STELLAR POPULATION SYNTHESIS BASED MODELING OF THE MILKY WAY USING ASTEROSEISMOLOGY OF 13,000 KEPLER RED GIANTS

The main Galactic stellar-population-synthesis model used in this paper is from the Galaxia code⁴ (Sharma et al. 2011). It uses a Galactic model based on the Besançon model by Robin et al. (2003) but with some modifications. Galaxia uses its own 3D extinction scheme to specify the dust distribution. We also apply a low latitude correction to the dust maps as in Sharma et al. (2014). The isochrones to predict the stellar properties are from the Padova database (Bertelli et al. 1994; Marigo et al. 2008). The unique feature of Galaxia is its novel star-spawning scheme which, unlike previous codes, does not discretize the spatial dimensions into multiple lines of sight. Instead, it generates a continuous three-dimensional distribution of stars.

Full details of the Galactic model are available in Robin et al. (2003) and Sharma et al. (2011), but here we summarize the main features. The Milky Way in Galaxia/Besançon consists of four major components: the thin disk, the thick disk, the bar shaped bulge, and the halo. Each component has its own initial mass function (IMF) and an analytic formula for the spatial distribution of stars. The thin disk is built up using a star formation history, while other components are assumed to be populations of fixed age. In the thin disk, the metallicity of stars is governed by an age–metallicity relation and the spatial distribution of stars is governed by an age scale-height relation. The full Besançon Galactic model is tuned to satisfy constraints from the Hipparcos mission (ESA 1997) and star counts from surveys in the optical and near-infrared bands (Robin et al. 2003). The Galactic potential is computed in a self-consistent manner taking into account the results from the Hipparcos mission.

In addition to Galaxia, we used the TRILEGAL⁵ Galactic stellar-population-synthesis model (Girardi et al. 2005). Unlike Galaxia, TRILEGAL cannot generate stars over a wide angular area. Therefore, we generated stars along 21 lines of sight pointing toward the centers of the 21 Kepler CCD-module foot prints on sky. We used TRILEGAL with the default settings but with binary stars turned off. For extinction, we used the extinction model of Galaxia. One noticeable difference between Galaxia and TRILEGAL is that Galaxia uses a constant star formation rate, while the default setting in TRILEGAL uses a two-step star formation rate, in which the rate between 1 and 4 Gyr is twice that at any other time.

2.1.1. Comparing the Synthetic Catalog Photometry to the KIC

Before we can start comparing the asteroseismic results with predictions from our Galactic models, we have to make sure that the photometry of stars in the KIC and the Galactic model predictions agree with each other. This is because the asteroseismic targets observed by Kepler were selected based on KIC parameters, which ultimately relied in its photometric entries. If the photometry of the Galactic model does not match with the KIC, the model will be fundamentally inconsistent with observations, preventing meaningful model-based inferences to be made. To proceed, we first compared the photometry of the synthetic stars with that of stars in the KIC and identified any differences. Next, we investigated the cause of any mismatch and derive transformation formulae to rectify it.

We generated a synthetic catalog of the Milky Way using the code Galaxia. Stars, both from the synthetic catalog and from the KIC, that lie within 8° from the center of the Kepler field and with magnitude $r\lt 14$ were selected for comparison. Note the KIC is expected to be complete to magnitudes even fainter than 14 mag in r. We then added photometric errors to the stars. For this we derived the following formulas to approximate the actual photometric errors in 2MASS and the KIC (Figure 4 in Brown et al. 2011):

$$\begin{eqnarray}&&{\sigma }_{{\rm{2MASS}}}=0.0125+0.01\mathrm{exp}(K-12.6)\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\sigma }_{{\rm{SDSS}}}=0.015+0.01\mathrm{exp}(r-16.0).\end{eqnarray} \tag{ 2 }$$

In Figure 1, we show the distribution of r-band magnitude and various colors for both the KIC (black) and the synthetic catalog (blue).⁶ The r magnitude and the color distributions match well, which is encouraging given that we do not enforce any fine tuning of the model. However, the distributions of $g-{\rm{ddo51}}$ and g − r (Figures 1(b), (d)) for the synthetic catalog are slightly redder for the giants (right most peaks). The discrepancy for $g-{\rm{ddo51}}$ is especially large. Either the g band or the ${\rm{ddo51}}$ band photometry, or both could be discrepant. A clue to the cause of the discrepancy is provided by results of Pinsonneault et al. (2012), who found systematic differences in the KIC $(u,g,r,i,z)$ photometry with respect to SDSS $(u,g,r,i,z)$ photometry, for a small part of the Kepler field which overlaps with the SDSS. Because the isochrones that we use are calibrated to the SDSS bands, this could partly explain the mismatch that we see with the KIC. To investigate this, in Figure 2, we show a copy of Figure 1(b), but we add (in green) the $g-{\rm{ddo51}}$ color after applying the Pinsonneault et al. (2012) transformations to the KIC stars. We see that this corrected KIC distribution matches the predictions of Galaxia (blue). Hence, a correction in the g band is enough to explain the mismatch seen in the distribution of $g-{\rm{ddo51}}$ color. The transformations proposed by Pinsonneault et al. (2012) were

