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The negative hydrogen ion H$^-$ is, almost without exception, treated in local thermodynamic equilibrium (LTE) in the modelling of F, G, and K stars, where it is the dominant opacity source in the visual spectral region. This assumption rests in practice on a study from the 1960s. Since that work, knowledge of relevant atomic processes and theoretical calculations of stellar atmospheres and their spectra have advanced significantly, but this question has not been reexamined. We present calculations based on a slightly modified analytical model that includes H, H$_2$, and H$^-$, together with modern atomic data and a grid of 1D LTE theoretical stellar atmosphere models with stellar parameters ranging from T$_\mathrm{eff} = 4000$ to 7000~K, $\log{g} = 1$ to 5 cm/s$^2$, and [Fe/H]$=-3$ to 0. We find direct non-LTE effects on populations in spectrum-forming regions, continua, and spectral lines of about 1-2% in stars with higher T$_\mathrm{eff}$ and/or lower $\log g$. Effects in models for solar parameters are smaller by a factor of 10, about 0.1-0.2%, and are practically absent in models with lower T$_\mathrm{eff}$ and/or higher $\log g$. These departures from LTE found in our calculations originate from the radiative recombination of electrons with hydrogen to form H$^-$ exceeding photodetachment, that is, overrecombination. Modern atomic data are not a source of significant differences compared to the previous work, although detailed data for processes on H$_2$ resolved with vibrational and rotational states provide a more complete and complex picture of the role of H$_2$ in the equilibrium of H$^-$. In the context of modern studies of stellar spectra at the percent level, our results suggest that this question requires further attention, including a more extensive reaction network, and indirect effects due to non-LTE electron populations., Comment: Accepted by Astronomy and Astrophysics. Replaced with new version following language editing, as well as correct font, and correction of some minor typos

Building on the success of previous missions, asteroseismic modelling will play a key role in future space-based missions, such as PLATO, CubeSpec, and Roman. Despite remarkable achievements, asteroseismology has revealed significant discrepancies in the physics of theoretical stellar models, which have the potential to bias stellar characterisation at the precision level demanded by PLATO. The current modelling strategies largely overlook magnetic activity, assuming that its effects are masked by filtering the so-called surface effects. Given the presence of activity cycles in multiple solar-like oscillators, and activity variations in a significant fraction of Kepler observations of main-sequence stars (Santos et al. 2019b, 2021, 2023), we measured the impact of magnetic activity on the asteroseismic characterisation of the Sun based on 26.5 years of GOLF and BiSON observations. While magnetic activity is partially absorbed in the treatment of surface effects, we found a discernible imprint of the activity cycle in the determination of the solar age. Notably, this imprint persists across both BiSON and GOLF datasets, with significant variations of up to 6.5% observed between solar minima and maxima. Considering that the Sun exhibits low levels of activity, our study underscores the looming challenge posed by magnetic activity for future photometry missions, and prompts a potential reevaluation of the asteroseismic characterisation of Kepler's most active targets., Comment: Accepted for publication as a Letter in Astronomy and Astrophysics

Context: Traditional one-dimensional (1D) hydrostatic model atmospheres introduce systematic modelling errors into spectroscopic analyses of FGK-type stars. Aims: We present an updated version of the STAGGER-grid of 3D model atmospheres, and explore the accuracy of post-processing methods in preparation for spectral synthesis. Methods: New and old models were (re)computed following an updated workflow, including an updated opacity binning technique. Spectroscopic tests were performed in 3D LTE for a grid of 216 fictitious Fe I lines, spanning a wide range in oscillator strength, excitation potential and central wavelength, and eight model atmospheres that cover the stellar atmospheric parameter range (Teff, log g, [Fe/H]) of FGK-type stars. Using this grid, the impact of vertical and horizontal resolution, and temporal sampling of model atmospheres on spectroscopic diagnostics was tested. Results: We find that downsampling the horizontal mesh from its original size of 240 x 240 grid cells to 80 x 80 cells, i.e. sampling every third grid cell, introduces minimal errors on the equivalent width and normalized line flux across the line and stellar parameter space. Regarding temporal sampling, we find that sampling ten statistically independent snapshots is sufficient to accurately model the shape of spectral line profiles. For equivalent widths, a subsample consisting of only two snapshots is sufficient, introducing an abundance error of less than 0.015 dex. Conclusions: We have computed 32 new model atmospheres and recomputed 116 old model atmospheres present in the original grid. The public release of the STAGGER-grid contains 243 models, excluding models with [Fe/H] = -4.00, and the processed snapshots can be used to improve the accuracy of spectroscopic analyses.

