Shadow of a dressed black hole and determination of spin and viewing angle
SShadow of a dressed black hole and determination of spin and viewing angle
Lingyun Yang and Zilong Li ∗ Center for Field Theory and Particle Physics and Department of Physics,Fudan University, 220 Handan Road, 200433 Shanghai, China (Dated: August 3, 2018)Shadows of black holes surrounded by an optically thin emitting medium have been extensivelydiscussed in the literature. The Hioki-Maeda algorithm is a simple recipe to characterize the shapeof these shadows and determine the parameters of the system. Here we extend their idea to the caseof a dressed black hole, namely a black hole surrounded by a geometrically thin and optically thickaccretion disk. While the boundary of the shadow of black holes surrounded by an optically thinemitting medium corresponds to the apparent photon capture sphere, that of dressed black holescorresponds to the apparent image of the innermost stable circular orbit. Even in this case, we cancharacterize the shape of the shadow and infer the black hole spin and viewing angle. The shapeand the size of the shadow of a dressed black hole are strongly affected by the black hole spin andinclination angle. Despite that, it seems that we cannot extract any additional information fromit. Here we study the possibility of testing the Kerr metric. Even with the full knowledge of theboundary of the shadow, those of Kerr and non-Kerr black holes are very similar and it is eventuallyvery difficult to distinguish the two cases.
PACS numbers: 04.70.-s, 95.30.Sf, 98.62.Js
I. INTRODUCTION
The direct image of a black hole surrounded by an op-tically thin emitting medium is characterized by the pres-ence of the so-called “shadow”, namely a dark area over abrighter background [1]. The shape of the shadow corre-sponds to the apparent photon capture sphere as seen bya distant observer and it is the result of the strong lightbending in the vicinity of the black hole. An accurateobservation of the shadow can potentially provide infor-mation on the spacetime geometry around the compactobject and test general relativity [2–6]. The interest inthis topic is today particularly motivated by the possibil-ity of observing the shadow of SgrA ∗ , the supermassiveblack hole candidate at the center of the Milky Way, withsub-millimeter very long baseline interferometry (VLBI)facilities [7].If the black hole has an optically thick and geometri-cally thin accretion disk, its apparent image is substan-tially different. However, even this “dressed” black holeis characterized by a shadow, whose shape is still deter-mined by the spacetime geometry around the compactobject and the viewing angle with respect to the line ofsight of the distant observer [8, 9]. In this case, the shapeof the shadow corresponds to the apparent image of theinner boundary of the accretion disk, which, under cer-tain conditions, should be located at the innermost stablecircular orbit (ISCO) [10].In the Kerr metric, the shape of the shadow is onlydetermined by the values of the black hole spin param-eter a ∗ = a/M = J/M , where M and J are the blackhole mass and spin angular momentum, and of the angle i between the spin and the line of sight of the distant ∗ Corresponding author: [email protected] observer. The size of the shadow on the observer’s skyis also regulated by the black hole mass and distance. InRef. [11], Hioki and Maeda proposed a simple algorithmto characterize the shape of the shadow of a Kerr blackhole surrounded by an optically thin medium in termsof a distortion parameter δ and of the shadow radius R . If we independently know the mass and the distanceof the object, the measurement of δ and R can providean estimate of the black hole spin and inclination angle.This algorithm has been later employed for the shadowof non-Kerr black holes as a general approach to char-acterize their shadow and get a rough estimate of thepossibility of testing general relativity with future VLBIobservations [12, 13].With the same spirit, here we introduce two parame-ters, α and β , to characterize the shape of the shadowof a dressed black hole. Even in this case, we show thatwe can eventually measure the black hole spin and theinclination angle. However, it seems we cannot do more.We study the possibility of testing the Kerr metric byconsidering a non-Kerr background with a spin and a de-formation parameter to quantify possible deviations fromthe