X-Ray Line Profile Variations during Quasar Microlensing

For illustration we use a simple model of the X-ray-emitting region of the quasar, consisting of a thin accretion disk in the equatorial plane of a Kerr black hole. We concentrate on the emission in the spectral vicinity of a single emission line, exemplified here by the typically most prominent iron Kα line. At a given point of the optically thick disk we describe the local rest-frame emission by a narrow line superimposed on a power-law continuum. We model the specific photon intensity of the continuum in the rest frame of an emitting point by

$$\begin{eqnarray}&&{I}_{\mathrm{em}}^{\mathrm{cont}}({E}_{\mathrm{em}};r)={I}_{0}^{\mathrm{cont}}\,{r}^{-q}\,{E}_{\mathrm{em}}^{-{\rm{\Gamma }}},\end{eqnarray} \tag{ 1 }$$

where E_em is the photon energy in the emission-point rest frame, Γ is the spectral power-law index, r is the radial Boyer–Lindquist coordinate of the emitting point, and ${I}_{0}^{\mathrm{cont}}$ is a constant. We assume a power-law decline of the emissivity with radius, parameterized by a radial decline index q (assumed energy independent).

The line width in the emission-point rest frame is negligible in comparison with the disk-integrated line width, which is dominated by gravitational redshift and the relativistic Doppler effect. Hence, we can safely model the iron line specific photon intensity in the emission-point rest frame by a delta-function profile,

$$\begin{eqnarray}&&{I}_{\mathrm{em}}^{\mathrm{Fe}}({E}_{\mathrm{em}};r)={I}_{0}^{\mathrm{Fe}}\,{r}^{-q}\,\delta [{E}_{\mathrm{em}}-{E}_{\mathrm{Fe}}],\end{eqnarray} \tag{ 2 }$$

where the rest-frame energy of the iron Kα line E_Fe = 6.4 keV and ${I}_{0}^{\mathrm{Fe}}$ is a constant.³ For simplicity we assume the same radial decline index q as for the continuum.

Next, we need to transform the emission-point specific intensities I_em(E_em; r) given by Equations (1) and (2) to the rest frame of an asymptotically distant observer viewing the disk under an inclination angle i. In order to connect a point (α, β) in the plane of the observer's sky to a corresponding emission point on the disk, we integrate backward along photon geodesics in the Kerr metric following Dovčiak et al. (2004). In this way we obtain numerically the Boyer–Lindquist emission radius r(α, β). The found emission point and its orbital velocity define the total energy shift (Doppler and gravitational) for a photon following the geodesic from the disk to the observer. The shift is described by the dimensionless g-factor,

$$\begin{eqnarray}&&g(\alpha ,\beta )=E/{E}_{\mathrm{em}},\end{eqnarray} \tag{ 3 }$$

where E is the photon energy in the rest frame of the observer. Note that r(α, β) and g(α, β) depend also on the disk inclination i and the black hole spin a. The specific photon intensity in the observer frame is then given by

$$\begin{eqnarray}&&{I}_{\mathrm{obs}}(E;\alpha ,\beta )={g}^{2}(\alpha ,\beta ){I}_{\mathrm{em}}[E/g(\alpha ,\beta );r(\alpha ,\beta )],\end{eqnarray} \tag{ 4 }$$

where we utilized the invariance ${I}_{\mathrm{obs}}/{E}^{2}={I}_{\mathrm{em}}/{E}_{\mathrm{em}}^{2}$ and Equation (3).

Observer-frame emission maps for the continuum and the line can be obtained by substituting for I_em from Equations (1) and (2), respectively. For the continuum we get

$$\begin{eqnarray}&&{I}_{\mathrm{obs}}^{\mathrm{cont}}(E;\alpha ,\beta )={I}_{0}^{\mathrm{cont}}{g}^{{\rm{\Gamma }}+2}(\alpha ,\beta ){r}^{-q}(\alpha ,\beta ){E}^{-{\rm{\Gamma }}},\end{eqnarray} \tag{ 5 }$$

i.e., the same power-law spectrum from any point (α, β), with amplitude spatially modulated by a combination of powers of the g-factor and the disk radial coordinate. For the iron line we get

$$\begin{eqnarray}&&{I}_{\mathrm{obs}}^{\mathrm{Fe}}(E;\alpha ,\beta )={I}_{0}^{\mathrm{Fe}}{g}^{3}(\alpha ,\beta ){r}^{-q}(\alpha ,\beta )\delta [E-{E}_{\mathrm{Fe}}\,g(\alpha ,\beta )],\end{eqnarray} \tag{ 6 }$$

a delta function from any point, shifted by the local g-factor, and with amplitude spatially modulated by a combination of powers of g and r. The extra power of the g-factor comes from rescaling the argument of the delta function.

For illustration we present in Figure 1 sample observer-frame plane-of-the-sky maps of the iron line specific intensity ${I}_{\mathrm{obs}}^{\mathrm{Fe}}(\alpha ,\beta )$. Both panels are computed for black hole spin a = 1 (corresponding to maximal rotation) and radial index q = 3. The counterclockwise rotating disk is viewed face-on (i = 0°) in the left panel and under inclination i = 70° in the right panel. The plots are centered on the black hole; the coordinates are marked in units of the gravitational radius ${r}_{{\rm{g}}}={GM}/{c}^{2}$, where M is the black hole mass. In the central region the disk is truncated at the innermost stable circular orbit (ISCO), which is located at the r = r_g horizon in the maximally rotating case. The color scale corresponds to the amplitude-modulating factor preceding the delta function in Equation (6), with values given in units of ${I}_{0}^{\mathrm{Fe}}\,{r}_{{\rm{g}}}^{-q}$. The plotted g-factor contours illustrate the relative shift of the delta-function line at the given position.

