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% Method  \section{Method}\label{method}  \subsection{Light Curve Analysis}  \emph{Kepler} photometry is available in 90-day quarters. When studying long-period eclipsing binaries, long time drifts and discontinuities can be dominated by instrumental effects. With the caveat that each individual light curve may require slightly different handling, we will remove instrumental effects from the signal with the goal of preserving eclipse profiles, stellar variability, and oscillations. To do this, we follow the approach in \citet{gau14} and work with the raw simple aperture photometry (SAP) from \emph{Kepler}, which is the integrated flux over each mask aperture.  A challenge arises for binaries with timescales longer than the 90-day \emph{Kepler} quarter. To combine all available quarters of light curve data into one time series, we first normalize the out-of-eclipse flux by dividing each quarter's data by the median flux with eclipses removed. To line up the ends of each ``chunk'' of the light curve, we proceed in one of two ways. When a gap is short with respect to photometric variability, we fit each side of the gap with a second order polynomial and extrapolate to the middle of the gap. The difference between both extrapolated values is then used to adjust the flux. When a gap is long with respect to photometric variability, we adjust the average flux of each ``chunk" so that they line up on either side of the gap.  Following this process, the only apparent instrumental feature that remains is a periodic modulation corresponding to \emph{Kepler}'s 372.5-day Earth-trailing orbit. Because all the time series have gaps, it is not possible to use Fourier filtering to remove this signal. Instead, we subtract a 372.5-day period sine curve fit to the data plus its first harmonic, which reduces the amplitude of this modulation to less than 0.5\% \citep{gau14}.  \subsection{Spectral Analysis: Radial Velocities}\label{specanal1}  To extract radial velocities from the spectra, we use the broadening function (BF) technique as outlined by \citet{ruc02}. The BF is a true linear de-convolution, while the more familiar cross-correlation function (CCF) is a non-linear proxy for the BF. It is therefore a preferred technique for isolating two line profiles that partially overlap. A template spectrum from a bright source, such as Arcturus, and a synthetic spectrum may both be considered. We first smooth the BF with a Gaussian to remove un-correlated, small-scale noise below the size of the spectrograph slit, and then fit Gaussian profiles to measure the location of the BF peaks in velocity space. The results of this technique applied to KIC 9246715 are shown in Figure \ref{bffig}.   \begin{figure}[h!]   \centering  \includegraphics[width=6in]{bfplot.png}  \caption{Raw radial velocities fit with broadening functions \citep{ruc02} for KIC 9246715. In all but two cases, two separate peaks---one corresponding to each star's motion---are clearly visible. The two ``single peak" cases occur when the stars' velocities are similar. We note that nearly all of the BFs show a small peak at a radial velocity of zero, which is an artifact from telluric absorption lines in Earth's atmosphere.}  \label{bffig}  \end{figure}  We note this example is a somewhat idealized case, as KIC 9246715 is an RG/EB with two red giants, yet only one shows solar-like oscillations. This makes it particularly straightforward to identify and extract double-lined radial velocities, because both stars are contributing about the same amount of light. If some systems have spectra so dominated by the red giant that the signal from the companion star cannot be seen, we will proceed with single-lined radial velocities, or use the techniques described in Section \ref{specanal2} to isolate each star's spectrum prior to extracting radial velocities.  \subsection{Spectral Analysis: Stellar Atmospheres}\label{specanal2}  For systems containing a red giant and a main sequence star, we will follow the method in \citet{fra13} and \citet{mac14}, which each use two tools for disentangling one star's spectrum from the other. First, we will use the two dimensional cross-correlation technique TODCOR \citep{maz92,maz94,zuc94}. This identifies a best-fit model spectrum template for each star. Second, we will employ the KOREL program \citep{had97} to decompose the observed composite spectrum into its constituent parts. KOREL assumes Keplerian motion in a binary system and computes the decomposed spectra together with optimized orbital elements. These two techniques should produce identical results through different means, for a "sanity check."  