Disentangling the Galactic Halo with APOGEE. II. Chemical and Star Formation Histories for the Two Distinct Populations

The first indications concerning a dual (or multiple) Galactic halo arose from the confrontation between the scenario of Eggen et al. (1962, ELS) and that of Searle & Zinn (1978, SZ). On one hand, we have the monolithic collapse, or free-fall, model of ELS, derived from various orbital-parameter versus ultraviolet-excess diagrams for 221 "well-observed" dwarf stars, which showed correlations suggesting a nearly free-fall collapse. On the other hand, we have the infall of "protogalactic fragments" proposed by SZ from the differences in the composition found between inner- and outer-halo globular clusters. Several reviews presented the strengths, weaknesses, and similarities of these two scenarios, such as Sandage (1986), Gilmore et al. (1989), and Majewski (1993).

It was suggested by some authors that these two contrasting views of halo formation were related to differences in the tracers themselves, halo field stars compared to globular clusters, or bias arising from their proper-motion-based selection (Mihalas & Binney 1981; Yoshii 1982; Norris et al. 1985; Chiba & Beers 2000). However, later studies using rele-vant observations discussed the implications and importance of combining such ideas in a dual-halo model for the Galaxy (Zinn 1993). For example, evidence for two Galactic halo components was found (Márquez & Schuster 1994; Carollo et al. 2007, 2010; Marín-Franch et al. 2009; de Jong et al. 2010; Jofré & Weiss 2011; Beers et al. 2012), using uvby–β photometry of halo field stars, globular clusters, and data from the Sloan Digital Sky Survey (SDSS; York et al. 2000) and the sub-program Sloan Extension for Galactic Understanding and Exploration (SEGUE; Yanny et al. 2009) and its extension, SEGUE-2.

Other studies revealed an even more complicated scheme for the Galactic stellar halo, with the discovery of streams, shells, clumps, tidal tails, debris, and the presence of correlated substructures of halo stars (e.g., Schlaufman et al. 2009, 2011, 2012; Carlberg et al. 2012; Koposov et al. 2012; Duffau et al. 2014; Slater et al. 2014; Carlin et al. 2016), pointing to a more chaotic dual, or even triple, component halo system (see Morrison et al. 2009), extrapolating beyond the ideas of SZ. Reviews concerning such observations and substructures for the stellar halo are given in Helmi (2008), De Lucia (2012), Belokurov et al. (2014), and Bernard et al. (2016).

Chemistry is considered a valuable tool to sort out the formation processes of the Galaxy. The chemical composition of stellar atmospheres resembles, in most cases, the composition of the interstellar medium (ISM) from which these stars formed. This ISM was chemically enriched by previous stellar populations that contributed their yields of elements (material synthesized by the star and ejected to the ISM) once they reached their last stages of evolution and died. The chemical species synthesized in the stellar interiors depend on the stellar properties, mainly the mass. Thus, the chemical composition measured in presently observed stellar atmospheres provides information about the properties of the previous stellar populations, such as the initial mass function (IMF) or the star formation rate (SFR) at early times—see Figure 1. These properties also allow us to constrain the processes that our Galaxy underwent during its early assembly.

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Signatures of a dichotomy in the α-to-iron ratios in halo stars were detected (Nissen & Schuster 1997; Fulbright 2002; Gratton et al. 2003; Ishigaki et al. 2013), related in some cases to the distance from the Galactic center (Ishigaki et al. 2010). Then, in a series of papers, Nissen & Schuster (2010, 2011—hereafter NS10, NS11; and Schuster et al. 2012) obtained high-precision (±0.01–0.04 dex) relative abundance ratios for 94 dwarf stars over the metallicity range −1.6 < [Fe/H] < −0.4, with 78 having halo kinematics according to the Toomre diagram, plus 16 thick-disk stars. Two groupings were clearly found in diagrams such as [Mg/Fe] versus [Fe/H] or [α/Fe] versus [Fe/H], with 56 high-α halo and thick-disk stars falling together, along with 38 low-α halo stars in the sample.

Clear separations between these two halo components were found for the elements Mg, Si, Ca, Ti, Na, Ni, Cu, and Zn with respect to iron, and also for [Ba/Y], all as a function of [Fe/H] (Ramírez et al. 2012 confirmed the same for [O/Fe]). In Schuster et al. (2012), it was shown that the high-α halo stars have ages higher by 2–3 Gyr than the low-α ones, and also smaller average values for the orbital parameters ${r}_{\max },{z}_{\max }$, and e_max. Again, some concordance with the ideas of ELS plus SZ were found (for example, via the in situ and accreted stellar populations of Zolotov et al. 2009, 2010; see also Tissera et al. 2013).

We should note that the distinction between two populations of stars for high [α/Fe] versus [Fe/H] has also been observed in between the thick and thin disk (Hayden et al. 2015), and also in other galaxies different than the Milky Way. Recently, Walcher et al. (2015) demostrated that early type galaxies present these two populations, associated with older (alpha-enhanced) and younger (alpha-deficient) populations. Actually, the star formation histories (SFHs) of both populations are different, with the first one presenting a more sharp one, with a smaller decay time. This result agrees with the strong age-[α/Fe] correlation found in both the Milky Way and other galaxies (e.g., Walcher et al. 2016).

Recent halo studies have benefited from the considerably larger SDSS stellar database with observations at numerous directions in the sky. In particular, the SDSS intermediate-resolution stellar spectra database has allowed for the exploration of chemical trends in halo stars over a broad range of distances, up to ∼50 kpc from the Galactic center, revealing gradients in [Fe/H], [Ca/H], and [Mg/H] (Fernández-Alvar et al. 2015). J. Yoon et al. (2017, in preparation) argue that the outer-halo gradient continues out to at least 80–100 kpc.

