Detailed Elemental Abundances in the M31 Stellar Halo: Low-Resolution Resolved Stellar Spectroscopy

While measurements of [$\alpha$/Fe] have been made in the stellar halo of the Milky Way, little is known about detailed chemical abundances in the stellar halo of M31. To make progress with existing telescopes, we apply spectral synthesis to low-resolution DEIMOS spectroscopy (R $\sim$ 2500 at 7000 \AA) across a wide spectral range (4500 \AA $<$ $\lambda$ $<$ 9100 \AA). By applying our technique to low-resolution spectra of 135 red giant branch (RGB) stars in 4 MW globular clusters, we demonstrate that our technique reproduces previous measurements from higher resolution spectroscopy. Based on the intrinsic dispersion in [Fe/H] and [$\alpha$/Fe] of individual stars in our combined cluster sample, we estimate systematic uncertainties of $\sim$0.11 dex and $\sim$0.04 dex in [Fe/H] and [$\alpha$/Fe], respectively. We apply our method to deep, low-resolution spectra of 14 RGB stars in the smooth halo of M31, resulting in higher signal-to-noise per spectral resolution element compared to DEIMOS medium-resolution spectroscopy, given the same exposure time and conditions. We find $\langle$[$\alpha$/Fe]$\rangle$ = 0.49 $\pm$ 0.31 dex and $\langle$[Fe/H]$\rangle$ = $-$1.53 $\pm$ 0.52 dex for our sample. This implies that the smooth halo field is likely composed of disrupted dwarf galaxies with truncated star formation histories that were accreted early in the formation history of M31.

1. INTRODUCTION Stellar chemical abundances are a key component in determining the origins of stellar halos of Milky Way (MW) like galaxies, providing insight into the formation of galaxy-scale structure. The long dynamical times of stellar halos allow tidal features to remain identifiable in phase space, in terms of kinematics and chemical abundances, for Gyr timescales. Stellar chemical abundances of stars retain information about star formation history and accretion times of progenitor satellite galaxies, even when substructures can no longer be detected by kinematics alone. In particular, measurements of metallicity 1 and α-element abundances provide a way of directly testing the hierarchical assembly paradigm central to ΛCDM cosmology, providing a fossil record of the formation E-mail: ie@astro.caltech.edu 1 We define metallicity in terms of stellar iron abundance, [Fe/H], where [Fe/H] = log 10 (n Fe /n H ) − log 10 (n Fe /n H ) environment of stars accreted onto the halo.
The [α/Fe] ratio serves as a useful diagnostic of formation history, given that it traces the star formation timescales of a galaxy (e.g., Gilmore & Wyse 1998). Type II supernovae (SNe) produce abundant α-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti), increasing [α/Fe], whereas Type Ia SNe produce Fe-rich ejecta, reducing [α/Fe]. While measurements of [α/Fe] have been made in the stellar halo of the MW, little is known about the detailed chemical abundances of the stellar halo of M31. A comparable understanding of the properties of the MW and M31 stellar halos is required to verify basic assumptions about how the MW evolved, where such assumptions are used to extrapolate MW-based results to studies of galaxies beyond the Local Group.
Although high-resolution (R 15,000), high signal-tonoise (S/N) spectra enables simultaneous measurements of a star's temperature, surface gravity, and individual element abundances based on individual lines, it is impractical to achieve high enough S/N for traditional high-resolution spectroscopic abundance analysis (e.g.,  for red giant branch (RGB) stars at the distance of M31 (783 kpc; Stanek & Garnavich 1998).
It is possible to obtain spectroscopic metallicity measurements of M31 RGB stars from medium-resolution spectra (R ∼ 6000) using spectral synthesis (e.g., Kirby et al. 2008a). This method leverages the entire spectrum's metallicity information simultaneously, enabling measurements of abundances from relatively low S/N spectra. Kirby et al. (2008bKirby et al. ( , 2010Kirby et al. ( , 2013 successfully measured [Fe/H] and [α/Fe] in MW globular clusters (GCs), MW dwarf spheroidal (dSph) satellite galaxies, and Local Group dwarf irregular galaxies, showing that abundances can be measured to a precision of ∼ 0.2 dex from spectra with S/N ∼ 15 Å −1 .
Only in 2014 has spectral synthesis been applied to individual RGB stars in the M31 system for the first time (Vargas et al. 2014a,b). Existing spectroscopic chemical abundance measurements in M31 are primarily based on metallicity estimates from the strength of the calcium triplet Koch et al. 2008;Kalirai et al. 2009;Richardson et al. 2009;Tanaka et al. 2010;Ibata et al. 2014;Gilbert et al. 2014;Ho et al. 2015). Vargas et al. (2014a)  However, the present spectroscopic sample size and measurement uncertainties of the And V data enable only qualitative conclusions about the chemical evolution of the dSph. Obtaining more quantitative descriptions of the chemical enrichment and star formation histories of the M31 system requires higher S/N spectroscopic data, which results in smaller uncertainties on abundance measurements. Only then can one-zone numerical chemical evolution models (Lanfranchi & Matteucci 2003Lanfranchi et al. 2006;Kirby et al. 2011b) be reliably applied to measurements to derive star formation and mass assembly histories.
Although Vargas et al. (2014a,b) demonstrated the feasibility of measuring [Fe/H] and [α/Fe] at the distance of M31, measuring [Fe/H] and [α/Fe] more precisely requires deep (∼6 hour) observations with DEIMOS using the 600 line mm −1 grating to yield higher S/N for the same exposure time and observing conditions. For magnitudes fainter than I 0 ∼ 21 (0.5 magnitudes below the tip of M31's RGB), sky line subtraction at λ > 7000 Å becomes the dominant source of noise in DEIMOS spectra observed with the 1200 line mm −1 grating. Given the access to blue optical wavelengths granted by the 600 line mm −1 grating, its spectra are less susceptible to the effects of sky noise. Additionally, using the 600 line mm −1 grating achieves higher S/N per pixel for stars as faint as I 0 ∼ 21.8.
Although using the 600 line mm −1 grating with DEIMOS results in a gain in S/N and wavelength coverage, it corresponds to a decrease in spectral resolution (∼2.8 Å FWHM, or R ∼ 2500 at 7000 Å, compared to ∼1.3 Å and R ∼ 5400 for 1200 line mm −1 ). Increasing the spectral range compensates for the decrease in spectral resolution, given the increase in the amount of available abundance information contained in the spectrum resulting from the higher density of absorption features at bluer optical wavelengths.
In this paper, we expand upon the technique first presented by Kirby et al. (2008a), applying spectral synthesis to low- resolution spectroscopy (LRS; R ∼ 2500) across a wide spectral range (λ ∼ 4500 -9100 Å). In § 2, we describe our data reduction and GC observations. § 3 and § 4 detail our preparations to the observed spectrum and the subsequent abundance analysis. This includes a presentation of our new line list and grid of synthetic spectra. In § 5, we illustrate the efficacy of our technique applied to MW GCs and compare our results to chemical abundances from high-resolution spectroscopy in § 6. We quantify the associated systematic uncertainties in § 7. We conclude by measuring [α/Fe] and [Fe/H] in a M31 stellar halo field in § 8 and summarize in § 9.
2. OBSERVATIONS We utilize observations of Galactic GCs (Table 1) taken using KeckII/DEIMOS (Faber et al. 2003) to validate our LRS method of spectral synthesis. For our science configuration (both for MW GCs and M31 observations, § 8.1), we used the GG455 filter with a central wavelength of 7200 Å, in combination with the 600ZD grating and 0.7" slitwidths. The spectral resolution is approximately ∼ 2.8 Å FWHM, compared to ∼ 1.3 Å FWHM for the 1200 line mm −1 grating used in prior observations (Kirby et al. 2010(Kirby et al. , 2013. The wavelength range for each spectrum obtained with the 600ZD grating is within 4100 Å − 1 µm, where we generally omit λ 4500 Å, owing to poor S/N in this regime and the presence of the G band. We also omit λ > 9100 Å, which extends beyond the wavelength coverage of our grid of synthetic spectra ( § 4.2).
To extract one-dimensional spectra from the raw DEIMOS data, we used a modification of version 1.1.4 of the data reduction pipeline developed by the DEEP2 Galaxy Redshift Survey (Cooper et al. 2012;Newman et al. 2013). Guhathakurta et al. (2006) provides a detailed description of the data reduction process. Modifications to the software include those of Simon & Geha (2007), where the pipeline was re-purposed for bright unresolved stellar sources (as opposed to faint, resolved galaxies). In addition, we include custom modifications to correct for atmospheric refraction in the twodimensional raw spectra, which affects bluer optical wavelengths, and to identify lines in separate arc lamp spectra, as opposed to a single stacked arc lamp spectrum.

