DETERMINING AGES OF APOGEE GIANTS WITH KNOWN DISTANCES

The APOGEE instrument is a high-resolution (R ∼ 22,500), fiber fed, near infrared (1.51–1.7 μm) spectrograph (J. Wilson 2015, in preparation) and is usually used with plug plates that feed 300 spectra from the SDSS 2.5 m telescope (Gunn et al. 2006). To observe single bright and nearby objects, such as well studied abundance calibrators, using the multi-object observing scheme of the Sloan 2.5 m telescope is an inefficient use of that valuable telescope time. Motivated by the need to access important bright abundance standard stars and Hipparcos stars, 10 fibers were installed to connect the APOGEE instrument to the NMSU 1 m telescope at the Apache Point Observatory (APO) about 50 yards away. The new fibers run underground from the APOGEE instrument port at the base of the 2.5 m to the NA2 port of the 1 m, where they are positioned at the focal plane of the NA2 port in a fixed linear configuration. This configuration allows for one science target fiber and nine sky fibers per observation.

An advantage of this new observing capability, hereafter 1 m+APOGEE, is the extended use of the APOGEE instrument during dark time, when other SDSS-III optical surveys are using the 2.5 m telescope, with minimal additional operational cost. The NMSU 1 m telescope is operated robotically during science observations as described by Holtzman et al. (2010), dramatically reducing the human-hours needed to complete these observations. The 1 m+APOGEE can observe 20–40 stars per night, depending on the target magnitude. Targets observed with the 1 m+APOGEE are typically limited to $H\lt 8$ because fainter targets require exposure times over an hour and are better suited for 2.5 m observations, and to $H\gt 0$ because such bright targets saturate the 1 m guide camera as well as the APOGEE instrument.

APOGEE spectra observed with the 1 m telescope have been found to be comparable to those taken with the 2.5 m telescope, and have been used to help calibrate the APOGEE survey data. A detailed description of these comparisons and calibration can be found in Holtzman et al. (2015).

As mentioned above, stellar age is a difficult parameter to determine, especially for red giant stars, which have similar atmospheric parameters across many ages. However, the age of a red giant star is determined by its mass. The mass of a star can be calculated from its luminosity, effective temperature, and surface gravity (see Section 4.1). The APOGEE instrument allows for effective temperature and surface gravity measurements to within 92 K and 0.11 dex, respectively, for spectra with signal-to-noise ratio ≈ 100 (Holtzman et al. 2015). Luminosities can be calculated for nearby stars with trigonometric parallax distances, such as those measured by the Hipparcos mission. Thus, stellar ages could be estimated for a sample of nearby red giant stars observed with APOGEE for which distance measurements are available. The 1 m+APOGEE capability is well suited to observe such a sample of bright nearby red giants, and these can also be directly compared to data from APOGEE survey observations of more distant, fainter stars.

Observations of 750 bright red giant stars were made between 2012 October and 2014 April using the 1 m+APOGEE. The sample was selected from the Hipparcos Catalog (van Leeuwen 2007) to have parallaxes measured to within 10%. A color cut of $(J-K)\gt 0.5$ and an absolute 2MASS H magnitude cut of ${M}_{H}\lt 2$ were used to select giants. Over 3500 targets meeting these criteria had a declination above −20, making them visible from APO. Observations were made of targets randomly selected each night from this final list. The observed sample is within 400 pc of the Sun covering all Galactic latitudes; it is displayed in Figure 1. These data are included in the SDSS DR12 (Alam et al. 2015).

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While the ages of main sequence turn-off (MSTO) stars and subgiants can be accurately determined through comparisons of measured parameters (e.g., temperature, surface gravity, luminosity) to model isochrones (e.g., Jørgensen & Lindegren 2005; Holmberg et al. 2009; Haywood et al. 2013), the measured properties of stars in more advanced stages of evolution, such as RGB and RC stars, are degenerate for stars of different ages and metallicities. Therefore, finding ages of giants from isochrone matching using these parameters is not as precise as it is for subgiants or MSTO stars. The red giant stage is brief compared to the main sequence lifetime of a star and the latter is determined primarily by the initial mass and metallicity of the star. Thus, the age of a RGB star can be constrained if the mass is known. We use stellar atmospheric parameters derived from high-resolution spectroscopy with accurately measured distances to calculate masses and estimate ages for local giant stars.

The mass of a star can be calculated from the photometrically and spectroscopically measured parameters using the following well-known relations:

$$\begin{eqnarray}&&L=4\pi {R}^{2}\sigma {T}_{{\rm{eff}}}^{4}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&g=\displaystyle \frac{{GM}}{{R}^{2}}\end{eqnarray} \tag{ 2 }$$