At present, another SDSS program, the APO Galactic Evolution Experiment (APOGEE; Majewski et al. 2017; Nidever et al. 2015), is gathering high-resolution, high signal-to-noise near-IR spectra to map the principal components of the Milky Way. With, eventually, a half million spectra, the APOGEE database is a very valuable sample to check previous findings, and to more completely investigate the chemical properties of stellar populations. Recently, investigations of metal-poor stars in the APOGEE database have shown signatures of the two chemically distinct populations revealed by Nissen & Schuster (Hawkins et al. 2015; Hayes et al. 2017, hereafter Paper I).

Encouraged by the possibilities of a chemical analysis of the two halo populations discovered in the APOGEE database, we aim to obtain information on the IMF, stellar yields, and SFH, or equivalently, SFR versus time, from the DR13 (Albareti et al. 2017) chemical abundances provided by APOGEE. This paper is organized as follows. The sample selection is discussed in Section 2. Section 3 describes the split of the sample into two populations, the derivation of the corresponding chemical trends, and the theoretical model from which we infer properties for each population. In Section 4, we relate our main results, and we discuss them in Section 5. Section 6 summarizes our conclusions.

APOGEE is an SDSS program (Eisenstein et al. 2011; Blanton et al. 2017) conceived to explore the structure of the Milky Way. The first APOGEE phase was in SDSS-III and collected data between 2011 and 2014 July, obtaining high-resolution (R ∼ 22,500) spectra with a typical signal-to-noise $\geqslant 100$ using a multiobject infrared spectrograph coupled to the 2.5 m SDSS telescope at the Apache Point Observatory (Gunn et al. 2006; Wilson et al. 2010). The targets map the Galactic disk, bulge, and halo (Zasowski et al. 2013). More than 143,000 objects were observed as part of that program.

The APOGEE Stellar Parameters and Abundances pipeline (ASPCAP) was developed to obtain stellar atmospheric parameters and chemical abundances from the H-band (1.5–1.7 μm), the spectral range covered by the APOGEE spectrograph. The methodology is based on the comparison with synthetic spectra in an N-dimensional parameter space, looking for the best fit with observations (more details in García Pérez et al. 2016). Abundances with accuracies ∼0.1 dex have been derived, and radial velocities have been determined with accuracies of ∼0.1 km s⁻¹ (Holtzman et al. 2015). DR12 (Alam et al. 2015) was the final SDSS-III data release. The thirteenth data release (DR13; Albareti et al. 2017) provides the final products of a reanalysis, after including several improvements to the pipeline. Chemical abundances of up to 26 chemical species are available for some stars, including the α-elements: O, Mg, S, Si, Ca, and Ti.

From this database, we want to draw a sample of halo stars. These objects clearly exhibit different kinematics from disk stars; objects with large heliocentric radial velocities have a high probability of belonging to this Galactic component.

In addition, the $l-\mathrm{GRV}/\cos (b)$ space (GRV is the radial velocity v_rad corrected for solar motion,¹¹ and l and b are the galactic longitude and latitude) can be used to isolate halo stars from disk stars. As performed by Hawkins et al. (2015) for the same purpose, we exclude from our sample those stars following a sinusoid of amplitude corresponding to the rotational velocity of the disk in the solar circle (220 km s⁻¹; Schönrich 2012) and a dispersion more than three times the dispersion of the same curve defined by disk stars.

This is the sinusoid expected to be drawn by objects rotating in the Galactic plane. Halo stars occupy randomly the GRV/cos(b) versus l space and, consequently, this selection criteria excludes not only disk stars but also stars belonging to the Galactic halo. However, we prefer to select only those objects with the highest confidence to be halo stars, even if our selection criteria is quite restrictive.

This selection to exclude disk stars works best in the case of objects at $| b| \lt 60^\circ$ (Majewski et al. 2012). Therefore, we measured the dispersion for stars at $| b| \lt 60^\circ$, having $[\mathrm{Fe}/{\rm{H}}]\gt 0.0$, which we expect to be dominated by disk stars. Thus, to explore stars with halo kinematics we select objects with v_rad > 180 km s⁻¹ and/or stars at $| b| \lt 60^\circ$ with an absolute values of GRV/cos(b) more than three times larger than that measured with disk stars in bins of $20^\circ$ in l.

The key to identifying different stellar populations by their chemistry is the accuracy and precision with which their chemical abundances are measured (Lindegren & Feltzing 2013). Both are also needed to infer parameters of the SFH from their chemical abundance trends. The random abundance uncertainties in the ASPCAP analysis vary as a function of ${T}_{\mathrm{eff}}$ and [Fe/H], as illustrated in Figure 2 of Bertran de Lis et al. (2016). They evaluated the [O/Fe] uncertainty as a function of these parameters by measuring the scatter observed in clusters. In light of these results, we select stars in the ${T}_{\mathrm{eff}}$ range at which the precision in [O/Fe] is the highest for the metallicity range covered by our halo sample (−2.5 < [Fe/H] < +0.5). We choose the interval $4000\lt {T}_{\mathrm{eff}}\lt 4500$ K, where the [O/Fe] uncertainties are σ[O/Fe] ≤ 0.02 dex, for −0.6 < [Fe/H] < +0.2, and increasing at lower metallicities. The empirical uncertainties calculated by ASPCAP are, on average, $\delta =0.05$ dex for [O/Fe] and [Mg/Fe], with a standard deviation $\sigma =0.03$ dex and 0.02 dex, respectively, and $\delta =0.04$ dex and 0.07 dex with $\sigma =0.02$ dex and 0.06 dex, for [Si/Fe] and [Ca/Fe], respectively.

We exclude stars with distances (adopted using the techniques of the Brazilian Participation Group—BPG; Santiago et al. 2016) from the Galactic center $r\leqslant 4\,\mathrm{kpc}$, in order to avoid bulge stars. We know that this selection cut may also exclude some halo stars located at this range of distances. However, we want to avoid any possible bulge contamination due to our goal of characterizing only the halo component of our Galaxy.

