Constraints on the ICM velocity power spectrum from the X-ray lines width and shift

Future X-ray observations of galaxy clusters by high spectral resolution missions will provide spatially resolved measurements of the energy and width for the brightest emission lines in the intracluster medium (ICM) spectrum. In this paper we discuss various ways of using these high resolution data to constrain velocity power spectrum in galaxy clusters. We argue that variations of these quantities with the projected distance R in cool core clusters contain important information on the velocity field length scales in the ICM. The effective length $l_{rm eff}$ along the line of sight, which provides dominant contribution to the line flux, increases with R, allowing one to probe the amplitude of the velocity variations at different spatial scales. In particular, we show that the width of the line as a function of R is closely linked to the structure function of the 3D velocity field. Yet another easily obtainable proxy of the velocity field length scales is the ratio of the amplitude of the projected velocity field (line energy) variations to the dispersion of the velocity along the line of sight (line width). Finally the projected velocity field can be easily converted into 3D velocity field, especially for clusters like Coma with an extended flat core in the surface brightness. Under assumption of a homogeneous isotropic Gaussian 3D velocity field we derived simple expressions relating the power spectrum of the 3D velocity field (or structure function) and the observables. The uncertainties in the observables, caused by stochastic nature of the velocity field, are estimated by making multiple realizations of the random Gaussian velocity field and evaluating the scatter in observables. If large scale motions are present in the ICM these uncertainties may dominate the statistical errors of line width and shift measurements.