Forecasting Chemical Abundance Precision for Extragalactic Stellar Archaeology

In many cases, there may be additional knowledge of the star's properties beyond the spectra in hand. For example, in an extragalactic context, we may know the distance to the star's host galaxy quite well and/or we may have photometry of the star. Such information can give external constraints on the luminosity, surface gravity, temperature, and even metallicity of a star, and can be used to improve the spectral fitting process. We now demonstrate how this information can be included in the CRLB calculation.

While the CRLB was initially derived in a frequentist context, a Bayesian equivalent of the CRLB can be formulated for application to scenarios in which prior information on the stellar labels is available. This is done by replacing the log-likelihood in Equation (1) with the full Bayesian probability:

$$\begin{eqnarray}&&\mathrm{ln}P(\theta | D)=\mathrm{ln}{\rm{\Pi }}(\theta )+\mathrm{ln}L(D| \theta ),\end{eqnarray} \tag{ 8 }$$

where Π(θ) is the prior on the stellar labels. This results in the following equation for the Bayesian FIM:

$$\begin{eqnarray}&&{F}_{\mathrm{Bayes}}={F}_{\mathrm{spec}}+{F}_{\mathrm{prior}}\end{eqnarray} \tag{ 9 }$$

Appendix A of Echeverria et al. (2016) presents a detailed derivations of Equation (9).

The first term on the right-hand side of the equation is the standard spectral gradient FIM found previously (Equation (6)). The second term on the right-hand side of the equation is the FIM of the prior and encapsulates the additional information included in the prior. It can be shown that for Gaussian priors with standard deviation ${\sigma }_{\mathrm{prior},\alpha }$ for each stellar label, the prior FIM is the diagonal matrix

$$\begin{eqnarray}&&{F}_{\mathrm{prior},\alpha \alpha }={\left(\displaystyle \frac{1}{{\sigma }_{\mathrm{prior},\alpha }}\right)}^{2}.\end{eqnarray} \tag{ 10 }$$

As a result, we can write the Bayesian CRLB of a stellar label, α, with Gaussian priors as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{Bayes},\alpha }=\sqrt{({F}_{\mathrm{spec}}+{F}_{\mathrm{prior}}{)}_{\alpha \alpha }^{-1}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&=\,{\left({\left[\displaystyle \frac{\partial f(\lambda ,\theta )}{\partial {\theta }_{\alpha }}\right]}_{\hat{\theta }}^{T}{{\rm{\Sigma }}}^{-1}{\left[\displaystyle \frac{\partial f(\lambda ,\theta )}{\partial {\theta }_{\alpha }}\right]}_{\hat{\theta }}+\displaystyle \frac{1}{{\sigma }_{\mathrm{prior},\alpha }^{2}}\right)}^{-1/2}.\end{eqnarray} \tag{ 12 }$$

As a check, we note that in the case of weak priors or strongly informative data, the CRLBs approach the value predicted by Equation (7), while in the case of strong priors or uninformative data, the CRLBs approach the standard deviation of the priors.

The CRLB can also be applied to the context in which multiple disjoint spectra of the same star exist across different wavelength ranges and resolutions, but are to be fit together. Such cases commonly arise for multiarmed spectrographs (e.g., the Low Resolution Imaging Spectrometer (LRIS) on Keck, Multi-Object Double Spectrographs (MODS) on the LBT, and DESI) and for echelle spectrographs, which observe multiple discrete orders of the stellar spectrum (e.g., GIRAFFE on the VLT).

Replacing the log-likelihood in Equation (2) with the sum of the log-likelihoods for each spectra and following through the previous derivation (Equations (5)–(7)) reveals that the relevant FIM for the joint fitting is simply the sum of the individual spectra's FIM. This is equivalent to concatenating the gradient spectra and covariance matrices of each observation together and using these combined quantities in Equation (7). This can be done for arbitrary combinations of stellar spectra provided that the covariance of overlapping wavelength ranges is properly accounted for (as done in Czekala et al. 2015), otherwise the number of independent information-carrying pixels is artificially inflated.

In this work, we limit our analysis to the stars predominantly accessible to spectroscopic campaigns of extragalactic stellar populations: metal-poor RGB stars. We also consider how the CRLBs vary from this fiducial star along several axes, including apparent magnitude, metallicity, and evolutionary phase as described below. The stellar labels used for these reference stars can be found in Table 1. Their position on the Kiel and Hertzprung–Russell diagrams can be seen in Figure 2.

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For each of the stellar targets considered in this work, we determine the effective temperature and surface gravity of the star using an isochrone from the MESA Isochrones and Stellar Tracks (MIST) project corresponding to the star's age, metallicity, and absolute magnitude (Paxton et al. 2011, 2013, 2015; Choi et al. 2016; Dotter 2016). As was done in Ting et al. (2017a), we assume a microturbulent velocity for each star using the  relationship between microturbulent velocity and surface gravity found by Holtzman et al. (2015):

$$\begin{eqnarray}&&{v}_{{\rm{turb}}}=2.478-0.325({\rm{log}}\,g)\,{\rm{km}}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 13 }$$

Fiducial Star—We adopt as our fiducial stellar reference a star that is roughly halfway up the RGB with a V-band absolute magnitude of ${M}_{V,\mathrm{Vega}}=-0.5$ (${M}_{g,\mathrm{AB}}\sim -0.2$). This choice splits the difference between the brighter but rarer stars at the tip of the RGB (TRGB) and the more numerous but fainter main-sequence turn-off (MSTO) stars. Furthermore, we assume that this fiducial star is 10 Gyr old, has a metallicity of $\mathrm{log}(Z/{Z}_{\odot })=-1.5$, and has solar abundance patterns.

Apparent Magnitude—As can be seen from Equation (7), the CRLB scales inversely proportional to the S/N of the spectrum. We consider our fiducial star with apparent magnitudes m_V = 18, 19.5, and 21, but at fixed stellar evolutionary phase, to avoid conflating the effects of S/N and the star's atmospheric parameters. This amounts to observing an identical star at distances of ∼50, 100, and 200 kpc, which are typical distances to nearby MW satellites. When not evaluating the effects of S/N on the chemical abundance precision, we assume the star is located at a distance of 100 kpc (m_V = 19.5).

Metallicity—We also investigate how the the information content of an RGB star's spectrum changes as its metallicity decreases from log Z = −1.5 to −2.5. Because the shape of the RGB changes as a function of metallicity, we make this comparison at fixed M_V instead of at fixed evolutionary phase. As a result, the lower-metallicity stars considered in this work are located farther down the RGB (i.e., have higher effective temperature and surface gravity; see Figure 2).

Evolutionary Phase—To isolate the effect of stellar evolutionary phase on the chemical abundance precision, we compare the CRLBs of our fiducial RGB star to that of an MSTO or RGB star of the same metallicity and apparent brightness.

Because the stars we consider in this work are so faint (m_V = 19.5), we limit our forecasts to instruments, both existing and planned, that can efficiently acquire spectra with modest S/N (>15 pixel⁻¹) in reasonable amounts of time (<1 night).

In practice, this includes instruments on ground-based telescopes with >5 meter apertures and large-aperture space telescopes. This excludes most of the spectrographs responsible for large MW surveys (e.g., RAVE, Steinmetz et al. 2006; SEGUE, Yanny et al. 2009; LAMOST, Luo et al. 2015; GALAH, de Silva et al. 2015; and APOGEE, Majewski et al. 2017) and most spectrographs with very high resolving powers (R > 50,000). We do not include any instruments with very low resolving powers (R < 1000), though there is reason to believe that the information content accessible to very low-resolution grism spectroscopy is still considerable (Bailer-Jones 2000).

Lastly, the line lists¹⁷ we use to generate synthetic spectra are limited in extent to wavelengths between 3000 Å and 1.8 μm. As such, we exclude instruments observing in the ultraviolet (UV) and infrared (IR) despite the significant chemical information that these wavelength regimes contain (e.g., García Pérez et al. 2016; Roederer 2019; Ting et al. 2019).

Even with the aforementioned restrictions, the list of spectrographs already on sky suitable for extragalactic stellar spectroscopy is extensive. As shown in Table 2, we consider 12 existing spectrographs at five world-class observing facilities as well as 9 spectrographs that will be coming online within the next decade. Each of these instruments features numerous choices of observing modes, dispersive elements, and other specifications. This flexibility enables a broad range of science, but makes an exhaustive evaluation of each observing configuration infeasible. Instead, we consider only the setups that we believe most relevant to acquiring precise chemical abundances in extragalactic stellar populations for a total of 41 configurations.¹⁸ For each observational setup, we attempt to use realistic wavelength coverage, wavelength sampling, and resolving power as reported either in literature or in design documents.

