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Eur. Phys. J. C (2014) 74:2900DOI 10.1140/epjc/s10052-014-2900-9
Regular Article - Theoretical Physics
Generalized Langevin equation with colored noise descriptionof the stochastic oscillations of accretion disks
Tiberiu Harko1,a, Chun Sing Leung2,b, Gabriela Mocanu3,c
1 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK2 Department of Applied Mathematics, Polytechnic University, Hong Kong, People’s Republic of China3 Faculty of Physics, Babes-Bolyai University, Cluj-Napoca, Romania
Received: 6 October 2013 / Accepted: 12 May 2014 / Published online: 27 May 2014© The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract We consider a description of the stochastic oscil-lations of the general relativistic accretion disks aroundcompact astrophysical objects interacting with their exter-nal medium based on a generalized Langevin equation withcolored noise and on the fluctuation–dissipation theorems.The former accounts for the general memory and retardedeffects of the frictional force. The presence of the memoryeffects influences the response of the disk to external ran-dom interactions, and it modifies the dynamical behavior ofthe disk, as well as the energy dissipation processes. Thegeneralized Langevin equation of the motion of the disk inthe vertical direction is studied numerically, and the verticaldisplacements, velocities, and luminosities of the stochas-tically perturbed disks are explicitly obtained for both theSchwarzschild and the Kerr cases. The power spectral distri-bution of the disk luminosity is also obtained. As a possibleastrophysical application of the formalism we investigate thepossibility that the intra-day variability of the active galacticnuclei may be due to the stochastic disk instabilities. Theperturbations due to colored/nontrivially correlated noiseinduce a complicated disk dynamics, which could explainsome astrophysical observational features related to diskvariability.
1 Introduction
In an astrophysical environment, compact general relativisticobjects like black holes or neutron stars are often surroundedby an accretion disk. The physics of accretion disks aroundcompact objects is thought to explain a number of obser-
a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
vations related to X-ray binaries or active galactic nuclei(AGN). The disks are usually considered as being composedfrom massive test particles that move in the gravitationalfield of the central compact object. Numerical simulationshave shown that the accretion-induced collapse of a whitedwarf may lead to a neutron star surrounded by a disk withmass up to 0.8M� [1]. Waves and normal-mode oscillationsin geometrically thin and thick disks around compact objectshave been studied extensively both within Newtonian grav-ity, and within a relativistic framework (see [2] for a reviewof accretion disk properties). It was also suggested that theintra-day variability (IDV) of AGNs can also be related tothe vertical oscillations of the accretion disks around blackholes [3,4].
Stochastic processes and methods play a fundamental rolein our understanding of many physical/astrophysical phe-nomena [5–7]. The study of the general relativistic oscil-lations of thin accretion disks around compact astrophysi-cal objects interacting with the surrounding medium throughnon-gravitational forces was considered in [8]. The inter-action of the accretion disk with the external medium wasmodeled via a friction force and a random force, respectively.The general equations describing the stochastically perturbeddisks were derived by considering the perturbations of tra-jectories of the test particles in equatorial orbits, assumedto move along the geodesic lines. By taking into account thepresence of a viscous dissipation and of a stochastic force thedynamics of the stochastically perturbed disks can be formu-lated in terms of a general relativistic Langevin equation. Thestochastic energy transport equation was also obtained. Thevertical oscillations of the disks in the Schwarzschild andKerr geometries were considered in detail, and the verticaldisplacements, velocities, and luminosities of the stochasti-cally perturbed disks were explicitly obtained for both theSchwarzschild and the Kerr cases by numerically integratingthe corresponding Langevin equations.
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The Langevin equation, used to describe the stochasticoscillations of the accretion disks in [8], provides a correctphenomenological and statistical description of the Brow-nian motion only in the long time limit, for times long ascompared to the characteristic relaxation time of the veloc-ity autocorrelation function [9,10]. In order to describe thedynamics of a homogeneous system without restriction on atime scale, one should generalize the Langevin equation byintroducing, instead of the simple friction term, a systematicforce term with an integral kernel [9,10]. The convolutionterm expresses the memory, or retardation effect, which canmake the accretion disk response to an external perturbingstochastic force quite complicated. As shown in [8], a simpleBrownian motion model for the disk oscillations producesa power spectrum distribution (PSD) P( f ) ∼ f −α , withα = −2. A completely uncorrelated interaction of the diskwith the external medium would produce white noise, with aPSD P( f ) = f 0 = const. However, the observational datafor a large percentage of AGN show that their optical IDV hasPSDs for which α is neither 0 nor −2, as shown by discreteFourier transform analysis [11], structure function analysis[12], and fractal dimension analysis [13]. Therefore, if thistype of variability is produced and has its origin in the accre-tion disks rotating around compact astrophysical objects, aconsistent mathematical description of the stochastic phe-nomena in accretion disks would require one to go beyondthe simple description of the complex accretion disk–cosmicenvironment interaction in terms of a simple friction and astochastic force, respectively. The Langevin type equationwith a damping term and stochastic force was used in [14] todescribe the stochastic oscillations on the vertical directionof the accretion disks around a black hole, and to calculate theluminosity and PSD for the oscillating disk. The stochasticresonance phenomenon in PSD curves for different param-eter values of viscosity coefficient, accretion rate, mass ofblack hole, and outer radius of the disk was also studied.The results show that the simulated PSD curves of luminos-ity for disk oscillation have the same profile as the observedPSD of black hole X-ray binaries, and the stochastic res-onance of accretion disk oscillation may be an alternativeinterpretation of the persistent low-frequency quasi-periodicoscillations.
A description of the stochastic oscillations of the gen-eral relativistic accretion disks around compact astrophys-ical objects based on the generalized Langevin equation,which accounts for the general retarded effects of the fric-tional force, and on the fluctuation–dissipation theorems, wasintroduced in [15]. The vertical displacements, velocities,and luminosities of the stochastically perturbed disks wereexplicitly obtained for both the Schwarzschild and the Kerrcases. The PSD of the simulated light curves was determinedand it was found that the spectral slope α has values thatcorrespond with observations. The theoretical predictions of
the model were compared with the observational data for theluminosity time variation of the BL Lac S5 0716+714 object.The influences of the friction parameter, spin parameter, andmass M of the central compact object on the stochastic reso-nance in PSD curves was discussed, by using the generalizedLangevin equation, in [16]. The results show that a largespin parameter can enhance the stochastic resonance phe-nomenon, but the larger the friction coefficient or the centralmass is, the weaker the stochastic resonance phenomenonbecomes. The simulated PSD curves of the output luminos-ity of stochastically oscillating disks have the same profile asthe observed PSDs of X-ray binaries. Hence the resonancepeak in the PSD curve can explain the quasi-periodic oscil-lations.
It is the purpose of the present paper to consider the mem-ory/retardation effects in the description of the stochasticoscillations of the particles in the accretion disks aroundcompact objects under the influence of the external envi-ronment, generating some random interactions between thedisk and the cosmic medium. More specifically, we considertwo astrophysical accretion disk–external medium interac-tion models that can be described by means of the Langevintype stochastic differential equations. If the black hole–accretion disk system is located in a dense cosmic environ-ment, like, for example, a stellar cluster, then the gravitationalforce exerted by the neighboring stars can be described as aneffective stochastic, time-dependent force [5,6]. The stochas-tic gravitational force appears due to the local fluctuations ofthe numbers of stars in the clusters, as shown in [5,6], wherethese processes have been studied in much detail. Thereforefor a massive central object–accretion disk system located ina dense stellar cluster, the accretion disk is subjected to twogravitational forces, one produced by the black hole, and asecond one representing an effective random force due to theneighboring stars. In this case the dynamical evolution of theaccretion disk can be described by means of the Langevinequation.
