  M87, the active galaxy at the center of the Virgo cluster, is ideal for
studying the interaction of a supermassive black hole (SMBH) with a hot,
gas-rich environment. A deep Chandra observation of M87 exhibits an
approximately circular shock front (13 kpc radius, in projection) driven by the
expansion of the central cavity (filled by the SMBH with relativistic
radio-emitting plasma) with projected radius $\sim$1.9 kpc. We combine
constraints from X-ray and radio observations of M87 with a shock model to
derive the properties of the outburst that created the 13 kpc shock. Principal
constraints for the model are 1) the measured Mach number ($M$$\sim$1.2), 2)
the radius of the 13 kpc shock, and 3) the observed size of the central
cavity/bubble (the radio-bright cocoon) that serves as the piston to drive the
shock. We find an outburst of $\sim$5$\times$$10^{57}$ ergs that began about 12
Myr ago and lasted $\sim$2 Myr matches all the constraints. In this model,
$\sim$22% of the energy is carried by the shock as it expands. The remaining
$\sim$80% of the outburst energy is available to heat the core gas. More than
half the total outburst energy initially goes into the enthalpy of the central
bubble, the radio cocoon. As the buoyant bubble rises, much of its energy is
transferred to the ambient thermal gas. For an outburst repetition rate of
about 12 Myrs (the age of the outburst), 80% of the outburst energy is
sufficient to balance the radiative cooling.
