Observational constraints on the origin of the elements – VII. NLTE analysis of Y ii lines in spectra of cool stars and implications for Y as a Galactic chemical clock Open Access

ABSTRACT

Yttrium (Y), a key s-process element, is commonly used in nucleosynthesis studies and as a Galactic chemical clock when combined with magnesium (Mg). We study the applicability of the previously assumed local thermal equilibrium (LTE) line formation assumption in Y abundance studies of main-sequence and red giant stars, and probe the impact of NLTE (non-LTE) effects on the [Y/Mg] ratio, a proposed stellar age indicator. We derive stellar parameters, ages, and NLTE abundances of Fe, Mg, and Y for 48 solar analogue stars from high-resolution spectra acquired within the Gaia-ESO survey. For Y, we present a new NLTE atomic model. We determine a solar NLTE abundance of A(Y)_NLTE = 2.12 ± 0.04 dex, 0.04 dex higher than LTE. NLTE effects on Y abundance are modest for optical Y ii lines, which are frequently used in Sun-like stars diagnostics. NLTE has a small impact on the [Y/Mg] ratio in such stars. For metal-poor red giants, NLTE effects on Y ii lines are substantial, potentially exceeding +0.5 dex. For the Gaia/4MOST/WEAVE benchmark star, HD 122563, we find the NLTE abundance ratio of [Y/Fe]_NLTE = −0.55 ± 0.04 dex with consistent abundances obtained from different Y ii lines. NLTE has a differential effect on Y abundance diagnostics in late-type stars. They notably affect Y ii lines in red giants and very metal-poor stars, which are typical Galactic enrichment tracers of neutron-capture elements. For main-sequence stars, NLTE effects on optical diagnostic Y ii lines remain minimal across metallicities. This affirms the [Y/Mg] ratio’s reliability as a cosmochronometer for Sun-like stars.

1 INTRODUCTION

Stellar age is a critical parameter for many fields in astronomy. The most common method for determining the age of a star is the method of fitting the stellar isochrones (e.g. Gallart, Zoccali & Aparicio 2005; Jørgensen & Lindegren 2005; Naylor 2009; Schönrich & Bergemann 2014; Kordopatis et al. 2023), or evolutionary tracks (Dotter et al. 2008; Serenelli et al. 2013; Gent et al. 2022). Although this method works well for coeval samples of stars, such as open and globular clusters (Naylor & Jeffries 2006; Barker & Paust 2018), determinations of ages of field Galactic stars are more difficult because of parameter degeneracies (e.g. Pont & Eyer 2004; Serenelli et al. 2013). Thus, recently several other methods have been explored in the literature. One of such methods is a so-called chemical clock and it is based on empirical correlations between the ages of stars obtained using classical methods (or in combination with asteroseismology) and measured abundance ratios from stellar spectra (see e.g. Bergemann et al. 2016; Martig et al. 2016; Ness et al. 2016; Silva Aguirre et al. 2018; Delgado Mena et al. 2019; Skúladóttir et al. 2019).

The ratio [Y/Mg] has also been proposed as one such potential chemical clock (da Silva et al. 2012). Mg is produced mainly by the core-collapse supernovae explosions of massive stars with some contribution from intermediate-mass stars (Timmes, Woosley & Weaver 1995; Woosley & Weaver 1995; Thielemann et al. 2000; Kobayashi et al. 2006; Karakas & Lattanzio 2014). Y is a slow neutron-capture (s-process) element. Its production mainly happens in the intermediate-mass stars during their asymptotic giant branch (AGB) phase (Busso, Gallino & Wasserburg 1999; Cristallo et al. 2011; Karakas & Lugaro 2016). Owing to the differences in characteristic time-scales of enrichment due to massive stars and AGB stars, the expectation is that [Y/Mg] should generally decrease with higher stellar age. Indeed, recently a well-defined correlation between [Y/Mg] and the stellar age was noted for solar twins in the local Galactic neighbourhood (da Silva et al. 2012; Nissen 2015; Tucci Maia et al. 2016; Jofré, Jackson & Tucci Maia 2020; Berger et al. 2022). The universality of the trend for stars in a larger Galactic volume was, however, questioned by Feltzing et al. (2017), who found that the trend breaks down for lower metallicity stars, [Fe/H] ≲ −0.5. The analysis of a large sample of stars by Casali et al. (2020) suggests that the [Y/Mg] and age relationship in not uniform with respect to different Galactocentric distances or star formation histories. This finding was attributed to the differences in star formation history of different Galactic populations coupled with metallicity dependence of s-process yields (Busso et al. 2001; Gallino et al. 2006; Karakas & Lugaro 2016) leading to distinct [Y/Mg] trends.

However, most of these studies assumed local thermal equilibrium (LTE) line formation for the Y abundance diagnostics. It has been well established that departures from LTE, the NLTE framework, lead to significant differences in the abundance analysis. Specifically, for other s-process elements such as Sr and Ba, NLTE effects are known to be of the order of around 0–0.2 dex, and up to 0.5 dex for some Sr lines (Bergemann et al. 2012b; Korotin et al. 2015; Gallagher et al. 2020). It is therefore of interest to quantify the effects of NLTE in Y, in order to assess the consequences for standard diagnostics in the context of Galactic Chemical Evolution (GCE; Prantzos et al. 2018; Kobayashi, Karakas & Lugaro 2020) and chemical tagging (Freeman & Bland-Hawthorn 2002; Hogg et al. 2016).

