r-process Abundance Patterns in the Globular Cluster M92

Nearly all of the data from this project come from the Keck Observatory Archive (KOA). We queried the archive using its web interface on 2020 February 5. We searched for all publicly available HIRES spectra within 30' of the center of M92. We also queried the archive again on 2023 July 18 to confirm that no additional spectra were added since our original query. The KOA provides both raw and extracted HIRES spectra. We used the extracted (1D, wavelength-calibrated, and sky-subtracted) spectra provided in FITS format on the archive.

We paired each spectrum with photometry from Gaia Data Release 3 (Gaia Collaboration et al. 2023) and the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). The coordinates given in the metadata from the KOA are not always precise enough to give an unambiguous match to a star in the catalog (G, BP, and RP magnitudes) and 2MASS catalog (K_s magnitude). In some cases, we looked up the star in the SIMBAD astronomical database (Wenger et al. 2000) by the identifier assigned by the HIRES observer. We used the coordinates of the star listed in SIMBAD to match to Gaia and 2MASS. Gaia measured a significantly different proper motion for one star in the HIRES archive (S1521) than the other M92 stars. We excluded S1521 from further analysis because we assumed that it is not a cluster member, and we verified that the other stars in our sample have proper motions consistent with cluster membership. We also excluded S4375 because it is faint (G = 18.80).

Table 1 lists the coordinates of the stars along with photometry. The last column of Table 1 gives the ${(\mathrm{BP}-{K}_{s})}_{0}$ color, where available. The magnitudes and photometry in the table were corrected for extinction and reddening assuming E(B − V) = 0.0191 (Schlafly & Finkbeiner 2011). The Gaia magnitudes were corrected following the color–extinction relations provided by Gaia Collaboration et al. (2018). The K_s magnitudes were corrected assuming A_K /E(B − V) = 0.310 (Fitzpatrick 1999).

