Warped Circumbinary Disks in Active Galactic Nuclei
Kimitake Hayasaki, Bong Won Sohn, Atsuo T. Okazaki, Taehyun Jung, Guangyao Zhao, Tsuguya Naito
aa r X i v : . [ a s t r o - ph . GA ] J un Warped Circumbinary Disks in Active Galactic Nuclei
Kimitake
Hayasaki , Bong Won Sohn , , Atsuo T. Okazaki , Taehyun Jung ,Guangyao Zhao , and Tsuguya Naito [email protected] ABSTRACT
We study a warping instability of a geometrically thin, non-self-gravitatingdisk surrounding binary supermassive black holes on a circular orbit. Such a cir-cumbinary disk is subject to not only tidal torques due to the binary gravitationalpotential but also radiative torques due to radiation emitted from an accretiondisk around each black hole. We find that a circumbinary disk initially alignedwith the binary orbital plane is unstable to radiation-driven warping beyond themarginally stable warping radius, which is sensitive to both the ratio of verticalto horizontal shear viscosities and the mass-to-energy conversion efficiency. Asexpected, the tidal torques give no contribution to the growth of warping modesbut tend to align the circumbinary disk with the orbital plane. Since the tidaltorques can suppress the warping modes in the inner part of circumbinary disk,the circumbinary disk starts to be warped at radii larger than the marginallystable warping radius. If the warping radius is of the order of 0 . − pc to 10 − pc for 10 M ⊙ black hole. We also discuss the possibility that the central objects of observedwarped maser disks in active galactic nuclei are binary supermassive black holeswith a triple disk: two accretion disks around the individual black holes and onecircumbinary disk surrounding them. Subject headings: accretion, accretion disks - black hole physics - galaxies: active- galaxies: evolution - galaxies: nuclei - gravitational waves - quasars: general-binaries:general Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Korea Department of Astronomy and Space Science, University of Science and Technology, 217 Gajeong-ro,Daejeon, Korea Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan Faculty of Management Information, Yamanashi Gakuin University, Kofu, Yamanashi 400-8575, Japan
1. Introduction
There is strong evidence that most galaxies harbor supermassive black holes (SMBHs)with mass 10 M ⊙ . M . M ⊙ at their centers (Kormendy&Richstone 1995). Hitherto,SMBHs have been found in 87 galaxies by observing the proper motion of stars bound bythe SMBHs or by detecting radiation emitted from gas pulled gravitationally by the SMBHs(Kormendy&Ho 2013). H O maser emission from active galactic nuclei (AGNs) in spiralgalaxies provides a strong tool to measure SMBH masses, because it shows a rotating diskon a subparsec scale with a nearly Keplerian velocity distribution around the SMBH. Thosedisks, so-called maser disks, have been observed at the centers of NGC 4258 (Miyoshi et al.1995), NGC 1068 (Greenhill&Gwinn 1997), NGC 3079 (Yamauchi et al. 2004), the Circinusgalaxy (Greenhill et al. 2003), UGC 3789 (Reid et al. 2009), NGC 6323 (Braatz et al. 2007),NGC 2273, NGC 6264, and some more objects (Kuo et al. 2011).Several maser disks show warped structure at the radii of the order of 0 . . . .
2. External Torques acting on the circumbinary disk
Let us consider the torques from the binary potential acting on the circumbinary disksurrounding the binary on a circular orbit. Figure 1 illustrates a schematic picture of ourmodel; binary black holes orbiting each other are surrounded by a misaligned circumbinarydisk. The binary is put on the x - y plane with its center of mass being at the origin in theCartesian coordinate. The masses of the primary and secondary black holes are representedby M and M , respectively, and M = M + M . We put a circumbinary disk around theorigin. The unit vector of specific angular momentum of the circumbinary disk is expressed 4 –by (e.g. Pringle 1996) l = cos γ sin β i + sin γ sin β j + cos β k , (1)where β is the tilt angle between the circumbinary disk plane and the binary orbital plane,and γ is the azimuth of tilt. Here, i , j , and k are unit vectors in the x , y , and z , respectively.The position vector of the circumbinary disk can be expressed by r = r (cos φ sin γ + sin φ cos γ cos β ) i + r (sin φ sin γ cos β − cos φ cos γ ) j − r sin φ sin β k (2)where the azimuthal angle φ is measured from the descending node. The position vector ofeach black