In this paper, we present a NLTE model atom of Y and apply the model in the analysis of spectra of Sun-like stars. In Section 2, we describe the observational data and the analysis of stellar parameters and ages. In Section 3, we provide the details on the statistical equilibrium calculations for Y and quantify the NLTE effects on the spectral lines of Y ii commonly used in the analyses of late-type stars. The abundance analysis of the solar analogue sample is presented in Section 4. The main results are described and compared to other literature studies in Section 5. We close the paper with conclusions in Section 6.

2 OBSERVATIONS AND STELLAR PARAMETERS

We use the optical high-resolution UVES spectra from the 4th public data release (DR) of the Gaia-ESO large spectroscopic survey.¹ (Gilmore et al. 2022; Randich et al. 2022). The spectra have the resolving power of |$\lambda /\Delta \lambda \approx 47\, 000$| and cover the optical wavelength range from 4770 to 6830 Å with a small beam splitter gap at 5774–5833 Å.

Stellar parameters, including T_eff, log g, and [Fe/H], and ages of stars, were determined using the sapp Bayesian pipeline (Stellar Abundances and atmospheric Parameters Pipeline) (Gent et al. 2022). The code adopts NLTE spectroscopic models as described in Kovalev et al. (2019) and the GARSTEC evolutionary tracks (Weiss & Schlattl 2008). In addition to spectra, we also include Gaia DR3 photogeometric distances (Bailer-Jones et al. 2021), as well Gaia DR3 (Gaia Collaboration 2023), 2MASS (Skrutskie et al. 2006), and APASS (Henden et al. 2015) photometry. The Bayesian analysis of stellar parameters closely follows the method outlined in Gent et al. (2022) (see their section 3.9 for the full procedure), with the only exception that the asteroseismic module is not used, since for none of our solar analogues global oscillation quantities are available. The main advantage of the sapp code is in its ability to combine all observational information available for stars, including spectra, photometry, and astrometry, in order to provide homogeneous estimates of fundamental stellar parameter in the same self-consistent comprehensive statistical framework. As such, the code does not assume the spectroscopic estimates to perform the analysis of ages, but rather provides T_eff, log g, [Fe/H], masses, and ages of stars from the combination of spectroscopic and photometric probability distribution functions (PDFs), the former assisted by grids of stellar spectra and the latter involving models of stellar evolution and parallax (for a similar, but not identical approach, see also Pont & Eyer 2004; Jørgensen & Lindegren 2005; Serenelli et al. 2013; Schönrich & Bergemann 2014). The result is the posterior PDF in the multidimensional parameter space, with the uncertainties reflecting the true shape of the individual distribution functions corresponding to individual observables. The final estimates of stellar parameters are computed from the moments of the posterior PDF. For the complete mathematical and numerical framework we refer the reader to Gent et al. (2022).

Since the scope of this work is to test the Y/Mg relation with age for solar analogues, out of the entire high-resolution Gaia-ESO sample, we have selected 48 stars that closely match the atmospheric parameters of the Sun, that is T_eff = 5777 ± 100 K and log g = 4.44 ± 0.1 dex. This sample comprises stars within a range of ages from 4 to 11 Gyr and heliocentric distances from 0.4 to 1.6 kpc. We did not impose any restrictions on [Fe/H], however, the choice on T_eff and log g naturally limits the sample to stars with [Fe/H] from ∼−0.6 to +0.1 dex. The spectral signal-to-noise ratio (SNR) ranges from 24 to 130 with a median of 48 per pixel. The errors of T_eff, log g, and [Fe/H] are on average 132 K, 0.1 dex, and 0.01 dex, respectively. Ages are accurate to |$\sim 30~{{\ \rm per\ cent}}$| (see also Serenelli et al. 2013), which is fully sufficient for the purposes of this paper, as we are not aiming to define an age–abundance relationship but only to investigate the impact of NLTE effects in the Y/Mg ratio regardless of the nature and slope of the correlation with age.

We note that, not only to enable a more comprehensive analysis of LTE and NLTE trends, but also since the model atom of Fe has been improved as discussed in Semenova et al. (2020), we have chosen to re-derive metallicities in LTE and in NLTE using the same approach as that adopted for Y and Mg, as described in Section 4 below. Comparison of the new NLTE metallicities with the NLTE sapp values suggests a very good agreement between the metallicity scales with the mean difference of only 0.05 dex.

3 NLTE ANALYSIS

The Y model atom was constructed using data from the Robert Kurucz's online data base (Kurucz et al. 1995).² The models contains 423 Y i and Y ii levels, and it is closed by the ground state of Y iii. This element only has one stable isotope, ⁸⁹Y and the hyperfine splitting is very small (Hannaford et al. 1982; Nissen et al. 2017), with the splitting less than 1 mÅ, and thus can be neglected in the line formation calculations. The model includes |$11\, 819$| radiative bound–bound transitions with the oscillator strengths from Biémont et al. (2011). The lines span the entire wavelength range from |$1000 \textrm {~}\lt \lambda \lt 100\, 000$| Å. Fig. 1 depicts the Grotrian diagram of the model atom, illustrating all bound-bound transitions. The radiative bound–free transitions are calculated using the hydrogenic recipe; this is not a significant source of uncertainty, because of the very large ionization potential of Y ii of 12 eV, which makes radiative ionization (the main source of NLTE effects in neutral ions, Bergemann & Nordlander 2014) extremely inefficient similar to other singly ionized species, such as Sr (Bergemann et al. 2012b) and Ba (Gallagher et al. 2020). For the lack of quantum-mechanical data, collisions with e⁻ and H atoms are described using the standard formulae from van Regemorter (1962), Seaton (1962), and Drawin (1968). To check the sensitivity of the NLTE corrections to collisional data, we performed additional calculations with scaling factors applied systematically to all rate coefficients, corresponding to the Y + H and Y + e⁻ reactions. The results are shown in Fig. 2. Clearly, decreasing the Drawin’s rates for bound–bound and bound–free transitions by a factor of 10 (SH = 0.1) or 100 (SH = 0.01) does not lead to any significant changes in the NLTE corrections, although, as expected the corrections slightly change, by ∼+0.03 to +0.05 dex at lower metallicity. Likewise, the variation of the scaling factor to electron collision rates, ES = 0.1, 0.01 leads to nearly identical results. The maximum change does not exceed 0.04 dex in metal-poor main-sequence model atmospheres. We note, however, that these classical collisional formulae typically overestimate the cross-sections of reactions associated with particle–particle collisions by several orders of magnitude (Barklem 2016; Bergemann et al. 2019, 2021) and consequently our results obtained with the reference model atom, which assumes unscaled rate coefficients (SH =1 and ES =1), can be viewed as conservative estimates of the NLTE effects in the diagnostic Y ii lines.

