Moving Corona and the Line Profile of the Relativistic Broad Iron Emission Line

Iron fluorescence emission lines from X-ray binaries and active galactic nuclei are important diagnostic tools for studying the physical processes near the event horizon of both the stellar-mass black holes in X-ray binaries and the supermassive black holes in active galactic nuclei. In this work, we investigate the line profile of the relativistic broad iron lines from the cool accretion disk of a black hole due to the asymmetric illumination of a moving corona, which moves away from the disk with a relativistic velocity. Both the off-axis location and the radial velocity of the moving corona are considered. Our results clearly show that the illumination and the line profile are dependent on the position and velocity of the corona, since the disk region below the corona receives more flux, which is the most important factor affecting the line profiles. As expected, if the corona is close to the receding part of the rotating disk, the red peak is enhanced, while the blue peak is weakened in the broad line profile, and the central energy of the emission line is low. Conversely, if the corona is close to the approaching part of the disk, the blue peak is strong and the central energy of the emission line is high, even higher than the intrinsic energy of the emission line. Due to the beaming effect of the moving corona, the corona with high velocity illuminates the outer region of the disk, which leads to the red peak disappearing and there being only one blue peak in the profile of the emission line.


Introduction
The X-ray emission lines observed in active galactic nuclei (AGNs) and the accreting black hole in the X-ray binaries are produced by X-ray reflection.According to the disk-corona model (Haardt & Maraschi 1993), the hard X-ray emission is from the hot corona near the black hole, that is, the supermassive black hole in the AGN or the stellar-mass black hole in X-ray binaries (Fabian et al. 2015).The accretion disk, which is relatively cold, is irradiated by the corona, causing the atoms in the surface layers to be photoexcited and produce fluorescence emission lines at definite frequencies (George & Fabian 1991).The iron Kα line at 6.4 keV is usually the strongest of these emission lines (Ross et al. 1999), which is observed to be broadened and asymmetric instead of intrinsically narrow due to the combination of relativistic effects such as Doppler beaming, gravitational lensing, gravitational redshift, and frame dragging (Fabian et al. 1989).The observation of the line profile of the X-ray emission provides the properties of the black hole and the information about the inner region of the accretion disk, such as the spin of the black hole, the dynamics, and the geometric structure of the disk and coronae.
The geometry of the corona determines the emissivity of the X-ray emission line in the accretion disk.Without much information about the geometry of the corona, it is common to model the emissivity of the emission line as a power law of radii of the accretion disk (Fabian et al. 1989) or a broken power law (Fabian et al. 2002).In the so-called lamppost model, the corona is assumed to be a point-like source at a certain height along the spin of the black hole (Laor 1991).In this model, the resulting disk illumination is expected to be axisymmetric.
Observations show that coronae are not always axisymmetric or stationary and that they might be connected with the jet that was first observed in BHB GX 399-4 (Hannikainen et al. 1998;Corbel et al. 2000) and seen also in the hard state of low-mass X-ray binaries (Gallo et al. 2003;Fender et al. 2004).Such observations suggest that coronae are actually the base of relativistic jets, which is the dominant region for the hard X-ray emissions by Comptonization processes (Markoff et al. 2005).
Recent observations of BHB MAXI J1820+070 suggest that there is a close relationship between the X-ray corona and the radio jet.The discovery of low-frequency quasi-periodic oscillations in the high-energy band strongly supports the jet precession model, which is also consistent with the physical picture of the hard X-ray-emitting region supported by multiwavelength results (Ma et al. 2021).The large lag of the low-frequency quasi-periodic oscillation in the high energy and the energy-related behaviors can be naturally explained as the result of a small-scale jet precession.The findings of the decrease of the reflection fraction in MAXI J1820+070 suggest that the corona is getting closer to the black hole and the coronal material might be outflowing faster (You et al. 2021).In the relativistic magnetohydrodynamics simulations, coronae are not static but instead are windy hot blobs blowing away from the inner regions of the accretion flow and are likely Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
In this paper, we investigate the effects of a hot corona deviating from the spin axis of the black hole and moving away from the black hole with a mildly relativistic velocity (Beloborodov 1999) on the line profile of the iron fluorescence emission line.

