The Marginally Stable Circular Orbit of the Fluid Disk around a Black Hole
aa r X i v : . [ a s t r o - ph . H E ] J un APS/123-QED
The Marginally Stable Circular Orbit of the Fluid Disk around a Black Hole
Lei Qian , Xue-Bing Wu , Li-Xin Li , Department of Astronomy, Peking University, Beijing, 100871, China Kavli Institute for Astronomy & Astrophysics, Peking University, Beijing, 100871, China (Dated: November 6, 2018)The inner boundary of a black hole accretion disk is often set to the marginally stable circularorbit (or the innermost stable circular orbit, ISCO) around the black hole. It is important for thetheories of black hole accretion disks and their applications to astrophysical black hole systems.Traditionally, the marginally stable circular orbit is obtained by considering the equatorial motionof a test particle around a black hole. However, in reality the accretion flow around black holesconsists of fluid, in which the pressure often plays an important role. Here we consider the influenceof fluid pressure on the location of marginally stable circular orbit around black holes. It is foundthat when the temperature of the fluid is so low that the thermal energy of a particle is muchsmaller than its rest energy, the location of marginally stable circular orbit is almost the same asthat in the test particle case. However, we demonstrate that in some special cases the marginallystable circular orbit can be different when the fluid pressure is large and the thermal energy becomesnon-negligible comparing with the rest energy. We present our results for both the cases of non-spinning and spinning black holes. The influences of our results on the black hole spin parametermeasurement in X-ray binaries and the energy release efficiency of accretion flows around black holesare discussed.
PACS numbers: 04.70.-s, 97.60.Lf, 98.35.Mp
I. INTRODUCTION
In the spacetime around a black hole, the circular mo-tion of a test particle is not always possible according togeneral relativity. There is a smallest radius on whichthe circular motion of a test particle is marginally sta-ble [1]. This radius is called marginally stable circularorbit. With a small perturbation, the circularly movingparticle at this radius will then plunge into the black holefreely. The properties of the transition from inspiral toplunge depend on the mass ratio η of the particle andthe black hole [2]. The transition would be less grad-ual (or more ”abrupt”) with smaller η (See also [3, 4]).This property is more relevant in the context of the in-spiral in a binary black hole system, when the two blackholes are just about to merge. When the self-force ofthe inspiralling particle is considered, the location of themarginally stable orbit is also modified [5].The marginally stable circular orbit is also importantin the estimate of energy release of the accretion to ablack hole. By calculating the binding energy of the cir-cular motion on this radius [6], one can estimate howmuch of the total energy can be released during the ac-cretion process. For the accretion flow around a blackhole, the energy release efficiency rangs from 5.6% (fornon-spinning or Schwarzschild black holes) to 42% (forextreme Kerr black holes). The location of the marginallystable circular orbit and the energy release efficiency areclosely related, both depending on the spin parameter ofthe black hole [7].In some applications of accretion disk model in astro-physical observations, the location of marginally stablecircular orbit is also crucial. Since the marginally stablecircular orbit depends on the spin of the central blackhole, it is possible to measure the spin of a black hole if the corresponding marginally stable orbit can be mea-sured. By fitting the observed soft state spectra of blackhole X-ray binaries, which are assumed to come from athin disk with its inner boundary at the marginally sta-ble circular orbit, one can derive the location of this orbitand estimate the spin