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Feedback by active galactic nuclei (AGNs) appears to be critical in balancingradiative cooling of the low-entropy gas at the centers of galaxy clusters andin mitigating the star formation of elliptical galaxies. New observations ofM87 enable us to put forward a comprehensive model for the physical heatingmechanism. Low-frequency radio observations by LOFAR revealed the absence offossil cosmic-ray (CR) electrons in the radio halo surrounding M87. This puzzlecan be resolved by accounting for the CR release from the radio cocoons and thesubsequent mixing of CRs with the dense ambient intracluster gas, whichthermalizes the electrons on a timescale similar to the radio halo age of 40Myr. Hadronic interactions of similarly injected CR protons with the ambientgas should produce an observable gamma-ray signal in accordance with the steadyemission of the low state of M87 detected by Fermi and H.E.S.S. Hence, wenormalize the CR population to the gamma-ray emission, which shows the samespectral slope as the CR injection spectrum probed by LOFAR, thereby supportinga common origin. We show that CRs, which stream at the Alfven velocity withrespect to the plasma rest frame, heat the surrounding thermal plasma at a ratethat balances that of radiative cooling on average at each radius. However, theresulting global thermal equilibrium is locally unstable and allows for theformation of the observed cooling multi-phase medium through thermalinstability. Provided that CR heating balances cooling during the emerging"cooling flow," the collapse of the majority of the gas is halted around 1 keV- in accordance with X-ray data. We show that both the existence of atemperature floor and the similar radial scaling of the heating and coolingrates are generic predictions of the CR heating model.

Toward a comprehensive model for feedback by active galactic nuclei: new insights from M87 observations by LOFAR, Fermi and H.E.S.S