#### 3.1. Integrals of Motion Defining the Boundary of Shadow

The basic parameters related to the integrals of motion on the photon geodesics are defined by [

1]

where the integrals of motion

E and

${L}_{z}$ are related to the Killing vectors

${K}_{t}$ and

${K}_{\varphi}$ in the axially symmetric geometry (

6).

The integral of motion

Q is related to the quadratic integral of motion

$\mathcal{K}$ given by [

1]

and related to the conformal Killing tensor

${K}_{\alpha \beta}$ existing in a spacetime of D-type of Petrov classification ([

1] and references therein).

The photon geodesics satisfy the equations

where dot denotes the derivative with respect to the affine parameter along a geodesic.

The innermost unstable photon orbits

$r=const$ which define the boundary of the shadow are defined by two equations

From these equations we find the relation between the parameters

$\eta $ and

$\xi $ on the orbits

and obtain the equation for

$\xi $
where

The determinant of Equation (

18) reduces to the form

The resulting general formulae for the integrals of motion

$\xi $ and

$\eta $ on the orbits read

For

$\mathcal{M}=M=const$ they coincide with those for the Kerr metric presented in [

1].

#### 3.2. Information About a Black Hole from Its Shadow

The celestial coordinates

$(x,y)$ related with the integrals of motion for the innermost orbits [

1], correspond to the impact parameters for the boundary of the gravitational capture cross-section as seen by an observer at infinity, since photons infalling with these impact parameters get to the innermost unstable photon orbits [

1,

2]

Here

${\theta}_{i}$ is the angular coordinate of an observer;

x is the apparent distance of the black hole image from the symmetry axis perpendicular to it;

y the apparent distance of the black hole image from its projection on the equatorial plane, perpendicular to it. Introduction of the impact parameters is shown in

Figure 2 Left, where

$\rho =\sqrt{{\rho}_{\perp}^{2}+{\rho}_{\left|\right|}^{2}}$.

To determine the asymmetry of the shadow we construct the fitting circle with the radius

R and the center at

$x={x}_{C}$, using the coordinates of the upper point of the shadow contour

$({x}_{t},{y}_{t})$ and of its left and right endpoints in the equatorial plane,

${x}_{L}$ and

${x}_{R}$. The asymmetry parameter

D is introduced as the distance between the right endpoints of the circle and of the shadow contour [

7] as shown in

Figure 2 Right, where

$R={x}_{C}-{x}_{L}$,

${R}^{2}={({x}_{C}-{x}_{t})}^{2}+{y}_{t}^{2}$ and

$D=2R-({x}_{R}-{x}_{L})$. Eliminating

${x}_{C}$, we obtain the radius of the fitting circle

and the distortion parameter

Equations (

22) and (

23) defining the contour of the shadow, can be written in the simpler form with using the integral

$\mathcal{K}$ given by (

14) in terms of

$\xi $ and

$\eta $, which reads

This transforms Equations (

22) and (

23) to the system of two equations

In the celestial coordinates

and Equation (

28) yield

Introducing the variables

we transform the system (

28) to

From the first equation we find

Differentiating equations of the system (

32) with respect to

u we obtain

which gives the equation for the derivative

$dy/dx$ (for

y and

x related by the curve for the boundary of the shadow)

Taking into account Equations (

31) and (

33) we obtain the derivative

$dy/dx$ on the shadow boundary dependently on the functions defining the unstable orbits forming the boundary

The upper point of the shadow contour

$r={r}_{t}$ is defined by

$dy/dx=0$ which gives

and Equation (

30) yields

As a result, the upper point of the shadow contour is determined by the system of Equations (

37) and (

38).

For an observer coordinate

${\theta}_{i}=\pi /2$ (

$s=1$) this system reduces to the simple form

Here and in what follows, distances are normalized to the mass parameter

M and

$f=r\mu \left(r\right)$, where

$\mu $ is the mass function

$\mathcal{M}\left(r\right)$ normalized to

M. For the Kerr black hole

$\mu \left(r\right)=1$ and we obtain

${r}_{t}=3;\phantom{\rule{3.33333pt}{0ex}}{x}_{t}=-2a\phantom{\rule{3.33333pt}{0ex}};\phantom{\rule{3.33333pt}{0ex}}{y}_{t}^{2}=27$ in agreement with the known results (see, e.g., [

1]).

The left and right points of the shadow contour correspond to

${y}^{2}=0$, and Equation (

30) gives the equation for the innermost orbit in the equatorial plane

Multiplying by

$[1+{\mu}^{\prime}]$, we obtain

Introducing the new variable

we obtain the cubic equation

For the equation in the canonical form

${w}^{3}+b{w}^{2}+cw+d=0$ with the coefficients

solutions are determined by the quantities

Types of solutions depend on the signs of

q and

S. For the constant mass the solutions should coincide with those for the Kerr black hole, for which

$q=1>0$ and

$S=4{a}^{2}(1-{a}^{2})\ge 0$. We can expect that the shape of a shadow for a regular black hole would not differ essentially from that for the Kerr black hole and assume that

$q>0$ and

$S\ge 0$. Then Equation (

43) has 3 real roots defined by

Taking into account (

42) we obtain the transcendental equation for

r
which gives the orbit radii forming the boundary of the shadow in the equatorial plane for an arbitrary regular axially symmetric metric

In the case of the Kerr metric

${\mu}_{k}=1$,

${\mu}_{k}^{\prime}=0$, and Equation (

48) yields

in agreement with [

1]. It follows that

$k=1$ applies to the retrograde orbit while

$k=2$ to the direct orbit. We can conclude that in general case the solutions specified by

$k=1$ and

$k=2$ in Equation (

48) represent the radii of the retrograde and direct orbits, respectively, for an arbitrary axially symmetric metric.

The shadow contour is calculated numerically for different values of the observer coordinate ${\theta}_{i}$. The spin of a black hole is determined from the relation between its spin and the distortion parameter for the boundary of its shadow.

In the equatorial plane the relation between the celestial coordinate

x and the orbit radius

r can be found by considering the innermost equatorial photon orbits which are specified by

are described by the function

$\mathcal{R}\left(r\right)$ in Equation (

15) which reduces to

and obey the system of equations in Equation (

16),

$\mathcal{R}\left(r\right)=0;\phantom{\rule{3.33333pt}{0ex}}d\mathcal{R}\left(r\right)/dr=0$ in the form

Ultimately we obtain for the celestial coordinate

xFor the Kerr geometry Equations (

54) and (

48) yield

${x}_{k}=6cos(\mathrm{arccos}(a/3)+2\pi /3)\pm a$ in agreement with the results presented in [

1].

For each value of

k in Equation (

48) we can now obtain

${r}_{k}$ and

${x}_{k}$ and determine the boundary of the shadow in the equatorial plane.

Equation (

54) with account of

${\mu}^{\prime}\ge 0$ for the considered class of metrics, gives the basic constraint

which suggests that (i) the shadow of a regular black should have to be smaller than that for the Kerr black hole, and (ii) the difference depends essentially on the density profile of a regular black hole.

Comparison of a black hole shadow with using de Sitter-Kerr metric for fitting its boundary, with the Kerr shadow can provide information about interior structure of a black hole.