The statistical equilibrium (SE) calculations are performed the multi1d code, which solves radiation transfer equation in one-dimensional (1D) plane-parallel geometry and the SE equations in the iterative process. The code was originally published by Carlsson (1986) and further updated by different research groups over the years to match the individual scientific needs. The updates performed in our group are described in Bergemann et al. (2019) and Gallagher et al. (2020). We make use of the MARCS 1D LTE model atmospheres (Gustafsson et al. 2008) that are available for the parameter space of late A and FGKM-type stars (2500 ≤ T_eff ≤ 8000 K), surface gravity (−1 ≤ log g ≤ 5 dex), metallicity (−5 ≤ [Fe/H] ≤ +1 dex) and microturbulence (ξ_t = 0, 1, 2, 5 km s⁻¹, where different temperature and surface gravity models have different available parameters chosen from these four options). The solar-metallicity opacity file, which contains the absorption properties of various atomic species, ions, and molecules in the atmosphere (referred to as ABSMET in MULTU1D), was used for all model calculations. This is because using opacity files with different metallicities had no impact on the line profiles of any diagnostic Y ii lines. The departure coefficients for the MARCS grids were calculated using a python wrapper to multi1d³ and these are available on our public data base that also includes other NLTE grids for the elements (Ba, Ca, Co, Fe, H, Mg, Mn, Na, Ni, O, Si, Sr, Ti) at https://keeper.mpdl.mpg.de/d/6c2033ef5c5d4c9ca8d1/.

The selection of diagnostic Fe, Mg, and Y lines is carried out on the basis of the Gaia-ESO linelist (Heiter et al. 2021), which represents a careful compilation of atomic and molecular data used in the Gaia-ESO survey analysis. The linelist was recently updated with new atomic data for several elements (Magg et al. 2022), including Mg, for which the oscillator strengths from Pehlivan Rhodin et al. (2017) are used. The list of Fe lines comprises 52 Fe i and 8 Fe ii lines with YY, YU, or UY flags in the Gaia-ESO linelist. These lines were selected through an initial fit applied to a larger sample of Fe lines. However, through a careful visual inspection we then removed the strong lines, lines resulting in poor fits (e.g. because of blends), or lines yielding vastly inconsistent abundances (possibly because of inaccurate gf values). For Y ii, the log gf values are taken from Biémont et al. (2011), who estimated the transition probabilities by combining the new level lifetimes measured using the technique of laser induced fluorescence with the theoretical branching fractions computed by means of pseudo-relativistic Hartree–Fock method with core-polarization effects. In Biémont et al. (2011), no errors are provided for individual transitions, however, the authors note that the uncertainties are expected to range from 5 to 15  per cent, with errors being larger for weaker transitions. Most of our diagnostic lines of Y ii are rather strong, as judged by the laboratory intensity value in column 3 of table 5 in Biémont et al. (2011). Therefore, here we conservatively adopt the uncertainty of 0.05 dex (⁠|${\approx} 11~{{\ \rm per\ cent}}$|⁠) for Y ii transitions, except the transitions at 5289 and 5728 Å, for which we adopt the oscillator strength error of 0.07 dex (⁠|${\approx} 15~{{\ \rm per\ cent}}$|⁠). Damping caused by elastic collisions with H atoms follows the quantum-mechanical data from Barklem, Piskunov & O’Mara (2000), for the 5711 Å Mg i line we employ the γ and σ values from Barklem (private communication). If these are not available, as is the case for the Y ii lines, the cross-sections are calculated from the theory of Anstee & O’Mara (1991, 1995).

Through a careful inspection of the line profiles, we found that out of the four diagnostic Y ii lines in the Gaia-ESO UVES data only the line at 4883 Å is sufficiently strong and clearly discernible from the other features in all spectra in our stellar sample. Therefore, the final abundance analysis of the solar analogues we primarily rely on this Y ii feature. However, the solar abundance analysis is carried out using a larger set of Y ii lines. Atomic data for the diagnostic Y ii and Mg i lines are provided in Table 1.

4 ABUNDANCE ANALYSIS

For the detailed abundance analysis, we use the turbospectrum-nlte (also known as tsv20) code (Gerber et al. 2023), which is an updated version of the original turbospectrum code (Alvarez & Plez 1998; Plez 2012). The code is open-source⁴ and allows computations of synthetic stellar spectra with NLTE effects for multiple chemical elements at once. To find the best-fitting abundances, we make use of the tsfitpy wrapper (Gerber et al. 2023) that relies on the Nelder–Mead optimization. For the purpose of this work, tsfitpy code was refactored, optimized, and several new features were added. One of the new features is the ability to fit macroturbulence (V_mac) and rotation for individual spectral lines. This is done using the L-BFGS-B (Limited-memory Broyden–Fletcher–Goldfarb–Shanno with bound consideration) algorithm (Byrd et al. 1995; Zhu et al. 1997) as a secondary step after fitting an abundance to break the degeneracy of fitting these two parameters. Additionally, the code was parallelized using the dask python package (Rocklin 2015; Dask Development Team 2016).

