Articles

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

, , and

Published 2012 June 22 © 2012. The American Astronomical Society. All rights reserved.
, , Citation Hélène M. L. G. Flohic et al 2012 ApJ 753 133 DOI 10.1088/0004-637X/753/2/133

0004-637X/753/2/133

ABSTRACT

We explore the connection between active galactic nuclei (AGNs) with single- and double-peaked broad Balmer emission lines by using models dealing with radiative transfer effects through a disk wind. Our primary goal is to assess the applicability of the Murray & Chiang model by making an extensive and systematic comparison of the model predictions with data. In the process, we also verify the original derivation and evaluate the importance of general relativistic effects. As the optical depth through the emission layer increases, the peaks of a double-peaked profile move closer and eventually merge, producing a single peak. The properties of the emission line profile depend as sensitively on the geometric parameters of the line-emitting portion of the disk as they do on the disk-wind parameters. Using a parameter range that encompasses the expected characteristics of the broad-line regions in AGNs, we construct a database of model profiles and measure a set of diagnostic properties. Comparisons of the model profiles with emission lines from a subset of Sloan digital Sky Survey quasars show that observed lines are consistent with moderately large optical depth in the disk wind and a range of disk inclinations i ≲ 45°. Including relativistic effects is necessary to produce the asymmetries of observed line profiles.

Export citation and abstract BibTeX RIS

1. INTRODUCTION

One of the defining characteristics of active galactic nuclei (AGNs) is the presence of broad emission lines in their optical spectra. These broad lines have full width at half-maximum (FWHM) that is typically greater than 5000 km s−1, far in excess of the thermal velocity (∼10 km s−1), which can be explained only by bulk motion of the emitting gas. The flux of the broad lines responds to variations in the flux of the continuum with a delay of the order of days to a few months (e.g., Peterson 1993; Korista et al. 1995; Peterson et al. 2004; Denney et al. 2010), indicating that the broad-line region (BLR) is located within the inner ∼0.01 pc of the central engine.

The width of the broad lines is a powerful tool for determining the mass of the supermassive black hole at the center of the AGN (e.g., Peterson & Wandel 2000; Onken & Peterson 2002; Kollatschny 2003). This method relies on the scaling relations determined by reverberation mapping (e.g., Bentz et al. 2009), which include a factor f dependent on the geometry and kinematics of the BLR. Currently, the value of f most commonly used was determined empirically by matching the mass obtained from reverberation mapping to that obtained with the M–σ* relation (Onken et al. 2004). However, with a better knowledge of the BLR structure and kinematics (including the role of radiative pressure; e.g., Marconi et al. 2008), one could calculate f (Netzer 1990). Thus, it is important to determine the geometry and kinematics of the BLR as well as how it depends on the black hole mass and accretion rate.

The theory steadily gaining support in the AGN community is that the accretion flow itself is the source of the broad lines. Studies of the response of the line profile to changes in the flux continuum indicate that the motion of the gas in the BLR of most AGNs is consistent with Keplerian motion (Bentz et al. 2009), in agreement with the disk model (Bentz et al. 2008 and Denney et al. 2010, however find examples of AGNs with signs of a free-falling BLR and outflowing BLR, which is hard to reconcile with the disk model). Studies also indicate that the BLR has a flattened geometry (e.g., Wills & Browne 1986; McLure & Dunlop 2002; Kollatschny 2003). Collin-Souffrin (1987) argued that the broad, low-ionization lines, such as Hα, Hβ, and Mg ii λ2798, could be produced by the surface layer of the dense accretion disk (n ∼ 1013–1015 cm−3) photoionized by a compact, central UV and X-ray source. In this context, the line-emitting region of the disk would have to be between a few hundred and a few thousand gravitational radii (rgGMBH/c2, where MBH is the mass of the black hole) from the center in order to explain the observed widths of the lines. The broad, high-ionization emission lines (C iii] λ1909 and C iv λ1549) and Lyα, however, cannot be produced efficiently in the dense accretion disk because the ionization parameter is not high enough. Therefore, a vertically extended, low-density, high-ionization accretion disk wind has been invoked (Shields 1977; Collin-Souffrin 1987) to boost the strengths of the UV lines relative to the Balmer lines in order to match the observations. The systematic blueshift of broad high-ionization emission lines with respect to broad low-ionization emission lines supports this theory (Gaskell 1982; Sulentic et al. 1995a). The origin of broad emission lines in the upper layer of an accretion disk and the associated wind is also compelling because the same wind scenario has been invoked to explain the broad, blueshifted absorption lines seen in the rest-frame UV spectra of a subset of AGNs (e.g., Murray et al. 1995; Proga & Kallman 2004).

