INCLINATION-DEPENDENT ACTIVE GALACTIC NUCLEUS FLUX PROFILES FROM STRONG LENSING OF THE KERR SPACETIME

X-ray emission is a defining characteristic of active galactic nuclei (AGNs); however, unlike optical emission, its origin is currently unclear. In contrast to X-ray binaries, standard AGN accretion disk theory predicts disk temperatures too low to emit X-rays. The X-ray emission is thought by many to be generated by inverse Compton scattering of UV photons emitted from the disk by hot electrons in a corona above/around the disk (see the review of Reynolds & Nowak 2003). However, both the geometry (e.g., its size, shape, and position, e.g., height above the accretion disk) and the physics (e.g., the mechanism for heating the electrons) of the corona are under debate (Haardt & Maraschi 1991; Haardt et al. 1994; Poutanen & Svensson 1996; Collin et al. 2003; Torricelli-Ciamponi et al. 2005; Trześniewski et al. 2011). Based on the Bayesian Monte Carlo microlensing analysis technique (Kochanek 2004; Kochanek et al. 2007), recent observations in gravitational microlensing have constrained the X-ray emission size of quasars to be of order 10 r_g (gravitational radius, GM/c²; Blackburne et al. 2006, 2011; Pooley et al. 2006; Morgan et al. 2008; Chartas et al. 2009; Dai et al. 2010), about one order of magnitude smaller than the optical emission region (Poindexter et al. 2008; Morgan et al. 2010; Mosquera et al. 2011). Chen et al. (2011, 2012b) detected energy-dependent X-ray microlensing in Q 2237+0305 and several other lenses for the first time, and their results suggest that the hard X-ray emission might come from regions smaller than the soft, while in RXJ 1131−1231 the effect is the opposite (Chartas et al. 2012). Morgan et al. (2012) found that the majority of the X-ray emission in QJ 0158−4325 comes from regions close to 6 r_g, i.e., in the vicinity of the last stable orbit of a Schwarzschild black hole.

For distances less than a few gravitational radii, the flux and profile of both the X-ray continuum and metal emission lines (in particular, the Fe Kα line) are strongly influenced by the gravitational field of the supermassive black hole at the center of an AGN. Due to the complexity of the Kerr metric (spacetime of a black hole with angular momentum, or spin), much work has focused on either the analytic treatment of the geodesics in a Kerr metric (Kraniotis 2004, 2005; Hackmann et al. 2010) or on developing fast ray-tracing codes (Karas et al. 1992; Dexter & Agol 2009; Vincent et al. 2011). There has also been much work studying AGN X-ray spectra under strong gravity (Fabian et al. 1989; Chen & Halpern 1989; Laor & Netzer 1989; Laor 1991; Hameury et al. 1994; Bromley et al. 1997; Fabian & Vaughan 2003; Beckwith & Done 2004; Fuerst & Wu 2004; Schnittman & Krolik 2010). The observed broad, skewed iron lines (e.g., MCG6−30−15; Tanaka et al. 1995) strongly suggest that the reflection component near the black hole contributes significantly to the total emissivity. Since the iron line profile strongly depends on the black hole spin, and the relativistic distortion of the iron emission line is much more easily measured than the broad X-ray continuum, most of these papers concentrated on the iron emission line. For example, Fuerst & Wu (2004) studied the radiation transfer of emission lines in the Kerr spacetime. However, less work has been done on the continuum emission. It has been found that strong lensing of an accretion disk by the central supermassive black hole is strongly dependent on inclination angles (Cunningham 1975), and for very large inclination angles higher-order images can contribute a significant fraction of the total luminosity (Beckwith & Done 2005). Polarization of the continuum emission near black holes was investigated in Connors et al. (1980) and is becoming an important tool in measuring the inclination angle of an accretion disk and the spin of its black hole (Dovčiak et al. 2004, 2008, 2011; McClintock et al. 2006; Li et al. 2009; Schnittman & Krolik 2009). Quasi-periodic oscillations (QPOs) have been observed in the X-ray light curves of black hole binaries and black hole candidates for more than a decade (Remillard & McClintock 2006); however, no convincing QPO signal has been detected for AGNs until very recently (Gierlinski et al. 2008; Middleton et al. 2009). High-frequency QPOs are often modeled in the context of strong gravity (the oscillation period is comparable to the Keplerian orbital period near the innermost stable orbit), and this can be used to accurately measure the black hole spin once the correct model of QPOs has been found (Wagoner et al. 2001; Schnittman 2005). The study of radiation transfer is crucial for these subjects, and Monte Carlo ray-tracing methods have been used extensively to solve (continuum or line) radiation transfer problems in AGNs (Agol & Blaes 1996; Laurent & Titarchuk 1999; Böttcher et al. 2003; Dolence et al. 2009; Schnittman & Krolik 2010). However, modeling of the AGN X-ray continuum emission was hampered by the lack of an accurate estimate of the size of the emission region. We study the influence of the strong field of the massive spinning black hole centering the AGNs on the X-ray continuum emission using the most recent constraints from quasar microlensing observations and our ray-tracing code. Despite the fact that there are nicely written public codes available for ray tracing in Kerr spacetime (e.g., Dexter & Agol 2009), we develop our own ray-tracing code (specially designed for backward ray tracing; see Section 3.2) with the future goal of solving the radiation transfer equation in Kerr spacetime using the characteristics method and operator perturbation algorithm (Hauschildt & Baron 2006; Chen et al. 2007; Baron et al. 2009a, 2009b).

In this paper, we study the inclination angle dependence of the flux profiles in the X-ray and the optical continuum of AGNs and their ratios caused by strong gravity, providing a reference for future usage. The X-ray-to-optical flux ratio, conventionally characterized by the parameter α_ox (Tananbaum et al. 1979), is empirically measured in many AGNs. Our inclination-dependent α_ox with the Kerr lensing and projection effects taken into account will provide a baseline model for interpreting the observed α_ox in AGNs. For example, assuming that quasars are observed at randomly distributed solid angles, if there exists strong inclination angle dependence for the X-ray-to-optical flux ratios caused by the strong gravity field, the flux ratios will be observed with larger fluctuations than the intrinsic ones. The mean value, scatter, and shape of the distributions might be changed by the strong gravity (see Section 6). Therefore, this can provide a test for different corona geometries, combined with a large number of measurements of α_ox from different AGNs.

Based on the unification model (Antonucci 1993), AGNs can be classified as Type I (normal) or Type II (obscured possibly by a torus) with ∼50% Type II AGNs locally at low redshifts (e.g., Simpson 2005; Maiolino et al. 2007; Gilli et al. 2007). There are suggestions that the fraction of Type II AGNs decreases with luminosity or increases with redshift (Hasinger 2008; Treister et al. 2009; Bianchi et al. 2012). Because emission of Type II AGNs is strongly modified by Compton-thick absorption, which makes it difficult to study the intrinsic α_ox, we do not compare our model predictions to Type II AGNs. In Type I quasars, a significant fraction (20%–40%) are broad absorption line (BAL) quasars (e.g., Dai et al. 2008; Shankar et al. 2008). These BAL quasars are only moderately obscured, and several studies show that their intrinsic α_ox are consistent with non-BAL quasars (Green & Mathur 1996; Brandt et al. 2000; Green et al. 2001; Gallagher et al. 2002). Therefore, we compare our model predictions with α_ox measurements in Type I quasars including BAL quasars. If the X-ray-to-optical magnification ratio R_X/O is different between normal quasars and BAL quasars, there might be bias in the current estimate of BAL absorption caused by ignoring the effect of strong gravity. We investigate these ideas in this paper.

In Section 2, we introduce the ray-tracing code for Kerr spacetime developed in this paper. In Section 3, we present the basic formalism for studying strong lensing of Kerr spacetime. In Section 4, we study the influence of strong gravity on the optical emission. In Section 5, we study strong lensing of the X-ray emission assuming three X-ray corona geometries. In Section 6, we compare the lensed α_ox of normal/BAL quasars with observations, and estimate possible errors in current measurements of BAL absorption caused by ignoring effects of the Kerr strong gravity. In Section 7, we draw our conclusions. Readers not interested in mathematical details are invited to skip Sections 2 and 3.2, and all or part of Section 5.