hole is given by r i = r i cos f i i + r i sin f i j ( i = 1 , , (3)where r i = ξ i a with ξ ≡ q/ (1 + q ) and ξ ≡ / (1 + q ). Here, q = M /M is the binarymass ratio and a is the semi-major axis of the binary. These and other model parametersare listed in Table 1. The gravitational force on the unit mass at position r on the circumbinary disk can bewritten by F grav = − X i =1 GM i | r − r i | ( r − r i ) (4)The corresponding torque is given by t grav = r × F grav = X i =1 GM i | r − r i | ( r × r i ) (5)We consider the tidal warping/precession with timescales much longer than local rotationperiod of the circumbinary disk. This allows us to use the torque averaged in the azimuthaldirection and over the orbital period: h T grav i = 14 π Z π Z π Σ | J | t grav dφd (Ω orb t ) = 38 ξ ξ Ω (cid:16) ar (cid:17) (cid:20) sin γ sin 2 β i − cos γ sin 2 β j (cid:21) , (6)where J ≡ r ΩΣ l , and Ω orb = p GM/a and Ω = p GM/r are the angular frequencies ofbinary motion and mean motion of circumbinary disk at r , respectively. Here, we used forthe integration the following approximations: | r − r i | − ≈ r − h r · r i r + O (( r i /r ) ) i M Primary black hole mass M Secondary black hole mass M Schwarzschild radius r S = 2 GM/c Binary mass ratio q = M /M Mass ratio parameters ξ = q/ (1 + q ), ξ = 1 / (1 + q )Binary semi-major axis a Orbital frequency Ω orb = p GM/a Orbital period P orb = 2 π/ Ω orb Tilt angle β Azimuth of tilt γ Azimuthal angle φ Shakura-Sunyaev viscosity parameter α Horizontal shear viscosity ν Vertical shear viscosity ν Ratio of vertical to horizontal shear viscosities η = ν /ν Mass-to-energy conversion efficiency ǫ Luminosities emitted from two accretion disks L , L Total luminosity L = L + L Binary irradiation parameter ζ = ( ξ L + ξ L ) /L For a small tilt angle β ≪
1, equation (6) is reduced to h T grav i ≈ ξ ξ Ω (cid:16) ar (cid:17) (cid:20) l y i − l x j (cid:21) , (7)where l x and l y can be written from equation (1) as l x = β cos γ and l y = β sin γ .The tidal torques tend to align the tilted circumbinary disk with the orbital plane (c.f.Bate et al. 2000). For e = 0, such a tidal alignment timescale is given by τ tid = sin β |h T grav i| ≈ π (cid:18) / ξ ξ (cid:19) (cid:16) ra (cid:17) / P orb , (8)where P orb = 2 π/ Ω orb is the binary orbital period. Since the inner edge of the circumbinarydisk is estimated to be ∼ . a (Artymowicz & Lubow 1994), the tidal alignment timescaleis longer than the binary orbital period. 6 – (cid:48) (cid:20) (cid:48) (cid:21) (cid:38)(cid:76)(cid:85)(cid:70)(cid:88)(cid:80)(cid:69)(cid:76)(cid:81)(cid:68)(cid:85)(cid:92)(cid:3)(cid:71)(cid:76)(cid:86)(cid:78) (cid:49)(cid:82)(cid:85)(cid:80)(cid:68)(cid:79)(cid:3)(cid:87)(cid:82)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:71)(cid:76)(cid:86)(cid:78)(cid:3)(cid:83)(cid:79)(cid:68)(cid:81)(cid:72)(cid:38)(cid:72)(cid:81)(cid:87)(cid:72)(cid:85)(cid:3)(cid:82)(cid:73)(cid:3)(cid:80)(cid:68)(cid:86)(cid:86) (cid:54)(cid:72)(cid:70)(cid:82)(cid:81)(cid:71)(cid:68)(cid:85)(cid:92)(cid:3)(cid:69)(cid:79)(cid:68)(cid:70)(cid:78)(cid:3)(cid:75)(cid:82)(cid:79)(cid:72)(cid:51)(cid:85)(cid:76)(cid:80)(cid:68)(cid:85)(cid:92)(cid:3)(cid:69)(cid:79)(cid:68)(cid:70)(cid:78)(cid:3)(cid:75)(cid:82)(cid:79)(cid:72) (cid:36)(cid:93)(cid:76)(cid:80)(cid:88)(cid:87)(cid:75)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:79)(cid:87)(cid:29)(cid:55)(cid:76)(cid:79)(cid:87)(cid:3)(cid:68)(cid:81)(cid:74)(cid:79)(cid:72)(cid:29)(cid:3)(cid:39)(cid:72)(cid:86)(cid:70)(cid:72)(cid:81)(cid:71)(cid:76)(cid:81)(cid:74)(cid:3)(cid:81)(cid:82)(cid:71)(cid:72) (cid:36)(cid:93)(cid:76)(cid:80)(cid:88)(cid:87)(cid:75)(cid:68)(cid:79)(cid:3)(cid:68)(cid:81)(cid:74)(cid:79)(cid:72)(cid:29) (cid:51)(cid:82)(cid:86)(cid:76)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:89)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:29) φ γ β y r z x Fig. 1.