Parameter constraints from shadows of Kerr–Newman-dS black holes with cloud strings and quintessence

Shadows of the Kerr–Newman-dS black hole surrounded by quintessence and a cloud of strings are investigated. For a spherically symmetric nonrotating black hole, its shadow is circular and its size is independent of an observation angle and a plane on which a circular photon orbit exists. The shadow sizes are significantly influenced by the parameters involving the cloud of strings, quintessence parameter, magnitude of quintessential state parameter, and cosmological constant. The black hole shadows increase with the cloud of strings and negative quintessential state parameter increasing or the quintessence parameter and cosmological constant decreasing. When the black hole is spinning and axially symmetric, the black hole shadow is dependent on the observation angle and the black hole spin. The effects of the parameters excluding the spin parameter and the observation angle on the sizes of black hole shadows in the rotating case are similar to those in the nonrotating case. The black hole shadows decrease as the black hole spins increase. When the observation angle in the range of 0 and π/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi /2$$\end{document} is large, the black hole shadow is deformed like the D shape for a high spin, but is close to a circle for a low spin. When the observation angle is small, the black hole shadow seems to be a circle regardless of the high or low spin case. Based on the Event Horizon Telescope observations of M87*, the constraint of the curvature radius is used to constrain these parameters. For slowly rotating black holes, the allowed regions of the parameters including the cosmological constant are given.


Introduction
The general theory of relativity predicts the existence of black holes in the universe.This prediction has been confirmed through lots of observation evidences.These evidences include the detections of the gravitational waves by LIGO [1] and the observations of the images of supermassive black hole M87* and SgrA* shadows by the Event Horizon Telescope (EHT) [2].
The shadow of a black hole is a black disk seen by an observer in the sky when the black hole is illuminated by a light source.This light source is distributed around the black hole but not between the observer and the black hole.The computation of black hole shadow is directly related to the study of photon regions, photon rings or spheres outside the event horizon of the black hole.For a Schwarzschild black hole with mass M , the bound photon orbits occur at r = 3M , and the critical impact parameter is ξ c = 3 √ 3M , which is the radius of observed photon ring or black hole shadow.In fact, the circle being the apparent shape of the shadow of a spherically symmetric black hole was first shown by Synge [3].Luminet [4] focused on the appearance of Schwarzschild black hole surrounded by an accretion disk.Two impact parameters are useful to determine the apparent positions of the shadow of a axially symmetric Kerr black hole, which was first investigated by Bardeen [5].The shadow of a rotating black hole is no longer circular.The spin of the black hole leads to the deviation of the shadow from a circle.There have been many other interesting studies concerning the shadows of Kerr-Newman black holes [6][7][8], Kerr-Newman-NUT Black Holes [9], and black holes surrounded by extra matter sources [10][11][12][13][14].The shadows of black holes in modified gravity have also been considered in numerous publications (see e.g.[15][16][17][18][19][20][21][22][23][24]).The obtained shadow images combined with the observations of M87* and Sgr A* shadows are helpful to test theories of gravity and to understand the geometrical structure of the event horizon and the parameters of black holes.
The length of a shadow boundary and a local curvature radius are two characterizations of a black hole shadow [25,26].The shadow boundary is a one-dimensional closed or open curve.For a spherically symmetric nonrotating black hole, the curvature radius of the black hole shadow is the radius of photon ring.For an axially symmetric nonvanishing spin black hole, the curvature radius has maximum and minimum values [27,28].The minimum and maximum of the curvature radius determine lower and upper bounds of the shadow size.Based on the observation of M87*, the black hole parameters can be constrained via the curvature radius.
In this paper we are interested in the study of shadows of Kerr-Newman-de Sitter (KNdS) black holes with quintessence and a cloud of strings [29][30][31].The cosmological constant associated with the vacuum energy is responsible for the accelerated expansion of the universe.This expansion is also due to quintessence dark energy [32,33].The universe is thought of as a collection of extended objects like one-dimensional strings instead of point particles [34,35].Fathi et al. [36] gave analytical expressions to the radii of planar and polar spherical photon orbits around a rotating black hole with quintessential field and cloud of strings.Critical orbits of particles and photons in the Schwarzschild black hole with quintessence and string cloud background spacetimes were investigated by Surya Shankar [37].Mustafa et al. [38] studied the influence of the cloud of string parameter and the quintessential parameter on the radius of the shadow of the Schwarzschild black hole and the weak defection angle.He et al. [39] considered the shadow and photon sphere of the Schwarzschild black bole in clouds of strings and quintessence with static and infalling spherical accretions.Effect of quintessential dark energy on black hole shadows was discussed by Singh [40].Atamurotov et al. [41] investigated the null geodesics and the shadow cast by the Kerr-Newman-Kiselev-Letelier black hole for different spacetime parameters consisting of the quintessence parameter, the cloud of string parameter, the spin parameter and the charge of the black hole.The metrics considered in the literature are parts of the KNdS black hole spacetimes.Now, we plan to focus on the shadows of KNdS black holes and constraining the black hole parameters through the curvature radius.
The paper is organized as follows.In Section 2, we introduce the null geodesic around the KNdS black holes and discuss circular and spherical photon orbits.In Section 3, we obtain the shadow curves observed by a locally rest observer, and analyze the local curvature radius for the black hole shadows.Then the parameters are constrained.Finally, we summarize our main results in Section 4.

