Appendix A Stationary and Axisymmetric Spacetimes

Transcription

1 Appendix A Stationary and Axisymmetric Spacetimes The aim of this appendix is to collect together some useful expressions for generic stationary and axisymmetric spacetimes and for the special case of the Kerr metric. More details can be found in Chap. 3 and in [2]. A.1 General Case The so-called canonical form of the line element of a generic stationary, axisymmetric, and asymptotically flat spacetime is 1 ds 2 = e 2ν dt 2 + e 2ψ (dφ ωdt) 2 + e 2λ dr 2 + e 2μ dθ 2, (A.1) where ν, ψ, ω, λ, and μ are functions of the coordinates r and θ and are independent of t and φ. In this class of spacetimes, it is always possible to introduce the so-called locally non-rotating observers [1]. These observers carry an orthonormal tetrad of 4-vectors which represent their locally Minkowskian coordinate basis vectors [3]. The orthonormal tetrad at any point of the spacetime is given by ( E μ (t) = e ν, 0, 0,ωe ν) ), E μ (r) (0, = e λ, 0, 0, ) E μ (θ) (0, = 0, e μ, 0, E μ (φ) (0, = 0, 0, e ψ), (A.2) and the corresponding basis of one-forms is 1 As pointed out in Sect. 3.1, this is not the most general stationary and axisymmetric spacetime, because even the g tr metric coefficient may be non-vanishing in the most general case. Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

2 306 Appendix A: Stationary and Axisymmetric Spacetimes ) ) E μ (e (t) = ν, 0, 0, 0, E μ (0, (r) = e λ, 0, 0, ) ( E μ (0, (θ) = 0, e μ, 0, E μ (φ) = ωe ψ, 0, 0, e ψ). (A.3) The metric of any locally non-rotating observer is η (a)(b) and is related to the metric g μν by η (a)(b) = E μ (a) g μν E ν (b), η(a)(b) = E μ (a) gμν E ν (b). (A.4) The relation between the components of vectors and dual vectors in the two reference systems is V (a) = E (a) μ V μ, V (a) = E μ (a) V μ. (A.5) For instance, the 3-velocity of a massive particle with respect to a locally nonrotating observer is v (i) = E μ (i) uμ E ν (t) u, (A.6) ν where u μ is the particle 4-velocity. In the case of time-like equatorial circular orbits (e.g. the motion of particles in a Novikov Thorne accretion disk), one finds v (r) = v (θ) = 0, v (φ) = e ψ ν (Ω ω), (A.7) where Ω = u φ /u t is the angular velocity of the particle. A.2 Kerr Metric In Boyer Lindquist coordinates, the line element of the Kerr metric reads ( ds 2 = 1 2Mr ) Σ + Σ Δ dr2 + Σdθ 2 + sin 2 θ dt 2 4Mar sin2 θ dtdφ Σ ( r 2 + a 2 + 2Ma2 r sin 2 θ Σ ) dφ 2, (A.8) with contravariant form ( ) 2 = A s ΣΔ + Δ Σ ( ) 2 4Mar ( )( ) t ΣΔ t φ ) ( ) 2 + Δ a2 sin 2 θ Σ θ ΣΔsin 2 θ ( r ( ) 2, (A.9) φ

3 Appendix A: Stationary and Axisymmetric Spacetimes 307 where Σ = r 2 + a 2 cos 2 θ, Δ = r 2 2Mr + a 2, A = ( r 2 + a 2) 2 a 2 Δ sin 2 θ. (A.10) With the notation of Eq. (A.1), we have e 2ν = ΣΔ A, e2ψ = A sin2 θ Σ, e 2λ = Σ Δ, e2μ = Σ, ω= 2Mar A. (A.11) The orthonormal tetrad carried by locally non-rotating observers is given by ( ) ) A E μ (t) = ΣΔ, 0, 0, 2Mar Δ, E μ (r) (0, = AΣΔ Σ, 0, 0, E μ (θ) = (0, 0, The associated basis of one-forms is ( ) ΣΔ E μ (t) =, 0, 0, 0 A ) 1, 0, E μ (φ) (0, = 0, 0, Σ Σ A 1 sin θ ), Σ, E μ (0, (r) = Δ, 0, 0 E μ (0, (θ) = 0, ( ) Σ,0, E μ (φ) = 2Mar sin θ, 0, 0, AΣ ). (A.12) ) A. (A.13) Σ A.3 Kerr Metric in Kerr Schild Coordinates In the Kerr Schild coordinates (t, r, θ, φ), the metric is regular at the event horizon. In these coordinates, the line element of the Kerr solution reads ( ds 2 = 1 2Mr ) dt 2 + 4Mr Σ Σ dtdr 4Mar sin2 θ dtdφ Σ + ( 1 + 2Mr Σ + sin 2 θ ) dr 2 2a ( 1 + 2Mr Σ ( r 2 + a 2 + 2Ma2 r sin 2 θ Σ where Σ = r 2 + a 2 cos 2 θ as before. ) sin 2 θdrdφ + Σdθ 2 ) dφ 2, (A.14)

4 308 Appendix A: Stationary and Axisymmetric Spacetimes If we indicate with (t KS, r KS,θ KS,φ KS ) the Kerr Schild coordinates and with (t BL, r BL,θ BL,φ BL ) the Boyer Lindquist ones, the relation between the two coordinate systems is dt KS = dt BL + 2Mr Δ drbl, dr KS = dr BL, dθ KS = dθ BL, dφ KS = Ma Δ drbl + dφ BL. (A.15) The radial coordinate of the even horizon is still r H = M + M 2 a 2, (A.16) but now neither g rr nor other metric coefficients diverge there. References 1. J.M. Bardeen, Astrophys. J. 162, 71 (1970). 2. J.M. Bardeen, W.H. Press, S.A. Teukolsky, Astrophys. J. 178, 347 (1972). 3. M. Nakahara, Geometry, Topology, and Physics (Institute of Physics Publishing, Bristol, 1990).

