THE VERTICAL METALLICITY GRADIENT OF THE MILKY WAY DISK: TRANSITIONS IN [α/Fe] POPULATIONS

Our SEGUE sample has a number of metal-poor stars that may be interlopers from the halo system. These stars can artificially shift the median [Fe/H] to lower values, affecting our measured vertical metallicity gradient. Limiting our sample to stars with [Fe/H] > −1.0 avoids contamination from halo stars, but it also may remove evidence of a metal-weak thick disk from our sample. As these stars are a small portion of our sample, their removal does not significantly change the determined vertical metallicity gradient. Thus, we limit our sample to stars with [Fe/H] above −1.0.

For each star that passes SEGUE target selection and our spectral-quality criteria, we determine the distances by matching them in (g − r)₀ and [Fe/H] to 10 Gyr isochrones from the empirically corrected Yale Rotation Evolution Code set (An et al. 2009). This technique, and an in-depth analysis of associated uncertainties, are described in detail in Schlesinger et al. (2012). Briefly, there are random distance errors arising from uncertainties in photometry, SSPP estimates of [Fe/H] and [α/Fe], and isochrone choice. The total random distance error is dominated by uncertainties in SSPP [Fe/H] estimates and ranges from around 18% for stars with [Fe/H] >−0.5% to 8% for more metal-poor stars. There are also systematic distance uncertainties from using 10 Gyr isochrones and the possibility of undetected binarity; these two uncertainties have a negligible effect on the distance of metal-poor stars, while the most metal rich stars have a systematic shift in distance of −3%, which is factored into our distance estimates.

Figure 3 shows the spatial coverage of our entire G-dwarf sample in Galactocentric radius projected onto the plane, R, and height above the plane, Z. The Sun's position is assumed to be at (R, Z) of (8.0, 0.0) kpc (Bovy et al. 2009). Whereas previous analyses of the vertical disk gradient have been limited in coverage, the SEGUE sample covers stars both close to and far from the plane of the Galaxy, including stars from both the proposed thin- and thick-disk components. This makes the SEGUE sample ideal for constraining the disk's vertical structure and examining the interconnectivity of the two proposed disk components.

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Due to the survey design and target-selection criteria of SEGUE, our G-dwarf sample, which is defined by color and magnitude cuts, is biased toward metal-poor stars (Schlesinger et al. 2012). Specifically:

• 1.  
  Other SEGUE target categories that are biased toward metal-poor stars overlap the color range of the SEGUE G dwarfs, biasing the color-selected sample in metallicity.
• 2.  
  SEGUE has the same limited number of spectroscopic fibers for each line of sight, regardless of its stellar density. Therefore, for lines of sight at large |b|, the spectroscopic sample is a higher fraction of the available photometric stars than for lines of sight at lower Galactic latitudes.
• 3.  
  The G-dwarf color cut selects a range of stellar masses, and thus a subset of the mass function, that varies with metallicity.

In Schlesinger et al. (2012), we utilized SDSS photometry from DR7 (Abazajian et al. 2009) to develop three weights for each target that account for these biases, such that the corrected sample reflects the true underlying structure of the Milky Way probed. Each individual spectroscopic target reflects the properties of similar photometric targets in SDSS and is weighted to reflect the number of similar stars. Since this work, the SDSS photometry has been improved, and the SSPP has been updated. Consequently, our stellar sample has changed. Using the algorithms described in Schlesinger et al. (2012), we have recalculated the target-type, r-magnitude, and mass-function weights for our DR9 SEGUE G-dwarf sample. Thus, our adjusted sample reflects the distribution of stars observed photometrically by SDSS along all of the SEGUE lines of sight.

We have also improved our weighting algorithm to better deal with anomalously large weights. Previously, the faintest stars often had large r-magnitude weights, as there was a single spectroscopic observation representing many photometric targets; we trimmed these objects from the sample using a magnitude cut (Schlesinger et al. 2012). For the DR9 sample, we use a more stringent S/N criteria; this results in individual spectroscopic observations representing numerous photometric objects at a range of magnitudes, rather than just the faint end. To account for this problem, we remove any SEGUE star from the sample that is alone in a 0.5 r-magnitude bin. This action removes less than 1% of stars from the sample.

Although many previous analyses have used the SEGUE G-dwarf sample, their target selection is different than our own. For example, Lee et al. (2011) select stars that are targeted by SEGUE as G dwarfs, whereas we use all stars that meet G-dwarf criteria, even if they are were assigned SEGUE fibers for a different reason. By selecting those stars that were targeted only as G-dwarfs, the sample will be biased against metal-poor stars. Due to the prioritization scheme for SEGUE targets, a star that fulfills the color cuts of both a "G-dwarf star" and "metal-poor star" is more likely to receive a fiber as a member of the latter category. In contrast, we include all stars that meet the color and magnitude criteria, regardless of why they were targeted, making the sample more complete. The analysis of Carrell et al. (2012) is similarly biased against metal-poor stars, as they select stars that meet only the SEGUE selection criteria of F, G, or K dwarfs, rather than stars that fulfill multiple criteria. Consequently, Carrell et al. (2012) will under-represent "interesting" F, G, and K targets, such as low-metallicity stars and other overlapping categories. This bias is particularly serious for the SEGUE K-dwarf category, which is allotted few SEGUE fibers (Schlesinger et al. 2012).

As discussed earlier, the Milky Way is often described as having a thin- and thick-disk component. Previous analyses of the disk chemistry divided their samples geometrically, kinematically, and/or chemically in order to examine these two structures. In this work, we have designed a series of different subsamples to examine the disk as a whole and in chemical subsets. In addition, we recreate the divisions of previous works for a useful comparison.

Although we have around 40,000 G-dwarf stars in our SEGUE sample, our individual subsamples consist of approximately 1000 stars each. In contrast to many previous analyses, we limit each subsample in distance such that it is volume-complete (Section 2.5). This approach dramatically decreases the number of spectroscopic targets for each subsample. However, our weighting scheme (Section 2.3) adjusts our sampling such that each spectroscopic target represents many photometric targets (Schlesinger et al. 2012). For example, our volume-complete Full Sample contains 1434 spectroscopic targets but reflects the properties of approximately 13,360 photometric targets.

2.4.1. Geometric Subsamples

The stellar number density with respect to height above the plane of the Milky Way disk is best fit by an exponential thin- and thick-disk component, with scale heights of around 300 pc and 900 pc, respectively (Gilmore & Reid 1983; Jurić et al. 2008). Due to the differences in scale height, we expect few thin-disk stars at large heights above the Galactic plane. To isolate the thick-disk population, previous works limited their samples to stars with |Z| ⩾ 1.0 kpc (Allende Prieto et al. 2006; Katz et al. 2011; Kordopatis et al. 2011; Chen et al. 2011; Carrell et al. 2012). We see a drop-off in star counts above heights of 1.0 kpc (Figure 4). Although it is unclear that a geometric division can effectively separate the disk components, not to mention whether the two are in fact separable, for the sake of comparison with literature results, we define a subsample that consists of all G-dwarf stars with |Z| greater than 1 kpc.

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2.4.2. Chemical Subsamples

Figure 5 shows the distribution of our weighted SEGUE G-dwarf sample in chemical space. As described earlier, and discussed at length in Schlesinger et al. (2012), when we correct for the selection biases in SEGUE, the distribution shifts more to metal-rich and α-poor values.

