Elemental Abundances in M31: [Fe/H] and [α/Fe] in M31 Dwarf Galaxies Using Coadded Spectra

The method for selecting candidate RGB stars in M31 is described in detail in Gilbert et al. (2006), Guhathakurta et al. (2006), Kalirai et al. (2006), and Tollerud et al. (2012). Shallow (∼1–2 hr) observations were obtained first (see Table 1), where stars likely to be RGB stars at the distance of M31 based on their color–magnitude diagram (CMD) position were given the highest preference during the slitmask design process. The target selection for the deep (∼6 hr exposure) masks was further refined using the shallow spectra to preferentially select stars highly likely to be M31 dSph members. Additional details regarding the design of the shallow masks can be found in Kalirai et al. (2009, 2010) and Tollerud et al. (2012). The imaging used for target selection for the shallow masks targeting And I, II, III, V, VII, IX, and XIV was obtained using the Washington M and T₂ filters, as well as the DDO51 intermediate filter (Ostheimer 2003; Kalirai et al. 2006; Tollerud et al. 2012; Beaton 2014). And X was discovered using photometry from the Sloan Digital Sky Survey (Adelman-McCarthy et al. 2006), and masks were designed using data from Zucker et al. (2007). Target selection for And XV was done using Canada–France–Hawaii Telescope archival imaging (Tollerud et al. 2012). The mask for And XVIII was designed using B- and V-band imaging from the Large Binocular Telescope (Beaton 2014).

For the slitmasks with deep observations, the photometry used to design the slitmasks was obtained from observations conducted with the Mosaic camera on the Kitt Peak National Observatory 4 m telescope (Beaton 2014; Kirby et al. 2020). Imaging was obtained in the Washington M and T₂ bands. The photometric transformations relations from Majewski et al. (2000) were used to transform photometry from the Washington M and T₂ magnitudes to Johnson–Cousins V and I. A star's position in (I, V − I) color–magnitude space was then used as part of the criteria to identify M31 RGB stars from MW field stars (see Section 4.1). In addition, photometry was obtained using the intermediate-width, surface gravity–sensitive DDO51 filter. Objects were assigned a DDO51 parameter, corresponding to the likelihood of the object being an RGB star, based on its position in (M–DDO51)–(M − T₂) color–color space (Palma et al. 2003). In general, objects with point-like morphology and high values of the DDO51 parameter were given the highest priority when constructing the spectroscopic slit masks for DEIMOS.

For five of these galaxies (And I, III, V, VII, and X), we have deep (∼6 hr total exposure time, $\langle {\rm{S}}/{\rm{N}}\rangle =15.3$ per angstrom) spectroscopic observations. For all 10 M31 dSphs, we have shallow spectroscopic observations (∼1–2 hr total exposure time, $\langle {\rm{S}}/{\rm{N}}\rangle =7.6$ per angstrom). We note that these shallow masks were originally designed for obtaining kinematic information only, while the deep masks were explicitly designed to get S/N high enough to determine abundances for individual stars. Details of the observations for each slitmask can be found in Table 1.

Spectra were obtained using the 1200 G (1200 l mm⁻¹) grating, with the OG550 order blocking filter. The observed spectra cover a spectral range of ∼6300 Å < λ < 9100 Å, with a resolution of ∼6000 at the central wavelength of ∼7800 Å. The spectral dispersion of this science configuration is 0.33 Å pix⁻¹, with a resolution of 1.2 Å FWHM.

One-dimensional science spectra were extracted from raw 2D DEIMOS spectra using the DEEP2 DEIMOS data reduction pipeline, contained within the spec2d IRAF package, developed by the DEEP2 Galaxy Redshift Survey¹⁰ (Cooper et al. 2012; Newman et al. 2013). Details on modifications made for better performance on bright stellar sources (versus faint galaxies for which the pipeline was written) can be found in Simon & Geha (2007). Line-of-sight velocities for each star were obtained by cross-correlating observed spectra with the stellar templates provided in Simon & Geha (2007). Table 2 provides a summary of the templates used and their associated stellar atmospheric parameters. To correct for the imperfect centering of a star in the slit, we use the relative positions of the Fraunhofer A band and other telluric absorption features (Simon & Geha 2007; Sohn et al. 2007).

Photometric effective temperature (${T}_{\mathrm{eff},\mathrm{phot}}$), surface gravity (log g), and photometric metallicity ([Fe/H]_phot) for individual stars are determined via linear interpolation between isochrones, assuming a single distance for a given object. Assuming an age of 14 Gyr and [α/Fe] = +0.3 dex, we find the best-fit Padova (Girardi 2016), Victoria–Regina (VandenBerg et al. 2006), and Yonsei–Yale (Demarque et al. 2004) isochrones, using a star's position in (I, V − I) space. We take the error-weighted mean of all three isochrone sets as our final measured ${T}_{\mathrm{eff},\mathrm{phot}}$, log g, and [Fe/H]_phot. While there are slight variations between the photometric values derived from the different isochrone sets, they do not significantly affect the abundance measurement (Kirby et al. 2008). Additionally, we justify a single age and α-enhancement by noting that, for our sample, ${T}_{\mathrm{eff},\mathrm{phot}}$ and log g are largely insensitive to variations in age and assumed [α/Fe] of the isochrone set. For individual stars, ${T}_{\mathrm{eff},\mathrm{phot}}$ is used as a first approximation in measuring spectroscopic T_eff, and log g is held constant throughout the spectroscopic abundance fitting procedure.