5 Appendix B (r,θ)-motion in the Kerr Spacetime We want to solve Eq. (6.34) D dr s r r e R = s θ i π/2 dθ, (B.1) Θ where R(r) = r 4 + ( a 2 λ 2 q 2) r 2 + 2M [ q 2 + (λ a) 2] r a 2 q 2, Θ(θ) = q 2 + a 2 cos 2 θ λ 2 cot 2 θ. (B.2) (B.3) s r = ±1 and s θ = ±1 according to the direction of propagation. The point of emission in the disk is at (r e,θ e = π/2). The observer is at (D, i), where D. The goal is to obtain r e in terms of the constants of motion λ and q: r e = r e (λ, q). (B.4) Equation (B.4) is necessary to calculate the Jacobian (6.30) in the transfer function. More details on the integration of photon orbits in the Kerr spacetime can be found, for instance, in [3 5], and references therein. B.1 θ-motion With the change of variable μ = cos θ, wehavetosolvetheintegral I μ = μobs where the disk is at μ = 0, the observer at μ = μ obs, and 0 dμ, (B.5) Ũ Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

6 310 Appendix B: (r,θ)-motion in the Kerr Spacetime Ũ = q 2 ( λ 2 + q 2 a 2) μ 2 a 2 μ 4. (B.6) Here we are interested in photons emitted from the disk in the equatorial plane (μ e = 0) and detected by the distant observer at μ obs = 0. For μ = 0, Ũ = q 2,so q 2 0 or the square root in the integral is not real. We can thus restrict our attention to the orbits with q 2 > 0. Orbits with q 2 0 do not connect the disk with the plane of the distant observer. Ũ can be written as Ũ = a 2 ( μ 2 + μ2)( μ 2 + μ2), (B.7) where μ 2 ± = ( λ 2 + q 2 a 2) + 4q 2 a 2 + ( λ 2 + q 2 a 2) 2. (B.8) 2a 2 For μ = 0, Ũ = q 2 = a 2 μ 2 + μ2, and therefore both μ2 + and μ2 are non-negative (positive in our case with q 2 > 0). μ 2 cannot exceed μ 2 + or from Eq. (B.7) Ũ would become negative. The integral of 1/ Ũ from some μ to μ+ can be written as (see Eq in [2]) μ+ μ dμ Ũ 1 = a ( ) cn 1 (cos ψ μ,κ μ ), (B.9) 2 μ μ 2 where cn 1 is the inverse of the Jacobian elliptic function cn, and ψ μ and κ μ are, respectively, the argument and the modulus of the Jacobian elliptic function cos ψ μ = μ, κμ 2 μ = μ μ μ 2. (B.10) The Jacobian elliptic functions can be defined as cn(x, k) = cos ϕ and sn(x, k) = sin ϕ, where x = ϕ 0 dθ 1 k2 sin 2 θ. (B.11) Let us note that cn 1 (cos ψ μ,κ μ ) = F(ψ μ,κ μ ), where F(ψ μ,κ μ ) is the incomplete elliptic integral of the first kind. The latter can be defined as F(ϕ, k) = ϕ 0 dθ 1 k2 sin 2 θ, (B.12) or, after a change of variables, as

7 Appendix B: (r,θ)-motion in the Kerr Spacetime 311 F(x, k) = x 0 dt (1 t 2 )( 1 k 2 t 2). (B.13) It follows that F(x, k) = sn 1 (x, k). It is easy to see that photons hitting the image plane of the distant observer with Y > 0 must have a turning point, which means that μ starts from 0, and its value first increases to reach the maximum value μ + and then decreases to μ obs.ify < 0, there is no turning point, so μ starts from 0 and monotonically increases to μ obs.in the presence of a turning point, the integral is I μ = μ+ 0 dμ Ũ μ+ dμ + μ obs Ũ (Y > 0). (B.14) In the absence of a turning point, the integral is I μ = μ+ 0 dμ Ũ μ+ dμ μ obs Ũ (Y < 0). (B.15) Equations (B.14) and (B.15) can be rewritten in a compact form as 1 [ I μ = a ( ) K (κμ ) + s Y cn 1 (cos ψ μ,κ μ ) ], (B.16) 2 μ μ 2 where s Y = 1 (respectively 0, 1) for Y > 0 (respectively = 0, < 0), ψ μ is evaluated at μ = μ obs, and K (κ μ ) is the complete elliptic integral of the first kind, namely K (κ μ ) = cn 1 (cos ψ μ = 0,κ μ ). (B.17) B.2 r-motion Let r 1, r 2, r 3, and r 4 be the four roots of the equation R(r) = 0. We write R(r) = (r r 1 )(r r 2 )(r r 3 )(r r 4 ). (B.18) The procedure for finding the roots of a quartic equation are discussed, for instance, in [1]. Since R(r ± ) = and R(r = 0) = a 2 q 2 0, at least two roots are real. We can distinguish two cases: (i) there are two complex roots and two real roots, or (ii) there are four real roots.

8 312 Appendix B: (r,θ)-motion in the Kerr Spacetime B.2.1 R(r) = 0 Has Two Complex Roots and Two Real Roots Let r 1 and r 2 be the two complex roots (r 1 = r2, where r 2 is the complex conjugate of r 2 ) and r 3 and r 4 the two real roots (assuming r 3 > r 4 ). Since R(0) = r 1 r 2 r 3 r 4 = r 1 2 r 3 r 4 = a 2 q 2 0, (B.19) r 3 must be non-negative and r 4 must be non-positive. Since R 0, the motion of the photon is in the region r r 3. Let us write the four roots as r 1 = u + iw, r 2 = u iw, r 3 = u + v, and r 4 = u v, where u, v, and w are real numbers and v is positive. The integration from r 3 to r > r 3 can be written as the inverse Jacobian elliptic function cn 1,see Eq in [2] r dr r 3 R = 1 AB cn 1 (cos ψ 2,κ 2 ), (B.20) where cos ψ 2 = (A B)r r 4 A + r 3 B (A + B)r r 4 A r 3 B, (B.21) κ2 2 = (A + B)2 (r 3 r 4 ) 2, 4AB (B.22) and A = (v 2u) 2 + w 2, B = (v + 2u) 2 + w 2. (B.23) B.2.2 R(r) = 0 Has Four Real Roots Let us assume that r 1 r 2 r 3 r 4.From R(r = 0) = r 1 r 2 r 3 r 4 = a 2 q 2 0, (B.24) it follows that r 4 0. Since R(r) 0, motion is allowed in the region r r 1, r 2 r r 3, and r r 4. In our case, we are interested in the region r r 1, because the observer is far from the black hole. For r 1 = r 2, the integral of 1/ R from r 1 to some r can be written in terms of the inverse Jacobian elliptic function sn 1,see Eq in [2]

