Stochastic oscillations of general relativistic disks

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by  Tiberiu Harko

Mon.Not.R.Astron.Soc.000,1–??(2002) Printed12January2012 (MNLATEXstylefilev2.2)
Stochastic oscillations of general relativistic disks
Tiberiu Harko1 ⋆ and Gabriela Mocanu2 †
1Department of Physics and Centerfor Theoretical and Computational Physics,
University of Hong Kong, Pok Fu Lam Road, Hong Kong, Hong Kong SAR, P. R. China
2Faculty of Physics, Department of Theoretical and Computational Physics, Babes-Bolyai University,Cluj-Napoca, Romania
2
1 12January2012
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2

n ABSTRACT
a We analyze the general relativistic oscillations of thin accretion disks around com-
J pact astrophysical objects interacting with the surrounding medium through non-

1 gravitational forces. The interaction with the external medium (a thermal bath) is
1 modeled via a friction force, and a random force, respectively. The general equations
describingthe stochasticallyperturbed disks arederivedby consideringthe perturba-

] tions of trajectories of the test particles in equatorial orbits, assumed to move along
c
q the geodesic lines. By taking into account the presence of a viscous dissipation and
- of a stochastic force we show that the dynamics of the stochastically perturbed disks
r canbe formulatedinterms ofa generalrelativistic Langevinequation.The stochastic
g
[ energytransportequationisalsoobtained.The verticaloscillationsofthe disksinthe
Schwarzschildand Kerrgeometries are consideredin detail, and they are analyzedby

