THE ABUNDANCES OF NEUTRON-CAPTURE SPECIES IN THE VERY METAL-POOR GLOBULAR CLUSTER M15: A UNIFORM ANALYSIS OF RED GIANT BRANCH AND RED HORIZONTAL BRANCH STARS

Special effort was made to employ the most recent laboratory measurements of oscillator strengths. When applicable, the inclusion of hyperfine and isotopic structure was done for the derivation of abundances. Tables 2 and 3 list the various literature sources for the transition probability data. Some species deserve special comment. The Fe transition probability values are taken from the critical compilation of Fuhr & Wiese (2006; note for neutral Fe, the authors heavily weigh the laboratory data from O'Brian et al. 1991). No up-to-date laboratory work has been done for Sr, and so, the adopted gf-values are from the semi-empirical study by Brage et al. (1998; these values are in good agreement with those derived empirically by Gratton & Sneden 1994). Similarly, the most recent laboratory effort for Y was by Hannaford et al. (1982). Yet these transition probabilities appear to be robust, yielding small line-to-line scatter.

A particular emphasis of the current work is the n-capture element abundances, for which a wealth of new transition probability data have become recently available. Correspondingly, the extensive sets of rare Earth gf-values from the Wisconsin Atomic Physics Group were adopted (Sneden et al. 2009; Lawler et al. 2009, and references therein). These data when applied to the solar spectrum yield photospheric abundances that are in excellent agreement with meteoritic abundances. For neutron-capture elements not studied by the Wisconsin group (which include Ba, Pr, Yb, Os, Ir, and Th), alternate literature references were employed (and these are accordingly given in the two aforementioned tables).

In the original version of the line transfer code MOOG (Sneden 1973), local thermodynamic equilibrium (LTE) was assumed and hence, scattering was treated as pure absorption. Accordingly, the source function, S_λ, was set equal to the Planck function, B_λ(T), which is an adequate assumption for metal-rich stars in all wavelength regions. However, for the extremely metal-deficient, cool M15 giants, the dominant source of opacity switches from H⁻_BF to Rayleigh scattering in the blue visible and ultraviolet wavelength domain (λ ≲ 4500 Å). It was then necessary to modify the MOOG program as the LTE approximation was no longer sufficient (this has also been remarked upon by other abundance surveys, e.g., Johnson 2002; Cayrel et al. 2004).

The classical assumptions of one-dimensionality and plane-parallel geometry continue to be employed in the code. Now with the inclusion of isotropic, coherent scattering, the framework for solution of the radiative transfer equation (RTE) shifts from an initial value to a boundary value problem. The source function then assumes the form¹¹ of S = (1 − )J + B and the description of line transfer becomes an integro-differential equation. The chosen methodology for the solution of the RTE (and the determination of mean intensity) is the approach of short characteristics that incorporates aspects of an accelerated convergence scheme. In essence, the short characteristics technique employs a tensor product grid in which the interpolation of intensity values occurs at selected grid points. The prescription generally followed was that from Koesterke et al. (2002 and references therein). The Appendix provides more detail with regard to the MOOG program alterations.

Prior to these modifications, for a low-temperature and low-metallicity star (e.g., an RGB target), the ultraviolet and blue visible spectral transitions reported aberrantly high abundances in comparison to those abundances found from redder lines. With the implementation of the revised code, better line-to-line agreement is found and accordingly, the majority of the abundance trend with wavelength is eliminated for these types of stars. Note for the RHB targets, minimal changes are seen in abundances with the employment of the modified MOOG program (as the dominant source of opacity for these relatively warm stars is always H⁻_BF over the spectral region of interest).

To obtain preliminary estimates of T_eff and log g for the M15 stars, photometric data from the aforementioned sources (K. M. Cudworth 2011, private communication; Buonanno et al. 1983; 2MASS) were employed as well as those data from Yanny et al. 1994. To transform the color, the color–T_eff relations of Alonso et al. (1999) were used in conjunction with the distance modulus ((m − M)₀ = 15.25) and reddening (E(B − V) = 0.10) determinations from Kraft & Ivans (2003). Note that an additional intrinsic uncertainty of about 0.1 dex in log g remains among luminous RGB stars owing to stochastic mass loss of order 0.1 dex. Consequently, initial masses of 0.8 M_☉ and 0.6 M_☉ were assumed for RGB and RHB stars, respectively. The photometric V and (V−K) values as well as the photometrically and spectroscopically derived stellar atmospheric parameters are collected in Table 1.

