Relation of X-ray activity and rotation in M dwarfs and predicted time-evolution of the X-ray luminosity

We present a sample of 14 M dwarf stars observed with XMM-Newton and Chandra, for which we also computed rotational periods from Kepler Two-Wheel (K2) Mission light curves. We compiled X-ray and rotation data from the literature and homogenized all data sets to provide the largest uniform sample of M dwarfs (302 stars) for X-ray activity and rotation studies to date. We then fit the relation between $L_{\rm x} - P_{\rm rot}$ using three different mass bins to separate partially and fully convective stars. We found a steeper slope in the unsaturated regime for fully convective stars and a nonconstant $L_{\rm x}$ level in the saturated regime for all masses. In the $L_{\rm x}/L_{\rm bol}-R_{\rm O}$ space we discovered a remarkable double gap that might be related to a discontinuous period evolution. Then we combined the evolution of $P_{\rm rot}$ predicted by angular momentum evolution models with our new results on the empirical $L_{\rm x} - P_{\rm rot}$ relation to provide an estimate for the age decay of X-ray luminosity. We compare predictions of this relationship with the actual X-ray luminosities of M stars with known ages from 100 Myr to a few billion years. We find remarkably good agreement between the predicted $L_{\rm x}$ and the observed values for partially convective stars. However, for fully convective stars at ages of a few billion years, the constructed $L_{\rm x}-$age relation overpredicts the X-ray luminosity because the angular momentum evolution model underpredicts the rotation period of these stars. Finally, we examine the effect of different parameterizations for the Rossby number ($R_{\rm O}$) on the shape of the activity-rotation relation in $L_{\rm x}/L_{\rm bol}-R_{\rm O}$ space, and we find that the slope in the unsaturated regime and the location of the break point of the dual power-law depend sensitively on the choice of $R_{\rm O}$.