Finally, we reject objects with ASPCAP flags possibly indicating poor estimates. We also reject targets in globular clusters in our Galaxy and in Andromeda—these show chemical abundances that strongly deviate from the chemical trends of field stars (Mészáros et al. 2015). Our sample is comprised of field stars. We do not expect to have included objects from dwarf spheroidal galaxies, because APOGEE only purposely targeted in DR13 the spheroidal Sagittarius. Most of their stars are cool M giants with ${T}_{\mathrm{eff}}\lt 4000$ K, and we rejected them by our ${T}_{\mathrm{eff}}$ selection criteria.

Each star of our final sample is assigned to belong to the Galactic halo merely based on its v_rad. A more robust attribution will be possible with Gaia parallaxes and proper motions very soon.

Within the α-elements derived by ASPCAP, S and Ti are less reliable. [Ti/Fe] derived from both neutral and ionized atomic lines shows a large dispersion, and differing trends in the [Ti/Fe] versus [Fe/H] space. In their evaluation of the ASPCAP products, Holtzman et al. (2015) warned about the reliability for this element's abundances. They found no trend of [Ti/Fe] with [Fe/H], which led them to suspect that some systematic error affected their measurement. [S/Fe] also exhibits a large dispersion. At low metallicities, the S i lines from which ASPCAP derives [S/Fe] abundances become weak and comparable to the noise level. The measured [S/Fe] abundances lead to enhanced values and are likely to be unreliable.

Our final sample comprises 175 stars. The 2MASS id as well as the main stellar atmospheric parameters and chemical abundances for the selected sample are shown in Table 1. The top panel of Figure 2 shows the comparison between [Mg/Fe] as a function of [Fe/H]. Two different chemical trends are clearly distinguishable. We split the two stellar populations (High-Mg and Low-Mg) following the same classification derived from the statistical analysis presented in Paper I from a larger sample, i.e., along [Mg/Fe] = −0.2[Fe/H]. In the bottom panel, we overplot the NS10 sample of halo stars with their [Mg/Fe] and [Fe/H] measurements. Their sample abundance trends follow very closely our results from APOGEE data.

Table 1.  Halo Stars Selected within the DR13 APOGEE Database, with the Stellar Atmospheric Parameters and Chemical Abundances Determined by ASPCAP and Used in This Work^a

2MASS ID	Population	${T}_{\mathrm{eff}}$	$\mathrm{log}g$	${v}_{\mathrm{rad}}$	$[\mathrm{Fe}/{\rm{H}}]$	$[{\rm{O}}/\mathrm{Fe}]$	$[\mathrm{Mg}/\mathrm{Fe}]$	$[\mathrm{Si}/\mathrm{Fe}]$	$[\mathrm{Ca}/\mathrm{Fe}]$
2M13590274+0118564	HMg	4486	1.0	259.790	−1.95	0.53	0.39	0.27	0.29
2M23161405+1257322	HMg	4435	1.1	−179.804	−1.86	0.30	0.49	0.66	0.16
2M10374221−1042328	HMg	4346	1.2	181.805	−1.34	0.43	0.31	0.12	0.43
2M11002833−1044050	HMg	4483	1.1	337.768	−1.28	⋯	0.31	0.40	0.38
2M11422622−1409451	HMg	4323	1.2	258.096	−1.20	0.36	0.24	0.20	0.33

Note.

^aFull table is available at the CDS.

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

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We investigate these two populations in the [X/Fe] versus [Fe/H] space for the other APOGEE-reliable α-chemical species determined by ASPCAP. The chemical abundances and their errors are displayed in Figure 3 (for the High-Mg population, upper panels, and for the Low-Mg population, lower panels). In order to better visualize the chemical trends and the differences between the two populations, we calculate the weighted mean [X/Fe] and its statistical error in [Fe/H] bins of 0.1 dex, with a minimum of five objects per bin.

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Figure 4 shows the [X/Fe] versus [Fe/H] for each population. The High-Mg population shows the largest enhancement of all the α-elements considered here. This broad separation is the reason we used this element as the primary discriminator of halo populations in Paper I. The enhancement level diminishes with O, Si, and Ca. For this reason, and to be consistent with the nomenclature in the first paper of this series, we refer to these populations as High-Mg (HMg) and Low-Mg (LMg).

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The [X/Fe] versus [Fe/H] trends (see Figure 4) can be divided into two parts, as we depict in Figure 1:

• 1.  
  The "thigh": This corresponds to the semi-plateau located between the lowest metallicity (${[\mathrm{Fe}/{\rm{H}}]}_{l}\sim -1.9$ and $\sim -1.4$ for the LMg and HMg populations, respectively) and the knee (${[\mathrm{Fe}/{\rm{H}}]}_{k}\sim -1.0$ and $\sim -0.4$ for the LMg and HMg populations, respectively). The ${[\mathrm{Fe}/{\rm{H}}]}_{k}$ is the metallicity at which the downward slope becomes steepest.
• 2.  
  The "shin": located between the metallicity of the knee and the largest metallicity, [Fe/H]_h.

The chemical trends observed for the HMg and LMg sub-samples in each one of these metallicity ranges are the result of a different chemical and SFH for each stellar population, as explained below.

We use the calculated weighted means, choosing bins of different sizes to determine the mean [Fe/H] at which the slope of the trend changes, corresponding to the knee of the population. We identify the HMg knee at [Fe/H] ∼ −0.4, and the LMg knee at [Fe/H] ∼ −1.

It is important to notice that not all the elemental abundances could be measured for every star. In some cases, it was not possible to determine the abundance of a particular element reliably due to the quality of the observations. For this reason, the number of stars in each sample slightly varies from one element to another, as well as the particular objects from which the means are calculated. This implies that the mean [Fe/H]_l, [Fe/H]_k, and [Fe/H]_h for each population are slightly different, depending on the chemical element that we consider. Table 2 shows the resulting various $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ for each α-element. Due to the low number of stars, the weighted means have large errors, in particular, at low metallicities, and they do not describe smooth chemical trends. For this reason, we perform a linear fit to the weighted [X/Fe] means in the "thigh" and in the "shin" metallicity ranges; see Figure 4.

