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 J121258.25-012310.1 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 an inclination of i = 85.7 ^ { \circ } \pm 0.5 ^ { \circ } , masses of M _ { \mathrm { WD } } = 0.439 \pm 0.002 M _ { \sun } and M _ { \mathrm { sec } } = 0.273 \pm 0.002 M _ { \sun } and radii R _ { \mathrm { WD } } = 0.0168 \pm 0.0003 R _ { \sun } and R _ { \mathrm { sec } } = 0.306 \pm 0.007 R _ { \sun } .
For GK Vir we find an inclination of i = 89.5 ^ { \circ } \pm 0.6 ^ { \circ } , masses of M _ { \mathrm { WD } } = 0.564 \pm 0.014 M _ { \sun } and M _ { \mathrm { sec } } = 0.116 \pm 0.003 M _ { \sun } and radii R _ { \mathrm { WD } } = 0.0170 \pm 0.0004 R _ { \sun } and R _ { \mathrm { sec } } = 0.155 \pm 0.003 R _ { \sun } .
The mass and radius of the white dwarf in GK Vir are consistent with evolutionary models for a 50 , 000 K 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 ( M _ { \mathrm { H } } / \mathrm { M } _ { \mathrm { WD } } \leq 10 ^ { -6 } ) .
Such a thin envelope has not been predicted by any 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 \sim 9 per cent larger than predicted .