One kinematic signature of accretion disks is a double-peaked profile of the emission lines (e.g., Horne & Marsh 1986 and references therein). The fact that some AGNs emit double-peaked low-ionization emission lines (Eracleous & Halpern 1994, 2003; Strateva et al. 2003) bolstered the theory of broad-line formation in the accretion disk and wind (the origin of this idea can be traced back to Shields 1977). Although other models for the origin of double-peaked emission lines (DPEL) have been suggested, they face significant challenges when compared to observations (see review in Eracleous & Halpern 2003). However, only a small fraction of AGNs have DPELs: 20% of radio-loud AGNs at z < 0.4 (Eracleous & Halpern 1994, 2003) and 4% of the Sloan digital Sky Survey (SDSS) quasars at z < 0.33 (Strateva et al. 2003). This observation raises the question: if broad emission lines are produced in the accretion disk, why are they not all double-peaked?

A few explanations for the rare occurrence of DPEL have been suggested in the context of the scenario of line production in the accretion disk: (1) geometrical reasons: large rout/rin (Dumont & Collin-Souffrin 1990) and/or small inclination (Corbin 1997), (2) radiative transfer effects through a disk wind (Chiang & Murray 1996; Murray & Chiang 1997, hereafter CM96 and MC97, respectively), (3) the geometry of the ionizing source and the scattering medium (Jackson et al. 1991), and (4) an additional BLR zone outside of the disk (Popović et al. 2004; Bon et al. 2009). A more detailed summary and discussion of these ideas can be found in Eracleous & Halpern (2003). It is worth noting that the broad Balmer emission lines of some AGNs have been observed to fluctuate between a double-peaked and a single-peaked profile: NGC 5548 (Peterson et al. 1999; Shapovalova et al. 2004; Sergeev et al. 2007), Pictor A (Halpern & Eracleous 1994; Sulentic et al. 1995b), and Ark 120 (Alloin et al. 1988; Marziani et al. 1992). The timescale to change the geometry of these systems (rout/rin and/or inclination) is much longer than the observed timescale of line profile variability, disfavoring geometrical changes as the cause of the observed variability. The line profile variability of these objects is uncorrelated with variability of the observed continuum flux of the source, in contradiction with predictions by Jackson et al (1991). Another issue with the third suggested explanation is that single-peaked lines are very difficult to produce in this scenario, which would lead to a higher fraction of double-peaked emitters than observed in the AGN population. The disk-wind model is the hypothesis that conflicts the least with observations and has added appeal since it also explains the broad absorption lines that are often seen in the rest-frame UV spectra of quasars.

The cases of NGC 5548, Pictor A, and Ark 120 strongly suggest that double-peaked and single-peaked emitters are connected.7 If we can unify these two populations of AGNs, then the conclusions drawn about the accretion disk structure of double-peaked emitters through long-term variability studies (e.g., Gezari et al. 2007; Storchi-Bergmann et al. 2003; Shapovalova et al. 2001; Gilbert et al. 1999; Eracleous et al. 1995; Flohic & Eracleous 2008; Jovanović et al. 2010; Lewis et al. 2010) can be applied also to single-peaked emitters. Moreover, since single-peaked emission lines represent the principal AGN population, understanding the connection between the two types of line profiles will go a long way toward solving the puzzle of the BLR.

Motivated by this, we consider the connection between the two profiles of low-ionization emission lines by employing models of radiative transfer of line photons through the emission layer in the accretion disk. Our primary goal in this paper is to compare the predictions of the specific model developed by CM96 and MC97 to the profiles of the Balmer emission lines observed in the spectra of quasars from the SDSS. As a first step of carrying out this exercise, we develop the model further to include relativistic effects.

A number of authors have presented calculations of line profiles from an accretion disk wind applied to other astrophysical objects. Numerical models for neutral hydrogen emission lines in young stars produce only double-peaked profiles because the low radial velocity gradient in the wind leads to relatively weak radiative line transfer effects (e.g., Sim et al. 2005). Also, analytic and numerical models have been applied to the profiles of high-ionization UV resonance in cataclysmic variables (CVs), most notably C iv (see Shlosman & Vitello 1993; Vitello & Shlosman 1993; Knigge et al. 1995; Knigge & Drew 1996). The calculation of Shlosman & Vitello (1993) follows the same methodology that we adopt here. These models assume a rotating, vertically extended, biconical wind above the inner disk (within ∼30 white dwarf radii from the center) and take the UV continuum to be a combination of blackbody emission from the same region of the disk and from the boundary layer. The line emission is a result of recombination and resonance scattering of continuum photons. As a result, when the system is viewed at small inclination angles relative to its axis, the wind is in front of the UV continuum source and the resulting line profiles appear double-peaked with a deep absorption trough at the line center. When the system is viewed at somewhat higher inclinations, the line profiles have P-Cygni profiles, while at inclinations close to edge-on there are no absorption troughs because the wind is not viewed against the continuum source.

