DOES SEGUE/SDSS INDICATE A DUAL GALACTIC HALO?

The basic idea of the C07, C10, and B12 distance calibrations is the strategy of Beers et al. (2000) to classify stars by their gravities and then sort them into different branches (luminosity classes). B12 still assign stars with log g > 4.0 into the main sequence branch and stars with log g < 3.5 into a subgiant branch. The intermediate objects with 3.5 < log g < 4.0 are considered to be "turn-off" stars placed on a sequence half way in magnitude between the dwarf and subgiant branches. For the clearly unphysical objects redward of the oldest possible turn-off point, B12 modified their approach by sorting stars with 3.5 < log g < 3.75 to the subgiant branch and objects with 3.75 < log g < 4.0 into the dwarf sequence.

This re-sorting, which affects nearly all metal-poor turnoff stars, does not resolve the general problem of assigning a wrong luminosity class to a fraction of the objects. In the unphysical position that is intermediate between the true relations, they all had an error of ∼1 mag. Now they are in permitted locations, but their intermediate surface gravity determinations imply errors of ∼2 mag (in both directions) for an unknown number of stars that are misidentified as dwarfs or subgiants. We know that there is a substantial number of misassignments both from the calibration on globular clusters (i.e., errors in derived log g in Lee et al. 2008b, 2008a) and from the distance statistics of Schönrich et al. (2012).

The absolute magnitudes on the three branches are taken from the calibrations of Beers et al. (2000), which are, however, formulated in Johnson bands B and V. Hence, they transform the Sloan Digital Sky Survey (SDSS) colors into Johnson colors via the linear relationships derived by Zhao & Newberg (2006). These neglect any non-linear terms like those found by Ivezić et al. (2007) and also any metallicity dependence, which are difficult to detect on the Zhao & Newberg (2006) sample of only 58 stars.

Beers et al. (2000) have only a coarse grid in metallicity for their color–magnitude relations, which comprises just three different metallicity values. How they perform interpolation/extrapolation between these three points in metallicity is not described by Beers et al. (2000) nor by B12, but from a comment in Ivezić et al. (2008), we conclude that the most likely scheme is cubic splines with the natural boundary condition.

As we cannot access the B12 distances, we cannot directly test our replica. However, we can compare to their older data, which are shown with red crosses in Figure 1. For this plot, we selected metal-poor objects with −2.4 < [Fe/H] < −2.1. All colors are transformed via the Zhao & Newberg (2006) formula into the Johnson–Cousins system. The absolute V-band magnitudes are taken from the old C10 data. Also shown in Figure 1 are BASTI isochrones (Pietrinferni et al. 2004, 2006) for ages of 1, 5, 10, and 13 Gyr and [Fe/H] = −2.27. Comparison with the isochrones shows again the unphysical "turn-off" objects from C07 and C10 as well as the surprising fourth sequence (labeled "unknown" in Figure 1) between dwarfs and turn-off stars. We cannot judge if the latter is addressed in their new calibration.

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Comparison to the published plots in B12 and the C10 data shows that the derived sequences (in particular the main sequence) are in the right place. However, the replicated data show a remaining stump of the turn-off sequence, which appears absent in Figure 2 from B12. B12 define a critical temperature T_crit = T_TO − 250 K, 250 K colder than the temperature of the isochrone turn-off, below which they re-sort stars on the turn-off branch into the two other sequences (see Equation (1) and Figure 1 in B12). The only possible solution to the discrepancy appears to be adding the 250 K instead of subtracting them. This gives the blue points in the bottom panel of Figure 1. We will adopt this solution throughout the paper.⁵

Apart from the well-agreeing sequence locations, both the old C10 sample (red points in Figure 1 and Figure 2 of B12 show significantly more magnitude scatter than our derivation. We cannot explain this scatter by metallicity differences, which should be small at such low metallicities. This is not a systematic shift but likely enhances the Lutz–Kelker bias in their sample.