9 Appendix B: (r,θ)-motion in the Kerr Spacetime 313 r dr r 1 R = 2 (r1 r 3 )(r 2 r 4 ) sn 1 (sin ψ 4,κ 4 ), (B.25) where sin ψ 4 = (r 2 r 4 )(r r 1 ) (r 1 r 4 )(r r 4 ), κ2 4 = (r 2 r 3 )(r 1 r 4 ) (r 1 r 3 )(r 2 r 4 ). (B.26) Again, sn 1 (sin ψ 4,κ 4 ) = F(ψ 4,κ 4 ) where F(ψ 4,κ 4 ) is the incomplete elliptic integral of the first kind. For the special case r 1 = r 2,wehave r dr r 1 R = 1 (r1 r 3 )(r 1 r 4 ) [ (r r3 )(r r 4 ) ln + r r 1 r1 2 + r ] 3r 4 + 2rr 1 (r r 1 ). (B.27) (r 1 r 3 )(r 1 r 4 ) B.2.3 Integration over r If R(r) = 0 has two complex roots and two real roots, the allowed region of interest for us is r r 3. In the case of four real roots, the allowed region from the emission point in the disk to the detection point is r r 1. The possible turning point will thus be at r t = r 3 in the first case, and at r t = r 1 in the second case. Let us define I r = dr re, Ir e = dr. (B.28) r t R r t R Necessary and sufficient condition for the existence of a turning point is I r < I μ, (B.29) where I μ is the integral defined in Eq. (B.16). In the presence of a turning point, r starts from r e, first decreases to r t, and then increases to. The integration over r along the photon path is rt dr dr I r = + = Ir + Ir e. (B.30) r e R r t R In the absence of a turning point, r monotonically increases from r e to and we have I r = dr = Ir Ir e. (B.31) r e R

10 314 Appendix B: (r,θ)-motion in the Kerr Spacetime B.3 Determination of the Emission Point r e We have to distinguish three cases: (i) R(r) = 0 has two complex roots and two real roots, (ii) R(r) = 0 has four real roots and r 1 = r 2, and (iii) R(r) = 0 has four real roots and r 1 = r 2. In the case (i), wehave I r = 1 AB cn 1 (cos ψ,κ 2 ), I e r = 1 AB cn 1 (cos ψ e,κ 2 ), (B.32) where cos ψ = A B A + B, cos ψ e = (A B)r e r 4 A + r 3 B (A + B) e r 4 A r 3 B. (B.33) If we plug these two expressions into Eqs. (B.30)or(B.31) and we impose I μ = I r, we obtain r e where r e = r 4 A r 3 B (r 4 A + r 3 B) cn(cos ψ 2,κ 2 ) (A B) (A + B) cn(cos ψ 2,κ 2 ), (B.34) cos ψ 2 = AB ( I μ I r ). (B.35) Since in cn(cos ψ 2,κ 2 ) the sign of ψ 2 does not matter, it is not necessary to distinguish the cases with (I μ > Ir ) and without (I μ < Ir ) turning point. In the case (ii), Ir and Ir e are, respectively, I r = 2 (r1 r 3 )(r 2 r 4 ) sn 1 (sin ψ,κ 4 ), I e r = 2 (r1 r 3 )(r 2 r 4 ) sn 1 (sin ψ e,κ 4 ), (B.36) with sin ψ = r2 r 4 r 1 r 4, sin ψ e = (r 2 r 4 )(r e r 1 ) (r 1 r 4 )(r e r 2 ) (B.37) Imposing I μ = I r, we obtain r e r e = r 1(r 2 r 4 ) r 2 (r 1 r 4 ) sn 2 (sin ψ 4,κ 4 ) (r 2 r 4 ) (r 1 r 4 ) sn 2 (sin ψ 4,κ 4 ), (B.38)

11 Appendix B: (r,θ)-motion in the Kerr Spacetime 315 where sin ψ 4 = 1 ( Iμ Ir 2 ) (r1 r 3 )(r 2 r 4 ). (B.39) Since the Jacobian elliptic function sn compares at the square, it does not matter if sin ψ 4 is positive or negative, so if there is or there is no turning point. Lastly, there is the case (iii), in which R(r) = 0 has four real roots and r 1 = r 2. Here, we do not have turning point, so I r = r e + ln dr = R { [ 1 ln 1 + (r1 r 3 )(r 1 r 4 ) [ (re r 3 )(r e r 4 ) r e r 1 + From I μ = I r, the solution for the emission radius r e is ] 2r 1 (r1 r 3 )(r 1 r 4 ) r1 2 + r ] 3r 4 + 2r 1 r e (r e r 1 ). (B.40) (r 1 r 3 )(r 1 r 4 ) r e = 1 [ r1 {r 2 3 r 4 γ r r ] 2 } 3r 4 r 1 (r1 r 3 )(r 1 r 4 ) { 1 [ γ 2r 1 (r1 r 3 )(r 1 r 4 )] 2 } 1, (B.41) where γ is γ = [ ] 2r [ 1 ] 1 + exp I μ (r1 r 3 )(r 1 r 4 ). (B.42) (r1 r 3 )(r 1 r 4 ) References 1. G. Birkhoff, S.S. Mac Lane, A Survey of Modern Algebra (Macmillan, New York, 1965). 2. P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer, Berlin, 1954). 3. S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon Press, Oxford, 1998). 4. J. Dexter, E. Agol, Astrophys. J. 696, 1616 (2009), arxiv: [astro-ph.he]. 5. L.X. Li, E.R. Zimmerman, R. Narayan, J.E. McClintock, Astrophys. J. Suppl. 157, 335 (2005), arxiv:astro-ph/

12 Appendix C AGN Classification Active galactic nuclei (AGN) are very bright galactic nuclei, powered by the mass accretion onto their central supermassive black hole. Figure C.1 shows the AGN classification, groups and subgroups, and the corresponding fraction of members. While it is thought that any normal galaxy has a central supermassive black hole at its center, only a small fraction of them host an AGN. In most galaxies, the central supermassive object is dormient, like SgrA, which has a luminosity of the order of 10 7 in Eddington units. About 93% of the galaxies are non-active. Among the 7% of the active galaxies, most of them are star-forming galaxies or low-ionization nuclear emission-line regions (LINERs). The latter are sometimes considered AGN. Proper AGN are relatively rare: they are in 0.5% of the active galaxies, which means only in 0.035% of all galaxies. AGN are mainly classified according to their luminosity and spectral features. It is thus useful to briefly review their possible spectral components: 1. Radio emission from jets with the typical spectrum from synchrotron radiation. 2. IR emission from the thermal spectrum of the accretion disk, which is reprocessed by gas and dust around the nucleus. This occurs when the accretion disk is obscured by gas and dust. 3. Optical continuum mainly from the thermal spectrum of the accretion disk, and in part from possible jets. 4. Narrow optical lines from cold material orbiting relatively far from the supermassive black hole. The orbital velocity of this material is 500 1,000 km/s. 5. Broad optical lines from cold material orbiting close to the supermassive black hole. The orbital velocity of this material is 1,000 5,000 km/s. The lines are broad due to Doppler boosting. 6. X-ray continuum from a hot corona and possible jets. 7. X-ray lines from fluorescence emission of the gas in the accretion disk illuminated by the X-ray continuum. The iron Kα line at 6.4 kev is usually one of the most prominent lines. Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