A Kerr black hole is fully characterized by two parameters, i.e., mass M and angular momentum a. In Boyer–Lindquist coordinates (Boyer & Lindquist 1967), the Kerr metric (Kerr 1963) with signature (−, +, +, +) can be written as

$$\begin{eqnarray} &&ds^2 =-\alpha ^2dt^2 +\frac{\Sigma ^2}{\rho ^2}\sin ^2\theta (d\phi -\omega dt)^2+\frac{\rho ^2}{\Delta }dr^2+\rho ^2d\theta ^2,\nonumber\\ \end{eqnarray} \tag{ 1 }$$

where t, r, θ, and ϕ are the four independent coordinate variables, and

$$\begin{eqnarray} \rho ^2 &\equiv & r^2+a^2\cos ^2\theta\nonumber\\ \ms \Delta &\equiv & r^2-2Mr+a^2\nonumber\\\ms \Sigma ^2 &\equiv & \rho ^2\left[(r^2+a^2)+\frac{2Mr}{\rho ^2}a^2\sin ^2\theta \right]\nonumber\\ \ms \alpha ^2 &\equiv & \frac{\rho ^2\Delta }{\Sigma ^2}\nonumber\\\ms \omega &\equiv & \frac{2aMr}{\Sigma ^2}. \end{eqnarray} \tag{ 2 }$$

The off-diagonal terms of the Kerr metric are characterized by ω, which represents the dragging of the inertial frame. A photon's geodesic motion can be described by a Hamiltonian ${\cal H}(x^a, p_a)$ defined on an eight-dimensional phase space (the index a runs from 0 to 3),

$$\begin{eqnarray} {\cal H} &\equiv &\frac{1}{2}g^{ab}(x)p_ap_b = -\frac{1}{2\alpha ^2}p_t^2-\frac{2Mar}{\rho ^2\Delta }p_t p_\phi\nonumber\\\ms\ms &&+ \frac{\Delta }{2\rho ^2}p_r^2+\frac{1}{2\rho ^2}p_\theta ^2+\frac{\Delta -a^2\sin ^2\theta }{2\rho ^2\Delta \sin ^2\theta } p_\phi ^2, \end{eqnarray} \tag{ 3 }$$

where g^ab is the inverse metric tensor and p_a is the covariant photon 4-momentum vector in the coordinate basis. To obtain photon trajectories in the Kerr spacetime, we solve the group of eight first-order Hamilton equations

$$\begin{eqnarray} \frac{dx^a}{d\xi }&=&\frac{\partial {\cal H}}{\partial p_a} \nonumber\\\ms \frac{dp_a}{d\xi }&=&{-}\frac{\partial {\cal H}}{\partial x^a} , \end{eqnarray} \tag{ 4 }$$

where ξ is the affine parameter (refer to Chandrasekhar 1983 for details), instead of the four second-order geodesic equations

$$\begin{equation} \frac{d^2x^a}{d\xi ^2}+\Gamma ^{a}_{\:bc}\frac{dx^b}{d\xi }\frac{dx^c}{d\xi }=0, \end{equation} \tag{ 5 }$$

where Γ^a_bc are the Christoffel symbols. The number of equations reduces to six since p_t and p_ϕ are motion constants, and we give them in the Appendix. These six first-order differential equations are much easier to solve numerically than the four second-order differential equations. There is another motion constant discovered by Carter (1968),

$$\begin{equation} {\cal C}= p^2_\theta +\cos ^2\theta \left(\frac{p^2_\phi }{\sin ^2\theta }-a^2p_t^2\right), \end{equation} \tag{ 6 }$$

which we use as an independent check for our ray-tracing code (the maximum relative error of $\cal C$ allowed in the ray tracing is less than 10⁻⁷). We solve Equation (4) numerically using a fifth-order Runge–Kutta algorithm with adaptive step size control (see, e.g., Press et al. 1992).

Let ν_o and ν_e be the photon frequency measured by a distance observer and an observer co-moving with the light source. We define the redshift factor

$$\begin{equation} g\equiv \frac{\nu _{\rm o}}{\nu _e} =\frac{(-u^ap_a)_{\rm o}}{(-u^ap_a)_{\rm e}}, \end{equation} \tag{ 7 }$$

where u^a_o and u^a_e are the 4-velocity of the distant observer and the source. For the distant observer, we choose

$$\begin{equation} u^a=\frac{1}{\alpha }(1,0,0,\omega), \:\:\: u_a=(-\alpha , 0, 0, 0) \end{equation} \tag{ 8 }$$

(the so-called zero-angular-momentum observer). An advantage of this choice is that the collection {u^a, ∂_r, ∂_θ, ∂_ϕ} forms an orthogonal basis and therefore the photon's spatial direction can be easily specified with respect to {∂_r, ∂_θ, ∂_ϕ}, whereas {∂_t, ∂_r, ∂_θ, ∂_ϕ} does not form an orthogonal basis (Bardeen et al. 1972). The 4-velocity of the light emitter is model-dependent. We assume that the source is circling around the z-axis, i.e., 

$$\begin{equation} u^a=\gamma (1, 0, 0,\Omega). \end{equation} \tag{ 9 }$$

The selection of Ω is not arbitrary, but constrained by the causality condition u^au_a = −1. If the source is on the accretion disk (e.g., optical emission), we assume a prograde⁵ Keplerian flow for the fluid (Bardeen et al. 1972), i.e., 

$$\begin{equation} \Omega =\Omega _K\equiv \frac{M^{1/2}}{r^{3/2}+a M^{1/2}}. \end{equation} \tag{ 10 }$$

If the source is not in the equatorial plane (e.g., the X-ray ball model in Section 5.2), we use the profile of Ruszkowski & Fabian (2000),

$$\begin{equation} \Omega =\left(\frac{\theta }{\pi /2}\right)\Omega _K+\left[1-\left(\frac{\theta }{\pi /2}\right)\right]\omega , \end{equation} \tag{ 11 }$$

where θ is the spherical polar angle in the Boyer–Lindquist coordinates (an interpolation between Ω_K and ω of the frame dragging, see Equation (2)).

Strong lensing effects of a Kerr black hole on the accretion disk and X-ray corona differ from standard linear gravitational lensing theory in a few important aspects. First, linear lensing theory requires a very small bending angle, i.e., GM/r₀ ≪ 1, where M and r₀ are the lens mass and the impact vector, respectively. This assumption breaks down for accretion disks whose inner cutoff can be as close as a few gravitational radii from the lensing black hole. Second, the source, lens, and observer are far away from each other in linear lensing theory, whereas for Kerr lensing, there are no well-separated lens and source planes, i.e., the source is very close to the lens. Third, the spacetime in the source plane is assumed to be flat in the linear theory, whereas for Kerr lensing, this is not true. Finally, since both the source and the observer are far away from the lens in the linear theory, there is no gravitational redshift caused by the lens, and the redshift comes solely from cosmology. This is incorrect for Kerr lensing. Since photons are emitted by sources moving with relativistic speeds in strongly curved spacetime, both the Doppler and gravitational redshifts have to be taken into account in Kerr lensing. Consequently, there is no simple lens equation relating the source and image positions through a single bending angle (the bending happens continuously instead of at a single point in the lens plane, as assumed in the linear lensing theory). We identify the source position for each ray arriving at the observer through backward ray tracing in the Kerr spacetime (see Section 3.2). The redshift effect is very important for computing observable quantities from the accretion disk around the supermassive black hole.

By definition, the monochromatic flux $F_{\nu _{\rm o}}$ measured by an observer O sitting in a (asymptotically) flat spacetime is

$$\begin{equation} F_{\nu _{\rm o}}\equiv \oint {I_{\nu _{\rm o}}\mu d\Omega _{\rm o}}= \int {I_{\nu _{\rm o}} d\Omega _{\rm o}}, \end{equation} \tag{ 12 }$$

where dΩ_o is the differential solid angle measured at the observer, μ is the angle cosine with respect to the normal of the detector window (see Figure 1), and can be safely dropped for sources very far away from the observer, i.e., the source is observed within a tiny solid angle. In standard lensing theory, there are no frequency redshifts between the source and observer (besides the cosmological redshift), therefore I_ν is a conserved quantity along the light path in case of no scattering and absorption, thus

$$\begin{equation} F_{\nu _{\rm o}}\equiv \int {I^{\rm obs}_{\nu _{\rm o}}(\mbox{\boldmath $\theta $})d\Omega _{\rm o}}= \int {I^{\rm source}_{\nu _{\rm o}}(\mbox{\boldmath $\beta $}) d\Omega _{\rm o}}, \end{equation} \tag{ 13 }$$

where $\mbox{\boldmath $\theta $}$ and $\mbox{\boldmath $\beta $}$ are the image and source positions, respectively. Since the spacetime in both the source and lens plane is assumed to be flat, the integral over solid angle can be easily transformed into two-dimensional surface integrals in the lens or source plane. To compute the flux magnification, we simply compute the ratio of the image area to the source area in the lens plane, i.e., 

$$\begin{equation} {\cal A}^{-1}\equiv \det \frac{\partial \mbox{\boldmath $\beta $} }{\partial \mbox{\boldmath $\theta $}}. \end{equation} \tag{ 14 }$$

This is equivalent to computing the ratios of the solid angles subtended by the image and the source at the observer. On the other hand, for Kerr lensing, we have

$$\begin{equation} F_{\nu _{\rm o}}\equiv \int {I^{\rm obs}_{\nu _{\rm o}}(\hat{\bf n})d\Omega _{\rm o}} = \int {g^3(\hat{\bf n}) I^{\rm source}_{\nu _{\rm e}}(x^a(\hat{\bf n}),p_a(\hat{\bf n})) d\Omega _{\rm o}}, \end{equation} \tag{ 15 }$$

where $\hat{\bf n}$ is the photon's three-dimensional direction at the observer, (x^a, p_a) is the photon's position and momentum at the emitter (found through backward ray-tracing), ν_e is the source frequency, and g is the redshift factor defined earlier. Clearly, the effect of Kerr lensing strongly depends on g, and consequently on the 4-velocity of the source. Solid angles (at the observer) and areas (in the lens or source plane) are often used interchangeably in linear lensing theory; however, in Kerr lensing theory, solid angles at the observer are more appropriate quantities to use since the spacetime around the source and lens is strongly curved.⁶

Zoom In Zoom Out Reset image size

3.1. Models of Optical and X-Ray Emission

We assume that the optical/UV emission comes from the accretion disk (Shakura & Sunyaev 1973) and is Planckian,

$$\begin{equation} B(\nu ,r)=\frac{2h}{c^2}\frac{\nu ^3}{\exp \left[\frac{h\nu }{k T(r)}\right]-1}. \end{equation} \tag{ 16 }$$