Photon motions near KNdS black holes with extra sources
At first, we introduce a KNdS black hole with quintessence and a cloud of strings.Then, a Hamiltonian for the description of photons moving around the black hole is provided.Finally, circular photon orbits and spherical photon orbits are discussed.

KNdS black hole metric
In Boyer-Lindquist coordinates (t, r, θ, φ), the KNdS black hole surrounded by quintessence and a cloud of strings is described by the following metric [29] where the related notations are defined as M , Q and a stand for the mass, electrical charge and specific angular momentum of the black hole, respectively.Q and a are given in the ranges of |Q| ≤ M and |a| ≤ M .In addition, α q is a positive quintessence parameter, and ω q is a quintessential state parameter which satisfies the condition −1 < ω q < −1/3 in a scenario of the accelerated expansion Universe.b c denotes a positive parameter measuring the intensity of the cloud of strings [29], and Λ is a positive cosmological constant.In fact, this metric is a solution of the Einstein field equation with cosmological constant, which can be obtained from the Newman-Janis transformation of the nonrotating black hole solution.The total stress-energy tensor in the nonrotating solution is a superposition of three extra sources including the quintessence, cloud of strings and electromagnetic field.See Refs.[37][38][39] for more information on the KNdS spacetime with quintessence and cloud strings.The speed of light c and the constant of gravity G are taken as geometrical units, c = G = 1.Now, let us consider the domains of outer communication.We formally write ∆ r as where r ± represent the outer and inner horizons of the black hole, r c denotes the cosmological horizon [16], and f (r) is a function of r.If the parameters Q, a, b c , α q , ω q and Λ are chosen appropriately, the equation ∆ r = 0 has three real roots r ± and r c . Figure 1 plots two parameter spaces for the existence of the real root r c , where the other parameters are given.Thus, light rays can reach a rest observer's eyes within the region r + ≪ r 0 ≤ r c , where r 0 denotes the distance of the observer to the black hole.

Hamiltonian formulism of photon motions
The motion of a photon around the black hole can be represented by the Lagrangian formulism where λ is not the proper time but is an affine parameter.Notice that ω q and b c are two dimensionless parameters.To make the Lagrangian dimensionless, we give scale transformations to the other quantities: r → rM , t → tM , a → aM , Q → QM , Λ → Λ/M 2 and α q → α q M 1+3ωq .λ is also measured in terms of the black hole mass, λ → λM .In this way, the mass factor M is eliminated or becomes 1 in the Lagrangian.
On the basis of the dimensionless Lagrangian, the photon has a covariant 4-momentum where ẋµ = ( dt dλ , dr dλ , dθ dλ , dφ dλ ) corresponds to the photon 4-velocity.Because the coordinates t and φ do not explicitly appear in the Lagrangian, their corresponding momenta are conserved.The conserved quantities are the photon energy E and angular momentum L: Here, 0 < E < 1, and L is one of the three possibilities of L > 0, L = 0 and L < 0. Through a Legendre transformation, the Lagrangian corresponds to a Hamiltonian formulism Since the Hamiltonian does not explicitly depend on the affine parameter λ, it is a third constant of motion.This constant is always identical to zero for the null geodesics: Set a generating function S(r, θ) = S r (r) + S θ (θ), where S r and S θ are functions satisfying the relations p r = ∂S r (r)/∂r and p θ = ∂S θ (θ)/∂θ.Noting Eqs.(11) and (12), we have the Hamilton-Jacobi equation where K is a Carter constant [42].Hence, we have two equations where R(r) and Θ(θ) are expressed as The equations of motion for the null geodesics are Two equations regarding ṫ and φ can be obtained from Eqs. ( 9) and (10).They are also parts of the null geodesic equations, but are not written because they are not used in this paper.
Several notable points are given here.(i ) The Carter constant K is a fourth constant of motion in the Hamiltonian system (11).Its existence is because the Hamilton-Jacobi equation ( 13) allows the separation of variables.Thus, the null geodesic is integrable and regular.(ii ) When the photon gives place to a test particle, the Hamilton-Jacobi equation still allows the separation of variables and then the Hamiltonian (11) has a Carter constant nonequal to K. (iii ) When an external electromagnetic field surrounds the black hole (i.e., it is included in the Hamiltonian ( 11)), the Carter constant is not present for the motion of a charged particle near the black hole, but it is if the cosmological constant vanishes, as was reported in Ref. [29].