With the use of the spectroscopic data analysis program SPECTRE (Fitzpatrick & Sneden 1987), the equivalents widths (EWs) of transitions from the elements Ti i/ii, Cr i/ii, and Fe i/ii were measured in the wavelength range 3800–6850 Å. The preliminary T_eff values were adjusted to achieve zero slope in plots of Fe abundance (log $(\epsilon _{\rm Fe\,{\mathsc{i}}}$)) as a function of excitation potential (χ) and wavelength (λ). The initial values of log g were tuned to minimize the disagreement between the neutral and ionized species abundances of Ti, Cr, and Fe (particular attention was paid to the Fe data). Lastly, the microturbulent velocities v_t were set so as to reduce any dependence on abundance as a function of EW. Final values of T_eff, log g, v_t, and metallicity [Fe/H] are listed in Table 1, as well as those values previously derived by Sneden et al. (2000a) for the RGB stars and Preston et al. (2006) for the RHB stars.

To conduct a standard abundance analysis under the fundamental assumptions of one-dimensionality and LTE, two grids of model atmospheres are generally employed: Kurucz–Castelli (Castelli & Kurucz 2003; Kurucz 2005) and MARCS (Gustafsson et al. 2008).¹² The model selection criteria were as follows: the reconciliation of the metallicity discrepancy between the RGB and RHB stars of M15, the derivation of (spectroscopic-based) atmospheric parameters in reasonable agreement with those found via photometry, and the attainment of ionization balance between the Fe i and Fe ii transitions. For the RHB targets, interpolated models from the Kurucz–Castelli and MARCS grids were comparable and yielded extremely similar abundance results. However, there are a few notable differences between the two model types for the RGB stars with regard to the P_gas and P_electron content pressures. Though beyond the scope of the current effort, it would be of considerable interest to examine in detail the exact departures between the Kurucz–Castelli and MARCS grids. To best achieve the aforementioned goals for the M15 data set, MARCS models were accordingly chosen.

For the RHB stars, the presently derived metallicities differ slightly from those of Preston et al. (2006): 〈[Fe_I/H]〉 = −2.69 (a change of Δ = −0.03) and 〈[Fe_II/H]〉 = −2.64 (a change of Δ = −0.04). However, for the RGB stars, the [Fe/H] results of the current study do vary significantly from those of Sneden et al. (2000a): 〈[Fe_I/H]〉 = −2.56 (a downward revision of Δ = −0.26) and 〈[Fe_II/H]〉 = −2.53 (a downward revision of Δ = −0.28). The remaining metallicity discrepancy between the RGB and RHB stars is as follows: $\Delta ({\rm RGB}-{\rm RHB})_{\rm Fe\,{\mathsc{i}}}$ = 0.13 and $\Delta ({\rm RGB}-{\rm RHB})_{\rm Fe\,{\mathsc{ii}}}$ = 0.11. Even with the employment of MARCS models and the incorporation of Rayleigh scattering (not done in previous efforts), the offset persists. Repeated exercises with variations in the T_eff, log g, and v_t values showed that this metallicity disagreement in all likelihood cannot be attributed to differences in these atmospheric parameters. As a further check, the derivation of [Fe_{I, II}/H] values was performed with a list of transitions satisfactorily measurable in both RGB and RHB spectra. No reduction in the metallicity disagreement was seen as the offsets were found to be: $\Delta ({\rm RGB}-{\rm RHB})_{\rm Fe\,{\mathsc{i}}}$ = 0.11 (with 45 candidate Fe_I lines) and $\Delta ({\rm RGB}-{\rm RHB})_{\rm Fe\,{\mathsc{ii}}}$ = 0.14 (with 3 candidate Fe ii_II lines).

