MAGELLAN/M2FS SPECTROSCOPY OF THE RETICULUM 2 DWARF SPHEROIDAL GALAXY^*

We use the software package MultiNest (Feroz & Hobson 2008; Feroz et al. 2009) to scan the parameter space and generate random samplings of the 15-dimensional posterior PDF. For a given parameter, X, we then calculate moments of the marginalized, 1D posterior PDF as follows. The first moment is the mean, $\bar{X}\equiv {N}^{-1}{\displaystyle \sum }_{i=1}^{N}{x}_{i}$, which we take to be the central value. The second moment is the variance, ${\sigma }_{X}^{2}\equiv {(N-1)}^{-1}{\displaystyle \sum }_{i\;=\;1}^{N}{({x}_{i}-\bar{X})}^{2}$, which we take to be the square of the $1\sigma$ credibility interval. The third moment is skewness, $S\equiv {N}^{-1}{\displaystyle \sum }_{i=1}^{N}{[({x}_{i}-\bar{X})/\sqrt{{\sigma }_{X}^{2}}]}^{3}$, which equals zero for a symmetric distribution. The fourth moment is kurtosis, $K\equiv {N}^{-1}{\displaystyle \sum }_{i=1}^{N}{[({x}_{i}-\bar{X})/\sqrt{{\sigma }_{X}^{2}}]}^{4}$, which distinguishes Gaussian distributions (K = 3) from "leptokurtic" ones with sharper peaks and fatter tails ($K\gt 3$) and "platykurtic" ones with broader peaks and weaker tails ($K\lt 3$).

For all 182 stacked M2FS spectra from Ret 2 targets, Figure 4 shows how moments of PDFs for physical parameters vary with a median signal-to-noise ratio (S/N) per pixel. As S/N grows, the PDFs become narrower and more Gaussian ($S\sim 0$, $K\sim 3$). In subsequent analysis we consider only the sample consisting of 37 observations for which the PDF for velocity has $\sigma \leqslant 5$ km s⁻¹, $| S| \leqslant 1$ and $| K-3| \leqslant 1$ (red points in Figure 4).

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In order to examine accuracy and precision, we apply the same analysis to 256 solar spectra that we acquired during evening twilight on the same night and with the same spectrograph configuration used for the Ret 2 observations. Reassuringly, parameter estimates from the collection of twilight spectra exhibit empirical scatter that is consistent with variances of the posterior PDFs. Among the 256 twilight spectra, standard deviations of central values for ${v}_{\mathrm{los}}$, ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ are 0.1 km s⁻¹, 48 K, 0.04 dex, and 0.03 dex, respectively. Furthermore, mean estimates show only mild systematic offsets with respect to known solar values: $\langle {v}_{\mathrm{los}}\rangle -{v}_{\mathrm{los},\odot }=-3.9$ km s⁻¹, $\langle {T}_{\mathrm{eff}}\rangle -{{T}_{\mathrm{eff}}}_{\odot }=10$ K, $\langle \mathrm{log}g\rangle -\mathrm{log}{g}_{\odot }=-0.03$ dex, and $\langle {[\mathrm{Fe}/{\rm{H}}]\rangle -[\mathrm{Fe}/{\rm{H}}]}_{\odot }=-0.18$ dex. Following W15, for stellar-atmospheric parameters we treat these offsets as zero-point errors and subtract them from raw estimates obtained for science targets. We also add the empirical scatter, in quadrature, to (square roots of) second moments for all observations. After applying these adjustments, our estimates of stellar-atmospheric parameters have median (minimum, maximum) random errors of ${\sigma }_{{T}_{\mathrm{eff}}}\;=$ 478 (61, 997) K, ${\sigma }_{\mathrm{log}g}\;=$ 0.63 (0.11, 1.27) dex, and ${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}\;=$ 0.47 (0.05, 0.85) dex.