$$\begin{eqnarray}&&{g}_{{\rm{SDSS}}}={g}_{{\rm{KIC}}}+0.0921({g}_{{\rm{KIC}}}-{r}_{{\rm{KIC}}})-0.0985+[0.055]\\ &&{r}_{{\rm{SDSS}}}={r}_{{\rm{KIC}}}+0.0548({r}_{{\rm{KIC}}}-{i}_{{\rm{KIC}}})-0.0383+[0.0]\\ &&{i}_{{\rm{SDSS}}}={i}_{{\rm{KIC}}}+0.0696({r}_{{\rm{KIC}}}-{i}_{{\rm{KIC}}})-0.0583+[0.02]\\ &&{z}_{{\rm{SDSS}}}={z}_{{\rm{KIC}}}+0.1587({i}_{{\rm{KIC}}}-{z}_{{\rm{KIC}}})-0.0597+[0.02].\end{eqnarray} \tag{ 3 }$$

A slight adjustment of zero-points (see square brackets) from the original Pinsonneault et al. (2012) transformations was needed to make the green curve match with the Galaxia prediction. Note slight adjustments of zero points are quite common when comparing photometry obtained from different sources.

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The transformations of Pinsonneault et al. (2012) convert KIC photometry to SDSS. However, for our analysis we require the inverse transformation, that is from SDSS to KIC colors. This is because the stars observed by Kepler were selected from the KIC photometry and not a re-calibrated version of it. Equation (3) can be written in a matrix form and then inverted to get the transformation formula for ${g}_{{\rm{KIC}}}$, but this would require knowing the magnitudes in three bands ${g}_{{\rm{SDSS}}},{r}_{{\rm{SDSS}}}$, and ${i}_{{\rm{SDSS}}}$. Instead, for our use, we derived a simpler formula,

$$\begin{eqnarray}&&{g}_{{\rm{KIC}}}={g}_{{\rm{SDSS}}}-0.25({g}_{{\rm{SDSS}}}-{\rm{ddo51}})+0.048,\end{eqnarray} \tag{ 4 }$$

that makes use of only two bands, ${g}_{{\rm{SDSS}}}$ and ${\rm{ddo51}}$. The formula is easily invertible to g_SDSS = g_KIC + (g_KIC − ddo51)/3 − 0.064. Moreover, because ${g}_{{\rm{SDSS}}}$ and ${\rm{ddo51}}$ are on different photometric systems, their errors are less likely to be correlated, which means the formula given by Equation (4) is more robust. The formula was derived by matching the ${g}_{{\rm{KIC}}}-{\rm{ddo51}}$ distribution of synthetic stars to those in the KIC. The red curve in Figure 2 shows the results after applying Equation (4) to the g-band photometry of the synthetic catalog, which makes the $g-{\rm{ddo51}}$ color distribution of synthetic stars agree nicely with that of the KIC (black). This result is copied onto Figure 1(b). The formula was also found to improve the agreement for the g − r color distribution, which is shown in Figure 1(d). Thus, we can now successfully reproduce the photometry of the KIC stars using Galaxia. For the rest of the paper we use the above formula to transform the g-band photometry of synthetic stars to compare the results of Galaxia with observations. In general, the formula should be valid for the version of Padova isochrones adopted in Galaxia. We note that Farmer et al. (2013) has also suggested transformations from SDSS to KIC photometry, but those transformations (like ours) are specific to the adopted stellar models and to the adopted bolometric corrections. We found their transformations to be inadequate to explain the mismatch of the $g-{\rm{ddo51}}$ color that we encounter.

As mentioned in the previous section, stellar targets of the Kepler mission were selected based on stellar parameters in the KIC, such as ${T}_{{\rm{eff}}},\mathrm{log}g,\mathrm{log}Z,M$, and R. These parameters were derived from photometry as part of the stellar classification project (hereafter SCP). As part of the SCP, Brown et al. (2011) developed a software code⁷ to generate stellar parameters for the stars in the KIC based on a Bayesian posterior maximization scheme. In order to reproduce the selection function of the observed targets, we used this SCP software code to generate KIC-equivalent stellar parameters for stars in the synthetic catalog.

To run the SCP code, one has to specify the photometric bands to be used and the photometric uncertainty in each band. While the KIC contains photometry in 10 bands, we chose to use only the following eight $(g,r,i,z,J,H,{K}_{s},{\rm{D51}})$ that are available for the majority of stars. We set the photometric uncertainty to 0.005 which was found to best reproduce the KIC stellar parameters when applied to KIC stars. The adopted photometric uncertainty is close to but smaller than the quoted typical uncertainty in the KIC of $\sigma =0.0175$ mag for $r\lt 14$ (Brown et al. 2011).

