THE EFFECT OF RADIATION PRESSURE ON EMISSION-LINE PROFILES AND BLACK HOLE MASS DETERMINATION IN ACTIVE GALACTIC NUCLEI

The basic equation of motion, ignoring the drag force, is

$$\begin{equation} a(r) = \frac{\sigma _T L_{\rm bol}}{\mu m_H c 4 \pi r^2} [M(r)-1/\Gamma ] -\frac{1}{\rho } \frac{dP_g}{dr} \,, \end{equation} \tag{ 4 }$$

where M(r) is the force multiplier, L_bol is the bolometric luminosity, μ is the average number of nucleons per electron, and Γ = L/L_Edd. The force multiplier depends on the gas composition and its level of ionization. An interesting case is a Compton-thin neutral cloud that absorbs all the ionizing radiation (a Compton-thin "block"). In this case, M(r) ≃ α(r)/(σ_TN_col), where N_col is the hydrogen column density and α(r) is the fraction of the bolometric luminosity which is absorbed by the gas. For such a "block," α(r) = L_ion/L_bol, but in general α(r) is radius dependent because of the changing column density and the level of ionization of the gas (see below).

Ignoring thermal pressure we obtain

$$\begin{equation} a(r)=\frac{L_{\rm bol} }{r^2} \left[ \frac{ 1.14 \times 10^{-11} \alpha (r) }{ N_{23} } - \frac{ 8.8 \times 10^{-13} }{ \Gamma } \right]\!, \end{equation} \tag{ 5 }$$

where N₂₃ = N_col/10²³. Thus, radiation pressure is the dominant force when

$$\begin{equation} \Gamma \ge 7.7 \times 10^{-2} \frac{ N_{23} }{ \alpha (r) }. \end{equation} \tag{ 6 }$$

The above expressions, including the one for the limiting Γ, include only radial terms and assume a pure radially dependent radiation pressure force. The calculations of real orbits, and the conditions for cloud escape, require their integration and will thus include the standard constants of motion (energy and angular momentum). Obviously, the conditions for escape depend on the cloud azimuthal velocity, v_θ, and can differ substantially from what is obtained by using Equation (6).

M08 and M09 derived similar expressions for the case of completely opaque clouds. According to them, radiation pressure dominates the cloud motion if

$$\begin{equation} \Gamma \ge 1.27 \times 10^{-2} b_{5100} N_{23} \end{equation} \tag{ 7 }$$

where b₅₁₀₀ = L_bol/L₅₁₀₀. The two expressions provide the same limiting Γ when

$$\begin{equation} \alpha (r) b_{5100} \simeq 6.1. \end{equation} \tag{ 8 }$$

A recent paper by Ferland et al. (2009), where mostly neutral, infalling clouds are considered, reaches basically identical conclusions.

BLR clouds are likely to be confined. The confining mechanism is not known, but high temperature gas and magnetic confinement have been proposed. The approach chosen here is consistent with the idea of magnetic confinement and some justifications for it are given in Rees et al. (1989). We adopt a simple model of numerous individual clouds that are moving under the combined influence of the BH gravity and radiation pressure force. Following Netzer (1990) and Kaspi & Netzer (1999), we assume that clouds retain their mass as they move in or out and the gas density changes with radius in a way that depends on the radial changes of the confining pressure.

Assume that the external pressure and hence the gas density in individual clouds are proportional to the radial coordinate, n_H ∝ r^−s. A reasonable guess that agrees with observations is 1 ⩽ s ⩽ 5/2 (Rees et al. 1989). This results in a radial dependence of the ionization parameter (the ratio of ionizing photon density to gas density), U ∝ r^{s − 2}. For spherical clouds, N_col ∝ r^−2s/3, R_c ∝ r^s/3, and A_c ∝ r^2s/3, where R_c is the cloud radius and A_c is its geometrical cross section. The line intensity contributed by a single cloud, (r), depends on its covering factor and the line emissivity j(r) which depends on the conditions in the gas,

$$\begin{equation} \epsilon (r) \propto j(r) A_c/r^2 \propto j(r) r^{2s/3-2}. \end{equation} \tag{ 9 }$$

In the real calculations, we ignore factors of order unity relating the mean cloud "size" and its mean column density since this is not known and requires different type of calculations.

The above considerations suggest that the importance of radiation pressure increases with the distance because of the dependence of N_col on r, i.e.,

$$\begin{equation} \Gamma _{\rm lim} \propto r^{-2s/3}/\alpha (r). \end{equation} \tag{ 10 }$$

Since Γ depends on the global accretion rate which has little to do with cloud properties, the more physical approach is to consider the case of a certain Γ and follow the cloud motion. The examples discussed below follow this approach.

In this work, we consider the following three types of clouds.

