Asteroseismology and Gaia: Testing Scaling Relations Using 2200 Kepler Stars with TGAS Parallaxes

Scaling relations for solar-like oscillations are based on the global asteroseismic observables ${\nu }_{\max }$, the frequency of maximum power, and ${\rm{\Delta }}\nu$, the average separation of oscillation modes with the same spherical degree and consecutive radial order. The relations are defined as follows (Kjeldsen & Bedding 1995):

$$\begin{eqnarray}&&{\rm{\Delta }}\nu \propto {\left(\displaystyle \frac{M}{{R}^{3}}\right)}^{1/2},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\nu }_{\max }\propto \displaystyle \frac{M}{{R}^{2}\sqrt{{T}_{\mathrm{eff}}}}.\end{eqnarray} \tag{ 2 }$$

Equations (1) and (2) can be rearranged to calculate radius as follows:

$$\begin{eqnarray}&&\displaystyle \frac{R}{{R}_{\odot }}\approx \left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{{\rm{T}}}_{\mathrm{eff},\odot }}\right)}^{1/2}.\end{eqnarray} \tag{ 3 }$$

We used ${{\nu }_{\max }}_{\odot }=3090$ $\mu \mathrm{Hz}$ and ${\rm{\Delta }}{\nu }_{\odot }=135.1$ $\mu \mathrm{Hz}$, the solar reference values for the SYD pipeline (Huber et al. 2011). Corrections for the ${\rm{\Delta }}\nu$ scaling relation (see Section 1) were calculated using asfgrid (Sharma et al. 2016).¹⁹ To calculate asteroseismic distances, we combined ${T}_{\mathrm{eff}}$ with the radius from Equation (3) to calculate luminosity, and then used the 2MASS K-band magnitude with bolometric corrections derived by linearly interpolating ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, $[\mathrm{Fe}/{\rm{H}}]$, and A_V in the MIST/C3K grid (C. Conroy et al. 2017, in preparation²⁰ ). To estimate A_V, we used the 3D reddening map by Green et al. (2015), as implemented in the mwdust package by Bovy et al. (2016). The derived distances, extinction values, and bolometric corrections were iterated until convergence.

Parallaxes can also be used to calculate luminosities (and hence radii), which can be compared to asteroseismic radii. To convert parallaxes into distances, we used an exponentially decreasing volume density prior with a length scale of 1.35 kpc (Bailer-Jones 2015; Astraatmadja & Bailer-Jones 2016). In practice, we implemented a Monte-Carlo method by sampling distances following the distance posterior distribution. For each distance sample, we calculated reddening given the 3D dust map, and combined this with samples for the apparent magnitude and ${T}_{\mathrm{eff}}$ (drawn from a random normal distribution with a standard deviation corresponding to the 1σ uncertainties) to calculate radii. The adopted bolometric corrections and ${T}_{\mathrm{eff}}$ values were identical to the ones used for the calculation of asteroseismic radii described above.

The resulting distributions were used to calculate the mode and 1σ confidence interval for radii derived from each Gaia parallax. We did not implement a more complex prior (e.g., based on a synthetic stellar population) due to the difficulty of reproducing the selection function of our sample, but note that the results in this paper do not heavily depend on the choice of distance prior.

The "direct method" for determining asteroseismic distances described in the previous section has the disadvantage that it relies on a reddening map, which may contain systematic errors. We therefore calculated a second set of asteroseismic distances and TGAS radii using isochrones and synthetic photometry, which allows reddening to be treated as a free parameter. We used isochrones from the MIST database (Paxton et al. 2011, 2013, 2015; Choi et al. 2016) to calculate a grid ranging in age from 0.5 to 14 Gyr with a stepsize of 0.25 Gyr and in metallicity from −2 to $+0.4$ dex in stepsizes of 0.02 dex. Interpolation was performed along equal evolutionary points in age and metallicity (Dotter 2016). For each model, we saved synthetic photometry in 2MASS JHK, Tycho ${B}_{T}{V}_{T}$, and Sloan griz, and calculated reddened photometry in each passband for a given V-band extinction A_V by interpolating the Cardelli et al. (1989) extinction law. Asteroseismic ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values for each model were calculated using Equations (1) and (2), both with and without the ${\rm{\Delta }}\nu$ scaling relation corrections by Sharma et al. (2016).