Model
The schematic of the model is shown in Figure 1.In the Boyer-Lindquist coordinates, the line element of the Kerr metric is written as (Bardeen et al. 1972) In the above equations, the geometrical units with G = c = M = 1 are used, where M is the gravitational mass of the black hole and a is the spin parameter of the black hole.The observer is assumed to be located at (r obs , θ obs , f obs ), where r obs = ∞ and f obs = 0°for convenience.We set the initial location of the X-ray corona at (r s , θ s , f s ), which is outside of the innermost stable circular orbit (ISCO), that is, r s  r ms (Miniutti & Fabian 2004), and we suppose the corona moves with a radial velocity.The emission from the corona is assumed to be instantaneous, and the photons are emitted before the corona moves substantially to alter its location (Fukumura & Kazanas 2007).
The accretion disk is assumed to be the standard disk model, where the cold accreting fluid is considered to be in Keplerian circular orbit around the black hole (Fabian et al. 1989).The disk is assumed to be geometrically thin and optically thick and illuminated by the X-ray source, producing the fluorescent iron line at 6.4 keV, which is mainly contributed from the region between the inner radius (r in = r ms ) and the outer radius (r out ).Using the ray-tracing method (Li et al. 2005;Dexter & Agol 2009; Dexter 2016, for instance), the line profile can be calculated with the emissivity of the emission line, which is assumed to be proportional to the flux absorbed per unit area of the disk (Laor 1991).
The four contravariant basis vectors of the observer at the local rest frame of the X-ray corona compose a matrix as (Yuan et al. 2009) ) .For each photon emitted from the corona, its fourmomentum can be calculated as p p e g = n a a m mn ( ) ( ) , where p ( α) is the four-momentum of the photon in the rest frame of the X-ray corona as p p 1, sin cos , sin sin , cos .5 The angles Θ and Φ correspond to the usual spherical-polar angles in the rest frame of the corona, defining the direction at which photons are emitted from the corona (Dabrowski & Lasenby 2001).Using the ray-tracing method, the location (r, f) where the photon with the four-momentum p ν from the source hit the disk can be calculated.In our work, the geokerr code is used to calculate the null geodesics (Dexter & Agol 2009; Dexter 2016).With the incident location for each photon, the illumination flux on the accretion disk can be calculated (Petrucci & Henri 1997).Photons emitted in the range of [Θ, Θ + dΘ] × [Φ, Φ + dΦ] impact the accretion disk with an area as (Wilkins & Fabian 2012) The ratio of the frequencies between the X-ray source and the accretion disk, that is, the factor of the gravitational redshift g s , can be calculated as where Ω is the coordinate angular velocity at the Keplerian circular orbit of the disk and γ Ω is the corresponding Lorentz factor (Bardeen et al. 1972).The photons are assumed to be emitted at equally spaced angles Θ and Φ.The illumination flux on the accretion disk is given by Lu & Yu (2001) and Dauser et al. (2013): where Γ = 2 is the power-law index of the spectrum of the primary X-ray from the corona since the emitted radiation is assumed as N s s n µ -G (Brenneman & Reynolds 2006).

Illumination Flux from Moving Corona
Figures 2 and 3 show the distribution of the illumination flux from the corona onto the accretion disk in the different models.As expected, the disk regions closer to the corona receive more flux since the X-ray corona deviates from the spin axis of the black hole, which results in the nonaxial symmetry of the emissivity of the emission line.For the black hole with low spin, the illumination flux at each certain azimuth still essentially follows the power law of the disk radii, while it is broken at the inner disk region of the black hole with high spin.The distribution of the illumination flux in the azimuthal angle is basically concentrated around f = 0°.For the corotating accretion disk, the illumination flux on the approaching side of the corona (180°< f − f s 360°) is generally more than that on the receding side (0°< f − f s 180°) due to the effect of the Doppler beaming (Yu & Lu 2000).When the source inclination is high, the source is lower above the disk, causing the distribution of photons to be more concentrated and the illumination to change more drastically.The maximum difference in the emissivity of the emission line at the different azimuths is more than one order of magnitude.
The spin of the black hole sensitively determines the sizes of the ISCO, which allows the illumination regions of the disk.In addition, a significant fraction of the photons is subject to the effect of the frame dragging of the rotating spacetime near the event horizon of the Kerr black hole, which causes the rotation of hard X-ray photons from the corona in the azimuthal direction before reaching the disk (Fukumura & Kazanas 2007).In the inner region of the accretion disk, photons are emitted in the direction of the spin of the black hole due to the dragging effect, causing the peak of the disk emissivity moving to the inside counterclockwise.There is a distinct peak of the illumination flux in the cases of f = 0°and 90°, as shown in Figures 3(B1) and (D1), which is because the corona with relativistic velocity emits more photons to the outer region of the disk.In the inner region for the case of f = 180°, more photons are received near the black hole due to the Doppler effect of the rotation of the accretion disk.