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In the symmetric face-on case in the left panel of Figure 1 all contours are redshifted (g < 1), with g → 1 far from the origin and g = 0 at the black hole horizon. The radius of the central gap is larger than r_g owing to the geodesic-focusing effect of the black hole. As indicated by the color scale, the intensity drops rapidly outward and increases toward the center, with a sharp drop to zero at the horizon due to its g-factor dependence.

In the inclined i = 70° case in the right panel, the g = 1 no-shift contour is plotted in bold, with blueshifted g > 1 contours to its left and redshifted g < 1 to its right. In the presented example the highest g = 1.304 blueshift comes from a point to the left of the hole, while the extreme g = 0 redshift comes from the horizon. Far from the central region the contours converge to the familiar "dipole" pattern of an inclined Keplerian disk. The radial decline of the intensity is distorted by its g-factor dependence. The intensity peaks just to the left of the horizon and drops to zero along the horizon.

In quasar microlensing the gravitational field of the local distribution of stars and continuous matter amplifies light passing through a particular region of the lens galaxy. The result is usually presented in the form of a map of the amplification of flux from a background point-like source as a function of its plane-of-the-sky position. A cut through the map along a given source trajectory yields the corresponding point-source light curve. An amplification map typically consists of areas with low amplification variation divided by a network of caustics, along which the amplification sharply diverges. The local structure of the caustics can be typically described by linear folds that connect sharply at cusps.

If the projected size of the source is smaller than the structure scale of the map, it is advantageous to use a simple local analytical approximation of the amplification instead of the full numerically computed map. This approach is justified in our case, since the X-ray-emitting region is limited to the innermost parts of the quasar accretion disk. We simulate a typical microlensing event by the crossing of a straight-line fold caustic, for which the point-source amplification

$$\begin{eqnarray}&&A({d}_{\perp })={A}_{0}\,+\,H[{d}_{\perp }]\,\sqrt{\displaystyle \frac{{d}_{0}}{{d}_{\perp }}}\end{eqnarray} \tag{ 7 }$$

depends only on the coordinate d_⊥ perpendicular to the caustic (Schneider et al. 1992). Here the parameter d₀ > 0 defines the strength of the caustic, the Heaviside function H[d_⊥ ≥ 0] = 1 and H[d_⊥ < 0] = 0, and A₀ is the baseline amplification. For points outside the caustic d_⊥ < 0 and the amplification is constant, while for points inside the caustic d_⊥ > 0 and the amplification diverges as the inverse square root of the separation from the caustic.

For a caustic oriented so that the direction pointing inside subtends an angle ψ from the α-axis we can write

$$\begin{eqnarray}&&{d}_{\perp }(\alpha ,\beta ;{\alpha }_{0},{\beta }_{0})=(\alpha -{\alpha }_{0})\cos \psi +(\beta -{\beta }_{0})\sin \psi \,,\end{eqnarray} \tag{ 8 }$$

where (α₀, β₀) are the coordinates of an arbitrary point on the caustic. For a caustic crossing the disk we can replace these by the velocity v_⊥ of its perpendicular motion and time t. We define the former in terms of the time t_g it takes the caustic to perpendicularly cross a distance corresponding to the gravitational radius r_g,

$$\begin{eqnarray}&&{v}_{\perp }=\displaystyle \frac{{r}_{{\rm{g}}}}{{t}_{{\rm{g}}}}\,{\sigma }_{\mathrm{enter}},\end{eqnarray} \tag{ 9 }$$

where σ_enter = 1 if the disk is entering the caustic (inside at $t\to \infty$) and σ_enter = −1 if the disk is exiting the caustic (outside at $t\to \infty$). The perpendicular coordinate can then be computed from

$$\begin{eqnarray}&&{d}_{\perp }(\alpha ,\beta ;t)=\alpha \cos \psi +\beta \sin \psi +\displaystyle \frac{t}{{t}_{{\rm{g}}}}\,{r}_{{\rm{g}}}\,{\sigma }_{\mathrm{enter}},\end{eqnarray} \tag{ 10 }$$

where time t = 0 when the caustic crosses the origin (i.e., the black hole position).

For an extended source such as a quasar accretion disk, we obtain the resulting microlensed flux F by convolving the amplification map with the intensity map of the source. Using the fold caustic approximation of the map given by Equation (7) we get

$$\begin{eqnarray}F(E;{\alpha }_{0},{\beta }_{0}) & = & {\theta }^{-2}{\int }_{\mathrm{disk}}A[{d}_{\perp }(\alpha ,\beta ;{\alpha }_{0},{\beta }_{0})]\\ & & \times \ {I}_{\mathrm{obs}}(E;\alpha ,\beta )\,d\alpha \,d\beta ,\end{eqnarray} \tag{ 11 }$$

where the position of the caustic is defined by (α₀, β₀) and the factor θ converts plane-of-the-sky coordinates (α, β) to angular units.