To infer model parameters, we followed the method by Serenelli et al. (2013) to integrate over all isochrone points to derive posterior distributions given a set of likelihoods and priors. Specifically, given any combination of a set of observables $x=\{{B}_{T}-{V}_{T}$, $g-r$, $r-i$, $i-z$, $J-H$, $H-K$, $\pi ,{T}_{\mathrm{eff}},[\mathrm{Fe}/{\rm{H}}],{\nu }_{\max },{\rm{\Delta }}\nu \}$ and model parameters $y=\{\mathrm{age},\ [\mathrm{Fe}/{\rm{H}}],\ \mathrm{mass},\ {{\rm{A}}}_{V}\}$, the posterior probability is

$$\begin{eqnarray}&&p(y| x)\propto p(y)p(x| y)\propto p(y)\displaystyle \prod _{i}\exp \left(-\displaystyle \frac{{({x}_{i}-{x}_{i}(y))}^{2}}{2{\sigma }_{x,i}^{2}}\right).\end{eqnarray} \tag{ 4 }$$

The likelihood function for π was calculated as (e.g., Bailer-Jones 2015)

$$\begin{eqnarray}&&p(\pi | d)\propto \exp \left[-\displaystyle \frac{1}{2{\sigma }_{\pi }^{2}}{\left(\pi -\displaystyle \frac{1}{d}\right)}^{2}\right],\end{eqnarray} \tag{ 5 }$$

where d is the model distance calculated given an absolute magnitude and A_V for each model, as well as the observed K-band magnitude. Probability distribution functions for each stellar parameter were then obtained by weighting $p(y| x)$ by the volume that each isochrone point encompasses in mass, age, metallicity, and A_V, and integrating the resulting distribution along a given stellar parameter (see Appendix A of Casagrande et al. 2011). For ease of computation, the integration was performed only for models within 4σ of the constraints set by the observables.

To calculate asteroseismic distances, we used as input the spectroscopic ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$, asteroseismic ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, ${B}_{T}{V}_{T}{griJHK}$ photometry, and a flat prior in age, resulting in posterior distributions for all stellar parameters as well as extinction and distance. To calculate TGAS radii, we replaced the asteroseismic observables with the TGAS parallax π, using a flat age prior and the same distance prior as adopted in the previous section.

3.3.1. Asteroseismic Parameters

Comparisons of different methods to measure asteroseismic parameters have yielded broadly good agreement (Hekker et al. 2011, 2012; Verner et al. 2011). The median scatter between the five methods in the APOKASC catalog (see Pinsonneault et al. 2014) is 0.5% in ${\rm{\Delta }}\nu$ and 1% in ${\nu }_{\max }$, which we added in quadrature to the formal uncertainties from the SYD pipeline (see Table 1) for the analysis described in the previous section.

To test the influence of systematic errors, we compared our asteroseismic distances calculated using the direct and grid-modeling method in Figure 2(a). The agreement is excellent, with a median offset of 0.2% and scatter of 2.6%. To test a variety of systematic errors that could enter the asteroseismic distance calculation, we compared our distances from the direct method with distances calculated using the Bellaterra Stellar Properties Pipeline (BeSPP, Serenelli et al. 2013), the BAyesian STellar Algorithm (BASTA, Silva Aguirre et al. 2015), as well as to literature values from the Stromgren Survey for Asteroseismology and Galactic Archeology (SAGA, Casagrande et al. 2014b) and Rodrigues et al. (2014). Three of these methods (BeSPP, BASTA, SAGA) used asteroseismic input values from the same pipeline but different isochrone grids, and one method used different asteroseismic input values and isochrone models (Rodrigues et al. 2014). The median offsets are ≈0.2% for BeSPP, 2.3% for BASTA, 0.1% for SAGA, and 1.8% for Rodrigues et al. (2014), with no strong systematic trends as a function of distance (see bottom panel of Figure 2(a)). We thus conclude that systematic differences between asteroseismic methods to calculate distances are of the order of a few percent.

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3.3.2. Extinction

3.3.3. Bolometric Corrections

To test the effect of systematic errors in bolometric corrections, we used the method by Stassun & Torres (2016a) to calculate bolometric fluxes by fitting spectral energy distributions to broadband photometry supplemented with a grid of ATLAS model atmospheres (Kurucz 1993). The SED fits used the same ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ values as input constraints, but reddening was left as a free parameter. We then used these bolometric fluxes with ${T}_{\mathrm{eff}}$ to calculate angular diameters which, combined with TGAS parallaxes, resulted in a set of stellar radii that could be directly compared with the radii calculated from TGAS parallaxes and bolometric corrections (see Section 3.1). Figure 3 shows a comparison between the two estimates. We observe good agreement, with a median difference of 0.7% and a scatter of ≈3%, and a small systematic trend with SED radii being larger by ≈1% for red giants (≈3–10 ${R}_{\odot }$).