— Configuration of a triple disk system composed of two accretion disks around theindividual black holes and a circumbinary disk surrounding them. There are two angles ( β, γ )which specify the orientation of the circumbinary disk plane with respect to the binary orbitalplane ( x - y plane). The azimuthal angle ( φ ) of an arbitrary position on the circumbinary diskis measured from the descending node. If there is an accretion disk around each black hole, the circumbinary disk can be illu-minated by light emitted from each accretion disk. The re-radiation from the circumbinary-disk surface, which absorbs photons emitted from these accretion disks, causes a reactionforce. This is the origin of the radiative torques. Below we take two accretion disks aspoint irradiation sources, because their sizes are much smaller than that of the circumbinarydisk. Note that negligible contribution arises from other radiation sources such as an accre-tion stream from the circumbinary disk towards each accretion disk (Hayasaki et al. 2007;MacFadyen & Milosavljevi´c 2008; Roedig et al. 2011; D’Orazio et al. 2013) and an inner rimof the circumbinary disk, because the mass-to-energy conversion efficiencies in these regionsare negligible in comparison with those in the inner parts of the accretion disks. Further-more, sin β would be larger than the dimensionless scale-height of each accretion disk, which 7 –is typically of the order of 0.01. If not, the radiation from the inner parts of the accretiondisks is shadowed and less flux will reach the circumbinary disk, except for the case that thecircumbinary disk is flaring.Since the surface element on the circumbinary disk is given in the polar coordinates by dS = ∂ r ∂r × ∂ r ∂φ drdφ = (cid:20) l − r (cid:18) − ∂β∂r sin φ + ∂γ∂r cos φ sin β (cid:19)(cid:21) rdrdφ, (9)the radiative flux at d S is given by dL = dL + dL = 14 π X i =1 L i | r − r i | | ( r − r i ) · d S || r − r i | , (10)where L is the sum of the luminosity of the radiation emitted from the primary black hole, L ,and that from the secondary black hole, L . Here, we assume that the surface element is notshadowed by other interior parts of the circumbinary disk. If we ignore limb darkening, theforce acting on the disk surface by the radiation reaction has the magnitude of (2 / dL/c )and is antiparallel to the local disk normal (Pringle 1996). The total radiative force on d S can then be written by d F rad = 16 πc X i =1 L i | ( r − r i ) · d S || r − r i | d S | d S | . (11)Consequently, the total radiative torque acting on a ring of radial width dr is given by d T rad = I r × d F rad = 16 πc I X i =1 (cid:20) L i | ( r − r i ) · d S || r − r i | (cid:21) r × d S | d S |≈ L πc r I | r · d S | r × d S | d S | + 12 πc r I X i =1 L i | ( r − r i ) · d S | ( r · r i ) r × d S | d S | , (12)where H | ( r − r i ) · d S | ( r × d S ) / | d S | = H | r · d S | ( r × d S ) / | d S | holds for β ≪ r∂β/∂r ≪ d T , of the right-hand side of equation (12) corresponds toequation (2.15) of Pringle (1996): d T = L c (cid:18) r ∂l y ∂r i − r ∂l x ∂r j (cid:19) dr (13)and the second term, which we call d T orb , is originated from the orbital motion of the binary.Here, we consider the radiation-driven warping/precession with timescales much longerthan the orbital period, as in the case of tidally driven warping/precession. The orbit-average 8 –of the torque d T rad is then given by h d T rad i = 12 π Z π d T rad d (Ω orb t ) ≈ d T + 12 π Z π d T orb d (Ω orb t )= L c (cid:26)(cid:18) − ζ (cid:16) ar (cid:17) l y + r (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l y ∂r (cid:19) i + (cid:18) ζ (cid:16) ar (cid:17) l x − r (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l x ∂r (cid:19) j (cid:27) dr, (14)where ζ ≡ ξ L + ξ L /L is a binary irradiation parameter. Note that ζ . ζ → L /L . q → ζ = 1 / q = 1. For r ≫ a or ζ = 0, h d T rad i is reducedto d T .From equation (14), the specific radiative torque averaged over azimuthal angle andorbital phase is given by h T rad i = 1 | J | πr h d T rad i dr = Γ r (cid:26)(cid:18) − ζ (cid:16) ar (cid:17) l y + r (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l y ∂r (cid:19) i + (cid:18) ζ (cid:16) ar (cid:17) l x − r (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l x ∂r (cid:19) j (cid:27) , (15)where Γ = L/ (12 π Σ r Ω c ). Assuming that L = ǫ ˙ M c with the mass-to-energy conversionefficiency ǫ and the mass accretion rate of the circumbinary disk ˙ M = 3 πν Σ, then thegrowth timescale of a warping mode induced by the radiative torque, r/ Γ, can be estimatedto be τ rad = 4 ǫα (cid:18) Hr (cid:19) − rc ∼ × (cid:18) . α (cid:19) (cid:18) . ǫ (cid:19) (cid:18) H/r . (cid:19) − (cid:18) rr S (cid:19) − / (cid:16) ra (cid:17) / P orb , (16)where ν = αc s H is the shear viscosity of the disk with the Shakura-Sunyaev viscosityparameter α , c s is the sound speed, and H is the scale-height of the circumbinary disk. Here, ǫ ≈ . ǫ = 0 .