The data from the M15 RGB and RHB stars originate from different telescope/instrument setups. Additionally, somewhat different data reduction procedures were employed for the two samples. Possible contributors to the iron abundance offset could be the lack of consideration of spherical symmetry in the line transfer computations and the generation of sufficiently representative stellar atmospheric models for these highly evolved stars (which exist at the very tip of the giant branch and/or have undergone He-core flash). Indeed, it is difficult to posit a single, clear-cut explanation for the disparity in the RGB and RHB [Fe/H] values. In an analysis of the GC M92, King et al. (1998) derived an average abundance ratio of 〈[Fe/H]〉 = −2.52 for six subgiant stars, a factor of two lower than the 〈[Fe/H]〉 value measured in the red giant stars. Similarly, Korn et al. (2007) surveyed turn off (TO) and RGB stars of NGC 6397 and found a metallicity offset of about 0.15 dex (with the TO stars reporting consistently lower values of [Fe/H]). They argued that the TO stars were afflicted by gravitational settling and other mixing processes and as a result, the Fe abundances of giant stars were likely to be nearer to the true value. While the TO stars do have T_eff values close to that of the M15 RHB stars, they have surface gravities and lifetimes that are considerably larger. Accordingly, it is not clear if the offset in M15 has a physical explanation similar to that proposed in the case of NGC 6397.

Within the RGB sample, the neutron-capture element abundances of K462 are consistently the largest whereas those of K583 are the smallest. The two RHB stars, B009 and B262, exhibit abundance trends similar to those of the RGB objects. The expected anti-correlations in the proton-capture elements (e.g., Na-O and Mg-Al) are seen. The greatest abundance variation with regard to the entire M15 data set is found for the neutron-capture elements. Indeed, the star-to-star spread for the majority of n-capture abundances is demonstrable for all M15 targets and is not likely due to internal errors.

Inspection of Table 4 data indicates that RHB stars generally have higher r-process element abundances than RGB stars (on average Δ[Elem/Fe]_RHB-RGB ≈ 0.3 dex). A sizeable portion of the discrepancy is attributable to the difference in the iron abundances as 〈[Fe/H]〉_RHB is approximately 0.12 dex lower than 〈[Fe/H]〉_RGB. The remaining offset is most likely a consequence of the small number of targets coupled with a serious selection effect. The original sample of RHB stars from Preston et al. (2006) was chosen as a random set of objects with colors and magnitudes representative of the red end of the HB. These objects were selected without prior knowledge of the heavy element abundances. On the other hand, the three RGB stars from Sneden et al. (2000a) were particularly chosen as representing the highest and lowest abundances of the r-process as predetermined in the 17 star sample of Sneden et al. (1997).

The finalized set of light element abundances include: C, O, Mg, Al, Si, Ca, and Sc_I/II. In general, an enhancement of these element abundances relative to solar is seen in the entire M15 data set.

An underabundance of carbon was found in one RHB and three RGB targets of M15 based on the measurement of CH spectral features. As the forbidden O i lines were detectable only in RGB stars, the average abundance ratio for M15 is 〈[O/Fe]〉_RGB = +0.75. This value is substantially larger than that found by Sneden et al. (1997). A portion of the discrepancy is due to the approximate 0.15 dex difference in the 〈[Fe_I/H]〉 values between the two investigations. The remainder of departure may be attributed to the adoption of different solar photospheric oxygen values: the current study employs log(O)_☉ = 8.71 (Scott et al. 2009), while Sneden et al. use log(O)_☉ = 8.93 (Anders & Grevesse 1989).

For the determination of the sodium abundance, the current study relies solely upon the D₁ resonance transitions. Table 4 lists the spuriously large spreads in the Na abundance for both the RGB and RHB groups. The Na D₁ lines are affected by the non-LTE phenomenon of resonance scattering (Asplund 2005b; Andrievsky et al. 2007), which MOOG does not take into account. Also, Sneden et al. (2000b) made note of the relative strength and line profile distortions associated with these transitions and chose to discard the [Na/Fe] values for stars with T_eff > 5000 K. Consequently, the sodium results from the current study are given little weight and are not plotted in Figure 3.

Aluminum is remarkable in its discordance: 〈[Al_I/Fe_I]〉 = 0.37 for one RHB and three RGB targets, whereas 〈[Al_I/Fe_I]〉 = −0.43 for five RHB stars. Now, the relative aluminum abundances for the RHB stars match well with the values found by Preston et al. (2006). Similarly, the Al abundances from the current analysis agree favorably with the RGB data from Sneden et al. (1997). Though relatively strong transitions are employed in the abundance derivation, the convergence upon two distinct [Al/Fe] values is nontrivial and could merit further exploration.

A decidedly consistent Ca abundance ratio is found for the RGB sample: $\langle[\rm Ca_{\rm I}/\rm Fe_{\rm I}]\rangle_{{\rm RGB}}= 0.29$; and also for the RHB sample: 〈[Ca_I/Fe_I]〉_RHB = 0.53. After consideration of the iron abundance offset, the RHB stars still report slightly higher calcium abundances than the RGB stars. Overall, a distinct overabundance of Ca relative to solar is present in the M15 cluster. Note that the Sc_I abundance determination was done for only one M15 star (and gives a rather aberrant result compared to the Sc_II abundance data from the other M15 targets).

