The Ages and Masses of a Million Galactic-disk Main-sequence Turnoff and Subgiant Stars from the LAMOST Galactic Spectroscopic Surveys

The LAMOST Galactic surveys (Deng et al. 2012; Zhao et al. 2012) have several components focusing on different yet related aspects of Galactic studies, namely surveys of the LAMOST Galactic halo (Deng et al. 2012), the Galactic anticenter (LSS-GAC; Liu et al. 2014), stellar clusters (Hou et al. 2013), and the Kepler fields (De Cat et al. 2015). A survey of very bright stars utilizing gray and bright lunar conditions is also included. The raw 2D spectra collected for all of the survey projects are processed uniformly with the LAMOST 2D reduction pipeline (Luo et al. 2015) to generate 1D spectra. Stellar parameters, including radial velocity ${V}_{{\rm{r}}}$, effective temperature ${T}_{\mathrm{eff}}$, surface gravity log g, and metallicity [Fe/H], are then derived from the 1D spectra with the LAMOST Stellar Parameter Pipeline (LASP; Wu et al. 2014). Both the 1D spectra and the LASP stellar parameters are publicly available via the LAMOST official data releases⁷ (Luo et al. 2012, 2015).

Since flux calibration by the default LAMOST 2D pipeline does not work well for plates of low Galactic latitudes, targeted by for example LSS-GAC spectra due to the unknown and significant extinction to the selected flux standard stars, an independent flux calibration pipeline has been developed at Peking University for LSS-GAC (Xiang et al. 2015c). A stellar parameter pipeline, LSP3, has also been developed at Peking University that delivers, in addition to ${V}_{{\rm{r}}}$, ${T}_{\mathrm{eff}}$, log g, and [Fe/H] yielded by LASP, values of [M/H], [α/M], [α/Fe], [C/H], [N/H], ${M}_{V}$, and ${M}_{{K}_{{\rm{s}}}}$, utilizing spectra processed with the LSS-GAC flux calibration pipeline (Xiang et al. 2015a; Li et al. 2016; Xiang et al. 2017a). Stellar parameters deduced with LSP3 for LSS-GAC targets, as well as values of extinction, distance, and orbital parameters inferred using the LSP3 stellar parameters, are publicly released as LSS-GAC value-added catalogs⁸ (Yuan et al. 2015b; Xiang et al. 2017b). Extensive examinations of stellar parameters yielded by LSP3 were carried out, and realistic parameter errors were assigned to each observation in a statistical way (Xiang et al. 2017a, 2017b). For spectra of signal-to-noise ratios (S/Ns) higher than 50, typical uncertainties of parameters of LSS-GAC DR2 are about 5 km/s for ${V}_{{\rm{r}}}$; 100 K for ${T}_{\mathrm{eff}}$; 0.3 mag for ${M}_{V}$ and ${M}_{{K}_{{\rm{s}}}}$; 0.1 dex for log g, [M/H], [Fe/H], [C/H], and [N/H]; 0.05 dex for [α/M] and [α/Fe]; 0.04 mag for $E(B-V)$; and 15% for distance (Xiang et al. 2017b).

Recently, we have applied the LSS-GAC flux calibration pipeline as well as Version 2 of the parameter determination pipeline LSP3 used to generate the LSS-GAC DR2 to all spectra of the LAMOST Galactic surveys collected by 2016 June. Basic stellar parameters, ${V}_{{\rm{r}}}$, ${T}_{\mathrm{eff}}$, $\mathrm{log}\,g$, and [Fe/H] yielded by the default LAMOST pipeline LASP, have been publicly released in 2016 December in the LAMOST DR4. Results from the LSP3 lead to a value-added catalog containing parameters derived from a total of 6.5 million stellar spectra of S/Ns higher than 10, for 4.4 million unique stars. The database is used to define the MSTO and subgiant star sample in the current work. Figure 1 plots the spatial distribution of stars in this value-added catalog.

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Accurate estimates of ${T}_{\mathrm{eff}}$ are essential for age estimation, particularly in avoiding significant biases and systematic errors in the results. There are two sets of ${T}_{\mathrm{eff}}$ estimates in the value-added catalog, both estimated using the MILES empirical spectral template library (Sánchez-Blázquez et al. 2006) but with two different algorithms, the weighted mean and the kernel-based principal component analysis (KPCA). Note that values of ${T}_{\mathrm{eff}}$ of the MILES template stars have been recalibrated using the color–temperature–metallicity relation of Huang et al. (2015), which is derived based on stars with direct, interferometric angular diameter measurements and Hipparcos parallaxes. The weighted-mean method works straightforwardly with well-controlled results and yields robust ${T}_{\mathrm{eff}}$ estimates, even for a spectral S/N as low as 10 (Xiang et al. 2015a, 2017b). On the other hand, due to the inhomogeneous distribution of the MILES stars in the parameter space, the weighted-mean estimates of ${T}_{\mathrm{eff}}$ suffer from the so-called clustering effects, at the level of a few tens to a hundred kelvin. While estimates of ${T}_{\mathrm{eff}}$ yielded by the KPCA method do not suffer from significant clustering effects, the robustness of the results rely heavily on the S/Ns of the spectra under analysis (Xiang et al. 2017a).

Since no direct external calibration of ${T}_{\mathrm{eff}}$ has been carried out for LSS-GAC DR2, some further examinations of the ${T}_{\mathrm{eff}}$ estimates are desirable, especially considering that there could be some potential offsets in absolute scale between temperatures inferred from the color–temperature–metallicity relation of Huang et al. (2015) and those of the theoretical isochrones. For this purpose, here we first define a photometric sample from the value-added catalog. We select stars with photometric errors smaller than 0.03 mag in the SDSS g and r bands, and with errors smaller than 0.04 mag in the 2MASS ${K}_{{\rm{s}}}$ band. We require the interstellar reddening $E(B-V)$ retrieved from the map of Schlegel et al. (1998, hereafter SFD98) to have a value smaller than 0.05 mag and that derived from the star-pair method in the value-added catalog to be smaller than 0.1 mag, considering that for stars near the Galactic plane the SFD98 map sometimes yields too small, unrealistic values. We further require the stars to have a distance larger than 300 pc as the SFD98 may have overestimated the reddening of stars of shorter distances. Stars observed with bad fibers of low throughputs or stars potentially affected by fiber crosstalk are discarded by requiring $\mathrm{BADFIBER}=0$ and ${\rm{S}}/{\rm{N}}\mbox{--}\mathrm{BRIGHTSNR}\gt -150$. Here "BRIGHTSNR" is the highest S/N of stars targeted by the five fibers adjacent to the fiber of the star of concern. Finally, we select only stars with a spectral S/N higher than 20. The above criteria lead to a total of 350,000 stars that form our photometric sample.

