New near-infrared period-luminosity-metallicity relations for RR Lyrae stars and the outlook for Gaia

We present results of the analysis of 70 RR Lyrae stars located in the bar of the Large Magellanic Cloud (LMC). Combining spectroscopically determined metallicity of these stars from the literature with precise periods from the OGLE III catalogue and multi-epoch $K_{\rm s}$ photometry from the VISTA survey of the Magellanic Clouds system (VMC), we derive a new near-infrared period-luminosity-metallicity (${\rm PL_{K_{\rm s}}Z}$) relation for RR Lyrae variables. In order to fit the relation we use a fitting method developed specifically for this study. The zero-point of the relation is estimated in two different ways: by assuming the value of the distance to the LMC and by using Hubble Space Telescope (HST) parallaxes of five RR Lyrae stars in the Milky Way (MW). The difference in distance moduli derived by applying these two approaches is $\sim0.2$ mag. To investigate this point further we derive the ${\rm PL_{K_{\rm s}}Z}$ relation based on 23 MW RR Lyrae stars which had been analysed in Baade-Wesselink studies. We compared the derived ${\rm PL_{K_{\rm s}}Z}$ relations for RR Lyrae stars in the MW and LMC. Slopes and zero-points are different, but still consistent within the errors. The shallow slope of the metallicity term is confirmed by both LMC and MW variables. The astrometric space mission Gaia is expected to provide a huge contribution to the determination of the RR Lyrae ${\rm PL_{K_{\rm s}}Z}$ relation, however, calculating an absolute magnitude from the trigonometric parallax of each star and fitting a ${\rm PL_{K_{\rm s}}Z}$ relation directly to period and absolute magnitude leads to biased results. We present a tool to achieve an unbiased solution by modelling the data and inferring the slope and zero-point of the relation via statistical methods.