As a second astrophysical process that can be modeled bya Langevin type stochastic differential equation we considerthe dynamical interaction between the accretion disk and theremnant matter from a star tidally disrupted by the centralblack hole. In this case, by assuming that the mass of thedisk is much smaller than the mass of the inflowing gas, onecan model the system as a single particle (the disk) locatedinto the hydrodynamical flow of the remnant gas. Under thesesimplifying assumptions the dynamics of the system can bedescribed again by a stochastic Langevin type equation.
Moreover, in the present analysis we assume that thestochastic perturbations of the disk (the particle), assumedto randomly interact with an external medium (fluctuatinggravitational force and gas remnants of a tidally disruptedstar), are described by the generalized Langevin equation,introduced in [17].
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Mathematically, the stochastic equation of motion of thedisk becomes a differential-integral equation, whose inte-gral kernel describes the memory effects. We use this equa-tion to provide a detailed description of the vertical stochas-tic oscillations of an accretion disk around a Schwarzschildblack hole and also of an accretion disk around a Kerr blackhole. The solution of the generalized Langevin equation isobtained numerically, and the basic physical parameters ofthe disk (vertical displacement, velocity, and luminosity) areobtained in a numerical form. As a possible application ofthe present formalism we investigate the possibility that theIDV of the AGNs has its origin in some random/stochasticeffects at the accretion disk level. In the present study we willfocus only on supermassive black holes, both static and rotat-ing, which can be described by the Schwarzschild and Kerrgeometries, and which are the best candidates to explain theelectromagnetic radiation emission properties of the AGNs,assumed to be powered by accretion of mass onto massiveblack holes, with masses of the order of 106 M� to 1010 M�.
The present paper is organized as follows. The generalizedLangevin equation, describing the stochastic oscillations ofthe accretion disks with memory is presented in Sect. 2. Twospecific examples of astrophysical interest of disk–externalmedium interaction that can be described by a Langevin typeequation are also considered. The dimensionless form of thedisk equation of motion, as well as the numerical proce-dure for solving the generalized Langevin equation, is pre-sented in Sect. 3. The stochastic oscillations of the accretiondisks in the presence of colored noise are studied for bothSchwarzschild and Kerr geometries in Sect. 4. The applica-tion of the present formalism for the study of the IDV’s isconsidered in Sect. 5. We conclude and discuss our results inSect. 6.
2 The generalized Langevin equation for stochasticoscillations of accretion disks
In the present section we consider the equation of motion ofthe stochastically oscillating accretion disks interacting withtheir external cosmic environment. We consider two specificcases of astrophysical interest, namely, the stochastic pertur-bations of the accretion disks, located in dense stellar clus-ters, by the gravitational fields of their neighboring stars, andthe interaction between the accretion disk with the gaseoushydrodynamic flow generated by the disruption of a star bythe tidal forces of the central supermassive black hole. Byassuming that the particles in the disk move on geodesiclines, we show first that the effects of a stochastic perturba-tion can be described, in the Newtonian approximation, by aLangevin type equation. The Langevin equation with mem-ory is introduced, and the connection between the generalizedLangevin equation and the fluctuating hydrodynamics [9,18]is also established.
2.1 Equation of motion of massive test particlesin an accretion disk
We model the accretion disk as a single thermodynamic sys-tem, consisting of a large number of particles, with overallenergy density ε and pressure p, respectively. The energy-momentum tensor Tμν of the disk is given by
Tμν = (ε + p)uμuν − pgμν, (1)
where the four-velocity uμ of the particles with mass m satis-fies the normalization condition uμuμ = 1, and the relationuμ∇νuμ = 0, respectively, where∇ν represents the covariantderivative with respect to the metric. We also introduce theprojection operator hμλ = gμλ − uμuλ, which satisfies therelation hμλuμ = 0. The equations of motion of the particlesin the disk follow from the equation of conservation of theenergy-momentum tensor, ∇μTμν = 0, and, after contractionwith hμλuμ, this leads to the equations
(ε + p)gμλuν∇νuμ − ∇νhν
λ = 0. (2)
Finally, contraction with gαλ gives the equation of motion ofthe fluid element of the disk as [19]
d2xμ
ds2 + �μαβ
dxα
ds
dxβ
ds= p,νε + p
hαν, (3)
where �μαβ are the Christoffel symbols associated to the met-ric, and a comma denotes the partial derivative with respectto the coordinate xν .
We consider the particles in the disk to be in contactwith the external environment (stochastic gravitational fieldof neighboring compact astrophysical objects, remnant gasfrom tidally disrupted stars flowing towards the central blackhole etc.), whose action on the disk can be described generallyby means of a stochastic force ξμ. If the disk is perturbed,a point particle in the nearby position x ′μ must satisfy thegeneralized equation of motion
d2x ′μ
ds2 + �′μαβ
dx ′α
ds
dx ′β
ds= p′
,ν
ε + ph
′αν + 1
mξμ, (4)
where ξμ is the external, random force, acting on the particle.The coordinates x ′μ are given by x ′μ = xμ + δxμ, whereδxμ is a small quantity. Thus, in the first approximation, weobtain for the Christoffel symbols
�′μαβ
(xσ + δxσ
) = �μαβ
(xσ
) + �μαβ,λ
(xσ
)δxλ, (5)
where �μαβ,λ = ∂�μαβ/∂xλ. Moreover,
u′ (x ′μ) = uμ(xμ
) + dδxμ
ds, (6)
g′μν (x ′α) = gμν(xα
) + ∇νδxμ + ∇μδxν, (7)
and
p′,ν
(x ′α) = p,ν
(xα
) + p,μ,λ(xα
)δxλ, (8)
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respectively. In the following we introduce the four-velocityof the perturbed motion as Vμ = dδxμ/ds. Vμ satisfies thecondition uμVμ = 0.
Therefore the equation of motion for δxμ becomes
d2δxμ
ds2 + 2�μαβuαV β +(
�μαβ,λuαuβ − p,ν,λh
αμ
ε + p
)
δxλ
= − p,νε + p
(∇νδxμ + ∇μδxν + uμV ν + uνVμ) + ξμ.
(9)
In obtaining Eq. (9) we have assumed that the matter con-tent of the disk can be modeled as a perfect fluid, and wehave neglected the possibility of energy dissipation, eitherby viscous processes, or by radiation emission, consideredto have a negligible effect on the disk structure and dynam-ics. Moreover, we have adopted a fully general relativisticapproach, which takes into account the modifications of thespacetime geometry in the disk due to the presence of thecentral supermassive black hole. For realistic astrophysicalsystems, the study of Eq. (9) can be done by using numericalmethods only. In the following sections we will obtain someapproximate, but still realistic, forms of Eq. (9), which allowfor a simpler numerical treatment of the problem.
2.2 The generalized Langevin equation
Since the self-gravity of the disk is negligibly small as com-pared to the gravitational field of the central supermassiveblack hole, the infinitesimal transformations x ′μ = xμ+δxμ
do not change the metric. Therefore the coordinate transfor-mations satisfy the Killing equations ∇νδxμ + ∇μδxν = 0.
Hence it follows that in this approximation the equationof motion of the disk particles perturbed by an external forceis given by
d2δxμ
ds2 + 2�μαβuαV β +(
�μαβ,λuαuβ − p,ν,λh
αμ
ε + p
)
δxλ
= − p,νε + p
(uμV ν + uνVμ
) + ξμ. (10)
In the case of a disk consisting of particles rotating withazimuthal angular velocity � = dφ/dt , the components ofthe four-velocity are given by uμ = ut (1, 0,�/c, 0).
