EFFECTS OF AN ACCRETION DISK WIND ON THE PROFILE OF THE BALMER EMISSION LINES FROM ACTIVE GALACTIC NUCLEI

The model we adopt assumes that the line-emitting region is a thin layer at the base of the wind (i.e., in the atmosphere of the disk that is starting to flow outward to form the wind), which is photoionized by a continuum source associated with the center of the disk. This assumption is justified by the photoionization calculations presented by Murray & Chiang (1998). These authors find that the Balmer line emissivity depends sensitively on density and peaks sharply in a narrow range of optimal densities for a given ionization parameter (this effect was previously emphasized by Baldwin et al. 1995). The density in the emission layer is high enough that collisions populate the first excited state of hydrogen and produce a significant optical depth in the Balmer lines. Above the emission layer, the density of the disk atmosphere drops sharply and the atmosphere becomes highly ionized. As a result, Balmer lines are not emitted efficiently and the optical depth to Balmer line photons above the emission layer is small (due to the low collision rate at these low densities). We have carried out a simple photoionization calculation of a static disk atmosphere to verify our assumptions and illustrate the structure of the emission layer, which we present in Appendix A. This calculation also confirms that the Balmer line optical depth in the emission layer is extremely high because the first excited state of hydrogen is populated by collisions.

The inner radius of the line-emitting portion of the disk is r_min ⩾ 100 r_g. Interior to this radius, the disk is hot and highly ionized so it does not emit Balmer lines efficiently. Moreover, our comparison with observed profiles in later sections of this paper shows that r_min ⩾ 100 r_g is necessary in order to reproduce the data. The UV/optical thermal continuum emission from the line-emitting region of the disk is very weak compared to what is produced at smaller radii and its spectrum peaks in the infrared band. Therefore, we neglect this local continuum, which has the important consequence that the line-emitting layer is never observed against a bright continuum source at any inclination and the Balmer lines always appear in emission, in agreement with observations.

The Balmer line photons emerging from the disk include a contribution from recombination and a contribution from resonance scattering of continuum photons from the inner disk. We do not make a distinction between these two mechanisms; we parameterize the volume emissivity as a power law in radius and assume that this parameterization captures the combination of the two effects. We also neglect emission, resonance scattering, or absorption of Balmer line photons from outflowing gas that is launched from the inner disk and passes over the line-emitting region (see the simulations of Proga & Kallman 2004, for example). This gas has relatively low density; therefore, it is highly ionized and collisional excitation of neutral hydrogen atoms is negligible (the ionization is accomplished by a combination of direct and multiply scattered continuum photons; see Sim et al. 2010). This medium cannot be neglected, however, when considering the propagation of high-ionization, resonance-line photons (e.g., C iv).

The electron scattering optical depth is negligible in the Balmer line emission layer (see Appendix A), which leads us to neglect electron scattering of line photons. However, this assumption may break down if a substantial amount of outflowing gas that originates in the inner disk passes over the outer disk (Proga & Kallman 2004). If such a substantial electron scattering medium exists, then it is also possible for energetic continuum photons from the inner disk to scatter downward and alter the ionization structure of the line-emitting portion of the disk, as we noted in the previous paragraph (see Sim et al. 2010). Exploring the importance of these effects requires Monte Carlo simulations of photon propagation coupled with a hydrodynamical model for the flow.

The model described below does not apply to high-ionization lines (e.g., C iv or other UV resonance lines). These lines are produced in a highly ionized layer that has a much larger vertical extent than the Balmer line emitting zone and also has a substantial vertical velocity and a non-Keplerian azimuthal velocity (see Murray et al. 1995). This extended emission zone can be observed against the bright continuum source of the inner disk for certain viewing angles. Therefore, a calculation of the profiles of high-ionization lines would have to adopt a different velocity field (see Hall & Chajet 2010) and take into account resonance scattering as well (see Higginbottom et al. 2012), since P-Cygni profiles are observed in the UV resonance lines of a substantial fraction of quasars.