13 318 Appendix C: AGN Classification 7% Galaxies 93% Active Galaxies 85% 0.5% Non-active Galaxies 15% Star-forming Galaxies 99% AGN LINERs 1% Radio-quiet AGN Radio-loud AGN Radio Galaxies Radio-loud Quasars Blazars OVV Quasars BL Lac Objects 95% Seyfert-ELFs Seyfert Galaxies NELGs Radio-quiet Quasars 30% 10% 60% Seyfert 1s Seyfert 2s Seyfert 1.5s Seyfert 1.8s Seyfert 1.9s Fig. C.1 Sketch of the AGN family and of its subgroups. This classification has to be taken with caution, because different authors may use slightly different classifications. The diagram shows also the fraction of members in each subgroup. AGN represent only 0.035% of the galactic nuclei. Most of the AGN are radio-quiet and belong to the class of Seyfert galaxies The AGN classification is sometimes confusing, some objects may not be easily associated to a specific group, and different authors may use different classifications. With reference to Fig. C.1, we see that AGN can be grouped into two classes, radio-quiet and radio-loud AGN. In radio-quiet AGN, the jet component is absent or negligible, so the radio luminosity is low. Radio-loud AGN have powerful jets. These jets may be powered by the black hole spin.

14 Appendix C: AGN Classification 319 Fig. C.2 Sketch of a radio-loud AGN according to the unified AGN model [1]. The black hole is surrounded by an accretion disk, which may be obscured by a dusty torus. The broad line region is close to the black hole and there are clouds orbiting with high velocity. The narrow line region is relatively far from the black hole and there are clouds moving at lower velocity. The arrows represent the lines of sight of the observer. Depending on the angle between the jet and the line of sight of the observer, the AGN can appear as a blazar, as a radio-loud quasar, or as a radio galaxy Radio-quiet AGN may be grouped into four classes: Seyfert extremely luminous far IR galaxies (Seyfert-ELFs), Seyfert galaxies, narrow emission line galaxies (NELGs), and radio-quiet quasars. The classification is based on a number of properties. For instance, Seyfert galaxies have an optical continuum and emission lines. Seyfert 1s have both narrow and broad emission lines, while Seyfert 2s have only narrow emission lines. Seyfert galaxies of type 1.5, 1.8, and 1.9 are grouped according to their spectral appearance. Radio-loud AGN can be grouped into three classes: radio galaxies, radio-loud quasars, and blazars. Blazars are characterized by rapid variability and by polarized optical, radio and X-ray emission. They are divided into BL Lacertae objects (BL Lac objects) and optically violent variable quasars (OVV quasars). OVV quasars have stronger broad emission lines than BL Lac objects. According to the unified AGN model [1], all AGN would essentially belong to the same class of objects. They look different because they are observed from different viewing angles. Figures C.2 and C.3 illustrate the idea of the unified AGN model. In the case of blazars, the jet would be along our line of sight. As the angle between the jet and our line of sight increases, we would have radio-loud quasars and then radio galaxies. In the case of radio-quiet AGN (Fig. C.3), we have a similar situation, even if there is no jet: depending on the viewing angle of the observer, the AGN can appear as a radio-quiet quasar, as a Seyfert 1 galaxy, or as a Seyfert 2 galaxy.

15 320 Appendix C: AGN Classification Radio-quiet Quasar Seyfert 1 Seyfert 2 Fig. C.3 As in Fig. C.2, in the case of a radio-quiet AGN. Depending on the viewing angle of the observer, the AGN can appear as a radio-quiet quasar, as a Seyfert 1 galaxy, or as a Seyfert 2 galaxy Reference 1. C.M. Urry, P. Padovani, Publ. Astron. Soc. Pac. 107, 803 (1995), arxiv:astro-ph/

16 Appendix D Jets Jets are a common feature of several astrophysical objects, including protostars, stars, neutron stars, and black holes. Jets are observed both from stellar-mass black holes in X-ray binaries and from supermassive black holes in galactic nuclei, see e.g. [5, 11, 21]. The exact mechanism responsible for the formation of black hole jets is currently unknown. D.1 Formation Mechanisms The two most popular mechanisms for the formations of black hole jets are the Blandford Znajek model [4] and the Blandford Payne model [3], both with a number of variants and extensions. There are also proposals of hybrid models, in which the two mechanisms can coexist [10]. In theblandford Znajek mechanism, magnetic fields thread the black hole horizon and can extract the rotational energy of the compact object [4]. This mechanism exploits the existence of the ergoregion via the Penrose process [15]. However, strictly speaking, the extraction of the rotational energy of a compact object may be possible even in the case of neutron stars in the presence of magnetic fields anchored on the surface of the body. In the original paper by Blandford and Znajek, the jet power P BZ was derived assuming a 1. In this case, one finds P BZ a 2. A more detailed analysis provides the following formula [18] P BZ = κ 16π Φ2 B Ω2 H f (Ω H). (D.1) κ is a numerical constant which depends on the magnetic field configuration; for instance, in the Kerr metric κ = for a split monopole geometry and for a parabolic geometry. Ω H is the angular frequency at the black hole horizon Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

17 322 Appendix D: Jets ( Ω H = g ) tφ (General case) (D.2) g φφ r=r H = a (Kerr metric). (D.3) 2r H Equation (D.2) assumes that the spacetime is stationary and axisymmetric and holds when the metric is written in the canonical form (see Sect. 3.1). Φ B is the magnetic flux threading the black hole horizon Φ B = π 2π 0 0 B r gdθdφ. (D.4) f (Ω H ) is a dimensionless function that takes into account higher order terms in Ω H, so f (Ω H ) 1 + c 1 M 2 Ω 2 H + c 2M 4 Ω 4 H + (D.5) where {c i } are numerical coefficients. For a Kerr black hole with a thin accretion disk, c 1 = 1.38 and c 2 = 9.2 [18]. The jet power eventually depends on the background metric, the angular frequency of the horizon Ω H, and the magnetic field. In the Blandford Payne mechanism, magnetic fields thread the accretion disk, corrotating with it [3]. Now the energy is provided by the gravitational potential energy of the accretion flow. The power of the jet can be written as P BZ ε L acc ln ( rout r in ), (D.6) where ε is the efficiency of the transformation of the binding energy of the accreting matter into jet power at the inner radius of the disk r in, and r out is the outer radius of the disk. D.2 Observations D.2.1 Stellar-Mass Black Holes In the case of black hole binaries, we observe two kinds of jets [5]. Steady jets manifest when a source is in the low-hard state. The jet is steady, typically not very relativistic, and may extend up to a few tens of AU. Transient jets are instead observed when a source switches from the hard to the soft state and crosses the jet line (see Sect. 4.5). These pc-scale jets appears as blobs of plasma emitting mainly in the radio band and are relativistic. They have features similar to the kpc-scale jets observed