Finally, we checked if our population synthesis model can match the distribution of stellar parameters in the KIC. The comparison is shown in Figure 3. We see that the SCP-generated stellar parameters in the synthetic catalog (red line) match well with that of the KIC (black line), which is a prerequisite to reproduce the selection function of the Kepler mission. As an aside, we compared the true stellar parameters (green line) of the synthetic catalog with that of their SCP estimates (red line). We see slight differences. The SCP code underestimates ${\rm{[Fe/H]}}$ by about 0.1 dex. The gravity of giants ($\mathrm{log}g\lt 3.5$) is overestimated and also has significant uncertainty (red peak at ∼2.6 is broadened and shifted). Finally, for stars hotter than 5200 K SCP underestimates ${T}_{{\rm{eff}}}$ by about 300 K. This means that the KIC underestimates the radii of the main-sequence stars. A similar effect was also reported by Verner et al. (2011) while comparing the asteroseismic radii with the KIC radii for a sample of subgiants and dwarfs. The underestimation of radius by KIC, has important implcations for detecting planets in habitable zones around these stars. Especially, this would lead to underestimation of planet radii when estimated using the transit method. However, this has no implication for the analysis of the red giant stars presented in this paper.

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Cool stars exhibit convectively driven oscillations whose frequency spectra show a pattern of peaks similar to that seen for the Sun. The near regular pattern is characterized by the so-called large frequency separation, ${\rm{\Delta }}\nu$, between overtone modes. The amplitude of the peaks is modulated by an envelope, which has a central frequency of maximum amplitude, ${\nu }_{{\rm{max}}}$. Theory indicates that ${\rm{\Delta }}\nu$ is related to the density of the star (Ulrich 1986), while ${\nu }_{{\rm{max}}}$ has been conjectured to depend upon the acoustic cut-off frequency of the atmosphere, and hence the surface gravity and the effective temperature of the star (Brown et al. 1991; Kjeldsen & Bedding 1995; Belkacem et al. 2011). This leads to the following approximate scaling relations:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\approx {f}_{{\rm{\Delta }}\nu }{\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{0.5},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{{\rm{max}}}}{{\nu }_{\mathrm{max},\odot }}\approx {f}_{{\nu }_{{\rm{max}}}}\displaystyle \frac{g}{{g}_{\odot }}{\left(\displaystyle \frac{{T}_{{\rm{eff}}}}{{T}_{\mathrm{eff},\odot }}\right)}^{-0.5}.\end{eqnarray} \tag{ 6 }$$

Here, ${f}_{{\rm{\Delta }}\nu }$ and ${f}_{{\nu }_{{\rm{max}}}}$ are correction factors, which we have introduced to quantify any deviation from the scaling relations. The correction factors are degenerate with the choice of solar reference values. We fix this by adopting a consistent definition for the solar reference values. Throughout the paper, the solar reference values that we use are ${\rm{\Delta }}{\nu }_{\odot }=135.1\;\mu {\rm{Hz}},{\nu }_{\mathrm{max},\odot }=3090\;\mu {\rm{Hz}}$, and ${T}_{\mathrm{eff},\odot }=5777\;{\rm{K}}$ and are adopted from Huber et al. (2011, 2013). These are the values of the Sun obtained using the Huber et al. (2009) pipeline, which was used by Stello et al. (2013). It is common practice to use such "method-specific" solar values, meaning the values returned from solar data when using the same method as used for the rest of the stellar sample. This practice is based on the assumption that method-specific systematic differences are the same for all stars (however see Hekker et al. 2013).

By rearranging the above equations, one can estimate mass and radius of a star from the above seismic observables and effective temperature as follows:

$$\begin{eqnarray}&&\displaystyle \frac{M}{{M}_{\odot }}\approx {\left(\displaystyle \frac{{\nu }_{{\rm{max}}}}{{f}_{{\nu }_{{\rm{max}}}}{\nu }_{\mathrm{max},\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{f}_{{\rm{\Delta }}\nu }{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}{\left(\displaystyle \frac{{T}_{{\rm{eff}}}}{{T}_{\mathrm{eff},\odot }}\right)}^{1.5}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{R}{{R}_{\odot }}\approx \left(\displaystyle \frac{{\nu }_{{\rm{max}}}}{{f}_{{\nu }_{{\rm{max}}}}{\nu }_{\mathrm{max},\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{f}_{{\rm{\Delta }}\nu }{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{{\rm{eff}}}}{{T}_{\mathrm{eff},\odot }}\right)}^{0.5}.\end{eqnarray} \tag{ 8 }$$

The temperatures of most of the stars observed by Kepler were estimated from photometry and have significant uncertainty and systematics that are not fully understood. In the following, we therefore investigate the particular combinations of ${\nu }_{{\rm{max}}}$ and ${\rm{\Delta }}\nu$ that comprise the seismic part of Equations (7) and (8). This will allow us to quantify the degree of error introduced into estimating mass and radius from ${\rm{\Delta }}\nu$ and ${\nu }_{{\rm{max}}}$ alone. Hence, we define the following two quantities,

$$\begin{eqnarray}&&{\kappa }_{M}={\left(\displaystyle \frac{{\nu }_{{\rm{max}}}}{{\nu }_{\mathrm{max},\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\kappa }_{R}=\left(\displaystyle \frac{{\nu }_{{\rm{max}}}}{{\nu }_{\mathrm{max},\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}.\end{eqnarray} \tag{ 10 }$$

To summarize, our strategy to compare asteroseismic parameters of synthetic stars with that of observations is as follows. We know M and R of the synthetic stars, which we use to estimate ${\rm{\Delta }}\nu$ and ${\nu }_{{\rm{max}}}$ using the scaling relations (Equations (5) and (6)). We then compute ${\kappa }_{M}$ and ${\kappa }_{R}$ of both the synthetic and observed stars (Equations (9) and (10)) and compare them.