Line Profile
The observed line profile depends on both the emissivity of the accretion disk and the observing direction.Figure 4 shows the profiles of the iron emission line observed at two observing inclinations (30°and 60°) in the case of a corona moving radially outward near the extreme Kerr black hole (a = 0.998).It is shown how the line profile changes with the azimuth of the corona.Due to the relativistic effect, the profile is not the narrow emission line as in the stationary rest frame but has a certain broadened and asymmetric shape and changes with the position of the corona and the observing direction.The profile shows single or double peaks as well as a cutoff at the blue band and a long wing extending toward the red band.
The flux of the emission line at the different energies comes from the different regions of the accretion disk.The red part of the line profile mainly comes from the inner region of the disk, from which the emitted photons get a strong gravitational redshift.The photons emitted near the ISCO contribute to the red wing, which is the reddest end in the line profile.The cutoff energy at the red end is sensitively dependent on the inner radius of the line-emitting region of the disk (Fabian et al. 2000).The radius of the ISCO is determined by the spin of the black hole and determines the minimum photon energy, which is also the energy that the red wing can extend.There is an accurate relationship between the cutoff energy at the red end and the spin of the black hole.Even if the position and motion of the corona are changed, the spin of the black hole can still be obtained by the cutoff energy at the red end of the observed line profile.
The peak at the red part of the line profile is closely related to the emissivity of the emission line of the inner region of the accretion disk moving away from the observer (f ∼ f obs + 90°).The photons emitted from this disk region are mainly affected by the Doppler shifting and gravitational redshift.As the azimuth of the corona increases along the direction of the spin of the black hole, the red peak of the line profile first increases, as shown in Figure 4(A).Then it moves to the blue part and gradually weakens, and in this case only one obvious peak can be observed in the line profile (Figure 4(B)).Finally, the peak increases again and moves to the blue end, resulting in the blue peak of the profile.It is shown in Figure 4(C) that a new red peak is formed at the end of the red wing.This can be explained by the fact that the changing line profile depends on the asymmetric distribution on the azimuth angle of disk emissivity.Different from the assumed lamppost model and the assumption of the power law of the primary hard X-ray, the region with the maximal disk emissivity is not always the innermost region of the accretion disk.When the corona azimuth is around 0°-90°, the emissivity of the inner region of the accretion disk moving away from the observer increases with the increase of the corona azimuth.When the corona azimuth is around 120°-210°, the emissivity gradually decreases, and the position of the maximal emissivity also moves to the outside of the accretion disk.When the corona azimuth is around 240°-330°, the position of the maximal emissivity in the disk moving away from the observer moves to the outside of the accretion disk and the emissivity at the inner region is low.The change of the observing inclination affects the peak energy and intensity of the red peak in the profile of the emission line.However, it has little effect on the trend of the variation of the line profile at the different azimuthal angles (f s ).For instance, as shown in Figure 4, the red peaks become more strong with the increase of the azimuthal angle, which is basically true in both cases of the observing inclinations (θ obs = 30°, 60°).
The peak at the blue part of the line profile is related to the emissivity of the region of the accretion disk moving toward the observer (f ∼ f obs − 90°).The photons emitted from this disk region are mainly affected by the strong Doppler shifting and Doppler beaming.As the azimuth of the corona increases along the direction of the spin of the black hole, the blue peak of the line profile first decreases and then increases.When the corona azimuth is around 120°, the blue peak of the line profile is weakest but still more than 15% of the maximum of the line profile.The change of the observing inclination also affects the energy and flux of the blue peak in the line profile.For a certain observing inclination, the cutoff energy of the blue peak is almost certain and stable with the different position and motion of the corona.
The line profiles for different inclinations of the corona and observer (15°, 30°, 60°) are shown in Figure 5.The observing inclination mainly affects the position of the peak in the line profile and the cutoff energy of the blue end.With a large observing inclination, the photons emitted from the accretion disk get strong Doppler shifting and Doppler beaming.As expected, the higher the energy of the peaks in the line profile, the higher the cutoff energy at the blue band of the emission line.The inclination of the corona mainly affects the intensity of the emission line at different energies.As the inclination of the corona increases and the corona is closer to the accretion disk, the red part of the line is enhanced, the blue part is weakened, and the peak of the emission line is narrower.This is because more photons emitted from the corona fall to the inner region of the accretion disk, resulting in a higher and more concentrated emissivity in the inner region of the disk.Photons with a strong gravitational redshift contribute to the strong red part and lead to a narrow peak of the line profile.When the inclination of the corona is large (θ s = 60°in the bottom panel of Figure 5), the blue part of the emission line is  The spin of a black hole is assumed to be a = 0.998.The corona radius, its inclination with respect to the spin axis of the rotating black hole, and its Lorentz factor are taken to be r s = 2, θ s = 30°, and β r = 0.3, respectively.Each of the colors shows the resulting profile for the different azimuth of the corona.The observing inclination θ obs = 30°for upper panels and 60°for lower panels.so weak that it is hardly observable.However, it can be found that the inclination of the corona makes little effect on the position of the red peak, which moves to the blue band as the observing inclination increases.Depending on the inclination of the corona, the observing inclination may be estimated to be small, with a large corona inclination according to the observed cutoff energy at the blue end of the line profile.Therefore, it is necessary to obtain the whole line profile, including the red and blue parts, to accurately estimate the observing inclination.
In the above discussion, the velocity of the corona is taken to be β r = 0.3.Although most studies of the corona region show that the corona has a mildly relativistic bulk velocity, we also calculate the profile of the emission line for the cases of the corona with higher radial velocities (see Figure 6).Contrary to the effect of corona inclination, the higher corona radial velocity results in a weaker red part in the profile of the emission line because the higher corona velocities cause more photons to be emitted into the outer region of the accretion disk, from where the emitted photons get a weak gravitational redshift.The radial velocity of the corona hardly affects the position of the peak in the line profile.However, with the strong relativistic velocity of the corona, as shown in the lower panel of Figure 6, there are more photons emitted to the outer region of the accretion disk, which is far from the central black hole, while only a small part of the inner disk region can receive the emission from the corona.The emissivity of the inner region of the accretion disk is weak, and there is only one distinct peak that can be observed on the line profile.Photons emitted from the outer region of the accretion disk have a weak Doppler shifting, and their frequencies almost remain constant.With the corona at some special azimuths (shown in Figure 6, they are 90°and 180°), the observed line profiles cut off around ν obs /ν 0 = 1.The line profiles in the case of the Schwarzschild black hole are also shown in Figure 7.There is only one distinct blue peak in the line profile in this case.Similar to the variation of the blue peak of the line profile in the case of the rotating black hole, with the increase of the azimuth of the corona, the blue peak of the broad emission line decreases first and then increases.In the case of the Schwarzschild black hole, there is a similar shape and intensity of the line profiles at f s = 0°and f s = 180°because the dragging effect of the Schwarzschild black hole on the surrounding spacetime is weaker than that of the rotating black hole.The distribution of photons from the corona received by the accretion disk is more symmetrical on two sides of the corona, and the emissivity distribution on the azimuth angle of the corona is more symmetrical.Unlike the case of the extreme Kerr black hole, no matter what the corona inclination is, the blue peak can always be observed.An accurate observing inclination can be obtained from the observed cutoff energy at the blue band.