The construction of the grid of synthetic spectra for the 1200 G grating is described in detail in Kirby et al. (2008, 2009) and Kirby (2011). To summarize, synthetic spectra are generated using modified ATLAS9 model atmospheres (Kurucz 1993; Sbordone et al. 2004; Sbordone 2005; Kirby 2011) without convective overshooting (Castelli et al. 1997). The line list for atomic transitions was sourced from the Vienna Atomic Line Database (Kupka et al. 1999), molecular lines from Kurucz (1992), and the hyperfine transition line list from Kurucz (1993), where the oscillator strengths were tweaked to match values from observed spectra of the Sun and Arcturus (Kirby 2011). The parameter ranges for the stellar atmospheric parameters cover old RGB stars, subgiants, and lower main-sequence stars: $3500\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 8000\,{\rm{K}}$, 0.0 ≤ log g ≤ 5.0, −5.0 ≤ [Fe/H] ≤ 0.0, and −0.8 ≤ [α/Fe] ≤ +1.2. Further details of the grid can be found in Kirby et al. (2008, 2009). Synthetic spectra are then generated using the LTE stellar synthesis code MOOG (Sneden 1973), with a resolution of 0.02 Å.

Here, we outline the pipeline we use to measure [Fe/H] and [α/Fe] for individual stars (Kirby et al. 2008; Escala et al. 2019), which we modify to measure abundances from coadded spectra as discussed in Section 4.

3.3.1. Telluric Correction

3.3.2. Initial Continuum Normalization

3.3.3. Abundance Masks

3.3.4. Measuring Spectroscopic T_eff, [Fe/H], [α/Fe]

To ensure membership for stars in our GC sample ($\langle {\rm{S}}/{\rm{N}}\rangle =158.13$ per angstrom), we use the (I, V − I) CMD to select only those stars that fall on the RGB or AGB. We also require that stars satisfy log g < 3.6 to ensure that they are giants.

For our sample of stars in M31 dSphs, we adopt the criteria used previously for analysis of these samples. We describe below the differences between the membership selection used for masks with deep observations versus those with shallow observations. For our M31 dSphs with deep (∼6 hr, $\langle {\rm{S}}/{\rm{N}}\rangle =15.3$ per angstrom) observations, we identify dSph members using a two-step process. First, we use the M31/MW membership classification scheme described by Gilbert et al. (2006) to identify stars that are likely to be RGB stars at approximately M31's distance, rather than foreground MW dwarf stars. This membership classification is based on a number of criteria including radial velocity, a star's position in ($M-\mathrm{DDO}51$) versus (M − T₂) color–color space, its position in (I, V − I) color–magnitude space, and the equivalent width of the Na i doublet at 8190 Å (λλ 8183,8195). We note that the membership classification for our dSphs assumes M31's distance modulus. Therefore, we consider stars to be a dSph member if they are more probable to be an M31 RGB star than an MW foreground star based on the above diagnostics. We make an exception for stars that are slightly too blue for the most metal-poor isochrone at the distance of M31 and would otherwise be classified as MW foreground stars. We include these stars in our membership selection because they are likely to belong to the M31 satellite if they pass additional velocity cuts (Gilbert et al. 2006). Second, we require that all stars that satisfy the M31 RGB criteria also explicitly fall within the expected radial velocity limits for a given M31 dSph, listed in Table 3. These velocity limits correspond to approximately the difference between the mean systemic velocity and three times the velocity dispersion for a given dSph (see Table 3 of Kirby et al. 2020).

Table 3.  Kinematic Properties of M31 dSphs

Object Name	$\langle {v}_{\mathrm{helio}}\rangle$	σv	Velocity Limits	Referencea
	(km s−1)	(km s−1)	(km s−1)
And I	$-{378.0}_{-2.3}^{+2.4}$	${9.4}_{-1.5}^{+1.7}$	$[-390,-340]$	1
And II	−192.4 ± 0.5			2
And III	$-348.0{\pm }_{-2.6}^{+2.7}$	${11.0}_{-1.6}^{+1.9}$	$[-370,-320]$	1
And V	−397.7 ± 1.5	${11.2}_{-1.0}^{+1.1}$	$[-440,-360]$	1
And VII	−311.2 ± 1.7	${13.2}_{-1.1}^{+1.2}$	$[-340,-260]$	1
And IX	−209.4 ± 2.5			3
And X	−166.9 ± 1.6	${5.5}_{-1.2}^{+1.4}$	$[-180,-140]$	1
And XIV	−480.6 ± 1.2			3
And XV	−323.0 ± 1.4			3
And XVIII	−332.1 ± 2.7			3

Note.

^aReferences. (1) Kirby et al. 2020; (2) Ho et al. 2012; (3) Tollerud et al. 2012.