For a caustic moving across the disk, the time dependence of the perpendicular coordinate d_⊥ is defined by Equation (10). For simplicity, we scale the observer-frame energy to the rest-frame energy of the iron line, defining g_obs ≡ E/E_Fe. If we now substitute for the observer-frame intensity from Equations (5) and (6), we obtain the observed spectrum as a combination of the continuum flux

$$\begin{eqnarray}{F}^{\mathrm{cont}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}};t) & = & {F}_{0}^{\mathrm{cont}}\,{g}_{\mathrm{obs}}^{-{\rm{\Gamma }}}{\int }_{\mathrm{disk}}A[{d}_{\perp }(\alpha ,\beta ;t)]\\ & & \times \ {g}^{{\rm{\Gamma }}+2}(\alpha ,\beta ){r}^{-q}(\alpha ,\beta )d\alpha \,d\beta \end{eqnarray} \tag{ 12 }$$

and the flux from the iron line

$$\begin{eqnarray}&&{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}};t)={F}_{0}^{\mathrm{Fe}}{\int }_{\mathrm{disk}}A[{d}_{\perp }(\alpha ,\beta ;t)]\\ &&\quad \times \ {g}^{3}(\alpha ,\beta ){r}^{-q}(\alpha ,\beta )\delta [{g}_{\mathrm{obs}}-g(\alpha ,\beta )]d\alpha \,d\beta .\end{eqnarray} \tag{ 13 }$$

The F₀ factors are combinations of constants, namely, ${F}_{0}^{\mathrm{cont}}={I}_{0}^{\mathrm{cont}}{\theta }^{-2}{E}_{\mathrm{Fe}}^{-{\rm{\Gamma }}}$ and ${F}_{0}^{\mathrm{Fe}}={I}_{0}^{\mathrm{Fe}}{\theta }^{-2}{E}_{\mathrm{Fe}}^{-1}$. Note that unlike the I₀ values, these constants have the same physical dimension. As seen from the dependence on g_obs in Equation (12), the continuum part of the microlensed spectrum retains the same power-law shape, with amplitude modulated by the time-dependent integral. This is a consequence of the decoupled spatial and energy dependence of ${I}_{\mathrm{obs}}^{\mathrm{cont}}$ in Equation (5).

On the other hand, the spatial and energy dependence of ${I}_{\mathrm{obs}}^{\mathrm{Fe}}$ in Equation (6) are coupled through the argument of the delta function. In addition, the spatial coordinates are coupled with time in the argument of the amplification in Equation (13). Hence, the iron line profile varies in amplitude as well as in shape during a microlensing event. Examining the delta function in Equation (13), we see that the line profile extends from g_min to g_max, where g_min and g_max are the minimum and maximum value, respectively, of the g-factor g(α, β) on the projected disk. For any value g_obs within this range, the delta function effectively reduces the two-dimensional integral to a line integral along the g-factor contour g(α, β) = g_obs.

It might seem straightforward to evaluate the integrals in Equations (12) and (13) by inverse ray shooting, after replacing the continuous energy dependence by a sequence of g_obs intervals. However, this method is poorly suited for obtaining good spectral resolution and achieving sufficient accuracy for studying the evolution of fine spectral effects. Its drawbacks include the following: (1) integrating each ray as a geodesic in the Kerr metric back to the disk to obtain r(α, β) and g(α, β) is time consuming; (2) two-dimensional ray shooting is poorly suited for computing one-dimensional contour integrals; (3) the integrands in Equations (12) and (13) are divergent along the caustic crossing the contours; (4) the small high-intensity region between the horizon and the g-factor peak (seen, e.g., in the right panel of Figure 1) contributes to all energies within the spectral line, so that one would need a high density of rays to obtain reliable results. Overcoming these drawbacks would require high numbers of rays integrated back toward the disk, making ray shooting computationally cumbersome and numerically inefficient in this scenario.

Instead, we compute the microlensed spectrum given by Equations (12) and (13) at a given time t using an adaptive grid integration algorithm. We divide the projected disk into a grid of square cells and evaluate r(α, β) and g(α, β) for the corners of the squares by geodesic integration. For the continuum we bilinearly interpolate the mostly slowly varying term g^Γ+2 r^−q in each square. We then combine the result with the expression for the amplification and integrate over one coordinate analytically, followed by numerical integration over the second coordinate. Next, we refine the grid by a factor of two in either dimension and repeat the calculation in the same way. Finally, we estimate the absolute error for each of the original squares and the relative error with respect to the cumulative flux from all squares. If the estimated error in some square is larger than the threshold, we divide it and each of its eight immediate neighbors again into four smaller squares. In this way we preserve continuity of the interpolation over the full disk. We repeat this algorithm until all relative errors are lower than a required threshold or a maximum number of iterations is exceeded. Note that for the continuum it is sufficient to evaluate one integral for all photon energies g_obs, due to the form of Equation (12).

For the iron line we have to evaluate the integral in Equation (13) separately for each chosen value of g_obs ∈ [g_min, g_max]. The procedure is similar to that used for the continuum, with a few alterations. In each square we bilinearly interpolate g³ r^−q and g_obs−g. We integrate only over the squares intersected by the interpolated g-factor contour g(α, β) =g_obs. Analytical integration over one coordinate is trivial, due to the presence of the delta function. Nevertheless, even in this case the integration over the second coordinate has to be performed numerically. For more details of the numerical procedure see Ledvina & Heyrovský (2015).

In order to verify the reliability of our results, we check that the spectra converge to the unlensed line profile when the caustic is far from the disk center (in the limits $t\to \pm \infty$). This profile, illustrated by the blue curves in Figures 2–7, can be also obtained by setting A[d_⊥(α, β; t)] = 1 in Equations (12) and (13). Its shape is in agreement with the iron line profiles that can be found in the literature, such as in Figure 2 of Laor (1991), Figure 5 of Fabian (2006), or Figures 7–9 of Dovčiak et al. (2004).