The effective temperature can be approximated as T(r)∝r^−3/4κ^1/4 where $\kappa =1-\sqrt{r_{\rm ISCO}/r}$ (r_ISCO is the innermost stable circular orbit; Bardeen et al. 1972). For convenience we set κ to unity since this factor only influences the innermost region of the accretion disk which does not contribute significantly to the total optical flux. Likewise we have ignored the general relativistic corrections to T(r) in the innermost regions (Novikov & Thorne 1973). From Equation (15), the observed monochromatic flux $F_{\nu _{\rm o}}$ at frequency ν_o is

$$\begin{eqnarray} F_{\nu _{\rm o}} &=& \int {g^3 B(\nu _{\rm e},r) d\Omega _{\rm o}} =\int { \frac{2h}{c^2}\frac{\nu _{\rm o}^3}{\exp \left[\frac{h\nu _{\rm o}}{gkT(r)}\right]-1}d\Omega _{\rm o}}\nonumber\\ &=& \int { \frac{2h}{c^2}\frac{\nu _{\rm o}^3}{\exp \left[\frac{1}{g}\frac{T_o}{T(r)}\right]-1}d\Omega _{\rm o}}\nonumber\\ &=&\int { \frac{2h}{c^2}\frac{\nu _{\rm o}^3}{\exp \left[\frac{1}{g}\left(\frac{r}{r_s}\right)^{\frac{3}{4}}\right]-1}d\Omega _{\rm o}}\nonumber\\ &=& \int {B(\nu _{\rm o},\frac{r}{g^{4/3}})d\Omega _{\rm o}}. \end{eqnarray} \tag{ 17 }$$

In the above derivations we have defined the so-called scale radius r_s (not the Schwarzschild radius) by hν_o ≡ kT_o ≡ kT(r_s). We assume that the accretion disk moves with the Keplerian flow, see Equation (10). We choose the inner cutoff of the optical emission disk at the innermost stable circular orbit, r_ISCO, where r_ISCO = 6 r_g, 4.23 r_g, and 1.24 r_g for a = 0, 0.5 M, and 0.998 M black holes, respectively (Bardeen et al. 1972). This last value corresponds to an extreme Kerr black hole (Thorne 1974).

The spectrum of the X-ray emission of AGNs is observed to follow a power law. We therefore assume a simple power-law model for the X-ray emission,

$$\begin{equation} I_\nu (\nu , r) \propto \frac{1}{r^n}\frac{1}{\nu ^{\Gamma -1}}, \end{equation} \tag{ 18 }$$

where r is the radial coordinate of the source, n specifies the steepness of the radial profile, and Γ is the photon index. Using Equation (18), we also assume that the X-ray emission is isotropic in the local rest frame of the X-ray source.⁷ The observed monochromatic flux is

$$\begin{eqnarray} F_{\nu _{\rm o}} &=& \int {g^3 I_{\nu _e}d\Omega _{\rm o}}=\int {\frac{g^3}{r^n\nu _e^{\Gamma -1}}d\Omega _{\rm o}}\nonumber\\ &=& \int {\frac{g^{\Gamma +2}}{r^n\nu _{\rm o}^{\Gamma -1}}d\Omega _{\rm o}} =\frac{1}{\nu _{\rm o}^{\Gamma -1}}\int {\frac{g^{\Gamma +2}}{r^n}d\Omega _{\rm o}}, \end{eqnarray} \tag{ 19 }$$

which follows a power law with the same photon index Γ and we have dropped the unimportant constant in Equation (18). The strong lensing correction is contained in the last integral in the equation above, and g depends only on the photon trajectory, or equivalently on the photon's direction of arrival $\hat{\bf n}$ at the observer, but not on its frequency because gravitational lensing is achromatic. Different radial profiles and sizes of X-ray emission regions give different strong lensing corrections. The observed bolometric flux (e.g., 1–50 keV, in the observer's frame) is

$$\begin{eqnarray} &&F\equiv \int _{\nu _{\rm min}}^{\nu _{\rm max}}{F_{\nu _{\rm o}}d\nu _{\rm o}} =\int _{\nu _{\rm min}}^{\nu _{\rm max}}{\frac{1}{\nu _{\rm o}^{\Gamma -1}}d\nu _{\rm o}}\times \int {\frac{g^{\Gamma +2}}{r^n}d\Omega _{\rm o}}.\quad\quad \end{eqnarray} \tag{ 20 }$$

Since the first integral in the above equation is independent of the second one, the strong lensing corrections to the integrated flux and the monochromatic flux are the same for the simple X-ray emission profiles considered in this paper.

We consider three different X-ray geometries: a thin X-ray disk immediately above the accretion disk moving with Keplerian flow, see Figure 2 (low X-ray disk model), a spherical corona around the black hole (X-ray ball model), and a thin disk above the accretion disk with different heights (high X-ray disk model). For the low X-ray disk model, we choose the inner cutoff at r_ISCO. We choose no inner cutoff for the high X-ray disk model. For the X-ray ball and high X-ray disk model, we assume that the X-ray source is moving with the Ruszkowski & Fabian (2000) profile, Equation (11); see Figures 3 and 4. We take the photon index Γ = 2, which is normal for quasar X-ray spectra. For each X-ray geometry, we evaluate three spin parameters: a = 0, 0.5 M, and 0.998 M.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.2. Backward Ray Tracing

For flat spacetime, light paths are straight lines, and there is no gravitational redshift. Ignoring the special relativistic Doppler shifts and aberration effect, we have g = 1. Therefore, Equations (17) and (19) can be easily carried out as a surface integral over the accretion disk or the X-ray source, i.e., 

$$\begin{equation} F_{\nu _{\rm o}}^{\rm O (flat)}=\frac{2\pi \cos \theta }{r_{\rm obs}^2}\int {B(\nu _{\rm o}, r)rdr}, \end{equation} \tag{ 21 }$$

and

$$\begin{eqnarray} &&F_{\nu _{\rm o}}^{\rm X (flat)}=\frac{1}{\nu _{\rm o}^{\Gamma -1}}\int {\frac{d\Omega _{\rm o}}{r^n}} = \frac{1}{\nu _{\rm o}^{\Gamma -1}}\frac{1}{r_{\rm obs}^2}\int {\frac{d{\cal A}_e^{\perp }}{r^n}}, \end{eqnarray} \tag{ 22 }$$

where θ is the inclination angle of the disk, r_obs is the radial coordinate of the observer, and $d{\cal A}_e^{\perp }$ is the differential (X-ray) source area orthogonal to the line of sight. For curved spacetime with relativistic flows, Equations (17) and (19) need to be computed numerically.

We choose a pencil beam with a rectangular cross-section, large enough to contain all rays arriving at the observer (Figure 1), around the central light ray toward the black hole, and divide the solid angle space into a uniform grid. Each pixel, labeled by (i, j), corresponds to an image area dA_{i, j} = dΩ_{i, j}r²_obs. Since we have used a uniform rectangular grid, dΩ_ij = dΩ is constant for each pixel. Therefore, to compute the image area, we need only count the number of pixels whose light rays end at the accretion disk. To compute the observed flux, we weight pixels by the integrands in Equations (17) and (19) for optical and X-ray emission, respectively. Equations (17) and (19) are written as integrals over solid angle at the observer; however, we emphasize that the integrands are evaluated at the emitter (near the black hole), and each piece of solid angle $d\Omega _{\rm o}(\hat{\bf n})$ is mapped to a photon's initial phase-space coordinate (x^a, p_a) on the accretion disk through backward ray tracing. For example, if a photon from the optical accretion disk arrives at the observer along direction $\hat{\bf n},$ we find its r coordinate on the accretion disk and redshift factor g through backward ray tracing (Figure 1). If g < 1 (redshifted), then we sample the temperature of the blackbody radiation at radius r/g^4/3 (>r, T(r/g^4/3) < T(r)). This approach takes into account gravitational light deflection, frequency redshift, and area distortion automatically. We do not use the popular transfer function method (Cunningham 1975), which only considers light emitted on the accretion disk (the equatorial plane), since we allow X-rays to be emitted outside of the equatorial plane.