For velocities, we do not adjust the zero point based on results from the twilight spectra. In our experience, M2FS velocities shift systematically as ambient temperature changes rapidly during twilight, and the observed offset between twilight and solar velocities reflects this phenomenon. As pointed out to us by the DES collaboration (J. Simon 2015, private communication), a temperature-dependent velocity shift continued, albeit at a slower rate, during our observations of Ret 2. In order to examine this effect, we compare velocity estimates that we obtain from fits to spectra obtained in individual science exposures. We consider only the 69 single-exposure spectra whose results satisfy the same quality-control criteria imposed on the stacked spectra (Section 3.1). Consistently with the effect noticed by the DES collaboration, we find that velocity estimates shift systematically over the three exposures and at different rates for the two spectograph channels. From 37 (32) spectra observed on M2FS's "blue" ("red") channel, we measure a weighted mean difference of $\langle {v}_{\mathrm{exp}3}-{v}_{\mathrm{exp}1}\rangle \;=\;$ 2.0 ± 0.3 km s⁻¹ (1.2 ± 0.3 km s⁻¹) between the first and third exposures. Thus the raw velocities that we estimate from the stacked frames have channel-dependent zero-point errors. In order to compensate for this effect, we subtract the appropriate channel-specific value of $\langle {v}_{\mathrm{exp}3}-{v}_{\mathrm{exp}1}\rangle$ from each of the raw velocities that we estimate from the stacked frames. We also add the corresponding error in our estimate of $\langle {v}_{\mathrm{exp}3}-{v}_{\mathrm{exp}1}\rangle$, in quadrature, to the error of our velocity estimates, after which our velocity sample has median (minimum, maximum) error ${\sigma }_{{v}_{\mathrm{los}}}\;=$ 1.4 (0.5, 3.2) km s⁻¹. This procedure is designed only to remove the dependence of the velocity zero point on channel, and does not necessarily correct for a global error in zero point. However, a comparison to an independent spectroscopic sample obtained with VLT/FLAMES (S. Koposov et al. 2015, in preparation), shows that the zero points are in agreement. For the 17 stars common to both samples, the mean velocity difference is $\langle {v}_{{\rm{M}}2\mathrm{FS}}-{v}_{\mathrm{VLT}}\rangle =0.3\;\pm$ 0.3 km s⁻¹.

In order to gauge repeatability of our estimates for physical parameters, we again consider results from our modeling of 69 single-exposure spectra whose results satisfy quality-control criteria. The top row of panels in Figure 5 compares parameter estimates from the first exposure with those in either of the two later exposures (all velocities from a given exposure and channel are shifted to remove the systematic drift in zero-point discussed above). The bottom row of panels in Figure 5 displays distributions of deviations with repect to weighted means—$\langle \bar{X}\rangle \equiv {\sum }_{i=1}^{N}({\bar{X}}_{i}/{\sigma }_{{X}_{i}}^{2}){/{\sum }_{i=1}}^{{N}_{\mathrm{obs}}}(1/{\sigma }_{{X}_{i}}^{2})$—normalized by propagated error. Gray curves show Gaussian distributions with the same integrated area as the histograms. The observed distributions are all narrower than the Gaussian curves, implying that the variances of the PDFs give credibility intervals that may be slightly overestimated.

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As a final check on reliability of our velocity estimates, we independently measure LOS velocities using IRAF's "fxcor" package. We cross-correlate each target spectrum from the stacked frame with a high-S/N M2FS spectrum that we acquired for the radial velocity standard star CPD-432527 (${v}_{\mathrm{los}}=19.7\;\pm$ 0.9 km s⁻¹; Udry et al. 1999) immediately before the Ret 2 observations. Reassuringly, we find good agreement with results from the Bayesian procedure described above. For the 37 spectra that satisfy quality-control criteria (Section 3.1), the mean deviation between (raw) velocity estimates is ∼0.1 ± 0.30 km s⁻¹. Furthermore, when we model the spectrum of CPD-432527 using our Bayesian procedure, we obtain ${v}_{\mathrm{los}}=19.7\;\pm$ 0.3 km s⁻¹, recovering the previously published value.

