Li detection in metal-poor stars: can 3D model atmospheres solve the second lithium problem?
The presence of Li in the atmospheres of metal-poor halo stars is usually inferred from the detection of a subtle extra depression in the red wing of the Li doublet line at 670.8 nm. However, as pointed out recently by Cayrel et al. (2007), the intrinsic line asymmetry caused by convective flows in the photospheres of cool stars is almost indistinguishable from the asymmetry produced by a weak Li blend on a (presumed) symmetric Li profile. Previous determinations of the Li/ Li isotopic ratio based on 1D model atmospheres, ignoring the convection-induced line asymmetry, must therefore be considered as upper limits. By comparing synthetic 1D LTE and 3D non-LTE line profiles of the \ionLii nm feature, we quantify the differential effect of the convective line asymmetry on the derived Li abundance as a function of effective temperature, gravity, and metallicity. As expected, we find that the asymmetry effect systematically reduces the resulting Li/Li ratios. Depending on the stellar parameters, the 3D-1D offset in Li/Li ranges between and . When this purely theoretical correction is taken into account for the Asplund et al. (2006) sample of stars, the number of significant Li detections decreases from 9 to 5 (2 criterion), or from 5 to 2 (3 criterion).
We also present preliminary results of a re-analysis of high-resolution, high S/N spectra of individual metal-poor turn-off stars, to see whether the second Lithium problem actually disappears when accounting properly for convection and non-LTE line formation in 3D stellar atmospheres. Out of stars, HD 84937 seems to be the only significant (2) detection of Li. In view of our results, the existence of a Li plateau appears questionable.
A systematic analysis of high-dispersion, high signal-to-noise (S/N) UVES and Keck spectra of about 30 bright metal-poor stars by Asplund et al. (2006) and Asplund & Melendez (2008) (henceforth A06 and A08, respectively) resulted in the detection of Li (at the level) in about one third of these objects. The average Li/Li isotopic ratio in the stars in which Li has been detected is about 4% and is very similar in each of these stars, defining a Li plateau at approximately LiH. A convincing theoretical explanation of this new Li plateau turned out to be problematic: the high abundance of Li at the lowest metallicities cannot be explained by current models of galactic cosmic-ray production, even if the depletion of Li during the pre-main-sequence phase is ignored (see reviews by e.g. Cayrel et al., 2008; Christlieb, 2008; Prantzos, 2010, 2012, and references therein).
A possible solution of this so-called second Lithium problem was proposed by Cayrel et al. (2007): noting that the spectroscopic signature of the presence of Li in the atmospheres of metal-poor halo stars is just a subtle extra depression in the red wing of the Li doublet, which is difficult to measure even in spectra of the highest quality, they point out that the intrinsic line asymmetry caused by convection in the photospheres of cool stars closely mimics the presence of Li at the level of a few percent if interpreted in terms of 1D, intrinsically symmetric blend components. As a consequence, the Li abundance derived so far by using 1D model atmospheres, ignoring the convective line asymmetry, are expected to be systematically too high.
We quantify the theoretical effect of the convection-induced line asymmetry on the resulting Li abundance by fitting a given synthetic profile both with a grid of 1D LTE and a grid of 3D non-LTE line profiles. The 1D-3D difference of the required Li/Li isotopic ratio measures the expected systematic error in Li/Li inherent to the standard 1D analysis. The synthetic spectra used for this differential comparison are based on a set of 3D hydrodynamical model atmospheres computed with the COBOLD code (Freytag et al., 2002, 2012) and a corresponding set of fully compatible 1D mixing-length models, respectively, as outlined in Sect. 2. The results of this investigation may be represented by an analytical approximation, giving the 3D correction of Li/Li as a function of effective temperature, gravity, and metallicity, for a parameter range that covers the stars of the A06 sample (Sect. 3). We also discuss the potential advantages and disadvantages of using so called ’calibration lines’ for fixing the residual line broadening, and show that the usage of such ’calibration lines’ is potentially dangerous, because the inferred broadening parameter shows considerable line-to-line variations. Depending on the choice of these lines, the resulting Li abundance may be systematically biased (Sect. 4).
