Brightest point in accretion disk and black hole spin: Implication to the image of black hole M87*

A dark region and a bright part of accretion disk are viewed on the first image of the supermassive black hole in the galaxy M87 obtained by the Event Horizon Telescope. We demonstrate that a viewed dark region is a silhouette of the black hole event horizon, while the outline (contour) of this silhouette is an equator of the event horizon globe. A dark silhouette of the black hole event horizon is placed within the expected position of the black hole shadow, which is not revealed on the first image. We calculate numerically a relation between the viewed position of the black hole silhouette and the brightest point in a thin accretion disk in dependence of the black hole spin. A spin of the supermassive black hole in the galaxy M87, derived from this relation, is $a=0.75\pm0.15$.


INTRODUCTION
The Event Horizon Telescope (EHT) recently presented for the first time an impressive image of the supermassive black hole in the galaxy M87 [1][2][3][4][5][6]. There are two specific features on this image: the dark silhouette and the bright region.
A rather obviously to suppose that the bright region is related with the emission of accreting matter around the supermassive black hole M87*. But what is a physical origin of the dark silhouette, viewed on the presented image? It depends on the black hole astrophysical environment.
The quite different case is a black hole image produced by the luminous non-stationary matter plunging into black hole in a region between the unstable circular photon orbit r ph and the black hole event horizon r ph . An evident example of the luminous non-stationary matter is the inner region of thin accretion disk between the Innermost Stable Circular Orbit r ISCO (see definition, e. g., in [23]) and the event horizon. A non-stationary luminous matter in this region is spiraling down toward the event horizon and produces a dark image of the event horizon silhouette instead of the black hole shadow [8,[24][25][26][27][28][29][30][31]. In addition, the outline (contour) of this silhouette is an image of the event horizon equator [32][33][34].
In this paper we calculate numerically a form of the event horizon silhouette of the rotating Kerr black hole by modeling the emission of non-stationary luminous matter spiraling down into black hole in the inner region of thin accretion disk at r ISCO ≤ r ≤ r h . The trajectories of photons are calculated in the geometric optic approximation and it is also supposed that accretion disk is optically thin. The event horizon silhouette is smaller than the black hole shadow and is placed on the celestial sphere within the awaited position of black hole shadow.
We identify the dark region on the EHT image with the event horizon silhouette and, respectively, the bright region with the brightest point in accretion disk. For a case of the supermassive black hole M87* we find a distance of the brightest point in the thin accretion disk from the center of event horizon silhouette in dependence of the black hole spin parameter a. In result, from this dependence we find that a spin of the supermassive black hole M87* is a = 0.75 ± 0.15.

PHOTON TRAJECTORIES IN KERR METRIC
We describe the trajectories of photons (null geodesics) in Kerr metric by using the standard Boyer-Lindquist coordinate system [35] with coordinates (t, r, θ, ϕ) and with units G = c = 1. Additionally we put mG/c 2 = 1. With these units a dimensionless radius of the black hole event horizon is r h = 1+ √ 1 − a 2 , where a spin parameter of the black hole 0 ≤ a ≤ 1.
Trajectories of particles with a rest mass µ in the Kerr space-time are determined by three constants of motion: a total energy E, a component of angular momentum parallel to symmetry axis (azimuth angular momentum) L and the Carter constant Q, which is related with the non-equatorial motion of particles [9,36]. Particle trajectories (geodesics) in the radial and latitudinal directions are defined, correspondingly, by a radial effective poten-tial R(r) = [E(r 2 +a 2 )−La] 2 −(r 2 −2r+a 2 )[µ 2 r 2 +(L−aE) 2 +Q] (1) and a latitudinal effective potential (2) Trajectories of photons (null geodesics with µ = 0) in the Kerr space-time are determined only by two dimensionless parameters, λ = L/E and q 2 = Q/E 2 . These parameters are related with the impact parameters on the celestial sphere α and β seen by a distant observer placed at a given radius r 0 >> r h (i. e., practically at infinity), at a given latitude θ 0 and at a given azimuth ϕ 0 (see, e. g., [7,37] for more details): where Θ(θ) is from (2). We use integral equations of motion for photons [9,36] for numerical calculations of the gravitational lensing by the Kerr black hole where the effective potentials V r and V θ are from Eqs. (17) and (2). The integrals in (4), and (5) are understood to be path integrals along the trajectory. The path integrals in (4), (5) and (6) are the ordinary ones for photon trajectories without the turning points: In the case of photon trajectories with one turning point θ min (λ, q) (an extremum of latitudinal potential Θ(θ)), equation (4) is written through the ordinary integrals as In general, a lensed black hole produces infinite number of images [37][38][39][40]. Except the very special orientation cases, the most luminous image is a so called direct or prime image, produced by photons which do not intersect the black hole equatorial plane on the way to a distant observer. Meantime, the secondary images (named also as the higher order images or light echoes), produced by photons which intersect the black hole equatorial plane several times. The energy flux from secondary images as a rule is very small with respect to one from direct image.

