Magellan/M2FS and MMT/Hectochelle Spectroscopy of Dwarf Galaxies and Faint Star Clusters within the Galactic Halo^*

The quantity and quality of imaging data available for selecting spectroscopic targets have evolved dramatically over the course of our observations. Our earliest spectroscopic targets were chosen based on our own two-filter photometry, which was limited to relatively central regions of the most luminous dwarf galaxies (e.g., Mateo et al. 2008). Others use more recent data sets from observational campaigns—e.g., the PRISTINE survey (Starkenburg et al. 2017)—that target individual systems. In one case—M2FS observations of the Reticulum II dwarf galaxy—we received targeting coordinates directly from the Dark Energy Survey's Milky Way working group, which had just discovered Reticulum II based on their then-proprietary photometric catalogs (The DES Collaboration et al 2015). Most recently, we select targets based on public data from large sky surveys—e.g., SDSS (Ahn et al. 2012), Pan-STARRS (Flewelling et al. 2020), DES (Abbott et al. 2021)—that provide multicolor photometry and, with the Gaia mission, precise and time-dependent astrometry over wide fields (Gaia Collaboration et al. 2016, 2023). In the special case of recent observations of star clusters at low Galactic latitude, we select targets based entirely on photometry and astrometry from Gaia (Pace et al. 2023).

One consequence of this progress is that our spectroscopic targeting criteria are heterogeneous, varying not only from system to system, but also across different fields and/or different epochs within a given system. Thus, we cannot provide a rigorous and consistent selection function that accounts for the sampling that produced the spectroscopic data sets presented herein. Instead, here we describe our general approach to selecting spectroscopic targets, and how that approach has evolved in response to advances in imaging surveys. In any case, our data products include coordinates of all observed spectroscopic targets regardless of data quality, allowing users to infer effective selection functions where necessary.

For nearly all of the stellar systems studied here, the member stars that are sufficiently bright for spectroscopy (magnitude G ≲ 21) are post-main-sequence stars on the red giant, subgiant, and horizontal branches. At distances ranging from tens to hundreds of kiloparsecs, stars at these evolutionary stages have broadband colors and magnitudes that are similar to those of late-type dwarf stars in the Galactic foreground. Our general strategy for target selection is first to use available photometry to identify these sequences of evolved stars along the line of sight to the system of interest, then to use additional information (e.g., parallax and proper motion), where available, to filter out likely foreground contaminants.

More specifically, since proper-motion data became available with Gaia's second data release (Gaia Collaboration et al. 2018b), we select spectroscopic targets according to the following procedure. First, we use wide-field survey photometry (e.g., SDSS, DES, and Pan-STARRS) to identify red giant, horizontal, and subgiant branch candidates as likely point sources (based on survey-specific criteria, e.g., requiring TYPE = 6 for SDSS photometry, ∣wavg_spread_model_r∣ < 0.003 for DES data) having g-band magnitudes and g − r colors within δ magnitudes of a best-fitting (by eye) theoretical isochrone (Dotter 2016). The tolerance $\delta =\sqrt{{\delta }_{\mathrm{err}}^{2}+{\delta }_{\min }^{2}}$ is set by the observational error, δ_err, associated with the photometric color, and a minimum tolerance that takes a typical value of ${\delta }_{\min }=0.2$ mag. Next we identify the photometrically selected stars for which Gaia measures a parallax that is unresolved (parallax angle is smaller than three times its observational error), and a proper motion that is consistent, given observational errors, with the systemic mean (e.g., Gaia Collaboration et al. 2018c; Pace & Li 2019). Given the list of prospective spectroscopic targets that pass these photometric and astrometric filters, we select randomly from those that lie within the available field of view of a given telescope pointing. For systems that extend beyond a single telescope pointing, our choice of pointing is based on competing interests in (1) observing large numbers of high-probability member stars, which favors central fields, (2) fairly sampling across the target system, which requires outer fields where member stars can be scarce, and (3) obtaining sufficient repeat measurements to gauge observational errors and intrinsic variability. Finally, we note that photometric and/or astrometric filter tolerances can be adjusted based on target density in order to make use of available fibers.

Figures 22 and 23 of the Appendix display sky positions, color–magnitude diagrams (CMDs), proper-motion coordinates, and our own measurements of metallicity, [Fe/H], versus (heliocentric) line-of-sight velocity, V_LOS, for spectroscopic targets toward each Galactic satellite that we observe. As discussed above, our actual target selection used a variety of different photometric data sets; however, for uniformity of presentation, the plotted CMDs all use Gaia's G-band photometry and integrated BP − RP spectra, with extinction corrections applied according to the procedure described by Gaia Collaboration et al. (2018a). Overplotted in the CMDs are theoretical isochrones (Morton 2015; Dotter 2016) computed for old (age = 10 Gyr) stellar populations and published values of metallicity for each object (e.g., McConnachie 2012). Ellipses in the sky maps have semimajor axes $a=2{R}_{\mathrm{half}}/\sqrt{1-\epsilon }$, where R_half is the projected half-light radius and ≡ 1 − b/a is the measured ellipticity. In the proper motion and [Fe/H] versus V_LOS panels, dashed lines indicate previously published values for systemic mean proper motions and velocities (Pace et al. 2022), where available.

The Michigan/Magellan Fiber System (M2FS; Mateo et al. 2012) is a fiber-fed, double spectrograph operating at the f/11 Nasmyth port of the Magellan/Clay 6.5 m telescope at Las Campanas Observatory, Chile. Each of the two M2FS spectrographs receives light from up to 128 fibers. A wide-field corrector provides good image quality over a field of diameter 30'. Fibers can operate at wavelengths between 3700 and 9500 Å, have entrance apertures of diameter 12, and tolerate center-to-center target separations as small as 12''. M2FS observers plug fibers by hand into masks that are machined at the Carnegie Observatories machine shop. Depending on choice of diffraction grating and order-blocking filters, M2FS offers a wide range of observing configurations, with spectral resolution ranging from ${ \mathcal R }\sim [0.2\mbox{--}34]\times {10}^{3}$ and wavelength coverage ranging from tens to thousands of angstroms.

For the vast majority of M2FS observations reported here, we use the high-resolution ("HiRes" hereafter) grating with both spectrographs, and with filters selected to pass light over a single order at 5130 ≲ λ ≲ 5190 Å. The most prominent feature in this region is the Mg i "b" triplet, with rest wavelengths of 5167.32, 5172.68, and 5183.60 Å. This region also contains many iron lines that enable a direct measurement of iron abundance. With these choices, we acquire single-order spectra for up to 256 sources per pointing, with resolving power ${ \mathcal R }$ ∼ 24,000. We bin the detector at 2 × 2 pixels², giving plate scale ∼0.065 Å pixel⁻¹ over the useful wavelength range.

For a small fraction of M2FS observations reported here, we use an alternative configuration that has at least one of the two spectrographs using a medium-resolution (henceforth "MedRes") grating that gives resolving power ${ \mathcal R }\sim 7000$. In order to cover the Mg triplet region, we use an order-blocking filter that passes light over the range 5115–5300 Å. Using the same 2 × 2 binning that we use with the HiRes grating, the MedRes observations have plate scale ∼0.2 Å pixel⁻¹ over the useful wavelength range.

During a typical observing night with M2FS, we take 100–200 zero-second "exposures" in order to measure the bias levels of the detectors in both spectrographs. We take between three and 10 exposures of the (scattered) solar spectrum during evening and/or morning twilight. For a typical science field, we expose for 1–3 hr, broken into two to five subexposures. Of the 256 available fibers, we assign ∼30 to regions of blank sky. Immediately before and after science exposures, and often between subexposures, we acquire calibration spectra of an LED source and then a ThArNe arc lamp, both of which are located at the secondary cage and illuminate the fibers at the focal surface. During daylight hours, we acquire sequences of hour-long "dark" exposures with both spectrographs' shutters closed.

Table 1 lists the instrument configuration, central field coordinates, date, total exposure time, and number of targets for all M2FS science fields observed for our program thus far. Including repeat observations, we have observed a total of 92 science fields with M2FS for this program—74 with both spectrographs using the HiRes grating, one with both using the MedRes grating, and 17 with one spectrograph in HiRes mode and the other in MedRes mode—for a total science exposure time of 0.68 megaseconds (Ms). We obtain acceptable M2FS HiRes spectroscopic measurements for ∼6600 unique sources within 18 different target systems, and we obtain acceptable M2FS MedRes measurements for ∼82 unique sources within five different systems. For ∼1400 M2FS sources, we have (up to 15 per source) multiple independent measurements.

Table 1. Log of M2FS Observations of Galactic Halo Objects

Instrument	Field Center		UT Date a	UT Start b	Exp. Time	${N}_{\exp }$	Ntarget	Object
	α2000 (deg)	δ2000 (deg)			(s)
M2FS HiRes	153.028333	−001.754667	2014-02-24	08:44:35	8900	5	218	Sextans
M2FS HiRes	100.746667	−050.848333	2014-02-25	03:01:36	5400	3	214	Carina
M2FS HiRes	100.610000	−051.082361	2014-02-25	05:04:05	5700	3	214	Carina
M2FS HiRes	153.684583	−001.500944	2014-02-26	08:00:05	6600	5	216	Sextans
M2FS HiRes	153.685000	−001.501000	2014-02-27	07:13:52	3600	3	216	Sextans
M2FS HiRes	153.292917	−001.604694	2014-02-28	07:46:28	3600	3	216	Sextans
M2FS HiRes	100.399167	−050.947139	2014-12-15	06:28:27	8100	3	218	Carina
M2FS HiRes	100.835417	−051.099611	2014-12-19	08:05:29	4500	3	207	Carina
M2FS HiRes	099.959583	−050.786417	2014-12-21	07:56:08	5400	3	187	Carina
M2FS HiRes	099.961667	−050.784722	2014-12-23	08:01:05	3600	3	177	Carina
M2FS HiRes	053.810000	−054.075444	2015-02-19	02:24:12	7200	3	186	Reticulum II
M2FS HiRes	153.292500	−001.601639	2015-02-22	06:50:05	7200	4	214	Sextans
M2FS HiRes/MedRes	343.062917	−058.493583	2015-07-18	09:40:45	9000	5	137	Tucana II

Notes.

^a YYYY-MM-DD format. ^b Universal time at start of first exposure; HH:MM:SS format.

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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Hectochelle is a fiber-fed echelle spectrograph at the f/5 focal surface of the MMT Observatory on Mt. Hopkins, Arizona, United States (Szentgyorgyi 2006). Hectochelle's optical fibers have entrance apertures of diameter 15, and are positioned robotically, allowing for simultaneous observation of up to 240 distinct sources. A wide-field corrector, coupled with an atmospheric dispersion compensator, gives a field of view of diameter 1°. Hectochelle spectra consist of a single diffraction order spanning ∼150 Å, with resolving power ${ \mathcal R }$ ∼ 32,000 at wavelength λ ∼ 5200 Å. We use Hectochelle's "RV31" order-blocking filter, which isolates the wavelength range 5150–5300 Å. We bin the detector by factors of 2 and 3 in the spectral and spatial dimensions, respectively, giving plate scale ∼0.10 Å pixel⁻¹.

Our observing strategy with Hectochelle is similar to that described above for M2FS. On a typical night, we acquire ∼100 zero-second bias "exposures," plus exposures of the scattered solar spectrum during evening and/or morning twilight. As with M2FS, for a given science field we acquire between 2 and 5 subexposures totalling 1–3 hr of integration time. Before and after science exposures, we acquire spectra of a ThAr arc lamp. Either before or after science exposures, we acquire the spectrum of a quartz lamp. The observatory staff acquires dark exposures regularly during daylight hours.

Table 2 lists the same information as Table 1, but for Hectochelle observations. With Hectochelle we have observed a total of 92 (including repeat observations) science fields for a total science exposure time of 1.42 Ms. We obtain acceptable measurements for ∼9700 unique sources within 21 target systems. For ∼2400 sources we have (up to 13 per source) multiple independent measurements.

