deletions | additions       diff --git a/Background.tex b/Background.tex  index e148bce..aa14563 100644  --- a/Background.tex  +++ b/Background.tex 
...
P-mode oscillations appear as spikes in an amplitude spectrum of a light curve that is sampled both frequently enough and for a sufficiently long duration. Figure \ref{rainbowmodes} shows oscillation spectra for stars with similar mass at different stages of stellar evolution. Each redder color corresponds to roughly an order of magnitude decrease in surface gravity.  \begin{figure}[h!]   \centering  \includegraphics[width=6.5in]{adaptedfig1.jpg}  \caption{Solar-like %\begin{figure}[h!]   %\centering  %\includegraphics[width=6.5in]{adaptedfig1.jpg}  %\caption{Solar-like  oscillation spectra of nine stars observed by \emph{Kepler}. Redder colors indicate later evolutionary phases, and correspondingly lower surface gravity. All stars shown have masses close to $1~M_{\odot}$. Those in the right panel are from short-cadence \emph{Kepler} data ($\sim 1$ min), and those on the left are from long-cadence mode ($\sim 30$ min). The top two stars on the right are main-sequence stars, the third and fourth stars down are subgiants, and the bottom star lies at the base of the RGB. The top two stars on the left are first-ascent RGB stars, and the bottom two are, respectively, RGB and RC stars that share similar surface properties. Figure adapted from \citet{cha13}.} \label{rainbowmodes}  \end{figure}  Compared to main-sequence stars, red giants oscillate with larger amplitudes and longer periods---several hours to days instead of minutes. There are two observable quantities that may be read directly from an oscillating star's power spectrum. The first is the large frequency separation, $\Delta \nu$, which is the separation in frequency space between modes of the same spherical order $l$. This is related to the sound speed, and therefore the average density of the star. As discussed in \citet{ulr86}, this suggests a scaling relation of the form  \begin{equation}\label{deltanu}  \langle \Delta \nu \rangle \propto \langle \rho \rangle^{1/2},  \end{equation}  \noindent where $\langle \rho \rangle \propto M/R^3$ is the average density of the star. It is generally reasonable to assume an average value for $\Delta \nu$ when considering p-modes of low radial order $n$. This is because geometric cancellation allows only low modes of spherical order $l$ to be observed, and we therefore can use an asymptotic approximation in the regime where $l/n \rightarrow 0$ \citep{chr10}.  The second quantity that may be read from a star's oscillation spectrum is the frequency of maximum amplitude, $\nu_{\rm{max}}$, which is the peak of a Gaussian envelope fit to the oscillation spikes. This relates to the acoustic cutoff frequency because both frequencies are determined by the near-surface properties. As shown in \citet{kje95}, this suggests the scaling  \begin{equation}\label{numax}  \nu_{\rm{max}} \propto g T_{\rm{eff}}^{-1/2},  \end{equation}  \noindent where $g \propto M/R^2$ is the stellar surface gravity and $T_{\rm{eff}}$ is the effective temperature.  It is possible to combine Equations \ref{deltanu} and \ref{numax} to derive stellar mass and radius for solar-like oscillators by scaling our Sun's properties as follows \citep{cha13}:  \begin{equation}\label{radius1}  \left( \frac{R}{R_\s} \right) \simeq \left( \frac{\nu_{\rm{max}}}{\nu_{\rm{max,\s}}} \right) \left( \frac{\Delta \nu}{\Delta \nu_\s} \right)^{-2} \left( \frac{T_{\rm{eff}}}{T_{\rm{eff,\s}}} \right)^{0.5}  \end{equation}  \begin{equation}\label{mass1}  \left( \frac{M}{M_\s} \right) \simeq \left( \frac{\nu_{\rm{max}}}{\nu_{\rm{max,\s}}} \right)^3 \left( \frac{\Delta \nu}{\Delta \nu_\s} \right)^{-4} \left( \frac{T_{\rm{eff}}}{T_{\rm{eff,\s}}} \right)^{1.5}.  \end{equation}  It is important to note that these scaling relations require careful measurements of not only $\Delta \nu$ and $\nu_{\rm{max}}$, but also an accurate determination of $T_{\rm{eff}}$. In addition, the forms of Equations \ref{radius1} and \ref{mass1} imply that asteroseismically estimated masses are inherently more uncertain than radii.  Asteroseismology is a powerful tool for studying evolved stars in particular. As a star evolves, its central regions contract, and the gravitational acceleration near the core increases along with the frequencies of g-modes. This is because the buoyancy frequency in the stellar interior increases. As a result, interior g-modes are able to occasionally interact with p-modes in the stellar envelope. These interactions appear in a star's oscillation spectrum and are called mixed modes \citep{mos12}. In contrast, main sequence solar-like oscillators only show pure p-modes.  