Table 1. Coordinates and Photometry

Star	Gaia Source ID	R.A. (J2000)	Decl. (J2000)	G0	(BP − RP)0	(BP − K)0	vr (km s−1)
III-13	1360410343884596096	17h17m2171	$+43^\circ 12^{\prime} 53\buildrel{\prime\prime}\over{.} 4$	11.56	1.56	3.39 a	−110.652
VII-18	1360384647101112832	17h16m3748	$+43^\circ 06^{\prime} 15\buildrel{\prime\prime}\over{.} 6$	11.68	1.50	3.25 a	−118.552
X-49	1360404399652677760	17h17m1280	$+43^\circ 05^{\prime} 41\buildrel{\prime\prime}\over{.} 8$	11.77	1.47	3.22 a	−132.417
III-65	1360405877121517568	17h17m1412	$+43^\circ 10^{\prime} 46\buildrel{\prime\prime}\over{.} 2$	12.01	1.41	3.09 a	−118.964
VII-122	1360405262943998208	17h16m5737	$+43^\circ 07^{\prime} 23\buildrel{\prime\prime}\over{.} 7$	12.02	1.41	3.09 a	−125.579
II-53	1360405808402036608	17h17m1306	$+43^\circ 09^{\prime} 48\buildrel{\prime\prime}\over{.} 3$	12.04	1.40	3.08 a	−124.641
XII-8 b	1360218208524710272	17h17m3171	$+43^\circ 05^{\prime} 41\buildrel{\prime\prime}\over{.} 4$	12.36	1.31	2.90 a	−119.156
V-45	1360408522818465536	17h16m4985	$+43^\circ 10^{\prime} 41\buildrel{\prime\prime}\over{.} 2$	12.42	1.31	2.91 a	−121.735
XI-19	1360216765415718528	17h17m1874	$+43^\circ 04^{\prime} 50\buildrel{\prime\prime}\over{.} 9$	12.48	1.30	2.88 a	−117.273
XI-80	1360404605811200256	17h17m1467	$+43^\circ 06^{\prime} 24\buildrel{\prime\prime}\over{.} 7$	12.58	1.28	2.86 a	−125.783
II-70 b	1360405812699772416	17h17m1653	$+43^\circ 10^{\prime} 44\buildrel{\prime\prime}\over{.} 9$	12.70	1.25	2.77 a	−111.765
IV-94 b	1360408694620020480	17h17m0587	$+43^\circ 10^{\prime} 17\buildrel{\prime\prime}\over{.} 2$	12.71	1.24	2.75 a	−119.801
I-67	1360404811969647616	17h17m2123	$+43^\circ 08^{\prime} 27\buildrel{\prime\prime}\over{.} 0$	12.96	1.21	2.72 a	−121.490
III-82	1360408728979771776	17h17m0806	$+43^\circ 10^{\prime} 44\buildrel{\prime\prime}\over{.} 9$	12.96	1.22	2.75 a	−121.772
IV-10	1360409695350418176	17h16m5772	$+43^\circ 14^{\prime} 11\buildrel{\prime\prime}\over{.} 4$	13.06	1.21	2.73 a	−118.071
XII-34 b	1360404747547336448	17h17m2157	$+43^\circ 07^{\prime} 40\buildrel{\prime\prime}\over{.} 8$	13.08	1.15	2.58 a	−115.400
IV-79	1360408385382348672	17h17m0080	$+43^\circ 10^{\prime} 25\buildrel{\prime\prime}\over{.} 1$	13.11	1.20	2.70 a	−121.686
IX-13	1360404163432211456	17h16m5612	$+43^\circ 04^{\prime} 07\buildrel{\prime\prime}\over{.} 3$	13.66	1.11	2.51 a	−126.466
VIII-24 b	1360404953706355200	17h16m5034	$+43^\circ 05^{\prime} 53\buildrel{\prime\prime}\over{.} 1$	13.79	1.03	2.31 a	−119.054
X-20	1360216662336507648	17h17m1333	$+43^\circ 04^{\prime} 13\buildrel{\prime\prime}\over{.} 4$	15.30	0.95	2.17 a	−118.018
S2710	1360404330933201536	17h17m1571	$+43^\circ 05^{\prime} 32\buildrel{\prime\prime}\over{.} 4$	15.32	0.95	2.16 a	−120.568
VI-90	1360408355320576128	17h16m5421	$+43^\circ 09^{\prime} 21\buildrel{\prime\prime}\over{.} 1$	15.42	0.95	2.15 a	−126.224
S2265	1360216593617025152	17h17m1361	$+43^\circ 03^{\prime} 35\buildrel{\prime\prime}\over{.} 7$	15.79	0.93	2.09 a	−120.836
VIII-45	1360405159864786304	17h16m5332	$+43^\circ 06^{\prime} 34\buildrel{\prime\prime}\over{.} 5$	15.86	0.92	2.12 a	−123.652
VII-28	1360384750180330496	17h16m4079	$+43^\circ 07^{\prime} 07\buildrel{\prime\prime}\over{.} 6$	15.92	0.91 a	2.17	−121.582
G17181_0638	1360404507029158656	17h17m1813	$+43^\circ 06^{\prime} 38\buildrel{\prime\prime}\over{.} 1$	16.32	0.89 a	2.08	−125.134
C17333_0832	1360406053217363456	17h17m3332	$+43^\circ 08^{\prime} 32\buildrel{\prime\prime}\over{.} 8$	16.81	0.87 a	2.05	−120.012
S3108	1360405808402045184	17h17m1524	$+43^\circ 09^{\prime} 53\buildrel{\prime\prime}\over{.} 9$	16.83	0.85 a	2.00	−118.782
S652	1360404232151697152	17h16m5789	$+43^\circ 04^{\prime} 40\buildrel{\prime\prime}\over{.} 4$	16.85	0.89 a	2.00	−127.370
S19	1360384750180328704	17h16m4126	$+43^\circ 07^{\prime} 08\buildrel{\prime\prime}\over{.} 5$	17.66	0.75 a	⋯	−130.271
D21	1360216215659937280	17h16m5878	$+43^\circ 02^{\prime} 04\buildrel{\prime\prime}\over{.} 1$	17.71	0.71 a	2.60	−127.547
S3880	1360406220716655616	17h17m2568	$+43^\circ 08^{\prime} 16\buildrel{\prime\prime}\over{.} 1$	17.79	0.71 a	⋯	−115.078
S4038	1360406255076061440	17h17m3299	$+43^\circ 09^{\prime} 21\buildrel{\prime\prime}\over{.} 1$	17.80	0.68 a	⋯	−123.615
S61	1360408144861994368	17h16m4528	$+43^\circ 07^{\prime} 00\buildrel{\prime\prime}\over{.} 8$	18.11	0.62 a	⋯	−124.981
S162	1360408179227211776	17h16m4811	$+43^\circ 07^{\prime} 35\buildrel{\prime\prime}\over{.} 6$	18.13	0.57 a	⋯	−123.328

Notes.

^a This is the color used for calculating T_eff. ^b AGB star.

A machine-readable version of the table is available.

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Table 2 gives the observing log organized by star. Most stars were observed on multiple dates. The name of the principal investigator (PI) and exposure times for each exposure are given for each date. The first row for each star includes the total exposure time and the S/N per pixel. The S/N is calculated from the pixels between 4650 and 4800 Å in the continuum-divided spectrum. We sigma clipped this spectral range by excluding pixels more than 3.0 standard deviations above the continuum or 0.5 standard deviations below the continuum. The asymmetry of the sigma clipping is intended to exclude strong absorption lines from the measurement of the S/N. Finally, the S/N is calculated as the inverse of the median absolute deviation of the spectrum from its mean.

Figure 1 shows the Gaia CMD for M92. The figure shows the stars with HIRES spectra in rainbow colors. The colors correspond to the effective temperatures (Section 3). The symbol shape corresponds to the evolutionary state. We determined the evolutionary state by close inspection of the CMD. A sufficiently zoomed-in CMD shows a clear delineation of the AGB from the RGB. (This distinction is not readily apparent in the zoomed-out CMD shown in Figure 1.) Our sample contains 35 stars: 24 RGB stars, 5 AGB stars, and 6 subgiants. While we report the abundances of the subgiants (Section 4.3), they do not play a role in our discussion (Section 6) because their abundances have larger uncertainties. In the tables and figure legends in this paper, AGB stars are indicated with asterisks next to their names. The small points in Figure 1 show stars within 7' of the cluster center and with proper motions within 1 mas yr^–1 of the median proper motion. The right axis of the figure gives absolute magnitudes assuming an apparent distance modulus (m − M)_V = 14.74 (VandenBerg et al. 2016), which we corrected for extinction to (m − M)₀ = 14.69.