3.3. Strong Lensing Correction to the X-Ray to Optical Flux Ratio

Let M_O and M_X be the strong lensing magnifications of the optical and X-ray emission; then the ratio between the observed monochromatic flux in the optical and X-ray band can be written as

$$\begin{equation} \frac{F^{\rm X}_{\nu _{\rm o}}}{F^{\rm O}_{\nu ^{\prime }_{\rm o}}} = \frac{M_{\rm X}}{M_{\rm O}} \frac{F^{\rm X(flat)}_{\nu _{\rm o}}}{F^{\rm O (flat)}_{\nu ^{\prime }_{\rm o}}} \equiv R_{\rm X/O}^{\rm (lens)}R_{\rm X/O}^{\rm (proj)} \left.\frac{F^{\rm X}_{\nu _{\rm o}}}{F^{\rm O }_{\nu ^{\prime }_{\rm o}}}\right|_{\theta =0}^{\rm (flat)} \equiv R_{\rm X/O}\left.\frac{F^{\rm X}_{\nu _{\rm o}}}{F^{\rm O }_{\nu ^{\prime }_{\rm o}}}\right|_{\theta =0}^{\rm (flat)},\vspace*{3pt} \end{equation} \tag{ 23 }$$

where we have defined the lensing magnification ratio,

$$\begin{equation} R_{\rm X/O}^{\rm (lens)}\equiv \frac{M_{\rm X}}{M_{\rm O}}, \end{equation} \tag{ 24 }$$

the (Euclidean) projection area ratio R^(proj)_X/O (with respect to the face-on case), and the total ratio

$$\begin{equation} R_{\rm X/O}\equiv R_{\rm X/O}^{\rm (lens)}R_{\rm X/O}^{\rm (proj)} \end{equation} \tag{ 25 }$$

which contains all inclination dependence (lensing plus projection). If the X-ray and optical emission regions are different in geometrical size, surface brightness profile, or are located at different places with respect to the central black hole, strong gravitational lensing of the Kerr black hole should magnify or demagnify these two emission sources differently. In particular, the inclination angle dependence of the magnification should be different for X-ray and optical emission, and the magnification ratio R^(lens)_X/O should be a function of θ. Among the three X-ray geometries considered, R^(proj)_X/O = 1 for the low and high X-ray disk model because the factor cos θ is canceled when considering both the optical and X-ray emission, and R_X/O purely reflects the differential lensing effect. For the X-ray ball model, R^(proj)_X/O = (1  +  cos θ)/2cos θ is inclination-dependent (see Section 5.2), where the projected X-ray emission area is non-vanishing at large inclination angles. Thus, R_X/O is affected by both lensing and projection effects for the X-ray ball model.

We investigate strong lensing of optical and X-ray emission by Kerr black holes in the next two sections, and pay particular attention to the lensing magnification ratios R^(lens)_X/O(θ). In Section 6 we compare the strong lensed modulated α_ox with observations.

Because of uncertainties in parameters such as the black hole mass and the accretion efficiency, the scale radius r_s, or the half light radius (i.e., radius of the circular region centering the peak brightness of the emission and containing half the total flux at a given wavelength) r_half = 2.45 r_s, cannot be precisely determined from the accretion disk theory. Here, we use the empirical results from recent quasar microlensing observations (Kochanek 2004; Pooley et al. 2006; Morgan et al. 2008; Eigenbrod et al. 2008; Chartas et al. 2009; Dai et al. 2010; Blackburne et al. 2011, Blackburne et al. 2012), and assume that r_half = 100 r_g, and take r_s = 41 r_g. We choose the outer boundary of the optical disk at r_disk = 200 r_g which is five times the scale radius r_s, and contains ∼80% of the monochromatic flux. We ignore the lensing correction for regions with r > 200 r_g, and assume that the flux magnification ratio is 1 beyond r_disk. Consequently, if the flux magnification is $M_{<200\,r_g}$ within r_disk, the total flux magnification M_O is then $M_{{\rm O}}=0.2+0.8\,M_{<200\,r_g}.$

We plot the strong lensing distortion to the image area and specific flux $F_{\nu _{o}}$ in Figure 5 for three different spin parameters, a = 0, 0.5 M, and 0.998M. Since the size of the optical emission region is relatively large (of an order 100 r_g), and the intensity profile (blackbody radiation with T∝r^−3/4; see Equation (16)), is not very steep, the distortion in either the image area or flux is not significant (close to 1) for AGNs observed nearly face-on, and the area/flux magnification curves M_O(θ) are flat until the inclination angle θ becomes large (≳ 75°, i.e., observed near edge-on). Since the black hole spin a contributes only in higher orders (e.g., ω ∼ aM/r³), we do not see a significant difference in the flux or image area distortion for different spin parameters for optical emission. If we increase the disk size r_disk, the magnification of image area M^area_O at all inclination angles will eventually approach 1, but the magnification in flux M^flux_O changes only slightly, since the majority of the emission is enclosed within r_disk and the effect of gravity is weak outside of r_disk.

Zoom In Zoom Out Reset image size

In this section, we consider three different X-ray models: the low X-ray disk model, the X-ray ball model, and the high X-ray disk model defined in Section 3.1. The results for these X-ray models as well as the optical emission model discussed in Section 4 are summarized in Table 1.

Table 1. Strong Lensing Flux Magnification for Optical and X-Ray Emission

Modela	Inclination Angle (deg)
	1°	15°	30°	45°	60°	675	70°	75°	80°	82°	85°	89°
Optical emissionb
a = 0.998, rdisk = 200 rg	0.97	0.97	0.98	0.99	1.02	1.04	1.05	1.09	1.15	1.19	1.31	2.45
a = 0.5,    rdisk = 200 rg	0.97	0.97	0.98	0.99	1.02	1.04	1.05	1.08	1.13	1.17	1.27	2.25
a = 0,        rdisk = 200 rg	0.97	0.97	0.98	0.99	1.02	1.04	1.05	1.07	1.12	1.16	1.26	2.17
Low X-ray disk
a = 0.998, n = 3, rdisk = 5 rg	0.05	0.06	0.09	0.17	0.38	0.63	0.76	1.15	1.96	2.57	4.35	22.79
a = 0.998, n = 0, rdisk = 5 rg	0.15	0.17	0.23	0.38	0.71	1.03	1.18	1.58	2.27	2.73	3.96	15.13
a = 0.998, n = 3, rdisk = 10 rg	0.11	0.12	0.15	0.24	0.46	0.70	0.83	1.20	1.94	2.49	4.10	20.62
a = 0.5,    n = 3, rdisk = 10 rg	0.34	0.36	0.44	0.62	0.94	1.20	1.30	1.59	2.03	2.31	3.04	9.54
a = 0,       n = 3, rdisk = 10 rg	0.41	0.44	0.52	0.69	0.98	1.20	1.29	1.53	1.89	2.12	2.74	8.25
a = 0.998, n = 0, rdisk = 10 rg	0.39	0.41	0.49	0.65	0.92	1.14	1.23	1.47	1.85	2.10	2.77	8.79
a = 0.5,    n = 0, rdisk = 10 rg	0.42	0.44	0.53	0.69	0.97	1.19	1.27	1.50	1.85	2;.07	2.66	7.94
a = 0,       n = 0, rdisk = 10 rg	0.44	0.47	0.55	0.72	0.99	1.19	1.28	1.49	1.83	2.04	2.61	7.71
X-ray ball
a = 0.998, rball = 3 rg	0.22	0.22	0.25	0.29	0.37	0.42	0.44	0.47	0.51	0.52	0.55	0.58
a = 0.5,     rball = 3 rg	0.14	0.15	0.17	0.20	0.27	0.32	0.35	0.40	0.46	0.50	0.54	0.62
a = 0,         rball = 3 rg	0.10	0.10	0.11	0.13	0.17	0.20	0.22	0.25	0.31	0.34	0.40	0.57
a = 0.998, rball = 5 rg	0.45	0.46	0.49	0.53	0.59	0.62	0.64	0.66	0.69	0.70	0.71	0.74
a = 0.5,     rball = 5 rg	0.42	0.43	0.45	0.50	0.56	0.60	0.62	0.65	0.68	0.69	0.71	0.74
a = 0,         rball = 5 rg	0.39	0.40	0.43	0.48	0.55	0.60	0.62	0.65	0.69	0.70	0.73	0.76
a = 0.998, rball = 10 rg	0.72	0.73	0.73	0.75	0.78	0.80	0.81	0.82	0.84	0.84	0.85	0.86
a = 0.5,     rball = 10 rg	0.71	0.72	0.72	0.74	0.77	0.79	0.80	0.82	0.83	0.84	0.85	0.86
a = 0,         rball = 10 rg	0.71	0.71	0.72	0.74	0.77	0.79	0.80	0.82	0.83	0.84	0.85	0.87
High X-ray disk
a = 0.998, zdisk = 3 rg	0.50	0.52	0.56	0.64	0.78	0.90	0.96	1.11	1.40	1.61	2.22	8.67
a = 0.5,      zdisk = 3 rg	0.49	0.50	0.55	0.63	0.77	0.89	0.94	1.10	1.40	1.61	2.24	8.84
a = 0.,        zdisk = 3 rg	0.47	0.49	0.53	0.61	0.76	0.88	0.94	1.10	1.40	1.62	2.27	9.09
a = 0.998, zdisk = 5 rg	0.58	0.59	0.62	0.68	0.79	0.89	0.94	1.07	1.34	1.54	2.13	8.35
a = 0.5,      zdisk = 5 rg	0.57	0.58	0.61	0.67	0.78	0.88	0.93	1.07	1.34	1.54	2.13	8.44
a = 0.,        zdisk = 5 rg	0.57	0.58	0.61	0.67	0.77	0.87	0.92	1.06	1.34	1.54	2.15	8.55
a = 0.998,  zdisk = 10 rg	0.71	0.71	0.73	0.76	0.84	0.91	0.95	1.06	1.27	1.44	1.93	7.14
a = 0.5,      zdisk = 10 rg	0.70	0.71	0.73	0.76	0.83	0.91	0.94	1.05	1.27	1.43	1.92	7.15
a = 0.,        zdisk = 10 rg	0.70	0.71	0.72	0.76	0.83	0.90	0.94	1.05	1.27	1.43	1.93	7.17

Notes. ^aTo obtain the inclination-dependent flux profile (see Figures 5, 9, 11, and 13), the geometrical projection effect should be included, i.e., a cos θ factor for optical emission and the two X-ray disk models, or a (1 + cos θ)/2 factor for the X-ray ball model. ^b$M_{\rm O}=0.2+0.8M_{<200\,r_g}$, where $M_{<200\,r_g}$ is the strong lensing magnification of the optical flux within r_disk = 200 r_g.

