Radiation from an emitter revolving around a magnetized non-rotating black hole

Radiation from an emitter revolving around a magnetized non-rotating black hole

Valeri P. Frolov vfrolov@ualberta.ca    Andrey A. Shoom ashoom@ualberta.ca    Christos Tzounis tzounis@ualberta.ca Theoretical Physics Institute, University of Alberta, Edmonton, AB, Canada, T6G 2E1
July 18, 2019
Abstract

One of the methods of study of black holes in astrophysics is based on broadening of the spectrum of radiation of ionized Iron atoms. The line K associated with Iron emission at 6.4 keV is very narrow. If such an ion is revolving around a black hole, this line is effectively broadened as a result of the Doppler and gravitational redshift effects. The profile of the broaden spectrum contains information about the gravitational field of the black hole. In the presence of a regular magnetic field in the vicinity of a black holes the characteristics of the motion of charged ions are modified. In particular, their innermost stable circular orbits become closer to the horizon. The purpose of this work is to study how this effect modifies the spectrum broadening of lines emitted by such an ion. Our final goal is to analyze whether the change of the spectrum profiles can give us information about the magnetic field in the black hole vicinity.

pacs:
04.20.Cv, 04.70.Bw, 04.70.-s, 04.25.-g Alberta-Thy-09-14

I Introduction

There are more and more evidences that astrophysical black holes exist. Black hole candidates (both of stellar mass and supermassive) are identified by demonstration that a large mass compact object is located in a region of sufficiently small size, that practically excludes other different from a black hole models. Accretion of matter onto a black hole produces intensive radiation. By means of this radiation black holes manifest themselves. In particular such radiation may contain information about properties of the spacetime in the vicinity of a black hole and may confirm that a compact object is really a black hole. (For a comprehensive review of modern status of black holes in astrophysics see, e.g., a nice review Na:05 ().)

One of the most useful methods is to use Iron K lines as probes of the black hole vicinity (see, e.g., ReBe:77 (); BrChMi (); ZaNuPaIn:05 (); Za:07 () and references therein). This line of Iron has an energy that depends on the state of the ionization of the atom and is in the range of keV. Such a spectral line is excited as a result fluorescence of K in a relatively cold accretion disc. The line is intrinsically very narrow. When such an ion is revolving around a black hole, a distant observer registers that the line broadened as a result of the Doppler and gravitational redshift effects. It has a characteristic asymmetric double-peaked shape. The form and the details of the broadened line depend on the parameters of the Keplerian orbit of the emitter.

There exist an important difference between Einstein and Newton gravity. Namely, for a particle moving in the gravitational field in the former case there exists the innermost stable circular orbit (ISCO). The radius of ISCO depends on the rotation parameter . For a non-rotating (Schwarzschild, ) black hole of mass the ISCO radius is . In the presence of rotation (in the Kerr metric) it is closer to the black hole horizon, so that in the limit of the extremal rotation () ISCO radius becomes . In principle, this allows one to use the method based on study of the spectral line broadening to determine a rotation parameter for a black hole. For a comparison with observations the broadened spectrum of Iron atoms emission should be additionally averaged over positions of the emitting atoms in the disk. This requires knowledge of the emissivity distribution on the disk (see, e.g., discussion in KaMi:04 ()). The K spectrum broadening from accretion disk around non-rotating and rotating black holes was calculated in LtoN95 (); CuBa:73 (); Cu:75 (); CaFaCa:98 (); FrKlNe:00 (); FuWu:04 (). A comprehensive review of the application of the fluorescent Iron lines as probe of properties of astrophysical black holes can be found in FaIwRe:00 (); ReNo:03 (); FuTs:04 (); Jo:12 ().

In this paper we would like to discuss another interesting aspect of spectral line broadening method, namely usage of Iron K lines as probes of the magnetic field in the black hole vicinity. There are (both theoretical and observational) evidences that magnetic field plays an important role in the black hole physics. Magnetic field seems to be essential to angular momentum transfer in accretion disks BaHa:98 (); Kr:01 (). Recent observations of the Faraday rotation of the radiation of a pulsar in the vicinity of a black hole in the center of the Milky Way (SgrA*) indicates that at a distance of few Schwarzschild radii there exists magnetic field of several hundred Gauss Ea:13 (). This supports emission models of SgrA* that requires similar magnetic field for explanation of the synchrotron radiation from a near horizon region (see, e.g., Fa (); Mo (); De ()).