We first demonstrate the overall character of changes in the spectrum by computing the total disk-integrated flux F = F^cont + F^Fe using Equations (12) and (13) for a face-on and an inclined disk entering a caustic (Figures 2 and 3, respectively).

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For either inclination the results are presented as a sequence of pairs of panels, showing the caustic position (left panel) and the corresponding spectrum (right panel). The caustic is plotted over a color map of the g-factor with the same contours as in Figure 1. The caustic position is marked by the dark-orange line, and its inner side is indicated by the light-orange band. Hence, the emission from the part of the disk lying to the upper left from the caustic is amplified, with highest amplification along the caustic. The part lying to the lower right from the caustic is unamplified. More exactly, "unamplified" here and in the following text means amplified only by the constant baseline amplification A₀. Our specific choice A₀ = 1 is consistent with the literal meaning, yielding no excess flux when the caustic lies far from the disk center ($| t| \to \infty$).

The spectrum in each right panel is calculated for the corresponding time shown in the label in units of t_g. The green curve represents the microlensed spectrum, plotted in terms of total flux in arbitrary units as a function of g_obs, the photon energy relative to E_Fe. For comparison we include the blue curve, which represents the spectrum of the same disk in absence of microlensing. The tick-mark spacing on the g_obs axis is equal to the g-factor contour spacing in the left panels, for easier interpretation of the iron line features.

3.1.1. Face-on Disk (i = 0°)

3.1.2. Inclined Disk (i = 70°)

The simulations in Sections 3.1.1 and 3.1.2 show that the most typical features generated in the microlensed iron line profile are peaks and edges. Their appearance and location clearly depend on the geometry of the g-factor contours in the vicinity of the caustic. In order to explore this connection in more detail, we present in Figures 4 and 5 zoomed-in details from Figure 3 around a sample peak and edge, respectively. For simplicity, we plot only the line flux F^Fe in the right panels, ignoring the power-law continuum.

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Peak evolution is illustrated in Figure 4. The middle row corresponds to details from the t = −2.53 t_g top right panels of the inclined-disk sequence in Figure 3. Shown in the right panel is the vicinity of the g_obs = 1.2 peak in the line profile up to the line edge at g_obs = 1.304. Enlarged in the left panel is the g-factor map region where the g = 1.2 contour corresponding to the peak approaches the caustic. Dotted contours correspond to minor ticks on the g_obs axis in the right panel; solid contours correspond to major ticks. The top and bottom rows correspond to times preceding and succeeding the middle row, respectively.

Inspecting the situation at t = −2.53 t_g in the middle row, we see that the caustic is exactly tangent to the g = 1.2 contour, which lies on its inner amplified side. The position of the peak thus corresponds to the contour with the longest segment in the highest-amplification region along the caustic. Contours for g-factor values on either side of the peak have gradually shorter segments in this region.

The situation at the two bracketing times supports this interpretation. In the top row at t = −2.63 t_g the caustic is tangent to the g = 1.223 contour (not plotted), and the peak thus appears at this higher energy. In the bottom row at t = −2.43 t_g the caustic is tangent to the g = 1.174 contour (not plotted), and the peak appears at this lower energy. As the caustic sweeps across the disk in this sequence, the peak moves across the line profile, appearing at the energy corresponding to the g-factor contour instantaneously tangent to the caustic. This interpretation also implies that the speed and sense of the energy shift of the peak are given by the dot product of the g-factor gradient at the tangent point and the velocity of perpendicular caustic motion.

Edge evolution in Figure 5 is illustrated by the behavior around the t = −12.5 t_g middle left panels of Figure 3. Shown in the middle right panel of Figure 5 is the vicinity of the g_obs = 1.2 edge in the line profile. The middle left panel shows the enlarged g-factor map region where the g = 1.2 contour approaches the caustic. The top and bottom rows again correspond to times preceding and succeeding the middle row, respectively.

Concentrating first on the middle row at t = −12.5 t_g, we see that the caustic is again exactly tangent to the g = 1.2 contour. However, this time the contour lies on its outer, unamplified side. The position of the edge thus corresponds to the last contour reaching the caustic. Contours for g-factor values farther on the inner side have a small segment in the high-amplification region, while contours for values farther on the outer side have no segment in the amplified region.

In the top row at t = −16.5 t_g the caustic is tangent to the g = 1.176 contour (not plotted), and the edge thus appears at this lower energy. In the bottom row at t = −8.5 t_g the caustic is tangent to the g = 1.238 contour (not plotted), and the edge appears at this higher energy. In analogy to the case of the peak, as the caustic sweeps across the disk in this sequence, the edge moves across the line profile, appearing at the energy corresponding to the g-factor contour instantaneously tangent to the caustic. The speed and sense of the energy shift of the edge are computed in the same way as for the peak.

Note that in this sequence the edge is descending since the g-factor increases outward from the tangent point. If it decreased outward from the tangent point (e.g., if we flipped the orientation of the gradient), the line profile would have an ascending edge. This can be seen, for example, at g_obs = 1.2 in the top row of Figure 2.

From the peak and edge analysis above we infer that the position of these spectral features is directly given by the energy of the g-factor contour tangent to the caustic. A contour tangent from the inner side of the caustic generates a peak, while a contour tangent from the outer side generates an edge. Extrapolating further, we expect that the local profile of these features should be described by local properties of the g-factor map of the accretion disk at the tangent point. We test this hypothesis analytically in Section 4.