The list of finalized Fe-peak element abundances consists of Ti_I/II, V_I/II, Cr_I/II, Mn, Co, and Ni. Due to RGB spectral crowding issues, derivations of [Ni_I/Fe_I] ratios are performed only for RHB stars.

Achievement of ionization equilibrium did not occur for any of the perspective species: Ti_I/II, V_I/II, or Cr _I/II. In consideration of the entire M15 data set, the best agreement between neutral and singly ionized species arises for titanium, with all 〈[Ti/Fe]〉 ratios being supersolar. The V_II relative abundances compare well with one another for the RGB and RHB targets (comparison for V_I is not possible as there are no RHB data for this species). Both of the RGB and RHB 〈[Cr_I/Fe_I]〉 ratios are underabundant with respect to solar and the neutral chromium values match almost exactly with one another (after accounting for the [Fe/H] offset). On the other hand, the worst agreement is found for Cr_I/II in RHB stars with Δ(II-I) = 0.47.

Subsolar values with minimal scatter were found for the 〈[Mn_I/Fe_I]〉 ratios in both the RGB and RHB stellar groups. However, in comparison to RGB stars, manganese appears to be substantially more deficient in RHB targets. The discrepancy may be attributed to both the RGB/RHB iron abundance disparity as well as the employment of the Mn_I resonance transition at 4034.5 Å for the RHB abundance determination. In particular, J. S. Sobeck et al. (2011, in preparation) have demonstrated that the manganese resonance triplet (4030.7, 4033.1, and 4034.5 Å) fails to be a reliable indicator of abundance. Consequently, the RHB abundance results for Mn_I are given little weight and are not plotted in Figure 4.

Finalized abundances for the light and intermediate n-capture elements include Cu, Zn, Sr, Y, Zr, Ba, La, Ce, and Pr. In general, the RGB element abundance ratios are slightly deficient with respect to the RHB values. Also, enhancement with respect to solar is consistently seen in all M15 targets for the elements Ce and Pr.

An extremely underabundant copper abundance relative to solar was found in the RGB stars: 〈[Cu_I/Fe_I]〉_RGB = −0.91. A similar derivation could not take place in the RGB targets as the Cu_I transitions were too weak. A large divergence between RGB and RHB stellar abundances exists for zinc. Detection of the Zn transitions was possible in only one RHB target, which could perhaps account for some of the discrepancy.

For the entire M15 data set, Y_II exhibits lower relative abundance ratios in juxtaposition to both Sr_II and Zr_II. With regard to the three average elemental abundances (of Sr, Y, Zr), moderate departures between the RGB and RHB groups are seen. Also, a large variation in the 〈[Sr_II/Fe_II]〉 ratio was found for the members of the RGB group.

Though different sets of lines are employed, the RGB and RHB 〈[Ba_II/Fe_II]〉 ratios are consistent with one another. A portion of the RHB abundance variation is due to the exclusive use of the resonance transitions in the determination (two lowest temperature RHB stars report quite high σ values; these strong lines could not be exploited in the RGB analysis). Notably for this element group, the greatest star-to-star abundance scatter was found for lanthanum: Δ_RGB = 0.46 and Δ_RHB = 0.61 (excluding the one RHB outlier). The relative cerium abundances also exhibit a wide spread in the RGB sample.

The list of finalized 〈[El/Fe]〉 ratios for the heavy n-capture elements is as follows: Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Hf, Os, Ir, Pb, and Th. All of these element abundance ratios are enriched with regard to the solar values. As shown in Figure 6, a larger abundance spread is found for this group in comparison to the other element groups.

Note that as T_eff increases, the strength of the heavy element transitions rapidly decreases and as a consequence, the use of these lines for abundance determinations in the warmest stars becomes unfeasible. It was possible to obtain robust abundances for Nd, Sm, Tb, and Tm in a single RHB target. On the other hand, abundance extractions for the species Os and Ir were done only in RHB stars (measurements of these element transitions were attainable as less spectral crowding occurs in these stars). Nonetheless, minimal line-to-line scatter is seen for the bulk of RGB and RHB n-capture abundances.