We emphasize that our estimates of stellar-atmospheric parameters come exclusively from our M2FS spectra and receive no input from photometry. In order to gauge consistency with photometrically derived parameters, Figure 6 compares our spectroscopic estimates of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ with photometric magnitude and the relations given by old (age = 12 Gyr), metal-poor Dartmouth isochrones (Dotter et al. 2008). For a given $i$-band magnitude, the isochrones (shifted according to distance modulus $m-M=17.4$; Koposov et al. 2015) predict temperatures and gravitites for stars on the red giant branch. With respect to the isochrone that most closely matches Ret 2's mean metallicity ([Fe/H] = −2.5; Section 4.2) our measurements for 17 probable Ret 2 members (Section 4.1) give ${\chi }^{2}\;=\;44.4$ and 73.9 for ${T}_{\mathrm{eff}}$ and $\mathrm{log}\ g$, respectively, indicating scatter that is significantly larger than expected to arise from our random measurement errors. However, these large values are driven almost entirely by two outliers (Ret2-080 and Ret2-178). If we exclude these two stars, we obtain ${\chi }^{2}=12.6$ and 20.8 for ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$, respectively, each with 15 degrees of freedom. One expects to obtain smaller values of ${\chi }^{2}$ with probability 0.37 and 0.86, respectively.

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The two outlying stars are the brightest in our sample of probable members. Thus their deviations from isochrone relations suggest that our spectroscopic estimates may give correlations between luminosity and ${T}_{\mathrm{eff}},\mathrm{log}g$ that are systematically too strong. Alternatively, we may have underestimated measurement errors for the two outlying stars. A definitive conclusion awaits the analysis of larger M2FS data sets. In the meantime, we note that if we were to exclude the two outlying stars from the chemodynamic analysis discussed below, our results would remain statistically identical (central values for our estimates of global velocity dispersion and mean metallicity change by less than $3\%$).

Table 1 lists results for the 37 spectra that satisfy the quality-control criteria discussed in Section 3.1. The online version of this article includes a supplementary table with results for all 182 observed stars. The first three columns of Table 1 list target ID and equatorial coordinates, followed by angular separation from Ret 2's center and g, r magnitudes from the photometry of K15, corrected for extinction using the dust maps of Schlegel et al. (1998). The seventh column lists median S/N per pixel. Columns 8–11 then list the results of our spectroscopic modeling in terms of the first four moments of PDFs for ${v}_{\mathrm{los}}$, ${T}_{\mathrm{eff}}$, $\mathrm{log}g$ and $[\mathrm{Fe}/{\rm{H}}]$. For each parameter, the quoted central value is the first moment (mean) and the credible interval is the square root of the second moment; both values have been adjusted according to the offsets and empirical scatter described in Section 3.2. Skewness and kurtosis are listed parenthetically as superscripts above the credible interval. The last column indicates whether or not the star satisfies membership criteria discussed in Section 4.1. For all observations, the first exposure began at heliocentric Julian date HJD = 2457072.542 days and the midpoint of the 3 × 40 minutes exposure occured at HJD = 2457072.579 days.

Four spectra from our sample merit special attention. Table 2 lists their coordinates and g, r magnitudes. The first two (Ret2-161 and Ret2-034) belong to the two BHB candidates and are displayed in the lower-right panels of Figure 2. Their spectra show well-sampled continua but no obvious absorption features, as expected given the greater degree of ionization at high temperature. As a result, model parameters for these stars are poorly constrained. While their velocities are loosely consistent with Ret 2 membership (see Section 4.1), the associated uncertainties are ∼30 km s⁻¹ and ∼260 km s⁻¹, such that neither spectrum satisfies our quality-control criteria (Section 3.1). On the other hand, their temperatures (${T}_{\mathrm{eff}}$ = 6890 ± 680 and 6880 ± 630 K), while also highly uncertain, are the two largest that we obtained for any of our 182 observed spectra. We conclude that these two stars remain strong BHB candidates, with confirmation requiring spectra that cover a larger range in wavelength.

The next special case (Ret2-097) appears in the upper panel of Figure 3. This "star" has color and magnitude placing it near the isochrone and above the horizontal branch in Figure 1. However, the spectrum is rich in broad and apparently double-valleyed absorption features. We model this spectrum as the superposition of two spectra of the form given by Equation (1). Allowing for a second set of stellar-atmospheric parameters and a second continuum polynomial, the resulting model has 25 free parameters. As shown in Figure 2, this double-star model gives a reasonably good fit to the spectrum. One of the sources has ${v}_{\mathrm{los}}\;=$ 54.8 ± 0.7 km s⁻¹, slightly below the range of velocities that we attribute to Ret 2 members (Section 4.1). Its strong surface gravity ($\mathrm{log}g\;=$ 4.54 ± 0.21) and rich metallicity ($[\mathrm{Fe}/{\rm{H}}]\;=$ −0.39 ± 0.08) are typical of foreground contamination. The other contributing star has ${v}_{\mathrm{los}}\;=$ 120.4 ± 0.7 km s⁻¹, $\mathrm{log}g\;=$ 4.65 ± 0.23 and is also metal-rich, with $[\mathrm{Fe}/{\rm{H}}]\;=$ −0.54 ± 0.10. We speculate that this source is a physical binary composed of two K-type main sequence stars at a distance of a few kiloparsecs. Assuming a mass ratio near unity, the center of mass has ${v}_{\mathrm{los}}\sim 100$ km s⁻¹, inconsistent with Ret 2 membership. Due to the anomalous nature of this spectrum, we do not include these results in Table 1 or consider them in subsequent analysis.