BLACK HOLE SHADOW
A black hole shadow is the gravitational capture crosssection of photons from the stationary luminous background placed at radial distance from black hole exceeding the radius of unstable photon circular orbit r ph (see definition in [23]) to form the image of black hole shadow.
A black hole shadow in the Kerr metric, projected on the celestial sphere and seen by a distant observer in the equatorial plane of the black hole, is determined from the simultaneous solution of equations R(r) = 0 and [rR(r)] ′ = 0, where the effective radial potential R(r) is from Eq. (17). The corresponding solution for the black hole shadow (for a distant observer in the black hole equatorial plane) in the parametric form (λ, q) = (λ(r), q(r)) is (see, e. g., [7,9] for more details).
Quite a different black hole image produced in the case of black hole highlighted by the non-stationary luminous matter plunging into black hole inside the radius of unstable photon circular orbit r ph .

EVENT HORIZON SILHOUETTE
A viewed dark event horizon silhouette is recovered by gravitational lensing of photons emitted in the innermost part of accretion disk adjoining the event horizon. In a geometrically thin accretion disk, placed in the equatorial plane of the black hole, there is an inner boundary for stable circular motion, named the marginally stable radius or the Inner Stable Circular Orbit (ISCO), r = r ISCO (see e. g., [23] for more details): where The corresponding values of parameters E and L for particles in the accretion disk co-rotating with the black hole at a circular orbit with a radius r are obtained from the We use a simple model for describing the non-stationary motion of small gas elements of accreting matter in the region at r h ≤ r ≤ r ISCO by supposing of a pure geodesic motion of the separate compact gas element (or compact gas clump) in the accretion flow with orbital parameters E and L from (12) and (13), corresponding to the radius r = r ISCO . We suppose also that a thin accretion disk is transparent and the energy flux in the rest frame of small gas elements is isotropic and conserved during their spiraling down into the black hole. See in Figure 1 parameters of photon trajectories λ and q, reaching a distant observer from the rings in accretion disk at r = 0.01r h and at r = r ISCO . It is shown the case of rotating black hole with spin a = 0.75 in the galaxy M87, corresponding to θ 0 = 17 • .
To calculate the energy shift of photons and energy flux from the lensed image of accretion disk it is needed to take into account both a red-shift in the black hole gravitational field and Doppler effect. It is convenient to use in this calculations the orthonormal Locally Non-Rotating Frames (LNRF) [23,41], for which the observers' world lines are r = const, θ = const, ϕ = ωt + const, where a frame dragging angular velocity ω = 2ar (r 2 + a 2 ) 2 − a 2 (r 2 − 2r + a 2 ) sin 2 θ .
The requested photon energy in the frame, comoving with the compact gas cloud at θ = π/2, is [34,38] E(λ, q) = In this equation the azimuth velocity V (ϕ) of a compact gas cloud with orbital parameters E, L and Q = 0 relative the LNRF, falling in the equatorial plane onto a black hole, is A corresponding radial velocity in the equatorial plane relative the LNRF is . (17) where R(r) is defined in (1) with Q = 0. The components of photon 4-momentum in the LNRF are Respectively, the energy shift (the ratio of photon frequency at infinity to a corresponding one in the rest frame of the compact gas clump) is g(λ, q) = 1/E(λ, q). This energy shift is used in numerical calculations of the energy flux from the accretion disk elements measured by a distant observer, following formalism by C. T. Cunnungham and J. M. Bardeen [37].

SPIN OF THE BLACK HOLE M87*
Model direct images of the inner non-stationary part of thin accretion disk in the region r rh ≤ r ≤ r ISCO , adjoining the black hole event horizon, are presented in Figures 2-5. The gravitational red-shift and Doppler effect are taken into account in these images. Local artificial colors of the thin accretion disk images are related with an effective local black-body temperature of the accreting gas elements. These numerical calculations demonstrate that a viewed brightest point in the thin accretion disk is always (for all values of black hole spin a) placed at radius r = r ISCO at the point corresponding to photon trajectory without turning point and with the maximum permissible azimuth angular momentum λ > 0.
The derived relation between position of the brightest point in the thin accretion disk and the viewed black hole silhouette in dependence of the black hole spin is shown in Figure 5. From comparison of this relation with the image obtained by EHT it is followed that spin of the supermassive black hole of M87* is a = 0.75 ± 0.15/. See in Figure 6 the superposition of M87* image and the modeled image of thin accretion disk in the case of a = 0.65.

CONCLUSIONS
We calculated numerically a form of the event horizon silhouette of the rotating Kerr black hole by modeling the emission of non-stationary luminous matter spiraling down into the black hole in the inner region of thin accretion disk at r ISCO ≤ r ≤ r h . The resulting form of the event horizon silhouette does not depend on the local emission of accretion disk and governed completely by the gravitational field of the black hole. In a case of the supermassive black hole M87*, viewed at the inclination angle 17 • , a dark silhouette on the EHT image is the southern hemisphere of the black hole event horizon. The contour of this silhouette is an equator of the event horizon.
The brightest point in accretion disk corresponds to the largest (positive) azimuth angular momentum λ of photon with the direct orbit, reaching a distant observer without the turning points. By using the first image of the black hole in the galaxy M87, obtained by the EHT, we find that the corresponding value of the black hole spin is a = 0.75 ± 0.15.