Table 2.  Spectroscopic Configurations Used in This Work

Telescope/Instrument	Spectroscopic	Wavelength	R	Sampling	Aperture	Section	Reference
	Configuration	Range (Å)	$(\lambda /{\rm{\Delta }}\lambda )$	(Pixels/FWHM)	(m)
Existing Instruments
Keck II/DEIMOSa	1200G	6500–9000	6500	4	10.0	4.1	[1]
	1200B	4000–6400	4000	4	10.0	4.2.1	[1]
	600ZD	4100–9000	2500	5	10.0	4.2.1	[1]
	900ZD	4000–7200	2500	5	10.0	4.2.1	[1]
Keck I/LRISa	600/4000	3900–5500	1800	4	10.0	4.2.1	[2]
	1200/7500	7700–9000	4000	5	10.0	4.2.1	[2]
Keck I/HIRESrb	B5 Decker	3900–8350	49000	3	10.0	4.3.1	[3]
	C5 Decker	3900–8350	35000	3	10.0	4.3.1	[3]
LBT/MODSa	Blue Arm	3200–5500	1850	4	11.8	4.2.2	[4]
	Red Arm	5500–10500	2300	4	11.8	4.2.2	[4]
Magellan/MIKEra	Blue (10 slit)	3500–5000	28000	4	6.5	4.3.1	[5]
	Red (10 slit)	5000–10000	22000	3	6.5	4.3.1	[5]
Magellan/M2FSd	HiRes	5130–5185	18000	3c	6.5	4.3.2	[6]
	MedRes	5100–5315	10000	3c	6.5	4.3.2	[6]
MMT/Hectochelled	RV31	5150–5300	32000	2	6.5	4.3.2	[7]
MMT/Hectospeca	270 mm−1	3900–9200	1500	5	6.5	4.2.2	[8]
	600 mm−1	5300–7800	5000	5	6.5	4.2.2	[8]
MMT/Binospeca	270 mm−1	3900–9200	1300	4	6.5	4.2.2	[9]
	600 mm−1	4500–7000	2700	3	6.5	4.2.2	[9]
	1000 mm−1	3900–5400	3900	3	6.5	4.2.2	[9]
VLT/MUSEe	Nominal	4800–9300	2500	3c	8.2	4.2.2	[10]
VLT/X-SHOOTERb	UVB (08 slits)	3000–5500	6700	5	8.2	4.3.1	[11]
	VIS (07 slits)	5500–10200	11400	4	8.2	4.3.1	[11]
	NIR (09 slits)	10200–18000	5600	4	8.2	4.3.1	[11]
VLT/FLAMES-UVESf	r580	4800–6800	40000	5	8.2	4.3.1	[12]
VLT/FLAMES-	LR8	4200–11000	6500	3c	8.2	4.3.2	[13]
VLT/GIRAFFEd	HR10	5340–5620	19800	3c	8.2	4.3.2	[13]
	HR13	6120–6400	22500	3c	8.2	4.3.2	[13]
	HR14A	6400–6620	28800	3c	8.2	4.3.2	[13]
	HR15	6620–6960	19300	3c	8.2	4.3.2	[13]
Future Instruments
JWST/NIRSpeca	G140M/F070LP	7000–12700	1000	3c	6.5	5.1	[14]
	G140M/F100LP	9700–18400	1000	3c	6.5	5.1	[14]
	G140H/F070LP	8100–12700	2700	3c	6.5	5.1	[14]
	G140H/F100LP	9700–18200	2700	3c	6.5	5.1	[14]
GMT/GMACSa	Blue Arm (LR)	3200–5500	1000	3	24.5	5.2	[15]
	Blue Arm (MR)	3700–5500	2500	3	24.5	5.2	[15]
	Blue Arm (HR)	4200–5000	5000	3	24.5	5.2	[15]
	Red Arm (LR)	5500–10000	1000	3	24.5	5.2	[15]
	Red Arm (MR)	6100–8900	2500	3	24.5	5.2	[15]
	Red Arm (HR)	6700–8300	5000	3	24.5	5.2	[15]
GMT/G-CLEFa	Med Res	3000–9000	35000	3	24.5	5.2	[16]
TMT/WFOSa	B1210	3100–5500	1500	3c	30.0	5.2	[17]
	B2479	3300–4750	3200	3c	30.0	5.2	[17]
	B3600	3250–4100	5000	3c	30.0	5.2	[17]
	R680	5500–10000	1500	3c	30.0	5.2	[17]
	R1392	5850–8400	3200	3c	30.0	5.2	[17]
	R2052	5750–7250	5000	3c	30.0	5.2	[17]
E-ELT/MOSAICa	HMM-Vis	4500–8000	5000	4	39.0	5.2	[18]
	HMM-NIR	8000–18000	5000	3	39.0	5.2	[18]
Subaru/PFSa	Blue Arm	3800–6300	2300	4	8.2	5.3	[19]
	Red Arm (LR)	6300–9400	3000	4	8.2	5.3	[19]
	Red Arm (MR)	7100–8850	5000	4	8.2	5.3	[19]
	NIR Arm	9400–12600	4300	4	8.2	5.3	[19]
MSEa	Blue Arm (MR)	3900–5000	5000	3	11.3	5.3	[20]
	Green Arm (MR)	5750–6900	5000	3	11.3	5.3	[20]
	Red Arm (MR)	7370–9000	5000	3	11.3	5.3	[20]
	All Arms (LR)	3600–13000	3000	3	11.3	5.3	[20]
Keck/FOBOSa	Proposed	3100–10000	3500	6	10.0	5.3	[21]
LAMOSTa		3700–9000	1800	3c	4.0	Appendix D	[22]
Mayall/DESIa	Blue Arm	3600–5550	2500	3	4.0	Appendix F	[23]
	Red Arm	5550–6560	3500	3	4.0	Appendix F	[23]
	Infrared Arm	6560–9800	4500	3	4.0	Appendix F	[23]

Notes. This table lists the spectroscopic configurations we adopt for computing the chemical abundance precision as well as the section in which those precisions are presented. For each instrument, we adopt a constant resolution and number of pixels per resolution element across the wavelength range indicated. The instruments listed here span a large range in wavelength coverage (3200 Å–1.8 μm), resolving powers (1000 < R < 49,000), and instrument designs.

^aLow-/Medium-Resolution Multi-Object Spectrograph. ^bSingle-Slit Multi-Order Echelle Spectrograph. ^cSampling information was not found so a nominal value of 3 pixels/FWHM is assumed. ^dMulti-Object Single-Order Echelle Spectrograph. ^eIntegral Field Unit Spectrograph. ^fMulti-Object Multi-Order Echelle Spectrograph.

References. [1] Faber et al. (2003), [2] Oke et al. (1995), [3] Vogt et al. (1994), [4] Pogge et al. (2010), [5] Bernstein et al. (2003), [6] Mateo et al. (2012), [7] Szentgyorgyi et al. (2011), [8] Fabricant et al. (2005), [9] Fabricant et al. (2019), [10] Bacon et al. (2010), [11] Vernet et al. (2011), [12] Dekker et al. (2000), [13] Pasquini et al. (2002), [14] Bagnasco et al. (2007), [15] DePoy et al. (2012), [16] Szentgyorgyi et al. (2016), [17] Pazder et al. (2006), [18] Jagourel et al. (2018), [19] Tamura et al. (2018), [20] MSE Science Team et al. (2019), [21] Bundy et al. (2019), [22] Cui et al. (2012), [23] DESI Collaboration et al. (2016a).

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Despite an extensive literature search, not all pertinent spectrograph details were readily available, and we had to make some assumptions. For example, for several instruments, the number of pixels per resolution element could not be found; in these cases we adopt a fiducial wavelength sampling of 3 pixels/FWHM as assumed in Ting et al. (2017a). For multiobject spectrographs (MOSs), we assume the nominal wavelength coverage for a star observed in the center of the instrument's field of view and ignore the variations in wavelength coverage incurred for off-center stars. Additionally, most instruments have wavelength-dependent resolving powers, usually decreasing toward the blue. The manner in which the resolving power changes across the spectrum, known as the line-spread function (LSF), depends on the star's position in the slit and can vary from slit to slit. For simplicity, we assume all instruments have a fixed LSF with a resolution approximately equal to the average across the entire spectrum.

Lastly, while we do compare and contrast the forecasted precision of these instruments, we emphasize that the "best" instrument is largely of a science-dependent nature. There are numerous trade-offs between field of view and multiplexing (see Table 3), radial velocity precision, and detailed chemical abundance measurements. Balancing them is a matter of their relative importance to the science at hand.

Table 3.  Field of View and Multiplexing of Instruments

Telescope/Instrument	Field of View	Nslits or Nfibers
Keck II/DEIMOS	16' × 40	100
Keck I/LRIS	60 × 78	40
Keck I/HIRESr	⋯	1
Magellan/MIKEr	⋯	1
Magellan/M2FS	300	250
MMT/Hectochelle	10	240
MMT/Hectospec	10	300
MMT/Binospec	160 × 150	150
VLT/MUSE	1'0 × 10	⋯
VLT/X-SHOOTER	⋯	1
VLT/FLAMES-UVES	250	8
VLT/FLAMES-GIRAFFE	250	130
LBT/MODS	60 × 60	50
JWST/NIRSpec	30 × 30	100
Mayall/DESI	28	5000
Subaru/PFS	13	2400
MSE	95	3250
Keck/FOBOS	200	1800
GMT/GMACS	74	100
GMT/GMACS+MANIFEST	20'	100 s
GMT/G-CLEF+MANIFEST	20'	40
TMT/WFOS	42 × 96	600
E-ELT/MOSAIC (HMM-Vis)	60	200
E-ELT/MOSAIC (HMM-NIR)	60	100

Note. N_slits (N_fibers) is the approximate number of slits (fibers) that an instrument can handle in a single pointing. This can be used as a rough estimate for the number of stars a spectrograph can observe simultaneously. In practice, of course, not all slits/fibers can be placed on stars because some may be required for guiding, alignment, or sky-subtraction, while others may go unused simply due to the distribution of stars in the field. Single numbers for the field of view (FoV) indicate the FoV's diameter, while pairs of numbers indicate the approximate rectangular dimensions of the FoV. For single-slit spectrographs, the FoV is irrelevant for resolved star spectroscopy. As an IFU, the multiplexing of MUSE depends on the density of stars in the field and the source extraction method employed.