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For M31 dSph stars with shallow (∼1–2 hr, $\langle {\rm{S}}/{\rm{N}}\rangle =7.6$) observations, we adopt the M31 dSph membership criteria described by Tollerud et al. (2012). This membership determination simultaneously evaluates the probability the star is an RGB star at the distance of the M31 system as well as a member of the dSph. It relies on three probabilistic criteria: the distance between a given star and the dSph to which it lies near, a star's position in (T₂, M − T₂) color–magnitude space, and the equivalent width of the Na i doublet. In addition, a "wide selection window around the dSphs systemic velocity" (Tollerud et al. 2012) is used to remove obvious outliers from their dSph membership. These four criteria are then used to calculate the probability of a star being a member of a given dSph (see Equation 1 of Tollerud et al. 2012), where stars having a membership probability greater than 0.1 are selected as dSph members. We include the systematic velocities of these dSphs with shallow observations in Table 3 for completeness.

We adopt two binning strategies, sorting stars by either photometric T_eff or photometric metallicity ([Fe/H]_phot), and then bin them such that each coadd has at least five stars. We find, as in Yang et al. (2013), that the final measured abundances are consistent when stars are grouped by either photometric temperature or metallicity, so for the purposes of our analysis, we simply use the photometric metallicity. The choice to bin by photometric [Fe/H] is motivated by the reasoning given in Yang et al. (2013), where they state that the synthetic spectra can account for a range of T_eff within a bin, but not [Fe/H] or [α/Fe]. For completeness, we include a brief summary of the following analysis done with stars sorted by photometric T_eff in Appendix B.

A group of spectra to be coadded are then processed using the same method described above (Section 3), where they are corrected for telluric absorption and shifted to the rest frame, and an initial continuum normalization is performed. Each spectrum in the group is then rebinned to the same spectral range (6300–9000 Å), with a wavelength spacing of 0.32 Å for the 1200 l mm⁻¹ grating, to ensure that all spectra are on the same pixel array. This rebinning is applied to both the flux and inverse variance arrays for each spectrum. We then perform a two-step sigma clipping for each pixel in the group of spectra to be added. If a pixel for any spectrum in the group deviates more than 10σ or 3σ (where σ is the standard deviation of the flux for a given pixel) from the median flux for the first and second step of the sigma clipping, respectively, the pixel is masked when adding that spectrum to the coadded spectrum.

Fluxes are coadded together, weighted by their inverse variance (1/σ²):

$$\begin{eqnarray}&&{\bar{x}}_{\mathrm{pixel},i}=\displaystyle \frac{{\sum }_{j=1}^{n}({x}_{\mathrm{pixel},{ij}}/{\sigma }_{\mathrm{pixel},{ij}}^{2})}{{\sum }_{j=1}^{n}1/{\sigma }_{\mathrm{pixel},{ij}}^{2}},\end{eqnarray} \tag{ 1 }$$

where ${x}_{\mathrm{pixel},{ij}}$ represents the flux of the ith pixel of the jth spectrum in a group, ${\sigma }_{\mathrm{pixel},{ij}}^{2}$ is the corresponding variance of ${x}_{\mathrm{pixel},{ij}}$, and n is the total number of spectra to be coadded. The inverse variance weighted flux of the ith pixel of the coadded spectra is then given by ${\bar{x}}_{\mathrm{pixel},i}$. The inverse variance of the coadded spectra is:

$$\begin{eqnarray}&&{\bar{\sigma }}_{\mathrm{pixel},i}={\left(\sum _{j=1}^{n}\displaystyle \frac{1}{\sigma {{}^{2}}_{\mathrm{pixel},{ij}}}\right)}^{-1/2}.\end{eqnarray} \tag{ 2 }$$

Figure 2 shows a small portion of the full spectral range for an example coadd, where the final coadded spectrum and the corresponding best-fit synthetic spectrum (red) are shown in the top panel, and the individual constituent spectra are shown in the lower panels.

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To ensure an accurate comparison between our observed coadded spectra and the synthetic spectral grid, we coadd synthetic spectra using the same procedure as the observed spectra (Equation (1)). During each abundance determination step, from the synthetic spectral grid, we select a group of synthetic spectra with temperatures and surface gravities that correspond to the T_eff and $\mathrm{log}g$ (determined from photometry) for each observed spectrum. The synthetic spectra are rebinned to the same wavelength range and pixel scale as the observed spectra, and are smoothed with a Gaussian filter to replicate the spectral resolution of the observed spectra, where the method to determine the spectral resolution is described in Section 3.3.4. The synthetic spectra are then weighted by the same inverse variance array (Equation (2)) as their corresponding observed counterparts to produce a coadded synthetic spectrum for each iteration of the fitting procedure.

The process for measuring abundances from coadded spectra varies slightly from the process outlined in Section 3. When finding the best fit between our coadded observational spectrum and coadded synthetic spectra, we use photometric T_eff and $\mathrm{log}g$ estimates from each observed star for their counterpart in the coadded synthetic spectrum in each iteration of the fit. We first fit only [Fe/H], keeping [α/Fe] fixed, and then fit [α/Fe], keeping [Fe/H] fixed, using the same masks described in Section 3.3.3 for these abundance measurements. We refine the continuum of the coadded spectrum as described in Section 3.3.4. We iterate over these three steps (fitting [Fe/H], [α/Fe], and continuum refinement) until the measured values for [Fe/H] [α/Fe] change by less than 0.001 dex on consecutive passes.