The first (Pop III) stars formed only out of H and He and were likely more massive than present-day stars. Massive Pop III stars in the range 140-260 M$_\odot$ are predicted to end their lives as pair-instability supernovae (PISNe), enriching the environment with a unique abundance pattern, with high ratios of odd to even elements. Recently, the most promising candidate for a pure descendant of a zero-metallicity massive PISN (260 M$_{\odot}$) was discovered by the LAMOST survey, the star J1010+2358. However, the key elements to verify the high PISN contribution, C and Al, were missing from the analysis. To rectify this, we obtained and analyzed a high-resolution VLT/UVES spectrum, correcting for 3D and/or non-LTE effects. Our measurements of both C and Al give much higher values (~1 dex) than expected from a 260 M$_{\odot}$ PISN. Furthermore, we find significant discrepancies with the previous analysis, and therefore a much less pronounced odd-even pattern. Thus, we show that J1010+2358 cannot be a pure descendant of a 260 M$_{\odot}$ PISN. Instead, we find that the best fit model consists of a 13 M$_{\odot}$ Pop II core-collapse supernova combined with a Pop III supernova. Alternative, less favoured solutions $(\chi^2/\chi^2_{\rm best}\approx2.3)$ include a 50% contribution from a 260 M$_{\odot}$ PISN, or a 40% contribution from a Pop III type Ia supernova. Ultimately, J1010+2358 is certainly a unique star giving insights into the earliest chemical enrichment, however, this star is not a pure PISN descendant., Comment: Accepted to ApJ Letters (3rd of May 2024). 1 table, 4 figures, and Appendix

Deep imaging of galaxy cluster fields have in recent years revealed tens of candidates for gravitationally lensed stars at redshifts $z\approx$ 1-6, and future searches are expected to reveal highly magnified stars from even earlier epochs. Multi-band photometric observations may be used to constrain the redshift, effective temperature $T_\mathrm{eff}$ and dust attenuation along the line of sight to such objects. When combined with an estimate of the likely magnification, these quantities may be converted into a constraint on the stellar luminosity and, for an adopted set of stellar evolutionary tracks, the initial stellar mass. Further characterization is, however, difficult without spectroscopic observations, which at the typical brightness levels of high-redshift lensed stars becomes extremely challenging for even the largest existing telescopes. Here, we explore what spectral features one can realistically hope to detect in lensed stars with peak brightness in the range 26-28 AB mag, $T_\mathrm{eff}=$ 4000-50 000 K and redshifts $z=$1-10, using spectroscopy with the James Webb Space Telescope (JWST) and the forthcoming Extremely Large Telescope. We find that a majority of detectable lines appear in the rest UV-range for stars with $T_\mathrm{eff}\geq$15 000 K. The strongest detectable spectral lines are the C IV $\lambda$ 1550 \r{A} line and the Si IV $\lambda\lambda$1393, 1403 \r{A}-doublet at $T_\mathrm{eff}=$30 000 K. For lower temperatures, the calcium H- and K-lines at $T_\mathrm{eff}=$6000 K are among the most readily detectable. In limited wavelength ranges, ELT is expected to provide more sensitive spectroscopic observations, and with higher resolution than JWST. We find that variations of both mass loss rate and metallicity lead to noticeable effects in the detectability of certain spectral lines with both JWST and ELT., Comment: 12 pages, 6 figures, accepted in A&A

