11institutetext: Department of Astronomy, Stockholm University, Albanova
University Center, 106 91 Stockholm, Sweden
11email<EMAIL_ADDRESS>22institutetext: Theoretical Astrophysics,
Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden 33institutetext: Stellar Astrophysics Centre, Department of
Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,
Denmark 44institutetext: ARC Centre of Excellence for All Sky Astrophysics in
3 Dimensions (ASTRO 3D), Australia 55institutetext: Research School of
Astronomy and Astrophysics, Australian National University, Canberra, ACT
2611, Australia 66institutetext: School of Physics and Astronomy, Monash
University, Clayton, Vic 3800, Australia

# Raising the observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.

C. Lagae Raising the observed metallicity floor with a 3D non-LTE analysis of
SDSS J102915.14+172927.9††thanks: Based on observations obtained at ESO
Paranal Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the
observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. A. M. Amarsi Raising the
observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. L. F. Rodríguez Díaz
Raising the observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. K. Lind Raising the
observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. T. Nordlander Raising
the observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. T. T. Hansen Raising the
observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio. A. Heger Raising the
observed metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.Raising the observed
metallicity floor with a 3D non-LTE analysis of SDSS
J102915.14+172927.9††thanks: Based on observations obtained at ESO Paranal
Observatory, programme 286.D-5045, P.I. P. Bonifacio.

(Accepted 2 March 2023; Received 23 December 2022)

###### Abstract

Context. The first stars mark the end of the cosmic dark ages, produce the
first heavy elements, and set the stage for the formation of the first
galaxies. Accurate chemical abundances of ultra metal-poor stars
($[\mathrm{Fe}/\mathrm{H}]<-4$) can be used to infer properties of the first
stars, and thus the formation mechanism for low-mass second generation stars
in the early universe. Spectroscopic studies have shown that most second
generation stars are carbon-enhanced. A notable exception is SDSS
J102915.14+172927.9, the most metal-poor star known to date, largely by virtue
of the low upper limits of the carbon abundance reported in earlier studies.

Aims. We reanalyse the composition of SDSS J102915.14+172927.9 with the aim to
provide improved observational constraints on the lowest metallicity possible
for low-mass star formation and constrain the properties of its Population III
progenitor star.

Methods. We developed a tailored three dimensional (3D) model atmosphere for
SDSS J102915.14+172927.9 with the Stagger-code, making use of an improved
surface gravity estimate based on the Gaia DR3 parallax. Snapshots from the
model were used as input in the radiative transfer code Balder to compute 3D
non-local thermodynamic equilibrium (non-LTE) synthetic spectra. These spectra
were then used to infer abundances for Mg, Si, Ca, Fe and Ni, and upper limits
on Li, Na and Al. Synthetic 3D LTE spectra were computed with Scate to infer
the abundance of Ti and upper limits on C and N.

Results. In contrast to earlier works based on 1D non-LTE corrections to 3D
LTE results, we are able to achieve ionisation balance for Ca I and Ca II when
employing our consistent 3D non-LTE treatment. The elemental abundances are
systematically higher than those found in earlier works. In particular, [Fe/H]
is increased by $0.57\,\mathrm{dex}$, and the upper limits of C and N are
larger by $0.90\,\mathrm{dex}$ and $1.82\,\mathrm{dex}$, respectively.

Conclusions. We find that Population III progenitors with masses
$10-20\,\mathrm{M}_{\sun}$ exploding with energy $E\lessapprox 3\cdot
10^{51}\,\mathrm{erg}$ can reproduce our 3D non-LTE abundance pattern. Our 3D
non-LTE abundances are able to better constrain the progenitor mass and
explosion energy as compared to our 1D LTE abundances. Contrary to previous
work, we obtain higher upper limits on the carbon abundance that are
“marginally consistent” with star formation through atomic line cooling, and
as such, prevent strong conclusions about the formation mechanism of this low
mass star.

###### Key Words.:

Atomic processes — Radiative transfer — Stars: atmospheres — Stars: abundances
— Stars: Population II — Stars: Population III

##  1 Introduction

The first stars (Population III stars, hereafter Pop III stars) in our
universe formed directly from Big Bang nucleosynthesis products: H, He and
traces of Li. These stars were the source of the first enrichment of the
interstellar and intergalactic medium, and set the stage for the formation of
the first galaxies and started the epoch of reionisation (Bromm & Yoshida,
2011).

Despite their importance, the details of star formation and the resulting
initial mass function (IMF) of Pop III stars are still uncertain. Initial
studies suggested that the lack of metals in star-forming clouds caused the
absence of an efficient cooling mechanism and suppressed the formation of low-
mass stars in the early universe. Hydrodynamic simulations based on cooling by
hydrogen molecules predicted stellar masses larger than
$>100\,\mathrm{M}_{\sun}$ (Bromm & Yoshida, 2011). More recent studies that
incorporate longer timescales, higher resolution, and improved physics such as
subsonic turbulence (Clark et al., 2011a), fragmentation of accretion disks
(Greif et al., 2011; Clark et al., 2011b), protostellar radiation feedback
(Hosokawa et al., 2011, 2016; Hirano et al., 2014), multiplicity (Stacy &
Bromm, 2013), and a combination of these (Susa, 2013; Susa et al., 2014; Stacy
et al., 2016) allow stars to form with initial masses of several solar masses
from $\lesssim 1\,\mathrm{M}_{\odot}$ up to $1000\,\mathrm{M}_{\sun}$. The
lack of direct observations of Pop III stars and expected scarcity of long-
lived low-mass ($M\lesssim 0.8\,\mathrm{M}_{\odot}$) Pop III stars in the
galaxy (Hartwig et al., 2015; Magg et al., 2019), however, makes it difficult
to directly constrain these predictions.

One way to indirectly study the properties of the first stars is to look at
metal-poor second-generation stars (Population II stars). These stars are born
from the gas that the Pop III stars enrich; moreover, studies claim that stars
with metallicity of up to $[\mathrm{Fe}/\mathrm{H}]<-3$ could, in principle,
be formed from a cloud enriched by a single supernova (Tominaga et al., 2007;
Nomoto et al., 2013; Keller et al., 2014; Frebel & Norris, 2015). Using a
semi-analytical model of early universe star formation, Hartwig et al. (2018)
found that $40\%$ of stars with
$-6\lessapprox[\mathrm{Fe}/\mathrm{H}]\lessapprox-4$ are enriched by only one
nucleosynthesis event (mono-enriched). Therefore, by comparing the chemical
compositions of ultra metal-poor stars (UMP, $\mathrm{[Fe/H]}<-4$, Beers &
Christlieb 2005) to theoretical yields of first-star core-collapse supernovae,
it is, in principle, possible to constrain the explosion properties and IMF of
the first stars – at least for single stars, although some questions about
asymmetric supernovae and dilution remain (Ezzeddine et al., 2019; Magg et
al., 2020). These types of analyses indicate that the progenitors of Pop II
stars typically have stellar masses in the range of
$10-100\,\mathrm{M}_{\sun}$ (Lai et al., 2008; Heger & Woosley, 2010; Joggerst
et al., 2010; Ishigaki et al., 2014, 2018; Tominaga et al., 2007, 2014; Placco
et al., 2015).

A related problem concerns the transition from Pop III star formation to low-
mass star formation that follows the currently known IMF peaking at $\sim
0.2-0.3\,\mathrm{M}_{\sun}$ (Kroupa, 2002; Chabrier, 2003; Bastian et al.,
2010). Two models have been proposed for this transition to predominantly low-
mass star formation: atomic fine-structure line cooling (Bromm & Loeb, 2003)
and dust-induced fragmentation (Schneider et al., 2003; Omukai et al., 2005;
Ji et al., 2014). Atomic fine-structure line cooling can occur if the gas is
enriched beyond critical abundances of C II and O I, often approximated as
$Z_{\mathrm{crit}}/Z_{\sun}\sim 10^{-3.5}$. On the other hand, dust-induced
fragmentation which requires dust formation in the ejecta from first star
supernovae, can already operate at critical abundances that are a factor
$10-100$ smaller: $Z_{\mathrm{crit}}/Z_{\sun}\sim 10^{-5}$ (Omukai et al.,
2005; Schneider et al., 2012a).

In the last decade, multiple large spectroscopic surveys have expanded our
sample of ultra metal-poor stars allowing us to make a first classification
(Beers et al., 1985; Beers & Christlieb, 2005; Christlieb et al., 2008). One
result of this work is that the fraction of carbon-enhanced metal-poor (CEMP,
$[\mathrm{C}/\mathrm{Fe}]>0.7$; Aoki et al. 2007 and Beers & Christlieb 2005)
stars increases for decreasing metallicity (Placco et al., 2014; Arentsen et
al., 2022). Below $[\mathrm{Fe}/\mathrm{H}]<-4.5$ we almost exclusively
observe CEMP-stars. Due to their carbon-enhanced nature, these stars all obey
the critical abundance necessary for atomic line cooling.

