GLOBAL AND NONGLOBAL PARAMETERS OF HORIZONTAL-BRANCH MORPHOLOGY OF GLOBULAR CLUSTERS^*

Since the early 1950s, metallicity has been considered the main parameter that determines the horizontal-branch (HB) morphology in globular clusters (GCs; e.g., Arp et al. 1952). Within a few years, evidence that some GCs with similar metallicity exhibit different HB morphologies suggested that at least a second parameter (2ndP) is required to properly characterize the HB morphology of GCs (e.g., Sandage & Wallerstein 1960; van den Bergh 1965). Since then, the so-called 2ndP problem has been widely investigated by many authors.

Several candidates have been suggested as possible 2ndPs, including mass loss (e.g., Peterson 1982; Catelan 2000), stellar rotation (e.g., Mengel & Gross 1976; Fusi-Pecci & Renzini 1978), planetary systems (e.g., Soker 1998; Siess & Livio 1999), magnetic fields (e.g., Rood & Seitzer 1981), and GC ellipticity (Norris 1983), but a comprehensive picture is still lacking. Age (e.g., Searle & Zinn 1978; Catelan & de Freitas Pacheco 1993; Lee et al. 1994), GC central density (e.g., Fusi Pecci et al. 1993), GC mass (e.g., Recio-Blanco et al. 2006), and helium abundance (e.g., Sandage & Wildey 1967; van den Bergh 1967) are among the best candidates. We refer the reader to the papers by Freeman & Norris (1981), Catelan (2009), Dotter et al. (2010), and Gratton et al. (2010) and references therein for reviews on HB stars and the 2ndP phenomenon in GCs.

The study of GCs has changed dramatically in recent years because of the overwhelming evidence for the existence of multiple stellar populations in GCs. In this way of thinking, a GC is made up of a first generation of stars, formed from the GC's primordial gas cloud, and at least one later generation, formed from a dilution of the primordial gas and the chemical yields of the high- and intermediate-mass stars of the first generation.

The possibility of GC self-enrichment, especially as it relates to enhanced helium, as the cause for the variation of the HB morphology has been investigated by several authors, as multiple stellar populations with different helium abundance can indeed explain features such as tails and multimodalities in the HBs of GCs (e.g., Ferraro et al. 1998; D'Antona et al. 2002, 2005; Piotto et al. 2007; Caloi & D'Antona 2008; Gratton et al. 2010). The idea of a connection between multiple stellar populations and HB morphology arose in the early 1980s, when pioneering papers showed that the cyanogen distribution is closely connected to the shape of the HB (e.g., Norris 1981; Norris et al. 1981; Smith & Norris 1993); this result has been confirmed by recent studies of HB stars.

In this context the GC M 4 is exemplary. High-resolution spectroscopy of red giant branch (RGB) stars reveals that this GC hosts two stellar populations with different Na and O abundances, while photometry reveals two RGBs in the U versus U − B color–magnitude diagram (CMD). The HB of M 4 is bimodal and well-populated on both sides of the RR Lyrae gap (Marino et al. 2008). The bimodality in Na and O is also present among the HB stars. Blue-HB stars belong to the second population and are O-poor and Na-rich, while red HB stars are first population (Marino et al. 2008, 2011b). Similar analyses of Na and O in HB stars in other GCs show that first-generation HB stars preferentially populate the reddest HB segment, while second-generation HB stars tend toward bluer colors (Villanova et al. 2009; Gratton et al. 2011, 2012, 2013; Lovisi et al. 2012; Marino et al. 2013). More recently (Marino et al. 2014) inferred from direct spectroscopic measurements that Na-rich HB stars of NGC 2808 are also strongly helium enhanced.

Several factors influence HB morphology, and it is difficult to disentangle the different effects. An important point in the study of the 2ndP is that the metric used to characterize HB morphology is not objective; the chosen way of representing the HB stars in a GC as a number has a nontrivial influence on the results of the investigation. Most studies to date adopt a single HB morphology metric. Different studies, using different metrics, can easily reach conflicting conclusions about the identity of the 2ndP.

Consider the following two examples. Recio-Blanco et al. (2006) defined their HB morphology metric as the maximum effective temperature along the HB and found that more massive GCs tend to have hotter HBs. Dotter et al. (2010) measured the median color difference between the HB and the RGB at the level of the HB (Δ(V − I)) from Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) photometry of 60 GCs and demonstrated that after the metallicity dependence is accurately removed, Δ(V − I) correlates most strongly with GC age.

Dotter et al. (2010, see their Figure 2) compared Δ(V − I) with the widely used HB Type index⁹ and the maximum effective temperature along the HB as defined by Recio-Blanco et al. (2006). The comparison shows that Δ(V − I) and HB Type are closely correlated but that Δ(V − I) and Recio-Blanco et al.'s maximum HB temperatures are not: they have very different sensitivities. It is not surprising that a metric sensitive to the extremes of the distribution correlates with a different (second) parameter than a metric sensitive to the center of the distribution. The problem is that 2ndP studies typically select only one HB morphology metric and the conclusions are influenced by that choice.

We suggest that a more effective way to proceed is to consider more than one HB morphology metric simultaneously. For maximum effect, these metrics should share a simple, common definition but not be closely correlated with each other. The motivation for this approach is not only based on the practicalities outlined above. Freeman & Norris (1981) argued that two parameters, one global and one local, may be needed to fully describe the observed variations in HB morphology. The local parameter is one that varies within a single GC; the global parameter is one that varies among the GC population.

The aim of this paper is an empirical investigation of the parameters governing the HB morphology of GCs, in the context of the classical 2ndP phenomenon. To do this we use the homogeneous high-accuracy photometry from GO 10775, the ACS Survey of Galactic GCs (PI: A. Sarajedini; Sarajedini et al. 2007), and from GO 11586 (PI: A. Dotter; Dotter et al. 2011), and additional photometry from HST to reinvestigate the HB morphology in GCs in light of the new findings on multiple stellar populations in GCs and of the global versus nonglobal parameter idea by Freeman & Norris (1981). The paper is organized as follows. Section 2 describes the observational data. Section 3 introduces the quantities adopted to describe the HB morphology and defines the new HB morphology parameters L1 and L2. Section 4 assembles a variety of GC parameters from the literature. Section 5 compares these parameters with L1 and L2. Section 6 discusses our findings in the context of similar studies in the literature. Finally, we summarize our findings in Section 7.