The accuracy of the scaling relations is still not fully quantified. While one can predict ${\rm{\Delta }}\nu$ from theoretically derived oscillation frequencies, it is currently not possible to do so for ${\nu }_{{\rm{max}}}$. In the next subsection we explore the corrections to the ${\rm{\Delta }}\nu$ scaling relation.

Analysis of the theoretical oscillation frequencies derived from stellar models predict that ${\rm{\Delta }}\nu$ does not perfectly scale as ${\rho }^{0.5}$ (Stello et al. 2009; White et al. 2011; Miglio et al. 2013a; Mosser et al. 2013). To quantify this deviation we use the correction factor

$$\begin{eqnarray}&&{f}_{{\rm{\Delta }}\nu }=\left(\displaystyle \frac{{\rm{\Delta }}\nu }{135.1\;\mu {\rm{Hz}}}\right){\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{-0.5}.\end{eqnarray} \tag{ 11 }$$

The value 135.1 μHz corresponds to our adopted choice for ${\rm{\Delta }}{\nu }_{\odot }$. The stellar models show that ${f}_{{\rm{\Delta }}\nu }$ varies with metallicity, Z, mass, M, and age, τ. To estimate ${f}_{{\rm{\Delta }}\nu }$ we ran a suite of stellar models for $-1.28\lt \mathrm{log}Z\lt 2.12$ (corresponding to $-3\;\lt$ [Fe/H] $\lt \;0.4$) and $0.8\lt M/{M}_{\odot }\lt 4.0$. For this we used the MESA (v6950) 1M_pre_ms_to_wdtest suite case (Paxton et al. 2011, 2013, 2015), but without rotation or mass loss invoked. We followed the approach by White et al. (2011) to derive ${\rm{\Delta }}\nu$ for each model, designed to mimic the way ${\rm{\Delta }}\nu$ is measured from the data. For this we used GYRE (Townsend & Teitler 2013) to calculate the required radial mode frequencies. An in-depth description of this model grid and some of its applications will be presented in future (D. Stello et al. 2016, in preparation).⁸ We estimated ${f}_{{\rm{\Delta }}\nu }$ along each stellar track ranging from the zero-age main sequence until the end of helium-core burning. The results were remeshed in a grid in $\mathrm{log}\;Z,M,{E}_{{\rm{state}}},{T}_{{\rm{eff}}}$ space with dimensions of $13\times 19\times 2\times 200$. The ${E}_{{\rm{state}}}$ is the evolutionary state, with 0 for pre helium-ignition evolution and 1 for post helium-ignition evolution. The age, $\tau$, is replaced by variables ${E}_{{\rm{state}}}$ and ${T}_{{\rm{eff}}}$, which are observable (Bedding et al. 2011). Then, using interpolation we computed the correction factor ${f}_{{\rm{\Delta }}\nu }$ for each synthetic star and used it to correct ${\rm{\Delta }}\nu$. From now on we denote this derived correction factor by ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$. We show ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ in Figure 4 for different mass and metallicity. Red giant branch stars require a larger relative correction (left panels, ${T}_{{\rm{eff}}}\lt 5000$ K) compared to the red clump stars (right panels). For the red giant branch stars, the correction factor has a strong dependence on both temperature and metallicity (see also White et al. 2011). There is also a dependence on mass. This can be seen as a shift of the green curves in panels (c) and (e) with respect to the dashed curve, which is the $M=1.0\text{}{M}_{\odot }$ equivalent from panel (a).

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In order to compare observations with Galactic model predictions, it is crucial that we know the selection function of the observed targets. However, so far it has not been possible to reproduce the selection function that generates the red giant sample observed by the Kepler mission. Previous attempts to characterize it can be found in Farmer et al. (2013) and Pinsonneault et al. (2014). The main source of information is the paper by Batalha et al. (2010), which describes the selection and prioritization of Kepler targets in general. The targets were classified into 13 priority groups based on their stellar parameters. The idea was to prioritize stars for which it was possible to detect exoplanets, and if possible terrestrial planets in habitable zones. The compiled list consisted of 261,636 stars, of which the top 150,000 stars were initially selected for observations. This included 5282 giants with $\mathrm{log}{g}_{{\rm{KIC}}}\lt 3.5$ and $r\lt 14.0$, but in the end 17,471 stars with these $\mathrm{log}{g}_{{\rm{KIC}}}$ and r limits were observed. In other words, more than 50% of the total Kepler red giant sample came from stars that were added later on by the mission and for which the exact selection procedure is not documented. Other minor factors that complicate the selection function are as follows. Numerous additional targets were observed through the proposals by the Guest Observer program and the Kepler Asteroseismic Science Consortium. Each of these had their own complex and often undocumented selection schemes. Moreover, all stars were not observed for the same number of quarters. The length of observation puts a limit on the range of $\mathrm{log}g$ over which oscillations can be detected and characterized (e.g., Stello et al. 2015a).