There is one notable exception: SDSS J102915.14+172927.9, hereafter “SDSS
J102915+172927” (Caffau et al., 2011). This is the star with the lowest metal-
mass fraction $Z$, currently known, in large parts due to a low upper limit on
its carbon abundance: $[\mathrm{C}/\mathrm{H}]\leq-4.5$. Its low carbon
abundance puts the star below the critical abundance for metal-line cooling,
providing support for the dust-induced formation model (Caffau et al., 2012;
Schneider et al., 2012b; Klessen et al., 2012).

Inferences about the Pop III progenitors and about the Pop II formation
mechanism, however, rely on stellar chemical abundances that are determined
via high-resolution spectroscopy. These analyses are strongly sensitive to the
approximations employed when modelling the synthetic spectra. The abundance
offsets introduced by the simplifying assumptions of one dimensional (1D)
hydrostatic atmospheres and local thermodynamic equilibrium (LTE) have been
shown to be particularly severe at lower metallicities (Amarsi et al., 2016b,
2022; Bergemann et al., 2012, 2019; Ezzeddine et al., 2017).

One source of these discrepancies is that, contrary to 3D simulations wherein
convection arises naturally, in 1D hydrostatic models convection must be
treated in an approximate way, for example by the mixing-length theory (Böhm-
Vitense, 1958). The 1D models also cannot make predictions on other intrinsic
3D hydrodynamic effects such as granulation and cooling by adiabatic
expansion. The first 3D models of metal-poor stellar atmospheres showed that
the absence of adiabatic cooling in 1D models leads to a severe overestimation
of the surface temperature stratification (Stein & Nordlund, 1998; Asplund et
al., 2005). Hence, in LTE, 3D synthetic spectral lines of neutral species that
are formed in these surface layers are often stronger than their 1D
counterparts. Equally important, non-LTE effects for minority species subject
to overionisation are expected to be larger in the steeper temperature
gradients of metal-poor stars (Bergemann et al., 2012). To mitigate the
problems, without attempting fully consistent 3D non-LTE synthesis, estimates
of non-LTE and 3D effects have often been made separately as in the case for
SDSS J102915+172927 (Caffau et al., 2011, 2012). The non-LTE effects, however,
in fact vary across the surface due to horizontal inhomogeneities (Asplund et
al., 2003; Nordlander et al., 2017). Hence attempting to correct for 3D and
non-LTE effects separately, often leads to new abundance offsets. Nordlander
et al. (2017) already showed that doing a full 3D non-LTE spectral synthesis
of SMSS0313-6708, the most iron-poor star currently known, can significantly
change abundance estimates. The full 3D non-LTE analysis increased the upper
limit on the Fe abundance for this star to $[\mathrm{Fe}/\mathrm{H}]<-6.53$ as
compared to the previous non-LTE upper limit $[\mathrm{Fe}/\mathrm{H}]<-7.52$
computed using averaged 3D ($\langle 3\mathrm{D}\rangle$) model atmospheres
(Bessell et al., 2015). Additionally, recent work on C and O (Amarsi et al.,
2016a), Li (Wang et al., 2021), and Fe (Amarsi et al., 2016b, 2022) also
demonstrated significant 1D LTE $-$ 3D non-LTE abundance corrections for
metal-poor stars.

In light of recent advancements in the field of modelling 3D model atmospheres
and 3D non-LTE spectral synthesis, we perform for the first time a fully
consistent 3D non-LTE abundance analysis of the most metal-poor star SDSS
J102915+172927. In $\S$2 we present the model atmosphere, radiative transfer
and model atoms used to produce 3D non-LTE synthetic spectra. The resulting
abundances are presented in $\S$3 and discussed in $\S$4; our conclusions
given in $\S$5.

##  2 Method

###  2.1 Observational data and stellar parameters

Observational spectra of SDSS J102915+172927 ($G=16.5$, Gaia Collaboration
2020) were obtained by the programme 286.D-5045 and principle investigator P.
Bonifacio, using the UVES spectrograph (Dekker et al., 2000) mounted on the
VLT. More information about the instrument settings and other details of the
observations can be found in Caffau et al. (2012). For this work, we
downloaded the reduced science spectra containing 14 individual exposures of
$3005\,\mathrm{s}$ each from the ESO advanced data product archive111Url:
http://archive.eso.org/eso/eso_archive_main.html. We find a signal-to-noise
ratio per pixel of approximately 35 at 650 nm for each individual exposure,
which agrees with the values found by Caffau et al. (2012). Each exposure was
aligned manually by adding a radial velocity shift such that the cores of the
H Balmer lines and the Ca II-doublet corresponded to the correct central
wavelength. Subsequently, co-adding all exposures resulted in the final
spectrum with a signal-to-noise ratio per pixel of $S/N\approx 20$ in the blue
region around $3500\,\AA$, $\approx 70$ around the green region $5200\,\AA$
and $\approx 35$ in the red region at $8500\,\AA$ of the spectrum.

Caffau et al. (2012) determined the stellar effective temperature
($T_{\rm\\!eff}$) from the $(g-z)$ colour obtained by Ludwig et al. (2008),
resulting in a value of $5811\pm 150\,\mathrm{K}$. For the surface gravity, it
was still unclear at the time of the original analysis whether SDSS
J102915+172927 is a dwarf or a subgiant star. Caffau et al. (2012) determined
a gravity of $\log g=4.0\pm 0.5\,\mathrm{dex}$, but not excluding a higher
gravity of $\approx 4.5\,\mathrm{dex}$. With the release of an improved
parallax by Gaia DR2 $\varpi=0.734\pm 0.07\,\mathrm{mas}$, Bonifacio et al.
(2018) resolved the gravity issue by comparing metal-poor isochrones from A.
Chieffi (Priv. Comms.) to their newly computed absolute $V$-magnitude using
Gaia DR2 data, concluding that SDSS J102915+172927 is a dwarf star. Sestito et
al. (2019) computed $T_{\mathrm{\\!eff}}=5764\pm 60\,\mathrm{K}$ and $\log
g=4.7\pm 0.1\,\mathrm{dex}$ by fitting MESA ischochrones, with
$\mathrm{[Fe/H]}=-4$, to observations using Gaia DR2 astrometric and
photometric data. The estimated error of using isochrones with a fixed
metallicity for stars that are more metal poor than $\mathrm{[Fe/H]}=-4$ is
expected to be small and accounted for by adding an additional error of
$0.01\leavevmode\nobreak\ \mathrm{mag}$ to the photometric uncertainities
(Sestito et al., 2019). Using the parallax from Gaia DR3 $\varpi=0.648\pm
0.06\,\mathrm{mas}$, Sestito (Priv. Comms.) arrived at
$T_{\mathrm{\\!eff}}=5811\pm 51\,\mathrm{K}$ and $\log g=4.68\pm
0.1\,\mathrm{dex}$. This effective temperature compares well to the original
derived value whereas the surface gravity is $0.7\,\mathrm{dex}$ higher but
corresponds to SDSS J102915+172927 being a dwarf star. In this work we used
the latest values derived by Sestito (Priv. Comms.) using Gaia DR3 data.

Table 1: Summary of the stellar parameters of SDSS J102915+172927 obtained in
previous literature studies and for the hydrodynamic models used in this work.

|  $T_{\mathrm{\\!eff}}$ [K] |  $\log g$ [dex] | [Fe/H]
---|---|---|---
Caffau et al. (2012) | $5811\pm 150$ | $4.0\pm 0.5$ | $-4.89\pm 0.10$
Sestito et al. (2019) $\&$ Gaia DR2 | $5764\pm 56$ | $4.69\pm 0.1$ | $\dots$
F. Sestito2 $\&$ Gaia DR3 | $5811\pm 51$ | $4.68\pm 0.1$ | $\dots$
Stagger 3D model atmosphere | $\approx 5810$ | $4.70$ | $-4$
MARCS 1D model atmosphere | 5811 | 4.68 | $-4$

2 Private communication

###  2.2 Model atmosphere

The tailored 3D “box-in-a-star” model atmosphere was computed for this work by
using the radiative-hydrodynamic Stagger-code (Nordlund & Galsgaard, 1995;
Nordlund et al., 2009), with later improvements by its user community (Magic
et al., 2013; Collet et al., 2018). Stagger solves the fluid conservation
equations (mass, energy, momentum) on a staggered Eulerian mesh and radiation
is included as heating and cooling terms in the energy conservation equation.
The size of the box is chosen such that it contains roughly ten granules and
covers the top of the convective zone, superadiabatic region, photosphere and
upper layers of the atmosphere. The surface gravity is fixed since the box
only represents a small region of the star. The radiative transfer is solved
in LTE, where the scattering contribution is included in the total extinction,
using an approximate opacity binning method (Nordlund, 1982; Skartlien, 2000;
Ludwig & Steffen, 2013; Collet et al., 2018) with 12 bins. Since computing the
full monochromatic solution in 3D is computationally unfeasible, all
wavelength points are sorted in 12 bins according to their formation depth
$\tau_{\mathrm{Ross}}(\tau_{\lambda}=1)$. For each bin a mean opacity
$\kappa_{i}$, containing both continuum and line opacity, is computed that is
subsequently used in the calculation of the mean intensity and radiative
heating rate:

$q_{\mathrm{rad}}=4\pi\rho\sum_{i}\kappa_{i}(J_{i}-B_{i})\,\,,$ (1)

with density $\rho$, $J_{i}$ the mean intensity for bin $i$ and $B_{i}$ the
Planck function at the local gas temperature. More details of the
implementation in Stagger can be found in Magic et al. (2013). The Equation of
State (EOS) is a customised version from Mihalas et al. (1988) constructed by
Trampedach et al. (2013), which includes the following species: H, He, C, N,
O, Ne, Na, Mg, Al, Si, S, Ar, K, Ca, Cr, Fe, Ni, $\mathrm{H}_{2}$, and
$\mathrm{H}_{2}^{+}$. Continuous absorption and scattering coefficients are
taken from Hayek et al. (2010) and line opacities from Gustafsson et al.
(2008).