Table 2.  Mean [Fe/H] and [X/Fe] Derived from the Two Stellar Populations at the Key [Fe/H] values

Population	Element	${[\mathrm{Fe}/{\rm{H}}]}_{l}$	${[{\rm{X}}/\mathrm{Fe}]}_{l}$	${[\mathrm{Fe}/{\rm{H}}]}_{k}$	${[{\rm{X}}/\mathrm{Fe}]}_{k}$	${[\mathrm{Fe}/{\rm{H}}]}_{h}$	${[{\rm{X}}/\mathrm{Fe}]}_{h}$
HMg	O	−1.46 ± 0.15	0.37 ± 0.03	−0.43 ± 0.02	0.20 ± 0.01	0.08 ± 0.09	0.09 ± 0.01
	Mg	−1.48 ± 0.13	0.34 ± 0.04	−0.43 ± 0.02	0.26 ± 0.01	0.09 ± 0.08	0.09 ± 0.03
	Si	−1.56 ± 0.14	0.30 ± 0.05	−0.43 ± 0.02	0.16 ± 0.03	0.08 ± 0.08	0.02 ± 0.02
	Ca	−1.51 ± 0.14	0.36 ± 0.09	−0.44 ± 0.02	0.14 ± 0.01	0.09 ± 0.07	0.03 ± 0.01
LMg	O	−2.11 ± 0.07	0.39 ± 0.03	−1.03 ± 0.01	0.21 ± 0.02	−0.65 ± 0.01	0.02 ± 0.00
	Mg	−2.10 ± 0.07	0.21 ± 0.05	−1.02 ± 0.01	0.11 ± 0.01	−0.65 ± 0.01	−0.00 ± 0.02
	Si	−2.12 ± 0.07	0.24 ± 0.06	−1.03 ± 0.01	0.21 ± 0.01	−0.65 ± 0.01	0.06 ± 0.01
	Ca	−2.08 ± 0.06	0.41 ± 0.13	−1.03 ± 0.01	0.18 ± 0.01	−0.65 ± 0.01	0.06 ± 0.02

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3.1. Chemical-evolution Model

As mentioned in the Introduction, we aim to obtain basic chemical histories for the HMg and LMg populations. In particular, we try inferring the upper mass (M_up) of the IMF, the integrated yields for massive stars (Y), the fractions of Type II supernovae (SN II) and Type Ia supernovae (SN Ia) (f_{SN II} and f_{SN Ia}, respectively) in each simple stellar population, and the efficiency (ν) of the SFR and duration (t_h) of the SFH.

In order to obtain general properties for each population, our simple chemical-evolution models are built based on the following assumptions:

• 1.  
  The HMg and LMg are two independent populations. Each population evolves in its own way, according to the trend described by the mean [X/Fe] values for O, Mg, Si, and Ca versus [Fe/H] (See Figure 4).
• 2.  
  We embrace the semi-instantaneous recycling approximation. In this approximation, after each burst of star formation, all massive stars explode as SN II, instantaneously enriching the ISM. SN Ia explode with a delay of about 1 Gyr after their progenitors are formed. For a similar prescription, see Franco & Carigi (2008) and Hernández-Martínez et al. (2011).
• 3.  
  A closed-box model that evolves from initial primordial gas. We assume one zone per population, and a continuous SFH.
• 4.  
  Each [X/Fe] versus [Fe/H] range represents a different evolutionary stage:

  • (a)  
    During the "thigh," only SN II contribute to the ISM enrichment.
  • (b)  
    During the "shin," SN II and SN Ia pollute the ISM. These SN II behave similarly to the SN II in the "thigh."

Based on assumptions 2 and 3, the chemical abundance by mass (X) of¹² an element in the ISM evolves between any two times, t₁ and t₂ ($\gt {t}_{1}$)

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{X}}={\rm{X}}({t}_{2})-{\rm{X}}({t}_{1})=-\langle {Y}_{{\rm{X}}}{\rangle }_{2-1}\mathrm{log}\displaystyle \frac{\mu ({t}_{2})}{\mu ({t}_{1})},\end{eqnarray} \tag{ 1 }$$

where Y_X is the synthesized mass fraction of element X ejected by dying stars, $\langle {Y}_{{\rm{X}}}{\rangle }_{2-1}$ represents the Z-average integrated yields between $Z({t}_{1})$ and $Z({t}_{2})$ (i.e., $Z({t}_{j})=0.02\times {10}^{{[\mathrm{Fe}/{\rm{H}}]}_{j}}$), $\mu (t)={M}_{\mathrm{gas}}(t)/{M}_{\mathrm{gas}}(0)$ is the gas consumption, M_gas represents the gas mass, and ${M}_{\mathrm{gas}}(0)$ is the initial gas mass (see Avila-Vergara et al. 2016).

For computing $\langle {Y}_{{\rm{X}}}\rangle$, we consider theoretical Z-dependent yields for SN II by Kobayashi et al. (2006) and for pre-SN by the Geneva group (see Robles-Valdez et al. 2013). We integrate these yields in mass over a Kroupa–Tout–Gilmore IMF (Kroupa et al. 1993). The integrated yields are calculated between $0.1\,{M}_{\odot }$ and M_up, where ${M}_{\mathrm{up}}=10$ to 40 in steps of 5 M_⊙. For SN Ia, we assume the Z-independent SN Ia yields by Iwamoto et al. (1999).