Central Energy and Line Flux
The central energy and the total flux of the relativistic broad emission line can be defined as follows (You et al. 2020): Time-dependent changes of the central energy are generally used to study the quasi-periodic oscillation in the X-ray flux of accreting stellar-mass black holes (Ingram et al. 2016;Nathan et al. 2022).In this model, the periodic changing azimuth of the corona results in the periodic changing central energy and flux of the emission line.There are significant variations of them with the corona azimuth, which depend on the different spin of black holes, the observing inclinations, inclinations, and Lorentz factors of the corona.
The variations of the central energy and flux of the emission line with the different inclinations of corona and observer (15°, 30°, 60°, 75°) are shown in Figure 8.It shows the central energy and flux of the line change periodically with the azimuth of the corona, which is caused by the asymmetric distribution of the emissivity of the accretion disk on the azimuth angle.In most cases, due to the strong gravitational redshift near the black hole, the central energy of the emission line is lower than that in the rest frame.
The average central energy of the emission line is mainly affected by the inclination of the corona and the observing inclination.Small corona inclinations and large observing inclinations result in the high average central energy.The large observing inclination leads to the strong effect of the Doppler beaming for the photons emitted from the accretion disk moving toward the observer.For the case of large corona inclination, more photons from corona fall to the inner region of the accretion disk, increasing the emissivity in this region of the disk.In the off-axis corona model, the emission lines usually have a stronger red wing on average than in the other on-axis models.When the observing inclination is small, the Doppler beaming is weak, and thus the inclination of the corona dominates the central energy of the emission line.When the observing inclination is large and the corona inclination is small, the result is that the average of the central energy of the line is slightly higher than the energy of the line in the rest frame (shown in Figure 8(G)).
The observing inclination also affects the degree of change in the central energy of the emission line.The large observing inclination results in the significant change of the central energy.For example, the difference between the maximal and minimal central energy (E E , max min ) in the case of a large observing inclination (75°) is about 4 times larger than the result in the case of a small observing inclination (15°), with the same position and radial velocity of the corona.
Figure 7.The line profile for the different inclinations and azimuths of the X-ray corona with a Schwarzschild black hole with the spin of the black hole taken to be a = 0.The radial location of the corona and its Lorentz factor of the radial motion are assumed to be r s = 6 and β r = 0.3, respectively.The upper/lower panels show the results for the different inclinations of the corona (θ s = 15°, 30°, and 60°), and the left/right panels show the results for the different observing inclinations (θ obs = 15°, 30°, and 60°).
Figure 8.The variation of the central energy E c (left panels) and flux F L (right panels) of the line with the different azimuth angle f s .The spin of the black hole is taken to be a = 0.998.The upper/lower panels show the results for the different observing inclinations (θ obs = 15°, 30°, 60°, and 75°).The different colors show the results for the different corona inclinations (θ s = 15°, 30°, 60°, and 75°).The radial location of the corona and its Lorentz factor of the radial motion are assumed to be r s = 2 and β r = 0.3, respectively.
Figure 9.The variation of the central energy E c (left panels) and flux F L (right panels) of the line with the different azimuth angle f s .The spin of the black hole is taken to a = 0.998.The upper/lower panels show the results for the different Lorentz factors of the corona (β r = 0, 0.5, and 0.9).The radial location and inclination of the corona are assumed to be r s = 2 and θ s = 30°, respectively.
The effect of the radial velocity of the corona on the change of the central energy of the emission line is shown in the left panels of Figure 9.The radial velocity of the corona has some effect on the average central energy.When the radial velocity of the corona is low, the higher velocity results in a higher average of the central energy.When the corona moves with an extreme relativistic velocity and the inclination of the corona is large, the average central energy of the emission line decreases slightly.This is because the high-velocity corona emits more photons into the outer region of the accretion disk, resulting in a lower emissivity in the inner region of the disk, and thus the blueshifted emission lines are observed.But when the corona moves with an extreme relativistic velocity and the inclination of the corona is large, more photons are emitted to the outer region of the accretion disk with the lower Kepler velocity.The photons emitted from this region receive a weak Doppler shifting.Their blueshift is not strong, and the ratio of their frequencies to the intrinsic frequency is usually around 1. The variation of the central energy is also affected by the radial velocity of the corona, which first decreases and then increases with the increase of the radial velocity.