Figure 2 plots the differences of LSP3 estimates of ${T}_{\mathrm{eff}}$ and photometric values inferred from $g-{K}_{{\rm{s}}}$ colors using the relation of Huang et al. (2015), as a function of ${T}_{\mathrm{eff}}^{\mathrm{phot}}$ or [Fe/H]. Here the $g-{K}_{{\rm{s}}}$ colors are dereddened using SFD98 values of $E(B-V)$ and the extinction coefficients of Yuan et al. (2013). The figure shows that for the temperature range 3800–6300 K, ${T}_{\mathrm{eff}}$ values yielded by both the weighted-mean and the KPCA methods are in good agreement with the photometric values, with differences $\lt 50$ K. At higher temperatures, the KPCA method yields values that are smaller than the photometric ones by 100–200 K, while the weighted-mean method yields results that are higher than the photometric values by 150 K for stars around 7000–7500 K. No systematic trends of differences with [Fe/H] are seen for ${T}_{\mathrm{eff}}$ yielded by the weighted-mean method in the [Fe/H] range −1.0 to 0.5 dex, the applicable metallicity range of the color–metallicity–temperature relation of Huang et al. (2015). However, for ${T}_{\mathrm{eff}}$ estimated with the KPCA method, there is a positive trend of difference with [Fe/H]. Nevertheless, the dispersions of the differences amount to only ∼95 K for both sets of LSP3 temperature estimates. Considering that a bias in ${T}_{\mathrm{eff}}$ estimates, especially a positive trend of ${T}_{\mathrm{eff}}$ with [Fe/H], may lead to some undesired systematics in the age estimates and in the distribution of stars in the age–metallicity space, we choose to correct for the biases, albeit small, in the KPCA ${T}_{\mathrm{eff}}$ estimates. The correction is done in the ${T}_{\mathrm{eff}}$–[Fe/H] plane by interpolating a grid of bias values created with the photometric sample.

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Figure 3 plots the logarithmic (base 10) number density of stars in the ${T}_{\mathrm{eff}}$–${M}_{V}$ diagrams for three metallicity bins. To better illustrate the potential systematic patterns, only stars with spectral S/Ns higher than 50 are shown. The Yonsei–Yale (Y²; Demarque et al. 2004) isochrones are overplotted. The figure shows that for stars in the solar-metallicity bin, the distribution in the ${T}_{\mathrm{eff}}$–${M}_{V}$ diagram is basically consistent with isochrones. In contrast, for stars in the two metal-poor bins, the distributions deviate from the theoretical isochrones. For example, most (88%) stars of ${M}_{V}\gt 4.0$ mag in the $-1.0\lt [\mathrm{Fe}/{\rm{H}}]\lt -0.9$ dex bin have temperatures of about 200 K lower than values of the isochrones of 14 Gyr, older than the dynamic age of the universe (13.8 Gyr, e.g., Planck Collaboration et al. 2016). The offsets are significant and cannot be caused by random errors of the stellar parameters only. The apparent [Fe/H]-dependent inconsistencies are undesired and could have severe impacts on our sample selection, age estimates, and subsequent statistical analysis. Similar deviates are seen when the isochrones of the Dartmouth Stellar Evolution Database (DESP; Dotter et al. 2008) or the PAdova and TRieste Stellar Evolution Code (PARSEC; Bressan et al. 2012) are used. We suspect that the offsets are caused by different temperature scales of the color–temperature–metallicity relation of Huang et al. (2015) and the theoretical isochrones. The color–temperature–metallicity relation of Huang et al. (2015) is based on "directly" measured temperatures, while the values of ${T}_{\mathrm{eff}}$ of theoretical isochrones depend on the stellar atmospheric models adopted. The figure also shows that, as mentioned above, values of ${T}_{\mathrm{eff}}$ yielded by the weighted-mean method suffer moderate clustering effects. In the current work, we thus adopt the KPCA estimates of ${T}_{\mathrm{eff}}$ in order to avoid any potential patterns in the age estimates due to the clustering effect in the temperature estimates.

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It is of particular interest to unravel the causes of the [Fe/H]-dependent discrepancies of temperature scales between the direct measurements and the theoretical isochrones, as it may help us understand the robustness of those currently widely used stellar atmospheric models, especially those of metal-poor stars. In fact, regardless of [Fe/H], Huang et al. (2015) have found an overall systematic difference of about 100 K between the temperatures given by their photometric relations and those derived from methods based on the stellar atmospheric models in the literature (e.g., Santos et al. 2004; Valenti & Fischer 2005; Casagrande et al. 2010). This important issue is, however, outside the scope of this paper. As a temporary remedy to avoid potential biases in our age estimation, here we have opted to adjust the temperature scale of the isochrones to match that of Huang et al. (2015). In doing so, we have implicitly assumed that the isochrone temperature scale is the same as that given by Casagrande et al. (2010) utilizing the infrared flux method (IRFM), which also relies on stellar atmospheric models. Figure 4 plots the differences of the IRFM ${T}_{\mathrm{eff}}$ of Casagrande et al. (2010) and those of Huang et al. (2015) for different metallicities. It is obvious that the differences depend on temperature and metallicity. At solar metallicity, the IRFM scale of Casagrande et al. (2010) gives a ${T}_{\mathrm{eff}}$ value that is ∼100 K higher than the values yielded by the relation of Huang et al. (2015) for a temperature of ∼5800 K. The difference is consistent with the finding of Huang et al. (2015). However, the trends of differences with metallicity and temperature shown in the figure cannot be ignored for robust and unbiased age estimation. We correct the isochrone temperatures to match the scale of Huang et al. (2015) for each metallicity $[\mathrm{Fe}/{\rm{H}}]\gt -1.2$ dex, but we leave the more metal-poor isochrones untouched. The disposition will not yield any inconsistency in our results because in the current work we are concerned only with disk stars of $[\mathrm{Fe}/{\rm{H}}]\gt -1.0$ dex, and most of our sample stars have [Fe/H] higher than −0.8 dex. Nevertheless, we have carefully examined the age estimation for stars of low metallicities. A test shows that if we instead calibrate the LSP3 spectroscopic temperatures to match the IRFM scale and estimate the ages using the isochrones without temperature corrections, the resultant ages do not deviate from the current estimates by any significant amounts ($\lt 1$ Gyr). Note that the IRFM temperature scale of Casagrande et al. (2010) is only applicable to a limited color range ($0.78\lt V-{K}_{{\rm{s}}}\lt 3.15$). To overcome the limitation, the isochrone grids of hotter temperatures are corrected for using the amount of temperature corrections at the boundary. This simplification again will not cause any significant impact on our results because hot (${T}_{\mathrm{eff}}\gt 7000$ K) stars are young (≲1 Gyr) and a 100–200 K difference in temperature will cause only very small changes in the age estimates. Finally, we have to point out that the IRFM temperature scale of Casagrande et al. (2010) is only appropriate for dwarf and subgiant stars but not for giant stars. Since we focus our work on MSTO and subgiant stars, this limitation does not affect the current work.