The lack of reliable atomic data can be a severe limitation in astrophysical modelling, in particular of events such as kilonovae that require information on all neutron-capture elements across a wide range of ionization stages. Notably, the presence of non-orthonormalities between electron orbitals representing configurations that are close in energy can introduce significant inaccuracies in computed energies and transition probabilities. Here, we propose an explicit targeted optimization method that can effectively circumvent this concern while retaining an orthonormal orbital basis set. We illustrate this method within the framework of small-scale atomic structure models of Au I, using the GRASP2018 multiconfigurational Dirac-Hartree-Fock atomic structure code. By comparing to conventional optimization schemes we show how a targeted optimization approach improves the energy level positioning and ordering. Targeted optimization also leads to better agreement with experimental data for the strongest E1 transitions. This illustrates how small-scale models can be significantly improved with minor computational costs if orbital non-orthonormalities are considered carefully. These results should prove useful to multi-element atomic structure calculations in, for example, astrophysical opacity applications involving neutron-capture elements., Comment: 14 pages, 3 figures, 6 tables; published in Journal of Physics B

Three-dimensional radiation-hydrodynamics (3D RHD) simulations of stellar surface convection provide valuable insights into many problems in solar and stellar physics. However, almost all 3D near-surface convection simulations to date are based on solar-scaled chemical compositions, which limit their application on stars with peculiar abundance patterns. To overcome this difficulty, we implement the robust and widely-used FreeEOS equation of state and our Blue opacity package into the Stagger 3D radiation-magnetohydrodynamics code. We present a new 3D RHD model of the solar atmosphere, and demonstrate that the mean stratification as well as the distributions of key physical quantities are in good agreement with those of the latest Stagger solar model atmosphere. The new model is further validated by comparing against solar observations. The new model atmospheres reproduce the observed flux spectrum, continuum centre-to-limb variation, and hydrogen line profiles at a satisfactory level, thereby confirming the realism of the model and the underlying input physics. These implementations open the prospect for studying other stars with different $\alpha$-element abundance, carbon-enhanced metal-poor stars and population II stars with peculiar chemical compositions using 3D Stagger model atmospheres., Comment: 24 pages, 20 figures, accepted for publication in A&A

Accurate and extensive atomic data are essential for spectroscopic analyses of stellar atmospheres and other astronomical objects. We present energy levels, lifetimes, and transition probabilities for neutral nitrogen, the sixth most abundant element in the cosmos. The calculations employ the fully relativistic multiconfiguration Dirac-Hartree-Fock and relativistic configuration interaction methods, and span the 103 lowest states up to and including 2s$^2$2p$^2$5s. Our theoretical energies are in excellent agreement with the experimental data, with an average relative difference of 0.07%. In addition, our transition probabilities are in good agreement with available experimental and theoretical data. We further verify the agreement of our data with experimental results via a re-analysis of the solar nitrogen abundance, with the results from the Babushkin and Coulomb gauges consistent to 2% or 0.01 dex. We estimated the uncertainties of the computed transition data based on a statistical analysis of the differences between the transition rates in Babushkin and Coulomb gauges. Out of the 1701 computed electric dipole transitions in this work, 83 (536) are associated with uncertainties less than 5% (10%)., Comment: 17 pages, 7 figures; Accepted for publication in The Astrophysical Journal Supplement Series

Star formation models predict that the metal-poor initial mass function (IMF) can be substantially different from that observed in the metal-rich Milky Way. This changeover occurs because metal-poor gas clouds cool inefficiently due to their lower abundance of metals and dust. However, predictions for the metal-poor IMF to date rely on assuming Solar-scaled abundances, that is, [X/O] = 0 at all [O/H]. There is now growing evidence that elements such as C and O that dominate metal line cooling in the ISM do not follow Solar scaling at low metallicities. In this work, we extend models that predict the variation in the characteristic (or, the peak) IMF mass as a function of metallicity using [C/O] ratios derived from observations of metal-poor Galactic stars and of H II regions in dwarf galaxies. These data show [C/O] < 0 at sub-Solar [O/H], which leads to a substantially different metal-poor IMF in the metallicity range where C I and C II cooling dominate ISM thermodynamics, resulting in an increase in the characteristic mass by a factor as large as 7. An important consequence of this difference is a shift in the location of the transition from a top- to a bottom-heavy IMF upwards by 0.5 $-$ 1 dex in metallicity. Our findings indicate that the IMF is very sensitive to the assumptions around Solar-scaled ISM compositions in metal-poor systems (e.g., dwarf galaxies, the Galactic halo and metal-poor stars) that are a key focus of JWST., Comment: 14 pages, 8 figures. Accepted by MNRAS

