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