Surface gravity, metallicity, bottom boundary entropy and a fiducial hyper-
viscosity parameter are the only free parameters in the Stagger simulations,
with effective temperature being an emergent property. As such, the surface
gravity was fixed to the value selected in $\S$2.1 and metallicity at
$\mathrm{[Fe/H]}=-4$ with alpha-enhancement $\mathrm{[\alpha/Fe]}=0.4$ typical
for stars at these low-metallicities (Mashonkina et al., 2017). The bottom
boundary entropy was modified iteratively in order to obtain the desired
effective temperature of $\sim 5811\,\mathrm{K}$. The H to He abundance ratio
is fixed at solar value while the other elemental abundance ratios with
respect to H are scaled down from their solar values, for example
$[\mathrm{C}/\mathrm{H}]=-4$ and $[\mathrm{N}/\mathrm{H}]=-4$ using solar
abundances from Asplund et al. (2009). A summary of the input and emergent
stellar parameters of the computed model are given in Table 1 together with
literature estimates. The model itself is set on a $240\times 240\times 240$
Cartesian grid that is more spatially refined near the surface to resolve the
steep photospheric temperature gradient. Information about the numerical
details of the code can be found in Magic et al. (2013) and Collet et al.
(2018).

In addition to the tailored 3D model atmosphere, a MARCS atmosphere,
interpolated from the MARCS grid (Gustafsson et al., 2008), was used to
compute 1D abundances. The 1D plane-parallel atmosphere has a metallicity of
$[\mathrm{Fe}/\mathrm{H}]=-4$ with similar $\alpha$, C, N, and O abundances as
the Stagger model and microturbulence of $1\,\mathrm{km/s}$. The exact value
of the microturbulence was relaxed for the post-processing spectrum synthesis
described in $\S$2.3. A comparison between the 1D MARCS and 3D Stagger model
is made in Fig. 1, where the temperature stratification of both is displayed
together with the average stratification of the 3D model
($\langle\mathrm{3D}\rangle$). This average is calculated by taking the mean
temperature of all vertical cross-sections at each depth point in optical
depth space. It showcases the impact of adiabatic cooling in 3D metal-poor
atmospheric models in the upper atmosphere where the temperature structure
differs significantly compared to the 1D model. The 3D model has a steeper
temperature gradient at the photosphere. Although the
$\langle\mathrm{3D}\rangle$-model is an improvement over the 1D model as it
accurately describes the temperature stratification of the full 3D model, it
loses information on the surface granulation and accompanying impact on
spectral line formation.

Figure 1: Temperature heatmap of the Stagger model with its corresponding
average $\langle 3\mathrm{D}\rangle$ of all vertical columns over optical
depth (red dashed). The 1D MARCS temperature stratification is shown in blue.

###  2.3 Spectral modelling

Table 2: 1D (non-)LTE abundance ($A(\mathrm{X})$, see Eq. 3) sensitivity to the uncertainties on $T_{\mathrm{\\!eff}}$, $\log g$ and $v_{\mathrm{mic}}$. Species | $T_{\mathrm{\\!eff}}+100\,\mathrm{K}$ | $T_{\mathrm{\\!eff}}-100\,\mathrm{K}$ | $\log g+0.2\,\mathrm{dex}$ | $\log g-0.2\,\mathrm{dex}$ | $v_{\mathrm{mic}}+0.5\,\mathrm{km/s}$ | $v_{\mathrm{mic}}-0.5\,\mathrm{km/s}$
---|---|---|---|---|---|---
Mg I | 0.08 | $-0.09$ | $-0.02$ | 0.05 | $-0.01$ | 0.01
Ca I | 0.09 | $-0.09$ | $-0.01$ | $0.05$ | $-0.01$ | 0.01
Ca II | $-0.04$ | $-0.12$ | $-0.03$ | $-0.12$ | $-0.06$ | 0.08
Si I | 0.07 | $-0.10$ | $-0.01$ | 0.01 | $-0.01$ | 0.01
Fe I | 0.10 | $-0.11$ | 0.01 | 0.05 | $-0.02$ | 0.03
Ti II | 0.07 | $-0.05$ | 0.08 | $-0.06$ | $-0.07$ | 0.09
Ni I | 0.11 | $-0.10$ | 0.01 | 0.02 | $-0.04$ | 0.07
C | 0.15 | $-0.15$ | $-0.05$ | $0.05$ | $\approx 0$ | $\approx 0$
N | 0.15 | $-0.15$ | $-0.15$ | $0.05$ | $\approx 0$ | $\approx 0$
Li I | 0.08 | $-0.07$ | 0.01 | 0.03 | $\approx 0$ | $\approx 0$
Na I | $0.07$ | $-0.06$ | $0.01$ | $0.02$ | $\approx 0$ | $\approx 0$
Al I | $0.07$ | $-0.08$ | $-0.01$ | $0.04$ | $\approx 0$ | $\approx 0$

The radiative transfer code Balder was used to compute 1D and 3D non-LTE
synthetic line profiles for the elements for which we have suitable atomic
models: Si (Amarsi & Asplund, 2017), Li (Wang et al., 2021), Ca, Mg (Asplund
et al., 2021), Na (Lind et al., 2011), Al (Nordlander & Lind, 2017) and Fe
(Amarsi et al., 2016b, 2022; Lind et al., 2017). The code itself is based upon
the original framework of Multi3D (Leenaarts & Carlsson, 2009). Amarsi et al.
(2016a, b) expanded the original code, in particular with an improved opacity
package BLUE, change in angle quadrature and implementation of frequency
parallelisation.

Balder solves the statistical equilibrium for user-specified trace elements
using the multilevel approximate lambda iteration pre-conditioning method of
Rybicki & Hummer (1992) on a Cartesian grid. The assumption is made that any
deviations from LTE do not impact the temperature stratification or the
densities of other elements. The Equation of State, background, and line
opacities are treated by the BLUE package (Amarsi et al., 2016b), in which the
latter is precomputed on a grid of wavelength, density and temperature. To
decrease the computational cost of the 3D calculations, the resolution of the
Stagger model was reduced to 60 grid points in the horizontal and 101 points
in the vertical direction. Nordlander et al. (2017) showed that this mesh
reduction impacts the derived abundances by at most $0.03\,\mathrm{dex}$. For
the 3D spectral synthesis, we sampled the temporal variation of the atmosphere
by computing line strengths from three different snapshots of the Stagger
model. These were chosen so as to maximise the difference in
$T_{\mathrm{\\!eff}}$ and, therefore, the difference in resulting line
profiles. We find mean abundance variations between snapshots of no more than
$0.02\,\mathrm{dex}$, which is in line with similar work done by Nordlander et
al. (2017).

For the other elements that we do not expect large deviations from non-LTE (Ti
II; Mallinson et al. 2022), or for which we do not have the atomic and
molecular data needed to solve the statistical equilibrium equations (Ni I, CH
and NH), we compute 1D and 3D LTE synthetic spectra using Scate (Hayek et al.,
2011). The necessary atomic line data (oscillator strengths, excitation
potential and damping parameters) are taken from Kurucz (2016) for Ti II and
Kurucz (2008) for Ni I, using the VALD database (Piskunov et al., 1995;
Ryabchikova et al., 2015).

A range of synthetic spectra were calculated with varying abundances
($\Delta=0.2\,\mathrm{dex}$) for the elements: Li I, CH, NH, Na I, Al I, Mg I,
Si I, Ca I, Ca II, Ti II, Fe I and Ni I, also listed in Table 3. These were
interpolated to the equivalent width ($W_{\lambda}$) measured in the observed
spectrum. The latter is found by fitting one or more, in the case of blended
lines, Gaussians to the observed line profiles. An overview of the identified
spectral lines and corresponding equivalent widths is given in Table
LABEL:t:EWobs. Caffau et al. (2012) report the equivalent widths for a
selection of lines in their Table 4, all of which agree with our values within
their error. For the elements with no visible lines in the UVES spectrum (Li,
Na, Al, CH, and NH), upper limits were estimated using either Cayrel’s formula
(Cayrel, 1988; Cayrel et al., 2004, for Li, Na and Al) or a reduced $\chi^{2}$
statistical test (CH and NH). In the second case we applied artificial
broadening to the synthetic spectra to simulate instrumental broadening with a
resolution of $38\,000$. Further details on the upper limit determination are
given in the respective results section of Li ($\S$3.1) and C ($\S$3.2).