In the following, we deduce an approximate form for the selection function of the Kepler giants by comparing the properties of the observed giant sample against that of all stars in the KIC. In Figure 5 we plot the KIC radius against the r-band magnitude of stars in the KIC. The majority of stars obey a selection of the form $({R}_{{\rm{low}}}\lt R\lt {R}_{{\rm{up}}}(r))$, where R is the stellar radius. The lower bound ${R}_{{\rm{low}}}=3.731{R}_{\odot }$ comes from the fact that the seismic analysis by Stello et al. (2013) only included stars with $\mathrm{log}{g}_{{\rm{KIC}}}\lt 3.45$. In the KIC, $\mathrm{log}{g}_{{\rm{KIC}}}$ and R almost follow a one-to-one relation, hence a limit on $\mathrm{log}{g}_{{\rm{KIC}}}$ implies a limit on R. We found that ${R}_{{\rm{up}}}(r)$ was well described by (see Figure 5)

$$\begin{eqnarray}&&{R}_{{\rm{up}}}(r)=\displaystyle \frac{3.7{R}_{{\rm{Earth}}}}{\sqrt{7.1{\sigma }_{{\rm{LC}}}/55.37}},\end{eqnarray} \tag{ 12 }$$

where ${\sigma }_{{\rm{LC}}}$ is the long cadence noise given by

$$\begin{eqnarray}&&{\sigma }_{{\rm{LC}}}=\displaystyle \frac{1}{c}\sqrt{c+7\times {10}^{6}\mathrm{max}{(1,{Kp}/14)}^{4}}\end{eqnarray} \tag{ 13 }$$

with $c=3.46\times {10}^{0.4(12-{Kp})+8}$ (Jenkins et al. 2010).

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The derivation of Equation (12) follows closely the scheme used by Batalha et al. (2010) to select and prioritize the planet search targets. The minimum detectable radius of a planet ${R}_{{\rm{planet,min}}}$ is given by

$$\begin{eqnarray}&&{R}_{{\rm{planet,min}}}=R\sqrt{7.1{\sigma }_{{\rm{tot}}}},\end{eqnarray} \tag{ 14 }$$

where the noise

$$\begin{eqnarray}&&{\sigma }_{{\rm{tot}}}=\displaystyle \frac{{\sigma }_{{\rm{LC}}}}{\sqrt{{N}_{{\rm{transit}}}{N}_{{\rm{sample}}}}}\end{eqnarray} \tag{ 15 }$$

depends on the number of transits, ${N}_{{\rm{transit}}}$, during the lifetime of the mission, ${t}_{{\rm{mission}}}$, and the number of photometric measurements per transit, ${N}_{{\rm{sample}}}$, with cadence interval, ${t}_{{\rm{cadence}}}$. ${N}_{{\rm{transit}}}$ and ${N}_{{\rm{sample}}}$ depend on mass, M, and radius, R, of the host star and on the semimajor axis, a, of the planet orbiting around it, as follows:

$$\begin{eqnarray}&&{N}_{{\rm{sample}}}=\displaystyle \frac{4}{\pi }\displaystyle \frac{2R}{{t}_{{\rm{cadence}}}}\sqrt{\displaystyle \frac{a}{{GM}}}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{N}_{{\rm{transit}}}=\displaystyle \frac{{t}_{{\rm{mission}}}}{2\pi a}\sqrt{\displaystyle \frac{{GM}}{a}}.\end{eqnarray} \tag{ 17 }$$

If the orbit radius is, say, $a=5R$, then for ${t}_{{\rm{mission}}}\;=$ 3.5 years and ${t}_{{\rm{cadence}}}\;=$ 30 minutes we have

$$\begin{eqnarray}&&{\sigma }_{{\rm{tot}}}=\displaystyle \frac{{\sigma }_{{\rm{LC}}}}{55.37}.\end{eqnarray} \tag{ 18 }$$

The condition that Kepler should be able to detect planets with radius greater than some predefined limit ${R}_{{\rm{planet,lim}}}$ means,

$$\begin{eqnarray}&&R\sqrt{7.1{\sigma }_{{\rm{tot}}}}\lt {R}_{{\rm{planet,lim}}},\end{eqnarray} \tag{ 19 }$$

using Equation (14). This puts the following limit on the radius of the host star

$$\begin{eqnarray}&&R\lt \displaystyle \frac{{R}_{{\rm{planet,lim}}}}{\sqrt{7.1{\sigma }_{{\rm{tot}}}({Kp})}}.\end{eqnarray} \tag{ 20 }$$

We found that ${R}_{{\rm{planet,lim}}}=3.7{R}_{{\rm{Earth}}}$ provides a good match to the upper envelope of the stellar radius for the red giant sample (Figure 5).