Table 2. Log of MMT/Hectochelle Observations of Galactic Halo Objects

Instrument	Field Center		UT Date a	UT Start b	Exp. Time	${N}_{\exp }$	Ntarget	Object
	α2000 (deg)	δ2000 (deg)			(s)
Hectochelle	152.064708	+012.349136	2005-04-01	05:06:30	9079	3	143	Leo I
Hectochelle	152.064708	+012.349136	2005-04-02	06:13:04	14400	4	143	Leo I
Hectochelle	259.425000	+058.049972	2005-04-02	10:52:57	8700	3	132	Draco
Hectochelle	152.166792	+012.274975	2006-04-20	05:53:41	7500	3	135	Leo I
Hectochelle	152.107875	+012.309992	2006-04-24	05:07:09	8100	3	135	Leo I
Hectochelle	168.355875	+022.149333	2006-04-25	05:12:38	8100	3	114	Leo II
Hectochelle	260.958333	+057.870000	2006-04-25	08:12:45	4846	5	107	Draco
Hectochelle	210.005667	+014.483664	2006-05-08	04:24:44	5400	3	191	Bootes I
Hectochelle	260.102667	+057.885250	2007-02-23	12:27:48	5400	3	120	Draco
Hectochelle	257.091792	+057.877306	2007-02-26	10:03:25	5400	3	139	Draco
Hectochelle	262.915167	+058.382108	2007-02-26	12:22:21	7200	4	145	Draco
Hectochelle	152.765458	−001.052389	2007-02-27	09:57:01	8400	4	203	Sextans
Hectochelle	259.407542	+057.775056	2007-02-27	12:05:59	5400	3	89	Draco

Notes.

^a YYYY-MM-DD format. ^b Universal time at start of first exposure; HH:MM:SS format.

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.

Download table as:  DataTypeset image

We begin by using the Astropy-affiliated package "ccdproc" (Craig et al. 2017) to perform standard corrections for overscan, bias, dark current, and gain. We apply all of these corrections independently to images from each of the two M2FS channels and, for a given channel, to each of the 1024 × 1028 (plus 128 × 128 overscan) image sections read out via each detector's four independent amplifiers. "ccdproc" replicates the tasks performed by the original IRAF (Tody 1986) package of the same name, but also calculates and stores a 2D array containing an estimate of the variance at each pixel.

For each amplifier on each detector and for each M2FS run individually, we generate an image of the master bias level, denoted B, by averaging (after iteratively discarding 3σ outliers at the pixel level) ≳100 zero-second (overscan-corrected) exposures. We generate an image of the master dark current rate, denoted D, by averaging (again with iterative 3σ outlier pixel rejection) the ≈250 3600 s dark exposures (after performing overscan correction and subtracting the run-dependent master bias image) taken over all M2FS runs ¹¹ involving observations presented here. For all individual exposures of interest, we then use "ccdproc" to perform overscan correction to obtain an image of raw counts, denoted C, and then to subtract estimates of the master bias and dark counts. Finally, "ccdproc" applies the appropriate gain correction (typically g ≈ 0.68 e⁻ ADU⁻¹) to obtain an estimate of counts in units of electrons:

$$\begin{eqnarray}&&\hat{N}=g(\hat{C}-\hat{B}-{t}_{\exp }\hat{D}),\end{eqnarray} \tag{ 1 }$$

where ${t}_{\exp }$ is exposure time, and we adopt the convention under which $\hat{X}$ denotes the estimate of X.

The variance estimated by "ccdproc" is by default the sum of the estimated gain-corrected count, $\hat{N}$, and the square of the read noise. One problem with this estimate is that for weak signals, read noise can dominate such that $\hat{N}$ and hence the estimated variance can be negative. Another problem with weak signals is that—even in the absence of read noise—the observed count skews toward values smaller than the expected count, and hence the variance, of a Poisson distribution. For example, for expected counts of 1, 10, and 100, random draws from Poisson distributions will be smaller than the expectation value with probabilities 0.37, 0.46, and 0.49, respectively, and larger than the expectation value with probabilities 0.26, 0.42, and 0.47, respectively.

Illustrating these problems and our ad hoc solution, Figure 1 depicts the mean value, from 10⁶ trials over a range of input signals, of ${\chi }_{1}^{2}\equiv {(S-{S}_{\mathrm{in}})}^{2}/{\hat{\sigma }}_{S}^{2}$, where S is a simulated observation, ${\hat{\sigma }}_{S}^{2}$ is an estimate of its variance, and S_in is the known input signal. In each trial, the simulated observation is S = S₀ + , where ¹² ${S}_{0}\sim {{ \mathcal P }}_{k}({S}_{\mathrm{in}})$ is drawn from the Poisson distribution with expected value equal to the input signal, and $\epsilon \sim {{ \mathcal N }}_{x}(0,{\delta }^{2})$ is drawn from the normal distribution with mean 0 and variance δ². In our simulation, we set δ equal to the typical M2FS read noise of σ_r = 2.6 e⁻, and we assume that any additional noise associated with, e.g., empirical estimation of bias and dark levels, is negligible.

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The black curve in Figure 1 indicates the mean values of ${\chi }_{1}^{2}$ that are calculated using the "true" variance, ${\hat{\sigma }}_{S}^{2}=\mathrm{Var}(S)={S}_{\mathrm{in}}+{\delta }^{2}$. As expected, use of the true variance gives mean ${\chi }_{1}^{2}$ values of unity; unfortunately, the true variance is inaccessible to the observer who does not know the input signal.

The blue curve in Figure 1 shows the result of estimating the variance as the observationally accessible—and commonly used—quantity ${\hat{\sigma }}_{S}^{2}=\max (S+{\delta }^{2},{\delta }^{2})$. The mean ${\chi }_{1}^{2}$ asymptotes to unity only at N_in ≳ 100. For δ ≲ S_in ≲ 100, the aforementioned bias toward S < S_in gives mean ${\chi }_{1}^{2}\gt 1$ as the true variance is underestimated. For the smallest signals, S_in ≲ δ, ${\chi }_{1}^{2}\lt 1$ as the $\max$ operation causes the true variance to be overestimated on average.

The red curve in Figure 1 shows the result of taking the variance to be

$$\begin{eqnarray}&&{\hat{\sigma }}_{S}^{2}=\max (S+2+{\delta }^{2},0.6{\delta }^{2}),\end{eqnarray} \tag{ 2 }$$

a formula that we found, via experiment, to bring mean values of ${\chi }_{1}^{2}$ closer to unity at all input signals specifically when δ ≈ 2.6e⁻; for other values of the Gaussian noise, the constants in Equation (2) would need to be redetermined.

Based on the above experiment, we use Equation (2) to estimate the variance at each pixel of the real M2FS images. In our application to real data, we take the Poisson component to be $S=\hat{N}+{t}_{\exp }\hat{D}$, the sum of estimated source (including background) and dark counts, and the Gaussian component to be ${\delta }^{2}={\hat{\sigma }}_{r}^{2}+{\sigma }_{\hat{B}}^{2}+{t}_{\exp }^{2}{\sigma }_{\hat{D}}^{2}$, the sum of contributions from the estimated read noise and noise associated with empirical estimates of bias and dark count levels.

The estimated M2FS read noise is typically ${\hat{\sigma }}_{r}\approx 2.6$ e⁻, as calculated from the mean standard deviation over all pixels within individual images contributing to the master bias frames. The master dark frame indicates a mean dark current rate of $\hat{D}\approx 2.0$ e⁻ hr⁻¹. The run-dependent master bias frames and the global master dark frame have typical uncertainties of ${\sigma }_{\hat{B}}\approx 0.15$ e⁻ and ${\sigma }_{\hat{D}}\approx 0.25$ e⁻ hr⁻¹, respectively, calculated as the standard deviations over the individual calibration frames divided by the square root of the number of calibration frames, and converted to units of electrons.

We reiterate that our application of Equation (2) represents an ad hoc solution to the problem of estimating variances of pixel counts directly from the data. It is tuned specifically to produce ${\chi }_{1}^{2}\approx 1$ at N_in ≲ 100 electrons, given M2FS-like read noise; at other levels of read noise, the form of Equation (2) would need to be redetermined. We note that there exist alternative solutions; e.g., Guy et al. (2023) developed a full model of the CCD image in order to estimate the variance at each pixel.

Finally, for each channel we stitch together the four independently processed sections read by each amplifier in order to obtain a single image of size 2048 (columns) × 2056 (rows) square pixels. Figure 2 displays examples of the stitched frames obtained for four types of exposures, with illumination by: LED (top left), twilight sky (top right), ThArNe lamp (bottom left), and target stars (bottom right). Single-order spectra appear as horizontal bands, each spanning 5130–5190 Å over columns 300–1400. Signals outside this range are contributed by light from adjacent orders, which we discard (see below).

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M2FS disperses light approximately along the direction parallel to rows in the stitched images, henceforth called the x-direction, where x is a continuous variable along the discrete "column" axis (see Figure 2). Adjacent spectra are offset approximately along the "row" axis, which we represent with continuous variable y. In order to identify and trace spectral apertures, we follow procedures similar to those performed by IRAF's "apall" package. For each science field, we operate on the corresponding stitched LED frame (top-left panel of Figure 2), as it contains sufficient counts to identify and trace most spectral apertures. For calibration exposures of standard stars or of twilight sky, counts are sufficiently high that we can operate directly on the stitched standard and twilight frames themselves. The top-right panel of Figure 2 displays the raw image obtained from an exposure taken during evening twilight.

We begin by bundling the central 20 columns (columns 1013–1032), effectively combining them by storing their mean count as a function of row number (y-value). Figure 3 displays a characteristic example of this function, which resembles an emission-line spectrum; but of course here the "lines" are central aperture illumination profiles. We use the astropy.modeling package to fit a Chebyshev polynomial that represents the "continuum" of this pseudospectrum. After subtracting the best-fitting model continuum, we use the find_lines_derivative function from the astropy.specutils package to identify aperture centers as local maxima.

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We use these centers to initialize Gaussian fits (again via the astropy.modeling package) to the "continuum"-subtracted pseudospectrum, restricting our fits to the 10 rows around the centers returned by the find_lines_derivative function, but refitting those centers under the Gaussian model. The fitted Gaussian functions then represent the aperture illumination profiles across the center of the stitched image. We repeat the this process for all 102 bundles of 20 (nonoverlapping) consecutive columns, allowing us to quantify how the centers and widths of aperture profiles vary with column number across the stitched image.

Next we inspect, by eye, the pseudospectrum along the central column bundle, as well as the Gaussian fit to each aperture profile along that central bundle. Since the apertures in both "blue" and "red" channels are known to follow a regular pattern of eight approximately evenly spaced groups, with each group containing 16 approximately evenly spaced apertures (see Figure 2), we can delete any obviously spurious aperture detections and insert artificial placeholders to represent (for book-keeping purposes) apertures corresponding to unassigned and/or broken fibers.

Then, for each visually confirmed aperture, we trace the full 2D shape by "marching" from the center to each edge of the useful region (columns ∼300–1400; see Figure 2) in the stitched image. We begin at the position whose x-coordinate is the median column number of the central column bundle, and whose y-coordinate is the fitted center of the aperture profile in that bundle. We then find the fitted aperture center in the adjacent bundle that has the smallest deviation in its y-coordinate. If the deviation has absolute value smaller than some threshold (we use 1.5 pixels), we step to a new position whose x-coordinate is the median column number of that adjacent bundle, and whose y-value is the fitted center of that bundle's aperture profile. We proceed in this manner either to the edge of the useful region in the image, or until three consecutive column bundles have no aperture whose center deviates from the current y-coordinate by less than the specified threshold. We then return to the center column and march, in the same manner, toward the opposite edge. We thus obtain a list of (x, y) positions that sample the aperture's 2D trace pattern. To these data we fit and store a fourth-order polynomial function, iteratively rejecting 3σ outliers. We also fit and store fourth-order polynomials to the stored amplitudes and standard deviations; these two functions then characterize the aperture profile as a function of x.

M2FS does not have an internal lamp that uniformly illuminates the detectors; all incident light travels through the fibers. In order to correct for random variations in pixel sensitivity within a given aperture, we use the previously fit (Section 3.2) polynomials that represent center, amplitude, and standard deviation of the LED aperture profile, all as functions of x, to generate a model 2D aperture image. At a given column within the aperture, we evaluate the fitted polynomials to specify the parameters (center, amplitude, and standard deviation) of the Gaussian aperture profile model. We integrate that model to estimate the expected count within each pixel along the column, including all rows whose centers are within three aperture profile standard deviations of the aperture center. Repeating this procedure at each column, we obtain a pixelated model of the 2D LED spectrum.