Download table as:  ASCIITypeset image

5.1. Low X-Ray Disk Model

We first consider a simple model where X-rays are emitted by a tenuous disk immediately above the quasar accretion disk and this X-ray disk moves with the same Keplerian flow as the optical emission disk. This is partially motivated by the so-called sandwich corona model (Haardt & Maraschi 1991, 1993; Ruszkowski & Fabian 2000). We choose the outer and inner cutoff at r_disk = 10 r_g and r_in = r_ISCO, respectively. For the intensity profile, we use Equation (18), with n = 3 (i.e., I_ν∝r⁻³, following the optical profile) or n = 0 (no radial dependence). For the case of an extreme Kerr black hole (a = 0.998 M), we also tried the case where r_disk = 5 r_g, which is not realistic for a = 0 or 0.5 M, considering the inner cutoff r_ISCO we choose. We show the lensing-distorted and redshifted images of such an X-ray disk in Figure 6 for observers at θ = 15°, 45°, and 75° (from nearly face-on to nearly edge-on), and three spins, a = 0, 0.5, and 0.998 M. All images in Figure 6 are distorted as a result of the lensing effect. Disks viewed at large inclination angle show stronger image distortion, and black holes with larger spins produce larger redshifts or blueshifts. Since the optical accretion disk (Section 4) is moving with the same Keplerian flow, the images for an optical disk are very similar, except that the image sizes are larger.

Zoom In Zoom Out Reset image size

In Figure 7, we show the amplification of the X-ray image area and its ratio with respect to the optical image area for each case. The curves in the two panels resemble each other at θ ≲ 75° since the optical area magnifications, M_O(θ), are nearly flat and very close to 1 in this range of inclination angle (Figure 5). At larger inclination angles, the optical disk area is also significantly magnified (Figure 5), and thus the corresponding curves in the two panels deviate from each other. The X-ray area amplification increases with inclination angle because more photons pass near the black hole at larger inclination angles, which are more strongly affected by lensing. The area amplification ratios also increase with inclination angle because the X-ray emission region is much smaller, resulting in stronger lensing.

Zoom In Zoom Out Reset image size

In Figure 8, we show the magnification of the observed flux F_ν as a function of inclination angle. For small inclination angles, the Doppler effect is unimportant, and the gravitational redshift reduces the observed specific flux through the invariance of I_ν/ν³. For large inclination angles, the amplification of image area increases rapidly (with θ) because of strong lensing (Figure 7), and Doppler blueshifted regions (approaching the observer) compete with Doppler redshifted regions (receding from the observer). The lensed specific flux begins to be greater than the unlensed flux, i.e., M_X > 1 at θ ≳ 70°. The model clearly predicts an inclination angle dependence of the magnification ratio R_X/O. Furthermore, the dependence of R_X/O on inclination angle is stronger for smaller X-ray disks, larger spins (smaller inner cutoff r_ISCO), or steeper radial profiles (more weight on the central part close to the black hole). If the sizes of the X-ray and optical emission regions constrained from microlensing observations are correct, then even the weakest inclination angle dependence among our models, i.e., the case in which a = 0, r_disk = 10 r_g, and flat radial profile, gives an increase of a factor of two in R_X/O when θ increases from 15° to 75° (see lower right panel of Figure 8). On the other hand, for the case a = 0.998 M, r_disk = 10 r_g, and I_ν∝r⁻³, R_X/O increases by a factor of 10 when θ increases from 15° to 75°. For the nearly edge-on case, R_X/O can increase by a further factor of 10 in extreme cases with small X-ray emission sizes and steep profiles.

Zoom In Zoom Out Reset image size

We plot the observed monochromatic flux modulated by strong lensing and geometrical projection as a function of inclination angle θ in Figure 9. For the case of an extreme Kerr black hole (a = 0.998 M) with steep radial profile, the strong lensing effect overwhelms geometrical projection, and consequently, the observed flux F_ν increases with θ. A comparison between the top and bottom panel of Figure 9 shows that the difference between different spins is more significant for the steeper radial profile case (i.e., n = 3).

Zoom In Zoom Out Reset image size

5.2. An X-Ray Ball Model

As the second geometrical model for the X-ray emission, we assume that the X-ray corona is a ball around the central black hole. Such a corona is often used in the literature (e.g., Schnittman & Krolik 2010) to study the X-ray polarization of AGNs. We furthermore assume that the accretion disk is opaque, and integrate over the half sphere above the accretion disk to obtain the observed X-ray flux. We test this X-ray ball model for three radii: r_ball = 3 r_g, 5 r_g, and 10 r_g. For the 4-velocity of the X-ray sphere, we use Equations (9) and (11) (Ruszkowski & Fabian 2000). We do not consider the radial motion of the X-ray source, and the X-ray sphere is differentially rotating around the polar axis. We show Ω(θ) for different r_ball and spins in Figure 3. Smaller r_ball gives larger Ω. For small to moderate inclination angles, Ω is dominated by ω, i.e., the frame dragging, and therefore a smaller a gives a smaller Ω, whereas for a large θ (near the accretion disk), Ω is dominated by the Keplerian flow Ω_K, and a smaller a gives a larger Ω for a prograde flow, see Equation (10) and Figure 2.

We show the lensing-distorted and redshifted image of the (half) X-ray ball above the equatorial plane in Figure 10 for an observer at θ = 75°, for three ball sizes, r_ball = 3 r_g, 5 r_g, and 10 r_g, and three spins, a = 0, 0.5 M, and 0.998 M. The images are taken in such a way that the fractional area of the image of the (half) X-ray sphere with respect to the area of the image would have been the same for each frame, were there no gravity. The area amplification is most significant for the r_ball = 3 r_g case (a factor ∼2.3 for θ = 75°) since it is closest to the black hole, and is therefore most influenced by the lensing effect, and least significant for r_ball = 10 r_g (about 20%). For the two a ≠ 0 cases, we observe that the image shape (not the color) is not mirror symmetric with respect to the polar axis; this is most obvious in the r_ball = 3 r_g case, because of frame dragging. The image is stretched more on the right-hand side (the receding side). Both the gravitational redshift and the Doppler red/blueshift effects are more significant for smaller r_ball (see also Figure 3). For example, for the moderate spin case (a = 0.5), the redshift factor g ≡ ν_o/ν_e lies between 0.63 and 1.26 for r_ball = 10 r_g and between 0.22 and 1.39 for r_ball = 3 r_g.

Zoom In Zoom Out Reset image size

For a distant observer (r_obs ≫ 1) at an inclination angle θ, the projected image area of a half sphere orthogonal to the line of sight is πr²(1 + cos θ)/2 when the spacetime is flat, whereas the lensed image area in Kerr spacetime has to be integrated numerically. Using Equation (19) the monochromatic flux amplification is found to be

$$\begin{equation} M_{\rm X} =\frac{\int {g^{\Gamma +2}d\Omega _{\rm o}}}{\int {d\Omega ^{\prime }_{\rm o}}}. \end{equation} \tag{ 26 }$$

(the 1/rⁿ factor drops out since it is a constant for the ball model, and therefore the results do not depend on the radial profile). We plot M_X and the lensing magnification ratio R^(lens)_X/O in Figure 11 (first row). As mentioned earlier, the amplification in the image area is largest for the r_ball = 3 r_g case (Figure 10). However, since the redshift effect dominates over the area amplification, the r_ball = 3 r_g case shows the smallest amplification in the observed flux (Figure 11, top left panel). As for the inclination angle dependence of flux magnification ratio R^(lens)_X/O, we find that the magnification ratio changes by less than 5% (15° ≲ θ ≲ 75°) for all three spins when r_ball = 10 r_g. For smaller r_ball, we find that the magnification ratio R^(lens)_X/O increases by ∼20%–40% for different spins when r_ball = 5 r_g and by ∼80%–130% for r_ball = 3 r_g with different spins. For large inclination angles, the magnification ratio R^(lens)_X/O decreases with θ, which is different from the X-ray disk model in the previous subsection (see Figures 7 and 8), and is mainly caused by the fact that in the case of no lensing the projected area of the thin optical disk approaches zero when θ approaches π/2 (the amplification of image area therefore approaches infinity) whereas that of the (half) X-ray ball approaches πr²/2 (the area magnification is therefore finite).

Zoom In Zoom Out Reset image size

Since the geometry of the X-ray ball model is different from that of the optical emission (a thin disk), the projection ratio R^(proj)_X/O = (1 + cos θ)/2cos θ is not a constant. Consequently both Kerr strong lensing and the projection effect contribute to the inclination angle dependence of the observed X-ray-to-optical flux ratio, and R_X/O ≠ R^(lens)_X/O (see Equation (25)). We plot R_X/O in Figure 11 (bottom right panel). R_X/O can increase by a factor of ∼10 for large inclination angle (θ ≳ 70°) where the major contribution comes from the geometrical projection instead of general relativity. We also show the observed flux as a function of θ in Figure 11 (bottom left panel). Similar to the low X-ray disk model, F_X can increase with θ for small ball radii.