Once each star's spectrum is isolated, we will use a spectrum synthesis technique with a tool such as MOOG \citep{sne73} to model stellar atmospheres and accurately derive $T_{\rm{eff}}$, $\log g$, and metallicity. This method, outlined by \citet{leh11} for pulsating B-stars, compares an observed spectrum with a grid of synthetic spectra and uses $\chi^2$ to measure the goodness of fit. We may also explore a Bayesian technique for deriving stellar atmosphere properties from spectra \citep{sch13}.   \subsection{Full Orbital Solutions}\label{orbit}  To combine radial velocities, parameters from atmosphere modeling, and light curves into a single physical solution, we will use the Eclipsing Light Curve (ELC) code \citep{oro00}. This code uses a genetic algorithm or Monte Carlo Markov Chain (MCMC) optimizers to simultaneously solve for a suite of stellar parameters. ELC is particularly well-suited to this analysis because it considers any set of input observables simultaneously and minimizes a global $\chi^2$ function. ELC is based on the Wilson-Devinney code \citep{wil71}. In the limiting case where the two stars are sufficiently separated as to be spherical in shape, ELC has a ``fast analytic mode'' that uses the equations in \citet{gim06}. We will use this option for most of the RG/EBs because they are well-detached binaries. In cases where we expect tidal forces to be significant, ELC can incorporate effects such as ellipsoidal variations via a full treatment of Roche geometry \citep{avn75}. A preliminary fit for the light curve and radial velocity curves of KIC 9246715 is shown in Figure \ref{elcfig}.  \begin{figure}[h!]   \centering  \includegraphics[width=4.8in]{markovfit_fixT1.png}  \caption{Model fit for KIC 9246715 using ELC with residuals. The top panels show the light curve, the middle panels show the radial velocities, and the bottom panels show a zoomed view of each eclipse. The red lines show the best-fit ELC model, black points show the observed \emph{Kepler} light curve, and blue points show observed radial velocities.}   \label{elcfig}  \end{figure}  In addition, ELC can be used to model star spots. It allows for up to two spots per star, and the user can specify spot latitude, longitude, size, and temperature scaling (constant, or radially dependent with a linear or Gaussian profile). Moving spots can be simulated by invoking non-synchronous rotation. We will split each RG/EB light curve into ``chunks'' when modeling spots, as we do not necessarily think a single spot would persist for several years. Each ``chunk'' will initially consist of one primary and one secondary eclipse, following the approach of, e.g., \citet{oro14}. This will help constrain the prevalence of spots on red giants during different epochs and distinguish between situations with small verses large spot covering fractions.  We recognize that star spots are not the only signature of stellar activity. We will also search spectra for any Ca II H \& K line emission, which could indicate the presence of faculae \citep{noy84}. These magnetically driven features are observed on the Sun even during solar minimum when sunspots are rare.  \subsection{Stellar Evolution Models}  To infer a binary system's past, we will assume the two stars evolved together and consider the evolution of radii with time. As discussed in \citet{mac14}, this is preferred for spectroscopic binaries over, e.g., the star's position in the H-R diagram, because radii are directly measured from the orbital solution while computing temperature and luminosity would rely on color transformations and bolometric corrections.   For a sample of oscillating RG/EBs and their otherwise similar non-oscillating counterparts, we will use the stellar evolution module of the Modules for Experiments in Stellar Astrophysics (MESA) code, called MESAstar \citep{pax11,pax13}, to model each red giant's evolution with appropriate microphysics. We will use our results from Sections \ref{specanal2} and \ref{orbit} as inputs, and select models such that both stars have at the same age their measured present-day masses and radii and the correct temperature ratio. Figure \ref{mesastar} shows example evolutionary tracks for several near-solar-mass stars, as calculated with MESAstar. This is just a small subset of what the program is capable of modeling.  \begin{figure}[h!]   \centering  \includegraphics[width=4.5in]{mesastar.png}  \caption{Evolution models from MESAstar for 0.9, 1.2, 1.5, and 2 $M_{\s}$ stars. The top panel shows the evolution tracks on a traditional Hertzsprung-Russell diagram while the bottom panel shows the same tracks in the central temperature/central density ($T_c - \rho_c$) plane. The solid red point is the location of each star on the zero-age main sequence, and the metallicity of all stars is set at $Z = 0.02$. In the bottom panel, the He core flash and subsequent evolution of the C/O core is shown. The dashed blue and heavy gray lines show two different constant electron degeneracies, and the dashed red line is a line of constant pressure relevant to the He core flash. Figure from \citet{pax11}.}  \label{mesastar}  \end{figure}