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Use a geometrically thin spherical shell with constant mass $M_{\mathrm{sh}}$ and assume that its dust transfers momentum efficiently to its gas. Define the ultraviolet and infrared opacities per unit total shell mass by ${\kappa }_{\mathrm{UV}}$ and ${\kappa }_{\mathrm{IR}}$. Its dust optical depths areFor a central ultraviolet luminosity $L$, the usual dusty radiation-pressure-driven shell model absorbs a fraction $1-e^{-{\tau }_{\mathrm{UV}}}$ of the incident momentum and adds an infrared trapping contribution ${\tau }_{\mathrm{IR}}L/c$. Neglecting swept-up mass, shell self-gravity, and other gravitating matter, its thin-shell momentum equation isBalancing these terms gives the critical luminosity of a dusty shell:This is a local force-balance luminosity, so it can depend on radius even when $L$ is constant.

The three useful limiting Eddington ratios areThe infrared limit assumes ${\tau }_{\mathrm{IR}}\gg 1$ and ultraviolet absorption; the single-scattering limit assumes ${\tau }_{\mathrm{UV}}\gg 1\gg {\tau }_{\mathrm{IR}}$; the ultraviolet-thin limit assumes ${\tau }_{\mathrm{UV}}\ll 1$ with infrared opacity negligible compared with ultraviolet opacity. In the ultraviolet-opaque region the two radiative contributions add, and the effective ratio is ${\Gamma }_{\mathrm{IR}}+{\Gamma }_{\mathrm{ss}}(R)$.

The ultraviolet dust transparency radius for this constant-mass shell isFor the terminal speed of a constant-mass dusty shell, approximate the ultraviolet absorption factor by $1$ inside $R_{\mathrm{UV}}$ and by ${\tau }_{\mathrm{UV}}$ outside. The infrared contribution outside that radius is neglected; it is small there if ${\kappa }_{\mathrm{IR}}\ll {\kappa }_{\mathrm{UV}}$. Define ${\Gamma }_{\mathrm{ss}}={\Gamma }_{\mathrm{ss}}(R_0)$, so this otherwise radius-dependent ratio in the paper's final formula is an initial value.

Inside $R_{\mathrm{UV}}$ the acceleration isIntegrating from $R_0$ to $R_{\mathrm{UV}}$ givesOutside $R_{\mathrm{UV}}$, $vdv/dR=G{M}_{\mathrm{BH}}({\Gamma }_{\mathrm{UV}}-1)/R^2$. Its integral to infinity supplies $2G{M}_{\mathrm{BH}}({\Gamma }_{\mathrm{UV}}-1)/R_{\mathrm{UV}}$. Combining the two integrals yieldsThis assumes that the outward trajectory actually reaches the transparency radius; a positive formal endpoint energy alone does not rule out an intermediate turning point.

Put $X=R_{\mathrm{UV}}/R_0=\sqrt{{\tau }_{\mathrm{UV}}(R_0)}$ and $v_{\mathrm{esc},0}^2=2G{M}_{\mathrm{BH}}/R_0$. The ratios obey ${\Gamma }_{\mathrm{UV}}={\Gamma }_{\mathrm{ss}}{X}^2$. The preceding expression becomesIf $X\gg 1$ and infrared work is subdominant,For a luminosity of order the initial single-scattering Eddington luminosity, ${\Gamma }_{\mathrm{ss}}\sim 1$, this implies $v_{\infty }/v_{\mathrm{esc},0}\sim \sqrt{2}{\tau }_{\mathrm{UV}}(R_0{)}^{1/4}\gg 1$. The shell continues intercepting approximately $L/c$ of momentum per unit time over a large distance while gravity weakens as $R^{-2}$. The resulting radiative work is not restricted to its initial gravitational binding energy: an Eddington luminosity is a local force threshold, not a universal cap on outflow speed.

If the unqualified $L_{\mathrm{Edd}}$ instead means the electron-scattering value, the statement needs an additional condition. For $L=L_{\mathrm{Edd},\mathrm{es}}$, ${\Gamma }_{\mathrm{UV}}={\kappa }_{\mathrm{UV}}/{\kappa }_{\mathrm{es}}$, andThus speeds greatly above escape velocity require ${\kappa }_{\mathrm{UV}}/{\kappa }_{\mathrm{es}}\gg \sqrt{{\tau }_{\mathrm{UV}}(R_0)}$, together with a viable launch. Large ultraviolet optical depth by itself does not prove the conclusion at fixed electron-scattering luminosity and opacity. The two interpretations must not be conflated.

For a swept shell, momentum conservation may be writtenThe first term is direct ultraviolet absorption plus trapped-infrared radiation pressure, the second is gravity from the black hole and host, and the last is external pressure. The derivative also accounts for the inertia of newly swept-up gas.

For a singular isothermal sphere,Outside the black hole's sphere of influence and with external pressure neglected, gravity is the constant force $4f_{\mathrm{gas}}{\sigma }^4/G$. The shell optical depths areThus the dust transparency radius and dimensionless optical depths areWith $R^{\prime }=R/R_{\tau }$, $t^{\prime }=t/t_{\tau }$, $t_{\tau }=R_{\tau }/(\sqrt{2}\sigma )$, andthe shell equation becomes

In the optically thick single-scattering regime, $1-e^{-{\tau }_{\mathrm{UV}}}\simeq 1$ and ${\tau }_{\mathrm{IR}}$ is neglected. Integrating the constant right-hand side givesThe zero-radius member has $R^{\prime }=\sqrt{\xi -1}{t}^{\prime }$ and hence the physical constant speed

As the shell becomes ultraviolet-thin,The net force becomes negative beyond the force-balance radiusbut inertia carries the shell farther before it stalls. Match the thick solution at $R^{\prime }=1$, where ${\dot{R}}^{\prime 2}=\xi -1$. In the thin approximation,Using $d/d{t}^{\prime }={\dot{R}}^{\prime }d/d{R}^{\prime }$ and integrating givesThe outer zero is the stalling radiusIncluding the optically thick travel time from a negligible launch radius, the dimensionless stalling time isSolving $d{t}_{\mathrm{stall}}^{\prime }/d\xi =0$ gives $\xi \simeq 1.248$, soFor $1<\xi <1+ϵ$, a stronger mildly supercritical source reaches transparency so much sooner that its total stalling time decreases even though it travels farther. Above this range, the increasing coasting distance dominates. Stalling means that loss of ultraviolet optical depth reduces radiation coupling below gravity; without renewed driving, the swept gas falls back or remains bound rather than escaping the halo.

In the infrared multi-scattering regime, the dominant force is $\xi {\tau }_{\mathrm{IR}}=a/R^{\prime }$ with $a=\xi {\kappa }_{\mathrm{IR}}/{\kappa }_{\mathrm{UV}}$. Seeking $R^{\prime }=A{t}^{\prime 2/3}$ ingives $A^3=9a/2$ and thereforeInfrared trapping supplies more than the single-scattering momentum while the shell is compact and optically thick. The speed nevertheless decreases as $t^{\prime -1/3}$ because the shell sweeps up mass and its infrared optical depth falls. Such driving can launch a powerful dusty radiation-pressure-driven shell, but propagation to halo scales requires enough integrated momentum before the shell becomes transparent; otherwise gravity eventually stalls it.