If the disk consists of dust particles, with negligible pres-sure, moving along the geodesic lines, the term containingthe pressure gradient can be neglected, and we obtain theequation of motion considered in [20]. In the adopted approx-imation of non-zero pressure, the deviation equation containssome new terms, which are proportional to the ratio of thesecond derivative of the pressure and the sum of the energydensity and pressure, respectively, and the product of thepressure gradient and the deviation velocity V μ, respectively.
In the non-relativistic case the friction force is given byf ifr =−νmvi , where ν is the friction coefficient, and vi are
the components of the non-relativistic velocity [10]. The rel-ativistic generalization of the friction force requires the intro-duction of the friction tensor νμα , given in [21–23] as
νμα = νm(δμα + VαVμ
). (11)
Therefore the friction force can be expressed as [21–23]
f μfr = −νμα(V α − uα
)
= −νm(δμα + VαVμ
) (V α − uα
). (12)
In the following we will consider only the vertical oscilla-tions of the disk, which for an arbitrary axisymmetric metricare described by the equation
d2δz
ds2 +[ω2⊥
(ut)2 − p,z,z
ε + pgzz
]δz
= −ν (δzα + VαV z) V α + ξ z, (13)
where
ω2⊥ = �ztt,z + 2�z
tφ,z�
c+ �z
φφ,z
(�
c
)2
, (14)
is the frequency of the gravitational oscillations of the disk[8].
In the following we assume that the gravitational effects,which enter in Eq. (9) via the derivatives of the Christoffelsymbols �μαβ,λ, are much stronger than the hydrodynamic
pressure effects, and therefore ω2⊥(ut )2 � p,z,zgzz/(ε+ p).By considering the non-relativistic limit of Eq. (10), the
perturbed stochastic geodesic equation describing the verti-cal oscillations of the disk is equivalent with the Langevintype equation [8]
d2δz
dt2 + cνdδz
dt+ c2ω2⊥δz = c2ξ z(t). (15)
Equation (15) practically represents a linear oscillator,with a viscosity term, and a random perturbing force. Inobtaining Eq. (15) we have assumed that the external per-turbative force is a function of time only, and we haveneglected its possible spatial coordinate dependence. A fun-damental assumption of non-relativistic Galilean physics isthe existence of a universal time t . Therefore, within the non-relativistic Langevin theory, it is quite natural to identify thisuniversal time t with the time parameter of the stochasticdriving process [23]. In general, the external stochastic noiseξ z can be represented as a function of a t-parameterizedtwo dimensional stochastic process, ξ z = ξ z [xμ(t), pμ(t)],where pμ is the four-momentum of the external source. Byassuming that the motion is non-relativistic, with v/c << 1,and | �p| << mc, where | �p| is the absolute value of themomentum, we have E ≈ mc2 ≈ constant, and con-sequently the dependence on the four-momentum of the
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stochastic force can be neglected. For a spatially inhomo-geneous external perturbing cosmic environment, the func-tion ξ z would also depend on z. However, if the geometricdimension of the external stochastic source is much largerthan the size of the disk, the inhomogeneities in the generatedstochastic force can be neglected. Hence, in the following theexternal noise ξ z is modeled by the one-dimensional standardhomogeneous Wiener process, with any possible momentumor space-like coordinate dependence neglected.
The Langevin equation (15) provides a correct phe-nomenological and statistical description of the Brownianmotion only in the long time limit, for times long as comparedto the characteristic relaxation time of the velocity autocor-relation function [9,10]. In order to describe the dynamics ofa homogeneous system without restriction on a time scale,one should generalize the Langevin equation by introduc-ing, instead of the simple friction term, a systematic forceterm with an integral kernel [9,10]. The convolution termexpresses the memory, or the retardation effects, and, in thespecific case of the stochastic oscillations of the accretiondisks, the effects of the non-zero disk pressure and energydensity.
In the case of the vertical oscillations of a thin accretiondisk, the generalized Langevin equation, introduced in [17],which takes into account the retarded effects of the frictionalforce, is given by
d2δz
dt2 +t∫
0
γ (t − τ)dδz(τ )
dτdτ + K ′(δz) = c2ξ z(t), (16)
where
γ (|t − τ |) = α1c
τdexp
{−|t − τ |
τd
}, (17)
and
K (δz) = c2ω2⊥2
(δz)2 , (18)
respectively. In Eq. (17) τd represents a characteristic diskmemory time, while α1 is interpreted as a disk dampingstrength. For |t − τ | τd ,γ (|t − τ |) ≈ α1 × c/τd , whilefor |t − τ | � τd , γ (|t − τ |) ≈ 0. From a physical pointof view the coefficient α1 describes the strength of energydissipation in the disk. The autocorrelation function of thestochastic force satisfies the condition
〈ξ z(t)ξ z(τ )〉 = 1
βγ (t − τ), (19)
where β is a constant.It is worthwhile to summarize now the physical assump-
tions, implicitly assumed by Eqs. (16), (17), and (19):
(a) The external cosmic environment interacting with theaccretion disk around a supermassive black hole is spa-
tially homogeneous and stationary; i.e., relaxation pro-cesses within the external environment occur on timescales much shorter than the relevant dynamical timescales associated with the motion of the accretion disk.The interaction with a spatially inhomogeneous non-stationary cosmic environment can be modeled by con-sidering friction and noise amplitude functions depend-ing on the spatial coordinates, and disk momenta.
(b) On a macroscopic level, the interaction between the diskparticles and the external cosmic environment is suffi-ciently well described by the generalized friction coeffi-cient with memoryγ (|t−τ |), and the stochastic Langevinforce ξ z(t).
(c) Stochastic collisions between the disk particles and theconstituents of the external environment occur virtuallyuncorrelated.
In the next section we will use Eq. (16) to study the verticaloscillations of the accretion disks around supermassive blackholes in both Schwarzschild and Kerr geometries.
2.3 Equation of motion of a thin disk in the presenceof a stochastic gravitational field
As a first example of the stochastic interaction between accre-tion disks around supermassive black holes and the cosmicenvironment we consider the situation in which the disk andits central object are a member of a larger stellar system,in which all components interact gravitationally. We assumethat at the center of the disk we have a compact object ofmass M . The force acting on the disk can be divided intotwo components, the gravitational force of the central object,which we denote by �FG , and the force �FR generated by theinfluence of the immediate local neighborhood, which mayconsist of ordinary stars and black holes. The force �FG isa smoothly varying function of position, while �FR can besubject to relatively rapid fluctuations [5,6]. The fluctuationin the external force is a direct consequence of the changingof the local stellar distribution that makes the influence ofthe near neighbors on the disk time variable. It can be shown[5,6] that the fluctuations in the force acting on the centralobject disk system, due to the changing local stellar distri-bution, do occur with extreme rapidity as compared to therate at which any of the other physical parameters change.Let p be the pressure of the matter in the disk. Thereforewe can write for the force acting on the disk the expression�F = �FG + �Fp + �Fvisc + �FR(t), where �Fp is the force due to
the pressure variation in the disk, �Fvisc is the force generatedby the viscous dissipative effects in the disk, and the fluctu-ating random force �FR , due to the gravitational interactionof the disk with the nearby stellar systems, can be consideredas a function of time only [5,6].
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In order to obtain the equation of motion of the disk inthe presence of the gravitational force of the central object,and of the external stochastic forces, we consider the disk tobe thin in the sense discussed in [24,25]. We approximatethe distribution of the surface density � by the expressions[26]
�(r) ={�0 = constant, Rin ≤ r ≤ Radj,
�0
(r
Radj
)−χ, Radj ≤ r ≤ Rout,
(20)
where Rin is the innermost radius of the disk, Radj is anadjustment radius in the disk, and Rout is the outer radius ofthe disk. The index χ of the surface density can be either 3/5or 3/4 [26].