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Since the MIST grid also uses ATLAS models, the above exercise is mostly sensitive to differences in deriving bolometric fluxes rather than systematic differences in model atmospheres. We therefore performed a second test by comparing distances calculated using the same seismic luminosity and reddening but bolometric corrections calculated from MARCS model atmospheres (Gustafsson et al. 2008) provided by Casagrande & VandenBerg (2014), as implemented in BASTA (see also the left panel of Figure 2). We observed an offset of ≈1% (with distances calculated using MARCS bolometric corrections being larger), which was approximately constant in distance. Based on these two tests, we conclude that systematic errors due to bolometric corrections are at the ≈1% level in radius and distance, which is small compared to the random uncertainties of TGAS parallaxes (see Figure 1).

The stellar classification software tools described above as well as all data to reproduce the results of this paper (Tables 1 and 2) are publicly available at https://github.com/danxhuber/isoclassify (Huber 2017). The tools can be used to derive posterior distributions for stellar parameters and distances given any input combination of asteroseismic, astrometric, photometric, and spectroscopic observables.

Table 2.  Derived Fundamental Properties, Distances, and Extinctions

KIC	Spectroscopy + Asteroseismology			Direct Method			Direct Method (${\rm{\Delta }}\nu$ corr)			Grid Modeling			Grid Modeling (Δν corr)			Grid Parallax			Ev	Src
	${T}_{\mathrm{eff}}$(K)	$\mathrm{log}g$	$[\mathrm{Fe}/{\rm{H}}]$	R(${R}_{\odot }$)	d (pc)	AV	R(${R}_{\odot }$)	d (pc)	AV	R(${R}_{\odot }$)	d (pc)	AV	R(${R}_{\odot }$)	d (pc)	AV	R(${R}_{\odot }$)	d (pc)	AV	—	—
1160789	4739 ± 86	2.307 ± 0.014	−0.340 ± 0.060	${10.84}_{-0.54}^{+0.54}$	${656}_{-42}^{+42}$	0.246	${9.85}_{-0.49}^{+0.49}$	${597}_{-38}^{+38}$	0.227	${11.29}_{-0.31}^{+0.38}$	${677}_{-19}^{+24}$	${0.133}_{-0.070}^{+0.070}$	${11.29}_{-0.31}^{+0.34}$	${679}_{-21}^{+25}$	${0.133}_{-0.070}^{+0.070}$	${11.92}_{-2.13}^{+7.10}$	${754}_{-147}^{+405}$	0.545	−1	apo
1161618	4763 ± 86	2.442 ± 0.009	−0.009 ± 0.060	${10.96}_{-0.31}^{+0.31}$	${806}_{-39}^{+39}$	0.085	${11.01}_{-0.31}^{+0.31}$	${810}_{-39}^{+39}$	0.086	${11.21}_{-0.07}^{+0.08}$	${800}_{-6}^{+6}$	${0.223}_{-0.070}^{+0.080}$	${11.21}_{-0.07}^{+0.09}$	${801}_{-6}^{+7}$	${0.223}_{-0.070}^{+0.070}$	${11.58}_{-2.68}^{+12.53}$	${808}_{-176}^{+881}$	0.108	1	apo
