MOA-2010-BLG-328: Keck and HST Expose the Limits of Occams Razor in Microlensing

We present high resolution follow-up data of the planetary microlensing event MOA-2010-BLG-328, using Keck and the Hubble. Keck data, taken 8 years after the event, reveal a strong lens detection enabling a direct measurement of lens flux and source-lens relative proper motion. We find the relative source-lens proper motion to be $\mu_{\rm rel, Hel} = 4.07 \pm 0.34\ \rm mas\ yr^{-1}$, with the lens being $\sim10$ times fainter than the source. The lens was very faint in the Hubble passbands, and the small lens-source separation of $\sim$35 mas made its detection difficult. However, we obtained estimates of the lens magnitudes in Hubble bands by constraining its location to match the Keck K-band detection. The original analysis by \citet{Furusawa2013} reports a degenerate light curve, with several viable models depending on higher-order effects. We attempt to break the degeneracy by remodeling the event using constraints from follow-up data. Our best fit model includes parallax, orbital motion, xallarap and the magnification of a source companion. Models omitting any of these are excluded. However, even with a lens detection the solution remains unclear, as the degeneracy between a nearby late M dwarf and a distant early M dwarf in the disk persists, and cannot be broken with NIR data alone. We conclude the lens is either a $\sim0.2\ M_{\odot}$ star at $2-3$kpc, or a $\sim0.5\ M_{\odot}$ star at $4-5$kpc. This study highlights the importance of multi-band data and comprehensive modeling to resolve microlensing degeneracies.