where σ is the Stefan Boltzman constant, and G is the gravitational constant. For this sample, the luminosities were calculated from apparent Tycho V-band magnitudes (V_T), Hipparcos distances (van Leeuwen 2007), and model-based bolometric corrections (BCs, description below). We use the Green et al. (2015) 3D dust map to get extinction estimates for our sample, but find that most of our sample is much closer than the minimum reliable distance recommended by Green et al. (2015). Including the extinction based on the minimum reliable distance results in $\lt 0.1$ dex age difference for most of the sample, and does not change any of our conclusions. Since the minimum reliable extinction distance is much farther than our sample, and those extinctions have a negligible effect on our conclusions, we do not include extinctions for any stars in our analysis. The BCs were determined using a 6th order polynomial fit to the PARSEC isochrone BCs as a function of temperature. The high order of the fit was chosen to minimize the residuals across a large temperature range, resulting in a BC uncertainty of 0.03 mag. The solar bolometric magnitude and luminosity were taken from Bahcall et al. (1995) as ${M}_{\mathrm{Bol},\odot }=4.77$ mag and ${L}_{\odot }=3.844\times {10}^{33}$ erg s⁻¹. These stellar luminosities were applied to Equation (1) along with the spectroscopic temperatures to calculate radii. The mass of each star was then calculated from the radius and the spectroscopic surface gravity using Equation (2). The derived masses are given in Table 1.

To calculate the mass error, we adopt the APOGEE reported errors for the spectroscopically derived parameters. The V_T magnitudes were taken from Anderson & Francis (2012) and have a typical uncertainty of 0.05 mag. The sample was selected to have distance errors $\lt 10\%$ to minimize the error contribution from the distance measurement. Due to the small errors in photometry and distance, the main source of error in the mass calculation comes from the ASPCAP uncertainty in surface gravity. This leads to an error in mass of 0.11 dex, about 30%.

We first derive ages from a simple RGB mass–age relation. The PARSEC isochrones were used to find a relation between log(mass) and log(age) for all evolved stars. It should be noted that we work in log₁₀(age), denoted as τ, because the isochrones are given in a log₁₀(age) grid. As the observed sample was selected for giants using a simple dereddened color cut, contributions from more advanced stages of evolution (e.g., RC and asymptotic giant branch (AGB)) are likely. As found in Section 3, 324 stars were identified as RC stars. The ages of such stars can also be constrained by their mass. We compare our derived mass to the current mass of the PARSEC isochrone points; therefore, age estimates for RC stars could have some additional uncertainty due to the mass loss prescription used in the isochrones. Due to the inclusion of RC stars, the mass–age relation was determined from all isochrone points with $\mathrm{log}\;g\lt 3.8$ instead of selecting only the RGB points from the PARSEC "stage" parameter. The relation is also metallicity dependent; therefore, it was fit for single metallicity populations with $-2.2\quad \lt \quad$[Fe/H]$\quad \lt \quad 0.6$ in steps of 0.1 dex. The mass–age relation for a single metallicity was derived by applying a quadratic fit to the isochrone points with the given metallicity. The derived relations are shown in Figure 4 for a sample of metallicities as indicated. The isochrones are given in log(age) from 9.0 to 10.1 in steps of 0.05 dex.

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Given the spectroscopic metallicity of each star, the corresponding mass–age relation was applied to determine the age from the mass. As the isochrones cover an age range of 100 Myr to 12.6 Gyr, we assign an age of 100 Myr to stars with a mass greater than 5 M_⊙. We also assign an age of 14 Gyr to stars less massive than 0.8 M_⊙, limited by the age of the universe. The distribution of ages determined through the simple mass–age relation is shown in Figure 5. The age values derived for each star are given in Table 1.

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The uncertainty in the age estimate is directly dependent on the error in the mass calculation. Using the derived mass–age relation, this results in a 0.38 dex uncertainty in log(age) for a $\mathrm{log}\;g$ uncertainty of 0.11 dex, which is a factor of two in age. The APOGEE errors in [M/H] are smaller than the isochrone [Fe/H] grid spacing and the dependence of the mass–age relation on [Fe/H] is weak compared to the isochrone grid spacing; therefore, the [M/H] errors have a negligible effect on the age uncertainty.

Although the spread in the mass–age relation for all evolved stars is mainly due to the metallicity, there is also a small spread in the possible isochrone age for a single mass and metallicity. This spread is due to the finite lifetimes of each evolutionary stage and the contamination from non-RGB isochrone points. The surface gravity, luminosity, and temperature of a single mass red giant star changes as it ascends the RGB. In addition, these parameters change as the star evolves through more advanced stages of evolution. The RGB lifetime is between a few to 10% of the total stellar lifetime; therefore, some age precision can be gained if the star's exact position along the RGB is known. Using the exact HR Diagram position also allows stars in more advanced stages of evolution, such as RC stars, to be identified and the ages to be more precisely determined based on their observed parameters. To get better age resolution, we test probabilistic isochrone matching techniques that use the individual measured parameters of each star instead of the single combined parameter of mass (see Sections 4.3 and 4.6).

If the atmospheric parameters are measured precisely, isochrone matching can be used to get some estimate of the age for each star, although not as precisely as for subgiants. As discussed in Section 4.1, if the mass of a RGB star is known then the age can be constrained. However, in this section we use the measured parameters needed to derive a mass and compare them directly to isochrones. We can apply constraints to more properties of the models if the luminosity, temperature, and surface gravity are explicitly used to find the best matching isochrone points rather than folded into the mass value. Typically, isochrone matching uses color or temperature, and luminosity to compare to observations. The precise surface gravity measured from the high-resolution APOGEE spectra allows the mass to indirectly constrain the isochrone matching. As can be seen in Figure 6, RGB and RC isochrones of different ages have a larger separation in absolute magnitude space than in surface gravity space. While surface gravity measurements provide the mass constraint, a measured distance is important in determining ages of red giant stars. For this sample we use both accurate absolute magnitudes from the Hipparcos data and the surface gravity from APOGEE high-resolution spectra to match to isochrones.