We then take the coadded spectrum with its refined continuum and fit for [Fe/H], taking the [α/Fe] value from the last iteration of the continuum refinement. We repeat this process for [α/Fe], keeping [Fe/H] constant from the previous step, and adopt this value as our final measured [α/Fe]. Finally, we refit for [Fe/H] using the final measured [α/Fe], and take the value of [Fe/H]from this step as our final measured [Fe/H]. After this round of fitting [Fe/H] and [α/Fe], we check the goodness of the fit. For our coadded abundance measurements, we apply the same quality criteria as in Section 3. In addition, we examine the χ² contours for [Fe/H] and [α/Fe] to ensure that the minimum is well-defined.

For our coadded spectra, we consider two sources of uncertainties on [Fe/H] and [α/Fe]. First, we estimate the fit (or statistical) uncertainty from the diagonals of the covariance matrix and the reduced chi-square value (χ²) of the fit. For either [Fe/H] or [α/Fe], we take into account the same spectral mask (Section 3.3.3) used in the fit, in order to only consider regions of the spectrum sensitive to either [Fe/H] or [α/Fe]. We also consider the effect of uncertainties on the photometric T_eff and $\mathrm{log}g$ values as part of the fit uncertainty. The process used to estimate this portion of the fit uncertainty is described in Section 5.1.1. Second, we consider the systematic uncertainties, which encompass all other sources of unknown uncertainty in our fits (Section 5.1.2).

5.1.1. Estimating Fit Uncertainties due to Uncertainties from Photometric Stellar Parameters

In measuring [Fe/H] and [α/Fe] for each star, we adopt values for T_eff and $\mathrm{log}g$ derived from photometry. These values are propagated through our pipeline to the comparison with synthetic spectra (Section 4.4). As we do not fit for T_eff or $\mathrm{log}g$, we estimate the component of the fit uncertainty due to uncertainties in these photometric stellar parameters. We assume the photometric T_eff and $\mathrm{log}g$ uncertainties are Gaussian, and draw 50 random samples from the covariant T_eff–$\mathrm{log}g$ distributions for each star in a given coadd. We then remeasure [Fe/H] and [α/Fe] for each of these samples for each coadd. We then take the standard deviation of the abundance measurements from all samples for a given coadd as the typical uncertainty due to the photometric T_eff and $\mathrm{log}g$(σ_MC), and add this uncertainty in quadrature to the fit uncertainty (σ_fit) to get our total fit uncertainty for both [Fe/H] and [α/Fe]:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{fit},\mathrm{total}}=\sqrt{{\sigma }_{\mathrm{fit}}^{2}+{\sigma }_{\mathrm{MC}}^{2}}\end{eqnarray} \tag{ 3 }$$

The magnitude of this uncertainty is small compared to the fit uncertainty, with median values of approximately 0.02 dex for [Fe/H] and 0.01 dex for [α/Fe].

5.1.2. Quantifying Systematic Uncertainties

We estimate the systematic uncertainties on our coadded abundance measurements by estimating the uncertainty term needed to enforce that the measurement from the coadd agrees with the weighted average within one standard deviation. We compute this using all validation coadds from all GCs:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\sum }_{i}{N}_{i}-1}{\sum }_{i=1}^{{N}_{\mathrm{GC}}}{\sum }_{j=1}^{{N}_{{\rm{i}}}}\displaystyle \frac{{\left({X}_{\mathrm{coadd},{\rm{i}},{\rm{j}}}-\langle {X}_{i}\rangle \right)}^{2}}{{\sigma }_{\mathrm{fit},\mathrm{coadd},{\rm{i}},{\rm{j}}}^{2}(X)+{\sigma }_{\mathrm{sys}}^{2}(X)}=1,\end{eqnarray} \tag{ 4 }$$

where i indicates a given GC, j indicates a given coadd for that GC, N_GC is the number of GCs in our sample (6), and N_i is the number of coadds in the ith GC. The coadded abundance value in a given GC for X = [Fe/H] or [α/Fe] is ${X}_{\mathrm{coadd},{\rm{i}},{\rm{j}}}$, ${\sigma }_{\mathrm{coadd},{\rm{i}},{\rm{j}}}{(X)}^{2}$ is the fit uncertainty for that measurement, and $\langle {X}_{{\rm{i}}}\rangle$ is the weighted average abundance from the coadds for a given GC. We then numerically solve for ${\sigma }_{\mathrm{sys}}^{2}(X)$, for both [Fe/H] and [α/Fe]. With our sample of GCs, we measure a systematic uncertainty of 0.02 dex for [Fe/H] and 0.07 for [α/Fe]. Figure 4 shows the uncertainty-weighted [Fe/H] and [α/Fe] distributions for the validation coadds in each GC (colored histograms) as well as for all GCs (black histogram). A Gaussian with standard deviation equal to the derived systematic uncertainty is overplotted. The width of this Gaussian corresponds to how well the expected uncertainties capture the total uncertainties in our sample. A best-fit Gaussian that is narrower than a unit Gaussian would indicate that we overestimate our uncertainties, while a best-fit Gaussian that is wider than a unit Gaussian would indicate that we underestimate our uncertainties.