We combine the Chen et al. (1989) and Chen & Halpern (1989) method for computing the emission line profile from an accretion disk (including relativistic effects in the weak-field limit) with the analytic prescription of CM96 and MC97 to include radiative transfer in a disk wind. Numerical models of winds (e.g., Proga et al. 2008; Everett 2005) have a more complete treatment of physical processes, but are computationally expensive. Here, we use analytic prescriptions in order to understand the effects of the wind properties on the line profile and its variability. We compute the line profile by numerically evaluating a modified version of Equation (8) of CM96

$$\begin{equation} f_{\nu } \propto \int \!\! \int \xi \; I_{\nu _e}(\xi,\phi ^{\prime }, \nu)\; D^3(\xi, \phi ^{\prime })\; \Psi (\xi, \phi ^{\prime })\; d\phi ^{\prime }\; d\xi \end{equation} \tag{ 1 }$$

over the surface of the disk. Here, ϕ' is the azimuthal angle in the disk frame (see Figure 1 of Chen et al. 1989) and ξ is the radial distance from the central black hole in units of the gravitational radius. The velocity structure of the disk is contained in the Doppler factor D(ξ, ϕ') and the light bending effects are included in Ψ(ξ, ϕ'). These factors were computed by Chen et al. (1989) for a circular disk and depend on the inclination i of the disk to the line of sight, measured away from the z'-axis.⁸

The specific intensity $I_{\nu _e}(\xi,\phi ^{\prime },\nu)$ contains information on the local line profile (assumed to be Gaussian), the disk emissivity function (ξ, ϕ'), and the radiative transfer of line photons:

$$\begin{equation} I_{\nu _e}(\xi,\phi ^{\prime }, \nu) \propto \beta (\tau _{\nu _e})\epsilon (\xi,\phi ^{\prime })\exp \left[-\frac{(\nu _e-\nu _0)^2}{2\sigma ^2}\right], \end{equation} \tag{ 2 }$$

where σ is the line broadening parameter, ν_e is the emitted frequency of the photon, ν₀ is the rest frequency of the line photon, and $\tau _{\nu _e}$ is the total optical depth along the line of sight. The factor $\beta (\tau _{\nu _e})=(1-e^{-\tau _{\nu _e}})/\tau _{\nu _e}$ represents the probability for a photon to escape the wind in the direction of the observer (the "directional escape probability" according to Hamann et al. 1993). Using the Sobolev approximation for flows whose velocity component to the line of sight is non-monotonic,⁹ the optical depth is (Rybicki & Hummer 1978, see also CM96)

$$\begin{equation} \tau = \frac{\kappa \,\rho \,\sigma }{| \bf {\hat{n}\cdot \Lambda \cdot \hat{n}} |}, \end{equation} \tag{ 3 }$$

where κ is the line absorption coefficient, ρ its density, σ is the local velocity dispersion in the emitting layer (thermal or turbulent), ${\bf \hat{n}}$ is the line of sight vector, and Λ is the symmetric strain tensor (e.g., Batchelor 1967). Assuming that the density, ρ, and the velocity dispersion, σ, can be expressed as power laws in ξ, one can rewrite Equation (3) as

$$\begin{equation} \tau (\xi,\phi ^\prime) = \frac{\tau _0}{|\bf {\hat{n}\cdot \Lambda \cdot \hat{n}}|} \left(\xi \over 1000\right)^{-\eta }, \end{equation} \tag{ 4 }$$

where τ₀ is a normalization parameter. Physically, τ₀ represents the total optical depth of the surface emission layer perpendicular to the disk plane at ξ = 1000 when that layer is static. In practice, the optical depth in the emission layer at ξ < 1000 can be several orders of magnitude higher than τ₀. As we will see below, the denominator in Equation (4) varies significantly with azimuth, which leads to large azimuthal variations of the optical depth. We introduce (ξ/1000)^−η to account explicitly for the primary dependence of ρ on radius. MC97 point out that ρ∝1/(r²v), whence we infer that η = 1.5. Also, Hamann et al. (1993) find that η = 2 results in a constant ionization parameter with radius. Based on the response time of different broad lines (with a range of ionization energies) to changes in the central ionizing continuum, Peterson et al. (1990) find that the ionization parameter decreases with radius, meaning that η < 2. Hence, we explore the dependence of the line profile on the value of η in Section 3.1.

