apartes de la conferencia de la sjg del 14 y 21 de enero de 2012: gravitational waves from ejection...

October 20, 2011 11:49 WSPC/INSTRUCTION FILE 00100

8th Alexander Friedmann International Seminaron Gravitation and CosmologyInternational Journal of Modern Physics: Conference SeriesVol. 3 (2011) 482–493c© World Scientific Publishing CompanyDOI: 10.1142/S2010194511001000

GRAVITATIONAL WAVES FROM EJECTION OF JETSUPERLUMINAL COMPONENTS AND PRECESSION OF

ACCRETION DISKS DYNAMICALLY DRIVEN BYBARDEEN-PETTERSON EFFECT

HERMAN J. MOSQUERA CUESTA,1,2 LUIS A. SANCHEZ,3 DANIEL ALFONSO PARDO,3,ANDERSON CAPRONI,4 ZULEMA ABRAHAM5 and LUIS HENRY QUIROGA NUNEZ6

1Departmento de Fısica, Universidade Estadual Vale do Acarau, Avenida da Universidade 850,Campus da Betania, CEP 62.040-370, Sobral, Ceara, Brazil

2Instituto de Cosmologia, Relatividade e Astrofısica (ICRA-BR)Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud 150, CEP 22290-180 Urca, Rio

de Janeiro, RJ, Brazil3Escuela de Fısica, Universidad Nacional de Colombia, Sede Medellın, A.A. 3840, Medellın,

Colombia4Nucleo de Astrofısica Teorica (NAT), Pos-graduacao e Pesquisa, Universidade Cruzeiro do Sul

Rua Galvao Bueno 868, Liberdade 01506-000, Sao Paulo, SP, Brasil5Instituto de Astronomia, Geofısica e Ciencias Atmosfericas, Universidade de Sao Paulo

Rua do Matao 1226, Cidade Universitaria, CEP 05508-900, Sao Paulo, SP, Brazil6Departamento de Fısica, Universidad Nacional de Colombia, Sede Bogota, A.A. 76948,

Bogota, D.C., Colombia

Received 1 July 2011Revised 11 July 2011

Jet superluminal components are recurrently ejected from active galactic nuclei, micro-quasars, T-Tauri star, and several other astrophysical systems, including gamma-rayburst sources. The mechanism driving this powerful phenomenon is not properly settleddown yet. In this article we suggest that ejection of ultrarelativistic components may beassociated to the superposition of two actions: precession of the accretion disk inducedby the Kerr black hole (KBH) spin, and fragmentation of the tilted disk; this last being

an astrophysical phenomenon driven by the general relativistic Bardeen-Petterson (B-P)effect. As fragmentation of the accretion disk takes place at the B-P transition radius, theincoming material that get trapped in this sort of Lagrange internal point will forciblyprecess becoming a source of continuous, frequency-modulated gravitational waves. Atresonance blobs can be expelled at ultrarelativistic velocities from the B-P radius. Thelaunching of superluminal components of jets should produce powerful gravitational wave(GW) bursts during its early acceleration phase, which can be catched on the fly bycurrent GW observatories. Here we compute the characteristic amplitude and frequencyof such signals and show that they are potentially detectable by the GW observatoryLISA.

Keywords: Active galaxies; galaxies nuclei; galaxies quasars; black hole physics.

Pacs: 97.10.Gz, 04.30.-w, 04.30.Nn

482

http://dx.doi.org/10.1142/S2010194511001000


Gravitational Waves from Accretion Disks 483

1. Introduction

Radio interferometric observations of active galactic nuclei (AGN – n radio galaxies,blazars, Seyfert I and II galaxies, etc.), micro-quasars and T-Tauri stars exhibitapparently stationary bright cores enshrouded by faint haloes, kiloparsec (kpc) jets,and superluminal parsec (pc) components that appear to travel following curvedtrajectories with spatially changing velocities. High spatial resolution observationsof AGN show that the superluminal ejected material is moving out from their verycentral regions: the core. The resultant structure is seen in the sky as pairs of curvedjets that extend from several pc, kpc, and even megaparsecs (Mpc) distance scalesfrom the AGN core. Jet material seems to be constituted of a plasma composedof electron-positron pairs plus some baryons. The jet morphology is diverse, someof them presenting features that resembles spirals in the sky. These characteristicswere interpreted by several authors as due to helical motion of the components alongthe jet. In an alternative view,1,2 it was proposed that such helicoidal appearance ofthe jet may be due to precession of the disk-jet structure. A few other proposals tocope with the observations are at disposal, but the actual mechanism driving thisphenomenology is nonetheless not yet conclusively established.

