deletions | additions       diff --git a/Beyond_a_stellar_evolution_model__.tex b/Beyond_a_stellar_evolution_model__.tex  index a6ee30a..6bc2125 100644  --- a/Beyond_a_stellar_evolution_model__.tex  +++ b/Beyond_a_stellar_evolution_model__.tex 
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For the MESA model described above with $M = 2.15 \ M_{\odot}$, we compute $\Delta \ln e = -1.6 \times 10^{-5}$ up until $t = 8.6 \times 10^8$ years (the age corresponding to $R \simeq 8 \ R_{\odot}$). Rewriting this as $\log [-\Delta \ln e] = -4.8$, a value clearly less than 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 twocoeval  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 to evolve past the tip of the red giant branch and well onto the red clump (with $R \simeq 30 \ R_\odot$ 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 both a red giant branch star of the proper radius ($R \simeq 8 \ R_\odot$) aged $8.6 \times 10^8$ years and with a slightly-too-large ($R = 8.5 \ R_\odot$) star just past of the tip of the red giant branch aged $8.8 \times 10^9$ years, but not with a significantly older and larger 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, a star spot present during one primary eclipse, 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.  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.