42 for an extreme Kerr blackhole case (e.g., see Kato, Fukue & Mineshige 2008). Since it is clear that τ rad ≫ P orb for ageometrically thin disk, our assumption for the orbit-averaged radiative torque is ensured.
3. Tilt angle evolution of circumbinary disks
In this section, we investigate the response of the circumbinary disk, which is initiallyaligned with the orbital plane, for external forces. The mass conservation equation is givenby ∂ Σ ∂t + 1 r ∂∂r ( r Σ v r ) = 0 , (17) 9 –where v r is the radial velocity. The angular momentum equation is given by(Papaloizou&Pringle1983) ∂ J ∂t + 1 r ∂∂r ( rv r J ) = 1 r ∂ G vis ∂r + | J | T ex , (18)where G vis represents the viscous torques of the circumbinary disk.The external torque T ex is written as the sum of tidal torques and radiative torques, T ex = h T grav i + h T rad i = (cid:26) (cid:16) ar (cid:17) (cid:20) ξ ξ Ω − ζτ rad (cid:21) l y + Γ (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l y ∂r (cid:27) i − (cid:26) (cid:16) ar (cid:17) (cid:20) ξ ξ − ζτ rad (cid:21) l x + Γ (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) ∂l x ∂r (cid:27) j . (19)The evolution equation for disk tilt is obtained from equation (17) and (18) as ∂ l ∂t + (cid:20) v r − ν Ω ′ Ω − ν ( r ΩΣ) ′ r ΩΣ (cid:21) ∂ l ∂r = ∂∂r (cid:18) ν ∂ l ∂r (cid:19) + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) ∂ l ∂r (cid:12)(cid:12)(cid:12)(cid:12) l + T ex (20)(Pringle 1996), where ν and ν are respectively the horizontal shear viscosity and the verticalshear viscosity, the latter of which is associated with reducing disk tilt. The primes indicatedifferentiation with respect to r . For simplicity, we adopted the same assumptions for thecircumbinary disk structure as in Pringle (1996) that v r = ν Ω ′ / Ω, r ΩΣ is constant, and ν is constant. Then, equation (20) can be reduced to ∂ l ∂t = 12 ν ∂ l ∂r + T ex , (21)where l · ∂ l /∂r = 0 is used.We look for solutions of equation (21) of the form l x , l y ∝ exp i ( ωt + kr ) with kr ≪ ∂/∂t with iω , ∂/∂r with ik , and ∂ /∂r with − k , we have the following set oflinearized equations: (cid:20) iω + ν k − ( A + ik B )( A + ik B ) iω + ν k (cid:21) (cid:18) l x l y (cid:19) = 0 , (22)where A = 32 (cid:16) ar (cid:17) (cid:20) ξ ξ Ω − ζτ rad (cid:21) , B = Γ (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) .
10 –The determinant of the coefficient matrix on the left hand side of equation (22) must vanishbecause of l = 0. The local dispersion relation is then obtained as ω = i ν k ± ( A + ik B ) = i (cid:26) ν k ± k Γ (cid:20) − ζ (cid:16) ar (cid:17) (cid:21)(cid:27) ± (cid:16) ar (cid:17) (cid:20) ξ ξ Ω − ζτ rad (cid:21) . (23)The imaginary part of ω corresponds to the excitation or damping of oscillation, whereasthe real part provides the local precession frequency due to the external torques. ´ ´ ´ ´ r (cid:144) a G r o w t h ti m e s ca l e @ y r D Ζ= Ζ= (cid:144) Ζ= (cid:144) Ζ= Fig. 2.— Growth timescale of the radiation-driven warping of a circumbinary disk with α = 0 . ǫ = 0 . H/r = 0 . M = 10 M ⊙ , and a = 10 r S . The black solid line, red dashedline, and red dotted line show the growth timescales with the binary irradiation parameters ζ = 0, 1 /
4, 1 /
2, and 1, respectively. The growth timescale with ζ = 0 corresponds to thatof the single black hole case.In order for the perturbation to grow, Im( ω ) must be negative. The growth conditionis given by 0 < k < ν (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) . (24)In terms of Γ bin ≡ Γ[1 − (3 / ζ ( a/r ) ], the growth timescale of the warping mode inducedby the radiative torques in the binary system is given by τ rad , bin = r Γ bin ≈ τ rad (cid:20) ζ (cid:16) ar (cid:17) (cid:21) . (25) 11 –Figure 2 shows the dependence of τ rad , bin on binary irradiation parameter ζ and r/a for amodel with α = 0 . ǫ = 0 . H/r = 0 . M = 10 M ⊙ , and a = 10 r S . The growth timescale τ rad , bin for ζ = 0 or r/a ≥ λ ≤ r , where λ = 2 π/k is the radialwavelength of the perturbation. The condition that the circumbinary disk is unstable to thewarping mode can be then rewritten as rr S ≥ π (cid:16) ηǫ (cid:17) (cid:20) − ζ (cid:16) ar (cid:17) (cid:21) − , (26)where η = ν /ν is the ratio of vertical to horizontal viscosities. Ogilvie (1999) derived therelationship between η and α : η = 2(1 + 7 α ) / ( α (4 + α )) by taking a non-linear effect ofthe fluid on the warped disk. The value of α consistent with X-ray binary observations isknown to be 0 . − . α in a gas-pressuredominated region of the disk (e.g. see Blaes 2013 and references therein). The range of η should therefore be η &
10 for α . .