The effect of the spin of the black hole on the change of the central energy of the emission line is shown in the left panels of Figure 10.A large spin of the black hole leads to a lower average central energy of the emission lines.When the spin of the black hole is not so large, the effect of the corona inclination on the average central energy is weak and the average central energy is high and does not change significantly.There is a strong blue peak on the line profile in this case.When the spin of the black hole is high, the central energy of the emission line decreases significantly with the increase of the spin and a distinct red wing can be observed on the line profile.This is because the radius of the ISCO decreases as the spin of the black hole increases and the emission region in the accretion disk can extend closer to the black hole.The relationship between the radius of the ISCO and the spin of the black hole is not linear, and the radius of the ISCO decreases faster when the spin of the black hole is large.There is also a difference in the range of the variation in the central energy of low-spin and high-spin black holes.When the spin of the black hole is less than 0.9, it has little effect on the variation range of the central energy.The difference between the maximum and the minimum of the central energy decreases slightly with the increase of the spin of the black hole, while its ratio to the average central energy E E E 0.07 max min ave -< ( ) hardly changes with the spin of the black hole.This value is usually above 0.16 for the black holes with higher spins.This is because the inner region of the accretion disk is under the strong dragging effect of the spacetime around the black hole with high spin, which makes the asymmetry of the disk emissivity become significant.
In the case of a large inclination of the corona, the asymmetry distribution of the emissivity in the accretion disk is obvious.The proportion of the rising and falling phases of the change of the central energy of the emission line is not equal in the whole period of the variation of f s .Observations of the changes in central energy can infer the asymmetric emissivity in the inner region of the accretion disk.In the case of an extreme Kerr black hole, the central energy of the line shows a slow rise and a rapid fall, which is more obvious when the observing inclination is large.It can be seen in Figure 8(G) that, for the observing inclination θ obs = 75°, when the inclination of the corona is larger than 30°, the fall stage of the central energy is only within 45% of the whole period, while its rise stage is more than 55% of the whole period.When the spin of the black hole is not so large, the central energy shows a rapid rise and a slow fall, which is more obvious when the corona inclination is large.It can be seen in Figure 10(C) that when the inclination of the corona is larger than 30°, the fall stage of central energy is more than 55% of the whole period, while its rise stage is within 45% of it.
With the same intrinsic luminosity of the corona, the effects of the different positions and motions of the corona on the flux of the emission line can be compared, as shown in the right panels of Figures 8, 9, and 10.The flux of the emission line is affected by the number of photons emitted from the corona that reach the accretion disk, the distribution of the photons illuminated on the disk, and the observing inclination that determines which part of the photon flux from the accretion disk is enhanced.
The flux of the emission line averaged by the azimuth angle of the corona is mainly affected by the observing inclination, the radial velocity of the corona, and the spin of the black hole.With the increase of the observing inclination, the photons emitted from the accretion disk receive strong blueshift and Doppler beaming, making the line flux increase.The high radial velocities also increase the average flux of the line.This is mainly due to the high velocity of corona allowing more photons to be emitted from the corona to the outer region of the accretion disk, producing more photons emitted from this region.These photons with the strong Doppler beaming contribute to a larger average flux of the line than those emitted from the inner disk region, which is closer to the black hole.The small spin of black holes also makes the average flux of the line increase due to the large radius of the ISCO.The variation range of line flux is roughly the same as the changing trend of the average of the line flux, since the average flux is mainly contributed by the part with the largest flux.
The asymmetry distribution of the emissivity in the accretion disk affects the rising and falling phases of the emission line flux, so that their proportions in the period are not exactly equal.In most cases, the change trend of the line flux is a rapid rise and a slow fall.Only with the extreme Kerr black holes and the large inclination of the corona (>60°) does the line flux show a slow rise and rapid fall, which is more obvious when the observing inclination is large.When the corona inclination is not large and the spin of the black hole is less than 0.9, the change of line flux is almost symmetrical.The proportion of the falling phase and the rising phase of the line flux in the whole period will not differ by more than 8%.
Additionally, changes in line flux always precede changes in the central energy of the emission line, indicating that the emission line becomes strong first and then hard.The phase difference between these two changes can be calculated.When the spin of the black hole is less than 0.9, the difference between the phase of the highest central energy and the phase of the flux peak is less than 10% of the period.This difference is smaller (within 5% of the period) when the spin of the black hole is smaller (a < 0.5).When the spin of the black hole is larger, it is generally more than 10% of the period and increases as the inclination and the velocity of the corona decrease.For the corona stationary at 15°inclination (Figures 10(A) and (B)), it is more than 30% of the whole period.