parameters of the black holes in X-ray binaries [8].In the theory of accretion flow, the marginally stablecircular orbit is also important. Due to its transitionalnature from the inspiralling region to the plunging re-gion, it is often believed that an accretion disk is torquefree at this radius. Although still being debated whenconsidering magnetic fields [9, 10], the torque free condi-tion on the marginally stable circular orbit serves as animportant inner boundary condition in many accretiondisk models.The self-gravity of the particle is not relevant in thecontext of accretion disks, but the material in a real ac-cretion disk is fluid rather than test particles. Whenconsidering fluid, the particles within it are interactingwith each other, and the pressure plays an importantrole in the dynamics. In the study of accretion disk the-ory, the marginally stable circular orbit for a test par-ticle is used as the inner boundary of an accretion diskin many models. More precisely, it has been shown byanalytical theory of fluid tori around black holes thatthe inner boundary of an accretion disk lies between themarginally stable circular orbit and the marginally boundorbit, while both expressions are for test particles [11–13].In this work, however, we focus on a different problem,that is, we try to investigate the marginally stable cir-cular orbit itself, with the influence from the pressure inthe fluid. In section 2, we briefly mention the marginallystable circular orbit in the case of a test particle. Thenwe consider a thin disk consisting of perfect fluid arounda black hole and discuss the marginally stable circularorbit in this case in section 3. In Section 4 we presenta brief discussion on our results. In this paper, we set c = G = M = 1, with c , G , and M the speed of light,gravitational constant, and the mass of the black hole,respectively. We use (+ − − − ) signature all through. II. MARGINALLY STABLE CIRCULAR ORBITOF A TEST PARTICLE
The metric of the spacetime outside a Kerr (spinning,non-charged) black hole is [1] ds = ρ ∆ A dt − A sin θρ ( dφ − ωdt ) − ρ ∆ dr − ρ dθ , (1)where ρ = r + a cos θ,A = ( r + a ) − ∆ a sin θ ∆ = r − r + a , ω = 2 arA , (2)and a is the spin parameter of the black hole. The co-variant form of the metric can be written as g tt = − ρ (cid:20) a sin θ − ( r + a ) ∆ (cid:21) , g rr = − ∆ ρ ,g θθ = − ρ , g φφ = − ρ (cid:18) θ − a ∆ (cid:19) ,g tφ = − aρ (cid:18) − r + a ∆ (cid:19) . (3)For a steady and axis-symmetric spacetime (e.g.Schwarzschild, Kerr), there are two apparent constantsof motion of a free particle with unit mass u t = E, u φ = − L (4)Since the metric is block diagonal, the normalization con-dition of the 4-velocity can be written as g tt u t u t + 2 g tφ u t u φ + g φφ u φ u φ + g rr u r u r + g θθ u θ u θ = 1(5)Consider the circular orbital motion on the equatorialplane ( θ = π/ u θ ≡ dθ/dτ = 0), the equation of motioncan be written as r (cid:18) drdτ (cid:19) = U r , (6)where U r = ( r + a r + 2 a r ) E + 4 arEL − ( r − r ) L − ( r − r + a r ) . (7)In order to maintain a circular orbit, dr/dτ and d r/dτ should both vanish, which imply U r = 0 , U ′ r = 0 , (8)where the prime denotes the derivative to r . The twoconstants of motion can be derived from the above twoequations as E = r / − r / + ar / ( r / − r / + 2 a ) / , (9) L = − r − ar / + a r / ( r / − r / + 2 a ) / . (10)For prograde motion, a >
0; for retrograde motion, a < U ′′ r ≤
0, and the marginal stable cir-cular orbit corresponds to U ′′ r = 0. Using the expressionfor E and L , we can derive the marginal stable orbit r ms = 3 + Z ∓ [(3 − Z )(3 + Z + 2 Z )] / , (11)where Z ≡ (cid:0) − a (cid:1) / [(1 + a ) / + (1 − a ) / ] ,Z ≡ (3 a + Z ) / . The upper sign and the lower sign are for prograde andretrograde motion of the particle, respectively. Note thatfor a Schwarzschild black hole, a = 0, r ms = 6. The ex-pression above has been widely adopted in various astro-physical studies of black hole systems. III. MARGINALLY STABLE CIRCULAR ORBITOF A PERFECT FLUID DISK