We use the analytic prescription of the velocity field of a disk wind by Murray et al. (1995), to calculate $Q\equiv \bf {\hat{n}\cdot \Lambda \cdot \hat{n}}$ (see Appendix B for details of the calculation):

$$\begin{eqnarray} Q &= & \sin ^2 i\left[\frac{\partial v_r}{\partial r}\cos ^2\phi ^{\prime }+\frac{3}{2}\frac{v_{\phi ^{\prime }}}{r}\sin \phi ^{\prime }\cos \phi ^{\prime }+\frac{v_r}{r}\sin ^2\phi ^{\prime } \,\right] \nonumber \\ &&+ \cos i \Bigg[\frac{\partial v_r}{\partial r}\left(\frac{1}{\sin \lambda } \sin i \cos \phi ^{\prime } + \cos i \right) \nonumber\\ &&+ \frac{v_{\phi ^{\prime }}}{2r\sin \lambda } \sin i \sin \phi ^{\prime } \,\Bigg], \end{eqnarray} \tag{ 5 }$$

where λ(r) is the opening angle of the wind from the surface of the disk (see Figure 1 of MC97) and $v_{\phi ^{\prime }}=(GM_{{\rm BH}}/r)^{1/2}$ is the rotational velocity. The radial velocity is given by

$$\begin{equation} v_r=v_\infty \left(1-\frac{\xi _f}{\xi }\right)^\gamma, \end{equation} \tag{ 6 }$$

where ξ_f is the launching radius of the wind streamline in units of r_g and v_∞ ≃ 4.7 ξ^−1/2_f (Murray & Chiang 1998). In our calculation of Q, we set ξ = ξ_f, which leads to the simplifications v_r = 0 and v_θ = 0. This assumption is justified because the bulk of the Balmer line emission comes from the base of the disk wind, where the density is the highest and the ionization parameter is relatively low. Typically, γ ≈ 1–1.3; we set γ = 1, following MC97, which leads to ∂v_r/∂r = v_∞/ξ ≠ 0 at ξ = ξ_f. Thus, we can rewrite the expression for Q from Equation (5) as

$$\begin{equation} Q = \xi ^{-3/2}\, Q_0 \,, \end{equation} \tag{ 7 }$$

where

$$\begin{eqnarray} Q_0 &=& \sin ^2 i\left(4.7\cos ^2\phi ^{\prime }+\frac{3}{2}\sin \phi ^{\prime }\cos \phi ^{\prime } \,\right) \nonumber\\ && + \cos i \left(\frac{4.7 \sin i \cos \phi ^{\prime }}{\sin \lambda } + 4.7\cos i + \frac{\sin i \sin \phi }{2\sin \lambda } \right). \nonumber\\ \end{eqnarray} \tag{ 8 }$$

As we note in Appendix B, our expression for Q in Equation (5) is different from that in Equation (15) of MC97. The difference is in the sign of the second term. An independent calculation (Hall & Chajet 2010; Chajet 2012; P. Hall et al. 2012, in preparation; L. Chajet et al. 2012, in preparation) confirms our own result and leads us to conclude that there is a sign error in the formula of MC97. In Appendix B, we compare our expression for Q with that of MC97, illustrate that the differences are small, and demonstrate that they lead to only slight differences in the profiles of the emission lines. Collecting all the terms together, we rewrite the final expression for the optical depth as

$$\begin{equation} \tau (\xi,\phi ^\prime) = \frac{\tau _0}{Q_0} \left(\xi \over 1000\right)^{3/2-\eta } \,, \end{equation} \tag{ 9 }$$

where Q₀ is defined by Equation (8), above. Equation (9) separates the primary radial dependence of the optical depth from its angular dependence. The function Q₀ depends implicitly on the radius through the wind opening angle, λ, but, as we illustrate in Section 3.1, below, this dependence is fairly weak.

Finally, the emissivity function (ξ, ϕ') can be any analytic or numerical prescription, from the simplest (e.g., a power law in ξ) to more complex models (e.g., logarithmic spiral arm and stochastically perturbed disk). In this paper, we use a power law of the form (ξ, ϕ')∝ξ^−q (motivated by the calculation of Collin-Souffrin & Dumont 1989, and our own calculations presented in Appendix A) in order to understand the influence of the model parameters on the line profile.