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One star, X-20, drew our attention for what appeared to be an unusually large potassium abundance (see Section 5.3). We asked members of the California Planet Survey (Howard et al. 2010) to observe X-20 in early 2021. A. Howard and H. Isaacson observed the star on 2021 March 26. They reduced the spectrum following the procedure described by Howard et al. (2010)

The KOA provides extracted spectra for each echelle order of each exposure. The spectra are in the heliocentric reference frame. Most stars had multiple exposures. In order to make the spectra suitable for measuring equivalent widths (EWs; Section 4.2), we combined all of the echelle orders and all of the exposures for each star into a single 1D spectrum. Stacking all of the component spectra required that we rebin them onto a common wavelength array. We used a logarithmically spaced wavelength array that encompassed the minimum and maximum wavelengths of all of the individual exposures. The bin size was chosen to be as fine as the finest wavelength bin in all of the individual orders. This ensured that we did not lose any information during the resampling.

2.3.1. Continuum Normalization

2.3.2. Spectral Coaddition

2.3.3. Radial Velocity Determination

Stellar parameters can be derived from photometry or spectroscopy. For example, T_eff can be measured from the photometric color of a star or by balancing the spectroscopic excitation equilibrium of one or more atomic species. The spectroscopic approach can be inaccurate when assuming LTE, especially for red giants (e.g., Frebel et al. 2013), which comprise the majority of our sample.

Photometric colors can be tied to (semi)direct measurements of temperature, which are not subject to non-LTE effects. The infrared flux method (IRFM) is a semidirect method of determining stellar temperatures. It uses the infrared flux and a model atmosphere to infer the star's angular diameter. The flux and diameter together give T_eff. González Hernández & Bonifacio (2009) provided empirical calibrations between 2MASS colors and temperatures derived from the IRFM for several hundred stars. Mucciarelli et al. (2021) extended these calibrations to Gaia colors and combinations of Gaia and 2MASS. The ${(\mathrm{BP}-{K}_{s})}_{0}$ color has the highest precision (${\sigma }_{{T}_{\mathrm{eff}}}=52$ K) because it has the longest dynamic range of the different filter combinations. Therefore, we adopted the ${(\mathrm{BP}-{K}_{s})}_{0}$ color temperature for stars brighter than about G₀ = 16. For fainter stars, the uncertainty in the K_s magnitude becomes the dominant error term in the temperature. Therefore, for fainter stars, we used the (BP − RP)₀ color temperature (${\sigma }_{{T}_{\mathrm{eff}}}=83$ K). We used Mucciarelli et al. (2021)'s color–temperature relation for dwarfs for stars marked "subgiant" in Figure 1 and the relation for giants otherwise. The relations require an assumption of metallicity, for which we used [M/H] = −2.41, which is approximately the average [Fe ii/H] of M92 that we eventually derived. The footnotes in Table 1 indicate which color was used to derive the T_eff for each star.

We estimated the uncertainty on T_eff by applying standard (uncorrelated) error propagation to the color–temperature polynomial. The relations also have an intrinsic scatter, which we added in quadrature to the random error.

Like temperature, the surface gravity can also be determined photometrically or spectroscopically. The spectroscopic method typically attempts to equalize the abundance derived from neutral and ionized absorption lines of the same element. However, overionization affects neutral absorption lines in an LTE analysis (Thévenin & Idiart 1999).

For stars of known distance, it is generally preferable to use photometric estimates of surface gravity. The surface gravity can be estimated from the Stefan–Boltzmann law in combination with the definition of surface gravity:

$$\begin{eqnarray}&&g=\displaystyle \frac{4\pi {GM}{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^{4}}{L}.\end{eqnarray} \tag{ 1 }$$

We assumed 0.75 M_☉, which is appropriate for ancient, metal-poor stars that have evolved past the main-sequence turnoff. We calculated the luminosity from the extinction-corrected Gaia G₀ magnitude, the distance modulus (m − M)₀ = 14.69 (see Section 2.1), and the bolometric corrections from Andrae et al. (2018).

We estimated the uncertainty on $\mathrm{log}g$ using standard propagation of error. We assumed that the uncertainty on the stellar mass was 0.1 M_☉. We further assumed that the errors on M, T_eff, and G₀ are uncorrelated.

Table 3 gives the model atmosphere parameters for each model: T_eff, $\mathrm{log}g$, and v_t (see Section 4.3). The table does not show [M/H] or [α/Fe] because these values are the same for every star.

