Influence of the convective mixing-length parameter α on the chemical abundances in the case of metal-poor giant HD 122563

In order to achieve sufficient precision and accuracy to determine the influence of α on line formation through the spectral synthesis method, observed spectra with high quality become critical. High Accuracy Radial velocity Planet Searcher (HARPS) (Mayor et al. 2003) at the European Southern Observatory (ESO) La Silla 3.6 m telescope observed stars with high resolution and high signal-to-noise ratio in order to search for exoplanets. Most of the nearby, bright stars were observed by HARPS. The resolution of spectra observed by HARPS is R ∼ 115 000. ESO has continued to release all HARPS science data in HAM/EGGS modes since the beginning of instrument operation in October 2003. All the released data are easily searched, checked and downloaded through the ESO Archive Science Portal¹

We selected the spectrum with highest signal-to-noise ratio (S/N = 318.5) of HD 122563 observed by HARPS. The spectrum was obtained on 2008 February 24 with exposure of 1500 s. This spectrum had been reduced automatically using the Data Reduction Software (DRS) pipeline developed by the HARPS Consortium. The spectrum ranges from 3800 Å to 6900 Å, which covers several strong metallic lines for our analysis. There is one spectral order (N = 115, from 5300 Å to 5330 Å) that was lost due to a gap between two CCD chips, which should, however, not affect our analysis,

In order to produce the stellar model grids of HD 122563, the basic stellar atmospheric parameters including effective temperature and surface gravity were derived through the following method. Effective temperature T_eff was obtained by fitting the wings of Balmer lines, Hα and Hβ. This analysis is based on the non-local thermodynamic equilibrium (NLTE) line formation for Hi implementing the method described in Mashonkina et al. (2008). Theoretical NLTE Hα and Hβ line profiles were calculated with the code Spectrum Investigation Utility (SIU) (Reetz 1999) with the departure coefficients calculated with a revised version of the code DETAIL (Butler & Giddings 1985). The updates are described in Mashonkina et al. (2008). The absorption profiles of Balmer lines were convolved with thermal, natural and Stark broadening, as well as self-broadening. For self-broadening, the Lorentz profile with a half-width computed considering the cross-section and velocity parameter from Barklem et al. (2000) was applied. In Figure 1 we display the fitting profiles of Hα and Hβ with the final T_eff. We derived the final effective temperature T_eff = 4600 ± 50 K which is consistent with previous study (Mashonkina et al. 2011).

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The surface gravity log g was estimated based on the latest and more precise parallax of 3.444 ± 0.063 mas, which was adopted from the Gaia Data Release 2 (DR2) catalog (Gaia Collaboration et al. 2018). The bolometric corrections for the absolute bolometric magnitude were taken from Alonso et al. (1999). We derived a final surface gravity log g of 1.40 ± 0.05 which is in good agreement with literature values such as Creevey et al. (2019)

As usual, during the modeling of stellar structure and evolution, the solar value of α was adopted, despite the diverse properties of stellar atmospheres. We need to calibrate the new α to replace the unified solar α for our sample in order to meet our analysis requirement.

3D theoretical model atmospheres in Magic et al. (2015) covered a wide range in stellar parameter space. Their paper performed functional fits of the mixing-length parameter f(x,y) with T_eff and log g for different metallicities individually. We have employed the model to predict the standard MLT α by Böhm-Vitense (1958, formulation in a 1D model. The stellar parameters were transformed with x = (T_eff – 5777)/1000 and y = log g – 4.44. The fitting function is

$$\begin{eqnarray}\begin{array}{ll}f(x,y)= & {a}_{0}+({a}_{1}+({a}_{3}+{a}_{5}x+{a}_{6}y)x+{a}_{4}y)x\\ & +{a}_{2}y.\end{array}\end{eqnarray} \tag{ 1 }$$

The coefficients a_i are listed in Table B.1 in Magic et al. (2015) for a grid of different metallicities. The a_i were calculated by interpolation for each star. The new α value of HD 122563 fitted from Equation (1) was α_BV = 2.0. Wu et al. (2015) used Balmer-line fitting and spectral energy distribution (SED) fits to determine the recalibrated α for several stars and derived α_CM = 1.0 for HD 122563, which was suitable for the Canuto & Mazzitelli (1991, 1992, CM hereafter) formulation. We finally adopted 2.0, which is based on the more common BV theory, as our input α value for the benchmark model. In the following analysis, the α of CM theory was also applied to calculate a grid of models. We compared the difference in abundances due to the two different α values.