The mass of the disk Md from Rin to Rout is given by
Md = 2π
Rout∫
Rin
�(r)rdr ≈ 2π�0 R2adj
2 − χ
(Rout
Radj
)2−χ,
χ �= 2, z Rin, (21)
where we have used the assumptions Rin Rout andRadj ≈ Rout, respectively. We assume that the disk as a wholeis perturbed, and that it deviates from the equatorial planeby a small distance δz. The restoring gravitational force FG
caused by the attraction of the central object is [26]
FG(δz) = GMδz
Rout∫
Rin
2πr�(r)(r2 + z2
)3/2 dr. (22)
After integration, using the density distribution given byEq. (20), we obtain
FG(δz) = 2πGM�0
Rin
(1 − χ
χ + 1
Rin
Radj
)δz. (23)
We define the mean vertical density of the disk as ρd =Md/h0, where h0 is the half-thickness of the disk. There-fore the vertical oscillations of the disk as a whole can bedescribed by the equation of motion
d2δz
dt2 + 1
MdFG(δz)+ 1
ρd
∂p
∂z+ Fvisc = 1
MdFR(t). (24)
As for the acceleration due to the vertical pressure distri-bution we can approximate it as apress = − (1/ρd) ∂p/∂z ≈−κz(r), where the vertical epicyclic frequency κz(r) dependson the horizontal radius r only. The simplest way to obtainsuch a relation is by considering a power series expansion ofthe pressure so that
p(r, z) = p(r, 0)+ ∂p(r, z)
∂z
∣∣∣∣z=0
z
+1
2
∂2 p(r, z)
∂z2
∣∣∣∣z=0
z2 + · · · (25)
The constant term in the acceleration can be eliminated by arescaling of the acceleration, and the epicyclic frequency canbe obtained as κz(r) = ∂2 p(r, z)/∂z2
∣∣z=0 /ρd . Therefore for
the equation of motion of the disk we obtain
d2δz
dt2 +(ω2
0 + κ2z
)δz + Fvisc = FR(t)
Md, (26)
where
ω20 = 4π2ν2
0
= (2 − χ)G M
Rin R2adj
(Rout
Radj
)χ−2 (1 − χ
χ + 1
Rin
Radj
). (27)
If some dissipative forces, like, for example, dissipation dueto radiation emission, or viscosity friction effects in the diskmatter, are also acting on the disk, one should include thedissipative forces in the equation of motion. The simplestphenomenological approach would be to assume that theviscous/heat dissipation is proportional to the velocity ofthe vertical oscillations, Fvisc ∝ δz/dt . However, a betterphenomenological description of the viscous friction can beobtained by assuming an integral kernel description of thedissipative processes. Therefore the equation of motion ofthe disk becomes
d2δz
dt2 +t∫
0
γ (t − τ)dδz(τ )
dτdτ
+(ω2
0 + κ2z
)δz = FR(t)
Md. (28)
In this model the disk as a whole undergoes stochastic oscilla-tions determined by the interaction between the disk, and thefluctuating stellar/black hole neighborhood. The equation ofmotion is given by the generalized Langevin equation (28).
In order to obtain the oscillations frequency of the disk weassume that the innermost radius of the disk Rin is locatedat Rin = 6GM/c2, corresponding to the innermost stableorbits in a general relativistic Schwarzschild geometry. Werepresent the adjustment radius of the disk Radj as Radj =f1 Rin and the outer radius of the disk as Rout = g1 Rin, wheref1 > 1 and g1 � 1 are constants. Then for the oscillationfrequency of the disk we obtain
ω0 = c3
6 f1G M
(g1
f1
)χ/2−1√
2 − χ
6
(1 − χ
χ + 1
1
f1
),
(29)
or
ω20 = 1.89 × 10−4 ×
(M
106 M�
)−2
× 2 − χ
f 21
(g1
f1
)χ−2
×(
1 − χ
χ + 1
1
f1
)s−2. (30)
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The period of the free disk oscillations is given by
T0 = 5.171 × 10−2 f1
(M
106 M�
)(g1
f1
)1−χ/2
×[
2 − χ
6
(1 − χ
χ + 1
1
f1
)]−1/2
s. (31)
In the present paper we will consider only the case ω20 � κ2
z ,and therefore our study will be limited to the considerationof the gravitational effects in the disk.
2.4 Stochastic equation of motion of accretion disksin a fluctuating environment due to stellar capture
Close encounters between stars from a nuclear cluster anda massive black hole lead to the disruption of stellar bodiesdue to the gravitational tidal forces of the black holes, andthe subsequent accretion of the remnant gas [27,28]. Theseastrophysical processes are among the likely mechanisms forfeeding black holes that are embedded in a dense stellar clus-ter. Stars on highly elongated orbits are susceptible to tidaldisruption and hence they provide a natural source of materialto replenish the inner disk. Such events can produce debrisand lead to recurring episodes of enhanced accretion activ-ity, with the in-falling matter interacting both gravitationallyand mechanically with the accretion disk. Intermediate-massblack holes have a high chance of capturing stars through tidalenergy dissipation within a few core relaxation times [27].The interaction between the accretion disk and the remnantgas falling towards the central black hole can be consideredas a random process, in which the particles of the disk interactwith the particles of the tidally disrupted star.
In order to model the interaction between the disk and theremnant matter from the disrupted star we assume that thedisk is thin, and its mass Md is much smaller than the massMF falling towards the black hole, Md MF , conditionvalid at least for finite time interval 0 < t < tF , where tF isthe in-falling time of MF . Hence, if this condition is satisfied,one can obtain a simplified description of the disk–remnantstellar matter interaction by considering the disk as a singlephysical system randomly moving in a fluid represented bythe matter accreted by the black hole. This simple physicaldescription allows the description of the dynamical evolutionof the disk in terms of a Langevin type equation, obtainedfrom the fluctuating hydrodynamic equations of motion ofthe interacting particle-fluid system.