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The bottom panels of Figure 3 compare the data and the isochrones after the temperature corrections. The plots show much better agreement, not only for the metal-poor bins but also for the solar-metallicity bin. Most of the MSTO stars are now encompassed by 14 Gyr isochrones. Nevertheless, for the metal-poor, low-temperature (${T}_{\mathrm{eff}}\lesssim 5300$ K) main-sequence stars, there are still some discrepancies between the data and the isochrones. Those offsets could either be due to possible different temperature scales of the isochrones and the IRFM calibration of Casagrande et al. (2010), or caused by overestimates of isochrone absolute magnitudes for those metal-poor, low-temperature stars. However, these remaining discrepancies are not expected to have any significant impact on our results because we focus on MSTO and subgiant stars, and our target selection criteria (Section 3) have effectively excluded those cool, metal-poor main-sequence stars.

We have chosen to use the Yonsei–Yale (${{\rm{Y}}}^{2}$) isochrones (Demarque et al. 2004) for the age estimation. The ${{\rm{Y}}}^{2}$ isochrones cover a wide range of stellar ages (0.001–20 Gyr), which is convenient for us to apply a Bayesian algorithm as biases induced by the abrupt age cutoff of the isochrone grids could be negligible. Moreover, the ${{\rm{Y}}}^{2}$ isochrones have grids of different [α/Fe], allowing us to make use of the [α/Fe] measurements to better constrain the ages. Grids of the ${{\rm{Y}}}^{2}$ isochrones are interpolated into a uniform set of grids of step 0.1 dex in [Fe/H] and [α/Fe], 0.1 Gyr in age for $\mathrm{age}\lt 1\,\mathrm{Gyr}$, 0.2 Gyr for $1\lt \mathrm{age}\lt 2\,\mathrm{Gyr}$, and 0.5 Gyr for $\mathrm{age}\gt 2\,\mathrm{Gyr}$, utilizing the interpolator provided by Demarque et al. (2004). The isochrones adopt the color table of Lejeune et al. (1998) and assign colors and magnitudes in UBVRIJHKLL'M photometric bands for each grid model. Here the UBV system is that of Buser (1978), of which the V band agrees well with that of Johnson (e.g., Bessell 2005), RI that of Bessell (1979), and JHKLL'M that of Bessell & Brett (1988).

There are also quite a few other sets of stellar isochrones that are widely used, such as the Dartmouth Stellar Evolutionary Database (DSEP; Dotter et al. 2008) and the PAdova and TRieste Stellar Evolution Code (Bressan et al. 2012). Different isochrones are based on more or less different stellar model assumptions and thus may lead to some different age estimates. Generally, the different isochrones yield stellar ages for MSTO stars with some systematic differences of the order of about 1 Gyr, along with dispersions at the same level (e.g., Haywood et al. 2013; Hills et al. 2015; Xiang et al. 2015b).

The observed properties of a star are largely determined by three parameters, namely age (τ), initial stellar mass (M), and chemical compositions (Z). In Bayesian theory, their joint (posterior) probability density function thus can be written as

$$\begin{eqnarray}&&f(\tau ,M,Z)={{Af}}_{0}(\tau ,M,Z)L(\tau ,M,Z),\end{eqnarray} \tag{ 3 }$$

where f₀ is the a priori density distribution, L is the likelihood function, and A is a normalization factor to ensure $\int \int \int f(\tau ,M,Z)d\tau {dMdZ}=1$.

Let $O$ denote the observed stellar parameters ${T}_{\mathrm{eff}}$, ${M}_{V}$, [Fe/H], and [α/Fe], and let $P$ denote the isochrone values given τ, M, and Z. The likelihood function L is then given by

$$\begin{eqnarray}&&L(\tau ,M,Z)=\displaystyle \prod _{i=1}^{n}\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{i}}\times \exp \left(-\displaystyle \frac{{\chi }^{2}}{2}\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{n}\,{\left(\displaystyle \frac{{O}_{i}-{P}_{i}(\tau ,M,Z)}{{\sigma }_{i}}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where n is the number of observables, and ${\sigma }_{i}$ is the Gaussian error of the ith observed parameter.

For the a priori density distribution, we adopt the same formula as used by Jørgensen & Lindegren (2005):

$$\begin{eqnarray}&&{f}_{0}(\tau ,M,Z)=\psi (\tau )\phi (Z| \tau )\xi (m| Z,\tau ).\end{eqnarray} \tag{ 6 }$$

Here $\psi (\tau )$ is the star formation history, $\phi (Z| \tau )$ is the metallicity distribution as a function of age, and $\xi (m| Z,\tau )$ is the IMF as a function of metallicity and age. In principle, the distributions of $\psi (\tau )$ and $\phi (Z| \tau )$ should be a function of position across the Milky Way. Considering that the star formation rate and the metallicity distribution as a function of age as well as of spatial position are not well known, we have adopted a flat distribution for both $\phi (Z| \tau )$ and $\xi (m| Z,\tau )$ to avoid potentially large biases in the resultant age estimates. The IMF is better known—the star formation process yields more low-mass stars than massive stars, and the number of stars as a function of mass can be generally well described by power laws or a log-normal distribution (e.g., Salpeter 1955; Kroupa 2001; Chabrier 2003). Here we have adopted the IMF form of Kroupa (2001):

$$\begin{eqnarray}&&\xi (m)\propto {m}^{-a},\end{eqnarray} \tag{ 7 }$$

where a = 0.3 for $m\lt 0.08\,{M}_{\odot }$, a = 1.3 for $0.08\lt m\,\lt 0.5\,{M}_{\odot }$, and a = 2.3 for $m\gt 0.5\,{M}_{\odot }$. We assume that the IMF is invariant with age and metallicity. For Galactic field stars, this may not be a bad assumption (e.g., Kroupa 2001; Kroupa et al. 2013), especially considering that our sample stars cover a rather limited mass range.

For each age, the joint probability is then evaluated using the isochrone grids with parameter values within ±3σ of the observed ones, and the age of the star of concern is then estimated by taking the mean of the distribution with the error given by the standard deviation. An alternative age estimate is obtained by taking the mode of the joint probability distribution. In the latter case, the error of the age is estimated by requiring that the 1σ error covers 68% of the area of the joint probability distribution. In some cases where the parameters are poorly estimated, the resultant joint probability has a broad distribution, peaking either near the young or old age cutoffs of the isochrones. As a consequence, the resultant mean ages tend to fall in the middle of the age interval of the isochrones, while the mode ages tend to have a value close to the upper or lower boundary of the age interval. A comparison of the two age estimates helps one to evaluate the quality of the age estimation. Such cases do not occur often, as most of our sample stars have well-determined parameters that fall within the suitable range of age estimation. The mean ages thus derived are analyzed in the sections below. The mass estimate is also taken as the weighted-mean value given by all isochrone grids within ±3σ of the observed stellar parameters. Here the weights for the mass estimates are the same as those for the age estimates. Note that for both age and mass estimation, effects from unevenly spaced age grids have been considered by multiplying the joint probability by ${\rm{\Delta }}\tau$, the age space of the grids.