Here, we explore this issue specifically for the Balmer lines, which are the strongest optical lines in the spectra of AGNs, taking the emission region to be a very thin layer on the surface of the outer accretion disk (see further discussion in Section 2.1). Thus, we make use of the CM96 and MC97 models which deal specifically with this geometry and provide a convenient, analytic prescription for the kinematics of gas. The location and geometry of the region emitting the Balmer lines are quite different from the location and geometry of the region emitting the UV resonance lines, which were the focus of previous studies (in AGNs as well as in CVs). As a consequence, the properties of the model Balmer line profiles are substantially different from those of the C iv line profiles described in the previous paragraph. Our exploration of the properties of the Balmer line profiles produced by these models and our systematic comparison with observed profiles is considerably more extensive than in any previous study. We also pay particular attention to relativistic effects and show that their inclusion is important in order to reproduce the properties of the observed line profiles.

In Section 2, we describe the model of CM96 and MC97 that we use to calculate line profiles for comparison with the observations. In the process, we examine the assumptions behind this model and discuss its limitations. We repeat the calculation of the line profiles, correcting a sign error in one of the fundamental formulae of MC97. We also extend the calculation to include relativistic effects and adopt an improved integration scheme. In Section 3, we explore the effect of different model parameters on the resulting line profiles and identify the conditions under which relativistic effects change the line profiles substantially. We show that single-peaked broad emission lines are naturally reproduced in this model and that a transition of an object from a double- to a single-peaked emitter and vice versa is possible, if the optical depth in the line-emitting skin of the disk (i.e., the base of the wind) changes significantly. This transition occurs for a relatively narrow range of wind optical depths, in agreement with the observation that only a few AGN seem to exhibit this phenomenon. We compare our model profiles to observations and use them to derive constraints on the inclination of the disk, disk size, and wind optical depth for the AGN population. Finally, we discuss the implications of these findings and give our conclusions in Section 4.

2. THE MODEL

2.1. Physical Picture, Assumptions, and Limitations

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 rmin ⩾ 100 rg. 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 rmin ⩾ 100 rg 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.

2.2. Method of Calculation

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

Equation (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 epsilon(ξ, ϕ'), and the radiative transfer of line photons:

Equation (2)

where σ is the line broadening parameter, νe 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)

Equation (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

Equation (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):

Equation (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

Equation (6)

where ξf is the launching radius of the wind streamline in units of rg and v ≃ 4.7 ξ−1/2f (Murray & Chiang 1998). In our calculation of Q, we set ξ = ξf, which leads to the simplifications vr = 0 and vθ = 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 ∂vr/∂r = v/ξ ≠ 0 at ξ = ξf. Thus, we can rewrite the expression for Q from Equation (5) as

Equation (7)

where

Equation (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

Equation (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 epsilon(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 epsilon(ξ, ϕ')∝ξq (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.

3. PROPERTIES OF THE MODEL LINE PROFILES

3.1. Effect of Model Parameters

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°, ξmin = 600, and ξmax = 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.

Figure 1.

Figure 1. Profiles of the C iv line using the CM96 expression for Q, no relativistic effects, and the same power-law disk and wind parameters as in their Figure 2. The single-peaked line profile (solid line) was calculated with τ0 = 10 and the double-peaked profile (dotted line) with τ0 = 10−7. This calculation differs from that shown in Figure 2 of CM96 only in the integration method. The differences in the resulting line profiles are negligible (see the discussion in Section 3.1 of the text).

Standard image High-resolution image

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 ξmin = 150, ξmax = 5000, λ(ξmin) = 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, ξmin, 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 ξmin 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.

Figure 2.

Figure 2. Grid of profiles of the Hα line without relativistic effects (solid) and with relativistic effects (dotted). From one panel to the other, only the inclination varies. The other model parameters are ξmin = 150, ξmax = 5000, λ(ξmin) = 10°, η = 0.5, q = 2.0, and τ0 = 10. The net redshift at the base of the line, produced by relativistic effects, is 50–70 Å (corresponding to 2300–3200 km s−1).

Standard image High-resolution image

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°, ξmin = 500, ξmax = 10, 000, q = 2.5, λ(ξmin) = 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.

Figure 3.

Figure 3. Profiles of the Hα line for different optical depths through the disk wind: τ0 = 10−7 (solid), 10−5 (dot), 10−3 (dash), and 10−1 (dash, dot). The other parameters are i = 60°, ξmin = 500, ξmax = 10, 000, q = 2.5, λ(ξmin) = 10°, and η = 2.

Standard image High-resolution image
Figure 4.

Figure 4. Maps of the value of Q over the disk using our expression derived in Section 2.2. The direction of ϕ' = 0 is toward the observer, at + along the x'-axis. The parameters used to construct this map are given in Section 3.1.

