Spectroscopy and Photometry of Multiple Populations along the Asymptotic Giant Branch of NGC 2808 and NGC 6121 (M4)*

A. F. Marino; A. P. Milone; D. Yong; G. Da Costa; M. Asplund; L. R. Bedin; H. Jerjen; D. Nardiello; G. Piotto; A. Renzini; M. Shetrone

doi:10.3847/1538-4357/aa7852

1. Introduction

It is now established that virtually all Galactic globular clusters (GCs) host two or more stellar populations with different chemical compositions: a first population of stars with the same chemical abundance as halo field stars at similar metallicity, and a second population(s) of stars enhanced in helium, nitrogen, and sodium and depleted in carbon and oxygen (e.g., Kraft 1994; Gratton et al. 2012). In the last decade, multiple stellar populations have been homogeneously studied in a large number of Galactic GCs by using both spectroscopy and photometry (e.g., Carretta et al. 2009; Piotto et al. 2015; Milone et al. 2017 and references therein). Most of these studies focused on stars along the red giant branch (RGB), the subgiant branch, and the main sequence (MS). Little attention, however, has been devoted to more evolved stages, like the asymptotic giant branch (AGB).

Spectroscopic work has shown that the AGB stars of the few studied GCs can exhibit star-to-star variations in light elements, in analogy with what is observed along the RGB. These conclusions are based on both early analysis of molecular bands (e.g., Smith & Norris 1993) and, more recently, high-resolution spectroscopy of light elements (e.g., Ivans et al. 1999, 2001; García-Hernández et al. 2015; Johnson et al. 2015).

On the photometric side, appropriate combinations of the ultraviolet and optical filters of the Hubble Space Telescope (HST ), such as the ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}=({m}_{{\rm{F}}275{\rm{W}}}-{m}_{{\rm{F}}336{\rm{W}}}$ )−( ${m}_{{\rm{F}}336{\rm{W}}}-{m}_{{\rm{F}}438{\rm{W}}}$ ) pseudo-color, have been proved to provide a powerful means to identify multiple populations along the entire color–magnitude diagram (CMD) of GCs, from the MS to the RGB and the horizontal branch (Milone et al. 2013). Recent work has used this index to investigate the AGB and found that the AGB of NGC 2808 hosts three distinct sequences (Milone et al. 2015a). Similarly, the AGBs of the GCs NGC 7089 and NGC 6352 are found to be inconsistent with a simple stellar population (Milone et al. 2015b; Nardiello et al. 2015b; for further photometric evidence, see Gruyters et al. 2017).

By contrast, there is some evidence that the population ratio, in terms of light-element chemical abundances, is different among RGB and AGB stars. Early analysis on cyanogen- (CN-) band strengths by Smith & Norris (1993) found a different distribution of CN-band strengths among AGBs and RGBs in NGC 6752 and M5. More recently, a lack of AGB stars with the highest sodium abundance observed on the RGB has been shown for NGC 6752 and NGC 6266 (M62; Campbell et al. 2013; Lapenna et al. 2015, 2016). In 47 Tuc, the AGB displays Na abundance variations similar to those in the RGB, but ≲20% of Na-rich RGB stars may not reach the AGB phase (Johnson et al. 2015). These results suggest that some stars enriched in the high-temperature H-burning products fail to ascend the AGB and have raised new interest in stellar populations in this evolutionary phase.

As the chemical enrichment in Na is indicative of a star belonging to a second stellar population in GCs, this means that the AGBs of some GCs may display a paucity of second-population stars compared to the RGB. Qualitatively, enrichment in helium among second-population stars, as observed in GCs (e.g., Milone et al. 2014), can account for the lack of the Na- (and He-) richest stars along the AGB. Indeed, He-enhanced stars have smaller envelope masses (and higher surface temperatures) on the horizontal branch, and, if the mass is low enough, they evolve directly to the white dwarf sequence through the so-called AGB-manqué phase. The lack of AGB stars in M62 is consistent with this scenario, as the populous second generation of this cluster has an extremely high helium abundance (Y ∼ 0.33; Milone 2015). Quantitatively, most GCs have internal helium variations, ${\rm{\Delta }}(Y)$ , of just a few hundredths (Milone 2015). Evolutionary models of low-mass stars suggest that, besides helium, the maximum sodium content expected on the AGB is a function of both metallicity and age. Younger stars are more massive on the horizontal branch and, hence, will have a better chance to climb the AGB even if they are helium-rich (e.g., Charbonnel & Chantereau 2016).

In this paper, we combine multiwavelength photometry from HST and ground-based telescopes with high-resolution spectroscopy from the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) of the Very Large Telescope (VLT) to further investigate the multiple populations along the AGB of the mildly metal-poor GCs NGC 2808 and NGC 6121 (M4).

The targets of this paper, NGC 2808 and M4, have been widely investigated in the context of multiple populations. NGC 2808 is one of the most massive and complex GCs of the Milky Way. Its "chromosome map," a photometric diagram introduced by Milone et al. (2015a) to separate the different stellar groups in GCs, hosts at least five distinct populations (e.g., Carretta 2015; Milone et al. 2015a). It exhibits stellar populations with very high helium content ( $Y\sim 0.32$ and $Y\sim 0.38$ ; D'Antona et al. 2005; Piotto et al. 2007; Marino et al. 2014) and extreme variations in light-element abundances, including O, Mg, Al, Si, and Na (Carretta et al. 2009; Carretta 2014).

In contrast, M4 is a much simpler GC. It hosts "only" two main populations (Marino et al. 2008; Lee et al. 2009; Milone et al. 2014, 2017) with relatively small variations in light elements (Ivans et al. 1999; Marino et al. 2008, 2011; Carretta et al. 2009; Villanova et al. 2012) and helium ( ${\rm{\Delta }}Y\sim 0.02$ ; Villanova et al. 2012; Nardiello et al. 2015a) compared to NGC 2808.

The AGBs of both NGC 2808 and M4 have recently been investigated by means of high-resolution spectroscopy. Wang et al. (2016) determined the sodium abundance of 31 AGB stars in NGC 2808 and concluded that this cluster hosts second-generation AGB stars and that the fraction of Na-rich AGB stars is higher than that observed on the RGB.

The presence of multiple populations along the AGB of M4 is quite a controversial issue. The chemical composition of 15 AGB stars in M4 has been recently investigated by MacLean et al. (2016) using the 2dF+HERMES facility on the Anglo-Australian Telescope. These authors suggested that the AGB is mostly populated by stars with low sodium and high oxygen abundance. However, these findings have been photometrically challenged by the recent work of Lardo et al. (2017), who analyzed the V versus ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}\ =$ (U − I)−(B − I) diagram of M4 and concluded that its broadened AGB is not consistent with a simple population.

The layout of the paper is as follows. Section 2 presents the photometric and spectroscopic data that we have analyzed, our chemical analysis is described in Section 3, our results on the chemical abundances of AGB stars are discussed in Section 4 in conjunction with the photometric properties, and Section 5 is a summary of our results.

2. Observations and Data Reduction

In order to study multiple stellar populations along the AGB of M4 and NGC 2808, we have combined information from both photometry and spectroscopy. The photometric and spectroscopic data sets are described in the following subsections.

2.1. The Photometric Data Set and Target Selection

We have used both HST and ground-based photometry of NGC 2808 and M4. The photometric and astrometric catalogs from HST were published by Piotto et al. (2015) and include stars in the innermost ( $\sim 2\buildrel{\,\prime}\over{.} 7\times 2\buildrel{\,\prime}\over{.} 7$ ) cluster regions. They were derived from images collected through the F275W, F336W, and F438W filters of the Ultraviolet and Visual Channel of the Wide Field Camera 3 (UVIS/WFC3) on HST. The data include stellar proper motions. Moreover, they also provide photometry of the images from Anderson et al. (2008), collected with the F814W band of the Wide Field Channel of the Advanced Camera for Surveys on HST. Only stars that, according to their proper motions, are considered cluster members are included in the analysis. This photometric data set has been used in several works (Milone et al. 2015a, 2017; Piotto et al. 2015; Simioni et al. 2016) to investigate stellar populations in NGC 2808 and M4, and we refer the reader to these papers for further details on the data and data reduction.

In addition, we have used the wide-field photometric catalogs from the database maintained by Peter Stetson that are derived from images collected with ground-based facilities (see Stetson 1987, 1994, 2000; Stetson et al. 2014). These data have been previously used by Monelli et al. (2013) to study multiple stellar populations along the RGB of both M4 and NGC 2808. We refer the reader to these papers for details on the data set and the method used to derive the photometry and astrometry. We have calculated relative proper motions by combining the positions of stars derived from images collected with the Wide Field Imager (WFI) of the MPI 2.2 m telescope in La Silla with those from the Gaia data release 1 (DR1; Lindegren et al. 2016).

The WFI data used to calculate stellar proper motions in the field of M4 consist of 6 × 100 + 3 × 10 s images collected on 2001 June 18 and 21 through the B band (program 69.D-05282). The photometry and astrometry of these images have been computed using the programs from Anderson et al. (2006), while proper motions have been derived as in Anderson et al. (2006) and Piotto et al. (2012). Both HST and ground-based photometry have been corrected for differential reddening following the recipe by Milone et al. (2012).