A careful reanalysis of individual objects is under way to see whether the second Lithium problem can be resolved when accounting properly for convection and non-LTE line formation in 3D stellar atmospheres. In Sect. 5 we present a preliminary analysis of eight individual metal-poor turn-off stars for which sufficiently high-resolution, high S/N spectra are at our disposal, fitting the observed \ionLii 670.8 nm feature with both 1D LTE and 3D non-LTE synthetic line profiles. As expected, the 3D analysis gives a systematically lower Li/Li ratio by roughly with respect to the 1D result. In most cases, we find that the detection of Li is not significant at the level.
2 3D model atmospheres and spectrum synthesis
The hydrodynamical atmospheres used in the present study are part of the CIFIST 3D model atmosphere grid (Ludwig et al., 2009). They have been obtained from realistic numerical simulations with the COBOLD code111http://www.astro.uu.se/~bf/co5bold_main.html (Freytag et al., 2002, 2012) which solves the time-dependent equations of compressible hydrodynamics in a constant gravity field together with the equations of non-local, frequency-dependent radiative transfer in a Cartesian box representative of a volume located at the stellar surface. The computational domain is periodic in and direction, has open top and bottom boundaries, and is resolved by typically 140140150 grid cells. The vertical optical depth of the box varies typically from (top) to (bottom). Further information about the grid of models used in the present study is compiled in Table 1. As indicated in columns (2) – (4), the grid covers the range K K, , . Radiative transfer is solved in or opacity bins (col. 5). Each of the models is represented by typically snapshots chosen from the full time sequence of the corresponding simulation (col. 6).
These representative snapshots are processed by the non-LTE code NLTE3D that solves the statistical equilibrium equations for a 17 level lithium atom with 34 line transitions, fully taking into account the 3D thermal structure of the respective model atmosphere (but neglecting the differential Doppler shifts implied by the hydrodynamical velocity field). The photo-ionizing radiation field is computed at frequency points between and 32 407 Å, using the opacity distribution functions of Castelli & Kurucz (2004) to allow for metallicity-dependent line-blanketing, including the H I–H and H I–H I quasi-molecular absorption near and Å, respectively. Collisional ionization by neutral hydrogen via the charge transfer reaction H() + Li() Li() + H is treated according to Barklem, et al. (2003). More details are given in Sbordone et al. (2010). Finally, 3D non-LTE synthetic line profiles of the \ionLii nm doublet are computed with the line formation code Linfor3D222http://www.aip.de/~mst/linfor3D_main.html, using the departure coefficients = provided by NLTE3D for each level of the lithium model atom as a function of geometrical position within the 3D model atmospheres. At this stage, the 3D hydrodynamical velocity field provides the differential line shifts along each line of sight. As demonstrated in Steffen et al. (2010b), non-LTE effects are very important for the 3D models of the metal-poor dwarfs considered here: they strongly reduce the height range of line formation such that the 3D non-LTE equivalent width is smaller by roughly a factor 2 compared to 3D LTE. Ironically, the line strength predicted by standard 1D mixing-length models in LTE are close to the results obtained from elaborate 3D non-LTE calculations. Nevertheless, when it comes to the determination of the Li/Li isotopic ratio, it is important to account for the convective line asymmetry, which requires the full 3D non-LTE line formation calculations.
Notes: averaged over selected snapshots; averaged over selected 3D non-LTE spectra
3 Theoretical 3D non-LTE correction of 1D LTE Li/Li determinations
As outlined above, the Li abundance is systematically overestimated if the intrinsic asymmetry of the Li line components is ignored. To quantify this bias theoretically, we rely on synthetic spectra. The idea is as follows: we represent the observation by the synthetic 3D non-LTE line profile of the Li line blend, computed with zero Li content. The non-thermal line broadening is naturally provided by the 3D hydrodynamical velocity field, which replaces the classical concept of micro- and macroturbulence, and also gives rise to a convective blue-shift and an intrinsic line asymmetry. Optionally, the line blend is subsequently broadened by rotation () and a Gaussian instrumental profile ( denoting its full width at half maximum). Now this realistic proxy of the Li line blend is fitted by a grid of ’classical’ 1D LTE synthetic line profiles. The 1D LTE line profiles are computed from so-called LHD models, 1D mixing-length model atmospheres that have the same stellar parameters and employ the same microphysics and radiative transfer scheme as the corresponding 3D model, assuming a mixing-length parameter of , and a depth-independent microturbulence of km/s.