Before investigating whether the mass and radius distributions inferred from asteroseismic observations match the Galactic model predictions, we first needed to verify that we have correctly reproduced the selection function. This was done by comparing the photometric properties of the two samples. In Figure 6 we show the distributions of r-band magnitude, colors, SCP-derived stellar parameters, the seismic observables $({\rm{\Delta }}\nu$, ${\nu }_{{\rm{max}}})$, and surface gravity of our giant sample, alongside synthetic stars from the Galactic model. The seismic observables for synthetic stars were calculated using asteroseismic scaling relations (Equations (5) and (6) with ${f}_{{\rm{\Delta }}\nu }={f}_{{\nu }_{{\rm{}}{\rm{max}}}}=1$). As before, the perfect match for the r-band distribution (Figure 6(a)) is by construction, but the good match of the color distributions (Figures 6(b)–(f)) and the other parameters gives confidence that the adopted selection function is correct. The SCP-derived stellar parameters also match well (Figures 6(g)–(i)). The slight difference in the SCP-derived gravity distribution is due to our inability to exactly reproduce SCP-derived gravity, as discussed in Section 2.2. Finally, the distribution of the seismic parameters $({\rm{\Delta }}\nu$, ${\nu }_{{\rm{max}}})$, and the inferred gravity also agree with the predictions of the Galactic model (Figures 6(j)–(l)).

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The distributions of ${\kappa }_{M}$ and ${\kappa }_{R}$ (Equations (9) and (10)) are shown in Figure 7. In Figures 7(a), (b) we do not apply a correction to either of the scaling relations (${f}_{{\rm{\Delta }}\nu }={f}_{{\nu }_{{\rm{}}{\rm{max}}}}=1$), while in Figures 7(c), (d) we set ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ (obtained from our grid; Section 2.3). In both cases the ${\kappa }_{M}$ distributions do not match. However, the use of ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ clearly reduces the discrepancy. In Figures 7(e), (f), we apply a correction to the ${\nu }_{{\rm{max}}}$ scaling relation by setting ${f}_{{\nu }_{{\rm{max}}}}=1.02$ to shift ${\kappa }_{M}$ to higher values improving the agreement with observations. However, the predicted distribution is still too broad (see Sections 3.4 and 4 for further discussion on these issues). We also investigated if choosing higher photometric uncertainty while generating the SCP parameters had any effect on our findings (se Section 2.2). We found that choosing a higher photometric uncertainty had no significant effect on the mass and the radius distribution of the synthetic stars.

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The isochrones in the Galactic model use ${Z}_{\odot }=0.019$. However, recent studies suggest a lower value of ${Z}_{\odot }\approx 0.014$ (Asplund et al. 2009; Caffau et al. 2011). To be consistent with the isochrones in the Galactic model we adopted ${Z}_{\odot }=0.019$ when computing ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ from our grid. We note that, decreasing Z_⊙ from 0.019 to 0.012 when computing ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ increases the predicted values of ${\kappa }_{M}$ by about 6%. This increase in ${\kappa }_{M}$ is similar to the increase in ${\kappa }_{M}$ that occurs when ${f}_{{\nu }_{{\rm{max}}}}$ is increased from 1.0 to 1.02.

We next study if the mismatch between observed and predicted ${\kappa }_{M}$ distributions has any dependence on temperature. We found that the mismatch is minimal if we restrict the sample to ${T}_{{\rm{eff,SCP}}}\lt 4700$ but becomes prominent with increasing temperature. This is shown in Figure 8 where we plot the ${\kappa }_{M}$ distributions for ${T}_{{\rm{eff,SCP}}}\lt 4700$ and ${T}_{{\rm{eff,SCP}}}\gt 4700$ K. ${\kappa }_{M}$ was computed with ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ and ${f}_{{\nu }_{{\rm{max}}}}=1.0$. Also, we use SCP-based temperatures because, for observed stars, we only have SCP-based temperatures. The cause behind such a temperature dependent trend is not yet clear. First, these are SCP-based temperatures, which may be slightly different from the actual temperatures. The age, metallicity, and temperature are correlated with each other. So, age and metallicity-based trends could also manifest themselves as temperature trends.

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In the previous section, we showed that the ${\kappa }_{M}$ distribution of the synthetic stars generated by Galaxia does not match the observed stars. We now investigate if the mismatch is for all stars in our sample, or whether it depends on metallicity. To do this we make use of the SAGA survey, which provides metallicities for an unbiased subset of red giants observed by Kepler.

The SAGA survey (Casagrande et al. 2014) provided Strömgren photometry for giants that have seismic parameters from Kepler observations. The current release comprises, 989 Kepler red giants based on observations covering a stripe centered at galactic longitude $l=74^\circ$ and galactic longitude ranging $7.6\lt b\lt 19.9$. In SAGA, Strömgren photometry was combined with broadband photometry, and the infrared flux method was used to obtain ${T}_{{\rm{eff}}}$ with a precision of 85 K and ${\rm{[Fe/H]}}$ with a precision of 0.18 dex. We expect the SAGA sample to be an unbiased subset of the full Kepler red giant sample because SAGA observed almost all stars down to V = 15 mag, while the seismic giants are all brighter than 14th mag in the r band (roughly corresponding to V = 14.35). Casagrande et al. (2014) reported the SAGA giant sample to be 95% complete with respect to the seismic targets. To check for any potential bias, we compared the distribution of metallicity, temperature and r-band magnitude of the SAGA giants with those of the synthetic stars satisfying the Kepler red giant selection function described in Section 3.1 (Figure 9). An uncertainty of 0.18 dex (similar to that of SAGA sample) was added to the metallicity of the synthetic stars. The good agreement between the synthetic and the observed distributions confirms that the SAGA sample is an unbiased sample of our Kepler red giant sample. No other large unbiased set of metallicities are currently available for the Kepler red giants.

