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In the semi-analytical work presented here the feedback from supermassive\nblack holes on galaxy clusters is investigated. In particular we aim at providing\nsimple diagnostics tools to constrain the characteristic velocities\nand spatial scales of the hot Intra Cluster Medium (ICM) motions. In\nthe so-called "cold core'' clusters these motions are believed to be\ndriven by the activity of a central black hole. The methods\ndeveloped here, together with present-day and future observations, are designed to help to solve the puzzle of cooling flow clusters (see section $1.3$)\nand understand better the AGN/gas interaction in smaller systems (down\nto individual galaxies).\\\\\n\nClusters of galaxies are the largest gravitationally bound systems in\nthe Universe: they are composed of hundreds to thousands of galaxies,\nmoving in a deep potential well set by the dominating dark matter. The\nwhole volume of clusters is filled with hot (temperature $\\sim\n10^7-10^8$~K) and rarefied (electron density $10^{-4}-10^{-1} {\\rm\ncm^{-3}}$) gas. In such a high-temperature regime even heavy elements\n(e.g. silicon, sulfur, iron etc.) are highly ionized up to [H]- or\n[He]-like ions and they emit in bright lines with energies from $\\sim\n0.7$ to $\\sim 8$ keV. Using X-ray observations one can reliable\nmeasure all the major gas properties: the temperature, density and abundance\nof heavy elements.\\\\\n\nA significant fraction of clusters (called "cool core'' clusters)\nshow distinct signatures in the central region: the gas temperature\ndrops inward, while the gas density increases. The central gas\nradiative cooling time in such clusters is much shorter than the age\nof the cluster and without any external source of energy the gas would cool\nwell below X-ray temperatures. However observations suggest that the\ngas temperature drops only to 1-2 keV. One plausible explanation of\nthis problem is that the activity of a central supermassive black hole\ndeposits large amounts of mechanical energy into the cluster gas and that this\nbalances the gas radiative losses. A direct implication of this\nhypothesis is that the hot gas is not at rest, but it is continuously stirred\nby the AGN activity.\\\\\n\nThe same class of cool core clusters is characterized by a centrally\npeaked distribution of the heavy elements abundance (usually measured\nusing the He-like iron 6.7 keV line). The peaked abundance profiles are\nlikely associated with the metals ejection by the stars of very massive\nelliptical galaxies, that are always present at the centers of these clusters.\nHowever the observed abundance distributions are significantly broader\nthan the central galaxy light profiles, suggesting that some gas motions\nare spreading the metals ejected from the galaxy. We treat this\nprocess in a diffusion approximation to derive, from the X-ray\nobservations, constraints on the characteristic velocities and\nspatial scales of the gas motions for a sample of cool core clusters and\ngroups (Chapters $2$ and $3$). The parameters derived from a simple\nsemi-analytic model are then compared with the results of numerical\nsimulations of the AGN/gas interaction in the cluster core (Chapter $4$).\\\\\nIn Chapter $5$ we discuss the impact of the gas motions on the width of\nthe strongest X-ray emission lines. Since the characteristic thermal\nvelocities of heavy ions (e.g. iron) are much smaller than the sound speed\nof the gas, the width of the lines sensitively depends on the presence\nof gas motions. We show that both the absolute value of the linewidth\nand its dependence on the projected distance from the cluster center\nprovide valuable diagnostics of the gas motions. Such measurements\nwill soon become possible with the launch of X-ray micro-calorimeters\nin space.\\\\\n\nThis work has been done in collaboration with E.Churazov, R.Sunyaev,\nH.B\\"ohringer, M.Br\\"uggen, W.Forman and E.Roediger.