Although true metal-free "Population III" stars have so-far escaped discovery, their nature, and that of their supernovae, is revealed in the chemical products left behind in the next generations of stars. Here we report the detection of an ultra-metal poor star in the Sculptor dwarf spheroidal galaxy, AS0039. With [Fe/H]$_{\rm LTE}=-4.11$, it is the most metal-poor star so far discovered in any external galaxy. Contrary to the majority of Milky Way stars at this metallicity, AS0039 is clearly not enhanced in carbon, with [C/Fe]$_{\rm LTE}=-0.75$ and A(C)=+3.60, making it the lowest detected carbon abundance in any star to date. It furthermore lacks $\alpha$-element uniformity, having extremely low [Mg/Ca]$_{\rm NLTE}=-0.60$ and [Mg/Ti]$_{\rm NLTE}=-0.86$, in stark contrast with the near solar ratios observed in C-normal stars within the Milky Way halo. The unique abundance pattern indicates that AS0039 formed out of material that was predominantly enriched by a $\sim$20$ M_\odot$ progenitor star with an unusually high explosion energy $E=10\times10^{51}$ erg. The star AS0039 is thus one of the first observational evidence for zero-metallicity hypernovae and provides a unique opportunity to investigate the diverse nature of Population III stars., Comment: Accepted in ApJL, 7 pages, 3 figures, 1 table, and an appendix (1 figure and 3 tables)

We establish a quantitative relationship between photometric and spectroscopic detections of solar-like oscillations using ab initio, three-dimensional (3D), hydrodynamical numerical simulations of stellar atmospheres. We present a theoretical derivation as proof of concept for our method. We perform realistic spectral line formation calculations to quantify the ratio between luminosity and radial velocity amplitude for two case studies: the Sun and the red giant $\epsilon$ Tau. Luminosity amplitudes are computed based on the bolometric flux predicted by 3D simulations with granulation background modelled the same way as asteroseismic observations. Radial velocity amplitudes are determined from the wavelength shift of synthesized spectral lines with methods closely resembling those used in BiSON and SONG observations. Consequently, the theoretical luminosity to radial velocity amplitude ratios are directly comparable with corresponding observations. For the Sun, we predict theoretical ratios of 21.0 and 23.7 ppm/[m/s] from BiSON and SONG respectively, in good agreement with observations 19.1 and 21.6 ppm/[m/s]. For $\epsilon$ Tau, we predict K2 and SONG ratios of 48.4 ppm/[m/s], again in good agreement with observations 42.2 ppm/[m/s], and much improved over the result from conventional empirical scaling relations which gives 23.2 ppm/[m/s]. This study thus opens the path towards a quantitative understanding of solar-like oscillations, via detailed modelling of 3D stellar atmospheres., Comment: 16 pages, 7 figures, accepted for publication in MNRAS

