deletions | additions       diff --git a/Beyond_a_stellar_evolution_model__.tex b/Beyond_a_stellar_evolution_model__.tex  index cb3af5b..42b0093 100644  --- a/Beyond_a_stellar_evolution_model__.tex  +++ b/Beyond_a_stellar_evolution_model__.tex 
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Beyond a stellar evolution model, it is important to consider how each star has affected the other over time. When the two stars in KIC 9246715 reach the tip of the red giant branch, they have radii of approximately $30 $25  \ R_\odot$, which is still significantly smaller than the periastron separation ($r_{\rm{peri}} = (1-e)~a = 137 \ R_\odot$). We never expect the stars to experience a common envelope phase, so this cannot be used to constrain the present evolutionary state. To estimate how tidal forces change orbital eccentricity, we follow the approach of \citet{ver95}. They use a theory of the equilibrium tide first proposed by \citet{zah77} to calculate a timescale for orbit circularization as a star evolves. It is important to note that \citet{ver95} assumed circularization would proceed by a small secondary star (main sequence or white dwarf) imposing an equilibrium tide on a large giant, while the situation with KIC 9246715 is more complicated. For a thorough review of tidal forces in stars, see \citet{ogi14}. 
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I(t) \equiv \int_0^t \left( \frac{T_{\rm{eff}}(t')}{4500 \rm{K}} \right)^{4/3} \left( \frac{M_{\rm{env}}(t')}{M_{\odot}} \right)^{2/3} \left( \frac{R(t')}{R_{\odot}} \right)^8 dt'. \nonumber  \end{equation}  For the MESA model described above with $M = 2.15 \ M_{\odot}$, we \revise{we  compute $\Delta \ln e = -1.6 -x.x \times 10^{-5}$ up until $t = 8.3 \times 10^8$ and $\Delta \ln e = -x.x  \times 10^{-5}$ up until $t = 8.6 9.3  \times 10^8$ years (the age ages  corresponding to $R \simeq 8 8.3  \ R_{\odot}$). Rewriting this these  as $\log [-\Delta \ln e] = -4.8$, a value clearly -x.x$ and $\log [-\Delta \ln e] = -x.x$, which are both  less than zero, zero},  indicates that the binary has \emph{not} had sufficient time to circularize its orbit, though it is possible the system's initial eccentricity was higher than the $e = 0.35$ we observe today. The two 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. Given another $2.4 \times 10^8$ years more time  to evolve past the tip of the red giant branch and well onto the red clump (with $R \simeq 30 25  \ R_\odot$ for the second time), $\log [-\Delta \ln e]$ becomes greater than zero and the expectation is a circular orbit. Therefore, the observed eccentricity is consistent with \revise{both a red giant branch with $R \simeq 8 \ R_\odot$ star  aged approximately $8.3 \times 10^8$ years and with a secondary red clump star just past the tip of the red giant branch with$R \simeq 8.5 \ R_\odot$  aged approximately $9.3 \times 10^9$ years}, but not with a significantly older and larger red clump star. years.}  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, \revise{residual scatter between light curve observations and the best-fit model during both eclipses}, a constraint on $v_{\rm{rot}} \sin i$ from spectra, and an asteroseismic period spacing consistent with a red clump star yet not clear enough to measure a robust core rotation rate.