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To find an age through isochrone matching, we evaluate the age probability density function (PDF) calculated using Bayesian estimation. Following the method described by Jørgensen & Lindegren (2005) the age PDF is given by

$$\begin{eqnarray*}&&f(\tau ,\zeta ,m)\propto {f}_{0}(\tau ,\zeta ,m)L(\tau ,\zeta ,m)\end{eqnarray*}$$

where f₀ is the prior density function and L is the likelihood function. Here, both functions depend on age, τ, metallicity, ζ, and mass, m. Integrating over mass gives

$$\begin{eqnarray}&&f(\tau ,\zeta )\propto \int {f}_{0}(\tau ,\zeta ,m)L(\tau ,\zeta ,m)S(\tau ,\zeta ,m)\;{dm}\end{eqnarray} \tag{ 3 }$$

where $S(\tau ,\zeta ,m)$ is the selection function we use to account for the fact that our sample contains only evolved stars. For giant stars the mass and age are non-separable, as seen from the mass–age relation in Section 4.2. We define the selection function as $S(\tau ,\zeta ,m)\quad =$ 1 for $\mathrm{log}\;g\lt 3.8$ and ${(J-K)}_{0}\gt 0.5,$ and 0 elsewhere. The prior density function, f₀, depends on the SFH ($\psi (\tau )$), the initial mass function (IMF, $\xi (m)$), and the metallicity distribution function (MDF, $\phi (\zeta )$). In this work we have the metallicity of the star measured to within 0.05 dex, which is small compared to the width of the MDF in the solar neighborhood; therefore, we simply assume a flat MDF across the measured metallicity uncertainty. We also assume that the IMF is independent of age and metallicity. We can therefore write Equation (3) as

$$\begin{eqnarray*}&&f(\tau ,\zeta )\propto \psi (\tau )\phi (\zeta )\int L(\tau ,\zeta ,m)S(\tau ,\zeta ,m)\xi (m)\;{dm}.\end{eqnarray*}$$

The likelihood function, $L(\tau ,\zeta ,m)$, is the likelihood that a given isochrone point matches the observed star based on a set of measured parameters. The likelihood function is calculated as the product of each observable parameter compared to an isochrone point assuming Gaussian uncertainties. To build the likelihood PDF, $L(\tau ,\zeta ,m)$ is summed over all isochrone points. Integrated over the mass this has the form

$$\begin{eqnarray}L(\tau ,\zeta ) & \propto & \displaystyle \sum _{i=1}^{n}\mathrm{exp}\left(\displaystyle \frac{-{({X}_{{\rm{1,obs}}}-{X}_{{\rm{1,iso}},i})}^{2}}{2{\sigma }_{X1,\mathrm{obs}}^{2}}\right)\\ & & \times \mathrm{exp}\left(\displaystyle \frac{-{({X}_{{\rm{2,obs}}}-{X}_{{\rm{2,iso}},i})}^{2}}{2{\sigma }_{X2,\mathrm{obs}}^{2}}\right)\\ & & \times \mathrm{exp}(...)\;S({\tau }_{i},{\zeta }_{i},{m}_{i})\;\xi ({m}_{i})\;{\rm{\Delta }}m\end{eqnarray} \tag{ 4 }$$

where X is the measured stellar parameter, ${\sigma }_{X,\mathrm{obs}}$ is the uncertainty in X, and n is the number of isochrone points. In practice we only sum over isochrone points within $3\sigma$ of all of the measured parameter value. We note that because the isochrones are given in 0.05 steps in log₁₀(age), we also work in log₁₀(age). For all equations given, τ denotes log₁₀(age). We assume a Chabrier lognormal IMF (Chabrier 2001) as is provided within the PARSEC isochrones.

The measured parameters considered were temperature (${T}_{{\rm{eff}}}$), metallicity ([M/H]), surface gravity ($\mathrm{log}\;g$), and absolute Tycho V-band magnitude (${M}_{{V}_{T}}$). Metallicity is included in all cases. The PARSEC isochrones have solar α-abundances; therefore, we derive an adjusted metallicity for each star in the observed Hipparcos sample to account for α-abundance. It has been shown that α-enhanced stars appear cooler than stars with solar α-abundance of the same age and [Fe/H] (see, e.g., Salaris et al. 1993). If ignored, this effect would result in an older age assigned to an α-enhanced star. As this is exactly the trend suggested by chemical evolution models of the thick disk populations, it is crucial to account for any α-enhancements when comparing to solar abundance isochrones. To correct for the temperature shift due to the stellar α-enhancement, an adjusted [M/H] was used to compare to the isochrone [Fe/H]. We use the correction described in Salaris et al. (1993). As our sample does not contain many α-enhanced stars, only a small adjustment is required. For 94% of the observed sample the adjustment to the metallicity is less than 0.1 dex, the PARSEC isochrone [Fe/H] step size. The effect on the age estimate is almost always less than 0.1 dex. We discuss below the combination of measured parameters that allows for the most accurate age determination of red giants. For the purpose of comparing to a direct age estimate from the mass (see Sections 4.1 and 4.2) we also consider the ages derived with this method using [M/H] and mass as the measured parameters.