• 1.  
  Very large column density clouds where radiation pressure force is negligible at all distances. Here virial cloud motion is a good approximation (the Ferland et al. 2009 infalling clouds belong to this category).
• 2.  
  Clouds for which radiation pressure is very important somewhere inside the "classical BLR" (e.g., inside the RM radius). Such clouds will escape the system on dynamical timescales and their contribution to the line profiles is small except for times immediately after a large increase in L_bol.
• 3.  
  Clouds for which radiation pressure is non-negligible but is not strong enough to allow escape. Such clouds are the ones discussed by M08 albeit without the radial dependence of N_col considered here. This case is the one most relevant to real BLRs and we discuss it in detail in the following section.

The modified equation of motion is obtained from Equation (5) by including the radial dependence of N_col. We define r₂₃ to be the distance where N₂₃ = 1. This gives

$$\begin{equation} a(r)=\frac{L_{\rm bol}}{r^2} \left[ \frac{ 1.14 \times 10^{-11} \alpha (r) }{ (r/r_{23})^{-2s/3} } - \frac{ 8.8 \times 10^{-13} }{ \Gamma } \right]. \end{equation} \tag{ 11 }$$

The column-density-dependent critical distance where the radiation force dominates the cloud motion is

$$\begin{equation} \frac{r}{r_{23}} \ge \left[ \frac{7.7 \times 10^{-2} }{ \alpha (r) \Gamma } \right] ^{3/2s}. \end{equation} \tag{ 12 }$$

For example, the case of s = 1 and α(r) = 0.5 gives a critical radius of r = 0.06Γ^−1.5r₂₃ for radially moving clouds. This dependence of r on Γ is the main motivation to suggest a new method for evaluating M_BH and r_BLR (Section 4). As explained, the critical radius should not be confused with the point of escape from the system. Non-radial velocity components (v_θ) that reflect the energy and angular momentum of the system will act to reduce this radius (see examples below).

The motion of BLR clouds with the above properties involves an acceleration term of the form

$$\begin{equation} a(r)=\frac{c_1 \alpha (r) }{ r^{2-2s/3} } - \frac{ c_2}{r^2 }, \end{equation} \tag{ 13 }$$

where

$$\begin{equation} c_1=1.14 \times 10^{-11} L_{\rm bol} r_{23}^{-2s/3} \end{equation} \tag{ 14 }$$

and

$$\begin{equation} c_2=8.8 \times 10^{-13}L_{\rm bol}/\Gamma. \end{equation} \tag{ 15 }$$

The radial potential is

$$\begin{equation} \Phi (r)= - \int _r^{r*} a(r) dr, \end{equation} \tag{ 16 }$$

where r* is the radius where Φ(r) = 0. Below we use this potential to calculate cloud orbits and line profiles. The energy and angular momentum terms that result from the above integration are included in the calculation by fixing the initial conditions, r, and the two velocity components at this location.

We carried out a series of calculations under a variety of conditions considered to be typical of different BLRs. Every model is calculated for assumed M_BH and Γ. This specifies L_bol and thus the potential Φ(r). The additional model parameters are as follows.

• 1.  
  The radial parameter s.
• 2.  
  The cloud column density normalization factor r₂₃.
• 3.  
  The initial radius r₀ and the initial velocity v₀ = v_θ(r₀). We assume that the orbits of clouds with very large N_col are ellipses of given eccentricities. r₀ is chosen to be the apogee of the orbit and v₀ (given below in units of the Keplerian velocity, v_Kepler) is determined from these conditions. This is a simple way to specify the angular momentum. In the examples below, we focus on those cases where the resulting FWHMs are consistent with the observations of the broad Hβ and C iv λ1549 emission lines but give details for several others.
• 4.  
  The initial ionization parameter U(r₀). We note that the exact value of the gas density, n_H(r), is less important. In the following, we assume n_H(r₀) = 10¹⁰ cm⁻² for all cases.
• 5.  
  The three-dimensional distribution of orbits. This is done in two steps. First, we calculate the motion of numerous identical clouds in a plane and then distribute many such planes in a spherical geometry specified by the inclinations of the planes to the line of sight. The profiles given below are only those for a line of sight which is perpendicular to the central plane of motion (if such a plane exists). All calculations assume a large enough number of clouds such that the predicted profiles are smooth (see Bottorff & Ferland 2000 for discussion and earlier references on this issue).

Physical properties that are not included in the present calculations are non-isotropic central radiation field, non-isotropic line emission, the photoionization of gas with a range of density and metallicity, different inclinations of the line of sight to the central plane of motion, large cloud covering factors in a specific direction, and central obscuration, e.g., by the accretion disk. Several of these are likely to be important in real BLRs but are beyond the scope of the present work.