18 Appendix D: Jets 323 in AGN and for this reason the black hole binaries producing transient jets are also called microquasars [11]. If the mechanism responsible for the formation of jets were the Blandford Znajek model, one may expect a correlation between black hole spin measurements and estimates of the jet power. Fender et al. [6] have studied spin measurements of black hole binaries reported in the literature and inferred from the continuum-fitting and the iron line methods. Their plots do not show any correlation between black hole spin and jet power. Narayan and McClintock [12] have proposed that the Blandford Znajek mechanism is responsible for the formation of transient jets. They selected some spin measurements obtained via the continuum-fitting method and used a different proxy for the jet power. They find a correlation between black hole spin measurements and jet power. For the moment, this is a controversial issue [14, 8]. Both conclusions are based on a small number of data with large uncertainty. Future observations should be able to increase the number of sources and decrease the uncertainty in these measurements, and provide a conclusive answer to this issue [8]. If some kind of jets were powered by the black hole spin, the combination of spin measurements and estimates of the jet power could be used to test the Kerr metric [1, 2, 16]. D.2.2 Supermassive Black Holes In the case of AGN, only a small fraction of them, around 1%, exhibit relativistic kpc-scale jets. One of the most spectacular example is Cygnus A, see Fig. D.1. Radio images of this object show two very collimated jets from the very center of the galaxy, Fig. D.1 Radio image of Cygnus A. The bright dot at the center is the location of the supermassive black hole, where the two relativistic jets are generated. The jets are stopped by the intergalactic medium, forming two giant lobes. Image courtesy of NRAO/AUI

19 324 Appendix D: Jets where it is supposed to be located its supermassive black hole. The two jets extend well outside the galaxy, for hundreds of kpc. Jets dominate the spectrum of a source at radio frequencies. There are apparently two distinct populations of AGN: radio-loud AGN and radio-quiet AGN [17]. This is particularly evident on the plane optical luminosity vs radio luminosity. For the same optical luminosity, radio-loud AGN have a radio luminosity 3 4 orders of magnitude higher than that of radio-quiet AGN. These two populations seem really to follow two different tracks with a gap between them. The origin of the radio-quiet/radio-loud dichotomy is not understood [17]. Some authors have also doubted about the actual existence of this dichotomy, suggesting it is a bias of observations. In the case of AGN with an accretion luminosity above 1% of the Eddington limit, the most natural interpretation is that their jets are the counterpart of the transient jets in black hole binaries. This conclusion may be supported by the consideration that microquasars show intermittent jets for a few percent of the time, which is similar to the fraction of radio-loud AGN [13]. The time scale of these systems is proportional to their mass, so intermittent jets in black hole binaries look like persistent jets in AGN. However, in the case of AGN with a luminosity below 1% of the Eddington limit, this explanation cannot work: black hole binaries with a low accretion luminosity are all radio-loud. Another popular interpretation is that the dichotomy is determined by the black hole spin. When the accretion luminosity is low, it turns out that radio-loud AGN are in elliptical galaxies, while radio-quiet AGN are mainly in spiral galaxies. Galaxies with different morphology have likely a different merger and accretion history. This, in turn, may have produced two populations of black holes, with high and low values of spin [20]. A difference in radio luminosity of 3 4 orders of magnitude between the two populations is impossible to explain if the jet power is proportional to Ω 2 H, but in the case of thick disks the jet power may scale as Ω 6 H [18]. If jets are powered by the rotational energy of the accreting compact object, it is possible to extract energy and have an accretion efficiency η>1, where L acc = ηṁ. Some observations indicate that some AGN may have η>1 [7, 9]. If these measurements are correct, the jet is extracting energy from the system, and it is likely that this is the rotational energy of the black hole; some version of the Blandford Znajek mechanism is working. While in the past some GRMHD simulations have not been able to find high efficiency from the formation of jets, more recently Tchekhovskoy et al. [19] have presented simulations in which η>1. References 1. C. Bambi, Phys. Rev. D 85, (2012), arxiv: [gr-qc]. 2. C. Bambi, Phys. Rev. D 86, (2012), arxiv: [gr-qc]. 3. R.D. Blandford, D.G. Payne, Mon. Not. R. Astron. Soc. 199, 883 (1982). 4. R.D. Blandford, R.L. Znajek, Mon. Not. R. Astron. Soc. 179, 433 (1977). 5. R.P. Fender, T.M. Belloni, E. Gallo, Mon. Not. R. Astron. Soc. 355, 1105 (2004), arxiv:astro-ph/

20 Appendix D: Jets R. Fender, E. Gallo, D. Russell, Mon. Not. R. Astron. Soc. 406, 1425 (2010), arxiv: [astro-ph.he]. 7. G. Ghisellini, F. Tavecchio, L. Foschini, G. Ghirlanda, L. Maraschi, A. Celotti, Mon. Not. R. Astron. Soc. 402, 497 (2010), arxiv: [astro-ph.co]. 8. J.E. McClintock, R. Narayan, J.F. Steiner, Space Sci. Rev. 183, 295 (2014), arxiv: [astro-ph.he]. 9. B.R. McNamara, M. Rohanizadegan, P.E.J. Nulsen, Astrophys. J. 727, 39 (2011), arxiv: [astro-ph.co]. 10. D.L. Meier, Astrophys. J. 548, L9 (2001), arxiv:astro-ph/ I.F. Mirabel, L.F. Rodriguez, Annu. Rev. Astron. Astrophys. 37, 409 (1999), arxiv:astro-ph/ R. Narayan, J.E. McClintock, Mon. Not. R. Astron. Soc. 419, L69 (2012), arxiv: [astro-ph.he]. 13. C. Nipoti, K.M. Blundell, J. Binney, Mon. Not. R. Astron. Soc. 361, 633 (2005), arxiv:astro-ph/ D.M. Russell, E. Gallo, R.P. Fender, Mon. Not. R. Astron. Soc. 431, 405 (2013), arxiv: [astro-ph.he]. 15. R. Penrose, Riv. Nuovo Cim. 1, 252 (1969) [Gen. Rel. Grav. 34, 1141 (2002)]. 16. G. Pei, S. Nampalliwar, C. Bambi, M.J. Middleton, Eur. Phys. J. C 76, 534 (2016), arxiv: [gr-qc]. 17. M. Sikora, L. Stawarz, J.P. Lasota, Astrophys. J. 658, 815 (2007), arxiv:astro-ph/ A. Tchekhovskoy, R. Narayan, J.C. McKinney, Astrophys. J. 711, 50 (2010), arxiv: [astro-ph.he]. 19. A. Tchekhovskoy, R. Narayan, J.C. McKinney, Mon. Not. R. Astron. Soc. 418, L79 (2011), arxiv: [astro-ph.he]. 20. M. Volonteri, M. Sikora, J.P. Lasota, Astrophys. J. 667, 704 (2007), arxiv: [astro-ph]. 21. J.A. Zensus, Annu. Rev. Astron. Astrophys. 35, 607 (1997).

