2018).

This study focuses on the first problem - the chemical abundance of oxygen in
the Sun. Over the past decades, several groups approached this problem from
various angles. Some groups attempted to measure the O abundances from the
solar photospheric spectrum (e.g. Grevesse & Sauval, 1998; Allende Prieto et
al., 2001; Asplund et al., 2004; Caffau et al., 2008; Pereira et al., 2009b;
Sitnova et al., 2013; Caffau et al., 2015; Socas-Navarro, 2015; Steffen et
al., 2015; Amarsi et al., 2018). Other groups derived the solar O abundances
from the high-ionised lines in the solar wind (e.g. Bochsler, 2007; Laming et
al., 2017). The latter can be, however, measured less accurately compared to
the photospheric analysis methods. In the seminal paper, Grevesse & Sauval
(1998) presented the O abundance of $8.83\pm 0.06$ dex, based on 1D LTE
methods. The most recent estimate, derived by means of a detailed NLTE
analysis with 3D radiation-hydrodynamics (RHD) simulations of solar
convection, is $\log{\rm A(O)}=8.69~{}\pm~{}0.03$ dex (Amarsi et al., 2018,
based on the 777 nm lines), which compares well within the uncertainties with
the estimate by Caffau et al. (2015, based on the forbidden 630 nm line)
($\log{\rm A(O)}=8.73\pm 0.02\pm 0.05$ dex). Neither of these estimates,
however, are satisfactory for the SSM calculations (Villante & Serenelli,
2020), as the internal structure of the present-day solar model - the sound
speed profile, the depth of the convective envelope - does not compare well
with independent constraints on its structure obtained by means of
helioseismology. Various scenarios have been put forward to explain the
mismatch, ranging from the under-estimated opacity in the interior (Bailey et
al. 2015, but see Nagayama et al. 2019) to energy transport by dark matter
particles (Vincent et al., 2015). The problem has, so far, not been
convincingly explained by any of these scenarios.

In this work, we present a new spectroscopic analysis of the solar
photospheric O abundance. We are motivated by the availability of new atomic
data and recent 3D magneto-hydrodynamic simulations (MHD) of near-surface
convection, the chromosphere, and the photosphere (Carlsson et al., 2016). Our
model atom of O relies on new data describing collisional excitation and
charge transfer with H atoms, photo-ionisation, and collisions between O i and
free electrons. We also analyse the uncertainty in the oscillator strengths of
the transitions observed. In addition, we develop a new model atom of Ni in
order to study the influence of the Ni blend in one of the diagnostic O i
features. Finally, we investigate the O i line formation using different types
of the solar model atmospheres, including the Stagger RHD model, but also two
3D MHD models computed self-consistently, with and without chromosphere, using
the Bifrost code. This is important for any study that requires precision
abundance diagnostics, such as, e.g., a detailed analysis of the Sun relative
to solar twins, comparative studies of exoplanet hosts, or spectroscopic age
indicators (e.g. Bedell et al., 2014; Buchhave & Latham, 2015; Bedell et al.,
2018; Lorenzo-Oliveira et al., 2018; Nissen et al., 2020).

The paper is organised as follows. Section 2 provides the details of
observational datasets used in the abundance calculations. In Sect. 3 we
describe the main properties of the atomic models of O (Sect. 3.1) and Ni
(Sect. 3.4), in particular, the calculations of new photo-ionisation cross-
sections and electron collision rates (Sect. 3.2), collisions with hydrogen
atoms (Sect. 3.3). In Sect. 3.5 we review the 1D and 3D (M)RHD model
atmospheres, the statistical equilibrium and radiation transfer codes (Sect.
3.6), and the details of abundance analysis (Sect. 3.7). We then describe the
results of our calculations in Sect. 4, addressing the following aspects: line
formation of O (Sect. 4.1) and Ni (Sect. 4.3), center-to-limb (CLV) variation
(Sect. 4.4), error analysis (Sect. 4.7), the solar O abundance (Sect. 4.6). We
perform a comparative analysis of our findings in the context of earlier
studies in the literature (Sect. 5). Finally Sect. 6 summarizes the
conclusions and provides an outlook for future work.

##  2 Observations

In this work, we make use of different spectroscopic data available for the
Sun.

Our primary dataset are the new spatially-resolved solar data obtained with
the Fourier Transform Spectrograph mounted on the Vacuum Vertical Telescope at
the Institut für Astrophysik Göttingen (hereafter, referred to as IAG data).
Light from the 50cm-siderostat is picked up by a fiber with an on-sky diameter
of 25" and sent to the FTS (for more information, see Reiners et al. (2016)
and Schäfer et al. (2020)). The FTS was operated in double-sided mode with a
spectral resolution of 0.024 cm-1, i.e.,
$R=\frac{\Delta\lambda}{\lambda}\approx 700\,000$ at $\lambda=6\,000$ Å). We
collected $123$ individual observations of the quiet Sun at 12 different
$\mu$-angles during April and August 2020. Each observation took approximately
10 minutes. We used the HITRAN database to identify telluric lines from H2O
and O2 and mask them in the spectra. Relative motion between the VVT and the
respective solar position was determined using the ephemeris code (Doerr,
2015) based on the NASA’s Navigation and Ancillary Information Facility SPICE
toolkit (Acton, 1996). We used the differential rotation law of the Sun
(Snodgrass & Ulrich, 1990) to correct the individual spectra for solar
rotation, and we subtracted barycentric and rotational motion from every
observed spectrum. Before addition, we normalized individual spectra and
corrected remaining radial velocity offsets between different exposures taken
at the same $\mu$-angles but at different positions on the Sun. We verified
that these offsets were within the range of pointing errors. Eventually, we
added spectra observed at the same $\mu$-angles. Our co-added spectra cover
the spectral range between $4700$Åand $7900$Åand have a signal-to-noise (S/N)
of 450 per resolution element at 6000Å. In our analysis we employ the $\mu$
angles111Here $\mu=\cos\theta$ and $\theta$ is the heliocentric viewing angle,
with the disc center corresponding to $\theta=0^{\circ}$.: $0.2$, $0.4$,
$0.6$, $0.8$, $1.0$.

We also employ high-resolution spatially resolved spectra (Pereira et al.,
2009a, b) obtained with the TRIPPEL (TRI-Port Polarimetric Echelle-Littrow
spectrograph) spectrograph mounted on the Swedish 1-m Solar Telescope
(hereafter, referred to as SST data) (Scharmer et al., 2003). These data have
a resolving power of $R=\lambda/\delta\lambda\approx 200\,000$ and cover the
wavelength regions around the O i triplet (7771 to 7775 Å) and [O I] (6300 Å)
lines.

In addition, we analyse the spatially resolved solar spectra obtained with the
Solar Optical Telescope on the Hinode satellite (Caffau et al., 2015). These
data are not affected by telluric absorption, which is especially critical in
the region around 630 nm. The HINODE data have a resolving power of
$R=200\,000$ and a S/N of $20\,000$. The pointings include the following $\mu$
angles: $0.38$, $0.70$, $0.86$, $0.95$, and $0.99$. For a more detailed
description of the observations and the data reduction, we refer the reader to
Caffau et al. (2015).

We complement the afore-mentioned data with the very high-quality, $R\approx
350\,000$, disc-center spectra obtained using the KPNO FTS instrument. These
data were released by Brault (1972) and a re-reduced version was published in
Neckel & Labs (1984).

Figure 1: Grotrian models of O (top panel) and Ni (bottom panel). The
diagnostic features used in the spectroscopic analysis are indicated using
bold blue and red lines.

The last resource of observational data in this work is the data collected by
Takeda & UeNo (2019) using the 60 cm Domeless Solar Telescope (hereafter,
referred to as DST data) with the Horizontal Spectrograph at Hida Observatory
of Kyoto University. In total, 31 different $\mu$-angles have been observed,
of which we were only interested in the disc-center intensity. The spectra
have a resolving power of $R\sim 140\,000$ and S/N of several hundreds.

##  3 Methods

###  3.1 O model atom

The model atom consists of $122$ energy states, with neutral oxygen
represented by 120 energy states and the singly-ionised oxygen by $2$ states,
the ground state and the first excited state. The energy levels, which
includes levels with principal quantum number $n\leq 30$ and orbital angular
momentum $l\leq 4$, were assembled from the NIST database (Kramida et al.,
2020). The original number of O i energy levels is $227$, however, we merge
the fine structure levels with the energy above $102\,700$ cm-1 (12.73 eV), so
that all states above the level $3s\,^{1}{\rm D}^{{\rm o}}$ are represented by
a term with the statistical weight $g$ obtained by summing the individual
statistical weights of the levels and the energy computed by weighted
averaging of fine structure level energies according to their $g$-value (cf.
Bergemann et al., 2012). As a result our O i model contains $35$ fine
structure states and $87$ terms. Fig. 1 (top panel) shows the Grotrian diagram
of O i, with the key diagnostic transitions in the solar spectrum indicated
with bold lines. The majority of lines in the O i system connect energy levels
with a very high , $E>10$ eV, excitation potential (relative to the ground
state).

The diagnostic O lines used in the abundance analysis are the triplet O i
lines at $777$ nm and the forbidden [O I] line at 630 nm (Table 1). The former
lines arise in electric dipole transitions from the 2p33s 5S${}^{o}_{2}$ level
at $9.15$ eV above the ground level. The line at 630 nm is a magnetic dipole
transition that arises from the ground level 2p4 3P2. This transition is also
observed in emission in the airglow spectra in the terrestrial atmosphere
(Rees, 1989; Slanger et al., 2011). Owing to its very low transition
probability ($\log gf=-9.72$, Storey & Zeippen 2000), the [O I] line is weak
even in the solar spectrum and its equivalent width (EW) measures only a few
mÅ (see Appendix). According to the NIST
database222https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html,
accessed on and before May 22, 2021. Note the link only works in the copy-and-
paste mode in a browser., the f-values of the 777 nm and the 630 nm
transitions have an uncertainty rating "A" ($\leq 3\%$, or better than 0.01
dex) and "B+" ($\leq 7\%$, or better than $0.03$ dex), respectively. However,
these uncertainty estimates from NIST appear to be highly overoptimistic. The
true uncertainties are likely significantly larger, as indicated by the
recently published new transition probabilities Civiš et al. (2018) and our
own calculations (Sect. 4.7). In this work, we have chosen to perform all
abundance calculations using the average of Hibbert et al. (1991) and Civiš et
al. (2018) f-values for the 777 nm lines ($\log gf=0.343,0.197,-0.025$ for the
7771, 7774, and 7775 Å lines, respectively), and Storey & Zeippen (2000) data
for the 630 nm lines. We furthermore use one half of the difference between
Hibbert et al. (1991) and Civiš et al. (2018) ($0.026$ dex) as a $1\sigma$
uncertainty, and adopt a 20% uncertainty ($0.08$ dex) on the f-value of the
magnetic dipole line. We return to this issue in Sect. 4.7. The parameters of
the diagnostic lines of O i are provided in Table 1.