We used the photometric catalogs obtained from GO 10775 and GO 11586 that include homogeneous photometric and astrometric measurements for 65 and 6 GCs, respectively. For each of them, the data set consists of one short and four or five long exposures in the F606W and F814W bandpasses. We excluded from GO 10775 three GCs: Pal 1 and E 3 for the lack of identifiable HB stars and Pal 2 for the extreme differential reddening. The details concerning the data, the data reduction, and the calibration are given in Anderson et al. (2008) and Dotter et al. (2011).

In their study of the HB in GCs, Dotter et al. (2010) emphasized the importance of properly accounting for the outer halo, where the 2ndP is more evident. To increase the number of outer-halo GCs, we have extended the GO 10775 and GO 11586 to six more GCs: AM-1, Eridanus, NGC 2419, Pal 3, Pal 4, and Pal 14. For both Pal 4 and Pal 14, we used 2 × 60 s F606W and 2 × 80 s F814W ACS/WFC images from GO 10622 (PI: Dolphin), while for NGC 2419 we used the F606W and F814W magnitudes published by di Criscienzo et al. (2011). These data have been reduced as already described in Anderson et al. (2008), Dotter et al. (2011), and di Criscienzo et al. (2011). For Pal 3 and Eridanus we used ground-based V, I photometry from Stetson et al. (1999) and for AM-1 Wide Field Planetary Camera 2 (WFPC2) photometry from Dotter et al. (2008) in F555W and F814W. Photometry for these three GCs has been transformed into F606W and F814W ACS/WFC bands by using the relationships given in Sirianni et al. (2005).

Photometry has been corrected for spatial photometric zero-point variation both due to differential reddening and small inaccuracies in the point-spread function model (Anderson et al. 2008). For most GCs we used the corrected magnitudes and colors published by Milone et al. (2012b; 59 GCs), Piotto et al. (2012; NGC 6715), di Criscienzo et al. (2011; NGC 2419), and Bellini et al. (2010; ω Centauri). For the remaining GCs we corrected the photometry following the procedure described in Milone et al. (2012b).

To investigate the HB morphology, we defined two quantities: L1, the color difference between the RGB and the coolest border the HB, and L2, the color extension of the HB.¹⁰

The procedure to determine L1 and L2 is illustrated in Figure 1 for the case of NGC 5904 (M 5). We selected by eye a sample of HB stars that we plotted as blue circles in the lower panel and a sample of RGB stars that we represented with red points. The RGB sample includes all the RGB stars with luminosity differing by less than ±0.1 F606W mag from the mean level of the HB (F606W_HB), where the F606W_HB values are taken from Dotter et al. (2010, Table 1). The histograms of the normalized m_F606W − m_F814W color distribution for the HB and RGB sample are shown in the upper panel and colored blue and red, respectively. We have defined two points on the HB, P_A and P_B, whose colors correspond to the fourth and the 96 percentile of the color distribution of HB stars. The color of the third point P_C is assumed as the median color of RGB stars; L1 is defined as the color difference between P_C and P_B and L2 as the color difference between P_B and P_A. Uncertainties on P_A, P_B, P_C, L1, and L2 are estimated for each GC by bootstrapping with replacements performed 1000 times on both the RGB and the HB. The error bars indicate one standard deviation (68.27th percentile) of the bootstrapped measurements. The colors of P_A, P_B, P_C, the values of L1 and L2, and the corresponding errors, are listed in Table 1.

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RR Lyrae have been observed at random phases, and some of them could lie outside the instability strip. Similarly to what has been done in previous papers on the second parameter (e.g.,  Gratton et al. 2010; Dotter et al. 2010), we included in our analysis only those RR Lyrae that are close to the instability strip. In the Appendix we investigate the impact of excluding RR Lyrae that might be out of the instability strip in the determination of L1 and L2 and conclude that this does not affect the conclusions of our paper.

In the next section we shall compare the L1 and L2 HB morphology indicators with the physical and morphological GC parameters described here. Metallicity ([Fe/H]), absolute visual magnitude (M_V), central velocity dispersion (σ_V), ellipticity (), central concentration (c), core relaxation timescale (τ_c), half-mass relaxation timescale (τ_hm), logarithm of central stellar density (ρ₀), central surface brightness (μ_V), and Galactocentric distance (R_GC) are extracted from the 2010 edition of the Harris (1996) catalog. The specific frequency of RR Lyrae variables (S_RR Lyrae) is taken from the 2003 edition of the Harris (1996) catalog. The fraction of binary stars have been measured by Milone et al. (2012b, 2012c) in the core of the GC ($f_{\rm bin}^{\rm C}$), in the region between the core and the half-mass radius ($f_{\rm bin}^{\rm C-HM}$), and outside the half-mass radius ($f_{\rm bin}^{\rm oHM}$). We also use age and helium measurements, and some indicators of light-element variations, as discussed in detail in Sections 4.1 and 4.2, respectively.

4.1. Age

4.2. Light-element and Helium Variations

In this section we investigate the correlations among L1, L2, and several physical and morphological parameters of their host GCs. Specifically, relations with metallicity, absolute magnitude, and age are discussed in Sections 5.1, 5.2, and 5.3, respectively. Section 5.4 investigates the correlations with the internal variations of the light elements and helium, while relations between L1 and L2 and the other parameters introduced in Section 4 are analyzed in Section 5.5.

When we compare two variables, as we do in the next section for L1, L2, and [Fe/H], we use the Spearman's rank correlation coefficient r to estimate the statistical dependence between the two. Uncertainties in r are estimated by means of bootstrapping statistics. We generated 1000 resamples of the observed data set, of equal size, and for each resample (1) (which is generated by random sampling with replacement from the original data set) we estimated r_i. We considered the dispersion of the r_i measurements (σ_r) as indicative of the robustness of r and provide the number of included GCs (N).