The CMDs of stars in the field of view of M4 and NGC 2808 from ground-based and HST photometry are shown in Figures 1 and 2, where we indicate (orange diamonds) our sample of 17 and 7 spectroscopically analyzed AGB stars, respectively. The remaining AGB stars have been studied from photometry only and are represented with black diamonds. We also show the vector-point diagram of proper motions obtained by combining the coordinates of stars from the Gaia and WFI catalogs that has been used to identify candidate cluster members and field stars. Our target AGB stars were carefully selected from CMDs derived from both ground-based and HST observations. Specifically, for NGC 2808, we have selected three stars on the three main AGB sequences identified by Milone et al. (2015a) by using the ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ index from HST photometry. The remaining four stars of NGC 2808 have been selected only from ground-based photometry by Stetson (2000). In the case of M4, multiwavelength HST photometry from Piotto et al. (2015) is available for six AGB stars, while the remaining 11 stars have been identified on the CMD obtained from ground-based photometry.

**Figure 1.** Left panels: B vs. B − I CMD of stars with radial distances less than 20' from the center of M4 (Stetson et al. 2014). Cluster members and field stars are colored black and aqua, respectively, and have been selected on the basis of their proper motions. The vector-point diagram of stellar proper motions obtained by combining information from *Gaia* catalogs and WFI images is plotted on the lower right side of the CMD. We also show a zoom of the CMD of the cluster members around the AGB and RGB. Right panel: ${m}_{{\rm{F}}275{\rm{W}}}$ vs. ${m}_{{\rm{F}}275{\rm{W}}}-{m}_{{\rm{F}}814{\rm{W}}}$ CMD of NGC 2808 cluster members from *HST* photometry (Piotto et al. 2015). The AGB stars in the central *HST* field are also included in the ground-based photometry. The photometrically selected AGB stars are indicated with filled diamonds in both CMDs, and those that were also observed spectroscopically are colored orange.
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**Figure 1.** Left panels: B vs. B − I CMD of stars with radial distances less than 20' from the center of M4 (Stetson et al. 2014). Cluster members and field stars are colored black and aqua, respectively, and have been selected on the basis of their proper motions. The vector-point diagram of stellar proper motions obtained by combining information from *Gaia* catalogs and WFI images is plotted on the lower right side of the CMD. We also show a zoom of the CMD of the cluster members around the AGB and RGB. Right panel: ${m}_{{\rm{F}}275{\rm{W}}}$ vs. ${m}_{{\rm{F}}275{\rm{W}}}-{m}_{{\rm{F}}814{\rm{W}}}$ CMD of NGC 2808 cluster members from *HST* photometry (Piotto et al. 2015). The AGB stars in the central *HST* field are also included in the ground-based photometry. The photometrically selected AGB stars are indicated with filled diamonds in both CMDs, and those that were also observed spectroscopically are colored orange.
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**Figure 2.** Same as Figure 1 but for NGC 2808. In this case, the CMD from ground-based photometry includes only stars with radial distances between 15 and 120 from the cluster center and does not include the AGB observed in the central *HST* field. The blue diamonds in the right panel mark candidate AGB-manqué stars.
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farcm — **Figure 2.** Same as Figure 1 but for NGC 2808. In this case, the CMD from ground-based photometry includes only stars with radial distances between 15 and 120 from the cluster center and does not include the AGB observed in the central *HST* field. The blue diamonds in the right panel mark candidate AGB-manqué stars.
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2.2. The Spectroscopic Data Set

Our data set consists of FLAMES/UVES spectra (RED580 setting; Pasquini et al. 2000) collected under the programs 093.D-0789 and 094.D-0455. Final spectra have been obtained by co-adding 2 × 2775 and 30 × 2775 s exposures for M4 and NGC 2808, respectively. The data were reduced using the UVES pipelines (Ballester et al. 2000), including bias subtraction, flat-field correction, wavelength calibration, sky subtraction, and spectral rectification. The spectra have a spectral coverage of ∼2000 Å, with the central wavelength at ∼5800 Å. Telluric subtraction has been performed using the European Southern Observatory (ESO) MOLECFIT tool (Kausch et al. 2015; Smette et al. 2015). The typical signal-to-noise ratio (S/N) for the final combined spectra around the [O i] λ6300 line ranges from S/N∼130 to ∼230 for M4 and from ∼160 to ∼250 for NGC 2808.

Radial velocities (RVs) were derived using the iraf@FXCOR task, which cross-correlates the object spectrum with a template. For the template, we used a synthetic spectrum obtained through MOOG (Sneden 1973) computed with a model stellar atmosphere interpolated from the Castelli & Kurucz (2004) grid, adopting parameters (effective temperature/surface gravity/microturbulence/metallicity) = (4900 K/2.0/2.0 $\mathrm{km}\,{{\rm{s}}}^{-1}$ /−1.20). Each spectrum was corrected to the rest-frame system, and observed RVs were then corrected to the heliocentric system. The mean heliocentric RVs of M4 and NGC 2808 are $\langle \mathrm{RV}\rangle$ =+69.5 ± 0.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (σ = 3.2 $\mathrm{km}\,{{\rm{s}}}^{-1}$ ) and $\langle \mathrm{RV}\rangle$ = +96.7 ±3.9 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (σ = 9.6 $\mathrm{km}\,{{\rm{s}}}^{-1}$ ), respectively. These values agree with those obtained from RGB stars in the same clusters, e.g., $\langle \mathrm{RV}\rangle$ = +70.6 ± 1 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for M4 (Marino et al. 2008) and $\langle \mathrm{RV}\rangle$ = +102.4 ± 1 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for NGC 2808 (σ = 9.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$ ; Carretta et al. 2006). As discussed in Section 2.1, our spectroscopic targets had already passed the membership selection criterion based on proper motions. In the end, all proper-motion members are also RV members, with the exception of M4 star 16235035–2632478 (Table 1). Based on the chemical abundances, discussed in the next sections, membership is further confirmed by the fact that all of the target stars have [Fe/H] consistent with the cluster mean metallicity, except 1623035–2632478, which was not analyzed.

Table 1. Coordinates, Radial Velocities, and Atmospheric Parameters of the AGB Stars Spectroscopically Analyzed in This Paper

ID (2MASS)	GC	R.A.	Decl.	RV	${T}_{\mathrm{eff}}$	log g	[Fe/H]	${\xi }_{{\rm{t}}}$	${T}_{\mathrm{eff}}$ (phot)	log g (phot)
		J2000	J2000	( $\mathrm{km}\,{{\rm{s}}}^{-1}$ )	(K)	(cgs)		( $\mathrm{km}\,{{\rm{s}}}^{-1}$ )	(K)	(cgs)
16233067–2629390	M4	16:23:30.70	−26:29:39.0	+72.88	4920	1.85	−1.18	1.81	4817	1.77
16233741–2638238	M4	16:23:37.44	−26:38:23.9	+66.88	4770	1.50	−1.22	1.72	4732	1.58
16234268–2631209	M4	16:23:42.71	−26:31:20.8	+74.01	5190	2.25	−1.07	1.78	5022	1.92
16233020–2633241	M4	16:23:30.23	−26:33:24.0	+67.26	4630	1.55	−1.19	1.91	4641	1.44
16235035–2632478	M4	16:23:50.38	−26:32:47.8	−48.01	⋯	⋯	⋯	⋯	⋯	⋯
16233142–2633110	M4	16:23:31.45	−26:33:10.9	+65.80	4400	1.25	−1.18	2.00	4445	1.18
16233846–2629235	M4	16:23:38.49	−26:29:23.5	+68.32	4780	1.75	−1.17	1.82	4701	1.63
16233535–2632225	M4	16:23:35.37	−26:32:22.5	+67.83	4450	1.20	−1.26	1.92	4497	1.27
16240858–2624552	M4	16:24:08.60	−26:24:55.2	+72.82	4680	1.60	−1.27	1.82	4623	1.46
16233477–2631349	M4	16:23:34.79	−26:31:34.9	+69.68	5150	2.50	−1.22	1.62	5023	1.92
16234740–2631463	M4	16:23:47.41	−26:31:46.3	+72.98	4500	1.35	−1.28	1.92	4536	1.32
16232988–2631490	M4	16:23:29.90	−26:31:49.0	+71.86	4530	1.40	−1.28	1.77	4561	1.39
16235375–2634426	M4	16:23:53.77	−26:34:42.6	+65.54	4530	1.40	−1.21	1.93	4547	1.31
16233667–2630397	M4	16:23:36.69	−26:30:39.7	+63.13	4470	1.23	−1.24	1.92	4508	1.28
16233614–2632015	M4	16:23:36.17	−26:32:01.5	+67.30	4570	1.75	−1.12	1.57	4570	1.31
16231672–2634279	M4	16:23:16.75	−26:34:28.0	+72.19	4500	1.40	−1.24	1.96	4435	1.20
16234085–2631215	M4	16:23:40.88	−26:31:21.5	+71.45	4550	1.60	−1.18	1.70	4587	1.48
16232114–2631598	M4	16:23:21.17	−26:31:59.7	+71.96	5090	2.47	−1.14	1.61	4964	2.00
09120251–6451001	NGC 2808	09:12:02.51	−64:51:00.2	+96.61	4750	1.55	−1.31	1.70	4773	1.66
09123016–6454129	NGC 2808	09:12:30.18	−64:54:12.9	+99.14	4770	1.78	−1.20	1.55	4796	1.54
09120852–6449107	NGC 2808	09:12:08.53	−64:49:10.7	+110.92	4630	1.55	−1.19	1.75	4690	1.55
09120213–6452243	NGC 2808	09:12:02.15	−64:52:24.4	+87.84	4860	1.55	−1.30	1.82	4788	1.62
09122027–6448450	NGC 2808	09:12:20.29	−64:48:45.1	+86.07	4500	1.15	−1.32	1.75	4579	1.36
09120665–6450253	NGC 2808	09:12:06.66	−64:50:25.4	+106.72	4800	1.60	−1.30	1.70	4887	1.80
09114655–6452144	NGC 2808	09:11:46.55	−64:52:14.5	+89.63	4400	0.90	−1.25	2.05	4420	1.07

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3. Chemical Abundance Analysis

Chemical abundances have been derived from a local thermodynamical equilibrium (LTE) analysis using MOOG (version 2013; Sneden 1973) and the α-enhanced model atmospheres of Castelli & Kurucz (2004). The line list and reference solar abundances are as in Marino et al. (2008), except Al for which we used solar abundances higher by 0.21.

