Average density in BAL X-ray absorbers

Astronomy over the weekend included fixing a factor of 2 numerical error in my most recent paper (which fortunately hasn't gone to press yet). A question from Joe Shields led me to think about density and column density limits in the outflow in FBQS J1408+3054. By crude analogy to a similar FeLoBAL with published high-resolution spectra and X-ray data (J0300+0048), the Fe II column density lower limit is uninteresting. But if the total column density in this object is Compton thick with N_H >= (10^24 cm^-2)/Z where Z is the metallicity relative to solar, then the average density of the X-ray absorber must be >~(10^4 cm^-3)/fZ where f is its filling factor along the line of sight. For example, if the X-ray absorber (which is likely not identical to the UV absorber) extends from the BH out to 1/10th the distance of the UV BAL gas, then its density has to be >~(10^5 cm^-3)/Z. If the X-ray absorber extends farther from the BH than the UV absorber, then f>1 in the above and the density can be lower. But too much of the X-ray absorber exterior to the UV absorber will cause Compton scattering of the quasar's UV continuum and remove it from sight.

Update: Morabito et al. 1011.4327 today reinforce the above by pointing out that in FeLoBALs studied to date, f<<1 because the X-ray absorber must be interior to the UV continuum emitting region or else it would Thomson scatter the UV continuum out of our line of sight. For the 1.4E8 Msun BHs in that paper, R_Sch = 4E13 cm and 20R_Sch=8E14 cm. For a column N_H=(1E25 cm^-2)/Z located interior to that radius, the average density is 1E25/8E14 = (1.25 x 10^10 cm^-3)/Z.