Figure 1 illustrates the orbits of three s = 1.2, r₀ = 10¹⁷ cm, Γ = 0.1, and v₀ = 0.5v_Kepler clouds moving under the influence of a 10⁸ M_☉ BH. The first is an ellipse typical of a cloud which is not affected by radiation pressure (e.g., r₂₃ = 1000r₀). This is shown by a thick solid line. Formally speaking, such clouds are Compton thick but this is of no practical implication since the only intention is to show a simple, gravity-dominated orbit. The second is a case where r₂₃ = 10r₀. Here, radiation pressure force is significant and acts to constantly change the direction of motion of the cloud. This results in a rotating planar orbit. The third orbit (dashed line) follows the trajectory of a smaller column density cloud (r₂₃ = 0.82r₀) that escapes the system. Regarding the second case, increasing Γ will increase the angle between two successive revolutions.

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We calculated various line profiles for the case of M_BH = 10⁸ M_☉, r₂₃ = 10r₀, v₀ = 0.5, and Γ in the range of 0.05 (negligible radiation pressure force) to 0.735 (just below escape). The bolometric luminosity in each of these is obtained from the combination of M_BH and Γ. We assume α(r) = 0.5 at all radii and (r) which takes into account only geometrical factors (i.e., constant j(r), see Equation (9)) and isotropic line emission. In terms of the total line emission, this is a reasonable approximation for lines like Hβ that reprocess roughly a constant fraction of the ionizing continuum radiation. Obviously, a large optical depth in Hβ will result in line emission anisotropy which is not considered here. It is not appropriate for lines like C iv λ1549 whose intensity is more sensitive to the level of ionization and the gas temperature. At this stage, we specifically avoid the use of a varying α(r) since the effect on the orbit can be significant even for small changes in this parameter (see below). The resulting profiles, assuming a complete spherical atmosphere (the entire ±π/2 radians range relative to the central plane), are shown in Figure 2. As expected, the profile becomes narrower with the increasing Γ, reflecting the fact that, as the luminosity increases, the cloud spends less and less time at small radii.

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The top part of Table 1 provides additional information about the calculations. For each profile, we give the FWHM in units of v_Kepler(r₀), the mean emissivity-weighted radius, 〈r〉/r₀, and the mass correction factor f (Equation (2)). The calculation of 〈r〉 is obtained by weighting the emissivity of the cloud and the time it spends at each radius. This is roughly equivalent to the observed RM radius. The mass correction factor is obtained by requiring fFWHM²〈r〉/G = M_BH. We also show (in the parentheses) the values of FWHM and f obtained for the case of a thick central disk which represents only a part of a spherical distribution. Here the cloud distribution corresponds to a width, relative to the central plane, of ±π/4 radians. The reduction in FWHM relative to the complete sphere is about a factor of 0.6 and there is a corresponding increase in f. Computed line profiles that are typical of this and similar geometries are shown in Figure 3.

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To explore models with different initial conditions, we computed two cases of planar orbits with the same orbital energy and different angular momentum. One such example is shown in Figure 4. The less eccentric case in the diagram corresponds to the orbit labeled with 2) in Figure 1 for which v₀ = 0.5. The more eccentric one assumes v₀ = 0.25 but with a non-zero radial velocity of v_r = 0.433. This results in a much narrower profile. In the middle part of Table 1, we report other cases where v₀ = 0.25 and v_r = 0. Such orbits are again very eccentric and the profiles are, indeed, much narrower. The corresponding values of f are now larger by a factor of 2–3 than those observed. Additional models (not shown here) with larger initial angular momentum give larger FWHM and smaller f. Obviously, some combination of all these is required to explain real observations.

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The bottom part of Table 1 shows the results of a set of line profile calculations carried out for smaller column density clouds. We chose r₂₃ = r₀ which corresponds to a factor of 6.3 decrease in N_col relative to the case shown at the top of the table. The scaling of FWHM between the two cases is simply done by the corresponding factor in Γ (i.e., the same FWHM for Γ smaller by a factor of 6.3). This illustrates the fact that in an atmosphere with a large range of column densities, there are always clouds that are close to being ejected from the system at large distances.

The changes in 〈r〉 for a given r₂₃ shown in Table 1 are due to the fact that as Γ increases, and radiation pressure is more important, the clouds spend more and more time away from the BH. This is noticeable for the case of r₂₃ = 10r₀ when Γ approaches 0.73 and for the case of r₂₃ = r₀ when Γ approaches 0.1.

As noted in Section 1, RM campaigns show that r_BLR(Hβ)∝L^{0.6 ± 0.1}_bol. It is interesting to note that this behavior is not very different from what is calculated here for the changes in 〈r〉 if we compare values over the range where Γ approaches its limiting value. However, it is not the case when Γ changes by similar factors close to the lower range shown in the table, where radiation pressure is negligible.

The values of f computed here should be compared with those determined observationally for selected AGN samples with measured σ*, in particular the Onken et al. (2004) and the Woo et al. (2010) AGN samples. The simulations illustrate how this factor depends on the BLR geometry, the distribution of Γ among objects in the sample, and the distribution of N_col in individual BLRs.