21 Appendix E Thick Accretion Disks The Novikov Thorne model reviewed in Chap. 6 describes geometrically thin and optically thick accretion disks around black holes. It assumes that the self-gravitation of the disk and the gas pressure are negligible, so the fluid elements follow the geodesics of the background metric. If the gas pressure cannot be neglected, the disk can be geometrically thick and the fluid elements do not follow the geodesics of the background metric any longer. This appendix briefly reviews the so-called Polish doughnut model [3, 1]. It is a simple model to describes thick accretion disks in which the self-gravitation of the disk is still neglected but the gas pressure is taken into account. E.1 Polish Doughnut Model Like the Novikov Thorne model for thin disks, even the Polish doughnut model can be formulated in a generic stationary, axisymmetric, and asymptotically flat spacetime. We can thus write the line element as ds 2 = g tt dt 2 + g rr dr 2 + g θθ dθ 2 + 2g tφ dtdφ + g φφ dφ 2, (E.1) where the metric elements are independent of the t and φ coordinates. The disk is modeled as a perfect fluid with purely azimuthal flow. Its energy-momentum tensor is thus T μν = (ρ + P)u μ u ν + g μν P, u μ = ( u t, 0, 0, u φ), (E.2) where ρ and P are, respectively, the energy density and the pressure of the fluid. From the definitions of the specific energy of the fluid element u t, its angular velocity Ω = u φ /u t, and its angular momentum per unit energy l = u φ /u t,we have the following relations Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

22 328 Appendix E: Thick Accretion Disks gtφ 2 u t = g ttg φφ, g φφ + 2lg tφ + l 2 g tt (E.3) Ω = lg tt + g tφ lg tφ + g φφ, (E.4) l = g tφ + Ωg φφ g tt + Ωg tφ. (E.5) It is worth noting that l is conserved for a stationary and axisymmetric flow in a stationary and axisymmetric spacetime [3]. The disk s structure can be inferred from the Euler equation, ν T μν = 0. One finds a μ = gμν + u μ u ν ν P, ρ + P (E.6) where a μ = u ν ν u μ is the fluid 4-acceleration. If the pressure is independent of the t and φ coordinates (which follows from the assumption that the spacetime is stationary and axisymmetric) and if the equation of state is barotropic, i.e. ρ = ρ(p), a μ can be written as a gradient of a scalar potential W (P) P dp a μ = μ W, W (P) = ρ(p ) + P. (E.7) After some algebra, it is possible to write Ω as a function of l, i.e. Ω = Ω(l), and integrate the Euler equation to get W 2 W = W in + ln u l t Ωdl u in + t l in 1 Ωl, (E.8) where W in, l in, and u in t are, respectively, the potential, the angular momentum per unit energy, and (minus) the energy per unit mass at the inner edge of the fluid configuration. Actually, in Eq. (E.8) W in, l in, and u t in can be replaced by the value of W, l, and u t at any other point of the fluid boundary. In the Newtonian limit, W reduces to the total potential, i.e. the sum of the gravitational potential and of the centrifugal one, and at infinity W = 0. If the metric of the spacetime is known, there is only one unspecified function, Ω = Ω(l), which characterizes the fluid rotation. In the zero-viscosity case, this function cannot be deduced from any equation, and it must be given as an assumption of the model. In this sense, the model is not very constrained as in the case of the Novikov Thorne one, in which the radial structure of the disk follows from the equations of conservation of rest-mass, energy, and angular momentum. 2 In the special case l = const., Ω is not constant, but Eq. (E.8) is still correct and the integral vanishes.

23 Appendix E: Thick Accretion Disks 329 Imposing a specific relation between Ω and l, we can find the equipotential surfaces W = const. <0, i.e. the surfaces of constant pressure describing the possible boundaries of the fluid configuration. One of these surfaces may have one (or more) sharp cusp(s), which may induce the accretion onto the compact object: like the cusp at the L 1 Lagrange point in a close binary system, the accreting gas can fill the Roche lobe and then be transferred to the compact object. The mechanism does not need the fluid viscosity to work, so the latter may be, at least in principle, very low. A particularly simple case is the configuration with l = const., which is marginally stable with respect to axisymmetric perturbations (the criterion for convective stability is simply that l does not have to decrease outward) [6, 5]. In this specific case, theintegralineq.(e.8) vanishes and W = ln( u t ) + const.. (E.9) In the Kerr spacetime, we find five qualitatively different configurations [1]. Figure E.1 shows these five possibilities in the case of a black hole with the spin parameter a = 0.8. We have: 1. l < l ms, where l ms is the angular momentum per unit energy of the marginally stable equatorial circular orbit (or ISCO). In this case, there are no closed equipotential surfaces and therefore there is no accretion disk (the angular momentum of the fluid is too low). 2. l = l ms. This is the limiting case between l < l ms and l ms < l < l mb.attheisco radius there is a flex (not a minimum) of W. The disk exists as an infinitesimally thin unstable ring at the ISCO. 3. l ms < l < l mb, where l mb is the angular momentum per unit energy of the marginally bound equatorial circular orbit. There is only one equipotential surface W = W cusp with a cusp on the equatorial plane. The cusp is located between the marginally bound and the marginally stable radius. Equipotential surfaces with W > W cusp do not represent any disk configuration, while those with W < W cusp describe the surface of non-accreting disks. Accretion starts when the gas fills the equipotential surface with the cusp. 4. l = l mb. This is the limiting case between l ms < l < l mb and l > l mb. The cusp is located in the equatorial plane and belongs to the marginally closed equipotential surface W = 0. Accretion is possible in the limit of a disk of infinite size. 5. l > l mb. The angular momentum of the fluid is too high and no accretion is possible. There are no equipotential surfaces W 0 with cusps. In non-kerr backgrounds, the picture may be slightly different [4]. For instance, there may not be marginally bound orbits and there may be disks with more than one cusp.