Standard image High-resolution image

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 < ξmax < 20000. For this, we fix the optical depth (τ0 = 10), the inclination (i = 60°), and the inner radius (ξmin = 500). To test the effect of η, we fixed q = 2.5, ξmax = 10, 000, and λ(ξmin) = 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 ξq − (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 ξmin = 500, relativistic effects can still produce skewed line profiles.

Figure 5.

Figure 5. Profiles of the Hα line for different η: η = 0.5 (solid, broadest), 1.0 (dotted), 1.5(dashed), and 2.0 (dot-dashed, narrowest). The other parameters are i = 30°, ξmin = 500, ξmax = 10, 000, q = 2.5, λ(ξmin) = 10°, and τ0 = 10.

Standard image High-resolution image

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

3.2. Properties of Model Profiles and Comparison with Observations

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: ξmin, ξmax, i, η, and τ. We compute the line profile for 100 ⩽ ξmin ⩽ 10100 in increments of 2000, 1000 ⩽ ξmax ⩽ 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 (vc(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 ξmin and ξmax (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 ξmin and two power-law relations: ξ−1/4min (dashed line) and ξ−1/2min (continuous line), both normalized to match the average FWHM at ξmin  ∼  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 ξmax instead of ξmin. 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.

Figure 6.

Figure 6. FWHM of the simulated profiles as a function of each free parameter of the disk-wind model. In each panel, the hashed area indicates the range of observed values in the Zamfir et al. (2010) sample. In the panels (a)–(c), the vertical bars indicate the 1σ range of FWHM among model profiles at the given disk-wind parameter. In panel (d), the FWHM of each modeled profile is plotted as a data point. The gray scale represents the value of ξmax, with the darker points having the smallest ξmax. The continuous line in panel (c) shows a sin i dependence, normalized to match the average value from the simulations at 60°.

Standard image High-resolution image
Figure 7.

Figure 7. FWHM of the simulated profiles as a function of ξmin, the inner radius of the line-emitting portion of the accretion disk. The continuous lines show a r−1/2 dependence, normalized to match the average value from the simulations at 6000ξmin, and the dashed line a r−1/4 dependence.

Standard image High-resolution image

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 ξmin and ξmax since the relativistic effects skewing the profile to the red diminish with increasing radius. Note that a full range of ξmin and ξmax 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.

Figure 8.

Figure 8. A.I. of the simulated profiles as a function of each free parameter of the disk-wind model. See Figure 6 for an explanation of the different symbols.

Standard image High-resolution image

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 ξmin, since the relativistic effects strengthen the profile wings at low radii. But the average K.I. decreases with ξmax, 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.

Figure 9.

Figure 9. K.I. of the simulated profiles as a function of each free parameter of the disk-wind model. See Figure 6 for an explanation of the different symbols.

Standard image High-resolution image

Figure 10 presents the variation of vc(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 vc(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 ξmin and ξmax, 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.

Figure 10.

Figure 10. vc(1/4)/FWQM of the simulated profiles as a function of each free parameter of the disk-wind model. See Figure 6 for an explanation of the different symbols.

Standard image High-resolution image

In previous paragraphs, we compared the data with individual model distributions of the A.I., K.I., FWHM, and vc(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 vc(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 (ξmin ≲ 2000). This restriction in turn leads to the constraint ξmax ≳ 5000, required in order to reproduce a wide range of values for vc(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. vc(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 ξmin ≲ 4000 produces a model vc(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.

Figure 11.

Figure 11. Histogram distributions of the A.I., K.I., and vc(1/4)/FWQM when we restrict our model parameters to τ0 ≳ 10−4, i ≲ 45°, ξmin ≲ 2000, and ξmax ≳ 5000. The dashed line shows the histograms of observed values adapted from Zamfir et al. (2010).

Standard image High-resolution image

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, ξmin = 100, and ξmax = 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, ξmin = 100, and ξmax = 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.

Figure 12.

Figure 12. Examples of simulated line profiles without relativistic effect (dashed line) and with relativistic effects (continuous line) overlaid upon observed line profiles (dotted line, adapted from Sulentic et al. 2009). The model parameters were chosen to have values within the restrictions used to produce Figure 11, namely, τ0 = 1, i = 30°, ξmin = 100, and q = 3. In the right panel, ξmax = 10, 000 and η = 1.5, producing a "double-Gaussian" profile (as dubbed by Sulentic et al. 2009). In the left panel, ξmax = 20, 000 and η = 2, producing a "Lorentzian" profile.

Standard image High-resolution image

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 vc(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 vc(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.

Figure 13.

Figure 13. vc(1/4)/FWQM and A.I. of the simulated profiles as a function of inclination when no relativistic effects are included.