Circular and spherical photon orbits
Given R(r) = 0 in Eq. ( 17), the energy is solved by Taking two impact parameters we define an effective potential Notice that E + in Eq. ( 23) is a function of r given by Eq. ( 21), and E in Eq. ( 23) is a certain given value of the energy.In terms of the two impact parameters, Eqs. ( 17) and ( 18) are rewritten as Using Eq. ( 18), Rahaman et al. [43] obtained another effective potential where λ = λ/E.

Circular photon orbits
If Θ(θ) = 0 in Eq. ( 25) for any time λ, then θ remains invariant and photon orbits are always lying on a certain two-dimensional plane θ = ϑ.In this case, η is given by The motions of photons on the plane are governed by the effective potential Let us take the parameters a = 0.5, Q = 0.2, b c = 0.01, α q = 0.01, ω q = −0.35,ξ = 8.4, and Λ = 1.02 × 10 −26 .Here, such a cosmological constant is a scaled value, labeled as Λ sca .
It corresponds to a realistic value of the astrophysical scenario Λ rea = c 4 /(M 2 G 2 ).If the black hole mass M is the Sun's mass M ⊙ , we have the realistic value Λ For the supermassive black hole candidate with mass M = 6.5 × 10 9 M ⊙ in the center of the giant elliptical galaxy M87, we have Λ rea = (M ⊙ /M ) 2 Λ 0 = 1.088 × 10 −26 m −2 .Thus, Λ sca = 1.02 × 10 −26 M −2 corresponds to the realistic cosmological constant Λ rea = 1.11 × 10 −52 m −2 , which was obtained from the Planck data [44][45].For simplicity, the subscripts such as sca are dropped in the scaled quantities like Λ sca .Fig. 2(a) plots the relation between the radial distance r and the effective potential V 1 or V e1 on the equatorial plane ϑ = π/2.When ξ < 8.4, the photon will fall into the black hole; but the photon will scatter to infinity when ξ > 8.4.
When ξ = 8.4, the photon will wind many times on a circular orbit.The circular orbit corresponds to the top point (i.e., the local maximum) of the effective potential V 1 or V e1 .It is clear that the circular photon orbit satisfies the conditions The circular photon orbit is unstable because The conditions ( 29) and ( 30) for the unstable circular photon orbit are equivalent to the following conditions [29] The impact parameter ξ for the unstable circular photon orbit in Fig. 1(a) is marked as ξ cp = 8.4.The circular photon orbit has a radius r cp = 2.4.
In fact, ξ cp and r cp are determined by Eq. ( 29) or (31) when another impact parameter η is given according to Eq. (27).Based on Eq. ( 31), ξ cp and η cp can be expressed in terms of r cp as where ∆ ′ r = d∆ r /dr.Because of Eq. ( 27), ξ cp and η cp also satisfy the relation equivalently, Thus, r cp can be solved from Eqs. ( 33), ( 34) and ( 36) for a given value ϑ.Then, ξ cp and η cp are determined through Eqs. ( 33) and (34).
In terms of Eqs. ( 33)-( 35), the radii of circular photon orbits on the equatorial plane for the Kerr black hole are expressed as where the upper sign "−" correspond to prograde orbits and the lower sign "+" to retrograde orbits.Then, ξ cp and η cp for the circular photon orbits on the equatorial plane can be given by Eqs. ( 33) and (34).When the Kerr black hole is surrounded by the extra sources, we have no way to provide an explicit expression of r cp .Eqs. ( 33), ( 34) and ( 36) must be solved through an iterative method, such as the Newtonian iterative method.In fact, r cp should be solved iteratively, and has two roots r + cp and r − cp with r + cp > r − cp for any angle ϑ in the interval 0 < ϑ < π.
If a = 0, the spacetime ( 1) is spherically symmetric and ξ cp cannot be given by Eq. ( 33) but can be given by Eq. (36).It is expressed as In this case, we use Eq. ( 31) to obtain For the Schwarzschild black hole, Eq. ( 39) has the solution r cp = 3, which corresponds to the radius of circular photon orbit in the Schwarzschild spacetime.Noting Eq. ( 34), we have η cp = 27.Hence, ξ cp = ±3 √ 3 on the equatorial plane is obtained from Eq. (38).For the non-Schwarzschild case, the solution r cp of Eq. ( 39) is solved by the Newtonian iterative method.Then, η cp is given by Eq. ( 34), and ξ cp is obtained from Eq. (38).It is clear that when the parameters b c , Q, Λ, α q and ω q are given, r cp and η cp are also determined.However, ξ cp is varied with a variation of ϑ.Particular for ϑ = π/2, ξ cp has the maximum Clearly, the path of the obtainment of r cp , ξ cp and η cp for the rotating case is somewhat unlike that for the nonrotating case.It is easier to obtain the solutions of r cp , ξ cp and η cp for the nonrotating case than those for the rotating case.