Applying Equation (1) to α and Fe, we derive

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{\rm{X}}({t}_{2})}{{\rm{\Delta }}{\rm{X}}({t}_{1})}=\displaystyle \frac{{\rm{X}}({t}_{2})-{\rm{X}}({t}_{1})}{\mathrm{Fe}({t}_{2})-\mathrm{Fe}({t}_{1})}=\displaystyle \frac{\langle {Y}_{{\rm{X}}}{\rangle }_{2-1}}{\langle {Y}_{\mathrm{Fe}}{\rangle }_{2-1}}\end{eqnarray} \tag{ 2 }$$

or, equivalently,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{\rm{X}}/{\rm{H}}}{{\rm{\Delta }}\mathrm{Fe}/{\rm{H}}}=\displaystyle \frac{{{\rm{X}}}_{2}/{\rm{H}}-{{\rm{X}}}_{1}/{\rm{H}}}{{\mathrm{Fe}}_{2}/{\rm{H}}-{\mathrm{Fe}}_{1}/{\rm{H}}}=\displaystyle \frac{\langle {Y}_{{\rm{X}}}{\rangle }_{2-1}}{\langle {Y}_{\mathrm{Fe}}{\rangle }_{2-1}}.\end{eqnarray} \tag{ 3 }$$

Equation (2) relates the abundance ratios (derived by ASPCAP from the observations) with the integrated yields (from theoretical yields and the IMF). For obtaining the best M_up values that reproduce the data, we apply Equation (2) on the "thigh," because during this range only SN II enrich the ISM.

Based on the data, we can obtain X/H (the fraction by number) from [X/Fe]−[Fe/H], taking into account the solar abundances by Asplund et al. (2005). These are the solar chemical abundances considered by ASPCAP in the generation of the synthetic spectra of the grids used to determined the elemental abundances from APOGEE spectra. We calculate X/H and Fe/H ratios from the values derived by the linear fit over the weighted [X/Fe] means shown in Figure 4. Then, we transform the X/H value by number to X/H by mass.

3.1.1. The "Thigh"

We infer the M_up of the IMF for each population from the "thigh." We use O abundances because, in the literature, chemical-evolution models cannot reproduce the [Mg/Fe]–[Fe/H] trend shown by stars of the solar vicinity. For example, according to Romano et al. (2010), Mg in halo stars is reproduced when models considered yields by Kobayashi et al. (2006) for supernovae and hypernovae, but with this yield combination Mg in disk stars does not fit, mainly in thick-disk stars.

Figure 5 shows the theoretical $\mathrm{log}({Y}_{{\rm{O}}}/{Y}_{\mathrm{Fe}})$ versus M_up for stellar initial metallicities equal to 0.001, 0.004, and 0.02 (cyan, magenta, and black lines). We add the yield ratios interpolated for the metallicities (Z_j) that corresponds to the [Fe/H] means in Figure 4 lower than the knee, ${[\mathrm{Fe}/{\rm{H}}]\leqslant [\mathrm{Fe}/{\rm{H}}]}_{k}$.

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We depict $\mathrm{log}\left(\tfrac{{{\rm{X}}}_{j+1}/{\rm{H}}-{{\rm{X}}}_{j}/{\rm{H}}}{{\mathrm{Fe}}_{j+1}/{\rm{H}}-{\mathrm{Fe}}_{j}/{\rm{H}}}\right)$ using horizontal lines. For convenience, we focus on the X/Fe that show lower errors. From the intersection between the theoretical yields and the measured abundances we infer the M_up. The inferred M_up ranges are located between vertical dotted lines (for the high- and low-α populations, upper and bottom panels, respectively). Based on the mean M_up values corresponding to the inferred M_up ranges, we compute the fraction of SN II (${\int }_{8}^{{M}_{\mathrm{up}}}$IMF(m)dm) for each simple stellar population and the integrated yields between $8\,{M}_{\odot }$ and M_up.

Considering that theoretical Fe yields for massive stars, $\langle {Y}_{\mathrm{Fe}}^{\mathrm{theo},\mathrm{II}}\rangle$, are well-computed and do not require correction, we employ Equation (2) to obtain the empirical yields, $\langle {Y}_{{\rm{X}}}^{\mathrm{emp},\mathrm{II}}\rangle$, needed to reproduce the observed [X/Fe] versus [Fe/H] between the fourth mean point and the knee, the most reliable range for the "thigh."

$$\begin{eqnarray}&&\langle {Y}_{{\rm{X}}}^{\mathrm{emp},\mathrm{II}}\rangle =\langle {Y}_{\mathrm{Fe}}^{\mathrm{theo},\mathrm{II}}\rangle \displaystyle \frac{{{\rm{X}}}_{k}/{\rm{H}}-{{\rm{X}}}_{4}/{\rm{H}}}{{\mathrm{Fe}}_{k}/{\rm{H}}-{\mathrm{Fe}}_{4}/{\rm{H}}}.\end{eqnarray} \tag{ 4 }$$

Figure 6 shows the correction factors, $\langle {Y}_{{\rm{X}}}^{\mathrm{emp},\mathrm{II}}\rangle /\langle {Y}_{{\rm{X}}}^{\mathrm{theo},\mathrm{II}}\rangle$, for each of the α-elements and populations, which are close to unity. Error bars are calculated from the minimum and maximum M_up (see Table 3) assumed in the computation of $\langle {Y}_{\mathrm{Fe}}^{\mathrm{theo},\mathrm{II}}\rangle$.

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Table 3.  Upper Mass Limit for the IMF Determined from the Observed [X/Fe] (see Figure 5), and the Subsequent f_{SN II}

Population	${M}_{\mathrm{up}}({M}_{\odot })$	fSN II (10−3)
HMg	26.4 ± 1.3	4.59 ± 0.1
LMg	17.9 ± 2.7	4.00 ± 0.4

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Oxygen presents the largest errors because O yields are the most stellar-Z and mass-dependent ones among the four APOGEE-reliable α-elements. The Mg correction factor is the most different between the populations, due to the high-Mg enhancement shown by the HMg population.