For perfect fluid, the energy-momentum tensor can bewritten as T µν = ( p + ε ) u µ u ν + pδ µν , (12)where u µ , p , and ε are the 4-velocity, pressure, and theenergy density respectively. It can be proved that thereare also two constants of motion [13]: p + εn u t = E , (13)and p + εn u φ = −L , (14)where n is the number density. Note that these two con-stants of motion represent no longer the energy and angu-lar momentum. Use the normalization of the 4-velocity,it is easy to show that r (cid:18) drdτ (cid:19) = U r , (15)where the effective potential U r = (cid:18) np + ε (cid:19) [( r + a r + 2 a r ) E + 4 ar EL− ( r − r ) L ] − ( r − r + a r ) . (16) A. The Schwarzschild Black Hole Case
We first consider the case of a thin fluid disk arounda non-rotating black hole (with spin parameter a = 0).The effective potential becomes U r = (cid:18) np + ε (cid:19) [ r E − ( r − r ) L ] − ( r − r ) . (17)In this case, the energy density ε consists of two parts:One is the rest energy (proportional to the number den-sity n ) and the other one is the thermal energy (assumedto be proportional to the pressure p ). Therefore we have ε = m n + 1 γ − p, (18)where m and γ are the rest mass of the particle and theratio of specific heat, both of which are constants. If thedependence of both number density and pressure on theradius are of power law forms, we can parameterize thefunction n/ ( p + ε ) as np + ε = B Cr − b , (19)where we have assumed p/ρ ∝ T ∝ r − b , where ρ ≡ m n is the density of the fluid. For the standard thin diskmodel [7, 14], b is 3 /
8, 9 /
10, 3 /
4, in the inner, middle andouter regions, respectively. For the self-similar solutionof an advection dominated accretion flow (ADAF, [15,16]), b equals to 1. However, one should note that thesedependence on radius r is only for the region where r ≫ r ms . When r ∼ r ms , b can be negative if the torquefree condition at r ∼ r ms is applied, which may be morerelevant in the case of real accretion flows around blackholes. The constant C is always positive.Since the constant B can be absorbed into E and L ,the effective potential can be rewritten as U r = (cid:18)
11 + Cr − b (cid:19) [ r E − ( r − r ) L ] − ( r − r ) . (20) In a normal fluid, the temperature is usually low andthe thermal energy is much smaller than the rest energy,that is 1 γ − p ≪ m n. (21)In another word, the constant C is very small ( C ≪ θ i ≡ kT i m i c ≈ . , (22) θ e ≡ kT e m e c ≈ . , (23)where k , T i , m i , T e , m e , c are the Boltzmann constant,ion temperature, ion mass, electron temperature, andspeed of light, respectively. In this case, C is not verylarge ( C <
1) but also non-negligible. We can expandthe marginal stable circular orbit as r ms = r ms, + C ∆ r ms . (24)In order to calculate the correction term ∆ r ms , we haveto know the first and second derivatives of the effectivepotential, Eq. (20). They are U ′ r = 2 bCr − ( b +1) (1 + Cr − b ) [ r E − ( r − r ) L ]+ 1(1 + Cr − b ) [4 r E − (2 r − L ] − (4 r − r )(25)and U ′′ r = 2 b ( b + 1) Cr − ( b +2) (1 + Cr − b ) [ r E − ( r − r ) L ]+ 6 b C r − b +1) (1 + Cr − b ) [ r E − ( r − r ) L ]+ 4 bCr − ( b +1) (1 + Cr − b ) [4 r E − (2 r − L ]+ 1(1 + Cr − b ) [12 r E − L ] − (12 r − r ) . (26)Setting Eq. (20) and above two equations to 0 ( U r = U ′ r = U ′′ r = 0), we get an equation after eliminating E and L ,( r − − b ( b + 1) Cr − b Cr − b ( r − − b C r − b (1 + Cr − b ) ( r − bCr − b Cr − b (4 r − r − (cid:20) − bCr − b Cr − b ( r − (cid:21) − r −
3) = 0 . (27)Keeping the terms of the order O ( C ) we can get∆ r ms = br − bms, { ( r ms, − − b ) r ms, + (2 b − − ( r ms, − r ms, − } (28)For a Schwarzschild spacetime, r ms, = 6, so we get r ms = r ms, + C ∆ r ms = 6 + 6 − b Cb (24 − b ) . (29) −0.5 0 0.5 1 1.5 2 2.5 32345678 b r m s FIG. 1: The dependence of the marginally stable circular orbit r ms in Schwarzschild spacetime on the power-law index b inthe case of C <