5.1. Metallicity

The left panel of Figure 2 shows L1 against the GC metallicity. An inspection of this plot reveals that all the metal-rich GCs have small L1 values and, hence, red HBs. At lower metallicities, there are GCs with almost the same iron abundance and yet different L1 values.¹¹ This reflects the basic 2ndP phenomenon. Indeed, if the all of the GCs followed the same relation in the L1 versus [Fe/H] plane, we would assume that metallicity alone is sufficient to determine L1. The fact that we observe clusters with the same [Fe/H] but different L1 values indicates that apart from metallicity, at least one more parameter is at work.

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Our finding that the analyzed GCs populate distinct regions in the L1 versus [Fe/H] plane and that the 2ndP phenomenon is absent among the majority of metal-rich GCs motivated us to define three groups of GCs as follows:

• 1.  
  The first group, G1, includes all the metal-rich GCs ([Fe/H] ⩾−1.0).
• 2.  
  The second, G2, is made of GCs with [Fe/H] <−1.0 and L1 ⩽ 0.4.
• 3.  
  The remaining GCs with L1 > 0.4 belong to G3.

Since the 2ndP phenomenon is absent among the majority of G1 GCs, we will also consider a group that includes all the GCs in G2 and G3, hereafter G2+G3. The statistical analysis presented in the following will be provided for all the GCs together, as well as for the different groups separately.

There is a significant anticorrelation between L1 and [Fe/H] among G2 GCs, with a Spearman's rank correlation coefficient r_G2 = −0.70 (σ_{r, G2} = 0.08, N_G2 = 38) that drops to −0.88 if we consider G1 and G2 GCs together, an even stronger correlation. The Spearman's correlation coefficients and the corresponding σ_r values are listed in Table 3 for the groups of GCs defined above. There is no correlation between L2 and [Fe/H], either for G1 or G3 GCs.

Table 3. Spearman's Rank Correlation Coefficients Indicating the Statistical Dependence between L1 (Columns 2–5), L2 (Columns 6–9), and Several Parameters of the Host GCs for G1, G2, G3, and G2+G3 GCs