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We find the systematic uncertainty for our GC sample to be on the order of their fit uncertainties. Therefore, we adopt the same systematic uncertainty floors as in Gilbert et al. (2019) for spectra measured with the 1200 G grating: 0.101 dex for [Fe/H], and 0.084 dex for [α/Fe]. The total uncertainty on our coadded abundance measurements (${\sigma }_{\mathrm{total},\mathrm{coadd}}(X)$) is then:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{total},\mathrm{coadd}}(X)=\sqrt{{\sigma }_{\mathrm{fit},\mathrm{coadd}}^{2}+{\sigma }_{\mathrm{sys}}^{2}},\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{\mathrm{fit},\mathrm{coadd}}^{2}$ is the fit uncertainty as described in Section 5.1.1, and ${\sigma }_{\mathrm{sys}}^{2}$ is the systematic uncertainty corresponding to the systematic uncertainty floor for either [Fe/H] or [α/Fe] as given above.

To test the accuracy of our pipeline, we compare the inverse-variance—weighted average [Fe/H] and [α/Fe] from measurements of individual spectra with that measured from coadded spectra. We would expect similar results if a coadded spectrum is truly representative of the weighted average of its constituents.

To compute the average inverse-variance—weighted abundance of a group of spectra, we define the following weights (ω_j), following Yang et al. (2013):

$$\begin{eqnarray}&&{\sigma }_{\mathrm{spec},j}^{2}={\left(\sum _{i=1}^{\mathrm{mpixel}}{({\sigma }_{\mathrm{pixel},{ij}})}^{-2}\cdot {M}_{\mathrm{element},X}\right)}^{-1},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\omega }_{j}(X)=\displaystyle \frac{1/{\sigma }_{\mathrm{spec},{\rm{j}}}^{2}}{\sum _{j=1}^{n}1/{\sigma }_{\mathrm{spec},{\rm{j}}}^{2}},\end{eqnarray} \tag{ 7 }$$

where mpixel is the total number of pixels in a spectrum, ${\sigma }_{\mathrm{pixel},{ij}}$ is the variance of the ith pixel of the jth spectrum in a group, and ${M}_{\mathrm{element},X}$ is the corresponding elemental mask for fitting X, where X is either [Fe/H] or [α/Fe]. The weighted variance for the jth spectrum is ${\sigma }_{\mathrm{spec},j}^{2}$, and ω_j(X) is the weight of either [Fe/H] or [α/Fe] for the jth star of n stars in a coadded group. The weighted average abundance is then:

$$\begin{eqnarray}&&{X}_{\mathrm{bin},\mathrm{wa}}=\sum _{j=1}^{n}{\omega }_{j}(X){X}_{j}=\displaystyle \frac{{\sum }_{j=1}^{n}{X}_{j}/{\sigma }_{\mathrm{spec},{\rm{j}}}^{2}}{{\sum }_{j=1}^{n}1/{\sigma }_{\mathrm{spec},{\rm{j}}}^{2}}.\end{eqnarray} \tag{ 8 }$$

The weighted uncertainty on the weighted average abundance is given as:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{bin},\mathrm{wa}}^{2}(X)={\omega }_{j}^{2}(X){\sigma }_{\mathrm{total},{\rm{j}}}^{2}(X),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\sigma }_{\mathrm{total},{\rm{j}}}(X)=\sqrt{{\left({\sigma }_{\mathrm{fit},{\rm{j}}}(X)\right)}^{2}+{\left({\sigma }_{\mathrm{sys}}(X)\right)}^{2}},\end{eqnarray} \tag{ 10 }$$

where we adopt the systematic uncertainties of 0.101 dex for [Fe/H] and 0.084 dex for [α/Fe] from Kirby et al. (2020), where the method used to measure these values is described by Kirby et al. (2010).

In Figures 5–7, we compare the weighted average [Fe/H] and [α/Fe] from measurements of individual spectra with their measurements from coadds for MW GCs, M31 dSphs with deep observations, and M31 dSphs with shallow observations, respectively. In each consecutive plot, we include the points from the previous plot (as gray points), for reference and to illustrate the range in [Fe/H] and [α/Fe] that each sample covers.

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5.2.1. MW GCs

5.2.2. M31 dSphs with Deep Observations

5.2.3. M31 dSphs with Shallow Observations

In Figure 7, we consider our sample of M31 dSphs for which we have shallow observations (colored points), with MW GCs and deep dSph samples for comparison in light gray and gray, respectively. This sample contains, on average, stars with much lower S/N spectra ($\langle {\rm{S}}/{\rm{N}}=11.7\rangle$, see Figure 3). For [Fe/H], we find an offset of 0.022 dex with a dispersion of 0.066 dex, and for [α/Fe], we find an offset of −0.053 dex, with a dispersion of 0.074 dex. There is no trend in difference with measured abundance for either [Fe/H] or [α/Fe]. We find, in general, that the difference between the weighted-average and coadded abundances for shallow M31 dSph spectra is consistent with those from deep observations, illustrating the performance of our method at the lower end of the S/N regime.