The [α/Fe] value of the Y² isochrone grids is limited in the range of 0.0–0.6 dex. As many stars have [α/Fe] values close to or smaller than 0.0 dex, an [α/Fe] cutoff of the isochrones causes a cutoff in the joint probability distribution function, which induces bias in the age and mass estimates. To avoid such bias, we opt not to use [α/Fe] when calculating the joint probability. Instead, we calculate ages for isochrones of [α/Fe] values of 0.0, 0.2, and 0.4 dex separately, and we then estimate the final age by linearly extrapolating (interpolating for stars with [α/Fe] between 0 and 0.4 dex) the results to match the observed [α/Fe].

As an examination of the method, we estimate stellar age and mass for a mock data set generated with a Monte Carlo simulation, and we compare the results with true values. To generate the mock data, 35 sets of isochrones are first selected, each with a given combination of τ, [Fe/H], and [α/Fe]. The isochrones cover an age range of 0.2–13.5 Gyr, with a step of 0.2 and 0.5 Gyr, respectively, for isochrones of ages below and above 1 Gyr. The isochrones cover an [Fe/H] range of −1.8 to 0.3 dex and an [α/Fe] range of 0.0–0.4 dex, with the older, more metal-poor isochrones having higher [α/Fe] values. For each set of isochrones, mock stars for a total mass of 50,000 M_⊙ are retrieved following the IMF of Kroupa (2001). Gaussian errors are added to the retrieved ${T}_{\mathrm{eff}}$, ${M}_{V}$, and [Fe/H], with dispersions of 130 K, 0.4 mag, and 0.15 dex, respectively. Note that these values of dispersions correspond to typical, not minimum, errors of the MSTO-SG sample stars.

MSTO and subgiant stars are selected from the mock catalog, and their ages and masses are estimated with the Bayesian method described above. Figure 7 plots the differences of measured and true ages as a function of either the former or the latter. The figure shows that the mean differences are close to zero at all ages except for the oldest stars. Ages of the truly oldest ($\gt 12$ Gyr) stars are systematically underestimated by ∼1 Gyr, while stars with the oldest measured ages are in fact generally 1–2 Gyr younger. The standard deviations of the differences increase from ∼0.6 Gyr at age 2 Gyr to 2.3 Gyr at age 8 Gyr, then flatten. There are a small fraction of young stars whose ages are significantly overestimated and a small fraction of old stars whose ages are significantly underestimated due to their large parameter errors. As a result, for stars of measured ages between approximately 4 and 9 Gyr, their true ages may spread over a wide range, although the number of stars having large age errors is expected to be small. Figure 8 plots a one-dimensional distribution of the measured ages for stars with the same true age, as well as a distribution of the true ages for stars within a given range of measured age. For a given true age, the distribution of measured ages is clearly non-Gaussian but exhibits a tail at the older end, a consequence of the uneven distribution of isochrones in the ${T}_{\mathrm{eff}}$–${M}_{V}$ plane. Stars with the oldest measured ages show a tail of small values in the distribution of their true ages, mainly caused by the cutoff of true age of isochrones at 13.5 Gyr. Typical percentage values of the mean differences are a few percent, with typical standard deviations of 25% for old stars and 35% for young stars.

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Figure 9 plots a comparison of estimated and true masses. The figure shows very good consistency, with small systematic differences ($\lt 0.05$ M_⊙) for the mass range 0.7–3.0 M_⊙, along with standard deviations of only ∼ 0.09 M_⊙, indicating typical relative mass errors smaller than 10%.

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Similar analyses were carried out for other sets of parameter errors corresponding to MSTO-SG sample stars having different spectral S/Ns. For example, for the parameter error set of 150 K, 0.5 mag, and 0.15 dex for respectively ${T}_{\mathrm{eff}}$, ${M}_{V}$, and [Fe/H], the results show that the standard deviations of age differences between the measured and true values are only slightly (≲5%) increased compared to the above quoted values. For the parameter error set of 100 K, 0.3 mag, 0.1 dex, which corresponds to spectral S/Ns ≳ 60, the standard deviations are only 20% for old stars and 25% for young stars. This is encouraging since more than one-third of the sample stars have parameter errors smaller than these values.

Stellar asteroseismology is suggested to yield log g with uncertainties smaller than 0.05 dex (e.g., Creevey et al. 2013; Hekker et al. 2013; Chaplin et al. 2014). We thus expect that stellar ages derived from asteroseismic log g measurements are more accurate than—and therefore can be used to test—the current estimates. For our sample, 230 stars have asteroseismic log g measurements available from the catalog compiled by Huber et al. (2014). For those stars, we have determined their ages using the asteroseismic log g measurements along with our spectroscopic estimates of ${T}_{\mathrm{eff}}$, [Fe/H], and [α/Fe]. Figure 10 compares ages estimated using ${M}_{V}$ ("Age of this work") with those based on the asteroseismic log g measurements. The figure shows good consistency between the two sets of age estimates. The mean value of percentage differences for the whole sample is quite small, along with a standard deviation that is only 14%. Nevertheless, for old stars, ages estimated using the approach of the current work tend to be underestimated by ∼1 Gyr compared to values derived with the asteroseismic log g, consistent with the results shown in Figure 7 for the mock data set. There is also a small fraction of stars for which the current age estimates are significantly larger than the estimates based on asteroseismic log g. Some of these stars are likely subgiant or red clump stars whose absolute magnitudes and effective temperatures from the LAMOST spectra are overestimated. Note that for both sets of age estimates, the same set of parameters ${T}_{\mathrm{eff}}$, [Fe/H], and [α/Fe] is used, so the differences seen in Figure 10 reflect the errors induced by the uncertainties in ${M}_{V}$ only.

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Figure 11 compares our mass estimates with those of Huber et al. (2014) for the 230 common stars, as well as with stellar masses directly inferred from the asteroseismic scaling relation for a subset sample of 17 stars whose scaling-relation-based masses have a propagated random error smaller than 0.2 M_⊙. The ${T}_{\mathrm{eff}}$ adopted in this work is used to infer masses from the scaling relation. Masses of Huber et al. are derived using asteroseismic log g, photometric ${T}_{\mathrm{eff}}$, and [Fe/H] utilizing multiple evolutionary tracks, but mainly the DESP tracks. Interestingly, although different sets of stellar parameters and isochrones have been used in deriving those stellar masses, the figure shows quite good agreement, with a mean difference of only 0.02 M_⊙ and a dispersion of 0.11 M_⊙. Nevertheless, at the high-mass end, there is a subset of stars for which our current mass estimates are systematically lower by 0.25 M_⊙. Those stars are also the outliers found in Figure 10. At the low-mass end, it seems that the current estimates yield masses ∼0.1–0.2 M_⊙ larger. A similar result is also seen in the comparison with masses derived from the scaling relation. For those stars, duplicate observations of LAMOST, as well as an examination of ${M}_{V}$ values derived using parallaxes from the Tycho-Gaia astrometric solution (TGAS; Michalik et al. 2015; Lindegren et al. 2016), suggest that our current estimates are robust. We thus suspect that the discrepancies are probably caused by random errors in asteroseismology-based mass estimates or by systematic errors in either the asteroseismic parameters or the asteroseismic scaling relation at the low-mass end. In fact, for those low-mass stars, masses inferred from the scaling relation have a typical propagated error of ∼0.2M_⊙, which is significantly larger than the errors of our current mass estimates (∼0.05 M_⊙).