Parameter	L1				L2
	G1	G2	G3	G2 + G3	G1	G2	G3	G2 + G3
[Fe/H]	r = −0.13	r = −0.70	r = 0.42	r = −0.47	r = −0.08	r = 0.19	r = 0.57	r = 0.24
	N = 15	N = 38	N = 21	N = 59	N = 15	N = 38	N = 21	N = 59
	σr = 0.26	σr = 0.08	σr = 0.18	σr = 0.11	σr = 0.31	σr = 0.16	σr = 0.13	σr = 0.13
MV	r = 0.09	r = −0.23	r = −0.08	r = 0.09	r = −0.57	r = −0.89	r = −0.71	r = −0.80
	N = 15	N = 38	N = 21	N = 59	N = 15	N = 38	N = 21	N = 59
	σr = 0.13	σr = 0.16	σr = 0.24	σr = 0.13	σr = 0.23	σr = 0.05	σr = 0.13	σr = 0.06
AGED10	r = 0.35	r = 0.76	r = −0.22	r = 0.72	r = 0.23	r = 0.28	r = −0.07	r = −0.06
	N = 15	N = 37	N = 21	N = 58	N = 15	N = 37	N = 21	N = 58
	σr = 0.20	σr = 0.08	σr = 0.20	σr = 0.08	σr = 0.29	σr = 0.18	σr = 0.24	σr = 0.16
AGEMF09	r = 0.38	r = 0.47	r = −0.05	r = 0.38	r = 0.30	r = −0.29	r = 0.17	r = −0.30
	N = 15	N = 26	N = 20	N = 46	N = 15	N = 26	N = 20	N = 46
	σr = 0.26	σr = 0.18	σr = 0.20	σr = 0.15	σr = 0.30	σr = 0.16	σr = 0.22	σr = 0.14
AGEDA05	r = 0.60	r = 0.72	r = −0.32	r = 0.57	r = −0.60	r = −0.50	r = 0.03	r = −0.49
	N = 5	N = 25	N = 11	N = 36	N = 5	N = 25	N = 11	N = 36
	σr = 0.37	σr = 0.13	σr = 0.36	σr = 0.13	σr = 0.40	σr = 0.14	σr = 0.36	σr = 0.13
AGEV13	r = 0.62	r = 0.73	r = −0.21	r = 0.68	r = 0.62	r = 0.01	r = −0.22	r = −0.34
	N = 12	N = 31	N = 18	N = 49	N = 12	N = 31	N = 18	N = 49
	σr = 0.20	σr = 0.09	σr = 0.25	σr = 0.10	σr = 0.21	σr = 0.20	σr = 0.25	σr = 0.16
AGED10a	r = 0.35	r = 0.76	r = −0.22	r = 0.74	r = 0.39	r = 0.30	r = −0.08	r = −0.08
	N = 14	N = 34	N = 21	N = 55	N = 14	N = 34	N = 21	N = 55
	σr = 0.20	σr = 0.08	σr = 0.20	σr = 0.07	σr = 0.26	σr = 0.19	σr = 0.24	σr = 0.17
AGEMF09a	r = 0.41	r = 0.40	r = −0.05	r = 0.33	r = 0.31	r = −0.29	r = 0.17	r = −0.30
	N = 14	N = 23	N = 20	N = 43	N = 14	N = 23	N = 20	N = 43
	σr =	σr = 0.21	σr = 0.20	σr = 0.16	σr = 0.30	σr = 0.16	σr = 0.22	σr = 0.14
AGEDA05a	r = 0.60	r = 0.71	r = −0.32	r = 0.53	r = −0.60	r = −0.53	r = 0.03	r = −0.47
	N = 5	N = 23	N = 11	N = 34	N = 5	N = 23	N = 11	N = 34
	σr = 0.37	σr = 0.14	σr = 0.36	σr = 0.15	σr = 0.40	σr = 0.14	σr = 0.36	σr = 0.13
AGEV13a	r = 0.62	r = 0.70	r = −0.21	r = 0.65	r = 0.62	r = 0.07	r = −0.22	r = −0.32
	N = 12	N = 28	N = 18	N = 46	N = 12	N = 28	N = 18	N = 46
	σr = 0.20	σr = 0.10	σr = 0.25	σr = 0.10	σr = 0.21	σr = 0.22	σr = 0.25	σr = 0.16
log(ΔY)	—	r = −0.30	—	r = −0.71	—	r = 0.70	—	r = 0.89
	N = 2	N = 5	N = 2	N = 7	N = 2	N = 5	N = 2	N = 7
	—	σr = 0.54	—	σr = 0.31	—	σr = 0.37	—	σr = 0.17
σV	r = −0.54	r = −0.07	r = 0.10	r = −0.02	r = 0.66	r = 0.79	r = 0.40	r = 0.46
	N = 6	N = 7	N = 13	N = 38	N = 6	N = 7	N = 13	N = 38
	σr = 0.38	σr = 0.22	σr = 0.30	σr = 0.15	σr = 0.29	σr = 0.12	σr = 0.28	σr = 0.14
c	r = 0.13	r = 0.08	r = 0.26	r = −0.10	r = 0.07	r = 0.54	r = 0.52	r = 0.55
	N = 14	N = 34	N = 16	N = 50	N = 14	N = 34	N = 16	N = 50
	σr = 0.25	σr = 0.17	σr = 0.23	σr = 0.14	σr = 0.31	σr = 0.14	σr = 0.23	σr = 0.10
μV	r = −0.01	r = −0.07	r = −0.23	r = −0.07	r = −0.65	r = −0.76	r = −0.68	r = −0.57
	N = 14	N = 37	N = 21	N = 58	N = 14	N = 37	N = 21	N = 58
	σr = 0.28	σr = 0.18	σr = 0.23	σr = 0.14	σr = 0.21	σr = 0.10	σr = 0.15	σr = 0.11
ρ	r = −0.09	r = 0.05	r = 0.29	r = 0.14	r = 0.69	r = 0.69	r = 0.56	r = 0.47
	N = 14	N = 37	N = 21	N = 58	N = 14	N = 37	N = 21	N = 58
	σr = 0.26	σr = 0.18	σr = 0.21	σr = 0.14	σr = 0.18	σr = 0.12	σr = 0.19	σr = 0.13
log (τc)	r = −0.31	r = 0.09	r = −0.28	r = −0.21	r = −0.15	r = −0.35	r = −0.8	r = −0.20
	N = 14	N = 37	N = 21	N = 58	N = 14	N = 37	N = 21	N = 58
	σr = 0.21	σr = 0.18	σr = 0.20	σr = 0.13	σr = 0.32	σr = 0.18	σr = 0.23	σr = 0.15
log (τh)	r = −0.29	r = 0.19	r = −0.21	r = −0.33	r = −0.28	r = −0.13	r = −0.23	r = 0.08
	N = 15	N = 37	N = 21	N = 58	N = 15	N = 37	N = 21	N = 58
	σr = 0.28	σr = 0.18	σr = 0.18	σr = 0.13	σr = 0.27	σr = 0.18	σr = 0.24	σr = 0.15
	r = −0.10	r = 0.45	r = 0.14	r = 0.07	r = 0.05	r = −0.05	r = −0.16	r = 0.09
	N = 12	N = 30	N = 17	N = 47	N = 12	N = 30	N = 17	N = 47
