Feedback by active galactic nuclei (AGNs) is essential for regulating the fast radiative cooling of low-entropy gas at the centers of galaxy clusters and for reducing star formation rates of central ellipticals. The details of self-regulation depend critically on the unknown contents of AGN-inflated bubbles. Observations of the Sunyaev-Zeldovich (SZ) signal of AGN bubbles provide us with the ability to directly measure the lobe electron pressure given a bubble morphology. Here we compute the SZ signal of jet-inflated bubbles in three-dimensional magnetohydrodynamical simulations of the galaxy cluster MS0735.6+7421 with the Arepo code, and compare our synthetic SZ results to inferences obtained with popular modelling approaches. We find that cutting out ellipsoidal bubbles from a double-beta pressure profile only matches the inner bubble edges in the simulations and fails to account for the emission of the shock-enhanced pressure cocoon outside the bubbles. This additional contribution significantly worsens the accuracy of the cut-out method for jets with small inclinations with respect to the line of sight. Also, the kinetic SZ effect of the bubbles, a previously neglected contribution, becomes relevant at these smaller inclinations due to entrainment and mixing of the intracluster medium with low-density jet material. Fortunately, the different signs of the kinetic SZ signal in opposite lobes allow modelling this effect. We present an approximate method to determine the jet inclination, which combines jet power and lifetime estimates, the stand-off distance between jet head and bow shock, and the kinetic SZ effect, thereby helping to correctly infer the bubble contents.
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