To infer the atmospheric parameters used in our chemical analysis, we took advantage of the high resolution and S/N of our UVES spectra and employed Fe lines. Specifically,(i) effective temperatures ( ${T}_{\mathrm{eff}}$ ) were derived by imposing the excitation potential equilibrium of the Fe i lines, (ii) surface gravities (log g) were set with the ionization equilibrium between the Fe i and Fe ii lines but allowing Fe ii abundances to be slightly higher than Fe i to take into account deviations from LTE (Bergemann et al. 2012; Lind et al. 2012), and(iii) microturbulent velocities ( ${\xi }_{{\rm{t}}}$ ) were set to minimize any dependence of Fe i abundances on equivalent widths (EWs).

In Table 1, we list our adopted spectroscopic parameters,together with ${T}_{\mathrm{eff}}$ obtained from the Alonso $(B-V)$ – ${T}_{\mathrm{eff}}$ calibrations (Alonso et al. 1999). We assume reddening of $E(B-V)=0.36$ (Harris 1996) and $E(B-V)=0.19$ (Bedin et al. 2000) for M4 and NGC 2808, respectively. Surface gravities have been derived by using the canonical relation where we have assumed a mass of 0.60 solar masses, temperatures and V magnitudes described above, and distance modulus of (m−M)V = 12.82 for M4, and (m−M)V = 15.59 for NGC2808. By comparing the spectroscopically derived atmospheric parameters with those from photometry, we get mean differences of $\langle {{T}_{\mathrm{eff}}}_{B-V}-$ ${{T}_{\mathrm{eff}}}_{\mathrm{Fe}}\rangle =\,-12$ ± 15 K, rms = 72 K, and $\langle \mathrm{log}\,{\text{}}{g}_{\mathrm{phot}}-$ $\mathrm{log}\,{\text{}}{g}_{\mathrm{Fe}}\rangle =\,-0.09\,\pm$ 0.04, rms = 0.21. The fact that our adopted ${T}_{\mathrm{eff}}$ and log g average values agree reasonably with the photometric ones based on recent reddening values gives us confidence in our adopted ${T}_{\mathrm{eff}}$ scales. By contrast, we notice that, for M4 AGBs, the discrepancy with the photometric temperatures is larger for hotter stars; specifically, for stars with ${T}_{\mathrm{eff}}$ > 4800 K, the photometric temperatures and gravities are lower by >100 K and >0.2, respectively.⁵ A variation in temperature of ±100 K will change the photometric surface gravities by around ±0.05–0.06 dex only. Hence, the discrepancy in ${T}_{\mathrm{eff}}$ alone is not enough to account for the lower photometric gravity in the higher-temperature stars. By changing the assumed mass by ±0.1 $M/{M}_{\odot }$ , log g varies by around ±0.08 dex. We are not able to explain why hotter stars display larger discrepancies between the spectroscopic and photometric parameters in M4, a feature that is not observed in NGC 2808, but note that in the latter case the observed range in ${T}_{\mathrm{eff}}$ is smaller (Δ ${T}_{\mathrm{eff}}$ = 460 K versus Δ ${T}_{\mathrm{eff}}$ = 790 K in M4).

To test the magnitude of the non-LTE effects on our metallicity values, assumed to be equal to the Fe i abundances, we derived the non-LTE corrections to the Fe i spectral lines from Lind et al. (2012) by using the inspect tool⁶ for one RGB (20766) and one AGB (16233142–2633110) star in M4. We found that both the RGB and the AGB stars have positive non-LTE corrections, 0.06 and 0.08 dex, respectively. Hence, metallicities in the non-LTE should be ∼0.08 dex higher in our AGB sample. Note that the AGB non-LTE correction is only marginally higher (by 0.021 ± 0.002) than that of the RGB.

For all the elements except O and Al, chemical abundances were obtained from the EWs derived from Gaussian fitting of isolated spectral lines. Oxygen and aluminum were derived from spectral synthesis of the lines [O i] λ6300 and λ6363 and the Al doublet λ6697 to account for blending with other spectral features. In the end, we have been able to infer chemical abundances for 16 elements, namely, O, Na, Mg, Al, Si, Ca, Sc (Sc i and Sc ii), Ti (Ti i and Ti ii), V, Cr (Cr i and Cr ii), Fe (Fe i and Fe ii), Co, Ni, Zn, Y ii, and Ce ii.

Estimates of the uncertainties in the chemical abundances have been obtained by rerunning the abundances, one at a time, varying ${T}_{\mathrm{eff}}$ /log g/[m/H]/ ${\xi }_{{\rm{t}}}$ by ±100 K/±0.20/±0.15/±0.30 $\mathrm{km}\,{{\rm{s}}}^{-1}$ . The uncertainties used in ${T}_{\mathrm{eff}}$ and log g are reasonable, as suggested by the comparison with the photometric values discussed above. As internal errors in [m/H] and ${\xi }_{{\rm{t}}}$ , we conservatively adopt ±0.15 dex and ±0.30 $\mathrm{km}\,{{\rm{s}}}^{-1}$ . In addition to the contribution introduced by internal errors in the atmospheric parameters, we estimated the contribution ( ${\sigma }_{\mathrm{fit}}$ ) due to the finite S/N, which affects the measurements of EWs and the spectral synthesis. The contribution due to EWs has been calculated by varying the EWs of spectral lines by ±4.5 mÅ. This value has been derived by comparing EWs from various exposures of the same stars. The variations in the abundances obtained by varying the EWs are then divided by $\sqrt{(N-1)}$ (where N is the number of available spectral lines). For the elements analyzed through spectral synthesis, we estimated the error in their chemical abundances by varying the continuum placement in the synthesis within a reasonable range. Variations in chemical abundances due to each contribution, plus the total error estimate obtained by summing all the different contributions in quadrature, are listed in Table 2.

Table 2. Sensitivity of Derived Abundances to the Uncertainties in Atmospheric Parameters, the Limited S/N ( ${\sigma }_{\mathrm{fit}}$ ), and the Total Error Due to These Contributions ( ${\sigma }_{\mathrm{tot}}$ )

	Δ ${T}_{\mathrm{eff}}$	Δlog g	Δ ${\xi }_{{\rm{t}}}$	Δ[m/H]	${\sigma }_{\mathrm{fit}}$	${\sigma }_{\mathrm{total}}$
	±100 K	±0.20	±0.30 $\mathrm{km}\,{{\rm{s}}}^{-1}$	0.15 dex
$[{\rm{O}}/\mathrm{Fe}]$	±0.02	±0.09	∓0.01	∓0.05	±0.06	0.12
$[\mathrm{Na}/\mathrm{Fe}]$	∓0.02	∓0.01	±0.03	∓0.01	±0.06	0.07
$[\mathrm{Mg}/\mathrm{Fe}]$	∓0.03	∓0.01	±0.00	∓0.00	±0.04	0.05
$[\mathrm{Al}/\mathrm{Fe}]$	±0.07	∓0.00	∓0.00	±0.01	±0.04	0.08
$[\mathrm{Si}/\mathrm{Fe}]$	∓0.08	±0.02	±0.04	±0.02	±0.07	0.12
$[\mathrm{Ca}/\mathrm{Fe}]$	±0.01	∓0.02	∓0.04	∓0.01	±0.01	0.05
$[\mathrm{Sc}/\mathrm{Fe}]$ i	±0.00	±0.01	±0.04	±0.01	±0.03	0.05
$[\mathrm{Sc}/\mathrm{Fe}]$ ii	±0.05	∓0.01	∓0.03	±0.01	±0.05	0.08
$[\mathrm{Ti}/\mathrm{Fe}]$ i	±0.06	∓0.01	∓0.01	∓0.01	±0.01	0.06
$[\mathrm{Ti}/\mathrm{Fe}]$ ii	±0.04	∓0.01	∓0.06	∓0.01	±0.03	0.08
$[{\rm{V}}/\mathrm{Fe}]$	±0.07	∓0.01	±0.02	∓0.01	±0.03	0.08
$[\mathrm{Cr}/\mathrm{Fe}]$ i	±0.04	∓0.01	∓0.02	∓0.01	±0.02	0.05
$[\mathrm{Cr}/\mathrm{Fe}]$ ii	±0.01	∓0.01	±0.02	∓0.01	±0.04	0.05
$[\mathrm{Fe}/{\rm{H}}]$ i	±0.10	±0.01	∓0.06	±0.00	±0.01	0.12
$[\mathrm{Fe}/{\rm{H}}]$ ii	∓0.06	±0.09	∓0.03	±0.04	±0.04	0.13
$[\mathrm{Co}/\mathrm{Fe}]$	∓0.01	±0.00	±0.05	±0.01	±0.04	0.07
$[\mathrm{Ni}/\mathrm{Fe}]$	∓0.02	±0.01	±0.03	±0.01	±0.01	0.04
$[\mathrm{Zn}/\mathrm{Fe}]$	∓0.13	±0.04	∓0.07	±0.02	±0.02	0.16
$[{\rm{Y}}/\mathrm{Fe}]$ ii	±0.06	∓0.01	∓0.09	∓0.00	±0.05	0.12
$[\mathrm{Ce}/\mathrm{Fe}]$ ii	±0.08	∓0.01	±0.01	±0.01	±0.07	0.11

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The average abundances and their corresponding uncertainties, defined as the rms from different spectral lines for the same species, are listed in Tables 3–5.