	σr = 0.32	σr = 0.14	σr = 0.27	σr = 0.15	σr = 0.33	σr = 0.20	σr = 0.27	σr = 0.15
log (RGC[kpc)	r = −0.08	r = −0.18	r = −0.29	r = −0.45	r = −0.50	r = −0.44	r = −0.50	r = −0.17
	N = 15	N = 38	N = 21	N = 59	N = 15	N = 38	N = 21	N = 59
	σr = 0.27	σr = 0.16	σr = 0.19	σr = 0.11	σr = 0.22	σr = 0.16	σr = 0.19	σr = 0.15
SRR Lyrae	r = −0.01	r = −0.26	r = −0.48	r = −0.66	r = −0.12	r = −0.68	r = −0.51	r = −0.32
	N = 14	N = 36	N = 20	N = 56	N = 14	N = 36	N = 20	N = 56
	σr = 0.27	σr = 0.17	σr = 0.17	σr = 0.11	σr = 0.22	σr = 0.19	σr = 0.24	σr = 0.13
${\it f}_{\rm bin}^{\rm C}$	r = 0.19	r = −0.03	r = −0.22	r = −0.18	r = −0.12	r = −0.68	r = −0.51	r = −0.32
	N = 9	N = 16	N = 15	N = 31	N = 9	N = 16	N = 15	N = 31
	σr = 0.40	σr = 0.30	σr = 0.26	σr = 0.18	σr = 0.36	σr = 0.15	σr = 0.25	σr = 0.17
${\it f}_{\rm bin}^{\rm C-HM}$	r = −0.09	r = −0.19	r = −0.18	r = −0.15	r = −0.67	r = −0.47	r = −0.56	r = −0.27
	N = 11	N = 24	N = 16	N = 40	N = 11	N = 24	N = 16	N = 40
	σr = 0.35	σr = 0.19	σr = 0.29	σr = 0.14	σr = 0.23	σr = 0.20	σr = 0.23	σr = 0.15
${\it f}_{\rm bin}^{\rm oHM}$	r = 0.49	r = −0.14	r = −0.07	r = −0.02	r = −0.75	r = −0.45	r = −0.69	r = −0.39
	N = 11	N = 18	N = 17	N = 35	N = 11	N = 18	N = 17	N = 35
	σr = 0.29	σr = 0.24	σr = 0.28	σr = 0.17	σr = 0.18	σr = 0.21	σr = 0.19	σr = 0.15
WRGB	—	r = −0.45	r = −0.08	r = −0.23	—	r = 0.50	r = 0.35	r = 0.30
	N = 3	N = 9	N = 9	N = 18	N = 3	N = 9	N = 9	N = 18
	—	σr = 0.31	σr = 0.35	σr = 0.22	—	σr = 0.35	σr = 0.37	σr = 0.23
IQR([O/Na])	—	r = −0.13	r = 0.00	r = −0.02	—	r = 0.82	r = 0.90	r = 0.41
	N = 4	N = 13	N = 8	N = 21	N = 4	N = 13	N = 8	N = 21
	—	σr = 0.26	σr = 0.38	σr = 0.21	—	σr = 0.13	σr = 0.17	σr = 0.21
RCN	—	r = 0.29	—	r = 0.13	—	r = 0.47	—	r = 0.38
	N = 2	N = 11	N = 3	N = 14	N = 2	N = 11	N = 3	N = 14
	—	σr = 0.30	—	σr = 0.27	—	σr = 0.24	—	σr = 0.27
R	r = 0.07	r = 0.17	r = −0.07	r = 0.11	r = 0.26	r = 0.19	r = −0.32	r = 0.07
	N = 8	N = 19	N = 7	N = 26	N = 8	N = 19	N = 7	N = 26
	σr = 0.40	σr = 0.26	σr = 0.39	σr = 0.19	σr = 0.35	σr = 0.24	σr = 0.42	σr = 0.22
fPOPI	—	r = 0.38	r = −0.32	r = −0.03	—	r = −0.12	r = −0.07	r = −0.02
	N = 4	N = 8	N = 7	N = 15	N = 4	N = 8	N = 7	N = 15
	—	σr = 0.39	σr = 0.36	σr = 0.29	—	σr = 0.38	σr = 0.36	σr = 0.27
Y(R')	r = −0.27	r = −0.16	r = 0.12	r = 0.07	r = 0.15	r = 0.44	r = −0.06	r = 0.04
	N = 11	N = 25	N = 17	N = 42	N = 11	N = 25	N = 17	N = 42
	σr = 0.36	σr = 0.23	σr = 0.28	σr = 0.15	σr = 0.34	σr = 0.21	σr = 0.26	σr = 0.16
L2	r = −0.35	r = 0.05	r = 0.10	r = −0.35	r = 1.00	r = 1.00	r = 1.00	r = 1.00
	N = 15	N = 38	N = 21	N = 59	N = 15	N = 38	N = 21	N = 59
	σr = 0.24	σr = 0.17	σr = 0.25	σr = 0.15	σr = 0.00	σr = 0.00	σr = 0.00	σr = 0.00
HBR	r = 0.43	r = 0.79	r = 0.48	r = 0.91	r = −0.24	r = 0.44	r = −0.04	r = −0.15
	N = 13	N = 37	N = 21	N = 58	N = 13	N = 37	N = 21	N = 58
	σr = 0.28	σr = 0.07	σr = 0.16	σr = 0.02	σr = 0.27	σr = 0.16	σr = 0.26	σr = 0.17
δM	r = −0.48	r = −0.36	r = 0.12	r = −0.19	r = 0.39	r = 0.66	r = 0.80	r = 0.48
	N = 12	N = 25	N = 17	N = 42	N = 12	N = 25	N = 17	N = 42
	σr = 0.24	σr = 0.20	σr = 0.29	σr = 0.30	σr = 0.30	σr = 0.14	σr = 0.15	σr = 0.13
ΔMmedian	r = −0.47	r = −0.68	r = 0.50	r = 0.00	r = 0.39	r = 0.63	r = 0.35	r = 0.14
	N = 11	N = 24	N = 17	N = 41	N = 11	N = 24	N = 17	N = 41
	σr = 0.28	σr = 0.10	σr = 0.19	σr = 0.17	σr = 0.30	σr = 0.17	σr = 0.22	σr = 0.17
Lt	—	r = −0.21	r = 0.68	r = −0.06	—	r = 0.62	r = 0.38	r = 0.44
	N = 4	N = 23	N = 11	N = 34	N = 4	N = 23	N = 11	N = 34
	—	σr = 0.19	σr = 0.21	σr = 0.19	—	σr = 0.17	σr = 0.33	σr = 0.18
log (Teff, MAX)	r = 0.10	r = 0.10	r = −0.14	r = 0.27	r = 0.71	r = 0.77	r = 0.90	r = 0.20
	N = 8	N = 17	N = 8	N = 25	N = 8	N = 17	N = 8	N = 25
	σr = 0.44	σr = 0.25	σr = 0.34	σr = 0.20	σr = 0.26	σr = 0.16	σr = 0.14	σr = 0.21
Δ(V − I)	r = 0.21	r = 0.66	r = 0.64	r = 0.76	r = 0.49	r = 0.57	r = 0.75	r = 0.10
	N = 15	N = 38	N = 21	N = 59	N = 15	N = 38	N = 21	N = 59
	σr = 0.26	σr = 0.12	σr = 0.14	σr = 0.08	σr = 0.23	σr = 0.14	σr = 0.14	σr = 0.14