Spectrum synthesis calculations in 1D contain microturbulent velocity
$v_{\mathrm{mic}}$ as an extra free broadening parameter to approximate the
convective velocity fields. This free parameter is usually determined by
flattening the abundances of individual Fe I and Fe II lines as a function of
reduced equivalent width $W_{\lambda}/\lambda_{0}$. Since we only detect Fe I
in the observed spectrum, we computed 1D (non-)LTE Fe I abundances for three
different values of microturbulence
$v_{\mathrm{mic}}=1,\,1.5\,\mathrm{and}\,2\,\mathrm{km/s}$ to obtain a final
best fitting value of $v_{\mathrm{mic}}=1.48\pm 0.25\,\mathrm{km/s}$. The
result of this analysis is shown in Fig. 2 where all detected Fe I lines are
used to determine $v_{\mathrm{mic}}$. This value is consistent with the
empirical analysis performed by Frebel et al. (2013), for a star with $\log
g=4.7$. In all subsequent spectral synthesis computations a value of
$v_{\mathrm{mic}}=1.5\,\mathrm{km/s}$ was used.

The stipulated abundance errors originate from manually placing the continuum
and from the noise of the spectrum. It is calculated from Cayrel’s formula
(Cayrel, 1988; Cayrel et al., 2004), which gives an error on the equivalent
width that propagates through to the abundance error, with a separate
contribution from the continuum placement. In addition, there is an error
coming from the uncertainties on $T_{\mathrm{\\!eff}}$, $\log g$ and
$v_{\mathrm{mic}}$, which are computed by shifting one stellar parameter at a
time in the MARCS atmosphere and performing an abundance analysis as described
above. The error is then equal to the abundance offset between the new model
and the model with correct stellar parameters. For example, the Mg I abundance
increases by $0.08\leavevmode\nobreak\ \mathrm{dex}$ when computed using a
MARCS atmosphere with a $T_{\mathrm{eff}}$ that is $100\leavevmode\nobreak\
\mathrm{K}$ hotter. These results are summarised in Table 2. Finally, the
total error becomes:

$\sigma^{2}_{\mathrm{tot}}=\sigma^{2}_{\mathrm{Cayrel}}+\sigma^{2}_{\mathrm{continuum}}+\sigma^{2}_{T_{\mathrm{\\!eff}},\log
g,v_{\mathrm{mic}}}\,\,$ (2)

with $\sigma_{\mathrm{continuum}}=\frac{1}{S/N}\cdot W_{\lambda}$.

Figure 2: 1D non-LTE Fe I abundances for each visible line in the spectrum
interpolated to a microturbulence of $v_{\mathrm{micro}}=1.48\,\mathrm{km/s}$.
The blue line shows the best linear fit through the data while the red dashed
line denotes the mean non-LTE abundance. The black solid line denotes the
cutoff in reduced equivalent width between saturated and non-saturated lines:
$\mathrm{log}[W_{\lambda}/\lambda_{0}]<-4.95$.

##  3 Results

All abundance results are summarised in Table 3 in the form:

$A(\mathrm{X})=\mathrm{log}(x)+A(\mathrm{H})\,\,,$ (3)

where $x$ is the number fraction of element X and $A(\mathrm{H})=12$. Whenever
an abundance is converted to the solar normalised abundance:
$[\mathrm{X}/\mathrm{H}]=A(\mathrm{X})-A(\mathrm{X})_{\sun}$, we used the
solar abundances from Asplund et al. (2021).

###  3.1 Lithium

The ${Li\textsc{I}}$ line at $6707.6\,\AA$ is not detected in the UVES
spectrum such that we could only determine an upper limit. We used Cayrel’s
formula (Cayrel, 1988), similar to Caffau et al. (2012), to determine a
$3\sigma$ upper limit. The resulting 3D non-LTE upper limit,
$A(\mathrm{Li})<1.06\pm 0.05$, is slightly higher than the original value,
$A(\mathrm{Li})_{\mathrm{Caffau+12}}<0.9$, but still well below the Spite
plateau, $A(\mathrm{Li})_{\mathrm{Spite}}\sim 2.2$ (Spite & Spite, 1982;
Sbordone et al., 2010; Meléndez et al., 2010), implying that the star has
undergone significant Li depletion, as expected based on its effective
temperature.

###  3.2 Carbon and nitrogen

Both C and N are undetected in the UVES spectrum, but the quality of the
spectrum allows the calculation of robust upper limits. To obtain these, we
computed 1D and 3D LTE synthetic spectra of the CH G-band at $4300\,\AA$ and
the NH-band at $3360\,\AA$ for a wide range of abundances
$A(\mathrm{C})=3.6-5.6$ and $A(\mathrm{N})=2.4-6.0$ in steps of
$\Delta=0.2\,\mathrm{dex}$. The upper limit was determined using a reduced
$\chi^{2}$-statistic where we ‘fitted’ the synthetic spectral bands to the
UVES spectrum. The null hypothesis is that our featureless observed spectrum
can be fully described by the synthetic spectrum, meaning that both are
indistinguishable. The upper limit is then defined as the synthetic spectrum
with abundance $A(\mathrm{C})$ that does not statistically describe the
observed featureless spectrum.

We start by masking out the blended ${Ti\textsc{ii}}$ and ${Fe\textsc{i}}$
line in the NH- and CH-band, respectively, and compute the reduced
$\chi^{2}_{\nu}$ for each synthetic spectrum:

$\bar{\chi}^{2}_{\nu}=\frac{1}{\nu}\sum_{\lambda}\Big{(}\frac{F_{\mathrm{obs}}(\lambda)-F_{\mathrm{synth}}(\lambda)}{\sigma_{\mathrm{continuum}}(\lambda)}\Big{)}^{2}\,\,,$
(4)

with $\nu$ the degrees of freedom equal to the number of wavelength points
minus 1 and $\sigma_{\mathrm{cont}}=S/N^{-1}$. For a significance level of
$\alpha=0.05$ we can look up the corresponding mean of the
$\chi^{2}$-distribution: $\chi^{2}_{{}_{\nu},\alpha}/\nu$ (Bognar, 2021),
where the probability that a randomly chosen $\chi^{2}$ from the distribution
function will be greater than $\chi^{2}_{{}_{\nu},\alpha}$ is equal to the
significance level, in this case $5\%$. It follows that the null hypothesis is
rejected for synthetic models that have
$\bar{\chi}^{2}_{\nu}>\chi^{2}_{{}_{\nu},\alpha}/\nu$, either because the
synthetic spectrum does not properly model the observed spectrum or due to a
statistically improbable excursion of probability $5\%$. This means that the
upper limit on the CH and NH abundance is equal to the synthetic spectrum for
which $\bar{\chi}^{2}_{\nu}=\chi^{2}_{{}_{\nu},\alpha}/\nu$, as any model with
a higher abundance do not statistically describe the featureless spectrum we
observe. A visualisation of the $\bar{\chi}^{2}_{\nu}$ values used to
determine the upper limits for C and N is shown in Figure 9.

Using this method, we increase the upper limit significantly for both
molecular bands: $A(\mathrm{C})=4.86\,\mathrm{dex}$ and
$A(\mathrm{N})=4.65\,\mathrm{dex}$, as compared to Caffau et al. (2012):
$A(\mathrm{C})_{\mathrm{Caffau}}=3.96\,\mathrm{dex}$ and
$A(\mathrm{N})_{\mathrm{Caffau}}=2.83\,\mathrm{dex}$. The corresponding
synthetic spectra are shown in Figs. 4 and 4 for N and C, respectively. In
addition, the new and old upper limits on C are shown together with a subset
of metal-poor stars from the SAGA database (Suda et al., 2008) in Fig. 5.

The large increase of the C and N upper limit can originate from multiple
sources such as: changes in surface gravity, spectrum synthesis code, and
atmosphere between the original and this work. In the case of the CH and NH
upper limits, however, we believe that the main discrepancy is likely due to a
difference in methodology when determining these limits. Caffau et al. (2012)
show in their Figs. 6 and 7 the molecular bands for different abundances
$A(\mathrm{C})=4.65,\,6.00\,\mathrm{and}\,A(\mathrm{N})=3.60,\,5.40$. From
visual inspection, their $A(\mathrm{C})=5.40$ spectrum corresponds well to our
synthetic spectrum with $A(\mathrm{C})=5.00$, which points towards a $\sim
0.4\,\mathrm{dex}$ abundance difference originating from the difference in
surface gravity, atmospheric model and spectrum synthesis code. We hypothesise
that the remaining difference is due to a difference in methodology. It is not
described in Caffau et al. (2012), however, how the upper limits are derived
such that a more direct comparison can not be made.

Figure 3: The UVES spectrum (black) zoomed in on the NH-band at $3360\,\AA$.
The synthetic spectrum corresponding to the upper limit derived in this work
is shown in blue, while the upper limit from Caffau et al. (2012) is shown in
red. The ${Ti\textsc{II}}$ at $3361.2\,\AA$ was excluded from the
$\chi^{2}$-analysis.

Figure 4: CH G-band at $4300\,\AA$. The ${Fe\textsc{I}}$ at $4307.9\,\AA$ was
excluded from the $\chi^{2}$-analysis.