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Accurate parallaxes from the Gaia TGAS catalog (Gaia Collaboration et al. 2016; Lindegren et al. 2016) provide independent determinations of absolute magnitudes and thus can be used to test our age estimates. A cross-identification of our value-added catalog with the TGAS catalog yields more than 0.3 million common stars, and about 50,000 of them have values of ${M}_{V}$ inferred from the TGAS parallaxes with errors smaller than 0.2 mag. For those common stars, Xiang et al. (2017b) have compared values of ${M}_{V}$ and distances with those derived utilizing the TGAS parallaxes, and they found very good agreement. Here we further derive stellar ages using ${M}_{V}$ inferred from the TGAS parallaxes and parameters ${T}_{\mathrm{eff}}$, [Fe/H], and [α/Fe] from the value-added catalog to test the robustness of our age estimates.

After applying an error cut of 0.2 mag for ${M}_{V}$ inferred from the TGAS parallaxes, 5258 unique stars in common with the TGAS catalog remain in our MSTO-SG star sample. Figure 12 compares our age estimates for these MSTO-SG stars with ages estimated utilizing ${M}_{V}$ inferred from the TGAS parallaxes. The figure shows good agreement, with a mean value of percentage differences close to zero and a standard deviation of only 14%. Similar to the results shown in Figure 7 for the mock data, there is a small fraction of stars for which our results seem to be significantly overestimated, probably due to the large uncertainties of their atmospheric parameter and absolute magnitude estimates. Note that for both sets of age estimates, the same values of atmospheric parameters ${T}_{\mathrm{eff}}$, [Fe/H], and [α/Fe] are used, so any discrepancies revealed by the comparison are likely mainly caused by the uncertainties in ${M}_{V}$ estimates only.

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Open clusters in the Milky Way are generally believed to form from a monolithic gas cloud on short timescales, so that member stars of a cluster belong to a single-aged population. Ages of cluster members thus provide an independent test of the robustness of our age estimation. For this purpose, a number of LAMOST plates have been designed to target open clusters of different ages utilizing gray nights reserved for monitoring the instrument performance. Together with data from the main surveys, we are able to select MSTO-SG stars in four open clusters, namely M35 (NGC 2168), NGC 2420, M67 (NGC 2682), and Berkeley32. These clusters cover an age range from ∼100 Myr to 6 Gyr.

A detailed description of member star identification for these open clusters will be presented elsewhere (Yang et al. in preparation). Briefly, for M67 and Berkeley32, member stars are identified by combining LAMOST radial velocities and UCAC4 proper motions (Zacharias et al. 2013). For M35 and NGC 2420, contaminations from background stars are so severe that kinematics alone is insufficient for robust member identification, so additional constraints from the distance moduli are used to discard background stars that deviate significantly ($\gt 1.5$ mag) from the peak values of the clusters. The numbers of member stars that pass our selection criteria of MSTO-SG stars are listed in Table 3. Note that here we have excluded some blue stragglers that also pass our selection criteria of MSTO-SG stars but whose ages may have been significantly underestimated (see Section 7). The measured ages, metallicities, extinction values, and distance moduli of the clusters, obtained by taking means of the individual member stars as well as age estimates in the literature, are also listed in Table 3.

Table 1.  Trajectories of the MSTO in the T_eff–M_V plane, ${M}_{V}^{\mathrm{TO}}={a}_{0}$ + a₁ × T_eff + ${a}_{2}\times {T}_{\mathrm{eff}}^{2}+{a}_{3}\times {T}_{\mathrm{eff}}^{3}$ + ${a}_{4}\times {T}_{\mathrm{eff}}^{4}$, and the Adopted Minimum Temperature of the MSTO of Isochrones, ${T}_{\mathrm{eff}}^{\mathrm{MINISO}}$

[Fe/H]	[α/Fe]	a0	a1	a2	a3	a4	${T}_{\mathrm{eff}}^{\mathrm{MINISO}}$ (K)
−1.0	0.30	32.9210	−0.00828	5.10153e−07	2.79169e−11	−2.79516e−15	5678
−0.9	0.28	24.8208	−0.00376	−4.27601e−07	1.13254e−10	−5.67469e−15	5637
−0.8	0.26	15.5995	0.00139	−1.50012e−06	2.11171e−10	−8.99138e−15	5594
−0.7	0.24	6.4951	0.00655	−2.58761e−06	3.11700e−10	−1.24379e−14	5546
−0.6	0.22	−3.6820	0.01233	−3.80997e−06	4.25071e−10	−1.63391e−14	5497
−0.5	0.20	−13.6972	0.01810	−5.04614e−06	5.41152e−10	−2.03809e−14	5450
−0.4	0.16	−24.1542	0.02418	−6.35876e−06	6.65373e−10	−2.47385e−14	5416
−0.3	0.12	−35.6453	0.03088	−7.81131e−06	8.03316e−10	−2.95945e−14	5377
−0.2	0.08	−46.9116	0.03755	−9.27544e−06	9.44030e−10	−3.46030e−14	5335
−0.1	0.04	−59.0560	0.04477	−1.08696e−05	1.09792e−09	−4.01032e−14	5268
0.0	0.00	−71.1208	0.05205	−1.24941e−05	1.25640e−09	−4.58219e−14	5253
0.1	0.00	−83.8769	0.05979	−1.42343e−05	1.42708e−09	−5.20100e−14	5179
0.2	0.00	−96.9268	0.06781	−1.60510e−05	1.60669e−09	−5.85678e−14	5175
0.3	0.00	−110.4560	0.07619	−1.79643e−05	1.79699e−09	−6.55526e−14	5130
0.4	0.00	−124.7290	0.08510	−2.00142e−05	2.00203e−09	−7.31135e−14	5096
0.5	0.00	−139.8100	0.09455	−2.21929e−05	2.22051e−09	−8.11890e−14	5287

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Table 3.  Age Estimates of Open Clusters

Cluster	${\mathrm{Age}}_{\mathrm{Liter}}$	${\rm{\Delta }}{\mathrm{Age}}_{\mathrm{Liter}}$ a	Age	σ(Age)	[Fe/H]	σ[Fe/H]	${(m-M)}_{0}$	$E(B-V)$	Number of Starsb
	(Gyr)	(Gyr)	(Gyr)	(Gyr)
M35	0.15	0.1–0.2	0.7	0.5	−0.14	0.19	9.75	0.35	395 (217)
NGC 2420	2.2	1.9–2.4	2.3	0.4	−0.31	0.09	11.70	0.04	34 (24)
M67	4.0	3.5–4.8	3.9	1.2	−0.04	0.08	9.80	0.03	982 (184)
Berkeley32	6.0	5.0–7.2	6.0	2.2	−0.44	0.10	12.44	0.21	18 (17)

Notes.