Notes. The values of σ_r, which provide an estimate of the robustness of r measurements, and the numbers of analyzed GCs (N) are also listed. ^aGCs with double SGB excluded from the analysis.

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Apart from NGC 6388 and NGC 6441, all G1 GCs host a purely red HB and have L2 values smaller than the majority of the other GCs. In G2 and G3, GCs metallicity is not responsible for the extension of the HB (L2) as shown in the right panel of Figure 2.

5.2. Absolute Magnitude

The left panel of Figure 3 shows that there is no significant correlation between L1 and the GC absolute luminosity for any of the groups of GCs defined above. In contrast, both G2 and G3 GCs exhibit significant anticorrelations between L2 and the absolute GC magnitude, which relates to the GC mass assuming all GCs have roughly the same mass-to-light ratio. This is shown in the right panel of Figure 3, where we plot L2 as a function of M_V. The Spearman's rank correlation coefficient is r_G2 = −0.89 (σ_{r, G2} = 0.05, N = 38) and r_G3 = −0.71 (σ_{r, G3} = 0.13, N_G3 = 21) for the G2 and G3 samples, respectively, r_{G2 + G3}=−0.80 (σ_{r, G2 + G3} = 0.06, N_{G2 + G3} = 59) for G2+G3 GCs.

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5.3. Age

Histograms of the age distributions for G1, G2, and G3 GCs are shown in the upper panels of Figure 4 for the age measurements from Dotter et al. (2010, 2011; left panel) and Marín-Franch et al. (2009; right panel) and in Figure 5 for the age measurements of De Angeli et al. (2005; left panel) and VandenBerg et al. (2013) and Leaman et al. (2013; right panel). On average, G3 GCs are systematically older than G2 GCs, and this result is independent of the adopted age scale. Specifically, independently from the four adopted age scales, G2 GCs are, on average, younger than G3 ones by ∼1 Gyr with G3 GCs clustered around the value of ∼13 Gyr and G2 GCs spanning a wider age interval. The mean ages of G2 and G3 GCs are listed in Table 4.

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Table 4. Average Ages for G2 and G3 GCs

Author	AgeG2	AgeG3
	(Gyr)	(Gyr)
	All
Dotter et al. (2010, 2011)	12.3 ± 0.1	13.2 ± 0.1
Marín-Franch et al. (2009)	12.8 ± 0.2	13.4 ± 0.2
De Angeli et al. (2005)	12.3 ± 0.3	13.2 ± 0.3
VandenBerg et al. (2013) and Leaman et al. (2013)	11.8 ± 0.1	12.5 ± 0.1
	Metal-rich sample
Dotter et al. (2010, 2011)	12.1 ± 0.2	12.9 ± 0.3
Marín-Franch et al. (2009)	12.7 ± 0.3	13.3 ± 0.1
De Angeli et al. (2005)	11.6 ± 0.5	12.8 ± 0.1
VandenBerg et al. (2013) and Leaman et al. (2013)	11.4 ± 0.2	12.3 ± 0.5
	Metal-intermediate sample
Dotter et al. (2010, 2011)	11.9 ± 0.2	13.0 ± 0.1
Marín-Franch et al. (2009)	12.5 ± 0.3	13.3 ± 0.3
De Angeli et al. (2005)	11.7 ± 0.5	13.0 ± 0.5
VandenBerg et al. (2013) and Leaman et al. (2013)	11.6 ± 0.1	12.2 ± 0.2
	Metal-poor sample
Dotter et al. (2010, 2011)	13.1 ± 0.1	13.3 ± 0.1
Marín-Franch et al. (2009)	13.0 ± 0.3	13.4 ± 0.3
De Angeli et al. (2005)	13.5 ± 0.3	14.0 ± 0.5
VandenBerg et al. (2013) and Leaman et al. (2013)	12.4 ± 0.1	12.7 ± 0.1

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The L1 is plotted as a function of GC age in the middle panels of Figures 4 and 5. There is a positive correlation between age and L1 for G2 GCs, with older G2 GCs having, on average, greater L1 values. The Spearman's coefficient is high r_G2 ⩾ 0.70, except when we adopt ages from Marín-Franch et al. (2009), indicating that in the latter case the significance level is low.¹²

As a further check we have divided GCs into three subgroups with almost the same metallicity. We have defined a metal-poor ([Fe/H] < −1.7), a metal-intermediate (−1.7 < [Fe/H] < −1.4), and a metal-rich ([Fe/H] > −1.4) group, and we investigate the age–L1 relation for GCs in each of them. Results are listed in Table 4. In all the cases, G3 GCs are systematically older even if, especially for metal-poor GCs, the statistical significance of the measured age difference is marginal, but we are limited by small number statistics. The fact that G3 GCs are systematically older than G2 GCs and the presence of significant correlation between age and L1 for G2 GCs indicate that GC age is partly responsible for the color distance between the RGB and the reddest part of the HB, with metallicity being the other parameter for L1 extension.¹³

There is no significant correlation between L2 and age as shown in the lower panels of Figures 4 and 5. GCs with a double or multimodal SGB, namely NGC 1851, NGC 6388, NGC 6656, and NGC 6715, have been excluded from the statistical analysis above. As discussed in Section 4.1, the large fraction of faint-SGB stars observed in these GCs can affect the age measurements. For completeness we provide in Table 3 the values of the Spearman's rank correlation coefficients to estimate the statistical significance of L1, L2, and age correlations, together with results for the whole sample of GCs. The main results of this section remain unchanged when the GCs above are included in the analysis.

5.4. Helium and Light Elements

As mentioned in Section 1, the recent findings that in some GCs, groups of stars with different light-element abundances populate different HB segments strongly suggest that certain aspects of HB morphology may be strictly connected with multiple populations. To further investigate this scenario, in Figures 6–8 we show the relations between L1 and L2 and those quantities indicating intracluster chemical variations, which we introduced in Section 4.2. As shown in Figures 6 and 7, there is no significant correlation between L1 and W_RGB, IQR([O/Na]), R_CN, or f_POPI and no correlation between L2 and W_RGB or f_POPI. No significant correlations are observed between L1 and R or Y(R') or between L2 and R or Y(R'). A mild correlation between L2 and Y(R') cannot be ruled out for G2 GCs (r_G2 = 0.44, σ_{r, G2} = 0.21).

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Figure 7 also shows that, on average, GCs with extended HBs have more extended Na–O anticorrelations, as demonstrated by the significant correlation between L2 and IQR([O/Na]) obtained for GCs in both G2 and G3. This result confirms the findings by Carretta et al. (2007) and Gratton et al. (2010). Among G2 GCs, those with large CN-strong and CN-weak populations (R_CN > 2) have, on average, a more extended HB. The small number of G2 and G3 GCs where R_CN measurements are available prevents us from making any strong conclusion regarding the significance of the correlation with L1 and L2.