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We now compare the ${\kappa }_{M}$ distribution for the SAGA sample with that from the Galactic model. The ${\kappa }_{M}$ values were computed using the correction factor ${f}_{{\rm{\Delta }}\nu }$ from the grid of stellar models in Section 2.4 and assuming ${f}_{{\nu }_{{\rm{max}}}}=1.02$. Figures 10(a)–(c) show that the mean ${\kappa }_{M}$ increases with increasing metallicity. However, the change in ${\kappa }_{M}$ is stronger in the Galactic model than in the data. The difference between the model and the data is largest for the low- and high-metallicity bins, which account for 50% of the sample. For $-0.5\lt [{\rm{Fe/H}}]\lt 0.0$, the predictions of the Galactic model agree with observations.

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Figure 10(d) shows the distribution of the SAGA sample for the full ${\rm{[Fe/H]}}$ range (black) alongside the full Kepler red giant sample (green). They agree with each other, which illustrates once again that the SAGA sample is an unbiased subset of the Kepler red giant sample.

We note that the analysis of the SAGA sample has its limitations. The SAGA sample has a metallicity uncertainty of 0.18 dex, which is relatively large. So, the suggested range of validity of the Galactic model, $-0.5\lt [{\rm{Fe/H}}]\lt 0.0$, is subject to this uncertainty in metallicity. The good match of ${\kappa }_{M}$ in $-0.5\lt [{\rm{Fe/H}}]\lt 0.0$ occurs only after an ad-hoc correction of 2% to ${\nu }_{{\rm{max}}}$. This correction is not a prescription for a global change to the scaling relation, but is a guide to quantify the differences between model predictions and observations. We note that it is possible that different groups of stars (e.g., red clump and red giant branch) have different correction factors that negate each other. This would make such global correction factor unphysical.

Clusters provide a unique opportunity to test the scaling relations because stars in a cluster have a negligible spread in age, metallicity and distance. NGC 6791 is one of the oldest and most metal-rich open clusters known. Brogaard et al. (2012) estimated the age, the helium content, and the mass of stars on the lower red giant branch using information from two eclipsing binaries along with color–magnitude diagrams of the cluster (see also Brogaard et al. 2011). The measured properties are given in Table 2.

Table 2.  Properties of the Open Cluster NGC 6791

	Brogaard et al. (2011, 2012)	Miglio et al. (2012)
Metallicity ${\rm{[Fe/H]}}$	0.29 ± 0.03 ± 0.08	⋯
Distance ${(m-M)}_{V}$	13.51 ± 0.06	⋯
Helium content Y	0.3 ± 0.01	⋯
Age	8.3 ± 0.4 ± 0.7
${M}_{{\rm{RGB}}}$	1.15 ± 0.02	1.23 ± 0.02
${M}_{{\rm{RC}}}$	⋯	1.15 ± 0.03

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Miglio et al. (2012) used asteroseismology from Kepler data to measure the masses of the red giant branch and red clump stars in NGC6791. Their estimates of masses for the red giant branch stars were higher than found by Brogaard et al. (2012). Using photometry, temperature, and distance modulus, Miglio et al. (2012) also measured the radii of the stars independently of asteroseismology. For the red giant branch stars, they found the radius estimated from photometry to be in agreement with those estimated by asteroseismology, but for the red clump stars they found the seismic estimate to be smaller by about 5%. Miglio et al. (2012) noted that the factor ${f}_{{\rm{\Delta }}\nu }$ suggested by their stellar models was higher for red clump stars compared to red giant branch stars. This prompted Miglio et al. (2012) to adopt ${f}_{{\rm{\Delta }}\nu }=1.0$ for red giant branch stars and 1.027 for red clump stars, which brought the seismic and non-seismic estimates of radius in agreement with each other. When, using ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ from our stellar models, we also find ${f}_{{\rm{\Delta }}\nu }$ to be higher for red clump stars (Figure 4). However, for NGC6791, on average the red giant branch stars have $\langle {f}_{{\rm{\Delta }}\nu }\rangle =0.974$ while the red clump stars have $\langle {f}_{{\rm{\Delta }}\nu }\rangle =0.999$ (close to unity). While the relative difference between the corrections for the two groups of stars is similar to what Miglio et al. (2012) found, our ${f}_{{\rm{\Delta }}\nu }$ values are lower, and hence our seismic estimates of radius, for both red giant branch and red clump stars, do not match the non-seismic estimates. However, a change of 0.15 mag in distance modulus, which has an uncertainty of 0.06 mag, is enough to bring the photometric and seismic estimates of radius in agreement.