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Chemical abundance is oftentimes associated with age (Wyse & Gilmore 1988; Bovy et al. 2012a; Haywood et al. 2013), as [α/Fe] and [Fe/H] proportions relate to the enrichment from SNe Type II and Ia, which contribute to the interstellar medium on different timescales (e.g., Maoz et al. 2011). Thus, examination of the vertical metallicity gradient with respect to [α/Fe] provides an indication of how disk structure varies for different epochs of star formation. We divide our sample into 0.1 dex bins of [α/Fe] to examine how the vertical metallicity gradient varies with α-enhancement. There are a few stars with [α/Fe] >+0.5, but not a sufficient number to populate their own bin, so we combine the two highest [α/Fe] bins into a single bin over 0.2 dex in [α/Fe]. Previous work by Bovy et al. (2012a, 2012b) divided the SEGUE G-dwarf sample into mono-abundance populations in [α/Fe] and [Fe/H]. In contrast, we do not limit our sample to 0.1 dex in [Fe/H] because we want to examine how metallicity varies over the volume of the disk.

Lee et al. (2011) detected bimodality in the raw SEGUE G-dwarf sample; they used the separation in [α/Fe] space to define an α-rich and α-poor sample, associated with the thick and thin disk, respectively (see their Figure 2). There are some important differences between our analysis and that of Lee et al. (2011). First, our sample of G dwarfs is different than that of Lee et al. (2011). As noted in Section 2.3, they select only stars targeted as G dwarfs by SEGUE, biasing their sample against low-metallicity stars. Secondly, we trim our sample to be volume-complete, as discussed at length in Section 2.5. While these actions do not necessarily affect the distribution in chemical space, they are important to consider when calculating the vertical metallicity gradients.

While we observe [α/Fe] bimodality in our bias-corrected sample, the thin/thick disk cuts of Lee et al. (2011) divide the metal-rich, α-poor peak of both our raw and weighted distributions, rather than clearly separating the sample into two α-populations (see Figure 5). The Lee et al. (2011) [α/Fe] cuts are not well suited to our observed bimodal distribution. This result is likely due to changes in the SSPP between DR8 and DR9. Examining the distribution of our sample in chemical space, we define our own thin/thick disk chemical subsamples based upon the observed bimodality at [α/Fe] = 0.28. We define all stars with [α/Fe] >+0.33 as our α-rich subsample, associated with the thick disk. Stars with [α/Fe] <+0.23 are our α-poor subsample, associated with the thin disk. Similar to Lee et al. (2011), we also include a "buffer" of 0.1 dex to prevent the two subpopulations from contaminating one another due to uncertainties in the SSPP [α/Fe] estimates. For the sake of comparison with the literature, we also determine the vertical metallicity gradients of the Lee et al. (2011) chemical subsamples.

In contrast to Lee et al. (2011), Bovy et al. (2012a, 2012b) found no bimodality in [α/Fe] for an adjusted sample of SEGUE G-dwarf stars. Similar to our technique, they correct for the SEGUE selection function and use all stars that fulfill the G-dwarf color cut, regardless of why it was targeted by SEGUE. However, they adjust the sample further, by using stellar-population modeling and an assumed star formation history to account for the limited lines of sight in SEGUE. As shown in Figure 1, the SEGUE survey probes 144 lines of sight. While we correct our spectroscopic sample such that it reflects the underlying populations for each of these lines of sight, Bovy et al. (2012a, 2012b) seek to model the disk as a whole, using assumptions about the Galactic disk structure to scale up the SEGUE lines of sight. As much of SEGUE focused on lines of sight at large Galactic latitude, it contains more metal-poor, "thick disk" stars than a sample that uniformly samples the disk. Thus, when they adjust their sample using Galaxy models, they scale down the metal-poor stars and scale up the metal-rich, finding a smooth decrease in number as [Fe/H] decreases. Such an approach can be strongly impacted by uncertainties in the underlying Galaxy model; thus, we limit ourselves to analyzing the underlying populations along the SEGUE lines of sight. While we observe bimodality in our SEGUE lines of sight after accounting for the local selection function, Bovy et al. (2012a, 2012b) find no bimodality when they adjust the sample to reflect chemistry beyond the SEGUE sampling.

Finally, recent work by Adibekyan et al. (2013) on the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) F-, G-, and K-dwarf sample identified a low-density region in the distribution of [α/Fe] versus [Fe/H]. Figure 5 displays their separation superposed on our distribution. Due to the low resolution of SEGUE spectra, any gap will be smeared out in our sample. Additionally, in contrast to HARPS, SEGUE is dominated by stars farther from the plane of the Galaxy. The separable thin-disk population identified in HARPS is not well sampled by SEGUE, and, consequently, will be difficult to detect. Thus, it is not surprising that their thin/thick separation is not observed in our sample of stars.

Previous works examined the vertical metallicity gradient over large distance ranges that are not volume-complete. For a given (g − r)₀ color, metal-rich stars are brighter than metal-poor stars. Thus, due to issues of survey completeness and saturation limits, the sample will be biased toward metal-rich stars at large distances and toward metal-poor at small distances. We specify distance limits for each of our subsamples such that they are volume-complete.

Our subsamples are typically defined in terms of [α/Fe]. For each of these, we examine the range of r₀ and [Fe/H] to determine an appropriate volume-complete distance range. The bright magnitude limit is set to r = 15, the saturation limit for SEGUE. We also institute a faint magnitude limit; if we do not trim the sample in magnitude, we occasionally include quite faint stars which result in an unreal distance range (e.g., the faint limit is closer than the bright limit). For the faint magnitude limit, we select the r₀ for which 85% of the sample is brighter, r₀ = 17.13, following the methodology of Schlesinger et al. (2012).

The most metal-poor stars with the least amount of α-enhancement define the faint distance limit; the most metal-rich stars with the maximum amount of [α/Fe] define the bright distance limit. Some of our subsamples have outliers in [Fe/H], which can lead to an unreal distance range, as with the magnitude range. To ensure that our distance limits reflect the overall parameters of the sample, rather than being skewed by outliers, we remove any stars in the extreme 5% of the distribution for each subsample.

For each of our defined subsamples, we generate 10 Gyr isochrones for the two possible chemical extremes of each subsample using the Dartmouth Isochrone Generator 2012 (Dotter et al. 2008); these correspond to the faint and bright distance limits. We then extract M_r at (g − r) of 0.48 and 0.55. Table 1 lists the properties of each subsample and its associated distance limits.

Schlesinger et al. (2012) restrict their full G-dwarf sample to distances between 1.59 and 2.29 kpc for volume-completeness. Our distance limits for the Full Sample are much more limited, between 1.447 and 1.614 kpc. We want to ensure that all of our stars have [α/Fe] measurements, which requires a S/N > 30, stricter than the S/N >10 constraint of Schlesinger et al. (2012). Not only does this S/N criteria limit our sample size, it also affects our faint magnitude limit. Whereas the sample in Schlesinger et al. (2012) has stars as faint as r₀ = 18.45, our limitation to r₀ = 17.13 greatly decreases our maximum distance limit, reducing the number of stars in each of our chemical subsamples.

As noted in Section 2.3, the SEGUE G-dwarf sample is biased toward metal-poor stars due to the survey's target-selection algorithm. Figure 6 demonstrates that accounting for these biases has a marked affect on both the slope and zero-point of the vertical metallicity gradients of the different subsamples. The slopes of the "raw" samples are listed in Table 2.