3. The equality of equation (26) is approximately heldat the marginally stable warping radius: r warp , bin r S ≈ r warp r S " ζ (cid:18) ar S (cid:19) (cid:30) (cid:18) r warp r S (cid:19) (27)in the case of r/a > ζ = O (0 . r warp r S = 8 π (cid:16) ηǫ (cid:17) , (28)which corresponds to the marginally stable warping radius for a single black hole (Pringle1996). The marginally stable warping radius substantially depends on η and the mass-to-energy conversion efficiency ǫ .Figure 3 shows the dependence of the marginally stable warping radius on the semi-major axis. While ǫ = 0 . ǫ = 0 .
42 is adopted in panel (b).Panels (a) and (b) thus correspond to the cases of a Schwarzschild black hole and a Kerrblack hole with maximum black hole spin parameter, respectively. In both panels, the blacksolid line and black dashed line show r warp , bin normalized by the Schwarzschild radius r S for M = 10 M ⊙ with η = 10 ( α = 0 .
27) and η = 50 ( α = 0 . τ rad , equals thetimescale for the disk to align with the orbital plane by the tidal torque, τ tid . This tidalalignment radius is given by r rad / tid r S = (cid:18) (cid:19) / (cid:18) ξ ξ ǫα (cid:19) / (cid:18) Hr (cid:19) − / (cid:18) ar S (cid:19) / (29) 12 –The growth of a finite-amplitude warping mode induced by the radiative torque can be sig-nificantly suppressed by the tidal torque in the region inside the tidal alignment radius. Thered solid and dashed lines show the tidal alignment radii with η = 10 and η = 50, respec-tively. The orange line shows the radius where the growth timescale of the radiation-drivenwarping mode equals the timescale in which the binary orbit decays by the gravitationalwave emission. The orbital decay timescale for a circular binary case is given by (Peters1964) τ gw = 58 1 ξ ξ (cid:18) ar S (cid:19) r S c (30)Equating equation (30) with equation (16), we obtain r rad / gw r S = 532 (cid:18) ǫαξ ξ (cid:19) (cid:18) Hr (cid:19) (cid:18) ar S (cid:19) . (31)Inside this orbital decay radius, the circumbinary disk can be warped before two SMBHscoalesce. The orange solid and dashed lines show the orbital decay radii with η = 10 and η = 50, respectively. The blue solid line and blue dashed line show the inner and outerradii of the circumbinary disk, respectively. The inner radius is assumed to be equal to thetidal truncation radius, where the tidal torque is balanced with the viscous torque of thecircumbinary disk (Artymowicz & Lubow 1994). In the case of a circular binary with a smallmass ratio, the tidal truncation radius is estimated to be ∼ . a .A gaseous disk around a SMBH in an AGN is surrounded by a dusty torus. Thegrains of the dusty torus are evaporated above the temperature 1500 K by the radiationemitted from the central source. The inner radius of the dusty torus should therefore bedetermined by the dust sublimation radius: r dust = 3 pc ( L/ erg s − ) / ( T / − . ,where T is the dust sublimation temperature (Barvainis 1987). Assuming that the AGNluminosity is the Eddington luminosity, the dust sublimation radius is rewritten as r dust =4 . × − ( M/ M ⊙ ) / pc with the adoption of T = 1500 K. Since the circumbinary diskshould be also inside the dusty torus in our scenario, the outer radius of the circumbinarydisk is given by r out r S ≈ . × (cid:18) M M ⊙ (cid:19) − / . (32)The shaded area between the two blue lines shows the whole region of the circumbinary disk.It is noted from the figure that the circumbinary disk is not warped by radiation-drivenwarping in the cases of η = 10 and 50 with ǫ = 0 . η = 50 with ǫ = 0 .