Five parameters are varied independently until the best fit is found ( minimization): in addition to the total Li+Li abundance, (Li), and the isotopic ratio, (Li)=Li/Li, which control line strength and line asymmetry, respectively, we also allow for a residual line broadening described by a Gaussian kernel with half-width , a global line shift, , and a global scaling of the continuum level, . For technical reasons, the rotational line broadening is not treated as a fitting parameter; it is fixed to the value used in the 3D spectrum synthesis.
Note that, although the five fitting parameters are correlated, each one has a distinctly different effect on the line profile. The broadening is fixed by the slope of the blue wing, which is not affected by the Li content, (see Fig. 1), while (Li) is fixed by the shape of the red wing. This allows to obtain unambiguously a unique solution although a given half width of the line blend can be obtained by different combinations of (Li) and , but with higher values than the best solution.
Finally, the 3D non-LTE correction of the Li/Li isotopic ratio is defined as the difference between the values of (Li) that provide the best fit () to the given profile using 1D LTE and 3D non-LTE profiles, respectively: (Li) = (Li) - (Li)333(Li) is zero by construction here. This differential correction is meant to be subtracted from the Li/Li isotopic ratio derived from the 1D LTE analysis to correct for the bias introduced by neglecting the intrinsic line asymmetry: (Li) = (Li) - (Li). Note that this correction procedure properly accounts for radiative transfer in the lines, including saturation effects.
We have determined (Li) according to the method outlined above for each of the 3D model atmospheres in our grid, considering two cases: (Li) and (Li) are the result of fitting the unprocessed ( km/s, km/s) and the broadened ( km/s, km/s) synthetic line profile, respectively. Case (B) is the more realistic one, while case (A) has been included only to show how the amount of instrumental broadening is affecting the amplitude of (Li). Note also that the 3D corrections given in previous studies by (Steffen et al., 2010a, b) correspond to case (A).
The results of this computationally very expensive procedure are given in cols. (8) and (9) of Table 1. They may be aproximated fairly well by the following polyonmial expression:
At given metallicity, the corrections are largest for low gravity and high effective temperature. They increase towards higher metallicity. Note that the corrections are strictly valid only for a Li abundance of (Li)=2.2. The results for [Fe/H]=0.0 are most uncertain, since they are based on fewer data points.
The analysis of A06 utilizes 1D LTE profiles computed from MARCS model atmospheres. Hence, the correction (Li) should be applied to their Li/Li isotopic ratios. The resulting downward corrections are typically in the range (Li) for the stars of their sample (cf. Fig. 2). After subtracting the individual (Li) for each of these stars, interpolated via Eq. (1) to their , , and [Fe/H], the mean Li/Li isotopic ratio of the sample is reduced from to . The center of the distribution of Li/Li is shifted essentially to zero (see Fig. 3).
Based on the original Li/Li isotopic ratios and their error bars as determined by A06, the number of stars with a Li detection above the and level is and , respectively, out of . After correction, the number of and detections is reduced to and , respectively. The detections after correction are HD 106038 and G020024, the remaining detections are HD 102200, HD 160617, and CD30 18140.
We note that HD 106038 survives as a detection only because of its amazingly small error bar of . Assuming a more typical error of , this object would not even qualify as a detection. HD 102200 seems to be a clear 2 detection ((Li) ), while CD30 18140 is a marginal case ((Li) ). The remaining objects, G020-024 and HD 160617, are discussed in more detail in Sect. 5, where their spectra are reanalyzed with 3D non-LTE line profiles; none of them provides convincing evidence for the presence of Li.
4 Fixing the residual line broadening by ’calibration lines’
So far we have derived all information from fitting the Li nm feature, for which we have a 3D non-LTE line formation model. In principle, the accuracy of the fitting procedure can be improved by fixing the value of the residual line broadening, , from additional unblended ’calibration lines’. This approach has been adopted by A06 (see also Lind et al., 2012, henceforth L12). This fact complicates the comparison with our results, and the correction procedure described in Sect. 3 explains only part of the differences. We show in the following that ’calibration lines’ may introduce additional uncertainties, unless their 3D non-LTE line formation is fully understood.
