 Solution

Moving radially outward from the quasar, the standard active-galactic-nucleus wind bubble contains: the freely expanding fast wind; a reverse shock that thermalizes it; a hot shocked-wind bubble; a contact discontinuity; a dense swept-up interstellar shell behind a forward shock; and finally the undisturbed interstellar medium.

Let $M_s(R)$ be the swept-up shell mass. If inverse-Compton and atomic radiative cooling remove the shocked wind's thermal energy faster than the bubble expands, the reverse shock is momentum driven andIf cooling is slow, the shocked wind remains hot and the bubble is energy driven:supplemented by the bubble energy equationThe hot bubble stores wind energy and performs work for much longer than the direct photon momentum-crossing time. Equating wind power with shell kinetic power gives the characteristic momentum boostwhen the shell is much slower than the nuclear wind.