The first systematic study on the Bardeen-Petterson (B-P) effect as the potentialdriving engine (mechanism) behind the observed dynamics of AGN was presentedin Ref. 3. (See also Refs. 4-7 in which we extend these results to the analysis ofsome other specific AGN sources, and to confront the B-P scenario for AGN withalternative mechanisms to explain the helicoidal motion of jet components and thedisk warping.) Therein we have shown that for a large sample of AGN the observeddisk and/or jet precession can be consistently explained as driven by the B-P effect.As described below, a torus encircling a rotating (Kerr) supermassive black hole(SMBH), as the state-of-the-art envisions AGN, may lead a rotating SMBH intoa dramatic dynamical stage, known as suspended accretion state (SAS),8 in whichmost of the SMBH spin energy can be released through gravitational radiation dueto the spin-disk magnetic field coupling. As SAS can develop only in a very compactblack hole (BH)-Torus structure,8 we argue in a paper related to this9 that in thecase of AGN the Bardeen-Petterson effect is a necessary and sufficient mechanismto produce SAS.a In that related paper9 we compute the emission of gravitationalwaves (GW) generated by the AGN torus dynamics as driven by the B-P effect.

In a tilted accretion disk orbiting around a rotating BH, the B-P effect createsa gap, or discontinuity, which is known as transition region or B-P radius. Matterarriving from the outer part of the disk can pile-up at this radius because of themagneto-centrifugal barrier creating a sort of Lagrange internal point. Consequently,the accumulated material can condensate in a blob or knot whereby becoming asource of continuous GW due to its orbital motion around the central BH at the

aIt worths to notice that the physical mechanism that could give origin to disk breaking andprecession in AGN and related astrophysical systems was not properly identified in Ref. 10.


484 H. J. Mosquera Cuesta et al.

B-P radius. It also radiates GW upon being launched into space at ultrarelativisticvelocity along the instantaneous jet direction (which explains why jets appear tobe helical, as any component ejection ramdonly points out different directions inspace). In this explanatory scenario for the observed AGN dynamics, we discuss thecharacteristics of those GW signals.

2. Bardeen-Petterson Effect in AGN

The frame dragging produced by a Kerr black hole, known as Lense-Thirring effect,leads a particle to precess (and nutate) if its orbital plane intersects the BH equato-rial plane, to which the BH angular momentum, �JBH, is perpendicular. The ampli-tude of the precession angular velocity decreases as the third power of the distance,i.e., ωL−T ∝ r−3, and becomes negligible at large distances from the KBH. If aviscous accretion disk is inclined with respect to the equatorial plane of the Kerrblack hole, the differential precession will produce warps in the disk. The inter-twined action of the Lense-Thirring effect and the internal viscosity of the accretiondisk drives the disk to break apart at a special distance from the KBH known asBardeen-Petterson radius, whilst the coupling of the gravitational rotating field tothe inner disk angular momentum enforces the spin axis of the inner accretion diskto align with the angular momentum axis of the Kerr black hole, in a time scalewhich depends on several parameters of the system. This is known as the Bardeen-Petterson effect11 and affects mainly the innermost part of the disk, while its outerparts tend to remain in its original configuration due to the short range of the Lense-Thirring effect. The transition region between these two regimes is referred to asB-P radius, and its exact location depends on the Mach number of the accretiondisk. (The attentive reader is addressed to Ref. 3 for further details and a list ofreferences on this astrophysical effect.)