Standard image High-resolution image

4. DISCUSSION AND CONCLUSIONS

Using the Sobolev approximation and an analytic disk-wind model, we have confirmed that an increase in the optical depth of an accretion disk wind significantly alters the profile of the broad emission lines produced by the accretion disk. Emission lines appear double-peaked for a low optical depth through the disk wind and single-peaked for large optical depth. The change in the optical depth of the disk wind is a plausible physical mechanism, which can lead to a transition from double-peaked to single-peaked in AGNs. The narrow layer, where the disk atmosphere starts to lift off the surface, is the main source of emissivity because its density is high. The radial velocity of this layer is negligibly small, but its velocity gradient is large, which affects the propagation of line photons. After accounting for an enhanced escape probability in certain directions, the apparent emissivity function is non-axisymmetric, producing single-peaked emission lines. We have improved the MC97 and CM96 models by including general relativistic effects and a numerical integration method without analytic approximations. The inclusion of general relativistic effects skews the line profile and the amount of redshift depends on the inner radius of the line-emitting region.

The opening angle of the wind has only a small effect on the line profile. The radial density structure, size of the line-emitting region, and inclination have a significant impact on the single-peaked line profile produced at high τ0. In order to produce line profiles from our model similar to those observed, restrictions in the inclination angle of the disk (i ≲ 45°), the disk size (ξmin ≲ 2000 and ξmax ≳ 5000), and the optical depth through the disk wind (τ ≳ 10−4) are favored by the data. A restriction in inclination angles is consistent with the requirement that the broad emission lines must not be hidden behind the obscuring torus (invoked in unification schemes).

Dramatic line profile variability from single-peaked to double-peaked, as observed in NGC 5548, Pictor A, and Ark 120 (see Section 1), requires variations of the optical depth by about one order of magnitude, with an initial τ0 close to 10−3. Such an event requires (1) a narrow range of the initial optical depth through the disk wind and (2) a change of the optical depth of about an order of magnitude, which explains why so few AGNs have a line profile varying from single-peaked to double-peaked. Hydrodynamical simulations by Proga et al. (2000) suggest that the optical depth through line-driven disk winds can change by a large amount in timescales of several years, allowing such a variation of the line profile.

The Sobolev approximation adopted in this work however may not reproduce the detailed effects of kinematics observed in simulations of disk winds and some of its limitations have been discussed in the literature (Hamann et al. 1993). For example, this approximation does not capture the presence of clumps or filaments in the wind, if the clumps have internal shear or if the clumps are optically thick. The hydrodynamical simulations of Proga et al. (2000) show that the knots in the disk wind are created by Kelvin–Helmholtz instabilities between the fast outflowing stream and the slow inflowing stream, but they do not specify if the knots themselves have shear. If they did, then that would slightly modify the expression for Q, which would then affect the line profile. Nevertheless, the profile will still change from double-peaked to single-peaked when τ0 increases. If τ0 becomes too high, then the clumps become optically thick and the escape probability will depend on the shape of the cloud instead of the velocity gradient (Junkkarinen 1983) and eventually lead to a line profile asymmetric toward the red side of the line (Hamann et al. 1993). Thus, if the Sobolev approximation breaks down, then the details of the line profile will differ.

The method presented here allows an exploration of the parameter space and a comparison with a large database of AGNs at a relatively low computational cost. Because parameters that describe the line-emitting portion of the disk and the disk wind can be efficiently disentangled, one can gain intuition about the most important physical properties by eliminating the parameters with the smallest impact. In this work, we have focused on the axisymmetric disk (as reflected by its density, velocity, and emission properties) but it is also possible to explore the role of non-axisymmetry in these properties on shaping the line profiles and causing them to vary. It is also possible to investigate the influence of the velocity field of the wind on the line profiles using different dynamical models (e.g., by Königl & Kartje 1994; Arav et al. 1994; Everett 2005). Similarly, this method allows comparison of model profiles with subsets of AGNs selected according to specific criteria (such as radio-loud and radio-quiet AGNs, for example) in order to elucidate the source of the disparities between different populations. While answers to many questions may not lie in the disk-wind model alone, the method presented here combined with findings from hydrodynamical models can help deepen our understanding of the physics of the BLR.

We re-iterate that our methodology is subject to a number of assumptions (see Section 2.1) and is applicable only to the calculation of Balmer line profiles. The validity of our simple models can be checked by calculating numerically the radiative transfer effects for a wind with the structure found from hydrodynamical simulations. Work in this direction is already underway, employing Monte Carlo radiative transfer codes that use the results of hydrodynamic simulations (see Sim et al. 2010; Higginbottom et al. 2012). Such a comparison would also allow us to relate the parameters of our analytical model to fundamental properties of the AGN, such as the black hole mass and accretion rate. This in turn would permit us to create a more realistic model population of AGNs to test whether the model line profile parameters match the observed ones.