Dividing the actual 2D LED spectrum by this model, we obtain the equivalent of a normalized "flat-field" spectrum. After repeating for each aperture, we divide the normalized "flat-field" frame into each individual stitched image whose random variations in pixel sensitivity we wish to correct (these include science, twilight, and arc-lamp exposures).

Having applied flat-field corrections to the stitched (science and calibration) images, next we estimate and remove scattered light. We first use the corresponding LED exposures (or bright standard star and/or twilight exposures) to mask the regions corresponding to the identified and traced spectral apertures. Specifically, we mask all pixels whose centers lie more than three standard deviations away from the center of the nearest aperture trace pattern, where the center and scale length are obtained by evaluating the polynomial functions fit to the aperture trace and aperture profile, respectively, at the pixel's x-coordinate (Section 3.2).

Returning to the frame of interest, we then fit a 2D fourth-order polynomial to the unmasked pixels, iteratively rejecting 3σ outliers, in order to estimate the contribution from scattered light. We remove scattered light by subtracting this function from the frame of interest.

In order to extract 1D spectra from each aperture, we collapse each column within the aperture into a single pixel regardless of the aperture trace pattern, thereby preserving independence between adjacent columns. This strategy would be optimal in the case that the spectral dispersion axis is exactly parallel to the detector's x-axis. In reality, the spectral apertures have nonzero curvature (Section 3.2); our procedure therefore results in some degradation of spectral resolution.

Let $\hat{N}(X,Y)$ and ${\hat{\sigma }}^{2}(X,Y)$ be the estimated count (in electrons) and estimated variance (in electrons²), respectively, at discrete pixel (X, Y). Let f(x, y) be the function that generates the 2D image of the spectrum—i.e., f(x, y)dx dy is the expected count within area element dx dy on the detector. Physically, the function f(x, y) is set by the intrinsic source (plus background) spectrum, the spectral resolution, and the geometry of both aperture and detector. We assume that, perpendicular to the spectral dispersion direction (i.e., the "spatial" direction, taken to be along the y-axis), the signal decays according to the Gaussian aperture profile whose parameters we evaluate from our polynomial fits described in Section 3.2, such that $f(y| x)={ \mathcal N }({y}_{0}(x),{\sigma }^{2}(x))$, where y₀(x) is the center of the aperture profile at dispersion coordinate x, and σ(x) is the standard deviation. ¹³

Under this model, the predicted count at discrete pixel (X, Y) is

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{mod}}(X,Y) & = & {\displaystyle \int }_{{X}_{1}}^{{X}_{2}}{dx}{\displaystyle \int }_{{Y}_{1}}^{{Y}_{2}}{dy}\,f(x,y)\\ & = & {\displaystyle \int }_{{X}_{1}}^{{X}_{2}}{dx}{\displaystyle \int }_{{Y}_{1}}^{{Y}_{2}}{dy}\,f(x)f(y| x))\\ & = & {\displaystyle \int }_{{X}_{1}}^{{X}_{2}}{dx}\,f(x){\displaystyle \int }_{{Y}_{1}}^{{Y}_{2}}{dy}\,{ \mathcal N }({y}_{0}(x),{\sigma }^{2}(x))\\ & = & {N}_{\mathrm{mod}}(X){\displaystyle \int }_{{Y}_{1}}^{{Y}_{2}}{ \mathcal N }({y}_{0}(X),{\sigma }^{2}(X)),\end{array}\end{eqnarray} \tag{ 3 }$$

where (X₁, Y₁) and (X₂, Y₂) are corners across the diagonal of the pixel. The count ${N}_{\mathrm{mod}}(X)$ is, by definition, the expectation value of f(x) = ∫p(x, y)dy at column X.

Within a given aperture, we take each of the N_row pixels in column X to be drawn independently from a normal distribution with mean predicted by Equation (3) and variance equal to the estimated value, ${\hat{\sigma }}^{2}(X,Y)$. We define

$$\begin{eqnarray}&&{\chi }^{2}\equiv \displaystyle \sum _{i=1}^{{N}_{\mathrm{row}}}\displaystyle \frac{{\left[\hat{N}(X,{Y}_{i})-{N}_{\mathrm{mod}}(X,{Y}_{i})\right]}^{2}}{{\hat{\sigma }}^{2}(X,{Y}_{i})}.\end{eqnarray} \tag{ 4 }$$

Minimizing χ² with respect to ${N}_{\mathrm{mod}}(X)$, we recover the "optimal" estimator of Horne (1986),

$$\begin{eqnarray}&&\hat{N}(X)=\displaystyle \frac{{\sum }_{i=1}^{{N}_{\mathrm{pix}}}\tfrac{\hat{N}(X,{Y}_{i}){I}_{i}}{{\hat{\sigma }}^{2}(X,{Y}_{i})}}{{\sum }_{i=1}^{N}\tfrac{{I}_{i}^{2}}{{\hat{\sigma }}^{2}(X,{Y}_{i})}}\end{eqnarray} \tag{ 5 }$$

where ${I}_{i}\equiv {\int }_{{Y}_{{1}_{i}}}^{{Y}_{{2}_{i}}}{dy}\,{ \mathcal N }\left(y0(X),{\sigma }^{2}(X)\right)$. Given the data in the 2D image, and the Gaussian aperture profile parameters fit to spectral apertures in the LED frame (Section 3.2), the estimator in Equation (5) is fully specified. For all science and calibration frames, we use Equation (5) to extract 1D spectra at every column of every aperture. We propagate the estimated variance as

$$\begin{eqnarray}&&{\hat{\sigma }}^{2}[\hat{N}(X)]={\left(\displaystyle \sum _{i=1}^{{N}_{\mathrm{pix}}}\displaystyle \frac{{I}_{i}^{2}}{{\hat{\sigma }}^{2}({X}_{,}{Y}_{i})}\right)}^{-1}.\end{eqnarray} \tag{ 6 }$$

We calibrate wavelengths using the 1D spectra extracted from exposures of the illuminated arc lamp containing thorium, argon, and neon ("ThArNe") gases. At the outset, for each individually extracted 1D ThArNe spectrum, we use a fifth-order polynomial to fit and subtract the continuum component, iteratively rejecting outliers at more than 5σ below the fit or more than 1σ above (the asymmetry effectively rejects pixels that sample emission features). To the continuum-subtracted spectrum, we then use the "find_lines_derivative" function from the astropy.specutils package to find emission features and estimate their centers in pixel space. Within the 10 pixels around the center of each identified emission line, we fit a Gaussian function and store the best-fitting center, standard deviation, and amplitude. The standard deviation quantifies the local spectral resolution.

Next we manually identify emission lines in a single 1D extracted ThArNe spectrum (i.e., the spectrum obtained in a single aperture), which thereafter serves as a template for automatically identifying emission lines in all other ThArNe spectra in all apertures in all ThArNe exposures acquired using the same M2FS configuration. Operating on the template spectrum only, we use NOIRLab's thorium–argon spectral atlas (Palmer & Engleman 1983) to visually identify individual emission lines interactively by eye. We store the atlas wavelength and pixel coordinate (from the Gaussian fit described above) of each line center.

Since we retain the pixelation native to the detector along the x-axis (column), we expect that the wavelength/pixel relationship will be unique for each aperture and—given small temporal changes in the aperture trace pattern—unique for each exposure. The next task, then, is to transfer our mapping of emission lines in the template ThArNe spectrum automatically to all individual nontemplate ThArNe spectra. For a given nontemplate spectrum, we begin by fitting a polynomial function that effectively distorts the template's pixel scale to bring the template's emission lines into alignment with those of the nontemplate spectrum. That is, letting T(X) and F(X) denote continuum-subtracted template and nontemplate ThArNe counts as functions of pixel number X, we find the order-m polynomial ${P}_{m}(x)={c}_{0}+{c}_{1}\left(\tfrac{x-{x}_{0}}{{x}_{s}}\right)+{c}_{2}{\left(\tfrac{x-{x}_{0}}{{x}_{s}}\right)}^{2}+\ldots +{c}_{m}{\left(\tfrac{x-{x}_{0}}{{x}_{s}}\right)}^{m}$ that minimizes the sum of squared residuals ${\sum }_{i=1}^{{N}_{\mathrm{pix}}}{(F({X}_{i})-{A}_{1}I({X}_{i}))}^{2}$, where ${x}_{0}\equiv 0.5\left({X}_{\max }+{X}_{\min }\right)$ is the midpoint of the template spectrum, ${x}_{s}\equiv 0.5\left({X}_{\max }-{X}_{\min }\right)$ is half the range of the template spectrum, and I(X) is the linear interpolation of $T\left({A}_{2}+x(1+{P}_{m}(x))\right)$ at X. We adopt m = 4; free parameters include the five polynomial coefficients and constants A₁, A₂.

We use the best-fitting model to transform the pixel coordinates of known emission lines in the template ThArNe spectrum to pixel coordinates in the nontemplate spectrum. To each emission line in the nontemplate spectrum, we assign the atlas wavelength of the nearest line in the transformed template spectrum, tolerating coordinate mismatches of ≤2 pixels. We then conduct the following iterative procedure: (1) using only the matched features (which typically number between 25 and 40 per nontemplate spectrum), we fit a fifth-order polynomial to the atlas wavelength as a function of pixel coordinate at the line center; (2) using this updated wavelength–pixel relation for the nontemplate spectrum, we assign atlas wavelengths to any as-yet unidentified emission lines in the nontemplate spectrum if their central wavelengths match those of as-yet unused template lines within a tolerance of ≤0.05 Å. After iterating up to 10 times, we save for each nontemplate ThArNe spectrum the pixel coordinates at atlas wavelengths of the identified emission lines, coefficients of the final polynomial fit to the wavelength–pixel relation, the number of emission features used in the wavelength–pixel fit, and the rms of residuals to the fit. For the HiRes (MedRes) configuration, over 34,698 (3248) nontemplate ThArNe spectra, the mean rms residual, after excluding those below the 1st percentile and those above the 99th, is 0.009 Å (0.023 Å), with standard deviation 0.001 Å (0.003 Å).

The next step is to use the wavelength–pixel relations obtained for the ThArNe spectra to estimate wavelengths at all pixels of each individual science frame exposure. Typically we obtain ThArNe calibration frames before and after each set of science exposures for a given target field, sometimes with an additional ThArNe exposure taken in between individual science exposures. These sequences let us quantify systematic shifts in the wavelength–pixel relationship that we expect to be due to flexure of the detector hardware and its sensitivity to temperature (as measured within the spectrograph cell) changes. Using one science field's set of ThArNe exposures as an example (observed with the HiRes grating), Figure 4 displays, across both detector arrays, the slopes d λ/dT that we fit to the wavelength/temperature relation at the location of each identified ThArNe emission line. We detect smooth variation across both detectors, with slope ranging from ∼0 to ∼0.04 Å K⁻¹.

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In order to compensate for these systematic drifts of the wavelength/pixel relation, for every pixel in the set of ThArNe exposures corresponding to a given science field, we fit a linear model for pixel wavelength as a function of time. Individual wavelengths are weighted by the inverse-square of the rms residual with respect to the fitted wavelength/pixel relation. For the time coordinate, we use the time at the exposure midpoint. At every pixel of a given science exposure, we then assign the wavelength obtained by evaluating the linear wavelength/time function at the temporal midpoint of the science exposure. In cases where multiple ThArNe exposures are not available for monitoring temperature and/or time dependence of the wavelength solution, we flag the corresponding catalog entries accordingly (see Section 5). The catalogs contain a column "n_wav_calibrations" that states the number of independent ThArNe exposures used in the wavelength calibration. Two other columns, "temp_min" and "temp_max," give the minimum and maximum spectrograph temperature, across the science subexposures. For the 221 spectra where n_wav_cal = 1 and temp_max−temp_min > 1 K (42 of which yield measurements passing our crude quality-control filter based on velocity uncertainty), we set flag wav_cal_flag = True in the M2FS catalogs.