In the Blandford-Znajec model a regular magnetic field in the black hole vicinity is postulated to explain black hole jets energetics Zn:76 (); BlZn (); MEM (). For example, in order to produce power of the magnitude erg/sec seen in the jets of supermassive (with mass ) rotating black holes, regular magnetic field of the order G is required MEM (). Another mechanism for energy extraction from a rotating black hole and formation of relativistic jet, based on the analogue of the Penrose mechanism for a magnetic field was proposed in KoShKuMe:02 (); Ko:04 (). The authors performed numerical simulations and demonstrated that the power in the jet emission as a result of such MHD Penrose process is of the same order as the estimate based on the Blandford-Znajek mechanism. In particular, for a strong magnetic field G around a stellar-mass () extremely rotating black hole the power of emission is estimated as erg/sec, which is similar to the power seen in gamma-ray bursts. Estimates based on the observed optical polarization for a number of active galactic nuclei gives the value G for the magnetic field at the horizon of the corresponding black holes Si:09 (); Pi:00 (); SiGn:13 ().

In this paper we discuss how a regular magnetic field in the vicinity of a black hole changes the parameters of the charged particle orbits. Our aim is to obtain the images of such orbits as they are seen by a far distant observer. We also study the affect the broadening spectrum of emission lines of Iron ions moving near nagnetized black holes. Let us mention that influence of a magnetic field on the distortion of the Iron K line profile was earlier discussed in Za:03 (). The authors focused on the splitting of lines of emission due to Zeeman effect. They demonstrated that this effect might be important if the magnetic field is of the order G. We consider completely different effect, which might exist at much weaker magnetic fields. Namely, we assume that an Iron ion, emitting radiation, revolves around a magnetized black hole. For simplicity we assume that the black hole of mass is non-rotating. The Lorentz force, acting on a moving charged emitter in the magnetic field, modifies its motion.

The analysis of the corresponding equations of motion shows that the position of the ISCO for charged particles is closer to the black hole horizon than ISCO radius for a neutral particle () AG (); FS (). This modification of the orbit is more profound in the case when the Lorentz force is repulsive. It is characterized by the dimensionless parameter

(1)

Here and are charge and mass of the charged particle, is the strength of the magnetic field, and is the mass of the black hole. (Here we use CGS system of units.) The parameter is proportional to the ratio of the cyclotron frequency of a charged particle in the magnetic field in the absence of gravity, to the Keplerian frequency of a neutral particle at ISCO in the gravitational field of the black hole. To estimate the value of this parameter one can consider a motion of a proton (mass g and charge ). Then for a stellar mass black hole, , this parameter takes value for the magnetic field G. For a supermassive black hole the corresponding field is G. If the charge of the ion is and its mass is the corresponding expression for the magnetic field parameter contains an additional factor . One can expect that for astrophysical black holes the parameter is large.

In fact, the magnetic field essentially modifies the orbits already when the parameter is of the order of 1. For the repulsive Lorentz force case the radius of ISCO in the strong magnetic field () can be located arbitrary close to the horizon. As a result, two new effects are present in the process of emission of an ion of Iron in magnetized black holes: (i) position of ISCO depends on the magnetic field, and (ii) even for a circular motion of the same radius, angular velocity of charged particles differs from the Keplerian angular velocity. In this paper we study how these effects modify the broadening of the K spectrum.

The paper is organized as follows. In Sec. II we remind the main formulas concerning the charged particle motion near a magnetized black hole. In Sec. III we collect results for ray tracing in the Schwarzschild geometry, which are required for construction of the orbit’s images and the calculations of the spectrum broadening. In Section IV we discuss the particle orbits images. In Sec. V. we derive an expression for a broadened spectrum for the sharp spectral lines emitted by the point-like source moving at the circular orbit near a magnetized black hole. The numerical results for the spectral function are presented in Sec. VI and discussed in Sec. VII. The latter section contains also general discussion of the results of the present paper and their possible extensions. Some auxiliary calculations are collected in appendices.

In this paper we use the sign conventions adopted in MTW () and units where .

Ii Charged particles in magnetized black holes

ii.1 Magnetized black hole

We consider a magnetized non-rotating black hole. We assume that magnetic field is weak, so that its back reaction on the black hole’s spacetime geometry can be neglected. The metric of the black hole is

(2)

where is the black hole’s gravitational radius, and

(3)

is the metric on a unit sphere . Such a spacetime has only one dimensional parameter, , and one can write the metric in the form

(4)

where and are dimensionless time and radius and . In what follows, we shall use this dimensionless form of the metric.