Table 3. Model Atmosphere Parameters

Star	Teff (K)	$\mathrm{log}g$ (cm s−1)	vt (km s−1)
III-13	4187 ± 51	0.48 ± 0.10	2.95 ± 0.21
VII-18	4275 ± 50	0.59 ± 0.10	2.96 ± 0.15
X-49	4301 ± 50	0.64 ± 0.10	2.58 ± 0.11
III-65	4392 ± 51	0.79 ± 0.10	2.56 ± 0.10
VII-122	4388 ± 51	0.79 ± 0.10	2.68 ± 0.10
II-53	4399 ± 51	0.81 ± 0.10	2.67 ± 0.10
XII-8*	4530 ± 51	1.01 ± 0.10	2.85 ± 0.17
V-45	4528 ± 52	1.03 ± 0.10	2.42 ± 0.06
XI-19	4546 ± 51	1.07 ± 0.10	2.54 ± 0.07
XI-80	4562 ± 51	1.12 ± 0.10	2.52 ± 0.10
II-70*	4636 ± 52	1.20 ± 0.10	2.47 ± 0.09
IV-94*	4652 ± 52	1.22 ± 0.10	2.62 ± 0.10
I-67	4675 ± 51	1.33 ± 0.10	2.37 ± 0.09
III-82	4648 ± 52	1.31 ± 0.10	2.35 ± 0.08
IV-10	4673 ± 51	1.37 ± 0.10	2.38 ± 0.08
XII-34*	4793 ± 52	1.43 ± 0.10	2.57 ± 0.09
IV-79	4690 ± 52	1.39 ± 0.10	2.30 ± 0.08
IX-13	4854 ± 53	1.70 ± 0.10	2.23 ± 0.09
VIII-24*	5043 ± 55	1.83 ± 0.10	2.34 ± 0.12
X-20	5185 ± 63	2.50 ± 0.10	1.87 ± 0.09
S2710	5191 ± 60	2.51 ± 0.10	1.88 ± 0.08
VI-90	5206 ± 65	2.55 ± 0.10	1.90 ± 0.11
S2265	5269 ± 73	2.73 ± 0.10	1.78 ± 0.09
VIII-45	5239 ± 72	2.74 ± 0.10	1.85 ± 0.10
VII-28	5276 ± 83	2.78 ± 0.10	1.78 ± 0.09
G17181_0638	5328 ± 84	2.96 ± 0.10	2.01 ± 0.22
C17333_0832	5373 ± 83	3.17 ± 0.10	1.67 ± 0.10
S3108	5413 ± 93	3.20 ± 0.10	1.89 ± 0.11
S652	5322 ± 83	3.17 ± 0.10	1.84 ± 0.18
S19	5793 ± 66	3.66 ± 0.10	1.81 ± 0.14
D21	5954 ± 74	3.73 ± 0.10	1.51 ± 0.12
S3880	5944 ± 77	3.76 ± 0.10	1.63 ± 0.19
S4038	6069 ± 85	3.80 ± 0.10	1.86 ± 0.33
S61	6308 ± 92	4.00 ± 0.10	1.25 ± 0.39
S162	6573 ± 98	4.08 ± 0.10	2.00 ± 0.37

A machine-readable version of the table is available.

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We used the same line list as Ji et al. (2020). The list is based mostly on Linemake (Placco et al. 2021a, 2021b) with updates as described by Ji et al. (2020). Some lines have hyperfine structure, which we also take from Linemake.

We developed a graphical utility called hiresspec to analyze each spectrum. The software fits Gaussian profiles to each absorption line in the line list using MPFIT (Markwardt 2012). The user examines each fit and has the option to change the fit by altering the wavelength bounds considered by MPFIT or the placement of the continuum. We gave special treatment to the two Mg b lines in the line list (5173 and 5184 Å). Because they often display obvious damping wings, we fit them with Voigt rather than Gaussian profiles. The EW for each line was calculated analytically from the amplitude and Doppler width (and Lorentzian width for the Voigt profile) determined in the fit. MPFIT also returns the 1σ uncertainty on the EW from the diagonal elements of the covariance matrix.

We computed upper limits on the EW for undetected lines. The upper limit was the strength of the Gaussian that was stronger than the observed spectrum by 3σ.

Table 4 gives the atomic data (wavelength, excitation potential (EP), and oscillator strength) for each line. It also gives each line's EW measured for each star. Only ten absorption lines and two stars are shown in the manuscript. The online version of the table includes all of the absorption lines and all of the stars.

Table 4. Line List with EWs and Abundances

				III-13				VII-18
Species	Wavelength	EP	$\mathrm{log}{gf}$	EW	Abundance	Weight		EW	Abundance	Weight
	(Å)	(eV)		(mÅ)				(mÅ)
O i	6363.78	0.020	−10.190	⋯	⋯	⋯		<1.5	<6.15	⋯
Na i	5682.63	2.100	−0.710	24.2	4.25	75.25		18.9	4.19	38.05
Na i	5688.20	2.100	−0.410	44.1	4.27	70.81		31.4	4.14	38.06
Na i	5889.95	0.000	0.110	⋯	⋯	⋯		408.0	4.48	19.66
Na i	5895.92	0.000	−0.190	⋯	⋯	⋯		341.9	4.33	7.61
Mg i	3986.75	4.350	−1.060	65.2	5.38	8.84		⋯	⋯	⋯
Mg i	4167.27	4.350	−0.740	120.5	6.05	7.07		74.8	5.39	20.69
Mg i	4571.10	0.000	−5.620	166.7	5.29	1.50		134.4	5.24	1.53
Mg i	4702.99	4.350	−0.440	150.6	5.77	6.85		91.3	5.06	20.19
Mg i	5172.68	2.710	−0.390	347.0	5.36	6.56		280.0	5.06	10.40