Advances in merged-beams instruments have allowed experimental studies of the mutual neutralisation (MN) processes in collisions of both Li$^+$ and Na$^+$ ions with D$^-$ at energies below 1 eV. These experimental results place constraints on theoretical predictions of MN processes of Li$^+$ and Na$^+$ with H$^-$, important for non-LTE modelling of Li and Na spectra in late-type stars. We compare experimental results with calculations for methods typically used to calculate MN processes, namely the full quantum (FQ) approach, and asymptotic model approaches based on the linear combination of atomic orbitals (LCAO) and semi-empirical (SE) methods for deriving couplings. It is found that FQ calculations compare best overall with the experiments, followed by the LCAO, and the SE approaches. The experimental results together with the theoretical calculations, allow us to investigate the effects on modelled spectra and derived abundances and their uncertainties arising from uncertainties in the MN rates. Numerical experiments in a large grid of 1D model atmospheres, and a smaller set of 3D models, indicate that neglect of MN can lead to abundance errors of up to 0.1 dex (26\%) for Li at low metallicity, and 0.2 dex (58\%) for Na at high metallicity, while the uncertainties in the relevant MN rates as constrained by experiments correspond to uncertainties in abundances of much less than 0.01~dex (2\%). This agreement for simple atoms gives confidence in the FQ, LCAO and SE model approaches to be able to predict MN with the accuracy required for non-LTE modelling in stellar atmospheres., Comment: Accepted by ApJ

The ensemble of chemical element abundance measurements for stars, along with precision distances and orbit properties, provides high-dimensional data to study the evolution of the Milky Way. With this third data release of the Galactic Archaeology with HERMES (GALAH) survey, we publish 678 423 spectra for 588 571 mostly nearby stars (81.2% of stars are within <2 kpc), observed with the HERMES spectrograph at the Anglo-Australian Telescope. This release (hereafter GALAH+ DR3) includes all observations from GALAH Phase 1 (bright, main, and faint survey, 70%), K2-HERMES (17%), TESS-HERMES (5%), and a subset of ancillary observations (8%) including the bulge and >75 stellar clusters. We derive stellar parameters $T_\text{eff}$, $\log g$, [Fe/H], $v_\text{mic}$, $v_\text{broad}$ & $v_\text{rad}$ using our modified version of the spectrum synthesis code Spectroscopy Made Easy (SME) and 1D MARCS model atmospheres. We break spectroscopic degeneracies in our spectrum analysis with astrometry from $Gaia$ DR2 and photometry from 2MASS. We report abundance ratios [X/Fe] for 30 different elements (11 of which are based on non-LTE computations) covering five nucleosynthetic pathways. We describe validations for accuracy and precision, flagging of peculiar stars/measurements and recommendations for using our results. Our catalogue comprises 65% dwarfs, 34% giants, and 1% other/unclassified stars. Based on unflagged chemical composition and age, we find 62% young low-$\alpha$, 9% young high-$\alpha$, 27% old high-$\alpha$, and 2% stars with $\mathrm{[Fe/H]} \leq -1$. Based on kinematics, 4% are halo stars. Several Value-Added-Catalogues, including stellar ages and dynamics, updated after $Gaia$ eDR3, accompany this release and allow chrono-chemodynamic analyses, as we showcase., Comment: 48 pages, 27 figures, 5 pages appendix, accepted for publication in MNRAS, comments welcome, GALAH+ DR3 can be found at https://docs.datacentral.org.au/galah/dr3/overview/, Catalogs can be downloaded from https://cloud.datacentral.org.au/teamdata/GALAH/public/GALAH_DR3/

An unprecedented number of exoplanets are being discovered by the Transiting Exoplanet Survey Satellite (TESS). Determining the orbital parameters of these exoplanets, and especially their mass and radius, will depend heavily upon the measured physical characteristics of their host stars. We have cross-matched spectroscopic, photometric, and astrometric data from GALAH Data Release 2, the TESS Input Catalog and Gaia Data Release 2, to create a curated, self-consistent catalog of physical and chemical properties for 47,285 stars. Using these data we have derived isochrone masses and radii that are precise to within 5\%. We have revised the parameters of three confirmed, and twelve candidate, TESS planetary systems. These results cast doubt on whether CTOI-20125677 is indeed a planetary system since the revised planetary radii are now comparable to stellar sizes. Our GALAH-TESS catalog contains abundances for up to 23 elements. We have specifically analysed the molar ratios for C/O, Mg/Si, Fe/Si and Fe/Mg, to assist in determining the composition and structure of planets with $R_p < 4R_\oplus$. From these ratios, 36% fall within 2 sigma of the Sun/Earth values, suggesting that these stars may host rocky exoplanets with geological compositions similar to planets found within our own Solar system., Comment: Submitted to MNRAS, comments from the community welcomed. GALAH-TESS catalog is available upon request from the corresponding authour

