Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 3

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.

For a dusty shell, direct ultraviolet absorption, infrared trapping, and incomplete ultraviolet absorption give the radiation forceBalancing it against gravity gives the general critical luminosity of a dusty shell

For a singular isothermal sphere,soandIn the infrared-thick limit $F_{\mathrm{rad}}\simeq {\tau }_{\mathrm{IR}}L/c$; in the single-scattering limit it is $L/c$; and in the ultraviolet-thin limit it is ${\tau }_{\mathrm{UV}}L/c$. Therefore

The swept shell has $M_s=Ar$, where $A=2f_{\mathrm{gas}}{\sigma }^2/G$. Its steady radial thin-shell momentum equation isLet $r_0$ be the launch radius and impose $v(r_0)=0$.

In the single-scattering regime, defineThe force difference is constant, and integration givesThe shell accelerates monotonically, but continuous sweeping of $M_s\propto r$ makes the speed approach the finite maximum

For the optically thin ultraviolet regime, define the launch Eddington factorSince $F_{\mathrm{rad}}={\tau }_{\mathrm{UV}}L/c\propto r^{-1}$ while the shell's gravitational force is constant, integration givesIt initially accelerates, reaches its maximum atand then decelerates, formally stalling at $r=r_0(2{\Gamma }_{\mathrm{UV}}-1)$. The contrast is physical: single-scattering transfers the same $L/c$ to the shell at every radius, whereas an ultraviolet-thin shell intercepts a fraction proportional to its declining optical depth, so the isothermal host's gravity eventually wins.