PREPARING THE SPECTRUM FOR ABUNDANCE MEASUREMENT 3.1. Telluric Absorption Correction
Unlike the red side of the optical (6300 -9100 Å), there is no strong telluric absorption in the bluer regions (4500 -6300 Å). As such, we do not make any corrections to the observed stellar spectra to take into account absorption from Earth's atmosphere in this wavelength range.
For the red (6100 -9100 Å), we correct for the absorption of Earth's atmosphere using the procedure described in Kirby et al. (2008a). We adopt HD066665 (B1V), observed on April 23, 2012 with an airmass of 1.081, using a long slit in the same science configuration ( § 2) as our data, as our spectrophotometric standard.

Spectral Resolution Determination
In contrast to Kirby et al. (2008a), who determined the spectral resolution as a function of wavelength based on the Gaussian widths of hundreds of sky lines, we assume a constant resolution, expressed as the typical FWHM of an absorption line, across the entire observed spectrum (∼4500−9100 Å). Owing to the dearth of sky lines at bluer wavelengths, the number of available sky lines is insufficient to reliably determine the resolution as a function of wavelength. As an alternative, we introduce an additional parameter, ∆λ, or the spectral resolution, into our chi-squared minimization, which determines the best-fit synthetic spectrum for each observed spectrum ( § 4.5).