1. The solid line and dashed line are for C = 0 . C = 0 .
3, respectively. The dotted horizontal linerepresents r ms = 6 in the test particle case. In the equation above, ∆ r ms ( b ) has its maximum 3 . b = 0 .
413 and local minimum − .
18 at b = 2 .
7, respec-tively. So the maximum of the marginally stable circularorbit is r ms,max = 6 + 3 . C, (30)corresponding to b = 0 .
413 and the local minimum is r ms,min = 6 − . C, (31) corresponding to b = 2 .
7. When b → ∞ , r ms →
6, andwhen b is negative, r ms < b (see Fig. 1). Therefore, inthis case when the thermal energy is smaller than the restenergy, the influence of the pressure on the marginallystable circular orbit is non-negligible, especially when b < . B. The Kerr Black Hole Case
In Kerr spacetime, the effective potential is expressedby Eq. (16). When C is small, we can do the sameexpansion as in Schwarzschild spacetime.In Kerr spacetime, the effective potential, and its firstand second order derivatives are U r = (cid:18)
11 + Cr − b (cid:19) [( r + a r + 2 a r ) E + 4 ar EL− ( r − r ) L ] − ( r − r + a r ) (32) U ′ r = 2 bCr − ( b +1) (1 + Cr − b ) [( r + a r +2 a r ) E +4 ar EL− ( r − r ) L ]+1(1 + Cr − b ) [(4 r + 2 a r + 2 a ) E + 4 a EL − (2 r − L ] − (4 r − r + 2 a r ) (33)and U ′′ r = − b ( b + 1) Cr − ( b +2) (1 + Cr − b ) [( r + a r + 2 a r ) E + 4 ar EL− ( r − r ) L ]+ 6 b C r − b +1) (1 + Cr − b ) [( r + a r + 2 a r ) E + 4 ar EL− ( r − r ) L ] + 4 bCr − ( b +1) (1 + Cr − b ) [(4 r + 2 a r + 2 a ) E +4 a EL − (2 r − L ] + 1(1 + Cr − b ) [(12 r + 2 a ) E − L ] − (12 r − r + 2 a ) (34)respectively. After some long but straightforward deduc-tions, we can get∆ r ms = 2 b ( b + 3) r − b ( r − r + a ) − br − b (4 r − r + 2 a ) − r E r E + 12 r E E ′ − r , (35)where r and E are expressed as r = 3 + Z ∓ [(3 − Z )(3 + Z + 2 Z )] / , (36)and E = r / − r / + ar / ( r / − r / + 2 a ) / , (37)respectively, where Z ≡ (cid:0) − a (cid:1) / [(1 + a ) / + (1 − a ) / ] ,Z ≡ (3 a + Z ) / . E ′ is the derivative of E with r . E is the correctionterm of E with order O ( C ).The behavior of the correction term is similar to that ofthe Schwarzschild case. This can be seen in Fig. 2. As anexample, Fig. 3 shows the dependence of the marginallystable circular orbit on b and C in Kerr spacetime with a = 0 . −0.5 0 0.5 1 1.5 2 2.5 3−5−4−3−2−1012345 b ∆ r m s FIG. 2: The dependence of the correction term ∆ r ms on thepower-law index b and spin parameter a . Dotted line: a = 0;solid line: a = 0 .