24 330 Appendix E: Thick Accretion Disks r sinθ l < l ms 4 3 W = 0 2 W < BH r cosθ r sinθ l = l ms 4 3 W = 0 2 W < BH r cosθ r sinθ l ms < l < l mb 4 3 W = 0 W = W cusp BH C r cosθ r sinθ l = l mb 4 3 W = W cusp = 0 2 W < BH C r cosθ r sinθ l > l mb 4 3 W = 0 2 W < BH C r cosθ Fig. E.1 Equipotential surfaces around a Kerr black hole. The spin parameter is a = 0.8, which implies l ms = and l mb = Top left panel l = and there is no disk. Top right panel l = l ms and it is possible an unstable ring at r = r ISCO. Central left panel l = 2.800, there is an infinite number of disks without cusp, and there is a disk with a cusp. Central right panel l = l mb, there is an infinite number of disks without cusp, and the disk with a cusp is that with W = 0. Bottom panel l = and there is no disk with a cusp. See the text for more details

25 Appendix E: Thick Accretion Disks 331 E.2 Evolution of the Spin Parameter As done in Sect. 6.4 for the thin disk case, we can now consider the evolution of the spin parameter of a black hole accreting from a thick disk. The argument is the same and Eq. (6.39) becomes da d ln M = 1 L in 2a, M E in (E.10) where E in and L in are, respectively, the specific energy and the specific angular momentum of the gas at the inner edge of the disk. Accretion from a thick disk is potentially more efficient to spin a black hole up [2]. If we consider Eq. (E.10) in the Kerr metric, we can see that the equilibrium spin parameter is still 1 but it can be reached after accreting a smaller amount of matter with respect to the thin disk case. If we take into account the radiation emitted by the disk and captured by the black hole, the counterpart of Eq. (6.43), we can potentially get closer to 1 than with a thin disk, because of the lower radiative efficiency. If we consider the extreme case with l = l mb, E in = 1 and η = 1 E in = 0. The situation in a realistic case is actually more complicated and it depends on the viscosity of the accreting fluid, but numerical calculations suggest that it is indeed possible to get a spin parameter very close to 1 [8]. E.3 Spectrum of an Ion Torus The Polish doughnut model can be provided with a microscopic model for the accreting gas and be used to model the spectrum of black holes with a geometrically thick disk. While it is a relatively simple model and more realistic scenarios would require GRMHD simulations, it may capture some key-features of the accretion spectrum of the source. For instance, the Polish doughnut model has been employed in [9, 10] to try to describe the spectrum of the accretion structure around SgrA. Straub et al. [9] and Vincent et al. [10] use the Polish doughnut model to describe a radiatively inefficient and advection dominated accretion flow like the one expected in the case of SgrA. They employ a gas pressure dominated optically thin 2-temperature plasma model for the microscopic physics. The model has seven parameters: the black hole spin of the background metric, the viewing angle of the observer, and five parameters for the accretion flow and its microphysics:

26 332 Appendix E: Thick Accretion Disks where background metric spin parameter observer viewing angle i ion torus dimensionless specific angular momentum λ magnetic to total pressure ratio β polytropic index n energy density at the center electron temperature at the center a ρ c T c λ = l l ms l mb l ms (E.11) ranges from 0 to 1. For a non-relativistic gas with negligible radiation pressure, the polytropic index is n = 3/2. The pressure P is the sum of the magnetic pressure P m and the gas pressure P gas, which, for simplicity, can be supposed to have a constant ratio P = P m + P gas, P m = B2 24π = β P. (E.12) B is the intensity of the magnetic field. The gas is described by a 2-temperature plasma, so the gas pressure is the sum of the ion and electron contributions. The pressure P, the energy density ρ, and the electron temperature T can be written as [9, 10] P = Kρ 5/3, (E.13) ρ = 1 [ (Kρ 1/n K n c + 1 ) ] ω n 1, (E.14) T = T c ( ρ ρ c ) 1/n, (E.15) where K = 1 ( ) Wc W s ρc 1/n exp 1, (E.16) n + 1 and W c and W s denote the scalar potential W evaluated, respectively, at the center and at the surface of the accretion disk. With this set-up, we can compute the spectrum of an ion torus. The observed specific flux is given by F obs (ν obs ) = I obs d Ω. (E.17)

27 Appendix E: Thick Accretion Disks 333 Fig. E.2 Left panel spectrum of an ion torus for the following set of parameters: a/m = 0.5, i = 60, λ = 0.3, β = 0.1, n = 3/2, ρ c = g/cm 3,andT c /T v = T v is the viral temperature. Right panel contribution from every radiative mechanism to the spectrum in the left panel bremsstrahlung emission (blue solid line), Comptonization of bremsstrahlung emission (violet dotted line), synchrotron radiation (red dashed line), and Comptonization of synchrotron radiation (green dashed-dotted line). From [7], reproduced by permission of IOP Publishing. All rights reserved The specific intensity I obs can be evaluated by integrating along every photon path the emission and absorption contributions I obs (ν e ) = g 3[ ] j e (ν e ) α(ν e )I e (ν e ) dl, (E.18) where j e and α are, respectively, the emission and absorption coefficient, dl = u μ k μ dτ is the infinitesimal proper length as measured in the rest frame of the emitter, and τ is the affine parameter of the photon trajectory. Neglecting absorption (α = 0) and assuming that the electromagnetic spectrum of the accretion structure is produced by bremsstrahlung, synchrotron processes, and inverse Compton scattering of both bremsstrahlung and synchrotron photons off free electrons in the medium, the emission coefficient can be written as j e (ν e ) = j brem e (ν e ) + j sync e (ν e ) + je Cbrem (ν e ) + je Csyn (ν e ). (E.19) Figure E.2 shows an example of spectrum from this model (left panel) and the contributions from every emission mechanism (right panel). References 1. M. Abramowicz, M. Jaroszynski, M. Sikora, Astron. Astrophys. 63, 221 (1978). 2. M.A. Abramowicz, J.P. Lasota, Acta Astron. 30, 35 (1980). 3. M. Kozlowski, M. Jaroszynski, M. Abramowicz, Astron. Astrophys. 63, 209 (1978). 4. Z.Li,C.Bambi,JCAP1303, 031 (2013), arxiv: [gr-qc]. 5. F. Daigne, J.A. Font, Mon. Not. R. Astron. Soc. 349, 841 (2004), arxiv:astro-ph/

28 334 Appendix E: Thick Accretion Disks 6. J.A. Font, F. Daigne, Mon. Not. R. Astron. Soc. 334, 383 (2002), arxiv:astro-ph/ N. Lin, Z. Li, J. Arthur, R. Asquith, C. Bambi, JCAP 1509, 09, 038 (2015), arxiv: [gr-qc]. 8. A. Sadowski, M. Bursa, M. Abramowicz, W. Kluzniak, J.-P. Lasota, R. Moderski, M. Safarzadeh, Astron. Astrophys. 532, A41 (2011), arxiv: [astro-ph.he]. 9. O. Straub, F.H. Vincent, M.A. Abramowicz, E. Gourgoulhon, T. Paumard, Astron. Astrophys. 543, A83 (2012), arxiv: [astro-ph.ga]. 10. F.H. Vincent, W. Yan, M.A. Abramowicz, A.A. Zdziarski, O. Straub, Astron. Astrophys. 574, A48 (2015), arxiv: [astro-ph.ga].

