THE SEGUE K GIANT SURVEY. II. A CATALOG OF DISTANCE DETERMINATIONS FOR THE SEGUE K GIANTS IN THE GALACTIC HALO

We start by recalling Bayes' theorem, cast in terms of the situation at hand:

$$\begin{eqnarray} && P({\rm \mathcal {DM}}\mid \lbrace m,c,{\rm [Fe/H]}\rbrace )= \nonumber\\ && \quad \times\frac{P (\lbrace m,c,{\rm [Fe/H]}\rbrace \mid {\rm \mathcal {DM}} )p_{{\rm prior}} ({\rm \mathcal {DM}} )}{P (\lbrace m,c,{\rm [Fe/H]}\rbrace )}. \end{eqnarray} \tag{ 1 }$$

Here, $P ({\rm \mathcal {DM}} \mid \lbrace m,c,{\rm [Fe/H]}\rbrace )$ is the pdf of the distance moduli, ($\mathcal {DM}=m-M$), and describes the relative probability of different ${\rm \mathcal {DM}}$, in light of the data, {m, c, [Fe/H]} (we use { } to denote the observational constraints, i.e., the estimates and the uncertainties for the observable quantities). $P (\lbrace m,c, {\rm [Fe/H]}\rbrace \mid \mathcal {DM})$ is the likelihood of ${\rm \mathcal {DM}}$ (e.g., ${\rm \mathscr{L}\, (\mathcal {DM})}$), and tells us how probable the data {m, c, [Fe/H]} are if ${\rm \mathcal {DM}}$ were true. The term $p_{{\rm prior}} (\mathcal {DM})$ is the prior probability for the $\mathcal {DM}$, which reflects independent information about this quantity, e.g., that the stellar number density in the Galactic halo follows a power law of r⁻³. The term P({m, c, [Fe/H]}) is a non-zero constant.

So, the probability of the ${\rm \mathcal {DM}}$ for a given star is proportional to the product of the likelihood of ${\rm \mathcal {DM}}$ and the prior probability of ${\rm \mathcal {DM}}$ (e.g., Equation (2)):

$$\begin{equation} P ({\rm \mathcal {DM}} \mid \lbrace m,c,{\rm [Fe/H]}\rbrace )\propto {\rm \mathscr{L}\, (\mathcal {DM})} p_{{\rm prior}} ({\rm \mathcal {DM}} ). \end{equation} \tag{ 2 }$$

The prior probability for the ${\rm \mathcal {DM}}$ in Equation (2) can be incorporated independently, and the main task is to derive ${\rm \mathscr{L}\, (\mathcal {DM})}$. In deriving ${\rm \mathscr{L}\, (\mathcal {DM})}$, we must in turn incorporate any prior information about other parameters that play a role, such as the giant-branch luminosity function, p_prior(M), or the metallicity distribution of halo giants, p_prior([Fe/H]). This is done via

$$\begin{eqnarray} \mathscr{L}\, (\mathcal {DM}) &=& \int \int p (\lbrace m,c,{\rm [Fe/H]}\rbrace \mid \mathcal {DM},M,{\rm FeH})\nonumber\\ && \times p_{{\rm prior}} (M)p_{{\rm prior}} ({\rm FeH})dM d{\rm FeH}. \end{eqnarray} \tag{ 3 }$$

Here we use FeH to denote the metallicity of the model, while we use [Fe/H] for the observed metallicity of the star.

The direct observables we obtain from SEGUE and the SSPP are the extinction corrected apparent magnitudes, colors, metallicities, and their corresponding errors (r₀, (g − r)₀, [Fe/H], Δr₀, Δ(g − r)₀, Δ[Fe/H]). Figure 1 shows the color–color diagram for our confirmed K giants, following application of the procedures described below. Figure 2 shows that the most common stars are the intrinsically fainter blue giants, as we would expect from the giant-branch luminosity function. Hereafter, we use c and m instead of (g − r)₀ and r₀ for convenience and generality in the expression of the formulas.

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In our analysis, we can and should account for three pieces of prior (external) information or knowledge about the RGB population that go beyond the immediate measurement of the one object at hand: $p_{{\rm prior}}(\mathcal {DM}), p_{{\rm prior}} (M)$, and p_prior([Fe/H]) (shown as Equation (1) and Equation (3)).