We do not apply heliocentric corrections to the calibrated wavelengths, which therefore include Doppler shifts due to the line-of-sight component of the observatory's velocity with respect to the barycentric rest frame. Instead we apply heliocentric corrections directly to the line-of-sight velocities estimated using the observed wavelengths (Section 4.1.2).

It is at this point that we identify and mask pixels in the extracted, wavelength-calibrated 1D science spectra that are affected by cosmic rays. To each science spectrum, we first fit the continuum level using a fourth-order polynomial, iteratively rejecting outliers more than 2σ below or 3σ above the fit, where σ is the rms value of the residuals. We then flag as a likely cosmic-ray signal any pixel value that exceeds the fitted continuum level by more than 5σ. In subsequent analysis, we mask these as well as the four nearest pixels. While this procedure will similarly mask bona fide emission features, we expect emission lines to be largely absent from the targeted stellar spectra over the observed spectral region.

We use the entire set of twilight exposures acquired during the observing run to estimate relative throughput as functions of fiber and wavelength. We begin by averaging, on a pixel-by-pixel basis within each aperture, the (3–10) 1D spectra from individual exposures during a given twilight sequence (i.e., the set of exposures taken during a given evening/morning twilight). When computing the mean, we weight the count in each pixel by its inverse variance. We then combine these nightly weighted-mean twilight spectra across all twilight observations within a given run, taking a new weighted mean count on a pixel-by-pixel basis within each aperture. This second averaging is unique to each science exposure, as the count in each pixel is weighted by the inverse-squared difference in time between the midpoint of the nightly twilight sequence and the midpoint of the science exposure.

In order to estimate the relative fiber throughputs that pertain to a given science exposure, we operate on the corresponding run-averaged twilight frame, where the dominant spectral features are solar absorption lines. Within each aperture, we fit a fourth-order polynomial to the mean twilight count as a function of wavelength, iteratively rejecting outliers more than 3σ above and more than 1σ below the fit in order to isolate the continuum component. For each pixel of a given science spectrum, we evaluate the twilight-continuum polynomials from all apertures at the wavelength of the pixel in the science spectrum. We then apply a wavelength-dependent throughput correction by dividing the count at each pixel by the ratio of the aperture's twilight-continuum level to the median level across all apertures.

A typical M2FS observation allocates 20–40 fibers to regions of blank sky, split approximately evenly among the two spectrographs. For each field and spectrograph, we combine the throughput-corrected sky spectra to obtain a median sky spectrum, and then subtract the mean sky spectrum from all of the throughput-corrected spectra for all targets observed with that spectrograph.

When combining individual sky spectra to obtain a single median spectrum, we must again contend with the fact that the wavelength–pixel relation is unique to each individual spectrum. Following Koposov et al. (2011), we interpolate all individual sky spectra onto a common wavelength grid that oversamples, with 10 times the number of pixels, the original spectrum. We then store the median sky spectrum in the oversampled space, and record the variance at each pixel as 2.198πMAD²/(2N_sky), where N_sky is the number of individual sky spectra and MAD is the median absolute deviation (Koposov et al. 2011). From each individual science spectrum, we then interpolate the median sky (and variance) spectrum onto the pixel scale of the science spectrum, letting us then perform the sky subtraction directly on a pixel-by-pixel basis.

The final step of our M2FS image processing is to combine, on an aperture-by-aperture basis, the spectra obtained in multiple exposures. For a given aperture, we combine spectra from multiple exposures by taking the weighted mean (sky-subtracted) count at each pixel. One drawback of this stacking on a pixel-by-pixel basis is that it can exacerbate the effect of temperature changes inside the spectrograph, which tend to cause the aperture trace pattern and wavelength/pixel relation to drift (Section 3.6). In order to compensate for this effect and assign wavelengths to individual pixels in the stacked spectra, we follow the same procedure described in Section 3.6, where we evaluate for each pixel the linear wavelength versus time relation determined from ThArNe exposures. For each pixel in the stacked spectrum, we adopt as the time coordinate the mean midpoint of the individual science exposures, weighted by the inverse variance of the sky-subtracted count.

All processed M2FS spectra are available for download from the Zenodo database at doi:10.5281/zenodo.7837922. For each frame of (up to) 128 spectra obtained on one of the spectrograph channels, a FITS file contains the pixel wavelengths (as calibrated to the observatory rest frame—i.e., not shifted to the heliocentric frame), the sky-subtracted counts and their variances, the sky spectrum that was subtracted, the pixel mask, and the best-fitting model spectrum (Section 4), plus various observational details (e.g., date, time, and spectrograph temperature of each individual exposure) and random samples from posterior probability distribution functions (PDFs) for model parameters inferred during analysis of the spectra (Section 4).

Figures 5–7 display examples of fully processed spectra acquired with M2FS HiRes, M2FS MedRes, and Hectochelle, respectively. Source magnitude increases from top to bottom. Left-hand and right-hand panels show spectra from stars measured to have weak and strong surface gravity, respectively, distinguishing the likely red giant stars within Galactic halo structures from dwarf stars in the Galactic foreground. Subpanels display residuals with respect to the best-fitting model spectra (Section 4.1.2), normalized by the propagated uncertainty in the observed count. In the top two panels, hash marks identify wavelengths of known Fe i, Fe ii, and Mg i absorption features that are listed in the database maintained by the Virtual Atomic and Molecular Data Centre Consortium, provided by the BASS2000 database (Aboudarham & Renié 2020).

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Given a continuum-normalized, zero-redshift template spectrum, T _θ (λ), corresponding to parameters θ ≡ (T_eff, $\mathrm{log}g$, [Fe/H], [Mg/Fe], σ_LSF), we compute a model stellar spectrum according to

$$\begin{eqnarray}&&M(\lambda )={P}_{l}(\lambda ){T}_{{\boldsymbol{\theta }}}\left(\lambda \left[1+z+{Q}_{m}(\lambda )\right]\right),\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{P}_{l}(\lambda ) & \equiv & {p}_{0}+{p}_{1}\left[\displaystyle \frac{\lambda -{\lambda }_{0}}{{\lambda }_{s}}\right]+{p}_{2}{\left[\displaystyle \frac{(\lambda -{\lambda }_{0})}{{\lambda }_{s}}\right]}^{2}\\ & & +\,\ldots +{p}_{l}{\left[\displaystyle \frac{(\lambda -{\lambda }_{0})}{{\lambda }_{s}}\right]}^{l}\end{array}\end{eqnarray} \tag{ 8 }$$

is an order-l polynomial that represents a smooth continuum component. In Equation (7), rest wavelengths of the template spectrum are modified according to source redshift (in the observatory rest frame), z ≈ V_LOS/c, and an order-m polynomial,

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{m}(\lambda ) & \equiv & \displaystyle \frac{{q}_{1}}{c}\left[\displaystyle \frac{\lambda -{\lambda }_{0}}{{\lambda }_{s}}\right]+\displaystyle \frac{{q}_{2}}{c}{\left[\displaystyle \frac{(\lambda -{\lambda }_{0})}{{\lambda }_{s}}\right]}^{2}\\ & & +\,...+\displaystyle \frac{{q}_{m}}{c}{\left[\displaystyle \frac{(\lambda -{\lambda }_{0})}{{\lambda }_{s}}\right]}^{m},\end{array}\end{eqnarray} \tag{ 9 }$$

which can apply nonlinear corrections to the wavelength–pixel relation. Note that we omit from Q_m (λ) a zero^th-order term, as it would be entirely degenerate with source redshift in Equation (7). We examine zero-point redshift errors via direct comparison to external data sets (Section 4.3).

We choose l = 5 and m = 2, which provide sufficient flexibility to fit the continuum shape and to accommodate low-order corrections to the wavelength solution. We adopt scale parameters ${\lambda }_{0}=\tfrac{1}{2}({\lambda }_{\max }+{\lambda }_{\min })$ and ${\lambda }_{s}=\tfrac{1}{2}({\lambda }_{\max }-{\lambda }_{\min })$ Å, such that −1 ≤ (λ − λ₀)/λ_s ≤ + 1 over the entire range of observed wavelengths. For M2FS HiRes, we use the range ${\lambda }_{\min }=5127$ Å to ${\lambda }_{\max }=5190\,\mathring{\rm A}$. For M2FS MedRes and Hectochelle, we use the range ${\lambda }_{\min }\,=5155$ Å to ${\lambda }_{\max }=5295\,\mathring{\rm A}$.

4.1.1. Template Spectra

We present a new high-resolution grid of template spectra spanning 5050 ≤ λ ≤ 5350 Å around the Mg i "b" triplet. It is sampled at Δλ = 0.05 Å intervals, yielding a resolving power of ${ \mathcal R }$ ≈ 104,000. We generate these template spectra using a recent version (2017) of the MOOG line analysis code (Sneden 1973; Sobeck et al. 2011). We interpolate model atmospheres from the ATLAS9 grid (Castelli & Kurucz 2004).

We generate line lists for the synthesis using the LINEMAKE code ¹⁴ (Placco et al. 2021). LINEMAKE creates an initial list of lines drawn from the Kurucz (2011) line compendia. It subsequently updates the transition probabilities, hyperfine splitting structure, and isotope shifts for lines with recent laboratory analysis (e.g., Lawler et al. 2009, 2017). LINEMAKE also incorporates recent laboratory work on molecules, including CH, CN, C₂, and MgH in this spectral range (Hinkle et al. 2013; Masseron et al. 2014; Ram et al. 2014; Sneden et al. 2014). The initial list includes more than 39,000 lines. We remove the weakest lines, ones contributing < 0.5% to the line-to-continuum opacity ratio, in a synthetic spectrum for a cool, metal-rich red giant (T_eff = 4000 K, $\mathrm{log}g$ = 0.0, [Fe/H] = +0.5). These lines contribute negligible absorption to stars that are warmer and/or more metal-poor. The final line list contains 17,884 lines.

As a proof of concept, we compare a small region of synthetic spectra generated using these tools with the observed spectra of the Sun and Arcturus (Kurucz et al. 1984; Hinkle et al. 2000) in Figure 8. We adopt the Holweger & Müller (1974) empirical model atmosphere for the Sun, and we adopt the Ramírez & Allende Prieto (2011) model atmosphere parameters for Arcturus (T_eff = 4286 K, $\mathrm{log}g$ = 1.66, microturbulence velocity parameter (v_t) = 1.74 km s⁻¹, and [Fe/H] = −0.52). We also adopt [Mg/Fe] = +0.37, [Si/Fe] = +0.33, and [Ti/Fe] = +0.24 in our synthesis of the Arcturus spectrum. We have empirically adjusted a small fraction (≈0.4%) of the $\mathrm{log}({gf})$ values in our final line list to better reproduce the 300 Å region of interest for the solar and Arcturus spectra. The overwhelming majority (75 of 77) of these changes are to lines without modern laboratory work, and most are relatively weak and thus will have a negligible impact on the fitting of metal-poor stellar spectra. The MADs for these regions of the solar and Arcturus spectra are 1.2% and 3.8%, respectively, demonstrating the general reliability of our method.

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We synthesize a grid spanning 3900 ≤ T_eff ≤ 7500 K in intervals of 100 K, 0.0 ≤ $\mathrm{log}g$ ≤ 5.0 [cgs] in intervals of 0.25 dex, −4.0 ≤ [Fe/H] ≤ +1.0 in intervals of 0.25 dex, and −1.0 ≤ [Mg/Fe] ≤ +1.4 in intervals of 0.20 dex. A few regions near the edge of this grid are excluded because they represent nonphysical combinations of parameters or they extend beyond the ATLAS9 grid. ATLAS9 models with α enhancement are adopted when [Mg/Fe] ≥ +0.1. The microturbulence velocity parameter is adopted as a function of $\mathrm{log}g$: v_t = 1.0 km s⁻¹ for dwarfs ($\mathrm{log}g$ ≥ 4.0), v_t = 2.0 km s⁻¹ for giants ($\mathrm{log}g$ ≤ 1.0), and varying linearly between these two points. The macroturbulence velocity is assumed to be 3.0 km s⁻¹ for dwarfs and subgiants ($\mathrm{log}g$ ≥ 3.0), 8.0 km s⁻¹ at $\mathrm{log}g$ = 0.0, and varying linearly between these two points. We adopt the solar values for carbon (¹²C/¹³C = 89/1) and magnesium (²⁴Mg/²⁵Mg/²⁶Mg = 79/10/11) isotope ratios. Our final grid contains a total of 186,071 model spectra, all of which we make publicly available at the Zenodo database at doi:10.5281/zenodo.7837922.