Using $L(\tau ,\zeta )$ as given in Equation (4), we assume a flat MDF over the uncertainty in [M/H] and integrate over metallicity to obtain the full age PDF for a single star:

$$\begin{eqnarray*}&&f(\tau )\propto \psi (\tau )L(\tau ).\end{eqnarray*}$$

The use of a grid of isochrones spaced in log(age) imposes a default prior of a flat SFH prior in log(age) or τ. We adopt a flat SFH prior in age by weighting each τ bin of the PDF by the linear age of that bin. This results in the full age PDF of a single star. In Section 4.6 we model the SFH as a parameterized function.

Using a sample of simulated stars, we examine which combination of measured parameters results in the most accurate age determination. The simulated sample was created by selecting random points from the PARSEC isochrones to create a sample with a flat distribution in age. The sample contains 1200 stars with parameters $3500\;{\rm{K}}\lt {T}_{{\rm{eff}}}\lt 5400\;{\rm{K}}$, $-1.0\lt [{\rm{Fe/H}}]\lt 0.7$, $3.8\lt \mathrm{log}\;g\lt 0$, and ${(J-K)}_{0}\gt 0.5$. This contains only evolved stars, covers a large range of metallicities, and includes the color cut imposed on the observational sample. Gaussian noise was introduced to the points based on the typical observational uncertainties of our sample, 92 K in ${T}_{{\rm{eff}}}$, 0.11 dex in $\mathrm{log}\;g$, 0.05 dex in [M/H], and 0.15 mag in ${M}_{{V}_{T}}$. From these parameters a spectroscopic mass was calculated just as is done with the observed sample. The disagreement in the true mass of the isochrones point and the mass calculated from the parameters with added noise is 38%, in agreement with the estimate based on observational errors in Section 4.1.

An age PDF was calculated for each star in this simulated sample using the method described in Section 4.3 for several combinations of measured parameters; however, [Fe/H] was included in all cases. To assign a single age to a star we take the mean of the age PDF. The age PDF for an evolved star is often not a Gaussian PDF, and can have more than one local maximum because these stars commonly have measured parameters similar to older or younger stars at a different stage of evolution (e.g., RC or AGB stars). The mean of the PDF is more sensitive to multiple peaks, which can introduce larger uncertainties in the age of a single star, but can result in a more accurate age distribution for a large sample. The difference in log(age) between the real and recovered age values for a few test parameter sets is shown in Figure 7. In these figures, a Gaussian is fit to the distribution of errors to determine the $1\sigma$ errors. Both the Gaussian fit and the σ value are indicated in the figure.

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The upper left panel of Figure 7 shows the age errors of a Bayesian isochrone matching analysis using [Fe/H] and M as the measured parameters. This analysis compares directly with the mass–age relation analysis from Section 4.2. The age errors here are generally smaller than the uncertainties from the mass–age relation due to the inclusion of priors; however, there is a tail of large age errors. If we compare this analysis with an analysis based directly on the individual parameters used to calculate the mass ([Fe/H], ${T}_{{\rm{eff}}}$, ${M}_{{V}_{T}}$, and $\mathrm{log}\;g$, lower right), the error in recovered ages is larger for the analysis based on the mass–age relation alone (upper left). This is due to the large uncertainty in the mass with no explicit constraints on other measured parameters. Analyses based on HR diagram position ([Fe/H], ${T}_{{\rm{eff}}}$, and ${M}_{{V}_{T}}$, lower left) and [Fe/H], ${T}_{{\rm{eff}}}$, and $\mathrm{log}\;g$ (upper right) illustrate the importance of including a constraint on luminosity. Due to the larger separation of RGB isochrones in ${M}_{{V}_{T}}$ than in $\mathrm{log}\;g$, see Figure 6, the distribution of age errors is always narrower when ${M}_{{V}_{T}}$ is included in the isochrone matching. The top panels contain no direct magnitude comparison and show large tails of recovered ages that are too old, indicating that the absolute magnitude gives much better age resolution in giants, particularly at young ages.

For the cases in which absolute magnitude is included in the isochrone matching and the full error distribution is smallest, the largest individual age errors occur in stars with $\mathrm{log}\;g\;\sim \;2.4$. This region of the HR diagram contains not only RGB stars, but also RC and AGB stars. F. Anders et al. (2015, in preparation) note that in their analysis, this region contains the most stars with double-peaked PDFs. An isochrone probability match based on all observed parameters ([Fe/H], ${T}_{{\rm{eff}}}$, ${M}_{{V}_{T}}$, and $\mathrm{log}\;g$) was found to give the smallest errors with a distribution width of 0.18 dex.

The distribution of recovered ages for each test case is shown in Figure 8 as the black solid line compared to the true age distribution of the sample as the dotted red line. It is again clear that when absolute magnitude is included in the isochrone matching, the youngest ages are recovered. However, the oldest ages are not recovered, even if ${M}_{{V}_{T}}$ is included. Figure 9 shows the age error as a function of true age. In this Figure it is apparent that the ages of young stars are over estimated, while the ages of old stars are significantly underestimated. We believe these biases are a consequence of taking the mean of the PDF to be the age of the star, as seen in Figure 10.