The prior probability of ${\rm \mathcal {DM}}$ reflects any information on the radial density profile of the Milky Way's stellar halo. Vivas & Zinn (2006) and Bell et al. (2008) indicated that the radial halo stellar density follows a power law ρ(r)∝r^α, with the best value of α = −3, and reasonable values in the range −2 > α > −4; this implies $p_{{\rm prior}}(\mathcal {DM})d\mathcal {DM}=\rho (r)4\pi r^2dr$, $p_{{\rm prior}}(\mathcal {DM}) \propto e^{({(3+\alpha)\log _e10}/{5})\mathcal {DM}}$. Quite fortuitously, the prior probability for ${\rm \mathcal {DM}}$ turns out to be flat for the radial stellar density of a power law of ρ(r)∝r⁻³. Given that the ${\rm \mathscr{L}\, (\mathcal {DM})}$ approximatively follows a Gaussian distribution with a mean of $\mathcal {DM}_0$ and standard deviation of $\Delta \mathcal {DM}$ (here $\Delta \mathcal {DM}$ is the error of ${\rm \mathcal {DM}}$), the $p_{{\rm prior}}(\mathcal {DM})$ will shift the estimate of $\mathcal {DM}_0$ by $({(3+\alpha)\log _e10}/{5})(\Delta \mathcal {DM})^2$, but with basically no change in $\Delta \mathcal {DM}$. Therefore, the shifts in the mean ${\rm \mathcal {DM}}$ caused by $p_{{\rm prior}}(\mathcal {DM})$ can be neglected for values of −2 > α > −4 (i.e., $({(3+\alpha)\log _e10}/{5})(\Delta \mathcal {DM})^2 \ll \Delta \mathcal {DM}$). In Section 3.6, we explicitly verify that different halo density profiles do not affect the distance estimation significantly using artificial data.

The prior probability of the absolute magnitude M can be inferred from the nearly universal luminosity function of the giant branch of old stellar populations. Specifically, we derive it from the globular clusters M5 ([Fe/H] = −1.4) and M30 ([Fe/H] = −2.13) (Sandquist et al. 1996, 1999), and from the Basti theoretical luminosity function with [Fe/H] = −2.4 and [Fe/H] = 0 (Pietrinferni et al. 2004). Figure 3 (top panel) shows that the luminosity functions for the RGBs derived from the globular clusters in the different bands are consistent with one another, and also with the Basti theoretic luminosity functions for the metal-rich and metal-poor cases. All the luminosity functions follow linear functions with similar slope, k = 0.32, as a result of the fact that the luminosity functions are insensitive to changes in the metallicity and color. According to p(M)dM = p(L)dL and M ∼ −2.5log L, the luminosity function p(M)∝10^kM means p(L)∝L^{−2.5k − 1}. We conclude that the luminosity function for the giant branch follows p(L)∝L^−1.8, shown as the dashed line in Figure 3.

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Our prior probability for [Fe/H] results from an empirical approach. In the SEGUE target selection, the K giants were split into four sub-categories: the l color K giants, the red K giants, the proper-motion K giants, and the serendipitous K giants. This suggests that one should adopt the overall metallicity distribution of each sub-category as the [Fe/H] prior for any one star in this sub-category (Figure 4). Figure 4 shows the metallicity distribution variation with apparent magnitude in the upper panel, and the four [Fe/H] priors in the lower panel. It can be seen that the four sub-categories have different metallicity distributions. For a star that has approximately the mean metallicity, this prior should leave ${\rm \mathscr{L}\, (\mathcal {DM})}$ unchanged, because the individual metallicity error is smaller than the spread of the p_prior([Fe/H]). However, for a star of seemingly very low metallicity, the prior implies that this has been more likely an underestimated metallicity of a (intrinsically) less metal-poor star, which would lead to an overestimated distance modulus.

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To obtain distance estimates, we determine an estimate of the absolute magnitude of each star, using its (g − r)₀ color and a set of giant-branch fiducials for clusters with metallicities ranging from [Fe/H] = −2.38 to [Fe/H] = +0.39, and then use the star's apparent magnitude (corrected for extinction using the estimates of Schlegel et al. 1998) to obtain its distance. We prefer to use fiducials, rather than model-isochrone giant branches wherever possible because isochrone giant branches cannot reproduce cluster fiducials with sufficient accuracy.