Kerr solution. Even introducing a third parameter todescribe the shadow shape, we fail to break the degener-acy between the spin and the deformation parameter ofthe background metric. We also introduce the function R ( φ ) to describe the entire shadow shape. Even with R ( φ ), we are not able to unambiguously distinguish aKerr and a non-Kerr black hole. A similar problem wasalready found in the case of the shadow of a black holesurrounded by an optically thin emitting medium [13],but in that case the shape and the size of the shadow donot change very much if the values of the physical pa-rameters of the system vary. For that of a dressed blackhole, both the shape and the size significantly changewith the spacetime geometry and the inclination angle. a r X i v : . [ a s t r o - ph . H E ] D ec Such a result can be probably understood by noting thatthe boundary of the shadow of a dressed black hole isthe apparent image of the ISCO, whose radius is just oneparameter. For a Kerr black hole, it is only determinedby the spin of the object. For a non-Kerr black hole, itdepends on both the spin and the deformation parame-ter and therefore there is a degeneracy between these twoquantities, in the sense that the same ISCO radius canbe obtained from different combinations of the spin andthe deformation parameter.We note that the results of our work have no implica-tions for future observations of SgrA ∗ , as this object hasno thin accretion disk. The shadow of a dressed blackhole can be potentially observed in the case of stellar-mass black holes in X-ray binaries with X-ray interfero-metric techniques. The stellar-mass black hole candidatewith the largest angular size on the sky should be thatin Cygnus X-1, which can be found in the state with athin accretion disk and it seems thus a good candidatefor this purpose. However, X-ray interferometric obser-vations will be unlikely possible in the near future.The content of the paper is as follows. In Section II,we briefly describe our approach to numerically computethe shadow of a dressed black hole. In Section III, we in-troduce the parameters α and β to characterize the shapeof the shadow and infer the black hole spin and its incli-nation angle with respect to the observer’s line of sight.In Section IV, we discuss how the detection of a shadowmay be used to test the Kerr metric and we introduce thefunction R ( φ ) to describe its whole boundary. Summaryand conclusions are reported in Section V. Throughoutthe paper, we employ units in which G N = c = 1. II. CALCULATION METHOD
The Kerr solution is a Petrov type-D spacetime andtherefore for a suitable choice of coordinates (e.g. inBoyer-Lindquist coordinates) the equations of motion areseparable and of first order. This can somehow simplifythe calculations of the black hole shadow. However, herewe use the general approach and therefore its extensionto non-Kerr backgrounds is straightforward. We use thecode described in Ref. [9]. Since we are only interestedin the calculation of the shape of the shadow, not in theintensity map of the direct image, we need to computethe photon trajectories from the image plane of the dis-tant observer to the vicinity of the black hole and checkwhether the photon hits the thin accretion disk or not.The set of points on the observer’s sky whose photonsdo not hit the disk forms the shadow of the black hole.Such photons can either hit the black hole or cross theequatorial plane between the inner edge of the disk andthe black hole and then escape to infinity.The initial conditions ( t , r , θ , φ ) for the photonwith Cartesian coordinates ( X, Y ) on the image plane of the distant observer are [9] t = 0 , (1) r = (cid:112) X + Y + D , (2) θ = arccos Y sin i + D cos i √ X + Y + D , (3) φ = arctan XD sin i − Y cos i . (4)The initial 3-momentum of the photon, k , is perpendic-ular to the plane of the image of the observer. The initialconditions for its 4-momentum are thus [9] k r = − D √ X + Y + D | k | , (5) k θ = cos i − D Y sin i + D cos iX + Y + D (cid:112) X + ( D sin i − Y cos i ) | k | , (6) k φ = X sin iX + ( D sin i − Y cos i ) | k | , (7) k t = (cid:113) ( k r ) + r (cid:0) k θ (cid:1) + r sin θ ( k φ ) . (8)In our calculations, the observer is located at D = 10 M ,which is far enough to assume that the background ge-ometry is flat. k t is thus obtained from the condition g µν k µ k ν = 0 with the metric tensor of a flat spacetime.The photon trajectory is