Throughout this paper, we use the Hα emission line profile as our working example, we set σ = 600 km s⁻¹, and we normalize all the line profiles to unit maximum, unless specified otherwise. Although it is desirable for some applications to normalize the line profiles by the continuum level, we are unable to do this in a meaningful way. This is because we do not have robust physical models for the AGN spectral energy distribution. As a result, (1) the relative normalization between the far-UV and X-ray continuum (which sets the line luminosity) and the optical continuum (which underlies the Balmer lines) is uncertain, and (2) the behavior of the continuum with inclination angle is uncertain. Moreover, the optical continuum source is much more compact than the emission-line region, therefore the optical continuum can suffer from extinction when the broad emission lines do not.

In this section, we investigate the differences in emission line profiles that arise from varying the model parameters and also from different implementations of the disk-wind model. We first confirm that our integration method gives results consistent with those in the literature. Since our calculation combines elements from CM96 and MC97, we do not expect to reproduce the results from either of those works exactly, yet we expect fairly close agreement. We compare our computed line profiles with Figure 2 of CM96, which demonstrates that the addition of an optically thick disk wind to an axisymmetric accretion disk transforms a double-peaked profile to a single-peaked profile. We adopt the same values as CM96 for the model parameters, namely, i = 75°, ξ_min = 600, and ξ_max = 90, 000. We also use their expression for Q. The remaining parameters, in our convention, are q = −2.5 and η = −0.6. We do not include relativistic effects, we calculate the line profile for the C iv line, and the lines are normalized to have the same integrated flux. Figure 1 shows the line profile with τ₀ = 10⁻⁷ (dotted line) and τ₀ = 10 (solid line) and demonstrates that we can reproduce the main features of the line profiles. In particular, the line is double-peaked with pronounced wings when τ₀ = 10⁻⁷ and becomes single-peaked with weaker wings when τ₀ = 10. There is a small difference in the shape of the peak of the single-peaked line which is largely a result of the different integration methods: we evaluate the double integral of Equation (1) over the surface of the disk numerically while they employ an approximation to reduce it to a single integral in the radial direction before evaluating it, thus our integration method is more accurate. Moreover, we use a narrow Gaussian as the local line profile while CM96 use a delta function, which produces a negligible difference between our results and theirs. Similarly, we are able to reproduce the single-peaked profile shown in Figure 3 of MC97.

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To illustrate how relativistic effects influence the line profile, we show in Figure 2 a grid of Hα line profiles showing the effects of relativity with increasing inclination. The model parameters are set to ξ_min = 150, ξ_max = 5000, λ(ξ_min) = 10°, η = 0.5, q = 2.0, and τ₀ = 10. When relativistic effects are taken into account for this combination of parameters, the wings of the profiles are shifted to the red by a few thousand kms⁻¹ and the profiles become skewed, while the FWHM remains approximately the same. As a consequence, the centroid of the profiles is also redshifted. The skewness and redshift appear more pronounced at smaller inclination angles because the FWHM of the profiles is smaller and the shifts from relativistic effects are comparable to the FWHM. At inclinations of 45° and 60°, we can also see that the top of the line profile is not flat: the blue side is somewhat stronger than the red side because of beaming of the light from the approaching side of the disk. The influence of relativistic effects on the line profiles is stronger when the inner radius of the line-emitting region, ξ_min, is small and/or the emissivity is heavily weighted toward the center of the disk (the radial emissivity profile of the disk depends on η, as we explain later in this section). The value of ξ_min we used in the examples of Figure 2 is similar to the value we use in Section 3.2 to reproduce the average observed line profiles, which have skewed and redshifted wings.

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In order to investigate the effects of varying the optical depth in the disk wind, we set the model parameters to the following values: i = 60°, ξ_min = 500, ξ_max = 10, 000, q = 2.5, λ(ξ_min) = 10°, and η = 2. We then produce line profiles with τ₀ increasing from 10⁻⁷ to 10⁻¹, by factors of 100, as shown in Figure 3. Line profiles with τ₀ < 10⁻⁷ are very similar to that with τ₀ = 10⁻⁷. Similarly, line profiles for τ₀ > 10⁻¹ are not significantly different from that with τ₀ = 10⁻¹. As the optical depth of the disk wind increases, the peaks of the line profile get closer together and eventually the line profile looks single-peaked. This is due to the fact that the escape probability through the wind is not isotropic in the optically thick case and the photons escape more easily in preferred directions. Figure 4 shows the map of Q over the surface of the disk for a large optical depth through the disk wind. In the large optical depth limit, the emissivity of the disk through the disk wind is proportional to Q. Thus, the emissivity function has a minimum when ϕ' ∼ 90°, 270° and a maximum when ϕ' ∼ 0°, 180°. Note that the maximum emission originates from the region in the disk where the velocity vector of the gas is nearly perpendicular to the line of sight. As a consequence, the wings and double peaks of the line profile are suppressed and the core appears strong in comparison. Despite the differences in methods and the somewhat different resulting line profiles, we arrive at the same general conclusion as CM96 and MC97 about the drastic effect of the wind optical depth on the line profile.