2.1. Kerr spacetime compactness: Modeling Qsr 3C 345 : Sgr A�

The disk dynamics caused by the B-P effect thus relates the total torque �T appliedto the accretion disk to the disk precession frequency ωprec or period Pprec, and itstotal angular momentum �Ld. The fundamental relation turns out to be (see Ref. 3for the derivation of this formula, and further details)

Pprec

MBH= KP

∫ ξout

ξmsΣd(ξ) [Υ(ξ)]−1

ξ3dξ∫ ξout

ξmsΣd(ξ)Ψ(ξ) [Υ(ξ)]−2 ξ3dξ

, (1)

where KP = 1.5×10−38 s g−1, Σd the disk surface density, ξ = r/Rg a dimensionlessvariable, ξout = Rout/Rg, Υ(ξ) = ξ3/2+a∗ and Ψ(ξ) = 1−

√1 − 4a∗ξ−3/2 + 3a2∗ξ−2,

with a∗ the spin parameter of KBH .As such, Eq. (1) is parameterized by the ratio of the precession period to the

black hole mass, which in fact is related to a fundamental invariant of the Kerrmetric (see discussion below). As one can see from Eq. (1), the precession period



depends on how the disk surface density Σd varies with the radius. In what followswe take as an AGN model the source quasar 3C 345. In the search for a disk modeldriven by the B-P effect able to reproduce both the observed period and the inferredmass of the BH in 3C 345, we have modelled the torus by using a power-law functionsuch as Σd(ξ) = Σ0ξ

s, and also an exponential function like Σd(ξ) = Σ0eσξ, where

Σ0, s and σ are arbitrary constants (see Fig. 1 for the results).One can verify that in the case of 3C 345 the numerical solutions matching

the observed Pprec/MBH ratio favors the spin parameter in the range (prograde)0.20 ≥ a∗ ≤ −0.35 (retrograde); for a power-law disk of index s = −2. Theseresults combined with the independent search for the size of the B-P radius tightly

1 2 3

log(Rout/Rg)

Power-law surface density - 3C 345

-1

-0.5

0

0.5

1

blac

k hi

le s

pin

0.9 1 1.1-1

-0.8

-0.6

-0.4

1 2 3

log(Rout/Rg)

Exponential-law surface density - 3C 345

-1

-0.5

0

0.5

1

blac

k ho

le s

pin

1 2 3

log(Rout/Rg)

Power-law surface density - Sgr A*

-1

-0.5

0

0.5

1

blac

k hi

le s

pin

1 2 3

log(Rout/Rg)

Exponential-law surface density - Sgr A*

-1

-0.5

0

0.5

1

blac

k ho

le s

pin

s =

2s

= 0

s =

-1

s = -2

s = -1

s = -2s = 0

s = 2s

= 2

s =

0

s = -1

s = -1

s = -2

s = -2

s = 2, s = 0

s =

2

s =

0

s =

-1

s = -2

s = -2

s = -1

s = 0

s = 2

1.1 Rm

s /Rg

1000 Rm

s /Rg

1.1 Rm

s /Rg

1.1 Rm

s /Rg

1.1 Rm

s /Rg

1000 Rm

s /Rg

1000 Rm

s /Rg

1000 Rm

s /Rg

Fig. 1. Isocontours (in logarithmic scale) of the spin-induced precession period normalised by theBH mass (in units of yr M−1

� ), for 3C 345 disk models with power-law and exponential densityprofiles (left and right panels, respectively). In this case, solutions compatible with observationsfavor a system with a∗ ≥ 0.20 and a∗ ≤ −0.35, correspondingly. Indeed, in this source a full

analysis shows that the spin of the black hole appears to be a∗ = −0.2, that is 3C 345 harboursa counter-rotating (w.r.t. the accretion disk) Kerr black hole. A similar plot is also shown for thesolution in the case of the KBH in Sgr A�. Notice the inner and outer radii limits imposed by theobservations.



constraint the BH spin in 3C 345 to the value a∗ � −0.20, which means that it iscounterotating w.r.t. disk angular momentum. It appears to be clear that the overallanalysis based on B-P effect also provides an insight on the physical properties ofits accretion disk.