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We assume the the flux covariance, Σ, is due entirely to photon noise and thus is a function solely of exposure time, instrument throughput, observing conditions, and the star's brightness, ignoring any uncertainty introduced by imperfect data reduction or continuum normalization.¹⁹ Whenever possible, we use the exposure time calculator (ETC) specific to each instrument listed in Table 4. This allows us to  adopt a flux covariance as specific as possible to each facility and accordingly compute realistic CRLBs. For instruments that do not have public ETCs, we scale the S/N from a similar instrument according to

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}\propto D{({nR})}^{-1/2},\end{eqnarray} \tag{ 14 }$$

where D is the effective aperture of the telescope, R is the instrument's resolving power, and n is the instrument's wavelength sampling.

Table 4.  ETC Configurations Used in This Work

Instrument	mV	texp	Airmass	Seeing	Slit Width/	Spatial × Spectral	Stellar	ETC
		(hr)			Fiber Diameter	Binning	Template
DEIMOS	18.0, 19.5, 21.0	1, 3, 6	1.1	075	075	1 × 1	G5V, K0V, K5V	1
LRISa	19.5	1	1.1	075	070	1 × 1	K0V	2
HIRESr (B5/C5)	19.5	6	1.1	075	086/110	2 × 2	K0V	3
MIKE	19.5	6	1.1	075	100	3 × 1	K0V	4
M2FS	19.5	6	1.1	075	120	2 × 2	K2V	5b
Hectochelle	19.5	6	1.1	075	100	2 × 3	K2V	5b
Hectospec	19.5	1	1.1	075	15	1 × 1	K0V	6
Binospec	19.5	1	1.1	075	10	1 × 1	K0V	6
MUSE	19.5	1	1.1	080	⋯c	(3 × 3) × 1	K2V	7
X-SHOOTER	19.5	6	1.1	075	080/070/	1 × 1	K2V	8
(UVB/VIS/NIR)					090
UVES	19.5	6	1.1	080	100	1 × 1	K2V	9
GIRAFFE	19.5	6	1.1	075	120	1 × 1	K2V	5
MODS	19.5	1	1.1	075	070	1 × 1	K2V	10
NIRSpec	21.0–26.0	6	⋯	⋯	02	1 × 1	K5III	11
PFS	19.5	1	1.1	075	105	1 × 1	K2V	12
MSE	19.5	1	1.0	075	080	1 × 1	K2V	13
FOBOS	19.5	1	1.1	075	080	1 × 1	K2V	14
GMACS	19.5, 21.0–26.0	1, 6	1.1	075	070	4 × 4	K0V, K5V	15
WFOS	19.5	1	1.1	075	075	1 × 1	K0V	14
MOSAIC (NIR/Vis)	19.5	1	1.1	075	080/060	1 × 1	K0I/V	10b/14b
G-CLEF	19.5	6	1.0	079	070	6 × 9	K2V	16

Notes. Exposure times are chosen to mimic realistic observing strategies for each instrument. Multiple apparent magnitudes, exposure times, and stellar templates are used with the fiducial 1200G grating on the Keck/DEIMOS spectrograph to investigate their effects on chemical abundance precision. Stellar templates are chosen to best match the stellar energy distribution of the relevant reference star. (1) DEIMOS ETC: http://etc.ucolick.org/web_s2n/deimos, (2) LRIS ETC: http://etc.ucolick.org/web_s2n/lris, (3) HIRES ETC: http://etc.ucolick.org/web_s2n/hires, (4) LCO ETC: http://www.lco.cl/scripts/lcoetc/lcoetc_sspec.html, (5) GIRAFFE ETC: https://www.eso.org/observing/etc/bin/gen/form?INS.NAME=GIRAFFE+INS.MODE=spectro, (6) SAO ETC v0.5: http://hopper.si.edu/etc-cgi/TEST/sao-etc, (7) MUSE ETC: eso.org/observing/etc/bin/gen/form?INS.NAME=MUSE+INS.MODE=swspectr, (8) X-SHOOTER ETC: https://www.eso.org/observing/etc/bin/gen/form?INS.NAME=X-SHOOTER+INS.MODE=spectro, (9) UVES ETC: https://www.eso.org/observing/etc/bin/gen/form?INS.NAME=UVES+INS.MODE=FLAMES, (10) MODS Instrumental Sensitivity: http://www.astronomy.ohio-state.edu/MODS/ObsTools/Docs/MODS1_InstSens.pdf, (11) JWST ETC: https://jwst.etc.stsci.edu/ (workbooks available upon request.), (12) PFS ETC and Spectrum Simulator: https://github.com/Subaru-PFS/spt_ExposureTimeCalculator, (13) MSE ETC: http://etc-dev.cfht.hawaii.edu/mse/, (14) FOBOS/WFOS ETC: https://github.com/Keck-FOBOS/enyo, (15) GMACS ETC v2.0: http://instrumentation.tamu.edu/etc_gmacs/, (16) G-CLEF ETC: http://gclef.cfa.harvard.edu/etc/.

^aThe LRIS ETC does not include the 1200/7500 grating throughput so the 1200/9000 grating throughput is used in its place. ^bS/N adapted from the ETC of a similar instrument according to Equation (14). ^cAs an IFU, MUSE does not have a definite fiber or slit size on the sky.

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For our S/N calculations we assume an airmass of 1.1 and a seeing of 075 (or as close to these values as possible with each ETC). We assume read-noise is negligible such that the S/N of a single one-hour exposure is the same as that of four 15 minute exposures stacked together.

Because not all ETCs provide the same stellar spectral energy distribution (SED), we use a K0I, K2V, or K0V spectral template (in preferential order when provided) to best match the SED of our fiducial RGB star. Additionally, we use a K0V spectral template for the RGB reference stars with ${\rm{log}}\,Z\leqslant -1.5$ and a K5V spectral template for the RGB stars with log Z > –1.5. For the log Z = –1.5 MSTO and TRGB reference stars we use G5V and K5III/K5V stellar templates respectively.

Once calculated by the ETC, the S/N is interpolated onto the same wavelength grid as the stellar spectra corresponding to that instrument's resolving power, spectral sampling, and wavelength range.

 Because most spectrographs are designed to slightly oversample the spectrum (≥3 pixels/FWHM), adjacent pixels are not completely uncorrelated, though most stellar abundance studies treat them as such (see however Czekala et al. 2015). For simplicity, we also assume no correlations between adjacent wavelength pixels so that we can write the covariance matrix of the normalized flux, Σ, as the diagonal matrix

$$\begin{eqnarray}{\rm{\Sigma }}=\left[\begin{array}{ccc}{\sigma }^{2}({\lambda }_{1}) & & \\ & \ddots & \\ & & {\sigma }^{2}({\lambda }_{N})\end{array}\right],\end{eqnarray} \tag{ 15 }$$

where ${\sigma }^{2}{({\lambda }_{i})=({\rm{S}}/{\rm{N}})}^{-2}$ is the variance in each pixel. A more accurate treatment of the pixel-to-pixel covariance would effectively reduce the number of independent information-carrying pixels in the spectrum, increasing the CRLB slightly—recall that the CRLB is proportional to n^−1/2, where n is the number of independent pixels per resolution element. A more in-depth analysis of pixel correlation and wavelength sampling is presented in Appendix C.

The large variety of resolving powers included in this work means that a universal "observing strategy" cannot be applied to all instruments. Instead, we consider separate observing setups for a fiducial spectrograph, low- and medium-resolution spectrographs (R < 10,000), high-resolution spectrographs (R > 10,000), and JWST/NIRSpec, which we describe below. A summary of all of the relevant assumptions used in the S/N calculation of each instrument is contained in Table 4.

Fiducial Spectrograph—To investigate the effects of exposure time, object brightness, and stellar evolutionary phase and metallicity, we adopt the 1200G grating on Keck/DEIMOS as our fiducial spectroscopic setup. We consider 1, 3, and 6 hr integration times and stars with m_V = 18, 19.5, and 21. For comparisons of metallicity and stellar evolutionary phase, we hold the integration time and apparent magnitude fixed at 1 hr and m_V = 19.5, respectively.