Figure 5: Upper limits on $[\mathrm{C}/\mathrm{Fe}]$ for SDSS J102915+172927
in 1D LTE (triangles) and 3D LTE (squares) for both the original limits
(Caffau et al., 2012, red), and the results of this work (blue). For
comparison, we show the $[\mathrm{C}/\mathrm{Fe}]$ abundance for a sample of
metal-poor stars from the SAGA database (Suda et al., 2008, black circles) and
the cutoff for carbon-enhancement at $[\mathrm{C}/\mathrm{Fe}]=0.7$ (dotted
line) from Aoki et al. (2007)

###  3.3 Sodium and aluminium

No Na or Al absorption features could be identified in the spectrum. We
therefore present upper limits for the two elements, in this star for the
first time, at the Na I line at $5891.6\,\AA$ and Al I line at $3961.5\,\AA$
using the same method as for Li. The resulting $3\sigma$-upper limits are
shown in Table 3.

###  3.4 Alpha elements

### Magnesium

We measured four Mg I lines which combine to a mean 3D non-LTE abundance of
$A(\mathrm{Mg})=3.18\pm 0.05$ that is marginally consistent with Caffau et al.
(2012), $A(\mathrm{Mg})=3.06\pm 0.12$. Sitnova et al. (2019) computed a 1D
non-LTE value of $A(\mathrm{Mg})=3.17\pm 0.07$ using the updated surface
gravity of $\log g=4.7$.

### Silicon

One Si I line was detected at $3905.5\,\AA$ for which we computed a 3D non-LTE
abundance of $A(\mathrm{Si})=3.42\pm 0.12$ that is smaller as compared to
Caffau et al. (2012), $A(\mathrm{Si})=3.55\pm 0.1$.

### Calcium

One Ca I line was detected at $4226\,{\AA}$ together with the Ca II H$\&$K
lines and the Ca II triplet at $\approx 8500\,{\AA}$. The Ca II H$\&$K and the
weakest line of the Ca II triplet were discarded due to being blended or too
weak for a reliable equivalent width measurement. Using the resulting lines,
we are able to recover ionisation balance between Ca I and Ca II in 3D non-
LTE:
$A(\mathrm{Ca})_{Ca\textsc{I}}-A(\mathrm{Ca})_{Ca\textsc{II}}=0.01\,\mathrm{dex}$
with a mean Ca abundance of $A(\mathrm{Ca})=1.87\pm 0.02$.

Caffau et al. (2012) obtained a 3D + non-LTE abundance of
$A(\mathrm{Ca})_{Ca\textsc{I}}=1.76\,\mathrm{and}\,A(\mathrm{Ca})_{Ca\textsc{II}}=1.35$.
Using an updated model atom, Sitnova et al. (2019) computed a 1D non-LTE Ca
abundance, using similar stellar parameters as in this work, of
$A(\mathrm{Ca})=1.82\pm 0.06$, where they fixed the Ca I/Ca II ionisation
equilibrium to derive the surface gravity.

### Titanium

We detected five Ti II lines in the UVES spectrum of which three were deemed
strong enough to determine a reliable equivalent width. The resulting mean 3D
LTE abundance $A({Ti\textsc{}})=0.62\pm 0.06$ is $+0.55\,\mathrm{dex}$ is
higher than the original value derived by Caffau et al. (2012). A similar
increase is seen in the 1D LTE case for which we obtain an abundance this is
larger by $+0.39\,\mathrm{dex}$. Following Table 2 we can attribute
approximately $0.2\,\mathrm{dex}$ to the change in surface gravity. The
remaining $0.19\,\mathrm{dex}$ difference for the 1D LTE abundance can not be
attributed to the different oscillator strengths used in this work, which are
higher than the ones reported in Caffau et al. (2012, Table 4), leaving the
origin of the discrepancy unknown.

###  3.5 Iron

The UVES spectrum contains 52 visible Fe I lines but no Fe II lines. For the
abundance computation a cutoff in reduced equivalent width of
$\mathrm{log}[W_{\lambda}/\lambda_{0}]<-4.95$ was used to remove lines that do
not fall on the linear part of the curve-of-growth (Vickers, 2000)222Chapter
“Stars, Spectroscopy of”, p.2199-2204, Figure 2. The remaining 46 Fe I lines
are shown in Fig. 2 to the left of the solid black line. For the 1D LTE case
we expect the only difference between our results and the one from Caffau et
al. (2012) to be the change in gravity ($\Delta\log g=+0.7\,\mathrm{dex}$), as
well as differences between the spectral synthesis codes and the model
atmosphere employed. We obtain a value of $A(\mathrm{Fe})=2.80\pm 0.15$ that
is comparable to the value found by Caffau et al. (2012),
$A(\mathrm{Fe})=2.87\pm 0.13$. This result is assuring since Fe I is not
strongly sensitive to surface gravity (see Table 2).

Our 1D non-LTE abundance, $A(\mathrm{Fe})=3.05\pm 0.13$, is comparable within
its error to Caffau et al. (2012), $A(\mathrm{Fe})=3.00\pm 0.13$. In addition,
Ezzeddine et al. (2017) recomputed the 1D non-LTE Fe abundance for this star,
using a surface gravity of $\log g=4.0$, and obtained a slightly higher value
of $A(\mathrm{Fe})=3.23\pm 0.14$. They based their analysis only on three Fe I
lines, however.

From our consistent 3D non-LTE analysis we derive an abundance of
$A(\mathrm{Fe})=3.28\pm 0.13$ that is $0.57\,\mathrm{dex}$ higher than the 3D
LTE + non-LTE value obtained by Caffau et al. (2012), $A(\mathrm{Fe})=2.71\pm
0.10$. We emphasise that Caffau et al. (2012) computed separate 1D non-LTE and
3D LTE abundance corrections and added these together to get a final 3D+non-
LTE abundance. Applying a similar method to our abundances would yield a value
of $A(\mathrm{Fe})_{\mathrm{3D+NLTE}}=2.97\pm 0.14$, which has a better
agreement with Caffau et al. (2012), but significantly lower than the fully
consistent 3D non-LTE result. Hence, this work demonstrates again that 3D and
non-LTE effects influence each other non-linearly and cannot be simply added
together for metal-poor stars.

This result supports the claim by Amarsi et al. (2016b) that metallicities of
the most-metal poor stars are often systematically underestimated. This is
because for Fe I lines, the non-LTE effects and the 3D effects tend to go in
the same direction, and can enhance each other in warm metal-poor stars.

###  3.6 Nickel

We detected nine usable Ni I lines in the UVES spectrum for which we computed
a mean 3D LTE abundance of $A(\mathrm{Ni})=1.67\pm 0.14$. This value is
$+0.34\,\mathrm{dex}$ higher than the original value. On the other hand, our
1D LTE abundance agrees to within $+0.03\,\mathrm{dex}$ with the value of
Caffau et al. (2012), $A(\mathrm{Ni})=1.65\pm 0.14$.

Table 3: Summary of the computed abundances ($A(\mathrm{X})$, see Eq. 3) and
upper limits together with 3D+non-LTE values from Caffau et al. (2012).

Species | 1D LTE | 1D non-LTE | 3D LTE | 3D non-LTE | $[\mathrm{X}/\mathrm{Fe}]_{\mathrm{3D\,non-LTE}}$ | $\mathrm{Caffau}_{\mathrm{3D+non-LTE}}$
---|---|---|---|---|---|---
${Mg\textsc{I}}$ | $2.90\pm 0.10$ | $3.09\pm 0.09$ | $2.93\pm 0.11$ | $3.18\pm 0.11$ | $-0.19$ | $3.05\pm 0.12$
$\mathrm{{Ca\textsc{I}}}$ | $1.58\pm 0.13$ | $1.72\pm 0.12$ | $1.57\pm 0.10$ | $1.88\pm 0.08$ | $-0.24$ | $1.79\pm 0.10$
$\mathrm{{Ca\textsc{II}}}$ | $2.14\pm 0.13$ | $1.87\pm 0.14$ | $1.96\pm 0.12$ | $1.87\pm 0.11$ | $-0.25$ | $1.38\pm 0.09$
${Si\textsc{I}}$ | $3.16\pm 0.13$ | $3.24\pm 0.12$ | $3.20\pm 0.12$ | $3.42\pm 0.12$ | $0.09$ | $3.56\pm 0.10$
${Fe\textsc{I}}$ | $2.80\pm 0.22$ | $3.05\pm 0.20$ | $2.70\pm 0.20$ | $3.28\pm 0.19$ | $\dots$ | $2.63\pm 0.10$
${Ti\textsc{II}}$ | $0.54\pm 0.18$ | $\dots$ | $0.62\pm 0.13$ | $\dots$ | $-0.17$ | $0.07\pm 0.16$
${Ni\textsc{I}}$ | $1.71\pm 0.22$ | $\dots$ | $1.67\pm 0.20$ | $\dots$ | $-0.38$ | $1.33\pm 0.11^{1}$
$\mathrm{C}$ | $<5.25\pm 0.14$ | $\dots$ | $<4.86\pm 0.15$ | $\dots$ | $<0.58$ | $<3.96^{1}$
$\mathrm{N}$ | $<5.13\pm 0.18$ | $\dots$ | $<4.65\pm 0.20$ | $\dots$ | $<1.00$ | $<2.83^{1}$
${Li\textsc{I}},\,3\sigma$ | $<1.02\pm 0.05$ | $<0.93\pm 0.05$ | $<1.00\pm 0.05$ | $<1.06\pm 0.05$ | $<1.99$ | $<0.90$
${Na\textsc{I}},\,3\sigma$ | $<1.55\pm 0.10$ | $<1.48\pm 0.10$ | $<1.58\pm 0.15$ | $<1.62\pm 0.15$ | $<-0.42$ | $\dots$
${Al\textsc{I}},\,3\sigma$ | $<1.51\pm 0.10$ | $<1.90\pm 0.10$ | $<1.51\pm 0.15$ | $<2.05\pm 0.15$ | $<-0.20$ | $\dots$

1 3D LTE abundance from Caffau et al. (2012)

##  4 Discussion

With our newly derived 3D non-LTE abundances, we can now address some of the
questions regarding the properties of Pop III stars and low-mass star
formation introduced in $\S$1.