The parameters of all other radiative bound-bound transitions in the O model
atom, including wavelength, f-values, damping constants, were taken from the
Kurucz 333http://kurucz.harvard.edu/atoms/0800/ database. There are $13\,483$
radiative transitions in the wavelength range from $666$ Å to $100\,000$ Å.
However, after merging the energy states and making a cut on the oscillator
strength ($\log gf=-10$), the number of radiative transitions is compressed to
$1364$. The chosen cut is needed to capture the resolved photoionisation and
recombination resonances, including the Rydberg Enhanced Recombination (Nemer
et al., 2019). These processes lead to a sequence of transitions that involve
the diagnostic O I lines, hence they are necessary for the accuracy of the SE
calculations. The lines long-ward of $20\,000$ Å are important, because they
represent a channel through which the majority of very high-excitation states
are connected and therefore, ensure their convergence to statistical
equilibrium (SE).

Most of the lines in the atomic model are represented by Voigt profiles with
$9$ frequency points, which is sufficient for the SE calculations. The
diagnostic O i lines are represented by $251$ frequency points. Van der Waals
damping, that is, broadening caused by elastic collisions with H atoms, is
included using the data from Barklem et al. (2000). It should be noted, though
that only $33$ of our lines are present in the Barklem et al. (2000) database,
therefore for the remainder of transitions we resort to the standard Unsöld
formalism (Unsöld, 1955). It is known that the Unsöld theory underestimates
line broadening, as it only applies to collisional broadening at large atomic
separations. However, this is not of a concern for O i, as our tests with
Unsöld damping constants scaled by a factor or $2$ and $0.5$ yield virtually
identical results for the SE and the profiles of diagnostic lines.

Table 1: Parameters of spectral lines used for the solar O abundance calculation. The wavelengths for the diagnostic lines are taken from the NIST database. References to the $\log gf$ values and H damping are also provided. Specie | $\lambda$ | $E_{\rm low}$ | $E_{\rm up}$ | Lower | Upper | $\log gf$ | vdW | Ref ($\log gf$) | Ref (vdW)
---|---|---|---|---|---|---|---|---|---
|  [Å] | [eV] | [eV] | level | level |  |  |  |
O |  |  |  |  |  |  |  |  |
O I | 7771.940 | 9.146 | 10.741 | $3s\,^{5}{\rm S}^{{\rm o}}_{\rm 2}$ | $3p\,^{5}{\rm P}_{\rm 3}$ |  $0.369$, $0.317$ | 453.234 |  Hibbert et al. (1991), Civiš et al. (2018) | Barklem et al. (2000)
O I | 7774.170 | 9.146 | 10.741 | $3s\,^{5}{\rm S}^{{\rm o}}_{\rm 2}$ | $3p\,^{5}{\rm P}_{\rm 2}$ |  $0.223$, $0.170$ | 453.234 |  Hibbert et al. (1991), Civiš et al. (2018) | Barklem et al. (2000)
O I | 7775.390 | 9.146 | 10.740 | $3s\,^{5}{\rm S}^{{\rm o}}_{\rm 2}$ | $3p\,^{5}{\rm P}_{\rm 1}$ |  $0.002$, $-0.051$ | 453.234 |  Hibbert et al. (1991), Civiš et al. (2018) | Barklem et al. (2000)
O I | 6300.304 | 0.000 | 1.967 | $2p^{4}\,{}^{3}{\rm P}_{\rm 2}$ | $2p^{4}\,{}^{1}{\rm D}_{\rm 2}$ | -9.72 |  | Storey & Zeippen (2000) | Unsöld (1955)
Ni |  |  |  |  |  |  |  |  |
Ni I | 6300.341 | 4.266 | 6.234 | $4p\,^{3}{\rm D}^{{\rm o}}_{\rm 1}$ | $4s^{2}\,{}^{1}{\rm S}_{\rm 0}$ | -2.11 |  | Johansson et al. (2003) | Unsöld (1955)

###  3.2 Photoionisation and electron collision rates

The standard source of photoionisation cross-sections is the Opacity Project
(OP) database TOPbase444http://cdsweb.u-strasbg.fr/topbase/topbase.html (Cunto
& Mendoza, 1992). However, the data were computed at a very coarse energy
resolution assuming the $LS$ coupling and they are available only for the
energy states with the principal quantum number of $n\leq 10$. These
shortcomings limit the applicability of the OP data to NLTE stellar atmosphere
problems.

Also the data that describe the rates of transitions caused by inelastic
collisions with free electrons are very sparse. The most recent source of this
information is Barklem (2007), who tabulated the rate coefficients for
transitions between the $7$ low-energy states in O i. They employed the
standard R-matrix method assuming LS coupling, which is known to deliver
relatively accurate results consistent with direct experimental values. These
sparse datasets are, however, not sufficient to describe the entire micro-
physics of interactions between O i atoms, radiation field, and electrons.
This is especially important, because the key diagnostic O i lines (the 777 nm
triplet) connect very high-excitation energy levels (Fig. 1). Therefore, we
opted to compute new photoionisation cross-sections and cross-sections for the
transitions caused by e$-$ collisions.

We computed new photoionisation cross-sections for all levels with a principal
quantum number up to $n=20$ (energies up to 109561 cm-1). These calculations
were carried out with the Breit-Pauli version of the code RMATRX (Berrington
et al., 1995). This is an implementation of the R-matrix method for atomic
scattering calculations. The atomic orbitals for the O+ target system were
obtained with the scaled Thomas-Fermi-Dirac-Amaldi central-field potential as
implemented in AUTOSTRUCTURE (Badnell, 1997). Our target representation
included eleven configurations $2s^{2}3p^{3}$, $2s^{2}3p^{2}3s$,
$2s^{2}3p^{2}3p$, $2s^{2}3p^{2}3d$, $2s3p^{4}$, $2s3p^{3}3s$, $2s3p^{3}3p$,
$2s3p^{3}3d$, $2s^{2}3p3s^{2}$, $2s^{2}3p3p^{2}$, and $2s^{2}3p3d^{2}$. This
expansion lets to $26$ close coupling LS terms and $39$ energy levels with
principal quantum number up to $n=20$.

We give a preference to the new photoionisation cross-sections over the
TOPbase values. First, our target and close coupling expansions are larger
than those of the OP. Second, the present cross-sections are in intermediate
coupling and account for relativistic corrections, as opposed to the LS-term
OP cross sections. Finally, we retained level-to-level partial cross-sections,
which allow for a reliable determination of recombination rates. Also, our
cross sections have the energy resolution of $2.2\times 10^{-4}$ Ryd, (about
$100\times$ better resolution than OP cross sections), which adequately
resolves the cross-section resonances. Fig. 2 compares our new values for two
selected energy levels with the TOPbase data. Our cross-sections include more
resonance structures, owing to the much larger close-coupling expansion. In
the case of the 3s 5So state, the TOPbase cross-section exhibits an abnormal
decreasing trend at higher energies and a sort of bulge at lower energies. Our
data show a correct high energy behaviour and reveal that the bulge is, in
fact, a series of resonances, which only show up as proper close-coupling
channels are included in the calculations. Regarding the cross-section for the
3p 5P1 level, the results in TOPbase and our data show very similar
qualitative behavior, but our cross-section accounts for more series of
resonances. It should be noted that TOPbase only provides data in $LS$
coupling, thus the cross-section show in Fig. 2 is actually the state’s cross-
section.

We also calculated new electron impact excitation rate coefficients using the
AUTOSTRUCTURE code (Badnell, 2011), which is based on the Breit Pauli
Distorted Wave method. All configurations of the form $2s^{2}2p^{4}$,
$2s^{2}2p^{3}nl$, $2s2p^{5}$, and $2s2p^{4}nl$, with $n\leq 30$ and $0\leq
l\leq 3$ were included in the calculations. For the rest of the O i system, we
complemented the R-matrix collision strengths with the rate coefficient
computed using the formulae from van Regemorter (1962).

Figure 2: New photoionisation cross-sections for the $3s\,^{5}{\rm S}^{{\rm
o}}_{\rm 2}$ and $3p\,^{5}{\rm P}_{\rm 1}$ levels of O i compared with the
Opacity Project data (here we take the cross-section in OP LS-coupling of the
5P for the 5P1 level).

###  3.3 Collisions with hydrogen atoms

Table 2: Scattering channels and the corresponding asymptotic energies, taken from NIST (Kramida et al., 2020). j | Scattering channels | Asymptotic energies
---|---|---
|  | eV
1 | O$(2p^{4}\,{}^{3}$P) + H$(1s)$ | 0.0000
2 | O$(2p^{4}\,{}^{1}$D) + H$(1s)$ | 1.9674
3 | O$(2p^{4}\,{}^{1}$S) + H$(1s)$ | 4.1897
4 | O$(2p^{3}3s\,^{5}$S∘) + H$(1s)$ | 9.1461
5 | O$(2p^{3}3s\,^{3}$S∘) + H$(1s)$ | 9.5214
6 | O$(2p^{4}\,{}^{3}$P) + H$(2s)$ | 10.2000
7 | O$(2p^{4}\,{}^{3}$P) + H$(2p)$ | 10.2000
8 | O$(2p^{3}3p\,^{5}$P) + H$(1s)$ | 10.7406
9 | O$(2p^{3}3p\,^{3}$P) + H$(1s)$ | 10.9888
10 | O$(2p^{3}4s\,^{5}$S∘) + H$(1s)$ | 11.8376
11 | O$(2p^{3}4s\,^{3}$S∘) + H$(1s)$ | 11.9304
12 | O$(2p^{3}3d\,^{5}$D∘) + H$(1s)$ | 12.0786
13 | O$(2p^{3}3d\,^{3}$D∘) + H$(1s)$ | 12.0870
14 | O$(2p^{3}4p\,^{5}$P) + H$(1s)$ | 12.2861
15 | O$(2p^{3}4p\,^{3}$P) + H$(1s)$ | 12.3589
16 | O$(2p^{3}3s\,^{3}$D∘)+H$(1s)$ | 12.5402
1 | O${}^{-}(2p^{5}\,{}^{2}$P) + H+ | 12.1500
2 | O${}^{+}(2p^{3}\,{}^{4}$S∘) + H-(1S) | 12.8641

Inelastic collisions with H atoms represent an important ingredient in NLTE
calculations, as they lead to excitation, de-excitation, ion-pair formation,
and mutual neutralization reactions. Detailed quantum-mechanical calculations
of the rate coefficients for O were recently presented by two independent
groups, Barklem (2018) and Belyaev et al. (2019). These studies rely on
different approaches. Barklem (2018) estimated the long-range electronic
structure of the OH molecule, by approximating the molecular wave function by
the linear combination of atomic orbitals (LCAO) of the atoms that form the
molecule, while the calculations of collisional dynamics were performed in the
framework of the multichannel model (Belyaev, 1993), we refer to this approach
as LCAO multichannel study. The O- \+ H+ scattering channel was not taken into
account. Independently, Mitrushchenkov et al. (2019) calculated the electronic
structure of the OH molecule using the multi-reference configuration
interaction (MRCI) method and estimated the rate coefficients for O + H, O+ \+
H- and O- \+ H+ collisional processes using the multi-channel model. Later on,
Belyaev et al. (2019), based on the MRCI electronic structure by
Mitrushchenkov et al. (2019), calculated the rate coefficients for inelastic
processes in the same transitions using the quantum hopping probability
current (QPC) method. The MRCI QPC study gives a more accurate description of
inelastic collisions than the LCAO multichannel study, because it uses a more
accurate electronic structure and take into account both the long- and short-
range non-adiabatic regions, see Belyaev et al. (2019) for the detailed
comparison.

