The Impact of Inelastic Collisions with Hydrogen on NLTE Copper Abundances in Metal-poor Stars

The stellar parameters of our program stars are adopted from Roederer & Barklem (2018), and they have also been used by Korotin et al. (2018). Briefly, the effective temperature T_eff was calculated from the Casagrande et al. (2010) metallicity-dependent color T_eff calibrations. The surface gravity log g was obtained from fundamental relations (Roederer & Barklem 2018; see formula (1)), which were based on the Hipparcos and Tycho-2 Gaia parallaxes and the masses estimated by Casagrande et al. (2010). It needs to be pointed out that the log g values calculated from the newly released Gaia EDR3 data are very similar to what is used here, and the differences are less than 0.02 dex. The metallicity [Fe/H] and microturbulence velocity ξ were determined via an iterative process with Fe i and Fe ii lines. For the reader's convenience, the stellar parameters are repeated in Table 1.

The copper lines used in this study are taken from Korotin et al. (2018), and the wavelength (λ), low excitation potential (E_low), and oscillator strengths (log gf) are listed in Table 2. It needs to be pointed out that we do not use all the lines, and the Cu i 2214 Å line is discarded in our analysis because it is blended with other lines for our program stars. For the optical copper lines, the ratio between the two copper isotopes (⁶³Cu and ⁶⁵Cu) is adopted as 0.69:0.31 (Asplund et al. 2009), and the hyperfine structures have been taken into account. Table 2 presents the mean wavelengths and average oscillator strengths. A detailed list of the wavelengths and oscillator strengths can be found in Shi et al. (2014, 2018). While for the UV lines, the hyperfine structures have not been considered, and the oscillator strengths are adopted from Korotin et al. (2018). As described in their work, the log gf values of the Cu ii lines are taken from Dong & Fritzsche (2005), and those for most neutral UV copper lines are taken from Kurucz (2011). In addition, for the Cu i 2024 Å and 2165 Å lines, the data are respectively from Lindgård et al. (1980) and Morton (1991).

For each star, we compute the 1D LTE MARCS atmospheric models, which are obtained by interpolation in the grid of plane-parallel MARCS models for 3.5 dex ≤ log g ≤ 5.5 dex (see Gustafsson et al. 2008, for details). For each individual model, the α enhancement is under consideration, as described below:

$$\begin{eqnarray*}[\alpha /\mathrm{Fe}]=\left\{\begin{array}{ccc}0.0 & \mathrm{if} & 0.0\lt [\mathrm{Fe}/{\rm{H}}]\\ 0.4\times | [\mathrm{Fe}/{\rm{H}}]| & \mathrm{if} & -1.0\leqslant [\mathrm{Fe}/{\rm{H}}]\leqslant 0.0\\ 0.4 & \mathrm{if} & [\mathrm{Fe}/{\rm{H}}]\lt -1.0.\end{array}\right.\end{eqnarray*}$$

For our NLTE calculations, we adopt the copper atomic model from Shi et al. (2014) as a basis. It contains 97 energy levels (96 Cu i levels and the Cu ii ground state) and 1089 line transitions of Cu i. The fine structures of the levels with low excitation energy are included, and the transition oscillator strengths and the photoionization cross sections from Liu et al. (2014) are introduced into our atomic model. To calculate the excitation of allowed and forbidden transitions by electron collisions, we use the formulae of van Regemorter (1962) and Allen (1973), respectively. And the formula of Seaton (1962) is used to calculate the collisional ionization rates.

A novelty of this study is that we take into account the mutual neutralization, ion-pair formation, excitation, and de-excitation inelastic processes in copper-hydrogen collisions according to the data from Belyaev et al. (2021) (hereafter the new model). Previously, for the calculation of the inelastic collisions with hydrogen, we used Drawin's formula (Drawin 1968, 1969), which was described by Steenbock & Holweger (1984) with a scaling factor of S_H = 0.1 following the suggestion of Shi et al. (2014) (hereafter the old model).

