IMPROVED LINE DATA FOR THE SWAN SYSTEM ¹²C¹³C ISOTOPOLOGUE

In stellar abundance analyses, one needs line lists with wavelengths, lower-state excitation energies, and transition probabilities (usually expressed as log gf values) for all relevant molecular and atomic absorbers. For C_2, we generated a line list suitable for input to the current version of the LTE synthetic spectrum code, MOOG (Sneden 1973). The relevant quantities for each line were derived for ¹²C₂ from the data in the table in Appendix A of Brooke et al. (2013) and for ¹²C¹³C from the data in Table 4. The resultant line list, containing nearly 184,000 transitions, can be obtained from the authors.⁹

Wavenumbers in air were computed from the vacuum wavenumbers with the Birch & Downs (1994) corrected version of Edlén's (1966) conversion formula, and inverted to produce air wavelengths in Angstroms. For lines with laboratory detections we adopted the measured wavenumbers, and used the predicted wavenumbers for the other transitions. We changed the lower-state energies E'' from units of cm⁻¹ to eV, a common convention in astronomy. Finally, we converted the Einstein spontaneous emission coefficient A^' to the commonly used gf via

$$\begin{equation} gf = f_{J^{\prime} \leftarrow J^{\prime\prime}} (2J^{\prime} + 1) \end{equation} \tag{ 3 }$$

The MOOG code follows the development of Schadee (1964) for diatomic molecules. His approach combined Saha and Boltzmann equations to form a transition lower-state electron number density in terms of the constituent atomic free number densities (after molecular equilibrium computations), their partition functions, and the molecular dissociation energy.

The C₂ dissociation energy is well determined (e.g., D₀ = 6.30 eV, Urdahl et al. 1991; 6.27 eV, Pradhan et al. 1994). We adopted D₀ = 6.27 eV. In the Sun, and most cool stars, the free number density of carbon depends almost entirely on the CO formation, that is

$$\begin{equation*} P({C}) = p({C}) + p({CO}) + {minor}\; {species} \end{equation*}$$

where P(C) is the fictitious (in the absence of any molecular or ionized species) pressure of carbon. The C₂ molecule is at most a trace constituent in typical, cool, stellar atmospheres, so its dissociation energy is unimportant for this equilibrium equation. In line-forming regions (τ ∼ −1) of the solar atmosphere, C₂ comprises about 0.01% of the total carbon, and in lower-pressure G–K giants the percentage of C₂ is orders of magnitude less.

Diatomic molecular partition functions mathematically cancel out in the Schadee (1964) line opacity equations. However, the partition functions for homonuclear ¹²C₂ and heteronuclear ¹²C¹³C differ by a factor of two because the parent molecule has half as many energy states as the isotopic form. This difference cannot be accounted for in MOOG's implementation of the Schadee formalism (D. L. Lambert 2013, private communication; see also the discussion in Cohen et al. 2006). Since C₂ is the only homonuclear molecule of interest in stellar spectroscopic investigations, we chose to make no general alteration to the synthesis code. Rather, like Cohen et al., we simply divided the gf values computed as above by a factor of two to mimic the partition function differences. This choice is reflected in our final C₂ line list, which contains all parent and isotopic lines from this study.

The major absorption bands of the C₂ Swan system occur in the 4500–5700 Å spectral region. However, in stars with abundance mixes like that of the Sun, the Swan system produces relatively weak absorption features that often are significantly contaminated with or completely masked by atomic lines. The strongest Swan bands, those of the Δv = 0 sequence, occur near 5100 Å, just where the strongest MgH A²Π–X²Σ⁺ bands are located (e.g., Hinkle et al. 2013 and references therein). Indeed, Swan system lines often serve as contaminants to MgH. As such, they increase the difficulty in deriving reliable Mg isotopic ratios from MgH, while not presenting enough isolated C₂ features for a reliable carbon abundance analysis. In order to make a cleaner test of our new line list, first we decided to generate synthetic spectra of the Swan bands to compare with high-resolution spectra of so-called "carbon-enhanced metal-poor" stars (CEMP; see Beers & Christlieb 2005).