We are grateful to Patrick Hall and Laura Chajet for very helpful discussions and for sharing with us the results of their calculations in advance of publication. We also thank the anonymous referee for many thoughtful comments. Finally, we are grateful to Drew Clausen for his help with the Cloudy photoionization calculations. H.M.L.G.F. acknowledges the partial support of the CONICYT-FONDECYT grant (No. 3110033), of the Joint Committee ESO-Chile, and of a Small Research Grant from the American Astronomy Society. Support for T.B. was provided by the National Aeronautics and Space Administration through Einstein Postdoctoral Fellowship Award No. PF9-00061 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060.

APPENDIX A: IONIZATION STRUCTURE OF THE DISK ATMOSPHERE

In this appendix, we describe the calculation of the vertical ionization structure of the disk atmosphere. Analogous calculations of photoionized accretion disks have been presented by other authors, most notably Collin-Souffrin & Dumont (1989) and Murray & Chiang (1998); however, those authors do not illustrate the vertical ionization structure of the disk atmosphere resulting from their calculations. Thus, the goals of our calculation are to verify the earlier calculations with a more modern code and, more importantly, to examine the vertical structure of the disk atmosphere, which is crucial for validating the assumptions made in our profile calculations.

To carry out our calculations, we used the photoionization code Cloudy (version 08.00, described by Ferland et al. 1998). To describe the disk structure, we adopted the model of Shakura & Sunyaev (1973), a black hole mass of 108M, and an accretion rate of 1% of the Eddington rate. The vertical density profile is Gaussian with a scale height that depends on the radius, h(r). The disk atmosphere is static, i.e., there is no outflowing wind. The disk is illuminated by a point-like continuum source located on its axis at height of 6 rg; at the radii of interest here, this is a very good approximation for illumination by a spherical source of radius 6 rg at the center of the disk. The ionizing continuum has a spectral energy distribution of a typical AGN: the big blue bump peaks at 1.5 Ryd, the optical/UV portion is described by a power law of index αo = 0.4 (where fν∝ν), the X-ray portion is also a power law with an index of αx = −1.9, and the relative normalization of the UV and X-ray portions is set by the parameter αox = −1.4. We adopt a range of radii between 500 and 20, 000 rg, which is typical of the line-emitting region we consider in this paper. The rays from the ionizing source strike the disk atmosphere obliquely. The photoionization calculation begins at 6 h(r) above the mid-plane, where the ionization parameter is typically high because of the low density, and stops when the hydrogen number density exceeds 1014 cm−3 or the temperature drops below 1000 K. The ionization parameter at 6 h above the disk midplane is log U > 0.7 regardless of radius, implying a neutral hydrogen fraction of <2 × 10−7. At the inner radius of the line-emitting portion of the disk, log U = 2.5 at z = 6 h. The stopping condition for the calculation is met at a height of 2.8 h above the disk mid-plain in the inner disk and at 3.8 h in the outer disk.

In Figure 14, we show the distribution of the Hα and Hβ line emissivity (power emitted per unit volume) as a function of radius and height in the disk. As the gray scale in these figures indicates, the emitting region is a thin layer near the surface of the accretion disk. The emissivity is sharply peaked in the vertical direction with the maximum occurring at ≈5 h at all radii in the disk. At r ≈ 500 rg, the emissivity-weighted width of the emitting layer (i.e., the standard deviation of the emissivity distribution) is ≈0.2 h, at r ≈ 10, 000 rg it is 0.3 h, and at r ≈ 20, 000 rg it is ≈0.1 h. The emissivity above the emitting layer drops abruptly because hydrogen becomes highly ionized as a result of the steep decline in the density with height. The emissivity below the emitting layer drops sharply because the flux of ionizing photons at this depth is severely attenuated. To evaluate the optical depth to Balmer line photons vertically through the emitting layer, we use the column density of hydrogen atoms in the first excited state (populated by collisions), which turns out to be 2.8 × 1016 cm−2 at r ≈ 500 rg, 3.3 × 1013 cm−2 at r ≈ 10, 000 rg, and 1.1 × 1012 cm−2 r ≈ 20, 000 rg. Thus, at the radii of interest, we obtain optical depths in the range 4.1 × 106–1.0 × 1011 at the center of the Hα line and 3.0 × 106–7.6 × 1010 at the center of the Hβ line. These optical depths were obtained assuming thermal broadening at a fiducial temperature of T = 104 K (or a thermal velocity dispersion σth ≈ 10 km s−1), thus they scale as T−1/2. If the local line profile is set by some other broadening mechanism, then the optical depth scales with velocity dispersion as σ−1. In comparison, the electron scattering optical depth through the emitting layer to the top of the disk atmosphere is negligible, τes ≲ 10−2. The Hα and Hβ flux emerging from the disk varies with radius as r−3, in agreement with earlier calculations by Collin-Souffrin & Dumont (1989).