5.3. High X-Ray Disk Model

As the last example of an X-ray emission model, we consider a thin X-ray disk of radius 10 r_g which is above the black hole (parallel to the equatorial plane) with height z = 3 r_g, 5 r_g, and 10 r_g, respectively. This is motivated by the light-bending model of Fabian & Vaughan (2003) where the authors tried to explain the rapid variability in the continuum observed for MCG6−30−15 using the fact that gravitational light bending by the strong field of the central black hole amplifies small changes in the X-ray source height (see also Miniutti et al. 2003; Miniutti & Fabian 2004). Intuitively, a larger z means greater distance of the X-ray disk from the black hole, and therefore less influence from the strong gravitational field, and this might give smaller inclination angle dependence in R_X/O when θ is not too large. We plot the angular velocity profile in Figure 4; the disk with the smallest height (closest to the black hole) has the largest velocity (for the same ρ, distance to the polar axis, and spin parameter a).

We show images of the lensed disk above the black hole as seen by an observer at θ = 75° in Figure 12 for three heights and three spins. The area amplification is most significant for the height z = 3 r_g case (a factor of ∼1.9 for all three spins), and least significant for the z = 10 r_g case (a factor about 1.5 for all three spins). The redshift effect (gravitational and Doppler) is also most significant for the z = 3 r_g case (see the first column of Figure 12). For example, for moderate spin a = 0.5, the g factor lies between 0.84 and 1.02 for z_disk = 10 r_g case, and between 0.58 and 1.16 for z_disk = 3 r_g case. Since points in the interior of the disk are closer to the black hole, they are more influenced by gravitational redshift. Since the Ω(ρ) profile (Ruszkowski & Fabian 2000) is not monotonic for this model (note that ρ increases with θ for fixed z_disk), the points on the disk with most significant Doppler blue/redshift can lie in the interior of the disk for small z_disk. For example, when z_disk = 3 r_g, the largest redshift occurs in the interior of the disk instead of on the boundary (Figure 12, first column).

Zoom In Zoom Out Reset image size

We choose no cutoff at the center of the disk, and use the radius-independent intensity profile (since the smallest disk height we used is 3 r_g, the dependence of the results on the radial profile should be much weaker than the low X-ray disk model in Section 5.1 where the case most sensitive to the radial profile is when a = 0.998 and r_ISCO = 1.24 r_g). This gives less weight to the central part of the disk which is closest to the black hole compared with the n = 3 (I_ν∝r⁻³) case, and will yield more conservative constraints. The projection effect (i.e., the cos θ factor) is the same for X-ray and optical emission, consequently R_X/O = R^(lens)_X/O. We show the magnification ratio R_X/O in Figure 13. Since the black hole spin contributes to the spacetime curvature only in higher orders, and the smallest X-ray disk height z we test is 3 r_g, we found that the results for different spins do not differ significantly. For 15° ≲ θ ≲ 75°, the magnification ratio R_X/O (flux) increases by about 85%, 55%, and 27% for z = 3 r_g, 5 r_g, and 10 r_g, respectively. Therefore, a thin X-ray disk of radius 10 r_g with the Ruszkowski & Fabian (2000) velocity profile is consistent with observed α_ox for normal quasars even when it is above the central black hole with a height as low as 3 r_g (R_X/O ∼ 2 corresponds to Δα_ox ∼ 0.1, well within the observed scatter of α_ox, see the next section). For high inclination angles (corresponding to BAL quasars), the magnification ratio R_X/O in observed monochromatic flux is larger than one (say, a factor of two; see Figure 13) which is similar to the low X-ray disk model (although less significant). This implies that the absorption in the X-ray band might be underestimated.

Zoom In Zoom Out Reset image size

In the so-called light-bending model (Miniutti et al. 2003; Fabian & Vaughan 2003; and Miniutti & Fabian 2004), the authors used the variation of the X-ray source height in a strong gravitational field as one interpretation for the observed large variability of the X-ray power-law component compared to that of the reflection component (in particular the fluorescent iron line emission). Choosing a ring-like source with distance ρ_s = 2 r_g from the z-axis and assuming that the source is co-rotating with the accretion disk with the same orbital velocity as the underlying disk, Miniutti & Fabian (2004) found that the observed flux from the (direct) power-law component can increase by a factor of ∼20 when the ring height z increases from 1 r_g to 20 r_g, or a factor of ∼5 when z increases from 3 r_g to 10 r_g (for inclination angle θ = 30°). For our disk model with r_disk = 10 r_g and Ruszkowski & Fabian (2000) velocity profile, we found that if such an X-ray disk is observed with an inclination angle θ = 30°, the observed X-ray flux can increase by a factor of ∼40% when the disk height changes from 3 r_g to 10 r_g (see Figure 13). Our results are qualitatively consistent with those of Miniutti & Fabian (2004). The quantitative difference, i.e., our models show much smaller variation in observed flux of the power-law component, is not surprising, since we have assumed a very different geometry for the source emission (a disk of size 10 r_g versus a ring with radius 2 r_g).

An important parameter characterizing the flux ratio between X-ray and optical emission is α_ox (Tananbaum et al. 1979)

$$\begin{equation} \alpha _{\rm ox}\equiv 0.3838 \log \frac{F_{\rm 2\, keV}}{F_{\rm 2500\, {\mathring{\rm A}}}}. \end{equation} \tag{ 27 }$$

Some authors use 3000 Å for the optical emission with the constant as 0.372, and this does not change the conclusion of this paper. There is evidence showing that the observed α_ox in quasars are strongly anti-correlated with quasar luminosities in both the optical and X-ray bands; however, no significant redshift evolution has been found (Strateva et al. 2005; Steffen et al. 2006; Lusso et al. 2010). The intrinsic scatter of α_ox is about 0.2–0.3 at a given luminosity (Steffen et al. 2006; Lusso et al. 2010). There are several factors contributing to this scatter, including quasar variability and intrinsic differences between quasars.

In this paper, we have investigated variations in R_X/O due to lensing and projection effects which are functions of the inclination angle θ (see Figures 8, 11, and 13), and therefore, the orientation effect also contributes to the distribution of α_ox. To test the significance of strong lensing effects on the observed X-ray-to-optical flux ratios, we separate the contributions to α_ox as

$$\begin{equation} \alpha _{\rm ox}= \alpha ^{\rm intr}_{\rm ox}+0.3838\log R_{\rm X/O}, \end{equation} \tag{ 28 }$$

where R_X/O includes the orientation-induced contribution and α^intr_ox represents other contributions such as spectral variability and intrinsic differences between quasars. We assume the intrinsic distribution of α^intr_ox to be Gaussian with mean value μ = −1.5 and scatter σ = 0.2. Assuming a uniform distribution for α^intr_ox, e.g., between (− 2, − 1), does not change our conclusions. The distributions of R_X/O are calculated for the three X-ray models we have considered, assuming observers are randomly distributed in all solid angles, i.e., a uniform distribution in the variable μ ≡ cos θ (0 < μ < 1). We use Monte Carlo simulations to combine the two distributions to obtain the distributions of α_ox using 500,000 realizations.

We assume that differences between normal quasars, BAL quasars, and obscured quasars are purely observational effects caused by different inclination angles. We furthermore assume that obscured quasars make up about 20% of the total population (i.e., Type I + Type II quasars; Hasinger 2008) since we are focusing on the high-luminosity regime, and BAL quasars account for 40% of Type I quasars (Dai et al. 2008; Shankar et al. 2008). Although the 20% Type II fraction might be lower than some other measurements, here we use this as a fiducial number to test the predictions of our model. In addition, some BAL quasars such as LoBAL and FeLoBAL quasars may have Compton thick absorptions, which can be classified as Type II quasars. The fraction of these objects is non-negligible, 4%–7% (Dai et al. 2012), which can compensate for our lower assumed Type II fraction. Consequently, we assume 0° < θ^Normal ⩽ 587 < θ^BAL ⩽ 785. We do not calculate the observed α_ox distributions for Type II quasars because their α_ox is hard to study due to the heavy Compton-thick absorption. This also avoids very large inclination angles where the thin disk approximation assumed for optical emission and the two X-ray disk models can be inaccurate (we have assumed zero thickness for the thin disk models; however, the finite thickness of the disks must be considered for nearly edge-on cases when computing unlensed quantities such as image area). The inclination dependence of R_X/O is the same as the strong lensing R^(lens)_X/O for the low and high X-ray disk models, because the projection factor cos θ is canceled out considering both the optical and X-ray emission. For the X-ray ball model, the projection effect, R^(proj)_X/O = (1  +  cos θ)/2cos θ, is not canceled out, and both lensing and projection effects contribute to the inclination-dependent flux ratios.