Reproducing the B12/C10 absolute magnitudes finally reveals the origin of the difference between C10 and the isochrones in SDSS colors: the Beers et al. (2000, hereafter B00) calibrations are brighter than the stellar models in Johnson colors: both the red points and the blue main sequence lie clearly above the isochrones in Figure 1. Which one is correct? Though B12 state that the Beers et al. (2000) distance calibration is based on Hipparcos parallaxes, this is only true for metal-rich stars. For metal-poor stars, there are hardly any reliable Hipparcos parallaxes and none for globular clusters. Yet, there would be indirect calibrations for metal-poor clusters by using the few Hipparcos parallaxes for very metal-poor stars. This was done, e.g., by Pont et al. (1998), who derived a distance modulus of μ₀ = 14.61 ± 0.08 for M92, compared to a μ₀ ∼ 14.57 derived via isochrone fitting from Harris et al. (1997). Instead of using these Hipparcos-derived parallaxes for their metal poor clusters, Beers et al. (2000) relied on a calibration of horizontal branch star magnitudes. Since they did not report their derived distance moduli, we cannot determine why their magnitude assignments deviate so far from well-calibrated models.

The Zhao & Newberg (2006) transformation from Johnson to SDSS is significantly bluer (by about 0.1 mag at low metallicities) than the predictions from model atmospheres entering the BASTI (Pietrinferni et al. 2004, 2006) models. This follows from its missing metallicity dependence, which forces a compromise between the different line-blanketing of metal-poor and metal-rich stars. The blue bias of the transformation for metal-poor stars helps to reduce the difference between the isochrones and the B00/B12 main sequence calibration in the SDSS system to about 0.3 mag, smaller than in the Johnson–Cousins system to which the B00/B12 calibration is native. Still we can see from Figure 2 that in contrast to the B00/B12 calibration the isochrones match the photometry for the cluster M92 very well on the main sequence with adopted standard reddening of 0.023 and a distance modulus of μ₀ = 14.6, in concordance with the Hipparcos-based calibration of M92.

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In summary, the bright bias of the B12 main sequence calibration in SDSS colors traces back to the calibrations of Beers et al. (2000) being up to 0.5 mag brighter than the isochrones in Johnson colors, mitigated to some extent by the transformations between the color systems.

As laid out in the Introduction, radial abundance trends are not a signature of halo duality and can be either caused by the specific accretion history (Cooper et al. 2010) or by kick-up of disk stars. As the Sun is at a significant distance from the Galactic center, such a radial gradient should in principle be reflected as well by a local gradient in altitude z.

B12 claim that the metallicity of halo stars decreases with altitude z (see their Figure 13 and corresponding discussion) down to a peak metallicity of [Fe/H] ∼ −2.2 at an altitude of z > 9 kpc. They identify this as a clear transition from an inner to an outer halo in their picture of a halo duality. Bias control in such an investigation is key: not only do we expect altitude-dependent disk contamination, but this sample exemplifies metallicity-dependent luminosity biases and selection effects. Here we will follow the standard route of statistical analysis. We derive bias controlled statistics in the next subsection and in the following parts provide the causal reasoning/backgrounds by analyzing the exact origin of the bias affecting the B12 findings.

3.1.1. Simple Statistical Analysis

In the upper half of Table 1, we see that under a linear fit we recover a strong decrease of metallicity with altitude in agreement with B12. The linear fit may not reflect a perfect assumption, but it gives a direct measure of the correlation between altitude z and metallicities. However, most of this decline is caused by the transition from disk to halo toward higher altitudes. To isolate the Galactic halo, we limit our sample to [Fe/H] < −1.4, where the disk contamination at lower altitudes should vanish in the present sample as evident in Figure 6 of Schönrich (2012). We can see this also from the red line in Figure 3—the curve has a kink at [Fe/H] ≈ −1 where the bulk of disk contamination ends. The weakening trend toward even lower metallicities is caused less by disk contamination than the subsequent narrowing of the available metallicity range. At those low metallicities, there is still a negative slope with a striking significance of 20σ. Is it real? We will show that this is not the case.

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Table 1. Linear Fits to the Declining Metallicity with Altitude both for the Entire Sample and Restricted to Likely Halo Objects with [Fe/H] < −1.4

Linear Fit Parameters
Parameter	All	σall	[Fe/H] < −1.4	σ[Fe/H] < −1.4
d[Fe/H]/dz	−0.1350	0.0019	−0.0338	0.0015
[Fe/H](z = 0)	−0.9389	0.0054	−1.7120	0.0054
d[Fe/H]/dz	−0.1657	0.0051	−0.0039	0.0048
d[Fe/H]/ds	0.0290	0.0045	−0.0272	0.0041
[Fe/H](z = 0)	−0.9555	0.0060	−1.7002	0.0057

Notes. In the upper half, we show results for the simple fit, while below we control for a distance dependence according to Equation (2).