An important point of the new calculation is the relatively small change in the value of FWHM listed in Table 1, only a factor of ∼1.5 over most of the range of Γ except very close to the limiting value. The changes in f are also small, only a factor of ∼1.3 over the same range in Γ. This seems to be in contradiction to the naive expectation that, for cases of increasing L, the term 〈r〉FWHM²/G will deviate more and more from M_BH (e.g., Equation (3)). There are two reasons for this behavior. First, for realistic cases where N_col depends on the cloud location, the mean emissivity distance and the velocity depend on L/M rather than on L. This suggests that very low and very high luminosity AGNs with similar Γ will react to radiation pressure force in a similar way. Second, for a planar motion, the changes in the radial potential Φ(r) do not affect the cloud velocity in a linear way. In fact, the mean orbital changes in v_θ are small enough such that the overall FWHM is very far from zero even for marginally escaping clouds. Moreover, the mean cloud location, 〈r〉, is increasing in reaction to the increasing radiation pressure term. The end result is that the product f〈r〉FWHM²/G, with a constant value of f, is always a reasonable approximation for M_BH with little dependence on the relative importance of gravity and radiation pressure force. We return to this issue in Section 4 where we suggest a new way to evaluate M_BH taking into account radiation pressure acceleration.

Finally, we note that while radiation pressure is negligible for very small values of Γ, the s-dependence of the cloud properties is still very important. For example, an s = 0 atmosphere gives constant column density clouds (similar to what was assumed in M08), yet the mean emissivity radius, the FWHMs of the emission lines, and the mass correction factor f in this case are always different from those of the s = 1.2 case, regardless of the column density. The reason is the dependence of the cloud cross section on s. For example, in the case of Γ = 0.01 (first entry in the bottom part of Table 1), the s = 0 case gives 〈r〉/r₀ = 0.39 (compared with 0.53 for s = 1.2) and FWHM/v_Kepler = 2.45 (compared with 1.51). The resulting f is therefore much smaller (0.42 compared with 0.76). Thus, the radial dependence of the cloud properties is important for all Γ.

The examples discussed above were normalized to give a typical r_BLR(Hβ) for AGNs with M_BH = 10⁸ M_☉ and Γ = 0.1. However, the computed line profiles cannot be directly compared with the observations of such sources for several reasons. First, we only consider a situation involving one type of clouds and neglect the possibility of different populations under different physical conditions in the same source. This applies to the distributions of both N_col and U(r). For example, Equation (1) and the constants given in Section 1 suggest that, in general, r_BLR(H_β)/r_BLR(C iv λ1549) ≃ 3. The question is whether cloud distributions like those considered in Table 1 can reproduce this ratio. Second, we did not take into account changes in α(r), the fraction of L_bol which is absorbed by clouds at various distances. This can be an important factor close to the BH where clouds become partly transparent. In this case, much of the Lyman continuum radiation is not absorbed and radiation pressure force is reduced. It can also affect medium to large column density clouds at large distances where α(r) approaches unity. For example, assuming α(r) = 0.75 instead of α(r) = 0.5 in the calculations of Table 1 results in a limiting value of Γ which is about 0.4 compared with Γ = 0.735 listed in the table.

To illustrate these effects, and to provide more realistic line profiles, we computed two grids of photoionization models for a range of column density and ionization parameter using the code ION (Netzer 2006). The first grid supplies calculated line intensities for Hβ and C iv λ1549 over a large range in U(r). Given r from the cloud motion simulation, we use the grid to compute j(r) and thus a more realistic (r). The second grid supplies the absorbed fraction, α(r), as a function of U(r) and N_col. Figure 5 shows part of the α(r) grid to illustrate the expected range in this parameter. We have not included the changes in gas density since they do not play a major role over the range of conditions considered here. We have also not considered anisotropy of the line emission which is bound to have an effect on the FWHM of some lines. Such modifications will be included in a forthcoming paper that is intended to present a comparison with observed line profiles.

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We tested a large number of single-zone models using the above grids of U(r) and α(r). The models cover a large range in angular momentum and BLR geometries. We have specifically investigated three cases of different eccentricity, defined by three values of v₀(r₀): 0.25, 0.5, and 0.75. These were calculated with different Γ and r₂₃. In general, it is easy to reproduce the observed I(C iv λ1549)/I(Hβ) but difficult to account, at the same time, for the emissivity-weighted radii of the two lines (Equation (1)) and the line width ratio. For example, the best case of the three, with v₀(r₀) = 0.25, gives I(C iv λ1549)/I(Hβ) = 4.4, 〈r〉(C iv λ1549)/〈r〉(Hβ) = 0.67, and FWHM(C iv λ1549)/FWHM(Hβ) = 1.43. The conclusion is that, within the range of parameters assumed here, there is no obvious way to explain all those properties when keeping with the idea of a single column density distribution (i.e., a single r₂₃).