Simulation of Observational Data
To check the effect of our off-axis model on the measurement of the source properties in observations, we use it to generate observational data and fit them with the standard on-axis lamppost model.In the calculation of each emission line, a large number of photons are generated, and their trajectories are tracked for a given coronal position and black hole spin.We generate the line profiles in the off-axis model with some selected parameters and create an XSPEC table model file.The version of XSPEC used in this work is 12.13.0g(Arnaud 1996).
The simulated observational data are generated in the offaxis corona model where the corona is off-axis and moving with a radial velocity.Then the spectra are fitted by using the standard lamppost model where the corona is stabilized on the axis of the black hole spin.To generate the observed spectra, the following model is used: where the fits file is the file for the off-axis model generated by the routine ftflx2tab in XSPEC.A power-law component is used to model the continuum from the corona observed directly.
We fix the photon index of the power-law component Γ = 1.6, the black hole spin a = 0.9, the radius of corona r s = 5, the inclination of corona θ s = 15°, the outer radius of the accretion disk r out = 400, and the inner radius r in = r ms .There are two cases of observing inclination, θ obs = 20°or 70°, and three cases of corona velocity, β r = 0.0, 0.2, or −0.2, representing the corona as stationary, moving outward, and moving inward, respectively.Eight spectra are generated for each case, for which the azimuth angles of the corona are taken from 0°to 315°with the space in 45°increments.
We assume the observation of a bright Galactic black hole X-ray binary system at z = 0.The energy flux is set to be 1 × 10 −8 erg cm −2 s −1 in the 1-10 keV energy range.The equivalent width of the emission line is set to be 250 eV.The fakeit command in XSPEC is utilized to simulate observations.The simulations use the instrumental response of the FPMA of NuSTAR (Harrison et al. 2013)  The impact of the off-axis and moving corona on the measurements of model parameters is significant.The azimuth angle of the corona affects the measurement of the black hole spin and the observing inclination, and the velocity of the motion of the corona affects the measurement of the height of the lamppost.In the case of both small and large observing inclinations, the measurement of the height of the lamppost is affected by the velocity of the corona.When the corona is stationary, the measured lamppost height is close to the input value.When the corona is moving outward, the measured heights of the lamppost are overestimated.When the corona is moving inward, the measured heights of the lamppost are underestimated.Because of the effect of the Doppler beaming on the moving corona, more photons irradiate in front of the moving direction of the corona.It produces a slight change in the emissivity of the emission line in the inner region of the The blue, orange, and green points represent the fitting results of the simulated data in the off-axis models with the input velocities of the corona β r = 0.0, 0.2, and −0.2, respectively, and the blue, orange, and green dashed lines represent the corresponding mean values.The black lines represent the values of the input parameters in the simulations; they are the input spin parameter a = 0.9, the photon index Γ = 1.6, the radius of corona r s = 5, and the inclination of the corona θ s = 15°.Points at the same f s are staggered for viewing convenience.The error bar on each parameter corresponds to the statistical uncertainty at the 90% confidence level.disk, which can be approximated by an on-axis corona with a higher (or lower) corona.
Measurements of the black hole spin rely on the assumption about the distribution of the emissivity in the accretion disk.In the case of a small observing inclination, the measured black hole spin varies with the azimuth of the corona.In the case of the corona with an inward motion, the measured spin is close to the input value.This indicates that the corona moving inward  produces a strong emissivity in the inner region of the accretion disk, so the red wing of the emission line is strong, which leads to the measured black hole spin by using the lamppost model that is very close to the black hole spin of the model.When the corona is stationary, the input values are within the error bar of the measurement results except for that at f s = 0°.When the  corona is moving outward, the measured spins are distributed over a large range and with a large statistical uncertainty.Among the eight measurement results corresponding to the azimuth of the corona, there are three measurements of the black hole spin whose error bar does not cover the input value.The error bar of the black hole spin is about 0.414-0.998 in the  case of f s = 315°.This is because the emissivity distribution produced by the outward moving corona in the inner region of the accretion disk is more asymmetric than that produced by the stationary or inward moving corona, which leads to a large difference in the shape of the line profile obtained in the offaxis model and the lamppost model, especially for the red wing of the emission line.Therefore, the measured spins of the black hole deviate from the input value with large uncertainties.
In the case of a small observing inclination, the azimuth of the corona has an obvious effect on the measurement of the observing inclination.When the corona is located near the region where the accretion disk is moving away from the observer, the measured observing inclination is close to the input value.When the corona is located near the region where the accretion disk is moving toward the observer, the measured observing inclination is accurate for models with an outwardly moving corona but is underestimated for models with a stationary or inwardly moving corona.This is because the inner region of the accretion disk is under the strong dragging effect by the black hole.Photons irradiating toward the inner region are bending at large angles around the black hole.When the azimuth of the corona f s is around 270°, photons emitted from the disk are more concentrated near the central frequency of the emission line than at the blue or red end.The energy at the peak of the emission line is lower than the energy to the maximum blueshift (see the peak of the emission line in Figure 4), resulting in a underestimated measured observing inclination.
In the case of a large observing inclination, both the measured black hole spin and the observing inclination are closer to the input values than those in the case of a small observing inclination.Around f s = 270°, the measurements of the black hole spin and observing inclination are relatively accurate in all cases of the motions of the corona.This is because the large observing inclination produces a strong blueshift from the accretion disk to the observer.Due to the effect of the Doppler beaming, the observed intensities of the emission from the different parts of the accretion disk are so inconsistent that the asymmetric emissivity distributions in the disk have less effect on the line profile.