^aReferences M35: von Hippel et al. (2002), Kalirai et al. (2003), Meibom et al. (2009); NGC 2420: Demarque et al. (1994), Twarog et al. (1999); M67: Demarque et al. (1992), Carraro et al. (1994), Dinescu et al. (1995), Fan et al. (1996), Richer et al. (1998), VandenBerg & Stetson (2004), Schiavon et al. (2004), Balaguer-Núñez et al. (2007), Sarajedini et al. (2009), Barnes et al. (2016); Berkeley32: Kaluzny & Mazur (1991), Richtler & Sagar (2001), Salaris et al. (2004), D'Orazi et al. (2006), Tosi et al. (2007). ^bIncluding duplicate observations, and the number of unique stars is shown in brackets.

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Figure 13 presents a direct comparison between the measured cluster ages and the literature values. The figure shows that our age estimates are largely in good agreement with the literature values, mostly derived by isochrone fitting of color–magnitude diagrams. The relatively large dispersion of ages of the individual member stars of Berkeley 32 is mainly due to the small number of member stars identified in this cluster. For the young open cluster M35, the relatively large deviation of our estimate from the literature values is caused by net overestimates of stellar ages for individual "MSTO" stars at low temperatures, as our sample selection criteria at low temperatures prefer both stars whose ${M}_{V}$ are underestimated due to random errors and background MSTO stars with old ages.

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As mentioned in Section 3, there are more than 400,000 duplicate observations for the MSTO-SG sample stars. As a sanity check, Figure 14 plots the differences of age estimates between those duplicate observations and the sample stars. Only duplicate observations that have S/Ns comparable (within 20%) to those of the sample stars are used in the comparison, and the comparisons are carried out for different S/N bins. The figure shows that for young (e.g., $\lt 6$ Gyr) stars, the duplicate observations yield ages in excellent agreement with those deduced from the default observations. For old stars, duplicate observations yield ages 1–2 Gyr younger, depending on the S/Ns. This deviation is due to the combination of the effects of the uneven distribution of the isochrones in the ${T}_{\mathrm{eff}}$–${M}_{V}$ plane and errors in the ${M}_{V}$ estimates. As a result of the uneven distribution of the isochrones in the ${T}_{\mathrm{eff}}$–${M}_{V}$ plane, deviation of the estimated ${M}_{V}$ from the true value of an "exact" MSTO star for a given effective temperature always yields an underestimated age. The dispersions of the differences are small, generally less than 20% at all ages. Note that both the default observations that define the sample and the duplicate observations contribute to the dispersions, and thus the random errors of ages estimated from the default observations as induced by the spectral noises are expected to be smaller than the dispersions (by a factor of ∼1.4).

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Figure 15 plots the differences of masses estimated from the default and the duplicate observations. The agreement is quite good. Typical dispersions are a few percent to 10%, depending on the S/Ns.

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Figure 16 plots the distributions of estimated ages and errors of the sample stars. The age distribution shows that there are more young stars than old ones in our sample. In particular, there is a peak of stars younger than 1 Gyr. Note that the distribution is a consequence of both the age distribution of the underlying stellar population and the selection effects of the observations as well as the sample definition, rather than simply the former. Both the observations and the S/N (at 4650 Å) cut prefer young stars as they are bright and blue and thus tend to have high S/Ns. Our selection criteria used to define sample stars in the HR diagram also prefer young stars. A detailed analysis of the selection effects of the sample is quite complicated and outside the scope of the current paper. The errors of age estimates vary sensitively with the S/N. For stars of S/Ns higher than 60, the relative errors of the age estimates are 20%–25%. For S/Ns around 20, the values can be as large as 45% or more. The median value of relative age errors for the whole sample is 34%. The numerous young stars in the sample contribute a significant part of this number. If only stars older than 2 Gyr are considered, the number drops to 30%. Figure 17 shows the distribution of median stellar age in 15 × 15 patches on the sky in the Galactic coordinate system. As expected, the figure presents a clear positive age gradient with increasing Galactic latitudes. Median stellar ages in the disk of $| b| \lt 10^\circ$ are younger than 2 Gyr, while at $| b| \gt 50^\circ$, the median age becomes older than 7 Gyr.

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The distributions of estimated masses and errors of the sample stars are shown in Figure 18. The masses cover a range of 0.7–3.0 M_⊙, peaking at 1.1 M_⊙. Typical errors are smaller than a few percent for low-mass stars ($\lt 1.5$ M_⊙) and are about 10% for more massive stars. The median value of the relative mass errors for the whole sample is 8%.

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Figure 19 plots the median age of sample stars in the individual mono-abundance bins of the [Fe/H]–[α/Fe] plane. To show potential patterns better, only stars of S/Ns higher than 50 are used to generate the figure because their atmospheric parameters, especially [α/Fe], are estimated with high quality. Since the precisions of the [α/Fe] estimates also vary significantly with effective temperatures, as hotter (younger) stars have less precision due to weaker spectral features of [α/Fe] indicators, we show results for stars of ${T}_{\mathrm{eff}}\lt 6500$ K only in the figure. After the ${T}_{\mathrm{eff}}$ cut, it is found that there is still a few percent of young (≲4 Gyr) stars whose [α/Fe] are artificially overestimated significantly ($\gt 0.2$ dex), which may cause fake features in the [Fe/H]–[α/Fe] patterns. Many of those stars are found to have weird spectra in the wavelength range used for the [α/Fe] estimation (4400–4600 and 5000–5300 Å), mainly due to artificial origin (e.g., contaminated by nearby bright stars or remains of cosmic ray removal), but some are also due to intrinsic origin (e.g., composite spectra). An effort to identify those weird spectra automatically is still in progress. As a remedy, here and below we replace those [α/Fe] by [α/Fe]₁ provided in the value-added catalog, which is estimated using spectral wavelength ranges of 3910–3980, 4400–4600, and 5000–5300 Å. Specifically, for stars with ${T}_{\mathrm{eff}}$ higher than 5800 K, if the [α/Fe] has a value larger than [α/Fe]₁ by 0.2 dex, then the [α/Fe]₁ is adopted. The usage of [α/Fe]₁ effectively reduces the number of young stars with significantly overestimated [α/Fe].