Cohen et al. (2006) conducted a high-resolution spectroscopic survey of 15 CEMP stars. Their spectra, obtained with the HIRES instrument, are available in the Keck Observatory archive. The observations were made as part of the program, "The first generation of stars in the Galaxy" (PI: J. Cohen), on 2002 September 27 and 28. Five 1200 s exposures were collected, using a 086 slit width, which yields a spectral resolution of R ≈ 45,000. Spectra cover the range between 3840 and 5330 Å. For more details about the observations, the interested reader is referred to Cohen et al. These archival spectra were reduced using the MAKEE¹⁰ spectrum reduction package, then continuum-normalized and shifted to rest-frame using IRAF.¹¹

All of the CEMP stars in the Cohen et al. (2006) survey are very metal-poor (–3.30 ⩽ [Fe/H] ⩽ −1.85), relatively warm (4945 K ⩽ T_eff ⩽ 6240 K) subgiants and giants (2.0 ⩽ log g ⩽ 3.7). These parameters combine to yield very weak atomic absorption lines in their spectra. However, the majority of CEMP stars are also enriched in neutron-capture (n-capture) elements, most noticeably in barium (Z = 56), the lanthanides (57 ⩽ Z ⩽ 71), and lead (Z = 82). Transitions arising from ionized n-capture species can be seen especially in the blue–violet spectral regions. Detailed abundance studies of these kinds of CEMP stars indicate that the dominant n-capture synthesis mechanism has been the slow "s-process" (e.g., Sneden et al. 2008 and references therein), so such stars are labeled CEMP-s.

Almost all very metal-poor halo stars have relative overabundances with respect to iron of the "α" elements Mg, Si, Ca, and Ti. The Cohen et al. (2006) sample is no exception, with 〈[Mg/Fe]〉 ∼ +0.5, but since the mean metallicity of their stars is 〈[Fe/H]〉 ∼ −2.1, the total magnesium abundance is still very small, thus absorption by MgH must be very weak or completely absent. Of course, carbon is very overabundant in CEMP stars; the mean enhancement in the Cohen et al. sample is 〈[C/Fe]〉 ∼ 1.9 (〈[C/H]〉 ∼ −0.2), making C₂ the dominant spectral feature near 5000 Å.

We inspected our set of Cohen et al. (2006) data, seeking stars with the largest C₂ absorption combined with the lowest ¹²C/¹³C ratios, in order to see the ¹²C¹³C isotopic band structures. We found clearly identifiable 1–0 and 2–1 P-branch band heads of this isotopologue in six stars, and 0–1 band heads in one star. For the 0–1 band region we added a spectrum of the CEMP subgiant, LP 625–44 (Aoki et al. 2006 and references therein), which was obtained by David Lai with Hamspec at Lick Observatory. From these spectra, we estimated the observed band head wavelength differences, λ(¹²C¹³C)–λ(¹²C₂). For this exercise, we defined the wavelength of deepest absorption to be the band head, which we estimated to be ∼0.3 Å blueward of the P-branch red–edge wavelength. From the wavelength differences, we computed ¹²C¹³C band head wavelengths using the corresponding ¹²C₂ wavelengths derived by Brooke et al. (2013). In Table 5 we list the ¹²C₂ band heads from that paper, our observed stellar and predicted ¹²C¹³C band heads, our predicted isotopic shifts, and those of Pesic et al. (1983). Clearly, our predicted band head positions are in good agreement with the observed ones.

We chose to concentrate on the star HE 0212–0557 to compare our predicted ¹²C¹³C line data with observations over larger wavelength ranges. For details about the model parameter and abundance determinations of this star, see Cohen et al. (2006). Briefly, they derived T_eff = 5075 K, log g = 2.15, v_micro = 1.8 km s⁻¹, and metallicity = [Fe/H] = −2.27. They reported abundances for 18 other elements. Highlights for our purposes are: [C/Fe] = 1.74, ¹²C/¹³C = 3–4, [α-element/Fe] = [〈Mg i, Ca i, Ti i, Ti ii〉/Fe] = +0.2, and [rare-earth-elements/Fe] = [〈Ba ii, La ii, Ce ii, Pr ii, Nd ii〉/Fe] = +2.2. The enormous overabundances of carbon and the rare-earth neutron-capture abundances definitely assign HE 0212–0557 to the CEMP-s category.