Therefore in the following we consider the accretion diskas a macroscopic system of mass Md in an incompressiblefluctuating fluid (the debris from stellar disruption). The posi-tion of a point on the surface of the accretion disk at time t isdescribed through the position vector �r1(t). We denote by �rthe position vector of a point in the fluid and by �R the posi-tion vector of the center of the accretion disk. The motion
of the fluid (the gaseous remnant of the star) in the presenceof an exterior (gravitational) force �Fext is described by thelinearized stochastic Landau–Lifshitz equations of motion[18,29–31],
ρm∂
∂t�u (�r , t) = −∇· ↔
P (�r , t)+ �Fext (�r , t) ,∣∣∣�r − �R (t)
∣∣∣ > r1, (32)
where↔P (�r , t) is the total fluid stress tensor, with components
Pi j = pδi j − η
(∂
∂riu j + ∂
∂r jui
)+�i j , (33)
where p is the pressure in the fluid, η is the shear viscositycoefficient of the matter falling towards the black hole, and↔� is the random stress tensor, with components �i j , whichhas the following stochastic properties [18,29]:⟨↔� (�r , t)
⟩= 0,
∣∣∣�r − �R (t)∣∣∣ > r1, (34)
⟨�i j (�r , t)�kl
(�r ′, t ′)⟩ = 2kB�η
(δikδ jl +δilδ jk − 2
3δi jδkl
)
× δ (�r − �r ′) δ(t − t ′
),
∣∣∣�r − �R (t)
∣∣∣ > r1, (35)
where the square brackets denote averages over an equilib-rium ensemble, kB is the Boltzmann constant and � is thetemperature of the external cosmic environment. We have
also denoted ∇· ↔P (�r , t) = ∂Pi j/∂x j . We assume that the
accreted gas is incompressible, and therefore its velocity sat-isfies the condition
∇ · �u = 0. (36)
The motion of the accretion disk, modeled as a singlephysical system, immersed in the fluid accreted by the centralblack hole in the presence of an external gravitational force�Fext (�r , t) is governed by the equation [9,18,29–31]
md�v(t)
dt= �F(t)+ �Fext((�r , t)
= −∫
S(t)
↔P (�r , t) · �ndS + �Fext(t), (37)
where↔P (�r , t) · �n = Pi j n j , and �n is the unit vector per-
pendicular to the thin accretion disk, and S(t) is the surfacearea of the accretion disk. Equations (32) and (37) simplyexpress the conservation of the momentum. In the fully lin-earized scheme one may neglect the time dependence of �R(t)and S(t) in these equations, and one may take the origin ofthe coordinates in the center of the accretion disk, so that�R(t) = 0. The stochastic boundary value problem associated
with the motion of the accretion disk of radius �r1(t) in the
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2900 Page 8 of 16 Eur. Phys. J. C (2014) 74:2900
fluid accreted by the black hole is specified by the boundarycondition
�v (�r1(t), t) = �V (t), (38)
where �r1(t) is the position vector of a point on the surface ofthe accretion disk at time t having the velocity �V (t). �r1(t) isgiven by �r1(t) = �r1 (0) + ∫ t
0�V (
t ′)
dt ′. The components ofthe force acting on the surface of the accretion disk due to thefluid falling towards the black hole are given by [9,18,29–31]
Fi (t) = −∫
S(t)
[pδi j − η
(∂
∂riu j + ∂
∂r jui
)]n j dS
−∫
S(t)
�i j n j dS. (39)
By estimating the perturbations of the velocities in thehydrodynamic flow of the gas remnant due to the presenceof the accretion disk it follows that the equation of motion ofthe perturbed accretion disk can be written in the form of ageneralized Langevin equation of the form [31]
Mdd�v(t)
dt= �Fext(t)+ m
t∫
0
γ(t − t ′
) �v(t ′)dt ′ + ξ(t), (40)
where the function γ(t − t ′
)represents a retarded effect of
the frictional force and ξ(t) is the random force generated bythe stochastic fluctuations in the fluid accreted by the blackhole. Assuming that the flow is unsteady, one should alsotake into account the so-called retarded viscous force effect,which is due to an additional term to the Stokes drag, relatedto the history of the particle acceleration. This additional dragforce, nowadays referred to as the Basset history force, hasthe form [9]
�FB = 6πr2pη
t∫
0
(t − t ′
)−1/2 d�v (t ′)
dt ′dt ′, (41)
where rp is the particle radius. However, in the present paperwe ignore the effects of the Basset history force on the accre-tion disk dynamics.
3 The numerical approach to the generalized Langevinequation
In the present section we write down the equation of motionof the stochastically oscillating accretion disks in a dimen-sionless form. The numerical procedure to solve the Langevinequation in the presence of a colored noise is also described.To characterize the luminosity of the stochastically oscillat-ing disks from a global point of view we will use the power
spectral distribution (PSD), whose main properties are brieflyreviewed.
3.1 Dimensionless form of the generalized Langevinequation
By introducing a set of dimensionless parameters (θ, σ, q)defined as
t = τdθ, τ = τdσ, δz = c2τ 2d
MDq, (42)
where MD is the mass of the disk, the equation of motion ofthe stochastically oscillating disk given by Eq. (16) can bewritten as
d2q
dθ2 + a
θ∫
0
e−|θ−σ | dq(σ )
dσdσ + bω2⊥q = ξ(θ), (43)
where we have denoted
a = α1cτd , b = c2τ 2d
M2 , ξ z = 1
MDξ(θ), (44)
and
ω⊥M = ω⊥, (45)
respectively, where M is the mass of the central compactobject.
For the luminosity calculation, the energy is the sum ofkinetic plus potential energy
E = 1
2
(dδz
dt
)2
+ K (δz) (46)
and the output luminosity is the time variation of the energy
L = −dE
dt= −dδz
dt
[d2δz
dt2 + K ′(δz)]. (47)
The luminosity, written in dimensionless form as
L = c4τd
M2d
L, (48)
has an evolution given by
L = −dq
dθ
(d2q
dθ2 + bω2⊥q
). (49)
3.2 The numerical procedure
To solve the dimensionless generalized Langevin equation(43), the algorithms already developed in the literature, see
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Eur. Phys. J. C (2014) 74:2900 Page 9 of 16 2900
e.g. [32], were used to obtain a set of three differential equa-tions as follows:
q = p, (50)
p = −bω2⊥q + z, (51)
z = −z − a p + ϒ(θ), (52)
where the noise ϒ has the property
〈ϒ(θ)ϒ(σ)〉 = Aδ(θ − σ), (53)
and the initial value of z is drawn from a distribution withsecond moment
〈z(0)2〉 = 2A. (54)
The dimensionless form of the disk luminosity is given by
L = −pz. (55)
The set of the stochastic linear differential equations (50)–(52) is solved by implementing the fourth order Runge–Kutta integrator developed in [32]. Time is discretized withtimestep h. At each timestep j +1 the values of the variablesin the set T j+1 = {q, p, z} are calculated as
x j+1 = xdetj (T j )+ x rand
j (T j ), (56)
i.e., as the sum of a deterministic part and a random part,where both parts depend on the full values of all the parame-ters in the set at the previous timestep. The deterministic partfollows by evolving the “normal” Runge–Kutta algorithm intime and has the form⎛
⎜⎜⎝
qdetj+1
pdetj+1
zdetj+1
⎞
⎟⎟⎠ = D
⎛
⎜⎜⎜⎝
q j
p j
z j
⎞
⎟⎟⎟⎠, (57)
where D is a matrix with components
D11 = −h2
2bω2⊥ + ω2⊥bh4
24
(bω2⊥ + a
), (58)
D12 = h − h3bω2⊥6
− h3a
6+ h4a
24, (59)
D13 = h2
2− h3
6− h4
24
(bω2⊥ + a − 1
), (60)
D21 = −hbω2⊥ + h3bω2⊥6
(bω2⊥ + a
)− h4abω2⊥
24, (61)
D22 = −h2
2
(bω2⊥ + a
)+ h3a
6
+h4
24
(b2ω4⊥ + 2abω2⊥ + a2 − a
), (62)
D23 = h − h2
2+ h3
6
(1 − bω2⊥ − a
)
+h4
24
(bω2⊥ + 2a − 1
), (63)
D31 = h2abω2⊥2
− h3abω2⊥6
+ h4abω2⊥24
(1 − bω2⊥ − a
),
(64)
D32 = −ah + h2
2− h3a
6
(1 − bω2⊥ − a
)
+h4a
24
(1 − bω2⊥ − 2a
), (65)
and
D33 = −h − h2
2(1 − a)+ h3
6(2a − 1)
+h4
24
(abω2⊥ + a2 − 3a + 1
), (66)
respectively.The random part is calculated as detailed in Section II.B
of [32], and for the case studied here it has the form
⎛
⎜⎜⎝
qrandj
prandj
zrandj
⎞
⎟⎟⎠=
⎛
⎜⎜⎝
0 0 1 −1
0 1 −1 −bω2⊥ − a + 1
1 −1 1 − a 2a − 1
⎞
⎟⎟⎠
⎛
⎜⎜⎜⎜⎜⎝
Z1
Z2
Z3
Z4
⎞
⎟⎟⎟⎟⎟⎠,
(67)
where the Zi ’s are linear combinations of four independentGaussian variables {a1, a2, a3, a4} ∈ N (0, h)
Z1 = a1, (68)
Z2 = h
(a1
2+ a2
2√
3
), (69)
Z3 = h2
(a1
3! + 2√
3a2
4! + a3√6!