1162746	4798 ± 86	2.356 ± 0.020	−0.478 ± 0.060	${10.97}_{-0.92}^{+0.92}$	${1452}_{-135}^{+135}$	0.251	${10.88}_{-0.92}^{+0.92}$	${1441}_{-134}^{+134}$	0.250	${11.02}_{-0.58}^{+0.80}$	${1435}_{-76}^{+114}$	${0.353}_{-0.090}^{+0.080}$	${11.02}_{-0.58}^{+0.80}$	${1435}_{-76}^{+114}$	${0.353}_{-0.080}^{+0.080}$	${7.52}_{-2.62}^{+44.45}$	${919}_{-399}^{+5788}$	0.282	1	apo
1163621	4959 ± 86	2.624 ± 0.008	−0.041 ± 0.060	${11.18}_{-0.29}^{+0.29}$	${1722}_{-79}^{+79}$	0.372	${11.34}_{-0.29}^{+0.29}$	${1747}_{-80}^{+80}$	0.372	${11.10}_{-0.23}^{+0.18}$	${1676}_{-34}^{+39}$	${0.383}_{-0.070}^{+0.080}$	${11.14}_{-0.15}^{+0.20}$	${1685}_{-27}^{+34}$	${0.373}_{-0.080}^{+0.080}$	${8.30}_{-3.03}^{+28.31}$	${1216}_{-348}^{+4349}$	0.385	1	apo
1294385	4825 ± 86	2.936 ± 0.006	0.030 ± 0.060	${6.92}_{-0.14}^{+0.14}$	${667}_{-29}^{+29}$	0.192	${6.71}_{-0.13}^{+0.13}$	${646}_{-28}^{+28}$	0.189	${6.93}_{-0.12}^{+0.16}$	${668}_{-12}^{+17}$	${0.073}_{-0.070}^{+0.070}$	${6.58}_{-0.12}^{+0.32}$	${633}_{-11}^{+31}$	${0.043}_{-0.090}^{+0.100}$	${7.81}_{-2.19}^{+27.34}$	${754}_{-209}^{+2612}$	0.287	−1	apo
1430163	6590 ± 77	4.226 ± 0.018	−0.050 ± 0.101	${1.52}_{-0.09}^{+0.09}$	${181}_{-12}^{+12}$	0.057	${1.47}_{-0.09}^{+0.09}$	${176}_{-12}^{+12}$	0.056	${1.48}_{-0.03}^{+0.03}$	${175}_{-4}^{+4}$	${0.103}_{-0.070}^{+0.070}$	${1.46}_{-0.02}^{+0.03}$	${174}_{-4}^{+4}$	${0.103}_{-0.070}^{+0.070}$	${1.53}_{-0.10}^{+0.11}$	${182}_{-11}^{+13}$	0.057	0	spc
1433803	4721 ± 86	3.081 ± 0.005	0.198 ± 0.060	${5.41}_{-0.10}^{+0.10}$	${405}_{-18}^{+18}$	0.152	${5.23}_{-0.10}^{+0.10}$	${392}_{-17}^{+17}$	0.149	${5.38}_{-0.09}^{+0.14}$	${403}_{-6}^{+10}$	${0.083}_{-0.080}^{+0.080}$	${5.14}_{-0.07}^{+0.09}$	${384}_{-6}^{+7}$	${0.053}_{-0.080}^{+0.080}$	${4.97}_{-0.59}^{+0.84}$	${375}_{-40}^{+60}$	0.147	0	apo
1435467	6326 ± 77	4.108 ± 0.004	0.010 ± 0.101	${1.72}_{-0.03}^{+0.03}$	${138}_{-5}^{+5}$	0.068	${1.69}_{-0.03}^{+0.03}$	${136}_{-5}^{+5}$	0.067	${1.70}_{-0.02}^{+0.02}$	${137}_{-2}^{+2}$	${0.003}_{-0.070}^{+0.070}$	${1.68}_{-0.01}^{+0.02}$	${135}_{-2}^{+2}$	${0.033}_{-0.080}^{+0.070}$	${2.23}_{-0.12}^{+0.12}$	${177}_{-7}^{+9}$	0.092	0	spc
1435573	4678 ± 86	2.304 ± 0.014	0.020 ± 0.060	${9.64}_{-0.58}^{+0.58}$	${1156}_{-83}^{+83}$	0.258	${9.36}_{-0.56}^{+0.56}$	${1122}_{-81}^{+81}$	0.255	${11.09}_{-0.20}^{+0.24}$	${1286}_{-23}^{+28}$	${0.283}_{-0.080}^{+0.080}$	${11.07}_{-0.18}^{+0.22}$	${1284}_{-20}^{+28}$	${0.283}_{-0.080}^{+0.080}$	${12.98}_{-4.02}^{+36.18}$	${1470}_{-345}^{+4308}$	0.316	−1	apo
1569842	4847 ± 86	3.039 ± 0.004	−0.317 ± 0.060	${5.26}_{-0.09}^{+0.09}$	${640}_{-27}^{+27}$	0.291	${5.03}_{-0.09}^{+0.09}$	${613}_{-26}^{+26}$	0.283	${5.26}_{-0.02}^{+0.04}$	${648}_{-4}^{+6}$	${0.153}_{-0.070}^{+0.070}$	${5.02}_{-0.06}^{+0.04}$	${618}_{-9}^{+6}$	${0.183}_{-0.070}^{+0.070}$	${6.36}_{-1.79}^{+34.96}$	${774}_{-216}^{+4208}$	0.416	0	apo