If we consider all three samples simultaneously, we find an average offset in [Fe/H] of 0.015 dex, with a dispersion of 0.074 dex, and for [α/Fe], we find an offset of 0.002 with a dispersion of 0.076 dex. Unlike Yang et al. (2013), we do not find any trend in average difference as a function of [Fe/H] or [α/Fe] for any of our samples (see their Figure 8). These offsets are approximately an order of magnitude lower than the systematic uncertainty floors (0.101 for [Fe/H], 0.084 for [α/Fe]). As a result, we conclude that our coaddition method accurately reproduces the same [Fe/H] or [α/Fe] measurement as the weighted average of the individual spectra in a group.

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In Figure 9, we present the [α/Fe]–[Fe/H] distributions for the 10 M31 dSphs in our sample, with measurements from individual stars indicated by transparent gray circles, measurements from validation coadds with solid circles, and measurements from science coadds with diamonds. Measurements made from shallow data are color-coded according to a galaxy's stellar mass, and measurements made from deep data are shown in gray. The dSphs are sorted approximately by mass, so that the relationship of decreasing average metallicity with decreasing mass is easily visible. As in Figure 8, we keep And V in the left-hand column, as it more relevant to compare it to the more massive satellites with a larger sample size. The less massive dSphs are more sparsely sampled, but we find they still follow roughly the same trend (see Section 7.3 for an in-depth discussion). We note that, for four dSphs in our sample (And XVIII, XV, XIV, and IX), our [α/Fe] measurements are the first available for these dwarf galaxies.

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For the majority of the dSphs in our sample, we also see a clear decrease in [α/Fe] as a function of [Fe/H] for individual stars (transparent gray points). Even where we have a significantly smaller sample of stars (e.g., And XIV, And X), there is still a visibly negative trend, which is further supported by the inclusion of our coadded abundances. We quantify these trends in Section 7.2. The validation coadd abundance measurements (circles) generally fall neatly within the center of the 2D [α/Fe]–[Fe/H] distribution of the individual stellar abundances, while the science coadded measurements have a wider spread in both [Fe/H] and [α/Fe]. This echoes what we have already found in Figure 8 and with the statistical tests performed in Section 6: the distributions of the validation sample and coadded measurements for a given dSph generally have the same mean, but differing variance.

Vargas et al. (2014a) measured [α/Fe] and [Fe/H] for individual stars in six of the dSphs in our sample (And I, II, III, V, VII, X), and so here we qualitatively compare our results with those presented in their Figure 4. We note that they did not cull their sample to remove stars with large uncertainties, whereas we have an uncertainty cut at 0.4 dex in both [α/Fe] and [Fe/H] for individual stars in our sample. In addition, they required that spectra in their sample have a minimum S/N of 15 per angstrom, while we have no minimum S/N requirement for stars in our sample. Finally, their sample consists of stars from slitmasks with shallow observations only. We find general agreement in the [α/Fe]–[Fe/H] distributions for the dSphs in common between our samples, with one notable exception. Vargas et al. (2014a) found a significant plateau in [α/Fe] for the metal-rich ([Fe/H] ≳ −1.5 dex) stars in And VII. In contrast, we find that, while we have a number of high-[α/Fe], metal-rich stars in And VII, they do not form a plateau. However, we also note that, while we measure a negative trend in the individual stars (Section 7.2), we do not measure a significant negative trend with our coadded measurements.

In Section 7, we presented [Fe/H] and [α/Fe] for 10 dSphs in M31. Nine of them have at least one measurement from individual stars, and all have coadded measurements that are representative of the distribution of individual abundance measurements. Five of these dSphs were also presented by Kirby et al. (2020): And VII, And I, And III, And V, and And X, where in addition to atmospheric [α/Fe], they also measure individual α-elements (Mg, Ca, Si, Ti) for a number of stars in their sample with sufficiently high S/N. In this section, we will focus on the SFH from coadded measurements in M31 dSphs at the lower end of the mass range of our sample (And V–And X), as the SFH of the higher-mass dSphs (And VII, II, I, and III) based on this chemical space were discussed at length in previous works (e.g., Vargas et al. 2014a; Kirby et al. 2020).

Vargas et al. (2014a) used the [α/Fe]–[Fe/H] plane to characterize the SFHs of the dSphs in their sample, and compared their results to those in the literature from photometric studies of their CMDs (e.g., Grebel & Guhathakurta 1999; da Costa et al. 2000; Da Costa et al. 2002; McConnachie et al. 2007; Weisz et al. 2014). To summarize the findings from photometric studies: the majority of M31 dSphs show evidence for an extended SFH from their CMDs, with prominent red horizontal branch (HB) stars. The presence of a red HB indicates stars that formed more recently at multiple metallicities, compared to the primarily metal-poor, ancient stars that populate the blue HB. Martin et al. (2017) reached similar conclusions for some of the less massive satellites (And XVIII, XV, IX, X) in our sample.

We find that our coadded measurements follow the same trend of decreasing [α/Fe] with [Fe/H] seen in individual stars. The more massive satellites in our sample (left column of Figure 9) illustrate this trend clearly, while the less massive satellites generally exhibit negative trends, albeit with smaller sample sizes. In general, a decreasing trend in [α/Fe] as a function of [Fe/H] indicates an extended SFH, which is consistent with what has been inferred from the presence of red HB stars in their CMDs.