In order to solve the statistical equilibrium and the coupled radiative transfer equations, we employ the revised DETAIL code (Butler & Giddings 1985), which is based on an accelerated lambda iteration method (Rybicki & Hummer 1991, 1992), to calculate the NLTE occupation numbers for the copper atomic model. Then, the obtained departure coefficients are used to compute the synthetic line profiles via the Spectrum Investigation Utility program (Reetz 1991).

The LTE and NLTE copper abundances in our six program stars are derived through the spectral synthesis method based on the Cu i and Cu ii lines. Abundance determination in this way is an iterative process and we change the copper abundances until the differences between the synthetic spectra and the observed ones are minimized.

Figure 1 illustrates the observed and synthetic line profiles of eight copper lines: the upper panel shows the profiles of four Cu i lines, and the lower panel presents the four Cu ii line profiles of HD 94028. In this figure, the LTE and NLTE profiles of the copper lines are indicated by the red dotted lines and the black solid lines, respectively. It needs to be pointed out that the LTE and NLTE profiles are synthesized with the same abundance for the Cu i line in the upper panel of Figure 1, and the LTE assumptions underestimate the Cu abundances.

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In Table 3, we present the [Cu/Fe] values derived from the line profile fitting of individual Cu i lines and the mean results along with the standard deviation. The LTE abundances of each star are shown in the first row, while the NLTE results based on the new and old copper atomic model are indicated in the second and third row, respectively. Considering that the copper abundances are mostly derived based on the UV copper lines, the uncertainty in abundance determination mainly comes from the indeterminacy of continuum determination, which can vary from 0.05 to 0.1 dex for a 1% shift from the true continuum flux level. Through inspection of Table 3, it can be found that the NLTE corrections, Δ_NLTE = [Cu i/Fe]_NLTE − [Cu i/Fe]_LTE, are evident for both copper atomic models, and the effects differ from line to line even for the same star. The NLTE corrections vary from 0.05 to 0.39 dex and 0.08 to 0.42 dex, for the new and old copper model, respectively. In Figure 2, the upper panel presents the NLTE corrections for resonance (filled pentagrams) and UV Cu i lines (open pentagrams) based on the new copper atomic model. It can be seen that the NLTE corrections decrease with increasing metallicity, which has been reported in previous studies (Yan et al. 2015; Andrievsky et al. 2018; Korotin et al. 2018; Shi et al. 2018; Xu et al. 2019, 2020). The NLTE corrections have no relation to the adopted Cu i lines. The lower panel shows the NLTE corrections for our two copper atomic models (filled circles denote the new model, while open circles represent the old one), and it indicates that the same feature (i.e., the NLTE corrections decrease with increasing metallicity) exists. Theoretically, if the [Fe/H] increases, UV line blocking will increase rapidly, and the overionization will become less important. Therefore, the NLTE correction, as expected, will decrease with increasing [Fe/H]. Meantime, it is clear that the NLTE corrections for the new model are lower than the old one, indicating that the application of accurate atomic data leads to a decrease in the departure from LTE.

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Figure 3 displays the mean differences between the NLTE Cu abundances derived from the Cu i lines for the new and old copper atomic models as functions of effective temperature, surface gravity, and metallicity.

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We find the difference in NLTE abundance depends on the metallicity with a range from −0.09 to −0.03 dex. The difference between the NLTE abundances derived with the new (accurate data) and old (approximate data) models diminishes with increasing metallicity, because, in more metal-rich stars, the collision with electrons become more important. Moreover, it can be seen that the differences are negative, which indicates that the application of accurate data for our program stars leads to lower NLTE Cu abundances.

In Table 4, we give the LTE copper abundances determined with individual Cu ii lines. The penultimate column shows the difference between the copper abundances derived from Cu i (NLTE) and Cu ii (LTE) lines for the new copper atomic model. The difference varies from −0.02 to 0.04 dex, which can be explained by the standard deviation in the mean results. Theoretically, as the copper atoms are mainly remaining in the ionization state in the metal-poor stars, the Cu ii lines are supposedly free of the NLTE influence. The nearly same abundance both from the lines of two ionization stages suggests that our new atomic model is valid.

