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We have constructed an analytical model of active galactic nuclei (AGN) feedback and studied its implications for elliptical galaxies and galaxy clusters. The results show that momentum injection above a critical value will eject material from low‐mass elliptical galaxies, and leads to an X‐ray luminosity,  L  X , that is  ∝σ 8−10  , depending on the AGN fuelling mechanism, where σ is the velocity dispersion of the hot gas. This result agrees well with both observations and semi‐analytic models. In more massive ellipticals and clusters, AGN outflows quickly become buoyancy dominated. This necessarily means that heating by a central cluster AGN redistributes the intracluster medium (ICM) such that the mass of hot gas, within the cooling radius, should be  ∝ L  X (&lt;  r  cool )/[ g ( r  cool )σ] , where   g ( r  cool )  is the gravitational acceleration at the cooling radius. This prediction is confirmed using observations of seven clusters. The same mechanism also defines a critical ICM cooling time of  ∼0.5 Gyr , which is in reasonable agreement with recent observations showing that star formation and AGN activity are triggered below a universal cooling time threshold.

Observable consequences of kinetic and thermal AGN feedback in elliptical galaxies and galaxy clusters