The abundances were determined via a strictly differential analysis, that is, relating the abundance from each individual line to the measurement from the solar spectrum, with respect to the solar UVES spectra from the Gaia-ESO benchmark library (Blanco-Cuaresma et al. 2014). One of the main advantages of the differential analysis is that one can minimize the errors caused by, e.g. atomic and molecular data, and for stars with very similar atmospheric parameters, this approach allows to partly counterbalance the physical limitations of the model atmospheres and line formation, such as the lack of 3D convection (e.g. Nissen & Gustafsson 2018). For the solar UVES spectrum, we obtain in NLTE, A(Fe) = 7.53 dex and A(Mg) = 7.55 dex, respectively. These estimates are fully consistent with the results by Bergemann et al. (2012a, 2017) and Magg et al. (2022), respectively. The detailed solar results for Y ii will be discussed in detail in the next section.

To estimate the systematic abundance error, we follow the approach adopted by Bergemann et al. (2017), in which the individual stellar parameters are varied according to their respective uncertainties and the change in the measured abundance ratios is derived by adding individual errors in quadrature:

using average errors on stellar parameters described in Section 2. This procedure yields systematic uncertainties σ_A(Y) = 0.04 dex and σ_A(Mg) = 0.09 dex. We also note that for the Sun, the primary source of the uncertainty in the abundance estimate is the quality of the oscillator strengths and physical limitations of models. This will be addressed in more detail in Section 5.2.

5 RESULTS

We begin the analysis of results with the discussion of NLTE effects in lines of Y ii. To allow for a more comprehensive understanding of NLTE effects, we also include a wider range of Y ii lines from the near-ultraviolet (near-UV) to the optical (Table A1), in order to make a more direct reference to studies of Y abundances in FGK-type stars in the literature.

We also limit the analysis to lines with the EW greater than 1 mÅ, as weaker features are not detectable in our observed spectra. The NLTE effects are quantified via the standard concept of the NLTE abundance correction (Bergemann & Nordlander 2014), which represents the difference in abundance that is required to match the equivalent width (EW) of a NLTE model line to that of the LTE line computed using the identical values of stellar parameters and Y abundance.

5.1 NLTE effects on Y ii lines

Fig. 3 shows the average NLTE corrections for two groups of Y ii lines. The lines are grouped by the excitation potential (EP) of their lower energy state, such that the low-EP features are those that originate from states with ∼0.1 eV and the corresponding spectral lines are visible in the near-UV and blue range of FGK-type stars, such as the lines at 3710.29, 3774.33, and 4398.01 Å. The medium-EP lines arise from the energy states around ∼1 eV, leading to a set of rather weak spectral lines in the optical; these are sometimes employed in the solar abundance analysis (Grevesse et al. 2015). Some representative Y ii lines of the latter group are the lines at 4883.68, 4900.12, and 5087.42 Å.

As seen in Fig. 3, overall the NLTE corrections to Y ii lines are typically positive and they increase with decreasing metallicity or increasing the T_eff of the model atmosphere. This is the consequence of progressively stronger UV radiation fields in more metal poor, or hotter, environments that enhances the overexcitation in the strong Y ii lines in the UV and in the optical wavelength ranges. At the solar metallicity, we find NLTE effects at the level of only ∼±0.05 dex for all models atmospheres and all Y ii lines. For the main-sequence and turn-off stars, the NLTE corrections for the low-EP lines typically do not exceed +0.2 dex even at [Fe/H] ≈−4. However, for red giant models, NLTE corrections may reach up to +0.75 dex at [Fe/H] ≈ −5. For the medium-excitation Y ii lines, such as those of multiplets 21 and 23, the NLTE corrections are typically much more modest and do not exceed 0.05 dex for the main-sequence stars and 0.15 dex for red giant models even at very low [Fe/H]. Interestingly, the change of slope of the NLTE correction is usually seen at [Fe/H] ∼ −1, which may have an impact on the GCE trend of [Y/Fe] (Kobayashi et al. 2020) or, for example, on the ratio of [Ba/Y] which is used to quantify the s-process efficiency (Travaglio et al. 2004; Prantzos et al. 2018).

To further examine the differences between the line formation of low- and medium-EP lines, we show the departure coefficients b_i as a function of optical depth in Fig. 4. Here |$b_i = b\frac{n_i \rm NLTE}{ n_i \rm LTE}$|⁠, i.e. the quantity represents the ratio of NLTE to LTE populations of the energy state i. We include four limiting cases: a red giant model (top) and a main-sequence model (bottom) with metallicities [Fe/H] = 0 and −3, in order to visualize the variations in line formation and their dependences on both EP and stellar metallicity. The black lines correspond to low-EP (3832 Å), while red ones correspond to medium-EP (4883 Å) lines. The formation heights of the cores of the diagnostic Y ii lines (log τ = 0 for line core) are shown with vertical lines.