As the SDSS imager saturates for stars brighter than g ∼ 14.5, almost none of the clusters observed by SDSS and used by An et al. (2008) have unsaturated giant branches. Therefore, we derived such fiducials, using the globular clusters M92, M13, and M71, and the open cluster NGC 6791, which have accurate ugriz photometry from the DAOphot reductions of An et al. (2008) for most of the stars; the giant branches can also be supplemented using the u'g'r'i'z' photometry of Clem et al. (2008). We transformed to ugriz using the transformations given in Tucker et al. (2006). Note that M71 is a disk globular cluster, and one of the few clusters in the north at this important intermediate metallicity. However, it has reddening that varies somewhat over the face of the cluster, making it a difficult cluster to work with. We use g−i instead of g−r to obtain more accurate estimates of the variable reddening map of M71, and use them to produce a better fiducial for this cluster (see H. L. Morrison et al., in preparation for details). We list our adopted values of [Fe/H], reddening, and distance modulus for each cluster in Table 1. In addition, we supplemented the fiducials with a solar-metallicity giant branch from the Basti α-enhanced isochrones (Pietrinferni et al. 2004). Figure 5 shows the four fiducials and the one theoretical isochrone. The color at a given M and [Fe/H], c(M, [Fe/H]), can then be interpolated from these color–magnitude fiducials.

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It is important to note that most of the halo K giants are α enhanced, except for a few giants close to solar metallicity (Morrison et al. in prep.). For less metal-poor K giants, the effect of [α/Fe] on luminosity is stronger (for instance, the difference of r-band absolute magnitude can be as large as 0.5 mag at the tip of the giant branch for an α-enhanced giant compared to one with solar-scaled α-element abundance but the same [Fe/H] value). The cluster fiducials and one isochrone we use for distance estimates when [Fe/H] < 0 are α enhanced, while above that metallicity, we assume gradual weakening of the α abundance, naturally introduced by the NGC 6791 (solar-scaled α abundances) fiducial line in the interpolation. In other word, when a giant's [Fe/H] is between solar and the NGC 6791 value, its α abundance is also assumed to be in between.

Given the sparse sampling of the M–(g − r)₀ space by the four isochrones, we need to construct interpolated fiducials. We do this by quadratic interpolation of c(M, [Fe/H]), based on the three nearest fiducial points in color, and construct a dense color table for given M and [Fe/H], which will be used for Equation (4). Extrapolation beyond the metal-poor and metal-rich boundaries and the tip of the RGB would be poorly constrained. Therefore, we use these limiting fiducials instead for the rare cases of stars with [Fe/H] < −2.38 or [Fe/H] > +0.39. Table 2 in the printed journal shows an interpolated fiducial with [Fe/H] = −1.18; the entire catalog of 20 interpolated fiducials with metallicity ranging from [Fe/H] = −2.38 to [Fe/H] = +0.39 is available in the online journal. While there is more than one way to interpolate the colors, such as quadratic or piecewise linear, we have checked and found that different interpolation schemes lead to an uncertainty less than ∼ ± 0.02 mag in color, due to the sparsity of the fiducials, which could be an additional source of error in $\mathcal {DM}$ estimates.

In addition, we have chosen not to assign distances to stars that lie on the giant branch below the level of the horizontal branch (HB). This is because the red HB or red clump (RC) giants in a cluster have the same color as these stars, but quite different absolute magnitudes, and the SSPP log g estimate is not sufficiently accurate to discriminate between the two options. We derive a relation between [Fe/H] and the (g − r)₀ color of the giant branch at the level of the HB, using eight clusters with ugriz photometry from An et al. (2008), with cluster data given in Table 3. The [Fe/H] and $(g-r)_0^{{\rm HB}}$ for the HB/RC of the clusters follow a quadratic polynomial, $(g-r)^{{\rm HB}}_0=0.087{\rm [Fe/H]}^2+0.39{\rm [Fe/H]}+0.96$, as shown in Figure 2. We then use this polynomial and its [Fe/H] estimate to work out, for each star, whether it is on the giant branch above the level of the HB. It turns out that more than half of the candidate K giants fall into the region of RGB–HB ambiguity.

Table 3. Metallicity and Color of the Red Horizontal Branch Onset for the Eight Clusters in An et al. (2008)

Name of Clusters	[Fe/H]	$(g-r)_0^{{\rm HB}}$
NGC 6791	+0.39	1.13
M71	−0.81	0.69
M5	−1.26	0.61
M3	−1.50	0.59
M13	−1.60	0.58
M53	−1.99	0.54
M92	−2.38	0.53
M15	−2.42	0.53

Note. The first column lists the names of the clusters; the next two columns provide the [Fe/H] of the clusters and the extinction corrected color (g − r)₀.

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To incorporate the errors of metallicities and colors, we envisage each star as a two-dimensional (2D) (error-) Gaussian in the color–metallicity plane, centered on its most likely value and the 2D Gaussian having widths of color errors and metallicity errors, respectively. Then, we can calculate the "chance of being clearly RGB" as the fraction of the 2D integral over the 2D error-Gaussian that is to the right of the line.

