ATOMIC DIFFUSION AND MIXING IN OLD STARS. III. ANALYSIS OF NGC 6397 STARS UNDER NEW CONSTRAINTS

We make an effort to employ the most realistic line-formation theory available. NLTE corrections are applied for Mg i (Gehren et al. 2004), Ca i (Mashonkina et al. 2007), and Cr i (Bergemann & Cescutti 2010). Additionally, we apply NLTE corrections to LTE-derived abundances for Na i (Lind et al. 2011a) and Li i (Lind et al. 2009a). The majority species Fe ii and Ti ii are modeled in LTE, as they have been shown to exhibit only minor NLTE effects (Mashonkina et al. 2011; Bergemann 2011).

We derive iron abundances from Fe ii lines only, as the modeled strengths of Fe i lines depend critically on NLTE and its modeling assumptions (specifically the inelastic H i collision factor S_H). As a diagnostic of this effect, we have also derived log g values from the iron ionization equilibrium on our new temperature scales. For this test, we employ NLTE corrections in the line-formation procedure following Korn et al. (2003), with rather efficient hydrogen collisions (S_H = 3). These spectroscopically derived log g values overall agree reasonably with those derived directly from photometry and distance. A significant difference is found on our hottest temperature scale, "200e," where log g values for the TOP stars are higher by 0.2 dex, placing them at log g = 4.2, which is just below the cluster TOP in our theoretical color–magnitude diagram (CMD; compare to Figure 1).

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3D effects on abundance trends in LTE were investigated in Paper I. Qualitative results implied stronger trends for Ti ii and Fe ii, but a weaker trend for Mg i.

For chromium, we find similar NLTE corrections for the high-excitation triplet line Cr i λ5206 in all groups, ranging from 0.31 dex at the RGB to 0.37 dex at the SGB. Bonifacio et al. (2009) predict very large differential 3D effects at low metallicities. At [Fe/H] = −3.0, these corrections comparing giants to dwarfs are on the order of δΔlog ε(Cr) = 0.4–0.5 dex for high-excitation lines. Preliminary 〈3D〉 NLTE modeling implies individual corrections of similar magnitude (M. Bergemann 2011, private communication), rendering comparisons between dwarfs and giants uncertain. Thus, we expect that our results may change significantly under more sophisticated modeling.

For lithium, we need not only consider differential modeling effects, but must also determine absolute abundances to compare with the predicted primordial abundance. The 3D NLTE treatment by Sbordone et al. (2010), using an eight-level model atom, results in corrections Δ(3D − 1D) = 0.03 dex for our TOP and SGB stars. In 1D, these NLTE corrections (ΔNLTE) are very near zero.

We have performed independent analyses following Lind et al. (2009a) on metal-poor stars similar to our sample: for the metal-poor dwarfs HD 84937 and G64-12, and the subdwarf HD 140283 representing well-studied Spite plateau stars similar to our TOP and SGB stars, as well as an additional model representing a star evolved somewhat beyond our sample (at T_eff = 5000 K, log g = 2.0, [Fe/H] = −2.0). In this analysis, 1D models were constructed from MARCS atmospheres (Gustafsson et al. 2008). Average 3D (〈3D〉) models were constructed from sequences of snapshots from radiation–hydrodynamic convection simulations using the Stagger Code (Stein & Nordlund 1998), with averaging over the 3D cubes on surfaces of equal τ₅₀₀ (R. Collet 2011, private communication). Results imply 〈3D〉 NLTE lithium abundances very similar to the corresponding 1D NLTE case. On the Spite plateau, we find ΔNLTE = −0.05 dex in 1D, with identical or slightly higher abundances in 〈3D〉. For evolved stars, the correction instead becomes positive. For the evolved model star investigated here, the NLTE abundance is lower in 〈3D〉 than in 1D. The NLTE effects in this star should be greater than in our bRGB and RGB stars, as it has significantly lower surface gravity. These results imply that hydrodynamical effects will not lead to significantly different results for our evolved stars, and that 1D LTE analyses indeed underestimate their lithium abundances.