The analysis was performed using plane-parallel, homogeneous (1D) local thermodynamic equilibrium (LTE) model atmospheres calculated with the code MAFAGS-OS (Grupp 2004a,b). MAFAGS-OS provides flexible selection of input parameters, which enabled us to perform the calculations with different atmospheric parameter grids. More importantly, this code allows for adjusting mixing-length parameter α with both BV and CM theory to treat convection.

Based on the T_eff and log g derived in Section 2.2 and new α value in Section 2.3, we calculated a grid of models to cover the steps of T_eff, log g and α individually for HD 122563. The details of the grid models are summarized in Table 1. We separate these models into six groups. Group 0 stands for the basic model with the stellar parameters that we adopt for HD 122563 (T_eff = 4600 K, log g = 1.40, [Fe/H] = –2.6, α = 2.0). This model was regarded as a standard benchmark. All the other models in the following groups were compared with this one to derive the abundance discrepancy. In the other five Groups, we calculated a grid with reasonable step sizes of one parameter and kept the other two parameters consistent with Group 0. The highest α was determined as 2.5. Higher values do not satisfy the MLT theory and the real physical picture. Group 4 and Group 5 were calculated with α_CM in order to investigate the difference between the two MLT α theories.

Table 2. Average Abundance Discrepancies in Different Stellar Model Grids

Species	|Δ[X/H]αBV|				$\overline{\Delta {[{\rm{X}}/{\rm{H}}]}_{\mathrm{log}g}}$			$\overline{\Delta {[{\rm{X}}/{\rm{H}}]}_{{T}_{{\rm{eff}}}}}$		Na
	–1.5	–1.0	–0.5	+0.5	0.2	0.1	0.05	100 K	50 K
Nai	0.020	0.020	0.030	0.010	0.090	0.045	0.020	0.185	0.085	2
Mgi	0.016	0.016	0.016	0.006	0.106	0.054	0.026	0.126	0.062	5
Ali	—	—	0.040	0.010	0.100	0.050	0.020	0.140	0.070	1
Sii	—	—	0.040	0.010	0.090	0.050	0.025	0.095	0.035	2
Cai	0.022	0.018	0.022	0.007	0.077	0.037	0.018	0.108	0.055	6
Scii	—	—	0.005	0.035	0.025	0.010	0.005	0.040	0.020	2
Tii	0.024	0.014	0.020	0.011	0.070	0.033	0.015	0.145	0.075	6
Tiii	0.019	0.022	0.013	0.017	0.020	0.011	0.003	0.026	0.011	15
Cri	0.000	0.010	0.006	0.018	0.124	0.060	0.030	0.180	0.098	5
Mni	—	—	0.025	0.013	0.138	0.070	0.035	0.157	0.078	4
Coi	—	—	0.020	0.005	0.090	0.045	0.025	0.160	0.080	2
Nii	0.030	0.030	0.027	0.020	0.093	0.047	0.023	0.167	0.083	3
Srii	—	—	0.020	0.010	0.060	0.030	0.020	0.070	0.030	1
Yii	—	—	0.040	0.010	0.010	0.000	0.000	0.070	0.030	1
Zrii	—	—	0.030	0.000	0.020	0.010	0.000	0.020	0.010	1
Baii	0.017	0.023	0.023	0.003	0.040	0.023	0.007	0.083	0.043	3
Total	0.019	0.019	0.024	0.012	0.072	0.036	0.017	0.111	0.054	59

^aTotal number of lines used for the corresponding species.

The lines adopted in our research were selected from the lists of several abundance analysis papers (Mashonkina et al. 2010; Zhao et al. 2016; Aoki et al. 2018). These papers provided a suitable metallic line list in the optical wavelength range for metal-poor stars, which could be applied to HD 122563. Most reliable and up-to-date atomic input data were adopted from the NIST ASD² as well as the above-mentioned references. The species identification, wavelength, excitation potential and gf values are included in Table 3. It should be noted that inaccurate atomic data could lead to unreliable derived abundances. However, in our study, we focus more on the abundance discrepancy due to the α influence instead of absolute abundance determination. Our analysis could counteract the uncertainty introduced by atomic data to a certain extent and thus the final result does not depend on the input atomic data.