Figure 14.

Figure 14. Distribution of the Hα and Hβ emissivity (power per unit volume of gas) resulting from our photoionization calculation. The plots show the vertical structure of the emitting layer on the surface of the disk. The details of the calculation are described in Appendix A. The scale in the radial and vertical directions is logarithmic (in units of the gravitational radius).

Standard image High-resolution image

APPENDIX B: CALCULATION OF Q

In this appendix, we show the details of the calculation of $Q\equiv {\bf \hat{n}\cdot \Lambda \cdot \hat{n}}$. We follow the method outlined in MC97 and CM96 and give all the steps in the calculation.

The line of sight vector ${\bf \hat{n}}$ is in the x–z plane at an angle i from the z-axis

Equation (B1)

Setting θ' = π/2, this simplifies to

Equation (B3)

Noting that Λ is a symmetric tensor, we can write the double scalar product, $Q={\bf \hat{n}\cdot \Lambda \cdot \hat{n}}$, in matrix notation as follows:

Equation (B4)

which yields

Equation (B5)

Batchelor (1967) gives the following expressions for the components of Λ in spherical coordinates:

Equation (B6)

Equation (B7)

Equation (B8)

Equation (B9)

Equation (B10)

Equation (B11)

where vr, v'θ, and v'ϕ are the velocity components of the disk wind.

Murray et al. (1995) give an analytical prescription for the disk-wind velocity field:

Equation (B12)

Equation (B13)

Equation (B14)

where γ ∼ 1.0–1.3, λ(r) is the opening angle of the wind and rf is the radius of the footprint of the wind stream. We set r = rf and θ' = π/2 since the bulk of the emission comes from close to the footprint of the stream, where the velocity gradient is highest. This assumption leads to the following simplifications: vr = 0 and $v_{\theta ^\prime }=0$. Moreover, azimuthal symmetry implies that ∂/∂ϕ' → 0. In order to calculate the velocity gradients in the θ' direction, we note that dz ≈ −rdθ' at r = rf, which leads to

Equation (B15)

At the footprint of the streamline, ∂r/∂z ≈ 1/sin λ (MC97). Thus, Equation (B15) becomes

Equation (B16)

Following MC97, we adopt γ = 1 for simplicity, which also leads to ∂vr/∂r ≠ 0 at r = rf. Equations (B6) to (B11) then become

Equation (B17)

Equation (B18)

Equation (B19)

Equation (B20)

Equation (B21)

Equation (B22)

We now insert the above expressions for the components of Λ in Equation (B5):

Equation (B23)

To arrive at the final expression for Q, which we adopt in this work, we note that the assumption that r = rf implies that vr(rf) = 0 and ∂vr/∂r = v/rf. Thus, we can rewrite Q as

Equation (B24)

where $v_\infty = 4.7v_{\phi ^\prime }$, according to Equation (2) of Murray & Chiang (1998). Moreover, the wind opening angle, λ, varies with radius according to λ(r) = λ(rmin)(rmin/r), where rmin is the inner radius of the line-emitting region of the disk.

Comparing our expression for Q in Equation (B23) to that in Equation (15) of MC97, we see a difference in the coefficient of ∂vr/∂r in the second term in square brackets: we have sin icos ϕ'/sin λ, while MC97 have $\sin i \cos \phi ^\prime \cot \lambda$. These two expressions are approximately equal, however, since λ is small. A more substantial difference is in the sign of the second term in square brackets: we have a "+ cos i" instead of the "−cos i" of MC97. This is a difference between our calculation and theirs that we cannot reconcile. An independent calculation by Patrick Hall and Laura Chajet (P. Hall & L. Chajet 2012, private communication; Hall & Chajet 2010; Chajet 2012; P. Hall et al. 2012, in preparation; L. Chajet et al. 2012, in preparation) confirms our result and leads us to conclude that this difference is due to a sign error in MC97.