Barklem (2018) tabulate the rate coefficients for the transitions between the
lowest $18$ terms in O i (up to $3s\,^{1}{\rm D}^{{\rm o}}$ at 102584 cm-1),
whereas Belyaev et al. (2019) provide the data for the lowest $11$ terms (up
to $4s\,^{3}{\rm S}^{{\rm o}}$ at 96147 cm-1). In this work, we complement the
latter data with new calculations for $5$ additional states of O i with
excitation energies from $12.0786$ eV to $12.5402$ eV, as well as two ionic
terms O+ \+ H- and O- \+ H+. All energy states included in the dynamical
calculations are provided in Table 2. For these $5$ additional states the
transition probabilities are computed using the asymptotic approach (Belyaev,
2013), including the non-adiabatic nuclear dynamical treatment accomplished by
the multi-channel approach. The calculations are performed within the
${}^{4}\Sigma^{-}$ molecular symmetry. Higher-lying covalent states are not
included in the calculations, because they either create ionic-covalent
avoided crossings at inter-nuclear distances greater than $\sim 100$ atomic
units, and hence the corresponding rates are negligible, or they do not create
avoided crossings at all. It should be noted that non-adiabatic transitions
between the higher-lying states are possible at short-range distances,
although short-range transitions do not lead to high rate coefficients. These
rates can be roughly estimated by means of the Kaulakys analytical approach
(Kaulakys, 1985, 1991). These datasets were recently computed by P.
Barklem555https://github.com/barklem/kaulakys, however, only for the lowest
$18$ energy states of O i that overlap with the states included in the LCAO
multichannel calculations. None of the available datasets accounts for the
fine structure, therefore, for the lack of a more accurate formulation, we
assign the same rate coefficient to each fine structure level within a given
term. We note that an alternative approach was proposed in the literature
(e.g. Osorio et al., 2015), however, also this approach involves an assumption
of the relative probability of transitions to different final spin states. Our
extensive tests suggest that reducing or increasing the quantum-mechanical
rate coefficients by a factor of $3$ (which corresponds to the average
multiplicity of O i) does not have any significant influence on the SE of
oxygen, in line with the findings of Bergemann et al. (2019), thus any further
arbitrary manipulation of the datasets is not justified.

Figure 3: Rate coefficients of the excitation ($\Delta E>0$) and ion-pair
formation processes in O $+$ H collisions at $T=6\,000$ K as a function of the
energy defect $\Delta E$ for the initial (top panel) and final (bottom panel)
channels $i=4$ and $i=5$ (Table 2). Three sets of data, Barklem (2018),
Belyaev et al. (2019), and Kaulakys (1991), are shown.

Fig. 3 (top panel) shows the data for some the transitions in common between
Barklem (2018), our new estimates, and the values estimated using the model of
Kaulakys (1985, 1991). Interestingly, there is no systematic difference
between the two approaches. For some transitions, Barklem (2018) provide
significantly, by up to $4$ orders of magnitude, lower rate coefficients. The
difference can be explained by the fact that the MRCI QPC study takes non-
adiabatic regions at small inter-nuclear distances into account. Due to the
transitions at a short range, the probability currents are redistributed
differently, as compared to the simplified picture given by including long
range non-adiabatic regions (as done in the LCAO multichannel study) only.
This also explain the difference in the rates for the important transitions,
$3s\,^{5}{\rm S}_{\rm o}$ \- $3s\,^{3}{\rm S}_{\rm o}$ and $3s\,^{5}{\rm
S}_{\rm o}$ \- $3p\,^{5}{\rm P}$. The rates computed using the Kaulakys (1985,
1991) recipe are typically much larger compared to the data computed using the
LCAO multichannel or the MRCI QPC studies. The differences are most
significant for transitions with $\Delta E\gtrsim 2.5$ eV. In this regime, the
Kaulakys model predicts the rate coefficients that are up to $10$ orders of
magnitude higher compared to the detailed quantum-mechanical calculations.
This is not surprising, as the model was developed to describe transitions
between high-energy Rydberg states.

Fig. 3 (bottom panel) illustrates the rate coefficients of ion-pair formation
processes computed using the MRCI QPC and the LCAO multichannel methods. There
is no significant difference for the majority of the ion-pair formation
processes, except the processes with $\Delta E<-2.5$ eV. But the general
behaviour of all ion-pair formation processes is very similar.

It is worth pointing out that the scattering channels $j=9-13$ belong to the
so-called ’optimal window’, in accordance with the general behavior found by
applying the simplified model (Belyaev & Yakovleva, 2017). The model predicts
the largest rates for mutual neutralization processes with final-state binding
energies in the vicinity of $-2$ eV, that corresponds to the excitation energy
of $11.65$ eV for the OH molecule. This optimal window is well understood
through the simplified model and it describes the mechanisms of inelastic
atomic collision processes due to long-range non-adiabatic regions created by
the ionic-covalent interaction.

Since the LCAO multichannel, MRCI QPC, and Kaulakys data represent physically
different approaches to the determination of H impact transition rates, we
proceed as follows. We create three NLTE atomic models of O and use them
independently in the statistical equilibrium and radiative transfer
calculations. One of the models includes the LCAO multichannel data only
(hereafter, the LCAO model atom). The other model relies on the LCAO
multichannel data co-added with Kaulakys rate coefficients (hereafter, the
LCAO$+$Kaul model), which is appropriate because the LCAO multichannel data
describe long-range interactions only and the Kaulakys data are expected to
compensate for the lack of short-range interactions. The third model
(hereafter, the QPC model) relies on the new rates computed using the MRCI
molecular structure and the hopping probability current method. Since in all
atomic models, less than $20$ energy states (out of $120$ O i states in the
model atom) are represented with detailed quantum-mechanical data, we connect
all other energy states by collision-induced transitions with the rate
coefficients computed using the scaled Drawin’s formula (Drawin, 1968). As we
show in Sect. 4.2, the only effect of this approach is that it ensures the
collisional coupling with the entire O i system and, therefore, helps to
greatly speed up the convergence in the very time-consuming 3D NLTE
calculations. Both atomic models also include the rates of charge transfer
between the O+ and H- ions, computed self-consistently using the methods
discussed above.

###  3.4 Ni model atom

Since one of the critical diagnostic O i lines is blended by Ni, we have opted
for performing a detailed study of NLTE effects in Ni. To the best of our
knowledge, this is the first analysis of the statistical equilibrium of Ni
with a comprehensive atomic model with 1D and 3D model atmospheres.

Similar to O, the atomic data for Ni i and Ni ii were adopted from the Kurucz
database666Taken from http://kurucz.harvard.edu/atoms/2800/, using the files
alitzen2800.dat, b2800e.com, and b2800o.com. The initial datasets comprise
$281$ energy levels and $9663$ radiative transitions for Ni i, and $716$
energy levels and $56\,193$ radiative transitions for Ni ii, respectively. The
Ni i levels are taken from the Litzèn et al. (1993) study, which combines
experimental measurements with theoretical calculations of the energy level
structure. We merge fine structure levels below $6.237$ eV and remove
transitions with $\log gf<-10$. The uppermost Ni I level in our model has the
energy of $7.34$ eV, whereas the first ionisation threshold (ground state of
Ni II) is located at $7.64$ eV. We note that there are only two known energy
levels of Ni I that are higher than our threshold for Ni I and are below the
1st ionisation threshold. One of them has the energy of 7.43 eV (59940.517
cm-1) and it represents one of the fine structure levels of the
3d8(3F)4s(4F)5p 5Do term. This level is, therefore, merged with the other
lower-energy FS levels of this term. The other level is the 3d8(3F)4s(4F)6s 5F
state with the energy of 7.43 eV (59862.611 cm-1). Owing to its small J-value
($J=5$) and a very small number of lines connecting it with the rest of Ni I,
we merge it with the Ni I state at 7.34 eV. We have verified that our NLTE
results do not depend on the treatment of these 2 levels. Our final model of
Ni comprises $538$ energy states and $2468$ spectral lines represented by
Voigt profiles with a $9$ point frequency quadrature. The first ionisation
threshold of Ni i is located at $7.64$ eV and that of Ni ii at $18.17$ eV. The
Grotrian diagram of Ni i is shown in Fig. 1 (bottom panel). Ni has a very
complex energy level system with many singlet, triplet, and quintet terms. It
should be noted that this diagram only shows energy levels with an LS
designation and it is not complete, as many energy levels in Ni i are
described in JcK coupling scheme (Litzèn et al., 1993).

Our model atom does not include isotopic splitting and hyperfine structure
(for the odd Ni components), as the shifts are very small. Using the data from
Johansson et al. (2003), we find, for example, that including the two main
isotopes of Ni changes the equivalent width of the Ni i line at $6300.342$ Å
by less than $0.5\%$. Therefore, we neglect isotopic shift in the SE
calculations of Ni and in the abundance analysis of O.

The photoionisation cross-sections were computed using the hydrogenic
approximation. The rates of collisional excitation by electrons and hydrogen
atoms were calculated using the standard formulae of van Regemorter (1962) and
Drawin (1968). We apply a scaling factor of $0.05$ to the rates that describe
the collisions of Ni i atoms with H, because the standard formulae are known
to severely over-estimate the rate coefficients compared to the detailed
quantum-mechanical data (Barklem, 2016). However, we investigate in Sect. 4
how this assumption influences the results. The rates of collisional
ionisation were computed using the Seaton (1962) formula.

###  3.5 Model atmospheres

Figure 4: Temperature structure of the STAGGER and Bifrost 3D (M)RHD models as
a function of logarithmic optical depth at 500 nm $\log\tau_{\rm 500}$. The
dashed lines show the T($\tau$) structure of the corresponding $\langle{\rm
3D}\rangle$ averages. See text

.

Figure 5: Temperature and vertical velocity distributions of a representative
Stagger (top) and Bifrost (bottom) model atmosphere snapshots. Both models
have the full horizontal resolution, but, for illustration, we do not show the
part of the atmosphere above $\log\tau_{\rm 500}\approx 0$. The location of
the horizontally averaged optical depth surface is indicated with a dashed
line.

We use several types of the solar model atmospheres computed under different
physical assumptions. The main difference between the types of models is the
treatment of geometry and convective energy transport, as described in the
subsequent sections.

For the purposes of comparison with previous calculations in the literature
(Asplund et al., 2004; Amarsi et al., 2018), we employ the solar MARCS model
also used in Bergemann et al. (2019). This is a 1D LTE line-blanketed model
atmosphere computed under the assumption of hydrostatic equilibrium.
Convective energy transport is parametrised using the mixing-length-theory
(MLT), with the MLT $\alpha$ constant set to $0.5$. The micro-turbulence
parameter is set to $1$ km s-1.