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Table 2. Vertical Metallicity Gradient for Different Subsamples

		Weighted	$\frac{d[{\rm Fe/H}]}{d|Z|}_{{\rm raw}}$	$\frac{d[{\rm Fe/H}]}{d|Z|}_{wt}$					σ					$\Delta \frac{d[{\rm Fe/H}]}{d|Z|}_{CE}$		$\frac{d[{\rm Fe/H}]}{d|Z|}$
Sample	Number	Number			Bootstrap	log g	[α/Fe]	Drand	Dsyst	Phot	Redden	Binary	RGC
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)	(18)
Full Sample	1434	13359	−0.252	−0.206	0.000	0.002	0.003	0.006	0.009	0.002	0.000	0.004	0.031	+0.037	0.020	−0.243	0.039
					0.042	0.004	0.001	0.009	0.006	0.002	0.006	0.002	−0.018		0.022		0.053
|Z| > 1 kpc	806	2080	−0.133	−0.102	0.096	0.032	0.033	0.082	0.112	0.022	0.018	0.026	0.012	+0.131	0.107	−0.233	0.209
					0.047	0.018	0.010	0.000	0.000	0.012	0.015	0.016	0.003		0.137		0.149
α-rich	1614	8996	−0.029	+0.083	0.038	0.006	0.002	0.008	0.004	0.002	0.006	0.001	0.002	+0.021	0.024	+0.063	0.047
					0.024	0.003	0.005	0.011	0.007	0.005	0.003	0.006	0.003		0.012		0.032
α-poor	2042	24550	−0.063	−0.094	0.020	0.003	0.001	0.001	0.000	0.001	0.002	0.000	0.035	−0.132	0.014	+0.038	0.043
					0.018	0.002	0.003	0.004	0.004	0.001	0.002	0.003	−0.029		0.011		0.037
Lee α-rich	1795	12008	−0.032	+0.025	0.027	0.003	0.001	0.010	0.003	0.001	0.003	0.007	−0.009	+0.045	0.016	−0.020	0.035
					0.024	0.006	0.007	0.009	0.005	0.003	0.006	0.005	0.002		0.015		0.033
Lee α-poor	1064	12802	−0.034	−0.078	0.059	0.003	0.000	0.002	0.000	0.000	0.001	0.000	0.041	−0.164	0.016	+0.087	0.074
					0.015	0.002	0.009	0.006	0.005	0.003	0.003	0.004	−0.054		0.015		0.059
α-bin 1	445	2874	+0.009	−0.038	0.039	0.005	0.003	0.003	0.001	0.002	0.002	0.002	0.024	−0.176	0.011	+0.138	0.048
					0.037	0.000	0.004	0.003	0.003	0.002	0.003	0.002	−0.025		0.023		0.051
α-bin 2	1448	18145	−0.008	−0.026	0.015	0.006	0.005	0.004	0.001	0.002	0.002	0.003	0.020	−0.053	0.018	+0.027	0.032
					0.018	0.001	0.006	0.004	0.003	0.002	0.003	0.002	−0.020		0.018		0.033
α-bin 3	1095	12996	−0.041	−0.063	0.026	0.007	0.010	0.014	0.000	0.010	0.021	0.004	0.022	+0.034	0.014	−0.097	0.048
					0.018	0.010	0.015	0.012	0.016	0.009	0.001	0.012	−0.022		0.016		0.045
α-bin 4	1261	9432	−0.000	+0.031	0.023	0.007	0.000	0.017	0.007	0.007	0.010	0.012	−0.001	+0.004	0.029	+0.027	0.045
					0.027	0.012	0.065	0.008	0.005	0.001	0.014	0.007	−0.016		0.035		0.083
α-bin 5	890	3085	+0.013	+0.008	0.061	0.012	0.022	0.012	0.005	0.006	0.000	0.011	0.017	+0.072	0.018	−0.064	0.073
					0.025	0.001	0.002	0.007	0.005	0.003	0.008	0.005	−0.004		0.020		0.035

Notes. The estimated vertical metallicity gradients for our different subsamples with their associated uncertainties. Each subsample uses the distance limits listed in Table 1; we have also determined the gradients for all subsamples over the distance range specified for the Full Sample; these values agree within 1σ. Column (2) lists the number of stars that fall within each subsample; Column (3) is the weighted number of stars after we have corrected for SEGUE target selection. Column (4) is the gradient measured for the raw stellar subsample. Column (5) lists the slope when we have adjusted for target-selection weights and the radial metallicity gradient, but not correlated errors. The change from correlated errors is listed in Column (15). Columns (6) through (14) list the uncertainty on the slope due to our different sources of uncertainty, as detailed in the Appendix. Column (17) presents our final estimated vertical metallicity gradient for each subsample, with the total uncertainty in Column (18).

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Although we have removed stars that are alone in their r magnitude bin from the sample, some of the G-dwarf stars still have anomalously large target-selection weights. Typically, these are stars in magnitude bins that have a small number of spectroscopic targets and a large number of photometric targets. Due to our stringent cuts in spectral S/N, these heavily weighted stars occur at a range of magnitudes. These anomalous weights will induce large wiggles in our vertical metallicity gradient, especially at low |Z| where the stellar density is high but the SEGUE sample size is small. For each subsample, we remove all stars with weights that lie 2σ from the mean weight. This procedure tends to remove more metal-rich stars, as these are more likely to be in high-density regions close to the plane. Although it has a small effect on the intercept, removing these anomalous weights does not significantly change the measured vertical metallicity gradient; rather it serves to smooth the structure with respect to |Z|. Most of the gradients measured for the untrimmed subsamples lie within 1σ of the values of the trimmed sample; all are within 2σ.

Our stellar sample ranges from 6 to 11 kpc in Galactocentric radius (Figure 3); over a volume-complete range, it is limited to 6.7–9.5 kpc. Thus, our sample is not well suited to determining the radial metallicity gradient. However, our sample is still affected by the Galactic radial metallicity gradient, which in turn will affect our estimates of the vertical metallicity gradient for different subsamples.

Cheng et al. (2012a) measured radial metallicity gradients for an unbiased low-latitude sample of SEGUE main-sequence turnoff stars, ranging from 6 to 16 kpc in Galactocentric radius. They also divided the sample into bins of [α/Fe] and |Z|, finding that stars with [α/Fe] >+0.2 showed a flat radial metallicity gradient at all |Z|. In contrast, the radial metallicity gradient of stars with [α/Fe] <+0.2 flattened with increasing |Z|, similar to the sample as a whole.

Based on each individual star's chemistry and Galactocentric radius, we adjust the stellar [Fe/H] using the slopes from Cheng et al. (2012b) such that they reflect the metallicity at the solar radius (see Figure 6). Other analyses have estimated the radial metallicity gradient (e.g., Maciel & Costa 2010; Boeche et al. 2013; Hayden et al. 2014), but we adopt the values from Cheng et al. (2012b) as they divide their sample in [α/Fe] and cover a similar range of |Z| with stars observed in situ. Figure 6 shows the changes in our vertical metallicity gradient when we remove the effect of radial metallicity structure is removed. The SEGUE G-dwarf sample is typically at larger |Z|, where the radial gradient is minimal. The largest change in [Fe/H], ≈ 0.05 dex, occurs at low |Z| and primarily affects the α-poor samples, which lie close to the Galactic plane.

In addition to corrections for SEGUE target selection and the radial metallicity gradient, we must also examine the systematic uncertainties in the vertical metallicity gradient stemming from correlated errors in estimates of [Fe/H] and distance. An underestimate in [Fe/H] will lead to an underestimated distance, and vice versa, moving targets into different bins of R, |Z|, and [Fe/H] (Schlesinger et al. 2012), which can induce an artificial slope in the derived vertical metallicity gradient.

To determine how correlated errors affect our vertical metallicity gradients, we must analyze a sample where we know the "true" underlying behavior. Thus, we simulate each SEGUE line of sight in our sample using the chemical and dynamical Galaxy model of Schönrich & Binney (2009a, 2009b). This model is notable for including radial migration in addition to simulating chemical evolution. Furthermore, it provides both [Fe/H] and [α/Fe] values. Schlesinger et al. (2012) found that this model did not accurately reflect the vertical metallicity structure of their observed sample, becoming more metal rich with increasing |Z|, whereas the SEGUE sample becomes more metal poor. Although the model does not recreate the observed distributions, the absolute metallicity structure is not of major consequence for the purpose of this error analysis. We are simply examining at the variation in the sample caused by correlated errors.