Note. ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$ were taken from APOGEE DR13 (SDSS Collaboration et al. 2016) for giants (src flag = "apo") and from Buchhave & Latham (2015) for dwarfs and subgiants (src flag = "spc"). $\mathrm{log}g$ was calculated from ${T}_{\mathrm{eff}}$ and ${\nu }_{\max }$ given in Table 1. "Ev" denotes the evolutionary state for non He-core burning stars (0), He-core burning stars (1), and stars with unknown evolutionary state (−1). Evolutionary state classifications were taken from Stello et al. (2013) and Vrard et al. (2016).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Figure 4 compares parallaxes from asteroseismology with those from TGAS for all 2200 stars in our sample. We show results without ${\rm{\Delta }}\nu$ correction applied, but note that the effects of this correction are small compared to the scatter (see Section 4.2). Qualitatively, the comparison shows good agreement over three orders of magnitude. The scatter is dominated by large TGAS uncertainties for distant, evolved stars, which cause a diagonal "edge" in the ratios (bottom panel) toward low parallax values due to TGAS data systematically scattering to lower values than asteroseismology. This is mainly caused by asteroseismic distances being an order of magnitude more precise: because the giant sample is magnitude limited, we observe a lack of small parallax values from asteroseismology.

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The qualitative agreement in Figure 4 appears to contradict De Ridder et al. (2016), who reported that asteroseismic and TGAS parallaxes are incompatible with a 1:1 relation for ≈900 giants from Rodrigues et al. (2014). To investigate this, we compare stars with parallaxes $\lt 5$ mas (corresponding roughly to the largest parallax in the sample by Rodrigues et al. 2014) on a linear scale in Figure 5. We indeed observe a deviation from the 1:1 relation, with seismic parallaxes being systematically larger. However, the larger sample used here, which covers the transition from red giants to main-sequence stars (Figure 1), demonstrates that this deviation appears to be significantly smaller than previously thought. Specifically, the TGAS parallax corrections derived from eclipsing binaries by Stassun & Torres (2016b), which indicated that TGAS parallaxes are too small (${\pi }_{\mathrm{TGAS} \mbox{-} \mathrm{EB}}=-0.25$ mas using the mean offset or ${\pi }_{\mathrm{TGAS} \mbox{-} \mathrm{EB}}=-0.39$ mas using an ecliptic latitude $\beta =55^\circ$) are significantly too large. There is also tension with the upper end of the Davies et al. (2017) correction (which predicts a similar offset to Stassun & Torres 2016b at ≈1.6 mas). We note that these results are not significantly affected by the small offset between our distances and Rodrigues et al. (2014) discussed in Section 3.3.1.

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In agreement with the combined results by Sesar et al. (2017), Jao et al. (2016), and Davies et al. (2017), we find that the absolute offset increases for larger parallaxes, which, on average, correspond to less evolved stars. This implies a stronger absolute systematic offset for main-sequence stars and subgiants, which is surprising given that scaling relations are generally thought to be more reliable for stars similar to the Sun. However, asteroseismic distances scale as ${{T}_{\mathrm{eff}}}^{2.5}$, which varies significantly for main-sequence and subgiant stars. Indeed, ${T}_{\mathrm{eff}}$ scales are often plagued by systematic offsets (e.g., Pinsonneault et al. 2012). In general, photometric ${T}_{\mathrm{eff}}$ scales from the infrared flux method (Casagrande et al. 2011) or open clusters (An et al. 2013) are systematically hotter than spectroscopic temperatures, though recent color-${T}_{\mathrm{eff}}$ calibrations are consistent with or cooler than spectroscopy (Huang et al. 2015). All ${T}_{\mathrm{eff}}$ scales rely on the accuracy of interferometric angular diameters (e.g., Boyajian et al. 2012a, 2012b; White et al. 2013), some of which have been suspected to be affected by systematic errors (Casagrande et al. 2014a). While efforts to systematically cross-calibrate angular diameters between different instruments are currently underway (e.g., Huber 2016), it is still unclear which ${T}_{\mathrm{eff}}$ scale is indeed most accurate.

To test the effect of changing the ${T}_{\mathrm{eff}}$ scale, we recalculated asteroseismic distances for dwarfs and subgiants using temperatures from the APOGEE pipeline (ASPCAP), and also using photometric ${T}_{\mathrm{eff}}$ values from the infrared flux method (IRFM, Casagrande et al. 2011) and Sloan photometry (SDSS, Pinsonneault et al. 2012) as listed in Pinsonneault et al. (2012). We note that that Pinsonneault et al. (2012) used $[\mathrm{Fe}/{\rm{H}}]=-0.2$ dex and extinction values from the KIC, which were shown to be overestimated compared to values derived from asteroseismology and spectroscopy (Rodrigues et al. 2014). Accounting for these differences would result in shifts of $\approx \,-20$ K for the SDSS and $\approx -65$ K for the IRFM scales, depending on the adopted initial ${T}_{\mathrm{eff}}$ and extinctions. Furthermore, the SDSS and IRFM scales are not entirely independent, since SDSS was calibrated to match IRFM for ${T}_{\mathrm{eff}}\gt 6000$ K. Re-deriving the SDSS and IRFM ${T}_{\mathrm{eff}}$ scales for the sample is beyond the scope of this paper, but we note that neither of these effects significantly change the conclusions below.