Low- and Medium-resolution Spectrographs—For spectrographs with R < 10,000, we consider the baseline observing strategy to be 1 hr of integration of our fiducial m_V = 19.5 RGB star. This is generally sufficient for spectrographs on 6+ meter telescopes to achieve ${\rm{S}}/{\rm{N}}\gt 15$ pixel⁻¹ across the optical spectrum. In this category, we include DEIMOS, LRIS, and FOBOS on Keck; Hectospec and Binospec on the MMT; the Multi Unit Spectroscopic Explorer (MUSE) on the VLT; LBT/MODS on the LBT; PFS on Subaru; MSE; the Multi-object Astronomical and Cosmological Spectrograph (GMACS) on the GMT; the Wide Field Optical Spectrometer (WFOS) on the TMT; and the Multi-Object Spectrograph (MOSAIC) on the E-ELT.

The GMACS ETC provides two sample settings, each of which assume a constant δλ across both the blue and red channels, resulting in wavelength-dependent resolutions. We choose the higher resolution setting (Δλ = 1.4) and scale the S/N at each pixel according to S/N ∝ R^−1/2 to match the constant resolving power we are attempting to emulate. Because ETCs do not yet exist for MOSAIC, we scale the S/N from GMACS for MOSAIC's visual high multiplex mode (HMM-Vis) and from NIRSpec for MOSAIC's near-infrared high multiplex mode (HMM-NIR) according to Equation (14).²⁰

High-resolution Spectrographs—Due to the higher dispersion and generally lower throughput of high-resolution spectrographs, a single hour of integration is insufficient to achieve adequate S/N (>15 pixel⁻¹) for an m_V = 19.5 RGB star. Instead we consider an integration of 6 hr (∼1 night of observing). Instruments in this category include the High Resolution Echelle Spectrometer (HIRES) on Keck; the Magellan Inamori Kyocera Echelle (MIKE) on Magellan and the Michigan/Magellan Fiber System (M2FS) on Magellan; Hectochelle on MMT; X-SHOOTER,²¹ GIRAFFE, and the Ultraviolet and Visual Echelle Spectrograph (UVES) on the VLT; and the GMT Consortium Large Earth Finder (G-CLEF) on GMT. M2FS and Hectochelle do not have public ETCs so we scale the average S/N from the GIRAFFE HR10 ETC according to Equation (14) and assume the S/N is roughly constant over the short wavelength range observed by these instruments.

JWST/NIRSpec—The strength of JWST/NIRSpec is its high sensitivity and high angular resolution. The most likely use case will be to acquire spectra in distant and/or crowded environments, which may require longer integration times than our fiducial 1 hr setup for ground-based, low-resolution instruments. Thus, for JWST only, we adopt a 6 hr of integration on an m_V = 21 TRGB star.²² This scenario is chosen to mimic the observation of bright stars in the disk of M31 or in a galaxy at the edge of the Local Group.

Beyond 1 Mpc—To investigate the distance to which JWST/NIRSpec and GMT/GMACS (as a representative ELT) can provide useful chemical measurements, we additionally hold the exposure time constant at 6 hr and systematically decrease the apparent magnitude of our target TRGB star from m_V = 21–26. This corresponds to observing a TRGB star at distances between 0.5 and 5 Mpc.

Our CRLBs formally represent the best achievable abundance precision via full spectral fitting, not necessarily what is obtained in practice (due to imperfect models, variable LSFs, masked or obscured features, etc.). It is therefore useful to compare our CRLB estimates to published abundance precisions from full spectral fitting to get a sense of how close current abundance measurements get to our predictions.

For an illustrative comparison, we select abundances measured by Kirby et al. (2018), who use a full spectral fitting technique (as opposed to EWs) for RGB stars in Local Group galaxies (Kirby et al. 2009). Because of the large variety in stellar targets and spectral quality, we make several cuts to the Kirby et al. (2018) sample in order to fairly compare the reported precision and our CRLBs. First, we consider only stars with T_eff between 4500 and 5000 K, log g between 1.7 and 1.9, and [Fe/H] between −2.0 and −1.0. Second, we consider only stars that were observed to  $35\,{\mathring{\rm A} }^{-1}\lt {\rm{S}}/{\rm{N}}\lt 65\,{\mathring{\rm A} }^{-1}$, which corresponds to roughly the mean S/N of a 1 hr exposure of a 19.5 mag star. These cuts leave the reported abundance precision of 33 stars.

Before we make a direct comparison, we modify our CRLB calculation to closely adhere to the choices made by Kirby et al. (2018). For example, log g and v_micro are not fit via spectroscopy, but held fixed at values determined by the star's photometry. This can lead to more precise recovery of abundances by removing their covariances with these labels. Similarly, only Fe, Ca, Ni, Si, Ti, Co, Mg, and Cr are fit, while all other abundances are fixed at the solar abundance value. These are not unreasonable assumptions because the information content of the spectra is dominated by these elements, and log g is typically better constrained with photometry than spectroscopy in extragalactic contexts where the distance is well constrained. We mimic this analysis by adopting a delta function prior on all stellar labels that are not fit for by Kirby et al. (2018).

In addition, Kirby et al. (2018) mask a handful of specific spectral regions that are contaminated by poorly modeled lines or strong telluric absorption features. Following Kirby et al. (2008) we mask 13 spectral regions including notable spectral features such as the Ca ii triplet (λλ6498, 8542, 8662) and the Mg i λ8807 line.

It is worth noting that there are several aspects of the method used by Kirby et al. (2018) that we cannot account for. First, they adopt a different set of stellar models and line lists than we do, albeit with similar 1D LTE assumptions (e.g., ATLAS9 vs. ATLAS12; see Kirby et al. 2010). Second, they fit stellar labels iteratively by looping through the labels and fitting each individually while holding the rest constant until convergence is achieved. It is possible that this approach may ignore some covariances between labels that are expected when all labels are fit simultaneously as assumed by the CRLB. Third, the specific wavelength coverage of each spectrum varies from the nominal depending on the star's location on DEIMOS's detector.

Lastly, we note that the chemical abundance uncertainties reported by Kirby et al. (2018) include both a statistical and systematic uncertainty component added in quadrature. Because CRLBs are purely a measure of statistical precision and not accuracy, we subtract out in quadrature the systematic component (of order 0.2 dex for Co and 0.1 dex for all other elements) to make a better one-to-one comparison with the literature uncertainties.

Figure 4 shows the reported precision of the 33 stars from Kirby et al. (2018) plotted with our D1200G CRLBs—both with and without adjustments to match their specific analysis We find that the abundances reported by Kirby et al. (2018) are within a factor of ∼2 of our corresponding CRLBs. The precisions reported for Fe (0.05 dex), Co (0.12 dex), and Cr (0.22 dex) are slightly less than our predicted precisions (0.06, 0.14, and 0.20 dex respectively). This may be due to a slight overestimation of the systematic uncertainty on these labels or the underestimation of label degeneracies as a result of the iterative fitting. The reported precision for Co, Mg, and Cr, are likely skewed to higher precision because only abundances recovered to better than 0.3 dex are reported, leaving only eight stars with Co abundances, one star with Mg abundances, and six stars with Cr abundances.

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The biggest difference between the CRLBs calculated previously and those calculated to mimic the analysis of Kirby et al. (2018) is in the forecasted uncertainty of Ca and Mg, which increased from 0.07 and 0.16 dex to 0.14 and 0.22 dex, respectively. This is the result of masking strong lines for these elements, which are both highly informative but challenging to model correctly. Fixing log g would have considerably improved the precision for Ca had the Ca i triplet not been masked due the feature's strong dependence on surface gravity. Instead, it only very slightly increases the precision of Fe and Ni from 0.06 and 0.09 dex to 0.05 and 0.08 dex, respectively, but otherwise does not change the CRLB substantially. From this comparison, we can see the importance of folding in these effects to our ability to estimate the expected precision.

 While the reported uncertainty for most elements is slightly higher than the CRLB, it is encouraging to see them within a factor of ∼2. There are several reasons why poorer precision in practice could be expected. Examples include poor model fidelity, imperfect calibrations, and masked or lost spectral regions (see Section 6.4 for further discussion). While future comparisons with abundance precisions from full-spectrum fitting are necessary to more completely understand the prospects of achieving the CRLB in practice, this comparison with D1200G illustrates that the CRLBs at least provide a realistic benchmark for spectroscopic abundance precision. In Appendix D, we perform an analogous comparison with LAMOST and find similar agreement between our CRLBs and the literature abundance precision.

We now consider how the CRLB changes as a function of metallicity. To do this we compare the CRLBs for RGB stars with log Z = −0.5, −1.0, −1.5, −2.0, and −2.5. In order to achieve similar observing conditions for each star, we make comparisons at fixed m_V instead of at fixed stellar phase (or fixed location on the RGB; see Figure 2). As a result of the RGB isochrone's metallicity-dependent morphology, T_eff and log g for these stars are all slightly different with more metal-poor stars having higher T_eff and log g (Table 1). The S/Ns for these stars are calculated for our fiducial observation of a 1 hr exposure of a star at 100 kpc (m_V = 19.5) and the configurations summarized in Table 4.

The CRLBs for the stars of various metallicity are plotted in Figure 5. As expected, the achievable abundance precision decreases toward lower metallicity as there are fewer and weaker absorption features. However, the dependence of precision with metallicity is not uniform across all elements. For example, the precision of Fe steadily decreases from ∼0.03 dex to ∼0.1 dex as the metallicity decreases from log Z = −0.5 to −2.5. The precision of V, however, decreases dramatically from ∼0.05 dex to ∼0.2 dex between log Z = −0.5 and −1.0 as a result of its absorption features being strongly temperature dependent. At even lower metallicities (and slightly higher T_eff), V features are nearly entirely absent.