We account for the finite spectral resolution of M2FS (resolving power ${ \mathcal R }$ ≈ 24,000 in our chosen configuration) and Hectochelle (${ \mathcal R }$ ≈ 32,000) by broadening each template spectrum via Gaussian kernel smoothing. We repeat for six different values of smoothing bandwidths: for modeling M2FS "HiRes" and Hectochelle spectra, we use σ_LSF = 0.06, 0.09, and 0.12 Å (resolving power ${ \mathcal R }$ ≈ 37,000, 24,000, and 18,000, respectively, at λ = 5200 Å). For modeling M2FS "MedRes" spectra, we use σ_LSF = 0.20 Å, 0.30 Å, and 0.40 Å (resolving power ${ \mathcal R }$ ≈ 11,000, 7400, and 5500, respectively).

Thus we obtain a library of "raw" synthetic stellar template spectra that discretely samples over a regular grid spanning a finite, 5D volume. We denote as ${T}_{{{\boldsymbol{\theta }}}_{0}}(\lambda )$ the raw template corresponding to grid point θ ₀ ≡ (T_eff, $\mathrm{log}g$, [Fe/H], [Mg/Fe], and σ_LSF). In order to evaluate models at arbitrary location (i.e., not necessarily at grid points), we combine the 2⁵ = 32 surrounding raw templates via 5D Gaussian kernel smoothing:

$$\begin{eqnarray}&&{T}_{{\boldsymbol{\theta }}}(\lambda )=\displaystyle \frac{{\sum }_{i=1}^{32}{T}_{{{\boldsymbol{\theta }}}_{0},i}(\lambda ){K}_{H}\left({{\boldsymbol{\theta }}}_{0,i}-{\boldsymbol{\theta }}\right)}{{\sum }_{i=1}^{32}{K}_{H}\left({{\boldsymbol{\theta }}}_{0,i}-{\boldsymbol{\theta }}\right)},\end{eqnarray} \tag{ 10 }$$

where ${K}_{H}({\boldsymbol{x}})\equiv \exp \left[-\tfrac{1}{2}{{\boldsymbol{x}}}^{T}{{\boldsymbol{H}}}^{-1}{\boldsymbol{x}}\right]$, and we adopt diagonal bandwidth matrix H = diag( h ◦ h ), with h = (300 K, 0.5, 0.25, 0.2, 0.03 Å) so that the smoothing bandwidth in each dimension equals the grid spacing. We note that, as a result of this nearest-neighbor smoothing, T _θ (λ) is not strictly a continuous function of θ and does not necessarily equal ${T}_{{{\boldsymbol{\theta }}}_{0}}(\lambda )$ when evaluated at grid points. Nevertheless, tests with mock spectra generated directly from templates indicate reliable recovery of input parameters (Walker et al. 2015a).

4.1.2. Inference

We estimate model parameters via Bayesian inference. Given observed spectrum S, the model specified by the free parameter vector θ has a posterior probability distribution

$$\begin{eqnarray}&&P({\boldsymbol{\theta }}| S)=\displaystyle \frac{P(S| {\boldsymbol{\theta }})P({\boldsymbol{\theta }})}{P(S)},\end{eqnarray} \tag{ 11 }$$

where P(S∣ θ ) is the conditional probability, given the model (or "likelihood"), of obtaining the observed spectrum, P( θ ) is the prior PDF for model parameters, and

$$\begin{eqnarray}&&P(S)\equiv \int P(S| {\boldsymbol{\theta }})P({\boldsymbol{\theta }})d{\boldsymbol{\theta }}\end{eqnarray} \tag{ 12 }$$

is the marginal likelihood. Assuming independence among the counts at all N_pix pixels, the spectrum has likelihood

$$\begin{eqnarray}&&P(S| {\boldsymbol{\theta }})=\displaystyle \prod _{i=1}^{{N}_{\mathrm{pix}}}{{ \mathcal N }}_{{S}_{i}}\left(M({\lambda }_{i}),{\sigma }_{i}^{2}\right),\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&{{ \mathcal N }}_{{S}_{i}}(M({\lambda }_{i}),{\sigma }_{i}^{2})\equiv \displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{i}^{2}}}\exp \left[-\displaystyle \frac{1}{2}\displaystyle \frac{{\left({S}_{i}-M({\lambda }_{i})\right)}^{2}}{{\sigma }_{i}^{2}}\right]\end{eqnarray} \tag{ 14 }$$

is the normal distribution, with mean M(λ_i ) equal to the model prediction (Equation (7)) for the count at the wavelength assigned to pixel i in the observed spectrum, and variance

$$\begin{eqnarray}&&{\sigma }_{i}^{2}\equiv {s}_{1}{\hat{\sigma }}_{{S}_{i}}^{2}+{s}_{2}^{2}\end{eqnarray} \tag{ 15 }$$

allows for a linear correction to the variance originally estimated for the observed count. In practice, given the fixed and discrete wavelength sampling of our template spectra, we evaluate (the logarithm of) Equation (13) after performing a linear interpolation of M(λ) onto the wavelengths assigned to pixels in the observed spectrum.

Our model contains 16 free parameters. Table 3 lists each parameter, along with the range over which the priors that we adopt are uniform and nonzero.

Table 3. Free Parameters and Priors of Spectral Model

Parameter	Prior	Description
VLOS/(km s−1)	uniform between −500, + 500	line-of-sight velocity
Teff/K	uniform between 3900, 7500	effective temperature
$\mathrm{log}g$	uniform between 0, 5	base-10 logarithm of surface gravity, cgs units
[Fe/H]	uniform between −4.0, + 0.5	iron abundance
[Mg/Fe]	uniform between −0.8, + 1.0	magnesium abundance
p0	uniform between a $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
p1	uniform between $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
p2	uniform between $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
p3	uniform between $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
p4	uniform between $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
p5	uniform between $-\max [S(\lambda )],+\max [S(\lambda )]$	polynomial coefficient (continuum; Equation (8))
q1/(km s−1)	uniform between −10, + 10	polynomial coefficient (wavelength solution; Equation (9))
q2/(km s−1)	uniform between −10, + 10	polynomial coefficient (wavelength solution; Equation (9))
σLSF/Å	uniform between 0.06, 0.12 (M2FS HiRes, Hectochelle)	bandwidth of Gaussian kernel to broaden line spread function
σLSF/Å	uniform between 0.2, 0.4 (M2FS MedRes)	bandwidth of Gaussian kernel to broaden line spread function
${\mathrm{log}}_{10}{s}_{1}$	uniform between −1, + 6	rescales observational errors (Equation (15))
${\mathrm{log}}_{10}{s}_{2}$	uniform between −2, + 2	adds to observational errors (Equation (15))

Note.

^a $\max [S(\lambda )]$ is the maximum value (discounting pixels flagged as cosmic rays) of the sky-subtracted spectrum.

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We use the software package MultiNest (Feroz & Hobson 2008; Feroz et al. 2009) to perform the inference. MultiNest implements a nested sampling algorithm (Skilling 2004) explicitly to compute the integral in Equation (12). As part of this procedure, it obtains a random sample from the posterior PDF (Equation (11)). These samples, for all of our M2FS and Hectochelle spectra, are provided along with the spectra at https://cmu.box.com/v/m2fs-hectochelle.

For convenience and simplicity of downstream analysis, we use simple statistics to summarize the full posterior PDFs. Specifically, we use MultiNest's random sampling of the PDF to estimate the mean, standard deviation, skew, and kurtosis of the marginal (1D) posterior PDF for each model parameter.

In previous work, we have used the summary statistics for posterior PDFs to define quality-control filters. For example, Walker et al. (2015a) discarded any observation for which the sampled marginal PDF for V_LOS has standard deviation >5 km s⁻¹, and/or skew and/or kurtosis deviating by more than one from the Gaussian value of zero. However, we find that the skew/kurtosis filters are approximately redundant with the cut on standard deviation alone. Therefore, in the analysis that follows, by default we discard only those observations for which the sampled marginal PDF for V_LOS has standard deviation >5 km s⁻¹. Of course, other users might reasonably choose other quality-control criteria, depending on scientific goals.

In our M2FS HiRes (M2FS MedRes, Hectochelle) sample, 8983 (189, 13328) spectra yield measurements that pass our simple quality-control filter. These spectra come from 6609 (82, 9678) unique sources, with 1330 (33, 2357) sources having multiple independent measurements. Figure 9 displays the distribution of number of independent measurements per star. As the number of independent measurements increases, the number of stars having that number of measurements declines approximately as a power law, with the M2FS sample containing stars having as many as 16 measurements, and the Hectochelle sample containing stars having as many as 14 measurements. In the M2FS sample, all stars having more than 10 measurements come from repeated observations of the Tucana II dwarf galaxy.

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We use the stars with repeat observations to fit models that specify the observational error associated with each measurement of each physical model parameter (V_LOS, T_eff, $\mathrm{log}g$, [Fe/H], [Mg/Fe]). For a given physical parameter, denoted here generically as X, we consider all pairs of independent measurements, X₁ and X₂, of the same sources. Following Li et al. (2019), we assume that deviations ΔX ≡ X₁ − X₂ are distributed as a mixture of two Gaussian distributions. The first has variance set by formal observational errors; the second, which allows for "outlier" measurements—including spurious measurements and/or cases of true variability, as with velocities measured for stars in binary systems—has constant variance ${\sigma }_{\mathrm{out}}^{2}$ that is unrelated to formal observational errors. That is, given zero-point offset μ_ΔX , variance ${\sigma }_{{\rm{\Delta }}X}^{2}\equiv {\sigma }_{{X}_{1}}^{2}+{\sigma }_{{X}_{2}}^{2}$ that is set by formal observational errors ${\sigma }_{{X}_{1}}$ and ${\sigma }_{{X}_{2}}$, outlier variance ${\sigma }_{\mathrm{out}}^{2}$ and outlier fraction f_out, the deviation between measurements 1 and 2 of a common source has probability

$$\begin{eqnarray}&&\begin{array}{l}P({\rm{\Delta }}X| \,{\mu }_{{\rm{\Delta }}X},{\sigma }_{{\rm{\Delta }}X}^{2},{f}_{\mathrm{out}},{\sigma }_{\mathrm{out}})\hspace{1.2in}\\ \quad =\,(1-{f}_{\mathrm{out}}){ \mathcal N }({\mu }_{{\rm{\Delta }}X},{\sigma }_{{\rm{\Delta }}X}^{2})+{f}_{\mathrm{out}}\,{ \mathcal N }(0,{\sigma }_{\mathrm{out}}^{2})\end{array}\end{eqnarray} \tag{ 16 }$$

where ${ \mathcal N }(\mu ,{\sigma }^{2})$ denotes the normal distribution with mean μ and variance σ². We assume μ_ΔX = 0 when comparing measurements from the same instrument, as in this section, but not when comparing measurements from different instruments, as in Section 4.3. We model the formal random errors as linear (in quadrature) functions of the standard deviations, denoted ${\sigma }_{{X}_{1,\mathrm{MN}}}$ and ${\sigma }_{{X}_{2,\mathrm{MN}}}$, obtained directly from MultiNest's random sampling of the marginal (1D) posterior PDF for parameter X. That is, we assume

$$\begin{eqnarray}&&\begin{array}{c}{\sigma }_{{X}_{1}}^{2}={s}^{2}+{k}^{2}{\sigma }_{{X}_{1,\mathrm{MN}}}^{2},\\ {\sigma }_{{X}_{2}}^{2}={s}^{2}+{k}^{2}{\sigma }_{{X}_{2,\mathrm{MN}}}^{2},\end{array}\end{eqnarray} \tag{ 17 }$$

and similar for all pairs of measurements obtained for common sources that deviate by amounts smaller than a threshold, ∣ΔX∣_out. We assume that deviations larger than ∣ΔX∣_out are contributed by spurious measurements, which we then exclude from our analysis (but not from the catalogs presented below). We take ∣ΔV_LOS∣_out = 100 km s⁻¹, ∣ΔT_eff∣_out = 2000 K, $| {\rm{\Delta }}\mathrm{log}g{| }_{\mathrm{out}}=2.5$ dex, ∣Δ[Fe/H]∣_out = 2.5 dex, and ∣Δ[Mg/Fe]∣_out = 1.0 dex. We assume that a single value of the error "floor," s, and a single value of scaling parameter, k, hold across the entire sample obtained with a given telescope/instrument. The total set of deviations, over all N_pair pairs of measurements, has likelihood

$$\begin{eqnarray}&&\displaystyle \prod _{i=1}^{{N}_{\mathrm{pair}}}\displaystyle \frac{P({\rm{\Delta }}{X}_{i}| {\mu }_{{\rm{\Delta }}X},{\sigma }_{{X}_{1}},{\sigma }_{{X}_{2}},{f}_{\mathrm{out}},{\sigma }_{\mathrm{out}})}{{\int }_{-| {\rm{\Delta }}X{| }_{\mathrm{out}}}^{+| {\rm{\Delta }}X{| }_{\mathrm{out}}}P({\rm{\Delta }}{X}_{i}| {\mu }_{{\rm{\Delta }}X}{\sigma }_{{X}_{1}},{\sigma }_{{X}_{2}},{f}_{\mathrm{out}},{\sigma }_{\mathrm{out}})\,d({\rm{\Delta }}X)}.\end{eqnarray} \tag{ 18 }$$