A rigorous determination of the europium relative abundance was performed for all M15 stars: 〈[Eu_II/Fe_II]〉_RGB = 0.53 and 〈[Eu_II/Fe_II]〉_RHB = 0.88. Despite the iron abundance offset, the largest departure between the two stellar groups is found for the element Ho_II. Further, the greatest star-to-star scatter in the heavy n-capture elements is seen for the [Yb_II/Fe_II] ratio: Δ_RGB = 0.55 and Δ_RHB = 0.58.

These new abundance results are now compared to those from the four prior CTG publications. For the majority of elements, the current data are in accord with the findings of Sneden et al. (1997, 2000a) and Preston et al. (2006). In this effort, abundance derivations are performed for 13 new species: Sc_I, V_II, Cu_I, Pr_II, Tb_II, Ho_II, Er_II, Tm_II, Yb_II, Hf_II, Os_I, Ir_I, and Pb_II. For elements re-analyzed in the current study, the abundance data have been improved with the use of higher quality atomic data, additional transitions, and a revised version of the MOOG program. A few large discrepancies in the [El/Fe] ratios do occur between the current study and the previous M15 efforts. These departures can be attributed to the employment of different [Fe/H] and solar photospheric values as well as the updated MOOG code. Accordingly, the results from the current analysis supersede those from the earlier CTG papers.

As in Sneden et al. (1997), the abundance behavior of the proton-capture elements appears to be decoupled from that of the neutron-capture elements. Notably for M15, significant spread in the abundances was confirmed for both Ba and Eu. The scatter of Δlog(Ba) = 0.48 and Δlog(Eu) = 0.90 from the current effort is in line with that of Δlog(Ba) = 0.60 and Δlog(Eu) = 0.73 from Sneden et al. (1997).

Comparison of the findings from the current study to those from Otsuki et al. (2006) has also been done and will be limited to the only star that the two investigations have in common, K462. Due to differences in the 〈[Fe/H]〉 values, the log () data of the two analyses are compared. The model atmospheric parameters for K462 differ somewhat between the current effort (T_eff/log g/v_t = 4400/0.30/2.00) and Otsuki et al. (T_eff/log g/v_t = 4225/0.50/2.25). However, the agreement in the abundances for the elements Y, Zr, Ba, La, and Eu is rather good between the two studies, with the exact differences ranging: 0.01 ⩽ ∣Δ(Otsuki − Current)∣ ⩽ 0.16. The largest disparity occurs for Sr, with both analyses employing the resonance transitions. As mentioned previously, these lines are not the most rigorous probes of abundance.

Sneden et al. (1997) claimed to have found a binary distribution in a plot of [Ba/Fe] versus [Eu/Fe], with eight stars exhibiting relative Ba and Eu abundances approximately 0.35 dex smaller than the remainder of the M15 data set. To re-examine their assertion, Figure 7 is generated, which plots [(Ba, La)/H] as a function of [Eu/H] for the entire data sample of the current study. It also displays the re-derived/re-scaled Ba, La, and Eu abundances for all of the giants from the Sneden et al. (1997) publication. No decisive offset is evident in either panel of Figure 7. For completeness, the EW data from Otsuki et al. (2006) were also re-analyzed and the abundances were re-determined. Again, no bifurcation was detected in the Ba and Eu data.¹³

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To further examine the anomalous light n-capture abundances in the M15 cluster, Figures 9 and 10 are generated. For the stars from the current effort and those from Otsuki et al. (2006), these two plots display the abundances of the n-capture elements (Sr, Y, Zr, and La) as a function of the [Ba/H] and [Eu/H] ratios, respectively. Moreover, the abundance results from five select field stars, which represent extremes in r-process or s-process enhancement, are plotted (CS 22892-052: Sneden et al. 2003, 2009; CS 22964-161: Thompson et al. 2008; HD 115444: Westin et al. 2000; Sneden et al. 2009; HD 122563: Cowan et al. 2005; Lai et al. 2007; HD 221170: Ivans et al. 2006; Sneden et al. 2009).

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In Figure 9, an anti-correlative trend is seen for Sr, Y, and Zr with Ba while no explicit correlative behavior is apparent for La. The correlation coefficient, r, is indicated in each panel. Likewise, La and Eu appear un-correlated in Figure 10. The [(Sr, Y, Zr)/Eu] ratios all exhibit anti-correlation with [Eu/H] in this figure. As shown, the elements Sr, Y, and Zr clearly demonstrate an anti-correlative relationship with both the markers of the s-process (Ba) and the r-process (Eu).