Let us first focus our attention on the top left panel in Fig. 4, which shows the solar metallicity, [Fe/H] = 0, model atmosphere of a red giant star. Both low- and medium-EP lines form in the outermost parts of the atmospheres, at log_τ500 ≲ −3. In these layers, the lower energy states of both lines, a³D and a³F, tend to be close to unity (b_i ≈ 1). However their upper energy states j, z³F* and z³D*, are significantly overpopulated with respect to the LTE values. Following Bergemann & Nordlander (2014), this implies that the line source function exceeds the Planck function |$S^{\rm l}/B^{\rm l}_{\nu } \approx b_j/ b_i \gt 1$|⁠. This results in line weakening under NLTE conditions compared to LTE and slightly positive NLTE corrections.

At low metallicity, [Fe/H] = −3 dex, another effect becomes relevant. Owing to less line blanketing, the more intense radiation field in the Wien’s regime leads to stronger radiative pumping in the transitions and correspondingly larger overpopulations of higher energy states in Y ii. This is manifested in an even larger source function versus Planck function imbalance, compared to the solar metallicity. But we also find a noticeable overpopulation of the medium-EP states, such as the a³F level, which represents the lower level of the medium-EP line. Consequently, the line opacity is slightly enhanced compared to LTE (κ^l ∼ b_i > 1), and the NLTE corrections decrease, also they remain positive. This effect of opacity is further amplified as metallicity decreases, and hence the NLTE corrections for the medium-EP lines change sign, eventually becoming negative at very low [Fe/H]. In contrast, the a³D state of Y ii remains nearly thermalized also at low metallicity, κ^l ≈ 1, and the progressively increasing |$S^{\rm l}/B^{\rm l}_{\nu }$| imbalance leads to steadily increasingly positive NLTE corrections for the low-EP Y ii lines over the entire metallicity range.

For the main-sequence model atmospheres, the b_i factors show a qualitatively different behaviour. At solar [Fe/H], the lower energy states of low- and medium-EP transitions remain close to unity throughout the entire range of optical depths, −3 ≲ log_τ500 ≲ 1. The ground state of the low-EP line is nearly thermalized at the depth of the core formation, which implies LTE line opacity. Although the line source function is slightly sub-thermal (as b_j/b_i < 1), the Y ii line is strong and the NLTE correction, although negative, remains close to zero. For the medium-EP line, the somewhat larger effect on S^l is counterbalanced by the weak depopulation of the lower energy state a³F, implying smaller opacity in NLTE. Also in this case, the net effect is that the NLTE correction is about zero. At low metallicity, [Fe/H] = −3, the Y ii lines become very weak and the NLTE corrections follow the same trend as that observed for low-[Fe/H] RGB models. The low-EP line is weakened under NLTE conditions, owing to decreased line opacity and increasing NLTE corrections. In contrast, the NLTE effects on the medium-EP line decrease, due to the progressively increasing overpopulation of its energy state.

To further investigate the change of slope for the medium-EP lines at [Fe/H] ≈ −1, in Fig. 5 we show the curve of growth (CoG) for the Y ii line 4883 Å for the model atmosphere of a star with T_eff = 6500 K and log g = 3 and different metallicities. The NLTE CoG is plotted in black, while the LTE CoG is in red. Since Fig. 3 was computed for [Y/Fe]  = 0, we focus our attention on that part of the graph as well. For [Fe/H] ≲ −3, the Y ii line is clearly on the linear part of the CoG, unless an extreme [Y/Fe] enhancement is adopted. However, at around [Fe/H] ≈ −1 (the exact value is a bit different for each line), the line undergoes a transition from the linear to saturated part of the CoG, and for higher metallicities this line is primarily saturated, even for sub-solar [Y/Fe] ratios. At this transition point, the LTE line becomes stronger leading to positive NLTE corrections. On the other hand, for other metallicity values, the LTE line is instead weaker giving rise to negative NLTE corrections. This curious behaviour results in a peak for NLTE corrections around [Fe/H] ≈ −1 as seen in the right part of Fig. 3.

To summarize, we find that the NLTE has a different effect on the near-UV and optical lines of Y ii. The lines emerging from the lowest excitation energy states tend to be weaker when calculated in NLTE, and the effect increases with decreasing [Fe/H] or increasing T_eff of the model atmosphere. In contrast, the lines emerging from the energy states above ∼1 eV are not so sensitive to NLTE, however, modest effects at the level of +0.1 dex can be expected depending on the metallicity, T_eff, and log g of the star.

5.2 The Sun

Grevesse et al. (2015) carried out a comprehensive study of the abundances of neutron-capture elements in the solar photosphere, including Y in LTE. They used nine Y ii lines (see Table A1). For most of these, log gf from Hannaford et al. (1982) was used. However, lines 4900, 5473, and 5728 were updated based on the lifetimes from Wannstrom et al. (1988) and Biémont et al. (2011). By carefully inspecting the line profiles, we found that four of those lines (4125, 4398, 5119, 5728) were very weak or were severely affected by blends. Therefore, we limited the abundance analysis to the five Y ii features (Table A1).

In Fig. 6, we show the observed solar Kitt Peak National Observatory FTS atlas (Kurucz et al. 1984) in comparison with the best-fitting NLTE Y ii models. Here, we assumed a constant macroturbulence V_mac = 3 km s⁻¹ and a microturbulence ξ_t = 1 km s⁻¹ from Bergemann et al. (2012a). We also include the diagnostic Y ii line at 4883.68 Å, as the feature is the primary diagnostic in the analysis of solar analogues. Based on these five Y ii lines, we obtain the average best-fitting solar Y abundance of A(Y)_LTE = 2.08 ± 0.04 dex and A(Y)_NLTE = 2.12 ± 0.04 dex. The values are significantly different, if we use the log gf values adopted by Grevesse et al. (2015), which are mainly based on older transition probabilities from Hannaford et al. (1982) although re-normalized to other sources of lifetimes for some of the lines. In this case, we obtain A(Y)_LTE = 2.19 ± 0.05 dex and A(Y)_NLTE = 2.22 ± 0.05 dex, which are also in agreement with the MARCS-based value A(Y)_LTE = 2.14 ± 0.06 dex by Grevesse et al. (2015).