The 3D model atmospheres employed in this paper were computed using two 3D RHD
codes. The STAGGER code (Nordlund & Galsgaard, 1995; Nordlund et al., 2009)
provides a well established 3D radiation-hydrodynamic simulation of sub-
surface stellar convection. The solar simulation used in this work (Collet et
al., 2011; Magic et al., 2013) encompasses the entire photosphere as well as
the upper part of the convective zone. Bifrost is a 3D (M)RHD code capable of
simulating the magnetic solar chromosphere (Gudiksen et al., 2011). We used a
Bifrost simulation ranging from $\approx-2.5$ Mm below the optical surface up
to $8$ Mm above. From one snapshot of this full-fledged "chromospheric"
simulation, we extracted a purely "photospheric" simulation by cutting off the
layers above $1$ Mm height. This simulation was given 15 minutes to re-adjust
to the new boundary. The horizontal extent of the STAGGER atmosphere is
$8$x$8$ Mm, whereas it is $12$x$12$ Mm for the Bifrost atmosphere. Yet both
are large enough to host at least $10$ granules at the surface, and have the
(x,y,z)-resolutions of 240x240x230 (STAGGER) and 512x512x512 (Bifrost),
respectively. The horizontal resolution of all models was scaled down to allow
for repeated NLTE calculations with modest computational cost. Our detailed
tests (Fig. 13, also see Sect. 4.7) allow us to conclude that (x,y)
resolutions in excess of (30,30) are not necessary, as already this
configuration preserves the main physical properties of line formation and
yields line profiles that differ by no more than $2\%$ from the full geometric
setup. This uncertainty is sub-dominant to other sources of error in the
abundance analysis and can be tolerated. Nonetheless, we account for this
error in the determination of the solar O abundance (Sect. 4.6). The vertical
resolution remains unaffected, namely 230 grid-points for the STAGGER
simulation and 512 or 192 grid-points for the chromospheric and photospheric
Bifrost simulations, respectively.

STAGGER and Bifrost codes differ in the input abundances and in the equation
of states (EOS). STAGGER is using a modified version of the EOS from Mihalas
et al. (1988) and abundances from Asplund et al. (2009). The Bifrost
simulation used here was originally made to make comparisons with older
simulations from Stein & Nordlund and used an EOS from Gustafsson (1973),
abundances and continuum opacities from Gustafsson et al. (1975) computed
using the Uppsala background opacity package (Gustafsson, 1973).

The temperature structure of the combined down-scaled snapshots of all models
is shown in Fig. 4. The dashed lines present the structure of the spatially
and temporally averaged ($\langle{\rm 3D}\rangle$) model atmospheres. The
averaged models were computed by interpolating the full 3D cubes to an equally
spaced optical depth scale and averaging over horizontal slices of equal
optical depth. Subsequently all variables were averaged temporally over all
spatially averaged snapshots. This was done directly for the temperature and
velocity field, but in the case of electron density temporal and horizontal
averaging was performed on logarithm of electron number density.

###  3.6 NLTE statistical equilibrium calculations in 1D and 3D

We use two SE codes, MULTI2.3 (Carlsson, 1992) and MULTI3D (Leenaarts &
Carlsson, 2009; Leenaarts et al., 2012), both updated as described in
Bergemann et al. (2019) and Gallagher et al. (2020). MULTI2.3 solves the
detailed equations of SE in a 1D plane-parallel geometry using the Accelerated
Lambda Iteration (ALI) method (Rybicki & Hummer, 1991, 1992). Radiation
transfer is solved by the method of long characteristics. All calculations in
this work were carried out using the local operator acting on the source
function. MULTI3D solves the radiative transfer equations in 3D geometry on a
Cartesian mesh via the short characteristics method (Kunasz & Auer, 1988). It
is also capable of solving radiative transfer column-by-column mode, that is
ignoring the influence of slanted rays. Extensive tests of the full 3D and the
column-by-column solution revealed that the differences between the both
approaches are negligibly small, especially for the Sun, as the line EWs
change by less than $0.5\%$ (see, e.g. the tests carried out by Bergemann et
al., 2019). Our calculation for the O lines also shows that the difference is
only $0.6\%$ (Fig. 6). Therefore, we employ the column-by-column radiative
transfer solution with MULTI3D in this work.

Figure 6: Line profiles of the O I 7771.94 Å line calculated using the column-
by-column 3D NLTE solver and the full 3D NLTE radiative transfer solver. The
difference in the line EW is only $0.6\%$.

In addition to continuum opacities, both MULTI2.3 and MULTI3D are capable of
including background line opacity in the form of a linelist. For O, this not
important, because the statistical equilibrium of the element is not sensitive
to line blanketing. For Ni i, however, a comprehensive treatment of line
blanketing across the entire UV and optical wavelength ranges is critical,
because the element, similar to other Fe-group species, is photoionisation-
dominated. Since MULTI2.3 is also capable of including sampled opacities, we
use the code to compute 3D NLTE corrections for Ni separately using the
column-by-column solver and apply them to 3D LTE calculations for Ni, in order
to get 3D NLTE line profiles of Ni i lines. This approach is needed, because
it is currently not feasible to run MULTI3D with a linelist that is
comprehensive and detailed enough, that is, that includes the bound-bound
transitions of all relevant absorbers across the entire range of frequencies
in the Ni model atom. We have verified (see also Gallagher et al., 2020), that
both codes provide identical results when the same initial conditions are
used. A detailed analysis of the influence of line blanketing in 3D NLTE
calculations for different chemical elements will be presented elsewhere
(Semenova et al. in prep).

###  3.7 Abundance calculations for O and Ni

The calculations of O abundance were carried out using the model atmospheres
described in Sect. 3.5. In the 3D analysis, we employ $13$ consecutive STAGGER
snapshots, as well as $10$ consecutive snapshots for each of the two Bifrost
model atmospheres (with and without chromosphere). All model atmospheres were
probed by $17$ rays at $5$ different angles, one ray being the disc center
intensity ($\cos{\theta}=\mu=1$) and the other $16$ rays going north, east,
south and west at inclinations $\mu=0.8,0.6,0.4,$ and $0.2$ in order to match
the IAG and SST observations.

We use this setup to compute O i line profiles for a dense grid of O
abundances, sampling the $\log{\rm A(O)}$ range from $8.50$ to $8.90$ dex in
3D and $8.50$ to $9.20$ dex in 1D. Since the diagnostic lines of O are not
strong, it is sufficient to assume an equidistant abundance spacing of $0.2$
dex. For Ni, we assume the meteoritic abundance of $\log{\rm A(Ni)}=6.23$ dex
from Lodders (2003). This quantity has an uncertainty of $0.04$ dex, and it
appears to be superior to the values inferred by spectroscopic methods.
Asplund et al. (2009) find the Ni abundance of $6.22\pm 0.04$ dex, using 3D
LTE modelling. This value is significantly higher than the 3D LTE value
recommended by Caffau et al. (2015) ($\log{\rm A(Ni)}=6.11\pm 0.04$ dex). The
1D LTE estimate by Wood et al. (2014), based on the analysis of 76 Ni i and Ni
ii lines, is $6.27\pm 0.06$ dex. We adopt the meteoritic Ni abundance in our
work, because of the afore-mentioned differences in the photospheric results
by different groups. We return to this issue in Sect. 5.

The region around the forbidden [O I] line is modelled by co-adding the
profiles of the O and Ni lines in 1D LTE, 1D NLTE, or 3D NLTE respectively and
fitting the combined line profiles to the observations. Comparing this
approximate approach to the exact procedure, in which the absorption
coefficients are co-added instead of the intensity or flux profiles, we find
that it provides an almost identical solution (differing in EW by less than
$0.5\%$). Our detailed approach to the model-data comparison and abundance
diagnostics is described in Sect. 4.4 and 4.6.

##  4 Results

In this section, we describe the results of our calculations using different
model atmospheres and model atoms. We start with a brief overview of the
formation properties of O i lines in Sect. 4.1, continue with the statistical
equilibrium of Ni in Sect. 4.3, discuss our results for the spatially-resolved
spectra in Sect. 4.4 and for different atomic models in Sect. 4.5, and comment
on the relevance of 3D (M)HD in the abundance calculations in Sect. 4.7.3. We
then summarise the methods used in the probabilistic abundance analysis and
present the final O abundances in Sect. 4.6. Finally, we compare our results
with recent estimates in the literature in Sect. 5 and draw conclusions.

Figure 7: Departure coefficients of the energy levels involved in the
diagnostic transitions of O i.

Figure 8: Model 1D LTE, 1D NLTE, and 3D NLTE line profiles of the O i lines at
7775.39 Å(top panel) and 6300.3 Å(bottom panel).

###  4.1 O line formation

The formation of O triplet lines at 7771, 7774, and 7775 Å has been a subject
of intense discussions over the past decades. Kiselman (1991) was among the
first to point out that for the solar model atmosphere the NLTE effects in the
diagnostic O lines, which connect the two high-excitation states, are
relatively insensitive to the detailed structure of the model atom and are
controlled by scattering in the lines. The formation of O triplet lines in the
solar photosphere was extensively discussed in Asplund et al. (2004), Steffen
et al. (2015) and Amarsi et al. (2018).

Our calculations confirm that the NLTE effects in the triplet lines are mostly
driven by photon losses in the lines (Bergemann & Nordlander, 2014) that leads
to the deviation of the line source function from the Planck function. The
overpopulation of the lower level comes at the expense of the ground state
population of O i, via a sequence of transitions that involve charge exchange
reactions with O ii, re-combinations to the high-excitation O i levels, and
spontaneous transitions to the lower levels. This is illustrated in Fig. 7,
which shows the departure coefficients $b_{i}$ of the lower $3s\,^{5}{\rm
S}^{{\rm o}}_{\rm 2}$ and upper $3p\,^{5}{\rm P}$ energy states, which connect
the diagnostic triplet lines, against the continuum optical depth at $5000$ Å.
Owing to the small Boltzmann factor, the line formation is restricted to a
very narrow range of optical depths ($-1$ < $\log(\tau_{500})$ < 0). In this
region, the ratio of the departure coefficients of the upper and lower energy
states drops below unity, $b_{j}/b_{i}<1$. As a consequence, the NLTE profiles
of the permitted O i lines at 777 nm come out stronger compared to the LTE
line profiles (Fig. 8 top panel). In contrast, the departure coefficients of
the levels involved in the transition (the [O I] line at 630 nm) are very
close to unity and the NLTE effects in the line profiles are negligibly small
(Fig. 8, bottom panel).

Figure 9: Model continuum intensity in the region of 777 nm is shown in panel
(a) in absolute units of W/cm2/Å. Ratios of NLTE to LTE EWs of the oxygen 777
nm triplet lines (panel b), oxygen 630 nm line (panel c) and nickel 630nm line
(panel d) computed using the Stagger model atmosphere.

