The empirical Gaia G-band extinction coefficient

The first Gaia data release unlocked the access to the photometric information of 1.1 billion sources in the $G$-band. Yet, given the high level of degeneracy between extinction and spectral energy distribution for large passbands such as the Gaia $G$-band, a correction for the interstellar reddening is needed in order to exploit Gaia data. The purpose of this manuscript is to provide the empirical estimation of the Gaia $G$-band extinction coefficient $k_G$ for both the red giants and main sequence stars, in order to be able to exploit the first data release DR1. We selected two samples of single stars: one for the red giants and one for the main sequence. Both samples are the result of a cross-match between Gaia DR1 and 2MASS catalogues; they consist of high quality photometry in the $G$-, $J$- and $Ks$-bands. These samples were complemented by temperature and metallicity information retrieved from, respectively, APOGEE DR13 and LAMOST DR2 surveys. We implemented a Markov Chain Monte Carlo method where we used $(G-Ks)_0$ vs $T_\mathrm{eff}$ and $(J-Ks)_0$ vs $(G-Ks)_0$ calibration relations to estimate the extinction coefficient $k_G$ and we quantify its corresponding confidence interval via bootstrap resampling method. We tested our method on samples of red giants and main sequence stars, finding consistent solutions. We present here the determination of the Gaia extinction coefficient through a completely empirical method. Furthermore we provide the scientific community a formula for measuring the extinction coefficient as a function of stellar effective temperature, the intrinsic colour $(G-Ks)_0$ and absorption.