The metric possesses four Killing vectors

(5)
(6)
(7)

The first one is the generator of time translations, while the other three are the generators of rotations.

We choose the magnetic field of the form

(8)

where (see, e.g., AG (); Wald ()). It is static and axisymmetric, and it is homogeneous at the asymptotic infinity with the strength and directed orthogonal to the equatorial plane . The electromagnetic field tensor has the following form:

(9)

ii.2 Equatorial motion of a charged particle

In this paper, we shall adopt a number of simplifying assumptions. We consider a freely moving single Iron ion, which emits a sharp spectral line, and we neglect its interaction with other matter, surrounding a black hole. Then a charged particle motion obeys the equation

(10)

Here is the particle 4-velocity, , is its dimensionless proper time, and are its electric charge and mass, respectively.

For the motion around the magnetized black hole there exist two conserved quantities associated with the Killing vectors Eq.(5): the specific energy and the specific generalized azimuthal angular momentum ,

(11)
(12)

Here the parameter characterizes the dimensionless strength of the magnetic field [see Eq.(1)].

It is easy to check that the -component of Eq.(10) allows for a solution . This is a motion in the equatorial plane of the black hole, which is orthogonal to the magnetic field. Here we restrict ourselves to this type of motion for which the conserved quantities Eq.(11) and Eq.(12) are sufficient for the complete integrability of the dynamical equations.

The complete integrability allows one to write the equations of motion in the equatorial plane in the following first order form

(13)
(14)

Here

(15)

is the effective potential.

For the motion in the equatorial plane the equations are invariant under the following transformations:

(16)

Thus, without loss of the generality, one can assume that the charge (and hence ) is positive. For a particle with a negative charge it is sufficient to make the transformation Eq.(16). According to the adopted convention, we have . The parameter can be positive or negative. For (sign ) the Lorentz force, acting on a charged particle, is repulsive, i.e., it is directed outward from the black hole. Following the paper AG (), we call such motion anti-Larmor motion. In the opposite case when (sign ) the Lorentz force is attractive, i.e., it is directed toward the black hole. We call it Larmor motion.

ii.3 Stable circular orbits (SCO’s)

The equatorial motion around the magnetized Schwarzschild black hole Eq.(2) was studied in detail in the papers FS (); FF (). We remind here some properties of the circular motion, following this paper. For the circular motion the radial coordinate is fixed .

Extrema of the effective potential are defined by the equation . We assume that , so that this is a local minimum and the corresponding circular orbit is stable. We call such an orbit a stable circular orbit or briefly SCO The equation allows us to find the parameter as a function of and ,

(17)

Accordingly, the specific energy of the particle at SCO is

(18)

Equations Eq.(14) determine the angular position of the particle . We choose the coordinate so that the motion with is counterclockwise and changes in the interval . The corresponding dimensionless angular frequency of the particle motion is

(19)

Substituting expressions Eq.(17) and Eq.(18) into Eq.(19) we find the angular frequency as a function of and . In this expression, for a fixed value of the parameter , the specific energy and the parameter are defined by the value of , which corresponds to minimum of the effective potential.

For greater than the ISCO radius this is a local minimum. If the specific energy is greater than the value of the potential at this minimum, the radial motion is an oscillation between the minimal and maximal values of the radius. As a result the motion with negative remains smooth, while for and large enough magnetic field the particle trajectory becomes curly. One can describe such a trajectory as a result of superposition of cyclotron rotation along small cycles and a slow drift motion of the center of the cycle around the black hole. One can expect that as a result of the synchrotron radiation such a trajectory would become more smooth and finally relax to a circular one. For more details concerning general type of motion in magnetized black holes, see FS ().

ii.4 Innermost stable circular orbits (ISCO’s)

For a given magnetic field there exist a minimal radius of SCO. For circular orbits with smaller radius the motion becomes unstable. The corresponding innermost stable circular orbit is known as ISCO. The ISCO radius is a simultaneous solution of the following two equations

(20)

Figure 1: Magnetic field of a magnetized black hole is a function of ISCO radius . Labels and stand for the anti-Larmor and Larmor orbit branches, respectively.

In this case, the parameter is not free and it depends on the value of . As a result, for ISCO we have,

(21)
(22)
(23)

Here

(24)
(25)

and and .