This formation of permitted O i lines is very similar in 1D NLTE and in 3D
NLTE (see also Asplund et al., 2004; Steffen et al., 2015; Amarsi et al.,
2018). The 777 nm lines are stronger in 3D NLTE compared to 1D LTE or 3D LTE,
that is, the abundances inferred from 3D NLTE line profiles are significantly
lower. The forbidden line is barely sensitive to NLTE effects, even in 3D
convective models. This is demonstrated in Fig. 9, which shows the absolute
continuum intensities (panel a) and the ratios of the NLTE and LTE equivalent
widths computed using the Stagger 3D model atmosphere for the 777 nm triplet
(panel b) and for the [O I] line at 6300 Å (panel c). The 777 nm lines are
stronger in 3D NLTE compared to 3D LTE across the entire simulation surface
and the NLTE effects are particularly prominent for the lines that form above
the inter-granular lanes. The forbidden [O I] line forms nearly in LTE,
although very weak departures from LTE (line weakening in NLTE) are present
above the inter-granular lanes.

###  4.2 O NLTE abundance corrections

Table 3 compares the results in terms of 1D NLTE and 3D NLTE abundance
corrections, that is the amount by which the abundance of O has be changed in
order to match the EWs of 1D NLTE and 3D NLTE line profiles to those of 1D LTE
line profiles. These quantities were computed for the disc-center intensity
and fluxes using the O abundance of 8.70 dex777We note that the correction is
almost identical, if the abundance of $8.90$ dex is adopted.. The abundance
corrections are not used in the abundance analysis, but are useful for their
didactic value.

Table 3: 1D NLTE and 3D NLTE abundance corrections for the disc center ($\mu=1$) intensities and fluxes of the diagnostic lines of O. The model atoms are defined as follows (Sect. 3.1): (a) LCAO data, (b) QPC data, and (c) LCAO data supplemented with the Kaulakys data. Line | Intensity ($\mu=1$) | Flux
---|---|---
| 1D NLTE | <3D> NLTE | 3D NLTE | LCAO
Å | a | b | c | a | b | c | a | b | c | 1D NLTE | <3D> NLTE | 3D NLTE
7771.940 |  $-$0.20 |  $-$0.17 |  $-$0.11 |  $-$0.16 |  $-$0.13 |  $-$0.06 |  $-$0.17 |  $-$0.15 |  $-$0.08 |  $-$0.28 |  $-$0.27 |  $-$0.31
7774.170 |  $-$0.18 |  $-$0.16 |  $-$0.10 |  $-$0.14 |  $-$0.11 |  $-$0.05 |  $-$0.15 |  $-$0.13 |  $-$0.07 |  $-$0.26 |  $-$0.24 |  $-$0.29
7775.390 |  $-$0.16 |  $-$0.14 |  $-$0.08 |  $-$0.11 |  $-$0.08 |  $-$0.02 |  $-$0.14 |  $-$0.12 |  $-$0.07 |  $-$0.23 |  $-$0.20 |  $-$0.27
6300.304 |  0.00 |  0.00 |  0.00 |  $+$0.04 |  $+$0.04 |  $+$0.04 |  $+$0.05 |  $+$0.05 | +0.05 |  0.00 |  $+$0.04 |  $+$0.04

Not surprisingly, the NLTE results and, consequently, the NLTE abundance
corrections depend strongly on the input atomic data, foremost on the rates of
H impact transitions. Using the quantum-mechanical data (LCAO or QPC), we
obtain the 1D NLTE corrections in the range from $-0.20$ dex (7771 Å line) to
$-0.16$ dex (7775 Å line) for the LCAO model, and $-0.17$ dex (7771 Å line) to
$-0.14$ dex (7775 Å line) for the QPC model. However, the results change
dramatically, if we include the Kaulakys data in the model, by co-adding them
with the LCAO datasets. In this case, the 1D NLTE corrections do not exceed
$-0.11$ dex for the 7771 line and $-0.08$ dex for the 7775 Å line. All other
quantities in the atomic models, such as the representation of H-impact
transition rates for the high-excitation energy states, electron collisions,
or photo-ionisation, do not influence the results at any significant level.
Likewise, the NLTE level populations and the profiles of the diagnostic lines
remain nearly identical, regardless of whether Drawin’s rates or a blanket
constant rate coefficient of $10^{-20}$ are assumed for the majority of the
uppermost states in O i (Fig. 15 in Appendix). As emphasised, however, it is
important to include these values to ensure rapid convergence, which is
important especially in extremely time-consuming 3D NLTE calculations.

The NLTE corrections computed using the spatially- and temporarily-averaged
$<$3D$>$ Stagger model atmosphere are less extreme. For the LCAO and QPC
atomic models (no Kaulakys data), the largest $<$3D$>$ NLTE corrections do not
exceed $-0.16$ dex and $-0.13$ dex (the 7771 Å line), respectively. The model
atom that includes LCAO and Kaulakys data returns the results that are not too
different from LTE: the largest NLTE correction amounts to $-0.06$ dex (7771 Å
line), whereas the 7775 Å line forms nearly in LTE, with the NLTE correction
of only $-0.02$ dex.

Interestingly, the 3D NLTE abundance corrections are very close to the
$<$3D$>$ NLTE results, although the former are slightly larger in absolute
value. For the LCAO model atom, the 3D NLTE corrections amount to $-0.17$ dex
for the 7771 Å line, $-0.15$ dex for the 7774 Å line, and $-0.14$ dex for the
7775 Å line. The latter line, which is the weakest of all three, is most
sensitive to the structure of the model atmospheres and to the properties of
the model atom. Its NLTE correction changes from $-0.16$ to $-0.02$, depending
on the details of modelling. The forbidden oxygen line forms almost in LTE,
but it shows a non-negligible sensitivity to convection. The 3D NLTE
correction for the 630 nm [O I] line amounts to $+0.05$ dex relative to 1D
LTE.

Table 3 also shows our 1D NLTE and 3D NLTE corrections for fluxes computed
using the LCAO model atom. These results can be directly compared to the
corresponding values from Asplund et al. (2004, their Tab. 2). In 1D NLTE,
they obtain $-0.24$ dex for the 7771 Å line, $-0.23$ dex for the 7774 Å line,
and $-0.20$ dex for the 7775 Å line. In 3D NLTE, they derive $-0.27$ dex for
the 7771 Å line, $-0.24$ dex for the 7774 Å line, and $-0.20$ dex for the 7775
Å line. Our values are in a good agreement with this study, although the
amplitude of our NLTE corrections is slightly larger. This can be explained by
the complexity of the atomic model. Our model includes 120 states of O i
coupled by a variety of processes (Sect. 3.1, whereas the atomic model
employed by Asplund et al. (2004) includes only $20$ O i states coupled by 43
radiative b-b transitions and electron collisions, and it neglects H-impact
collisions. Since the model atom from that study was not published, to check
the influence of model atom completeness we performed a sequence of
calculations in 1D NLTE (using the MARCS and $<$3D$>$ model atmosphere),
progressively reducing the complexity of the model atom down to $22$ levels
coupled by $57$ radiative b-b transitions. We find that the model that
includes $41$ oxygen states, with O i closed by the 4f 3F term (102968 cm-1)
reproduces exactly the results obtained using the complete model. However,
atomic models smaller than this limit lead to more modest NLTE corrections. A
22-level atom underestimates the NLTE corrections by $0.07$ dex, resulting
into 1D NLTE corrections of the order $-0.21$ (7771) to $-0.16$ dex. The
residual differences can be explained by Asplund et al. (2004) not including
H-impact collisions in their model and using the older version of the Stagger
model (M. Asplund priv. comm).

In summary, our findings are similar to those of the previous studies of 1D
NLTE and 3D NLTE line formation of O i lines in the solar atmosphere. As we
will show below, however, the details of the O i model atom (primarily, the H
collision data) and the approach to the statistical analysis of spectral
lines, lead to quantitatively different results, when it comes to the analysis
of the solar photospheric O abundance.

###  4.3 Ni line formation

The statistical equilibrium of Ni was previously investigated by Bruls (1993).
Using a model atom with $19$ energy levels and $81$ radiative transitions,
they found that Ni iis subject to over-ionisation and line pumping, the
processes that generally favour under-population of energy levels and cause
weakening of spectral lines compared to LTE. Over-ionisation and pumping are
driven by strong non-local UV radiation field and, similar to other Fe-group
species (Bergemann et al., 2012), influence the levels with excitation
energies of $\sim 3$ to $4$ eV. Ding et al. (2002) used a simplified 10-level
version of the model developed by Bruls (1993) in order to study the NLTE line
formation properties of the at $6767.8$ Å line, which is typically used in
helioseismology. Vieytes & Fontenla (2013) developed a larger model atom,
including $61$ energy levels and $401$ radiative transitions, and applied the
model atom in the calculations of the solar broad-band spectrum. None of these
studies considered the formation of the Ni i feature at 6300.341 Å that
contaminates the [O I] line.

Our calculations confirm that Ni i behaves as a classical minority element
with numerous important ionisation edges in the UV, which is subject to over-
ionisation by strong non-local radiation field. As seen in the diagram of the
departure coefficients (Fig. 10), the majority of energy levels in Ni i are
under-populated. The states with excitation energies of $2$ to $4$ eV are
over-ionised, whereas the higher-lying Ni i states are collisionally coupled
with the latter and therefore display similar NLTE effects. The ground state
of Ni ii is nearly thermalised across the entire optical depth scale, but
higher-lying Ni ii states are over-populated, closely resembling the behaviour
seen in other Fe-group elements, such as Cr, Ti, Mn, and Co.

Table 4: 1D and 3D NLTE abundance corrections (in dex) for the disc center intensities of the Ni I line at 6300.341 Å. Sbf the scaling factor to the collisional ionisation rates computed using the Drawin formula. Line | LTE | NLTE
---|---|---
3D | 1D | 3D
|  | $0$ | $100$ | $1000$ | $0$ | $100$ | $1000$
6300.341 | 0.06 | 0.15 | 0.08 | 0.02 | 0.21 | 0.14 | 0.08
Figure 10: NLTE departure coefficients of energy levels in Ni i. The levels
involved in the 6300.341 Å transition are highlighted. Figure 11: 1D LTE, 1D
NLTE, and 3D NLTE line profiles of the Ni lines at 6300.34 Å (top panel) and
5157.98 Å (bottom panel). The 6300.34 Å feature blends with the forbidden O I
line at 6300.304 Å.

Table 4 presents the 3D LTE, 1D NLTE, and 3D NLTE corrections for the key Ni
line at 6300.34 Å. The line is clearly sensitive to convection, with the 3D
LTE estimate being higher by $0.06$ dex compared to 1D LTE. The NLTE values
were computed using three atomic models of Ni, which differ only in the
scaling factor to Drawin’s ionisation rates. For the excitation rates, we used
the scaling factor of 0.05 (Sect. 3.4). Clearly, assuming no H-impact ionising
collisions (S${}_{\rm{bf}}=0$) or increasing them by a factor of $100$
(S${}_{\rm{bf}}=100$) has a significant influence on the NLTE correction,
which changes from $+0.15$ dex to $+0.08$ dex (MARCS) and from $+0.21$ dex to
$+0.14$ dex (3D Stagger) respectively. Assuming very large - likely
unrealistically too large - Drawin’s collisional rates
(S${}_{\rm{bf}}=1\,000$), the difference between 3D NLTE and 3D LTE
effectively vanishes. The scaling factors of S${}_{\rm{bf}}=0$ and
S${}_{\rm{bf}}=1\,000$ provide the maximum (hereafter, Ni-max) and minimum
(hereafter, Ni-min) limits on the NLTE effects in Ni lines, respectively. The
former approach assumes no H-impact collisional ionisation and the latter
results in nearly thermalised (LTE) level populations. It has to be kept in
mind, though, that the NLTE effects in Ni are very likely larger and thus, the
O abundance might increase. We include the uncertainty related to the
amplitude of NLTE effects in Ni into the final estimate of O abundance (Sect.
3.4).