3.1.2. The "Shin"

Again, we apply Equation (2), but now for the "shin" (between the knee and the ${[\mathrm{Fe}/{\rm{H}}]}_{h}$), taking into account the same correction factors for integrated yields of massive stars ($\langle {Y}_{{\rm{X}}}^{\mathrm{emp},\mathrm{II}}\rangle$) obtained in the "thigh" range. As previously mentioned, SN II and SN Ia contribute during the "shin" to the ISM enrichment in alpha and Fe elements. Therefore Z-average integrated yields ($\langle {Y}_{{\rm{X}}}{\rangle }_{k-h}$) between $Z({t}_{k})$ and $Z({t}_{h})$ are

$$\begin{eqnarray}&&\langle {Y}_{{\rm{X}}}{\rangle }_{k-h}=\langle {Y}^{\mathrm{emp},\mathrm{II}}{\rangle }_{k-h}+{f}_{\mathrm{SN}\mathrm{Ia}}\times {y}_{{\rm{X}}}^{\mathrm{Ia}},\end{eqnarray} \tag{ 5 }$$

where y_X^Ia is the SN Ia yield for a specific element, and f_{SN Ia} is the fraction of SN Ia that is contributed during the "shin." In this range, the contribution of low- and intermediate-mass stars are not considered, either because these stars do not produce the α-elements considered in this paper, or their yields are negligible compare with the SN II and SN Ia yields.

Substituting $\langle {Y}_{{\rm{X}}}{\rangle }_{k-h}$ and $\langle {Y}_{\mathrm{Fe}}{\rangle }_{k-h}$ in Equation (3), we obtain f_{SN Ia}. Figure 7 shows the f_{SN Ia} values (upper panel) and $\tfrac{{f}_{\mathrm{SN}\mathrm{Ia}}}{{f}_{\mathrm{SN}\mathrm{II}}}$ for each α-element and population. Figure 8 shows the percentage contribution of SN Ia and SN II to the α-element enrichment.

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Finally, this basic model also allows us to estimate the efficiency of the SFR and the times when $M/{{\rm{H}}}_{k}$ and $M/{{\rm{H}}}_{h}$ occur. For that, in Equation (1), we assume the following:

• 1.  
  The SFR is proportional to the M_gas, with efficiency ν, $\mathrm{SFR}(t)=\nu M\mathrm{gas}(t)$.
• 2.  
  ν is constant during the entire evolution.
• 3.  
  ${t}_{k}=\mathrm{delay} \mbox{-} \mathrm{time}$ for $\mathrm{SN}\,\mathrm{Ia}=1\,\mathrm{Gyr}$, and
• 4.  
  ${\rm{X}}({t}_{j})\sim {{\rm{X}}}_{j}/{\rm{H}}\times {H}_{\odot }$, where ${H}_{\odot }=0.7392$ (Asplund et al. 2005).

Therefore,

$$\begin{eqnarray}&&{M}_{\mathrm{gas}}(t)={M}_{\mathrm{gas}}(0){e}^{-\nu (1-R)t},\end{eqnarray} \tag{ 6 }$$

where R is the fraction of the mass ejected into the ISM by the dying stars.

From t = 0 until t_k, a left-infinite "thigh," we find that

$$\begin{eqnarray}&&{\rm{X}}({t}_{k})=\langle {Y}_{{\rm{X}}}\rangle \nu (1-R){t}_{k},\end{eqnarray} \tag{ 7 }$$

and we obtain ν. The bottom panel of Figure 6 shows the ν values for α-elements and each population.

Focusing on the "shin," we compute t_h from

$$\begin{eqnarray}&&{\rm{X}}({t}_{h})-{\rm{X}}({t}_{k})=\langle {Y}_{{\rm{X}}}{\rangle }_{k-h}\nu (1-R)({t}_{h}-{t}_{k}).\end{eqnarray} \tag{ 8 }$$

Finally, we calculate the SFR for each population:

$$\begin{eqnarray}&&\mathrm{SFR}(t)=\nu {M}_{\mathrm{gas}}(0){e}^{-\nu (1-R)t}.\end{eqnarray} \tag{ 9 }$$

We compute R using stellar ejecta by Kobayashi et al. (2006) and Karakas et al. (2009). Before the knee, only massive stars contribute to the ISM, and R = 0.061. After the knee, massive and intermediate-mass stars die, and R = 0.158. We derive the SFR(t) assuming ${t}_{k}=1\,\mathrm{Gyr}$ and ${M}_{\mathrm{gas}}(0)=1\,{M}_{\odot }$. These choices are motivated to easily obtain the SFHs for other t_k and ${M}_{\mathrm{gas}}(0)$ values. Figure 9 exhibits the resulting SFR(t) for the HMg and LMg, in blue and red, respectively. We show the SFH for each population considering ν obtained from Fe, due to the familiar time–metallicity relation.

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Table 4 presents the times when the final enrichment occurs for the analyzed elements for each population.

4.1. Chemical Trends

The resulting trends for both populations are characterized by [X/Fe] decreasing with [Fe/H]. From the weighted [X/Fe] means, we identify the knee in the distribution at approximately −1.0 in the case of the LMg population and approximately −0.4 for the HMg population. Figure 4 reveals that there is a gap in the weighted mean abundance ratios between the two populations. This separation is lower for [Si/Fe] and [Ca/Fe] than for [O/Fe] and [Mg/Fe]. The latter exhibits the largest difference.

It is important to notice that our sample of halo stars detected in APOGEE covers a metallicity range that is broader than those in previous works. The HMg population includes objects with $[\mathrm{Fe}/{\rm{H}}]\gt -0.4$. Other halo studies had not taken account of stars at larger metallicities because of the difficulty of distinguishing them from disk stars without precise kinematical data. The accurate radial velocities measured in APOGEE allow us to distinguish these halo objects at $[\mathrm{Fe}/{\rm{H}}]\gt -0.4$. The halo sample revealed at such high [Fe/H] shows a significant decreasing trend of [X/Fe] with [Fe/H]. This trend was suggested by a handful of objects in NS97 at [Fe/H] ∼−0.4, but it is now well-established in this work.