Finally, the bottom panel of Figure 3 displays a spectrum (Ret2–019) that is difficult to classify given the limited spectral range. Here we consider a few possibilities. First, the absorption features may have molecular origin, specifically from MgH, which has a bandhead at ∼5200 Å and extends blue-ward by ∼200 Å. However, the features in our spectrum are too wide, too deep and too irregularly spaced to match the typical appearance of that band (Weck et al. 2003). In addition, MgH absorption becomes evident typically in stars of spectral types K0 or later, much redder than the de-reddened color ($g-r\sim 0.17$) of this source. Alternatively, this spectrum may reveal the presence of a complex absorption line system along the LOS to a remote active galactic nucleus. The line widths and extreme depths appear similar to numerous line complexes reported by Srianand et al. (2010) for various absorption lines in Lyα systems. Furthermore, the source's blue color is consistent with that of a background quasi-stellar object (QSO) at moderate redshift. For a number of plausible UV lines that may produce the observed absorption features, the implied redshift of the absorber is $z\gtrsim 2.5$, with the background source obviously more distant. The nearest known extragalactic source in the NED catalog is detected in the UV, by GALEX, and is 24 arcsec away from our target. A definitive classification of this source will require spectroscopy with broader wavelength coverage.

Figure 7 also helps to distinguish bona fide members of Ret 2 from contaminants in the Galactic foreground. We expect Ret 2 members to exhibit a relatively narrow range in velocity, lower metallicities, lower surface gravities, and to be clustered nearer Ret 2's center than are their foreground counterparts. Figure 7 clearly shows a population of stars having these characteristics. Red boxes in Figure 7 enclose measurements that cluster near the centers of Ret 2's distributions for ${v}_{\mathrm{los}}$, $\mathrm{log}g$, $[\mathrm{Fe}/{\rm{H}}]$ and position, as determined by eye. We make no selection based on color, magnitude or temperature because the full ranges for these quantities are consistent with Ret 2 membership. We count 17 stars that are inside all red boxes. Table 1 lists a membership status of "Y" for these stars, and "N" for stars that lie outside any of the red boxes.

In previous work we have employed an expectation-maximization (EM) algorithm in order to estimate model parameters in the presence of contamination (Walker et al. 2009b). Under the assumption that velocity dispersion is spatially uniform, our EM algorithm provides estimates of membership probabilities for all individual stars. Reassuringly, when we apply the EM algorithm to the Ret 2 data in Table 1, the sum of membership probabilities is 16.7, with 16 of the 17 stars inside our red boxes receiving membership probability $\gt 0.99$. The remaining star, Ret2-142, receives membership probability 0.7. While Ret2-142 is the target nearest Ret 2's center (R = 1.2 arcmin) and has velocity (${v}_{\mathrm{los}}=65.2\pm 2.9$ km s⁻¹) similar to Ret 2's mean, its metallicity ($[\mathrm{Fe}/{\rm{H}}]=-0.80\pm 0.66$) and surface gravity ($\mathrm{log}g=4.27\;\pm$ 0.60) are both the largest of any probable member of Ret 2. However, Ret-142 is also the faintest ($r\;=\;19.98$) probable member, so it should have the strongest surface gravity. In any case, the EM algorithm's estimates of means and dispersions of velocity and metallicity distributions agree well with those obtained in the independent analysis that we present in Section 4.2.