Ultimately, we are left with 6036 stars with more than a 45% chance of being clearly on the RGB, above the level of the HB. Of these, 5962 stars have more than a 50% chance of being RGB, 5030 stars have more than a 68% chance of being RGB, and 3638 stars have more than a 90% chance of being RGB. In addition, 216 satisfy the target criteria for red K giants, 506 the criteria for proper-motion K giants, and 4246 the l color K-giant criteria. Another 1068 were serendipitous identifications—stars targeted in other categories which nevertheless were giants. Figure 2 shows the distribution of the apparent magnitudes, r₀, and metallicities, [Fe/H], along with the color, (g − r)₀.

Besides the contamination from HB/RC stars, we need to consider possible contamination of our sample by asymptotic giant branch (AGB) stars because it is not possible for us to distinguish RGB from AGB stars with our spectra. While the difference in absolute magnitude can be large (reaching ∼0.8 mag, implying a 40% distance underestimate at the blue end of our giant color range), the proportion of our giants that are on the AGB is relatively small. We used both the luminosity function of Sandquist & Bolte (2004) for the globular cluster M5 and evolutionary tracks from Basti isochrones for old populations of metallicity [Fe/H] = −2.6 and [Fe/H] = −1.0 to estimate the percentage of stars that are on the AGB. We find that for the most metal-poor stars, around 10% will be AGB stars, while for stars with [Fe/H] close to [Fe/H] = −1.0 the fraction is near 20%. For less metal-poor stars with [Fe/H] > −1.0, the expected fraction of AGB stars becomes larger than 20%, but in the SEGUE K-giant sample, less than 10% of stars have [Fe/H] > −1.0.

For any given star, the observables are its apparent magnitude and associated error, (m_i, Δm_i), its color and error (c_i, Δc_i), and its metallicity and error ([Fe/H]_i, Δ[Fe/H]_i). The ${\rm \mathcal {DM}}$ and the data are linked through the absolute magnitude M via: $m_i=M+\mathcal {DM}_i$ and the fiducial c(M, FeH), which we presume to be a relation of negligible scatter. Now we can incorporate the errors of the data and the specific priors on the stellar luminosity and metallicity distribution when calculating ${\rm \mathscr{L}\, (\mathcal {DM})}$ (see Equation (3)).

In practice, the errors on color, apparent magnitude, and metallicity can be approximated as Gaussian functions, in which case $p (\lbrace m,c,{\rm [Fe/H]}\rbrace \mid \mathcal {DM},M,{\rm FeH})$ (see Equation (3)) is modeled as a product of Gaussian distributions with mean and Delta (c_i, Δc_i), (m_i, Δm_i), and ([Fe/H]_i, Δ[Fe/H]_i):

$$\begin{eqnarray} && p (\lbrace m,c,{\rm [Fe/H]}\rbrace _i\mid \mathcal {DM},M,{\rm FeH})=\frac{1}{\sqrt{2\pi }\Delta c_i} \nonumber\\ && \quad \times \exp \left (-\frac{(c(M,{\rm FeH})-c_i)^2}{2(\Delta c_i)^2}\right) \times \frac{1}{\sqrt{2\pi }\Delta m_i} \nonumber\\ && \quad \times \exp \left (-\frac{(\mathcal {DM}+M-m_i)^2}{2(\Delta m_i)^2}\right) \times \frac{1}{\sqrt{2\pi }\Delta {\rm [Fe/H]}_i} \nonumber\\ && \quad \times \exp \left (-\frac{({\rm FeH}-{\rm [Fe/H]}_i)^2}{2(\Delta {\rm [Fe/H]}_i)^2}\right). \end{eqnarray} \tag{ 4 }$$

For Equation (3), we use the priors p_prior(M), based on the luminosity function of the giant branch, p(L)∝L^−1.8 (Figure 3), and p_prior([Fe/H]), based on the metallicity distributions of the K-giant sub-categories (Figure 4).