Our estimates of mass, using ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$, are shown in Figure 11, which match well with estimates of Brogaard et al. (2012; dashed line). When estimating the average mass of the red giant branch stars, we restricted them to be hotter than the coldest red clump star (hence less luminous than red clump stars) to avoid giants whose evolutionary state is ambiguous.

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Epstein et al. (2014) studied asteroseismic masses of metal-poor stars observed by Kepler and APOGEE that were part of the APOKASC sample (Pinsonneault et al. 2014). Nine stars were found with $[{\rm{M}}/{\rm{H}}]\lt -1$. Seven of them were labeled as halo stars and the other two as thick disk stars, based on their kinematics. By making some assumptions on $[\alpha /{\rm{Fe}}]$ and age of these stars, Epstein et al. (2014) predicted their masses. They adopted a range of 0.2–0.4 for $[\alpha /{\rm{}}\;{\rm{Fe}}]$, 10–13.77 Gyr for the age of the halo and 8–13.77 Gyr for the age of the thick disk. An approximate reproduction of these predicted masses by us is shown as the shaded region in Figure 12. Next, Epstein et al. (2014) measured the mean offset of seismic masses from these predictions, and found them to be ${\rm{\Delta }}M=0.17\pm 0.05{M}_{\odot }$. They used ${\nu }_{\mathrm{max},\odot }=3140\;\mu {\rm{Hz}}$, which is the method-specific value for the OCT method (Hekker et al. 2010) used by them. This choice is equivalent to ${f}_{{\nu }_{{\rm{max}}}}=1.016$ in our terminology (Equation (6)), because we use ${\nu }_{\mathrm{max},\odot }=3090\;\mu {\rm{Hz}}$. Our results, adopting the OCT values for the stars, with ${f}_{{\nu }_{{\rm{max}}}}=1.02$ are shown in Figure 12 and Table 1. The results, both with and without ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ correction, are shown (options 2 and 4). We find that adopting the ${f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ correction lowers the mass on average by 0.15 ${M}_{\odot }$ and brings ${\rm{\Delta }}M$ closer to the expected value of zero. If the OCT method really has method specific systematics then the above results do not suggest any additional correction to the ${\nu }_{{\rm{max}}}$ scaling relation besides that required to acccount for a higher value of ${\nu }_{\mathrm{max},\odot }$. For 8 out of 9 metal-poor giants we also have ${\rm{\Delta }}\nu$ and ${\nu }_{{\rm{max}}}$ estimates using the SYD pipeline, for which ${\nu }_{\mathrm{max},\odot }=3090\;\mu {\rm{Hz}}$. Results for these are also shown in Table 1. Here, option 4 is the best, this supports correction to both the ${\rm{\Delta }}\nu$ and the ${\nu }_{{\rm{max}}}$ scaling relations. A correction to the ${\nu }_{{\rm{max}}}$ scaling relation was also suggested by Epstein et al. (2014), based on two metal-poor stars, ν Indi (Bedding et al. 2006) and KIC 7341231 (Deheuvels et al. 2012).

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Detached eclipsing binaries can be used to measure the mass and radius of each stellar component independently of asteroseismology, and hence offer the opportunity to test the asteroseismic scaling relations. Of the 13 candidate eclipsing binaries with a pulsating red giant component currently found in the Kepler data (Gaulme et al. 2013), KIC 8410637 and KIC 9246715 are the only ones that have mass and radius measured from the analysis of the eclipses in the light curves. It can be seen from Table 1 that correcting the scaling relations, with ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ and ${f}_{{\nu }_{{\rm{max}}}}=1.02$, brings the seismic masses and radii in better agreement with the dynamical masses and radii. The correction factor ${f}_{{\rm{\Delta }}\nu }$, which depends upon the evolutionary state of a star, is different for a red giant branch star and a red clump star. Currently we do not have any information about the evolutionary state of these stars. In Table 1 the values corresponding to a red clump star are shown in square brackets. It can be seen that for both the stars the dynamical estimates of mass and radius favor them to be red giant branch stars rather than red clump stars.

Long-baseline interferometry in optical/near-infrared wavelengths from instruments like CHARA (ten Brummelaar et al. 2005) allows one to measure angular diameters of stars with 1%–2% accuracy (Huber et al. 2012). If distance is known, one can convert angular diameter to radius. However, accurate reliable distances, e.g., from parallax measurements, are currently only available for bright nearby stars. HD185351, with V = 5.18, is the third brightest star observed by Kepler (Johnson et al. 2014) and is also nearby with a parallax-based distance of 40.83 ± 0.36 pc. By combining this distance with the interferometric measurement of the angular diameter, Johnson et al. (2014) estimated the radius to be $4.97\pm 0.07{R}_{\odot }$. This estimate is slightly larger than the asteroseismic estimate of $5.35\pm 0.2{R}_{\odot }$ based on uncorrected scaling relations. With corrections, ${f}_{{\rm{\Delta }}\nu }={f}_{{\rm{\Delta }}\nu }^{{\rm{Grid}}}$ and ${f}_{{\nu }_{{\rm{max}}}}=1.02$, the asteroseismic estimate of radius is $5.23\pm 0.2{R}_{\odot }$. which agrees better with the interferometric and parallax-based measurement.