We consider all pairs of measurements that both satisfy our crude quality-control criterion (velocity error ≤5 km s⁻¹) for common sources, excluding measurements from sources listed in Gaia's (DR3) catalog of RR Lyrae variables (see Section 4.4). This selection gives N_pair = 6830 for M2FS HiRes, N_pair = 259 for M2FS MedRes, and N_pair = 6301 for Hectochelle. We use MultiNest to perform the inference. For each of the five physical parameters we infer from spectra, Table 4 lists the prior for each of the four parameters of our error model, as well as the mean and standard deviation of the marginal posterior PDF.

Table 4. Summary of Posterior PDFs for Parameters of the Model Used to Adjust Observational Errors (Section 4.2)

Quantity	s	k	fout	σout
	(Floor)	(Multiplier)	(Outlier Fraction)	(Outlier Std. Dev.)
M2FS HiRes
Vlos	0.57 ± 0.01 km s−1	0.86 ± 0.02	0.10 ± 0.00	24.30 ± 0.69 km s−1
Teff	58.59 ± 2.13 K	0.91 ± 0.01	0.08 ± 0.01	424.43 ± 24.79 K
${\mathrm{log}}_{10}[g]$	0.12 ± 0.02	0.86 ± 0.04	0.13 ± 0.06	0.61 ± 0.38
[Fe/H]	0.06 ± 0.00	1.18 ± 0.02	0.17 ± 0.02	0.29 ± 0.02
[Mg/Fe]	0.04 ± 0.01	0.74 ± 0.03	0.48 ± 0.01	0.34 ± 0.01
M2FS MedRes
Vlos	0.59 ± 0.78 km s−1	1.38 ± 0.14	0.12 ± 0.03	116.61 ± 170.64 km s−1
Teff	10.16 ± 19.39 K	1.00 ± 0.10	0.18 ± 0.10	507.14 ± 519.59 K
${\mathrm{log}}_{10}[g]$	0.03 ± 0.02	0.46 ± 0.05	0.09 ± 0.05	4.24 ± 2.43
[Fe/H]	0.12 ± 0.08	1.34 ± 0.15	0.19 ± 0.12	0.51 ± 0.55
[Mg/Fe]	0.03 ± 0.02	0.80 ± 0.07	0.19 ± 0.10	0.97 ± 1.50
MMT/Hectochelle
Vlos	0.39 ± 0.01 km s−1	0.94 ± 0.02	0.06 ± 0.00	27.64 ± 1.14 km s−1
Teff	0.61 ± 1.00 K	1.17 ± 0.01	0.06 ± 0.01	216.25 ± 26.55 K
${\mathrm{log}}_{10}[g]$	0.03 ± 0.01	1.05 ± 0.02	0.10 ± 0.02	0.42 ± 0.04
[Fe/H]	0.01 ± 0.00	1.20 ± 0.02	0.16 ± 0.01	0.37 ± 0.02
[Mg/Fe]	0.01 ± 0.00	1.11 ± 0.02	0.23 ± 0.02	0.38 ± 0.02

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For M2FS HiRes (MedRes), we infer error "floors" of ${s}_{{V}_{\mathrm{LOS}}}$ = 0.57 ± 0.01 km s⁻¹ (0.59 ± 0.78 km s⁻¹), ${s}_{{T}_{\mathrm{eff}}}$ = 58.59 ± 2.13 K (10.16 ± 19.39 K), ${s}_{\mathrm{log}g}$ = 0.12 ± 0.02 (0.03 ± 0.02), s_[Fe/H] = 0.06 ± 0.00 (0.12 ± 0.08), and s_[Mg/Fe] = 0.04 ± 0.01 (0.03 ± 0.02). For Hectochelle, the floors are all lower, presumably as a benefit of wider spectral coverage, with ${s}_{{V}_{\mathrm{LOS}}}$ = 0.39 ± 0.01 km s⁻¹, ${s}_{{T}_{\mathrm{eff}}}$ = 0.61 ± 1.00 K, ${s}_{\mathrm{log}g}$ = 0.03 ± 0.01, s_[Fe/H] = 0.01 ± 0.00, and s_[Mg/Fe] = 0.01 ± 0.00. The inferred scaling parameters are scattered around unity, in several cases (including the velocity measurements for M2FS HiRes and Hectochelle) consistent with a value of unity within the 99% credible interval. The outlier fraction tends to comprise ≲10% of the samples, except for the measurements of [Fe/H] and [Mg/Fe], where the outlier fractions reach ∼20%–50%. Analyses of chemical abundance distributions may therefore benefit from stricter sample selection criteria than our fiducial one, which is based solely on the formal error in V_LOS.

Figure 10 shows distributions of pair-wise measurement deviations normalized by combined measurement errors, with the combined measurement error calculated from the standard deviations of the posterior PDF originally sampled by MultiNest, ${\sigma }_{{\rm{\Delta }}{X}_{\mathrm{MN}}}=\sqrt{{\sigma }_{{X}_{1,\mathrm{MN}}}^{2}+{\sigma }_{{X}_{2,\mathrm{MN}}}^{2}}$ (black histograms), and from the formal errors returned by the best-fitting error model, ${\sigma }_{{\rm{\Delta }}X}=\sqrt{{\sigma }_{{X}_{1}}^{2}+{\sigma }_{{X}_{2}}^{2}}$ (red histograms). By design, the latter are generally closer to the standard normal distribution (solid black curves). In our data catalogs, the columns "X_error" list the errors for observable "X" after performing the adjustment of Equation (17), with mean values of error model parameters listed in Table 4. The columns "X_error_raw" list the pre-adjusted values obtained directly from the posterior sampled by MultiNest.

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We compare our M2FS and Hectochelle catalogs directly to each other and to large spectroscopic data sets that are previously published and/or in progress. Our primary goal is to detect and quantify systematic differences, e.g., zero-point offsets. The top panels of Figures 11 and 12 compare velocities and stellar-atmospheric parameters, respectively, that we measure with M2FS HiRes, M2FS MedRes, and Hectochelle, for all stars that appear in at least two instrument-specific samples. In both figures, the bottom three rows of panels compare our M2FS and Hectochelle measurements to those from external catalogs by Walker et al. (2009, "W09" hereafter), Kirby et al. (2010, "K10" hereafter), SDSS's APOGEE project (Abdurro'uf et al. 2022, DR17), and the Hectochelle in the Halo at High Resolution Survey (Conroy et al. 2019, "H3" hereafter).

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W09's catalog includes 8855 line-of-sight velocities measured for 7103 unique sources toward the dwarf spheroidal galaxies Carina, Fornax, Sculptor, and Sextans. The W09 spectra were acquired using the Michigan-MIKE Fiber System (Walker et al. 2007), a precursor to M2FS at Magellan that operated at similar spectral resolution over a similar spectral range. The W09 catalog has 1440 sources in common with our current M2FS HiRes sample, 10 sources in common with our M2FS MedRes sample, and 194 sources in common with our Hectochelle sample. While W09 measure spectroscopic indices for iron and magnesium absorption features, they do not measure the set of stellar-atmospheric parameters that we have in our current samples. Our comparisons to W09's catalog are therefore limited to line-of-sight velocities.

K10 measured T_eff, $\mathrm{log}g$, [Fe/H], and [Mg/Fe] for ∼3000 stars in eight of the Milky Way's dSph satellites. The K10 catalog has 115 (all HiRes mode) and 326 stars in common with our M2FS and Hectochelle samples, respectively. The K10 spectra have resolving power ${ \mathcal R }\sim 6500$ near the calcium triplet at λ ∼ 8500 Å, probing a different wavelength range at lower resolution than the other catalogs considered here. In contrast to our estimates of T_eff and $\mathrm{log}g$, which rely entirely on information contained in the spectrum, K10 incorporated stellar photometry into their estimate of T_eff and used photometry alone to estimate $\mathrm{log}g$. The K10 catalog does not list measurements of V_LOS; therefore, our comparisons to K10's catalog are limited to stellar-atmospheric parameters.

The APOGEE catalog, from the 17th data release of SDSS (DR 17; Abdurro'uf et al. 2022), includes line-of-sight velocities and stellar-atmospheric parameters measured from high-resolution (${ \mathcal R }$ ∼ 22,500 over ∼1.5–1.7 μm in wavelength) spectra obtained for ∼650,000 stars in the Milky Way and a few of its dwarf galaxy satellites. We select all sources from the APOGEE DR17 "allstar" catalog for which the APOGEE Stellar Parameters and Abundances Pipeline (ASPCAP) returns measurements for all of T_eff, $\mathrm{log}g$, [Fe/H], and [Mg/Fe] (ASPCAP lists separate measurements of [Mg/Fe] and [α/Fe]; we use only the former for purposes of direct comparison). We then discard any sources for which the "RV_FLAG" bitmask has the "RV_SUSPECT" bit set, and/or the "ASPCAPFLAG" bitmask has the "STAR_WARN" bit set. After applying these filters and then removing stars for which we measure [Fe/H]< −2.5 (i.e., below the minimum metallicity of APOGEE's template spectra), there are 117 APOGEE stars in common with our M2FS HiRes sample, two stars in common with our M2FS MedRes sample, and 94 in common with our Hectochelle sample. For a given star, we take the mean APOGEE velocity as given by the "VHELIO_AVG" parameter, with observational error given by "VERR."

Finally, the H3 Survey (Conroy et al. 2019) is ongoing, using the same MMT/Hectochelle configuration that we do. H3 is designed to map the Galactic stellar halo, targeting ∼2 × 10⁵ halo stars down to a magnitude limit of r ≲ 18. H3's and our spectra are acquired and modeled independently, but processed using the same CfA pipeline discussed at the beginning of Section 3. The H3 team models individual spectra using the software package MINESweeper (Cargile et al. 2020), which simultaneously fits isochrone models to stellar magnitudes measured from broadband photometry. H3's incorporation of photometric information provides additional power to constrain stellar-atmospheric parameters, while also giving capability to infer spectro-photometric distances to individual sources.

The faint end of H3's sample overlaps only slightly with the bright end of ours, leaving relatively few stars common to both surveys. In order to provide a more meaningful basis for comparison, the H3 team applied their MINESweeper analysis directly to our Hectochelle spectra from four different fields in the Sextans dSph galaxy (P. Cargile 2023, private communication). While not part of the actual H3 survey, this comparison "H3" sample contains 77 sources common to our M2FS HiRes sample, one source common to our M2FS MedRes sample, and 767 sources common to our Hectochelle sample.