Accurate line formation computations and abundance determination were performed with SIU. First we reprocessed the HARPS spectrum of HD 122563. The wavelength was recalibrated and flux normalization was performed manually in a 5 Å window for each individual line. We also recalibrated the wavelength of the spectrum to match the laboratory wavelength. We utilized the spline function for continuum rectification.

We applied SIU to calculate line profiles with selected stellar atmospheric models and produced synthetic spectra which overlap with the observed spectrum. To determine the abundance, we degrade the theoretical profiles with observational uncertainties by convolving them with a profile that combines instrumental broadening with a Gaussian profile, rotational broadening and broadening by macroturbulence with a radial-tangential profile. Finally, the best line profile and abundance were determined through adjusting the element abundance by minimizing the χ² between the observed and synthetic spectra. Figure 2 presents an example of the best synthetic spectrum fit to the Cai λ6162 Å line calculated by SIU with benchmark model Group 0. We also evaluated the equivalent width for each line with SIU. During the spectral synthesis calculation, we eliminated bad lines, for example, heavily blended lines that are nearly unseparable and lines too weak to measure with high accuracy. The equivalent widths and abundances derived with the basic model (Group 0) for each line are presented in Table 3.

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Table 3. Atomic line data, equivalent widths (EWs) and derived abundances for each metallic line.

Species	λ Å	Eexc eV	log gf	EW mÅ	log
Nai	5889.96	0.00	0.11	178.7	4.29
Nai	5895.93	0.00	–0.19	159.6	4.29
Mgi	4571.09	0.00	–5.40	79.7	5.51
Mgi	4702.99	4.35	–0.46	72.7	5.44
Mgi	5172.68	2.71	–0.39	226.2	5.64
Mgi	5183.60	2.71	–0.17	252.8	5.64
Mgi	5528.41	4.35	–0.51	76.5	5.44
Ali	3961.52	0.01	–0.33	156.8	3.96
Sii	3905.53	1.91	–1.10	200.4	5.73
Sii	4102.94	1.91	–2.99	80.9	5.63
Cai	4226.73	0.00	0.24	258.2	4.09
Cai	4454.78	1.94	0.26	75.6	4.28
Cai	5588.75	2.53	0.36	47.8	3.88
Cai	6122.22	1.89	–0.32	67.7	4.19
Cai	6162.17	1.90	–0.09	82.8	4.25
Cai	6439.08	2.53	0.39	63.6	4.08
Scii	4246.82	0.32	0.24	125.3	1.45
Scii	4415.56	0.60	–0.67	80.1	1.20
Tii	3998.64	0.05	–0.06	72.7	2.76
Tii	4533.25	0.85	0.48	50.8	2.47
Tii	4534.78	0.84	0.28	42.8	2.51
Tii	4981.73	0.85	0.51	59.3	2.63
Tii	4991.06	0.84	0.45	55.8	2.63
Tii	5210.39	0.05	–0.88	44.3	2.43
Tiii	4028.34	1.89	–0.96	49.8	2.83
Tiii	4394.05	1.22	–1.78	62.1	2.53
Tiii	4395.03	1.24	–1.93	50.4	2.73
Tiii	4399.77	1.24	–1.19	90.8	2.73
Tiii	4417.72	1.16	–1.19	93.9	3.32
Tiii	4418.33	1.24	–1.97	50.8	3.35
Tiii	4443.79	1.08	–0.72	120.2	2.83
Tiii	4444.56	1.12	–2.24	48.2	2.80
Tiii	4450.48	1.08	–1.52	86.7	3.43
Tiii	4464.45	1.16	–1.81	69.5	2.78
Tiii	4468.51	1.13	–0.60	127.4	2.73
Tiii	4470.86	1.16	–2.02	50.2	2.73
Tiii	4501.27	1.12	–0.77	107.3	3.23
Tiii	4533.97	1.24	–0.53	100.6	2.66
Tiii	4571.97	1.57	–0.32	113.3	2.71
Cri	4254.33	0.00	–0.11	112.6	3.40
Cri	4274.80	0.00	–0.23	108.2	3.45
Cri	4289.72	0.00	–0.37	104.2	3.50
Cri	5206.04	0.94	0.02	88.3	3.15
Cri	5208.42	0.94	0.17	92.6	3.15
Mni	4030.75	0.00	–0.47	132.8	3.70
Mni	4033.06	0.00	–0.62	125.6	3.68
Mni	4034.48	0.00	–0.81	112.9	3.62
Mni	4041.35	2.11	0.28	42.7	3.65
Coi	4118.77	1.05	–0.47	78.2	3.00
Coi	4121.31	0.92	–0.30	89.3	3.20
Nii	3807.14	0.42	–1.22	111.1	4.22
Nii	3858.29	0.42	–0.95	116.6	4.38
Nii	5476.90	1.83	–0.89	79.9	3.62
Srii	4077.72	0.00	0.15	170.5	0.70
Yii	3950.35	0.10	0.49	48.1	–0.08
Zrii	3998.97	0.56	–0.52	40.1	0.30
Baii	4554.03	0.00	0.17	93.8	–0.20
Baii	4934.07	0.00	–0.15	88.3	–0.20
Baii	6141.71	0.70	–0.08	42.3	–1.00

