Can Radio Sources Offset Cooling?

Assuming X-ray shell and radio bubble are in pressure equilibrium, the total energy output of the radio source, including the work done on compressing the gas is E ~ 5/2 PV (with  = 5/3).
Compare with luminosity of cooling gas

Assuming X-ray shell and radio bubble are in pressure equilibrium, the total energy output of the radio source, including the work done on compressing the gas is E ~ 5/2 PV (with  = 5/3).