Conclusion
In this work, we study the line profile of the iron fluorescence emission line in the off-axis corona disk model of a black hole in which the corona deviates from the spin of the black hole axis and moves away from the black hole with a relativistic velocity.The effects of both the position and velocity of the corona on the line profile are investigated.The illumination of the accretion disk by the corona and the profile of the emission line are dependent on the position and motion of the corona, since the disk region below the corona receives more flux, which is the most important factor to affect the line profiles.The distribution of the emissivity in the accretion disk around a black hole with high spin significantly breaks the power law and is asymmetric in azimuth, which results in a large difference in the line profiles observed in the different observing inclinations.
In the off-axis model, the central energy of the emission line is lower than that in the on-axis model, which means the blue peak on the line profile is weaker and the red peak is stronger.With the large observing inclination, it can be observed that the red peak is stronger than the blue peak.The relationship between the cutoff of the energy at the blue band of the line and the observing inclination is broken when the disk emissivity is extremely asymmetric.This is usually when the inclination of the corona is large or the corona velocity reaches the extreme relativistic velocity.
The central energy and flux of the emission line show periodic changes with the azimuth of the corona.The peak of the emission line flux always comes earlier than the peak of its central energy, which indicates that the emission line follows the behavior of increasing first and then becoming hard.
Our current model is limited by the statistical accuracy of the photons received on the accretion disk and the computational speed of ray tracing, which makes it difficult to use to analyze the data.Also, in the analysis of a real spectrum, it is necessary to calculate the whole reflection spectrum, not only the iron line.In future work, we will improve the model so that it can be used for real observations.Currently the most widely used relxill model is based on relline and rellinelp models (García et al. 2014).Both of these models are valid for the axisymmetric distribution of emissivity.By adding azimuth-related calculations to the model, the emission line profiles with the nonaxial symmetry of the emissivity can be quickly calculated.Finally, by using the angle-dependent reflection code to assign the reflected spectrum for each region on the accretion disk, the model will be extended to the calculation of the whole reflection spectrum.
= −(1 − 2r/Σ) is the time-time component of the Kerr matrix and β r is the relative physical radial velocity of the local rest frame with respect to the Boyer-Lindquist coordinate frames and the Lorentz factor 1