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Figure 19 shows clear patterns in the distribution of median stellar ages across the [Fe/H]–[α/Fe] plane. Generally, more metal-poor and α-enhanced stars have older ages than metal-rich and α-poor ones, consistent with previous findings of high-resolution spectroscopy of solar-neighborhood stars (e.g., Haywood et al. 2013; Bergemann et al. 2014). The figure further reveals several interesting features. First, the most metal-poor ($[\mathrm{Fe}/{\rm{H}}]\lt -0.5$ dex) and α-enhanced ($[\alpha /\mathrm{Fe}]\,\gt 0.2$ dex) stars are dominated by stars older than 10 Gyr. Second, stars of intermediate-to-old ages (5–8 Gyr) show a contiguous distribution across the whole metallicity range from −1.0 to 0.5 dex and exhibit a clear and sharp demarcation from younger and more α-poor ([α/Fe] ≲ 0.0) stars. Third, on the relatively α-poor ([α/Fe] ≲ 0.0) part of the distribution, stellar ages exhibit a gradient with [Fe/H]: the median ages decrease from ∼7–8 Gyr at [Fe/H] of −0.8 dex to 1–2 Gyr at supersolar metallicities.

Figure 20 plots the stellar number density distribution in the [Fe/H]–[α/Fe] plane for stars in different age bins. Here a lower S/N cut of 50 is adopted. Rather than imposing a temperature cut of 6500 K as done for Figure 19, here stars of ${T}_{\mathrm{eff}}\lt 7500$ K are adopted. This is because the hot (${T}_{\mathrm{eff}}\gt 6500$ K) stars are mainly distributed in the 0–2 Gyr bin and thus do not have an impact on the [Fe/H]–[α/Fe] patterns for older stellar populations. The figure shows that for all individual age bins, stars exhibit wide distributions in the [Fe/H]–[α/Fe] plane, implying that in a given mono-abundance bin of [Fe/H] and [α/Fe], stars could have an extensive age distribution, especially for bins of intermediate abundances (e.g., $-0.5\lesssim [\mathrm{Fe}/{\rm{H}}]\lesssim 0$, 0 ≲ [α/Fe] ≲ 0.1 dex). Nevertheless, the figure demonstrates a clear temporal evolution trend of [Fe/H]–[α/Fe] sequences. Stars in a relatively young ($\lt 8$ Gyr) age bin are distributed along a single sequence with relatively low [α/Fe] (≲0.1 dex), while in the age bin of 10–14 Gyr, stars are distributed mainly along a sequence of high [α/Fe] (≳0.1 dex), but with a weak extension to low [α/Fe] values (0.0 dex) at solar metallicity. Both the low-α and high-α sequences are presented in the age bin of 8–10 Gyr. As the age increases from 0–2 Gyr to 8–10 Gyr, [α/Fe] values of the lower-α sequence at solar metallicity increase from about −0.1 dex to about 0.0 dex. Note that the [Fe/H]–[α/Fe] sequence of the youngest stars (0–2 Gyr) exhibits a steeper slope, which is probably due to problematic [α/Fe] estimates for such young (hot) stars. From 8–10 Gyr to 10–14 Gyr, it seems that the high-α sequence extends to lower metallicity (by 0.1–0.2 dex) and slightly (≲0.05 dex) higher [α/Fe] values. Such a double-sequence feature is consistent with the widely suggested thin and thick disk sequences (e.g., Fuhrmann 1998; Bensby et al. 2003; Lee et al. 2011; Haywood et al. 2013; Hayden et al. 2015). Our results thus suggest that the Galactic thin disk became a prominent structure at 8–10 Gyr ago, while the Galactic thick disk formed at an earlier epoch and was almost quenched at about 8 Gyr ago.

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Figure 21 plots the density distribution of the sample stars in the age–[α/Fe] and age–[α/H] planes. Here the [α/H] is converted from [α/Fe] and [Fe/H]. Only stars of ${\rm{S}}/\mathrm{Ns}\gt 50$ and ${T}_{\mathrm{eff}}\lt 7500$ K are used to ensure reliable [α/Fe] estimates. Moreover, as has been discussed above, the current [α/Fe] estimates for hot (young) stars are likely problematic, so we further discard stars younger than 2 Gyr. Note that despite these measures, some young (e.g., $\lt 4$ Gyr) stars with problematic [α/Fe] estimates (e.g., $\gt 0.2$ dex) still remain in the figure. The figure shows two sequences in the age–[α/Fe] plane. Stars younger than 8 Gyr belong to a sequence of lower [α/Fe] values, and the [α/Fe] slowly increases with age in an approximately linear manner with a slope of ≲0.02 dex/Gyr. At the older end, the low-α sequence extends to an age older than 10 Gyr. There is also a sequence with higher [α/Fe], which has an almost constant [α/Fe] value of about 0.25(±0.05) dex for stars older than 10 Gyr. At the younger end, the high-α sequence extends to ∼8 Gyr, when it connects with the low-α sequence, consistent with the results from Figure 20. The presence of two age–[α/Fe] sequences either suggests the existence of two distinct phases of formation history of the Galactic disk (e.g., Haywood et al. 2013; Xiang et al. 2015a) or is a natural consequence of a continuous disk formation process (e.g., Schönrich & Binney 2009b). Whatever processes have caused the multiple age–[α/Fe] relations, it seems that 8–10 Gyr is a special epoch in the disk formation history.

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The age–[α/H] plane exhibits a significant lack of old ($\gt 8$ Gyr), α-rich ($\gt 0.0$) stars, leading to a negative age–[α/Fe] sequence at early time. At the younger end, it seems that the sequence extends to ∼6 Gyr, when the [α/H] reaches a maximum value of 0.3–0.4 dex. At any given age younger than 8 Gyr, the [α/H] exhibits a wide distribution. Nevertheless, it seems that stars younger than 5 Gyr follow an overall negative age–[α/H] sequence, rather than a flat one. This younger sequence has a median [α/H] of about −0.3 dex at an age of 5 Gyr and reaches a median [α/H] value of −0.2 dex at 2 Gyr. At the intermediate age range of 5–8 Gyr, overlaps of the two sequences seem to have smoothed the negative age–[α/H] trends. At the high-[α/H] end, the contours show positive slopes, probably a natural consequence of the overlapping of the two sequences.