Currently, there seems to be no strong reason in favour of older gf values, which were adopted by Grevesse et al. (2015), see also a comprehensive discussion of the improvements in Biémont et al. (2011). Hence, it is likely that the fact that our photospheric abundance of Y based on Biémont et al. (2011) atomic data is slightly lower compared to the meteoritic value, 2.17 ± 0.04 dex (Lodders, Palme & Gail 2009), is due to the lack of 3D effects in our spectrum synthesis calculations. Indeed, if we include the 3D abundance corrections obtained by comparing the 3D LTE with 1D LTE estimates from Grevesse et al. (2015), which amount to +0.07 dex (in the sense that 3D abundance are higher compared to 1D), then we would obtain A(Y)_LTE = 2.15 ± 0.08 dex and A(Y)_NLTE = 2.19 ± 0.08 dex, in agreement with meteoritic values, however, with a somewhat larger scatter compared to our 1D estimates. The much increased error may imply that combining the solutions obtained 1D LTE, 1D NLTE, and 3D LTE is not a reliable approach, and full 3D NLTE calculations would be necessary to further constrain the solar Y abundance, which we defer to a subsequent work in the series.

5.3 HD 122563

An interesting test of the NLTE model atom is to perform the analysis of the excitation–ionization balance of an element. This test is particularly relevant in the regime, where the NLTE corrections show a different sign and/or amplitude (e.g. Bergemann et al. 2012b). However, no suitable Y i lines are available spectra of FGK type stars, also very few observed spectra have the quality needed to reliably isolate Y ii lines of different multiplets.

To test the excitation balance of Y, we have opted for the analysis of the very high-resolution (R  = |$190\, 000$|⁠) and SNR (660.8) ESPRESSO spectrum⁵ of the well-studied metal-poor red giant HD 122563. The wavelength coverage extends to 3800 Å that allows probing the low- and medium-EP Y ii lines. Based on Bergemann et al. (2012a), Afşar et al. (2016), Creevey et al. (2019), and Sneden et al. (2021), we chose the following stellar parameters for the fit T_eff = 4600 K, log g = 1.39 dex, [Fe/H]_LTE = −2.60 dex, and [Fe/H]_NLTE = −2.55 dex. Here, the surface gravity is taken from the asteroseismic analysis by Creevey et al. (2019). The T_eff is in excellent agreement with the interferometric value (Karovicova et al. 2020), T_eff = 4635 ± 34 K. We adopt ξ_t of 2.2 km s⁻¹ for the analysis based on Afşar et al. (2016) and Sneden et al. (2021).

The best-fitting models are compared with ESPRESSO observations in Fig. 7. Out of all low-EP lines in Table A1, only the feature at 3832 Å was of a sufficient quality to extract a reliable abundance measurement. Of the medium-EP lines, the three features at 4883, 4900, and 5087 Å are suitable. Owing to the exquisite data quality, the internal error of the fit is very small. Clearly the models yield a good agreement with the data, although we emphasize that the quality of the fit is equally good in LTE and NLTE.

In Fig. 8 (top panel), we show the excitation balance of Y ii, with the abundance of Y derived from each the four diagnostic Y ii features a function of the EP of the lower energy level of each transition. The [Y/H] ratio is useful because it accounts for the differences in the adopted [Fe/H], that is LTE or NLTE. The possibly most interesting feature of this figure is that NLTE significantly improves the excitation balance of Y ii. Specifically, in LTE, we find a large difference between the medium-EP and the low-EP lines, with the latter yielding a 0.28 dex lower Y abundance and thus A(Y)_LTE = −1.02 ± 0.13 dex. In NLTE, remarkably, the difference reduces to 0.06 dex and the low-EP line now yields a slightly higher abundance compared to the medium-EP features. Thus, we obtain a nearly perfect excitation balance in NLTE, with the average abundance of A(Y)_NLTE = −0.89 ± 0.04 dex or [Y/Fe] = −0.55 ± 0.04 dex. For comparison, literature values for HD 122563 include Aoki et al. (2005), who obtained A(Y)_LTE = −0.84 ± 0.24 dex, and Honda et al. (2006) with A(Y)_LTE = −0.93 ± 0.09 dex. These measurements are in full agreement with our LTE value.

Finally, in Fig. 8 (bottom panel) we also illustrate the results in the plane of line EW–[Y/Fe], where the latter is the parameters of a primary interest in GCE studies (Travaglio et al. 2004; Prantzos et al. 2018; Kobayashi et al. 2020). Whereas the figure is conceptually identical to the other panel, it is useful, because it highlights the bias incurred by LTE assumption in the GCE modelling of [Y/Fe]. Also in this figure, for consistency, we use LTE value of [Fe/H] for LTE A(Y) and NLTE value of metallicity for the NLTE estimate of A(Y). Most relevant, in this respect, is the vastly underestimated [Y/Fe] abundance ratio obtained in LTE from the near-UV low-EP line. The LTE [Y/Fe] value obtained from this feature is −0.84 dex, which is 0.28 dex lower compared to the LTE value obtained from the optical Y ii lines. As we show the stellar parameters of HD 122563 are very well constrained, with T_eff known to 35 K (Karovicova et al. 2020) and log g to 0.01 dex (Creevey et al. 2019). This implies that Y abundance studies that rely on medium-resolution and/or low-SNR spectra, which do not allow for a reliable diagnostics of weak spectral features, may contain a strong systematic bias. Fortunately, the medium-EP optical Y ii lines are not affected by this problem, so that rather reliable results can also be obtained from LTE measurements. However, this requires spectra with a very high SNR and high resolving power.

