Modelling the emergence and evolution of the rotation-activity relation

Main-sequence stars follow a well-defined rotation-activity relation. There are two primary regimes: saturated, where the fractional X-ray luminosity $\log(L_{\rm X}/L_*)$ is approximately constant, and unsaturated, where the fractional X-ray luminosity decreases with increasing Rossby number (or decreasing rotation rate). Pre-main sequence (PMS) stars have a larger scatter in $\log(L_{\rm X}/L_*)$ than main-sequence stars, are observed to have saturated levels of X-ray emission, and do not follow the rotation-activity relation. We investigate how PMS stars evolve in the rotation-activity plane and the timescale over which the X-ray rotation-activity relation emerges. Using observational data of $\sim$600 stars from four PMS clusters, stellar internal structure models, a rotational evolution model, and observed X-ray luminosity trends with age, we simulate the evolution of the PMS stars in the rotation-activity plane up to ages of 100 Myr. Our model reproduces the rotation-activity relation found for main-sequence stars, with higher-mass stars beginning to form the unsaturated regime from around 10 Myr. After $\sim$25 Myr, the gradient of the unsaturated regime matches that found for main-sequence stars. For stars of mass greater than 0.6 M$_{\odot}$, the maximum age by which a star has left the saturated regime correlates with when the star leaves the PMS. We find that an intra-cluster age spread is a key factor in contributing to the observed scatter in $\log(L_{\rm X}/L_*)$, particularly for ages < 10 Myr.