composition and its size, we choose to adopt the simplification of assigning a uniform sublimation temperature to all the dust in our calculations. The sublimation radius may be estimated as

$$\begin{aligned}
r_\mathrm{sub} &\approx \sqrt{\frac{L}{4 \pi \sigma_{\mathrm{SB}}T_{\mathrm{sub}}^4}}\nonumber \\
&= 0.62 \mathrm{\;pc} \left(\frac{L}{10^{46} \mathrm{\; erg s}^{-1}} \right)^{1/2}\left( \frac{T_\mathrm{sub}}{1400 \mathrm{\; K}}\right)^{-2} \; .\end{aligned}$$

When the gas distribution surrounding the SMBH is not isotropic, $r_\mathrm{sub}$ may vary with angle. Within this radius, electron scattering dominates the opacity, and the usual Eddington limit applies.

Once the intrinsic photons from the accretion disk encounter dust in the surrounding gas, they are absorbed and the energy is re-emitted at infrared wavelengths. If the gas is also optically thick to the infrared, then the re-emitted radiation will continue to be absorbed and re-emitted in a random-walk pattern until it exits the optically thick region. Along the way, momentum will be imparted by the photons to the gas multiple times. In this scenario it is possible for the radiation to transfer momentum to the gas at a rate that exceeds $L_\mathrm{Edd}/c$. For a spherically symmetric problem, this “boost” factor to the infrared radiation force is exactly the infrared optical depth of the gas, which can be shown as follows: 