Partitioning the Outburst Energy of a Low Eddington Accretion Rate AGN at the Center of an Elliptical Galaxy: the Recent 12 Myr History of the Supermassive Black Hole in M87

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.