Figure 1 shows a relation between the value of the magnetic field and the radius of the corresponding ISCO. Labels and stand for anti-Larmor and Larmor orbits, respectively. Specific energy at ISCO orbits as a function of the ISCO radius for both types (anti-Larmor and Larmor) of motion is presented in Figure 2. The next plot (Figure 3) shows the angular velocity at ISCO as a function of its radius . For small () both branches and approach the same value , which is the Keplerian ISCO angular velocity for a non-magnetized black hole.


Figure 2: Specific energy of a charged particle at ISCO in a magnetized black hole as a function of ISCO radius . Labels and stand for the anti-Larmor and Larmor orbit branches, respectively.

The asymptotics of these functions for anti-Larmor ISCO and large are111In this paper, we focus on the anti-Larmor orbits which can be arbitrary close to the horizon of the black hole. However, similar expressions can be easily obtained in the large limit for Larmor orbits. For example, one has (26) (27) (28)

(29)
(30)
(31)

In the limit of the strong magnetic field () (branch ) and (branch ).


Figure 3: Angular velocity as a function ISCO radius in a magnetized black hole. Labels and stand for the anti-Larmor and Larmor orbit branches, respectively.

ii.5 Circular motion in the rest frame basis

Let us introduce a local rest frame

(32)
(33)

Figure 4: Velocity of a charged particle at the ISCO in a magnetized black hole at a function of ISCO radius . Labels and stand for the anti-Larmor and Larmor orbit branches, respectively.

The four-vector of velocity for the circular motion with (dimensionless) angular velocity can be written as follows

(34)
(35)

Here (which can be either positive or negative) is the velocity of the particle with respect to a rest frame, and is the corresponding Lorentz gamma factor. A simple analysis shows, that the velocity at the anti-Larmor ISCO remains close to in the entire interval of ISCO radii and, hence, this motion is not very relativistic. For the opposite direction of motion, the velocity for ISCO changes from , in the absence of the magnetic field, up to 1 for very large magnetic field (see Figure 4). For more details see FF ().

ii.6 Near-horizon orbits

Denote by a proper distance from the black hole’s horizon

(36)

At the horizon and in its vicinity one has

(37)

where is the surface gravity of the black hole. We consider now a limit when is fixed, while . We consider a space region near the equatorial plane which has the size in the orthogonal to direction much less than . Denote , and , then the metric Eq.(2) in such a domain can be written as follows

(38)

This is the Rindler metric. In the spacetime domain covered by the Rindler coordinates the ISCO is described by a simple equations: ,  const,  const. Such a particle moves in the direction with a constant velocity [see Eq.(34)]. The equation of the charged particle motion Eq.(10) implies

(39)

This equation has a simple meaning: In order to stay at  const, the gravitational attraction force [the left-hand side of Eq.(39)] must be compensated by the repulsive Lorentz force [the right-hand side of Eq.(39)]. Simple calculations give

(40)
(41)

The magnetic field in the Rindler domain has only one component

(42)

so that the Lorentz force acting on a particle moving in the direction with the velocity is

(43)

As expected, it is directed along the axis, that is away from the horizon. After substitution of the relations (40) and (41) into Eq.(39)] one obtains

(44)

Since close to the horizon , one gets

(45)

This formula correctly reproduces the asymptotic relation (29).

ii.7 Main properties of anti-Larmor particle motion in magnetized black holes

Let us summarize. The ISCO radius for charged particles in the magnetized black holes is smaller than in both the cases of anti-Larmor and Larmor orbits. However, for the Larmor orbits in the presence of the magnetic field the specific energy for such orbits is larger than for ISCO with , while for anti-Larmor orbits decreases with the magnetic field. This means that for a given direction of the magnetic field and direction of the particles motion the behavior of charged particles with opposite charges is qualitatively different. The ‘anti-Larmor’ particles can continue their circular motion and after losing their energy and angular momentum they can reach their ISCO located close to the black hole. Particle with the opposite charge after passing through the radius must fall into the black hole. This provides one with quite interesting mechanism of charge separation in the magnetized black holes. In the present paper we discuss only motion of the anti-Larmor particles near magnetized black holes. In order to simplify formulas from now on we omit the indices in the expressions similar to Eq.(21)–Eq.(23).