42, since themarginally stable warping radii are outside of the circumbinary disk. On the other hand, 13 –
10 100 1000 10 a (cid:144) r S r (cid:144) r S (a) ¶ = 0.1
10 100 1000 10 a (cid:144) r S r (cid:144) r S r out r in Τ rad =Τ GW Τ rad =Τ GW Τ rad =Τ tid Τ rad =Τ tid Η= Η= (b) ¶ = 0.42 Fig. 3.— Characteristic radii of the warped circumbinary disk around binary SMBHson a circular orbit with ζ = 1, q = 0 .
1, and M = 10 M ⊙ . While ǫ = 0 . ǫ = 0 .
42 is adopted in panel (b). In both panels, the black solid line andblack dashed line show the marginally stable warping radii with η = 10 ( α = 0 .
10) and η = 50 ( α = 0 . η = 10 and η = 50, respectively. The orange lines show the orbital decay radius where the growthtimescale of the radiation-driven warping is equal to the orbital decay timescale due to thegravitational wave emission. The orange solid and dashed lines show the orbital decay radiiwith η = 10 and η = 50, respectively. While the blue solid line represents the inner radiusof the circumbinary disk r in /a ≈ .
7, the blue dashed line represents the outer radius of thecircumbinary disk r out /r S ≈ . × ( M/ M ⊙ ) − / . The shaded area between the bluesolid and dashed lines represents the whole region of the circumbinary disk.the circumbinary disk is warped in the case of η = 10 with ǫ = 0 .
42. In this case, themarginally stable warping radius corresponds to that of a single black hole at a/r S . . Ifthe circumbinary disk around binary SMBHs is warped by radiative torques, the semi-majoraxis of the binary is predicted to be in a range of a min . a . a max , where a min is equal tothe semi-major axis at the intersection point between r warp and r gw / rad in panel (b), whichis given as a min r S = " (cid:18) ξ ξ ǫα (cid:19) (cid:18) Hr (cid:19) − (cid:18) r warp r S (cid:19) / (33)by equating equation (28) with equation (31), and a max is equal to the semi-major axis at 14 –the intersection point between r rad / tid and r out in panel (b), which is given as a max r S = (cid:18) (cid:19) / (cid:18) ǫαξ ξ (cid:19) / (cid:18) Hr (cid:19) (cid:18) r out r S (cid:19) / (34)by equating equation (29) with equation (32). r (cid:144) a P r ece ss i on ti m e s ca l e @ y r D Τ p ,tot Τ p ,tid Τ p ,rad Fig. 4.— Precession timescale of the warped circumbinary disk around binary SMBHs ona circular orbit with ζ = 1, q = 0 . ǫ = 0 . α = 0 . H/r = 0 .
01, and M = 10 M ⊙ . Theblack solid line, blue dotted line, and red dashed line show the precession timescales from theradiative torque, from the tidal torque, and from the sum of those two torques, respectively.The local precession frequency Ω p , tot of the linear warping mode is obtained from equa-tion (23) by Ω p , tot = Re( ω ) = − (cid:16) ar (cid:17) (cid:20) ξ ξ Ω − ζτ rad (cid:21) = − (Ω p , tid − Ω p , rad ) , (35)where Ω p , tid = 34 (cid:16) ar (cid:17) ξ ξ Ω , Ω p , rad = 32 (cid:16) ar (cid:17) ζτ rad .
15 –The radius where Ω p , tid is balanced with Ω p , rad is given by r Ω rad / tid r S ∼ . × (cid:18) ξ ξ / (cid:19) (cid:18) . ǫ (cid:19) (cid:18) . α (cid:19) (cid:18) ζ (cid:19) (cid:18) H/r . (cid:19) . (36)Since r Ω rad / tid ≫ r out , the tidal precession frequency is higher than the radiative precessionfrequency. Thus, the circumbinary disk slowly precesses in the retrograde direction.Figure 4 shows the dependence of the precession timescales on the circumbinary diskradius normalized by the semi-major axis. The black solid line, the red dashed line, andthe blue dotted line show the precession timescale for the radiative torques τ p , rad = 1 / Ω p , rad ,tidal torques τ p , tid = 1 / Ω p , tid , and total torque τ p , tot = 1 / Ω p , tot , respectively. The precessiontimescale is much longer than the orbital period.
