deletions | additions       diff --git a/Beyond_a_stellar_evolution_model__.tex b/Beyond_a_stellar_evolution_model__.tex  index aeecd4e..0235b22 100644  --- a/Beyond_a_stellar_evolution_model__.tex  +++ b/Beyond_a_stellar_evolution_model__.tex 
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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} = f \left( \frac{L}{M_{\rm{env}}R^2} \right)^{1/3} \frac{M_{\rm{env}}}{M} q \frac{M_2}{M}  \frac{M+M_2}{M} \left( \frac{R}{a} \right)^8. \end{equation}  Here, $f$ is a factor of order unity \citep{ver95}, \citep{ver95} and  $M_{\rm{env}}$ is the mass of the convective envelope of the primary star, and $q$ is the mass ratio. star.  We integrate this expression over the lifetime of the star to estimate the total expected change in orbital eccentricity, $\Delta \ln e$. We assume $a$ is constant, $f = 1$, and further assume no mass loss. Because KIC 9246715 is a well-separated binary, we can separate the integral into a part that is independent of the orbit and a part that must be integrated over time: 
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\Delta \ln e = -1.7 \times 10^{-5} \ f \ {\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},  \end{equation}  where $q$ is the mass ratio and  \begin{equation}  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'. 
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