Spherical photon orbits
As is demonstrated above, the effective potential (28) governs the radial motion of photons on the two-dimensional plane.If Θ is not always identical to zero for any time λ, then θ varies in the range 0 < θ < π with time λ in Eq. (25).Note that η satisfying Eq. ( 25), i.e. θ = ϑ as a solution of Θ(θ) = 0, is possible at some time, but is impossible at any other times.Without loss of generality, θ satisfying the condition Θ(θ) ≥ 0 is arbitrarily given in the range 0 < θ < π.In this case, the motion of photons is not lying on the two-dimensional plane but is lying in the three-dimensional space.Eq. ( 23) or ( 26) is still the effective potential in the three-dimensional space, labeled as V 2 or V e2 .The top point of the effective potential V 2 or V e2 in Fig. 2(b) corresponds to a spherical photon orbit.The conditions for the existence of spherical photon orbit are still the same as Eqs.( 29) and (30) (or Eqs. ( 31) and ( 32)) for the existence of circular photon orbit.An explicit difference between the spherical photon orbit and the circular photon orbit is that Eq. ( 27) is not satisfied for the spherical photon orbit, whereas it is for the circular photon orbit.The two impact parameters ξ sp and η sp for the spherical photon orbit are consistent with the expressions of ξ cp and η cp for the circular photon orbit in Eqs. ( 33) and ( 34), but do not satisfy Eq. (36).The values of ξ cp are based on a = 0.If a = 0, then the spacetime (1) is spherically symmetric, and the above-mentioned circular photon orbits are present.ξ cp is given in Eq. ( 38) rather than Eq.(33).
As shown in Fig. 2(b), the two parameters ξ sp = 1 and η sp = 24 can cause the photon to wind many times on the spherical photon orbit with radius r sp = 2.79.When η sp is fixed and ξ is slightly larger than ξ sp , the photon comes close to this spherical orbit from infinity, but goes back to infinity.If ξ is slightly smaller than ξ sp , then the photon gets into the horizon from infinity.
Using Eq. ( 32), we can know that for the case of a = 0, r sp is constrained in the range r − cp ≤ r sp ≤ r + cp , where r − cp and r + cp are the radii of circular photon orbits on the plane θ = ϑ.Note that ϑ may not be π/2.As the spherical radius r sp ranges from r − cp to r + cp , an infinite number of points (ξ sp , η sp ) are obtained from Eqs. (33) and (34).

Parameter constraints based on black hole shadows
A black hole shadow is observed by an observer in a zero angular moment observer (ZAMO) reference frame [5].Then a local curvature radius for the boundary of black hole shadow is discussed.Finally, the constraint of curvature radius is used to constrain the parameters.