For any K giant with {m_i, c_i, [Fe/H]_i}, we can then calculate ${\rm \mathscr{L}\, (\mathcal {DM})}$ by computing the integral of a bivariate function (Equation (3)) over dM and dFeH, using iterated Gaussian quadrature. As described in Section 3.2, the $p_{{\rm prior}}(\mathcal {DM})$ is taken as a constant for a halo density profile of ρ(r)∝r⁻³, so $P (\mathcal {DM}\mid \lbrace m,c,{\rm [Fe/H]}\rbrace )\propto \mathscr{L}\, (\mathcal {DM})$. Then, the best estimate of ${\rm \mathcal {DM}}$ is at the peak of $\mathscr{L}\, (\mathcal {DM})$, and its error is the central 68% interval of ${\rm \mathscr{L}\, (\mathcal {DM})}$ (i.e., $({\mathcal {DM}_{84\%}-\mathcal {DM}_{16\%}})/{2}$).

To speed up the determination of the integral in Equation (3), we look up c(M, [Fe/H]) in a pre-calculated and finely sampled color table, instead of an actual interpolation. This approach can provide a consistent c for given M and [Fe/H], if the pre-prepared color table is suitable. We use a color table, c(M, [Fe/H]), of size 6500 × 4140, with −3.5 < M < 3 and −3.58 < [Fe/H] < +0.56.

To test whether our approach leads to largely unbiased $\mathcal {DM}$ estimates, a simulated data set was generated in order to mimic the SEGUE K-giant sample. As mentioned in Section 2, there are four sub-categories of K giants, and they have different distributions of [Fe/H], so the simulated stars were generated independently to mimic each sub-category. First, we produced a set of randomly generated values of distance, luminosity, and [Fe/H], following a halo stellar density profile of ρ(r)∝r⁻³, the luminosity function p(L)∝L^−1.8, and the metallicity distribution of each category of K giants, to cover similar ranges of $(\mathcal {DM}, M, {\rm [Fe/H]})$ as our sample of K giants. Then the apparent magnitudes and colors (m, c) were calculated from $(\mathcal {DM}, M, {\rm [Fe/H]})$ and fiducials. Gaussian errors were added to directly observable quantities (m, c, [Fe/H]), with variances taken from observed SEGUE K giants with similar (m, c, [Fe/H]). Finally, a simulated star was accepted only if its magnitude and color fall within the selection criteria of the pertinent K-giant sub-category.

A total of 6036 simulated stars were generated according to the above procedure. The distributions in r magnitude and in color are displayed in Figure 6, along with those of SEGUE K giants, showing that the simulated sample is a reasonable match, except for some apparent incompleteness at the faint end in the "real" data.²³ This sample was then analyzed using the same approach for estimating the ${\rm \mathcal {DM}}$ that was applied to actually observed SEGUE K giants, using the known-to-be-correct priors. When considering the difference between the calculated distance modulus of each star and its true value, divided by the distance modulus uncertainty, $(\rm \mathcal {DM}_{{\rm cal.}}-\mathcal {DM}_{{\rm true}})/\rm \sigma _{\mathcal {DM}_{{\rm cal.}}}$, we should then expect a Gaussian of mean zero and a variance of unity. Indeed, we find a mean of −0.09 and a variance of 0.94 for the case of using luminosity and metallicity priors, but a mean of −0.14 and a variance of 0.95 for the case of neglecting the metallicity prior, and a mean of 0.17 and a variance of 0.95 for the case of neglecting the luminosity prior. Note that these are in units of $\sigma _{\mathcal {DM}_{{\rm cal.}}}$, which is typically 0.35 mag; therefore, any systematic biases in distance will be of order 1%. Using both priors should lead to unbiased distance estimates.

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For the actual SEGUE data, the priors are not known perfectly, as we do not know the overall density profile of the halo, particularly at large distances (Deason et al. 2011; Sesar et al. 2011). Previous work indicated that the halo radial stellar density follows a power law ρ(r)∝r^α, with reasonable values of −2 > α > −4 (Vivas & Zinn 2006; Bell et al. 2008). To test the influence of assuming a different ρ(r), we made two sets of simulated data following ρ(r)∝r⁻² or ρ(r)∝r⁻⁴, according to the above procedure, and then applied the Bayesian approach to estimate $\mathcal {DM}_{{\rm cal.}}$ by using a halo stellar density profile of ρ(r)∝r⁻³, the luminosity function of p(L)∝L^−1.8, and the metallicity distribution of each category as priors. Considering the distribution of $(\rm \mathcal {DM}_{{\rm cal.}}-\mathcal {DM}_{{\rm true}})/\rm \sigma _{\mathcal {DM}_{{\rm cal.}}}$, we found a mean of −0.14 and a dispersion of 0.95 for the ρ(r)∝r⁻² case and a mean of −0.08 and a dispersion of 0.93 for the ρ(r)∝r⁻⁴ case, very similar to the ρ(r)∝r⁻³ case, implying that the exact form of the prior for the halo density profile does not affect our results significantly. This is consistent with Burnett & Binney (2010), who also concluded that approximate priors in the analysis of a real sample will yield reliable results.