We also tested a case of M_BH = 10⁸ M_☉, Γ = 0.1 and two distinct cloud populations in inner and outer zones with some overlap between the two. In this case, the initial conditions for the two populations are decoupled from each other, but the changes in density, column density, and ionization parameter follow the same pattern with the same s = 1.2 density law. The FWHMs of both emission lines were calculated under the assumption of a thick central spherical sector with clouds occupying a region of ±π/4 radians relative to the central plane. The inner zone clouds have r₀ = 5 × 10¹⁶ cm and U(r₀) = 10⁻¹ and the outer zone clouds r₀ = 3 × 10¹⁷ cm and U(r₀) = 10^−2.5. The starting velocity in both zones is v₀ = 0.5 at the appropriate r₀. In both zones r₂₃ = 3r₀. We followed the cloud motion and calculated, in each zone, the line intensity ratio, I(C iv λ1549)/I(Hβ), the line FWHMs, and the emissivity-weighted radii. These numbers are listed in Table 2 where we also show the properties of the combined spectrum which is calculated under the assumption of equal contributions to Hβ from both zones. The emissivity-weighted radii for the two zones are given in units of the RM radii of the two lines (Equation (1)). This very simple two-zone model gives results that are in good agreement with the observations of many low-to-intermediate luminosity AGNs. Figure 6 is a graphical summary of these results. The left and central panels show Hβ and C iv λ1549 profiles for the inner and outer zones, again assuming isotropic line emission, and the right panel shows the combined two-zone profiles.

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In conclusion, the simple single-zone models explored here cannot reproduce all the observed properties: line intensity ratio, mean emissivity radii, and FWHM ratio. The main reason is that the starting conditions fix the cloud orbit, and hence the line emissivity and FWHM. Simple two-zone models like the ones presented here can account for most observed properties of the Hβ and C iv λ1549 lines. In particular, they can account for the mean line ratio, the mean emissivity-weighted radii, and the mean relative FWHM of the Hβ and C iv λ1549 lines measured in various RM samples. Obviously, such simple models are not intended to explain all the observed line profile properties that can differ from one object to the next and contain additional components (see some obvious examples for complex C iv λ1549 profiles in Richards et al. 2002 and Sulentic et al. 2007). Fitting these is deferred to a forthcoming paper.

The above calculations allow us to investigate the intensity, the width, and the shape of the broad emission lines and to evaluate various methods used to estimate M_BH. We defer the discussion of specific observed line profiles to a future paper.

Assume a system of clouds with a given total amount of gas and a large range of column densities. Such a system will eventually break into three: virialized clouds, non-virialized bound clouds, and escaping clouds. The third group will not contribute significantly to the observed line emission for more than several dynamical times. The relative contribution of the first and second groups to the line emission depends on the cloud mass distribution. A sudden increase in L_bol will increase the importance of radiation pressure and will remove more gas from the system. A decrease in L_bol will drive the system closer to virial equilibrium. A new gas supply, e.g., from a disk wind, will produce bound as well as unbound clouds. All aspects of this general scenario must be considered when evaluating the observed line profiles and the various methods developed to use them in estimating M_BH.

A major objective of the present paper is to evaluate the accuracy and the normalization of various M_BH estimators in type-I AGNs. The results presented in Tables 1 and 2 suggest the following.

• 1.  
  Every AGN is likely to contain a large number of clouds with a large range in N_col. This can be the result of a broad cloud mass distribution and/or due to cloud motion in a radial-pressure-dependent environment with a positive value of s. A given Γ results in a lower limit on N_col at a given location for a given orbit eccentricity. Under such conditions, there are always some clouds, e.g., those that are very close to the BH, for which radiation pressure is negligible. For others, radiation pressure can be very important.
• 2.  
  For a small enough N_col, the effective r_BLR depends on both Γ and N_col. Under these conditions, the BH mass itself is an important factor in determining r_BLR. To illustrate this, consider two AGNs with identical spectral energy distribution (SED), L_bol, BLR geometry, N_col distribution and inclination to the line of sight. The effective r_BLR in the two is the same provided they harbor identical BHs. Different r_BLR will be measured if the two BHs have different masses despite the fact that L_bol is the same in both. This is the result of the larger Γ in the smaller BH AGN. The effect may not be recognized in a large sample of sources and can, in fact, be attributed to a large intrinsic scatter in the L_bol–r_BLR relationship. Any derived L_bol–r_BLR relationship will depend on the properties of the sources in the chosen RM sample, in particular on the distribution of Γ.
• 3.  
  Assuming a range in N_col in every AGN, the M08 suggestion to include a luminosity-dependent term in the calculation of M_BH (Equation (3)) is not in accord with our calculation that indicates that r_BLR and FWHM depend on L/M and not on L.