Applying SEGUE and subsample selection criteria to the model, we randomly assign each star an uncertainty in metallicity, based on the expected SSPP errors. We then determine the resulting change in distance for each model star, and shift its distance, R, and Z accordingly. We run this procedure 20 times to produce 20 iterations of the simulated SEGUE fields. For each iteration, we perform a least-squares fit to the vertical metallicity gradient with the correlated uncertainties in [Fe/H] and distance and compare the median slope to the true model vertical metallicity gradient for each subsample. This calculation reveals the degree to which correlated errors may change our measured gradients. We also examine the variation in the metallicity gradient with correlated errors to estimate the 1σ uncertainty in this correction. For the vertical metallicity gradient of each subsample, we subtract the estimated induced slope from our value of the weighted vertical metallicity gradient. Table 2 lists the corrections, and their associated uncertainty, for correlated errors for each subsample. For a given Δ[Fe/H], the magnitude difference between metal-rich isochrones is larger than that between metal-poor for the main sequence. Consequently, the change in estimated distance due to Δ[Fe/H] will be larger at the metal-rich end, leading to a larger uncertainty from correlated errors in the vertical metallicity gradient. Figure 6 presents the change in the vertical metallicity gradients when we account for correlated errors; these corrections vary for different subsamples, but are typically around ±0.08 dex kpc⁻¹.

To estimate the uncertainties in our vertical metallicity gradients, we combine information from a bootstrap analysis (with replacement) and a Monte Carlo analysis. Using these two techniques, we quantify the uncertainty in [Fe/H] at each |Z| and the error on the measured slope for each chemical subsample. To determine the total uncertainty in median [Fe/H] at each height, we combine the errors from the following sources: sample selection, photometry, extinction, random and systematic distance, [α/Fe], radial metallicity gradient correction, binarity, and log g. We discuss our methodology in detail in the Appendix. Uncertainties from our bootstrap analysis, related to variation in sample selection, contribute the largest error at each |Z| for the different subsamples, around 0.06 dex. Close to the plane of the Galaxy, there are fewer SEGUE stars and a high stellar number density. Consequently, our uncertainties in [Fe/H] are larger at low |Z|.

We also determine the variation in slope due to the individual sources of error in our sample, namely sample selection, log g, [α/Fe], distance, photometry, reddening, undetected binarity, and radial metallicity gradient corrections (Table 2). The results from our bootstrap analysis produce the largest shifts in slope, around ±0.04 dex kpc⁻¹, indicating that much of the uncertainty is driven by sample selection.

Using our boxcar-smoothing technique and various systematic corrections, we have measured a vertical metallicity gradient of −0.243$^{+0.039}_{-0.053}$ dex kpc⁻¹ for the volume-complete Full Sample of SEGUE G dwarfs from 0.27 to 1.62 kpc in |Z| (bottom panel of Figure 7, Table 2). This volume-complete sample contains stars with metallicity in the range −1.000 < [Fe/H] <−0.035 and α-enhancement 0.00 < [α/Fe] < +0.55. At low heights, the stars are metal-rich ([Fe/H] ≈ −0.3); as |Z| increases, the sample becomes increasingly metal-poor, leveling off around an [Fe/H] = −0.55.

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There are some "wiggles" in [Fe/H] around |Z| of 0.6 and 0.8 kpc. These "wiggles" are well within our error bars and are a manifestation of large target-selection weights for a few stars at this height; although larger than average, these weights are still within our 2σ limit. This "bend" is not seen in the raw sample (see Figure 6), and when we use a more stringent cut on the target-selection weights, the feature disappears. We do not believe that this behavior shows any peculiar structure at this height but rather arises from our small sample size close to the plane of the Galaxy. As indicated in Figure 7 and discussed in the Appendix, the uncertainties at these low heights are larger than at higher |Z|.

We have also divided the SEGUE G-dwarf sample into bins of [α/Fe], each of which probes a different volume-complete distance range (Table 1). Examining these subsamples provides a more detailed picture of the disk and does not presuppose a thin/thick disk structure. The top panel of Figure 7 shows the measured vertical metallicity gradient for our volume-complete [α/Fe] subsamples after our corrections for target-selection biases and correlated errors have been applied. These values are also listed in Table 2.

Our lowest α-bin subsample, with 0.0 < [α/Fe] < +0.1, has a positive vertical metallicity gradient, +0.138$^{+0.048}_{-0.051}$ dex kpc⁻¹. While the weighted vertical metallicity gradient itself is quite flat, with a slope of 0.009 dex kpc⁻¹, factoring in the correlated errors shifts the slope in the positive direction. As noted in Section 3.3, even small changes in [Fe/H] at the metal-rich end will result in large changes in distance. Thus, correlated errors have a larger affect on the gradient of this sample than the other α-bins. Note also that this subsample has the smallest number of stars and is in a region of the Galaxy that is not as well sampled by the SEGUE survey. Thus, the magnitude of this gradient is uncertain.

As the sample becomes increasingly α-rich, the vertical metallicity gradient changes. For α-bin 2, with +0.1 < [α/Fe] < +0.2, the slope is +0.027$^{+0.032}_{-0.033}$ dex kpc⁻¹. α-bin 4 shows a similar value, with a slope of +0.027$^{+0.045}_{-0.083}$ dex kpc⁻¹. In contrast, for α-bins 3 and 5, we estimate slopes of −0.097$^{+0.048}_{-0.045}$ and −0.064$^{+0.073}_{-0.035}$ dex kpc⁻¹, respectively. Although the sign and extent of the vertical metallicity gradients for the α-subsamples varies, each shows minimal change in median [Fe/H] with respect to height above the Galactic plane. With the exception of α-bin 1, the measured vertical gradients are generally consistent with one another within the expected uncertainties; they are also consistent within 2σ with a flat metallicity gradient. Thus, while we see a strong vertical metallicity gradient over the disk as a whole, individual α-populations show little variation in the median [Fe/H] with height above the Galactic plane.

We also examine the vertical metallicity gradients of the α-bins over the volume-complete distance range for the Full Sample, from 1.447 to 1.614 kpc, which covers from 0.28 to 1.58 kpc in |Z| (Figure 8). These two sets of vertical metallicity gradients provide slightly different information. The gradients discussed above, with each subsample covering the distance range specified by Table 1, allow us to examine each individual [α/Fe] population over as much of the disk as possible with SEGUE. Namely, we can inspect the variations in [Fe/H] over the largest possible disk volume. This approach also includes the largest possible number of stars for each subsample. In contrast, when we limit the subsamples to distances between 1.447 and 1.614 kpc, we can directly compare the different gradients with one another (Table 3). The bias in metallicity with respect to volume coverage is minimized (Section 2.5), and we can examine the interplay of different chemical populations over a limited disk volume. With the exception of α-bin 1, the gradients of each subsample over the two distance cuts are consistent within 1σ and a slope of 0.0 within 2σ. As with our other distance cuts, the gradient of each α-bin subsample is much flatter than that of the disk as a whole. Of the different bins, the sample with [α/Fe] between +0.2 and +0.3 exhibits the strongest gradient over the Full Sample distance range; this population covers the bimodal break in [α/Fe], which may explain some of this structure (see Figure 5).