For comparison, we discarded stars with $\pi \lt 1.5$ mas to avoid the "edge" bias that arises from large uncertainty differences discussed above. The average difference between the coolest (ASPCAP) and hottest (IRFM) ${T}_{\mathrm{eff}}$ scale is ≈270 K. The results in Figure 6 demonstrate that the hotter ${T}_{\mathrm{eff}}$ scales bring better agreement between asteroseismic and TGAS parallaxes, particularly for $\pi \lesssim 10$ mas. Specifically, the median offset over the whole sample reduces by more than a factor of 2 from 5.8 ± 0.6% for the coolest ${T}_{\mathrm{eff}}$ scale (ASPCAP) to 2.0 ± 0.7% for the IRFM. Figure 6 also shows the proposed corrections by Stassun & Torres (2016b) derived from eclipsing binaries. The −0.25 mas correction, which was the main result of the study, provides a good match to the data for $\pi \gtrsim 5$ mas and spectroscopic ${T}_{\mathrm{eff}}$ scales, but is overestimated for $\pi \lesssim 5$ mas for all ${T}_{\mathrm{eff}}$ scales. The correction including an ecliptic latitude dependence is overestimated for $\pi \lesssim 10$ mas for all ${T}_{\mathrm{eff}}$ scales.

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In summary, our analysis demonstrates that offsets between TGAS parallaxes, asteroseismology, and eclipsing binaries are likely smaller than previously reported for $\pi \lesssim 5\mbox{--}10$ mas (≳100–200 pc), and can be at least partially compensated by systematic errors in ${T}_{\mathrm{eff}}$ scales for dwarfs and subgiants. Residual differences are small fractions rather than absolute offsets, and are ≈2% for the hottest ${T}_{\mathrm{eff}}$ scales. This conclusion is consistent with Silva Aguirre et al. (2017) and Jao et al. (2016), who found agreement with the offset by Stassun & Torres (2016b) for nearby dwarfs for which ≈2% produces a −0.25 mas offset. These results imply that previously proposed TGAS parallax corrections are overestimated for $\pi \lesssim 5\mbox{--}10$ mas (≈90%–98% of the TGAS sample). We note that this difference is most likely due to the larger sample size used in this study, rather than systematic differences in the adopted methods or distance scales. The above results also provide empirical evidence that hotter ${T}_{\mathrm{eff}}$ scales (such as the infrared flux method) are more accurate than cooler, spectroscopic estimates. Importantly, this conclusion assumes that there are no strong systematic errors in TGAS and asteroseismic distances.

Comparing radii instead of parallaxes reduces the ${T}_{\mathrm{eff}}$ dependence (from ${{T}_{\mathrm{eff}}}^{2.5}$ to ${{T}_{\mathrm{eff}}}^{1.5}$) and allows a more direct test of a fundamental parameter predicted by scaling relations. Figure 7 compares asteroseismic and TGAS radii for all stars with a TGAS parallax measured to better than 20%, which approximately corresponds to the limit where the distance ratios are not heavily influenced by the exponentially decreasing volume density prior (see Section 3.1) or artifacts introduced by large differences in random errors (see Section 4.1). The overall agreement is excellent, empirically demonstrating that asteroseismic radii from scaling relations without any corrections are accurate to at least ≈10% for stars ranging from $\approx 0.8$ to $10\,{R}_{\odot }$. The color-coding in Figure 7 furthermore demonstrates that there are no strong biases in asteroseismic radii as a function of metallicity.

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To illustrate this further, Figure 8 shows the ratios as a function of ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, $[\mathrm{Fe}/{\rm{H}}]$, and TGAS radius, both with and without applying the ${\rm{\Delta }}\nu$ correction by Sharma et al. (2016). In addition to the raw data (small symbols), we also show median bins (large symbols). We have tested that spatial correlations between asteroseismic and TGAS parallaxes (Zinn et al. 2017) do not significantly affect these median values or their uncertainties for the typical spatial separations of stars contributing to a given bin (≈15). We also show 68% confidence intervals calculated by bootstrapping a local-quadratic nonparametric regression using pyqt-fit.²¹

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We observe no significant trends with metallicity for $[\mathrm{Fe}/{\rm{H}}]=-0.8$ to $+0.4$ dex (Figure 8(b)). Intriguingly, however, the ratios show a trend as a function of TGAS radius (Figure 8(d)): stars near the main sequence (∼1–1.5 ${R}_{\odot }$) show no offset, while the seismic radii of subgiants (∼1.5–3 ${R}_{\odot }$) are too small by ≈5%–7%. The offset reduces for low-luminosity red giants, before increasing for high-luminosity red giants ($\gtrsim 10{R}_{\odot }$). The ${\rm{\Delta }}\nu$ scaling relation correction slightly reduces these deviations (blue triangles). The upturn for high-luminosity red giants ($\gtrsim 10{R}_{\odot }$) in Figure 8(d) is artificially introduced by large uncertainties of TGAS radii in a magnitude-limited sample, similar to the "edge" bias for parallaxes in the bottom panel of Figure 4. The underestimated seismic radii for subgiants, however, cannot be explained by such an effect.