In another example, Sneden et al. (2003) investigated a metal-poor red giant star CS 22892−052 (T_eff = 4700 K, log g = 1.0, and [Fe/H] = −3.10 dex), using 12 low-EP and 3 medium-EP lines of Y ii. They reported an [Y/Fe] ratio lower by 0.15–0.30 dex compared to that of Sr and Zr, and noting that Y was deviating significantly from the scaled Solar system abundance distribution for neutron-capture species from Burris et al. (2000). From the analysis of our data presented in Fig. 3, we find the average NLTE correction for the stellar parameters and line selection from Sneden et al. (2003) would amount to ∼+0.25 dex, thereby bringing the abundance of Y into agreement with the other first peak s-process elements. Hence, NLTE effects in Y could potentially be relevant within the context of the problem of a ‘significant deviation from solar neutron-process curve for some lighter neutron-capture elements’, as described by these authors.

In summary, we show that NLTE resolves the problem of excitation imbalance between different Y ii lines in the spectrum of a metal-poor red giant HD 122563. Whereas medium-EP Y ii lines, specifically the optical features at 4883, 4900, and 5087 Å, can be reliably modelled in LTE, we find that the low-EP lines, such as the near-UV Y ii line at 3832 Å (e.g. Sneden et al. 2003; Lai et al. 2008; Cohen et al. 2013; Tanrıverdi & Baştürk 2016), require NLTE modelling for an unbiased estimate of Y abundance in the spectra of metal-poor stars.

5.4 Solar analogues and the [Y/Mg] chemical clock

Finally, we explore the consequence of NLTE effects in the Y lines on the use of this element as a Galactic chemical clock. Fig. 9 shows our NLTE [Y/H] ratios as a function of stellar ages for our sample of solar analogues. To remind, these stars are selected strictly on T_eff and log g to be similar to the Sun, but the [Fe/H] values are unconstrained. The correlation is strong, suggesting that the Y abundance increases with decreasing stellar age. The solution of the best-fitting line is given by [Y/H] = 0.574 − 0.101(± 0.012) × age [Gyr] with r = −0.77, where r is Pearson product–moment correlation coefficient.

Now we can also investigate how the [Y/Mg] ratio – the ‘chemical clock’ – behaves as a function of age. For LTE values of [Y/Mg], the correlation with stellar age is weak with r = −0.48 and the resulting least-squares fit yields [Y/Mg] = 0.320 − 0.051(± 0.014) × age [Gyr]. The slope is noticeably higher compared to other studies (such as slope of −0.0404 ± 0.0019 dex Gyr⁻¹ in Nissen 2015). The weak correlation is not surprising, given that our stellar sample probes a substantial spatial volume and their chemical elemental enrichment would also vary. Fig. 10 shows the NLTE [Y/Mg] ratios as a function of age, with Mg determined self-consistently in NLTE. While the core objective of this study is not to directly investigate the Y abundance as a’chemical clock’, but to assess the potential impact of NLTE on this quantity, it is still important to comment on the increased scatter associated with the observed data. As outlined in Section 4, systematic uncertainties of abundances derived from stellar parameters are larger for Mg than for Y, possibly leading to the increased scatter observed in the [Y/Mg]–age plane. Nevertheless, the combined [Y/Mg] systematic uncertainty is around 0.13 dex, which is not large enough to explain the [Y/Mg] scatter of approximately 0.20–0.30 dex. Considering the age uncertainties of 30  per cent, it is unclear if the total scatter originates from any physical factors or from combined uncertainties. However, as our focus is primarily on the NLTE effect on this relation, we concentrate our analysis on this specific question, and encourage more detailed studies of the [Y/Mg]–age correlation using ages obtained using other complementary quantities, such as asteroseismic measurements.

The combined NLTE corrections for [Y/Mg] do not exceed 0.08 dex for all stars in the sample. Thus, the equation has a similar although a bit smaller slope with equation [Y/Mg] = 0.268 − 0.045(± 0.013) × age [Gyr] and a similar r = −0.46. This would imply that NLTE effects do not significantly influence the [Y/Mg] ratio for stars with T_eff and log g similar to the Sun. Therefore, the applicability of this ratio as a’chemical clock’ for Sun-like stars would not be substantially biased by neglecting NLTE effects.

6 CONCLUSIONS

In this study, we perform a detailed analysis of NLTE effects in the diagnostic lines of Y ii that are typically used in the chemical abundance calculations for late-type stars. We develop a new NLTE model atom of Y and explore the sensitivity of the resulting NLTE effects to the still poorly known collisional data.

We find that NLTE effects in the Y ii lines depend on stellar parameters and on the atomic properties of lines, however, the NLTE corrections are robust with respect to assumptions on H and e⁻ collisions. For the solar photosphere, we obtain the NLTE Y abundance of A(Y)_NLTE = 2.12 ± 0.04 dex, which is 0.04 dex higher compared to the LTE value. The absolute value is slightly lower compared to the abundance of Y in C i chondrites, however, this difference could be either caused by the quality of transition probabilities and/or by 3D effects. Applying the fiducial 3D corrections from Grevesse et al. (2015) to our NLTE value, we obtain a solar photospheric Y abundance A(Y)_NLTE = 2.19 ± 0.08 dex consistent with meteorites, however, with a double line-to-line scatter, which may indicate the need for full 3D NLTE calculations to put the solar Y abundance on a firmer ground.