calculated by solving the secondorder geodesic equations with the fourth order Runge-Kutta-Nystr¨om method, as described in [9]. The trajec-tory is numerically integrated backwards in time to checkwhether the photon hits the black hole, hits the disk, orcrosses the equatorial plane between the inner edge of thedisk and the black hole and then escapes to infinity. Wenote that some photons may cross the equatorial planebetween the inner edge of the disk and the black hole andthen hit either the disk or the black hole. We also notethat we employ the usual set-up, in which the disk is onthe equatorial plane and the inner edge of the disk is atthe ISCO radius.Fig. 1 shows some examples of our calculations. Inthe left panel, the inclination angle is fixed to i = 60 ◦ and we change the value of the spin parameter to seeits effect on the shape of the shadow. The impact of a ∗ is definitively different from the case of the shadowof a black hole surrounded by an optically thin emittingmedium. In the case of a dressed black hole, the size ofthe shadow is very sensitive to the black hole spin, as itcould have been expected from the fact the ISCO radiusranges from 6 M for a non-rotating Schwarzschild blackhole to M for a maximally rotating Kerr black hole anda corotating disk, while the ISCO is at 9 M when thedisk of the maximally rotating Kerr black hole is coun-terrotating. The right panel in Fig. 1 shows the effect ofthe inclination angle. Here the black hole has spin pa-rameter a ∗ = 0 . i = 0 ◦ , 30 ◦ ,60 ◦ , and 80 ◦ . Now the size of the shadow is roughly thesame, while the shape changes significantly. If we lookat the shadows of black holes surrounded by an optically (cid:239) (cid:239) (cid:239) (cid:239) X/M Y / M a/M (cid:239) (cid:239) (cid:239) (cid:239) X/M Y / M i ° ° ° ° FIG. 1. Examples of shadows of dressed black holes. In the left panels, the inclination angle is i = 60 ◦ and we show the effectof the spin parameter a ∗ . In the right panel, the spin parameter is a ∗ = 0 . i . thin emitting medium (see e.g. the figures in [11]), it iseasy to conclude that both the shape and the size of theshadow of a geometrically thin and optically thick diskare much more sensitive to the values of the spin and ofthe inclination angle. (cid:239) (cid:239) (cid:239) (cid:239) X/M Y / M (cid:95)(cid:96) C AB
FIG. 2. Definition of the parameter α and β to characterizethe shape of the shadow of a dressed black hole. III. DETERMINATION OF SPIN ANDVIEWING ANGLE
In this section, we want to introduce two parametersto characterize the shadow of a dressed black hole, tobe used to infer the values of its spin and viewing an-gle. Unlike the shadow of a black hole surrounded by anoptically thin emitting medium, in general our shadowshave not an axis of symmetry and therefore we need toadopt a slightly different approach with respect to thatin Ref. [11].As first step in our algorithm, we find the “center” C of the shadow (see Fig. 2). It reminds the center of massof a body and its coordinates ( X C , Y C ) on the sky aregiven by X C = (cid:82) ρ ( X, Y ) XdXdY (cid:82) ρ ( X, Y ) dXdY ,Y C = (cid:82) ρ ( X, Y ) Y dXdY (cid:82) ρ ( X, Y ) dXdY , (9)where ρ ( X, Y ) = 1 inside the shadow and 0 outside. Oncewe have C , we can determine the distance of C from everypoint of the boundary of the shadow. If A and B are thepoints on the boundary respectively with the maximumand minimum distance from C , we call α the distance AC and β the distance BC , as shown in Fig. 2.In the case of the Hioki-Maeda algorithm for theshadow of a black hole surrounded by an optically thinemitting medium, we determine two parameters: theshadow radius in units of the apparent size of the gravi-tational radius on the observer’s sky, R/M , and the (di-mensionless) distortion parameter, δ . With these twoquantities we can infer the black hole spin parameter, . . . . . . . . a/M i [ d e g ] . . . . . . . . . . . . . . . a/M i [ d e g ] FIG. 3. Contour maps of α/M (left panel) and of α/β (right panel) in the plane spin parameter vs inclination angle for theKerr metric. See the text for more details. (cid:239) (cid:239) (cid:239) (cid:239) X/M Y / M (cid:161) (cid:239)
0 0.60.70.80.911.1 (cid:113) R / (cid:95) a/M (cid:47) /2 (cid:47) (cid:47) /2 2 (cid:47) FIG. 4. Left panel: impact of the deformation parameter (cid:15) on the shape of the shadow of a dressed black hole. The inclinationangle is i = 70 ◦ , the spin parameter is a ∗ = 0 .