However, neglecting the luminosity and metallicity priors, as has often been done in previous work (Ratnatunga & Bahcall 1985; Norris et al. 1985; Beers et al. 2000), would lead to a mean systematic distance modulus bias of up to 0.1 mag compared to accounting for both priors shown as Figure 7, which is also illustrated by the K-giant sample in Figure 11 and discussed in Section 4.2. Therefore, only an approach with explicit priors will lead to unbiased distance estimates.

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The most immediate results of the analysis in Section 3.4 are estimates of the distance moduli and their uncertainties from $P (\mathcal {DM}\mid \lbrace m,c,{\rm [Fe/H]}\rbrace )$ (Figure 8). At the same time, we obtain estimates for the intrinsic luminosities by $M_{r}={r_0}-\mathcal {DM}_{{\rm peak}}$, distances from the Sun, and Galactocentric distances by assuming R_☉ = 8.0 kpc. This results in the main entries in our public catalog for 6036 K giants: the best estimates of the distance moduli and their uncertainties ($\mathcal {DM}_{{\rm peak}}$, $\Delta \mathcal {DM}$), heliocentric distances and their errors (d, Δd), Galactocentric distances and their errors (r_GC, Δr_GC), the absolute magnitudes along with the errors (M_r, ΔM_r), and other parameters. In addition, we describe the $P (\mathcal {DM}\mid \lbrace m,c,{\rm [Fe/H]}\rbrace )$ by a set of percentages, which are also included in Table 4; the complete version of this table is available in the online journal.

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Table 4. List of 6036 K Giants Selected from SDSS DR9

R.A. (J2000)	Decl. (J2000)	r0	Δr0	(g − r)0	Δ(g − r)0	RV	ΔRV	Teff	[Fe/H]	Δ[Fe/H]	log g	${\rm \mathcal {DM}}_{{\rm peak}}$
(deg)	(deg)	(mag)	(mag)	(mag)	(mag)	(km s−1)	(km s−1)	(K)				(mag)
154.7659	−0.8354	17.257	0.040	0.965	0.028	43.7	2.5	4626	−0.71	0.16	3.28	18.13
174.6570	−0.9330	17.036	0.041	0.590	0.035	147.0	3.6	5259	−1.50	0.15	2.59	16.37
189.9634	1.0202	17.078	0.040	0.558	0.028	122.7	6.1	5296	−1.89	0.19	2.99	16.41
196.0723	−0.5404	16.891	0.041	0.526	0.030	−169.5	4.2	5158	−2.12	0.15	1.75	16.03
205.9106	−0.3442	18.067	0.041	0.631	0.031	86.5	4.5	4901	−1.23	0.21	1.98	17.68
${\rm \mathcal {DM}}_{5\%}$	${\rm \mathcal {DM}}_{16\%}$	${\rm \mathcal {DM}}_{50\%}$	${\rm \mathcal {DM}}_{84\%}$	${\rm \mathcal {DM}}_{95\%}$	$\Delta {\rm \mathcal {DM}}$	Mr	ΔMr	d	Δd	rGC	ΔrGC	PaboveHB
(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(kpc)	(kpc)	(kpc)	(kpc)
17.62	17.85	18.21	18.62	18.83	0.38	−0.88	0.39	42.35	7.89	45.51	7.78	1.00
15.43	15.81	16.33	16.79	17.07	0.49	0.67	0.49	18.78	4.11	20.55	3.79	0.67
15.56	15.90	16.39	16.86	17.14	0.48	0.66	0.48	19.17	4.18	19.27	3.82	0.75
15.00	15.40	15.96	16.46	16.76	0.53	0.86	0.53	16.05	3.79	15.64	3.31	0.51
17.00	17.29	17.68	18.07	18.32	0.39	0.39	0.39	34.38	6.24	31.74	6.07	0.66