To more visually verify our adopted copper atomic model, we present the abundance differences as a function of [Fe/H] in Figure 4. The upper panel shows the differences for the resonance and UV lines based on the new model. Obviously, not only for the resonance lines but also for the UV lines, the same conclusion can be drawn as described in the above paragraph. The lower panel presents the differences for our new and old models. It can be seen that the difference is negligible for both models when the errors of abundances are considered. Besides, it is noted that although the inelastic collisions with hydrogen are obtained with approximate formulas for the old model, our old atomic model is also valid to analyze the NLTE copper abundances to some extent.

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Several groups have derived copper abundances for metal-poor stars under both LTE and NLTE assumptions. We compare our results with those from other works and discuss the reasons for the differences.

Yan et al. (2015, 2016) derived the copper abundances of a large number of metal-poor stars with the Cu i lines at 5105.5, 5218.2, and 5782.1 Å. Their results are in good agreement with ours, and the average difference between our NLTE abundances and theirs is −0.02 ± 0.03 for the two common stars.

Andrievsky et al. (2018) developed a model atom for Cu and investigated the NLTE effects of Cu i lines in very metal-poor stars. We have three objects in common with theirs, i.e., HD 84937, HD94028, and HD 140283. For HD 84937, their NLTE result is 0.15 dex higher than ours, while it is 0.13 for HD 94028. For HD 140283, both results agree well (0.03 dex), since similar stellar parameters have been adopted in these two works, which cannot explain the differences. Besides, we note that both studies used different log gf values for the five common optical lines, and their values are generally higher than ours of 0.11 ± 0.07 dex. Theoretically, it will lead to lower [Cu/Fe] ratios compared to ours. However, they derived higher NLTE copper abundances. The reason is that the different copper atomic model adopted in NLTE calculations for both works, e.g., for the inelastic collisions with hydrogen, Andrievsky et al. (2018) used the Drawin formula, while our work uses the data from Belyaev et al. (2021).

Roederer & Barklem (2018) made a new test of copper abundances in late-type stars using ultraviolet Cu ii lines and concluded that LTE underestimated the Cu abundances obtained from Cu i lines. As their work has no NLTE results, we compare their Cu ii LTE abundances with ours. For HD 19445, HD 76932, and HD 94028, the results of both works agree well. While for HD 160617, our LTE result is 0.16 dex higher than theirs, which can be explained by the difference of continuum determination. However, for HD 84937 and HD 140283, our results are 0.30 and 0.49 dex higher, respectively. We note that the numbers of copper lines used by both works are different. It is noted that, except for the eight common copper lines from Roederer & Barklem (2018), we include another 11 UV lines. For the eight common lines, Roederer & Barklem (2018) adopted their log gf values from the NIST Atomic Spectra Database, which are higher than ours. While our log gf values are mainly adopted from previous literature (see Section 3.2), and have been tested by Korotin et al. (2018), the average difference of the log gf values is 0.08 ± 0.06 dex for the eight common lines. Thus, the difference between the log gf values can lead to a discrepancy of about 0.1 dex in copper abundance. Moreover, the statistical uncertainties include contributions from the uncertainty of line fitting and log gf values have been listed in Table 7 of Roederer & Barklem (2018), and the values are 0.19 and 0.29 dex. Considering the abovementioned factors, the large differences can be explained.