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Table 3. Vertical Metallicity Gradient for Different Subsamples with 1.447 kpc < D < 1.614 kpc

Sample	Number	Weighted	$\frac{d[{\rm Fe/H}]}{d|Z|}_{{\rm raw}}$	$\frac{d[{\rm Fe/H}]}{d|Z|}_{wt}$	σ									$\Delta \frac{d[{\rm Fe/H}]}{d|Z|}_{CE}$		$\frac{d[{\rm Fe/H}]}{d|Z|}$
		Number			Bootstrap	log g	[α/Fe]	Drand	Dsyst	Phot	Redden	Binary	RGC
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)	(18)
Full Sample	1434	13359	−0.252	−0.206	0.000	0.002	0.003	0.006	0.009	0.002	0.000	0.004	0.031	+0.037	0.020	−0.243	0.039
					0.042	0.004	0.001	0.009	0.006	0.002	0.006	0.002	−0.018		0.022		0.053
|Z| > 1 kpc	806	2080	−0.133	−0.102	0.096	0.032	0.033	0.082	0.112	0.022	0.018	0.026	0.012	+0.131	0.107	−0.233	0.209
					0.047	0.018	0.010	0.000	0.000	0.012	0.015	0.016	0.003		0.137		0.149
α-rich	528	4067	−0.002	+0.128	0.060	0.012	0.004	0.013	0.004	0.003	0.001	0.007	0.013	+0.035	0.014	+0.093	0.066
					0.051	0.003	0.008	0.023	0.018	0.006	0.016	0.015	−0.022		0.025		0.072
α-poor	538	5871	−0.085	−0.136	0.048	0.006	0.001	0.002	0.005	0.002	0.001	0.004	0.037	−0.125	0.067	−0.011	0.091
					0.028	0.001	0.007	0.014	0.003	0.001	0.009	0.003	−0.041		0.031		0.062
Lee α-rich	648	5422	−0.071	+0.014	0.088	0.014	0.008	0.017	0.010	0.002	0.007	0.009	0.005	+0.046	0.017	−0.032	0.094
					0.041	0.007	0.010	0.011	0.013	0.012	0.010	0.010	−0.007		0.019		0.053
Lee α-poor	345	3528	+0.027	−0.126	0.082	0.004	0.005	0.007	0.006	0.003	0.002	0.004	0.070	−0.129	0.040	+0.003	0.116
					0.041	0.001	0.010	0.013	0.007	0.005	0.007	0.003	0.004		0.034		0.057
α-bin 1	137	860	−0.008	−0.001	0.076	0.016	0.019	0.010	0.009	0.009	0.004	0.007	0.066	−0.190	0.037	+0.189	0.112
					0.077	0.012	0.021	0.021	0.013	0.000	0.015	0.011	−0.048		0.043		0.108
α-bin 2	287	3415	−0.043	−0.024	0.044	0.010	0.014	0.020	0.007	0.005	0.011	0.011	0.027	+0.020	0.054	−0.044	0.082
					0.078	0.006	0.015	0.019	0.012	0.003	0.007	0.005	−0.067		0.090		0.139
α-bin 3	381	3546	−0.042	−0.089	0.040	0.008	0.025	0.033	0.012	0.005	0.018	0.011	0.013	+0.080	0.053	−0.169	0.084
					0.024	0.007	0.021	0.016	0.011	0.008	0.004	0.008	−0.013		0.054		0.069
α-bin 4	398	3956	−0.003	−0.003	0.089	0.018	0.000	0.000	0.012	0.007	0.003	0.023	0.002	−0.044	0.055	+0.040	0.110
					0.041	0.011	0.092	0.052	0.051	0.007	0.034	0.019	−0.009		0.070		0.149
α-bin 5	216	1092	+0.063	+0.044	0.099	0.015	0.057	0.049	0.008	0.012	0.016	0.032	0.013	0.121	0.037	−0.078	0.137
					0.118	0.030	0.010	0.013	0.025	0.005	0.020	0.016	0.045		0.038		0.141

Notes. Same as Table 2 except here all subsamples are limited to distances between 1.447 and 1.614 kpc, i.e., are volume-complete over the distance range of the Full Sample. Note that the number of stars decrease significantly for most samples, increasing the overall uncertainties on the slope.

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Examining the vertical metallicity gradient for each subsample over the volume-complete distance range of the Full Sample reveals that different α populations are present at different heights (Figure 8). For example, while there are a negligible number of stars with [α/Fe] <+0.1 above |Z| = 1 kpc, there are also few stars with +0.4 < [α/Fe] < +0.6 below this height. Each [α/Fe] population probes a different range of |Z| heights. We discuss the repercussions of this result in Section 5.3.

The early picture of the disk of the Milky Way consists of an α-rich thick disk and an α-poor thin disk (e.g., Fuhrmann 1998, 2011). For the sake of comparison with previous analyses, we divide our sample of SEGUE G-dwarfs into two larger bins of [α/Fe] to examine the vertical structure of these two proposed components. If there are two separable components, with distinct formation and evolution processes, they may exhibit different vertical metallicity structure.

First, we divide the sample according to the chemical separation defined by Lee et al. (2011). Using our boxcar smoothing, we measure the corrected vertical metallicity gradients for these subsamples over a volume-complete distance range listed in Table 1. As with the smaller [α/Fe] bins, the gradients flatten significantly compared to that over the disk as a whole. The α-rich stars have a slope of −0.020$^{+0.035}_{-0.033}$ dex kpc⁻¹. The α-poor sample has a slope of 0.087$^{+0.074}_{-0.059}$ dex kpc⁻¹ (Table 2). Both of these subsamples exhibit little change in median [Fe/H] with increasing distance from the Galactic plane.

As noted earlier, the Lee et al. (2011) cuts were based on the [α/Fe] versus [Fe/H] distribution of their raw G-dwarf sample. However, these distributions change when we account for the SEGUE target-selection biases (Figure 5, Section 2.4.2). We have defined our own α-rich and α-poor populations based on the unbiased distribution in chemical space and determined vertical metallicity gradients over a volume-complete distance range. Stars with [α/Fe] ⩾ +0.33, associated with the thick disk, have a vertical metallicity gradient of +0.063$^{+0.047}_{-0.032}$ dex kpc⁻¹ (Figure 7). This value is larger than that obtained when using the Lee et al. (2011) criteria. The vertical metallicity gradient of the α-poor subsample, with [α/Fe] ⩽ +0.23, is +0.038$^{+0.043}_{-0.037}$; this measurement is in agreement with the Lee et al. (2011) subsample value within the uncertainties. The measured vertical metallicity gradient changes slightly as we vary our cuts in chemistry. However, they show consistent overall behavior. Namely, they are all significantly flatter than the gradient over the disk as a whole and indicates that there is little change in the typical [Fe/H] with increasing |Z| over a small range in [α/Fe].

Figure 7 shows that the α-rich subsample shows a median [Fe/H] of ≈ −0.6 from |Z| of 0.5–1.6 kpc. The α-poor subsample is more complex, with a large amount of structure below |Z| of 0.75 kpc. While some of this complexity is due to anomalous weights (Section 4.1), the variation is also due to our small sample size of stars at these low heights. With a limited number of targets, uncertainties from bootstrapping increase (Appendix A.2). Between 0.75 and 1.5 kpc in |Z|, where this population is better sampled by SEGUE, the α-poor stars show little variation in the median [Fe/H], around −0.225.

When we limit the distance range of our subsamples to 1.447–1.614 kpc, we measure gradients that are consistent within 1σ with those using the distance limits in Table 1 (Table 3, Figure 8). This change significantly decreases the sample size, resulting in larger uncertainties, particularly at low |Z| for the α-poor sample, a population which is not as well sampled by SEGUE. Both the α-rich and α-poor subsamples continue to display little change in their median [Fe/H] with respect to height. As seen in the α-bin analysis, different α-samples dominate at different heights above the plane. There are few α-rich stars below |Z| = 0.7 kpc and few α-poor above |Z| = 1.3 kpc. The vertical metallicity gradient of the full sample is in good agreement with the α-poor sample at low |Z|, where these stars dominate the population, and shifts to match the α-rich sample at high |Z|. We discuss this topic further in Section 5.3.