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In statistics, a well-known phenomenon is the omitted variable bias. Let there be a causal connection between a quantity X and a quantity Y. However, we measure the dependence of X on another variable Z, which is not causally connected to X but is correlated with Y. In this case, we will infer an artificial correlation between Z and X after neglecting Y. The standard solution to this problem is controlling for the omitted variable by including it into a simultaneous fit of X against Y and Z. An omitted variable bias is demonstrated if the trend gets significantly altered or even vanishes after inclusion of the additional variable. Note, however, that the reverse case is not true: finding a trend with inclusion of one omitted variable does not prove its reliability, since there might be missing variables, or the wrong functional form is used in the fit.

In our case, the three quantities are distance s, altitude |z|, and metallicity [Fe/H]. In any pencil-beam study centered on the Sun, the average altitude |z| will by the geometry rise almost linearly with distance s, and hence, the two quantities are strongly correlated. The strong poleward bias (i.e., most stars are at high Galactic latitudes b) of the SEGUE sample exacerbates this correlation. At the same time, [Fe/H] is correlated with distance: stars with lower metallicity are brighter than more metal-rich stars of similar mass and age, so that metal-poor stars can be seen further away. In a general sample selection, this competes with the color shift: metal-rich stars are redder, so that at fixed color metal-rich main sequence stars are brighter than their metal-poor counterparts. For the subgiant branch, we only have the first effect, since metal-poor subgiant stars are in general brighter than metal-rich ones. In addition, the gravities of stars, as determined by the SEGUE pipeline, are significantly lower on the metal-poor end (for [Fe/H] < −1 dex the "main sequence" density ridge slopes by ∼0.4 dex in gravity per  dex in metallicity), and the observational error is larger (see the section below), making them more prone to be selected for the turn-off or subgiant branches in the B12 scheme with its fixed gravity cuts.

Thus, in order to claim a trend of metallicities with altitude, one must prove that this trend does not come from a selection effect in distance s. A simple solution is expanding the fit equation:

$$\begin{equation} {[{\rm Fe}/{\rm H}]}_i = (d{[{\rm Fe}/{\rm H}]}/dz)\cdot z_i + \epsilon _i \end{equation} \tag{ 1 }$$

to

$$\begin{equation} {[{\rm Fe}/{\rm H}]}_i = (d{[{\rm Fe}/{\rm H}]}/dz) \cdot z_i + (d{[{\rm Fe}/{\rm H}]}/ds) \cdot s_i + \epsilon _i, \end{equation} \tag{ 2 }$$

where i is the index running over the stellar sample, (d[Fe/H]/dz) and (d[Fe/H]/ds) are the free fit parameters, measuring the correlation between metallicity and altitude z as well as distance s, respectively, and _i is the individual deviations that are to be minimized.

The statistics for this test are shown in the lower half of Table 1 and are visualized in Figure 3. Along the x-axis, we vary an upper metallicity limit, selecting only stars with [Fe/H] < [Fe/H]_lim. Indeed there is a strong trend of metallicity with distance. Simultaneously, the physical trend of metallicity with altitude when including disk stars is enhanced (blue error bars), while the altitude trend d[Fe/H]/dz for halo stars vanishes.

The bias can also directly be seen in Figure 4, where we bin the sample in altitude z and in-plane distance (x² + y²)^0.5. To account for a possible radial metallicity gradient, stars with R < R_{G, ☉}, i.e., inside the solar galactocentric radius, are given a negative sign. As apparent from the upper panel of Figure 4, the stars with lower gravities dominate the longer distance range as they are sorted into the turn-off or subgiant sequences (with limits log g = 4.0, 3.75, and 3.5). Since the DR8 pipeline has a strong correlation between metallicity and gravity (see Figure 3 in SAC11), it comes as no surprise that we measured a strong distance bias in metallicity. This is evident in the lower panel of Figure 4, where we plot the mean metallicity of stars with pure halo metallicities, i.e., [Fe/H] < −1.4. Notice that instead of a vertical trend, the lower metallicity bins populate concentric rings around the Sun, associated with the boundaries of turn-off and subgiant star domination in the sample.

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3.1.2. Detailed Distance Dependence

For a thorough analysis of the found distance bias, we plot the profile of the mean metallicity versus distance in the DR9 sample with green error bars in Figure 5. The metallicity profile is highly unphysical a priori and is proof by itself for the distance bias. It is, however, useful to nail down the origin of this odd structure.