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Figure 10 shows two examples of the age PDFs from the simulated sample. The top panel is a $\tau =8.35$ star and the bottom panel is a $\tau =9.95$ star. For young stars, the PDF often has a tail toward older ages. Taking the mean of this distribution can push the age of a very young star older. For the older ages, the bias toward younger ages is likely strongly driven by the isochrone grid edge. The PDF of the older star is peaked at old ages, but is truncated at the isochrone grid edge, pushing the mean of the PDF toward younger ages. In addition, ${\rm{\Delta }}\tau$ is not linear with Δ observational uncertainty, and the separation between isochrones in ${T}_{{\rm{eff}}}$, ${M}_{{V}_{T}}$, and $\mathrm{log}\;g$ decreases with increasing age. This results in larger uncertainties in age at older ages for the same uncertainty in ${T}_{{\rm{eff}}}$, ${M}_{{V}_{T}}$, or $\mathrm{log}\;g$. We test analyzing a simulated sample using uncertainties that are an order of magnitude smaller than our typical observational uncertainties. Figure 11 shows that when the imposed uncertainties are very small, the distribution of recovered ages matches the true age distribution of the input sample. From this we conclude that the bias in recovered age is a consequence of taking the mean on the PDF, the nonlinear relation between τ and the measured parameters, and because the isochrone grid edge truncates the PDFs at τ of 10.1. This bias is most significant at the oldest age bin.

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Using the isochrone matching method described above, we determine individual ages for the sample of nearby giant stars observed with the 1 m+APOGEE. The measured parameters used to compare to the isochrones were the α-adjusted [M/H] described in Section 4.3, ${T}_{{\rm{eff}}}$, ${M}_{{V}_{T}}$, and $\mathrm{log}\;g$. The individual stellar ages are given in Table 1. The age distribution of the sample is shown in Figure 12. This distribution has a peak at $\tau =9.1$ and at $\tau =9.6$. Note that this would be expected even for a population of giants with a constant SFH due to the evolutionary rate of stars of different masses. The evolutionary rate of high mass stars is higher at younger ages and the distribution of RC lifetimes has an excess at ∼2 Gyr ($\tau =9.3$) with respect to the RGB. This results in an age distribution that is skewed toward younger ages. Figure 13 shows the expected age and metallicity distribution for a sample of evolved stars with our selection function, $S(\tau ,\zeta ,m)$. In this figure, $N(\tau ,\zeta )$ is calculated from the isochrones as

$$\begin{eqnarray*}&&N(\tau ,\zeta )=\int \xi (m)S(\tau ,\zeta ,m)\;{dm}.\end{eqnarray*}$$

We also calculate $N(\tau )$ assuming a MDF for the solar neighborhood similar to that found by Hayden et al. (2015). There is a strong peak in $N(\tau )$ at τ of 9.2 from RC and AGB stars, and a smaller peak at τ of 9.6 from RGB stars. The age distribution of the observed sample is has a similar behavior to this expected age distribution based on our selection function and the isochrones. We do note that the age of the peaks in the observed sample are slightly offset from the theoretical prediction, complicating analysis of the underlying population. However, the peaks are within 0.1 dex of the theoretical prediction, which is a reasonable result given the uncertainty of 0.18 dex in the simulated sample.

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We use the variance of the age PDFs as the uncertainty in the age for each star. The mean uncertainty in the age using this Bayesian method is 0.15 dex, with a spread of about 0.07 dex. This about half the uncertainty in age using the mass–age relation.

The derived ages described above in Section 4.3 assume a flat SFH in τ with no metallicity dependence. However, we do not know the true SFH of this sample. In this section we explore a Bayesian hierarchical modeling approach to determining a more realistic prior, ${f}_{0}(\tau ,\zeta ,m)$, for this sample. The hierarchical modeling approach uses the likelihood age PDFs from Equation (4) to determine the parameters of a model SFH. In this way we use the full age PDF of each star in the whole sample to inform our SFH rather than independently finding individual stellar ages assuming a flat SFH. Although one could adopt a SFH based on previous knowledge of the solar neighborhood, a hierarchical modeling approach allows the data to inform the SFH of the given sample. A Bayesian hierarchical analysis could be applied to a sample outside the solar neighborhood, where there are fewer known constraints on the SFH.