For a given observable quantity, X, we infer zero-point offsets for each of the above catalogs simultaneously. We begin by constructing vectors of deviations, ΔX ≡ X₁ − X₂, and corresponding errors, ${\sigma }_{{\rm{\Delta }}X}=\sqrt{{\sigma }_{X,1}^{2}+{\sigma }_{{X}_{2}}^{2}}$, for all pairs of sources common to different catalogs "1" and "2." We loop over all possible combinations of catalogs, such that a star appearing at least once in all six catalogs will have 10 pairs of measurements; ¹⁵ within a given catalog, multiple measurements of the same source are replaced by the inverse-variance-weighted mean value.

We assume that, for a given observable, X, the pair-wise deviations, ΔX, follow a Gaussian distribution, with standard deviation σ_ΔX and pair-dependent mean, ${\mu }_{{\rm{\Delta }}X}=\overline{{\rm{\Delta }}{X}_{1}}-\overline{{\rm{\Delta }}{X}_{2}}$, that is specified by the difference in mean offsets (from some standard zero-point) of catalogs 1 and 2. This model and the corresponding likelihood function can be specified by Equations (16) and (18), respectively, only now with the outlier fraction assumed to be f_out = 0 and the cataloged observational errors taken at face value (s = 0, k = 1). In order to guard against catastrophic outliers, as described in Section 4.2, we discard pairs with deviations in excess of ∣ΔV_LOS∣_out = 100 km s⁻¹, ∣ΔT_eff∣_out = 2000 K, $| {\rm{\Delta }}\mathrm{log}g{| }_{\mathrm{out}}=2.5$ dex, ∣Δ[Fe/H]∣_out = 2.5 dex, and ∣Δ[Mg/Fe]∣_out = 1.0 dex. Four free parameters specify zero-point offsets: ${\overline{{\rm{\Delta }}X}}_{{\rm{M}}2\mathrm{FS}}$, ${\overline{{\rm{\Delta }}X}}_{\mathrm{Hecto}}$, ${\overline{{\rm{\Delta }}X}}_{{\rm{H}}3}$, and ${\overline{{\rm{\Delta }}X}}_{{\rm{W}}09}$ (if X = V_LOS), ${\overline{{\rm{\Delta }}X}}_{{\rm{K}}10}$ (if X = T_eff, $\mathrm{log}g$, [Fe/H], or [Mg/Fe]).

Given the APOGEE catalog's size and widespread use across different subfields, we choose that catalog to define the absolute zero-point, assuming ${\overline{{\rm{\Delta }}X}}_{\mathrm{Apo}}=0$ for all X. Table 5 lists offsets, relative to the APOGEE zero-point, that we infer (again via MultiNest, as in Section 4.2) for each observable and each catalog. Positive offsets, $\overline{{\rm{\Delta }}X}\gt 0$, imply that a catalog's zero-point is more positive than APOGEE's. For each catalog named in column 1, columns 2–7 identify the number of pairs of sources in common with each of the other individual catalogs.

Table 5. Zero-point Offsets (with Respect to APOGEE DR17) Inferred for M2FS, Hectochelle, and External Data Sets (Section 4.3)

Sample a	N1	N2	N3	N4	N5	N6	N7	$\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}$ b	$\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}$	$\overline{{\rm{\Delta }}\mathrm{logg}}$	$\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}$	$\overline{{\rm{\Delta }}[\mathrm{Mg}/\mathrm{Fe}]}$
M2FS HiRes	⋯	180	26	1440	115	77	117	0.52 ± 0.06	−142 ± 7	−0.52 ± 0.02	−0.26 ± 0.01	0.19 ± 0.01
M2FS MedRes		⋯	4	10	0	1	2	0.04 ± 0.59	−323 ± 18	−0.40 ± 0.07	−0.60 ± 0.08	0.15 ± 0.03
Hectochelle			⋯	194	326	767	94	−0.13 ± 0.05	−220 ± 4	−0.50 ± 0.01	−0.25 ± 0.01	−0.01 ± 0.01
W09				⋯	⋯	91	281	−0.15 ± 0.06	⋯	⋯	⋯	⋯
K10					⋯	25	75	⋯	−173 ± 2	−0.26 ± 0.01	−0.18 ± 0.01	0.24 ± 0.02
H3 c						⋯	22	−0.31 ± 0.05	−68 ± 4	−0.20 ± 0.01	−0.00 ± 0.01	0.23 ± 0.01

Notes.

^a Samples: 1 = M2FS HiRes; 2 = M2FS MedRes; 3 = Hectochelle; 4 = Walker et al. (2009); 5 = Kirby et al. (2010); 6 = H3; 7 = APOGEE DR17. ^b A value $\overline{{\rm{\Delta }}X}\equiv \overline{X-{X}_{7}}\gt 0$ implies a zero-point that is more positive than that of the APOGEE catalog. ^c The "H3" sample that we use here is from the H3 team's analysis of a subset of ∼750 spectra from our program.

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Examining the results for our M2FS and Hectochelle samples, we find that, whereas the Hectochelle sample shows little velocity offset with respect to APOGEE (${\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}}_{\mathrm{Hecto}}$ = −0.13 ± 0.05 km s⁻¹), the M2FS HiRes sample is systematically offset by ${\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}}_{{\rm{M}}2\mathrm{FS},\mathrm{HiRes}}$ = 0.52 ± 0.06 km s⁻¹. The M2FS MedRes sample shows no significant offset, with ${\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}}_{{\rm{M}}2\mathrm{FS},\mathrm{MedRes}}$ = 0.04 ± 0.59 km s⁻¹, but with a large uncertainty reflecting the fact that the M2FS MedRes sample has relatively few stars in common with the other samples. For most stellar-atmospheric parameters, both the M2FS HiRes and Hectochelle samples show statistically significant offsets from APOGEE. The offsets in surface gravity (${\overline{{\rm{\Delta }}\mathrm{log}g}}_{{\rm{M}}2\mathrm{FS}\,\mathrm{HiRes}}$ = −0.52 ± 0.02 and ${\overline{{\rm{\Delta }}\mathrm{log}g}}_{\mathrm{Hecto}}$ = −0.50 ± 0.01) and metallicity (${\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}}_{{\rm{M}}2\mathrm{FS}\,\mathrm{HiRes}}$ = −0.26 ± 0.01 and ${\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}}_{\mathrm{Hecto}}$ = −0.25 ± 0.01) are similar for both samples, while the difference in temperature offsets (${\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}}_{{\rm{M}}2\mathrm{FS}\,\mathrm{HiRes}}$ = −142 ± 7 Kand ${\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}}_{\mathrm{Hecto}}$ = −220 ± 4 K) likely reflects the different wavelength coverage of the different instruments/configurations. However, the smaller temperature offset of the H3 sample (${\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}}_{{\rm{H}}3}$ = −68 ± 4 K), which uses the same Hectochelle configuration that we do, also implicates differences in analysis procedure as a source of systematic error. Finally, while our Hectochelle sample shows good agreement with APOGEE in terms of the magnesium abundance (${\overline{{\rm{\Delta }}[\mathrm{Mg}/\mathrm{Fe}]}}_{\mathrm{Hecto}}$ = −0.01 ± 0.01), the M2FS sample is offset by ${\overline{{\rm{\Delta }}[\mathrm{Mg}/\mathrm{Fe}]}}_{{\rm{M}}2\mathrm{FS}}$ = 0.19 ± 0.01.

Perhaps most eye-catching among the external comparisons are those involving surface gravity in the H3 catalog (bottom row, second column of Figure 12). The H3 surface gravities are multimodal at $\mathrm{log}g$ ≳ 2. This feature is likely real (i.e., reflecting a true multimodality among the observed high-gravity stars) and detectable here because of H3's simultaneous fitting of isochrone and spectral models. The modes at $\mathrm{log}g$ ∼ 4.5, $\mathrm{log}g$ ∼ 3.7, and $\mathrm{log}g$ ∼ 2.5 correspond to the main sequence, subgiant, and horizontal branches, respectively, all of which are confined to distinct ranges of surface gravity in isochrone space. H3's fitting of isochrone models to broadband photometry effectively requires these evolutionary stage to be separated, giving rise to the observed multimodality in $\mathrm{log}g$ space.

The primary lesson we take from all of these external comparisons is that zero-point offsets among all of the independent data sets are common at the level of a few ×0.1 km s⁻¹ in line-of-sight velocity, ∼100 K in effective temperature, and a few ×0.1 dex in surface gravity, metallicity, and magnesium abundance. Offsets of these magnitudes are perhaps not surprising, given the variety of spectral resolutions, wavelength ranges, and analysis techniques employed. We acknowledge that our M2FS+Hectochelle results for individual stars are susceptible to systematic errors at these levels.

In the M2FS (HiRes and MedRes) and Hectochelle catalogs presented below, we subtract from each individual measurement of V_LOS, T_eff, $\mathrm{log}g$, [Fe/H], and [Mg/Fe] the zero-point offset listed in Table 5, such that the catalogs are effectively shifted to the APOGEE zero-point. Table columns labeled "X" list values of observable "X" after shifting to the Apogee zero-point. Columns labeled "X_raw" list the original values—i.e., before applying the zero-point correction.

After applying the zero-point corrections, we compare our current M2FS and Hectochelle catalogs to measurements that we have previously published for subsets of the current samples—including stellar targets in the dwarf galaxies Draco, Reticulum II, Tucana II, Grus I, Crater II, Leo II, Ursa Minor, Hydrus I, and Fornax (Walker et al. 2015a, 2015b, 2016; Caldwell et al. 2017; Spencer et al. 2017, 2018; Koposov et al. 2018; Pace et al. 2021). Despite using the same raw M2FS+Hectochelle spectra, the previously published measurements can differ systematically from current ones even before applying zero-point corrections, as they are derived using an entirely different library of synthetic template spectra. Specifically, the previously published measurements are based not on the library we introduce in Section 4.1.1, but instead on a library that was designed originally for use with the SDSS Segue Stellar Parameter Pipeline ("SSPP" Lee et al. 2008). The SSPP library is computed over a fixed grid in T_eff $\mathrm{log}g$ and [Fe/H], and assumes a monotonic relationship between α-element abundance and [Fe/H]. Experimenting with three independent libraries of synthetic template spectra, Walker et al. (2015a) observed library-dependent zero-point offsets as large as $\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}\sim 0.5$ km s⁻¹, $\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}\sim 300$ K, $\overline{{\rm{\Delta }}\mathrm{log}g}\sim 0.7$ dex, and $\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}\sim 0.5$ dex.

The previously published M2FS HiRes, M2FS MedRes, and Hectochelle data sets contain 1265, 33, and 3008 sources, respectively, from our current samples. Comparing these measurements directly to the current ones, we find that the previously published M2FS HiRes (M2FS MedRes) measurements are offset from current (raw, i.e., before applying an offset to the APOGEE zero-point) values by $\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}$ = −0.52 ± 0.04 km s⁻¹ (−2.14 ± 0.71 km s⁻¹), $\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}$ = 174 ± 3 K (124 ± 38 K), $\overline{{\rm{\Delta }}\mathrm{log}g}$ = 0.43 ± 0.01 dex (0.16 ± 0.07 dex), and $\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}$ = 0.19 ± 0.01 dex (0.22 ± 0.08 dex), where positive values imply that the current measurements are, on average, larger than the previously published ones. The previously published Hectochelle measurements show offsets of similar magnitude, with $\overline{{\rm{\Delta }}{V}_{\mathrm{LOS}}}$ = 0.65 ± 0.02 km s⁻¹, $\overline{{\rm{\Delta }}{T}_{\mathrm{eff}}}$ = −181 ± 1 K, $\overline{{\rm{\Delta }}\mathrm{log}g}$ = −0.23 ± 0.00 dex, and $\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}$ = −0.18 ± 0.00 dex. We notice that these offsets with respect to current values are similar to, or smaller than, the zero-point shifts that were applied to raw measurements in previously published works (see Walker et al. 2015a, 2015b for details). Those shifts were determined empirically, based on observed offsets between known solar values and values measured from high signal-to-noise ratio (S/N) spectra acquired during twilight exposures. Specifically, the previously published M2FS measurements include zero-point shifts (i.e., quantities that were added to raw measurements) of ΔV_LOS = 0 km s⁻¹, ΔT_eff = −69 K, ${\rm{\Delta }}\mathrm{log}g=-0.09$ dex, and Δ[Fe/H] = +0.20 dex, while the previously published Hectochelle measurements include shifts of ΔV_LOS = −0.81 km s⁻¹, ΔT_eff = +303 K, ${\rm{\Delta }}\mathrm{log}g\,=+0.63$ dex, and Δ[Fe/H] = + 0.48 dex. Based on these direct comparisons, then, we find that our switch to the new template library (described in Section 4.1.1), followed by our new zero-point calibration based on external comparisons, results in relatively small offsets from previous values.