Table 3. Derived Chemical Abundances for O, Na, Mg, Al, Si, and Ca in M4 and NGC 2808 AGBs

ID (2MASS)	$[{\rm{O}}/\mathrm{Fe}]$	rms/#	$[\mathrm{Na}/\mathrm{Fe}]$	rms/#	$[\mathrm{Mg}/\mathrm{Fe}]$	rms/#	$[\mathrm{Al}/\mathrm{Fe}]$	rms/#	$[\mathrm{Si}/\mathrm{Fe}]$	rms/#	$[\mathrm{Ca}/\mathrm{Fe}]$	rms/#
M4
16233142–2633110	+0.34	0.04/2	+0.25	0.13/3	+0.51	0.07/3	+0.34	0.04/2	+0.49	0.01/3	+0.19	0.09/13
16233477–2631349	+0.57	0.02/2	+0.09	0.12/3	+0.46	0.09/2	+0.27	0.06/2	+0.42	0.04/3	+0.32	0.09/14
16233535–2632225	+0.34	0.04/2	+0.37	0.09/3	+0.53	0.00/2	+0.38	0.03/2	+0.51	0.04/3	+0.20	0.08/13
16233614–2632015	+0.56	0.02/2	+0.24	0.07/3	+0.44	0.02/3	+0.23	0.00/2	+0.42	0.06/3	+0.31	0.09/13
16233741–2638238	+0.24	0.04/2	+0.27	0.14/3	+0.48	0.09/2	+0.28	0.02/2	+0.46	0.01/3	+0.21	0.11/12
16234085–2631215	+0.53	0.01/2	+0.02	0.06/3	+0.61	0.00/2	+0.50	0.05/2	+0.56	0.02/3	+0.21	0.10/13
16234268–2631209	+0.40	0.01/2	+0.32	0.16/4	+0.35	−/1	+0.25	0.04/2	+0.47	0.02/3	+0.30	0.10/11
16234740–2631463	+0.53	0.03/2	+0.09	0.06/3	+0.54	0.02/3	+0.31	0.02/2	+0.51	0.04/3	+0.18	0.09/13
16235375–2634426	+0.37	0.04/2	+0.36	0.08/3	+0.49	0.04/3	+0.31	0.03/2	+0.53	0.06/2	+0.18	0.10/13
16240858–2624552	+0.58	0.02/2	−0.07	0.10/3	+0.47	0.06/2	+0.20	0.04/2	+0.45	0.07/3	+0.17	0.09/13
16231672–2634279	+0.46	0.01/2	+0.29	0.06/3	+0.51	0.02/2	+0.30	0.01/2	+0.51	0.05/3	+0.17	0.10/13
16232114–2631598	+0.54	0.13/2	+0.13	0.07/4	+0.45	0.06/2	+0.19	0.04/2	+0.46	0.07/3	+0.30	0.08/13
16232988–2631490	+0.50	0.01/2	+0.02	0.08/3	+0.55	0.01/2	+0.32	0.04/2	+0.51	0.03/3	+0.19	0.09/13
16233020–2633241	+0.35	0.04/2	+0.36	0.19/4	+0.51	0.05/2	+0.34	0.04/2	+0.52	0.03/3	+0.20	0.10/12
16233067–2629390	+0.50	0.06/2	+0.00	0.13/4	+0.52	0.22/2	+0.17	0.01/2	+0.52	0.02/2	+0.22	0.11/ 9
16233667–2630397	+0.36	0.01/2	+0.34	0.06/3	+0.52	0.01/3	+0.35	0.01/2	+0.51	0.04/3	+0.19	0.09/12
16233846–2629235	+0.44	0.04/2	+0.17	0.09/3	+0.48	0.06/2	+0.23	0.01/2	+0.46	0.05/3	+0.20	0.09/12

mean	+0.45		+0.19		+0.50		+0.29		+0.49		+0.22
±	0.02		0.04		0.01		0.02		0.01		0.01
rms	0.10		0.14		0.05		0.08		0.04		0.04

NGC 2808
09120665–6450253	+0.38	0.03/2	−0.05	0.07/4	+0.28	0.01/2	−0.16	0.00/2	+0.32	0.09/3	+0.26	0.09/14
09120251–6451001	+0.27	0.07/2	+0.11	0.11/4	+0.24	0.04/2	+0.30	0.04/2	+0.23	0.02/3	+0.22	0.10/11
09120213–6452243	−0.19	0.09/2	+0.27	0.07/4	+0.08	0.14/2	+0.70	0.01/2	+0.30	0.03/3	+0.17	0.13/12
09114655–6452144	−0.09	0.08/2	+0.30	0.10/4	+0.10	0.02/3	+0.68	0.03/2	+0.29	0.05/3	+0.18	0.11/10
09122027–6448450	+0.21	0.01/2	+0.14	0.09/4	+0.30	0.05/3	+0.02	0.01/2	+0.25	0.06/3	+0.22	0.11/13
09123016–6454129	+0.34	0.04/2	−0.09	0.11/4	+0.28	0.01/3	−0.24	0.07/2	+0.24	0.04/3	+0.31	0.10/14
09120852–6449107	+0.43	0.04/2	−0.08	0.13/4	+0.36	0.06/2	−0.16	0.04/2	+0.30	0.14/2	+0.30	0.08/14

mean	+0.19		+0.09		+0.23		+0.16		+0.28		+0.24
±	0.09		0.06		0.04		0.15		0.01		0.02
rms	0.22		0.15		0.10		0.37		0.03		0.05

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Table 4. Derived Chemical Abundances for Sc, Ti, V, and Cr in M4 and NGC 2808 AGBs

ID (2MASS)	$[\mathrm{Sc}/\mathrm{Fe}]$ i	rms/#	$[\mathrm{Sc}/\mathrm{Fe}]$ ii	rms/#	$[\mathrm{Ti}/\mathrm{Fe}]$ i	rms/#	$[\mathrm{Ti}/\mathrm{Fe}]$ ii	rms/#	$[{\rm{V}}/\mathrm{Fe}]$	rms/#	$[\mathrm{Cr}/\mathrm{Fe}]$ i	rms/#	$[\mathrm{Cr}/\mathrm{Fe}]$ ii	rms/#
M4
16233142–2633110	⋯	−/0	+0.01	0.11/6	+0.22	0.06/15	+0.30	0.15/5	−0.12	0.30/13	−0.12	0.14/10	−0.05	0.04/2
16233477–2631349	−0.02	−/1	+0.14	0.04/6	+0.32	0.08/15	+0.36	0.08/5	+0.01	0.11/10	−0.06	0.10/ 7	−0.01	0.02/2
16233535–2632225	−0.07	−/1	−0.01	0.07/5	+0.23	0.05/16	+0.21	0.06/4	−0.03	0.08/12	−0.16	0.11/ 7	−0.08	−/1
16233614–2632015	+0.04	−/1	+0.15	0.06/6	+0.29	0.06/15	+0.28	0.03/4	+0.07	0.06/12	−0.04	0.10/ 9	−0.05	−/1
16233741–2638238	−0.09	0.23/6	−0.12	0.23/6	+0.22	0.09/13	+0.27	0.15/5	−0.16	0.18/12	−0.13	0.10/ 7	−0.23	−/1
16234085–2631215	−0.05	−/1	+0.02	0.06/6	+0.23	0.06/14	+0.25	0.07/5	−0.05	0.04/12	−0.15	0.14/ 6	−0.16	−/1
16234268–2631209	−0.01	0.22/5	−0.04	0.22/5	+0.30	0.07/10	+0.30	0.10/5	−0.01	0.10/ 4	−0.10	0.09/ 6	−0.09	−/1
16234740–2631463	−0.04	−/1	−0.01	0.06/6	+0.25	0.04/13	+0.27	0.12/5	−0.01	0.08/13	−0.15	0.10/ 7	−0.12	−/1
16235375–2634426	−0.12	−/1	+0.04	0.04/6	+0.23	0.05/16	+0.25	0.11/5	−0.01	0.08/13	−0.15	0.10/ 8	−0.09	−/1
16240858–2624552	−0.09	−/1	+0.00	0.06/6	+0.23	0.06/15	+0.27	0.10/5	−0.06	0.07/12	−0.15	0.10/ 6	−0.12	−/1
16231672–2634279	−0.05	−/1	+0.01	0.05/6	+0.24	0.05/15	+0.27	0.07/5	−0.03	0.08/13	−0.14	0.14/ 6	−0.13	−/1
16232114–2631598	+0.16	0.07/6	+0.13	0.07/6	+0.32	0.08/14	+0.30	0.02/4	−0.01	0.07/ 9	−0.05	0.06/ 7	+0.03	0.05/2
16232988–2631490	−0.12	−/1	−0.02	0.06/6	+0.21	0.05/16	+0.28	0.10/5	−0.08	0.07/13	−0.15	0.08/ 6	−0.12	−/1
16233020–2633241	+0.06	0.08/5	+0.02	0.08/5	+0.23	0.05/12	+0.29	0.16/5	−0.10	0.22/12	−0.11	0.17/ 6	−0.16	−/1
16233067–2629390	−0.10	0.14/6	−0.13	0.14/6	+0.25	0.06/14	+0.27	0.17/5	−0.10	0.21/10	−0.11	0.10/ 6	−0.11	−/1
16233667–2630397	−0.04	−/1	−0.01	0.07/6	+0.22	0.04/15	+0.29	0.13/5	−0.04	0.10/13	−0.14	0.11/ 6	−0.07	−/1
16233846–2629235	−0.11	−/1	−0.02	0.03/6	+0.23	0.07/15	+0.30	0.14/5	−0.07	0.07/12	−0.17	0.09/ 8	−0.14	−/1

mean	−0.04		+0.01		+0.25		+0.28		−0.05		−0.12		−0.10
±	0.02		0.02		0.01		0.01		0.01		0.01		0.01
rms	0.07		0.07		0.03		0.03		0.05		0.04		0.06