)
, (70)
Z4 = h3
(a1
4! + a2√
3
40+ a3
24√
5+ a4
120√
7
)
. (71)
3.3 The power spectral distribution
As a parameter allowing a global characterization of thestochastic behavior of the disk we use the slope of the powerspectral distribution (PSD) of the physical parameters of thedisk. The numerical values of the slopes of the PSD pro-vide insight to the nature of the mechanism leading to theobserved variability. In statistical analysis, if X is some fluc-tuating stationary quantity, with mean μX and variance σ 2
X ,then the autocorrelation function for X is defined as [33,34]
RX (τ ) = 〈(Xs − μX ) (Xs+τ − μ)〉σ 2
X
, (72)
where Xs is the values of X measured at time s and 〈〉 denotesaveraging over all values s. The PSD is defined based on thecorrelation function as [33,34]
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2900 Page 10 of 16 Eur. Phys. J. C (2014) 74:2900
P( f ) =+∞∫
−∞RX (τ )e
−ı2π f τdτ, (73)
and it is straightforward to see its importance in terms of the“memory” of a given process. For example, if X is the B band(optical) magnitude of the disk, the slope of the PSD of a timeseries of X provides insight into the degree of correlation theunderlying physical process has with itself. The system needsadditional energy to fluctuate and this mechanism is histori-cally best explained for Brownian motion, in which case theenergy is thermal. As an example of the connection betweenthe process generating the light curve and the observed spec-tral slope, a Brownian motion process would produce a PSDof the shape P( f ) ∼ f −2, while a completely uncorrelatedevolution of a system would produce white noise, i.e., a PSDof the shape P( f ) = f 0 = const. From this it is importantto remember that the slope of the PSD can be regarded asan indicator of the type of underlying process generating thelight curves. As already stated in Sect. 1, the observed IDVlight curves have spectral slopes that do not fit in the simplemodels of Brownian motion or uncorrelated evolution. How-ever, in the case of experimental curves, one cannot knowbefore-hand if the process is purely stationary.
The experimental and simulated light curves are analyzedin this paper with the software .R, which produces spectralslopes in accordance with the algorithm described in [35].
In order to trust an interpretation in which the spectralslope represents a sign of correlation in the signal, the inves-tigated signal should be stationary. By definition, a stationarystochastic process is a process whose properties are constantin time. More precisely, the mean value and the variance ofsuch a signal are time independent and the autocorrelationof the signal at different times depends only on the time dif-ference and not on the actual time at which the correlation isevaluated.
In our model, the perturbing noise is a stationary stochasticprocess, as can easily be seen because of its prescribed prop-erty Eq. (56). To show that the resulting light curve L = − pzis stationary, one should look at Eqs. (56)–(71). The stochas-tic part of the signals involved are given by linear combina-tions of Gaussian random variables (the Zi s); therefore, theirproperties are stationary. We thus expect that our interpreta-tion of the spectral slope is reasonably accurate.
4 Colored stochastic oscillations of the generalrelativistic disks
In the present section we will obtain the basic physical param-eters (displacement, velocity, and luminosity), of the verti-cally oscillating accretion disks in both static Schwarzschildgeometry, and in the rotating Kerr geometry, respectively.
4.1 Colored stochastic disk oscillations for theSchwarzschild geometry
The general form of a static axisymmetric metric can be rep-resented as [36]
ds2 = −e2U c2dt2 + e−2U[e2ζ
(dρ2 + dz2
)+ ρ2dφ2
].
(74)
For the Schwarzschild solution the functions U and ζ aregiven by
U = 1
2ln
√ρ2 + (M + z)2 +
√ρ2 + (M − z)2 − 2M
√ρ2 + (M + z)2 +
√ρ2 + (M − z)2 + 2M
,
(75)
ζ = 1
2ln
[√(M + z)2 + ρ2 +
√(M − z)2 + ρ2
]2 − 4M2
4√(M + z)2 + ρ2
√(M − z)2 + ρ2
,
(76)
where M is the mass of the central compact object in geo-metrical units [36]. The radial frequency at infinity�2⊥ of theparticles in the disk, and the proper free oscillation frequencyω2⊥ of the disk can be obtained as [8,37,38]
�2⊥ = e4U−2ζ
1 − ρU,ρU,zz, (77)
and
ω2⊥ = e2U−2ζ
1 − 2ρU,ρU,zz, (78)
respectively. In the case of the static Schwarzschild geometry,the free vertical oscillation frequency of general relativisticdisks can be written as [8,38]
ω2⊥ (M, ρ) = M
ρ2√
M2+ρ2−2M2(
M+√M2+ρ2
) , (79)
where M is expressed in natural units as M = G M/c2, andρ is the distance from the black hole center in the radial planeof the disk.
If we express ρ in terms of the mass of the central objectas ρ = nM , then
ω2⊥ (n) = 1[√
1 + n2(n2 − 2
) − 2] . (80)
For M = 108 M�, n = 10, ω2⊥ = 0.001 and fixed valuesof the integration step h = 1, dimensionless initial displace-ment q(0) = 0, dimensionless velocity p(0) = 0.01 andb = 4, we analyzed two cases. All trajectories are obtainedby mediating over 1000 stochastic trajectories (for which thevariable z0 is redrawn from its distribution).
First for fixed A = 0.0015 we varied a and obtainedthe behavior of the dimensionless displacement, velocity and
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Eur. Phys. J. C (2014) 74:2900 Page 11 of 16 2900
Fig. 1 Vertical displacement ofthe stochastically oscillatingdisk in the Schwarzschildgeometry for different values ofa and A: A = 0.0015 anda = 0.03 (red curve), a = 0.1(green curve), a = 0.001 (blackcurve)—left figure, anda = 0.03 and A = 1.5 (redcurve), A = 0.15 (green curve),and A = 0.015 (blackcurve)—right figure
0 200 400 600 800 1000
0.05
0.00
0.05
0.10
timestep
q
0 200 400 600 800 10000.10
0.05
0.00
0.05
0.10
timestep
q
Fig. 2 Velocity of the diskoscillations for theSchwarzschild geometry fordifferent values of a and A:A = 0.0015, and a = 0.03 (redcurve), a = 0.1 (green curve),a = 0.001 (black curve)—leftfigure, and a = 0.03 andA = 1.5 (red curve), A = 0.15(green curve), A = 0.015 (blackcurve)—right figure
0 200 400 600 800 1000
0.10
0.05
0.00
0.05
0.10
timestep
p
0 200 400 600 800 1000
0.15
0.10
0.05
0.00
0.05
0.10
0.15
timestep
p
Fig. 3 Luminosity of thestochastically oscillating disk inthe Schwarzschild geometry, fordifferent values of a and A:A = 0.0015 and a = 0.03 (redcurve), a = 0.1 (green curve),a = 0.001 (black curve)—leftfigure, and a = 0.03 andA = 1.5 (red curve), A = 0.15(green curve), A = 0.015 (blackcurve)—right figure
0 200 400 600 800 1000
0.00
0.01
0.02
0.03
0.04
timestep
L
0 200 400 600 800 1000
0.00
0.01
0.02
0.03
timestep
L
luminosity. Then for fixed a = 0.03 we varied A and obtainedthe behavior of the same physical parameters. The results ofthe numerical simulations of the displacements, velocitiesand luminosities for the stochastic oscillations of the accre-tion disks are presented in Figs. 1, 2, and 3, respectively.