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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4.2.1. Potassium

We measured potassium from the resonance line K i λ7699. This region of the spectrum lies in the red tail of the telluric A band. The most prominent telluric feature that affects the potassium line is an electronic transition of O₂ at a vacuum wavelength of 7697.96 Å. This line arises from the PQ branch in the 0–0 transition of the $b\,{}^{1}{{\rm{\Sigma }}}_{g}^{+}\to X\,{}^{3}{{\rm{\Sigma }}}_{g}^{-}$ series of O₂ in rotational level K = 31 (Babcock & Herzberg 1948). Fortuitously, the line is always accompanied by the P branch of the same transition at 7698.98 Å, a wavelength free of stellar features. The strength ratio of the lines is a constant because they arise from the same electronic energy level, and the lines are weak (typically 30 mÅ) and therefore on the linear portion of the curve of growth. The strength ratio is the ratio of their oscillator strengths (gf). We computed the oscillator strengths from their Einstein coefficients and level degeneracies given by the HITRAN molecular database (Yu et al. 2014; Gordon et al. 2022): gf(7697.96)/gf(7698.99) = 0.959.

We decontaminated the K i line by subtracting the bluer of the O₂ lines. First, we modeled the redder O₂ line, which does not overlap with stellar features, as a Gaussian. The Gaussian width is generally narrower than the width of the K i line because the broadening effects in the Earth's atmosphere are less than in the star. We fit a Gaussian centered at the observed (geocentric) wavelength, but we allowed the central wavelength, strength, and FWHM to vary. Then, we created a model of the bluer O₂ line by shifting the Gaussian in wavelength and multiplying the strength by 0.959. We subtracted both Gaussian models from the observed spectrum.

Figure 2 shows three examples of the potassium spectral region. The dotted spectra are shown before O₂ subtraction, and the solid spectra are subtracted. We discuss the potassium abundances further in Section 5.3.

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We used ATLAS9 (Kurucz 1993) model atmospheres in conjunction with MOOG (Sneden 1973) to compute an abundance for each EW. We computed the model atmosphere by interpolating between the grid points of Kirby (2011)'s grid of ATLAS9 models. We used T_eff and $\mathrm{log}g$ as determined in Sections 3.1 and 3.2. We started with an initial assumption of v_t based on a linear relation with $\mathrm{log}g$ (Kirby et al. 2009). We assumed [M/H] = −2.41 and [α/Fe] = +0.41. These values are consistent with past abundance estimates for M92 (Sneden et al. 1991) as well as our estimates. Because M92 has been shown repeatedly to have no metallicity variation (Sneden et al. 1991; Shetrone 1996; Cohen & McCarthy 1997; Sneden et al. 2000b; Shetrone et al. 2001; Cohen 2011), we kept [M/H] fixed at −2.41. ⁸ Some of the α elements, especially Mg, are known to have large variations, but we also kept [α/Fe] fixed to keep the analysis as consistent as possible between stars.

We calculated the abundance of each absorption line with the 2021 update to MOOG. ⁹ This version includes the Sobeck et al. (2011) update, which separates the source function from the Planck function to allow Rayleigh scattering to be a significant source of opacity, which is important at λ < 4500 Å in metal-poor giants, such as the majority of our M92 sample. Table 4 includes the abundance for each line. The abundances are corrected for non-LTE effects (Section 4.5) where appropriate.

We also computed abundances from upper limits on EWs. For atomic species with no detected lines and more than one upper limit, we took the abundance from the absorption line with the most stringent abundance limit. For the upper limits, Table 4 includes only the upper limit used in computing the limit on the abundance.

Some absorption lines display isotopic splitting, which we took from Linemake. For the transition metals, we used the solar system isotope distribution (Asplund et al. 2009). For neutron-capture elements, we used the solar r-process isotope distribution (Simmerer et al. 2004; Sneden et al. 2008).

We refined the initial guess at v_t by minimizing the trend of the abundance of Fe i lines with the reduced width (RW = EW/λ). We considered only lines with logRW < −4.5 because strong lines tend to have damping wings, which may not be well represented by our Gaussian fits. Furthermore, very strong lines (like very weak lines) are not very sensitive to v_t and are therefore not very useful at determining v_t . We fit a line with least-squares regression to the abundance versus logRW. We used MPFIT to minimize the slope of this line by varying v_t . Table 3 gives the final values of v_t for each star.

We also propagated the uncertainty on the EW (described in Section 4.2) to the uncertainty on the abundance of each line. We ran MOOG twice: once with the EWs perturbed upward by the EW uncertainty and another time with the EWs perturbed downward. We took the propagated error on the abundance (e_i ) on absorption line i as the average of the absolute values of the changes in the abundance from the upward and downward perturbations.