Continuum Normalization
It is necessary to normalize the observed flux by its slowly varying stellar continuum in order to meaningfully compare to synthetic spectra for the abundance determination ( § 4.5). To obtain reliable abundances from spectral synthesis of lowand medium-resolution spectra dominated by weak absorption features, the continuum determination must be accurate (Shetrone et al. 2009;Kirby et al. 2009). This is particularly important for bluer wavelengths, where absorption lines are so numerous and dense that we cannot define "continuum regions" (Kirby et al. 2008a) in the blue. Instead, we utilize the entire observed spectrum, excluding regions with strong telluric absorption and bad pixels, to determine the continuum for 4500 − 9100 Å. In contrast to Kirby et al. (2008a), we do not utilize continuum regions at redder wavelengths (6300 -9100 Å), despite the fact that they can be reliably defined, to maintain consistency in the continuum normalization method between each wavelength region of the observed spectrum.
We determined the initial continuum fit to the raw observed spectrum, which we shift into the rest frame, using a thirdorder B-spline with a breakpoint spacing of 200 pixels, excluding 5 pixels around the chip gap and at the start and stop wavelengths of the observed spectrum. In all steps, we weighted the spline fit by the inverse variance of each pixel in the observed spectrum. We performed sigma clipping, such that pixels that deviate by more than 5σ (0.1σ) above (below) the fit are excluded from the subsequent continuum determination, where σ is the inverse square root of the inverse variance array. We did not perform the fit iteratively beyond the above steps, given that our stringent criterion to prevent the numerous absorption lines from offsetting our continuum determination can eliminate a significant fraction of the pixels from subsequent iterations of the fit.
To further refine our continuum determination, we recalculate the continuum fit iteratively in the initial step of the abundance analysis ( § 4.5). Once we have found a best-fit synthetic spectrum, we divide the continuum-normalized observed spectrum by the best-fit synthetic spectrum to construct a "flat noise" spectrum, which captures the higher order terms in the observed spectrum not represented in the fit. We fit a third-order B-spline with a breakpoint spacing of 100 pixels to the flat noise spectrum, excluding 3σ deviant (above and below the fit) pixels, dividing the continuum-normalized observed spectrum by this fit. The modified continuumnormalized spectrum is then used in the next iteration of the continuum refinement until convergence is achieved ( § 4.5).

Custom Pixel Masks
In addition to wavelength masks corresponding to a particular abundance ( § 4.4), we constructed a pixel mask for each analyzed observed spectrum. Typically excluded regions include 5 pixels on either side of the chip gap between the blue and red sides of the CCD, areas with improper sky line subtraction, the region around the Na D1 and D2 lines (5585 -5905 Å), and other apparent instrumental artifacts. Table 2 includes a summary of prominent spectral features in DEIMOS spectra between 4100 -6300 Å, where wavelength ranges given in parenthesis indicate regions that are masked. For example, we excluded 10 Å regions around Hγ (4335 -4345 Å) and Hβ (4856 -4866 Å). MOOG does not incorporate the effects of non-local thermodynamic equilibrium and thus cannot properly model the strong Balmer lines. If necessary, we also masked regions where the initial continuum fit failed, most often owing to degrading signal-to-noise as a function of wavelength at bluer wavelengths ( 4500 Å). As for the red (6300 -9100 Å), we adopted the pixel mask from Kirby et al. (2008a), which excludes spectral features such as the Ca II triplet, Hα, and regions with strong telluric absorption.

Signal-to-Noise Estimation
We estimate S/N per Angstrom for objects observed with the 600 line mm −1 grating from wavelength regions of the spectrum utilized in the initial continuum determination ( § 3.3). Given that we cannot define continuum regions for wavelengths blueward of 6300 Å, we calculate the S/N after the continuum refinement process ( § 4.5). We estimate the noise as the deviation between the continuum-refined observed spectrum and the best-fit synthetic spectrum and the signal as the best-fit synthetic spectrum itself. The S/N estimate per pixel is the mean of the S/N as a function of wavelength calculated from these quantities, where we exclude pixels that exceed the average noise threshold by more than 3σ. To convert to units of per Angstrom, we multiply this quantity by the inverse square root of the pixel scale (∼ 0.64 Å for the 600 line mm −1 grating).

CHEMICAL ABUNDANCE ANALYSIS
Here, we present a new library of synthetic spectra in the range 4100 -6300 Å. In this section, we describe our procedure for spectral synthesis in the blue, where we use our new  -(Top) A comparison between high-resolution spectra (Hinkle et al. 2000) (blue) of the Sun (left) and Arcturus (right) and synthetic spectra (orange) generated using the blue line list ( § 4.1). Both spectra are smoothed to the expected resolution of the DEIMOS 600ZD grating (∼ 2.8 Å). For a description of synthetic spectrum generation, see § 4.2. (Bottom) The difference between the observed and synthetic spectra for the Sun and Arcturus. To improve the agreement between the synthetic and observed spectra, we have manually vetted the line list, adjusting the oscillator strengths of discrepant atomic transitions as necessary. In this process, we have favored the Sun over Arcturus, thus the larger residuals between the observed and synthetic spectra for the latter star (which has a lower effective temperature). grid in conjunction with the red grid of Kirby et al. (2008a) to measure abundances across an expanded optical range (4100 -9100 Å).