After deriving the synthetic spectrum of each line with the benchmark model, we employed other models in Group 1 to derive the abundances with different α values. Finally, the abundance discrepancy was derived. The abundance discrepancy due to α_BV variation was defined as Δ [X/H]_αBV = [X/H]_αGroup1 – [X/H]_{αBV = 2.0}. Here, the α value in Group 1 was 0.5, 1.0, 1.5 and 2.0 as we introduce in Section 2.4. We used similar definitions for Δ [X/H]_Teff and Δ [X/H]_{log g} to represent the impact of T_eff and log g on abundance determination (Groups 2 and 3). We also calculated Δ [X/H]_αCM which implemented another MLT of CM formulation for comparison (Groups 4 and 5). The final Δ [X/H] was derived with line-by-line differential analysis. We analyzed the metallicity discrepancy for several iron lines both under LTE and NLTE assumptions in our previous work (Song et al. 2020). Even though some line profiles could be fitted better under the NLTE assumption, the influence of convection on NLTE effects could be ignored with the line-by-line differential analysis, so we did not take NLTE effects into account in this work.

Based on the line list in Section 2.5, we analyzed 59 metallic lines for the following species to evaluate the influence of α on abundance determination:

• –  
  neutral atoms: Nai, Mgi, Ali, Sii, Cai, Tii, Cri, Mni, Coi, Nii;
• –  
  ionized atoms: Scii, Tiii, Srii, Yii, Zrii, Baii.

All these lines were thoroughly studied for the metal-poor giant HD 122563. In order to derive the best fitting profile for each line, the additional tunable broadening parameters were convolved by a Gaussian function for each input model. The abundance discrepancy of each line is listed in Table 4 in the columns marked by Group 1-0. The largest Δ [X/H]_αBV values are 0.05 dex (Cai λ4226 Å with α = 0.5 and Scii λ4246 Å with α = 2.5). Most of the Δ [X/H]_αBV values are lower than 0.04 dex, which suggest that the variation in α only results in a negligible degree of difference in abundances derived from metallic lines.

We also investigated the impact of α on the shape of the line profile (Fig. 3). The wings and core of weak lines are both formed in deep layers, where convection affects the wings and core in the same way. For weaker lines with equivalent widths less than 150 mÅ, α influences on the whole line profile are equal and there are no changes in the line shape of the final best fitted synthetic spectra after abundance rectification. For stronger lines, the core forms in layers shallower than those that form the wings, so convection influences the wings and core of the strong lines differently. The effects of α mostly influence the wings more than core of the line profile. However, it hardly affects the final abundance determination.

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When deriving abundances from absorption lines, it is assumed that the line strength is directly related to the abundance of the element. We plot the Δ [X/H]_αBV versus equivalent width in order to investigate the influence of α on the strength of the line (Fig. 4). For the metal-poor star HD 122563, most of the lines are weak. Only limited lines (e.g., Mg Ib, Na D) are strong lines. There is no obvious relationship between line strength and abundance discrepancy. Both neutral lines and ionized lines experience little change in HD 122563.

