Meredith L. Rawls edited Discussion.tex  almost 9 years ago

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We now re-estimate $\nu_{\rm{max}}$ and $\Delta \nu$ for the oscillation spectrum in the same way as \citet{gau14}, but by using the whole \textit{Kepler} dataset (Q0--Q17). Differences with respect to previous estimates are negligible, as we find $\nu_{\rm{max}} = 106.4 \pm 0.8$ and $\Delta \nu = 8.31 \pm 0.01 \ \mu \rm{Hz}$. Because the ELC results yield $T_2/T_1=0.989$ (Table \ref{table1}) and the stellar atmosphere analysis gives $T_1 = 4990 \pm 90 \ \rm{K}$ and $T_2 = 5030 \pm 80 \ \rm{K}$ (Section \ref{parameters}), we assume an effective temperature $T_{\rm{eff}} = 5000 \pm 100 \ \rm{K}$ in the asteroseismic scaling equations. To determine mass, radius, surface gravity, and mean density, we use the scaling relations after correcting $\Delta \nu$ for the red giant regime \citep{mos13}. In essence, instead of directly plugging the observed $\Delta \nu_{\rm{obs}}$ into Equations \ref{density} and \ref{gravity}, we estimate the asymptotic large spacing $\Delta \nu_{\rm{as}}$ as follows: $\Delta \nu_{\rm{as}} = \Delta \nu_{\rm{obs}} (1 + \zeta)$, where $\zeta = 0.038$. With this correction of the large spacing, we obtain $M = 2.17 \pm 0.12 \ M_{\odot}$ and $R = 8.26 \pm 0.16 \ R_{\odot}$. In terms of mean density and surface gravity, which independently test the $\Delta \nu$ and $\nu_{\rm{max}}$ relations, respectively, we find $\bar{\rho}/\bar{\rho}_{\odot} = (3.862 \pm 0.009) \times 10 ^{-3}$ and $\log g = 2.942 \pm 0.007$. A comparison of key parameters determined from all our different modeling techniques is in Table \ref{table2}.  The asteroseismic mass and surface gravity are consistent with those from the ELC model for both stars, while the asteroseismic radius is only consistent with Star 2. Neither star's mean density agrees with the asteroseismic value, but Star 2 is much closer than Star 1. Overall, our asteroseismic analysis suggests the oscillating star is Star 2. However, we cannot definitely conclude this without considering the temperature dependence of the scaling relations. From \citet{gau13}, \citet{gau14}, and the present work, asteroseismic masses and radii were reported to be $(1.7 \pm 0.3 \  M_\odot, 7.7 \pm 0.4 \  R_\odot)$ and $(2.06 \pm 0.13 \  M_\odot, 8.10 \pm 0.18 \  R_\odot)$, and now we have $(2.17 \pm 0.12 \  M_\odot, 8.26 \pm 0.16 \  R_\odot)$. Among these, $\nu_{\rm{max}}$ does not vary much ($102.2, 106.4, 106.4 \ \mu\rm{Hz}$ respectively) while $\Delta \nu$ varies even less ($8.3, 8.32, 8.31 \ \mu\rm{Hz}$), while the assumed temperatures were 4699 K (from the KIC), 4857 K (from \citet{hub14.2}), and 5000 K (this work). Even if temperature is the least influential parameter on stellar masses and radii in the asteroseismic scalings, we are at a level of precision where errors on temperature dominate the global asteroseismic results. Generally speaking, asteroseismic scaling laws tend to better match mean density measurements than surface gravity measurements because of the latter's temperature dependence. %\begin{equation} \label{radeq}