The mass and radius of the M-dwarf in the short period eclipsing binary RR Caeli

We present new photometry and spectroscopy of the eclipsing white dwarf - M-dwarf binary star RR Cae. We use timings of the primary eclipse from white-light photo-electric photometry to derive a new ephemeris for the eclipses. We find no evidence for any period change greater than Pdot/P ~ 5E-12 over a timescale of 10 years. We have measured the effective temperature of the white dwarf, T_WD, from an analysis of two high resolution spectra of RR Cae and find T_WD = (7540 +- 175)K. We estimate a spectral type of M4 for the companion from the same spectra. We have combined new spectroscopic orbits for the white dwarf and M-dwarf with an analysis of the primary eclipse and cooling models for helium white dwarfs to measure the mass and radius of the M-dwarf. The mass of the M-dwarf is (0.182 - 0.183) +- 0.013 Msun and the radius is (0.203 - 0.215) +- 0.013 Rsun, where the ranges quoted for these values reflect the range of white dwarf models used. In contrast to previous studies, which lacked a spectroscopic orbit for the white dwarf, we find that the mass and radius of the M-dwarf are normal for an M4 dwarf. The mass of the white dwarf is (0.440 +-0.022) Msun. With these revised masses and radii we find that RR Cae will become a cataclysmic variable star when the orbital period is reduced from its current value of 7.3 hours to 121 minutes by magnetic braking in 9-20 Gyr. We note that there is night-to-night variability of a few seconds in the timing of primary eclipse caused by changes to the shape of the primary eclipse. We speculate as to the possible causes of this phenomenon. (Abridged)