We first produced synthetic spectra with the MOOG code, generating input quantities as follows. A model atmosphere for HE 0212–0557 with the Cohen et al. (2006) parameters T_eff, log g, [Fe/H], v_micro was calculated by interpolating in the ATLAS model grid (Kurucz 2011 and references therein).¹² We constructed the line lists beginning with the ¹²C₂ and ¹²C¹³C lines described above, and added atomic lines from Kurucz's line compendium.¹³ For most atomic lines, we adopted the transition probabilities from the Kurucz database, but for neutron-capture elements (particularly the rare-earth ions) we adopted the gf's determined in a series of papers by the Wisconsin atomic physics group (Sneden et al. 2009 and references therein). The computed spectra were then convolved with Gaussian functions that empirically accounted for the combined effects of the Keck spectrograph slit function and stellar macroturbulence. Finally, we compared the synthetic spectra to the observed HE 0212–0557 spectrum, which had been corrected for its apparent radial velocity and continuum-normalized.

Some illustrative spectra of ¹²C₂ and ¹²C¹³C bands in HE 0212–0557 are provided in Figures 1 and 2. We show spectra for small wavelength ranges that include the 0–0 and 1–1 band heads in Figure 1. Absorption by C₂ dominates the spectrum blueward of 5165 Å. Isotopic wavelength shifts of the Δv = 0 bands are very small; the ¹²C₂ and ¹²C¹³C band heads are indistinguishable. Nevertheless, it is easy to see that the observed spectrum must include large contributions from ¹²C¹³C lines. We varied the carbon isotopic ratio and the total carbon abundance until the best synthetic/observed spectrum match was achieved, deriving log ε(C) = 8.12 ± 0.05 ([C/Fe] = 1.8) and ¹²C/¹³C ≈ 4–5. Both of these results are in excellent accord with those of Cohen et al. (2006).

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Panel (a) of Figure 1 also includes a 10 Å region redward of the 0–0 band head, which is almost devoid of C₂ absorption. Several prominent features here are due to Fe i, but there are also two of the three Mg i b lines, which are labeled in the figure. In solar abundance mixtures the Mg i lines dominate the 5160–5190 Å spectral region. Their relative weakness compared to C₂ is one of the defining characteristics of an extreme CEMP star; HE 0212–0557 is a near-textbook example of such an object.

In Figure 2 we show small wavelength sections that cover the 1–0 and 2–1 band heads. Happily, the isotope shifts for the Δv = +1 sequence are large; the ¹²C¹³C band heads are about 8 Å to the red of their ¹²C₂ counterparts. This allows a much easier estimation of the isotopic ratio. Our derived total carbon abundance (log ε(C) = 8.09 ± 0.05) and ¹²C/¹³C ratio (4.5 ± 0.5) are indistinguishable from those obtained from the Δv = 0 bands. In panel (a) of this figure we have labeled three prominent rare-earth, ionized-species transitions. These lines would be extremely weak in metal-poor stars without the enormous overabundances of the n-capture elements of HE 0212–0557. There are other n-capture absorptions that appear among the C₂ bands; their inclusion in our syntheses improves the quality of the match to the observed spectrum. In panel (b) the ¹²C₂ 2–1 band head is easily identified since it dominates the absorption in the 4712–4716 Å region. One can also find the ¹²C¹³C 2–1 band head at 4722 Å, but this task is made much more difficult by the strong absorption lines in this region due to 1–0 ¹²C₂ and ¹²C¹³C.