Line List
We constructed a line list of wavelengths, excitation potentials (EPs), and oscillator strengths (log g f ) for atomic and molecular transitions in the spectral range covering 4100 -6300 Å for stars in our stellar parameter range (T eff > 4000 K). We queried the Vienna Atomic Line Database (VALD; Kupka et al. 1999) and the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2016) for all transitions of neutral or singly ionized atoms with EP < 10 eV and log g f > -5, supplementing the line list with molecular (Kurucz 1992) and hyperfine transitions (Kurucz 1993). All Fe I line oscillator strengths from Fuhr & Wiese (2006) are included in the NIST database.
To produce agreement between the synthetic and observed spectra, we vetted the line list by manually adjusting the oscillator strengths of aberrant atomic lines as necessary. We preferred the Sun over Arcturus in this process, given that Arcturus is cool giant with stronger molecular absorption features (e.g., the G band) that are more difficult to match. For features absent from the line list, which could not be resolved by considering lines with log g f < −5, we included Fe I transitions with EPs and log g f to match the observed strength in both the Sun and Arcturus. The final blue line list contains 132 chemical species (atomic, molecular, neutral, and ionized), including 74 unique elements and 2 molecules (CN and CH). In total, the line list contains 53,164 atomic line transitions and 58,062 molecular transitions. Figure 1 illustrates a comparison between the Hinkle spectra and their syntheses for the Sun and Arcturus. At the expected resolution of the DEIMOS 600ZD grating (∼ 2.8 Å), the mean absolute deviation of the residuals between the observed spectra and their syntheses across the wavelength range of the line list are 8.3 × 10 −3 and 2.2 × 10 −2 for the Sun and Arcturus respectively. . We show an example spectrum (black) over the wavelength range 4500 -6300 Å, where we corrected the spectrum for telluric absorption ( § 3.1) and performed an initial continuum normalization ( § 3.3). We do not show spectral regions with wavelengths below 4500 Å, since low S/N generally prevents utilization of the observed spectrum in this wavelength range. The spectrum is for a star in the globular cluster NGC 2419. Spectral regions sensitive to Mg, Ca, and Si are shown as highlighted ranges in magenta, blue, and green respectively. The atmospheric value of [α/Fe] is measured using the union of the Mg, Ca, and Si spectral regions. Step T e f f (K) 3500 5600 100 5600

Synthetic Spectra
for T e f f ≤ 4100 K, certain stellar atmosphere models fail to converge when solving for molecular equilibrium in each atmospheric layer. Synthetic spectra with [Fe/H] < −4.5 exist for a majority of T e f flog g pairs for T e f f ≤ 4100 K, but our grid is complete for all parameter combinations only above [Fe/H] = −4.5 in this regime.
We employ the ATLAS9 (Kurucz 1993) grid of model stellar atmospheres, with no convective overshooting (Castelli et al. 1997). We base our grid on recomputed (Kirby et al. 2009 and references therein) ATLAS9 model atmospheres with updated opacity distribution functions, available for [α/Fe] = 0.0 and +0.4 (Castelli & Kurucz 2004). We adopt the solar composition of Anders & Grevesse 1989, except for Fe (Sneden et al. 1992). The elements considered to be αelements are O, Ne, Mg, Si, S, Ar, Ca, and Ti.
For stellar parameters between grid points, we linearly interpolated to generate model atmospheres within the ranges 3500 K < T e f f < 8000 K, 0.0 < log g < 5.0, −4.5 < [Fe/H] < 0.0, and -0.8 < [α/Fe] < +1.2. A full description of the grid is presented in Table 4. Here, [α/Fe] represents a total αelement abundance for the atmosphere, which augments the abundances of individual α-elements without distinguishing between their relative abundances. In total, the grid contains 316,848 synthetic spectra.
We generated the synthetic spectra using MOOG (Sneden 1973), an LTE spectral synthesis software. MOOG takes into account neutral hydrogen collisional line broadening (Barklem et al. 2000;Barklem & Aspelund-Johansson 2005), in addition to radiative and Stark broadening and van der Waals line damping. The most recent version (2017) includes an improved treatment of Rayleigh scattering in the continuum opacity (Alex Ji, private communication). The resolution of each generated synthetic spectrum is 0.02 Å.

Photometric Constraints
To reduce the dimensionality of parameter space and to optimize our ability to find the global chi-squared minimum in the parameter estimation ( § 4.5), we constrained the effective temperature and surface gravity of the synthetic spectra by available photometry for red giant stars in our sample. The photometric effective temperature is estimated using a combination of the Padova (Girardi et al. 2002), Victoria-Regina (VandenBerg et al. 2006), and Yonsei-Yale (Demarque et al. 2004) sets of isochrones, assuming an age of 14 Gyr and an α-element abundance of 0.3 dex. If available, we also employed the Ramírez & Meléndez (2005) color temperature. We adopted a single effective temperature (T eff,phot ) and associated uncertainty (σ Teff,phot ) from an average of the isochrone/color temperatures for each star.
We determined the photometric surface gravity in a similar fashion. However, no color-log g relation exists, so we could not include this additional source for the photometric surface gravity. Unlike the effective temperature, we did not solve for log g using spectral synthesis techniques, as the errors on the photometric surface gravity are negligible when the distance is known. Additionally, LRS and MRS spectra cannot effectively provide constraints on its value owing to the lack of ionized lines in the spectra. Thus, we held log g fixed in the abundance determination.