Finally, after having applied the zero-point calibration as discussed above, we compare our measurements of [Fe/H] and [Mg/Fe] directly to previously published abundance measurements derived from high-resolution spectra acquired for relatively small samples of individual stars in dSph galaxies. The external samples come from observations with the HIRES spectrograph at the Keck Telescopes (Shetrone et al. 2001; Fulbright et al. 2004; Cohen & Huang 2009, 2010; Frebel et al. 2010), the High Dispersion Spectrograph at the Subaru Telescope (Sadakane et al. 2004; Aoki et al. 2009), the UVES (Shetrone et al. 2003; Norris et al. 2010; Tafelmeyer et al. 2010; Lucchesi et al. 2020), the X-Shooter (Starkenburg et al. 2013) spectrographs at the Very Large Telescope, and the MIKE spectrograph at Magellan (Simon et al. 2015). Figure 13 displays the comparisons.

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Comparing [Fe/H] metallicities (left panel of Figure 13), we find generally good agreement with the high-resolution studies. The bulk of measurements are consistent with a small offset such that our values may be systematically metal-rich by ∼0.1 dex, with no significant dependence on additional stellar-atmospheric parameters like T_eff or $\mathrm{log}g$. At the very metal-poor end, however, our measurements for two stars (both in the Sculptor dSph galaxy) with previously published values [Fe/H] ≲ −4 (Tafelmeyer et al. 2010; Simon et al. 2015) both come in at [Fe/H] ≳ −3 in our work, disagreeing with the previous measurements at the ∼2σ level. One of these stars, Scl07-50, has been identified (based on the previous measurements) as the most metal-poor star known in an external galaxy (Tafelmeyer et al. 2010). It is potentially concerning that our measurements do not reproduce this result. However, we note that our library of template spectra includes only metallicities [Fe/H] ≥ −4, and that our applied offsets of $\overline{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}$ imply that the minimum metallicity that we can in principle measure is [Fe/H] ∼ −3.75. The relatively large uncertainties on our measurements of these two stars imply that the mean values will be correspondingly larger than this minimum. We expect, therefore, that users may choose to apply stricter quality-control filters (e.g., a threshold in formal uncertainty) when analyzing chemical abundances, especially when working near the limits of our metallicity scale.

Comparing [Mg/Fe] abundances (right panel of Figure 13), we see what is perhaps the opposite problem, as our template library extends to lower [Mg/Fe] than is allowed in some previous studies. At solar and higher values of [Mg/Fe], we find generally good agreement with the results of previous high-resolution studies. At subsolar abundance, our measurements of [Mg/Fe] tend to be lower than those previously reported. Again, we expect that users may want to tighten quality-control filters when analyzing chemical abundances; we note that requiring our measurement of [Fe/H] to have uncertainty smaller than 0.5 dex would remove from the comparison sample all but one of the stars for which we measure [Mg/Fe] to be subsolar.

We perform one additional external cross-check on our metallicity measurements, fitting our spectral models (Section 4) to archival Hectochelle spectra acquired during observations of globular and open star clusters. These observations, performed by other investigators (including the H3 team), used the same spectrograph configuration and processing pipeline that we employ for our own Hectochelle spectra. Figure 14 displays histograms of [Fe/H] that we obtain for each of the clusters M3, M13, M67, M71, M92, and M107, which span a range of −2.2 ≲ [Fe/H] ≲ 0 in metallicity. For each cluster, we keep only stars for which our measurements have velocity error <5 km s⁻¹, metallicity error <0.5 dex, and—in order to reduce contamination from nonmember sources—$\mathrm{log}g$ < 3 and V_LOS within 10 km s⁻¹ of the systemic mean tabulated by Harris (1996), except for M67, for which we adopt the spectroscopic mean velocity and metallicity measured by Pace et al. (2008). Figure 14 shows the resulting distributions of [Fe/H] observed toward each cluster, with clear peaks associated with cluster members. We find good agreement with the previously published mean metallicities, giving confidence that our calibrated zero-point is accurate.

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Our target selection filters (Section 2) are designed to isolate primarily red giant stars in the Galactic halo substructures of interest, with contamination contributed mainly by dwarf stars in the Galactic foreground. Our spectral templates are designed to fit individual stars within the limited range of stellar-atmospheric parameters identified in Section 4.1.1, which can accommodate the vast majority of selected targets. Nevertheless, we expect our target selection filters to admit various kinds of anomalous sources for which our templates may provide relatively poor fits—e.g., carbon-enhanced stars, unresolved galaxies, and quasars.

In order to identify anomalous sources systematically, first we look for cases where the observed spectrum, S(λ), exhibits relatively large residuals with respect to the best-fitting model spectrum, M(λ). For each individual spectrum in our M2FS (top) and Hectochelle (bottom) samples, the top two panels of Figure 15 plot the mean value of ${\chi }^{2}\equiv {\sum }_{i=1}^{{N}_{\mathrm{pix}}}{({S}_{i}-M({\lambda }_{i}))}^{2}/\mathrm{Var}[{S}_{i}]$ as a function of the median S/N, where the mean and median are evaluated over all N_pix unmasked pixels. The variance spectrum, Var[S], is the original one, uncorrected by the linear rescaling parameters inferred as part of the spectral fit (see Equation (15)), as the rescaled variance will be inflated to compensate for template mismatch. For both M2FS and Hectochelle, we find that the mean value of χ² is approximately constant at median S/N ≲ 10, with characteristic values of χ²/pix ∼ 1.0 for M2FS and χ²/pix ∼ 1.5 for Hectochelle, suggesting that the uncertainties in pixel counts estimated by the Hectochelle pipeline tend to be underestimated by ∼20%. We reiterate that, by design, our linear rescaling of the raw variances (Equation (15)) brings the typical values to χ²/pix ∼ 1.

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Figure 15 also reveals that mean χ² values rise steadily at S/Ns ≳ 10. One contribution to this behavior comes from the fact that our polynomial model for the continuum spectrum is fixed at order l = 5 (Section 4), limiting ability to fit details of the continuum structure that become apparent only at high S/N. In order to flag anomalous spectra despite the steady rise in χ² with S/N, we identify outliers above the smooth S/N-dependent curves drawn in both panels of Figure 15. The curves are broken power laws of the form ${\chi }^{2}/\mathrm{pix}\,={a}_{1}{(1+({\rm{S}}/{\rm{N}})/{a}_{2})}^{3}$, with (a₁, a₂) = (1.2, 25) for M2FS and (4.0, 75) for Hectochelle. For all anomalous spectra identified in this way, we set the flag chi2_flag = True in the data catalogs (Section 5). We identify 60 such anomalous M2FS HiRes spectra, 114 anomalous M2FS MedRes spectra, and 131 anomalous Hectochelle spectra having median S/N ≥ 1 per pixel. For sources having at least one observation that passed our quality-control filter, we set the flag "any_chi2_flag = True" if the spectrum from any of the individual accepted observations has chi2_flag = True. There are 44 such sources in our M2FS HiRes catalog (not necessarily the same as those that have S/N ≥ 1), 28 in our M2FS MedRes catalog, and 41 in our Hectochelle catalog.

Figure 16 displays representative examples of these anomalous spectra, some types of which have already been identified and discussed in previous M2FS papers by Walker et al. (2015a) and Song et al. (2019, 2021). The top two M2FS spectra (left-hand panels) and the top Hectochelle spectrum (right-hand panels) are from stars showing various levels of carbon enhancement, with the Swan (1857) C₂ bandhead clearly visible near 5165 Å. The second (from top) Hectochelle spectrum is dominated by emission lines, presumably from a distant star-forming galaxy; a few tens of similar spectra are among the χ² outliers in our Hectochelle sample but are not, due to our masking of strong emission-like features (Section 3.7), in our M2FS sample. The third (from top) row of spectra are from cool M dwarf stars, with the TiO bandhead visible near 5170 Å. The bottom row of spectra are from known quasars, previously measured to have redshifts of z ∼ 3.7 (Boutsia et al. 2021, left) and z ∼ 3.4 (Pâris et al. 2014, right).

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Following Song et al. (2021), we obtain a cleaner sample of carbon stars by comparing the median flux across the bandpass 5160–5167 Å, denoted W₅₁₆₃ to the median flux across 5176–5183 Å, denoted W₅₁₈₀. The bottom two panels of Figure 15 plot the ratio W₅₁₆₃/W₅₁₈₀ as a function of median S/N. We identify as candidate carbon stars those sources for which the flux ratio falls below the curves drawn in the bottom two panels of Figure 15; spectra that satisfy this criterion have flag carbon_flag = True in the data catalogs (Section 5). The M2FS HiRes sample contains 37 sources that have at least one spectrum that is flagged as carbon enhanced and has S/N ≥ 1; the M2FS MedRes sample contains one such source, and the Hectochelle sample contains 144 such sources. For sources having at least one observation that passed our quality-control filter, we set the flag "any_carbon_flag" = True if the spectrum from any of the individual accepted observations has carbon_flag = True. There are 37 such sources in our M2FS HiRes catalog, zero in our MedRes catalog, and 88 in our Hectochelle catalog.

Our samples also contain sources that the Gaia (DR3) database flags as photometrically variable ("phot_variable_flag" = "VARIABLE") in the main source catalog, and/or lists in dedicated variability tables for active galactic nuclei (AGNs; variability table "vari_agn") or RR Lyrae ("vari_rrlyrae"). Our spectroscopic catalogs list for each source the value of Gaia's phot_variable_flag, and also set flags gaia_agn = True, gaia_rrl = True if the source appears in the corresponding variability tables. Considering only those having at least one spectrum with S/N ≥ 1, our M2FS HiRes, M2FS MedRes, and Hectochelle samples contain 551, 3, and 764 sources, respectively, which Gaia flags as photometric variables in the main source catalog, with 75, 1, and 363 sources appearing in Gaia's dedicated AGN table. For all but 3, 0, and 6 of these sources, our M2FS HiRes, M2FS MedRes, and Hectochelle observations do not yield measurements that pass our quality-control criteria.

Finally, considering only those sources having at least one M2FS HiRes, M2FS MedRes, or Hectochelle observation that passed our quality-control filter, 292, 0, and 40, respectively, are listed in Gaia's dedicated RR Lyrae table. While we can obtain good fits to the spectra of RR Lyrae, our repeat measurements detect the intrinsic line-of-sight velocity variability of these pulsating stars. For each of our sources that have multiple spectroscopic measurements that pass our quality-control filter, histograms in Figure 17 show distributions of the ratio of the weighted standard deviation (about the weighted mean) of the measured V_LOS, T_eff, $\mathrm{log}g$, [Fe/H], and [Mg/Fe] to the weighted mean error. This ratio is a measure of intrinsic variability of the source. Red (blue) histograms represent sources that are (are not) listed in Gaia's (DR3) RR Lyrae variability table (vari_rrlyrae). The ratios for RRL stars generally track those of the non-RRLs for the atmospheric parameters T_eff, $\mathrm{log}g$, [Fe/H], and [Mg/Fe]. For V_LOS, however (leftmost panel of Figure 17), the RRLs exhibit dramatically larger scatter than the non-RRLs, directly reflecting the rates at which the pulsating stars expand and contract. Users who are interested in the observed stars as dynamical tracers will need to take into account this source of intrinsic velocity variability.

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Of course, there are sources of intrinsic variability other than pulsation—e.g., binary star systems—for which we do not necessarily have a diagnostic classification a priori. For all stars having multiple independent measurements that pass our quality-control filter, we identify sources exhibiting potentially intrinsic variability as those for which the ratio of weighted standard deviation to weighted mean error exceeds a value of 3, regardless of whether the source is classified as RRL. In our data catalogs (Section 5), we set the flag "X_variable_flag" = True for such cases, where X can be any of the observables "vlos," "teff," "logg," "feh," or "mgfe."