deletions | additions       diff --git a/Beyond_a_stellar_evolution_model__.tex b/Beyond_a_stellar_evolution_model__.tex  index 05f6919..3c04768 100644  --- a/Beyond_a_stellar_evolution_model__.tex  +++ b/Beyond_a_stellar_evolution_model__.tex 
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From \citet{ver95}, the timescale $\tau_c$ on which orbital circularization occurs is given by  \begin{equation}  \frac{1}{\tau_c} \equiv \frac{\rm{d} \ln e}{\rm{d}t} \simeq -1.7f  {\left( \frac{T_{\rm{eff}}}{4500 \rm{K}} \right)}^{4/3} \left( \frac{M_{\rm{env}}}{M_{\odot}} \right)^{2/3} \frac{M_{\odot}}{M} \frac{M_2}{M} \frac{M+M_2}{M} \left( \frac{R}{a} \right)^8 \ \rm{yr}^{-1}, \end{equation}  where$f$ is a dimensionless factor of order unity,  $M$, $R$, and $T_{\rm{eff}}$ are the mass, radius, and temperature of a giant star with dissipative tides, $M_{\rm{env}}$ is the mass of its convective envelope, $M_2$ is the mass of the companion star, and $a$ is the semi-major axis of the binary orbit. We integrate this expression over the lifetime of KIC 9246715 to estimate the total expected change in orbital eccentricity, $\Delta \ln e$. We assume $a$ is constant, $f = 1$, constant  and that there is no mass loss. Because KIC 9246715 is a well-separated detached  binary, we can separate the integral into a part that is independent of the orbit and a part that must be integrated over time:%To estimate the change in eccentricity over the lifetime of KIC 9246715, we integrate over the orbit circularization timescale:  %\begin{equation}\label{circ}  %\Delta \ln e = \int_0^t \frac{\rm{d}t'}{\tau_c(t')}.  %\end{equation}  %We assume the stars are sufficiently separated to make $a$ constant over time, and we further assume no mass loss. We can then use Equations 5 and 6 from \citet{ver95} to calculate the expected change in orbital eccentricity:  \begin{equation}\label{tide1}  \Delta \ln e = \int_0^t \frac{\rm{d}t'}{\tau_c(t')} \simeq -1.7 \times 10^{-5} \ {\left( \frac{M}{M_{\odot}} \right)}^{-11/3} \ q(1+q)^{-5/3} \ I(t) {\left( \frac{P_{\rm{orb}}}{\rm{day}} \right)}^{-16/3}, 
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The two coeval stars in KIC 9246715 have very similar masses, radii, and temperatures, so this rough calculation is valid both for Star 1 acting on Star 2 and vice versa. In contrast, given just another $0.4 \times 10^8$ years to evolve past the red giant branch toward the red clump, $\log [-\Delta \ln e]$ becomes greater than zero and the expectation is a circular orbit. Therefore, the observed $e > 0$ is consistent only with a younger red giant branch star, and not with an older red clump star.  Tidal forces also tend to synchronize a binary star's orbit with the stellar rotation period, generally on shorter timescales than required for circularization \citep{ogi14}. Hints of KIC 9246715's stellar rotation behavior are present throughout this study: quasi-periodic light curve variability on the order of half the orbital period (Section \ref{discuss}), a star spot present during one primary eclipse event only (Section \ref{segment}), a constraint on $v \sin i$ from spectra (Section \ref{parameters}), and asteroseismic period spacing consistent with red clump stars yet not clear enough to measure a robust core rotation rate (Section \ref{discuss}). While full tidal circularization has not occurred, it is clear that modest tidal forces have played a role in the evolution of KIC 9246715, and may be linked to the absence or weakness of solar-like oscillations. Future studies of RG/EBs with different evolutionary histories and orbital configurations will help explore this connection further.