[Fe/H] and [α/Fe] Regions
In order to increase the sensitivity of the synthetic spectrum fit to a given abundance measurement, we constructed -An example of a continuum-normalized observed spectrum (light blue) and its best-fit synthetic spectrum (orange). The observed spectrum corresponds to the same object in Figure 2. We show the portion of the fit utilizing our new blue grid of synthetic spectra (∼ 4700 -6300 ). The highlighted regions (grey) correspond to our standard mask ( § 3.4), which excludes lines such as Hβ, Na D1 and D2 from the fit. We adopt the parameters of the best-fit synthetic spectrum for the observed spectrum, T eff = 4296 K, log g = 0.71 dex, [Fe/H] = -1.98 dex, [α/Fe] = 0.18 dex, and ∆λ = 2.66 Å FWHM. We measure χ 2 ν = 1.64 for the quality of the fit across the full wavelength range (∼ 4700 -9100 Å), based on the regions of the spectrum used to measure [Fe/H] ( § 4.4). The normalized residuals (dark blue) between the continuum-refined observed spectrum and best-fit synthetic spectrum are also shown. The residuals have been scaled by the inverse variance of the observed spectrum and the degree of freedom of the fit, such that each pixel represents the direct contribution to χ 2 ν .
wavelength masks that highlight regions that are particularly responsive to changes in [Fe/H] and [α/Fe]. We employed the same procedure as Kirby et al. (2009) to make the masks, starting with a base synthetic spectrum for each combination of T eff (3500 -8000 K in steps of 500 K) and log g (0.0 -3.5 in steps of 0.5 dex). We assumed a bulk metallicity [Fe/H] = -1.5 and solar [α/Fe] for the atmosphere. We then generated synthetic spectra with either enhanced or depleted values of individual element abundances (Fe, Mg, Si, Ca, and Ti) for each T eff -log g pair and compare to the base synthetic spectra, identifying wavelength regions that differ by more than 0.5%. In the determination of the [Fe/H] and [α/Fe] wavelength regions, we smoothed all synthetic spectra used to an approximation of the expected resolution of the 600ZD grating (∼ 2.8 Å) across the entire spectrum (4100 -9100 Å). We then compared the spectral regions for each element against the line list and high signal-to-noise (S/N > 100) spectra of cool (T eff < 4200 K) globular cluster stars, eliminating any regions that do not have a corresponding transition in the line list or an absorption feature in the spectra. Although our measurements reflect the atmospheric value of [α/Fe], we constructed the associated wavelength mask from the regions sensitive to changes in the individual elements Mg, Si, and Ca. We excluded Ti from the [α/Fe] mask owing to the prevalence of regions sensitive to Ti at bluer optical wavelengths, such that we cannot meaningfully isolate its elemental abundance. Figure 2

Parameter Determination from Spectral Synthesis
Here, we outline the steps involved in our measurement of atmospheric parameters and elemental abundances from spec-tral synthesis of low-resolution spectra. Our method is nearly identical to that of Kirby et al. (2009), excepting our introduction of an additional free parameter, ∆λ, the resolution of the observed spectrum. We use a Levenberg-Marquardt algorithm to perform each comparison between a given observed spectrum and a synthetic spectrum. We weight the comparison according to the inverse variance of the observed spectrum. In each step, the synthetic spectra utilized in the minimization are interpolated onto the observed wavelength array and smoothed to the fitted observed resolution, ∆λ, prior to comparison with a given observed spectrum. 3. Iterative continuum refinement. After a best-fit synthetic spectrum is determined according to steps 1 and 2, we refine the continuum normalization according to § 3.3. We perform the continuum refinement iteratively, enforcing the convergence conditions that the difference in parameter values between the previous and current iteration cannot exceed 1 K, 0.001 dex, 0.001 dex, and 0.001 Å for T eff , [Fe/H], [α/Fe], and ∆λ respectively. If these conditions are not met in a given iteration, the continuum-refined spectrum is used to repeat steps 1 and 2 until convergence is achieved. If the maximum number of iterations (N iter, max = 50) is exceeded, which occurs for a small fraction of observed spectra, we do not include the observed spectra in the subsequent analysis.  ) for the same data set as Figure 4. The lack of a trend between the two quantities for each GC implies that our chemical abundance analysis is robust to systematic covariance in these parameters.
NGC 2419 is a luminous outer halo GC located ∼ 90 kpc away from the Galactic center (Harris et al. 1997

Membership
For all subsequent analysis in this section, we utilize only stars that have been identified as RGB star members by Kirby et al. (2016). We removed asymptotic giant branch stars from our GC sample that Kirby et al. (2016) manually selected from color-magnitude diagrams. Membership is defined using both radial velocity and metallicity criteria based on MRS, such that any star whose measurement uncertainties are greater than 3σ from the mean of either radial velocity or metallicity is not considered a member. The colors and magnitudes of member stars must also conform to the cluster's giant branch.

Metallicity
As described in § 4.5, we measure metallicity from spectral regions sensitive to variations in [Fe/H]. In addition to membership criteria ( § 5.1), we further refine our sample by requiring that the 5σ contours in each of the four fitted parameters ( § 4.5) identify the minimum. This condition is effectively equivalent to requiring that a given star has sufficient S/N, a converged continuum iteration, and overall high enough quality fit (χ 2 ν ) to produce a reliable abundance measurement. We illustrate our results for [Fe/H] in the form of metallicity distribution functions (Figure 4) (Kacharov et al. 2013), and −2.34 dex (Sneden et al. 2000) 2 respectively. We present a detailed comparison to HRS abundances in § 6.
As another example of our ability to reliably recover [Fe/H], we show [Fe/H] vs. spectroscopically determined T eff in Figure 5 for all GCs. In a nearly mono-metallic population like a GC, correlation of metallicity with other fitted parameters, such as T eff , would indicate the presence of systematic effects. Because T eff is strongly covariant with [Fe/H], the fitting procedure might erroneously select a lower value of [Fe/H] and T eff in order to match spectral features. Figure 5 presents evidence against any such correlation.  Figure 6, no such systematics are present in our data for each GC.

α-element
6. COMPARISON TO HIGH-RESOLUTION SPECTROSCOPY 6.1. High-Resolution Data Given the variety in approaches of HRS studies of the MW GCs listed in Table 1, we provide a summary of the stellar parameter determination and abundance analysis in each case. For all GCs, membership is determined based on radial velocities.  (2004) atmospheric models. Mg was measured from a single line (5711 Å). They determined T eff from excitation equilibrium and surface gravities from T eff , extinction-corrected bolometric magnitude, and the known distance to the cluster. The measurement uncertainties are a combination of the random error (based on the number of lines used in the abundance analysis for a given element) and a component that reflects the error from adopted stellar atmosphere parameters. For the latter component, we adopt the larger, more conservative errors that reflect averages based on the entire GC sample of Kacharov et al. (2013).
We emphasize that the HRS abundances do not include true estimates of systematic uncertainty, e.g., resulting from limitations from the selected grid of model atmospheres or line list. Additionally, we are comparing our homogenous LRS abundances to an inhomogenous HRS sample. As a result, some of the differences among the HRS studies can be attributed simply to different abundance measurement tools and techniques.