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Below log Z = −0.5, the CRLBs for T_eff and log g remain constant, or even improve. This seemingly counterintuitive result is due to increasingly prominent Paschen lines redward of 8200 Å with increasing temperature. These lines are very sensitive to the star's T_eff and log g, allowing for precise measurements of these atmospheric parameters despite the lower metallicities.

Just as a star's spectral gradients vary as a function of metallicity, it also varies as a function of atmospheric structure (i.e., log g, T_eff, and v_micro). As a result, we expect the achievable abundance precision at varying stellar phases to be different even at fixed metallicity and apparent magnitude. While we focus our analysis on a typical RGB star, stars from the MSTO to the tip of the red giant branch (TRGB) are also targets of extragalactic studies.

Here, we consider the CRLBs for the log Z = −0.5 RGB star considered previously with that of an MSTO and TRGB star at the same metallicity (see Table 1). We once more consider a 1 hr integration of a m_V = 19.5 star with the relevant ETC configuration in Table 4.

The CRLBs of each of these stellar phases are plotted in Figure 6, illustrating that the chemical abundance precision is best for TRGB stars and worst for MSTO stars (all other things being equal). While only 3 elements can be measured to better than 0.3 dex from the spectrum of the MSTO star, 10 elements can be measured to this precision in the RGB star, and 19 in the TRGB star. For a fixed element the precision is roughly two times better for the TRGB star than the RGB star and another two times better than the MSTO star.

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These differences are expected because the absorption features of hot subgiants are significantly weaker than for cool giants. This is especially true for elements like C, N, and O, which are measured primarily from molecular features that are pronounced in TRGB stars but practically nonexistent in MSTO stars. Similarly, Fe, Si, Mg, Al, and other elements whose abundances affect a star's atmospheric structure leave a larger signature in cool, low surface gravity stars than hot, high surface gravity stars.

Recovering T_eff and log g, on the other hand, can be done more precisely in MSTO stars, due to the strong dependence of the Paschen lines on the star's atmospheric parameters.

For stars with secure distances (as members of external galaxies typically are), photometry can be used to constrain T_eff and log g to roughly ±100 K and ±0.15 dex, respectively (Kirby et al. 2009; Casagrande et al. 2011; Heiter et al. 2015). Knowledge of log g and Equation (13) can also constrain v_micro to roughly ±0.25 km s⁻¹ (Holtzman et al. 2015). We can incorporate these photometric estimates as priors on our spectroscopically determined labels as shown in Section 2.2.1. To do so we adopt Gaussian priors on these parameter with standard deviations equal to their photometric uncertainties. We once more consider a 1 hr observation of our fiducial log Z = –1.5 RGB star at 50, 100, and 200 kpc.

Figure 7 shows the results of the CRLBs assuming Gaussian priors. For reference, we include the CRLBs from Figure 3 (top), which assume uniform priors.

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For the highest S/N case (at 50 kpc; S/N ∼ 75 pixel⁻¹), the precision on T_eff and log g from D1200G spectroscopy alone is significantly better than the priors. The priors therefore contribute negligible additional information, and the CRLBs only minimally improve.

However, in the lowest S/N case (at 200 kpc; S/N ∼ 10 pixel⁻¹), T_eff, log g, and v_micro are substantially less constrained by the spectroscopy compared to the priors and so nearly all of the information about these stellar labels are coming from the prior. As a result, use of these priors improve the precision of T_eff, log g, and v_micro by factors of 2–6 compared to the uniform prior case.

In addition, because spectral gradients of T_eff and log g are covariant with the spectral gradients of elements like Fe, Ca, and Ni, priors that better constrain T_eff and log g also lead to improved precision on these chemical abundances. For example, in the case of our faintest star, the Fe, Ca, and Ni abundance precision improves by ∼50% when Gaussian priors on T_eff and log g are included. We expect the inclusion of photometric priors to have more impact when the spectral gradients of different labels are more covariant (i.e., for low-resolution spectra with heavily blended lines and spectra with very limited wavelength coverage and few absorption lines).

On the Keck/DEIMOS spectrograph there are several options that provide access to wavelengths bluer than 5000 Å. As listed in Table 2, the 900ZD, 600ZD, and 1200B gratings all provide bluer wavelength coverage, but slightly lower resolution, compared to the D1200G setup. These gratings have already enabled abundance determinations not possible from red-optical spectroscopy, such as the measurement of α elements in the M31 halo (Escala et al. 2019b) and Ba in several dwarf galaxies (Duggan et al. 2018). The 1200B grating is a recent addition to DEIMOS's grating collection and has not been used to measure stellar abundances at the time of this paper's writing.

In addition to DEIMOS, the Keck telescopes also host the LRIS MOS, which operates using separate red and blue channels. The 600/4000 grism on the blue arm boasts impressive blue throughput compared to DEIMOS gratings²⁸ , while the 1200/7500 grating on the red arm provides coverage around the Ca ii triplet (Table 2). While LRIS has only ever been used for very limited stellar abundance determinations (Shetrone et al. 2009; Lai et al. 2011), it is nonetheless a promising instrument, particularly given the demonstrated success of LAMOST.

To quantify the information content accessible in the blue optical by these instrumental setups, we calculate their CRLBs given a 1 hr exposure of our fiducial log Z = –1.5 RGB star at 100 kpc and the relevant ETC configurations for each instrument from Table 4.

The forecasted abundance precision for each element is presented in Figure 8. Despite their lower resolving powers, instruments with bluer wavelength coverage provide more precise measurements of more elements than D1200G. For example, the 1200B grating on DEIMOS and the 600/4000+1200/7500 LRIS setup enable the recovery of 21 and 22 elements respectively to better than 0.3 dex—about twice that from comparable red-optical spectroscopy at fixed integration time and stellar type. This includes eight r- and s-process elements (Y, Ce, La, Zr, Ba, Sr, Pr, and Eu), which have most, if not all of their absorption features at wavelengths shorter than 5000 Å and are thus largely inaccessible to D1200G and other longer wavelength spectrographs. Information about C and N comes primarily from C₂, CH, and CN absorption bands between 4000 and 5000 Å and to a lesser extent from CN bands between 7000 and 9000 Å.

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D1200G does provides comparable or better precision for Fe, Ni, Si, and Co, which have many lines at wavelengths longer than ∼6500 Å, as well as for Ca, Na, and K, which have strong features in the red optical.²⁹

LRIS's improved precision is due to a combination of its exceptional throughput down to 3900 Å and the additional wavelength coverage provided by its red arm.³⁰ However, it is important to remember that LRIS has roughly half the field of view and half the multiplexing as DEIMOS (Table 3), meaning that it may ultimately be less efficient for some elements, when the number of stars is included in the calculation.

As a reminder, the DEIMOS 600ZD and 900ZD gratings and the LRIS 1200/7500 grating all oversample their spectra with 5 pixels/FWHM. If the pixels in these spectra are not completely independent as we assume here, the CRLBs we present may be slightly more precise than would be expected in practice (see Section 6.4.3).

We now turn our attention to blue-sensitive instruments on facilities other than the Keck Telescopes, which include MODS on the LBT, MUSE on the VLT, and Hectospec and Binospec on the MMT.

MODS, like LRIS, operates at low resolution (R ∼ 2000) across the optical spectrum with a red and a blue arm, and modest multiplexing (Tables 2 and 3). Other than a recent study on a chemically peculiar ultra metal-poor star in the dwarf galaxy Canes Venatici I (Yoon et al. 2019), MODS has not been utilized for stellar chemical abundance measurements.

While MUSE is not technically an MOS but rather an integral field unit (IFU), it can nonetheless be used effectively for low-resolution multi-object resolved star spectroscopy in crowded fields. MUSE has already been used to conduct several campaigns for both stellar radial velocity and chemical abundance measurements in globular clusters (e.g., Husser et al. 2016; Kamann et al. 2016, 2018; Latour et al. 2019), in dwarf galaxies (e.g., Voggel et al. 2016; Alfaro-Cuello et al. 2019; Evans et al. 2019), and in NGC 300 (Roth et al. 2018; McLeod et al. 2020).

Hectospec, in comparison to MODS, MUSE, and the spectrographs on Keck, has a very large field of view (1° × 1°), which makes it a powerful instrument for spectroscopic observations of very extended stellar populations. For example, Carlin et al. (2009) used Hectospec to measure the kinematics and bulk metallicity of stars in the disrupted MW dwarf galaxy Boötes III. Binospec is a new, complementary MOS to Hectospec with very high throughput, but a significantly smaller field of view and a more limited multiplexing capability (Table 3). Both Hectospec and Binospec have a number of gratings that allow for a range in wavelength coverage and resolving power. We examine a few setups we consider to be most applicable to extragalactic stellar spectroscopy (see Table 2 for specifics).

