The three hundred. Cluster dynamical states and relaxation period

We introduce a new parameter λDS to quantify the dynamical state of galaxy clusters and test it using simulations from The Three Hundred cluster zoom suite. λDS is a combination of three previously used dynamical state measures, namely virial ratio, centre of mass offset, and substructure mass fraction, crafted to assume a double-Gaussian distribution, thereby yielding a natural division between relaxed and unrelaxed clusters where the Gaussians cross. Using dark matter-only simulations, we identify the optimal separator to be λDS = 3.424. We test this same criterion on two sets of fully hydrodynamical The Three Hundred runs (Gadget-X and GIZMO-SIMBA), and find only a weak dependence on the input baryonic physics. We correlate the evolution of λDS with the mass accretion history and find that halo mass changes of $\frac{\Delta M_{200}}{M_{200}} \lesssim 0.12$ do not typically alter the dynamical state. We examine the relaxation period, defined as the time taken to return to relaxation after becoming disturbed, and find a correlation between this relaxation period and the strength of halo mass change $\frac{\Delta M_{200}}{M_{200}}$. By fitting this correlation, we show that the relaxation period can be estimated from $\frac{\Delta M_{200}}{M_{200}}$ (even for multiple mass accretion events) with good accuracy.