Theoretical models predict that star-to-star light-element variations observed in GC stars are associated with helium differences that lead to HB stars with different masses because of the well-known inverse relationship between helium abundance and stellar mass for fixed metallicity and age (e.g., Ventura & D'Antona 2005, and references therein).¹⁴ The relation between L1, L2, and the maximum internal helium difference measured from MS studies is plotted in Figure 8. The tight correlation between L2 (r_{G2 + G3} = 0.89, σ_{r, G2 + G3} = 0.17, N_{G2 + G3} = 7) and the small corresponding value of σ_r for G2+G3 clusters confirms theoretical indications that helium-enhanced stellar population are responsible of the HB extension.

5.5. Relationships with Other Parameters of the Host Globular Clusters

Figure 9 shows other monovariate relations involving L1. There is no significant correlation between L1 and central velocity dispersion, King (1962) model central concentration, central brightness, central density, core and half mass relaxation time, GC ellipticity, Galactocentric distance, and binary fraction. This is confirmed by the values of the Spearman's correlation coefficient listed in Table 3.

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In Figure 10 we see that L2 correlates with ρ and σ_V for G2 and G3 GCs and anticorrelates with μ_V and $f_{\rm bin}^{\rm C, C-HM, oHM}$ for each group of GCs, even if the anticorrelation is less or not significant for the G2+G3 sample. These results are not unexpected as these quantities also correlate with GC mass (Djorgovski & Meylan 1994).

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According to Gratton et al. (2010), in the ADS database there are more than 200 papers dedicated to the 2ndP phenomenon. Hence, any comparison here with the wide literature on the HB 2ndP can only be very far from complete. In this section we discuss some of the more relevant results. We refer the reader to review papers (e.g.,  Freeman & Norris 1981; Catelan 2009) and references therein for a complete view on this topic.

As already mentioned in Section 1, works on HB morphology in GCs make use of different HB metrics. In Figures 11 and 12, we compare L1 and L2 with other quantities used to parameterize HB morphology. The parameter to describe HB morphology that is mostly used in literature is the HB Type index or HBR (see Section 1). Figures 11 and 12 compare L1 and L2 with HBR. There is a linear correlation between L1 and HBR for G2 GCs, and then HBR saturates for G1 and for G3 GCs; L2 does not correlate with HBR.

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The wide literature on the 2ndP includes several works, similar to the present investigation, that are based on a fully observational approach, together with others that also use a series of theoretical assumptions. A recent example of the latter is the paper by Gratton et al. (2010), in which the authors used HST/WFPC2 and ground-based photometry of about 100 GCs to derive median and extreme colors and magnitudes of stars along the HB. They used isochrones and HB evolutionary models to transform these colors into median and extreme masses of stars on the HB and adopted the median mass loss (ΔM_median = M_RGB − M_median) and the difference between the median and the minimum HB masses (δM = M_median − M_min, where M_RGB, M_median, and M_min are RGB, median, and minimum HB masses, respectively) as parameters of the HB morphology. To determine ΔM_median and δM, Gratton et al. (2010) assumed for each cluster the value of metallicity and age from Carretta et al. (2009). They find that the median mass loss correlates with metallicity and suggest that if the mass-loss law they used is universal, age is the 2ndP. They conclude that age can explain the behavior of the median HB when it is coupled with a given mass-loss law that is a linear function of [Fe/H]. Gratton et al. also suggest that at least another parameter is needed to explain the HB morphology in GCs and argue that He abundance is the most likely candidate. They show that star-to-star helium variations, when combined with a small random quantity, can reproduce the HB morphology, thus supporting the results of other authors (e.g., D'Antona et al. 2002; D'Antona & Caloi 2008; Dalessandro et al. 2013). They find that the HB extension correlates with the interquartile of the Na–O anticorrelation, as previously noticed by the same group of authors (Carretta et al. 2007). Figures 11 and 12 show that δM correlates with L2 in G2 and G3 GCs, while there is no significant correlation between L2 and ΔM_median. The relation between L1 and ΔM_median is similar to that of L1 and [Fe/H]. This reflects the tight correlation between ΔM_median and metallicity.

Fusi Pecci et al. (1993) analyzed 53 GCs and found that the net length (L_t) of the HB and the presence and extent of blue tails are correlated with the GC density and concentrations, with more concentrated or denser GCs having bluer and longer HB morphologies. A correlation between HB morphology and absolute magnitude has been also detected by Recio-Blanco et al. (2006), who analyzed the CMDs of 54 GCs obtained from homogeneous HST WFPC2 data (Piotto et al. 2002) and concluded that the maximum effective temperature (T_{eff, MAX}) encountered along the HB correlates with M_V, with more-luminous GCs having also more-extended HBs.

As discussed in Section 1, the way an HB morphology metric is defined influences the outcome. It explains why some studies conclude that mass and/or He content are the main driver of HB morphology, while others indicate age as the main 2ndP. The definition of the two parameters L1 and L2 and their comparison with other quantities commonly used to parameterize the HB morphology (like L_t, T_{eff, MAX}, and Δ(V − I)) may help to shed some light on this controversy. Figure 11 shows that on the one hand there is no significant correlation between either L_t or T_{eff, MAX} and L1. On the other hand, among G2 GCs, L1 correlates with Δ(V − I), and G3 GCs have, on average, larger L1 values than G2 GCs (indicative of older ages in G3 than G2). From Figure 12 we note that L2 correlates with T_{eff, MAX} for both G2 and G3 GCs, and a correlation between L2 and L_t is also observed for G2 GCs. Figure 12 reveals no significant correlation between L2 and Δ(V − I). Finally, in Figure 13 we compare the two parameters introduced in this paper and show that L2 and L1 are not significantly correlated.

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We conclude that both the metric defined by Fusi Pecci et al. (1993) and Recio-Blanco et al. (2006), as well as L2, are sensitive to some properties (possibly helium variations) of the HB morphology but lack sensitivity to others (such us age). In contrast, Δ(V − I) and L1 are more sensitive to different properties of the HB (e.g.,  metallicity and age). The use of a pair of parameters, such as L1 and L2, can provide a more exhaustive description of the HB morphology than one alone.

