  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% of the input values. We demonstrated the efficiency of our carefully
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.
