# Sound wave generation by a spherically symmetric outburst and AGN Feedback in Galaxy Clusters

## TL;DR

This paper models the evolution of spherical outbursts in galaxy clusters, focusing on how energy is partitioned into sound waves and shocks, and applies the model to real clusters like Perseus and M87.

## Contribution

It introduces a comprehensive model for outburst evolution, linking intrinsic properties to observables, and validates it against realistic cluster profiles with good accuracy.

## Key findings

- Sound waves carry up to 12% of injected energy in instantaneous outbursts.
- Energy partition varies with outburst duration, approaching zero in slow piston limit.
- Model parameters closely match numerical simulations for Perseus and M87 clusters.

## Abstract

We consider the evolution of an outburst in a uniform medium under spherical symmetry, having in mind AGN feedback in the intra cluster medium (ICM). For a given density and pressure of the medium, the spatial structure and energy partition at a given time $t_{age}$ (since the onset of the outburst) are fully determined by the total injected energy $E_{inj}$ and the duration $t_b$ of the outburst. We are particularly interested in the late phase evolution when the strong shock transforms into a sound wave. We studied the energy partition during such transition with different combinations of $E_{inj}$ and $t_b$. For an instantaneous outburst with $t_b\rightarrow 0$, which corresponds to the extension of classic Sedov-Taylor solution with counter-pressure, the fraction of energy that can be carried away by sound waves is $\lesssim$12% of $E_{inj}$. As $t_b$ increases, the solution approaches the "slow piston" limit, with the fraction of energy in sound waves approaching zero. We then repeat the simulations using radial density and temperature profiles measured in Perseus and M87/Virgo clusters. We find that the results with a uniform medium broadly reproduce an outburst in more realistic conditions once proper scaling is applied. We also develop techniques to map intrinsic properties of an outburst $(E_{inj}, t_b$ and $t_{age})$ to the observables like the Mach number of the shock and radii of the shock and ejecta. For the Perseus cluster and M87, the estimated $(E_{inj}, t_b$ and $t_{age})$ agree with numerical simulations tailored for these objects with $20-30\%$ accuracy.