4. Application to observed maser disks in AGNs
In this section, we discuss the application of our proposed model to a warped maser disksystem. There is observational evidence for disk warping in the maser disks at the center ofNGC 4258 (Herrnstein et al. 2005), Circinus (Greenhill et al. 2003), NGC 2273, UGC 3789,NGC 6264, and NGC 6323 (Kuo et al. 2011). We assume that these maser disks start to bewarped at the innermost maser spot radii, which we call the observed warping radii, in thefollowing discussion.From equation (28), the marginally stable warping radius for an extreme Kerr blackhole with η = 10 is estimated to be r warp r S ∼ . × (cid:16) η (cid:17) (cid:18) . ǫ (cid:19) . (37)Equation (37) is a good approximation to the marginally stable warping radius for a binarySMBH as long as a/r S . , as seen in the solid black line at panel (b) of Figure 3.In order for the maser disks to be warped, the marginally stable warping radius must beless than not only the outer radius of the circumbinary disk but also the observed warpingradius. Otherwise, radiation-driven warping is unlikely as a mechanism to explain the warpedstructure of these maser disks. We adopt this condition in order to examine whether ourmodel is appropriate for the observed warped maser disks.Table 2 summarizes the results of applying our model to observed warped maser disks.The first and second columns denote the name and observed black hole mass of each targetsystem, respectively. The third and fourth columns represent the observed warping radius, r obswarp and outer radius of the circumbinary disk, respectively. The outer radius is obtained by 16 –equation (32). The fifth to sixth columns denotes the inferred semi-major axis for each targetsystem, if the observed warped maser disk is a circumbinary disk around binary SMBHs andthe observed warping radius is larger than the marginally stable warping radius given byequation (37). Since their observed warping radii also are smaller than the outer radius ofthe circumbinary disk, they intersect with two lines of r rad / tid and r rad / gw , respectively. It isclear from panel (b) of Figure 3 that the semi-major axis at the intersection point between r obswarp and r rad / tid provides the maximum value of the inferred semi-major axis, whereas thesemi-major axis at the intersection point between r obswarp and r rad / gw gives the minimum valueof the inferred semi-major axis. Each semi-major axis is then obtained by equating eachobserved warping radius with equations (29) and (31) as a obsmin r S = " (cid:18) ξ ξ ǫα (cid:19) (cid:18) Hr (cid:19) − r obswarp r S ! / , (38) a obsmax r S = (cid:18) (cid:19) / (cid:18) ǫαξ ξ (cid:19) / (cid:18) Hr (cid:19) r obswarp r S ! / , (39)where we adopt that H/r = 0 .
01 and q = 0 . ξ ξ = 10 / η = 40 and ǫ = 0 .
42, while beingsmaller than the outer radius of the circumbinary disk. On the other hand, all systems,except for NGC 2273, satisfy the same condition but for η = 10 and ǫ = 0 .
42. The radiation-driven warping can thus be a promising mechanism for explaining the warped structureof the observed maser disks in these systems. There is also a possibility that the centralmassive objects are binary SMBHs with the semi-major axis on several tens of milliparsec tosub-milliparsec scales. However, it is difficult to distinguish, solely by the current analysis,whether the central object is a single SMBH or binary SMBHs. To do so, independenttheoretical and observational approaches are needed.
Table 2: Application to observed warped maser disks. The first column denotes the name of each target system. Thesecond and third columns show the black hole mass and innermost-maser-spot radius of each target system, respectively(see Greenhill et al. (2003); Herrnstein et al. (2005); Kuo et al. (2011); Kormendy&Ho (2013)). The fourth columnrepresents the outer radius of the circumbinary disk, which is given by equation (32). The fifth and sixth columnsdenote the semi-major axes estimated by equations (38) and (39), respectively. The seventh and final columns indicatethe corresponding orbital periods. Note that the marginally stable warping radii are estimated by equation (37) to be4 . × r S for ( η, ǫ ) = (10 , .
42) and 7 . × r S for ( η, ǫ ) = (40 , . M [M ⊙ ] r obswarp [ r S ] r out [ r S ] a obsmin [ r S ] a obsmax [ r S ] P obsmin [yr] P obsmax [yr]NGC 4258 3 . × . × . ×
216 7 . × . × − . . × . × . ×
464 3 . × . × − . × . × . × − − − − UGC 3789 1 . × . × . ×
250 1 . × . × − . . × . × . ×
251 1 . × . × − . × . × . ×
286 3 . × . × −
151 18 –
5. Summary and Discussion
We have investigated the instability of a warping mode in a geometrically thin, non-self-gravitating circumbinary disk induced by radiative torques originated from two accretiondisks around interior black holes. Here, the two accretion disks are regarded as point irra-diation sources for simplicity. We have derived the condition where the circumbinary diskis unstable to the warping mode induced by the radiative torques and the timescales of pre-cession caused by both tidal and radiative torques for a small tilt angle ( β ≪ r/a &
8, the growth timescale of the warping mode in the binary SMBH case isreduced to that of the single SMBH case.2. The marginally stable warping radius substantially depends on both the ratio of thevertical to horizontal shear viscosities η and the mass-to-energy conversion efficiency ǫ . The marginally stable warping radius in the binary SMBH case is reduced to thatof the single SMBH case for r ≫ a .3. For a small tilt angle ( β ≪ M ⊙ are likely to have a binary separation on 10 − pc to 10 − pc scales.6. The circumbinary disk can precess due to both tidal torques and radiative torques.While the radiative torques tend to precess the circumbinary disk in the progradedirection, the tidal torques tend to precess it in the retrograde direction. Since theformer precession frequency is much lower than the latter precession frequency, the 19 –circumbinary disk slowly precesses in the retrograde direction. The precession timescaleis much longer than the orbital period. Therefore, it is unlikely that the periodic lightvariation due to the warped precession could be detected.In this paper, we have studied warping of circumbinary disks where disk self-gravity isnegligible. A few warped maser disks are, however, thought to be massive to be comparableto the black hole mass (e.g., Wardle&Yusef-Zadeh 2012). The self-gravitating force in sucha massive disk makes the velocity profile deviate significantly from the Keplerian one. Inaddition, the dominant origin of both the horizontal and vertical shear viscosities, on whichthe condition of the radiation-driven warping is sensitive, is the self-gravitating instability ofthe disk. However, little is known about how the self-gravitating force affects disk warpingin a geometrically thin, self-gravitating circumbinary disk consistent with the maser diskobservations. Further observational and theoretical studies are necessary.We have assumed that the binary is on a circular orbit. There are, however, theoreticalindication that the orbital eccentricity increases by the interaction between binary SMBHsand their circumbinary disks (Armitage & Natarajan 2005; Hayasaki 2009). In the ideallyefficient binary-disk interaction case, the orbital eccentricity is driven up to ∼ .