In this method, we model the prior, ${f}_{0}(\tau ,\zeta ,m| a)$, as a function with free parameter(s) a. The PDF for a given parameter a is then

$$\begin{eqnarray*}&&p(a| {\rm{data}})\propto p({\rm{data}}| a)\;p(a).\end{eqnarray*}$$

This can be re-written in terms of the model prior and the likelihood distributions from Section 4.3,

$$\begin{eqnarray*}&&\propto p(a)\displaystyle \int p({\rm{data}}| \tau ,\zeta ,m)\;p(\tau ,\zeta ,m| a)d\tau \;d\zeta \;{dm}\\ &&\propto p(a)\displaystyle \int p({\rm{data}}| \tau ,\zeta ,m)\;N(a)\\ &&\times {f}_{0}(\tau ,\zeta ,m| a)d\tau \;d\zeta \;{dm}\end{eqnarray*}$$

where N(a) is a model normalization term defined such that

$$\begin{eqnarray*}&&N(a)\int {f}_{0}(\tau ,\zeta ,m)\;S(\tau ,\zeta ,m)\;d\tau \;d\zeta \;{dm}=1.\end{eqnarray*}$$

This normalization term ensures that the total probability that a star in the sample is a giant given the model prior is unity. This is necessary as the sample is defined to contain only giants. Recall from Section 4.3 that the prior depends on the SFH, the IMF, and the metallicity distribution, as well as the selection function, $S(\tau ,\zeta ,m),$ to account for a sample of all giants. As we are modeling the SFH it now also depends on the parameter(s) a of the model. Therefore,

$$\begin{eqnarray*}p(a| {\rm{data}}) & \propto & p(a)\displaystyle \int p({\rm{data}}| \tau ,\zeta ,m)\;\phi (\zeta )\;\xi (m)\\ & & \times S(\tau ,\zeta ,m)\;N(a)\;\psi (\tau | a)\;d\tau \;d\zeta \;{dm}.\end{eqnarray*}$$

Here $p({\rm{data}}| \tau ,\zeta ,m)$ is the likelihood function $L(\tau ,\zeta ,m)$ for an individual star, so we can use this function and take the product over all stars:

$$\begin{eqnarray*}p(a| {\rm{data}}) & \propto & p(a)\displaystyle \prod _{i}\displaystyle \int {L}_{i}(\tau ,\zeta ,m)\;\phi (\zeta )\;\xi (m)\\ & & \times N(a)\;\psi (\tau | a)\;{S}_{i}(\tau ,\zeta ,m)\;d\tau \;d\zeta \;{dm}.\end{eqnarray*}$$

Now recalling Equation (4), we can integrate over mass and use $L(\tau ,\zeta )$, and also over metallicity, assuming a flat metallicity distribution:

$$\begin{eqnarray}&&p(a| {\rm{data}})\propto p(a)\displaystyle \prod _{i}\int {L}_{i}(\tau )\;N(a)\;\psi (\tau | a)\;d\tau .\end{eqnarray} \tag{ 5 }$$

We have already computed ${L}_{i}(\tau )$ for each star in Section 4.3; therefore, we can use these PDFs to constrain the parameter a of the model prior. The likelihood PDFs were calculated under assumed SFH, IMF, and metallicity distribution priors. The assumed IMF and MDF are the same as in Section 4.3. The effect of using a grid of isochrones in log(age) is a flat SFH prior in τ. However, the prior being modeled here is the SFH, which will be a function of τ; therefore, we do not need to explicitly remove the prior imposed by the isochrone grid. The model SFH priors also have free parameter(s) a, $\psi (\tau | a)$.

We test this hierarchical modeling method on a mock sample of stars generated to have an underlying Gaussian SFH, with a mean age of $\tau =9.0$ and an age dispersion of $\sigma =0.4$, and selected to be only giants using our selection function. We find that the hierarchical modeling accurately recovers the mean age and age dispersion of the simulated sample. The model SFH was found to have a mean age of 9.0 and an age dispersion of 0.37, very similar to the input sample. This demonstrates that the hierarchical modeling method is able to correctly recover the underlying SFH, assuming the model function is a good representation of the true SFH.

We test a few simple models for the SFH of the observed Hipparcos sample, starting with a flat SFH in log(age), τ, given by

$$\begin{eqnarray*}\psi (\tau | {\tau }_{{\rm{min}}},{\tau }_{{\rm{max}}})=\left\{\begin{array}{ll}1 & {\tau }_{{\rm{min}}}\leqslant \tau \leqslant {\tau }_{{\rm{max}}}\\ 0 & {\rm{elsewhere}}\end{array}\right.\end{eqnarray*}$$

where ${\tau }_{{\rm{min}}}$ and ${\tau }_{{\rm{max}}}$ are the free parameters. To determine the values of ${\tau }_{{\rm{min}}}$ and ${\tau }_{{\rm{max}}}$ we calculate the probability of the parameter value given the data (Equation (5)) for a grid of ${\tau }_{{\rm{min}}}$ and ${\tau }_{{\rm{max}}}$ values. We find the most likely parameters to be ${\tau }_{{\rm{min}}}$ of 8.05 and ${\tau }_{{\rm{max}}}$ of 10.1. This is a reasonable range of ages for a sample of solar neighborhood giants; however, the SFH is likely more complicated. We also test models for a linear SFH in age, and a Gaussian SFH in log(age). We find that the linear SFH model cannot recover a likely slope while maintaining a positive SFR across the full age range.

The Gaussian model has the form

$$\begin{eqnarray*}&&\psi (\tau | \mu ,\sigma )=\displaystyle \frac{1}{\sigma \sqrt{2\pi }}\mathrm{exp}\left(\displaystyle \frac{{(\tau -\mu )}^{2}}{2{\sigma }^{2}}\right)\end{eqnarray*}$$

where the mean log(age), μ, and log(age) dispersion, σ, are the free parameters. The value of these parameters was determined through a similar grid search as the previous model, resulting in a most likely μ of 9.25 and σ of 0.39. This is consistent with the distribution of ages found in Section 4.3 using the assumed flat SFH. This is also consistent with the expected age distribution of a sample of giants in the solar neighborhood as shown in Figure 13.