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The local minimum just outside 4 kpc that we saw in Figure 4 divides two relatively metal-rich peaks, and beyond 8 kpc, the metallicity drops to a value of [Fe/H] ∼ −2.2. To trace the origin of this bias, we use a mock sample: we assume a halo with a simple 1/R³ density profile and a spatially constant metallicity distribution to populate the color–magnitude space with the 13 Gyr BASTI isochrones⁶ out to 40 kpc. We also assume a constant reddening of 0.15 mag in the g-band. This is not meant to be a fully realistic representation of the Galactic halo and the survey, but it suffices for demonstration purposes.

On this sample, we mimic the SEGUE observations. To match their color distribution, we select only stars with 0.12 < (g − r)₀ < 0.55. The blue bias of the sample is modeled by multiplying with exp (− ((g − r)₀ − 0.38)/0.045) redward of (g − r)₀ = 0.38. The magnitude distribution of SEGUE is dominated by the division into faint (g₀ ≳ 17) and bright (g₀ ≲ 17) observing plates. For our metal-poor subsample, the observed magnitude distribution ρ_g(g₀) is very well described by

$$\begin{eqnarray} && \rho _g(g_0) =\nonumber \\ &&\quad \left\lbrace {\begin{array}{@{}lll}64.2(g_0 - 14.4)^2 - 1) && {\rm for}\; 15.4 < g_0 \le 17 \\ 10.6(g_0 - 14.4)^2 - 1) && {\rm for}\; 17 < g_0 \le 18.5 \\ 0 && {\rm otherwise}. \end{array}} \right. \end{eqnarray} \tag{ 3 }$$

We bin the virtually observed sample in magnitude and weight each bin to achieve the observed magnitude distribution. The resulting metallicity profile is shown with a green line. This green line, which purely shows the selection bias of the sample, already captures the features of the observed data. Now we have to replicate the actual observations with their actual errors. We compare the DR9 results to gravity determinations from Allende Prieto et al. (2008), fitting a linear trend for the gravity difference depending on metallicity:

$$\begin{equation} \Delta _{\log {g}} = \delta _0 + \delta _{{[{\rm Fe}/{\rm H}]}} \cdot {[{\rm Fe}/{\rm H}]}, \end{equation} \tag{ 4 }$$

where Δ_log g = log g_SSPP − log g_AP. The resulting fit parameters are δ_[Fe/H] = 0.10 ± −0.05 and δ₀ = (− 0.24 ± −0.07) dex, i.e., the SEGUE DR9 results have −0.25 dex lower gravity at solar metallicity than Allende Prieto et al. (2008), and this difference widens by 0.1 dex in gravity per  dex lower metallicity. Then we "correct" the SEGUE DR9 data by this metallicity-dependent offset and measure the trend in the absolute values of the residuals. This gives a slope of −0.16 ± 0.03 dex in gravity per dex in metallicity, with a zero point offset of (0.14 ± 0.04) dex, i.e., the dispersion at solar metallicity is about 0.16 dex, widening to about 0.3 dex at [Fe/H] ∼ −1.

While selection effects bar us from a direct assessment of the gravity errors depending on metallicity near the turn-off, the trend in mean metallicity for the main sequence can be clearly identified in Figure 6, where we plot the temperature–gravity plane of extremely metal-poor stars from the entire SEGUE sample as selected in Schönrich (2012) with red dots. Red squares highlight the stars ending up in the B12 sample selection. The hot bias of the latter is mostly caused by the blue bias of the photometric and reddening calibration star selection. In comparison to stellar model expectations (BASTI isochrones depicted with blue lines), the main sequence and subgiant branch gravities are too low by about half a dex, in good agreement with the findings from comparing to Allende Prieto et al. (2008). Though the main sequence is recognizable, a confident separation of stars into sequences with the criteria of B12 (black lines) is impossible. In the following, we will use δ_log g = −0.1 + 0.2[Fe/H] and σ_log g = 0.15 − 0.12[Fe/H] as a rough estimate. The resulting metallicity profile is plotted in Figure 5 with dark red.