Black hole shadows
The above-mentioned points (ξ sp , η sp ) from the spherical photon orbits are used to study the black hole shadows.
In order to obtain an image of the black hole, we introduce celestial coordinates in an observer's sky.Assume that the static observer locally stays at point (r 0 , θ 0 ) in the ZAMO reference frame, where the observer can determine the image points.The observer basis {ê t , êr , êθ , êφ } can be expressed in terms of the coordinate basis {∂ t , ∂ r , ∂ θ , ∂ φ } as [46][47][48] where êν (µ) is a transform matrix satisfying the relation g µν e µ α e ν β = η αβ with η αβ being the Minkowski metric.In general, it is convenient to choose the observer located in the frame Since ê(t) •∂ φ = 0, the observer in this local rest frame has zero angular momentum at infinity.In this sense, the frame is called the ZAMO reference frame, representing the zero angular momentum observer.The four-momentum p µ of a photon by its projection onto êµ is locally measured by On the basis of Eqs.(41)(42)(43)(44), the four-momentum p µ can be rewritten as where ζ = êt (t) and γ = êφ (t) .Suppose the observer has a radial distance r 0 in Fig. 3.The inclination angle between the symmetrical axis of the black hole and the direction to the observer is θ 0 .The 3-vector p is the photon's linear momentum with three components p (r) , p (θ) and p (φ) in the orthonormal basis {ê t , êr , êθ , êφ } [46][47][48]: The observation angles (α, β) are introduced by Thus, we have An image point is described by celestial coordinates (x, y) [48]: Because x and y satisfy the relation For the nonrotating case of a = 0, Eq. (60) becomes Obviously, the black hole shadow is a standard circle with the radius R sh .The Schwarzschild black hole shadow has its radius R sh = 3 √ 3. The shadow size is independent of the angles ϑ and θ 0 .The result is still present when the extra sources such as the quintessence, cloud strings, cosmological constant and black hole charge are included in the Schwarzschild spacetime.This is because r cp is given by Eq. ( 39) that does not depend on the angles ϑ and θ 0 , and η cp is obtained from Eq. ( 34) that does not contain the angles ϑ and θ 0 .In fact, r cp depends on the parameters b c , α q , ω q and Λ. η p or R sh is also determined by these parameters.Table 1 lists the values of r cp and R sh for several different combinations of the parameters.The radius of circular photon orbit increases as each of the parameters b c α q , and |ω q | increases.However, a variation of Λ does not affect the radius of circular photon orbit because r cp in Eq. ( 39) is independent of Λ.The shadow R sh increases as either parameters b c or ω q increases.This result is due to the increase of the black hole gravitational field.On the contrary, as parameters α q and Λ increase, the gravitational field of the black hole is weakened, and the size of the black hole shadow decreases.
For the rotating case a = 0, the celestial coordinates (x, y) are obtained unlike those for the nonrotating case.If the observer stays at the equatorial plane θ 0 = π/2, the celestial coordinates have the relation Notice that η p in Eq. ( 62) is unlike that in Eq. ( 61).η p in Eq. ( 61) is dependent on the radius of spherical photon orbits (i.e. the radius of circular photon orbits) and has only one invariant value.Therefore, the shadow for the nonrotating black hole is a standard circle.However, there are two photon circular orbits on a plane in Eq. ( 62), and the radii of spherical photon orbits are arbitrarily given between the two radii of photon circular orbits.That is, η p in Eq. ( 62) is varied.In this situation, Eq. ( 62) cannot be thought of as a circle for the rotating black hole.
For the Kerr spacetime, the radii of photon circular orbits on the equatorial plane are r ∓ cp in Eq. (37).As the radii of photon spherical radii r p range from r − cp to r + cp , ξ ∓ p and η ∓ p can be given by Eqs.(33) and (34).In this way, all celestial coordinate points on the shadow are obtained.If the observer is not located on the equatorial plane (i.e.θ 0 = π/2), we can still use the method but have to discard imaginary points during the calculations.When θ 0 is consistent with ϑ, the celestial coordinates are no longer given by the spherical photon orbits with the radii given in the range (r − cp , r + cp ) of Eq. (37).They should be determined by the spherical photon orbits, whose radii are arbitrarily given in the range (r − cp , r + cp ).Here, r − cp and r + cp are the radii of photon circular orbits for satisfying Eqs. ( 33), ( 34) and ( 36) on the planes ϑ in Fig. 4 (a) and (c).The smallest radius of photon circular orbit r D and the largest radius of photon circular orbit r R are provided.Another path for the obtainment of r D and r R is to solve the equation y(r cp ) = 0, where r cp corresponds to the radii of unstable photon circular orbits on the planes θ 0 in Fig. 4 (b) and (d).In fact, the celestial coordinate y is identical to zero for the critical photon circular orbits when η cp in Eq. ( 35) is substituted into Eq.( 55).This fact is why the radii of photon circular orbits are obtained by solving the equation y(r cp ) = 0.The two methods give the same numerical results to the circular orbit radii r D and r R on the planes ϑ = θ 0 in Table 2. Fig. 5 plots the points (x D , 0) and (x R , 0), which are determined by the prograde and retrograde photon orbits in the plane ϑ = θ 0 .The complete black hole shadow is made of the two points and other points associated with unstable spherical photon orbits between the radii of two circular photon orbits in the plane ϑ = θ 0 .
The above demonstrations introduce several methods for the computation of the Kerr black hole shadows.These methods are still suitable for the computation of black hole shadows when the quintessence, cloud strings, cosmological constant and black hole charge are included in the Kerr spacetime.

