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A precision study of two eclipsing white dwarf plus M dwarf binaries

We use a combination of X-shooter spectroscopy, ULTRACAM high-speed photometry and SOFI near-infrared photometry to measure the masses and radii of both components of the eclipsing post common envelope binaries SDSS J1212-0123 and GK Vir. For both systems we measure the gravitational redshift of the white dwarf and combine it with light curve model fits to determine the inclinations, masses and radii. For SDSS J1212-0123 we find a white dwarf mass and radius of 0.439 +/- 0.002 Msun and 0.0168 +/- 0.0003 Rsun, and a secondary star mass and radius of 0.273 +/- 0.002 Msun and 0.306 +/- 0.007 Rsun. For GK Vir we find a white dwarf mass and radius of 0.564 +/- 0.014 Msun and 0.0170 +/- 0.0004 Rsun, and a secondary star mass and radius of 0.116 +/- 0.003 Msun and 0.155 +/- 0.003 Rsun. The mass and radius of the white dwarf in GK Vir are consistent with evolutionary models for a 50,000K carbon-oxygen core white dwarf. Although the mass and radius of the white dwarf in SDSS J1212-0123 are consistent with carbon-oxygen core models, evolutionary models imply that a white dwarf with such a low mass and in a short period binary must have a helium core. The mass and radius measurements are consistent with helium core models but only if the white dwarf has a very thin hydrogen envelope, which has not been predicted by evolutionary models. The mass and radius of the secondary star in GK Vir are consistent with evolutionary models after correcting for the effects of irradiation by the white dwarf. The secondary star in SDSS J1212-0123 has a radius ~9 per cent larger than predicted.