Similarly small Δ(3D − 1D) corrections result from these preliminary 3D and 〈3D〉 NLTE lithium abundance analyses. For this reason, we feel safe in applying 1D NLTE corrections, but keep in mind the larger systematic uncertainty in the abundances of evolved stars.

We compare the effective temperatures deduced from spectroscopic Hα analyses to those from three different photometric calibrations in Table 2. All photometric calibrations indicate a range ΔT_eff(TOP − RGB) similar to or somewhat more narrow than the spectroscopic analysis. Working on these temperature scales would generally strengthen the previously identified trends.

The Casagrande et al. (2010) calibrations for dwarfs and subgiants indicate higher effective temperatures for the TOP by 110 ± 30 K, but are not applicable to giants. Assuming that the empirical calibrations of Alonso et al. (1996) hold on the RGB, this would allow an increase in ΔT_eff(TOP − RGB) by 100 K. Note, however, that this is not the most likely interpretation of the hotter temperatures stemming from the Casagrande et al. (2010) calibration. Rather, the difference is largely due to the calibration of the infrared bands, resulting in an absolute offset, rather than the inclusion of interferometric radii for dwarfs, which would not affect the giants and thus give a differential effect (L. Casagrande 2011, private communication).

On the SGB, the effective temperatures are similarly higher by 115 ± 100 K, indicating no significant change in ΔT_eff(TOP − SGB). The errors are, however, fairly large, as the calibration sample is rather thin at this evolutionary stage (see Figure 13 of Casagrande et al. 2010).

The Hα analysis employs the broadening theory of Ali & Griem (1966). Direct comparison to that of Barklem et al. (2000) indicates only minor deviations, affecting ΔT_eff(TOP − RGB) by 20 K, and increasing ΔT_eff(TOP − SGB) by some 40 K.

Starting from the spectroscopic temperature scale, we introduce three hypothetical temperature scales. For each, photometric log g values are determined as described in Section 2. In the most probable scenario, we shift all effective temperatures by 100 K, to match the expected absolute offset introduced in the Casagrande et al. (2010) calibration. We refer to this temperature scale as "100s."

Additionally, we create two expanded temperature scales. The first retains ΔT_eff(TOP − SGB) in reasonable agreement with both photometry and spectroscopy, expanding the temperature scale linearly by a fixed amount for each stellar group as determined by its mean temperature. We expand ΔT_eff(TOP − RGB) by 100 K, keeping temperatures fixed at the RGB, and refer to this temperature scale as "100e." The increase in parameter ΔT_eff(TOP − SGB), 40 K, is in agreement with the improved Hα broadening, and does not violate the photometric indicators. This scenario represents a realistic but conservative case, relative to "100s." The temperature scale "200e" is created by the same method, with ΔT_eff(TOP − RGB) expanded by 200 K. This temperature scale, while poorly supported by our photometric and spectroscopic analyses, is meant to represent the case where atmospheric parameters are tuned to flatten the abundance trends.

The stellar parameters on each temperature scale are shown together with theoretical isochrones at three different assumed ages in Figure 1. The WDCS prefers an age of 12.0 ± 0.5 Gyr (Hansen et al. 2007; Kowalski 2007), which the models effectively bracket. This age constraint is well fulfilled by all three temperature scales. Additionally, the spectroscopic values are compatible within their error bars.

An additional constraint comes from the observed cluster CMD (see Paper I, Figure 2). We expect our TOP stars to lie some 100–200 K cooler than the cluster TOP (Paper I). This criterion disfavors the hot "200e" temperature scale as well as the spectroscopic determination—indicating a cluster too young or old, respectively—while "100s" and "100e" are both fully compatible.

For traditional stellar models computed without atomic diffusion, the lack of gravitational settling of helium results in a significantly older cluster. Then, only "200e" seems compatible with the WDCS age constraint, as well as the cluster CMD. From the cluster CMD constraint, "100s" and "100e" would both prefer ages greater than 13.5 Gyr.