SEGUE provides a large sample with accurate spectroscopic stellar parameters. We can adjust the sample such that it is unbiased in chemistry, reflecting the underlying stellar populations over an extensive volume of the disk. Here we compare our results to values from the recent literature. Our results are consistent with many of these analyses, such as Ivezić et al. (2008). Others show more discrepant values for the vertical metallicity gradient, whether due to limited sample size and/or spatial coverage (e.g., Kordopatis et al. 2011) or issues related to target-selection biases (e.g., Carrell et al. 2012). Of particular interest are comparisons with recent results from APOGEE (Hayden et al. 2014) and RAVE (Boeche et al. 2014).

5.1.1. Ivezić et al. (2008)

5.1.2. Kordopatis et al. (2011)

Kordopatis et al. (2011) use a sample of ≈700 stars along a single line of sight to examine the vertical metallicity gradient from 1 kpc ⩽|Z| ⩽ 4 kpc. These stars have low-resolution (R ≈ 6500) spectra and are selected based on their V magnitude and inclusion in the Ojha et al. (1996) sample, which provides proper motion information. Note that the Ojha et al. (1996) sample uses a (B − V) color criteria to identify F and G stars for analysis.

By limiting their |Z| range to above 1 kpc, Kordopatis et al. (2011) seek to isolate the thick disk population, which has a scale height of around 900 pc (Jurić et al. 2008). Although the efficacy of this approach is unclear, we have analysed a subsample of the Full Sample with |Z| > 1 kpc. Accounting for target-selection biases and correlated errors, the slope is −0.233$^{+0.209}_{-0.0149}$ dex kpc⁻¹ (Table 2). Figure 9 compares their measured vertical metallicity gradient to ours for |Z| ⩾ 1 kpc. Their sample has a slope of −0.14 ± 0.05 dex kpc⁻¹ between 1 and 4 kpc in |Z|. Due to the large uncertainties on our vertical metallicity gradient for this subsample, the two gradients are consistent with one another within 1σ. However, Figure 9 indicates that their sample is more metal-rich than ours at all heights.

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Although they do not include stars below |Z| of 1 kpc in their gradient calculation, there are some in their sample; using their low |Z| points, we calculate the vertical metallicity gradient for the disk as a whole for their entire sample. Applying a linear least-squares analysis, this vertical metallicity gradient is −0.124 dex kpc⁻¹ for their full sample between 0 and 4 kpc in |Z| (Figure 10). This slope is less than that of our Full Sample (−0.243$^{+0.039}_{-0.053}$ dex kpc⁻¹), and, as with the slopes above |Z| of 1 kpc, their points are more metal-rich than ours.

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We suspect that the variation in the vertical metallicity gradient between their analysis and ours stems from sample differences, i.e., they are more limited in size and spatial coverage, while our sample is larger and multi-directional. Using simulated lines of sight from the model of Schönrich & Binney (2009a, 2009b), we compare the vertical metallicity gradient of the model as a whole to that of subsamples comparable to that of Kordopatis et al. (2011). We manufacture 10 bootstrap with replacement versions of the Schönrich & Binney (2009a, 2009b) model of the different SEGUE lines of sight. First, we determine the gradients for this simulation using our boxcar-smoothing technique and a least-squares analysis; the uncertainty is from the variation in the gradients over the 10 iterations. This is our "control" gradient. We repeat this analysis, limiting the sample used to determine the gradient to 700 randomly selected stars, comparable to the sample size of Kordopatis et al. (2011). Finally, we limit the sample in both number and lines of sight. SEGUE does not contain the particular lines of sight in Kordopatis et al. (2011), so we isolate stars in the same area of the sky. Applying the limits associated with our Full and |Z| > 1 kpc samples, we find that limitations in coverage and number will affect the measured vertical metallicity gradients in a non-negligible way, likely creating the differences between our values and those of Kordopatis et al. (2011). In addition, there may be some selection biases in the sample due to the color criteria used by Ojha et al. (1996).

5.1.3. Katz et al. (2011)

5.1.4. Chen et al. (2011)

5.1.5. Carrell et al. (2012)

5.1.6. Allende Prieto et al. (2006)

5.1.7. Hayden et al. (2014)

5.1.8. Boeche et al. (2014)

In models of thick disk formation by Brook et al. (2004, 2005) and Bournaud et al. (2009), the galaxy accretes gas-rich material early on. This behavior prompts a burst of star formation which creates the thick disk either in situ or over a dynamically short time period, resulting in a vertically-uniform thick disk. Their simulations show a narrow range in [α/Fe] associated with the simulated thick disk; most cover less than 0.15 dex, comparable to our α-bin subsamples. They predict no vertical metallicity gradient for these simulations, in agreement with our analysis of α-subsamples, suggesting that the disk is well mixed chemically over |Z| for a given epoch of star formation.

If the initial disk exhibits a vertical metallicity gradient, it can be smoothed by radial migration, which will mix up stars of various chemical compositions at different |Z| heights. However, if there is no initial vertical metallicity gradient, radial migration can induce one, as older stars have more time, and thus opportunity, to move outward in R and |Z|. The predictions from these simulations vary significantly depending on initial parameters.

Loebman et al. (2011) form a thick-disk component via radial migration in an N-body simulation. Their model is not designed to recreate the Milky Way and is not calibrated with existing observational data. This radial migration will create a vertical metallicity gradient for the disk as a whole of ≈ −0.18 dex kpc⁻¹, just outside our quantified uncertainties (Figure 10). Their gradient is more metal rich than ours, which is not surprising for a model independent of observations.

In contrast to Loebman et al. (2011), the simulations of Bird et al. (2013) indicate that the chemical structure of the disk is largely set by in situ star formation within a rotating, collapsing gas cloud; although secular heating and radial migration are present, they have a sub-dominant effect on the chemical trends. This result suggests that our metal-poor, α-enhanced stars formed in large scale-height populations, rather than scattering to larger |Z| over time. The effect of radial migration on our sample is currently unclear due to the wide range of predicted vertical metallicity structure. As these simulations improve, they will have more detailed chemical information. The added parameter of [α/Fe] will allow one to better distinguish between different formation models.

As |Z| increases, we expect to detect more and more α-enhanced stars relative to α-poor (e.g., Bovy et al. 2012a, 2012b; Schlesinger et al. 2012). Figure 11 shows the MDF of the Full, α-rich, and α-poor samples with respect to |Z|. These three samples cover a distance range where all are volume-complete (1.447 kpc < d < 1.614 kpc); the error bars reflect uncertainties from a bootstrap (with replacement) analysis over 100 iterations (see Appendix A.2). To ensure that the number of stars in each of our α subsamples sum to the total number of stars, we define our α-rich stars as having [α/Fe] > +0.28 and α-poor as [α/Fe] < +0.28 dex for this comparison. This cut is motivated by the distribution shown in Figure 5. At low |Z|, the population is approximately 70% metal-rich, α-poor stars. As |Z| increases, this percentage decreases; at |Z| ≈ 1 kpc, the total sample is approximately 50% α-poor, metal-rich stars. Above 1.25 kpc, the G-dwarf sample is dominated by metal-poor, α-enhanced stars (75%). As the height above the Galactic plane increases, the disk transitions between [α/Fe] populations, manifesting itself as a strong vertical metallicity gradient.

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We compare the vertical metallicity gradient of the full sample and an α-rich and α-poor subsample in the lower panel of Figure 8. Each of these gradients cover the same volume-complete distance range. As seen in Figure 11, at low |Z| there are few α-rich stars; Figure 8 shows that the vertical metallicity gradient of the full sample is aligned with the α-poor vertical metallicity gradient. At heights above ≈0.8 kpc, the proportion of α-rich stars increases (as reflected in the cumulative distributions of Figure 11). The gradient of the Full Sample shifts to lower median [Fe/H]. Above 1.4 kpc, there are few stars in the α-poor subsample, and the gradient of the Full Sample is aligned with the α-rich.