29 Appendix F Astrophysical Constants Quantity Symbol, Equation Value Speed of Light in Vacuum c ms 1 Newton s Constant G N (8) m 3 kg 1 s 2 Boltzmann Constant k B (13) JK (78) 10 5 ev K 1 Stefan Boltzmann Constant σ SB (21) 10 8 Wm 2 K 4 Permeability of Free Space μ 0 4π 10 7 NA 2 Permittivity of Free Space ε 0 = 1/(μ 0 c 2 ) Fm 1 Electron Charge Magnitude e (35) C Electron Mass m e (40) kg Proton Mass m p (74) kg Classical Electron Radius r e = e 2 /(4πɛ 0 m e c 2 ) (27) m Thomson Cross Section σ Th = 8πre 2/ (13) m 2 Astronomic Unit AU m Parsec (1 AU/1 arc sec) pc m Solar Mass M (2) kg Solar Luminosity L W erg s 1 Schwarschild Radius of the Sun r Sch = 2G N M c (2) 10 3 m Gravitational Radius of the Sun r g = G N M c (1) 10 3 m Characteristic Time τ = r g /c G N M c s Characteristic Frequency ν = 1/τ 1/(G N M c 3 ) Hz Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

30 Appendix G Glossary Active galactic nucleus (AGN). Compact region at the center of a galaxy powered by the mass accretion onto a supermassive black hole. The term is also used to indicate the central supermassive black hole. See Appendix C about the AGN classification. Black hole binary (BHB). Binary system in which one of the two objects is a stellar-mass black hole. Binary black hole. Binary system in which both objects are black holes. The term can be used to indicate either a binary system of two stellar-mass black holes or a binary system of two supermassive black holes. Blazar. A class of particularly bright quasars. According to the unified AGN model, blazars are those radio-loud AGN in which the relativistic jet points in the direction of Earth. See Appendix C for more details. Bolometric luminosity. Total electromagnetic luminosity of an object, namely the electromagnetic luminosity integrated over all wavelengths. Broad line region (BLR). It is the region close to a supermassive black hole, where cold material orbits with a velocity of 1,000 5,000 km/s. This material produces broad lines because of the Doppler boosting. Corona. This term is used to indicate some kind of hot, usually optically-thin, electron cloud which enshrouds the central disk and acts as a source of X-rays. See Chap. 8 and Fig. 8.1 for more details. Eddington luminosity. It is the maximum luminosity for an object. Such a luminosity is reached when the pressure of the radiation luminosity balances the gravitational force towards the object. The Eddington luminosity of an object of mass M is (assuming a ionized gas of protons and electrons) L Edd = 4πG ( ) N Mm p c M = erg/s, (G.1) σ Th M Springer Nature Singapore Pte Ltd C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, DOI /

31 338 Appendix G: Glossary where G N is Newton s gravitational constant, m p is the proton mass, c is the speed of light, and σ Th is the electron Thomson cross section. In the special case of a black hole, the Eddington luminosity sets the maximum accretion luminosity. If we write L Edd = η r Ṁ Edd c 2, where η r is the radiative efficiency, the Eddington accretion rate is ( )( ) 0.1 M Ṁ Edd = g/s η r M ( )( ) 0.1 M = M /yr. (G.2) η r M Effective area. It is somehow a measurement of the efficiency of the optics and the detector of an instrument at detecting photons. It is usually measured in units of cm 2. See Sect for more details. Hardness. It is the ratio between the luminosities in the hard and soft X-ray bands, but the exact definition may vary. For instance, it may be the ratio between the luminosity of the 6 10 kev band and of the 2 6 kev band, i.e. H = L 6 10 kev /L 2 6 kev. Hard X-ray band. There is not a universal definition of hard X-ray band, so the exact energy range must be specified. In some contexts, it may indicate, for instance, the 6 10 kev band. In other contexts, it may be used for the X-ray spectrum above 10 kev. Hawking radiation. If quantum effects are taken into account, black holes radiate (almost) like a blackbody with the temperature T BH = κ/(2π), where κ is the surface gravity at the event horizon. Such a radiation is called the Hawking radiation. For a Schwarzschild black hole, the temperature is T BH = 1 8π M = ( M M ) ev. (G.3) The luminosity of a black hole due to Hawking radiation depends on the black hole mass and the particle content of the theory. The black hole luminosity can be written as ( ) 2 L BH = σ 4π R 2 TBH 4 M erg/s, (G.4) M where σ is some constant that takes into account the particle content and also depends on the black hole mass. The constant σ would be the Stefan Boltzmann constant if the black hole emission were exactly that of a blackbody. A black hole emits any particle of the theory (not only electromagnetic radiation) and the spectrum of the emitted particles deviates from the blackbody one due to the mass and the spin of the particles, the finite size of the black hole, etc. The value of L BH is extremely low for M M, and the Hawking radiation can be unlikely detected from astrophysical

32 Appendix G: Glossary 339 black holes, even in the future. This radiation causes the evaporation of the black hole: the black hole emits particles and its mass decreases. The evaporation time can be obtained from dm dt = L BH = 4π R 2 σ T 4 BH = σ 4 4 π 3 1 M 2. (G.5) Integrating by parts, we find (assuming for simplicity σ = const.) τ evap = τevap 0 dt σ M 4 4 π 3 0 M 2 d M ( M M ) 3 yrs. (G.6) High-mass X-ray binary (HMXB). X-ray binary with a high-mass (M > 10 M ) stellar companion. HMXBs are often persistent sources, because mass accretion onto the compact object originates from the wind of the companion star, which is a relatively stable process. The compact object may be either a black hole or a neutron star. High-soft state. It is one of the two historical states of a black hole binary. The spectrum is soft and dominated by the thermal component of the accretion disk. See Sect. 4.5 for more details. Geometrically thin/thick disk. An accretion disk is geometrically thin (thick) if the disk opening angle is h/r 1(h/r 1), where h is the semi-thickness of the disk at the radial coordinate r. Innermost stable circular orbit (ISCO). It is the circular orbit separating unstable and stable circular orbits. See Sects. 3.1 and 3.2 for more details. Low-hard state. It is one of the two historical states of a black hole binary. The spectrum is hard and dominated by the power-law component. See Sect. 4.5 for more details. Low-mass X-ray binary (LMXB). X-ray binary with a low-mass (M < 3 M ) stellar companion. LMXBs are often transient sources, because mass accretion onto the compact object occurs via Roche lobe overflow from the companion star, which is not a stable process. The compact object may be either a black hole or a neutron star. Marginally stable circular orbit. It is the circular orbit separating unstable and stable circular orbits. It is equivalent to innermost stable circular orbit (ISCO). See Sects. 3.1 and 3.2 for more details. Narrow line region (NLR). It is the region far from a supermassive black hole, where cold material orbits with a velocity of 500 1,000 km/s. This material produces narrow lines because the effect of the Doppler boosting is weak. Optical band. Spectrum of the electromagnetic radiation in the wavelength range nm (energy range ev). See Table 5.1 for more details.