1; dashed line: a = 0 .
5; dash-dotted line: a = 0 . IV. DISCUSSION
The marginally stable circular orbit can be treated asthe inner boundary of an accretion disk around a blackhole in some cases. This is important because it is cru-cial to study the structure of accretion disks. It is alsoimportant when calculating the radiation of a standard −0.5 0 0.5 1 1.5 2 2.5 322.533.544.555.5 b r m s FIG. 3: The dependence of the marginally stable circular orbit r ms in Kerr spacetime with a = 0 . b in the case of C <
1. The solid line and dashed line are for C = 0 . C = 0 .
3, respectively. The dotted horizontal linerepresents r ms = 4 .
233 in the test particle case for a = 0 . accretion disk, where it is usually assumed that there isno radiation from the region inside the marginally stablecircular orbit.However, the widely accepted marginally stable circu-lar orbit is based on the test particle assumption. Whenthe temperature in an accretion disk is not very high(namely the thermal energy is much smaller than restenergy), the expression of the marginally stable circularorbit for a test particle is accurate enough. But when thetemperature is so high that the thermal energy is non-negligible comparing to the rest energy, pressure may in-troduce some corrections to the location of the marginallystable circular orbit.If the torque free boundary condition is applied at theinner boundary of accretion disks, the temperature nearthe marginally stable circular orbit would decrease withthe decreasing of radius, which is different from the regionfar from the inner boundary. Assuming the dependenceof temperature on radius near the marginally stable cir-cular orbit is a power-law T ∝ r − b , then b <
0. As canbe seen from Fig. 2, the correction term changes signif-icantly with b when b is negative. As an example, forthe Schwarzschild case, when C = 0 . b = − .
2, themarginally stable circular orbit is r ms = 5.The correction to the marginally stable circular or-bit not only influence the inner boundary of accretiondisks, but also affect the energy release of the accretionflow around black holes. In Schwarzschild spacetime, thebinding energy E (see Eq. 9) at a certain radius r is [1] E = r − r / ( r − / , (38)which equals to 0 . r = r ms = 6, so the correspond-ing efficiency is 1 − . . r = 5 the efficiency is 5 . E of a test particle co-rotating ina circular orbit at r is described by Eq. (9) The depen-dence of the efficiency 1 − E on radius r is shown in Fig. 4.Note that it is not monotonic, when the radius r < r ms ,the efficiency is also lower than that at the marginallystable circular orbit. − E FIG. 4: The dependence of the efficiency 1 − E on radius r .The solid line, dashed line anddash-dotted line correspond tothe cases of a =0, 0.5 and 0.9, respectively. However, not only the true location of inner boundaryof accretion disks is different from marginally stable cir-cular orbit as shown by the analytic models [11–13], butas [17] also mentioned, in a disk where the advection isimportant, it is not proper to use the marginally stablecircular orbit as its inner boundary to calculate the ra-diation of the disk. The reason is that in an advectiondominated flow, there is no ”equilibrium” of the circularmotion, and the condition based on which the marginallystable circular orbit is derived is not fulfilled. But thewidely used technique to measure black hole spin by fit-ting thermal soft state spectra [8, 18] is all right becausein a standard thin disk, the marginally stable orbit is al-most the same as for the test particle case, which is usedfor the measurement of the black hole spin parameters inX-ray binaries.One thing should be noted is that our treatment isbased on a perfect fluid disk. However, a real accretiondisk consists of viscous fluid. Another point that mayworth further consideration is the influence of pressureon the ”abruptness” of the transition from inspiral toplunge regions. Because this property will heavily affectthe inner boundary condition of accretion disks, if thetransition is somehow less ”abrupt”, the fluid disk maycontinue further in beyond the marginally stable circular orbit. However, this is out of the scope of this work. Wehope to tackle this problem in a future work.
Acknowledgments
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