In the framework of 1D models, , represents the combined effect of macroturbulence and instrumental broadening (for given microturbulence and rotational velocity) and is expected to be independent of the spectral line. In general, an average value of from several calibration lines is then used for the analysis of the Li line, thus reducing the number of free fitting parameters, and hence the formal errors of the fitting results.
|HD 74000||-2.05||ESO3.6 / HARPS||C07|
|G271162||-2.25||VLT / UVES||N00|
|HD 84937||-2.40||CFHT / GECKO||C99|
|G020024||-1.89||VLT / UVES||A06|
|HD 140283||-2.40||SUBARU / HDS||A04|
|HD 160617||-1.76||ESO3.6 / HARPS||M12|
|G6412||-3.24||Keck / HIRES||A08|
|G2754||-3.21||VLT / UVES||N09|
Cayrel et al. (2007),
Nissen et al. (2000),
Cayrel et al. (1999),
Asplund et al. (2006),
Aoki et al. (2004),
L. Monaco and G. Lo Curto (2012, priv. comm.), Asplund & Melendez (2008), P.E. Nissen (2009, priv. comm.),
S/N estimated from the (re-reduced) spectra actually used in this work, may differ from literature values.
We have simulated this procedure using synthetic lines only. Our selection of calibration lines is compiled in Tab. 3. The lines have been selected to have similar wavelength, equivalent width, and excitation energy (relative to the ionization continuum) as \ionLii nm. Synthetic line profiles generated from 3D model d3t63g40mm20n01 (assuming LTE) represent the observed line profiles and are fitted with 1D LTE profiles from the corresponding LHD model to derive . The results are given in col. (5) of Tab. 3. These numbers have to be compared with the value of obtained by fitting the 3D non-LTE Li profile of the same 3D model with 1D LTE profiles from LHD models, km/s (last row of Tab. 3). When fixing to the values implied by the individual calibration lines, the resulting Li/Li isotopic ratio changes by (Li) with respect to the value obtained when treating as a free fitting parameter.
We find that, even under these idealized conditions, (Li) is not constant but varies from line to line (see col. (6) of Tab. 3). This is an indication that 1D micro / macro model is not perfectly adequate to describe the 3D hydrodynamical velocity field. Moreover, since (Li) is positive for all of our calibration lines, the use of calibration lines leads to an additional overestimation of the Li isotopic ratio by up to %. The total error of the 1D analysis is obtained by adding the corrections from Tables 1 and 3, (Li) = (Li) + (Li).
In a further experiment, we have fitted both the calibration lines and the Li feature with 3D LTE line profiles. By construction, the derived is then equal to of the assumed instrumental profile, and is identical for all calibration lines. The Li isotopic ratio obtained from fitting the 3D non-LTE Li feature with 3D LTE profiles and fixed is (Li)=, compared to if is a free fitting parameter. The basic reason for this alarming result is that the half width of the Li line is significantly smaller in LTE than in non-LTE, which must be compensated by a higher Li content. This effect may explain why A06 find higher Li/Li isotopic ratios in their 3D LTE analysis (with respect to 1D LTE, their Table 5).
We note that the above results are consistent with previous findings by Steffen et al. (2010b), who analyzed the spectrum of HD 74000 with a subset of 6 clean \ionFei calibration lines, and obtained significantly higher Li/Li isotopic ratios compared to the analysis without calibration lines. The same behavior is seen by L12, if they fit both the calibration lines and Li with 3D LTE profiles. Using instead non-LTE profiles for fitting both \ionCai and \ionLii lines leads to consistent Li/Li isotopic ratios for HD 84937.
5 Analysis of observed spectra: 3D non-LTE versus 1D LTE line fitting
With the results given in Sect. 3, the recommended method of analysis of the \ionLii nm doublet in an observed spectrum is as follows. The first step is to produce a grid of 1D LTE synthetic line profiles as a function of (Li), (Li), using any standard 1D mixing-length model with the correct stellar parameters (, , [Fe/H]). This grid is then used to find the best fit to the observed line profile, as described above (no calibration lines). Finally, the resulting (Li) is corrected for 3D effects by subtracting the differential correction (Li) defined in Sect. 3, interpolated to the actual stellar parameters according to Eq. (1) and Table 2.
