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Appendix A: Ionic column-density measurements

The column-density associated with a given ion as a function of the radial velocity v is defined as: (A.1)where f_j, λ_j and ⟨ τ_j(v) ⟩ are respectively the oscillator strength, the rest wavelength and the average optical depth across the emission source of the line j for which the optical depth solution is derived (see Edmonds et al. 2011). The optical depth solution across a trough is found for a given ion by assuming an absorber model. As shown in Edmonds et al. (2011), the major uncertainty on the derived column-densities comes from the choice of absorption model. In this study we investigate the outflow properties using column-densities derived from three common absorber models.

Assuming a single, homogeneous emission source of intensity F₀, the simplest absorber model is the one where a homogeneous absorber parameterized by a single optical depth fully covers the photon source. In that case, known as the apparent optical depth scenario (AOD), the optical depth of a line j as a function of the radial velocity v in the trough is simply derived by the inversion of the Beer-Lambert law: τ_j(v) = −ln(F_j(v) /F₀(v)), where F_j(v) is the observed intensity of the line.

Early studies of AGN outflows pointed out the inadequacy of such an absorber model, specifically its inability to account for the observed departure of measured optical depth ratio between the components of typical doublet lines from the expected laboratory line strength ratio R = λ_if_i/λ_jf_j. Two parameter absorber models have been developed to explain such discrepancies.

The partial covering model (e.g. Hamann et al. 1997; Arav et al. 1999, 2002, 2005) assumes that only a fraction C of the emission source is covered by absorbing material with constant optical depth τ. In that case, the intensity observed for a line j of a chosen ion can be expressed as (A.2)Our third choice are inhomogeneous absorber models. In that scenario, the emission source is totally covered by a smooth distribution of absorbing material across its spatial dimension x. Assuming the typical power law distribution of the optical depth τ(x) = τ_maxx^a (de Kool et al. 2002; Arav et al. 2005, 2008), the observed intensity observed for a line j of a chosen ion is given by (A.3)Once the line profiles have been binned on a common velocity scale (we choose a resolution dv = 20 km s^-1, slightly lower than the resolution of COS), a velocity dependent solution can be obtained for the couple of parameters (C,τ_j) or (a,τ_max) of both absorber models as long as one observes at least two lines from a given ion, sharing the same lower energy level. Once the velocity dependent solution is computed, the corresponding column density is derived using Eq. (A.1) where ⟨ τ_j(v) ⟩ = C_ion(v)τ_j(v) for the partial covering model and ⟨ τ_j(v) ⟩ = τ_max,j(v) / (a_ion(v) + 1) for the power law distribution. Note that the AOD solution can be computed for any line (singlet or multiplet), without further assumption on the model, but will essentially give a lower limit on the column-density when the expected line strength ratio observed is different from the laboratory value.

© ESO, 2015