Table 5 gives the resulting value of [Fe/H] measured from Fe i lines with logRW < −4.5. The table also gives the slope of the abundances measured from Fe i lines versus logRW. This slope is a diagnosis of microturbulence, as discussed above. The slopes are essentially zero because v_t was tuned to minimize these slopes. The third column of the table gives the slope of the abundance versus EP (in units of eV). This slope is a diagnosis of T_eff. Positive (negative) slopes indicate that the value of T_eff is too low (high). Some deviation from zero is expected because of the deficiencies in the assumption of LTE. The last column of the table is the difference in the mean iron abundance between neutral and singly ionized Fe lines. This difference is a diagnosis of $\mathrm{log}g$. Positive (negative) differences indicate that $\mathrm{log}g$ is too low (high). As with T_eff, some deviation from zero is expected in an LTE analysis, especially due to the non-LTE overionization effect in cool giants (Thévenin & Idiart 1999).

Table 5. Abundance Trends

Star	(Fe i/H)	d /dlogRW	d /d(EP)	(Fe i) − (Fe ii)
III-13	−2.60	+0.00	−0.02	−0.11
VII-18	−2.65	−0.00	+0.01	−0.22
X-49	−2.56	+0.00	+0.00	−0.10
III-65	−2.55	+0.00	−0.03	−0.10
VII-122	−2.60	−0.00	+0.00	−0.12
II-53	−2.59	+0.00	−0.02	−0.10
XII-8*	−2.71	−0.00	−0.05	−0.14
V-45	−2.61	+0.01	−0.05	−0.07
XI-19	−2.49	−0.00	−0.04	−0.04
XI-80	−2.52	−0.00	−0.03	−0.05
II-70*	−2.58	−0.00	−0.04	−0.01
IV-94*	−2.62	+0.00	−0.03	−0.06
I-67	−2.65	+0.00	−0.04	−0.05
III-82	−2.53	+0.00	−0.04	−0.07
IV-10	−2.44	+0.00	−0.03	−0.00
XII-34*	−2.52	−0.00	−0.05	−0.04
IV-79	−2.54	+0.00	−0.04	−0.04
IX-13	−2.41	+0.00	−0.04	+0.01
VIII-24*	−2.47	−0.00	−0.05	−0.03
X-20	−2.36	−0.00	−0.05	+0.06
S2710	−2.36	+0.00	−0.05	+0.05
VI-90	−2.43	+0.00	−0.05	+0.01
S2265	−2.42	+0.00	−0.06	+0.01
VIII-45	−2.49	−0.00	−0.04	−0.03
VII-28	−2.38	+0.00	−0.05	−0.01
G17181_0638	−2.50	+0.00	−0.05	+0.02
C17333_0832	−2.41	+0.00	−0.04	−0.03
S3108	−2.47	−0.00	−0.05	−0.00
S652	−2.46	+0.00	−0.04	−0.08
S19	−2.43	+0.00	−0.06	−0.01
D21	−2.29	−0.00	−0.07	+0.03
S3880	−2.46	+0.00	−0.02	−0.13
S4038	−2.45	+0.00	−0.00	−0.03
S61	−2.60	−0.00	−0.05	−0.11
S162	−2.33	−0.00	−0.07	+0.07

A machine-readable version of the table is available.

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Some elements are better suited to abundance measurements by spectral synthesis than by EWs. This is especially true for measurements from molecular features (like CH), intrinsically broad atomic features (like Li i λ6707), and features contaminated by blends. We computed the abundances of Li and C by spectral synthesis.

We used Linemake to create line lists for spectral synthesis. For Li, the wavelength range was 6704.7–6710.9 Å, and the Li was purely ⁷Li (no ⁶Li). For C, the wavelength range was 4273.9–4333.0 Å. The ¹²C/¹³C ratio changes with evolution along the RGB. We used $\mathrm{log}g$ as a proxy for evolution, and we used the following prescription (Keller et al. 2001; Kirby et al. 2015) for the isotope ratio:

$$\begin{eqnarray}\begin{array}{cc}{}^{12}{\rm{C}}{/}^{13}{\rm{C}}=50 & \,{\rm{if}}\,{\rm{log}}g\gt 2.7,\\ {}^{12}{\rm{C}}{/}^{13}{\rm{C}}=63\,{\rm{log}}g-120 & \,{\rm{if}}\,2.0\lt {\rm{log}}g\leqslant 2.7,\\ {}^{12}{\rm{C}}{/}^{13}{\rm{C}}=6 & \,{\rm{if}}\,{\rm{log}}g\leqslant \mathrm{2.0.}\end{array}\end{eqnarray} \tag{ 2 }$$

We used MPFIT to find the best-fitting abundance. We then calculated a continuum correction. For Li, the continuum correction was the median of the residual (the observed spectrum divided by the best-fit spectrum). For the CH feature, which spans a longer wavelength range, the continuum correction was a spline with a breakpoint spacing of 10 Å and sigma clipping at ±2σ. We divided the observed spectrum by the continuum correction and then remeasured the abundance. We iterated the continuum correction until the abundance changed by less than 0.001 between iterations.