In order to characterize the internal chemodynamics of Ret 2, we consider a simple model in which the velocities and metallicities of the 17 member stars are normally distributed about means that can vary systematically with position in order to account for velocity and metallicity gradients. Specifically, we assume that the mean LOS velocity, ${\mu }_{{v}_{\mathrm{los}}}$, is a function of projected position ($R,\theta$),

$$\begin{eqnarray}&&{\mu }_{{v}_{\mathrm{los}}}(R,\theta )\equiv \langle {v}_{\mathrm{los}}\rangle +{k}_{{v}_{\mathrm{los}}}R\mathrm{cos}(\theta -{\theta }_{{v}_{\mathrm{los}}}),\end{eqnarray} \tag{ 2 }$$

where $\langle {v}_{\mathrm{los}}\rangle$ is the mean velocity at the center, ${k}_{{v}_{\mathrm{los}}}\equiv | d{\mu }_{{v}_{\mathrm{los}}}/{dR}|$ is the magnitude of maximum velocity gradient and ${\theta }_{{v}_{\mathrm{los}}}$ (measured from north of Ret 2's center and opening to the east, in equatorial coordinates) specifies its direction. We assume that mean metallicity, ${\mu }_{[\mathrm{Fe}/{\rm{H}}]}$, is a function only of separation from the center,

$$\begin{eqnarray}&&{\mu }_{[\mathrm{Fe}/{\rm{H}}]}(R)\equiv \langle [\mathrm{Fe}/{\rm{H}}]\rangle +{k}_{[\mathrm{Fe}/{\rm{H}}]}R,\end{eqnarray} \tag{ 3 }$$

where $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ is the mean metallicity at the center and ${k}_{[\mathrm{Fe}/{\rm{H}}]}\equiv d{\mu }_{[\mathrm{Fe}/{\rm{H}}]}/{dR}$ specifies the magnitude of any (isotropic) metallicity gradient.

Under these assumptions, the joint probability density for observables ${v}_{\mathrm{los}}$ and $[\mathrm{Fe}/{\rm{H}}]$, for a star at position $(R,\theta )$, is

$$\begin{eqnarray}p\left({v}_{\mathrm{los}},[\mathrm{Fe}/{\rm{H}}]| R,\theta \right) & = & \frac{{(2\pi )}^{-1}}{\sqrt{\left({\sigma }_{{v}_{\mathrm{los}}}^{2}+{\delta }_{{v}_{\mathrm{los}}}^{2}\right)\left({\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}+{\delta }_{[\mathrm{Fe}/{\rm{H}}]}^{2}\right)}}\\ & & \times \mathrm{exp}\left[-\frac{1}{2}\frac{{\left({v}_{\mathrm{los}}-{\mu }_{{v}_{\mathrm{los}}}\right)}^{2}}{{\sigma }_{{v}_{\mathrm{los}}}^{2}+{\delta }_{{v}_{\mathrm{los}}}^{2}}\right.\\ & & -\left.\frac{1}{2}\frac{{\left([\mathrm{Fe}/{\rm{H}}]-{\mu }_{[\mathrm{Fe}/{\rm{H}}]}\right)}^{2}}{{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}+{\delta }_{[\mathrm{Fe}/{\rm{H}}]}^{2}}\right],\end{eqnarray} \tag{ 4 }$$

where ${\sigma }_{{v}_{\mathrm{los}}}$ and ${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}$ are velocity and metallicity dispersions, respectively, and ${\delta }_{{v}_{\mathrm{los}}}$ and ${\delta }_{[\mathrm{Fe}/{\rm{H}}]}$ are measurement errors. Given a vector of free parameters ${\boldsymbol{\theta }}\equiv (\langle {v}_{\mathrm{los}}\rangle ,{\sigma }_{{v}_{\mathrm{los}}},{k}_{{v}_{\mathrm{los}}},{\theta }_{{v}_{\mathrm{los}}},\langle [\mathrm{Fe}/{\rm{H}}]\rangle ,{\sigma }_{[\mathrm{Fe}/{\rm{H}}]},{k}_{[\mathrm{Fe}/{\rm{H}}]})$, a data set consisting of $N$ observations of Ret 2 members, $D\equiv {\{({R}_{i},{\theta }_{i},{v}_{\mathrm{los},i},{[\mathrm{Fe}/{\rm{H}}]}_{i})\}}_{i=1}^{N}$, has likelihood

$$\begin{eqnarray}&&{\mathcal{L}}(D| {\boldsymbol{\theta }})=\displaystyle \prod _{i=1}^{N}p({v}_{\mathrm{los},i},{[\mathrm{Fe}/{\rm{H}}]}_{i}| {R}_{i},{\theta }_{i}).\end{eqnarray} \tag{ 5 }$$