5, and we change the deformation parameter (cid:15) . Right panel: examples of profileof the boundary of the shadow. Here we assume the Kerr metric, the inclination angle is i = 60 ◦ , and we show the profile fora few different spin parameters. a ∗ , and the angle between the spin axis and the line ofsight of the distant observer, i . Here we have the samesituation. α/M is the counterpart of R/M , while α/β plays the role of the Hioki-Maeda distortion parameter δ .In Fig. 3, we show the contour maps of α/M (left panel)and α/β (right panel). α/M is mainly determined by theblack hole spin, while the effect of the inclination angle i is smaller. On the contrary, α/β is very sensitive to the exact value of i and it is affected only weakly by a ∗ . If wecan determine both α/M and α/β , we can infer a ∗ and i . We note that in our case of the shadow of a dressedblack hole we could estimate the inclination angle i (withsome uncertainty) without knowing the apparent size ofthe gravitational radius on the observer’s sky. a/M (cid:161) (cid:239) FIG. 5. Contour map of S ( a ∗ , (cid:15) , i ) in which the referencemodel is a Kerr black hole with spin parameter 0.7 and ob-served from an inclination angle 60 ◦ . Here i = 60 ◦ is fixed.See the text for more details. IV. TESTING THE KERR METRIC
If we characterize the shape of the shadow of a blackhole with two parameters, their determination can beused at most to infer two physical parameters of the blackhole, like the spin and the inclination angle. In this sec-tion we want to figure out if the detection of the shadowof a dressed black hole can test the Kerr metric. To dothis, we relax the assumption of the Kerr background andwe consider a metric more general than the Kerr solution.The metric is now described by the black hole mass M ,the spin parameter a ∗ , and, in the simplest case, a de-formation parameter that quantifies possible deviationsfrom the Kerr metric. We want to see if it is possibleto measure the shadow and infer the three parameters ofthe system.As example, we consider the Johannsen-Psaltis met-ric [14], whose line element reads ds = − (cid:18) − M r Σ (cid:19) (1 + h ) dt + Σ(1 + h )∆ + a h sin θ dr +Σ dθ − aM r sin θ Σ (1 + h ) dtdφ + (cid:34) sin θ (cid:18) r + a + 2 a M r sin θ Σ (cid:19) + a (Σ + 2 M r ) sin θ Σ h (cid:35) dφ . (10)Here Σ = r + a cos θ , ∆ = r − M r + a , and, in thesimplest version with only one deformation parameter, h is h = (cid:15) M r Σ . (11) (cid:15) is the “deformation parameter” and it is used to quan-tify possible deviations from the Kerr geometry. Thecompact object is more prolate (oblate) than a Kerr blackhole for (cid:15) > (cid:15) < (cid:15) = 0, we exactly recoverthe Kerr solution. The impact of the deformation param-eter on the black hole shadow is shown in the left panelin Fig. 4, where a ∗ and i are fixed and we change (cid:15) .First, we tried to identify a third parameter to charac-terize the shape of the shadow of a dressed black holes.We studied a few options (area of the shadow, distancebetween C and other points on the boundary, etc.). How-ever, we failed, in the sense that we did not find threeparameters of the shadow to infer the three physical pa-rameters of the black hole because of the degeneracy be-tween a ∗ and (cid:15) .To increase our chances of success, we map the wholeshadow boundary. We introduce the function R ( φ ) de-fined as the distance between C and the boundary of theshadow, starting from α , i.e. R ( φ = 0) = α , where φ is the angle between the the segment AC determining α and the segment between C and the point under consid-eration. If we do not have an independent measurementof the black hole mass and distance, we can only measurethe actual shape of the shadow (not the size) and there-fore we can measure R/α . Some examples are shown inthe right panel in Fig. 4, where we have only consideredKerr black holes with inclination angle i = 60 ◦ and wevary the spin.With the use of R , we can compare the shadow ofdifferent black holes. To quantify the similarity betweentwo systems, we use the following estimator S ( a ∗ , (cid:15) , i ) = (cid:88) k (cid:18) R ( a ∗ , (cid:15) , i ; φ k ) α ( a ∗ , (cid:15) , i ) − R ref ( φ k ) α ref (cid:19) , (12)where R ( a ∗ , (cid:15) , i ; φ k ) is the function R at φ = φ k of theblack hole under consideration, α ( a ∗ , (cid:15) , i ) is its α , while R ref ( φ k ) and α ref are, respectively, the the function R at φ = φ k of some reference black hole and the correspond-ing α . With this estimator, we are simply consideringthe least squares method for the normalized radius of theshadow and therefore the “best fits” is obtained when thesum of the squared residuals is minimum.As example, we consider a reference black hole with a ∗ = 0 . (cid:15) = 0 (Kerr black hole), and i = 60 ◦ . Fig. 5shows S with i = 60 ◦ for every black hole. In the leftpanel in Fig. 6, we show S with i free and we haveselected the inclination angle that minimizes S . In theright panel, we show the values of the inclination anglesof the left panel. As we can see from Figs. 5 and 6,there is a strong correlation between the spin a ∗ and thedeformation (cid:15) and it seems it is very difficult to infer theactual values of the physical parameters of the system.If we have an independent measurement