We quantitatively investigate the SFHs from coadded measurements in our dSphs by measuring the slope in [α/Fe]–[Fe/H] space (d[α/Fe]/d[Fe/H]), where [α/Fe] acts as a proxy for stellar age and [Fe/H] provides a measure of the chemical enrichment. To measure d[α/Fe]/d[Fe/H], we use a marginalized posterior distribution with flat priors to fit a line to data with uncertainties on both the dependent and independent axes, as presented by Hogg et al. (2010) and described in detail for a similar application in Kirby et al. (2019).

We use a Markov chain Monte Carlo, implemented with the emcee Python package (Foreman-Mackey et al. 2013), to generate a distribution of models that fit our data while also taking uncertainties into account. Rather than fitting a line with a slope (m) and intercept (b), we parameterize the line of best fit with an angle (θ), and the perpendicular distance of the line from the origin (b_⊥), where ${b}_{\perp }\equiv b\,\cos (\theta )$ (Hogg et al. 2010). This parameterization avoids the preference for shallow slopes when there is a flat prior on the slope. The log likelihood is defined as:

$$\begin{eqnarray}&&\mathrm{ln}\,{\mathscr{L}}=-0.5\sum _{i=1}^{N}\displaystyle \frac{{{\rm{\Delta }}}_{i}^{2}}{{{\rm{\Sigma }}}_{i}^{2}}-\mathrm{ln}(\sqrt{2\pi }{{\rm{\Sigma }}}_{i}),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}={[\alpha /\mathrm{Fe}]}_{i}\,\cos \theta -{[\mathrm{Fe}/{\rm{H}}]}_{i}\,\sin \theta -{b}_{\perp },\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{i}^{2}=\delta [\alpha /\mathrm{Fe}{]}_{i}^{2}\,{\cos }^{2}\theta +\delta [\mathrm{Fe}/{\rm{H}}{]}_{i}^{2}\,{\sin }^{2}\theta ,\end{eqnarray} \tag{ 13 }$$

where for star i, δ[Fe/H]_i and δ[α/Fe]_i represent the total measurement uncertainties for [Fe/H] and [α/Fe], respectively. Using the emcee package, we sample the probability distribution and transform our parameters back into the more familiar m and b using m = tan(θ) and $b={b}_{\perp }/\cos (\theta )$. For both θ and b_⊥, we take the median of the resulting distribution as our measured value, and the 16th and 84th percentiles (1σ) as the uncertainties on our measured values.

We measure slopes for both the individual measurements (transparent gray points in Figure 9), as well as the slope for all validation and science coadds combined. The measured slopes and their associated uncertainties are given in Table 7. We find that, for the majority of our sample, the slopes from measured individual abundance measurements and coadds roughly agree, within ∼1–2σ. We interpret this finding as another indication that, in general, abundance measurements from our coadded spectra are representative of the overall abundance measurements from individual spectra.

Table 7.  Mass–Metallicity Relation and Abundance Trends in M31 dSphs

Object Name	log(M*/M⊙)	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	m (individual)	m (coadd)
And VII	7.21 ± 0.12*	−1.26 ± 0.003	−0.17 ± 0.07	+0.01 ± 0.09
And II	6.96 ± 0.08	−1.47 ± 0.006	−0.72 ± 0.14	−0.64 ± 0.16
And I	6.86 ± 0.40*	−1.36 ± 0.004	−0.65 ± 0.10	−0.73 ± 0.15
And III	6.27 ± 0.12*	−1.65 ± 0.005	−0.73 ± 0.11	−0.42 ± 0.10
And V	5.81 ± 0.12*	−1.76 ± 0.003	−0.45 ± 0.06	−0.40 ± 0.08
And XVIII	5.90 ± 0.30	−1.33 ± 0.02	⋯	−0.01 ± 0.25
And XV	5.89 ± 0.145	−1.43 ± 0.42	⋯	−0.43 ± 0.21
And XIV	5.58 ± 0.25	−2.23 ± 0.01	−0.16 ± 0.29	−0.80 ± 0.82
And IX	5.38 ± 0.44	−2.03 ± 0.01	⋯	−0.12 ± 0.26
And X	5.08 ± 0.03*	−2.52 ± 0.27	−0.60 ± 0.21	−0.32 ± 0.25

Notes. Masses indicated with asterisks are from Kirby et al. (2020), and are the product of the stellar mass-to-light ratios from Woo et al. (2008) and luminosities from Tollerud et al. (2012). Masses without asterisks are from Kirby et al. (2013) and references therein.

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We find evidence for slopes in [α/Fe]−[Fe/H] from our coadded measurements at the 2σ significance level for all of our low-mass galaxies with the exceptions of And VII, And IX, and And XVIII. For And VII, we interpret this as a consequence of the presence of metal-rich, high-[α/Fe] stars, which would flatten a negative trend in [α/Fe]–[Fe/H]. Such metal-rich, high-[α/Fe] stars are absent from our And IX sample, and we discuss below potential reasons for the lack of a significant trend for this dSph. Finally, we note that the slope for And XVIII was measured using only four coadded points.