Following Richard et al. (2002, 2005), a sparse grid of isochrones has been computed with effects of atomic diffusion. Three different ages and six different efficiencies of turbulent mixing are considered. At 11.5 and 12.5 Gyr, a metallicity of [Fe/H] = −2.1 was used. The 13.5 Gyr models were reused from Paper I, with [Fe/H] = −2.0. This has minor effects on, e.g., the stellar mass at a given evolutionary stage, but is negligible for the evolution of surface abundances. Turbulent mixing is prescribed ad hoc with an efficiency parameterized by a reference temperature T₀ determining the behavior in both strength and depth (see Section 3.2 of Richard et al. 2005 for more details). We use the shorthand Tx for models using log T₀ = x, and refer to the magnitude of this parameter as the turbulent-mixing efficiency. The T5.95 and T6.25 models, respectively, represent the lowest and highest turbulent-mixing efficiencies that seem compatible with the flatness of the Spite plateau of lithium.

As noted in Paper I, the bRGB and RGB groups show some scatter in the magnesium abundance. Magnesium is known to anticorrelate with aluminum in globular clusters (see Gratton et al. 2004), but no lines of the latter are detected in our spectra. We resort to using sodium as a proxy via its well-studied O–Na anticorrelation (e.g., Carretta et al. 2009; Gratton et al. 2004), and the related Li–Na anticorrelation which is known to be significant only for the most strongly polluted stars (Lind et al. 2009b).

From a large sample of RGB stars, Carretta et al. (2009) have determined that some 25% ± 13 } of stars in NGC 6397 belong to the first generation. The independent determination by Lind et al. (2011b) confirms this result. They identify for their sample of turnoff stars mean abundances for the first and second generations as log ε(Na) ≈ 3.65 and 4.05, respectively. For their combined sample of giants and diffusion-corrected dwarfs (+0.20 dex), the levels increase to log ε(Na) = 3.84 ± 0.10 and 4.23 ± 0.13, respectively. As our sample size is insufficient for statistical analyses of the abundance trends, we rely on comparisons with the data and results of Carretta et al. (2009) and Lind et al. (2008, 2009b, 2011b) for our population determinations. We perform the sodium abundance analysis on our original spectroscopic Hα temperature scale, as its good agreement with previous photometry makes it directly comparable with the cited studies, giving us comparable abundance scales. This assumption is further supported by the agreement of stellar parameters for stars in common with the other studies.

For our sample of six RGB stars, we find a strong anticorrelation between magnesium and sodium, identifying two out of our six stars as belonging to the first generation in this cluster with log ε(Na) ≈ 3.9. Individual measurements of the very weak lithium line give a relatively large scatter (σ ≈ 0.10 dex), seemingly anticorrelated to sodium abundances at rather low significance.

As indicated in Paper I, our sample of bRGB stars shows a bimodal distribution of magnesium abundances (Δ ≈ 0.17  ±  0.05). No anticorrelation is detected, as the very weak sodium line seems to exhibit only mild scatter (σ ≈ 0.06 dex), without correlation to the former. Applying different effective temperatures from various photometric calibrations does not alter this result. Lithium shows significant scatter in line with expectations from an evolutionary scatter within the group (the cooler stars have more strongly diluted surface layers, with lower lithium abundances). Absolute sodium abundances (log ε(Na) ≈ 4.2) indicate that all five stars belong to the second generation.

Within the SGB and TOP groups, no significant scatter is detected for magnesium. Sodium abundances identify both SGB stars as borderline second generation (log ε(Na) ≈ 4.0), while all five TOP stars clearly belong to the second generation (log ε(Na) > 4.0).

We account for pollution effects by attempting to restore the magnesium abundances of the first-generation stars. We have directly identified first-generation stars in the RGB group, while for the other groups we must extrapolate from comparisons with the aforementioned studies. We apply upward corrections by 0.08, 0.07, 0.11, 0.11 dex to the group-averaged magnesium abundances for the RGB, bRGB, SGB, and TOP group, respectively. We propagate estimated uncertainties of the latter three corrections into those of the abundance determinations.