For unevolved – main-sequence and subgiant – stars with metallicities in the range −4 ≲ [Fe/H] ≲ +0.5, the NLTE corrections are within +0.2 dex for all diagnostic Y ii lines and typically, for the parameter space of the Galactic disc stars they do not exceed ±0.1 dex. Especially, medium-EP lines of multiplets 21 and 23 are almost unaffected by NLTE. Since these lines form the main Y diagnostics in optical high-resolution spectra of FGK-type stars (e.g. Bensby et al. 2014; Nissen 2015; Tucci Maia et al. 2016), we conclude that no significant biases are associated with LTE-based abundance derived from these features, as long as one does not aim at Y abundances precise to better than ≈0.1 dex.

For red giants, the NLTE effects are more severe and may exceed 0.5 dex for the low-EP Y ii lines at low metallicities, [Fe/H] ≲ −3 dex. Such spectral lines are typically used in the abundance studies of very- and ultra-metal-poor stars in the Galactic halo (e.g. Johnson 2002; Sneden et al. 2003; Roederer et al. 2014b), indicating that the LTE approximation is not suitable in this regime and may lead to substantial biases in [Y/Fe] abundance ratios. For the Gaia/4MOST benchmark star, HD 122563, we find a significant excitation imbalance in LTE, with [Y/Fe]_LTE = −0.60  ±  0.11 dex, and the lower EP line yielding a much lower Y abundance compared to medium-EP features. The problem is resolved in NLTE, with [Y/Fe]_NLTE = −0.55 ± 0.04 dex, and a very good agreement between all diagnostic Y ii lines.

Finally, we show, through a detailed LTE and NLTE abundance analysis of 48 solar analogues with high-quality Gaia-ESO spectra, that the [Y/Mg] ratio in Sun-like stars is not significantly affected by NLTE, confirming the reliability of [Y/Fe] ratio as a potential Galactic chemical clock, at least from the perspective of line formation of Y and Mg. Similar to previous literature studies (e.g. da Silva et al. 2012; Nissen 2015; Feltzing et al. 2017; Jofré et al. 2020), we also find a significant correlation between [Y/Fe] and stellar ages, however, the correlation of [Y/Mg] with ages is far less pronounced, which could possibly be due to less strict limits on our definition of solar analogues, such as the lack of [Fe/H] as a selection criterion (see also Feltzing et al. 2017), or a larger Galactic volume probed by our stellar sample.

ACKNOWLEDGEMENTS

Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme ID(s) 072.D-0019(B), 072.D-0309(A), 072.D-0337(A), 072.D-0406(A), 072.D-0507(A), 072.D-0742(A), 072.D-0777(A), 073.C-0251(B), 073.C-0251(C), 073.C-0251(D), 073.C-0251(E), 073.C-0251(F), 073.D-0100(A), 073.D-0211(A), 073.D-0550(A), 073.D-0695(A), 073.D-0760(A), 074.D-0571(A), 075.C-0245(A), 075.C-0245(C), 075.C-0245(D), 075.C-0245(E), 075.C-0245(F), 075.C-0256(A), 075.D-0492(A), 076.B-0263(A), 076.D-0220(A), 077.C-0655(A), 077.D-0246(A), 077.D-0484(A), 078.D-0825(A), 078.D-0825(B), 078.D-0825(C), 079.B-0721(A), 079.D-0178(A), 079.D-0645(A), 079.D-0674(A), 079.D-0674(B), 079.D-0674(C), 079.D-0825(B), 079.D-0825(C), 079.D-0825(D), 080.B-0489(A), 080.B-0784(A), 080.C-0718(A), 081.D-0253(A), 081.D-0287(A), 082.D-0726(A), 083.B-0083(A), 083.D-0208(A), 083.D-0671(A), 083.D-0682(A), 083.D-0798(B), 084.D-0470(A), 084.D-0693(A), 084.D-0933(A), 085.D-0205(A), 086.D-0141(A), 087.D-0203(B), 087.D-0230(A), 087.D-0276(A), 088.B-0403(A), 088.B-0492(A), 088.C-0239(A), 088.D-0026(A), 088.D-0026(B), 088.D-0026(C), 088.D-0026(D), 088.D-0045(A), 089.D-0038(A), 089.D-0298(A), 089.D-0579(A), 090.D-0487(A), 091.D-0427(A), 092.D-0171(C), 092.D-0477(A), 093.D-0286(A), 093.D-0818(A), 094.D-0363(A), 094.D-0455(A), 171.D-0237(A), 187.B-0909(A), 188.B-3002(A), 188.B-3002(B), 188.B-3002(C), and 188.B-3.

Based on observations collected at the European Southern Observatory under ESO programme ID 0103.D-0118 and data obtained from the ESO Science Archive Facility with DOI(s) under https://doi.org/10.18727/archive/21.

This work has made use of the VALD data base, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.

NS and MB acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 949173).

MB is supported through the Lise Meitner grant from the Max Planck Society. We acknowledge support by the Collaborative Research Centre SFB 881 (projects A5, A10), Heidelberg University, of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

We would like to express our appreciation to the anonymous reviewer for their insightful comments and suggestions, which significantly contributed to improving the quality of this paper.

DATA AVAILABILITY

All work presented in this study relies on publicly available data acquired within the Gaia-ESO survey. We also make use of the publicly available ESPRESSO spectrum of HD 122563.

Footnotes

References

APPENDIX A: TABLE WITH Y ii LINES