Figure 3 shows the syntheses for the CH feature for stars of various effective temperatures.

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We considered non-LTE corrections for most lines in the line list. These corrections are so large for some elements, like Al and K, that the corrections are required for any meaningful interpretation of the abundances. The corrections for some elements, like Mg, are moderate. Finally, the corrections for some elements, like O (when measured from the forbidden lines), are negligible.

We measured Li i abundances from the marginally resolved doublet at 6708 Å. The line shape and a blend with a nearby Fe i line required that we synthesize the absorption line rather than fit a simple Gaussian (see Section 4.4). Lind et al. (2009) computed non-LTE corrections to this feature, and we applied them to the abundance computed from the synthesis.

We considered only the [O i] λ λ6300,6364 forbidden doublet. The non-LTE corrections are negligibly small for both lines (Amarsi et al. 2015; Bergemann et al. 2021). The main uncertainty arises from the oscillator strengths (e.g., Storey & Zeippen 2000).

We measured Na i abundances from the Na D resonance doublet and another doublet at 5683 and 5688 Å. We applied the Lind et al. (2011) corrections to each of these lines.

For Mg i, we applied the corrections computed by Bergemann et al. (2017). We used the online interface provided by the MPIA database of non-LTE corrections. ¹⁰ The interface interpolates non-LTE corrections from a precomputed grid based on stellar parameters. We chose the corrections computed with the 1D MARCS spherical model atmospheres (Gustafsson et al. 2008). The correction grids for any other options for model atmospheres did not include the full range of stellar parameters spanned by our sample. The typical correction was +0.2 dex, in the sense that the non-LTE abundance is higher than the LTE abundance.

Nordlander & Lind (2017) computed 1D non-LTE corrections to Al i lines, including both lines used in our study. We read the corrections for Al i λ3962 from their Figure 13 based on each star's T_eff and $\mathrm{log}g$. They did not provide a similar plot for Al i λ3944, but the corrections for both lines are similar. Therefore, we applied the corrections for the redder line to the bluer line. Corrections ranged from +0.5 dex for the base of the RGB to +1.1 dex for the coolest giants.

The MPIA database showed that corrections for Si i (Bergemann et al. 2013) are negligible. Furthermore, the corrections to Ca i are very small (Mashonkina et al. 2007). We applied no corrections to these elements.

We used the K i λ7699 corrections of Reggiani et al. (2019). We read the corrections appropriate for each star's T_eff and $\mathrm{log}g$ from their Figure 12. Although the non-LTE correction for this line is very large in warm, metal-rich giants, the corrections at low metallicity are typically −0.2 dex.

Non-LTE corrections for the transition metals are complicated. For example, Bergemann (2011) gave Ti i corrections in the range +0.4 to +1.0 dex. Ti ii corrections are typically less than +0.1 dex, as would be expected for ionized lines in red giants. As a trial, we applied these corrections to the Ti lines in our study. For stars at the tip of the RGB, the difference between the Ti i and Ti ii abundances decreased, and the slope of the Ti i abundances with EP approached zero. However, both of these diagnostics worsened after applying the non-LTE correction for the majority of the stars in our sample, with the difference between neutral and ionized abundances as large as +0.7 dex. Similarly, we applied the Bergemann et al. (2012b) corrections to the Fe i and Fe ii lines. The corrections are typically +0.1 dex, much smaller than for Ti. The corrections did not improve the slope of Fe i abundances with EP, and they exacerbated the differences between the Fe i and Fe ii abundances. Therefore, we chose not to apply non-LTE corrections to transition metals. Importantly, our line list excludes commonly used Mn i resonance lines because they are particularly affected by non-LTE corrections (Bergemann et al. 2019). All of the Mn transitions used here have EPs of at least 2 eV.

We also considered non-LTE corrections for neutron-capture elements. For example, Bergemann et al. (2012a) computed corrections for Sr i and Sr ii, and Korotin et al. (2015) computed corrections for Ba ii. The typical corrections for ionized lines in our study are less than 0.1 dex. Therefore, we applied no corrections to ionized lines of neutron-capture elements. However, Sr i λ4607 requires a correction of about 0.4 dex. However, Bergemann et al. (2012a) did not compute corrections for $\mathrm{log}g\lt 2.2$. As a result, we omitted Sr i from our study. All of the Sr abundances reported here are based on Sr ii.

The abundances of some elements display a trend with stellar luminosity (or surface gravity) even after the application of non-LTE corrections. Figure 4 shows these trends. Some elements, especially carbon, are expected to show a trend with luminosity. We discuss the carbon trend with luminosity in Section 5.1.

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Trends with luminosity for heavier elements could result from atomic diffusion, including gravitational settling and radiative acceleration (or levitation). The effect could be strong for very metal-poor GCs, like M92 (Richard et al. 2002). However, it is expected to be strongest at the main-sequence turnoff.

Giant stars have deep convective envelopes that remix the abundances to levels close to their initial surface values. Observations of metal-poor GCs show only small differences in the abundances of heavy metals, like Fe, between the main-sequence turnoff and the giant branch (Cohen & Meléndez 2005; Korn et al. 2007; Lind et al. 2008). As a result, we do not expect to see any trends in abundance for elements heavier than oxygen for the giants in our sample.