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The general structure is the same both with perfect distances and with the replicated B12 sample. The underlying metallicity distribution is constant in space, but the measurement induces strong Malmquist-like biases. The red sample cut-off leads to a metallicity rise at the smallest distances: metal-poor stars on the main sequence are, at fixed color, less luminous than their metal-rich counterparts and can thus enter the sample at smaller distances. Just beyond 4 kpc the metallicity drops. At this point, the turn-off—which essentially gives the maximum luminosity of the bulk of stars—reaches the faint cut-off of the sample. Since metal-poor populations have a brighter turn-off and subgiant branch, they will dominate the far part of the sample. The 4 kpc drop is connected to the first magnitude limit of g₀ < 17 in the considered sample. Further out, the fainter sample selection dominates and gives rise to a second metallicity peak, followed by a drop beyond 8 kpc, where the sample only contains subgiants and giants. The effect is again exacerbated by the red cut-off, which further reduces the relative number of metal-rich stars in the sample. Compared to perfect distances (blue line), the B12 distances together with the pipeline errors (dark red line) enhance the metallicity drop around s = 8 kpc.

While the dark red line approximates the qualitative structure of the observations remarkably well, there are mild quantitative differences. In particular, the metallicity of the nearest stars is overestimated. This is certainly not a "real" trend, but it derives from our rough assumptions like a fixed reddening, or the rough shape of the metallicity distribution. Indeed the exact shape of the replica depends quite strongly on the assumed pipeline errors, age structure, initial mass function, and much more. Had we used "perfect" gravities and metallicities on our mock sample, the overall structure would be shifted massively inward as depicted by the light red line in Figure 5: the gravity cuts of the B12 distances are deep in the subgiant regime, and hence, on clean gravity determinations, they would lead to a prevalence of distance underestimates. Any further refinement on the mock catalogue would hence primarily serve to understand the SEGUE errors and selection rather than advancing the discussion on halo structure.

Two important points should be taken away from this section.

• 1.  
  In a sample that is not volume complete, the metallicity distribution cannot be explored without extensive bias correction. This is apparent from the fact that the unbiased sample (green line) already showed significant metallicity effects. Even if we had perfect distances, we would suffer from both the magnitude and color selection and hence a spatially dependent bias would remain.
• 2.  
  Our understanding of the detailed distance dependence of metallicities in the sample complements the statistical analysis from the previous section. It confirms that the data are consistent with no change in mean metallicity throughout the extent of the sample out to an altitude of about z = 10–15 kpc. However, the statistical analysis provides a more robust value, since it filters the distance dependence without requiring knowledge of the detailed errors and biases in the sample. It shows that at [Fe/H] < −1.4 dex the data are consistent with a constant metallicity. Any vertical trend in these data is less than a third of the B12 claim, i.e., smaller than ∼0.01 dex kpc⁻¹.

3.1.3. BHB Star Metallicity Trend with Galactocentric Radius

B12 make a similar claim for BHB star metallicities, now not as a function of altitude but as a function of galactocentric distance. The effect is very small (less than −0.01 dex kpc⁻¹; see Figure 7), and it is nearly impossible to reliably disentangle it from a distance bias. A clean separation further out in the halo is not reliable, because distance from the Sun becomes degenerate with the distance from the Galactic center. Additionally, the signal to noise ratio will affect the observed width of the metallicity distribution and related parameters. Wilhelm et al. (1999) report not only a larger error in measured line-widths at lower signal-to-noise ratios, but also a significant systematic shift due to issues with continuum placement. Hence, it is reasonable to expect some metallicity shift with signal-to-noise ratio and hence distance. The Wilhelm et al. (1999) method is also vulnerable to small color errors as small as 0.03 mag, which are expected at large distances (high observed magnitude) and at low latitudes (reddening uncertainty), reporting a σ_[Fe/H] of up to 0.72 dex. Furthermore, distances to BHB stars depend significantly on their metallicity with metal-poor stars being about ∼0.3 mag brighter Pietrinferni et al. (2004); Fermani & Schönrich (2013a) in the g-band, such that the precautions of the above subsection fully apply.

