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Meredith L. Rawls deleted file Method.tex
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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}