###  4.1 Supernova yield comparison

We compared our 3D non-LTE abundances with yields of core-collapse supernovae
of non-rotating Pop III stars Heger & Woosley (2010), updated in 2012 for
missing low-mass models333Data is available at https://starfit.org, using the
StarFit code (Heger et al., 2023). Specifically, we used the Python version
(0.17.12) of StarFit444Data and routines are available at
https://pypi.org/project/starfit/ to find the best match between an input
abundance pattern and supernova model employing a reduced $\chi^{2}$-algorithm
($\bar{\chi}^{2}$) (Heger & Woosley, 2010, see their Section 7). Upper limits,
however, are now treated by StarFit as in Eq. (6) of Magg et al. (2020) — they
contribute to $\bar{\chi}^{2}$ similar to a measurement mismatch if violated
and are basically ignored when well fulfilled.. The one-dimensional mixing
$\&$ fallback models (‘*.S4.*’) cover the following range of progenitor star
masses $9.6$–$100\,\mathrm{M}_{\sun}$, as well as piston-driven explosion
energies $E_{51}=0.3$–$10$ (with $E_{51}=10^{51}$ erg) and hydrodynamic mixing
fractions $f_{\mathrm{mix}}=0$–$0.25$ (see Heger & Woosley 2010 for details).
In addition, the fitting code employs a dilution factor $f_{\mathrm{dil}}$
that accounts for mixing of the supernova ejecta with the interstellar medium
(Big Bang composition). This free parameter essentially shifts the whole
supernova abundance pattern up or down. Li is excluded from the fitting
procedure and C, N, Al, and Na are treated as upper limits. For Na and Al,
more strict $1\sigma$ upper limits are calculated with Cayrel’s formula
(Cayrel, 1988; Cayrel et al., 2004), to use in the supernova fitting:
$A(\mathrm{Na})_{1\sigma}<1.12\pm 0.15$ and $A(\mathrm{Na})_{1\sigma}<1.54\pm
0.15$.

Figure 6: The top panel shows the 3D non-LTE abundances (red squares) together
with all supernovae models that have $\bar{\chi}^{2}<3$ (solid grey lines),
the best fit is shown in blue. The Bottom Panel show the progenitor mass and
explosion energy for the same best-fitting models as shown in the Top Panel,
together with their respective $\bar{\chi}^{2}$ values.

The best fitting model has a progenitor mass of $M=10.9\,\mathrm{M}_{\sun}$,
an explosion energy of $E_{51}=0.6$ and a mixing efficiency of
$f_{\mathrm{mix}}=0.01$, and matches with
$\bar{\chi}^{2}_{\mathrm{best}}=1.03$. The variance of the best-fitting models
($\bar{\chi}^{2}<3$) was analysed by selecting for each mass-energy pair the
$f_{\mathrm{mix}}$ that has the lowest $\bar{\chi}^{2}$. The reasoning is that
mixing acts as a free parameter in the one-dimensional explosion models.
Moreover, the effect of mixing is small at masses $\sim 10\,\mathrm{M}_{\sun}$
and mainly affects elements with $Z>20\,(\mathrm{Ca})$. The resulting
distribution in mass and energy, and corresponding abundance patterns are
shown in Fig. 6. We find that only a narrow range of progenitor masses,
$M=10-20\,\mathrm{M}_{\sun}$, and explosion energies, $E_{51}=0.3-3$, are able
to fit the abundance pattern of SDSS J102915+172927.

The difference between the best fit and yields from models with higher
progenitor masses is shown in Fig. 7. At higher masses the supernova
underproduces heavy elements (Ti, Fe, Ni) and overproduces Mg, Si, and Ca
compared to the best fit model.

Applying the same fitting routine to our 1D LTE abundances results in a
broader mass $M\approx 10-30$ and energy range $E_{51}=0.3-10$ for all models
with $\bar{\chi}^{2}<3$. This is explained by the lower 1D LTE Fe abundance
that allows for models with higher progenitor mass to achieve good fits.
Already in Fig. 7 it is noticeable that most supernova yields have lower Fe
abundances than our 3D non-LTE result.

Figure 7: 3D non-LTE abundances from this paper (red squares) are shown
together with the best fitting progenitor supernova-model in blue and three
other models with a higher progenitor mass (grey): $M=50$, $70$,
$100\,\mathrm{M}_{\sun},\,E_{51}=0.3$ and $f_{\mathrm{mix}}=0.25$.

After the original discovery of SDSS J102915+172927, there have been several
papers comparing its abundance pattern to theoretical yields of Pop III
supernovae. The following literature results all use the original abundances
from Caffau et al. (2012). Similar to this work, Placco et al. (2015) used the
supernova models of Heger & Woosley (2010) to find a best fitting progenitor
mass of $M=10.6\,\mathrm{M}_{\sun}$ with explosion energy $E_{51}=0.9$, which
is in agreement with our work. Schneider et al. (2012b) used the Pop III core-
collapse supernova models by Limongi & Chieffi (2012) to find a best fit for
progenitors with mass $M=20\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ 35\leavevmode\nobreak\ \mathrm{M}_{\odot}$
and explosion energy $E_{51}=1$. Tominaga et al. (2014) compared three core-
collapse SNe models taken from Iwamoto et al. (2005); Tominaga et al. (2007)
with varying mass ($25\,\mathrm{and}\,40\,\mathrm{M}_{\sun}$) and mixing
efficiency with a sample of metal-poor stars including SDSS J102915+172927.
Their best match is a progenitor star with $M=25\,\mathrm{M}_{\sun}$ without
mixing enhancement. Using the same Pop III supernova and hypernova models as
Tominaga et al. (2014), Ishigaki et al. (2014) found a better agreement with a
hypernova of $M=40\,\mathrm{M}_{\sun}$, $E_{51}=30$ and mixing
$\mathrm{log}f=-0.9$, where they attribute the obtained explosion energy to
the relatively high [Si/Ca] ratio. These papers find a higher progenitor mass
and explosion energy than this work. We note, however, that the data sets of
Iwamoto et al. (2005) and Tominaga et al. (2007) only cover a comparably
limited progenitor mass range. In summary, based on our new analysis and using
the model set of Heger & Woosley (2010), we find that the observed abundance
pattern of SDSS J102915+172927 can be well explained by a typical single star
supernova (typical mass range $10$–$20\,\mathrm{M}_{\odot}$ and typical
explosion energy, $E_{51}=0.3-3$) of a compact Pop III star (low mixing,
$f_{\mathrm{mix}}=0.01$, as compared to red super-giants, consistent with the
hydrodynamical models of Joggerst et al. 2009).

###  4.2 Mean alpha abundance

Using our new 3D non-LTE abundances, we computed the mean alpha-abundance
using a simple mean of: $[\mathrm{Si}/\mathrm{Fe}]$,
$[\mathrm{Mg}/\mathrm{Fe}]$, and $[\mathrm{Ca}/\mathrm{Fe}]$. We find that the
mean alpha enhancement of SDSS J102915+172927 decreases substantially compared
to the original work:

$[\alpha/\mathrm{Fe}]_{\mathrm{Caffau}}=0.49\,\rightarrow\,[\alpha/\mathrm{Fe}]_{\mathrm{Lagae}}=-0.11\;,$
(5)

also shown in Fig. 8. The observed change from alpha-enhanced
($[\alpha/\mathrm{Fe}]\sim 0.4$) to sub-solar alpha abundance is mainly driven
by the relative large increase in Fe abundance, compared to the alpha-
elements, when going from 1D LTE to 3D non-LTE. Amarsi et al. (2022) found
that for warm metal-poor subgiants ($[\mathrm{M}/\mathrm{H}]=-3$) the change
in iron abundance,
$A({Fe\textsc{I}})_{\mathrm{1DLTE}}-A({Fe\textsc{I}})_{\mathrm{3DnonLTE}}$,
can be as large as $0.4-0.5\,\mathrm{dex}$. 3D non-LTE abundance changes for
the alpha elements are expected to be smaller. Hence, it is not unrealistic to
expect similar shifts in alpha-enhancement for other ultra metal-poor stars,
shifting the mean alpha-to-iron ratio of the stars shown in Fig. 8 downwards.
This would imply that the yields of Pop III supernovae would have produced a
lower $[\alpha/\mathrm{Fe}]$ than previously thought. Verification of this
claim, however, would require a consistent 3D non-LTE abundance analysis of
extremely-metal poor stars ($[\mathrm{Fe}/\mathrm{H}]<-3$).

