A toy model for gas sloshing in galaxy clusters

We apply a toy model based on 'pendulum waves' to gas sloshing in galaxy clusters. Starting with a galaxy cluster potential filled with a hydrostatic intracluster medium (ICM), we perturb all ICM by an initial small, unidirectional velocity, i.e. an instantaneous kick. Consequently, each parcel of ICM will oscillate due to buoyancy with its local Brunt-Vaisala (BV) period, which we show to be approximately proportional to the cluster radius. The oscillation of gas parcels at different radii with different periods leads to a characteristic, outward-moving coherent pattern of local compressions and rarefactions; the former form the sloshing cold fronts (SCFs). Our model predicts that SCFs (i) appear in the cluster centre first, (ii) move outwards on several Gyr time-scales, (iii) form a staggered pattern on opposite sides of a given cluster, (iv) each move outwards with approximately constant speed; and that (v) inner SCFs form discontinuities more easily than outer ones. These features are well known from idealized (magneto)hydrodynamic simulations of cluster sloshing. We perform comparison hydrodynamic + N-body simulations where sloshing is triggered either by an instantaneous kick or a minor merger. Sloshing in these simulations qualitatively behaves as predicted by the toy model. However, the toy model somewhat overpredicts the speed of sloshing fronts, and does not predict that inner SCFs emerge with a delay compared to outer ones. In light of this, we identify the outermost cold front, which may be a 'failed' SCF, as the best tracer of the age of the merger that set a cluster sloshing.