Local curvature radius
The boundary of a black hole shadow can describe some properties of the black hole.It is a one-dimensional closed curve in the celestial coordinates.From the viewpoint of differential geometry, the curve has its length and local curvature radius.The curvature radius in Ref.
[57] is written as Utilizing the symmetry of black hole shadow, the authors of [54] discussed several characteristic points along the boundary curve of the Kerr black hole shadow in Fig. 5.These points are D, R, B and T .The Kerr black hole parameters can be constrained in terms of the characteristic points and curvature radius.The points T and B are determined by The points D and R are governed by Since the shadow curve has the Z2 symmetry, the vertical diameter ∆y of the shadow is ∆y = 2y T .
The horizontal diameter is given by The KNdS black hole shadows are plotted for different parameters in Figs. 6.The gravitational radius from the EHT observations [2,25] is expressed as where l is the distance from the observer to the black hole, i.e. l = r 0 .Using the gravitational radius, we can compute the curvature radii at the points T , D and R, which correspond to ℜ T , ℜ D and ℜ R , respectively.The values ∆x, ∆y, ℜ T , ℜ D and ℜ R for the black hole shadows in Figs. 6 are listed in Table 3.We take the observation angle θ 0 = 17 • , which is the angle between the approaching jet from the central radio source in M87* and the line of sight [49].Because a highly charged dilaton black hole is ruled out by the measurements of the Event Horizon Telescope [6], smaller values are given to the black hole charges in our work.
The shadow size increases as either parameter b c or ω q increases, as is shown in Fig. 6, where a = 0.9, θ 0 = 17 • and Q = 0.2.This result is due to the increase of the black hole gravitational field.On the contrary, the gravitational field of the black hole is weakened and the size of the black hole shadow decreases as parameters α q and Λ increase.In addition, the curvature radii at the characteristic points ℜ T , ℜ D and ℜ R have small changes with the increases or decreases of these parameters.Consequently, the shadow remains nearly circular.

Constraints of the parameters
Based on the 2017 EHT observations of M87*, the radius of the shadow is constrained in the range 4.31M ≤ r sh,A ≤ 6.08M. (69) Equivalently, the constraint to the curvature radius was given in [27][28] by The curvature radius may have a maximum value and a minimum value.It has two local maximum values at r p = r D and r p = r R .The local maximum at r p = r D is larger than that at r p = r R .Hence we have ℜ max = ℜ(r D ).The minimum curvature radius ℜ min = ℜ(r T ) corresponds to the well of these curves.Moreover, ℜ min and ℜ max give lower and upper bounds to the size of the shadow.In other words, ℜ min should not decrease below 4.31M and ℜ max should not increase beyond 6.08M.
Finally, we utilize the constraint of the curvature radius to restrict the parameters.The allowed region of the cloud strings b c in Fig. 7(a) is down the red curve corresponding to the maximum curvature radius ℜ max when the black hole spin a ranges from 0 to 1 and the other parameters are given.Similarly, the allowed region of the quintessence parameter α q in Fig. 7(b) and the cosmological constant Λ in Fig. 7(d) is down the blue curve corresponding to the minimum curvature radius ℜ min when the black hole spin a ranges from 0 to 1 and the other parameters are given.However, the allowed region of the quintessential state parameter ω q in Fig. 7(c) is the upper region of the blue curve.For any spin a ∈ [0, 1], the allowed regions of the parameters are 0 ≤ b c < 0.15 in Fig. 7(a), 0 ≤ α q < 0.18 in Fig. 7(b), −0.376 < ω q < −1/3 in Fig. 7(c), and 0 ≤ Λ < 5 × 10 −21 in Fig. 7(d).