The multi-year RM campaign of NGC 5548 is the best example to test some of these ideas in a specific source. The campaign has been described and analyzed in numerous papers, and the ones most relevant to the present study are Peterson et al. (1999) and Gilbert & Peterson (2003).

Figure 7 shows the variations in L₅₁₀₀ and time lag (in this case the centroid of the CCF) in NGC 5548. Each point represents a full observing season which is typically ∼300 days long. The data are taken from the recent compilation of Bentz et al. (2009) which provides the best galaxy-subtracted flux at 5100 Å. The uncertainty on L₅₁₀₀ is basically the range of this quantity over the observing season. This is of the same order as the variation from one season to the next. As clearly seen from the diagram, r_BLR(Hβ) lags the continuum in such a way that more luminous phases are associated with longer lags. This has been noted in earlier publications (e.g., Gilbert & Peterson 2003). Figure 8 shows t(lag) versus L₅₁₀₀ for the same data set. While the uncertainties are large, some correlation, with a slope of 0.5–1, is evident. An earlier version of the diagram, with fewer points, is shown in Peterson et al. (1999).

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For NGC 5548, M_BH ≃10⁸ M_☉ and r_BLR(Hβ) ≃20 l.d.. Thus, the dynamical time is of order 6 years and the time it takes to change r_BLR by 50% (e.g., Table 1) is approximately 3 years. This seems to be compatible with the changes in L₅₁₀₀ and t(lag) in Figure 7, thus some adjustment of r_BLR(Hβ) due to the effects discussed in this work is possible. The measured L₅₁₀₀, with a bolometric correction factor of about 10, indicates a mean Γ of about 0.02. The bottom part of Table 1 provides approximate parameters for such a case. Any successful model of NGC 5548 should account for the behavior shown in Figure 7, as well as for the observed FWHMs and luminosities of both Hβ and C iv λ1549. While the full investigation is deferred to a future paper, we consider here the predicted lags for Γ = 0.005, 0.01, 0.02 assuming a BH mass of 10⁸ M_☉ and two families of clouds: one with r₂₃ = 10r₀ and v₀ = 0.5, and one with r₂₃ = 0.093r₀ and v₀ = 0.25. The second assumed family of clouds results in pseudo-orbits of higher eccentricity that, as explained earlier, are more strongly affected by radiation pressure. In both cases, the predicted lags for Γ = 0.02 are consistent with the observed values (log t(lag) ≈ 1.34 at log L₅₁₀₀ ≈ 43.48). However, the calculated slope of log t(lag) versus L₅₁₀₀ is flatter than observed.

As argued earlier, a single family of BLR clouds cannot provide a full explanation to the observed spectrum of many AGNs. This must applied to NGC 5548. (To appreciate the complexity of this case see the various components considered by Kaspi & Netzer 1999 to explain only the variable line intensities.) The simple examples considered here suggest that the dynamical scaling of the BLR in NGC 5548, due to radiation pressure force, is an additional, physically motivated mechanism that must be added to any cloud model when attempting to explain the observed variations in t(lag).

Current BH mass estimates utilize RM-based measurements of r_BLR, measured FWHMs (or an equivalent velocity estimator) of certain broad emission lines, and Equation (2). The normalization constant f is obtained by comparing M_BH obtained in this way with the mass obtained from the M–σ* method. Having examined a large range of cloud orbits and line profiles under various conditions, and the corresponding values of the effective r_BLR, we are now in a position to evaluate the merits of this method.

We consider three general possibilities. The first is the case where all AGNs contain BLR clouds with a wide column density distribution. A randomly chosen object will have in its BLR some clouds that are affected by radiation pressure force and others that are not. This is the case for any Γ. The cloud dynamics and the observed line profiles reflect the (unknown) column density distribution. Our calculations suggest that an RM sample drawn randomly from such an AGN population can be safely used to determine the best value of f by comparing the derived M_BH with the M–σ* method. This is justified by the fact that M_BH ∝ 〈r〉FWHM² even if radiation pressure force is important (see Table 1). The observed FWHMs are, indeed, smaller than those that would have been observed if all clouds had extremely large column densities. This, however, has no practical implication since the column densities are not known and f is simply a normalization factor that serves to bring two completely different methods of estimating M_BH into agreement. Mass estimates obtained in this way are reliable provided the properties of the RM sample represent well the population properties.

The second case reflects a situation where the cloud column density distribution is, again, very broad but part of the population is underrepresented in the RM sample. For example, the RM sample may contain mostly sources with Γ ∼ 0.1, while the overall distribution of Γ is much wider. In this case, the normalization factor f will reflect only the properties of the measured sources and its use will provide poor mass estimates for cases with much larger or much smaller accretion rates. This may well be the case in the RM sample which is most commonly used (Kaspi et al. 2000; Bentz et al. 2009) that contains only very few AGNs with Γ>0.3. The numbers in Table 1 enable us to evaluate the resulting deviations in the estimated M_BH. For example, if we use the first part of the table and assume a source with a certain L_bol and Γ = 0.1, we find that the mass of a similar L_bol source with Γ = 0.7 will be underestimated by a factor of 1.11/0.75.