From Bayes' theorem, given the data, the model has posterior probability distribution function (PDF)

$$\begin{eqnarray}&&p({\boldsymbol{\theta }}| D)=\frac{{\mathcal{L}}(D| {\boldsymbol{\theta }})p({\boldsymbol{\theta }})}{p(D)},\end{eqnarray} \tag{ 6 }$$

where $p({\boldsymbol{\theta }})$ is the prior PDF and $p(D)\equiv \int {\mathcal{L}}(D| {\boldsymbol{\theta }})p({\boldsymbol{\theta }})d{\boldsymbol{\theta }}$ is the "evidence."

Again we use MultiNest to scan the seven-dimensional parameter space and to draw random samples from the posterior PDF. Figure 8 displays random samplings from PDFs for the means and dispersions. Table 3 summarizes the results. For each of the seven free parameters, the second column lists boundaries inside which the prior is uniform and nonzero (outside these boundaries, the prior probability is zero). The third column lists median-likelihood values and credibility intervals that enclose the central 68% (95%) of area under the posterior PDF.

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Table 3.  Summary of Probability Distribution Functions for Chemodynamical Parameters

Parameter	Prior	Posterior	Description
$\langle {v}_{\mathrm{los}}\rangle$ (km s−1)	uniform between −500 and +500	${64.34}_{-1.18(-2.51)}^{+1.21(+2.48)}$	mean velocity at center
${\sigma }_{{v}_{\mathrm{los}}}$ (km s−1)	uniform between 0 and +500	${3.59}_{-0.7(-1.24)}^{+0.95(+2.23)}$	velocity dispersion
${k}_{{v}_{\mathrm{los}}}$ (km s−1 arcmin−1)	uniform between 0 and +10	${0.46}_{-0.27(-0.43)}^{+0.4(+1.02)}$	magnitude of maximum velocity gradient
${\theta }_{{v}_{\mathrm{los}}}$ ($^\circ$)	uniform between −180 and +180	$-{91.51}_{-65.32(-84.1)}^{+216.75(+268.35)}$	direction of maximum velocity gradient
$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	uniform between −5 and +1	$-{2.58}_{-0.33(-0.65)}^{+0.34(+0.74)}$	mean metallicity at center
${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}$	uniform between 0 and +2	${0.49}_{-0.14(-0.25)}^{+0.19(+0.43)}$	metallicity dispersion
${k}_{[\mathrm{Fe}/{\rm{H}}]}$ (dex arcmin−1)	uniform between −1 and +1	${0.01}_{-0.06(-0.13)}^{+0.06(+0.12)}$	magnitude of metallicity gradient

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We find that Ret 2 has mean LOS velocity $\langle {v}_{\mathrm{los}}\rangle \;=$ ${64.3}_{-1.2}^{+1.2}$ km s⁻¹, with a resolved velocity dispersion of ${\sigma }_{{v}_{\mathrm{los}}}\;=\;$ ${3.6}_{-0.7}^{+1.0}$ km s⁻¹. We obtain a mean metallicity of $\langle [\mathrm{Fe}/{\rm{H}}]\rangle \;=\;$ $-{2.58}_{-0.33}^{+0.34}$, with a dispersion of ${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}\;=$ ${0.49}_{-0.14}^{+0.19}$. Taken at face value, the estimated metallicity dispersion is five times larger than is exhibited by any known globular cluster (Willman & Strader 2012). However, we note that the estimated metallicity dispersion is also similar to the median credibility interval for our metallicity estimates. Koposov et al. (2011) demonstrate that, in such cases, estimates of intrinsic dispersion can be extremely sensitive to systematic over- or under-estimation of measurement errors. Given the behavior in the bottom-right panel of Figure 5 (see Section 3.3 for discussion), we suspect that our metallicity errors are systematically overestimated, which would imply that our estimate of Ret 2's intrinsic metallicity dispersion is underestimated. In any case, this estimate should be interpreted with caution and will need to be confirmed using spectra spanning a larger range of wavelengths.

