Toward mapping turbulence in the intracluster medium III. Constraints on the turbulent power spectrum with Athena/X-IFU

Context. Future X-ray observatories with high spectral resolution and imaging capabilities will enable measurements and mappings of emission line shifts in the intracluster medium (ICM). Such direct measurements can serve as unique probes of turbulent motions in the ICM. Determining the level and scales of turbulence will improve our understanding of the galaxy cluster dynamical evolution and assembly, together with a more precise evaluation of the non thermal support pressure budget. This will allow for more accurate constraints to be placed on the masses of galaxy clusters, among other potential benfits. Aims. In this view, we implemented the methods presented in the previous instalments of our work to characterize the turbulence in the ICM in a feasibility study with the X-IFU on board the future European X-ray observatory, Athena. Methods. From idealized mock observations of a toy model cluster, we reconstructed the second-order structure function built with the observed velocity field to constrain the turbulence. We carefully accounted for the various sources of errors to derive the most realistic and comprehensive error budget within the limits of our approach. With prior assumptions on the dissipation scale and power spectrum slope, we constrained the parameters of the turbulent power spectrum model through the use of MCMC sampling. Results. With favourable assumptions, we were able to retrieve the injection scale, velocity dispersion, and power spectrum slope, with 1sigma uncertainties better than  15 set framework to constrain the turbulence in the ICM from high-resolution X-ray spectroscopic observations, paving the way for more in-depth investigation of the optimal required observing strategy within a more restrictive observational setup with the future X-IFU instrument.