Next we discuss the generation of GWs through the B-P mechanism. To thispurpose, we first stress that the compactness of a rotating BH-Torus system [Eq. (1)],an example of a Kerr space-time, the ratio of rotational energy to linear size of thesystem is an invariant feature, say C, under rescaling of the BH mass. Formally, itreads ∫ EGW

0

fGW dE = C �∑

i

fGWi ∆Ei . (2)

Because of this, one can expect that a physics, essentially similar to that developedby van Putten et al.12 for the case of stellar-mass BHs as sources of Gamma-RayBursts and GWs, can take place in this size-enlarged version describing the SMBH-Torus in quasar 3C 345. Indeed, according to Ref. 13 a similar physics seems todominate the source Sgr A∗, which is said to harbour a SMBH-torus system inthe center of our Galaxy. In the case of Sgr A∗, the rise time for flaring shouldvary from half of, to at least a few times, the period of the vertical oscillations.This parameter has been estimated to be on the order of ∆PF

vert = 350 − 1000 s.Meanwhile, the total duration of the flare (F ) should be about a few times theperiod of the radial oscillations, a parameter which has been estimated to be onthe range ∆PF

rad = 40 − 120 min! These timescales have been measured by bothChandra14 and XMM-Newton X-ray Telescopes.15,16

3. GWs from Ultra Relativistic Acceleration of Jet Components inAGN Dynamically Driven by B-P Effect

The mechanism driving the ejection of jet components from AGN is not settled yet.Several possibilities have been suggested. Here we propose that the ejection processcould benefit of the existence of a Lagrange internal point (in the static limit ofthe Kerr metric) created by the B-P effect at the transition radius. This is a place-ment wherein a force-free blob of accreted, piled-up material can be readily ejected.Among possible mechanisms driving the launching of superluminal components onecan quote the instability of the vertical epicyclic oscillation of velocity V31 at radiusr31, i.e. the 3:1 resonance which can provide energy enough to seed an accretion-driven outflow from the AGN core. Nonetheless, several other alternative sources ofenergy to power the component ejection process are also gravitational. To quote oneof those, one can consider BH-torus systems exhibiting the Aschenbach effect.16 Insuch systems, one could benefit of the ∼ 2% reduction in the disk orbital velocitybetween radii rmin and rmax around the KBH, which could amount as much as 4.3MeV/nucleon. However, here we do not discuss further long this issue since it is outof the scope of this paper.



Back to the mainstream of this section, to describe the emission of GWs duringthe early acceleration phase of an ultra relativistic blob of matter ejected from theAGN core, i.e. piled-up mass that was initially trapped at the transition B-P region,one can start by linearizing Einstein’s equations (hereafter we follow Ref. 17)

Gµν = −kTµν , (3)

since we expect that the gravitational field produced by the ejection process atthe B-P radius must be weak. To this purpose we use the expansion of the metrictensor: gµν � ηµν+hµν where ηµν defines the Minkowsky metric, and hµν representsthe space-time perturbation produced by the launching of the mass blob. In whatfollows we assume that the gµν metric has signature (− + ++), we use Cartesiancoordinates rλ = t, ri with ri = x1 = x, x2 = y, x3 = z, and we also use geometricunits in which G = c = 1, so though k = 8π. We stress in passing that thisGW signal is characterized by a gravitational wave memory, which means that theamplitude of the GW strain does not go back to zero as far as there is still availableenergy for the acceleration phase to go on. The GW signal will remain in that stageuntil gravitational energy is available in the radiating source (the launched blob).Considering the harmonic (Lorenz) gauge hαν

′ν = 0, one obtains

�hµν = h βµν′β = 16πTµν , (4)

where hµν = hµν − 12ηµνhα

α. Notice that one can figure out hereafter the knot as a“particle” of mass Mb moving along the world line rλ(τ) (with τ the proper time)and having an energy-stress tensor

Tµν(x) = Mb

∫VµVνδ(4)[x − r(τ)] dτ, (5)

where V α = drα/dτ is the particle 4-vector velocity.Plugging this energy source into Eq. (4) leads to the retarded solution (a gener-

alization of the Lienard-Wiechert solution)

hµν = 4 MbVµ(τ)Vν (τ)

−Vλ · [x − r(τ)]λ

∣∣∣∣∣τ=τ0

. (6)

This result is to be evaluated at the retarded time, which corresponds to the inter-section time of rα(τ) with the observer’s past light-cone. Notice that the term in thedenominator, −Vλ · [x− r(τ)]λ, containing the 4-velocity, is responsible for the nonvanishing amplitude (“memory”) of the GW signal emitted during the launching ofthe blob from the AGN core.