This broad range in metallicity reveals the knee for both populations. This fact allows us to compare the level of [X/Fe] in stars from each population formed before and after the main contribution of SN Ia to the ISM. Consequently, we are able to well-establish whether the HMg population is in fact α-enhanced relative to the LMg population. Since the contribution by SN Ia of the α-elements Si and Ca is larger than that of Mg, we want to confirm that the enhancement observed in the [X/Fe] versus [Fe/H] space is also detectable in [X/H] versus [Fe/H]. From the weighted means, we obtain the α-to-hydrogen ratios by subtracting the corresponding mean [Fe/H]. Figure 10 shows the [X/H] mean abundances as a function of [Fe/H] for each population in the "thigh." We see that the HMg population reaches higher [X/H] values than the LMg population at [Fe/H] ∼ −1. In addition, the former has its knee shifted to a higher [Fe/H] and includes stars with higher [Fe/H] than the latter (which do not show stars at $[\mathrm{Fe}/{\rm{H}}]\gt -0.6$ dex—see Table 1). This implies that the HMg population is also metal-enriched relative to the LMg population.

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As detected in previous works, we see that the separation between the two populations depends on the element considered. Besides, although all the APOGEE-reliable α-elements show a decrease of [X/Fe] with [Fe/H], the slope differs from one element to another, especially at the lowest metallicities. This is, however, expected since the yields from the very massive progenitor stars for these very metal-poor objects are different for each element. However, the slope of [Mg/Fe] with [Fe/H] from the knee up to ${[\mathrm{Fe}/{\rm{H}}]}_{h}$ in the LMg is less steep than the slope at the same range in metallicity observed for [O/Fe]. This is not expected at all, considering the current yields of Mg and Fe for SN Ia, which predict a very low contribution of [Mg/H] and [O/H], and greater contributions of [Si/H] and [Ca/H]. Consequently, the slope would be similar to that for [O/Fe] and steeper than that observed for [Si/Fe] and [Ca/Fe]. This is not what we observed from our LMg chemical trends. [O/Fe], [Si/Fe], and [Ca/Fe] exhibit similar slopes, which are steeper than for [Mg/Fe].

4.2. Inferences from the Chemical-evolution Models

We have explored and modeled the chemical evolution of the two halo populations seen clearly in APOGEE data, as described in Paper I.

Halo stars selected by their large radial velocities and a non-disk-like motion in the GRV/cos(b) versus l space within the APOGEE DR13 database cover a metallicity range −2.2 < [Fe/H] < +0.5. This broad range in metallicity reveals the knee for the two halo populations. The metal-rich side of the HMg population was unexplored in previous works (Nissen & Schuster papers; Hawkins et al. 2015; Paper I). The population with higher α-to-iron ratios was truncated at −0.4 or lower metallicities in their samples, and the high-[α/Fe] trend with metallicity was described as flat and constant. Our work reveals that this population also shows a steeper decreasing trend at higher metallicities, similar to the decrease observed in the NS10 low-α population, which they ascribed to an increase of iron abundance in the ISM from the contribution of Type Ia supernovae.

NS10 and later works tried to explain the chemical differences observed between populations in the metallicity range −1.6 < [Fe/H] < −0.4 in terms of the contribution of SN Ia for the low-α population due to a lower SFH. The SFH of the high-α population should have been faster in order to reach larger metallicities without the contribution of SN Ia. Kobayashi et al. (2014) pointed out that these chemical differences observed between populations could be accounted for by yields from massive stars between 10 and 20 solar masses. They suggested that there could be a difference in the IMF that led to the differences detected in the chemical abundances.

However, it is important to notice that they were trying to explain chemical differences in objects that would have formed from an ISM enriched before and after the pollution by SN Ia (the high-α and low-α population, respectively). Since we detect the metallicity value at which the contribution of SN Ia became relevant, we are able to compare those objects from each stellar population, which had the same kind of progenitors. Therefore, we are able to clarify their hypothesis.

On the one hand, we see that the metallicity range analyzed by Nissen & Schuster (−1.6 < [Fe/H] < −0.4) comprises objects before the knee for the HMg population and before and after the knee for the LMg population. This fact implies that the differences observed are, at least partially, due to the contribution of SN Ia, as NS10 suggested. On the other hand, the inference of the IMF upper mass limit from objects before the knee for both populations lets us ascertain whether there is, in addition, a difference in the IMF between the populations. We obtain that there is actually a difference in the IMF. Besides, we derive that the upper mass limit for the LMg population is between 10 and 20 solar masses, as pointed by Kobayashi et al. (2014). Therefore, we conclude that the chemical differences previously detected by NS are due to the combination of a difference in the IMF as well as the con-tribution of SN Ia for the LMg stars at metallicities lower than −0.4, at which point there was not yet a contribution of SN Ia for the HMg population.

The parameters inferred from these two different chemical trends lead us to two populations with different SFHs:

• 1.  
  one population with an IMF weighted to more massive stars, and an SFR more intense and extended in time, and
• 2.  
  a second population with a top-lighter IMF, and a lower and shorter SFH.

The SFR(t) from the HMg stars behaves similarly to that of the inner Galactic disk ($r\sim 4$ kpc) during the last 10 Gyr, while the SFR(t) from the LMg stars resembles that of the intermediate Galactic disk ($r\sim 8$ kpc) during the last 7 Gyr (Carigi & Peimbert 2008). Both regions of the Milky Way disk are explained assuming an inside-out scenario (see Carigi & Peimbert 2008, 2011). Therefore, we also may explain the two halo populations as resulting from an inside-out scenario for halo formation, where first the HMg stars formed in the inner halo, and immediately after the LMg stars formed in the outer halo. The IMF for the inner halo needs to be top-heavier to match its α-enhancement, and the outer halo requires a dynamically disrupted component to reproduce the retrograde mean orbit observed by NS10.