Figure 9 shows the CRLBs for our fiducial RGB star (log Z = –1.5, m_V = 19.5) and a 1 hr exposure. For these observing conditions, MODS is forecasted to recover up to 30 individual elements to better than 0.3 dex. MODS's precision can be attributed to two key factors: its large, nearly 12 meter effective aperture and its throughput below 4000 Å, which together achieve S/N of >40 pixel⁻¹ down to 4000 Å and >10 pixel⁻¹ down to 3500 Å. As discussed in Section 4.2.1, these regions become increasingly information rich due to the high densities and strengths of absorption features of many elements.

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There are a few specific elements that are worth examining in more detail. Just as with the blue-optimized spectrographs on Keck, the constraints on C and N abundances come predominantly from absorption bands at wavelengths bluer than 5000 Å and (to a lesser extent) between 8000 Å and 1 μm. MODS's sensitivity across both of these ranges leads exceptional recovery of C and N compared to the other instruments analyzed here. MUSE and the 600 gratings of Hectospec and Binospec do not push nearly as blue (or red) and thus recover C and N abundances less precisely or not at all.  While the 270 grating on Hectospec and the 270 and 1000 gratings on Binospec do include most of the blue carbon features, they miss most of the blue nitrogen features and (with the exception of the 270 grating on Binospec) achieve an S/N in this region roughly half that of MODS. As a result they also do not recover C and N as precisely as MODS.

In addition to C and N, MODS is also able to recover O to better than 0.2 dex because of strong OH absorption features below 3500 Å and the important role of O in the CNO molecular network (Ting et al. 2018).

Again, it is worth highlighting the precision capable of these blue-optimized spectrographs for heavy r- and s-process elements  Nd, Ce, Zr, La, Sr, Y, Eu, Ba, Pr, Dy, Gd, and Sm (in order of decreasing precision for MODS). In addition to those seen in Figure 8, the ability to recover  Nd, Dy, Gd, and Sm is the direct result of MODS blue sensitivity (discussed further in Section 6.1). A few of these are recoverable by  MUSE, Hectospec, or Binospec, but measurement is made more difficult due to lower S/N and smaller wavelength coverage.

Given the smaller light-collecting power of MMT, it is reasonable that Hectospec and Binospec are forecasted to recover fewer elemental abundances and at larger uncertainties. It is nonetheless still interesting to look at them in greater detail and compare the various Hectospec and Binospec settings. Generally Binospec's higher throughput leads to higher precision measurements, but this of course comes with a diminished field of view and fewer fibers for stars.

Similarly, the increased abundance precision of MODS, MUSE, and other Keck spectrographs is also modulated by much reduced fields of view. The choice between these instruments then ultimately comes down to weighing the importance of detailed abundance patterns versus the importance of a large sample size to the desired science.

We remind the reader that the Hectospec configurations oversample their spectra with 5 pixels/FWHM. If the pixels in these spectra are not completely independent as we assume here, the CRLBs we present may be slightly more precise than would be expected in practice (see Section 6.4.3).

High-resolution spectroscopic observations of stars provide precise radial velocities and are the gold standard for chemical abundance determinations. Because high-resolution spectroscopy provides spectra with fewer blended absorption features, spectral abundance determinations preferentially use clean, isolated lines that can be fit with EW methods over blended lines, which require spectral synthesis techniques. By not fitting the star's entire spectrum simultaneously, some of the spectrum's chemical information goes un-utilized. By calculating the CRLBs for several high-resolution spectrographs, we illustrate the chemical information that can be accessed through full-spectrum-fitting techniques.

In the context of extragalactic studies, two commonly used single-slit echelle spectrographs are Magellan/MIKE and Keck/HIRES. Both instruments provide high-resolution spectra across the entire optical regime and have been used extensively for abundance measurements in MW globular clusters (e.g., Boesgaard et al. 2000, 2005; Venn et al. 2001; Koch & Côté 2010) and in nearby dwarf galaxies (e.g., Shetrone et al. 1998; Frebel et al. 2014, 2016; Koch & Rich 2014; Ji et al. 2016a, 2016b, 2016c, 2020).

We also consider two spectrographs on the VLT: UVES and X-SHOOTER. UVES is a high-resolution spectrograph with a more limited wavelength coverage (only 4800–6800 Å) but is capable of observing up to eight stars at a time when connected with the Fibre Large Array Multi Element Spectrograph (FLAMES) fiber feed.³¹ It has been used to observe RGB stars in MW globular clusters (e.g., Alves-Brito et al. 2006) and in nearby dwarf galaxies (e.g., Shetrone et al. 2003; Letarte et al. 2006; Hill et al. 2019; Lucchesi et al. 2020). X-SHOOTER has also been used to measure abundances of bright stars in dwarf galaxies (Starkenburg et al. 2013; Spite et al. 2018) and provides slightly lower resolution than MIKE, HIRES, and UVES but significantly higher throughput and broader wavelength coverage.³²

As discussed in Section 3.1.3, a 1 hr exposure of a m_V = 19.5 RGB star is typically insufficient for high-resolution spectrographs to overcome the read-noise-limited regime of faint object spectroscopy. Instead we consider a more realistic 6 hr (∼1 night) of integration, which yields S/N > 15 (10) pixel⁻¹ at 4500 Å and S/N > 20 (20) pixel⁻¹ at 7500 Å for HIRES (MIKE) when adopting the ETC configurations in Table 4.

Figure 10 shows the CRLBs for HIRES, MIKE, FLAMES-UVES, and X-SHOOTER. As expected, high-resolution spectra provide very precise detailed chemical abundance patterns.  HIRES, MIKE, and X-SHOOTER are forecasted to measure a dozen elements to nearly 0.01 dex and over 30 elements to better than 0.3 dex. UVES, with its smaller wavelength coverage and lower S/N (5–10 pixel⁻¹), is still forecasted to recover over 20 elements. This high precision is predicted despite the low S/N (<20 redward of 4500 Å) of these observations, demonstrating the potential power of full spectrum fitting applied to high-resolution spectra. While at low S/N any given absorption feature might be only weakly informative, the ensemble of all spectral features still provides strong constraints on the chemical abundances of a star.

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The chemical information for many of the elements in Figure 10 can be traced to the same large numbers of features below ∼5000 Å as previously discussed in Section 4.2. While these absorption features are still subject to blending, the higher resolution of these instruments increases the rms depth of the absorption feature and alleviates degeneracy between elements. This results in increased abundance precision over low-resolution instruments at fixed wavelength coverage. We can see this effect when comparing the CRLBs of the two HIRES settings, which have the same wavelength coverage but different resolving powers—the CRLBs scale with resolving power σ_CRLB ∝ R^−1/2 as expected for instruments with the same wavelength range.³³

In addition to elements previously discussed in Sections 4.1 and 4.2, HIRES can recover the abundances of neutron-capture elements Sm, Er, Tb, and Os to better than 0.3 dex. At R ∼ 50,000 there are nearly 100 Sm lines with gradients >5% dex^–1 and over 30 lines with gradients of 10%–30% dex^–1 in the HIRES wavelength range—all of which are below 4500 Å. The same spectrum has ∼15 (5) absorption lines with gradients of >5% dex^–1 (10% dex^–1) absorption lines for Er (Tb) blueward of 5000 Å. Os can be recovered to ∼0.3 dex from no more than five absorption lines with >5% dex^–1 gradients.

MIKE's bluer wavelength coverage is largely offset by its lower resolving power (R ∼ 28,000) and very low S/N (<5 pixel⁻¹) below 5000 Å. Nevertheless, MIKE achieves slightly better precision for Tb and  Er, which have two to three times more lines between 3500 and 3900 Å than they do at wavelengths longer than 3900 Å. MIKE's recovery of N is aided by strong molecular absorption bands at λ3550 and λ3800 and another in the red at λ9150. Its  higher precision for Al and S compared to HIRES is the result of additional atomic absorption lines beyond 8500 Å and its higher S/N  in the red.

X-SHOOTER, despite its lower resolution (R ∼ 10,000), recovers  most elements as precisely as, if not better than, MIKE and HIRES. For C, N, and O, X-SHOOTER can achieve precisions two to three times better than MIKE and HIRES as a result of its larger wavelength coverage. It is sensitive to both the CNO molecular bands in the blue optical and the NIR molecular features beyond 1 μm. Si, Mg, Na, Al, K, and S also have a handful of absorption features in the NIR, enabling one to two times higher precision with X-Shooter than MIKE and HIRES.  Furthermore, because the NIR is generally less dense with absorption features, the gradients for these elements are less degenerate with other stellar labels and can thus be more precisely recovered.

The comparatively lower precision of FLAMES-UVES can be attributed to its shorter (and redder) wavelength coverage, which does not include nearly as much of the high-information density spectral regions as the other spectrographs considered here. Furthermore, the S/N is roughly two to three times lower than that of MIKE or HIRES. Depending on the desired science, however, the multiplexing capabilities of UVES may more than make up for its lower throughput and wavelength coverage.

At low S/N (e.g., 5 pixel⁻¹), there may be a concern that the assumptions of Gaussianity, which underlies the CRLB, may not be valid. However, we show in Appendix E that the CRLBs are robust to the level of ∼0.01 dex down to S/N ∼ 5 pixel⁻¹. Thus, we believe non-Gaussianity to have a minimal impact on the CRLBs, especially compared to other practical limitations (e.g., model fidelity) that make it difficult to fully realize the precision forecasted by the CRLBs.