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We also investigate the effects of varying other model parameters within their physically plausible range, namely, 0.5 < η < 2.0, 5 < λ < 25°, 2 < q < 3, and 1000 < ξ_max < 20000. For this, we fix the optical depth (τ₀ = 10), the inclination (i = 60°), and the inner radius (ξ_min = 500). To test the effect of η, we fixed q = 2.5, ξ_max = 10, 000, and λ(ξ_min) = 10°. As we show in Figure 5, decreasing η produces profiles with larger FWHM and a more extended and pronounced red wing. This is because at high optical depths the radial emissivity profile of the disk varies as ξ^{−q − (3/2) + η} (since β∝1/τ for large τ), and thus, decreasing η weights the emissivity more heavily toward the center of the disk where relativistic effects are more important. This exercise also illustrates that for inner disk radii as large as ξ_min = 500, relativistic effects can still produce skewed line profiles.

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To test the effect of varying λ(ξ_min), we set q = 2.5, ξ_max = 10, 000, and η = 2. The FWHM of the line profile does not vary with λ(ξ_min), but an increase in λ(ξ_min) leads to a slight redshift of the line profile, amounting to 4 Å (180 km s⁻¹) for λ(ξ_min) between 5° and 30°. A decrease in q leads to a narrowing of the line profile. With η = 2, λ(ξ_min) = 10°, and ξ_max = 20, 000, we find an FWHM of ∼6200 km s⁻¹ with q = 2 and an FWHM ∼ 7300 km s⁻¹ with q = 3. Finally, as ξ_max increases, the FWHM decreases. With η = 2, λ(ξ_min) = 10°, and q = 2.5, we find an FWHM of ∼14, 000 km s⁻¹ with ξ_max = 2000 and an FWHM ∼ 6600 km s⁻¹ with ξ_max = 10, 000. Also, a lower ξ_max produces a larger redshift of the centroid of the line (8 Å between the two test cases above). Hence, η and ξ_max are the parameters, other than τ₀, that influence the line profile the most significantly, with the largest impact in the FWHM.

In order to determine whether our wind model can reproduce observed emission line profiles, we created a database of model line profiles spanning a large parameter space. Since we determined that the opening angle of the wind, and emissivity power-law index had little influence on the line profile, we set the values of these parameters to λ = 15° and q = 2.5, respectively. This leaves five parameters: ξ_min, ξ_max, i, η, and τ. We compute the line profile for 100 ⩽ ξ_min ⩽ 10100 in increments of 2000, 1000 ⩽ ξ_max ⩽ 21000 in increments of 5000, 5 ⩽ i ⩽ 85 in increments of 10, 0.5 ⩽ η ⩽ 2.0 in increments of 0.5, and 5 × 10⁻⁷ ⩽ τ ⩽ 5 × 10⁻¹ in increments of one order of magnitude, for a total of 5544 profiles.

In order to compare the resulting line profiles with observed line profiles, we determined the FWHM and full width at quarter maximum (FWQM) for each line profile. We also computed the asymmetry index (A.I.), kurtosis index (K.I.), and centroid shift at quarter maximum (v_c(1/4)) defined in Equations (3)–(5) of Marziani et al. (1996). We compare these indices for the modeled line profiles with those of the Zamfir et al. (2010) sample, which consists of the Hβ line profiles of ∼470 low-redshift quasars extracted from SDSS (Data Release 5). While we selected the Hα line to model, the formalism used here should be applicable to other low-ionization broad permitted lines. For example, La Mura et al. (2007) found that Hα and Hβ had similar emission line profiles, so the comparison of our simulated Hα line with observed Hβ line is valid.