Lithium depletion and enrichment in the cosmos is not yet well understood. To help tighten constraints on stellar and Galactic evolution models, we present the largest high-resolution analysis of Li abundances A(Li) to date, with results for over 100 000 GALAH field stars spanning effective temperatures $5900\,\mathrm{K} \lesssim \rm{T_{eff}} \lesssim7000\,\mathrm{K}$ and metallicities $-3 \lesssim \rm[Fe/H] \lesssim +0.5$. We separated these stars into two groups, on the warm and cool side of the so-called Li-dip, a localised region of the Kiel diagram wherein lithium is severely depleted. We discovered that stars in these two groups show similar trends in the A(Li)-[Fe/H] plane, but with a roughly constant offset in A(Li) of 0.4 dex, the warm group having higher Li abundances. At $\rm[Fe/H]\gtrsim-0.5$, a significant increasing in Li abundance with increasing metallicity is evident in both groups, signalling the onset of significant Galactic production. At lower metallicity, stars in the cool group sit on the Spite plateau, showing a reduced lithium of around 0.4 dex relative to the primordial value predicted from Big Bang nucleosynthesis (BBN). However, stars in the warm group between [Fe/H] = -1.0 and -0.5, form an elevated plateau that is largely consistent with the BBN prediction. This may indicate that these stars in fact preserve the primordial Li produced in the early Universe., Comment: 5 pages, 2 figures, Accepted for publication by MNRAS

Older GCE models predict [K/Fe] ratios as much as 1 dex lower than those inferred from stellar observations. Abundances of potassium are mainly based on analyses of the 7698 $\AA$ resonance line, and the discrepancy between models and observations is in part caused by the LTE assumption. We study the statistical equilibrium of KI, focusing on the non-LTE effects on the $7698 \ \AA$ line. We aim to determine how non-LTE abundances of K can improve the analysis of its chemical evolution, and help to constrain the yields of models. We construct a model atom that employs the most up-to-date data. In particular, we calculate and present inelastic e+K collisional excitation cross-sections from the convergent close-coupling and the $B$-Spline $R$-matrix methods, and H+K collisions from the two-electron model. We constructed a fine grid of non-LTE abundance corrections that span $4000<\teff / \rm{K}<8000$, $0.50<\lgg<5.00$, $-5.00<\feh<+0.50$, and applied the corrections to abundances from the literature. In concordance with previous studies, we find severe non-LTE effects in the $7698 \ \AA$ line, which is stronger in non-LTE with abundance corrections that can reach $\sim-0.7\,\dex$. We explore the effects of atmospheric inhomogeneity by computing a full 3D non-LTE stellar spectrum of KI for a test star. We find that 3D is necessary to predict a correct shape of the resonance 7698 $\AA$ line, but the line strength is similar to that found in 1D non-LTE. Our non-LTE abundance corrections reduce the scatter and change the cosmic trends of literature K abundances. In the regime [Fe/H]$\lesssim-1.0$ the non-LTE abundances show a good agreement with the GCE model with yields from rotating massive stars. The reduced scatter of the non-LTE corrected abundances of a sample of solar twins shows that line-by-line differential analysis techniques cannot fully compensate for systematic modelling errors., Comment: 18 pages, 15 Figures

