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We compare the gravitational potential profiles of the elliptical galaxies NGC 4486 (M87) and NGC 1399 (the central galaxy in the Fornax cluster) derived from X‐ray and optical data. This comparison suggests that the combined contribution of cosmic rays, magnetic fields and microturbulence to the pressure is ∼10 per cent of the gas thermal pressure in the cores of NGC 1399 and M87, although the uncertainties in our model assumptions (e.g. spherical symmetry) are sufficiently large that the contribution could be consistent with zero. In the absence of any other form of non‐thermal pressure support, these upper bounds translate into upper limits on the magnetic field of ∼10–20 μG at a distance of 1–2 arcmin from the centers of NGC 1399 and M87. We show that these results are consistent with the current paradigm of cool cluster cores, based on the assumption that active galactic nuclei regulate the thermal state of the gas by injecting energy into the intracluster medium. The limit of ∼10–20 per cent on the energy density in the form of relativistic protons applies not only to the current state of the gas, but also essentially to the entire history of the intracluster medium, provided that cosmic ray protons evolve adiabatically and that their spatial diffusion is suppressed.

Measuring the non‐thermal pressure in early‐type galaxy atmospheres: a comparison of X‐ray and optical potential profiles in M87 and NGC 1399