Figures 9 and 10 collectively imply that the production of the light neutron-capture elements most likely did not transpire via the classical forms of the s-process or the r-process. This finding is not novel. The abundance survey of halo field stars by Travaglio et al. (2004) previously established the decoupled behavior of the light n-capture species to both Ba and Eu. Further, they postulated that an additional nucleosynthetic process was necessary for the production of these elements (Sr, Y, Zr) in metal-deficient regimes (coined the Lighter Element Primary Process, LEPP).

The overabundances of Sr and Zr (see Figure 8) could have been the result of a small s-process contribution to the M15 protocluster environment. To investigate this possibility, an abundance determination is performed for Pb, a definitive main s-process product. The upper panel of Figure 11 illustrates the synthetic spectrum fits to the neutral Pb transition at a wavelength of 4057.8 Å in the M15 giant, K462. An upper limit of log (Pb) ≲ −0.35 can only be established for this star. For the remaining two RGB targets, upper limits were also determined and, accordingly for all three, the average values of log (Pb) ≲ −0.4 and 〈[Pb/Eu]〉 ≲ −0.15 were found.

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The lower panel of Figure 11 plots [Pb/Eu] as a function of [Eu/Fe] for the three M15 RGB stars and the five previously employed halo field stars. In a recent paper, Roederer et al. (2010) suggest that detections of Pb and enhanced [Pb/Eu] ratios should be strong indicators of main s-process nucleosynthesis. In turn, they contend that non-detections of Pb and depleted [Pb/Eu] ratios should signify the absence of nucleosynthetic input from the main component of the s-process (see their paper for further discussion). With the abundances of 161 low-metallicity stars ([Fe/H] < −1), Roederer et al. empirically determined a threshold value of [Pb/Eu] = +0.3 for minimum asymptotic giant branch (AGB) contribution. As shown in the figure, the M15 giants lie below this threshold and, accordingly, are likely devoid of main s-process input. Thus, in the case of the M15 GC, the light neutron-capture elements presumably originated from an alternate nucleosynthetic process (e.g., ν − p process; Fröhlich et al. 2006; high entropy winds; Farouqi et al. 2009).

The upper panel of Figure 12 displays the evolution of the [Mg/Fe] abundance ratio with [Fe/H] for all M15 targets as well as for a sample of hundreds of field stars. For this figure, halo and disk star data have been taken from these surveys: Fulbright (2000), Reddy et al. (2003), Cayrel et al. (2004), Cohen et al. (2004), Simmerer et al. (2004), Barklem et al. (2005), Reddy et al. (2006), François et al. (2007), and Lai et al. (2008). As shown, the scatter in the [Mg/Fe] abundance ratio is fairly small: Δ([Mg/Fe])_MAX ≈ 0.6 dex for all stars under consideration and Δ([Mg/Fe])_MAX ≈ 0.1 dex for the M15 data set. In the metallicity regime below [Fe/H] ≲ −1.1, the roughly consistent trend of [Mg/Fe] abundance ratio is due in part to the production history for these elements: magnesium originates from hydrostatic burning in massive stars while iron is manufactured by massive star, core-collapse SNe. If the short evolutionary lifetimes of these massive stars are taken into context with the abundance data, it would seem to indicate that the core-collapse SNe are rather ubiquitous events in the Galactic halo. Accordingly, the products that result from both stellar and explosive nucleosynthesis of massive stars should be well mixed in the interstellar and intercluster medium. The apparent downward trend in the [Mg/Fe] ratio, in the metallicity region with [Fe/H] ≳ −1.1, is due to nucleosynthetic input from Type Ia SNe, which produce much more iron in comparison to Type II events.