If better abundance determinations for, e.g., sodium or nitrogen were available, one could, in principle, reject polluted stars in the abundance analyses of sensitive elements. Unfortunately, it seems that none of our TOP and SGB stars belong to the first generation, and thus may well represent a slightly decreased abundance in lithium. The magnitude of this effect may be gleaned from the observed difference among dwarfs in the large sample of Lind et al. (2009b), ⩽0.05 dex. Future studies could apply the identification method based on the nitrogen-sensitive photometric index c_y when selecting their targets (see Lind et al. 2011b and references therein).

On our preferred temperature scale, "100s," ΔT_eff is not altered, thus affecting abundance trends only by second-order effects. For magnesium, the corrections for pollution weaken the abundance trend to Δlog ε(RGB − TOP) = 0.18 ± 0.08, which is significant on the 2σ level. The trend in iron, meanwhile, is significant on the 3σ level with Δlog ε(RGB − TOP) = 0.16 ± 0.05. Calcium and titanium both show mildly significant trends, Δlog ε(RGB − SGB) = 0.12 ± 0.07. Lithium displays a significant upturn at the SGB, Δlog ε(SGB − TOP) = 0.13 ± 0.07, in line with the independent results of Lind et al. (2009b). The evolution of chromium is consistent with no variation.

Comparing to Paper I, magnesium abundances have been most significantly altered by correcting for cluster-internal pollution. This decreases Δlog ε(RGB − TOP) by 0.03 dex, while increasing the uncertainty of this measure (from 0.07 to 0.08 dex). The altered temperature scale has not affected abundance trends in neither magnesium nor lithium. Iron abundances are now based upon ionic lines, which are not temperature sensitive. The absolute abundances of iron and titanium are affected by the new, increased log g values, with insignificant differential effects. The trend in calcium has strengthened by the combined differential effects from T_eff and log g by 0.04 dex.

Expanding the temperature scale, i.e., increasing parameter ΔT_eff(TOP − RGB), implies higher abundances for the TOP stars. This leads to higher inferred abundances in temperature sensitive elements, which decreases parameter Δlog ε(RGB − TOP). This tends to flatten the observed abundance trends in Paper I, which all exhibited Δlog ε(RGB − TOP) > 0.

On the expanded temperature scale "100e," trends in magnesium and iron are both significant on the 2σ level. Titanium and chromium exhibit similar but weakly significant trends of opposite sign. Calcium abundances are consistent with no variation. The upturn in lithium abundances at the SGB remains, Δlog ε(SGB − TOP) = 0.10 ± 0.07.

On the hottest temperature scale, "200e," abundance trends by construction flatten significantly. The variation in iron, however, remains significant with Δlog ε(RGB − TOP) = 0.11 ± 0.05. Chromium too deviates strongly, due singly to the behavior on the RGB. We note that while most signatures flatten, Δlog ε(TOP − SGB) of titanium actually increases somewhat on this temperature scale. This could be counteracted by arbitrarily increasing log g values on the SGB, while inducing a complementary effect on iron. Flattening the iron trend further would similarly require increasing log g values at the TOP, with the drawback of equally strengthening the trend of titanium. Hence, this scenario results in nearly optimally flattened abundance variations.

In Figure 2, we compare observed abundance trends on the shifted temperature scale "100s" to stellar-structure predictions, as introduced in Section 2.3. Results on the expanded temperature scales are presented in Figures 3 and 4. The dashed horizontal line represents the initial composition of the models, adjusted to the observed abundance level. The precise choice of age, 11.5 versus 12.5 Gyr, does not significantly influence the strength of the predicted evolutionary trends. Differences are larger when comparing to 13.5 Gyr models, in the sense that those predict stronger abundance trends for a given turbulent-mixing efficiency. The influence of age is most significant on the cluster CMD (see Section 2.2). From the discussion there, we compare "200e" to the 11.5 Gyr isochrones, and the other temperature scales to the 12.5 Gyr isochrones, where more models are available.