Figure 6 shows how the FWHM varies with the different model parameters. The FWQM behaves like the FWHM and is not shown. In panel (a), we see that the average FWHM decreases between 10⁻⁵ < τ₀ < 10⁻², but stays constant outside of this range. This behavior was discussed in Section 3.1 and illustrated in Figure 3. In Section 3.1, we mentioned that the FWHM decreases with η and panel (b) of Figure 6 shows that the average FWHM indeed falls with increasing η, but the large standard deviation means that the other model parameters also influence the FWHM significantly. Panel (c) shows that the average FWHM has a shallow peak at a large inclination (∼75°). For a simple disk model, we would expect the FWHM and FWQM to have a sin i behavior if all the other model parameters are kept constant. We indeed observe such a behavior of the average FWHM until ∼70°, where the disk wind is oriented directly into the line of sight. This disrupts the sin i behavior, leading to a fall in FWHM instead of a continued rise. Finally, the FWHM decreases with ξ_min and ξ_max (see panel (d) of Figure 6). The dependence of the average FWHM on the inner or outer radius of the disk in our model is flatter than expectations from Keplerian motion. This is illustrated in Figure 7, where we show the dependence of the FWHM on ξ_min and two power-law relations: ξ^−1/4_min (dashed line) and ξ^−1/2_min (continuous line), both normalized to match the average FWHM at ξ_min  ∼  6000. We find that ξ^−1/4 better describes the behavior of the average FWHM. A similar behavior is observed when plotting the average FWHM versus ξ_max instead of ξ_min. A direct comparison of the simulated values of FWHM with the range of FWHM observed by Zamfir et al. (2010; hashed area in Figure 6)) does not allow us to place useful constraints on either of the model parameters due to the large dispersions of the simulated values.

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Figure 8 shows the behavior of the A.I. when varying the model parameters. A positive A.I. means that the line profile is skewed toward the red, while a negative value indicates a skewness toward the blue. Panel (a) shows that the average A.I. decreases with τ₀. This behavior arises because the double-peaked line profiles are more skewed toward the red than single-peaked line profiles. Panel (b) shows that the average A.I. rises with η, and the large dispersion indicates that the other model parameters contribute more to the A.I. than η. In panel (c), we see that the average A.I. changes non-monotonically with the inclination of the accretion disk. It first rises until ∼35°, then decreases until ∼75° where it rises sharply. This behavior can be explained by a close inspection of Figure 2 (in particular, the dashed line profiles that include relativistic effects). From i = 10° to i = 30°, the peak of the line profile develops an asymmetry and shifts toward the blue, which increases A.I. At larger inclinations, the peak of the line does not shift as significantly with inclination, but the red wing disappears leading to a fall in the A.I. When the line of sight is so close to the disk plane that i > 90° = λ, the red wing of the line profile becomes stronger, increasing the A.I. In panel (d), the average A.I. falls until ∼0 with ξ_min and ξ_max since the relativistic effects skewing the profile to the red diminish with increasing radius. Note that a full range of ξ_min and ξ_max introduces a significant dispersion in the A.I. In all panels, the dispersion of the A.I. of simulated profiles is large, which leads to significant overlap with the observed values. Thus, the A.I. alone does not lead to stringent constraints on the model parameters.

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Figure 9 illustrates the influence of the model parameters on the K.I. A line profile has a higher K.I. when it has suppressed wings and a steep rise toward the core. Panel (a) shows a slight decrease of the average K.I. with τ₀, as expected from Figure 3, which demonstrates that the wings of the line profile are more prominent at large τ₀. Panel (b) shows that an increase in η leads to a rise in the average K.I. In panel (c), the average K.I. has a shallow peak at ∼60°, resulting from the increase in line core size with rising inclination, which stalls at intermediate inclinations (Figure 2). Finally, the average K.I. slightly falls with ξ_min, since the relativistic effects strengthen the profile wings at low radii. But the average K.I. decreases with ξ_max, since this leads to the weakening of the core of the line. The K.I. appears to be very sensitive to inclination, and therefore, by comparing the model values of K.I. with the values observed by Zamfir et al. (2010), we can put a constraint on the inclination: i ≲ 45°, based on panel (c). However, we are not able to place stringent constraints on the other model parameters using the K.I. alone.