The PSD of luminosity for the cases a = 0.03, A =0.0015 and a = 0.03 and A = 1.5 are shown in Fig. 4,respectively. The PSD curves were obtained by using the .Rsoftware [35].
4.2 Stochastic disk oscillations in the presence of colorednoise in the Kerr geometry
The line element of a stationary axisymmetric spacetime isgiven by the Weyl–Lewis–Papapetrou metric [39–44]
ds2 = e2(ψ1−ψ2)(
dρ2 + dz2)
+ ρ2e−2ψ2 dφ2
−e2ψ2 (cdt − ψ3dφ)2 , (81)
where ψi , i = 1, 2, 3 are functions of z and ρ only. Themetric functions generating the Kerr black hole solution inaxisymmetric form are given by
ψ1 = 1
2ln(R+ + R−)2 − 4M2 + (
J 2/ϑ2)(R+ − R−)2
4R+ R−,
(82)
ψ2 = 1
2ln(R+ + R−)2 − 4M2 + (
J 2/ϑ2)(R+ − R−)2
(R+ + R− + 2M)2 + (J 2/ϑ2
)(R+ − R−)2
,
(83)
ψ3 = − J M
ϑ2
(R+ + R− + 2M)[(R+ − R−)2 − 4ϑ2
]
(R+ + R−)2 − 4M2 + (J 2/ϑ2
)(R+ − R−)2
,
(84)
where M and J are the mass and the specific angular momen-tum of the compact object, R± =
√ρ2 + (z ± ϑ)2, and
ϑ = √M2 − J 2.
By defining ρ = nM as the radial distance from theorigin in the plane of the disk, and the angular momen-
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Fig. 4 PSD of the luminosity for the Schwarzschild case, for a = 0.03 and A = 0.0015 (left figure). If a fit according to P( f ) ∼ f α is attempted,α = −1.356. PSD of luminosity for a = 0.03 and A = 1.5 (right figure). If a fit according to P( f ) ∼ f α is attempted, α = −1.455
Fig. 5 Vertical displacement ofthe stochastically oscillatingdisk in the Kerr geometry forA = 0.0015 and a = 0.03 (redcurve), a = 0.1 (green curve),a = 0.001 (black curve)—leftfigure, and for a = 0.03 andA = 1.5 (red curve), A = 0.15(green curve), A = 0.015 (blackcurve)—right curve 0 200 400 600 800 1000
0.05
0.00
0.05
0.10
timestep
q
0 200 400 600 800 1000
0.05
0.00
0.05
timestep
q
tum of the compact object as J = k M , and denotingδ = M
√n2 + 1 − k2, with n > 0 and k ∈ [0, 1], the fre-
quency of the free vertical oscillations of the general rel-ativistic disks in the Kerr geometry can be represented as[8,45]
ω2⊥ = ω21 (n, k)
ω22 (n, k)
, (85)
where
ω21 (n, k) = 3n4 + n2
√1 + δ
×[2k (5 + 3δ)+ 3
√1 + δ
(6 + 3δ + δ2
)]
− 2 (1 + δ)3/2 ×[
− 4k(2 + δ)
+√1 + δ
(−8 + 3δ2 + δ3
) ], (86)
and
ω22 (n, k) = (1 + δ)
[n2 − δ (1 + δ) (3 + δ)
]2
×{
δ −[k + (1 + δ)3/2
]2 [4 (1 + δ)+ n2 (3 + δ)
]
[n2 − δ (1 + δ) (3 + δ)
]2
+ 4k[k + (1 + δ)3/2
]
n2 − δ (1 + δ) (3 + δ)− 1
}
, (87)
respectively.By adopting for the mass of the central object a value
of M = 108 M�, and by fixing n = 10, k = 0.9, andω2⊥ = 0.0012, respectively, and considering for the stepand the initial conditions the values h = 1, q(0) = 0,
p(0) = 0.01, b = 4, we have analyzed two cases. Alltrajectories are obtained by mediating over 1000 stochas-tic trajectories (for which the variable z0 is redrawn from itsdistribution).
For fixed A = 0.0015 we varied a and obtained the behav-ior of the dimensionless displacement, dimensionless veloc-ity, and dimensionless luminosity. Then, for a = 0.03, wevaried A and obtained the behavior of the same physicalparameters of the disk. The results of the numerical simula-tions are presented in Figs. 5, 6 and 7, respectively.
The PSDs of the luminosity for the cases a = 0.03,A = 0.0015 and a = 0.03, A = 1.5, respectively, are shownin Fig. 8. Values for the spectral slope obtained for a = 0.03,A = 0.0015 and different configurations of the system pro-ducing the simulated light curves are given in Table 1. Thefourth column in Table 1 is the Bayesian probability that thesource actually behaves like in the attempted fit, where pB
close to 1 suggests that the fit is correct. The definition andcalculation of pB is quite lengthy, but we sketch it here andrefer the reader to [35] for full detail. In cases such as investi-gating data from an AGN, when there is only one light curveavailable (call it �xobs), where reproducibility of the experi-ment is not an option, the .R software with the Bayes scriptoffers the possibility to assess whether or not the recordeddata were produced by a process behaving like a null hypoth-esis, H0. For this purpose, the script simulates numerous setsof data based on the statistical properties of �xobs, stores themin �x rep and treats these new data sets as they would be multi-ple measurements of the same process. Afterwards it defines�(�x), called test statistics, which is a real-valued function of
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Eur. Phys. J. C (2014) 74:2900 Page 13 of 16 2900
Fig. 6 Velocity of thestochastically oscillating disk inthe Kerr geometry, forA = 0.0015 and a = 0.03 (redcurve), a = 0.1 (green curve),a = 0.001 (black curve)—leftfigure, and for a = 0.03 andA = 1.5 (red curve), A = 0.15(green curve), A = 0.015 (blackcurve)—right figure 0 200 400 600 800 1000
0.10
0.05
0.00
0.05
0.10
0.15
timestep
p
0 200 400 600 800 1000
0.150.100.050.000.050.100.15
timestep
p
Fig. 7 The luminosity of thestochastically oscillating disk inthe Kerr geometry forA = 0.0015 and a = 0.03 (redcurve), a = 0.1 (green curve),a = 0.001 (black curve)—leftfigure, and for a = 0.03 andA = 1.5 (red curve), A = 0.15(green curve), A = 0.015 (blackcurve)—right figure 0 200 400 600 800 1000
0.00
0.01
0.02
0.03
0.04
0.05
timestep
L
0 200 400 600 800 1000
0.00
0.01
0.02
0.03
0.04
timestep
L
Fig. 8 PSD’s of the luminosity of the stochastically oscillating disk inthe Kerr geometry. In the left figure, a = 0.03 and A = 0.0015. If a fitaccording to P( f ) ∼ f α is performed, α = −1.65. In the right figure
a = 0.03 and A = 1.5. If a fit according to P( f ) ∼ f α is performed,α = −1.619
Table 1 Values for the spectral slope of the luminosity curves, obtainedby using the .R software, for a = 0.03, A = 0.0015, and different massand angular momentum configurations of the black hole–accretion disksystem producing the simulated light curves. The values given in thecolumn for −α represent the mean values of the parameter calculatedfrom the series simulated with .R software (as explained in the text),followed by [±standard deviation]k n −α pB
0 10 1.356 [±0.05] 0.749
0.9 10 1.650 [±0.05] 0.064
0.5 7 1.480 [±0.05] 0.288
0.5 8 1.372 [±0.05] 0.755
0.5 9 1.499 [±0.05] 0.717
0.9 7 1.387 [±0.05] 0.289
0.9 8 1.369 [±0.05] 0.079
0.9 9 1.562 [±0.09] 0.087
the data chosen such that extreme values are unlikely whenthe null hypothesis H0 is true. With these parameters, theBayesian probability is defined as
pB(�x) = Pr{�(�x rep) ≤ �(�xobs)|�xobs, H0
}, (88)
where Pr = {x |y} means probability of event x given thatevent y has occurred.