Finally, we do not detect significant gradients in either velocity or metallicity. Our estimate of ${k}_{{v}_{\mathrm{los}}}\;=\;$ ${0.5}_{-0.3}^{+0.4}$ km s⁻¹ arcmin⁻¹ is consistent with zero at the $\lesssim 2\sigma$ level. On the other hand, values as large as ∼1 km s⁻¹ arcmin⁻¹ are similarly allowed. The steepest allowed gradients are directed along position angle ${\theta }_{{v}_{\mathrm{los}}}\sim -90$°, roughly ∼20° from Ret 2's morphological major axis (K15, DES15). Our estimate of ${k}_{[\mathrm{Fe}/{\rm{H}}]}\;=$ ${0.01}_{-0.06}^{+0.06}$ dex arcmin⁻¹ is consistent with zero at the $\sim 1\sigma$ level, but also allows gradients as steep as ±0.15 dex arcmin⁻¹ within $\sim 2\sigma$. We notice degeneracies between our estimates of dispersions and gradients for both velocity and metallicity distributions, such that larger gradients correspond to smaller dispersions. However, our estimates for individual parameters (Table 3) naturally include such effects, as the 1D PDFs for individual parameters are obtained by marginalizing over all other dimensions of the parameter space.

K15 use photometric data to show that Ret 2 occupies a region of structural parameter space that is populated by objects of ambiguous classification, somewhat intermediate between well-established globular clusters and dwarf galaxies (Gilmore et al. 2007, cf. K15's Figure 17). Specifically, with projected halflight radius ${R}_{{\rm{h}}}=32\pm 1$ pc, Ret 2 is larger than nearly all globular clusters and smaller than nearly all known galaxies. Moreover, its absolute magnitude, ${M}_{V}=-2.7\pm 0.1$, would place Ret 2 among the least luminous members of either population. In these regards, Ret 2 and many of its newly discovered siblings are similar to "ultrafaint" satellites Segue 1, Segue 2, and Willman 1, as well as to the extended globular clusters Pal 14 and Crater (Belokurov et al. 2014; Laevens et al. 2014, Mateo et al. 2015).

Our spectroscopic results provide new leverage that can settle the question of Ret 2's nature. For Galactic globular clusters as well as the dwarf spheroidal satellites of the Milky Way and M31, the top panel of Figure 9 plots mean metallicity against luminosity. While globular clusters show no obvious trend, it is well-known that dwarf galaxies follow a luminosity/metallicity relation (Mateo 1998; Tolstoy et al. 2009; Kirby et al. 2013). Given the low mean metallicity that we estimate from the M2FS spectra, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle \;=$ $-{2.58}_{-0.33}^{+0.34}$, we place Ret 2 squarely onto the galactic relation (large black square in Figure 9).

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The lower panel of Figure 9 shows another well-known scaling relation that distinguishes dwarf galaxies from globular clusters. Specifically, the mass-to-light ratios (M/Ls) of dwarf galaxies are anti-correlated with luminosity, such that gravitational potentials in the least luminous galaxies all seem to be dominated by dark matter (Mateo et al. 1993; Martin et al. 2007; Simon & Geha 2007; Walker et al. 2007). In contrast, the stellar kinematics of globular clusters generally do not require an internal dark matter component. In order to see these trends, we plot ${R}_{{\rm{h}}}{\sigma }_{{v}_{\mathrm{los}}}^{2}/({{GL}}_{V})$ against luminosity, where ${R}_{{\rm{h}}}$ is halflight radius, ${\sigma }_{{v}_{\mathrm{los}}}$ is LOS velocity dispersion, ${L}_{V}$ is total V-band luminosity and G is Newton's constant. The combination of macroscopic observables on the vertical axis has dimensions of M/L⁵ and is therefore sufficient to highlight the different behavior of dwarf galaxies and globular clusters. Again we find that Ret 2 follows the galactic relation. Moreover, assuming dynamic equilibrium and negligible contamination from binary stars, Ret 2 has among the highest dynamical M/Ls of any known object. The crude mass estimator of Walker et al. (2009a) implies that the dynamical mass enclosed within Ret 2's projected halflight radius is $M({R}_{{\rm{h}}})\approx 5{R}_{{\rm{h}}}{\sigma }_{{v}_{\mathrm{los}}}{}^{2}/(2G)\;=\;$ ${2.4}_{-0.8}^{+1.4}\times {10}^{5}$ ${M}_{\odot }$, and the associated M/L is $\approx 2M({R}_{{\rm{h}}})/{L}_{V}\;=$ ${467}_{-168}^{+286}$ in solar units.