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We can also test for another influence on the trend: the Galactic disk. B12 cut in galactocentric radius, while for their BHB sample they did not discuss and examine the dependence on the altitude above the Galactic plane. In particular, the lower latitudes of the sample will carry a disk contamination that evokes a spurious trend. In addition, it is not known how far contamination by the bulge extends into the Galactic halo, and the sample contains a significant number of stars at low galactocentric distances and relatively low altitudes (R ∼ 5 kpc, |z| ⩽ 6 kpc). Note also that the disk contamination can exceed the normal geometric limits of the disk because of a significant contamination with blue stragglers that have distance overestimates (Sirko et al. 2004). In our determination of the radial metallicity trend in the sample, we hence vary the lower limit of the altitude |z| as plotted in Figure 7. Indeed, for the Wilhelm et al. (1999) metallicities, we encounter a very mild negative trend of metallicity with galactocentric radius R for the entire sample. However, the trend diminishes successively while we move up the cut to censor regions of likely disk contamination. Further, the trend noted by B12 only exists when using the Wilhelm et al. (1999) metallicities. If we use instead the pipeline adopted metallicities from SEGUE DR8 (blue lines in Figure 7), the trend starts at a very small value and becomes already insignificant at a limiting altitude of z_lim ∼ 3 kpc. We note, however, that there is an increase in the metallicity spread noticed by an increase in the number of both objects with [Fe/H] < −2.5 and [Fe/H] > −1.5 at larger distances. The dispersion increase might be real, but we cannot cleanly disentangle it from a natural increase in the observational error.

In summary, we cannot detect any reliable downtrend of mean metallicity with galactocentric radius that is robust against a disk contamination.

The observed azimuthal velocity distribution is frequently used to decompose an observed data set into single components. SAC11 and Schönrich & Binney (2012) emphasized the fact that a typical disk velocity distribution is not symmetric and has a long tail toward lower velocities, providing new and physically motivated fitting functions to replace the unphysical Gaussian functions.

B12 tried to circumvent this problem by examining the velocity distribution of stars that they can by metallicity classify as halo stars and again attempt a Gaussian decomposition. However, this does not address the fundamental failure of decompositions in terms of Gaussian distributions to characterize inherently non-Gaussian distributions, which was one of our main concerns raised in SAC11.

• 1.  
  There is expectation that the halo azimuthal velocity distribution should be Gaussian. Dynamic friction may influence the shape, as well as the relaxation time for single halo stars is longer than a Hubble time, so a fully thermalized halo velocity distribution cannot be expected.
• 2.  
  The observational error distribution that has to be folded with the underlying velocity distributions to obtain the final observed velocity distribution is highly non-Gaussian. While the non-Gaussianity caused by using photometric parallaxes has been known at least since Strömberg (1927), it is commonly known as a Lutz & Kelker (1973) bias, who demonstrated its existence for geometric parallaxes. It is inevitable that the B12 sample is affected by this bias, and correction for it is essential, which however was not done by B12 even though it was one of the key concerns we had articulated in SAC11.

The non-Gaussian error on distances translates directly into a non-Gaussian deformation of the velocity distribution. The long tail of distance overestimates translates to a long tail of the velocity distribution away from the solar velocity, i.e., into the retrograde regime. This was already pointed out by Strömberg (1927) and more recently by Ryan (1992). This distortion is not only caused by distance overestimates: distance underestimates and misidentifications of evolved stars as dwarfs yield a distribution skewed in the same way, as they create a narrower shifted onto the prograde side. An attempt to fit such a left-skewed distribution at moderate data precision with Gaussians will usually produce an unphysical split into a strong component centered around the maximum and a weaker component out in the long tail.

In addition, proper motion errors themselves are highly non-Gaussian (though usually symmetric) due to a major magnitude dependence of astrometric precision (Munn et al. 2004; Dong et al. 2011). These proper motion errors multiply with the varying distance and angle terms, so that velocity errors generated by them are highly non-Gaussian.

A typical outcome of trying to fit data with such unphysical functional components is large scatter in parameters and unphysical parameters (see Johnson & Faunt 1992) that cannot be properly explained by just the moderate distance shifts between the different methods. This is exactly what can be observed in the results of C10 and B12. Figure 8 shows the mean azimuthal velocities of their low (i.e., more retrograde and identified as an outer halo) component on the x-axis against the mean azimuthal velocities of their high component (more prograde, their inner halo component) on the y-axis under their different selections and applied distance scales. The red error bars are their general dwarf samples, while the green error bars restrict those samples to stars that in their analysis reach altitudes in excess of 5 kpc. Unsurprisingly, the latter samples favor more retrograde components, because they have a bias toward stars that are further away and have a high proper motion impact on V_ϕ (and have hence lower accuracy in their kinematics), as well as stars with distance overestimates. Blue error bars depict the full sample with their new distance calibration. Clearly, the exact rotational velocities for the two claimed halo components differ wildly depending on the chosen distance prescriptions and how exactly the analysis of C07, C10, and B12 is carried out.