The top panel of Figure 8 allows further study of the variation in chemistry with height. Stars with [α/Fe] between 0.0 and +0.1 are rarely present above 1 kpc. In contrast, stars with +0.4 < [α/Fe] < +0.6 have few members below this height. To further examine this point, we determine the mean |Z|, and σ_|Z|, for bins of 0.025 dex in [α/Fe] (Figure 4). Each bin contains a different number of targets, as indicated in the histogram in the top panel of Figure 4. There is a clear increase in mean |Z| with α-enhancement, from around 0.5 kpc at [α/Fe] = 0.0–1.0 kpc at [α/Fe] = +0.5. In addition, the spread in |Z| increases with α-enhancement (bottom panel of Figure 4). It is unclear whether σ_|Z| is flat for [α/Fe] between 0.00 and 0.15 and 0.3 and 0.5, with a transition region in the middle, or smoothly transitions over the entire [α/Fe] range. We leave further investigation of this for a future study.

As expected from investigation of the MDFs, the strong vertical gradient measured for the total sample reflects the transition from α-poor, metal-rich stars at low |Z| to α-rich, metal-poor stars at high |Z|. Similarly, Figure 8 shows a mix of stars from [α/Fe] of 0.1–0.6 above |Z| of 1 kpc. Consequently, we are not surprised to find a negative vertical metallicity gradient above this height that is comparable to that of the sample as a whole.

The α-abundance ratio for a given stellar population is intrinsically linked with the various timescales of chemical evolution. Due to changes in the relative contribution to the interstellar medium by SN II and SN Ia ejecta, stars formed earlier in the Galaxy have higher [α/Fe], whereas more recently-born stars have lower [α/Fe]. However, the purity of this relationship is unclear. The high-resolution analysis of 189 F- and G-dwarf by Edvardsson et al. (1993) found significant scatter in the chemistry of disk stars formed at the same time. With improved ages and chemical abundances, Haywood et al. (2013) find a less-scattered relationship between age and chemistry for over 1000 FGK dwarfs in the solar neighborhood, albeit with a number of significant outliers. Further complicating matters, Minchev et al. (2013) use a chemodynamical model to find that, while there is a clear relationship between chemistry and age for stars from a particular birth radius, radial migration can destroy all evidence of it.

If we assume that each value of [α/Fe] is associated with a certain stellar age, our results suggest that the vertical metallicity gradient of the disk as a whole reflects changes in the chemical structure over a range of ages, while individual α-bins reveal the behavior of a particular epoch of star formation. We find consistently small vertical metallicity gradients for our α-subsamples, consistent with predictions of the chemodynamical model of Minchev et al. (2013). First, this means that each "age" of stars is associated with a particular median [Fe/H], regardless of its vertical location in the disk. Second, the consistency over the range of [α/Fe] implies that each epoch of star formation creates similar vertical metallicity structure. Many models suggest that different disk-formation processes will produce different vertical chemical structure (Sections 1 and 5.2); the consistency of our vertical metallicity gradient with increasing [α/Fe] suggests that there is not a dramatic change in the dominant formation mechanisms over time.

We can also consider the links between age and typical |Z| height. As described in Section 5.3, stars with low [α/Fe] are typically close to the plane of the Galaxy, whereas stars with enhanced [α/Fe] appear at a larger mean |Z|, in addition to having a larger range of |Z| values (Figures 4 and 8). This indicates that stars that formed late remain at low |Z|, in contrast to stars that formed much earlier. These stars could be born in situ at a wider range of |Z| than young stars, or they could extend their range of heights through various dynamical effects.

Recently, Bird et al. (2013) analysed a high-resolution hydrodynamic simulation of a Milky Way-like galaxy, which included an active merger phase at z > 3, secular heating, and radial migration. By examining the evolution of the simulation with respect to different epochs of star formation, they determined that the disk structure formed "upside-down." Old stellar populations formed in a vertically extended and radially compact structure; subsequent star formation shows a smooth transition to increasingly vertically compact and radially elongated spatial configurations as time progresses. Stinson et al. (2013) have performed a cosmological hydrodynamical simulation of a Milky-Way-like galaxy. Consistent with Bird et al. (2013), they find that older, more metal-poor and α-rich stars exhibited short scale lengths and large scale heights, while younger, more metal-rich stars had long scale lengths and short scale heights. The chemodynamical disk model of Minchev et al. (2013) also made similar predictions. These analyses fit well with the smooth increase in scale heights with decreasing [Fe/H] and increasing [α/Fe] found by Bovy et al. (2012a). Similarly, when we assume that chemistry and age are intrinsically linked, these simulations are consistent with the variation in |Z| for our different α-subsamples and the increase in mean |Z| with respect to α-abundance (Figures 4 and 8).

Using high-resolution spectroscopy of a sample of ≈850 FGK dwarfs in the solar-neighborhood, Adibekyan et al. (2013) reported a gap in the [α/Fe] versus [Fe/H] distribution (see their Figure 1). Using this stellar sample, Haywood et al. (2013) have proposed two distinct epochs of Milky Way star formation which give rise to the thick and thin disk, respectively. However, they are careful to note that two epochs of star formation do not necessarily give rise to two distinct disk components. Defining the "thick disk" as consisting of stars with ages greater than 8 Gyr, they detect an age–metallicity and age–σ_W relationship. These correlations mean that older, metal-poor stars will extend farther from the Galactic plane than younger, metal-rich stars, giving rise to a vertical metallicity gradient.

Due to the low resolution of SEGUE spectra, uncertainties in chemical abundances will smear out any [α/Fe] versus [Fe/H] gap that may be present (Haywood et al. 2013); thus, we cannot separate our sample using their thin/thick disk criteria (Figure 5). However, their analysis indicates that stars with [Fe/H] < −0.5 and [α/Fe] above ∼0.05 should have ages larger than 8 Gyr. Thus, our α-rich subsample, which has stars with −1.0 < [Fe/H] < −0.45 and [α/Fe] > +0.33, is comparable to their "thick disk" sample isolated by age. While they predict a negative vertical metallicity gradient for these old stars, we find a negligible change in [Fe/H] with respect to increasing |Z|. This result suggests that the age–metallicity and age–σ_W relationships are not as clean and well defined throughout the in situ disk as they find for their solar neighborhood sample.

Numerous analyses of disk chemistry find a separation in [α/Fe] versus [Fe/H] space (e.g., Fuhrmann 1998, 2011; Lee et al. 2011; Adibekyan et al. 2013; Hayden et al. 2014; Anders et al. 2014; Ramírez et al. 2013), which is oftentimes invoked to support the separable thin/thick disk paradigm. Schönrich & Binney (2009a), Haywood et al. (2013), and Minchev et al. (2013) discuss that a gap in chemical space can be a naturally occurring feature from different epochs of star formation and does not necessarily indicate two separate stellar populations. The latter picture is supported by the well-defined break in chemistry identified by Haywood et al. (2013) around 8 Gyr. However, the cleanliness of the age–chemistry relationship is unclear; the kinematically defined thin and thick disk in Ramírez et al. (2013) show significant overlaps in [O/Fe] and [Fe/H] abundance for stars over a range of ages. While we observe chemical bimodality in our sample of G dwarfs (Figure 5), it is different than the gap observed in other data sets and at least partially stems from the line-of-sight coverage of SEGUE (Section 2.4.2).