5 Comparison with AGN IDV data
Extensive observational and theoretical efforts have beenmade in order to explain intra-day variability (IDV) in someclasses of active galactic nuclei (AGN) [3,46–49]. IDV mani-fests as the fast change (in less than one day) in the luminosityoutput of an object, and for supermassive black holes IDVoccurs in the optical domain. The PSD of these light curvesare found to be nontrivial [11–13].
In the present paper we assume that the IDV emissionoccurs from the disk of a supermassive black hole–accretiondisk system [3,4], and it is due to the oscillations of the diskperturbed by a sudden interaction with the external environ-ment. Such a transient interaction can be due to the stochastic
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2.650 2.655 2.660 2.6650
10
20
30
40
50
60
Log L
No.
of
data
poin
ts
θ
Fig. 9 Test of the log-normal distribution of the optical IDV flux dataof BL Lac S5 0716+714 [48]. If the bins (the distribution of the data)can be fitted by the continuous curve (a normal distribution with thesame mean and variance as the data), the distribution of the data islog-normal. The data were recorded in the B filter, on JD 2454824
gravitational force generated by the fluctuations of the starnumber in the near neighborhood of the black hole–accretiondisk system, if the system is located in a dense stellar cluster,or to the interaction of the disk with the debris produced bythe tidal disruption of a star captured by the central blackhole.
For BL Lac S5 0716+714, an AGN with M ∈ [107.68 M�,108.38 M�] [50], a PSD analysis performed on IDV data in theBVRI bands has shown that α varied between approximately−1.5 and −2.7 for that set of data [51].
There are indications that the relationship between theroot-mean-square-deviation (rms) and the flux provides moreinformation regarding the source-process of fast variability,at least in X-Ray Binaries [52]. For these objects, for whichIDV is exhibited in the X-Ray domain, it was found that thisrelationship is linear. However, a test performed on opticalIDV for BL Lac S5 0716+714 on observational data fromdifferent epochs showed that the rms-flux relation was notlinear [53].
In order to test if the optical IDV flux data of BL Lac S50716+714 are log-normally distributed, in Fig. 9 we comparethe observational data with the normal distribution. The binsrepresent the logarithm of the observational data, and the
smooth line is a superimposed Gaussian with the same meanand variance as the data. If the two would have coincided(i.e. if the flux would have been log-normally distributed),the rms-flux relation would have been linear. We find thatthe simulated data, obtained from the generalized Langevinequation description of the luminosity of the stochasticallyoscillating disks, also exhibit this feature, i.e., that the rms-flux relation is not linear, for the Schwarzschild nor the Kerrcase (an example is shown in Fig. 10).
The model presented in this work is thus consistent withobservational data, from the point of the statistics exhibitedby the light curves. This is very encouraging and supportsthe idea that perturbations due to colored/nontrivially corre-lated noise consistently explain the statistical properties ofthe observed IDV light curves.
6 Discussions and final remarks
In the present paper we have introduced a mathematicaldescription of the stochastic vertical oscillations of the accre-tion disks based on the generalized Langevin equation withcolored noise and memory effects. From an astrophysicalpoint of view this approach may be relevant for modeling ofaccretion disks around compact general relativistic objectsrandomly interacting with the cosmic environment surround-ing the disk. Events like the encounter of a star, or a massivecompact object, by the central black hole, or the interactionbetween the disk and the cosmic dust resulting from the tidaldisruption of a star captured by the central black hole, maytrigger such interactions, which are essentially random innature. In our description we have assumed that the oscil-lation velocities of the massive test particles composing thedisk are small, and therefore they can be described by theNewtonian equation of motion. Hence in our approach wehave assumed that the motion of the disk is non-relativisticin the sense that the velocity δz/dt of the particles is muchsmaller than the speed of light, δz/dt c. However, since
12 10 8 6 4 2 00
20
40
60
80
100
120
140
Log Lθ
No.
of
data
poin
ts
14 12 10 8 6 4 2 00
20
40
60
80
100
120
Log Lθ
No.
of
data
poin
ts
Fig. 10 Test of the log-normal distribution of the light curves gen-erated by the stochastically oscillating accretion disks. The simulateddata are represented by the bins. The continuous superimposed line is anormal distribution with the same mean and variance as the simulated
data. Left panel Schwarzschild case, with A = 0.0015, a = 0.03 andn = 10. Right panel Kerr case, with A = 0.015, a = 0.03, n = 10 andk = 0.9
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the gravitational field in the vicinity of the black hole isstrong, we have adopted for the oscillation frequency of thedisk in the gravitational field the general relativistic values.The cases of the static and rotating black holes were con-sidered independently, and the effect of the rotation on thedisk luminosity was explicitly considered. The analysis ofthe PSD of the luminosity curves has shown that in the pres-ence of memory effects and colored noise nontrivial valuescan be obtained for the spectral slope. Therefore the pres-ence of the colored noise/memory effects could allow a bet-ter description of the astrophysical processes involving ran-dom/stochastic factors.
In the present paper we have considered that accretiondisks are composed from massive test particles forming afluid system, and moving on non-geodesic lines around thecentral black hole. By considering the equation of motion ofthe disk perturbations, and assuming that the gravitationaleffects dominate the hydrodynamic ones, in the first orderof approximation the effects of the non-zero pressure andenergy density give in the equation of motion a correctionterm proportional to the magnitude of the vertical perturba-tions δz. In the non-relativistic limit the disk pressure effectsand the external gravitational perturbation terms can be com-bined in a single, effective coefficient, proportional to thevertical displacement, and which determines the frequencyof the disk oscillations. In order to construct realistic physicalmodels one should assume an equation of state for the diskmatter components, and adopt some particular disk struc-tures. Since there is no general consensus on the physicalproperties of the accretion disks, there is no guarantee thatadopting a particular disk model would produce observation-ally relevant results. On the other hand, in the presence of astrong gravitational field the gravitational effects dominatethe pressure effects, which can be neglected with a very goodapproximation. In this case the motion of the particles in thedisk can be considered as taking place on a geodesic curve.This approximation is certainly valid for particles located inthe inner region of the disk. As for the viscous dissipativeeffects, in the present paper we have assumed that they canbe described by a single integral kernel term, whose func-tional form has been fixed phenomenologically, so that thefluctuation–dissipation theorems can be satisfied. However,some essential features of the accretion disks dynamics canbe obtained, and analyzed, even by the present simplifiedtheoretical model.
As a possible astrophysical application of our model wehave compared a set of IDV observational light curve datawith the results of the simulations of the vertical stochasticoscillations of the accretion disks, described by the general-ized Langevin equation. Some common features of the the-oretical model and observations, namely, the possibility ofobtaining nontrivial PSD values, and the deviation in bothcases of data from a log-normal distribution were identified.
Hence the generalized Langevin equation with colored noisecould provide a powerful tool in the description of the phys-ical processes in the vicinity of compact general relativisticobjects.
Acknowledgments We would like to thank to the anonymous refereefor comments and suggestions that helped us to significantly improveour manuscript. GM is supported by a grant of the Romanian NationalAuthority of Scientific Research, Program for Research-Space Tech-nology and Advanced Research-STAR, project number 72/29.11.2013.
Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.Funded by SCOAP3 / License Version CC BY 4.0.
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