Open cluster members are coeval and share the same initial bulk chemical compositions. Consequently, differences in surface abundances between members of a cluster that are at different evolutionary stages can be used to study the effects of mixing and internal chemical processing. We carry out an abundance analysis of seven elements (Li, O, Na, Mg, Al, Si, Fe) in 66 stars belonging to the open cluster M67, based on high resolution GALAH spectra, 1D MARCS model atmospheres, and, for the first time for a large sample of M67 stars, non-local thermodynamic equilibrium (non-LTE) radiative transfer. From the non-LTE analysis, we find a typical star-to-star scatter in the abundance ratios of around 0.05 dex; this scatter is slightly but systematically larger when LTE is assumed instead. We find trends in the abundance ratios with effective temperature, indicating systematic differences in the surface abundances between turn-off and giant stars; these trends are more pronounced when LTE is assumed. However, in the non-LTE analysis, most of the element trends have been flattened. Two species are exceptions to this behaviour, namely Al and Si, which both clearly display remaining trends in the non-LTE analysis. We comment on the possible origin of these trends, by comparing them with recent stellar models that include atomic diffusion., Comment: 14 pages, 10 figures; resubmitted to MNRAS

We determine Mg abundances in 6 Gaia benchmark stars using theoretical one-dimensional (1D) hydrostatic model atmospheres, as well as temporally- and spatially-averaged 3D model atmospheres (<3D>). The stars cover a range of Teff from 4700 to 6500 K, log g from 1.6 to 4.4 dex, and [Fe/H] from -3.0 dex to solar. Spectrum synthesis calculations are performed in local thermodynamic equilibrium (LTE) and in non-LTE (NLTE) using the oscillator strengths recently published by Pehlivan Rhodin et al. We find that: a) Mg abundances determined from the infrared spectra are as accurate as the optical diagnostics, b) the NLTE effects on Mg I line strengths and abundances in this sample of stars are minor (although for a few Mg I lines the NLTE effects on abundance exceed 0.6 dex in <3D> and 0.1 dex in 1D, c) the solar Mg abundance is 7.56 +/- 0.05 dex (total error), in the excellent agreement with the Mg abundance measured in CI chondritic meteorites, d) the 1D NLTE and <3D> NLTE approach can be used with confidence to analyse optical Mg I lines in spectra of dwarfs and sub-giants, but for red giants the Mg I 5711 A line should be preferred, e) low-excitation Mg I lines are sensitive to the atmospheric structure; for these lines, LTE calculations with <3D> models lead to significant systematic abundance errors. The methods developed in this work will be used to study Mg abundances of a large sample of stars in the next paper in the series., Comment: 25 pages, accepted for publication in ApJ

Models of star formation in the early universe depend on the details of accretion, fragmentation and radiative feedback. Different simulations predict different initial mass functions of the first stars, ranging from predominantly low mass (0.1-10 Msol), to massive (10-100 Msol), or even supermassive (100-1000 Msol). The mass distribution of the first stars should lead to unique chemical imprints on the low-mass second and later generation metal-poor stars still in existence. The chemical composition of SMSS0313-6708, which has the lowest abundances of Ca and Fe of any star known, indicates it was enriched by a single massive supernova. However, even weak spectral lines may be affected by strong 3D and NLTE effects in metal-poor stars. If these effects are ignored or treated incorrectly, errors in the inferred abundances may significantly bias the inferred properties of the polluting supernovae. We redetermine the chemical composition of SMSS0313-6708 using 3D NLTE radiative transfer to obtain accurate abundances for Li, Na, Mg, Al, Ca and Fe. The model atoms employ realistic collisional rates, with no calibrated free parameters. We find significantly higher abundances in 3D NLTE than 1D LTE by 0.8 dex for Fe, and 0.5 dex for Mg, Al and Ca, while Li and Na are unaffected to within 0.03 dex. In particular, our upper limit for [Fe/H] is now a factor ten larger, at [Fe/H] < -6.53 (3 sigma), than previous estimates based on <3D> NLTE (i.e. using averaged 3D models). This higher estimate is due to a conservative upper limit estimation, updated NLTE data, and 3D-<3D> NLTE differences, all of which lead to a higher abundance determination. We find that the revised chemical composition of SMSS0313-6708 matches supernova yields for massive progenitors of 20-60 Msol exploding with low energies (1-2 x 10^51 erg), as well as progenitors of 10 Msol with very low explosion energies (< 10^51 erg)., Comment: 17 pages, 10 pages; Accepted for publication in A&A