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To assess the halo velocity distribution with an appropriate error analysis, we make the crude assumption of an underlying Gaussian distribution and fold it with the following error terms.

• 1.  
  The proper motion errors, which can be treated as approximately Gaussian for single stars. We calculate the distribution of velocity errors resulting from the positions and proper motion errors of each single star and fold it onto the velocity distribution.
• 2.  
  A correction for distance errors. Those are highly non-Gaussian if we assume a Gaussian magnitude error. We use values of σ_V = 0.35 mag and σ_V = 0.5 mag, corresponding to a distance error of approximately 17% and 26%.
• 3.  
  A smaller correction for velocity crossovers via the distance uncertainty as discussed by Schönrich et al. (2012), which should be approximately symmetric in velocity space. We keep this at a conservative 7 km s⁻¹.

Figure 9 ⁷ shows the azimuthal velocity distributions of stars with metallicities of [Fe/H] < −1.9. For our fits, we use the mean velocity, the width, and the normalization of the underlying Gaussian term as free parameters. For both values of σ_V, we include also a case where we model the identification errors that will occur in the sample. For this case, we use both an underestimated and an overestimated fraction, each comprising 5% of the total sample and having an additional 1.2 mag offset. This is a rather conservative estimate, regarding the fact that a main-sequence-subgiant identification error amounts to ∼1–3 mag (cf. Figure 1). Similarly, the number of misidentified stars is chosen far on the low side. Most objects are near the turn-off and very hot and metal-poor. For these stars, gravity determinations are very difficult, due to the disappearance of suitable temperature and pressure broadened wings at the typical signal to noise ratio and spectral resolution for SEGUE. This renders the correct identification of luminosity classes very difficult. This is also shown by their distance statistics. The linear distance estimator from Schönrich et al. (2012) flags an average 35% distance overestimate⁸ at 7σ significance for the subgiants with [Fe/H] < −1.9, opposing a fair average distance for the declared dwarfs. For the latter, the misidentified subgiants balance the overestimated main sequence distances.

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We show the underlying velocity distribution fit for the latter model. The differences are subtle, but a higher σ_V expands the left wing while shifting the peak a little bit into the prograde regime; whereas the identification errors mostly show up at the broader wings. The comparison to the unfolded distribution (black) demonstrates how much more the left (retrograde) wing is enhanced by the symmetric magnitude errors. Neglecting the observational error asymmetry would drive at least one other (artificial) component in a Gaussian decomposition, as indeed we argue has happened in C07, C10, and B12.

The determinations for the underlying parameters of the velocity distribution are dominated by systematic errors: The dispersion values vary between 98 km s⁻¹ for σ_V = 0.35 mag without identification errors and 91 km s⁻¹ for σ_V = 0.5 mag accounting for identification errors. Since we likely underestimate the uncertainties, the real azimuthal velocity dispersion of the halo should be around 90 km s⁻¹ or markedly lower. The mean azimuthal velocity varies between 11 km s⁻¹ and 14 km s⁻¹, consistent with a non-rotating or at best weakly prograde halo. However, this is a gross underestimate of the systematic uncertainty. The value shifts by 18.5 km s⁻¹ when distances are shifted by 10%, in addition to the uncertainty in the solar azimuthal velocity of another 8 km s⁻¹. Note also that as a consequence of their stark distance overestimates the subgiants display a mean azimuthal velocity about 60 km s⁻¹ lower than the dwarf stars at the same metallicity. These problems can be partly remedied by better quality cuts and also by applying the distance corrections of Schönrich et al. (2012) and by using robust estimators. However, this has been already done by Schönrich (2012) for the local sample and Fermani & Schönrich (2013b) for BHB stars, who found a non-rotating or at best weakly prograde Galactic halo.

Within the expected magnitude and identification uncertainties, the wings of the distribution can be fully explained by just the observational errors, without having to deviate from the simplistic underlying Gaussian velocity distribution (which would not yet imply the presence of a second component). Hence, the azimuthal velocity distribution offers not the least hint of any halo duality.

B12 argue that their retrograde halo stars are real, because their distribution in the proper motion plane is different from the rest of the sample.