NGC 2808
09120665–6450253	+0.03	0.05/6	+0.00	0.05/6	+0.20	0.07/14	+0.24	0.11/5	−0.11	0.09/13	−0.10	0.12/8	−0.12	0.02/2
09120251–6451001	−0.03	−/1	−0.05	0.06/6	+0.18	0.07/14	+0.13	0.11/5	−0.12	0.05/11	−0.11	0.13/7	−0.12	0.01/2
09120213–6452243	+0.04	0.05/6	+0.01	0.05/6	+0.14	0.06/13	+0.22	0.13/5	−0.14	0.11/12	−0.16	0.09/7	−0.10	0.07/2
09114655–6452144	−0.08	−/1	+0.01	0.09/5	+0.15	0.08/15	+0.20	0.13/5	−0.07	0.09/13	−0.11	0.13/6	−0.14	−/1
09122027–6448450	−0.15	−/1	+0.02	0.05/6	+0.17	0.05/15	+0.24	0.11/5	−0.12	0.04/12	−0.12	0.10/8	−0.07	0.02/2
09123016–6454129	−0.09	−/1	+0.06	0.03/5	+0.23	0.09/16	+0.32	0.11/4	−0.07	0.07/12	−0.09	0.08/7	−0.05	−/1
09120852–6449107	+0.12	0.06/6	+0.08	0.06/6	+0.24	0.09/14	+0.34	0.09/5	−0.04	0.10/13	−0.07	0.10/7	⋯	−/0

mean	−0.02		+0.02		+0.19		+0.24		−0.10		−0.11		−0.10
±	0.03		0.02		0.01		0.03		0.01		0.01		0.01
rms	0.09		0.04		0.04		0.07		0.03		0.03		0.03

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Table 5. Derived Chemical Abundances for Fe, Co, Ni, Zn, Y, and Ce in M4 and NGC 2808 AGBs

ID (2MASS)	$[\mathrm{Fe}/{\rm{H}}]$ i	rms/#	$[\mathrm{Fe}/{\rm{H}}]$ ii	rms/#	$[\mathrm{Co}/\mathrm{Fe}]$	rms/#	$[\mathrm{Ni}/\mathrm{Fe}]$	rms/#	$[\mathrm{Zn}/\mathrm{Fe}]$	rms/#	$[{\rm{Y}}/\mathrm{Fe}]$ ii	rms/#	$[\mathrm{Ce}/\mathrm{Fe}]$ ii	rms/#
M4
16233142–2633110	−1.18	0.08/75	−1.14	0.09/6	−0.03	0.02/4	−0.01	0.08/28	+0.19	−/1	+0.26	0.16/4	+0.13	−/1
16233477–2631349	−1.22	0.08/68	−1.20	0.08/8	+0.07	−/1	+0.01	0.10/28	+0.09	−/1	+0.39	0.09/4	+0.31	−/1
16233535–2632225	−1.26	0.08/88	−1.24	0.10/8	−0.07	0.06/4	−0.03	0.08/30	+0.16	−/1	+0.29	0.18/3	+0.17	−/1
16233614–2632015	−1.12	0.07/92	−1.08	0.07/9	−0.05	0.06/4	−0.03	0.06/32	+0.19	−/1	+0.32	0.15/4	+0.09	−/1
16233741–2638238	−1.22	0.08/67	−1.19	0.18/9	−0.05	0.05/3	−0.03	0.10/24	+0.34	0.07/2	+0.23	0.11/4	+0.01	−/1
16234085–2631215	−1.18	0.09/84	−1.15	0.08/8	−0.07	0.09/4	−0.02	0.07/31	+0.25	−/1	+0.36	0.15/3	+0.35	−/1
16234268–2631209	−1.07	0.09/45	−1.04	0.14/8	−0.11	−/1	−0.01	0.08/14	+0.24	−/1	+0.38	0.06/4	+0.25	−/1
16234740–2631463	−1.28	0.08/89	−1.25	0.10/9	−0.03	0.04/4	−0.02	0.06/30	+0.31	0.18/2	+0.21	0.14/3	+0.15	−/1
16235375–2634426	−1.21	0.08/87	−1.18	0.07/9	−0.03	0.06/4	−0.02	0.07/28	+0.23	−/1	+0.27	0.13/4	+0.11	−/1
16240858–2624552	−1.27	0.07/85	−1.24	0.08/9	−0.06	0.08/4	−0.04	0.06/30	+0.20	0.03/2	+0.16	0.12/4	+0.07	−/1
16231672–2634279	−1.24	0.08/86	−1.20	0.07/9	−0.07	0.05/3	−0.02	0.07/30	+0.17	−/1	+0.25	0.11/4	+0.13	−/1
16232114–2631598	−1.14	0.07/67	−1.11	0.07/9	−0.03	0.08/2	+0.00	0.09/28	+0.16	−/1	+0.33	0.13/4	+0.18	−/1
16232988–2631490	−1.28	0.07/83	−1.25	0.08/9	−0.04	0.05/4	−0.04	0.07/31	+0.33	0.15/2	+0.28	0.16/4	+0.13	−/1
16233020–2633241	−1.19	0.07/66	−1.15	0.10/7	−0.02	0.04/3	−0.01	0.07/27	+0.44	0.07/2	+0.21	0.15/3	+0.08	−/1
16233067–2629390	−1.18	0.07/61	−1.15	0.15/9	−0.02	0.05/3	+0.01	0.11/23	+0.30	0.01/2	+0.14	0.09/4	+0.02	−/1
16233667–2630397	−1.24	0.08/82	−1.21	0.07/9	−0.04	0.05/4	−0.01	0.07/27	+0.23	−/1	+0.30	0.12/4	+0.06	−/1
16233846–2629235	−1.17	0.08/78	−1.14	0.06/8	−0.04	0.02/3	−0.03	0.09/30	+0.16	−/1	+0.17	0.10/4	+0.13	−/1

mean	−1.20		−1.17		−0.04		−0.02		+0.23		+0.27		+0.14
±	0.01		0.01		0.01		0.01		0.02		0.02		0.02
rms	0.06		0.06		0.04		0.01		0.08		0.07		0.09

NGC 2808
09120665–6450253	−1.30	0.07/68	−1.27	0.06/9	−0.02	0.07/3	−0.07	0.07/25	−0.01	−/1	−0.13	0.15/3	−0.17	−/1
09120251–6451001	−1.31	0.08/71	−1.24	0.07/9	−0.17	0.07/2	−0.10	0.09/26	+0.13	0.22/2	−0.16	0.06/4	−0.11	−/1
09120213–6452243	−1.30	0.07/67	−1.27	0.05/8	−0.12	0.18/2	−0.07	0.13/26	+0.03	−/1	−0.04	0.12/3	−0.13	−/1
09114655–6452144	−1.25	0.08/81	−1.21	0.06/9	−0.15	0.00/3	−0.13	0.09/27	+0.08	−/1	−0.01	0.14/3	−0.15	−/1
09122027–6448450	−1.32	0.09/81	−1.29	0.06/9	−0.17	0.01/4	−0.10	0.09/25	+0.14	−/1	−0.03	0.10/4	−0.06	−/1
09123016–6454129	−1.20	0.08/75	−1.17	0.09/9	−0.13	0.04/2	−0.12	0.08/23	+0.21	0.16/2	+0.11	0.10/4	+0.08	−/1
09120852–6449107	−1.19	0.08/80	−1.15	0.07/9	−0.06	0.01/3	−0.04	0.11/19	+0.20	0.04/2	+0.07	0.16/3	⋯	−/0

mean	−1.27		−1.23		−0.12		−0.09		+0.11		−0.03		−0.09
±	0.02		0.02		0.02		0.01		0.03		0.04		0.04
rms	0.05		0.05		0.05		0.03		0.08		0.09		0.08

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4. The Chemical Composition of Multiple Populations along the AGB

An illustration of our derived chemical abundances for AGB stars in M4 and NGC 2808 is plotted in Figure 3. The most obvious difference we observe among the AGBs of the two GCs is a much higher range in Al for NGC 2808. M4 AGBs exhibit higher mean abundances for O, Mg, Si, Zn, and the neutron-capture elements Y and Ce. In analogy to what is observed among RGB stars, significant scatter is observed in both clusters for the elements involved in the high-temperature H-burning, namely, O, and Na, plus Al, and Mg in the case of NGC 2808. This fact demonstrates that the AGBs of both M4 and NGC 2808 host multiple stellar populations. Direct evidence of AGB stars with different Na and O abundances in M4 is further provided in Figure 4, where we compare the spectra of two stars with similar atmospheric parameters but different [O/Fe] and [Na/Fe]. We note that the [Fe/H] values are all consistent with cluster membership. The higher spreads in the Zn, Y ii, and Ce ii abundances are most likely just the result of the small number of lines and larger errors. Indeed, the error estimates for these elements listed in Table 2 are relatively high, though we cannot exclude the possibility that they are overestimated, as they are higher than our observed rms values.