Notes. The first two columns list the position (R.A., Decl.) for each object. The magnitudes, colors, and their errors are provided in the next four columns: the subscript 0 means extinction corrected and the errors are corrected for measurement errors on the photometry and reddening (see Section 2). The heliocentric radial velocities and their errors are listed in the next two columns. The next four columns contain the stellar atmospheric parameters and the errors in the metallicities as a relation of S/N. Effective temperatures and surface gravities are not used in our work, and they are all published in SDSS DR9, so we recommend interested readers download their errors from CasJob. The ${\rm \mathcal {DM}}$ at the peak and (5%, 16%, 50%, 84%, 95%) confidence of ${\rm \mathscr{L}\, (\mathcal {DM})}$ are listed in the next six columns. ${\rm \mathcal {DM}}_{{\rm peak}}$ is the best estimate of the distance modulus for the K giant. The $\Delta _{{\rm \mathcal {DM}}}$ is the uncertainty of the distance modulus, which is calculated from $({\rm \mathcal {DM}}_{84\%}-{\rm \mathcal {DM}}_{16\%})/2$. The last seven columns are absolute magnitude and distances calculated from ${\rm \mathcal {DM}}_{{\rm peak}}$, assuming R_☉ = 8.0 kpc (i.e., $M_{r}={r_0}-{\rm \mathcal {DM}}_{{\rm peak}}$, $d=10^{(({\mathcal {DM}+5})/{5})}$), and the chance of being clearly RGB.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Figure 9 illustrates the overall properties of the ensemble of distance estimates. The top two panels show the mean precisions of 16% in Δd/d and ±0.35 mag in ${\rm \mathcal {DM}}$; these panels also show that the fractional distances are less precise for more nearby stars because they tend to be stars on the lower part of the giant branch, which is steep in the CMD, particularly at low metallicities.

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The bottom panel of Figure 9 shows that the mean error in M_r is ±0.35 mag, and that faint giants have less precise intrinsic luminosity estimates. Figure 10 (upper panel) shows the distribution of K giants on the CMD. There are more stars in the lower part of the giant branch, which is consistent with the prediction of the luminosity function. The lower panel of Figure 10 shows that stars in the upper part of the RGB have more precise distances than those in the lower part of the CMD because the fiducials are much steeper near the sub-giant branch. This is equivalent to the fact that the fractional distance precision is higher for the largest distances (see Figure 9).

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The giants in our sample lie in the region of 5–125 kpc from the Galactic center. There are 1647 stars beyond 30 kpc, 283 stars beyond 50 kpc, and 43 stars beyond 80 kpc (see the 5 red giants beyond 50 kpc in Battaglia et al. 2005, 16 halo stars beyond 80 kpc in Deason et al. 2012, and no BHB stars beyond 80 kpc in Xue et al. 2008, 2011). Our sample comprises the largest sample of distant stellar halo stars with measured radial velocities and distances to date.

In this section we briefly analyze how important the priors actually were in deriving the distance estimates. For each star, the evaluation of Equation (3) using Equation (4) and the interpolated fiducials results in ${\rm \mathscr{L}\, (\mathcal {DM})}$ (Figure 8), i.e., the likelihood of the distance modulus, before folding in an explicit prior on ${\rm \mathcal {DM}}$, but after accounting for the priors on M and [Fe/H] (Equation (3)). In this section we present some example ${\rm \mathscr{L}\, (\mathcal {DM})}$, but first explore the systematic impact on ${\rm \mathcal {DM}}$ of neglecting the M and [Fe/H] priors.

When estimating the distance to a given star, without the benefit of external prior information, one would evaluate Equation (3) presuming that p_prior(M) and p_prior([Fe/H]) are constant.

To test the impact of p_prior(M), we estimate the distances for two cases: (1) no priors applied (p_prior(M) = 1 and p_prior([Fe/H]) = 1) and (2) only the prior on the luminosity function applied (p(L)∝L^−1.8 and p_prior([Fe/H]) = 1). The distance modulus estimate that neglects the explicit priors is denoted as DM₀, while the distance modulus with only p_prior(M) applied is marked as DM_L. The top of the left panel of Figure 11 illustrates the importance of including the "luminosity prior," by showing the systematic difference in ${\rm \mathcal {DM}}$ that results from neglecting it. For stars near the tip of the giant branch, the bias is very small, but for stars near the bottom of the giant branch, the mean systematic bias of neglecting the luminosity prior information is 0.1 mag with systematic bias as high as ∼0.25 mag in some cases.

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To test the impact of p_prior([Fe/H]), we estimate the distances where only the metallicity prior was applied, and mark the relevant ${\rm \mathcal {DM}}$ as DM_[Fe/H]. Compared with the distance modulus with no priors applied, DM₀, we find p_prior([Fe/H]) can correct a mean overestimate of 0.03 mag on the ${\rm \mathcal {DM}}$ for the metal-poor stars and a mean underestimate of 0.05 mag on the ${\rm \mathcal {DM}}$ for the metal-rich ones, but the neglect of the metallicity prior causes a smaller bias in ${\rm \mathcal {DM}}$ than neglecting the luminosity prior (0.05 mag versus 0.1 mag at mean), as shown in Figure 11 (middle of left panel). The distance modulus bias caused by neglecting both priors is presented in the bottom panel of Figure 11. Neglecting both priors causes a mean bias of 0.1 mag and a maximum bias of 0.3 mag in distance modulus.