Under special circumstances the tangent contour cannot be classified as reaching the caustic from the inner or outer side, leading to spectral features and their sequences other than the peaks and edges described in Section 3.2. One such case involves the point-like contour at the g-factor maximum, such as the one crossed by the caustic in the bottom left panels of Figure 3. Another case occurs when the tangent point is an inflection point of the contour, as shown in the middle right panels of Figure 3.

Spectral evolution during the crossing of the g-factor maximum is shown in Figure 6, detailing the situation around the t = −3.86 t_g panels of Figure 3. In the top row at t = −5.00 t_g the maximum lies outside the caustic, which is tangent from its outer side to the g = 1.291 contour and thus forms an edge in the line profile. In the middle row at t = −3.86 t_g the caustic directly crosses the maximum, forming a prominent asymmetric peak at the line edge. In the bottom row at t = −3.00 t_g the maximum lies inside the caustic, which is now tangent from its inner side to the g = 1.275 contour (not plotted) and thus forms a regular peak in the line profile.

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In general, the crossing of a maximum of the g-factor is associated with a change from an edge to a peak (or vice versa) via a strong asymmetric peak. In the disk emission models used in this study the only local maximum (and the only local extremum) is the global maximum of the g-factor. Hence, the asymmetric peak occurs only at the high-energy edge of the line.

Figure 7 illustrates the spectral change during the tangent crossing of a g-factor contour inflection point, detailing the situation around the t = −1.93 t_g panels of Figure 3. Note that the presented sequence occurs over a very short time interval of 0.08 t_g, so that the shifting caustic position is barely noticeable in the left column. Each of the three line profiles includes a small peak at g_obs = 1.020 unrelated to the inflection crossing, from a contour tangent internally to the caustic beyond the area plotted in the left column. In the following we discuss the other line profile features.

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In the top row at t = −1.98 t_g the caustic is tangent from its inner side to the g = 1.0137 contour (not plotted), forming a peak in the line profile. To the upper right of this tangent point the caustic is crossed from inside by the g = 1.0068 contour (not plotted), which turns back to touch the caustic from outside to the lower left of the first contour's tangent point, forming an edge in the line profile. As the caustic advances to the inflection point at t = −1.93 t_g in the middle row, the peak and edge merge to form a prominent asymmetric peak at g_obs = 0.9996, equal to the value of the g-factor contour tangent at its inflection point. In the bottom row at t = −1.90 t_g no contour is tangent to the caustic in the vicinity of the crossed inflection point. As a result, the prominent peak disappears, leaving behind a bump at g_obs = 0.9938 corresponding to the contour with the longest segment in the high-magnification region along the caustic.

To summarize, the tangent crossing of a g-factor contour inflection point is associated with a change from an adjacent edge and peak pair via a strong asymmetric peak to a bump that eventually vanishes. Studying the full set of g-factor contours and plotting their inflection points yields curves running across the contours in the map (see further in Section 4.4). The caustic may cross any of these points at different energies. However, we stress that the described sequence with the strong peak occurs only if the caustic is tangent to the local g-factor contour passing through the point.

Finally, we note that the small bump visible at g_obs = 1.02 in the top right panels of Figure 3 has a similar origin to the bump in the bottom row of Figure 7. It arises outside the plotted area from a nontangent contour closely approaching the high-amplification region along the caustic.

A contour tangent from inside the caustic (i.e., from negative y in our example) requires g_xx/g_y > 0, as can be seen from Equation (17). This configuration is illustrated in the top left panel of Figure 8, in which the parabolic contours are plotted over the source divided into concentric rings, with the outer radius normalized to unity. The g(x, y) = g₀ contour passes through the center.

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The remaining simple integration yields an inverse hyperbolic function with the type depending on the sign of (g₀−g_obs)/g_y: inverse hyperbolic sine for positive, inverse hyperbolic cosine for negative. The integration limits for contours near g₀ depend on the position of the contour, as seen on the sketch in Figure 8. The contours farther inside the caustic extend from edge to edge of the source, while those protruding outside the caustic have to be integrated in two segments extending from the edge to the caustic, omitting the central part. The x coordinate of the intersection of these contours with the caustic can be obtained by setting the denominator in the integrand of Equation (21) equal to zero. The intersection with the edge of the source can be obtained by substituting from Equation (19) in x² + y² = ρ².

Rather than providing all the bulky but straightforward intermediate expressions explicitly, we present here the leading-order term when ${g}_{\mathrm{obs}}\to {g}_{0}$, i.e., the line profile in the vicinity of g₀. Interestingly, this term has the same form for all the specific cases:

$$\begin{eqnarray}{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}}) & \approx & -I\,\sqrt{\displaystyle \frac{2\,{d}_{0}}{{g}_{y}\,{g}_{{xx}}}}\,\mathrm{ln}| {g}_{\mathrm{obs}}-{g}_{0}| \\ & = & \ -P\sqrt{{d}_{0}}\,\mathrm{ln}| {g}_{\mathrm{obs}}-{g}_{0}| ,\end{eqnarray} \tag{ 22 }$$

where the constant $I={F}_{0}^{\mathrm{Fe}}\,{r}_{\mathrm{circ}}^{-q}\,{g}_{0}^{3}$ is independent of the contour geometry and of the caustic properties. We may separate the caustic strength parameter from the contour geometry and define the peak strength

$$\begin{eqnarray}&&P\,=\,I\sqrt{2/| {g}_{y}\,{g}_{{xx}}| },\end{eqnarray} \tag{ 23 }$$

which depends purely on the local disk properties, including the local shape of the g-factor contour.