Figure 22 plots the density distribution of the sample stars in the age–[Fe/H] plane. To ensure small uncertainties in age and [Fe/H] estimates in order to better illustrate systematic trends, only stars of ${\rm{S}}/\mathrm{Ns}\gt 50$ and ${T}_{\mathrm{eff}}\lt 8000$ K are shown. Stars younger than 1 Gyr are discarded for completeness (in [Fe/H]) reasons. That is, as the temperature of an MSTO star depends sensitively on both age and metallicity, a ${T}_{\mathrm{eff}}$ cut of 8000 K discards more metal-poor, young ($\lt 1$ Gyr) stars than metal-rich ones, thus leading to undesired trends in the age range 0–1 Gyr. The figure shows a wide range of [Fe/H] at all ages younger than 8 Gyr. At the older end ($\gt 8$ Gyr), there is an obvious lack of metal-rich stars, yielding a relatively tight age–[Fe/H] correlation. The patterns are in good agreement with previous findings for stars in the solar neighborhood (e.g., Haywood et al. 2013; Bergemann et al. 2014). The relatively tight age–[Fe/H] correlation for old disk stars implies that at any given time, the interstellar medium forming the stars was relatively well mixed. On the other hand, the broad range of [Fe/H] values for young disk stars at a given age suggests a more complicated chemical enrichment history. As the sample stars cover a large volume, one possible cause of the broad [Fe/H] distribution is the existence of both radial and vertical [Fe/H] gradients for mono-age stellar populations (Xiang et al. 2015b). However, it is also found that even in a limited volume, for instance, the solar neighborhood, the age–[Fe/H] relation for young ($\lt 8$ Gyr) stars still exhibits a broad distribution. The inevitable presence of mixing of stars born at different positions (thus with different values of [Fe/H]) caused by stellar radial migration (e.g., Sellwood & Binney 2002; Roškar et al. 2008; Schönrich & Binney 2009a; Loebman et al. 2011) has certainly played a role in such [Fe/H] broadening. Whereas for very young (e.g., ∼1 Gyr) stars, the broad [Fe/H] distribution is probably largely caused by a sustained star-formation process via accreting metal-poor gas from outside the disk, as the timescale is too short for radial migration to make a great impact.

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In addition to the above qualitative patterns in agreement with the previous findings, the current large sample also reveals several interesting features. First, rather than a "flat" age–[Fe/H] relation, as suggested by the previous studies (e.g., Bergemann et al. 2014), young ($\lt 5$ Gyr) disk stars seem to exhibit a negative overall trend of [Fe/H] with age, similar to that found for [α/H]. Few studies exist on such a possible negative age–[Fe/H] trend for the young disk stars due to the limited size of the stellar sample available previously. A further, more careful analysis shows that the slopes of the age–[Fe/H] relations of young stars vary with Galactocentric radius. In the outer disk, the negative age–[Fe/H] relation becomes steeper. The observed age–[Fe/H] relation of the Galactic disk is thus similar to the age–[α/H] relation, composed of at least two negative sequences, one for old (≳8 Gyr) stars and another for young (≲5 Gyr) stars. At the intermediate age range of 5–8 Gyr, mixing of stars that follow the two separate sequences makes the trend less distinct. These two distinct sequences of age–[Fe/H] relation, if confirmed, may provide important constraints on the chemical enrichment history of the Galactic disk—it is possible that they result from two different global chemical enrichment processes of the Galactic disk. Interestingly, utilizing about 20,000 subgiant stars selected from the LAMOST DR2, Liu et al. (2015) find evidence of a "narrow stripe" of stars alongside the "main stripe" of stars in the age–[Fe/H] plane. They interpret those "narrow stripe" stars as migrators. However, we note that their sample is less complete than ours in the age–[Fe/H] plane as their sample lacks both metal-rich and metal-poor young ($\lt 4$ Gyr) stars (see their Figure 7). Second, there is a considerable fraction of intermediate-aged (5–8 Gyr) stars of $[\mathrm{Fe}/{\rm{H}}]\gt 0.3$ dex. As a result, the density contours at the metal-rich end show positive slopes. One possible origin of these positive trends is that those metal-rich stars are migrators from the inner disk: the older the stars, the farther from the inside, as they have a longer time to reach their current positions (Loebman et al. 2011). Alternatively, a natural explanation of those positive trends is the mixing of stars from the two sequences of age–[Fe/H] relations: the intermediate-aged stars of $[\mathrm{Fe}/{\rm{H}}]\gt 0.3$ dex belong to the older sequence, whereas the young, metal-rich stars are mainly composed of stars following the younger sequence. In addition, there are a considerable number of young ($\lt 4$ Gyr), metal-poor stars in our sample; some of them can be as metal-poor as ($\lesssim -0.6$ dex) the oldest ($\gt 10$ Gyr) ones. The origin of those young yet metal-poor stars, as well as that for the young [α/H]-poor stars in Figure 21, needs to be further studied. Finally, we note that the distribution also shows a number of substructures or overdensities whose genuineness and origins remain to be further studied.

Figure 23 plots the median age of stars at different positions across the R–Z plane of the Galactic disk. Here R is the projected Galactocentric distance in the disk midplane, and Z the height above the disk midplane. The top panel presents results from all of the sample stars. Generally, the data exhibit negative age gradients in the radial and positive age gradients in the vertical direction. At small heights, the outer disk of $R\gtrsim 9\,\mathrm{kpc}$ is dominated by young (≲2 Gyr) stars, which reach larger heights above the disk plane at the farther disk, which exhibits a strong flare in median stellar age. The inner disk ($R\lesssim 9$ kpc) exhibits a positive vertical age gradient for $| Z| \lesssim 1\,\mathrm{kpc}$, while at larger heights above the disk plane, old (≳10 Gyr) stars dominate the population with no significant vertical gradients. However, for many bins near the boundary of the R–Z plane covered by the sample stars, the stellar populations are dominated by unexpected young to intermediate-aged stars. At large height, e.g., $Z\gt 2\,\mathrm{kpc}$, those unexpected features are likely caused by blue stragglers whose ages have been artificially underestimated, as will be discussed in Section 7. Those stars are usually hot and bright and thus can be detected at large distances. To reduce the contaminations of blue stragglers, in the bottom panel of Figure 23, we present the age distribution after excluding stars of ${T}_{\mathrm{eff}}\gt 7000$ K. The result shows much more clean patterns, and the unexpected young populations at large heights in the inner disk now largely disappear. On the other hand, since intrinsically young stars are also discarded by the temperature cut, the outer disk exhibits systematically older ages compared to those shown in the top panel. Nevertheless, the overall structures and patterns remain unaffected.

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A radial age gradient of the geometrically thick disk was also presented by Martig et al. (2016a) using giants from the APOGEE survey. A flaring young stellar disk in the outer part has been observed previously via star counts (e.g., Derriere & Robin 2001; López-Corredoira et al. 2002; López-Corredoira & Molgó 2014) and is well reproduced by simulations (e.g., Narayan & Jog 2002; Rahimi et al. 2014; Minchev et al. 2015) as a suggested consequence of a weaker restoring force at the outer Galactocentric radii. Nevertheless, Figure 23 demonstrates the first explicit picture of disk flare in stellar age, which will provide further constraints on disk flare models.

Note, however, that our results are no doubt affected by some selection effects since the sample is a magnitude-limited one. Younger stars tend to be brighter and thus probe a larger volume than older, fainter stars. The unexpectedly young stellar ages near the boundary of the R–Z plane at small heights of the inner disk are likely due to such selection effects. The age distribution in the outer disk is probably also suffering from severe biases due to selection effects. A detailed and quantitative study of the selection effects of our sample stars is beyond the scope of this paper and will be presented elsewhere.