Fig. 11 shows the line profiles of the 6300 Å and 5157.98 Å Ni i line,
computed using the model atom with collisions scaled by S${}_{\rm{bf}}=1000$
and $\log{\rm A(Ni)}=6.23$ dex. The latter spectral line was used in the
analysis of the solar Ni abundance in Scott et al. (2015) and it appears to be
relatively unblended compared to other Ni i features. Both lines are clearly
weaker in 3D LTE compared to 1D LTE, and in 3D NLTE the difference is even
larger. The weakening of Ni I lines in 3D LTE was also demonstrated by Scott
et al. (2015), although no NLTE calculations were performed in that study. To
the best of our knowledge, Bruls (1993) and Vieytes & Fontenla (2013) are the
only studies of NLTE effects Ni i in the solar spectrum, but they do not
provide estimates of line EWs and NLTE abundance corrections in 1D and 3D, so
a more quantitative comparison cannot be carried out.

Finally, we note that for Ni i, no quantum-mechanical estimates of photo-
ionisation cross-sections, H and e$-$ collision rates are available. So we
have to resort to classical recipes, but we caution that these recipes tend to
under-estimate the effects of over-ionisation and line pumping, as detailed
NLTE studies of other similar species demonstrated (e.g. Mn: Bergemann et al.
2019; Ti: Sitnova et al. 2020). Therefore our present calculations can be
viewed as a conservative scenario, yet the actual NLTE effects in the Ni i
lines are expected to be larger.

Figure 12: Center-to-limb variation of the combined oxygen triplet for
different model atom/atmosphere combinations. 1D is represented by the Marcs
solar model and 3D is represented by the Stagger solar model.

###  4.4 Center-to-limb variation of oxygen abundances

We begin with the analysis of O abundances obtained for different $\mu$ angles
and for different diagnostic lines of O. The best-fit 3D NLTE and 1D LTE
models are compared with the spatially-resolved observations of the Sun in
Fig. 16.

Figure 12 shows our abundance estimates obtained in 1D LTE, 1D NLTE, and 3D
LTE for each of the $5$ studied inclinations in the IAG and SST data (Sect.
2). Ideally the slope $\alpha$ of the overdrawn trend line would equal zero,
meaning that a given model returns self-consistent results at each angle. Low
$\alpha$ values are therefore, to a first order, a good indicator of a more
reliable result. Additionally, Fig. 18 (Appendix) shows the observed EWs
against the model predictions for 1D LTE, 1D NLTE, and 3D NLTE (LCAO$+$K).
This figure is useful for its pedagogical content, to illustrate the failure
of standard 1D or 3D LTE modelling. However, we do not use the actual observed
EWs in the abundance analysis (see Sect. 4.6).

Clearly, the 1D LTE assumption does not yield reliable abundance estimates for
the O i triplet (Fig. 12, top panels), as there is a strong negative
correlation between the individual 1D LTE abundance estimates and the viewing
angle, ranging from $\log{\rm A(O)}\approx 8.8$ dex at the disc center to
$\log{\rm A(O)}\approx 9.3$ dex at the limb ($\mu=0.2$). Also the 3D LTE
results are sub-optimal: the slope is slightly smaller compared to 1D LTE,
nonetheless the abundances range from $\log{\rm A(O)}\approx 8.85$ dex at
$\mu=1$ to $\approx 9.10$ dex at $\mu=0.2$, with a significant systematic
bias. 1D NLTE results are similar to 3D LTE in terms of the $\mu$ \- $\log{\rm
A(O)}$ slope, however, the disc-center 1D NLTE abundance (here shown for the
model atom LCAO$+$Kaul), $\log{\rm A(O)}=8.66$ dex, is in a much better
agreement with the 3D NLTE disc-center abundance obtained using the Stagger
model, $\log{\rm A(O)}=8.70$ dex. However, the 3D NLTE approach clearly
outperforms the analyses in that it leads to significantly stronger profiles
of the triplet lines at the limb, thereby improving the agreement with the
observations (see also Kiselman & Nordlund, 1995; Asplund et al., 2004). The
results obtained using the IAG and SST data are similar in terms of their CLV
abundance slopes.

The forbidden [O I] line (Fig. 12, bottom panels) is less sensitive to the
model atmosphere and to the O model atom, as the angle-dependent abundances
are similar in 1D LTE and in 1D NLTE. However, all 3D NLTE and 3D LTE values
are about $0.10$ dex higher compared to 1D results. The differences between
various 3D NLTE CLV results are entirely due to the differences in the
predicted strength of the Ni blend at 6300.34 Å. The SST results show peculiar
undulations, with abundances at $\mu=1$ and $\mu=0.4$ being significantly
lower compared to other angles. The origin of this systematic is difficult to
pin-point. It is possible that the subtraction of fringing and other
instrumental artefacts in these data lead to residual systematic biases. In
fact, it is clearly seen in the Fig. 2 of Pereira et al. (2009a) that the O I
line unfortunately coincides with a minimum in the fringe pattern used to
model the continuum of the SST data. The O abundances based on the IAG data at
different angles are more internally consistent, supporting the choice of
these observations as currently the most reliable dataset.

###  4.5 Influence of O+H collisions

It is useful to look more closely into the abundances computed from different
NLTE model atoms of O. In this respect, the most critical reactions are
inelastic collisions between O and H atoms, which have a direct influence on
the statistical equilibrium of O i.

As shown in Sect. 3.3, our new rate coefficients that describe the transitions
in O i caused by collisions with H are different from the previous estimates
by Barklem (2018). The latter quantities (LCAO) are on average smaller
compared to the data computed using the QPC method. However, if we co-add the
data from Barklem (2018) with the data from Kaulakys (1991), in order to
account for the lack of short-rang interactions in the LCAO approach (as
recommended by Amarsi et al., 2018), the net result is that for the majority
of energy levels the combined LCAO$+$Kaulakys rates are higher compared to the
LCAO and to the QPC data.

As a result, the O abundances computed from the 777 triplet lines using the
three atomic models show a clear systematic difference (Fig. 12, top panels).
The LCAO$+$Kaulakys model atom leads to a $\approx+0.1$ (disc center) to
$\approx+0.15$ dex (limb) higher solar O abundance compared to the LCAO and
QPC models. This is because the latter two models, owing to the less efficient
collisional thermalisation they produce, yield larger NLTE effects in the 777
nm O i lines. However, neither of the atomic models can be given a preference,
based on the CLV - abundance slope, because the moduli of the slopes are very
similar. This suggests that the CLV of O lines alone is not sufficient as a
metric to distinguish between the three NLTE atomic models. In other words,
each of the model atoms - LCAO, QPC, or LCAO$+$Kaulakys - allows us to achieve
internally consistent results, which are, however, systematically different
with respect to each other.

For completeness, we note that the choice of collisional data in the O model
atom is of no relevance in modelling the 630 nm oxygen line, as the
populations of its energy levels are very close to thermal and the NLTE
effects are very minor.

###  4.6 Model-data comparison

We determine the best-fit abundances by employing the $\chi^{2}$ statistics,
comparing the observed data with the model line profiles. Line profiles for a
fine grid of abundances are produced and for each observation we find the most
probable abundance by minimising the reduced $\chi^{2}$:

$\chi^{2}=\frac{1}{N_{\rm pix}-N_{\rm free}}\sum_{i}^{N_{\rm
pix}}\left[\frac{f_{\rm obs,i}-f_{\rm model,i}}{\sigma_{\rm
obs,i}}\right]^{2},$ (1)

where $N_{\rm pix}$ is the number of wavelength points and $N_{\rm free}$ is 1
as the oxygen abundance is the only free parameter. The observational error
$\sigma_{f_{\rm obs}}$ represents the uncertainty caused by the limited
signal-to-noise ratio of the data.

For the 777 nm triplet lines, we mask the observations to the region of
$\pm$0.5 Å from the line centers and a few surrounding blends (Cr 1 at 7771.76
Å, Ti 1 at 7771.41 Å, Ti 1 at 7772.16 Å, and Sc 1 7772.56 Å), modelled in 1D
LTE, are subtracted from the observations. These blends are weak and they are
entirely contained in the absorption features associated with the triplet
lines. Since the exact position of the line centers varies between different
atlases and $\mu$-angles, the models are slightly blue- or red-shifted, so
that the synthetic line centers match the observed data.

The 630 nm [O I] line is located in a region, which contains a number of
blends, Si 1 ($6299.599$ Å), Sc 2 ($6300.698$ Å), and Fe 1 ($6301.500$ Å).
However, models are woefully inadequate to reproduce the observations in this
region (Fig. 19). So to estimate the contribution due to blends, that is to
de-blend the O line, we adopted the following approach. We fitted arbitrary
Voigt profiles to each of the blending features, allowing for the variation of
the amplitude, $\gamma$ and $\sigma$ parameters in the Gaussian and Lorentzian
profiles. We also allowed for parabolic bisectors in order to deal with the
asymmetry of both lines. This fit (Fig. 20) was then used to determine the O
abundance from the de-blended O$+$Ni feature using a mask with the width of
$\pm 0.3$ Å. The [O I] line was modelled using a grid of oxygen abundances,
while the Ni was modelled with a fixed abundance of 6.23 dex. The meteoritic
value has an uncertainty of $0.04$ dex, but this uncertainty is entirely
covered by the difference in the Ni blend contribution caused by adopting
different NLTE atomic models of Ni.

###  4.7 Systematic errors

The model (systematic) error $\sigma_{f_{\rm model}}$ comprises several
sources, including the errors caused by the oscillator strengths and damping
constants, as well as the errors associated with the limited geometric
resolution of the 3D model atmosphere. These errors are not included in the
$\chi^{2}$ calculations (Eq. 1), but are considered separately in the
abundance analysis (see Table 5).

####  4.7.1 Atomic data

The error caused by the uncertainty of oscillator strengths is fully
correlated with the abundance error. As described in Sect. 3.1, we use the
Hibbert et al. (1991) data for the 777 triplet lines and the Storey & Zeippen
(2000) data for the 630 nm forbidden line. However, recently new $f$ values
for allowed transitions were presented by Civiš et al. (2018). These data are
based on the Quantum Defect Theory (QDT) approach and, for the 777 nm triplet
lines they are 12% ($-0.05$ dex) lower than the $f$-values provided by Hibbert
et al. (1991) and recommended by NIST. The difference between the $f$-values
from the two sources is four times the NIST estimated uncertainty in the
transition.