Although the Gaussian model finds the overall SFH of the whole sample, there are more individual star PDFs that differ largely from the average fit than would be expected from purely Gaussian wings. This suggests we need a more complex model to account for these outlier stars.

Motivated by recent work that demonstrates that α-abundance may correlate more closely with age than [Fe/H] (see, e.g., Haywood et al. 2013, and Figure 15 below) we test an α-dependent Gaussian SFH model. The SFH model is applied to a subsample of stars with a single α-abundance, and most likely parameters determined for each abundance bin. Even with an α-dependent model we find stars with age PDFs that are significantly inconsistent with the most likely SFH model for the given α-abundance. We therefore test a uniform+Gaussian α-dependent SFH model, which consists of a Gaussian SFH plus a constant SFH for some fraction of outlier stars. This model is given by

$$\begin{eqnarray*}&&\psi (\tau | \mu ,\sigma )=\displaystyle \frac{(1-A)}{\sigma \sqrt{2\pi }}\mathrm{exp}\left(\displaystyle \frac{{(\tau -\mu )}^{2}}{2{\sigma }^{2}}\right)+A\times C\end{eqnarray*}$$

where A is the outlier fraction and C is a constant. This allows the single α-abundance population to be fit by a Gaussian SFH while also allowing for a small fraction of outlier stars. Although the pure Gaussian models are sufficient to examine the mean age of the α-abundance dependent populations, when determining ages for individual stars, the age PDFs for outlier stars are significantly modified by the SFH prior. The Gaussian+uniform SFH model is needed for the SFH prior used in an empirical Bayesian analysis to determine individual star ages. We find an outlier fraction of $7.5\%$ is consistent with our sample; however, the value of the outlier fraction has a very small effect on the individual stellar ages. When applied to the whole sample independently of abundance, the Gaussian+uniform model results in a mean τ of 9.25 and a dispersion of 0.37. This is very similar to the results of the pure Gaussian model suggesting that the constant term does not have a large impact on the fit to the dominant population.

Our sample has only 64 stars with [α/M] $\quad \gt \quad 0.1$; therefore, we use single α-abundance bins independent of [M/H]. The SFH model could be applied to a larger sample of stars for monoabundance populations such as those examined by Bovy et al. (2012).

The α-dependent SFH model provides some useful constraints on the SFH of single α-abundance populations in the solar neighborhood, see Section 5.1. To determine individual stellar ages for the Hipparcos sample, we adopt the α-dependent, Gaussian+uniform SFH models as the SFH prior in an empirical Bayesian analysis, as was done in Section 4.3 assuming a flat SFH in age. For this analysis we construct a full age PDF in log age and again assign a single age to an individual star as the mean of the PDF. The individual stellar ages are given in Table 1. The full likelihood, Bayesian, and empirical Bayesian PDFs, as well as the hierarchically modeled SFH, for each star is available (doi:10.5281/zenodo.34693).

The bottom panel of Figure 12 shows the distribution of ages found for the Hipparcos sample. Using the empirical Bayesian method we again find a peak in the age distribution at 9.1 and a peak at 9.55. The behavior of this distribution is consistent with the expected distribution for our selection function; however, the exact age of the peaks is slightly shifted from the theoretical prediction in Figure 13 by about 0.1 dex. Again, this offset makes analysis of the underlying population difficult, but is a reasonable distribution given the uncertainty in age. As in Section 4.5, we use the variance of the individual age PDF as the uncertainty in the age. For this analysis the mean uncertainty is 0.1 dex with a spread of about 0.05 dex. This is a smaller age uncertainty than was found using the uninformed Bayesian analysis in Section 4.3.

The α-dependent Gaussian+uniform SFH models give a most likely mean τ and dispersion for each population of stars with a single α-abundance. The α bins used to fit the models were chosen to have at least 15 stars in each bin, with a minimum bin width of 0.02 dex in [α/M]. Figure 14 shows the mean τ of the Gaussian SFH model as a function of α-abundance. The τ dispersion of each model is indicated by the error bars. The mean age increases strongly with α-abundance, even at solar α-abundance and young ages. Recent work from the GES has shown a large spread in α-abundance at a given age with a very shallow increase in α-abundance (Bergemann et al. 2014). Other studies in the solar neighborhood have found little trend in α-abundance at young ages and a steep increase in α-abundance with age above about 8 Gyr (e.g., Haywood et al. 2013; Ramírez et al. 2013; Bensby et al. 2014). The hierarchical modeling shows that there is a correlation between α-abundance and age across the full abundance range. We note also that the dispersion in the SFH model is smaller for the more α-rich populations, and larger for the solar-α populations. This suggests that while the overall [α/M] decreased with time in the Galactic disk, some solar α-abundance stars formed early on, while the formation of α-enhanced stars was limited to early times.