Based on the high-resolution, high signal-to-noise ratio UVES spectra, Shi et al. (2018) determined the copper abundances of 29 metal-poor stars. There are four common stars between both studies. For HD 76932 and HD 84937, both results agree well, while for HD 140283 and HD 160617, their abundances are 0.41 and 0.33 dex lower than ours. For these two stars, we note that the adopted stellar parameters are different in both works, which will lead to the large differences. For HD 140283, both studies used similar T_eff and log g values, while their [Fe/H] and ξ values are 0.18 and 0.2 dex higher than ours. For HD 160617, although both works used the same ξ (1.5 km s⁻¹), their T_eff and log g are 110 K and 0.11 dex lower than our values, and their [Fe/H] ratio is 0.11 dex higher than ours. When we used the same parameters as Shi et al. (2018) to derive the copper abundances for the two stars, the differences will decrease to 0.13 and 0.10 dex, which can be explained by the impact of continuum determination.

Using the same stars and stellar parameters with Roederer & Barklem (2018), Korotin et al. (2018) checked the consistency between the copper abundance for ultraviolet Cu ii lines and Cu i lines, and showed that their NLTE results of the Cu i lines removed disagreement between these two sets of lines presented in the former work. For the six common stars, the average difference between their [Cu I/Fe] values and ours is 0.14 ± 0.07 dex, while it is 0.05 ± 0.11 dex between their [Cu II/Fe] values and our results. As we note they adopted the same copper atomic model with Andrievsky et al. (2018), and their [Cu I/Fe] ratios in NLTE assumptions are systematically higher than ours, which is due to the difference between adopted copper atomic models.

The [Cu/Fe] ratios play an important role in revealing the nature of Cu nucleosynthesis and constraining the GCE model. In Figure 5 we illustrate the Cu abundances derived in this study. The left panel of Figure 5 compares the NLTE [Cu/Fe] ratios determined from the Cu i lines with a representative selection of previous works from the literature. The references are listed in the caption of Figure 5. It can be seen that our NLTE Cu abundances are consistent with the results of Shi et al. (2018), excluding the group of stars with low-α ratios as discussed by Shi et al. (2018). Meanwhile, our [Cu/Fe] ratios are systematically lower than those from Andrievsky et al. (2018) and Korotin et al. (2018), and we suggest it could be ascribed to the difference of adopted copper atomic models as discussed in Section 4.4. Moreover, through the whole metallicity range of our program stars, the features of the [Cu/Fe] trend have no large difference between the results when using the old and new models. There is only a downward shift in the [Cu/Fe] ratios with respect to that of the old model; namely the application of accurate collisional data for our sample stars leads to a lower NLTE copper abundance.

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The right panel of Figure 5 presents the [Cu/Fe] ratios derived from Cu ii lines, and we can see that our LTE Cu abundances are overall in good agreement with the results in the literature. Although our sample stars are small, a feature of the [Cu/Fe] trend can still be found: it increases with increasing metallicity when ∼−2.0 < [Fe/H] < ∼−1.0 dex, which indicates that copper behaves as a secondary (metallicity-dependent) element. The rise of [Cu/Fe] ratios with increasing metallicity is most likely due to the metallicity-dependent Cu yields from the weak s process in massive stars (SN ii progenitors; Bisterzo et al. 2004). While in the metallicity range of [Fe/H] < ∼−2.0 dex, the feature is not clear due to the small sample of stars.

Many groups (e.g., Sneden et al. 1991; Timmes et al. 1995; Goswami & Prantzos 2000; Mishenina et al. 2002; Kobayashi et al. 2006; Romano & Matteucci 2007; Romano et al. 2010; Kobayashi et al. 2011, 2020) have attempted to model the Galactic evolution of copper. In Figure 5, we overplot the behavior of the [Cu/Fe] trend predicted by the GCE model from Kobayashi et al. (2020). We note that our [Cu/Fe] ratios are consistent with the model predictions in the metallicity range ∼−2.0 < [Fe/H] < ∼−1.0 dex. However, when [Fe/H] < ∼−2.0 dex, the model predictions cannot reproduce the [Cu/Fe] ratios. It needs to be pointed out that our sample of stars with metallicity less than ∼−2.0 dex, is small, and more extremely metal-poor stars are needed in future work to explain the discrepancy.