Our atmospheric models where calculated using alpha elements enhanced by
$\mathrm{[\alpha/Fe]}=+0.4$, contrary to our findings that SDSS J102915+172927
has a sub-solar alpha abundance. At these low metallicities H is expected to
be the main electron donor such that small changes in metal mass fraction are
not expected to significantly impact the atmosphere. To test this, we compared
two MARCS models ($T_{\mathrm{eff}}=5750\leavevmode\nobreak\ \mathrm{K}$,
$\log g=4.5$, $\mathrm{[Fe/H]}=-2.5$, $v_{micro}=1\mathrm{km/s}$) that are
alpha enhanced ($\mathrm{[\alpha/Fe]}=+0.4$) and alpha normal
($\mathrm{[\alpha/Fe]}=+0.0$), resulting in maximum temperature differences of
$15\leavevmode\nobreak\ \mathrm{K}$ throughout the atmosphere.

Figure 8: Similar to Fig. 5 but now for the mean $[\alpha/\mathrm{Fe}]$
abundance.

###  4.3 Impact on Pop II star formation theories

In the original work, Caffau et al. (2012) concluded that the low carbon upper
limit of SDSS J102915+172927 excluded metal-line cooling as a possible
mechanism of its star formation. In addition, the total metal mass fraction
made SDSS J102915+172927 the most-metal poor star known to date. Using the new
3D non-LTE abundances derived in this work, we first make an estimate of the
metal mass fraction $Z$:

$\frac{Z}{X}=\sum_{\mathrm{i=element}}\Big{[}10^{A(i)-12}\cdot
A_{\mathrm{w}}(i)\Big{]}\;,$ (6)

where $A(i)$ is the abundance and $A_{\mathrm{w}}$ the atomic weight of
element $i$. Subsequently, the normalisation $X+Y+Z=1$ allows to compute the
total metal mass fraction $Z$:

$Z=\frac{1-Y}{1+\Bigg{(}\sum_{\mathrm{i=element}}\Big{[}10^{A(i)-12}\cdot
A_{\mathrm{w}}(i)\Big{]}\,\Bigg{)}^{-1}}\,\,,$ (7)

where $Y=0.2477$ is the primordial He mass fraction (Peimbert et al., 2007).
Using equation 7, we computed two estimates of the mass fraction, $Z$: one
where we restricted the sum over all elements to the species that Caffau et
al. (2012) reported (including upper limits), to make a consistent comparison,
and one that used the abundances from the best fitting Population III
supernova model.

In the first case we included oxygen as $[\mathrm{O}/\mathrm{Fe}]=+0.6$ and Sr
with the upper limit as derived in (Caffau et al., 2012). This results in a
total metal content of $Z\lessapprox 1.38\cdot 10^{-6}$ or
$Z/\mathrm{Z}_{\sun}\lessapprox 9.8\cdot 10^{-5}$, that is approximately a
factor two higher than the original estimation
$Z_{\mathrm{Caffau+12}}/\mathrm{Z}_{\sun}=5\cdot 10^{-5}$. In the second case,
applying the abundances of all species from Li to Zn of the best fitting
supernova model, we obtain a value of $Z_{\mathrm{SNe}}\approx 9.1\cdot
10^{-7}$ or $Z_{\mathrm{SNe}}/\mathrm{Z}_{\sun}\approx 6.6\cdot 10^{-5}$. Even
though this estimate includes more elements, the significantly lower N
abundance of the supernova model results in an overall lower mass fraction
compared to the first case.

Turning now to the possibility of low mass star formation through the
mechanism of metal-line cooling. It is theorised (Bromm & Loeb, 2003) that
this will only occur if there is a sufficient amount of C and O available in
the star-forming cloud. Indeed, metal-line cooling is able to produce low-mass
stars if the transition discriminant (Bromm & Loeb, 2003; Frebel et al.,
2007):

$D=\mathrm{log\Big{[}10^{[\mathrm{C}/\mathrm{H}]}+0.3\cdot
10^{[\mathrm{O}/\mathrm{H}]}\Big{]}}\,\,,$ (8)

is greater than $D\geq D_{\mathrm{crit}}=-3.5\pm 0.2$. Using our new upper
limit on C and $[\mathrm{O}/\mathrm{Fe}]=+0.6$, we obtain a transition
discriminant $D\leq-3.6\pm 0.15$ that is marginally comparable to the critical
value needed for metal-line cooling, contrary to the original result of Caffau
et al. (2011) $D\leq-4.2$. Hence, our result cannot exclude either metal-line
cooling or dust-induced fragmentation as the underlying mechanism in the
formation of SDSS J102915+172927.

##  5 Conclusions

In this work we have performed a fully consistent 3D non-LTE abundance
analysis of SDSS J102915+172927, known as the most-metal poor star. For this
purpose we employed a tailored 3D atmospheric model using improved stellar
parameters and up-to-date atomic data to calculate synthetic spectra. The
primary outcome is that the resulting Fe-abundance is $+0.57\,\mathrm{dex}$
higher than the original value, showcasing the importance of performing
consistent 3D non-LTE calculations for ultra metal-poor stars.

The increase in metallicity, together with new upper limits on C and N, has
important implications regarding progenitor Pop III properties and low-mass
star formation in the early universe. First, we find that the upper limit of
the total metal content of the star increases by a factor 2. Together with the
increased carbon upper limit, we find that the abundances of the star is
consistent with formation both through metal-line cooling and dust-induced
fragmentation. Observations with higher $S/N$ are necessary to improve the
upper limit on C and to draw stronger conclusions. Secondly, our new 3D non-
LTE abundances provide stronger constraints on the mass of the progenitor Pop
III star as well as its explosion energy, assuming that SDSS J102915+172927 is
mono-enriched, as compared to a 1D LTE analysis. In particular, we find that
progenitors with mass $M=10-20\,\mathrm{M}_{\sun}$ and explosion energy
$E_{51}=0.3-3$ are able to reproduce the abundance pattern. The best fit is a
Pop III progenitor with mass $M=10.9\,\mathrm{M}_{\sun}$ that exploded with
energy $E_{51}=0.6$. These are typical masses and explosion energies for core
collapse supernovae (e.g., Müller, 2020) and no hypernova model is required.
Lastly, the strong increase in $A(\mathrm{Fe})$ coupled with relatively
smaller changes in alpha-element abundances changes the status of the star
from alpha-enhanced to a star with sub-solar alpha abundance.

Following the conclusion of Nordlander et al. (2017), and in light of upcoming
large scale surveys which will provide dozens of new ultra metal-poor stars,
it is critical for future studies to apply full 3D non-LTE abundance
calculations whenever possible.

###### Acknowledgements.

We thank the anonymous referee for their comments, which have improved the
manuscript. We gratefully thank Federico Sestito for providing updated stellar
parameters for SDSS J102915.14+172927.9 using Gaia DR3 data. This work has
made use of the VALD database, operated at Uppsala University, the Institute
of Astronomy RAS in Moscow, and the University of Vienna. CL and KL
acknowledge funds from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement No.
852977). KL also acknowledges funds from the Knut and Alice Wallenberg
foundation. AMA gratefully acknowledges support from the Swedish Research
Council (VR 2020-03940). LFRD acknowledges support from the Carlsberg
Foundation (grant agreement CF19-0649). TTH acknowledges support from the
Swedish Research Council (VR 2021-05556). Parts of this research were
supported by the Australian Research Council (ARC) Centre of Excellence (CoE)
for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013. AH wad supported, in part, by the ARC CoE for Gravitational Wave
Discovery (OzGrave) project number CE170100004 and acknowledges software
development support from Astronomy Australia Limited’s ADACS scheme (Project
IDs AHeger_2022B, AHeger_2023A). This work is based on data obtained from the
ESO Science Archive Facility. The computations were performed on resources
provided by the Swedish National Infrastructure for Computing (SNIC) at UPPMAX
under project SNIC2022/5-351 and PDC under project PDC-BUS-2022-4, and at the
Centre for Scientific Computing, Aarhus: http://phys.au.dk/forskning/cscaa/.

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##  Appendix A $\bar{\chi}^{2}$-values of the upper limit determination

Figure 9: Overview of the $\bar{\chi}^{2}$ values corresponding to the
different synthetic spectra of CH and NH. The red star represents the
interpolated abundance where $\bar{\chi}^{2}\approx\chi^{2}_{\alpha}/\nu$,
also highlighted with a vertical dashed line.