Figure 1 .
Figure1.Schematic of the disk-moving corona model.The black circle indicates the location of the black hole.The z-axis follows the direction of the spin.The accretion disk is considered to be in Keplerian circular orbit around the black hole with an angular velocity Ω.The green ball indicates the location of the moving corona, which is set at (r s , θ s , f s ) and assumed to be moving with a radial velocity β r .The observer is assumed to be in the x-z plane, considered infinitely far away from the black hole.The inclination angle between the observer's line of sight and the spin axis of the black hole is θ obs .

Figure 2 .
Figure 2. The illumination flux on the accretion disk by the corona at the different radii and with the different inclinations near the black holes with the different spins.Panels (A) and (C) show the results for the case a = 0 and the location of the corona (r s , θ s , f s ) = (6, 30°, 0°) and (6, 60°, 0°), and their projections are represented by the green triangles.Panels (B) and (D) show the results for the cases a = 0.998 and the location of the corona (r s , θ s , f s ) = (2, 30°, 0°) and (2, 60°, 0°), respectively.The Lorentz factor of the corona is taken to be β r = 0.3.

Figure 3 .
Figure 3.The illumination flux as a function of the radii and azimuthal angle in the accretion disk.Panels (A)-(D) correspond to the results shown in the panels of Figure 2.

Figure 4 .
Figure4.The line profile for the corona located at the different azimuthal angles.The spin of a black hole is assumed to be a = 0.998.The corona radius, its inclination with respect to the spin axis of the rotating black hole, and its Lorentz factor are taken to be r s = 2, θ s = 30°, and β r = 0.3, respectively.Each of the colors shows the resulting profile for the different azimuth of the corona.The observing inclination θ obs = 30°for upper panels and 60°for lower panels.

Figure 5 .
Figure5.The line profile for the different inclinations, azimuths of X-ray corona, and observing inclinations.The spin of the black hole is taken to be a = 0.998.The upper/lower panels show the results for the different inclinations of the corona (θ s = 15°, 30°, and 60°), and the left/right panels show the results for the different observing inclinations (θ obs = 15°, 30°, and 60°).The different lines in each panel show the results for f s = 0°, 90°, 180°, 270°.The radial location of the corona and its Lorentz factor of the radial motion are assumed to be r s = 2 and β r = 0.3, respectively.

Figure 6 .
Figure 6.The profiles of the emission line with the different Lorentz factors of the moving corona.The model parameters are the same as those in the Figure 5(D), (E), and(F) (r s = 2, θ s = 30°, a = 0.998, θ obs = 15°, 30°, and 60°) but with the different Lorentz factor of the radial motion of the corona β r = 0.5 (upper panels) and β r = 0.9 (lower panels).

Figure 10 .
Figure10.The variation of the central energy E c (left panels) and flux F L (right panels) of the line with the different azimuth angle f s .The upper/lower panels show the results for the different spin of black holes (a = 0, 0.5, and 0.8).The corona is located at the ISCO for each black hole.The inclination of the corona and its Lorentz factor of the radial motion are assumed to be θ s = 30°and β r = 0.3, respectively.
in the 3-79 keV energy band.Ancillary response and background files are selected, assuming a circular extraction region with a 60″ radius centered 60″ off axis.The exposure time for each observation is 40 ks.The simulated observational data are analyzed with the following model(Dauser et al. 2013):The parameters of the line energy, inner radius, outer radius of the disk, and redshift are fixed at E line = 6.4 keV, r in = r ms , r out = 400, and z = 0.The photon index of the power law and emission line model are linked in fitting.All other parameters are treated as free parameters.The best-fit values of the parameters are shown in Figures11 and 12.The residuals in terms of σ produced in fitting the spectra are shown from Figures 13 to 18.The reduced chisquare red 2 c is close to 1 for all observations.The simulation data for the off-axis model can be fitted using the lamppost model, and the values of the measured model parameters are distributed around the input values.

Figure 11 .
Figure11.The best-fit value of the lamppost height h, black hole spin a, observing inclination angle θ obs , photon index Γ of the power-law component, and reduced chisquare red 2 c from the simulated observational data with the observing inclination θ obs = 20°as a function the azimuth angle of the corona in the standard lamppost model.The blue, orange, and green points represent the fitting results of the simulated data in the off-axis models with the input velocities of the corona β r = 0.0, 0.2, and −0.2, respectively, and the blue, orange, and green dashed lines represent the corresponding mean values.The black lines represent the values of the input parameters in the simulations; they are the input spin parameter a = 0.9, the photon index Γ = 1.6, the radius of corona r s = 5, and the inclination of the corona θ s = 15°.Points at the same f s are staggered for viewing convenience.The error bar on each parameter corresponds to the statistical uncertainty at the 90% confidence level.

Figure 12 .
Figure 12.The same as Figure 11, but the input inclination angle in the model is θ obs = 70°.

Figure 13 .
Figure 13.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 20°and corona velocity β r = 0.

Figure 14 .
Figure 14.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 20°and corona velocity β r = 0.2.

Figure 15 .
Figure 15.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 20°and corona velocity β r = −0.2.

Figure 16 .
Figure16.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 70°and corona velocity β r = 0.

Figure 17 .
Figure 17.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 70°and corona velocity β r = 0.2.

Figure 18 .
Figure18.The residuals in terms of σ for the simulation of observations with the observing inclination θ obs = 70°and corona velocity β r = −0.2.