The results for the three X-ray models are shown in Table 2 and Figures 14–16. For each X-ray model, we generate the distribution of the observed α_ox, and compute its mean, standard deviation, and skewness, i.e., 〈α_ox〉, $\sigma _{\alpha _{\rm ox}},$ and $\cal S,$ respectively, through Monte Carlo simulation. The extra scatter added to α_ox by the Kerr strong lensing (also by different geometries for the X-ray ball model) is insignificant. We have assumed α^intr_ox to be Gaussian with σ = 0.2 whereas the largest σ in the observed α_ox is less than 0.24 (corresponding to σ^(lens) ⩽ 0.13) among all models considered in Table 2. Because the scatter of α_ox is constrained in normal quasars (non-BAL quasars, 0° < θ^Normal ⩽ 587 in our model), the differential lensing amplification between optical and X-ray emission is small at these small inclination angles. Therefore, to match the observed scatter in α_ox, 0.2–0.3, we have to assume that most of the scatter comes from other factors not related to the orientation effect, such as spectral variability or intrinsic differences between quasars. The skewness of the simulated α_ox is also small with the largest case about 0.03. This is not surprising since the distribution of α_ox is the convolution of α^intr_ox with zero skewness and R_X/O, where we assumed a larger scatter in α^intr_ox. If the measured intrinsic scatter of α_ox in normal quasars were smaller, the skewness of the distribution would be a powerful tool to constrain the corona geometry. Given the large measured scatter in non-BAL quasars, which requires a larger α^intr_ox, it is probably not realistic to try to discriminate corona geometries based on the extra scatter or skewness of the lensed distribution of α_ox.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 2. α_ox Distributions after Kerr Strong Lensing

Modela	Normal QSOb			BALQSOb			Normal+BAL QSOb			Δ〈αox〉c
	〈αox〉c	σ	$\cal S$	〈αox〉	σ	$\cal S$	〈αox〉	σ	$\cal S$
Low X-ray disk
a = 0.998, n = 3	−1.76	0.211	0.003	−1.55	0.208	0.004	−1.67	0.234	0.03	0.21
a = 0.5,    n = 3	−1.60	0.205	0.001	−1.47	0.202	0.000	−1.55	0.214	0.002	0.13
a = 0,       n = 3	−1.58	0.204	0.001	−1.47	0.202	0.000	−1.53	0.210	0.001	0.11
a = 0.998, n = 0	−1.59	0.204	0.001	−1.48	0.202	0.001	−1.54	0.210	0.002	0.11
a = 0.5,    n = 0	−1.57	0.203	0.001	−1.47	0.201	0.000	−1.53	0.209	0.001	0.10
a = 0,       n = 0	−1.57	0.203	0.001	−1.47	0.201	0.000	−1.53	0.208	0.001	0.10
X-ray balld
a = 0.998, rball = 3 rg	−1.68	0.204	0.000	−1.54	0.205	0.003	−1.63	0.216	0.014	0.14
a = 0.5,     rball = 3 rg	−1.75	0.205	0.000	−1.58	0.207	0.006	−1.68	0.222	0.029	0.17
a = 0,         rball = 3 rg	−1.82	0.204	0.001	−1.66	0.208	0.007	−1.75	0.221	0.035	0.16
a = 0.998, rball = 5 rg	−1.58	0.202	0.001	−1.47	0.203	0.002	−1.54	0.209	0.011	0.11
a = 0.5,     rball = 5 rg	−1.59	0.202	0.001	−1.48	0.204	0.002	−1.55	0.210	0.013	0.11
a = 0,         rball = 5 rg	−1.60	0.203	0.001	−1.48	0.204	0.002	−1.55	0.212	0.015	0.12
High X-ray disk
a = 0.998, z = 3 rg	−1.58	0.201	0.000	−1.52	0.201	0.000	−1.55	0.203	0.001	0.06
a = 0.998, z = 5 rg	−1.57	0.201	0.000	−1.52	0.201	0.000	−1.55	0.202	0.001	0.05
a = 0.998, z = 10 rg	−1.54	0.200	0.000	−1.52	0.200	0.000	−1.53	0.201	0.000	0.02

Notes. ^aFor the X-ray ball model, we test only for case r_ball = 3 r_g, and 5 r_g considering the relative flatness of R_X/O(θ) for r_ball = 10 r_g (Figure 11). For the high X-ray disk model, we consider only the a = 0.998 case because of the similarity of the R_X/O(θ) curves between different spins (Figure 13). ^bWe assume that obscured (Type II) quasars are about 20% of quasar population and BAL quasars count for 40% of Type I quasars. The intrinsic distribution of α_ox is assumed to be Gaussian with μ = −1.5 and σ = 0.2. 〈α_ox〉, σ, and $\cal S$ are respectively the mean, standard deviation, and skewness (μ₃/σ³) of α_ox distribution. ^cΔ〈α_ox〉 ≡ 〈α_ox〉^BAL − 〈α_ox〉^Normal, the difference in mean of α_ox between BALQSOs and normal quasars. ^dThere exists extra scatter of α_ox (besides strong gravity) due to projection effect of different geometries of X-ray (half sphere) and optical emission (thin disk). We have included this effect into our Monte Carlo simulation.

Download table as:  ASCIITypeset image

Nevertheless, we find that the mean of α_ox, 〈α_ox〉, can be shifted significantly from the intrinsic values. For example, for the low X-ray disk model with the extreme spin and the steep radial profile, 〈α_ox〉^Normal is smaller than 〈α_ox〉^intrinsic (equal to −1.5) by ∼0.25. The difference in α_ox between normal and BAL quasars (see the last column of Table 2) can be as large as ∼0.2. For the low X-ray disk and the X-ray ball model, the difference in 〈α_ox〉 between normal and BAL quasars is ⩾0.1. For the high X-ray disk model, the splitting of 〈α_ox〉 between BAL quasars and non-BAL quasars is ⩽0.06. Therefore, comparing large samples of observed, absorption corrected α_ox between BAL quasars and non-BAL quasars, it may be possible to differentiate corona geometry using the orientation effects. Here, the absorptions in BAL quasars should be determined from spectral properties without assuming the same SEDs for BAL quasars and non-BAL quasars.

A number of studies show that BAL quasars and normal quasars have the same intrinsic spectral energy distribution, if the observed soft X-ray weakness of BAL quasars is caused by intrinsic absorption with column densities ∼10²²–10²⁴ cm⁻² (e.g., Green et al. 2001; Gallagher et al. 2002). For all three X-ray models considered in this paper, we found 〈α_ox〉^BAL > 〈α_ox〉^normal (Table 2). This implies that on average X-ray emission is boosted more for BAL quasars than for normal quasars. Therefore, the aforementioned high column densities might still be an underestimate of the intrinsic X-ray absorption of BAL quasars. For example, let us consider a BAL quasar at z = 2 with $\hbox{${N}_{\rm H}$}=2\times 10^{23}\ \hbox{cm$^{-2}$}$ determined to correct the α_ox of this BAL quasar to be the same as non-BAL quasars. If R^BAL_X/O/R^normal_X/O = 2 between a BAL quasar and a normal quasar based on our model calculations, corresponding to Δα_ox ∼ 0.11 and a typical value based on Table 2, then an absorption of column density ∼10²⁴ cm⁻² is needed for the BAL after including the strong lensing boost of the X-ray emission for BAL quasars. Even the smallest shift in 〈α_ox〉 between BAL and normal quasars, i.e., 0.02 (corresponds to R_X/O  ∼ 1.13, see Table 2), can cause an underestimate of the intrinsic absorption by $\Delta \hbox{${N}_{\rm H}$}= 2\times 10^{23}\ \hbox{cm$^{-2}$}$. It is therefore important to consider the effects of the Kerr strong gravity in order to accurately measure the X-ray absorptions in BAL quasars.

We have developed a ray-tracing code for the Kerr metric based on numerical integrations of the first-order Hamilton equations for massless free particles in curved spacetime using an adaptive fifth-order Runge Kutta algorithm. This is the first step toward developing a fully general relativistic three-dimensional radiation transfer code for the Kerr spacetime with arbitrary flows using characteristic methods and operator perturbation (Baron et al. 2009b). We derive the formalism for studying the strong lensing of the optical and X-ray emission by Kerr black holes. We find that the redshift effect caused by strong gravity plays an important role in computing the observed flux. We do not consider the cosmological redshift, but to include the cosmic redshift is trivial: in Equations (17) and (19), replace the strong lensing redshift factor g by g/(1 + z_s), where z_s is the cosmological redshift of the quasar.⁸

Using the sizes of the optical and X-ray emission regions measured by recent quasar microlensing observations (Kochanek 2004; Pooley et al. 2006; Morgan et al. 2008; Eigenbrod et al. 2008; Chartas et al. 2009; Dai et al. 2010; Blackburne et al. 2011, 2012), we investigate the effects of the strong gravity field of the central Kerr black hole on the optical and X-ray continuum emission. In particular, we calculate the correction to the X-ray-to-optical flux ratio caused by differential lensing distortions for the X-ray and optical emission. We assume the standard thin disk model (Shakura & Sunyaev 1973) for optical emission and find that the effect of the strong gravity is not very significant on the optical emission, since its emission size, ∼100 r_g, is about 10 times larger than the X-ray emission (Figure 5). The effect of strong gravity on optical emission is important only for large inclination angles. We assume a simple power law and test three simple geometries for the X-ray emission: an X-ray disk immediately above the accretion disk with inner cutoff at r_ISCO moving with the Keplerian flow, an X-ray ball surrounding the central black hole moving with Ruszkowski & Fabian (2000) velocity flow, and an X-ray disk above the black hole with nonzero height moving with Ruszkowski & Fabian (2000) velocity flow. We find that the redshift effect is more important for X-ray emission than for optical emission (see, e.g., Equations (17) and (19), Figures 5 and 11). Unlike optical emission, the effect of Kerr strong lensing on the observed X-ray flux can be important even for moderate inclination angles (i.e.,  for normal quasars). Among the three X-ray models tested in this paper, the low X-ray disk model shows the largest variation in the X-ray-to-optical magnification ratio R_X/O (as large as a factor of 10 for normal quasars, and another factor of 10 for BAL+obscured quasars). Scatter by a factor of 10 (corresponds to Δα_ox = 0.384) is still marginally consistent with current observations of α_ox (see Equation (3) of Steffen et al. 2006) for normal quasars. The strong inclination angle dependence makes R_X/O a random variable with nontrivial distribution, and the observed distribution of α_ox is therefore modulated by that of R_X/O. In particular, α_ox for a BAL quasar can be greater than that of a normal quasar by ∼0.1–0.2 (see Table 2). Intrinsic absorption with column density ∼10²⁴ cm⁻² is needed in order to balance this boost of X-ray emission by Kerr strong lensing. Comparing with current estimates of the BAL absorption column density (∼10²²–10²⁴ cm⁻²; e.g., Green et al. 2001; Gallagher et al. 2002), we find that it is important to include the Kerr strong gravity in order to correctly quantify the BAL absorption.

