We study the emission of X-rays from lobes of FR-II radio galaxies by inverse Compton scattering of microwave background photons .
Using a simple model that takes into account injection of relativistic electrons , their energy losses through adiabatic expansion , synchrotron and inverse Compton emission , and also the stopping of the jet after a certain time , we study the evolution of the total X-ray power , the surface brightness , angular size of the X-ray bright region and the X-ray photon index , as functions of time and cocoon size , and compare the predictions with observations .
We find that the radio power drops rapidly after the stopping of the jet , with a shorter time-scale than the X-ray power .
The X-ray spectrum initially hardens until the jet stops because the steepening of electron spectrum is mitigated by the injection of fresh particles , for electrons with \gamma \geq 10 ^ { 3 } .
This happens because of the concurrence of two times scales , that of the typical jet lifetimes and cooling due to inverse Compton scattering ( \sim 10 ^ { 7 \hbox { - - } 8 } yr ) , of electrons responsible for scattering CMB photons into keV range photons ( with \gamma \sim \sqrt { 1 { keV } / kT _ { CMB } } ) .
Another finding is that the ratio of the X-ray to radio power is a robust parameter that varies mostly with redshift and ambient density , but is weakly dependent on other parameters .
We also determine the time-averaged ratio of X-ray to radio luminosities ( at 1 keV and 151 MHz ) and find that it scales with redshift as \propto ( 1 + z ) ^ { 3.8 } , for typical values of parameters .
We then estimate the X-ray luminosity function of FR-II radio galaxies and estimate the number of these diffuse X-ray bright objects above a flux limit of \sim 3 \times 10 ^ { -16 } erg cm ^ { -2 } s ^ { -1 } to be \sim 25 deg ^ { -2 } .