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This analysis is useful for general trends in the disk populations with time; however, we would also like to examine how the individual stars deviate from that overall trend. The ages for individual stars determined using the α-dependent Gaussian+uniform model SFH as a prior in an empirical Bayesian analysis are shown as a function of stellar α-abundance in the bottom panel of Figure 15. The tight relation between α-abundance and age dictated by the SFH model is recovered. However, there is still some spread in α at all ages. Using the Bayesian ages, top panel of Figure 15, the relation between [α/M] and age is still present, confirming the motivation to use an α-dependent model SFH. The spread in α is much larger for the Bayesian ages than for the empirical Bayesian ages, particularly older than 1 Gyr. Although the trend between [α/M] and age is consistent with previous work, our sample does not have stars past 11 Gyr as is seen in some studies of the Galactic disk (e.g., Haywood et al. 2013; Bensby et al. 2014; Bergemann et al. 2014). This is likely due to our method of taking the mean of the age PDF and the isochrone grid edge of 12.6 Gyr, creating a bias against the oldest stars, as discussed in Section 4.3.

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Although the empirical Bayesian ages include an α-dependent prior, there are still a few outlier stars, suggesting that these stars are robustly very young or very old for their α-abundance. Recently, young α-rich stars have been found using asteroseimology in the Kepler (Martig et al. 2015) and CoRoT (Chiappini et al. 2015) fields. We find six stars in our sample that are α-rich ([α/M] $\gt \quad 0.13$) and strongly constrained to be younger than $\tau =9.75\;\sim \;6\;{\rm{Gyr}}$. We also find seven stars with [α/M] $\lt \quad 0.1$ and older than $\tau =9.75\;\sim \;6\;{\rm{Gyr}}$.

Previous work has relied on the [α/Fe] versus [Fe/H] plane to study the evolution of the thick and thin disk populations (e.g., Fuhrmann 1998; Prochaska et al. 2000; Reddy et al. 2006; Haywood et al. 2013; Bensby et al. 2014; Hayden et al. 2015), finding the thick disk to be older and α-enhanced, and the thin disk to be younger with solar α-abundance. We examine the [α/M] versus [M/H] relation of this sample, shown in Figure 16, with the added dimension of age, indicated by the color. Our sample shows a separation of low-α and high-α abundance populations as is typical of the local disk. The high-α track is older than the low-α track; however, the low-α track does contain 16 stars older than 6 Gyr. The ages of the high-α track are consistent with the age of the thick disk determined from the white dwarf luminosity function (see, e.g., Leggett et al. 1998; Reid 2005). The low-α population shows a "banana"-shaped distribution as is seen by Adibekyan et al. (2012) in their sample of FGK stars, Nidever et al. (2014) in a sample of RC stars observed by the APOGEE survey, and Bensby et al. (2014) in the Ti abundances of local FG dwarfs and subgiants.

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The presence of an age–metallicity relation is a basic prediction of simple stellar and galactic evolution models; as stars evolve they enrich the galaxy with new metals so that successive generations of stars are more metal rich. Recent spectroscopic studies of stars in the solar neighborhood have found a deviation from this expected age–metallicity relation. Edvardsson et al. (1993), Haywood et al. (2013), Bensby et al. (2014), and Bergemann et al. (2014) all find a ∼1.0 dex spread in metallicity at all ages, with a flat relation at ages younger than 8 Gyr and the predicted decrease in metallicity with increased age present only in stars older than 8 Gyr. The age–metallicity relation derived from our sample is shown in Figure 17. In agreement with recent studies, we find a weak age–metallicity relation at younger ages with a steeper decrease in metallicity at around 6 Gyr and a spread in metallicity across all ages. Our results agree with previous work to suggest that the chemical evolution of the Milky Way disk was more complicated than a simple closed box model.

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The red points in Figure 17 indicate the mean [M/H] and standard deviation in [M/H] for age bins of 0.2 dex. This is consistent with a very shallow age–metallicity relation below ∼6 Gyr. More interestingly, we note that the scatter in the metallicity is smaller for stars with ages $\lt 1\;{\rm{Gyr}}$. This could reflect the effects of radial migration on the MDF of the solar neighborhood. Young stars that formed near the solar neighborhood should have a similar metallicity, while star than formed elsewhere could have a different metallicity. These older stars have had time to migrate to the solar neighborhood, causing a spread in the MDF of older stars.

As full kinematic information is available for this sample, we examine trends in kinematics with age. We find that the spread in velocity increases with age in all velocity components, in agreement with the GCS results from Holmberg et al. (2009). Figure 18 shows the resulting velocity dispersion as a function of age with this sample. In this figure, each age bin contains equal numbers of stars, 50 stars each. We find that the velocity dispersion increases with age for all velocity components. The line indicates the best fit power law to the data. The highest and lowest age bin is excluded from the fit. The indices of the power law for the U, V, W, and total velocities are 0.30, 0.39, 0.44, and 0.36, respectively. The fits are normalized to 26.0, 20.2, 12.4, and 36.7 km s⁻¹ at 1 Gyr for the U, V, W, and total velocity components, respectively. The spread around the exponential fit is 4, 3, 4, and 5 km s⁻¹ for the U, V, W, and total velocity components, respectively. Visually, the trends and magnitude of the velocity dispersion with age are consistent with the improved GCS results (Holmberg et al. 2009), and the power law indices agree within 0.1 for all velocity components.

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