Regarding the second case, it is important to note that underestimates and overestimates of M_BH are equally likely. Consider again an RM sample where, for most sources, Γ = 0.1. This results in a certain value of f which takes into account the effect of radiation pressure force in some of these sources (see the bottom part of Table 1). Assume a second, randomly selected AGN sample with a similar BH mass distribution but a typical Γ which is much smaller than 0.1. Most measured FWHMs in this sample are broader than those in the RM sample because radiation pressure force is not as effective in reducing the cloud velocity. Using the value of f derived for the RM sample will result in overestimating M_BH in the second sample. The lower part of Table 1 gives some idea about the magnitude of this effect, e.g., an overestimate by a factor of 1.01/0.76.

The third case is similar to the first one except that large luminosity variations, on timescales that are not too different from the BLR dynamical time, are occurring in most sources, including those selected for RM monitoring. Table 1 shows that, like the first case, the deduced f well represents the population because 〈r〉 follows the variations in L_bol. The mean M_BH in such a sample is recovered albeit with a larger uncertainty.

Given the above considerations, we now investigate an alternative method to calculate M_BH. The method takes into account the effect of radiation pressure force on the cloud motion, and the results will be compared to those obtained by the old method (Equation (2)) and by the M08 method.

Our new calculations indicate that the emissivity-weighted r_BLR depends both on the (large) range in L across the entire AGN population, as well as on short timescale changes in r_BLR in individual sources. The first of these depends roughly on L^1/2 and is a manifestation of the observational fact that the ionization parameter, U(r), and the SED are not changing much with source luminosity. The second reflects changes in the BLR structure due to the reaction of various column density clouds to the (changing) radiation pressure force. This depends on both L_bol and M_BH. This is seen, for example, in Equation (12) for the critical radius where clouds can escape the system and also in the calculations of Table 1. It is therefore reasonable to assume that r_BLR is given by an expression of the form

$$\begin{equation} r_{\rm BLR}= a_1 L^{\gamma } + a_2 (L/M)^{\delta } \,\,, \end{equation} \tag{ 17 }$$

where a₁ and a₂ are constants and L is a measure of the source luminosity, e.g., L₅₁₀₀ if r_BLR = r_BLR(Hβ). Obviously, the above approximation is not unique and one can assume other dependences that are consistent with the line profile calculations, e.g., a dependence of FWHM on L/M.

The idea of introducing a second, luminosity-dependent term into the calculation of M_BH is not new. In particular, M08 suggested an expression for M_BH which depends on both L^1/2 and L/N_col (Equation (3)). Assuming all AGNs obey the same relationship, and N_col is the same in all, the M08 expression leads to extremely large values of M_BH for the most luminous AGNs. The reason is the linear dependence of M_BH on L at very high luminosities combined with the calibration of the relationship at small L, typical of the M–σ* sample of Onken et al. (2004). The additional consequence of this approach is an upper limit of Γ ∼ 0.1 in many high luminosity, large BH mass sources. In their later work, M09 considered the possibility that N_col can differ from one source to another but is still constant for all clouds in a given BLR. This would result in smaller M_BH and larger Γ in some high luminosity sources since in some BLRs, N_col can exceed 10²³ cm⁻² by a large factor thus reducing the importance of radiation pressure force.

The limitation of the M08 mass estimate is the detachment of L from M_BH. As shown here, this is not the case in more realistic BLRs, especially those where the masses of the clouds are conserved. In such cases, the location of the outer clouds that still contribute to the line profiles depends on L/M and the three-dimensional velocities of the marginally bound clouds are such that the product r_BLRFWHM² is not very different from what is found in pure gravity-dominated systems. Moreover, for pressure-confined clouds, the dependence on N_col is likely to be different in different parts of the BLR. Thus, we are looking for an expression that will reflect, properly, all these effects and will allow for the possibility of a range of column densities in every source. We also want to avoid biasing in the derivation of M_BH in the limits of very large or very small L and to retain the experimental results that r_BLR ∝ L^γ with γ = 0.6 ± 0.1.