##  Appendix B Linelist

1]

Table 4: Equivalent widths of the measures spectral lines together with respective oscillator strength (log $gf$) and lower level excitation potential ($E_{\mathrm{low}}$). References to the atomic data are given in $\S$ 2.3. Ion | $\lambda$ | $W_{\mathrm{UVES}}$ | log $gf$ | $E_{\mathrm{low}}$
---|---|---|---|---
| $[\AA]$ |  $[$mÅ$]$ |  | $[\mathrm{eV}]$
${Li\textsc{I}}$ | 6707.6000 | $22\pm 2$ | $0.174$ | 0.000
${Na\textsc{I}}$ | 5891.6000 | $22\pm 2$ | $-0.194$ | 0.000
${Mg\textsc{I}}$ | 3832.2900 | $28\pm 4$ | $-0.339$ | 2.712
| … | … | $0.138$ | …
${Mg\textsc{I}}$ | 3838.2900 | $36\pm 4$ | $-1.515$ | 2.717
… | … | … | $0.409$ | …
… | … | … | $-0.339$ | …
${Mg\textsc{I}}$ | 5172.7140 | $11\pm 1$ | $-0.363$ | 2.712
${Mg\textsc{I}}$ | 5183.6340 | $18\pm 2$ | $-0.168$ | 2.717
${Al\textsc{I}}$ | 3961.5000 | $22\pm 2$ | $-0.333$ | 0.014
${Si\textsc{I}}$ | 3905.5460 | $17\pm 2$ | $-1.040$ | 1.909
${Ca\textsc{I}}$ | 4226.7500 | $22\pm 2$ | $0.244$ | 0.000
${Ca\textsc{II}}$ | 8542.0900 | $93\pm 3$ | $-0.463$ | 7.813
${Ca\textsc{II}}$ | 8662.1410 | $77\pm 3$ | $-0.723$ | 7.806
${Ti\textsc{II}}$ | 3372.7926 | $36\pm 9$ | 0.280 | 0.012
${Ti\textsc{II}}$ | 3361.2121 | $37\pm 8$ | 0.410 | 0.028
${Ti\textsc{II}}$ | 3349.4022 | $39\pm 11$ | 0.540 | 0.049
${Fe\textsc{I}}$ | 4383.5444 | $31\pm 2$ | $0.208$ | 0.859
${Fe\textsc{I}}$ | 4325.7615 | $11\pm 2$ | $0.006$ | 1.608
${Fe\textsc{I}}$ | 4307.9016 | $12\pm 2$ | $-0.072$ | 1.557
${Fe\textsc{I}}$ | 4271.7599 | $18\pm 2$ | $-0.173$ | 1.485
${Fe\textsc{I}}$ | 4202.0288 | $10\pm 2$ | $-0.689$ | 1.485
${Fe\textsc{I}}$ | 4071.7375 | $13\pm 2$ | $-0.008$ | 1.608
${Fe\textsc{I}}$ | 4063.5936 | $20\pm 2$ | $0.062$ | 1.557
${Fe\textsc{I}}$ | 4045.8119 | $28\pm 2$ | $0.284$ | 1.485
${Fe\textsc{I}}$ | 3930.2963 | $26\pm 2$ | $-1.491$ | 0.087
${Fe\textsc{I}}$ | 3927.9194 | $19\pm 2$ | $-1.522$ | 0.110
${Fe\textsc{I}}$ | 3922.9112 | $19\pm 2$ | $-1.626$ | 0.052
${Fe\textsc{I}}$ | 3920.2574 | $8\pm 2$ | $-1.734$ | 0.121
${Fe\textsc{I}}$ | 3899.7070 | $19\pm 2$ | $-1.515$ | 0.087
${Fe\textsc{I}}$ | 3895.6559 | $14\pm 2$ | $-1.668$ | 0.110
${Fe\textsc{I}}$ | 3886.2818 | $39\pm 2$ | $-1.055$ | 0.052
${Fe\textsc{I}}$ | 3878.5728 | $32\pm 2$ | $-1.379$ | 0.087
${Fe\textsc{I}}$ | 3878.0177 | $10\pm 2$ | $-0.896$ | 0.958
${Fe\textsc{I}}$ | 3859.9110 | $61\pm 2$ | $-0.698$ | 0.000
${Fe\textsc{I}}$ | 3856.3711 | $26\pm 2$ | $-1.280$ | 0.052
${Fe\textsc{I}}$ | 3841.0475 | $11\pm 2$ | $-0.044$ | 1.608
${Fe\textsc{I}}$ | 3840.4372 | $17\pm 2$ | $-0.497$ | 0.990
${Fe\textsc{I}}$ | 3834.2221 | $15\pm 2$ | $-0.269$ | 0.958
${Fe\textsc{I}}$ | 3827.8222 | $12\pm 2$ | $0.094$ | 1.557
${Fe\textsc{I}}$ | 3825.8805 | $36\pm 2$ | $-0.024$ | 0.915
${Fe\textsc{I}}$ | 3824.4432 | $29\pm 2$ | $-1.342$ | 0.000
${Fe\textsc{I}}$ | 3820.4249 | $49\pm 2$ | $0.157$ | 0.859
${Fe\textsc{I}}$ | 3815.8396 | $23\pm 2$ | $0.237$ | 1.485
${Fe\textsc{I}}$ | 3812.9642 | $5.3\pm 1.5$ | $-1.047$ | 0.958
${Fe\textsc{I}}$ | 3787.8799 | $11\pm 2$ | $-0.838$ | 1.011
${Fe\textsc{I}}$ | 3767.1914 | $12\pm 2$ | $-0.382$ | 1.011
${Fe\textsc{I}}$ | 3763.7886 | $27\pm 2$ | $-0.221$ | 0.990
${Fe\textsc{I}}$ | 3758.2326 | $37\pm 4$ | $-0.005$ | 0.958
${Fe\textsc{I}}$ | 3749.4848 | $32\pm 4$ | $0.190$ | 0.915
${Fe\textsc{I}}$ | 3748.2618 | $30\pm 4$ | $-1.008$ | 0.110
${Fe\textsc{I}}$ | 3745.8991 | $21\pm 4$ | $-1.336$ | 0.121
${Fe\textsc{I}}$ | 3745.5608 | $50\pm 4$ | $-0.767$ | 0.087
${Fe\textsc{I}}$ | 3737.1312 | $57\pm 4$ | $-0.572$ | 0.052
${Fe\textsc{I}}$ | 3727.6187 | $17\pm 4$ | $-0.601$ | 0.958
${Fe\textsc{I}}$ | 3722.5627 | $26\pm 4$ | $-1.280$ | 0.087
${Fe\textsc{I}}$ | 3719.9344 | $63\pm 4$ | $-0.424$ | 0.000
${Fe\textsc{I}}$ | 3705.5654 | $23\pm 4$ | $-1.321$ | 0.052
${Fe\textsc{I}}$ | 3647.8422 | $32\pm 3$ | $-0.141$ | 0.915
${Fe\textsc{I}}$ | 3631.4629 | $33\pm 3$ | $0.0001$ | 0.958
${Fe\textsc{I}}$ | 3618.7675 | $35\pm 4$ | $-0.003$ | 0.990
${Fe\textsc{I}}$ | 3608.8589 | $28\pm 4$ | $-0.090$ | 1.011
${Fe\textsc{I}}$ | 3586.9843 | $14\pm 3$ | $-0.795$ | 0.990
${Fe\textsc{I}}$ | 3581.1927 | $62\pm 4$ | $0.415$ | 0.859
${Fe\textsc{I}}$ | 3565.3787 | $26\pm 4$ | $-0.133$ | 0.958
${Fe\textsc{I}}$ | 3490.5735 | $30\pm 4$ | $-1.105$ | 0.052
${Fe\textsc{I}}$ | 3475.4499 | $29\pm 4$ | $-1.054$ | 0.087
${Fe\textsc{I}}$ | 3465.8603 | $26\pm 5$ | $-1.192$ | 0.110
${Fe\textsc{I}}$ | 3440.9883 | $53\pm 5$ | $-0.958$ | 0.052
${Fe\textsc{I}}$ | 3440.6054 | $53\pm 6$ | $-0.673$ | 0.000
${Ni\textsc{I}}$ | 3619.3910 | $20\pm 4$ | $-0.137$ | 0.423
${Ni\textsc{I}}$ | 3524.5360 | $50\pm 6$ | $-0.157$ | 0.025
${Ni\textsc{I}}$ | 3510.3320 | $11\pm 4$ | $-0.807$ | 0.212
${Ni\textsc{I}}$ | 3492.9570 | $27\pm 6$ | $-0.407$ | 0.109
${Ni\textsc{I}}$ | 3461.6540 | $37\pm 9$ | $-0.517$ | 0.025
${Ni\textsc{I}}$ | 3458.4600 | $23\pm 6$ | $-0.377$ | 0.212
${Ni\textsc{I}}$ | 3446.2590 | $28\pm 7$ | $-0.547$ | 0.109
${Ni\textsc{I}}$ | 3414.7650 | $46\pm 8$ | $-0.167$ | 0.025
${Ni\textsc{I}}$ | 3392.9860 | $31\pm 8$ | $-0.677$ | 0.025
CH | G-band 4300 |  |  |
NH | NH-band 3360 |  |  |
Table 4: continued