Specifically, B12 (p. 16) argued that "if the retrograde signature were indeed created in the manner suggested by S10 alone (and the stars we assign to a highly retrograde tail were otherwise identical in their kinematic properties to the rest of the low-metallicity halo stars), we would a priori expect their observed proper motions to be drawn from the same parent population as those not in the tail."

The strategy to refute this argument is to construct a mock sample and to show that distance errors produce many false retrograde objects, while these objects display a remarkable difference in proper motion space compared to the rest of the sample.

We note first that even with a single Gaussian-shaped halo velocity distribution centered at zero, we would expect 0.5 to 1% of all stars to have V_ϕ < −200 km s⁻¹. Hence, we do not claim that all strongly retrograde stars are spurious but that the majority of observations in this region is a consequence of observational errors. Further, it is easier to move a star by observational errors from −100 km s⁻¹ to −200 km s⁻¹ than from 50 km s⁻¹. Hence, the "a priori" claim of B12 that a Lutz–Kelker bias draws stars equally from the entire parent population is a priori wrong.

Distance errors actually tend to enhance the contrasts between the thought-to-be retrograde tail population and the parent population, especially in a sample like SEGUE, where the poleward bias makes proper motions the dominant part in the azimuthal velocity measurements. Distance errors in a measurement affect exclusively the part of motion relative to the Sun that is carried by the proper motions.⁹ Thus, the presence of distance errors is only able to sweep stars into spurious retrograde orbits when these stars have a considerable proper motion, i.e., the likelihood of a star being wrongly sorted into retrograde motion vanishes at the origin (0,0) in the proper motion plane and generally increases with proper motion. In turn, the density contrast between stars with high and low V_ϕ around the origin of the proper motion plane will commonly intensify in the presence of distance errors.

Let us examine this effect in a mock sample depicted in Figure 10. Following Schönrich et al. (2012), we use a sample of 25000 halo stars with a Gaussian distribution with zero mean velocity and dispersions (σ_U, σ_V, σ_W) = (150, 75, 75) km s⁻¹. If we plotted the entire sample, it would be fully centered around the origin of the proper motion plane. To make the test resemble more the SEGUE sample, we only select stars toward the Galactic anticenter (cos l < 0) and in the northern Galactic hemisphere (b > 0). Again this is not supposed to be a realistic Galaxy, nor a good match to SEGUE, which has a narrower and more ragged angle distribution.

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The result, assuming a perfect measurement without any errors, is shown with blue dots in the top panel of Figure 10. Red data points depict the strongly retrograde stars with V_ϕ < −200 km s⁻¹. Since there are only ∼25 such stars, black crosses highlight in addition the stars with V_ϕ < −150 km s⁻¹ to visualize the spatial distribution. In the middle panel, we apply typical errors for a sample, using 3 mas yr⁻¹ in both proper motion components, 5 km s⁻¹ in the line-of-sight velocities, combined with a systematic distance error of +15% plus a random distance error of 25%. As a consequence, the number of stars deemed to be retrograde multiplies by a factor of ∼10. For the reasons laid out above, the spurious members are at high proper motion, while virtually no false detections get added at lower proper motions, where the proper motions are just too small to change the velocities considerably (as a back-of-the-envelope calculation, consider that 10 mas yr⁻¹ at 2 kpc distance correspond to ∼100 km s⁻¹ in transverse velocity, which have to be multiplied with the geometrical factor (<1) and the fractional distance error).

To consider a really extreme case, we assign completely random distances with 0.7 kpc < s < 5 kpc to each star in the bottom panel. Though virtually all of our detections are wrong, the contrast between the sample distributions is actually enhanced (the relative density around the origin in the retrograde sample has measurably diminished) with similar asymmetric distribution in the proper motion plane as the case without any observational errors but obviously with many more spurious retrograde objects and with more pronounced asymmetry in azimuthal velocities. Such an asymmetry was argued by B12 to clearly demonstrate that the highly retrograde stars must be drawn from a different parent distribution than the stars with less pronounced rotation. The above illustration plainly shows that no such conclusion can be inferred from their proper motions. In particular, one can not directly interpret structure in the proper motion plane together with kinematic cuts.

The argument turns even in the opposite way: the prominent proper motion structure seen by B12 would, if anything, indicate that many of their highly retrograde objects are artificial.

To summarize, none of the arguments of B12 bear up under closer scrutiny. We cannot find even a hint of a halo duality in these statistics; instead, this discussion serves as a prime example for the importance of sound error analysis and the danger of hidden biases.