of the nucleus that contribute a substantial fraction to the gas column density, and the force integral can be well-approximated by setting its lower limit to $r_\mathrm{sub}$.

To get a sense of the sort of momentum deposition rates that have been observed, consider the case of Mrk 231. This system features an outflow of neutral gas with velocities in the range 360-900 km s$^{-1}$ and a mass-loss rate estimated at 420 solar masses per year [@Rupke2011]. The momentum flux in the outflow, estimated by multiplying the mass loss rate by the velocity, is between 2.6 to 6.5 times $L/c$ where $L$ is measured to be $1.1 \times 10^{46}$ ergs s$^{-1}$. The kinetic luminosity of the outflow, on the other hand, is estimated at $7.3 \times 10^{43}$ ergs s$^{-1}$, less than 1% of the bolometric AGN luminosity.

Modeling the force from radiation pressure, and predicting by what factor it exceeds $L/c$, becomes a difficult problem to tackle analytically in the absence of spherical symmetry, the presence of clumps, and with an accounting for dust sublimation. For these reasons, we turn here to three-dimensional radiative transfer calculations using the wavelength-dependent Monte Carlo radiative transfer code SEDONA [@Kasen2006]. Given that the radiative diffusion 