In this paper we exploit both recent observational findings and ideas provided in the early 1980s to investigate the relation between HB morphology and various properties in GCs. These new findings come from studies on multiple stellar populations in GCs that show that the position of a star along the HB is connected to its chemical composition. First-generation stars populate the cooler side of the HB, and second-generation (He-enriched) stars populate the hotter side.

Freeman & Norris (1981) suggested that apart from metallicity, at least two parameters are needed to explain the HB morphology. One of these should be a global parameter that varies from GC to GC and the other a nonglobal parameter that varies within the GC. Driven by this idea we defined two new parameters to describe the HB morphology: L1, which indicates the distance between the RGB and the coolest part or the HB, and L2, which measures the color extension of the HB. Our analysis reveals that L1 depends on GC age and metallicity, while L2 correlates with the GC luminosity (hence, the mass) and the range of He content (ΔY).

These results suggest that, along the lines suggested by Freeman & Norris, age and metallicity are the main global parameters of the HB morphology of GCs, while GC mass is related to the HB extension. Works on multiple stellar populations in GCs show that more massive GCs exhibit, on average, larger internal helium variations, ΔY, than less massive GCs; ΔY is positively correlated with L2 and GC mass, though this analysis is limited to a small number of GCs at present. This makes it very tempting to suggest that internal star-to-star helium variation, associated with GC mass and the presence of multiple populations, is the main nonglobal parameter.¹⁵ The use of two quantities L1 and L2, that share a common definition (Section 3) but are sensitive to different phenomena allow us to discriminate the effects of global and nonglobal parameters on the HB morphology.

We thank F. D'Antona and J. Lattanzio for useful discussion and suggestion. We are grateful to P. Stetson for providing unpublished light curves of RR Lyrae. A.P.M., J.E.N., and H.J. acknowledge the financial support from the Australian Research Council through Discovery Project grant DP120100475. J.E.N. is also supported by the Australian Research Council through Discovery Project grant DP0984924. A.D. and M.A. acknowledge support from the Australian Research Council (grant FL110100012). G.P. and S.C. acknowledge financial support from PRIN MIUR 2010–2011, project "The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies," prot. 2010LY5N2T, and from PRIN-INAF 2011 "Multiple Populations in Globular Clusters: their role in the Galaxy assembly." Support for this work has been provided by the IAC (grant 310394) and the Education and Science Ministry of Spain (grants AYA2007-3E3506, and AYA2010-16717).

To investigate the impact of excluding RR Lyrae that might be out of the instability strip in the determination of L1 and L2, we have simulated a number of CMDs for different choices of the fraction of RR Lyrae (f_V), red-HB (f_R), and blue-HB (f_B) stars. We assumed that RR Lyrae are distributed along the whole instability strip and that all the RR Lyrae ab have the same light curve (this corresponds to the light curve observed for V27 in M 4, P ∼ 0.612 day, and is one of the RR Lyrae ab in M 4). We assumed for all the RR Lyrae c the light curve of V40 in M 4, P ∼ 0.299 day. The reason why we have chosen these two RR Lyrae is that their amplitude and period are typical of RR Lyrae ab and c. The light curves have been kindly provided by Peter Stetson. They are based on more than 1000 observations in B, V, and R bands and have been converted into F606W and F814W by using the color–temperature relations by Dotter et al. (2008). We choose V40, for which we have a light curve with the largest amplitude available to us, to maximize possible effects on the determination of L1 and L2.

F606W and F814W magnitudes have been simulated at different phases to account for the fact that F606W and F814W images are taken at different times. We assumed f_R = N × (f_B + f_R) (N = 0.0, 0.04, 0.10, 0.50, 0.90, 0.96, 1.0) and f_V = M × (f_B + f_R + f_V) (M = 0,0.10,0.25,0.50) and simulated 1000 CMDs for each combination of f_B, f_V, f_R. For each CMD we have calculated L1_I, L2_I and L1_II, L2_II. These are the values of L1 and L2 obtained when all RR Lyrae lie within the instability strip and when RR Lyrae are at random phase, respectively. The differences ΔL1 = L1_I − L1_II and ΔL2 = L2_I − L2_II are maximal in the case of an HB made of RR Lyrae only (ΔL1 = 0.10, ΔL2 = 0.22). Large difference are also detected in the case of an HB with a very small fraction of red HB stars and a large fraction of RR Lyrae variables (f_R = 0 or f_R = 0.04). We obtain ΔL1 = ΔL2 ∼ 0.08 when assuming f_V = 0.50; ΔL1 = ΔL2 ∼ 0.04 for f_V = 0.25; and ΔL1 = ΔL2 ∼ 0.01 for f_V = 0.1. We obtain similar results for ΔL2 and ΔL1 ∼ 0 when the blue HB hosts a very small fraction of stars (f_B = 0 and f_B = 0.04). The ΔL1 and ΔL2 get closer to zero for larger values of f_B and f_R.

According to the literature values (Lee et al. 1994; Harris 1996, 2010 edition; Gratton et al. 2010, and references therein), extreme cases of f_V ⩾ 0.25 and f_R ⩽ 0.1 are not present among the clusters studied in this paper, which suggests that any error related to the RR Lyrae phase should be smaller than ∼0.03–0.04 mag.

As a further test, for each GC studied in this paper we assumed the corresponding values of f_B, f_V, and f_R (Lee et al. 1994; Harris 1996, 2003 edition, and references therein) and simulated 10,000 CMDs. For most GCs we found ΔL1 < 0.02 and ΔL2 < 0.02; ΔL1 is greater than 0.03 mag only in a few GCs, namely NGC 4590, NGC 7078, and NGC 5466. The ΔL2 exceeds 0.03 mag also in the cases of Pal 3 and Rup 106. Both ΔL1 and ΔL2 never exceed 0.04 mag. Our tests suggest that the uncertainties on L1 and L2 measurements due to the random phase of RR Lyrae should be negligible for our purposes. This conclusion is similar to that of Gratton et al. (2010), who showed that RR Lyrae should not affect the determination of the median colors of HB stars.