of the blackhole mass and distance, we can measure R in units of a/M (cid:161) (cid:239) a/M (cid:161) (cid:239) FIG. 6. Left panel: as in Fig. 5 with i free in the fit. Right panel: contour map of the values of the inclination angle i thatminimizes S in the left panel. See the text for more details. a/M (cid:161) (cid:239) a/M (cid:161) (cid:239) FIG. 7. Left panel: as in Fig. 5 for S . Right panel: as in the left panel in Fig. 6 for S . See the text for more details. gravitational radii. In this case, the estimator for thecomparison of two shadows is S ( a ∗ , (cid:15) , i ) = (cid:88) k (cid:0) R ( a ∗ , (cid:15) , i ; φ k ) − R ref ( φ k ) (cid:1) M . (13)Now the least squares method is for the absolute radiusof the shadow. The results for constant i and free i arereported in Fig. 7, respectively in left and right pan-els. Even in the case of the measurement of R , it seemswe cannot test the Kerr metric because of the degen-eracy between the spin and the deformation parameter.A simple explanation is probably that the shadow of adressed black hole corresponds to the apparent image of the ISCO. The radius of the ISCO is just one parameter.If we have a Kerr black hole, it is only determined by thespin of the object and the measurement of the shadowcan be used to infer the spin. In the case of a blackhole with a spin and a possible non-vanishing deforma-tion parameter, the same ISCO radius can be obtainedwith many different combinations of a ∗ and (cid:15) , namelythere is a degeneracy. The measurement of the shape ofthe shadow cannot thus give an independent estimate of a ∗ and (cid:15) .To test the Kerr metric, the shadow constrain shouldbe combined with other observations that are not primar-ily sensitive to the position of ISCO in order to breakthe parameter degeneracy. While we have not investi-gated this point, we can expect that continuum-fittingand iron line measurements are not good to do it becausethey are strongly affected by the ISCO radius [15]. Ob-servations like measurements of QPOs [16] or estimate ofthe jet power [17] sounds more promising because basedon other properties of the spacetime. V. CONCLUDING REMARKS
The shadow of a black hole is the dark area appearingin the direct image of the accretion flow. The shadows ofblack holes in general relativity and in alternative theo-ries of gravity surrounded by an optically thin emittingmedium have been extensively discussed in the literatureand the interest on the topic is particularly motivatedby the possibility of observing the shadow of SgrA ∗ withVLBI facilities in the next few years. In this paper, wehave studied the shadow of a dressed black hole, namelya black hole surrounded by a geometrically thin and op-tically thick accretion disk. Even these shadows can bepotentially observed, but we will probably need to waitfor a longer time because the sources are Galactic stellar-mass black holes in X-ray binaries, whose angular size onthe sky is about five orders of magnitude smaller. X-rayinterferometric techniques may observe these shadows,but there are no scheduled missions at the moment.The boundary of the shadow of a dressed black holecorresponds to the apparent image of the inner edge ofthe disk, which, under certain conditions, should be lo-cated at the ISCO radius. Following the spirit of theHioki-Maeda algorithm [11], we have introduced the pa-rameters α and β to characterize the shape of the shadow.In the standard set-up with a Kerr black hole, the bound-ary of the shadow only depends on the black hole spinand inclination angle with respect to the line of sight ofthe distant observer, like the shadow of a black hole sur-rounded by an optically thin emitting medium. However,unlike the latter case, both the shape and the size signifi- cantly change if we vary a ∗ and i . This may suggest thatthe shadow of a dressed black hole is more informativethan that of a black hole surrounded by an optically thinemitting medium, but this is not what we have eventuallyfound.If the mass and the distant of the black holes areknown, the measurement of α/M and α/β can be used toinfer the black hole spin parameter a ∗ and the inclinationangle i . Actually the two estimates are very weakly cor-related, because α/M is mainly sensitive to a ∗ while α/β is essentially determined by i . As a result, if the mass andthe distant of the black holes are not known and one canonly measure α/β , it is still possible to get an estimateof the inclination angle i with some uncertainty.As second step, we have checked if an accurate de-termination of the shadow of a dressed black hole canbe used to test the Kerr metric. In the simplest case,the system is now defined by three physical parameters(spin parameter, deformation parameter, viewing angle).While it is an easy job to infer a ∗ and i in the standardset-up, it seems we cannot test the Kerr metric becauseof a degeneracy between the spin and the deformationparameter. Even the full knowledge of the boundary ofthe shadow, which we have here described with the func-tion R ( φ ), cannot do it. In other words, the shadow ofa dressed black hole is very sensitive to two parameters,the spacetime geometry around the compact object andthe inclination angle, but not more. ACKNOWLEDGMENTS
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