All the above can be achieved by assuming that r_BLR is given by Equation (17) and requiring that M_BH ∝ r_BLRFWHM². For the sake of simplicity, we assume γ = 0.5 and δ = 1 and substitute Equation (17) into the mass expression. This leads to a simple quadratic equation in M_BH with the following solution:

$$\begin{equation} M_{\rm BH}=\frac{1}{2} a_1 L^{1/2} \rm FWHM^2 \left[ 1 + \sqrt{\left(\vphantom{\left(1+ \frac{4 a_2}{a_1^2 \rm FWHM^2}\right)}\right.}\left.1+ \frac{4 a_2}{a_1^2 \rm FWHM^2}\right) \right], \end{equation} \tag{ 18 }$$

where a₁ and a₂ are the same ones used in Equation (17) except for a common multiplicative constant which depends on the units of r_BLR, L₅₁₀₀ and M_BH. For example, using the measure parameters for the Hβ line, L = L₅₁₀₀, FWHM = FWHM(Hβ), the constant multiplying a₁ and a₂ in Equation (17) is 10^16.123 when M_BH is measured in M_☉, L₅₁₀₀ in units of 10⁴⁴ erg s⁻¹ and r_BLR in cm.

We used Equation (18) and the Woo et al. (2010) sample to find a₁ and a₂ for 29 AGNs with measured σ*. The list is an extension of the one used by Onken et al. (2004) that contains only 16 sources. We have supplemented the data in Woo et al. by data from Bentz et al. (2009) on r_BLR and L₅₁₀₀ where this information was missing. First, we performed a χ² analysis on M_BH(RM) versus M–σ* using the parameters recommended by Gültekin et al. (2009). This gave f = 1.0 which is consistent with the values found by Onken et al. (2004) and Woo et al. (2010)³

Next, we carried out a χ² minimization to solve for a₁ and a₂ in Equation (18). Since the minimization involves the error estimate on L₅₁₀₀, and since this error is not very well defined given the combination of observational uncertainly and the intrinsic scatter in L₅₁₀₀ over several long RM campaigns, we decided to adopt a uniform value of ΔL₅₁₀₀/L₅₁₀₀ = 0.3. We also assume a minimum of 0.1 to Δ(σ*)/σ* and a minimum of 0.05 on Δ(FWHM)/FWHM. Our results depend slightly on these assumptions.

The best values obtained in this procedure are a₁ = 4.1, a₂ = 7.1 × 10⁷, and χ²/ν = 1.73. Extensive tests show that the χ² changes very little if a₁ or a₂ are changing by up to 10%. This is the result of some degeneracy between a₁ and a₂ (see Equation (18)). The average deviation between the new mass estimates and those obtained by the M–σ* method is 0.31 dex. There is a weak dependence of the deviation on the line width (larger deviation for larger FWHM(Hβ)) which is marginal given the small number of sources in the sample. The corresponding number for the deviation of masses obtained directly from the RM measurements and the above value of f is 0.36 dex. Thus, the new method is, indeed, superior in this respect. Obviously, it is not surprising to find such an improvement when adding a new free parameter to the model.

To compare the various mass estimates more thoroughly, we calculated M_BH in three different ways: the old method (Equation (2)) with f = 1.0, the M08 method (Equation (3)) with f₁ = 0.56 and f_g = 10^7.7 (as in M08), and the new method (Equation (18)) with the above a₁ and a₂. For the M08 method, we followed the M09 recommendation and assumed a log-normal distribution of N_col with a mean of 10²³ cm⁻² and a large standard deviation of 0.5 dex. We also calculated r_BLR in the old (Equation (1)) and new (Equation (17)) ways.

Figure 9 compares two mass ratios, M_BH(new)/M_BH(old) (red points) and M_BH(new)/M_BH(M08) (black points), in a large simulated AGN sample. The sample covers, uniformly, the luminosity range L₅₁₀₀ = 10⁴³–10⁴⁷ erg s⁻¹ and the simulations assume a Gaussian, luminosity-independent distribution of FWHM(Hβ) with a mean of 4500 km s⁻¹ and a variance of 1500 km s⁻¹. The diagram shows that the new and old estimates are similar at all L_bol but M_BH(M08) deviates from both, by a large factor, at both low and high luminosities. Moreover, the slight deviation between the new and old methods at the very high luminosity end, by up to about 0.2 dex in M_BH, is most likely due to the fact that the procedure used to obtain a₁ and a₂ is based on a sample of 29 mostly low-to-intermediate luminosity AGNs while the simulations reach a much larger value of L_bol. A comparison of the estimated r_BLR (Equation (1) and (17)) leads to similar conclusions.

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We also made a similar test on the Netzer & Trakhtenbrot (2007) sample using all three methods. The luminosity range in this case is smaller but the FWHM(Hβ) distribution is more typical of observed AGNs. The results (not shown here) are very similar to those of the simulated sample.

In conclusion, the new method for estimating M_BH gives results that do not deviate much from the old method which is based on a single constant f. This is true at both high and low luminosities and over a large range in FWHM. Obviously, the range of parameters tested here (s, orbit eccentricity, several types of cloud distributions, etc.) is rather limited, and more extensive modeling is required to confirm these results. However, it is our opinion that the main limitation of the M_BH determination methods remains observational and is related to the fact that the present AGN M–σ* sample is small (29 sources) and cannot possibly represent the entire range of properties, mostly Γ, observed in AGNs.