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For the shifted temperature scale "100s," the models of least efficient turbulent mixing, T5.95–6.0, find excellent agreement with the observed trends in lithium, calcium, iron, chromium, and titanium. For calcium, the predicted trend is somewhat stronger than observed, and for magnesium this is most apparent for the abundance at the SGB. Lithium, titanium, and iron all exhibit trends favorable of weak efficiency. Taking into account the stronger upward adjustment required to match the TOP and SGB abundances of lithium, the disagreement with the more uncertain RGB stars lessens when the T6.0 models are chosen. Hence, as in Paper I, this remains the optimal model, which gives the best overall agreement with observations.

Models with moderately strong efficiency, T6.09 and T6.15, find mild disagreement with the predicted upturn in lithium abundances at the SGB, where instead a flat behavior is predicted. Similarly, the evolutionary trend of iron becomes too weak. At less than 2σ disagreement for all elements, these models, however, can not be excluded.

Models with very strong turbulent-mixing efficiency, T6.25, disagree with the evolutionary trend of iron on the 2σ level. Even worse, they disagree on the 3σ level with the evolutionary trend of lithium, where a strong downturn is predicted at the SGB. This is because, with turbulent mixing this strong, gravitationally settled material is brought to sufficiently hot layers that lithium is destroyed. As the surface convection zone expands inward, it mixes with depleted material, thus diluting the surface abundance further. Notably, this causes additional 0.15 dex of lithium destruction, which naturally carries into the evolution along the RGB.

Hence, models with turbulent-mixing efficiencies in the range T5.95–6.15 agree well with observations, with a preference for the weaker efficiencies. A strong turbulent-mixing efficiency, T6.25, can be excluded for our sample of stars at this temperature scale from its predictions for both iron and lithium.

On the expanded temperature scale "100e," the above results hold, with a possible preference for slightly stronger turbulent-mixing efficiency. The sole exception is chromium, where the TOP–bRGB range exhibits no variation, but Δlog ε(TOP − RGB) disagrees on the 2σ level, with similar magnitude but opposite sign to predictions. This result may, however, be significantly altered by 〈3D〉 NLTE modeling (see Section 2.1).

On the strongly expanded temperature scale "200e," trends by design flatten further, while not significantly altering the above conclusions. A weak efficiency of turbulent mixing is still compatible with all observed trends, except for magnesium which clearly prefers somewhat higher efficiency, and chromium, discussed previously.

In Figure 5, we show that the initial lithium abundance of the cluster as inferred from our optimal model (T6.0), log ε(Li)_init = 2.57 ± 0.1, is compatible with the predicted primordial abundance based on WMAP data (Cyburt et al. 2010; log ε(Li)_BBN = 2.71 ± 0.06). Adopting the mutual error bars, the values differ by 0.14 ± 0.11 dex, or 1.2σ. If instead the T5.95 model were adopted, the initial abundance would further increase by 0.04 dex to log ε(Li)_init = 2.61. Modeling in 〈3D〉 NLTE may further increase the abundance slightly, as would the assumption of some cluster-internal lithium destruction. Adopting a cluster age of 12.0 or 11.5 Gyr decreases the predicted surface depletion slightly, which reduces the inferred initial abundance by at most 0.01 dex.

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The model of high-efficiency model (T6.25) is not compatible with the lithium abundances on the TOP and SGB. The deep mixing causes additional destruction of lithium by 0.15 dex. When calibrating solely on the RGB, this would result in the inferred initial abundance log ε(Li) = 2.60. Note, however, that lithium abundances on the RGB are uncertain from the modeling perspective as regards both line-formation theory (see Section 2.1) and stellar-evolution models (as indicated by a possible offset in T_eff or insufficient depletion of lithium along the SGB, as seen in Figure 5, and likewise in Lind et al. (2009b)). These problems must be taken into account when considering the analysis of Mucciarelli et al. (2012). Indeed, these stars are less sensitive to the precise efficiency of atomic diffusion. But additional destruction of lithium occurs in models with strong turbulent mixing, an effect also affecting RGB stars, leading to underestimated initial abundances. We note in passing that their mean lithium abundance on the RGB for this cluster, log ε(Li) = 1.09 ± 0.1, agrees with ours on the temperature scale "100s," log ε(Li) = 1.12 ± 0.1.