In order to understand the uncertainties of the $f$-values we carried out new
atomic calculation. We used the code AUTOSTRUCTURE (Badnell, 2011) to do
multiple calculations of increasing complexity. We started with the same
configuration expansion as used by Storey & Zeippen (2000). Further, we added
additional configurations with the same orbitals with principal quantum number
n$\leq 4$ and configurations with n=5. In each case we tried different orbital
optimization schemes. From the different calculations we find a scatter in the
$f$-value of the 777 nm transitions of about $10\%$ with a tendency for the
rate to be closer to lower value of Civiš et al. (2018) than to the Hibbert et
al. (1991) value. Therefore, for the 777 lines we adopted the average of both
values as our central value, and the associated $f$-value error (2 $\sigma$)
is assumed to be the difference between the both quantities.

The $f$-value for the magnetic dipole line at 630 nm is much more difficult to
pin point. Such weak lines are generally very sensitive to cancellation
effects among different contributions from configuration interaction
representations of the atomic levels. The main challenge is not the
convergence of the expansion, but the sensitivity to how the orbitals are
optimized. The NIST recommended transition probability for this line is
$5.63\times 10^{-3}$ s-1 with the uncertainty flag B$+$ (better than $7\%$),
which is adopted from Baluja & Zeippen (1988) and Fischer & Saha (1983). The
most recent calculations by Storey & Zeippen (2000) yield $6.45\times 10^{-3}$
s-1. This is a difference of $\sim 15\%$, which is more than twice the
estimated uncertainty in this rate by NIST. The results of our own
calculations show the rate varies widely between $\sim 3\times 10^{-3}$ s-1
and $\sim 1\times 10^{-2}$ s-1, with the best value being $\sim 8\times
10^{-3}$ s-1. As stated earlier, we use the Storey & Zeippen (2000) rate in
our analysis, however, to account for the extreme sensitivity of the
$f$-values to the details of orbital optimisation, we assume a more realistic
uncertainty of 20%.

Finally, we investigated the uncertainty associated with damping, by
performing spectrum synthesis calculations with a modified, by $\pm 10\%$,
value of the damping constant. This, however, has a negligible influence on
the O triplet and [O I] lines and is therefore neglected.

####  4.7.2 Geometric resolution

We also take into account a systematic error associated with the limited
geometric resolution of 3D model atmospheres used in radiative transfer
calculations. This correction is estimated as follows. The two families of
photospheric 3D atmospheres employed in this paper have (x,y,z)-resolutions of
240x240x230 (STAGGER) and 512x512x192 (Bifrost-phot). We restrain from re-
sampling the vertical resolution, but down-sample the number of vertical
columns, ensuring that we have enough columns to maintain the ratio of
intergranular lanes to granules covering the entire simulation. We converted a
single STAGGER snapshot to horizontal resolutions 5x5, 10x10, 20x20, 30x30 and
120x120 and computed equivalent widths of the diagnostic O i lines for each
down-sampled snapshot. Similarly, equivalent widths for a photospheric Bifrost
snapshot with different horizontal resolutions between $10$x$10$ and
$160$x$160$ were computed. These calculations were carried out in LTE, because
our previous analysis (Bergemann et al., 2019) confirmed that the scaling is
very similar in LTE and in NLTE.

Figure 13: Influence of the geometrical resolution (nx,ny) of 3D STAGGER model
on the measured equivalent widths. Here nx is kept constant at 5. ny varies
from 5 to 240, where 240x240 corresponds to the full resolution of the model.
The vertical dashed line shows our default geometric resolution (30,30)
adopted in 3D NLTE radiative transfer calculations.

We find that the horizontal resolution of (x,y)$=(30,30)$ offers a reasonable
compromise between the computational expense of 3D NLTE modelling and the
physical realism, leading to a small ($\sim 1-2\%$) systematic error in the
line strength compared to that obtained the full geometric setup. To correct
for this small bias, we extract the geometric resolution correction (Table 5)
by comparing the EWs of lines computed at our nominal resolution and at the
highest-possible resolution. This correction is then separately applied to the
abundance determined at each $\mu$ angle. The results are shown in Fig. 13.
The disc-center and limb results are not very sensitive to this effect, but at
$\mu=0.8$ and $\mu=0.6$, the EW error caused by using the horizontal
resolution of (x,y)$=(30,30)$ can be as large as $3\%$. For the disc-center,
the abundance correction for the diagnostic O lines amounts to $0.01$ dex.

Table 5: Errors in O abundance calculations. The errors caused by the chromosphere and limited geometric resolution in 3D RT calculations are strictly systematic and they are directly applied to the resulting abundances. We note that these errors refer to the estimates at the disc center. See text. Line | f-value | Ni blend | Observations | Chromosphere | Resolution
---|---|---|---|---|---
Å | dex | dex | dex | dex | dex
7771.940 |  $\pm$ 0.026 | - |  $\pm$ 0.015 | +0.007 | +0.010
7774.170 |  $\pm$ 0.026 | - |  $\pm$ 0.015 | +0.007 | +0.010
7775.390 |  $\pm$ 0.026 | - |  $\pm$ 0.015 | +0.005 | +0.010
6300.304 |  $\pm$ 0.08 |  $\pm$ 0.03 |  $\pm$ 0.035 | -0.003 | +0.011

####  4.7.3 Influence of the chromosphere

We do not use the Bifrost models to derive the absolute solar O abundance
(Sect. 3.5), but only to quantify the influence of the chromosphere, that is,
the re-adjustment of the photospheric structure under the influence of the
chromosphere, on the line formation and on the O abundances. This is done by
comparing the results calculated with two types of Bifrost models: (a) the
full simulation covering the photosphere and the chromosphere, and (b) the
strictly photospheric simulation (Sect. 3.5). Each simulation represents an
extended time series, from which we extract $10$ snapshots for the detailed
radiative transfer. The average difference between the $T_{\rm eff}$ values of
the snapshots is $~{}10$ K.

Comparing the O abundances derived using the two Bifrost model sequences in 3D
NLTE, we find we find that the abundance differences are not large (Table 5).
The chromosphere weakens the 777 triplet lines that corresponds to the
increase of O abundance by $\sim+0.01$ dex at the disc center and $+0.02$ dex
at the limb relative to a pure photospheric simulation. The 6300 [O I] line is
relatively unaffected by the chromosphere, and the O abundance inferred from
this line is only $0.003$ dex lower. We note that even ignoring the snapshot
with the largest $T_{\rm eff}$ difference ($16$ K) does not alter the estimate
of chromospheric correction by more than $0.001$ dex.

Whereas the chromospheric correction for the O lines is small, the presence of
a chromosphere may have a larger influence on the spectral lines that form
higher up in the layers close and above the temperature minimum, such as the
Ni 6767 Å helioseismology line.

Table 6: Final 3D NLTE abundance estimates obtained from the 777 nm and 630 nm oxygen lines. | 777 nm | 630 nm |
---|---|---|---
IAG | 8.739 | 8.771 |
SST | 8.716 | 8.737 |
Hinode | - | 8.748 |
DST | - | 8.799 |

###  4.8 Final O abundance

Table 6 provides our estimates of the photospheric O abundance computed in 3D
NLTE from different observational datasets, where the results were corrected
for the chromospheric and spatial resolution effects (Sects. 4.7.2, 4.7.3).
The individual results obtained in 1D LTE, 1D NLTE, 3D NLTE with different
atomic models are provided in Tab. 7 (Appendix). The EWs of the corresponding
lines are tabulated in Tab. 8 (Appendix).

To compute the final estimates of O abundance and its uncertainties from
different observational data and different atomic models, we proceed as
follows. For both 777 and 630 nm lines, we adopt the central values as derived
from the IAG data using the LCAO$+$K model atom of O, because of the superior
resolving power of these observations and more consistent estimates of
abundances derived from the spatially-resolved spectra taken at different
pointings across the solar disc 888We note that no optimisation by hand was
involved in the analysis.. The LCAO$+$K (with Kaulakys data) atomic model,
furthermore, provides a flatter CLV abundance slope, compared to the LCAO or
QPC collisional data, but also from the perspective of atomic physics, the
Kaulakys data appear to be necessary, as they are expected to compensate for
the lack of short-range interactions in the quantum-mechanical calculations.
We also adopt the $S_{\rm bf}=100$ model atom of Ni, as this NLTE model
provides the average of the plausible 3D NLTE Ni blend contributions to the [O
I] feature that corresponds to the average between the extremely high,
moderate, and negligible collision rates (Sect. 4.3).

To estimate the final combined uncertainties, we resort to a Monte Carlo
simulation and make use of the central limit theorem, which states that the
normalised sum of random uncorrelated variables closely follows a normal
distribution, regardless of the shape of the distribution function of
individual variables. This is a commonly used approach in case when the shapes
of uncertainties are not known, as in our case. The errors associated with the
analysis (see Tab. 5) are used to generate three uniform distributions (large
samples of 1 million points each), representing (a) the uncertainty of the
oscillator strengths ($f$-value), (b) the uncertainty caused by the Ni blend
(630 nm line) and that caused by the O$+$H collisional data (777 nm lines),
and (c) the uncertainty caused by using different observed spectra. The
uncertainty caused by the collisional data is taken to be the difference
between the QPC and LCAO$+$K results. The latter is assumed to be one half of
the difference between the SST (min) and the IAG (max) results for the 777 nm
line, and that between the SST (min) and DST (max) results for the 630 nm
line. These errors are co-added and one standard deviation ($1\sigma$) of the
resulting distribution (Fig. 14) is adopted as the final uncertainty. We note
that the final combined distribution of errors for the 777 nm line is very
close to normal, whereas for the 630 nm line there is a small deviation owing
to a large dominant uncertainty source. However, this has no influence on the
conclusions.

Figure 14: Final combined distribution function for the solar photospheric
abundance of oxygen. See Sect. 4.8. A normal distribution is overplotted with
a thin line, for comparison.

As a result, our 3D NLTE value derived from the 777 and 630 nm lines is
$\log{\rm A(O)}=8.739\pm 0.027$ dex and $\log{\rm A(O)}=8.771\pm 0.053$ dex,
respectively. Assuming that the results are independent, we can combine them
to obtain the final photospheric O abundance value of $\log{\rm A(O)}=8.75\pm
0.03$ dex.

We note that if we were to use the old Hibbert et al. (1991) $f$-values and
the SST data, our results based on the LCAO$+$K model atom of O would be in
excellent agreement with the 3D NLTE estimate by Amarsi et al. (2018) who
relied on the same input. Likewise, our 1D NLTE result based on the MARCS
model is in agreement with that by Sitnova & Mashonkina (2018).

##  5 Discussion

Our estimates of O abundances based on the permitted 777 nm triplet lines and
on the 630 forbidden [O I] line are in a good agreement within the respective
uncertainties. The triplet lines are, however, sensitive to the input physics
of the O model atom, whereas the forbidden line is sensitive to the input
physics of the Ni model atom, primarily to the collisional data. Also, the
$f$-values carry a significant source of uncertainty, as the direct
experimental verification of the transition probabilities for the 777 nm and
630 nm is not possible. These errors are associated with the physical
limitations of theory of atomic physics, molecular physics, and collisional
dynamics, and, therefore, their true distributions are unknown. Nonetheless,
considering all sources of error, it is very plausible that the photospheric O
abundance is close to $\log{\rm A(O)}=8.75$ dex.