Figure 2 shows that for anti-Larmor ISCO the specific energy can be arbitrary small. This means that for large value of the energy , which can be extracted from a charged anti-Larmor particle before it reaches ISCO can be close to the proper energy of the particle. Thus, the efficiency of the energy extraction in the magnetized black holes can be high and exceed the efficiency of the Kerr black hole.

Anti-Larmor ISCO particles have velocity (as measured by local rest observer) which faintly depends on the value of the magnetic field (see Figure 4). It has the value for both the limits and . Moving with such velocity the particle returns to the local rest observer after time in his/her local time. Because of the time delay, a distant observer would see this motion as slowed down by his/her clocks. As a result of this effect, the observed angular velocity at the ISCO tends to zero in the strong field limit (see Figure 3).

Iii Null rays

iii.1 Conserved quantities and equations of motion

A distant observer receives information from an emitter revolving around the black hole by observing its radiation. Two different types of the observations are of the most interest: (1) Study of the images of the emitter orbits and (2) Study of the spectral properties of the observed radiation. The theoretical technics required for these two problems are slightly different. However, in both the cases one needs at first to perform similar calculations. Namely, one needs to integrate equations for the light propagation in the Schwarzschild geometry. This is a well studied problem. Many results concerning ray tracing as well as the study of the narrow spectral line broadening in the Schwarzschild spacetime can be found in the literature (see, e.g. FZ () and references therein). Since the magnetic field does not affect the photons propagation one can use similar technics for our problem. However, there are two new features of the problem. Namely, (1) the radius of the emitter can be less than , the ISCO radius for a neutral article, and (2) even if the charged emitter is at the same orbit as a neutral one, its angular velocity differs from the Keplerian velocity. For this reason one should perform the required calculations and adapt them to a new set-up of the problem.

In this section, we briefly remind main useful formulas concerning the null rays propagation in the Schwarzschild spacetime and fix notations used later in this paper.

Geodesic equation for a null ray is

(46)

where and is an affine parameter. For the symmetries Eq.(5)-Eq.(7) there exist three commuting integrals of motion

(47)
(48)
(49)

Let us remind that, we use the dimensionless quantities. In particular, this means that the ‘physical’ energy is . In what follows, it is convenient to use the following quantities:

(50)

iii.2 Motion in the equatorial plane

The motion of a ray (as well as the motion of any particle) in the Schwarzschild geometry is planar. One can always choose this plane to coincide with the equatorial plane. For such a choice and . Thus, one has only one conserved quantity, . To make notations brief we denote it simply as . The equation of motion in the equatorial plane can be written in the following first order form:

(51)
(52)

For outgoing rays, when increases along the trajectory, , and for incoming rays. For fixed value of the impact parameter the radial turning point (if it exists) is determined by the condition

(53)

The evolution of the angle along the trajectory can be found from the following equation:

(54)

Figure 5: Motion of a photon in the equatorial plane. The photon emitted at propagates to a distant observer along a trajectory without radial turning points (a direct ray). The ray emitted at is indirect. It at first moves to the black hole and only after it passes through a radial turning point it propagates to the distant observer.

In what follows, we shall use the following function:

(55)

The integral Eq.(55) can be written in terms of the elliptic function of the first kind 222Here we use the definition . One has

(56)

where,

(57)
(58)
(59)

Consider a ray emitted at the radius that reaches the infinity. Denote by the angle between the direction to the point of emission and the direction to the point of observation. It is easy to show that a null ray can have no more than one radial turning point. The emitted ray either propagates to infinity with monotonic increase of , or it at first moves to the smaller value of and only after that goes to infinity. In the former case the bending angle is

(60)

In the latter case one has

(61)

Here is a radius of the radial turning point.

iii.3 Integrals of motion and impact parameters

We are interested in study of propagation of photons emitted by an object revolving around a magnetized black hole in the plane orthogonal to the magnetic field. Using the freedom in the rigid rotations, it is convenient to choose the spherical coordinates so that this plane coincides with the equatorial plane . For such a choice, in the general case, the plane determined by the trajectory of the emitted photon will be tilted with respect to the equatorial one. There still remains freedom in rotation in the direction, preserving the plane connected with the charged particle motion, which we shall fix later.