For our stars in And VII with individual measurements, we find a negative slope in d[α/Fe]/d[Fe/H] that roughly corresponds with that found by Kirby et al. (2020), and this negative slope is significantly more shallow compared to other galaxies in the sample. Moreover, using our coadded measurements, we do not find evidence for a negative correlation. These findings, along with the presence of high-[α/Fe], metal-rich stars, are in contrast to the SFH from shallow Hubble Space Telescope (HST) photometry (Weisz et al. 2014), where they find And VII to have an ancient SFH. To resolve this disparity, deeper spectroscopic observations of And VII are needed, as noted by Kirby et al. (2020).

The slopes that we measure quantitatively agree with Kirby et al. (2020), where the shallower slopes of the more massive galaxies indicate higher star formation rates over an extended period. They compare the observed metallicity distribution function with chemical evolution models (Kirby 2011; Kirby et al. 2013) to find a best-fit chemical evolution model, and compare these best-fit models with the slopes they measure in [α/Fe]–[Fe/H]. They find that galaxies with shallower slopes are incompatible with a simple Leaky Box Model (Schmidt 1963; Talbot & Arnett 1971), and are more suggestive of the Pre-Enriched Model (Pagel 1997), or a model where gas accretion prolongs star formation (Accretion Model; Lynden-Bell 1975). In the Leaky Box model, a galaxy can lose gas at a rate equal to that of its star formation activity, but it does not acquire any new gas. The fact that we find more massive galaxies to have shallower slopes in [α/Fe]–[Fe/H] indicates they are more likely to hold onto their gas, or to have had some accretion of gas, likely owing to the depth of their gravitational potential wells in comparison to the less massive dSphs (Tolstoy et al. 2009; Letarte et al. 2010; Kirby 2011). As a result, these galaxies have a less efficient SFH, where SNe Ia, which produce more Fe than α elements, dominate over SNe II.

We take the presence of a slope in our lower-mass galaxies as further evidence of an extended SFH in these galaxies, as also evidenced by the presence of red HB stars in the CMDs. In particular, And XIV is interesting because it hosts a more balanced (i.e., a more equal number of red HB and blue HB stars) HB (Martin et al. 2017) in comparison to the other dSphs in the low-mass regime. From our spectroscopic abundance measurements, we find that it is more metal-poor than one would naively expect following the mass–metallicity relation (MZR, Section 7.3). These spectroscopic measurements corroborate the evidence from photometric data that And XIV hosts a more substantial fraction of old, metal-poor stars in comparison to other dSphs of similar mass.

We also find tentative evidence for a slope in the [α/Fe]–[Fe/H] distribution for And IX (m = −0.12 ± 0.26). Using HST photometry, Martin et al. (2017) found And IX was relatively metal-rich for its stellar mass, but they noted that their sample likely had some contamination from the M31 halo. Such contamination would flatten the slope. However, even considering our strict membership criteria, we still find that And IX is more metal-rich than other dSphs of similar mass. We note that the slope (d[α/Fe]/d[Fe/H]) we measure from coadded stars in And IX is significantly shallower compared to other dSphs with similar stellar masses. This difference in the slope may be due to the presence of metal-rich stars (which are undetected in our sample of And IX stars, unlike for And VII), large uncertainties in [α/Fe], or simply a lack of spectroscopic metallicity measurements at the metal-poor end of the distribution. As the slope from coadded measurements is calculated from only five stars, we refrain from drawing further conclusions.

Kirby et al. (2013) presented the MZR for dwarf galaxies in the Local Group, fit using 15 MW dSphs. In their analysis, they found from coadded spectra that the M31 dSphs in their sample roughly corresponded to the same relation, indicating that satellite galaxies of the MW and M31 share the same MZR, which is roughly consistent with the MZR measured from more massive galaxies. However, for the majority of their M31 dSphs, the average metallicity was calculated by coadding stars in only a few bins, and in some cases, only one bin was used for a given dSph. Kirby et al. (2020) update this result by remeasuring the mean [Fe/H] from measurements from individual stars for five M31 dSphs with deep spectroscopic observations (And I, III, V, VII, X). They find that their new values agree within 0.2 dex for all dSphs in their sample except for And VII, which agrees within 0.25 dex.

We update this finding by combining both the individual measurements and the coadded measurements for [Fe/H]for each dSph. To do this, we take the uncertainty-weighted average of the individual and coadded measurements, where the total uncertainty for individual stars is given in Section 3.3.4, and the total uncertainty for our coadded measurements is given by Equation (5). Stellar mass and mean [Fe/H] are given in Table 7 for each dSph in our sample. Stellar masses for M31 dSphs are the same as in Kirby et al. (2013, 2020), where the mass-to-light ratios from Woo et al. (2008) are multiplied by luminosities presented in Tollerud et al. (2012) and McConnachie (2012).

Figure 10 shows the MZR from our sample of M31 dSphs, compared to the values from Kirby et al. (2013) and Kirby et al. (2020). Overall, we find that our measurements (black triangles) are consistent with the mean [Fe/H] from either coadded measurements (Kirby et al. 2013) or individual measurements (Kirby et al. 2020). We note that, while And X and XIV seem to deviate from the MZR fit from MW dSphs, our measurements are consistent within ∼1–2σ uncertainty. Using our updated sample, which includes coadded abundances from stars that did not have high enough S/N to derive individual measurements, we demonstrate that we can robustly determine the average metallicity of a given M31 dSph by properly accounting for these stars.

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