This metric perturbation when transformed to the Lorenz gauge becomes hµν =hµν − 1

2ηµν hαα; or equivalently

hµν =4 Mb

−Vλ · [x − r(τ)]λ

(Vµ(τ)Vν (τ) +

12ηµν

). (7)

Equation (7) must be furtherly transformed into the transverse - traceless (TT )gauge, i.e., hµν −→ hTT

µν ; which is the best suited to discuss the GW detector



response to that signal. We invite the attentive reader to follow the detailed analysisof the detector response as given in Ref. 17. Such analysis shows that the maximumGW strain in the detector is obtained for a wave vector, �n, orthogonal to thedetector’s arm. In this case, the GW amplitude then reads

hmax =γ Mb β2

D

(sin2 θ

1 − β cos θ

), (8)

where β = |�v|/c, with |�v| the particle’s 3-D velocity, θ the angle between �v and �n,D the distance to the source and γ the Lorentz expansion factor.

This result, obtained first in Ref. 17, shows that the GW space-time perturbationis not strongly beamed in the forward direction �n, as opposed to the electromagneticradiation, for instance in gamma-ray bursts (GRBs). Instead, the metric perturba-tion at the ultra-relativistic limit (which is the case of superluminal components ofAGN jets, but not for the run away galactic pulsars discussed in Ref. 18) has a direc-tional dependence which scales as 1 + cos θ. Indeed, because of the strong beamingeffect, the electromagnetic radiation emitted from the source, over the same timeinterval, is visible only over the very small solid angle (θ ∼ γ−1)2, whereas the GWsignal is observable over a wide solid angle; almost π radians. Besides, the observedGW frequency is Doppler blueshifted in the forward direction, and therefore theenergy flux carried by the GWs is beamed in the forward direction, too.

Table 1. Physical properties of the C7 componentejected from quasar 3C 345 for Julian Date 1992.05,as given in Hirotani et al. (2000).19

N(e−) Ang. Size Diameter Masscm−3 mas pc h−1 M� h−3

0.5 - 11 0.20 ± 0.04 0.94 0.66 −→� 1

The two asymptotic forms of the GW amplitude read (always for γ � 1)

hmax(θ � 1) � 4 γ Mb β2

D

(θ2

γ−2 + θ2

), (9)

hmax(θ � γ−1) � 2 γ Mb β2

D(1 + cos θ) . (10)

Thus, if one applies this result to the information gathered from the quasar 3C 345and for its C7 component (see Table 1 and Ref. 9), then the mass of the ejectedcomponent is Mb � 1 M�. Besides, if one assumes a distance D = 150 Mpc,20

(instead of 2.5 Gpc quoted by earlier references cited by Ref. 9), the angle betweenthe line-of-sight vs. jet component axis ΘC7↔

l.o.s = 2◦, the velocity ratio β ∼ 1, and



the Lorentz factor γ = 20,14 then one can see that the GW amplitude amounts

h3C 345D=150 Mpc � 1.3 × 10−20 . (11)

This is a GW amplitude which lies within the current strain sensitivity of LISA.b

Meanwhile, because of the invariant compactness of Kerr metric the azimuthalfrequency scales inversely proportional to the KBH mass. That is,

fGW

∣∣∣∣3C 345

fGW

∣∣∣∣SgrA∗

=MBH

SgrA∗

MBH3C 345

. (12)