While p-modes are evenly spaced in frequency in the asymptotic approximation, allowing $\Delta \nu$ to be easily observed, g-modes are analogously evenly spaced in period and characterized by $\Delta \Pi$. Observing mixed modes in evolved stars gives a clear way to distinguish core-He-burning red clump (RC) stars from shell-He-burning stars ascending the red giant branch (RGB). This is possible because g-mode period spacings are substantially higher for RC stars than for those ascending the RGB, and is an excellent example of how asteroseismology lets us observe stellar interiors \citep{bed11}.  \subsection{Why Don't \emph{All} Stars With Convective Envelopes Oscillate?}\label{why}  In principle, there are two reasons why a star such as a red giant with a convective outer layer might not show solar-like oscillations. First, there could be a physical mechanism that is preventing oscillation modes to become excited in the first place. Second, these modes could initially exist, but could subsequently not be allowed to resonate or be damped by physical processes in the star.  \citet{gau14} suggest that the second reason may be the case for several RG/EBs, and that the inhibiting mechanism is related to shorter orbital periods and stellar activity, such as magnetism manifested in star spots. They speculate that tidally locked binaries may inhibit oscillations altogether through a combination of tidal forces and strong surface activity. Tidal forces are strongest when two stars are close, and tend to change orbits over time, making them more circular and synchronous with rotation (i.e., tidally locked). Tides may also affect the shape of a star's convection cavity, which could change the gravitational acceleration in certain regions and prevent modes from resonating. Differing magnetic field strengths may also affect oscillations, as stellar dynamo is linked to plasma rotation and could also conceivably alter a star's convection cavity. In addition, spots in the Sun are known to locally damp p-mode oscillations.  Figure \ref{pvar_porb} shows two examples of how oscillation signatures vary. On the left are 10 RG/EBs with significant stellar activity detected during eclipses. Because the measured periods of variability are resonances of each system's orbital period, \citet{gau14} conclude that stellar activity is related to tidal interactions, which together inhibit oscillation strength. On the right is an example of the absence of oscillation modes. The Gaussian envelope of solar-like oscillations is clearly present in both blue power spectra, but strikingly absent in the black power spectrum.  \begin{figure}[h!]   \centering  \includegraphics[width=3.0in]{pvar_porb.png}  \includegraphics[width=3.4in]{oscillate_ornot.png}  \caption{A hint of why some RG/EBs oscillate less than others. The left panel shows variability period (i.e., light curve variations due to star spots) verses orbital period. Larger symbols have higher amplitudes of stellar activity, and darker circles have stronger solar-like oscillations. Blue lines indicate resonances of $P_{\rm{var}}/P_{\rm{orb}}$. The right panel shows three power spectra for different RG/EBs, similar to those shown in Figure \ref{rainbowmodes}. The light blue has $\nu_{\rm{max}} \sim 4 \ \mu \rm{Hz}$ and the dark blue has $\nu_{\rm{max}} \sim 80 \ \mu \rm{Hz}$. However, the star with its power spectrum plotted in black clearly lacks solar-like oscillations. Figure from \citet{gau14}.}  \label{pvar_porb}  \end{figure}  \subsection{Stellar Evolution in Binary Stars}  When stars of different mass form in a binary system, they are destined to evolve at different times. In close binaries, the situation can be dramatic: when one star expands to become a red giant, it can engulf its companion into a common envelope and affect the system's orbit, mass ratio, and ultimate fate. For the RG/EBs identified by \citet{gau13} and \citet{gau14}, however, such a history seems unlikely. These systems are all well detached; preliminary light curve modeling suggests that the sum of the two stars' radii never exceeds $\sim 30\%$ of the orbital semi-major axis.  On the other hand, many RG/EBs have eccentric orbits, as can be seen in Figure \ref{light_curves_all}. The five systems with the shortest periods appear to be synchronized in phase and in rotation with respect to the binary orbit, and have nearly circular orbits. This is unsurprising. In non-crowded environments, stars in a binary generally form at the same time, and evolve together. Given physical parameters such as mass, radius, and metallicity for an evolved red giant, and additional constraints on evolutionary stage from asteroseismology, it is possible to piece together a star's ``life story'' in the context of a binary system.  \begin{figure}[h!]   \centering  \includegraphics[width=4.0in]{light_curves_all.png}  \caption{\emph{Kepler} light curves for \citet{gau14}'s sample of RG/EBs, ordered from longest to shortest orbital period (top to bottom). Eccentric orbits, which appear as unevenly spaced primary and secondary eclipses, are clearly more prevalent for longer-period systems. The system names given in red are those which do not exhibit solar-like oscillations.}  \label{light_curves_all}  \end{figure} %\label{rainbowmodes}  %\end{figure}