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Photodynamical Modeling of the Fascinating Eclipses in the Triple-Star System KOI-126

We explore the fascinating eclipses and dynamics of the compact hierarchical triple star system KOI-126 (KIC 5897826). This system is comprised of a pair of M-dwarf stars (KOI-126 B and C) in a 1.74 day orbit which revolve around an F-star (KOI-126 A) every 34 days. Complex eclipse shapes are created as the M stars transit the F star, due to two effects: (i) the duration of the eclipse is a significant fraction of the M-star orbital period, so the prograde or retrograde motion of the M stars in their orbit lead to unusually short or long duration eclipses; (ii) due to 3-body dynamics, the M-star orbit precesses with an astonishingly quick timescale of 1.74 years for the periastron (apsidal) precession, and 2.73 years for the inclination and nodal angle precession. Using the full Kepler data set, supplemented with ground-based photometry, plus 29 radial velocity measurements that span 6 years, our photodynamical modeling yields masses of $M_{A} = 1.2713 \pm 0.0047 M_{\odot}$ (0.37%), $M_{B} = 0.23529 \pm 0.00062 M_{\odot}$ (0.26%), and $M_{C} = 0.20739 \pm 0.00055 M_{\odot}$ (0.27%) and radii of $R_{A} = 1.9984 \pm 0.0027 R_{\odot}$ (0.14%), $R_{B}= 0.25504 \pm 0.00076 R_{\odot}$ (0.3%), and $R_{C} = 0.23196 \pm 0.00069 R_{\odot}$ (0.3%). We also estimate the apsidal motion constant of the M-dwarfs, a parameter that characterizes the internal mass distribution. While not particularly precise, we measure a mean apsidal motion constant, $\overline{k_{2}}$, of $ 0.046^{+0.046}_{-0.028}$, which is approximately 2-$σ$ lower than the theoretical model prediction of 0.150. We explore possible causes for this discrepancy.