Abundance Comparison
We find reasonable agreement in [Fe/H] for the 9 stars common between both data sets, within the 1σ uncertainties. In order to perform the comparison, we have shifted the HRS abundances (Cohen & Meléndez 2005;Asplund et al. 2009) to the same solar abundance scale as the LRS abundances. Figure 7 shows no correlation between the LRS and HRS metallicity measurements within each of NGC 2419 and NGC 1904, which is expected for monometallic globular clusters.
In order to compare our [α/Fe] measurements to an analogous HRS quantity, we construct [α/Fe] HRS based on a weighted sum of Mg, Si, and Ca elemental abundances. To derive the weights, we start with a reference synthethic spectrum defined by T eff = 4400 K, log g = 1.0 dex, [Fe/H] = −1.8 dex, which correspond to mean parameter values from HRS studies of NGC 2419, NGC 1904, NGC 6864, and NGC 6341, and [α/Fe] = 0. We assume a spectral resolution of ∆λ = 2.8 Å and interpolate the synthetic spectrum onto a wavelength array with spacing equal to the pixel scale of the 600 line mm −1 grating (∼ 0.64 Å). Next, we enhance/deplete the α-element abundance by 0.1, 0.2, and 0.3 dex, calculating the sum of the absolute difference between the reference and enhanced/depleted synthetic spectrum in each case. For each α-element, we utilize only the relevant wavelength regions Between LRS and HRS data sets, we have common measurements for 9 stars. Although some scatter is present between data sets, they exhibit broad agreement within the 1σ uncertainties (δ sys = 0.02 dex, Eq. 3). No correlation exists between the LRS and HRS measurements, which we expect for a monometallic GC. ( § 4.4) and spectral coverage that corresponds to our data set (4500 -9100 Å). Additionally, we exclude contributions from masked wavelength regions ( § 3.4). We adopt the normalized average value of the summed absolute flux differences as our final weight for a given element, i.e.,  (2) In Figure 8, we utilize Eqs. 1 and 2 to directly compare HRS and LRS α-element abundances in NGC 2419 and NGC 6864. We emphasize that [α/Fe] HRS represents only an approximation to the atmospheric value of [α/Fe], given the fundamental differences between the HRS and LRS methods ( § 6.1).
Despite this apparent offset in the cluster means, we find that a star-by-star comparison of [α/Fe] and [Fe/H] shows that our LRS results are consistent with those from HRS within the random uncertainties. We prove this consistency by calculating what additional error term would be required to force the LRS and HRS measurement to agree within one standard deviation. The relevant equation is where represents a given elemental abundance, such as [Fe/H] or [α/Fe], δ is the corresponding statistical uncertainty on the measurement, i is an index representing a given star in common between both the HRS and LRS data sets, and N is the total number of common stars. We do not find a solution for δ sys in the case of [   7. QUANTIFICATION OF SYSTEMATIC UNCERTAINTY The total uncertainty on fitted parameters is composed of two components added in quadrature, the statistical (fit) uncertainty, δ fit , and a systematic component, δ sys . The fit uncertainty is calculated according to the reduced chi-squared value (χ 2 ν ) and the diagonals of the covariance matrix of the fit (σ ii ), i.e., σ ii (χ 2 ν ) 1/2 . We calculate χ 2 ν using only the regions of the observed spectrum utilized in the fit, e.g., in the case of [Fe/H], we use the wavelength regions sensitive to [Fe/H] ( § 4.4) and not excluded by the pixel mask ( § 3.4). The systematic component encapsulates uncertainty intrinsic to our method, owing to sources such as the linelist ( § 4.1), assumptions involved in spectral synthesis ( § 4.2), details of our method, such as the continuum normalization ( § 3.3) and fitting procedure ( § 4.5), and covariance with other fitted parameters. 3 7.1. Metallicity Because most GCs, including those in our sample, are nearly monometallic (Carretta et al. 2009a 10.-Probability distributions used to determine the systematic uncertainty, as in Figure 9, except for the case of [α/Fe]. We show the distributions for NGC 1904 (cyan), NGC 6864 (magenta), NGC 6341 (green), and all three clusters (blue) (69 stars). We find that δ([α/Fe])sys = 0.039 dex.
is zero, i.e., in the inequality allows stars to be considered members even if they fall outside of the allowed metallicity range, as long as some part of their 1σ confidence intervals falls within the range.
After performing this sigma clipping for each individual cluster, we subtract the mean cluster metallicity from each star's measurement of [Fe/H], and we solve for the intrinsic dispersion based on the combined sample (Eq. 4). We obtain a systematic uncertainty in [Fe/H] of δ[Fe/H] sys = 0.105 dex (Table 6) based on 117 stars. We present an illustration of this method in Figure 9, where we show the probability distributions for the total-error-weighted metallicity of each cluster, in addition to the the combined GC sample. The fact that the combined distribution is well-approximated by a Gaussian with σ = 1 indicates that the calculated systematic uncertainty sufficiently accounts for the observed metallicity spread.
Thus, the total error is, In general, the statistical fit uncertainty is negligible compared 2018 Oct 11 0.60 1.49 2400 37 a Slitmasks indicated "a" and "b" are identical, except that the slits are titled according to the median parallactic angle at the approximate time of observation.
to the systematic error for GCs. However, this will not be the case for M31, given the low value of the expected S/N.