We use five clusters (M13, M71, M92, NGC 6791, and Berkeley29) to test the precision of the distance estimates because they have spectroscopic members observed in SEGUE. Berkeley29 (Be29) is a comparatively young open cluster with an age of 3–4 Gyr (Sestito et al. 2008), younger than our adopted fiducials (10–12 Gyr). This illustrates how distances could be in error as a result of an incorrect age prior. M71 is a disk globular cluster. Because of its low Galactic latitude (less than 5° from the disk plane) and relatively circular orbit, separation of genuine M71 members from field stars is much more difficult than for halo clusters (M13 and M92). The analysis concerning the membership of stars in the stellar clusters will be reported in detail in the Appendix of H. L. Morrison et al. (in preparation).

In general, we identify cluster membership using proper motion and radial velocity. The proper motions provide a membership probability for each star (Cudworth 1976, 1985; Cudworth & Monet 1979; Rees 1992), and then the radial velocities are used for further membership checks, as described in detail in H. L. Morrison et al. (in preparation).

Based on the CMD of the clusters, we select giant members with (g − r)₀ > 0.5 and above the sub-giant branch of the clusters to test the distance precision. The range of S/N for the spectra of the cluster giant stars is [10, 70], and the S/N range for K-giant spectra is [10, 120]. Here we do not apply the very stringent criterion to eliminate HB/RC stars described in Section 3.4, because this criterion also culls many lower RGB stars that are useful for the test. Figure 12 shows there are some HB/RC stars or AGB stars that can help verify how distances would be in error for the non-RGB stars. Furthermore, we do not use the members with |[Fe/H]_member − [Fe/H]_GC| > 0.23 dex because of the strong sensitivity of the distances to metallicity errors.

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We estimate the ${\rm \mathcal {DM}}$ for each selected member RGB star, adopting the luminosity prior of p(L)∝L^−1.8, and a Dirac delta function centered at the literature cluster metallicity, [Fe/H]_GC, as the metallicity prior. Figure 13 shows the difference between our individual ${\rm \mathcal {DM}}$ estimate for each selected member RGB star and the literature ${\rm \mathcal {DM}}_{\rm GC}$ (shown in Table 1) for M13, M71, M92, and NGC 6791, respectively. Since M71 has significant differential reddening and fewer members, it is not a suitable cluster with which to verify the distance errors to little-reddened, usually more metal-poor halo giants. However, all four M71 members exhibit less than 0.2 mag scatter from ${\rm \mathcal {DM}}_{\rm GC}$, as shown in Figure 13. The RGB members show consistent distances with the literature value derived by main-sequence fitting within 1σ, but the distance moduli are underestimated by a maximum 1.24 mag for the non-RGB stars. Fortunately, the criterion to eliminate HB/RC stars adopted in Section 3.4 is sufficiently stringent to cull all HB/RC stars and many lower-RGB stars. As mentioned previously, there is a relative paucity of AGB stars in the SEGUE K-giant sample.

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The mean values of $({\rm \mathcal {DM}}-{\rm \mathcal {DM}}_{\rm GC})$ for the RGB members at different color ranges are within ±0.1 mag. Compared to the typical error of 0.35 mag in ${\rm \mathcal {DM}}$, our estimates of ${\rm \mathcal {DM}}$ are reasonably precise.

Figure 14 shows the distributions of Be29 members around their fiducials. Be29 members are far from the interpolated fiducial based on our old fiducials. The old fiducials lead to totally wrong distance estimates for relatively young giants in Be29, as shown in Figure 15. If the age prior is wrong, the distance estimates are unreliable. The derived errors on the distance moduli of the K giants are only valid if the ages of the K giants are older than 10 Gyr.

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In addition, we calculate the distances with flat priors (which means no priors), and find that neglecting the priors would lead to biases in distance of (6%, 6%, 3%, 0.7%) from the literature values for NGC 6791, M71, M13, and M92. However, we only find biases in distance of (0.7%, 2%, 2%, 0.4%) the literature values for NGC 6791, M71, M13, and M92 when using both priors. Therefore, neglecting the priors would lead to biased distance estimates.