The result in Equation (22) shows the logarithmic character of the peak generated at g₀, the g-factor of the contour tangent to the caustic from inside. This result corroborates the behavior seen in the numerical simulations in Section 3.2 and Figure 4.

In the top row of Figure 8 the spectra in the two right panels show the numerically integrated excess flux generated by the caustic. In this example g_y = −0.1 (the g-factor decreases upward in the y-direction) and g_xx = −0.3. In the middle panel the color-coded widening profiles from blue to yellow correspond to increasing the source size from the small inner circle to the full outer circle. In the right panel the color-coded increasing profiles from yellow to blue correspond to an annular source starting from the outer ring, with the inner radius gradually shrinking to zero. Both plots clearly show that the region of the source directly at the tangent point is responsible for the (logarithmic) divergence at g₀.

If we swap signs of both relevant g-factor derivatives, the shape of the contours in the left panel remains the same; only the g-factor increases upward in the y-direction, and the spectra are mirrored vertically around g₀.

The g₀ contour is tangent from outside the caustic (i.e., from positive y in our example) if the derivatives fulfill the condition g_xx/g_y < 0. This configuration is illustrated in the bottom left panel of Figure 8, with the g(x, y) = g₀ contour passing through the center.

The remaining simple integration in Equation (21) depends on the sign of (g₀−g_obs)/g_y: we get inverse hyperbolic sine for positive, zero for negative. In the former case the integration limits for contours near g₀ extend from the caustic to the caustic, as seen on the sketch in Figure 8. In the latter case the contours lie entirely outside the caustic, hence the zero contribution.

For both cases we can write the leading-order term when ${g}_{\mathrm{obs}}\to {g}_{0}$ in the form

$$\begin{eqnarray}{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}}) & \approx & I\,\sqrt{\displaystyle \frac{2\,{d}_{0}}{-{g}_{y}\,{g}_{{xx}}}}\,\pi \,H\left[\displaystyle \frac{{g}_{0}-{g}_{\mathrm{obs}}}{{g}_{y}}\right]\\ & = & \ P\sqrt{{d}_{0}}\,\pi \,H\left[\displaystyle \frac{{g}_{0}-{g}_{\mathrm{obs}}}{{g}_{y}}\right],\end{eqnarray} \tag{ 24 }$$

where H is the Heaviside step function, and the constant I has the same definition as in Section 4.1. By separating the caustic strength parameter from the contour geometry, we get the same parameter P defined in Equation (23), depending purely on the local disk properties, including the local shape of the g-factor contour. Instead of peak strength, here it appears as a measure of the step size.

The result in Equation (24) shows the step-like character of the edge generated at g₀, the g-factor of the contour tangent to the caustic from outside. This result corroborates the behavior seen in the numerical simulations in Section 3.2 and Figure 5.

In the bottom row of Figure 8 the spectra in the two right panels show the numerically integrated excess flux generated by the caustic. In this example g_y = 0.1 (the g-factor increases upward in the y-direction) and g_xx = −0.3. In the middle panel the color-coded widening profiles from blue to yellow correspond to increasing the source size from the small inner circle to the full outer circle. In the right panel the color-coded increasing profiles from yellow to blue correspond to an annular source starting from the outer ring with the inner radius gradually shrinking to zero. Both plots show that the region of the source directly at the tangent point is responsible for the step-like high-energy edge at g₀.

Just like in Section 4.1, if we swap signs of both relevant g-factor derivatives, the shape of the contours in the left panel remains the same; only the g-factor decreases upward in the y-direction, and the spectra are mirrored vertically around g₀, with the step forming a low-energy edge.

For a caustic crossing the contour at the origin under a general angle, we substitute the general expressions from Equations (14) and (19) in Equation (20) and obtain the excess flux in the form

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}})={F}_{0}^{\mathrm{Fe}}\,{r}_{\mathrm{circ}}^{-q}\,\sqrt{{d}_{0}}\,\displaystyle \frac{{g}_{\mathrm{obs}}^{3}}{| {g}_{y}| }\\ \,\times \ {\int }_{x \mbox{-} \mathrm{range}}\sqrt{\displaystyle \frac{{g}_{y}}{{g}_{y}\,x\,\cos (\psi -\xi )-(\tfrac{1}{2}\,{g}_{{xx}}\,{x}^{2}+{g}_{0}-{g}_{\mathrm{obs}})\sin (\psi -\xi )}}\,{dx}.\end{array}\end{eqnarray} \tag{ 25 }$$

where the integration range is limited to the positions on the circular source with a positive expression under the square root. The expression in the denominator of the integrand can be rearranged by collecting the terms depending on x to form a complete square, yielding

$$\begin{eqnarray}&&{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}})={F}_{0}^{\mathrm{Fe}}\,{r}_{\mathrm{circ}}^{-q}\,\sqrt{{d}_{0}}\,\displaystyle \frac{{g}_{\mathrm{obs}}^{3}}{| {g}_{y}| }\\ &&\quad \times \,{\int }_{x \mbox{-} \mathrm{range}}\sqrt{\displaystyle \frac{-{g}_{y}/\sin (\psi -\xi )}{\tfrac{1}{2}\,{g}_{{xx}}\,{[x-{x}_{\mathrm{tp}}(\psi )]}^{2}+{g}_{\mathrm{tp}}(\psi )-{g}_{\mathrm{obs}}}}\,{dx},\end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray}{g}_{\mathrm{tp}}(\psi ) & = & {g}_{0}-\displaystyle \frac{{g}_{y}^{2}}{2\,{g}_{{xx}}}{\cot }^{2}(\psi -\xi ),\,\,\mathrm{and}\\ & & {x}_{\mathrm{tp}}(\psi )=\displaystyle \frac{{g}_{y}}{{g}_{{xx}}}\cot (\psi -\xi )\end{eqnarray} \tag{ 27 }$$

are the g-factor of the contour that is tangent to the caustic and the position of the tangent point, respectively.