α-element Abundance
To determine the systematic uncertainty in [α/Fe], δ([α/Fe]) sys , we calculate the intrinsic dispersion in the clusters, analogously to Eq. 4. Whereas it is generally reasonable to assume that GCs have negligible spread in [Fe/H], the assumption of zero intrinsic variation in [α/Fe] must be evaluated individually for each cluster. We exclude NGC 2419 from our combined GC sample in this case, given that abundance analysis of HRS has detected a significant spread in Mg (Cohen et al. 2011;. Although NGC 6864 possesses chemically distinct populations, O is the only α-element that exhibits significant variation within the cluster, as opposed to Mg, Si, or Ca (Kacharov et al. 2013). NGC 6341 is not known to possess α-element variations (Sneden et al. 2000), with the caveat that no recent Mg abundances from HRS have been published to our knowledge. We therefore construct our combined GC sample from NGC 1904, NGC 6864, and NGC 6341 to compute δ([α/Fe]) sys , obtaining a value of 0.039 dex (Table 6) from 69 stars. Figure 10 illustrates that the adopted error floor in [α/Fe] describes the data well. We anticipate a smaller value of δ([α/Fe]) sys relative to δ([Fe/H]) sys , given that the systematic effects (uncertainties in the line list, atmospheric parameters, continuum normalization, etc.) that impact [Fe/H] tend to similarly affect [α/H]. Therefore, the net effect on the [α/Fe] ratio is zero to first order.

THE STAR FORMATION HISTORY OF THE
STELLAR HALO OF M31 We apply our spectral synthesis technique to spectra of individual RGB stars in the stellar halo of M31. We select a field with no identified substructure as an example. We will apply our method to additional stellar halo fields in future work.

Halo Field Observations
The field, f130_2, is located at 23 kpc in projected radius along the minor axis of M31, and was first observed and characterized by Gilbert et al. (2007) using the Keck II/DEIMOS 1200 line mm −1 grating. We selected it owing to its proximity to the 21 kpc halo field of Brown et al. (2007), for which Brown et al. (2009) presented catalogs of deep optical photometry obtained using the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope. Table 7 summarizes our observations of the M31 stellar halo field, which we observed with the same configuration as described in § 2. The total exposure time was 5.8 hours. Following Cunningham et al. (2016), we designed two separate slitmasks for the single field, with the same mask center, mask position angle, and target list, but with differing slit position  (Vargas et al. 2014b). We plot the latter sample of metal-poor halo stars (grey) (S/N 15 Å −1 ) over our data set for comparison (S/N ∼ 8−22 Å −1 ).
angles. Switching slitmasks in the middle of the observation allows us to approximately track the change in parallactic angle over the course of the night. This technique mitigates flux losses due to differential atmospheric refraction (DAR), which disproportionately affects blue wavelengths. Thus, it is especially important to consider DAR when observing with the 600 line mm 1 grating, which covers a wider spectral range than any other DEIMOS grating.