It is interesting to discuss our results in comparison with the other recent
estimates of the solar photospheric O abundance. Amarsi et al. (2018) and
Asplund et al. (2021) found $\log{\rm A(O)}=8.69\pm 0.03$ dex and $8.69\pm
0.04$ dex, respectively. The former estimate is based on the 777 nm line,
while the latter also includes the 630 nm and selected molecular OH lines. In
principle, our results and those by this group are not inconsistent within
their combined uncertainties. Our value is in a much better agreement with the
values proposed by Caffau et al. (2008), Caffau et al. (2015), and Steffen et
al. (2015), who analysed different O lines, including several forbidden [O I]
lines, the 777 triplet, and the near-IR O I features. Another recent study of
the solar photospheric O abundance is that by Cubas Armas et al. (2020). Their
analysis is based on a semi-empirical method of determining the abundance by
inverting spatially-resolved observed solar spectra. Their estimate, based on
the forbidden [O I] line at 630 nm, is $\log{\rm A(O)}=8.80\pm 0.03$ dex, also
consistent with our results for this feature.

##  6 Conclusions

We present a re-analysis of the solar photospheric O abundance using new
atomic data for O and different 3D radiation hydro-dynamical models of solar
atmospheres. The NLTE model atom includes novel rates of collisions between O
and H atoms, computed using the quantum hopping probability current method
(Belyaev et al., 2019), increasing the number of O i terms with accurate
collisional data to $16$. The R-matrix method for atomic scattering
calculations (Berrington et al., 1978) is employed to compute new
photoionisation cross-sections and the rates describing transitions caused by
inelastic processes in collisions of O atoms with free electrons. We also
present a new comprehensive model atom of Ni, which, for the lack of detailed
quantum-mechanical data, is still based on classical formulae for
photoionisation and collisional reaction rates. We use the 1D MARCS solar
model atmosphere, the 3D Stagger model (Collet et al., 2011; Magic et al.,
2013), and two version of 3D MHD models computed self-consistently, with and
without the chromosphere, using the Bifrost code (Carlsson et al., 2016).

The model atoms are used to performed 1D LTE, 1D NLTE, and 3D NLTE radiative
transfer calculations and abundance analysis of O lines in the solar spectrum.
We focus on the least blended lines of O i at 7771, 7774, 7775 Å, and we also
consider the forbidden O i line at 6300 Å. We also perform 3D NLTE
calculations of the Ni i blend at 6300.341 Å, which contributes about $25\%$
of the entire absorption feature. The observational data are taken from
different sources. Our primary source of data are the new spatially-resolved
spectra taken with the IAG FTS instrument. The data have a resolving power of
$R\sim 700\,000$, much higher compared to all previously available solar data.
Similar to previous studies, we also include the spectra obtained with the
ground-based Swedish Solar Telescope (Pereira et al., 2009a), the Domeless
Solar Telescope (Takeda & UeNo, 2019), and with the space-based Solar Optical
Telescope on the HINODE satellite (Caffau et al., 2015).

We obtain the solar O abundance of $\log{\rm A(O)}=8.739\pm~{}0.027$ dex and
$\log{\rm A(O)}=8.771\pm~{}0.053$ dex based on the 777 nm triplet and 630 nm
forbidden [O I] lines, respectively. Our final combined value is $\log{\rm
A(O)}=8.75\pm~{}0.03$ dex, which is close to the estimate proposed in the
compilation of the solar abundances by Caffau et al. (2008, $8.76\pm 0.07$
dex), but slightly higher than the recent value by Asplund et al. (2021,
$8.69\pm 0.04$ dex). The differences between our and previous results can be
explained by the updated transition probabilities for the 777 lines, new
higher-resolution observations collected with the IAG FTS facility, 3D NLTE
modelling of the Ni blend, and a correction for a systematic bias caused by
the chromospheric back-heating. These effects add up to increase the solar
abundance by about $+0.05$ dex relative to the value proposed by Asplund et
al. In addition, we confirm that using the HINODE space-based data, which were
employed by Caffau et al. (2015), we recover a slightly higher abundance
compared to the value obtained by using the SST data chosen by Asplund et al.
(2021).

Our results thus suggest that the question of whether the solar photospheric O
abundance is low or high, that is closer to the values from Grevesse & Sauval
1998, is still open. There are remaining controversies associated with
modelling of the permitted 777 nm lines and of the forbidden [O I] line.
First, the large systematic differences between two sources of H-impact
collisional data (Kaulakys 1991 and Belyaev et al. 2019) must be resolved.
Second, the differences between the new lower $f$-values from Civiš et al.
(2018) and the older values from Hibbert et al. (1991) must be settled. Until
then, we recommend to use the average of the values from both sources. Our
results suggest that the indirect influence of the chromosphere on the O lines
in the solar spectrum is not large. However, it is desirable to perform new
self-consistent 3D R(M)HD simulations of the solar atmosphere, including
chromosphere, with state-of-the-art micro-physics and Non-LTE radiative
transfer.

## Acknowledgements

We are grateful to the anonymous referee for the helpful remarks and
suggestions on the manuscript. We thank Elisabetta Caffau for providing the
spectra taken with Solar Optical Telescope on the Hinode satellite. We are
grateful to Martin Asplund and Hans-Guenther Ludwig for valuable discussions
and comments on the manuscript. SAY, YVV and AKB gratefully acknowledge
support from the Ministry of Education (the Russian Federation), project
FSZN-2020-0026. MB and ES are supported through the Lise Meitner grant from
the Max Planck Society. We acknowledge support by the Collaborative Research
centre SFB 881 (projects A5, A10), Heidelberg University, of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation). ME is supported
through the SPP 1992 _Exoplanet Diversity_ grant RE 1664/17-1. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 949173).

## Data Availability

The O model atoms (QPC, LCAO, LCAO$+$K) and the error map representing the
blends around the 630.0 nm O$+$Ni feature are available at
https://keeper.mpdl.mpg.de/d/da74c98c1c9c4477b67f/

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##  Appendix A Appendix

Figure 15: Profiles of the 7771 Å line computed in NLTE using three different
atomic models with the collisional rate coefficients for the upper 100 states
uniformly set to $10^{-20}$, assumed to follow the Drawin’s recipe, and scaled
by a factor of 10 relative to Drawin. Figure 16: 3D NLTE profiles of the 777
nm and 630 nm lines computed using the Stagger 3D model atmosphere and the
LCAO + Kaulakys model atom, compared with the spatially-resolved IAG
observations of the Sun for different $\mu$ angles (top to bottom). Figure 17:
1D LTE profiles of the 777 nm and 630 nm lines computed using the MARCS 1D
model atmosphere and the LCAO + Kaulakys model atom, compared with the
spatially-resolved IAG observations of the Sun for different $\mu$ angles (top
to bottom).

Figure 18: Comparison of observed EWs of the 7771.94 and 6300.3 Å features with the predictions of 1D LTE, 1D NLTE, 3D NLTE models. We note that the O abundances are determined using the more correct full profile fitting method, not from the line EWs. Figure 19: Spectral lines surrounding the oxygen + nickel blend at 630 nm at the disc center $\mu=1.0$. Figure 20: Top panel shows the Voigt profiles used to de-blend of the 630 nm [O I]$+$Ni feature. In addition, we show the classical technique, in which a local quasi-continuum is fit directly through the local maxima of the relative flux (here, just for demonstration we employ the dark grey points). The grey lines in the bottom panel depict the integration limits of the equivalent widths stated in the legend. These EW limits are used for demonstration only, but not for the abundance estimate (see Sect. 4.6). Table 7: Individual abundance estimates for different diagnostic lines of O (777 triplet and 630 forbidden features). We note that the modelling of the 630 nm Ni I line is fully consistent with the modelling of O lines. That is, Ni is computed in 1D LTE if 1D LTE is adopted for O, and in 3D NLTE if 3D NLTE is used for O. |  | 1D LTE | 3D LTE | 1D NLTE | 3D NLTE | 3D NLTE | 3D NLTE
---|---|---|---|---|---|---|---
|  |  |  | (LCAO+K) | (LCAO) | (QPC) | (LCAO+K)
Spectrum | Line |  |  |  |  |  |
IAG | 777 | 8.803 | 8.876 | 8.691 | 8.643 | 8.671 | 8.739
| 630 (Ni-max) |  |  |  | 8.795 | 8.795 | 8.795
| 630 (Ni-ref) | 8.661 | 8.743 | 8.671 | 8.771 | 8.771 | 8.771
| 630 (Ni-min) |  |  |  | 8.743 | 8.743 | 8.743
SST | 777 | 8.784 | 8.856 | 8.671 | 8.621 | 8.648 | 8.716
| 630 (Ni-max) |  |  |  | 8.763 | 8.763 | 8.763
| 630 (Ni-ref) | 8.622 | 8.705 | 8.632 | 8.737 | 8.737 | 8.737
| 630 (Ni-min) |  |  |  | 8.706 | 8.706 | 8.706
Hinode | 777 |  |  |  |  |  |
| 630 (Ni-max) |  |  |  | 8.773 | 8.773 | 8.773
| 630 (Ni-ref) | 8.633 | 8.719 | 8.643 | 8.748 | 8.748 | 8.748
| 630 (Ni-min) |  |  |  | 8.720 | 8.720 | 8.720
DST | 777 |  |  |  |  |  |
| 630 (Ni-max) |  |  |  | 8.822 | 8.822 | 8.822
| 630 (Ni-ref) | 8.699 | 8.774 | 8.708 | 8.799 | 8.799 | 8.799
| 630 (Ni-min) |  |  |  | 8.774 | 8.774 | 8.774
Neckel | 777 | 8.793 | 8.866 | 8.682 | 8.635 | 8.662 | 8.729
| 630 (Ni-max) |  |  |  | 8.788 | 8.788 | 8.788
| 630 (Ni-ref) | 8.652 | 8.734 | 8.662 | 8.763 | 8.763 | 8.763
| 630 (Ni-min) |  |  |  | 8.734 | 8.734 | 8.734
KPNO | 777 | 8.797 | 8.869 | 8.682 | 8.632 | 8.660 | 8.728
| 630 (Ni-max) |  |  |  | 8.803 | 8.803 | 8.803
| 630 (Ni-ref) | 8.671 | 8.751 | 8.680 | 8.779 | 8.779 | 8.779
| 630 (Ni-min) |  |  |  | 8.751 | 8.751 | 8.751
Table 8: EWs estimates of spectral lines obtained using 3D NLTE line profiles of the LCAO + Kaulakys model atom. All equivalent widths in mÅ. Specie | line [Å] |  $\mu$ = 1.0 |  $\mu$ = 0.8 |  $\mu$ = 0.6 |  $\mu$ = 0.4 |  $\mu$ = 0.2
---|---|---|---|---|---|---
O I | 7771.94 | 84.25 | 76.67 | 69.39 | 60.67 | 47.26
O I | 7774.17 | 74.48 | 67.55 | 60.70 | 52.52 | 40.28
O I | 7775.39 | 59.26 | 53.30 | 47.15 | 39.88 | 29.49
O I | 6300.30 | 3.83 | 4.09 | 4.60 | 5.11 | 5.17
Ni I | 6300.34 | 1.49 | 1.56 | 1.66 | 1.72 | 1.72