To derive properties of such photons we write the corresponding equations

(62)
(63)
(64)

The turning points of the motion, where , are determined by the condition . Denote these angles by and . Denote by angle between the normal to the tilted plane and the normal to the equatorial plane. One has , so that and

(65)

In what follows, we shall consider rays emitted by a revolving body which propagate to infinity, where an observer is located. In order to characterize asymptotic properties of these rays, which are directly connected with observations one can proceeds as follows. Denote by and the asymptotic angles for the ray trajectory. The angles of displacement of the photon in the and directions are and . These angles decrease as . Multiplying them by and taking the limit one obtains the dimensionless impact parameters

(66)
(67)

Consider a unit sphere with the coordinates and denote a plane tangent to it at the point by . We call it the impact plane. Denote by and unit vectors in directed along the coordinate lines of and , correspondingly. We call the vector

(68)

the impact vector. Its norm is

(69)

One also has

(70)

iii.4 Asymptotic data for null rays

For study outgoing null rays it is convenient to rewrite the Schwarzschild metric Eq.(3) in the retarded time coordinates

(71)
(72)

where . The conformal metric, Eq.(72), is especially convenient for describing the asymptotic properties of null rays ar . This metric is regular at the infinity , so that the 3D surface with the coordinates is nothing but the future null infinity for our spacetime. Rays with the same asymptotic parameters are asymptotically parallel in the ‘physical’ spacetime with the metric Eq.(71). To fix a ray in such a beam one needs two additional parameters, namely the impact vector Eq.(68). Thus, a point at together with the impact vector uniquely specify a null ray which reaches infinity. We call these five parameters the asymptotic data.

Equations (63) are equivalent to the following set of equations:

(73)
(74)
(75)

We remind that for the outgoing ray . For given position at and the impact vector one can determine the integrals of motion and . For given asymptotic data one can integrate equations (73)–(75) back in time, from the starting point , and to restore the complete null ray trajectory.

Iv Orbit’s images

iv.1 Angular relations

Denote by , , an event of the radiation of a quantum by the emitter revolving around the black hole. This quantum is registered by a distant observer . Here, is the moment of the retarded time when the ray arrives to the observer, and is the angle between his/her position and the direction orthogonal to the plane . We use the freedom of rigid rotations around the axis and put the angle at the point of observations equal to zero.

For a discussion of the photons trajectories it is convenient to use a unit round sphere shown in Figure 6, which allows one to represent motion of photons and the emitter in the 2D sector. We embed this sphere in a flat 3D Euclidean space, so that a point on the surface of the sphere is uniquely determined by a unit vector with the origin at the center of the sphere. We call this 2D space an angular space. Motion of the emitter is represented by the equator of the sphere, while orbits of photons, since they are planar, are represented by large circles. We use the same letters and as earlier to denote a position of the points of the emission and the observer location in the angular space.


Figure 6: Angular definitions.

A trajectory of the photon emitted at and arriving to the observer is represented by a large circle, passing through these two points (see Figure 6). We denote by the angle between the vectors and from the center of the unit sphere to the points and , respectively. We call the bending angle. It changes from , when to when . We call such rays with primary to distinguish them from secondary rays, that make one or more turns around the black hole before they reach the observer. The characteristic property of these rays is that after the emission they move at first below the equatorial plane. The brightness of the secondary images generated by such rays is greatly suppressed. That is why we do not consider them in our paper.

As one can see from the Figure 6, the angle for the primary rays emitted in the interval monotonically decreases from the point of emission to . For the rays emitted from the other part of the circle the angle at first decreases. It increases after passing through its minimal value (an angular turning point ). Equation (66) implies that in the former case the coordinate of the image on the impact plane , while for the latter one . This means that the image of the part of the emitter trajectory lying in the half-plane with positive is located in the lower half of the impact plane , while the part with has the image in the upper half of the impact plane . Two points with are images of the radiation send by the emitter when its crosses the -axis, where . We denote by an angle between the direction of the photon motion and the velocity of the emitter. The angle is connected with the angle [see Eq.(65)] as follows: .

A simplest way to find relations between angles which will be used later is to consider a spherical triangle on a unit sphere. Denote by , and its angles, and by , and the length of the sides of the triangle, opposite to , and , respectively. Then, one has

(76)
(77)

For example, consider the spherical triangle (see Figure 6). It has the angles (at ), (at ) and (at ). The length of the sides opposite to the apexes , , and of this triangle are , , and , respectively. Using Eq.(76) and Eq.(77) one obtains

(78)

These equations together with the expressions Eq.(66)–Eq.(68) allow one to determine the impact vector in terms of the total angular momentum , the inclination angle and the position angle of the emitter

(79)
(80)

One also has

(81)
(82)

These relations, besides the inclination angle of the orbit and angular position of the emitter, , contain only one unspecified parameter .