Comparison with Equation (21) shows that the resulting profiles will be very similar. However, the peak or edge will appear at g_tp instead of g₀, as long as the tangent point x_tp lies within the circular source. The type of feature is now given by the sign of g_xx sin(ψ − ξ)/g_y.

A peak occurs if the contour is tangent from inside the caustic, which requires g_xx sin(ψ − ξ)/g_y < 0. The leading-order term when ${g}_{\mathrm{obs}}\to {g}_{\mathrm{tp}}$ gives a flux excess

$$\begin{eqnarray}&&{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}})\approx -I\,\sqrt{\displaystyle \frac{-2\,{d}_{0}}{{g}_{y}\,{g}_{{xx}}\,\sin (\psi -\xi )}}\,\mathrm{ln}| {g}_{\mathrm{obs}}-{g}_{\mathrm{tp}}| ,\end{eqnarray} \tag{ 28 }$$

where the constant I has the same form as in Section 4.1. Changing the caustic orientation away from the tangent ψ = ξ − π/2 shifts the peak position g_tp and boosts the strength of the peak. This increase can be explained by the contour staying longer near the caustic.

An edge occurs if the contour is tangent from outside the caustic, with g_xx sin(ψ − ξ)/g_y > 0. The leading-order term when ${g}_{\mathrm{obs}}\to {g}_{\mathrm{tp}}$ gives a flux excess

$$\begin{eqnarray}{\rm{\Delta }}{F}^{\mathrm{Fe}}({g}_{\mathrm{obs}}{E}_{\mathrm{Fe}}) & \approx & I\,\sqrt{\displaystyle \frac{2\,{d}_{0}}{{g}_{y}\,{g}_{{xx}}\,\sin (\psi -\xi )}}\,\pi \\ & & \times \ H\left[\displaystyle \frac{{g}_{\mathrm{obs}}-{g}_{\mathrm{tp}}}{{g}_{y}}\,\sin (\psi -\xi )\right].\end{eqnarray} \tag{ 29 }$$

The dependence on caustic orientation is similar here, with the amplitude of the step increasing when rotating the caustic away from the tangent ψ = ξ − π/2.

A sequence of numerically integrated flux excess profiles for different caustic orientations is shown in Figure 9, rotating gradually from ψ = ξ − π/2 in the first column to ψ =ξ + π/2 in the last column. The orientation in the first column produces a peak at g₀, as seen in the top row of Figure 8. In the second column the internally tangent contour is marked by the dot-dashed line in the upper sketch. In this case g_tp > g₀, as can be seen from the peak position in the bottom panel. In the third column no contour is tangent to the caustic within the circular source, and thus the corresponding profile has no caustic-related feature. In the fourth column an externally tangent contour is marked by the dot-dashed line in the top panel. The corresponding edge can be seen at the position g_tp > g₀ in the bottom panel. In this case the edge looks like a peak from its red side. However, this is merely an artifact of the cutoff at the edge of the circle, since contours with lower g_obs would approach the caustic outside the circular area. Finally, the last column shows the edge at g₀ demonstrated in detail in the bottom row of Figure 8.

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We can now safely conclude that the peaks or edges generated in the line profile by the microlensing caustic directly correspond to g-factor contours tangent to the caustic from its inner or outer side, respectively.

The peak strength P defined by Equation (23) is obtained by multiplying two factors. The intensity factor I, which depends on the assumed disk emissivity law, is equal up to a conversion constant to the color maps shown in Figure 1 for two inclinations. The square-root factor in Equation (23) depends purely on the local shape of the g-factor contours, which is in turn given by the geometric setup of a Keplerian disk orbiting in the equatorial plane of a Kerr black hole.

In Figure 10 we present maps of the peak strength P for six disk inclinations i ranging from the face-on 0° to a nearly edge-on 85° orientation. We construct the maps by taking each point on the projected disk, extending the caustic through it as a tangent of the local g-factor contour, computing the local parabolic approximation of the contour following Equation (17) with the caustic defining the x-axis, and using the obtained g-factor derivatives to compute the peak strength P from Equation (23). As discussed in the previous sections, the caustic generates a peak or edge in the line profile when the contour lies on its amplified or unamplified side, respectively. The computed P values correspond to either feature. The obtained maps can tell us which regions of the projected disk may produce prominent peaks or edges when crossed by a favorably oriented caustic, and which regions may yield only weak features.

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For the face-on disk the map is axially symmetric. With increasing inclination we can see changes in the geometry of the g-factor contours. Up to inclination 10° there is no prominent change in peak strengths (the g-factor maximum lies far from the black hole). Increasing the inclination further up to 30° leads to some g-factor contours locally changing from convex to concave. This generates a curve in the map with divergent peak strength, since g_xx vanishes at g-factor contour inflection points (see description in Section 3.3).