Conclusions
In this paper, we focus on the motion of photons around the KNdS black hole surrounded by quintessence and a cloud of strings.Due to the existence of the Carter constant, unstable circular photon orbits on a two-dimensional plane not limited to the equatorial plane and unstable spherical photon orbits in the three-dimensional space can be present.The conditions for the existence of circular photon orbits are basically consistent with those for the existence of spherical photon orbits.However, only one difference between them is only that the angle θ always remains invariant for the circular photon orbits and is varied with time for the spherical photon orbits.The two impact parameters can be determined by these circular photon orbits and spherical photon orbits.The radius of circular photon orbit increases as each of the parameters involving the cloud of strings b c , quintessence parameter α q , and quintessential state parameter |ω q | increases.However, a small variation of the cosmological constant Λ does not typically affect the circular photon orbit radius.
For the nonrotating case with the spherical symmetry, the black hole shadows are circular and their sizes are independent of the observation angles and the planes on which photon circular orbits exist.For the rotating case with the axialsymmetry, the black hole shadow is dependent on the observation angle.In both cases, small changes of the parameters excluding the spin parameter exert similar influences on the sizes of black hole shadows.
Table 1: Values of rcp and R sh of circular photon orbits on the equatorial plane for each of the parameters bc, αq, ωq and Λ in the nonrotating case.The combinations of the other parameters are those of Fig. 6 (a)-(d    In the green regions for rc > r 0 , light rays can reach a rest observer's eyes; but they cannot in the blue regions for rc < r 0 .The gray regions represent rc as imaginary numbers.a q =0.01 w q =-0.35 L=1.02´10 (-26)      Figure 4: Critical photon orbits obtained from different methods.The green curves represent the circular photon orbits that determine the characteristic points T introduced in Ref. [48].(a): For the Kerr black hole with a = 0.5, circular photon orbits stay in all planes θ = ϑ.(b): For the Kerr black hole with a = 0.5, all observed angles correspond to circular photon orbits.(c): For the Kerr black hole with a = 0.9, circular photon orbits exist in all planes θ = ϑ.(d): For the Kerr black hole with a = 0.9, circular photon orbits on all observed angles are given.Λ=10 (-21)   Λ=10 (-20)   a=0.9 θ 0 =17 o Q=0.2 b c =0.01 α q =0.01 ω q =-0.35

Figure 1 :
Figure 1: (a): The domain of outer communication on the parameters a and bc with Q = 0.2, αq = 0.01, ωq = −0.35 and Λ = 1.02 × 10 −26 .(b): The domain on the parameters a and αq with Q = 0.2, bc = 0.01, ωq = −0.35 and Λ = 1.02 × 10 −26 .(c): The domain on the parameters a and ωq with Q = 0.2 and bc = 0.01, αq = 0.01 and Λ = 1.02 × 10 −26 .(d): The domain on the parameters a and Λ with Q = 0.2, bc = 0.01, αq = 0.01 and ωq = −0.35.Assume that the distance from the observer to the black hole is r 0 = 10 10 M .In the green regions for rc > r 0 , light rays can reach a rest observer's eyes; but they cannot in the blue regions for rc < r 0 .The gray regions represent rc as imaginary numbers.

Figure 2 :
Figure 2: Effective potentials for the motion of photons around the black holes.The maximum values correspond to unstable circular photon orbits or unstable spherical photon orbits.

Figure 5 :
Figure 5: The characteristic points of the Kerr black hole shadow.The red circle denotes the shape of the shadow for the spin a and observation angle θ 0 .The characteristic points D, R, T and B respectively correspond to the left, right, top, and bottom points of the shadow.∆x and ∆y denote the horizontal and vertical diameters of the shadow.

Figure 6 :
Figure 6: The black hole shadows for different parameters.

Table 2 :
), respectively.Numerical comparison of circular photon orbits r D and r R in Fig.3.

Table 3 :
Shadows for different black hole parameters.The horizontal diameter (HD) (µas) is ∆x, and the vertical diameter (VD) (µas) is ∆y.The shadow diameter (SD) (µas) is d, and the gravitational radius is θg ≈ 3.8µas.The other parameters of the shadows are those of Figs.5 and 6.