**Figure 4.** Comparison between the spectra of the AGB stars 16234740–2631463 (blue) and 16233667–2630397 (red) in M4. These stars have similar atmospheric parameters but different sodium and oxygen abundances.
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A visual comparison between AGB and RGB abundances in the two analyzed clusters is shown in Figure 5. This comparison reveals that AGBs and RGBs exhibit similar distributions for most elements. The comparison does not extend beyond Ni, because Zn, Y ii, and Ce ii abundances are not available for the RGB stars studied in Marino et al. (2008) and Carretta (2014, 2015).

Remarkable exceptions are the distributions of oxygen, sodium, magnesium, and aluminum for NGC 2808, for which we observe differences in range between RGB and AGB stars. To properly compare these variations, we have calculated the difference, δ, between the 90th and 10th percentiles of the distribution of these elements for both RGB and AGB stars. We have also associated an uncertainty with each measurement that has been calculated by means of bootstrapping, as in Milone et al. (2014), and is indicative of the robustness of the δ determinations. For NGC 2808, a large and significant difference between the δ values for RGB and AGB stars has been derived for oxygen: ( ${\delta }_{\mathrm{RGB}}^{[{\rm{O}}/\mathrm{Fe}]}=0.94\pm 0.03$ , ${\delta }_{\mathrm{AGB}}^{[{\rm{O}}/\mathrm{Fe}]}\,=0.60\pm 0.09$ ). For Na, Mg, and Al, the differences in the δ values are also large, but the significance is lower: ( ${\delta }_{\mathrm{RGB}}^{[\mathrm{Na}/\mathrm{Fe}]}\,=0.58\pm 0.05$ , ${\delta }_{\mathrm{AGB}}^{[\mathrm{Na}/\mathrm{Fe}]}=0.40\pm 0.16$ ), ( ${\delta }_{\mathrm{RGB}}^{[\mathrm{Mg}/\mathrm{Fe}]}=0.46\pm 0.18$ , ${\delta }_{\mathrm{AGB}}^{[\mathrm{Mg}/\mathrm{Fe}]}=0.18\pm 0.05$ ), and ( ${\delta }_{\mathrm{RGB}}^{[\mathrm{Al}/\mathrm{Fe}]}=1.13\pm 0.13$ , ${\delta }_{\mathrm{AGB}}^{[\mathrm{Al}/\mathrm{Fe}]}\,=0.77\pm 0.40$ ). Furthermore, as is apparent from Figure 5, the results indicate that, in NGC 2808, stars with the largest abundance of Na and Al and the lowest O and Mg content are clearly absent in the analyzed sample of AGB stars. This suggests that the stellar population in NGC 2808 with extreme chemical composition avoids the AGB phase.

In contrast, the distributions of sodium and oxygen for RGB and AGB stars in M4 are quite similar, even though the mean Na for AGBs is a bit lower.⁷ For this GC, we find ( ${\delta }_{\mathrm{RGB}}^{[{\rm{O}}/\mathrm{Fe}]}=0.26\pm 0.02$ , ${\delta }_{\mathrm{AGB}}^{[{\rm{O}}/\mathrm{Fe}]}=0.24\pm 0.06$ ) for oxygen and ( ${\delta }_{\mathrm{RGB}}^{[\mathrm{Na}/\mathrm{Fe}]}=0.43\pm 0.02$ , ${\delta }_{\mathrm{AGB}}^{[\mathrm{Na}/\mathrm{Fe}]}=0.37\pm 0.07$ ) for sodium. Furthermore, there is no indication of any abundance distribution differences between the RGB and AGB stars for the other elements measured; this is also the case for NGC 2808, except for the differences already mentioned.

In the next subsections, we present results of the light-element (anti-)correlations and the connection between chemical abundances and photometric properties of AGBs in M4 (Section 4.1) and NGC 2808 (Section 4.2) in the context of multiple stellar populations.

4.1. The AGB of M4

The upper panels of Figure 6 compare the positions in the [Na/Fe] versus [O/Fe] and the [Al/Fe] versus [Mg/Fe] planes of the AGB stars analyzed in this paper and the RGB stars from Marino et al. (2008). The AGB stars of M4 clearly exhibit the Na–O anticorrelation, similar to what has been observed along the RGB. Both the RGB and the AGB clearly show two main groups of Na-poor/O-rich and Na-rich/O-poor stars, although the AGB stars could either not reach the highest Na abundances observed on the RGB or be shifted to lower values. There is no evidence for a Mg–Al anticorrelation along the RGB or AGB, but we note that the Na-rich AGB and RGB stars are slightly more Al-rich than the Na-poor stars. The fractions of Na-poor/O-rich RGB and AGB stars are the same within 1σ. Specifically, the 42% ± 5% of RGB stars (37 out of 88) and the 53% ± 13% of AGB stars (9 out of 17) are considered Na-poor.

**Figure 6.** Upper panels: sodium vs. oxygen (left) and aluminum vs. magnesium (right) for AGB stars in M4. The lower panels show the V vs. ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ (left) and ${m}_{{\rm{F}}438{\rm{W}}}$ vs. ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ (right) pseudo-CMDs for M4 stars. The AGB and RGB stars analyzed spectroscopically in this paper and in Marino et al. (2008) are represented with large-filled and small symbols, respectively. The two populations of Na-poor/O-rich and Na-rich/O-poor RGB and AGB stars are shown as blue circles and red triangles, respectively. AGB stars that have not been analyzed spectroscopically are represented with black diamonds. To put the Al abundances on the same scale for AGBs and RGBs, we subtract 0.21 from the RGB abundances (see Figure 5).
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In contrast to our results, MacLean et al. (2016) suggested that, in their sample of M4 AGB stars, only first-generation stars were present, and that, as a result, the M4 AGB lacks second-generation O-depleted, Na-rich stars. There are 10 stars in common between our M4 AGB-star sample and that of MacLean et al. (2016). For these stars, we find that our Na abundances are 0.09 ± 0.01 dex (standard error of the mean) higher and our O abundances are 0.06 ± 0.03 dex lower than the abundances listed in MacLean et al. (2016). These differences are consistent with the slightly different stellar parameters adopted. Our temperatures are 65 K cooler (σ 55 K), and our gravities are 0.13 dex lower (σ 0.19 dex) than their values. We also have different analyzed spectral features: they use the 777 nm O triplet and the 568 nm Na doublet, while we use the forbidden O lines and the Na doublets at ∼568 and ∼616 nm. Intriguingly, if we plot the [Na/Fe] and [O/Fe] values from MacLean et al. (2016) against each other solely for these 10 AGB stars, they clearly fall into two distinct groups, as do our abundances for the same stars (see Figure 6, upper left panel), with one group having lower [O/Fe] and higher [Na/Fe] than the other. The same stars occupy each group in both samples, with one group having mean ([O/Fe], [Na/Fe]) of (0.55, −0.05) and the second having (0.45, 0.22) with the MacLean et al. (2016) abundances. This suggests that the O–Na anticorrelation is indeed present in the MacLean et al. AGB star sample. The mean abundance differences between the two groups are essentially identical to those of our abundances: (Δ[O/Fe], Δ[Na/Fe]) is (−0.10, +0.27) for the MacLean et al. abundances and (−0.17, +0.26) for those presented here. We suggest that the conclusion of MacLean et al. (2016) that the M4 AGB lacks second-generation stars may not be valid.

The apparent discrepancy with MacLean et al. (2016) might be reconciled by possible systematically lower Na in the AGB than in the RGB counterpart. Our sample seems to suggest a systematic difference in the same direction, though much less pronounced. A similar phenomenon was already noticed by Ivans et al. (1999) for Na and Smith & Norris (1993) for the indices S(3839) (mostly sensitive to N) and W(G) (sensitive to C). Both studies report internal abundance dispersion on the AGB as on the RGB but with different abundances. Smith & Norris discussed possible causes for these discrepancies, including the C $\to \,{\rm{N}}$ processing occurring in the RGB envelopes, which could modify the CN surface abundances on the AGB. Furthermore, the same authors showed through synthetic spectra computations that the CN-band strengths on the AGB are less pronounced than those on the RGB due to different atmospheric parameters, causing the lower S(3839) indices observed in AGB stars. This discussion enlightens that Na and O abundances should better represent the primordial abundances of AGB stars, although there might be some evolutionary effect also on these elements. We suggest that star-to-star elemental internal variations are a much more reliable tool when we attempt a comparison between AGB and RGB stars in the context of multiple stellar populations.

