The Mass of the White Dwarf Companion in the Self-Lensing Binary KOI-3278: Einstein vs. Newton

KOI-3278 is a self-lensing stellar binary consisting of a white-dwarf secondary orbiting a Sun-like primary star. Kruse and Agol (2014) noticed small periodic brightenings every 88.18 days in the Kepler photometry and interpreted these as the result of microlensing by a white dwarf with about 63$\%$ of the mass of the Sun. We obtained two sets of spectra for the primary that allowed us to derive three sets of spectroscopic estimates for its effective temperature, surface gravity, and metallicity for the first time. We used these values to update the Kruse and Agol (2014) Einsteinian microlensing model, resulting in a revised mass for the white dwarf of $0.539^{+0.022}_{-0.020} \, M_{\odot}$. The spectra also allowed us to determine radial velocities and derive orbital solutions, with good agreement between the two independent data sets. An independent Newtonian dynamical MCMC model of the combined velocities yielded a mass for the white dwarf of $0.5122^{+0.0057}_{-0.0058} \, M_{\odot}$. The nominal uncertainty for the Newtonian mass is about four times better than for the Einsteinian, $\pm 1.1\%$ vs. $\pm 4.1\%$ and the difference between the two mass determinations is $5.2 \%$. We then present a joint Einsteinian microlensing and Newtonian radial velocity model for KOI-3278, which yielded a mass for the white dwarf of $0.5250^{+0.0082}_{-0.0089} \, M_{\odot}$. This joint model does not rely on any white dwarf evolutionary models or assumptions on the white dwarf mass-radius relation. We discuss the benefits of a joint model of self-lensing binaries, and how future studies of these systems can provide insight into the mass-radius relation of white dwarfs.