In the following, however, we disregard this differential approach, and present instead preliminary results of fitting the observed \ionLii nm spectra of eight halo turn-off stars (see Table 4) directly with our grids of 3D non-LTE and 1D LTE synthetic line profiles, computed from 3D COBOLD models and 1D LHD models, respectively. As described above, five free fitting parameters ((Li), (Li), , , ) are varied independently to find the best fit. Formally, we fix the rotational velocity to = km/s, noting that the derived Li/Li isotopic ratio is insensitive to this assumption (see Steffen et al., 2010b).
|HD 74000||d3t63g40mm20n01||3D NLTE||—|
|HD 84937||d3t63g40mm20n01||3D NLTE|
|HD 140283||d3t57g37mm20n01||3D NLTE|
|HD 160617||d3t59g40mm20n02||3D NLTE||—|
The results are presented in Table 5. The errors quoted for and (Li) are the formal confidence intervals due to the finite S/N of the fitted spectra. They are computed in the usual way (see e.g. Nissen et al., 2000; Asplund et al., 2006) by finding the distance of the parameter of interest from is optimum value such that , fixing and minimizing over the remaining free fittting parameters. Here is defined as
where and are the observed and synthetic flux at wavelength point , and =(S/N), with S/N taken from Tab. 4. Note that this is a lower limit to the real error, which may have several other significant contributions (see discussion in García Pérez et al., 2009).
As expected, the 3D analysis yields systematically lower Li/Li isotopic ratios by up to 1.8%. For comparison, we list in the last column the 1D LTE results of A06, A08, and the 3D non-LTE results of L12 from their case (a) for the stars in common with our sample. In some cases, the agreement is very good (G271–162, HD140283), even though the analysis is based on different observational data. In other cases, we obtain significantly lower Li/Li ratios (HD 160617, G64–12). The most striking discrepancy, this time in the opposite direction, is seen for HD 84937.
G020–024 is a special case. We have retrieved the spectra obtained by A06 from the ESO archive and re-reduced them with the current UVES pipeline. We note that all six sub-exposures show an unexplained ’perturbation’ in the lower red wing of the Li nm line that cannot be fitted properly, neither with 1D LTE nor with 3D non-LTE model spectra. The best 3D non-LTE fit suggests a very high Li content near (see Fig. 4), even exceeding the 1D LTE result by A06 of . In view of the poor quality of the fit, we consider this result as doubtful. It is worth mentioning that G020–024 is listed as a suspected binary in Fouts (1987); moreover, A06 report that this star shows an unusually large discrepancy between the photometric and the H-based . Possibly, G020–024 is a spectroscopic binary whose components are of similar spectral type.
Disregarding G020–024, we are left with one formal detection (HD 84937), one detection (G275–4), and five non-detections (HD 74000, G271–162, HD 140283, HD160617, G64–12) out of the eight stars, when considering the 3D non-LTE results only. In 1D LTE, G271–162 would turn into a formal detection (see Tab. 5).
The Li/Li isotopic ratio derived by fitting the Li I doublet with 3D non-LTE synthetic line profiles is shown to be 1% to 2% lower than what is obtained with 1D LTE profiles. This result implies that only out of the stars of the Asplund et al. (2006) sample would formally remain significant () Li detections when subjected to a 3D non-LTE analysis.
In another theoretical case study we have demonstrated that the difference between 3D non-LTE and 1D LTE results increases even more if we would rely on additional ’calibration lines’ to fix the residual line broadening, as advocated e.g. by A06, A08, L12. The number of possible Li detections is thus further reduced. We conclude that the usage of additional ’calibration lines’, even if carefully selected, introduces additional uncertainties rather than reducing the error of the analysis, unless the 3D non-LTE line formation is fully understood for all involved lines.
Finally, we have analyzed available high quality spectra of eight turn-off halo stars, both with 1D LTE and 3D non-LTE modeling. In most cases, the evidence for the presence of Li is not significant. Only in the case of HD 84937 it is difficult to deny the signature of Li. Surprisingly, L12 no longer find any significant evidence for the presence of Li in this object, which since almost two decades has been considered as an undisputed Li detection. From the results reported by L12, it is unclear whether the reason for this disturbing discrepancy is related to the new superior observational data or to the improved method of analysis.
We conclude that 3D model atmospheres can indeed help to solve the second lithium problem. In view of the 3D non-LTE results reported in this work (and by L12), it seems that the presence of Li in the atmospheres of galactic halo stars is rather the exception than the rule, and hence does not necessarily constitute a cosmological problem.
Acknowledgements.We thank L. Monaco and G. Lo Curto for re-reducing the HARPS spectrum of HD 160617, P.E. Nissen for providing us with a carefully reduced UVES spectrum of G275-4, and W. Aoki for allowing us to use his HDS spectrum of HD 140283.
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