To further compare the AGB with the RGB multiple-populations pattern in M4, we take advantage of our photometry. It is well known that the two groups of RGB stars with different chemical compositions populate distinct sequences in the CMD or pseudo-CMD of M4 made with an appropriate combination of ultraviolet and optical filters (e.g., Marino et al. 2008; Monelli et al. 2013). In the lower right panel of Figure 6, we reproduce the ${m}_{{\rm{F}}438{\rm{W}}}$ versus ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-CMD of M4 from Piotto et al. (2015), where the two main populations of M4 are clearly visible along the MS and RGB. The two groups of Na-rich and Na-poor stars selected in the upper left panel clearly correspond to the two main RGBs observed in this pseudo-CMD that we have obtained from HST photometry. We note that AGB stars span a much wider interval in ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ than what we expect from observational errors only. Moreover, although HST photometry is available for only six stars spectroscopically analyzed in this paper, Na-rich AGB stars have smaller ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ values than Na-poor AGB stars with the same luminosity, in close analogy with what has been observed along the RGB.

The lower left panel of Figure 6 plots the V versus ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ pseudo-CMD from ground-based photometry. This diagram has been recently used by Lardo et al. (2017) to show that both first- and second-generation stars climb the AGB of M4. Our diagram confirms that the ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ spread for the AGB is much larger than what we expect from photometric uncertainties only and is comparable to the ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ spread of RGB stars with similar luminosities. As already shown by Monelli et al. (2013), the two populations of Na-rich and Na-poor RGB stars are distributed along the two distinct RGB ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ sequences of this cluster. The populations of Na-poor/O-rich and Na-rich/O-poor AGB stars exhibit a similar behavior, occupying a different location on the V versus ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ plane. These facts confirm that both the ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ and ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ pseudo-colors are efficient in identifying multiple stellar populations along the AGB of GCs and suggest that the stars of both main populations of M4 climb the AGB.

4.2. The AGB of NGC 2808

Studies based on high-resolution spectroscopy of RGB stars have revealed that NGC 2808 exhibits very extended Na–O and Al–Mg anticorrelations, as shown in the upper panels of Figure 7. Here, the gray circles represent the [Na/Fe] versus [O/Fe] and [Al/Fe] versus [Mg/Fe] for RGB stars from Carretta (2014, 2015). Photometry has shown that the MS and RGB of NGC 2808 host at least five stellar populations, named A–E, identified by means of the "chromosome map" tool that is able to maximize the separation between stellar populations with different chemical content. The chromosome map of the NGC 2808 RGBs published in Milone et al. (2015a) is reproduced in the lower right panel of Figure 8, with the different populations represented by different colors. Populations A–E have distinct combinations of light elements/helium abundances (Milone et al. 2015a). (i) Populations E and D have a very high helium content ( $Y\sim 0.38$ and $Y\sim 0.32$ , respectively) and extreme abundances of N, O, Na, Mg, and Al. They are clearly separated from the remaining stars in both the [Na/Fe]–[O/Fe] and the [Al/Fe]–[Mg/Fe] planes and correspond to the groups of stars with $[{\rm{O}}/\mathrm{Fe}]\lt -0.2$ and $-0.2\lt [\mathrm{Mg}/\mathrm{Fe}]\lt 0.2$ shown in Figure 7. (ii) Stellar populations B and C, which have low sodium and aluminum content, are not clearly distinguishable in the diagrams of Figure 7. (iii) There is no spectroscopic information on population-A stars, which, according to multiwavelength photometry, have light-element abundances similar to those of populations B and C.

**Figure 7.** Upper panels: sodium–oxygen and magnesium–aluminum anticorrelation for RGB stars (gray circles; Carretta 2014, 2015) and AGB stars (colored symbols). Lower panels: V vs. ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ (left) and ${m}_{{\rm{F}}438{\rm{W}}}$ vs. ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ (right) pseudo-CMDs of NGC 2808 stars. AGB stars are indicated with black diamonds, while groups 1, 2, and 3 of AGB stars (observed spectroscopically), selected on the basis of their position in the [Na/Fe] vs. [O/Fe] plane, are represented with blue dots, red triangles, and magenta stars, respectively. As in Figures 5 and 6, Al abundances of RGB stars have been decreased by 0.21 dex due to different assumed solar abundances. Smaller shifts have been applied to Na and Mg, whose RGB values have been shifted by −0.05 and +0.10 dex, respectively.
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**Figure 8.** Upper panels: the ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ histogram distribution for AGB (left) and RGB (right) stars in NGC 2808. Lower panels: the ${\rm{\Delta }}{C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ vs. ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ pseudo two-color diagram, or chromosome map, of AGB (left) and RGB (right) stars in NGC 2808 from Milone et al. (2015a). AGB star groups 1, 2, and 3 are shown in blue, red, and magenta, respectively, while large colored symbols indicate our spectroscopic AGB targets. The five populations of RGB stars, A–E, are shown in green, orange, yellow, cyan, and aqua, respectively. Outliers not assigned to any population are represented by black points in both the AGB and RGB chromosome map.
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**Figure 8.** Upper panels: the ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ histogram distribution for AGB (left) and RGB (right) stars in NGC 2808. Lower panels: the ${\rm{\Delta }}{C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ vs. ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ pseudo two-color diagram, or chromosome map, of AGB (left) and RGB (right) stars in NGC 2808 from Milone et al. (2015a). AGB star groups 1, 2, and 3 are shown in blue, red, and magenta, respectively, while large colored symbols indicate our spectroscopic AGB targets. The five populations of RGB stars, A–E, are shown in green, orange, yellow, cyan, and aqua, respectively. Outliers not assigned to any population are represented by black points in both the AGB and RGB chromosome map.
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In this section, we compare the chemical abundances and photometry of AGB stars with the results of multiple RGBs from the literature and attempt to connect the populations along the AGB, RGB, and MS. In the upper panels of Figure 7, we compare our Na–O and Al–Mg anticorrelations obtained from the AGB stars with those on the RGB. This comparison clearly reveals that, among the analyzed AGB stars, [Na/Fe] anticorrelates with [O/Fe], while [Al/Fe] anticorrelates with [Mg/Fe], qualitatively confirming that the AGB exhibits the same chemical pattern as the RGB. The obvious difference between AGBs and RGBs is that AGBs do not reach extremely O-poor and Mg-poor abundances, as RGBs do.

Comparing with the chemical composition observed on the chromosome map by Milone et al. (2015a), the two AGBs with $[{\rm{O}}/\mathrm{Fe}]\lt 0.0$ have chemical compositions consistent with population-D RGB stars and are represented with magenta stars in Figure 7. Two out of the five AGB stars with $[{\rm{O}}/\mathrm{Fe}]\gt 0.0$ have slightly higher sodium and aluminum abundances than the remaining AGB stars. They share the same chemical composition as population-C RGB stars and are represented by red triangles. The blue circles represent the remaining three stars with [Na/Fe] < 0.0, which have chemical abundances similar to those of population-B RGB stars. For simplicity, in the following, we refer to the AGB stars shown in blue, red, and magenta as groups 1, 2, and 3.

The most extreme RGB stars in terms of chemical composition, e.g., those with the highest He and lowest O, are those belonging to population E on the chromosome map. None of the seven analyzed AGB stars belong to this group. This could be due to either the small number of studied AGB stars or the lack of stars with extreme chemical compositions along the AGB. By assuming that population E includes 15% of the total number of NGC 2808 stars (e.g., Simioni et al. 2016), the probability that the lack of AGB population-E stars is due to the small statistical sample, inferred from Monte Carlo simulations, is 0.32. Therefore, without additional AGB stars spectroscopically analyzed, we cannot draw a firm conclusion. By accounting for the radial distribution of different stellar populations, we know that population-E stars are more centrally concentrated than the other stars of NGC 2808, and the fraction of population-E stars with respect to the total number of cluster stars ranges from 21% ± 3% for radial distance $R\lt 0.6$ ' to 9% ± 5% for R > 5 farcm 5 (Simioni et al. 2016). By assuming these extreme values for the fraction of population-E stars, we find that the probability that the lack of AGB population-E stars is due to the small statistical sample is 17% and 50%, respectively. These numbers support the previous conclusion that, due to the small number of analyzed stars, we cannot draw any strong conclusions about the lack of population-E stars along the AGB from spectroscopy only. Nevertheless, population-E RGB stars are likely to evolve to become the hottest Horizontal Branch (HB) stars, which will then fail to reach the AGB. So, even if the statistics are poor, the evidence is consistent with the suggestion that population E does not reach the AGB.

Hence, to further investigate multiple populations along the AGB of NGC 2808, we combine information from both spectroscopy and photometry. As shown in the lower right panel of Figure 7, the AGB of NGC 2808 splits into three distinct sequences in the ${m}_{{\rm{F}}438{\rm{W}}}$ versus ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-CMD. We have analyzed the spectra of one star in each of the three sequences and found that the three stars belong to groups 1, 2, and 3, as previously defined. Therefore, they have distinct light-element content, with the Na abundance decreasing from the lowest to the highest value of ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ .

Similar to the AGB, the RGB of NGC 2808 exhibits three main branches in the ${m}_{{\rm{F}}438{\rm{W}}}$ versus ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-CMD (Milone et al. 2015a; Piotto et al. 2015). Most of the difference in the ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-color among stars with the same magnitude is interpreted as the effect of light-element variation in the spectrum of the star. Thus, we expect that the stars in the three AGB photometric sequences, shown in the lower right panel of Figure 7, are associated with the three corresponding main RGBs.