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We also plot Δ [X/H]_αBV versus excitation potential (E_exc) for different α models (Fig. 5). Most of our metallic lines are concentrated in low excitation potential regions. The E_exc values of most lines are lower than 3 eV. Only two magnesium lines exhibit higher E_exc with 4.35 eV. There is no obvious trend between Δ [X/H]_αBV and excitation potential. Kučinskas et al. (2013) demonstrated that abundance corrections affected by convection grew larger with increasing excitation potential for red giant stars. However, the abundance correction for most chemical species with excitation potential lower than 4.5 eV was negligible. In figure 3 of their paper, the abundance correction was lower than 0.05 dex as well. It is also hard to distinguish the abundance discrepancy of different species in Figure 5. Our results from the 59 lines concur with the theoretical study from Kučinskas et al. (2013). The low-excitation lines usually form in the upper atmospheric layers, which are hardly affected by the internal convection zone.

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We calculated the average abundance discrepancy for each chemical species

$$\begin{eqnarray}&&|\Delta {[\rm{X}/ \rm{H}]}_{{\alpha }_{{\rm{BV}}}}|=\displaystyle \frac{\displaystyle \sum _{}|\Delta {[X/H]}_{{\alpha }_{{\rm{BV}}}}|}{n},\end{eqnarray} \tag{ 2 }$$

where n is the number of lines. Results for all species are listed in Table 2. The average value of the discrepancy rarely exceeded 0.04 dex. It is hard to distinguish the difference between neutral atoms and ionized atoms. The average abundance discrepancies of all the species we investigated in HD 122563 were 0.019, 0.019, 0.024 and 0.012 dex, which corresponded to α_BV = 0.5, 1.0, 1.5 and 2.5, respectively. It should be noted that α = 0.5 and 1.0 might be far from the true convection property of HD 122563; α = α_{BV = 2.0} ± 0.5 is more reasonable in the real physical picture. In our previous study (Song et al. 2020), the influence of α on the iron abundance correction was at a level of 0.03 dex for HD 122563. We derived consistent abundance discrepancies for other chemical species as well.

The abundance uncertainties caused by error in T_eff and log g are also evaluated to compare with the influence of α. The results with Δ T_eff = ± 50, 100 K and Δ log g = ± 0.05, 0.1, 0.2 are listed in Table 4. The average abundance uncertainties are also presented (Table 2).

The abundances obtained from strong lines are more affected by uncertainties in stellar parameters than from weak lines (Jofré et al. 2019). We find that the temperature sensitivity of spectral lines originating from neutral atoms is common among strong lines. The absorption in the wings of strong lines is sensitive to pressure in the atmosphere. The abundance corrections due to varying T_eff and log g grew larger with increasing line strength. By changing log g by 0.2 dex or T_eff by 100 K, all abundances obtained from strong lines changed by 0.1 dex or more, which clearly illustrates that Δ [X/H] increased as effective temperature and surface gravity increased for all species we analyzed (Fig. 6). The neutral atoms deviated by a larger extent than the ionized atoms. For all species, the average discrepancies caused by variations in T_eff and log g were higher than the discrepancies due to α variation.

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As we introduced in Section 2.3, we calculated two types of mixing-length parameter α models, one from the most commonly used BV theory and the other from the improved CM theory. The values for Δ[X/H]α_BV and Δ[X/H]α_CM are listed in Table 5 for comparison. Abundance discrepancies between two α models for each line were also calculated, as defined by,

$$\begin{eqnarray}&&\Delta {[{\rm{X}}/{\rm{H}}]}_{\alpha =2}={[{\rm{X}}/{\rm{H}}]}_{{\alpha }_{{\rm{CM}}=2}}-{[{\rm{X}}/{\rm{H}}]}_{{\alpha }_{{\rm{BV}}=2}}.\end{eqnarray} \tag{ 3 }$$

The largest Δ [X/H]_{α = 2} is the specific line Sii 4102 Å which is up to 0.07 dex; some discrepancies are at a level of 0.05 dex. This discrepancy reflects the difference between the two α models with α = 2 and is expected because the α values in the two theories stand for the different efficiencies of convective energy transport. As Wu et al. (2015) suggested, α = 1 in CM theory best represents HD 122563, and we additionally calculated Δ [X/H] between α_{CM = 1} and α_{BV = 2}. The recalculated Δ [X/H] did not exceed 0.04 dex. The real difference between two recalibrated α is modest.

The largest abundance discrepancy by α_CM variation is 0.05 dex. Most Δ[X/H]α_CM are smaller than 0.04 dex, which is consistent with Δ[X/H]α_BV. Despite the slight difference between the BV and CM theory parameterization of the mixing-length parameter, the influence of the two α values on abundance determination is similar for HD 122563.