On the other hand, our results are also consistent with massive satellites reaching and populating more inner regions within the host galaxy during its formation (Tissera et al. 2014; Amorisco 2017). The more massive the satellites are able to continue star formation after they enter the virial radius of the host galaxy. This implies that their SFR would be more extended in time. The less massive satellites would not be able to survive inside the galactic potential, which means that they would be likely to populate the outer regions. They will be disrupted and their star formation would cease. Their SFR before the disruption would be lower because of their lower masses.

Recent results have shown that the top-mass end of the IMF may vary from galaxy to galaxy and across the galaxies to explain the dark-matter and baryonic mass-to-light ratio (Cappellari et al. 2012; Lyubenova et al. 2016). Even more, variations are also found across each galaxy, with clear correlations with the stellar metallicity (Martín-Navarro et al. 2015). An IMF dominated at early times by high-mass stars would also produce an enhanced [Mg/Fe] (Martín-Navarro et al. 2015), presenting old stellar populations and being produced by a strong and short star formation event (Walcher et al. 2015). This agrees with our results indicating that HMg stars were formed in a stronger star formation event, with a shorter decline time than LHg ones.

Figure 11 shows the space distribution of our sample, using distances from the BPG (Santiago et al. 2016). From the xy, yz, and xz planes (in Galactic coordinates), we see that the HMg population is mainly confined at inner regions of the halo. The bottom right panel shows the distance from the Galactic plane, z, as a function of distance from the Galactic center, r. The HMg stars are concentrated nearer the Galactic plane ($| z| \,\leqslant 5$ kpc) and the LMg stars reach to larger distances from the center of the Galaxy ($r\geqslant 15$ kpc) and from the Galactic plane. This is consistent with our previous conclusions.

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The abundance dispersion is larger for the LMg population than for the HMg population. It is also larger than the errors in the measurements. This fact suggests that this LMg sample is comprised of several populations, i.e., stars formed from environments with different previous enrichments. Moreover, the dispersion may also be caused by the stochastic pollution by massive stars, as the stochastic effects are more relevant when massive stars die in a metal-poor ISM inside small satellites (Carigi & Hernandez 2008).

We assume a simple chemical-evolution model that is able to reproduce the chemical trends observed. We do not need to claim for inflows or outflows. However, a kinematical analysis with the precise data provided in the following Gaia data releases will help to reveal the origin of the stars and better clarify the stellar populations comprised in our sample and their chemical trends. A more complex chemical-evolution model might be necessary then. It is also necessary to establish with simulations the accretion history of the Galaxy, and better establish whether the chemical trends observed in halo stars could be the result of an inside-out scenario or different kinds of satellites accreted at different times from the host halo of the Galaxy.

We evaluate chemical trends in the [X/Fe] versus [Fe/H] space from 175 stars selected within the DR13 APOGEE database, at $4000\lt {T}_{\mathrm{eff}}\lt 4500$ K, for which abundances of the α-elements O, Mg, Si, and Ca calculated by ASPCAP have the highest accuracy (mean uncertainties ∼0.05 dex). We infer the IMF upper mass limit, fractions of SN II and SN Ia relative to the total stellar population, and the star formation efficiency, following a closed-box model of the chemical evolution of the Galaxy under a semi-instantaneous recycling approximation.

We obtain the following:

• 1.  
  Two populations are distinguishable for each α-element in the [X/Fe] versus [Fe/H] space. Their [α/Fe] versus [Fe/H] trends are in agreement with those found by NS10 and NS11 for halo stars.
• 2.  
  The metallicity range covered extends, at both the low- and high-metallicity limits, beyond that analyzed for the two halo populations previously. This is the first time that these two populations are analyzed over the range $-2.2\,\lt [\mathrm{Fe}/{\rm{H}}]\lt +0.5$.
• 3.  
  Both populations exhibit a decreasing α-to-iron abundance trend associated with Fe enrichment of the ISM by SN Ia. This change in the slope (at −0.4 and −1.0 for the HMg and LMg populations, respectively) is also observed for all the α-elements examined, except Ti.
• 4.  
  Thus, we compare stars before the knee and beyond the knee, i.e., objects formed from an ISM with the contribution of only SN II and objects from an ISM with the additional contribution of SN Ia. This permits a proper comparison between both populations, in order to clarify whether one population is α-enhanced with respect to the other. We corroborate that the population with higher α-to-iron values revealed by NS10 is in fact α-enhanced with respect to the other. Besides, this population is also metal-enriched with respect to the LMg population.
• 5.  
  According to our closed-box model, more massive stars contribute to the ISM where the HMg formed with respect to the LMg population, which implies an IMF weighted to a higher upper mass limit.
• 6.  
  There is no significant difference between the two populations regarding the contribution of SN Ia to enrich the ISM from which the populations formed.
• 7.  
  The SFR was higher in the HMg population, decreases more steeply with time, and was longer than the SFR(t) inferred for the LMg population. The latter was lower at early times, more constant, and shorter.

E.F.A. acknowledges support from DGAPA-UNAM postdoctoral fellowships. L.C. is thankful for the financial support provided by CONACyT of México (grant 241732), by PAPIIT of México (IG100115, IA101215, IA101517), and by MINECO of Spain (AYA2015-65205-P). W.J.S. is thankful for the financial support provided by PAPIIT of México (IN103014). C.R.H. and S.R.M. acknowledge NSF grants AST-1312863 and AST-1616636. C.A.P. is thankful to the Spanish Government for funding for his research through grant AYA-2014-56359-P. T.C.B. acknowledges partial support from grant PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science Foundation. D.A.G.H. was funded by the Ramn y Cajal fellowship number RYC-2013-14182. D.A.G.H. and O.Z. acknowledge support provided by the Spanish Ministry of Economy and Competitiveness (MINECO) under grant AYA-2014-58082-P. B.T., J.F.-T., and D.G. gratefully acknowledge support from the Chilean BASAL Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) grant PFB-06/2007. S.V. gratefully acknowledges the support provided by Fondecyt reg. 1170518.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is http://www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.