Table 4 summarizes the observed properties of Ret 2, based on the photometric results of K15 and the spectroscopic results presented in this work. Where photometric results from DES15 differ from those of K15 (e.g., for absolute magnitude), we list the DES15 results as well.

Table 4.  Summary of Reticulum 2's Observed Photometric and Spectroscopic Properties

Quantity	Value	Reference	Notes
R.A. at center	${\alpha }_{{\rm{J}}2000}\;=\;$03:35:42	K15 a	⋯
Decl. at center	${\delta }_{{\rm{J}}2000}=-$54:02:57	K15	⋯
Galactic longitude	l = 266.2958 deg	K15	⋯
Galactic latitude	$b=-49.7357$ deg	K15	⋯
Distance modulus	$m-M=17.4\pm 0.2$	K15	⋯
Distance from Sun	$D\sim 30$ kpc	K15	⋯
Absolute magnitude	${M}_{V}=-2.7\pm 0.1$ (−3.6 ± 0.1)	K15 (DES15)	⋯
Exponential scale length	${R}_{{\rm{e}}}={3.37}_{-0.13}^{+0.23}$ arcmin	K15	semimajor axis
Ellipticity	$e=1-(b/a)={0.59}_{-0.03}^{+0.02}$	K15	⋯
Position angle	PA $\;=\;71\pm 1$ deg	K15	⋯
Projected halflight radius	${R}_{{\rm{h}}}={3.64}_{-0.12}^{+0.21}$ arcmin	K15	${R}_{{\rm{h}}}\approx 1.68{R}_{e}\sqrt{1-e}$
Projected halflight radius	${R}_{{\rm{h}}}={32}_{-1.1}^{+1.9}$ pc	K15	⋯
Systemic line of sight velocity	${v}_{\mathrm{los}}\;=$ ${64.3}_{-1.2}^{+1.2}$ km s−1	this work	solar rest frame
Systemic line of sight velocity	${v}_{\mathrm{los}}=-90.9$ km s−1	this work	Galactic rest frame, given solar motion measured by Schönrich et al. (2010)
Internal velocity dispersion	${\sigma }_{{v}_{\mathrm{los}}}\;=$ ${3.6}_{-0.7}^{+1.0}$ km s−1	this work	⋯
Velocity gradient	${k}_{{v}_{\mathrm{los}}}\;=$ ${0.5}_{-0.3}^{+0.4}$ km s−1 arcmin−1	this work	⋯
PA of velocity gradient	${\theta }_{{v}_{\mathrm{los}}}\;=$ $-{92}_{-65}^{+217}$ deg	this work	⋯
Mean metallicity	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle \;=$ $-{2.58}_{-0.33}^{+0.34}$ dex	this work	⋯
Metallicity dispersion	${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}\;=$ ${0.49}_{-0.14}^{+0.19}$ dex	this work	similar to median metallicity error
Metallicity gradient	${k}_{[\mathrm{Fe}/{\rm{H}}]}\;=$ ${0.01}_{-0.06}^{+0.06}$ dex arcmin−1	this work	⋯
Mass enclosed within ${R}_{{\rm{h}}}$	$M({R}_{{\rm{h}}})$ = ${2.4}_{-0.8}^{+1.4}\times {10}^{5}$ ${M}_{\odot }$	this work	$M({R}_{{\rm{h}}})\approx 5{R}_{{\rm{h}}}{\sigma }_{{v}_{\mathrm{los}}}{}^{2}/(2G)$; assumes equilibrium, negligible binary stars
Mass-to-light raio	$\Upsilon \;=$ ${467}_{-168}^{+286}$ ${M}_{\odot }/{L}_{\odot }$	this work	$\Upsilon \approx 2M({R}_{{\rm{h}}})/{L}_{V}$; assumes equilibrium, negligible binary stars

Note.

^aUnless otherwise noted, K15 and DES15 report similar values.

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