UVES and the UVB arm of X-SHOOTER oversample their spectra with 5 pixels/FWHM. If the pixels in these spectra are not completely independent as we assume, the CRLBs may not be as precise as we present here (see Section 6.4.3).

Another approach to high-resolution spectroscopy involves using order-blocking filters that block all but one order of the echelle spectrum. Doing so allows for improved multiplexing, but limits the observed wavelength to a small window of 50–300 Å. Historically, the primary application of these instruments for extragalactic archaeology has been the efficient measurement of precise radial velocities in dwarf galaxies (e.g., Walker et al. 2007, 2009b), but these spectra clearly contain chemical information as well.

We consider three such high-resolution, single-order, fiber-fed MOS: VLT/FLAMES-GIRAFFE, MMT/Hectochelle, and Magellan/M2FS. Due to the nature of order blocking in these instruments, there is great flexibility in deciding what small portion of spectrum to observe. In this work, we will only look at spectral regions targeted by existing observations and save a detailed analysis of the optimal wavelength windows for a future paper. For M2FS, this includes a "HiRes" and a "MedRes" setting around the Mg i b triplet (λλ5183, 5172, 5167), which have been used for membership determination and [Fe/H] measurement in several MW satellites (e.g., Walker et al. 2007, 2009b, 2015a, 2016). The RV31 order-blocking filter was used on Hectochelle for similar purposes (e.g., Walker et al. 2009b, 2015b; Spencer et al. 2017) and is also utilized by the H3 MW halo survey (Conroy et al. 2019a, 2019b). On FLAMES-GIRAFFE, five setting have been used by the DART (Dwarf Abundances and Radial Velocities Team) program to measure various abundances and radial velocities in Local Group dwarf galaxies: LR8, HR10, HR13, HR14A, and HR15 (e.g., Hill et al. 2019; Theler et al. 2019). Details for all of these instruments and settings can be found in Table 2.

Just as with the previous high-resolution CRLBs, we consider 6 hr of integration on our log Z = –1.5 RGB star at 100 kpc and the ETC configurations in Table 4.

Figure 11 shows the forecasted precision for these single-order echelle spectrographs. As expected, the limited wavelength coverage of these setups severely reduces their chemical abundance recovery compared to the full-optical, high-resolution spectrographs presented in Figure 10. Even most low-resolution spectrographs can achieve comparable or better abundance recovery in a fraction of the time as presented in Figures 8 and 9. This is because the information content scales proportionally with the square root of the number of absorption features. A smaller wavelength range means fewer lines for a given element and worse precision.

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Nevertheless, given the narrow wavelength range covered by these orders and the low S/N (∼15–30 pixel⁻¹), it is promising that more than a handful of elements beyond Fe can be recovered to better than 0.3 dex. We first consider the abundance precision for M2FS and Hectochelle (Figure 11; top), which cover 5100–5300 Å. This narrow region of the spectrum contains numerous absorption lines of Fe, and to a lesser extent also of Ni, Ti, Co, Cr, and Nd, which enable their recovery. All three filters were designed to include the Mg I b triplet and as a result Mg can also be measured.  There are also a few (<5) strong (∼10%–30% dex^–1 at R ∼ 32,000) lines each for Ca, Sc, Y, and Cu in this wavelength range that enable the M2FS MedRes configuration and Hectochelle to recover these elements. Hectochelle achieves slightly higher precision due to its higher resolving power. Because M2FS's  HiRes filter has a more limited wavelength range,  it misses a considerable fraction of these lines and thus cannot measure these abundances as precisely.

Next we consider GIRAFFE, which has several orders that span the entire optical spectrum. Fe, Ca, Ni, Ti, and Co all have numerous strong lines (>10% dex^–1 at R ≳ 20,000) below 7000 Å, enabling their recovery by all high-resolution order-blocking filters. Mn, however, has the majority of its strongest lines between 5300 and 5600 Å and is thus  only recovered by HR10. The same is approximately true for Y and Nd. Ba has two moderate absorption features (>10% dex^–1 at R ≳ 20,000) at λ6143 and λ6499 in the HR13 and HR14A filters, respectively, but is  better recovered in HR14A because of the filter's higher S/N and resolving power.  The combination of throughput and resolution enables HR14A to achieve higher precision for its recoverable elements than the other individual filters, though its redder wavelength coverage precludes it from measuring elements whose lines reside primarily at wavelengths bluer than 6000 Å.

For reference, we also include in the bottom of Figure 11 the CRLB for the combined analysis of all five GIRAFFE orders as was done in Hill et al. (2019). It is clear that by combining the many information-carrying absorption features across all orders provides a significant improvement in the possible stellar label precision and enables the measurement of elements that no individual filter alone could recover (e.g., N and La). However, to achieve the S/N and abundance precision found here, would require 6 hr of integration on each of the five GIRAFFE orders for a total of 30 hr of integration. Still, it is useful to compare this precision to that of low-resolution MOS and high-resolution single-slit echelle spectrographs. While low-resolution. blue-optical spectroscopy can achieve similar precision abundance determinations for a similar number of stars in a small fraction of the time, the kinematic information in these observations is limited—at R ∼ 2000, the precision of radial velocity measurements is only σ_RV ∼ 150 km s⁻¹, which is good enough for membership determination, but not for detailed kinematic studies. In contrast, R ∼ 20,000 spectra yield σ_RV ∼ 5 km s⁻¹, which are precise enough for stellar multiplicity determinations, orbit reconstruction, and dark matter mass measurements. Furthermore, these high-resolution observations will be less prone to systematics incurred by model imperfections in blended lines.

A drawback to high-resolution, single-slit echelle spectrographs is the amount of time required to build up large samples of stars. In 30 hr of integration time, assuming 6 hr per pointing and ignoring overheads, HIRES, MIKE, and X-SHOOTER could observe five stars, while five echelle orders (6 hr each) could be acquired by GIRAFFE for ∼100 stars. Ultimately, the choice of instrument and observing strategy is highly dependent on the science case and whether higher abundance precision or a larger sample size is most valuable and whether precise radial velocities are needed. However, in the specific case of chemodynamical studies of dwarf galaxies, where both chemical and kinematic information are desired for a large number of stars, it may be worth trading in full optical coverage for specific wavelength regions and higher multiplexing.

We first consider the three low-resolution spectrographs, WFOS, MOSAIC, and GMACS, which all enable observations of 100+ stars across the full optical spectrum at resolving powers between R ∼ 1000 and R ∼ 5000. The configurations we consider are listed in Table 2. As in Section 4.2, we assume a 1 hr observation of our fiducial log Z = –1.5 RGB star with m_V = 19.5 and the ETC configurations in Table 4.

Figure 13 presents the CRLBs for these ELT spectrographs. We predict that all three optical ELT spectrographs are capable of measuring 30–40 elemental abundances to better than 0.3 dex. In addition to all Fe-peak elements and most α elements, this includes 22 neutron-capture elements spanning all three r- and s-process peaks. Of these, 12, 9, and 8 can be recovered to better than 0.1 dex by GMACS (G3), MOSAIC (HMM-VIS), and WFOS (B2479/R1392), respectively.

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Many of these elements have only weak features below 4000 Å, which necessitate high S/N in the blue optical and near-UV for their recovery. Tb and Tm, for example, have ∼20 absorption lines with 1%–3% dex^–1 gradients at R ∼ 3500, but nearly all are found at wavelengths shorter than 4000 Å. Similarly, Pd, Os, and Hf have fewer than 10 absorption lines of similar strengths, which are also predominantly located blueward of 4000 Å. The strongest line of Th is at λ4019 with a gradient of ∼1.5% dex^–1, while ∼20 weaker (0.5%–1.0% dex^–1) features exist between 3100 and 4000 Å. Despite the limited chemical information, spectrographs on ELTs are capable of measuring these elements because their large-aperture telescopes and blue wavelength coverage can achieve S/N ∼ 100 at 4000 Å.

The informative power of blue-optical spectroscopy can be further seen in the comparatively poorer abundance recovery of MOSAIC's HMM-Vis and HMM-NIR settings. Because the optical arm only extends to 4500 Å, it cannot capitalize on the information-rich, near-UV stellar spectrum. The NIR is expected to recover even fewer abundances than the optical arm due to the lower information density beyond 8000 Å. Nevertheless there are some elements (e.g., Ca, Si, Sr, O, Al, and S) whose absorption features are better observed in the NIR. CN absorption in the red and NIR also allow for recovery of C and N to a similar degree to what can be done with spectra down to 4500 Å. We note, however, that because the JWST NIRSpec ETC was repurposed to provide S/N in the NIR for MOSAIC, the S/N used here does not include the effects of troublesome NIR telluric features. As a result, we expect the abundance precision of MOSAIC's HMM-NIR spectra to be noticeably worse in practice.

