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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 t₀₆₄ (since the onset of the outburst) are fully determined by the total injected energy E₈₍₉ and the duration tb 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 E₈₍₉ and tb. For an instantaneous outburst with tb 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 12% of E₈₍₉. As tb 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 (E₈₍₉, tb and t₀₆₄) 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 (E₈₍₉, tb and t₀₆₄) agree with numerical simulations tailored for these objects with 20-30\% accuracy.
Tang et al. (Wed,) studied this question.