We have determined that the vertical metallicity gradient of the disk as a whole reflects the different scale heights of each [α/Fe] population. There is no break in the behavior of the disk with respect to vertical coverage and chemistry, and the typical |Z| increases as the sample becomes more α-enhanced (Figure 4). This result aligns with the "upside-down" Galaxy formation model of Bird et al. (2013), in which the thin and thick disk form during the smooth, continuous collapse of the star-forming gas reservoir, and the observations of a smoothly varying disk in Bovy et al. (2012a). In addition, every star-forming epoch, which we associate with our bins in [α/Fe], has consistent vertical metallicity structure. As noted in Carraro et al. (1998, p. 1051), "stars or clusters with different ages are not necessarily expected to trace the same metallicity gradient." The consistency over [α/Fe] suggests similar star formation processes over the development of the disk. There is no clear discrepancy in vertical chemical structure between the proposed thin- and thick-disk components.

Although our stellar sample supports a picture of a smoothly varying disk structure, it does not allow us to rule out either the multi- or single-component scenarios of Galactic disk development. However, our analysis of the vertical behavior of the Milky Way disk provides valuable constraints for Galaxy models experimenting with different evolutionary pathways, some with two distinct populations and others with a complex single population. We eagerly await upcoming data from Gaia-ESO (Gilmore et al. 2012), the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (Cui et al. 2012), GALAH (Zucker et al. 2012), and APOGEE (Ahn et al. 2014), which may be better able to constrain the disk chemical and dynamical structure.

There are two uncertainties that stem from our radial metallicity gradient correction. First, uncertainties in a star's chemistry and distance will result in a change in [Fe/H]. This effect is included as part of the uncertainties for each individual parameter, discussed in Appendix A.3. Second, there are uncertainties on the radial metallicity gradients themselves, which will change the vertical gradient on a larger scale. We examine the vertical gradients for each subsample using the extreme positive and negative radial metallicity slope values reported by Cheng et al. (2012b).

At low |Z|, the uncertainties in the radial metallicity gradient are large, resulting in a larger uncertainty in the median [Fe/H] at this height (Δ[Fe/H] ≈ 0.04 dex, see Figure 12). Above |Z| = 0.8 kpc, variations in [Fe/H] from the radial metallicity gradient are negligible. The uncertainty in the radial metallicity gradient at low |Z| also contributes uncertainty to our measured slopes. We determine the change in slope that occurs when we use the extreme values of the radial metallicity gradient, a mean change of 0.02 dex kpc⁻¹ over the different subsamples.

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We employ a bootstrap analysis to examine the variation in the sample arising from uncertainties in atmospheric parameters, photometry, and target-selection weights. For our bootstrap, we randomly select G dwarfs, sampling with replacement, and compare the vertical metallicity gradients for 100 iterations of the sample. At a given |Z|, these uncertainties typically change the median [Fe/H] by 0.02 dex. At low |Z|, the stellar density is high and our sample size is small (≈120 stars with |Z| ⩽ 0.5 kpc), resulting in uncertainties of around 0.03 dex in [Fe/H] (see Figure 3). Our α-poor subsamples are generally at low |Z|; the bootstrapping errors for these samples are typically larger, with a maximum Δ[Fe/H] of 0.1 dex.

We quantify the change in slope due to uncertainties in sample selection by comparing the gradient for each bootstrapped iteration to our measured value, determining the 1σ variation in slope. This approach typically causes the largest uncertainty in the gradient for each of the subsamples, as listed in Table 2.

We use a Monte Carlo analysis of the Galaxy model of Schönrich & Binney (2009a, 2009b) to examine the uncertainty in the vertical metallicity gradient due to errors in individual parameters (see Section 3.3). For each source of SEGUE uncertainty, we convolve the expected uncertainty for each star (based on its S/N, [Fe/H], etc.) with a Gaussian 20 times to create 20 "mock catalogs." We then apply the target-selection criteria, distance, and chemical cuts on the modeled lines of sight, modified to include uncertainties, to simulate each of our subsamples and examine how the resulting vertical metallicity gradient changes. For more details about modeling these uncertainties, see Schlesinger et al. (2012). For each |Z|, we examine the [Fe/H] values, determining the 1σ variation from the original model value. We also examine the variation in the vertical metallicity gradient slope over each iteration, determining the 1σ uncertainty from the underlying model value.

SEGUE uses color and magnitude cuts to identify the G-dwarf sample. Although the uncertainty in SDSS photometry is typically 2%–3%, stars could be erroneously included/excluded in the sample, as explored with our bootstrap analysis (Appendix A.2). These small photometric uncertainties have little effect on the vertical metallicity gradient; the uncertainty in slope due to photometry is around 0.005 dex kpc⁻¹. Undetected binarity could also shift the properties for a given star. However, Schlesinger et al. (2010) found that undetected binarity has little to no effect on [Fe/H] and photometry, implying that resulting uncertainties from binarity will be negligible. Using the methodology described in Schlesinger et al. (2012), we simulate the change in stellar properties due to an undetected secondary. This has little effect on the vertical metallicity gradient, contributing an uncertainty of 0.006 dex kpc⁻¹ and a Δ[Fe/H] at a given height of less than 0.01 dex. Finally, extinction can artificially shift stars in and out of the G-dwarf color and magnitude range. Values from Schlegel et al. (1998) assume that each star lies behind the full amount of line-of-sight dust; this can lead to a reddening overcorrection, scattering blue stars in and red stars out of the appropriate color range, creating a metallicity bias. Cheng et al. (2012a) estimate the effect of differential reddening on SEGUE main-sequence turnoff stars over lines of sight at a range of extinctions. They demonstrated that in areas of low extinction, like the <0.2 mag in our different lines-of-sight, using Schlegel et al. (1998) reddening estimates will not induce a bias in our sample against metal-rich stars by preferentially removing nearby stars. Following the methodology of Schlesinger et al. (2012), we model the reddening with respect to distance for each line of sight using the data of Cheng et al. (2012a) in a Monte Carlo analysis and find that uncertainty from reddening leads to a mean error on the slope of around 0.007 dex kpc⁻¹. It is similarly small when examining the change in [Fe/H] at a given |Z|, less than 0.01 dex.

Uncertainties associated with the estimates of atmospheric parameters are typically larger. The errors for the SSPP log g are ±0.4 dex for S/N of 25; for [α/Fe] it is ±0.10 at this S/N. These uncertainties can shift stars in and out of our G-dwarf sample, and our chemically defined subsamples. Our Monte Carlo analysis of the Schönrich & Binney (2009a, 2009b) model indicates that errors in log g and [α/Fe] change the vertical metallicity gradients of our subsamples by less than 0.02 dex in [Fe/H] at a given |Z| (see Figure 12). These uncertainties also have a small affect on the measured slope (Table 2).

Schlesinger et al. (2012) performed an in depth analysis of the random and systematic distance errors using an isochrone-matching technique on this G-dwarf sample. The random distance uncertainty stems from errors in photometry, [Fe/H], [α/Fe], and isochrone choice. There is also a systematic distance uncertainty from the age assumptions and undetected binarity. For metal-rich stars ([Fe/H] > −0.5), the random uncertainty is ≈18% and the systematic uncertainty is −3%. More metal-poor stars have a random distance uncertainty of ≈8% and a negligible systematic shift.

Uncertainties in distance will shift stars in and out of our subsamples. In addition, they will affect a star's |Z|, and thus the vertical metallicity gradient. Using the same Monte Carlo technique explained above, the random distance uncertainties change the median [Fe/H] by approximately 0.02 dex at a given |Z| and contribute around ±0.01 dex kpc⁻¹ of uncertainty to our measured slopes. The systematic distance uncertainties have a smaller effect, ±0.01 dex in [Fe/H] at a given |Z| and a ±0.007 dex kpc⁻¹ uncertainty in slope.