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In a similar vein, the lower panel of Figure 12 plots [Eu/Fe] as a function of [Fe/H] and demonstrates that as the metallicity decreases, the spread in the [Eu/Fe] abundance ratio increases enormously.¹⁵ By contrast, the scatter in the M15 [Eu/Fe] ratios is large and comparable to the spread of the halo field at that metallicity. Specifically in the metallicity interval −2.7 ⩽ [Fe/H] ⩽ −2.2, the scatter in the [Eu/Fe] ratio is found to be σ = ±0.27 for the nine stars of the M15 sample and similarly for the 23 halo giants, the associated scatter is σ = ±0.33. This variation in the relative europium abundance ratio (as first detected by Gilroy et al. 1988 and later confirmed by others, e.g., Burris et al. 2000; Barklem et al. 2005) indicates an inhomogeneous production history for Eu and other corresponding r-process elements. These elements likely originate from lower mass SNe and their production is not correlated with that of the alpha elements (Cowan & Thielemann 2004). Furthermore, it seems that nucleosynthetic events which generated the r-process elements were rare occurrences in the early Galaxy. As a consequence, these elements were not well mixed in the interstellar medium and intercluster medium (Sneden et al. 2009). Note that r-process enhancement seems to be a common feature of all GCs (e.g., Gratton et al. 2004). On the other hand, the scatter in select r-process element abundances, as found in M15, is not.

The essential approach to the solution of radiative transfer in the MOOG program has been altered with the employment of short characteristics and the application of an accelerated lambda iteration (ALI) scheme. Original development of the short characteristics (SC) methodology in the context of radiative transfer was done by Mihalas et al. (1978). Improvement of the SC approach in the explicit specification of the source function (at all grid points) was made by Olson & Kunasz (1987) and Kunasz & Auer (1988). As a supplemental reference, the current version of MOOG draws upon the concise treatment of Auer & Paletou (1994). The implementation of the ALI technique within the framework of radiative transfer and stellar atmospheres was first done by Werner (1986) and subsequently refined by both, e.g., Rybicki & Hummer (1991) and Hubeny (1992). The main SC and ALI prescription followed by the MOOG code is that from Koesterke et al. (2002, and references therein). Since the text below is a general description and pertains to the specific coding in MOOG, the reader should consult the aforementioned references as they contain significantly more information.

The primary modification to MOOG is the creation and incorporation of four new subroutines. The names and purposes of the new subroutines are as follows: AngWeight.f, which determines the Gaussian weights and integration points; Sourcefunc_scat_cont.f, which incorporates both a scattering and an absorption component to compute the source function and resultant flux for the continuum; Sourcefunc_scat_line.f, which incorporates both a scattering and an absorption component to compute the source function and resultant flux for the line; and, Cdcalc_JS.f, which calculates the final line depth via the emergent continuum and line fluxes. To accommodate these additions, several key subroutines were also revised. Further details and the publicly available MOOG code may be found at the Web site: http://www.as.utexas.edu/~chris/moog.html. Note that MOOG still retains the capacity to operate in pure absorption mode with the source function set simply to S = B.

Ongoing and future improvements to the MOOG code include the incorporation of the Lee & Kim (2004) formulation for Rayleigh scattering (for atomic hydrogen), the employment of a further discretization with regard to frequency, and the implementation of spherically symmetric geometry in the solution of radiative transfer.

To remind the reader, in the visible spectral range, the two principal sources of opacity in stellar atmospheres are the bound-free absorption from the negative hydrogen ion (H⁻) and Rayleigh scattering from neutral atomic hydrogen. The standard expression (e.g., Gray 1976) for the H⁻_BF absorption coefficient is

$$\begin{equation} \kappa \simeq 4.1458 \times 10^{-10} \alpha _{{\rm BF}} P_{e} \Theta ^{5/2} 10^{0.754\Theta }, \end{equation} \tag{ A1 }$$

where α_BF is the bound-free atomic absorption coefficient (which has frequency dependence), P_e is the electron pressure, and Θ = 5040/T (note that Equation (A1) is per neutral hydrogen atom).

With regard to Rayleigh scattering, the scattering cross section of radiation with angular frequency ω incident upon a neutral H atom is given by the Kramers–Heisenberg formula in terms of atomic units as

$$\begin{equation} \frac{\sigma (\omega)}{\sigma _{T}}= \left(\frac{\omega }{\omega _{1}} \right)^{4} \left| A_{0} + A_{2} \left(\frac{\omega }{\omega _{1}} \right)^{2} + A_{4} \left(\frac{\omega }{\omega _{1}} \right)^{4}+ \cdots \right|^2, \end{equation} \tag{ A2 }$$

where σ_T is the Thompson scattering cross section and ω₁ is the angular frequency corresponding to the Lyman limit. Numerical calculations of the A_i coefficients and the generation of an exact expression for Equation (A2) have been done by Dalgarno & Williams (1962) and more recently by Lee & Kim (2004).