In other words, it is consistent with the Rayleigh criterion: in Kerr space-time, theenergy per unit of angular momentum is small at large distances. Nonetheless, thecharacteristic acceleration time scale for the vertical launch of a given jet componentduring a flare is not properly known. A few AGN have such parameter reasonablyestimated. For instance, Sgr A� accretion disk has a period of < ∆P osc

vert >� 500 s forthe vertical oscillations, while the period for the radial oscillations is < ∆P osc

radial >�60 min. Thus, the rise time for flaring must be either half of, or a few (≤ 0.1) timesthe period of vertical oscillations,14,15 while the total duration of a flare must bea few (≤ 0.1) times the period of radial oscillations. In our studies we associatethis time scale with the interval over which the peak acceleration of (and thus themaximum amplitude of the associated GW signal from) any ejected component isreached.

In this respect, since the B-P transition radius can be figured out as a Lagrangeinternal point from which each component is force-free ejected, then one can thinkof that the characteristic time for such acceleration must be a “universal constant”,except for some extra factors on the order of 1 (see below the discussion on equipar-tition of energy between a mass inhomogeneity in the disk and the torus mass).Indeed, regarding observations of most jets in the whole class of objects under anal-ysis here, it appears that the distance from the host source that a given jet canreach does not depend on the mass of the system, but rather on the conditionsin the circumjet medium, as density, for instance. That is, on ideal astrophysicalconditions, all jets might reach the same distance from their host source.c

Thus, based on the Kerr compactness [Eq. (12)], and on the observational factthat the mass of the pc scale accretion torus in a number of AGN is, in average, lesthan a factor of 10 smaller than the mass of the SMBH in their host system, one canclaim that the most likely astrophysical parameter playing some significant role ina pc scale torus make up by 1H1 plasma is the sound speed (which involves severalthermodynamical properties). It scales as the square root of the torus density, ρ, and

b Such GW signal would also be detected by Earth-based interferometers, but see next the dis-cussion on the frequency of this signal.cOne can invoke the Equivalence Principle to assert the existence of a universal time scale, despitethat the KBH in 3C 345 weighs 103 (M�) times more than Sgr A�.



also dictates the angular frequency of the warps in the disk,21 which hereby fixesthe linear momentum to be delivered to a given jet component during its ejection.Such parameter depends on both disk height scale and radial size, which in the B-Peffect scenario depends on both the spin parameter and the disk viscosity. (See theastrophysical foundations of this discussion in Ref. 9.)

Under this premise, one can take the Sgr A� flaring time scale as such “universalconstant”. In this way, one can estimate the GW frequency associated to the ejectionof the C7 jet component of 3C 345

fGW =1

< ∆P oscvert >

� 1500 s

= 2 × 10−3 Hz

(ρdSgrA∗

ρd3C345

)1/2

. (13)

This frequency is clearly within the band width of LISA (see Fig. 2). Thus, byputting the information from both parameters into the strain spectral density dia-gram of the LISA GW observatory one can claim that the GW signals from accel-eration of ultra relativistic jet components in this AGN 3C 345 can be detected.

Hence, by extending this result to each process of ejection of components fromAGN; with the characteristic (typical) mass of any ejected component being aboutMb � 1 M�, one concludes that as far as the rise time (acceleration time scale) islower than 105 s, which corresponds to the lowest frequency limit of LISA, such GWsignals would be detectable for distances upto nearly the Hubble radius, i.e. 3 Gpc.Indeed, to perform a more consistent analysis, or a better characterization, of theGW signal produced during the early acceleration time of the ejected component,one can model the growth with time of the GW amplitude through a power-law ofthe sort h(t) = h0(t− t0)n, where t0 defines the time elapsed till the strain achievesits “final” steady-state, where no more GW radiation is emitted. In this way, one

,

Fig. 2. Parameter space, fixed by X-ray observations and the compactness of Kerr space-time,for the GW signals expected from acceleration of jet components in AGN 3C 345, and Sgr A�,and its potential detectability by current GW observatories.



can evolve the GW waveform (Fourier transform of h(t)) so as to compare it withthe strain-amplitude noise curve of LISA. This analysis will be performed elsewhere(see the forthcoming paper in Ref. 9).