Our stated errors for [C/Fe] and ¹²C/¹³C simply reflect uncertainties in matching the observed spectrum. Strong C₂ Swan absorption occurs throughout the 4500–5200 Å region in HE 0212–0557. This makes accurate continuum placement somewhat difficult, especially near the C₂ band heads. However, our synthetic spectra were computed over about 100 Å each in the Δv = 0 and +1 wavelength regions, and we derived consistent carbon abundances and isotopic ratios in all of the synthesized regions. Note that our ±0.05 error in total carbon abundance is very small because of the great sensitivity of the "double-metal" C₂ absorption to the carbon abundance. In these illustrative synthetic spectrum computations, we have not attempted to estimate the effects of model atmosphere parameters on [C/Fe]; see Cohen et al. (2006) for an extensive uncertainty analysis. However, we should re-emphasize statements from many past isotopic investigations: derived ¹²C/¹³C ratios are almost totally insensitive to errors in T_eff, log g, and [M/H] because the molecular parameters (dissociation and excitation energies, transition probabilities) are nearly identical for ¹²C₂ and ¹²C¹³C.

Our HE 0212–0557 spectrum does not extend into the wavelength region of the Δv = −1 bands, which cover about 5400–5640 Å; the ¹²C₂ 0–1 band head lies at 5635 Å. These bands are somewhat weaker than the Δv = 0 and +1 sequences, and the isotopic shifts are to the blue. For example, the ¹²C¹³C 0–1 band head lies at 5625 Å. We synthesized this region to compare with the spectrum of another Cohen et al. (2006) star, HE 1150–0428, which has a very low ¹²C/¹³C ratio. We detected the ¹²C¹³C band head, and confirmed that our line list accurately matches the wavelengths of the observed spectrum. We concur with the Cohen et al. derived abundances ([C/Fe] ≈ +2.4 for [Fe/H] ≈ −3.3, and ¹²C/¹³C ≈ 4) in HE 0212–0557, but have no new information to add on the Δv = −1 band sequence.

The very bright K-giant star Arcturus has been studied at high spectral resolution for decades. The atmospheric parameters of this mildly metal-poor star are well known (T_eff = 4300 K, log g = 1.50, v_micro = 1.5, [Fe/H] = −0.50; e.g., Bergemann et al. 2012). Its spectrum is used as a template for studies of other cool stars, which sometimes become differential analyses with respect to Arcturus.

The C₂ Swan bands are very weak in the Arcturus spectrum, and their lines are often completely masked by strong atomic lines. Additionally, in discussing new line data for the MgH molecule, Hinkle et al. (2013) included an estimate of the Mg isotopic fractions in Arcturus. Thus, unlike the CEMP-s star discussed in Section 4.2, the MgH lines mixed in with the C₂ Δv = 0 system in Arcturus make a carbon abundance analysis very challenging.

In Figure 3 we show synthesized and observed 0–0 and 0–1 Swan band heads in Arcturus. The observed spectrum is the very high resolution and signal-to-noise Arcturus atlas of Hinkle et al. (2000). The synthetic spectra were generated in the same manner as those of HE 0212–0557 discussed in Section 4.2. Both band heads are severely compromised with contamination by other species. In panel (a), the combined ¹²C₂ and ¹²C¹³C 0–0 band head near 5165.3 Å is detectable, but obviously there are cleaner C₂ features away from the band head that are better carbon abundance indicators. In panel (b), the much weaker ¹²C₂ 0–1 band head near 5636.5 Å is adequate for abundance studies because the atomic contamination is weak. However, we do not show the ¹²C¹³C band head near 5625.9 Å because it is completely buried underneath several strong atomic lines. Given the overall metallicity of Arcturus, ¹³C would need to be much more abundant to be detectable with confidence in this spectral region. Even less promising are the ¹²C₂ and ¹²C¹³C 0–1 band heads at 4737 and 4744 Å; both are very weak and severely blended in the Arcturus spectrum. The best options for determining the carbon abundance from C₂ in this star (and others like it) appear to be individual transitions of the 0–0 band and the 0–1 band head. The ¹²C/¹³C ratios in ordinary cool giants are probably better determined from other molecular features, e.g., the CN red system.

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