6. Thisis because the binary orbital angular momentum is mainly transferred to the circumbinarydisk when the binary is at the apastron. The saturation value of the orbital eccentricity isestimated by equating the angular frequency at the inner radius of the circumbinary diskwith the binary orbital frequency at the apastron (Hayasaki et al. 2010; Roedig et al. 2011).In addition, more enhanced periodic light variations are expected in eccentric binary SMBHsby interaction with the circumbinary disk (Hayasaki et al. 2007, 2008) than in the circularbinary case (MacFadyen & Milosavljevi´c 2008; D’Orazio et al. 2013). Such periodic lightcurves provide an independent tool to evaluate whether the central object of the warpedmaser disk is binary SMBHs or a single SMBH. We will examine the effect of the orbitaleccentricity on the radiation-driven warping of the circumbinary disk in a subsequent paper.For simplicity, we have also assumed that the circumbinary disk is initially alignedwith the binary orbital plane ( β ≪ ∼ a . Since the marginallystable warping radius is substantially larger than the inner radius of the circumbinary disk,the shape of the cavity gives little influence on the warping condition.Probing gravitational waves (GWs) from individual binary SMBHs with masses & M ⊙ with Pulser Timing Arrays (PTAs) (Lommen&Backer 2001; Sesana et al. 2009) alsogives a powerful tool to determine if the central object surrounded by the warped maserdisk is binary SMBHs or a single SMBH. For a typical PTA error box ( ≈
40 deg ) in thesky, the number of interloping AGNs are of the order of 10 for more than 10 M ⊙ blackholes if the redshift range is between 0 and 0 . . − for inspiral GWs and . − for memoryGWs associated with the final mergers (Seto 2009). Since they are three to four orders ofmagnitude less than the current PTA sensitivity of & − , it is unlikely for GWs to bedetected from the currently identified warped maser disk systems. If the total mass of binarySMBHs is more massive, however, the characteristic amplitudes of the GW signals could belarge enough to be detected with future planned PTAs such as the Square Kilometer Arraywith & − sensitivity. It will therefore be desired to identify warped maser disks aroundthe central massive objects with masses & M ⊙ in nearby AGNs.We have also discussed the application of the warped circumbinary disk model to theobserved warped maser disks in Table 2. In the case of the marginally stable warping radiuswith η = 40 ( α ≈ .
1) and ǫ = 0 .
42, only the Circinus meets the condition that the marginallystable warping radius is less than both the observed warping radius and the dust sublimationradius of AGN which is assumed to be equal to the outer radius of the circumbinary disk. Inthis case, the resultant inferred semi-major axis is between 6 . × − pc and 2 . × − pc.On the other hand, it is unlikely that the warped structure of the maser disks at the center ofother five systems originates from radiative torque, even if their central objects are a single 21 –SMBH. The condition in question substantially depends on the observed warping radius andvalues of η and ǫ in the marginally stable warping radius. Further theoretical argumentsabout an appropriate treatment of ǫ and η , and observations to measure the warping radiimore precisely in the existing maser disks, are desirable. Acknowledgments
The authors thank the anonymous referee for fruitful comments and suggestions. Theauthors also thank Nicholas Stone for his carefully reading the manuscript and helpful com-ments. KH is grateful to Jongsoo Kim for helpful discussions and his continuous encourage-ment. BWS and THJ are grateful for support from KASI-Yonsei DRC program of KoreaResearch Council of Fundamental Science and Technology (DRC-12-2-KASI). This work wasalso supported in part by the Grants-in-Aid for Scientific Research (C) of Japan Society forthe Promotion of Science [23540271 TN and KH, 24540235 ATO and KH].
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