Nonetheless, we do observe abundance trends with luminosity. We interpret these trends as artifacts of our measurements. For example, they could be uncorrected non-LTE effects. The main goal of our study is to quantify star-to-star abundance variations. Therefore, we take the conservative approach of removing the trends with luminosity. We do not fit the luminosity trends of elements that are known to vary in GCs: Li, C, O, Na, Mg, Al, and K. For other elements, we fit the abundance of each element as a function of luminosity. For most elements, the function is a quadratic. For elements measured over a limited range of luminosity (Ce, Nd, and Sm), the fit is linear. The luminosity correction is a subtraction of this line or quadratic, normalized such that the correction is zero at M_G,0 = 0. We used LEO-Py (Feldmann 2019) to perform a "censored" fit, which takes into account upper limits. Figure 4 shows the fitted luminosity trends. We applied the luminosity correction only to elements with red trend lines in the figure. Elements not shown in the figure were not corrected.

Error analysis in high-resolution spectroscopy can sometimes be simplistic or arbitrary. For species with multiple absorption lines, one common approach is to report the standard deviation divided by the square root of the number of lines. For species with just one or a few lines, sometimes arbitrary uncertainties—such as 0.1 or 0.2 dex—are reported. Some of this guesswork is unavoidable because the uncertainties on abundances inherit difficult-to-quantify uncertainties on predecessor variables, such as oscillator strengths, deficiencies in the model atmosphere, or non-LTE corrections.

McWilliam et al. (1995, 2013) improved on the standard error analysis by accounting for uncertainties in atmospheric parameters, including covariance from correlated parameters, like T_eff and $\mathrm{log}g$. Accounting for covariance lessens the severity of the spurious correlations between abundance ratios that could be introduced by errors in atmospheric parameters (Roederer & Thompson 2015). Ji et al. (2020) extended this framework to the abundance measurements themselves (the mean abundance from all the lines of a single species) rather than merely their uncertainties. Ji et al. (2020) documented their approach in their Appendix B, which we followed in this work.

The framework requires that we know the correlations between the atmospheric parameters. The correlation between T_eff and $\mathrm{log}g$ is a direct consequence of the way in which surface gravity is computed. Specifically, $g\propto {T}_{\mathrm{eff}}^{4}$ (Equation (1)). We computed the Pearson correlation coefficients between the three unique pairs of T_eff, $\mathrm{log}g$, and v_t from their measured values (Table 3). The value of [M/H] was identical for all stars in our sample, so we copied the correlation coefficients that involve [M/H] from Ji et al.(2020).

Following Appendix B of Ji et al. (2020), the correlation matrix (ρ) is multiplied into a vector (δ) containing the uncertainties in the atmospheric parameters for the star. Table 3 gives the uncertainties in T_eff, $\mathrm{log}g$, and v_t . We used 0.05 dex as the uncertainty in [M/H]. This number is the weighted standard deviation of the Fe ii abundances. The matrix ρ and the vector δ can be used together with the e_i (the abundance uncertainties propagated from the EW uncertainties) to compute a matrix called $\widetilde{{\rm{\Sigma }}}$:

$$\begin{eqnarray}&&\widetilde{{\rm{\Sigma }}}=\mathrm{diag}({e}_{i}^{2}+{s}_{{\rm{X}}}^{2})+\delta \rho {\delta }^{T}.\end{eqnarray} \tag{ 3 }$$

The variable s_X is a systematic uncertainty applied to every line of a given species. We calculated the s_X values according to the procedure described by Ji et al. (2020). As a modification to that procedure, we forced the minimum value of s_X to be 0.1. This minimum value approximates the hard-to-estimate systematic errors we discussed at the beginning of Section 4.7. The difference in our procedure is that we apply the minimum systematic error to each line rather than the final abundance measurements, which have no forced minimum.

This framework permits negative weights. In effect, a negative weight on an absorption line pushes the average abundance away from the abundance measured for that line. Negative weights are a result of the assumption that line strengths behave linearly with changes in abundance and in stellar parameters. This assumption is not necessarily true for strong lines. We resolve this conundrum by iteratively omitting lines with negative weights, then recomputing $\widetilde{{\rm{\Sigma }}}$ and the new weights.

Table 4 includes the weights for each absorption line. The table omits lines that were rejected for having negative weights.

Ji et al. (2020) also described how to calculate the error in abundance ratios. Most lines in our study vary in the same direction with changes in the atmospheric parameters. As a result, the error on most abundance ratios is smaller than simply adding the individual elements' errors in quadrature. We also calculated the covariance between two different abundance ratios. The figures in Section 5 show the covariances as ellipses.

Table 6 gives the abundances and errors for each star. Fe i and Fe ii are given as [Fe/H]. The other elements are shown as ratios relative to iron ((X/Fe)). The neutral species are shown relative to Fe i, and the ionized species are shown relative to Fe ii. The columns labeled N give the number of lines used in the abundance determinations. Elements measured with spectral synthesis are reported with "syn" in this column.