Sample Selection
The observed field, at a M31 galactocentric radius of 23 kpc, includes a non-negligible contamination fraction of Milky Way foreground dwarf stars. In order to identify secure M31 members, we used a likelihood-based method ) that relies on three criteria to determine membership: the strength of the Na I λλ8190 absorption line doublet, the (V, I) color-magnitude diagram location, and photometric versus spectroscopic (Ca II λλ8500) metallicity estimates. Following Gilbert et al. (2007), we excluded radial velocity as a criterion to result in a more complete sample. In total, we identified 37 M31 stellar halo members (20 I 0 22.5) in this field out of 106 targets.
We required that our abundance measurement technique determined the abundances reliably ( § 5.2): δ([Fe/H]) < 0.5 and δ([α/Fe]) < 0.5. We also required that the 5σ χ 2 contours in each of the four fitted parameters ( § 4.5) identify the minimum. Both of these criteria effectively mimic a S/N cut (S/N 8 Å −1 ). Lastly, we manually screened member stars for molecular TiO bands between 7055−7245 Å (Cenarro et al. 2001;Gilbert et al. 2006), where affected stars exhibit a distinctive pattern. Stars with strong TiO absorption tend to be more metal-rich ([Fe/H] −1.5), have red colors ((V − I) 0 > 2.0), and can also show unusual χ 2 contours in [α/Fe]. We omitted 3 M31 member stars that meet the (V − I) 0 color criterion and show spectral evidence of strong TiO absorption. In total, this reduces the sample size to 14 stars (S/N ∼ 8−22 Å −1 ), for which we present a summary of stellar parameters and chemical abundances in Table 8.  Figure 11) over the same metallicity range (−2.5 dex [Fe/H] −1.5 dex) suggests the lack of a significant radial trend with [α/Fe] in M31 stellar halo fields absent of substructure. We also find that our 23 kpc field is on average 0.2 dex more metal rich than the outer halo Vargas et al. (2014b) measurements (see Appendix A for a discussion of potential selection effects). In combination with the approximately constant value of [α/Fe] with both [Fe/H] and radius, this may indicate that we are probing the same extended halo component, which is metal-poor, α-enhanced, and underlies substructure at all radii Gilbert et al. 2012;Ibata et al. 2014).
Given the low luminosity of the smooth halo component (L Ibata et al. (2014) inferred that it would consist of many low luminosity structures accreted at early times. In terms of star formation history (SFH), high α-element abundances indicate that the stellar population in f130_2 is characterized by rapid star formation and is dominated by the yields of Type II supernovae. Recognizing that the outer regions ( 20 kpc) of the stellar halo are most likely formed via accretion Cooper et al. 2010;Tissera et al. 2012), we infer that the disrupted dwarf galaxies that were the progenitors of this field likely had short SFHs. Their SFHs could have been truncated by accretion onto M31. Interestingly, the slightly lower average α-element abundance (0.28 dex) of Vargas et al. (2014b) could suggest that the outer halo is composed of progenitors with more extended chemical evolution as compared to the inner halo. If true, this would be in accordance with expectations from hierarchical buildup of the stellar halo Font et al. 2008). However, we cannot draw a robust conclusion on this matter given that the average α-element abundances, similar to the case of [Fe/H], between Vargas et al.'s 2014b sample and our sample are consistent at the 1σ level, which is compounded by limited sample sizes.
Our inferred SFH for f130_2 qualitatively agrees with the trend derived from deep photometry in a nearby HST/ACS field located at 21 kpc along the minor axis. The mask centers of the fields are separated by 6.33 arcmin on the sky, or 1.44 kpc, assuming a distance to both fields of 783 kpc (Stanek & Garnavich 1998). Using the Brown et al. (2006) method of comparing theoretical isochrones to color-magnitude diagrams, Brown et al. (2007) derived a SFH for the ACS field, assuming [α/Fe] = 0. They found a wide range of stellar ages and metallicities, providing support for an accretion origin, as opposed to early monolithic collapse. The field exhibits evidence for an extended SFH, with the majority of stellar ages between ∼8−10 Gyr, with a small but non-negligible ( 5%) population of stars with ages 8 Gyr. The wide range of metallicity (−2.3 < [Fe/H] < −0.7 dex) that we find in this work is consistent with a multiple progenitor hypothesis. If the nearby ACS field is representative of f130_2, this implies a composition for f130_2 of intermediate-age system(s) that had elevated star formation rates, quenched at latest 8 Gyr ago.
Comparing our average α-element abundance to that of other systems, we find that, in general, they are similarly α-enhanced. [α/Fe] for the 23 kpc M31 halo field agrees with that of M31 GCs (0.37 ± 0.16 dex) within 20 kpc of the galactic center (Colucci et al. 2009). Additionally, the metal-poor MW halo possesses elevated α-element abundance ratios of approximately +0.4 dex (Venn et al. 2004;Cayrel et al. 2004;Ishigaki et al. 2012;Bensby et al. 2014), which is comparable to our result.
Drawing comparisons to M31 dwarf galaxies is less straightforward, given that their average α-element abundance varies from approximately solar to highly α-enhanced (∼0.5 dex) (Vargas et al. 2014a). This may indicate a range of star formation timescales for these systems, where some are dominated by old stellar populations ( 10 Gyr ago) and others possess intermediate-age (∼10-7 Gyr ago) stars, although the systematic uncertainties on their SFHs at early times are large (Weisz et al. 2014 Tollerud et al. 2012). The latter case is in accordance with abundance trends found in MW dwarf spheroidal galaxies (Shetrone et al. 2001Tolstoy et al. 2003;Venn et al. 2004;Kirby et al. 2009Kirby et al. , 2011a and systems with more extended SFHs. In terms of α-enhancement and SFH, our field resembles old M31 dSphs, although it is possible that f130_2 contains intermediate age stars (Brown et al. 2007). Vargas et al. (2014a) inferred that a present-day stellar halo constructed from M31 dwarf galaxies would be metal-rich, where [Fe/H] ∼ −0.7 dex (−1.4 dex) for their full sample (old dwarf galaxies only), with a distinct α-element abundance pattern as compared to the MW halo. Given the similarly flat [α/Fe]-[Fe/H] trend between f130_2 and And VII, and the similar [Fe/H] and [Fe/H] range between f130_2 and old M31 dSphs, it is possible that the progenitors of f130_2 were composed of systems similar to And VII. In order to meaningfully test if systems similar to present day M31 dwarf galaxies could have contributed to the smooth halo component, or whether the α-element abundance pattern of the smooth halo differs from that of the MW, we would require larger sample sizes across more halo fields. 9. SUMMARY In an effort to increase the amount of available high-quality data in M31, we have developed a method of measuring [Fe/H] and [α/Fe] from low-resolution spectroscopy of individual RGB stars. We applied our technique to a field in M31's smooth stellar halo component.
The primary advantages of utilizing low-resolution spectroscopy are (1) the substantial increase in wavelength coverage (from ∼ 2800 Å with MRS to ∼ 4600 Å with LRS) available to constrain the abundances and (2) the accompanying increase in S/N per pixel for the same exposure time and observing conditions. To make spectral synthesis of DEIMOS LRS a reality, we generated a new grid of synthetic spectra spanning 4100 − 6300 Å based on a line list we constructed for bluer optical wavelengths. We find the following results: 5. Given its high α-enhancement, we surmise that the smooth halo field is likely composed of disrupted dwarf galaxies with elevated star formation rates and truncated SFHs, accreted early in the formation history of M31.
In future work, we will measure [Fe/H] and [α/Fe] from ∼6 hour observations of individual RGB stars in additional M31 halo and tidal stream fields with deep HST photometry (Brown et al. 2006), with the goal of deriving chemicallybased star formation histories.