  We present results from a very deep (650 ks) Chandra X-ray observation of the
galaxy group NGC~5813, the deepest Chandra observation of a galaxy group to
date. Earlier observations showed two pairs of cavities distributed roughly
collinearly, with each pair associated with an elliptical shock front. The new
observations confirm a third pair of outer cavities, collinear with the other
pairs, and reveal an associated outer outburst shock at ~30 kpc. This system is
therefore unique in exhibiting three cavity pairs, each associated with an
unambiguous AGN outburst shock front. The implied mean kinetic power is roughly
the same for each outburst, demonstrating that the average AGN kinetic
luminosity can remain stable over long timescales (~50 Myr). The two older
outbursts have larger, roughly equal total energies as compared with the
youngest outburst, implying that the youngest outburst is ongoing. We find that
the radiative cooling rate and the mean shock heating rate of the gas are well
balanced at each shock front, suggesting that AGN outburst shock heating alone
is sufficient to offset cooling and establish AGN/ICM feedback within at least
the central 30 kpc. This heating takes place roughly isotropically and most
strongly at small radii, as is required for feedback to operate. We suggest
that shock heating may play a significant role in AGN feedback at smaller radii
in other systems, where weak shocks are more difficult to detect. We find
non-zero shock front widths that are too large to be explained by particle
diffusion. Instead, all measured widths are consistent with shock broadening
due to propagation through a turbulent ICM with a mean turbulent speed of ~70
km/s. Finally, we place lower limits on the temperature of any volume-filling
thermal gas within the cavities that would balance the internal cavity pressure
with the external ICM.