The next panel with 70° inclination already shows the g-factor maximum inside the plotted area. At this point the peak strength diverges as well, here due to the vanishing of g_y (see description in Section 3.3). Note also the increased peak strength around coordinates (15 r_g, 10 r_g). In this area the contours are very flat, and a further increase to i = 75° generates here two new inflection points on each affected contour. Thus, at inclination 75° we can see three different regions where peak strength P diverges: two inflection curves (curves connecting the inflection points of g-factor contours) and the g-factor maximum. The last panel with inclination 85° displays a third inflection curve formed above the horizontal axis extending from the center to the left.

The obtained plots clearly show that for higher inclinations one may expect stronger variations in generated peaks or edges as the caustic progresses across the disk. Note, however, that the requirement of tangency implies that for higher inclinations strong generated features occur for specific caustic orientations.

Finally, we recall that the peak strength is modulated by the intensity factor I, which affects the amplitude but not the occurrence or spectral position of the peak. A different emissivity law would change the color maps but not the contours shown in Figure 1. This would lead primarily to a relative boost or suppression of peak strength in the central versus outer regions of the disk.

In Section 4.1 we derived an analytical expression for the line profile in the vicinity of the microlensing-generated peak. Here we study its applicability by evaluating how accurately it fits the line flux in a microlensed spectrum computed following Equation (13), as described in Section 2.3.

We perform tests for caustic tangent points taken from a regular 512 × 512 grid spanning an α, β ∈ (−30 r_g, 30 r_g) square in the plane of the sky. For each point we calculate first and second derivatives of the g-factor with respect to α, β. Using the notation introduced in Equation (8), these define the orientation of the caustic tangent to the g-factor contour g(α, β) = g₀ passing through the point,

$$\begin{eqnarray}&&(\cos \psi ,\sin \psi )=\pm ({g}_{\alpha },{g}_{\beta })/\,{[{g}_{\alpha }^{2}+{g}_{\beta }^{2}]}^{1/2},\end{eqnarray} \tag{ 30 }$$

where the sign is chosen so that the angle ψ defining the inner side of the caustic points toward the center of curvature of the contour (identified using the second derivatives). We then compute the microlensed line flux F^Fe(g) from Equation (13) for a disk with inclination i = 70° and a caustic with d₀ = 25 r_g and A₀ = 1, using Equation (8) for the distance d_⊥.

The analytical model we use to fit the peak region,

$$\begin{eqnarray}&&{F}_{{\rm{m}}}^{\mathrm{Fe}}(g)=-{C}_{1}\,\mathrm{ln}| g-{g}_{{\rm{m}}0}| +{C}_{2}+{C}_{3}(g-{g}_{{\rm{m}}0}),\end{eqnarray} \tag{ 31 }$$

is based on the result from Equation (22) superimposed on a linear background. We obtain the model parameters C₁, C₂, C₃, g_m0 by a least-squares fit to the line flux F^Fe(g) performed over the spectral interval [g₀ − Δg, g₀ + Δg]. We measure the goodness of fit as a function of increasing half-width Δg by the relative approximation error

$$\begin{eqnarray}&&\epsilon ({\rm{\Delta }}g)=\sqrt{\displaystyle \frac{\int {[{F}_{{\rm{m}}}^{\mathrm{Fe}}(g)-{F}^{\mathrm{Fe}}(g)]}^{2}\,dg}{\int {F}^{\mathrm{Fe}}{(g)}^{2}\,dg}},\end{eqnarray} \tag{ 32 }$$

where we integrate from g₀ − Δg to g₀ + Δg. We present the obtained results as a function of tangent point position in the form of two plots in Figure 11.

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In the left panel of Figure 11 we present a color map of the half-width of the fitted interval at which the error (Δg) reaches 1%. For better orientation we plot the contour Δg = 0.05. Clearly, for tangent points in most of the plot the spectrum can be fitted over an even wider interval with 1% accuracy. Note that there are regions where the map changes abruptly. This is caused by the dependence (Δg) not being monotonic. In one pixel there might be a local maximum just above 1% and in a neighboring one just under 1%, causing a jump in the width of the fitted interval.

In the complementary right panel we present a color map of the relative error (0.05) obtained for the fixed half-width Δg = 0.05. Here we plot the contour  = 1%, which should closely correspond to the contour in the left panel. The small differences are caused by the nonmonotonic dependence (Δg) mentioned above. The plot shows that for most tangent points in the studied area the spectrum can be fitted over the interval [g₀ − 0.05, g₀ + 0.05] with better than 1% accuracy. For the iron Kα line this interval corresponds to ±0.3 keV around the microlensing-generated peak.

In both panels of Figure 11 we see several areas inside the plotted contours where the quality of the fit decreases. Close to the horizon the g-factor contours change rapidly within a small area; hence, our approximation has limited validity. The area at upper right is caused by very flat g-factor contours, not adequately approximated by parabolas. The band running from α = −10 r_g at the lower edge to α = −10 r_g at the upper edge is caused by the presence of two very close peaks, which cross each other in the middle of the band. An outer band running from α = −25 r_g at the lower edge to α = −18 r_g at the upper edge surrounds the inflection curve, at which the contours are not parabolic. Finally, the area to the left of the center around the maximum of the g-factor has elliptic contours, with parabolic approximation possible only in very small sections.

We conclude that with the exception of these areas, microlensing-generated peaks from most parts of the central accretion disk are described by the model in Equation (31) with an accuracy better than 1% within a g-factor interval of at least ±0.05 around the peak.