We explored the effect of the different expressions for Q on the line profiles, starting with a comparison of two-dimensional maps of the value of Q0 (the azimuthal part of Q) over the surface of the disk. In Figure 15, we compare maps of the value of Q0 for the disk parameters that we adopt in the illustrations of Section 3.1 (a relatively "small" disk, with ξmaxmin = 10). The maps are very similar; the main differences are seen at the smallest radii and small projected velocity (i.e., along the line to the observer), as we illustrate in Figure 16. In Figures 17 and 18, we show another comparison between our version of Q0 and that of MC97, using the parameters that give a good match to the average Hβ line profile of SDSS quasars (see Section 3.1 and Figure 12; this is a "large" disk with ξmaxmin = 200). The main difference between the two version of the maps is that the MC97 version is front-back symmetric relative to the direction of the observer, while our version is not. In our two examples, the asymmetry corresponds to a factor of 1.5–2 at the inner radius and declines with radius following the decline in λ(ξmin). By exploring the ξmini–λ parameter space, we find that the front-back asymmetry becomes more pronounced as i and λ(ξmin) increases. In particular, the maximum asymmetry amounts to a factor of three at the most extreme reasonable values of these parameters, i.e., i = 60° and λ(ξmin) = 20°. To illustrate the influence of the expression for Q on the line profiles, we compare in Figure 19 the profiles resulting from the two different expressions for the disk dimensions used in Figures 15 and 17. In the first example, we choose parameters that make the difference between the line profiles as pronounced as possible. In the second example, we use the model parameters required to fit the average Hβ line profile of SDSS quasars (see Section 3.1 and Figure 12). Our overall conclusion from this comparison is that our expression for Q yields qualitatively very similar line profiles. For specific choices of models parameters, the difference can be as big as what is shown in the top panel of Figure 19. However, for the parameters required for the models to match the observations, the profiles resulting from the two different expressions for Q are nearly indistinguishable.

Figure 15.

Figure 15. Maps of the value of Q0 (the azimuthal part of Q) over the disk using the MC97 formula (left) and our expression (right). The direction of ϕ' = 0 is toward the observer, at + along the x'-axis. The properties of disk and wind are the same as those used for our exploration of the properties of the line profiles in Section 3.1, i.e., ξmin = 500, ξmax = 5000, i = 30°, and λ(ξmin) = 10°.

Standard image High-resolution image
Figure 16.

Figure 16. Variation of Q0 with azimuthal angle at two different radii for the model shown in Figure 15. The top panel compares our map of Q0 (solid line) with that of MC97 (dashed line) at the inner radius of the line-emitting portion of the disk, while the lower panel shows the same comparison at a larger radius. The observer's direction corresponds to ϕ/2π = 0.5. The main difference is that the MC97 version of Q0 is front-back symmetric while our version shows an asymmetry by a factor of ≈1.5 at the inner radius, which becomes smaller at larger radii.

Standard image High-resolution image
Figure 17.

Figure 17. Maps of the value of Q0 over the disk using the MC97 formula (left) and our expression (right). The direction of ϕ' = 0 is toward the observer, at + along the x'-axis. The properties of disk and wind are the same as those required to fit the average Hβ line profile of SDSS quasars (see Section 3.1 and Figure 12), i.e., ξmin = 100, ξmax = 20, 000, i = 30°, and λ(ξmin) = 15°.

Standard image High-resolution image
Figure 18.

Figure 18. Variation of Q0 with azimuthal angle at two different radii for the maps shown in Figure 17. The top panel compares our map of Q0 (solid line) with that of MC97 (dashed line) at the inner radius of the line-emitting portion of the disk, while the lower panel shows the same comparison at a larger radius. The observer's direction corresponds to ϕ/2π = 0.5. The main difference is that the MC97 version of Q0 is front-back symmetric while our version shows an asymmetry by a factor of ≈2 at the inner radius, which becomes smaller at larger radii.

Standard image High-resolution image
Figure 19.

Figure 19. Profiles of the Hα line using the MC96 expression for Q (solid line) and our expression for Q (dotted line). In the upper panel, the disk dimensions are the same as in Figure 15, while other parameters are chosen to make the difference between the profiles as pronounced as possible. In particular, ξmin = 500, ξmax = 5000, i = 60°, λ(ξmin) = 20°, σ = 600 km s−1, η = 0.5, q = 2.5, and τ0 = 107. In the lower panel, we use the model parameters required to fit the average Hβ line profile of SDSS quasars (see Section 3.1 and Figure 12), i.e., ξmin = 100, ξmax = 20, 000, i = 30°, λ(ξmin) = 15°, σ = 600 km s−1, η = 2.0, q = 3.0, and τ0 = 1. Relativistic effects are included in both examples. In the latter example, the profiles resulting from the different formulae for Q are indistinguishable.

Standard image High-resolution image

Footnotes

  • N. Murray suggested that these transitions may be the result of changes in the optical depth of the emitting layer at the base of the accretion disk wind. This suggestion was made in a presentation at the workshop "Mass Outflow in Active Galactic Nuclei: New Perspectives," held at The Catholic University of America, 2001 March 8–10. No corresponding paper was published in the proceedings of the workshop (Crenshaw et al. 2002).

  • The notation Ψ(ξ, ϕ') was introduced by Eracleous et al. (1995). This corresponds to the function g(D) used by Chen et al. (1989).

  • In the context of the Sobolev approximation, photons of a given frequency are absorbed within the line transition over a narrow spatial range and nonlocal radiative effects can be ignored.

  • 10 

    Zamfir et al. (2010) shows two distinct histograms for 260 population A AGNs and 209 population B AGNs. We combine these two histograms into one.

Please wait… references are loading.
10.1088/0004-637X/753/2/133