We confirmed that the radius trend in Figure 8(d) is independent of the distance prior, reddening, method for calculating asteroseismic observables, or adopted ${T}_{\mathrm{eff}}$ scales (see Figure 9). Note that we have excluded giants with $R\gt 10{R}_{\odot }$ from this comparison to remove the bias discussed above. Specifically, we used a flat distance prior, reddening measured from the grid-modeling method described in Section 3.2, as well as ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values from the COR pipeline (Mosser & Appourchaux 2009). Adopting IRFM ${T}_{\mathrm{eff}}$ values for dwarfs and subgiants instead of the default SPC scale reduced the offset for subgiants by ≈2% (magenta symbols Figure 9). We added this value in quadrature in the subsequent analysis to account for ${T}_{\mathrm{eff}}$-dependent systematics.

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To put the TGAS radius comparison into context, Figure 10 also shows results from eclipsing binaries (Gaulme et al. 2016) and interferometry (Huber et al. 2012; White et al. 2013; Johnson et al. 2014). The interferometry sample is sparse for subgiants, but does not strongly contradict the ≈5% bias for subgiants from TGAS. For giants, our results are compatible with Gaulme et al. (2016), though the ${\rm{\Delta }}\nu$-corrected results are in slight tension with their predicted 5% offset. Either way, the TGAS results imply that the ≈5% radius bias reported by Gaulme et al. (2016) does not seem to extend the regime of low-luminosity red giants, which are prime targets for studies of exoplanets orbiting asteroseismic hosts (Grunblatt et al. 2016). A larger interferometric sample (T. R. White et al. 2017, in preparation) as well as spectrophotometric angular diameters in combination with Gaia parallaxes (S. K. Grunblatt et al. 2017, in preparation) will allow us to confirm and quantify the trends in Figure 10. Table 3 lists the median binned ratios shown in Figure 10, which may be used to estimate systematic errors in seismic radii from scaling relations.

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Table 3.  Median Binned Ratios between Gaia and Seismic Radii

${R}_{{\text{}}\mathrm{Gaia}}({R}_{\odot })$	${R}_{{\text{}}\mathrm{Gaia}}/{R}_{\mathrm{seismo}}$	${R}_{{\text{}}\mathrm{Gaia}}/{R}_{\mathrm{seismo},{\rm{\Delta }}\nu \mathrm{corr}}$
0.90	1.007 ± 0.038	1.008 ± 0.035
1.16	1.010 ± 0.024	1.014 ± 0.024
1.50	1.018 ± 0.023	1.015 ± 0.022
1.93	1.049 ± 0.023	1.036 ± 0.023
2.48	1.077 ± 0.024	1.069 ± 0.024
3.20	1.041 ± 0.030	1.031 ± 0.029
4.12	0.971 ± 0.030	0.977 ± 0.029
5.30	0.966 ± 0.036	0.986 ± 0.037
6.83	0.966 ± 0.035	1.004 ± 0.035
8.79	1.008 ± 0.030	1.039 ± 0.030

Note. ${R}_{\mathrm{seismo},{\rm{\Delta }}\nu \mathrm{corr}}$ corresponds to seismic radii derived using the ${\rm{\Delta }}\nu$ scaling relation correction by Sharma et al. (2016; i.e., blue symbols in Figure 10). Uncertainties include a 2% systematic error due to different ${T}_{\mathrm{eff}}$ scales.

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Models of red giants lead us to expect a systematic difference in the ${\rm{\Delta }}\nu$ scaling relation as a function of the evolutionary state due to the changes in their interior sound-speed profile after the onset of He-core burning (Miglio et al. 2012). However, the degree and even the sign of this difference is not yet fully settled. For example, Miglio et al. (2012) showed that applying the ${\rm{\Delta }}\nu$ correction to red clump stars improves the agreement with independent radii measured in clusters, while the results by Sharma et al. (2016) implied that the largest effect of the ${\rm{\Delta }}\nu$ correction applies for ascending RGB stars. Previous samples to empirically test scaling relations have been too small to decide this question.