The fact that the AGB of NGC 2808 is not consistent with a simple population is further supported by the V versus ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ pseudo-CMD from ground-based photometry plotted in the lower left panel of Figure 7, where the ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ dispersion of AGB stars is significantly larger than that expected from observational errors only. In contrast with the RGB, where there is a clear correlation (anticorrelation) between the ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ value of a star and its sodium (oxygen) abundance (Monelli et al. 2013), the small number of analyzed AGB stars prevents us from reaching any strong conclusions about the possible correlation between the light-element abundance of an AGB star and its ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ pseudo-color. Nevertheless, we note that Na-poor AGB stars have smaller ${C}_{{\rm{U}},{\rm{B}},{\rm{I}}}$ values than Na-rich AGB stars, in close analogy with what we have observed in M4.⁸

In the lower left panel of Figure 8, we plot the ${\rm{\Delta }}{C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ versus ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ pseudo-CMD, or chromosome map, of AGB stars in NGC 2808 that we derive by extending to AGB stars the method that we previously introduced for the RGB (Milone et al. 2015a). The lower right panel of the figure reproduces the same diagram derived for RGB stars and marks the main populations, A–E, of NGC 2808 with different colors. In contrast to the observations of MS and RGB stars, where at least five stellar populations are present in the chromosome map, the lower left panel of Figure 8 reveals only three groups of AGB stars. For example, relative to the number of population-D stars on the AGB, there are many fewer stars in the population-E region. Similarly, there is an apparent dearth of AGB stars in the population-A region.

We followed the recipe by McLachlan & Peel (2000) to derive the groups of AGB stars that are statistically significant. Briefly, we determined the maximum-likelihood fit to various numbers of groups and calculated the optimal number of groups by using the Bayesian information criterion (BIC) penalized-likelihood measure for model complexity. To this aim, we varied the size and shape of the distinct groups of AGB stars. For each combination of shape and size, we assumed a number N of populations from 1 to 8 and calculated a BIC value. The most likely explanation corresponds to BIC = 79.2 and N = 4 with the assumption that the groups have equal shape and variable volume and orientation (VEV). Similarly, the second-best BIC value (BIC = 74.2) corresponds to N = 4 and stellar groups with variable shape, volume, and orientation (VVV). The resulting three main groups of stars are shown in blue, red, and magenta in Figure 8. The fourth stellar group includes six outliers that are shown in black. The results support the visual impression that the AGB of NGC 2808 hosts only three main stellar populations, strengthening the idea that stars with extreme He and O do not evolve through the AGB phase.

We note that the chromosome maps of RGB and AGB stars reveal significant differences. Specifically, only one group of stars with ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}\sim 0.0$ is present along the AGB, in contrast with what we observe along the RGB, where we clearly distinguish the two groups of population-A and population-B stars. It remains unclear whether population-A stars do not exist along the AGB or if they are mixed with the population-B stars.

Moreover, while we clearly observe population-D and population-E stars along the RGB and MS, only one group of AGB stars with large values of ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ is present. To investigate the presence of population-E AGB stars in the upper panels of Figure 8, we compare the ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ histogram distribution of AGB stars and RGB stars. As discussed in Milone et al. (2015a; see their Figure 3), the RGB of NGC 2808 is consistent with three main peaks that correspond to populations E and D and the group of populations A+B+C. In contrast, along the AGB, we distinguish a dominant peak associated with stellar groups 1 and 2, as well as a stellar tail mostly due to group 3.

If group 3 includes populations D and E, the fraction of group 3 stars with respect to the total number of AGB stars should be consistent with the ratio between population D+E RGB stars and the total number of RGB stars. We find that group 3 includes 27% ± 5% of the total number of AGB stars. This value significantly differs from the ratio of RGB D+E stars with respect to the total number of AGB stars, which is 50% ± 1%. In contrast, the ratio between RGB D stars and the number of RGB A+B+C+D stars is 30% ± 1% and is consistent within 1σ with the fraction of group 3 AGB stars with respect to the total number of AGB stars. These results are consistent with the lack of population-E stars along the AGB.

5. Summary and Conclusions

We provide a photometric and spectroscopic investigation of multiple populations along the AGB of the Galactic GCs NGC 2808 and M4. Our study is based on (i) high-resolution spectroscopy from FLAMES@VLT, (ii) multiwavelength photometry from the HST UV survey of Galactic GCs and ground-based telescopes, and (iii) proper motions derived by combining stellar positions from the Gaia DR1 and positions derived from images collected with the WFI@MPI 2.2 m telescope.

In NGC 2808, we have identified three main stellar populations of AGB stars that populate three AGB sequences in the ${m}_{{\rm{F}}438{\rm{W}}}$ versus ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-CMD and the ${{\rm{\Delta }}}_{C{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ versus ${{\rm{\Delta }}}_{{\rm{F}}275{\rm{W}},{\rm{F}}814{\rm{W}}}$ pseudo two-color diagram, or chromosome map. The three populations of AGB stars include 41%, 32%, and 27% of the total number of AGB stars and have different O, Na, Mg, and Al abundances. This evidence of multiple populations of AGB stars in NGC 2808 adds to the recent finding by Wang et al. (2016) based on the distribution of Na in the same cluster. By combining information from this paper and the literature, we followed multiple stellar populations along the different evolutionary phases from the MS to the HB and AGB of NGC 2808.

Recent papers show that NGC 2808 hosts five main populations, named A–E, that have been detected along the MS and RGB using multiwavelength photometry and correspond to stellar populations with different helium and light-element abundances (Carretta 2015; Milone et al. 2015a). Unfortunately, there are no spectroscopic studies on population-A stars. On the AGB, we find in this paper that group 1 AGB stars mostly correspond to population B, while group 2 AGB stars are the progeny of population C. Population-D stars are enhanced in helium up to $Y\sim 0.32$ and have low oxygen and high sodium abundances. We have shown that population-D stars climb the AGB and define the sequence of group 3 AGB stars in the ${m}_{{\rm{F}}438{\rm{W}}}$ versus ${C}_{{\rm{F}}275{\rm{W}},{\rm{F}}336{\rm{W}},{\rm{F}}438{\rm{W}}}$ pseudo-CMD.

We did not find any spectroscopic evidence for population-E stars with extreme helium and light-element abundances along the AGB, although the small number of analyzed stars prevents us from reaching strong conclusions on the basis of spectroscopy only. However, this idea is strengthened by the analysis of the chromosome map of AGB stars: by comparing the relative numbers of stars along the distinct AGBs and RGBs, we concluded that the fraction of group 3 AGB stars with respect to the total number of AGB stars is not consistent with the presence of population E along the AGB. The possibility that population-E stars avoid the AGB phase is further supported by the presence of evolved stars that are clearly visible in the ${m}_{{\rm{F}}275{\rm{W}}}$ versus ${m}_{{\rm{F}}275{\rm{W}}}-{m}_{{\rm{F}}814{\rm{W}}}$ CMD of Figure 2 (blue diamonds) and that have been interpreted by Castellani et al. (2003) as AGB-manqué stars. Since population-E stars have extreme helium abundances ( $Y\sim 0.38$ ; D'Antona et al. 2005; Piotto et al. 2007; Milone et al. 2015a), our findings support the prediction from stellar evolution that He-rich stars in stellar populations avoid the AGB phase and evolve as AGB-manqué stars (e.g., Greggio & Renzini 1990; D'Cruz et al. 2000; Brown et al. 2001; Moehler et al. 2004; Gratton et al. 2010; Chantereau et al. 2016).

Specifically, we note that NGC 2808 is considered quite a young GC (age = 11.5 ± 0.75 Gyr; Dotter et al. 2010; Milone et al. 2014). The lack of stars with extreme helium abundances along the AGB of NGC 2808 would be in agreement with the conclusion by Charbonnel & Chantereau (2016), who predicted that the internal AGB helium spread of a GC with [Fe/H] = −1.15 and age = 11.5 is smaller than ${\rm{\Delta }}Y\ \sim$ 0.09. The GC M4 has a metallicity and age (12.50 ± 0.50 Gyr) similar to that of NGC 2808 (although NGC 2808 seems slightly younger; Marín-Franch et al. 2009). In contrast, in the context of multiple populations, M4 looks much less complex than NGC 2808. This cluster hosts two main populations of stars with different C, N, O, and Na abundances that have been observed along the MS and RGB (e.g., Marino et al. 2008; Piotto et al. 2015). The Na-poor/O-rich stars are slightly enhanced in helium by ${\rm{\Delta }}Y\sim 0.02$ (Nardiello et al. 2015a) with respect to the primordial value; they populate the red HB, while the blue HB stars are more depleted in oxygen and have higher in sodium (Marino et al. 2011). Despite the possible presence of systematics between the abundances of AGB and RGB stars, we find that the chemical abundance dispersions of AGB stars in M4 are not consistent with a simple stellar population and provide both photometric and spectroscopic evidence that stars belonging to different populations ascend the AGB of this cluster.

In conclusion, while the extremely He-rich (Na-rich/O-poor) population of the RGB very likely misses the AGB phase in NGC 2808, we do not find any strong evidence for a lack of some of the RGB populations on the AGB in M4. Except for the lack of the extremely He-enhanced population of NGC 2808, the number ratios of second-population AGB stars are similar to those observed on the RGB in both GCs. These results suggest that only a high level of He enrichment, such as that in the extreme population of NGC 2808, is able to make a star avoid the AGB phase. At a given metallicity and age, He seems to be the main parameter controlling evolution toward the AGB.

We are grateful to the referee for several suggestions that have improved the quality of this manuscript. We thank P. B. Stetson, who has kindly provided us with the ground-based photometric catalogs of M4 and NGC 2808; Ben MacLean for supplying the positions of his M4 AGB stars to enable a cross-match with our sample; and Simon Campbell for useful discussion. This work has been supported by the Australian Research Council (grants DE160100851, DE150101816, DP150100862, FT140100554, and FL110100012).