We note that for normal quasars, since the strong lensing distortion in the optical emission is not significant given the larger sizes of the optical emission region (of order 10² r_g), the current microlensing results need only minor modifications in order to incorporate the effect of strong gravity. As can be seen from Figure 5, the strong lensing effect is significant for optical emission only when the inclination angle θ is high (observed nearly edge-on). Abolmasov & Shakura (2012) studied three high-amplification events from optical observations of SBS J1520+530, and Q 2237+0305 using straight-line caustic crossing, and found that a model incorporating the strong lensing of the central black holes improves the fit; however, it requires high inclination angles (θ > 70°) not typical for normal quasars. This is qualitatively consistent with our results (see Section 4).

For X-ray emission, if the size of the X-ray emitting region is of order 10 r_g (or even smaller, given the fact that most current microlensing observations give no lower limit on the X-ray emission sizes) and is centered around the central black hole, then the X-ray emission will be strongly lensed by the central black hole before it is microlensed by stars in a lens galaxy much farther away. Therefore, it might be important to add a strong lensing piece to the microlensing code which in principle should give more accurate constraints on the source sizes, in particular, the X-ray emission sizes. Furthermore, since Kerr strong lensing depends on important parameters such as the spin of the black hole, the inclination angle of the accretion disk, and the geometry of the corona, it is possible to extract this valuable information by combining the Kerr and microlensing models. A first attempt along this direction will be presented in a companion paper (B. Chen et al. 2012a).

Of course, models such as a razor-thin X-ray disk above the accretion disk with an isotropic intensity profile may be too simplistic. More complicated and physically solid models should be tested in the future. But these simple models still show the importance of studying the effect of the strong gravity if the X-ray source is indeed very small and very close to the central black hole. Although we have used the innermost stable circular orbit as a natural cutoff for the optical emission and the first X-ray model, it is possible to extend the emission region to within r_ISCO with the gas following the so-called plunging trajectories along geodesics (Krolik 1999; Agol & Krolik 2000) and strong gravity plays an even more important role—the effects shown in this paper might be more significant. A radiation transfer code based on the ray-tracing code developed in this paper is being developed by the PHOENIX group and will be published in the future.⁹

We thank Ronald Kantowski and Junxian Wang for helpful discussions. We also thank the anonymous referee for the careful review of this work. B.C. and X.D. acknowledge support for this work provided by the National Aeronautics and Space Administration through Chandra Award Number GO0-11121B, GO1-12139B, GO2-13132A issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. B.C. and X.D. acknowledge support for program number HST-GO-11732.07-A provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy. B.C. and E.B. acknowledge NSF AST-0707704, and US DOE Grant DE-FG02-07ER41517 and Support for Program number HST-GO-12298.05-A provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.

We give the equations of motion (six first-order differential equations) used in our backward ray-tracing code. Inserting Equation (3) into Equation (4), the equations of motion follow after straightforward but tedious algebra. For the four configuration-space variables (t, r, θ, ϕ), we have

$$\begin{eqnarray} \frac{dt}{d\lambda }&=&{-}\frac{p_t}{\alpha ^2}- \frac{2Mar}{\rho ^2\Delta }p_\phi \cr \frac{dr}{d\lambda }&=&\frac{\Delta }{\rho ^2}p_r\cr \frac{d\theta }{d\lambda }&=&\frac{p_\theta }{\rho ^2}\cr \frac{d\phi }{d\lambda }&=&\left(\frac{\Delta -a^2\sin ^2\theta }{\rho ^2\Delta \sin ^2\theta }\right)p_\phi -\frac{2Mar}{\rho ^2\Delta }p_t, \end{eqnarray} \tag{ A1 }$$

and for the two momentum-space variables p_r, and p_θ (p_t and p_ϕ are constants of the motion),

$$\begin{eqnarray} \frac{dp_r}{d\lambda }&=&\frac{1}{2}\frac{\partial }{\partial r}\left(\frac{1}{\alpha ^2}\right)p_t^2+ \frac{\partial }{\partial r}\left(\frac{2Mar}{\rho ^2\Delta }\right)p_t p_\phi\nonumber\\\ms &&-\frac{1}{2}\frac{\partial }{\partial r}\left(\frac{\Delta }{\rho ^2}\right)p_r^2-\frac{1}{2}\frac{\partial }{\partial r}\left(\frac{1}{\rho ^2}\right)p_\theta ^2\nonumber\\\ms &&-\frac{1}{2}\frac{\partial }{\partial r}\left(\frac{\Delta -a^2\sin ^2\theta }{\rho ^2\Delta \sin ^2\theta } \right) p_\phi ^2, \end{eqnarray} \tag{ A2 }$$

and

$$\begin{eqnarray} \frac{dp_\theta }{d\lambda }&=&\frac{1}{2}\frac{\partial }{\partial \theta }\left(\frac{1}{\alpha ^2}\right)p_t^2+ \frac{\partial }{\partial \theta }\left(\frac{2Mar}{\rho ^2\Delta }\right)p_t p_\phi\nonumber\\\ms &&-\frac{1}{2}\frac{\partial }{\partial \theta }\left(\frac{\Delta }{\rho ^2}\right)p_r^2-\frac{1}{2}\frac{\partial }{\partial \theta }\left(\frac{1}{\rho ^2}\right)p_\theta ^2\nonumber\\\ms &&-\frac{1}{2}\frac{\partial }{\partial \theta }\left(\frac{\Delta -a^2\sin ^2\theta }{\rho ^2\Delta \sin ^2\theta } \right) p_\phi ^2, \end{eqnarray} \tag{ A3 }$$

where

$$\begin{eqnarray} \frac{\partial }{\partial r}\left(\frac{1}{\alpha ^2}\right)&=&\left(\frac{2M}{\Delta \rho ^2}\right)\left(\frac{a^4-r^4}{\Delta }-\frac{2r^2a^2\sin ^2\theta }{\rho ^2}\right)\nonumber\\\ms \frac{\partial }{\partial \theta }\left(\frac{1}{\alpha ^2}\right)&=&\frac{4M a^2 r(a^2+r^2)}{\Delta \rho ^4}\sin \theta \cos \theta , \end{eqnarray} \tag{ A4 }$$

$$\begin{eqnarray} \frac{\partial }{\partial r}\left(\frac{\Delta }{\rho ^2}\right)&=&\frac{2}{\rho ^2}\left(r-M-\frac{r\Delta }{\rho ^2}\right)\nonumber\\\ms \frac{\partial }{\partial \theta }\left(\frac{\Delta }{\rho ^2}\right)&=&\frac{2}{\rho ^4}a^2\Delta \sin \theta \cos \theta , \end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray} \frac{\partial }{\partial r}\left(\frac{1}{\rho ^2}\right)&=&{-}2\frac{r}{\rho ^4}\nonumber\\\ms \frac{\partial }{\partial \theta }\left(\frac{1}{\rho ^2}\right)&=&2\frac{a^2}{\rho ^4}\sin \theta \cos \theta , \end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray} \frac{\partial }{\partial r}\left(\frac{2Mar}{\rho ^2\Delta }\right)&=& {-}\frac{\partial g^{03}}{\partial r} =\frac{2Ma[\rho ^2(a^2-r^2)-2r^2\Delta]}{\rho ^4\Delta ^2}\nonumber\\\ms \frac{\partial }{\partial \theta }\left(\frac{2Mar}{\rho ^2\Delta }\right)&=& {-}\frac{\partial g^{03}}{\partial \theta } =\frac{4Mra^3\sin \theta \cos \theta }{\rho ^4\Delta }, \end{eqnarray} \tag{ A7 }$$

and

$$\begin{eqnarray} &&\frac{\partial }{\partial r}\left(\frac{\Delta -a^2\sin ^2\theta }{\rho ^2\Delta \sin ^2\theta } \right) = \frac{\partial g^{33}}{\partial r}\nonumber\\\ms &&\quad\quad=\frac{2a^2\sin ^2\theta [r\Delta +(r-M)\rho ^2]-2r\Delta ^2}{\rho ^4\Delta ^2\sin ^2\theta }\nonumber\\\ms &&\frac{\partial }{\partial \theta }\left(\frac{\Delta -a^2\sin ^2\theta }{\rho ^2\Delta \sin ^2\theta } \right) =\frac{\partial g^{33}}{\partial \theta }\nonumber\\\ms &&\quad\quad=\frac{-2\cos \theta [\rho ^2(\rho ^2-2Mr)+2Mra^2\sin ^2\theta ]}{\rho ^4\Delta \sin ^3\theta }.\quad\quad \end{eqnarray} \tag{ A8 }$$