TGAS parallaxes allow us to test the dependency of the scaling relation correction on evolutionary state. To separate RGB and red clump stars, we used classifications based on mixed mode period spacings by Stello et al. (2013) and Vrard et al. (2016). Figure 11 shows parallaxes (left panels) and radii (right panels) both with (bottom) and without (top) applying the ${\rm{\Delta }}\nu$ scaling relation correction by Sharma et al. (2016). The samples in each panel are separated into RGB (blue circles) and red clump stars (red triangles). Note that we relaxed the fractional parallax uncertainty cut to <40% to include more red clump stars in the sample. Due to this relaxed cut, the median bins were offset from the local-quadratic fit, and we thus adopted mean bins for consistency. However, the conclusions below are not unaffected by whether mean or median bins are used.

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While the scatter is too large to determine whether the RGB or red clump stars agree better with TGAS, there is tentative evidence that the ${\rm{\Delta }}\nu$ correction provides a improvement for RGB stars. Specifically, the weighted mean offset reduces from 5.4 ± 1.3% to 2.7 ± 0.7% in parallax and from −3.1 ± 1.4% to −1.0 ± 1.5% for radius. The corrections for red clump stars are negligible, as expected. We conclude that TGAS parallaxes are not precise enough to decide how the ${\rm{\Delta }}\nu$ correction depends on evolutionary state, but provide tentative evidence (at the ≈2σ level) that the Sharma et al. (2016) corrections improve the accuracy of seismic distances and radii.

TGAS provides a first glimpse of the potential of Gaia to measure distances, and vast precision improvements are expected for upcoming data releases. Since asteroseismic and TGAS distances agree to within a few percent over several orders of magnitude, it is interesting to explore the complementary nature of Gaia and asteroseismology to measure distances to galactic stellar populations. To investigate this, we calculated the expected end-of-mission Gaia parallax precision for seismic Kepler targets using the Gaia performance model:²²

$$\begin{eqnarray}{\sigma }_{\pi }/\mathrm{mas} & = & \sqrt{(-1.631+680.766{\rm{z}}+32.732{{\rm{z}}}^{2})}\\ & & \times (0.986+(1-0.986)V-{I}_{C}),\end{eqnarray} \tag{ 6 }$$

with

$$\begin{eqnarray}z=\left\{\begin{array}{ll}0.0685 & \mathrm{for}\quad G\lt 12.1\\ {10}^{0.4(G-15)} & \mathrm{otherwise}\end{array}\right.\end{eqnarray} \tag{ 7 }$$

Here, ${\sigma }_{\pi }$ is the predicted end-of-mission parallax uncertainty averaged over the sky. The Gaia G-band magnitude and Johnson-Cousins V − I color were calculated from KIC gri photometry (Table 1) using the following relations (Jordi et al. 2006, 2010):

$$\begin{eqnarray}G & = & (-0.0662-0.7854(g-r)-0.2859{(g-r)}^{2}\\ & & +0.0145{(g-r)}^{3})+g,\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}V-I=\left\{\begin{array}{ll}0.675(g-r)+0.364\, & \mathrm{for}\quad g-r\lt 2.1\\ 1.11(g-r)-0.52\, & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 9 }$$

To account for the sky-position dependency of parallax uncertainties due to the Gaia scanning law, we interpolated the recommended scaling factors²³ for the ecliptic coordinates of each Kepler target. This yielded on average ∼28% smaller uncertainty than the uncertainties calculated from Equation (6).

Figure 12 compares the distance uncertainty from asteroseismology to the expected end-of-mission Gaia precision for stars with asteroseismic distances from this work, Rodrigues et al. (2014), Casagrande et al. (2014b), and Mathur et al. (2016).²⁴ Remarkably, asteroseismology will provide more precise distances than the best Gaia performance for stars beyond 3 kpc. This is because the asteroseismic sensitivity does not depend strongly on apparent magnitude and hence distant, high-luminosity red giants still yield precisions of a few percent out to tens of kiloparsecs (Mathur et al. 2016). Asteroseismology will therefore be critical to extend the reach of Gaia to distant stellar populations, particularly if combined with spectroscopy, which simultaneously allows us to constrain interstellar extinction (Figure 2). Current and future opportunities to detect oscillations in distant red giants outside the Kepler field include the K2 Mission (Howell et al. 2014), targets with one-year coverage near the ecliptic poles observed by TESS (Ricker et al. 2014), red giants in the bulge observed with WFIRST (Gould et al. 2015), and red giants observed with PLATO (Rauer et al. 2014).

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