The H⁻_BF and Rayleigh scattering opacity contributions depend on temperature and metallicity (and, to some extent, on the surface gravity). Rayleigh scattering also has a λ⁻⁴ dependence, and, as a consequence, it greatly influences blue wavelength transitions. For the majority of stars (such as dwarfs and subgiants), H⁻_BF is the dominant opacity source in the visible spectral regime. However, for low-temperature, low-metallicity giants, the Rayleigh scattering contribution becomes comparable to and even exceeds that from H⁻_BF in the ultraviolet and blue visible wavelength regions. Therefore, to accurately determine the line intensity with the correct amount of flux and opacity contribution for all stellar types and over a wide spectral range, isotropic, coherent scattering must be considered.

The source function is then written as S = (1 − )J + B (where is the thermal coupling parameter, J is the mean intensity, and B is the Planck function). To commence with the formal solution of the radiation transfer equation, a fundamental assumption is made in that the source function is specified completely in terms of optical depth. After some mathematical manipulation, the RTE becomes

$$\begin{equation} \mu ^{2} \frac{d^{2}j}{d\tau ^{2}} = J - S, \end{equation} \tag{ A3 }$$

where μ is the directional cosine and τ is the optical depth. Equation (A3) is an integro-differential equation (and subject to boundary conditions). To obtain the numerical solution of Equation (A3), a discretization in angle and optical depth is necessary. As a consequence, the solution is simplified and a Gaussian quadrature summation is done instead of an integration. Though it is eventually possible to evaluate Equation (A3) in a single step, the use of an iterative method to arrive at a solution is preferred as it is computationally faster than a straightforward approach. For the MOOG program, the ALI technique is employed with the application of a full acceleration. In the context of ALI scheme, the transfer equation takes the form of J = Λ[S], where Λ represents the matrix operator. Through the concept of preconditioning, ALI allows for the efficient, iterative solution of a (potentially) large system of linear equations. The main steps of the iterative cycle are the evaluation of J = Λ[S], the computation of the ΔS quantity, and the corresponding adjustment to the source function.

From a general viewpoint, radiative transfer can be thought of as the propagation of photons along a ray on a two-dimensional grid. Note that the number of rays corresponds to the number of quadrature angles. Determination of radiation along a ray is done periodically at ray segments, or short characteristics. Essentially, SC start at a grid point and proceed along the ray until a cell boundary is met. At these cell boundaries, the intensity, opacity, and source function values are established. Then with the knowledge of the cell boundary intensities, the intensity at other non-grid points can be calculated.

Specifically with regard to MOOG, the intensity determination is a function of depths (i) and angles (j). It is performed for both an inward (i − 1) and an outward (i + 1) direction. Along the characteristic, the opacity quantity is assumed to be a linear function. In effect, the intensity for the ray can be expressed as

$$\begin{equation} I = Ie^{-\Delta \tau (i)} + \int S(\tau)e^{-\tau }d\tau. \end{equation} \tag{ A4 }$$

The optical depth step Δτ can be thought of as the path integral of the opacity along the characteristic (the entire, involved definition of the Δτ quantity is found in the Sourcefunc_scat_* subroutines). Now, the evaluation of Equation (A4) requires the interpolation of the source function. A linear interpolation is sufficient to satisfy the various boundary conditions. Interpolation over [S(i), S(i∓1)] then entails

$$\begin{equation} \int S(\tau)e^{-\tau }d\tau = S(i)w_{1}(i) + S(i \mp 1)w_{2}(i). \end{equation} \tag{ A5 }$$

The weights are given by the relations

$$\begin{equation} w_{0}(i) = (e^{-\Delta \tau (i)}-1)/\Delta \tau (i), \end{equation} \tag{ A6 }$$

$$\begin{equation} w_{1}(i) = 1 + w_{0}(i), \end{equation} \tag{ A7 }$$

$$\begin{equation} w_{2}(i) = -e^{-\Delta \tau (i)}-w_{0}(i). \end{equation} \tag{ A8 }$$

These weights are found by recursion. The use of linear interpolation does not generate significant error (as normally would occur) due to the optically thin nature of the boundary layer. The expression for the mean intensity, J, subsequently becomes

$$\begin{equation} J(i) = J(i) + 0.5w_{\rm Gau}(j)I, \end{equation} \tag{ A9 }$$

where the w_Gau are the Gaussian quadrature weights (these are distinct from the weights of Equation (A5)). The summation over all depth points and angles(/rays) is necessarily performed. The SC formal solution of the transfer equation then proceeds in an iterative manner.