Figure 13 (top) illustrates the trade-offs in S/N, wavelength coverage, and resolution at a fixed number of detector pixels for three different GMACS gratings. As predicted by Ting et al. (2017a), the abundance precision of a detector with fixed pixel real estate under the assumption of uniform distribution of chemical information is relatively invariant of the resolving power. Of course, there are slight differences in the expected precision of the gratings. For many elements, G2 (R = 1000) performs more poorly than the higher resolution gratings, which is likely due to strongly blended lines at R = 1000 and the resulting increased covariance between elements. It is also apparent that the chemical information is not uniformly distributed; there are several abundances (e.g., Cr, C, Ba, Al, Dy, Gd, and K) which the G4 grating recovers noticeably worse if not at all because the absorption features of these elements lie outside of its reduced wavelength coverage. These elements are predominantly those with few strong features that lie below 4200 Å. Similar conclusions can be drawn from a comparison of the three WFOS grating combinations.

Here, we consider G-CLEF, a GMT first-light, fiber-fed echelle spectrograph. While it is primarily optimized for very high-resolution (R ∼ 100,000) single-slit spectroscopy across the optical, it will also feature a MOS mode that will combine modest multiplexing, Keck/HIRES-like spectra, and a 24.5 meter aperture telescope that will dramatically increase the feasibility of high-resolution spectroscopy of stars beyond the immediate vicinity of the Local Group (see Tables 2 and 3 for details). We calculate the S/N using the G-CLEF ETC given the same observational conditions used for the forecasting of existing high-resolution instruments (see Table 4).

Figure 14 shows the CRLBs of G-CLEF with the HIRES 10 CRLBs for comparison. We forecast that G-CLEF observations will recover 30 elements to better than 0.1 dex (and nearly 40–0.3 dex) similar to HIRES and the other single-slit high-resolution spectrographs analyzed previously in Section 4.3.1. In addition to achieving HIRES-like abundance recovery, G-CLEF's multiplexing enables the simultaneous observation of up to 40 stars at a time. This dramatically increasing the feasibility of high-resolution studies of substantial numbers of stars in extragalactic systems (for both chemistry and kinematics).

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The reason G-CLEF does not achieve substantially better abundance precision than its 10 meter class analogs appears to be largely a consequence of G-CLEFs lower predicted throughput. Despite having a much larger light-collecting power, G-CLEF acquires roughly the same S/N as Keck/HIRES at wavelengths shorter than 6000 Å where most of the chemical information resides. G-CLEF achieves higher S/N (∼35 pixel⁻¹ compared to ∼20 pixel⁻¹) at longer wavelengths, but this only yields small improvements in abundance precision. Furthermore, G-CLEF's bluer wavelength coverage is at S/N < 5 pixel⁻¹ and thus provides little additional information.

Model fidelity is a fundamental assumption inherent in all problems of parameter estimation. The CRLB of stellar spectra is no exception to this as the gradient spectra used in the above calculations are strongly dependent on the physical assumptions and spectral line lists that underpin any spectral synthesis model. It is important to keep in mind that the CRLB makes no claims about the accuracy of stellar label measurements, merely the possible precision. Nevertheless, incomplete or incorrect line lists will leave out or misplace spectral information, while models that assume 1D atmospheres in local thermodynamic equilibrium (LTE) may incorrectly predict the spectral response to varying stellar labels for non-LTE lines. It is thus important to strive for consistency and consider the CRLBs calculated using the models relevant to the spectral fitting that will be conducted. While comparing CRLBs of different models is a valuable exercise to evaluate systematics in the predicted CRLBs, this should not be done to pass judgment on model quality.

A common practice in full-spectrum fitting is the masking of spectral regions that are known to be poorly fit by the spectral model to avoid introducing potential systematics into the analysis. Often the poor fit is due to non-LTE effects, but may also be the result of 3D effects, poorly calibrated oscillator strengths, or an incomplete (or incorrect) line list (see Nissen & Gustafsson 2018 and references therein). When these regions are masked, so too is the information that it holds. In such a case the appropriate CRLB should be calculated with gradient spectra masked in the same regions (as we do in Section 4.1.1), resulting in a higher uncertainty for the stellar labels. We note, however, that because information adds in quadrature, masking 90% of the lines only worsens the CRLB by a factor of ∼3. For a more thorough analysis of the CRLB's dependence on masked regions, see Ting et al. (2017a).

Another underlying challenge for our CRLBs is the assumption that the continuum can be perfectly determined. In the red-optical and near-infrared region of the spectrum, lines are sufficiently sparse that even at R ∼ 2000 identifying the continuum and dividing it out is routine. Unfortunately, the many absorption features in the blue optical and UV, make it challenging to define a stellar continuum. Instead, a pseudo-continuum is defined using a polynomial function (or some smoothing kernel) and divided out, potentially introducing systematics or additional uncertainty in the normalized flux that will worsen the precision. By similarly normalizing the model spectra (instead of using the true continuum), any systematics introduced through imperfect normalization can be minimized.

Knowledge of the instrumental LSF is necessary to fit observed spectra with model spectra at the same resolving power. In this work, we have assumed a constant LSF. However, in practice, the LSF is not always known to great precision and can vary from object to object depending on where in the field of view the star lies. Use of the wrong LSF is thus another means by which systematics may be introduced into the fitting of stellar labels. Ting et al. (2017a) showed that at least at moderate resolution (R ∼ 6000) and high S/N (>200), mismatched LSFs only bias stellar label recovery for differences in broadening greater than 10 km s⁻¹ and is unlikely to affect the measurement precision. Spectral fitting at lower resolving powers should be even less sensitive to mismatches in LSF.

In addition, when using rest-frame synthetic spectra, it is necessary to properly determine and correct for the radial velocity of stars. As with the continuum normalization and LSF, we have not quantified the uncertainty in stellar labels that is introduced when the radial velocity is fit simultaneously with other stellar labels. We expect any changes in the CRLBs to be small given that radial velocity is unlikely to correlate with other stellar labels. We will pursue this analysis in a future study.

Even with perfect spectral models, continuum normalization, and instrument characterization, fully extracting the chemical information content of a spectrum requires fitting the full wavelength range (as opposed to measuring EWs) for all stellar labels simultaneously. This is particularly important at low and moderate resolution to account for the degeneracies between labels introduced by blended spectral features. In practice, this can be computationally challenging owing to the high dimensionality of the stellar label space and the large runtimes needed to generate even 1D LTE stellar atmospheres.

Despite these challenges, the future of extragalactic stellar spectroscopy looks bright as steady progress is being made in all of the aforementioned areas. Attempts to incorporated non-LTE and 3D effects into stellar atmosphere and radiative transfer models have been undertaken by a number of groups (e.g., Caffau et al. 2011; Bergemann et al. 2012; Amarsi et al. 2016). Several groups have committed to further refining line lists through the identification of unknown (or misplaced) lines in stellar spectra (e.g., Shetrone et al. 2015; Andreasen et al. 2016) and the improved calibration of transition oscillator strengths (e.g., Pickering et al. 2001; Aldenius et al. 2007; Pehlivan Rhodin et al. 2017; Laverick et al. 2018). Lastly, full spectrum-fitting techniques have made major strides with spectral "emulators" trained through data-driven (e.g., the Cannon; Ness et al. 2015), ab initio (e.g., the Payne; Ting et al. 2019), or combined (e.g., The DD-Payne; Xiang et al. 2019) methods, which bypass the computationally expensive stellar atmosphere and radiative transfer calculations.

The above challenges to achieving the precision predicted by the CRLBs should not dissuade from the use of CRLBs. Instead, the precision forecasted by the CRLBs provide strong motivation for the continued efforts toward understanding stars, their atmospheres, and their spectra.

Implicit in the derivations of Equations (1), (2), and (5) was that of Gaussian likelihoods and uncertainties. When these conditions are not met, the CRLB will inaccurately predict measurement errors and degeneracies between stellar labels. In such situations, a more accurate estimate of the achievable precision can be found using Bayesian sampling techniques. A comparison of the CRLB and the precision predicted by HMC sampling in the low-S/N limit is performed in Appendix E, and we find it robust down to an S/N of 5 in the case of D1200G (assuming a constant S/N with wavelength).

Throughout this study we simplify our analysis by setting the correlation between adjacent pixels to zero when calculating the CRLBs.³⁹ In practice, however, most spectrographs are designed to oversample their spectra such that the number of pixels per resolution element is larger than the Nyquist sampling (∼2 pixels/FWHM).⁴⁰ As a result, adjacent pixels will show some correlation and not be truly independent as we have assumed.

While this is unlikely to make a large difference for most spectrographs, which only slightly oversample their spectra (3–4 pixels/FWHM), the pixel-to-pixel correlation of spectrographs that more highly oversample (e.g., Hectospec, FLAMES-UVES, FOBOS, and some DEIMOS and LRIS gratings) may be nonnegligible in practice. If instead we believe that only 2 pixels per resolution element are informative then the CRLBs should be a factor of $\sqrt{2}$ ($\sqrt{3}$) larger than presented for spectrographs with a sampling of 4 (6) pixels/FWHM as the CRLBs scale as n^−1/2. More realistically, additional sampling beyond the Nyquist limit will yield pixels that are still informative, just less so than wholly independent pixels. Thus, we expect the increase in the CRLB to be considerably less than a factor of $\sqrt{2}$ ($\sqrt{3}$) when the correlation of adjacent pixels are taken into account. In Appendix C, we present an illustrative example of the impact of wavelength sampling and pixel-to-pixel correlation on the CRLBs.