iv.2 Map between equatorial and impact planes

Let us now consider the radial equation (62). For (domain in Figure 7) the ray trajectory has a radial turning point. Such a ray reaches the minimal radius that is always greater than , and after goes to infinity again. These rays always cross the equatorial plane in the black hole exterior. For the ray trajectory does not have a radial turning point, so that being traced back from they reach the horizon of the black hole. The radial ray, with , does not cross the equator before it enters the black hole. The same property have rays reaching in some vicinity of this ray with sufficiently small impact parameter (domain in Figure 7). They enter the black hole before they cross the equatorial plane. The rays with impact parameters outside the region cross the equatorial plane first. In both the cases, and one must put , since the radial turning points are absent.


Figure 7: Impact plane

Figure 8: The boundary between the regions and for different values of the observer’s angle : Curve 1: ; Curve 2: ; Curve 3: . A large circle is a curve .

Using Eq.(60) one can find the boundary between the regions and from the following relation:

(83)

(Let us remind that the rays in this domain do not have a radial turning point.) Figure 8 presents solutions of this equation for different values of the inclination angle .

Let us summarize this part. For a fixed position of the observer at infinity and the moment of arrival of the rays there exist a one-to-one correspondence between the region of the impact plane and the region of the equatorial plane, located outside the black hole. We call this map

(84)

iv.3 Direct and indirect rays

Consider a ray connecting a point on the equatorial plane and a distant observer . If such a ray does not have a radial turning point we call it direct. In the opposite case we call it an indirect ray (see Figure 5). If the radius of the emitter orbit is small enough all the rays from it to the distant observer are direct. For a larger radius of the orbit there exist such a value of the angle that for the part one has only direct rays, while for and the rays are indirect. We denote the critical inverse radius which separates these two cases by . For the radial turning point is located at on the equatorial plane. For this case one has

(85)

The critical value is a function of the inclination angle . To find it let us denote by the following integral

(86)

By change of the variable this integral can be rewritten in the form

(87)

Figure 9 show a plot of this function.


Figure 9: Function . It monotonically grows with from at and becomes infinite at .

The function is determined by the relation

(88)

The plot of is shown in Figure 10. monotonically increases from 0 (at ) to its maximal value at equal to

(89)

Figure 10: The critical inverse radius as a function the inclination angle .

Consider now a circular orbit with . The following equation determines an angle on such an orbit which separates its two parts (with direct and indirect rays):

(90)

It is convenient to combine the relations Eq.(60) and Eq.(61). We introduce the functions which is defined as follows:

(91)

where is defined by Eq.(53), , and function is defined by Eq.(87). Note that for numerical computations of it is more convenient to consider as a parameter.

The following equation:

(92)

establishes a relation between position (angle ) of the emitter on the orbit with the inverse radius and the angular momentum of the photon that reaches a distant observer with the inclination angle . In this relation one need to choose sign for a direct trajectory and for an indirect one.

The corresponding image on the impact plane can be found by using Eq.(79) and Eq.(80). By integrating equation (75) one obtains a relation between the time of emission, , and the retarded time of observation, , at .

In conclusion of this section let us give examples of the images of orbits on the impact plane. These images for the inclination angle and the inverse radius of the orbit equal to are shown in Figure 11.


Figure 11: Images of some orbits corresponding to . The innermost curve is the image of the orbit, it is formed by direct null rays. The next curve is the image of the orbit, it is formed by direct null rays as well. And the outermost curve is the image of the orbit. The lower part of the curve is formed by direct null rays. The upper part of the curve is formed by indirect rays. The circle represents the rim of the black hole shadow.

V Spectral broadening

v.1 Photon momentum and conserved quantities

In what follows, we use the following orthonormal tetrad at the point of emission :

(93)

The first of the vectors, , is in the direction of the Killing vector . The second vector is along the radial direction, while the last two vectors are tangent to the surface  const and  const. We choose to lie in the photon orbit plane and directed from the point of emission to the point of observation . The last vector is uniquely defined by the condition that the tetrad Eq.(93) is right-hand oriented. The unit vector in the equatorial plane and tangent to the orbit of the emitter can be written in the form

(94)

where

(95)

Consider a photon with the impact parameter . Denote by the plane of its orbit, and by the Killing vector generating rotations preserving . Then the momentum of the photon at the moment when it pass the radius can be written in the form

(96)

One has

(97)