4. GWs from a Mass Orbiting the SMBH at the Transition Radiusin B-P Dominated AGN

In the case of AGN driven by the B-P effect and suspended accretion any intrinsicmass-inhomogeneity Mb in the torus, say a lumpiness or blob of mass in a warptrapped inside the B-P transition region, (which we assume to be eventually ejectedas a jet component), generates a continuous GW signal. The typical strain can beestimated by using the Peters-Mathews weak field, quadrupole formula,22 whichhas been successfully applied to describe the dynamics of the PSR 1913+16 binarypulsar, and several other similar systems. Hence at the B-P radius a trapped massproduces a luminosity in GWs given as23

LGW = 2.5 × 1057 erg s−1

[MBH

109 M�

]5(10 Rg

RB−P

)5 [Mb

1 M�

]2(109 M�MBH

)2

, (14)

where(

Mb

MBH

)is related to the chirp mass.

In this respect, the observations of the component C7, and the results of ourB-P modeling of 3C 345 shown in Fig. 1, together with the observational parametersgiven in Table 1, lead to infer that Mb ∼ 0.66 M�. Hereafter, we approximate thismass inhomogeneity to � 1 M� in the computations below.

Thus, from this estimate and by recalling the relation between the GW ampli-tude h and the GW luminosity LGW one can compute the average (modulated)

Fig. 3. Frequency-modulated waveform of the GW signal from accretion disks dominated by B-Peffect.



amplitude of each of the GW polarization modes h+ and hx, which then reads9,10

< h+max >� 6 × 10−22 and < h+

min >� 2 × 10−22 (15)

and

< hxmax >� 4 × 10−22 and < hx

min >� 2 × 10−22 , (16)

which were obtained for a distance D = 2.45 Gpc to quasar 3C 345. Meanwhile,for the most quoted distance to 3C 345, D = 150 Mpc, the amplitudes given aboveshould be multiplied by a factor of 16.7. Be aware of, that in order to search forthis type of signals LISA should integrate it over several periods, which in the caseof 3C 345 amounts to several years.

By analyzing the amplitudes of these strains one can conclude that such GWsignals (see Fig.3) are broad-band and frequency-modulated.9 This feature is dueto the fact that both h+, hx have the same minimum value of its strain, while thecorresponding maximum strain differs by a factor of 1.5.

Meanwhile, the angular frequency of this GW signal can be computed as theKeplerian frequency at the Bardeen-Petterson radius, which in Ref. 3 we inferredto be RB−P = 14Rg, with Rg = GMBH/c2. In this way, one gets

fGW � M1/2

R3/2B−P

� 1.25 × 10−7 Hz. (17)

Therefore, with these characteristics in the case of 3C 345 such GW signals may notbe detectable by LISA. Nonetheless, the result is quite stimulating since observationsof AGN allow to infer that the masses for SMBHs range between 106 ≤ MBH ≤ 109

M�, which means that for SMBHs with masses MBH ≤ 107 M� the GW signal froma typical mass inhomogeneity Mb � 1 M� orbiting around it, would be detectableby LISA. As an example, the dynamics around the SMBH in Sgr A� would clearlystrike within the amplitude and frequency range of sensitivity (strain-amplitudenoise curve) of the planned LISA space antenna. Certainly, a large sample of otherAGN with similar characteristics as those of Sgr A� are relatively close-by to us(D ≤ 100 Mpc). Thus, their continuous GW signals would be detectable with LISA.

5. Conclusions

From the above analysis one can conclude that any AGN for which the disk and/orjet precession are driven by the Bardeen-Petterson effect becomes a natural envi-rons for a suspended accretion state to develop. This B-P effect, in turn, driveseach AGN to become a powerful source of bursts of gravitational waves from theultrarelativistic acceleration of jet components, and of continuous broad-line grav-itational radiation from both the orbiting of blobs of mass at the B-P radius; andthe precession of the distorted, turbulent accretion disk. All such GW signals couldbe detected by GW observatories like LISA. One also notices that the prospectivecoeval detection of gravitational and electromagnetic radiations from these AGN



may decisively help in picturing a consistent scenario for understanding these cos-mic sources.

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