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Figure 10 presents the variation of v_c(1/4)/FWQM with the various model parameters. Panel (a) shows that this parameter rises with τ₀, which is mainly driven by the decrease of the FWQM with τ₀. In panel (b), the average v_c(1/4)/FWQM falls with η but the dispersion in values is large. The centroid shift over FWQM depends non-monotonically on the inclination of the accretion disk. It first falls until ∼45°, then rises until ∼75° where it sharply falls again. We interpret this as the blueshifting of the centroid dominating the decrease in FWQM at low inclination, but becoming less significant compared to the FWQM past ∼45°. At large inclination, when i > 90° − λ, the shift of the centroid suddenly decreases. Finally, panel (d) shows that the average centroid shift over FWQM evolves to 0 with ξ_min and ξ_max, since relativistic effects displace the centroid at low radii. A comparison of the model values with the values quoted by Zamfir et al. (2010) does not allow us to constrain the model parameters.

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In previous paragraphs, we compared the data with individual model distributions of the A.I., K.I., FWHM, and v_c(1/4)/FWQM. We found that the K.I. allows us to constrain the inclination to i ≲ 45°, while the rest of the modeled properties do not yield stringent limits because they are characterized by a large dispersion. Here, we discuss additional constraints that these properties place on the parameter space collectively. For a given range of inclination, we find that τ₀ ≳ 10⁻⁴ are favored by the data. This follows from the properties of the Zamfir et al. (2010) sample which comprises only a handful of AGNs with DPELs. In the following steps, we compare the observed and modeled distributions of A.I. and v_c(1/4)/FWQM, given the allowed range of i and τ₀. We find that the A.I. distribution is satisfactorily reproduced for smaller values of inner radius (ξ_min ≲ 2000). This restriction in turn leads to the constraint ξ_max ≳ 5000, required in order to reproduce a wide range of values for v_c(1/4)/FWQM. Figure 11 presents the distributions of the model line profile parameters after applying these restrictions to the model parameters. The dashed histograms are produced from the data presented in Zamfir et al. (2010).¹⁰ We find a very good agreement between the model and observed K.I. histograms. The ranges of model and observed A.I. are comparable, but the average model A.I. is somewhat larger than the observed value. v_c(1/4)/FWQM is slightly low compared to observations. A finer adjustment of the limits of the model parameters would likely produce histogram distributions in even better agreement with observations. For example, a limit of ξ_min ≲ 4000 produces a model v_c(1/4)/FWQM distribution in better agreement with the observations, but a low average A.I. and too many line profiles with large K.I.

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We illustrate in Figure 12 that the disk-wind model can produce realistic line profiles. We plot the average observed line profile of the two types of AGNs as given in Zamfir et al. (2008) (see Sulentic et al. 2009, for the definition of the two populations). Population A AGNs have an FWHM < 4000 km s⁻¹ and low-ionization broad lines are best described by a Lorentz profile. We were able to produce a model line profile in agreement with the observed profile with a choice of model parameters within the constraints discussed above, namely, τ₀ = 1, i = 30°, η = 2, ξ_min = 100, and ξ_max = 20, 000. The line profile of population B AGNs is best described by a double-Gaussian with an FWHM > 4000 km s⁻¹. We were able to produce a similar model profile with τ₀ = 1, i = 30°, η = 1.5, ξ_min = 100, and ξ_max = 10, 000. The simulated line profiles are not intended to be a fit to the observed line profiles, but to demonstrate that average model parameters can produce realistic line profiles.

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Finally, in order to determine whether realistic line profiles can be produced without the inclusion of relativistic effects, we created a second database of simulated line profiles: without relativistic effects. We explored how the different line profile parameters depend on the wind model parameters. We find that the behavior of the K.I. and FWHM with model parameters is the same whether relativistic effects are included or not. However, as illustrated in Figure 13, the behavior of the A.I. and v_c(1/4)/FWQM against a varying inclination is radically different when relativistic effects are not included. In order to produce realistic values for these line profile parameters, high inclinations are favored, which is at odds with the requirement for small inclinations from the behavior of the FWHM. Thus, without relativistic effects, realistic distributions of A.I. and v_c(1/4)/FWQM cannot be reproduced. This is also illustrated in Figure 12, where we plot simulated line profiles without relativistic effects. Without these effects, it is impossible to reproduce the wing asymmetry observed in the double-Gaussian profile.

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