1 numerically integrating the corresponding Langevin equations. The vertical displace-
v
ments, velocities and luminosities of the stochastically perturbed disks are explicitly
8
obtained for both the Schwarzschild and the Kerr cases.
1
2 Key words: Accretion– accretiondisks; black hole physics: gravitation:instabilities
2
.
1
0
2
1 INTRODUCTION An exact formulation of the physics of Barnett relax-
1
ation, based on a realistic kinetic model of the relaxation
: The simplest physical model describing the random mo-
v mechanism which includes the alignment of the grain an-
tion of a non-relativistic Brownian particle is given by the
i gular momentum in body coordinates by Barnett dissipa-
X Langevin equation, which can be written as (Coffey et al.
tion, disalignment by thermal fluctuations, and coupling of
r 2004)
the angular momentum to the gas via gas damping was in-
a
dp troducedinLazarian & Roberge(1997).TheFokker-Planck
= νp+K+ξ(t), (1)
dt − equation for the measure of internal alignment was solved
where p is the (non-relativistic) momentum, and the three by using numerical integration of the equivalent Langevin
terms of the right-hand side of Eq. (1) correspond to the equationforBrownian rotation.Itwasalso shown thatina
friction, to an external force K, and to a stochastic force ξ, steadystate,energyistransferredbacktothegrainmaterial
respectively. ν is a coefficient called the friction coefficient. at an equal rate by Barnett dissipation.
Due to the friction term, the particle in Brownian motion
loses energy to the medium, but simultaneously gains en-
A simple physical model to describe the dynamics of
ergy from the random kicks of the thermal bath, modeled
a massive point-like object, such as a black hole, near
by the random force. The random force, which we assume
the center of a dense stellar system was developed in
to be a white noise, satisfies the conditions ξ(t) = 0 and
h i Chatterjee et al.(2002).Thetotalforceonthemassivebody
ξ(t)ξ(t′) =kδ(t t′), respectively, where k is a constant.
h i − canbeseparatedintotwoindependentparts,oneofwhichis
Theseparationoftheforceintofrictionalandrandomparts
theslowly varyinginfluenceoftheaggregate stellar system,
is merely a phenomenological simplification - microscopi-
and the other being therapidly fluctuatingstochastic force
cally, the two forces have the same origin (collision with
duetodiscreteencounterswithindividualstars.Themotion
the external medium constituents). The Langevin equation
oftheblackholeissimilar tothatofaBrownian particlein
has a large range of applications in physics, astrophysics,
a harmonic potential, and its dynamics can be analyzed by
engineering, biology etc. (Coffey et al. 2004).
using an approach based on theLangevin equation. By nu-
merically solving theLangevin equation onecan obtain the
⋆ E-mail:[email protected] average values, time-autocorrelation functions, and proba-
† E-mail:[email protected] bilitydistributions oftheblack hole’s position andvelocity.
(cid:13)c 2002RAS
2 T. Harko and G. Mocanu
By using this model a lower limit on the mass of the black androtatingKerrgeometries.Inthesecasestheequationsof
hole Sgr A*in theGalactic centercan be derived. motionofthediskcanbeformulatedasLangevinequations,
A Langevin–typetreatment of themotion of acharged describingtheinteraction between thedisk andan external
particle in the inter-galactic medium magnetic field, which thermalbath.Bynumericallyintegratingthecorresponding
allows to estimate both the average and of the root mean Langevinequationweobtaintheverticaldisplacements,ve-
square time-delay for particles of given energy, was intro- locities,andtheluminositiesofthestochasticallyoscillating
ducedinVietri et al.(2003).Thismodelwascomparedwith disks.
a scenario where the particles are accelerated at internal The present paper is organized as follows. In Section 2
shocks. The generalized thermodynamics and the collapse we briefly review the motion of the test particles in ax-
of a system of self-gravitating Langevin particles exhibit- isymmetricgravitationalfields.InSection3weconsiderthe
inganomalous diffusion ina spaceof dimension D was con- stochastic perturbations of the geodesic trajectories of test
sidered in Chavanis & Sire (2004). The equilibrium states particlesmovingaroundacentralcompactobject.Thegen-
correspond to polytropic distributions. The dynamical sta- eral system of perturbed equations for stochastic disks is
bility of stellar systems, gaseous stars and two-dimensional obtained in Section 4. The energy transfer in stochastically
vortices was studied by using a thermodynamical analogy. perturbed disks is discussed in Section 5. The vertical mo-
There are many processes in the accretion of mate- tionsofthedisksareconsideredinSection6,anditisshown
rial onto an active galactic nuclei that occur on relatively that they can be described by a Langevin type equation.
short timescales as compared to the cooling time of the The disk oscillation frequencies are also obtained for both
X-ray emitting gas in and around elliptical galaxies and the Schwarzschild and the Kerr metrics. The vertical os-
galaxy clusters. Accretion rates onto active galactic nuclei cillations of the stochastically perturbed disks are consid-
are likely to be extremely variable on short timescales. Us- ered in Section 7, and the vertical displacements, velocities
ing a Langevin type equation Pope (2007) has shown that, and luminosities are obtained, by numerically integrating
for a simple feedback system, this can induce variability in the corresponding Langevin equation, for both the static
theactivegalacticnucleipoweroutputthatisofmuchlarger SchwarzschildandrotatingKerrgeometries.Wediscussand
amplitude,andpersistsforlongertimescales,thantheinitial concludeour results in Section 8.
fluctuations.Animplicationofthisresultisthatrichgalaxy
clusters are expected to show the largest and longest-lived
fluctuations.Stochastic variations in theaccretion ratealso 2 MOTION OF TEST PARTICLES IN
mean that the active galactic nuclei injects energy across a STATIONARY AXISYMMETRIC
wide range of timescales. GRAVITATIONAL FIELDS
Accretion disks around compact objects have become
The metric generated by a rotating axisymmetric compact
standard models for a number of astrophysical phenom-
generalrelativisticobjectcanbegenerallygivenincylindri-
ena like X-ray binaries or active galactic nuclei. The disks
cal coordinates (ct,ρ,φ,z) as
are usually considered as being composed from massive
test particles that move in the gravitational field of the ds2=gtt(cdt)2+2gtφcdtdφ+gφφdφ2+gρρ dρ2+dz2 ,(2)
central compact object. Waves and normal-mode oscilla-
whereallthemetricfunctionsarefunctions(cid:0)ofρandz(cid:1)only.
tions in geometrically thin and thick disks around com-
Theequatorialplaneisplacedatz =0.Foragiventimelike
pact objects have been studied extensively both within worldline xµ(s) with four velocity uµ = dxµ/ds and az-
Newtonian gravity (see Kato (2001) 2001 for a re-
imuthal angular velocity Ω= dφ/dt, the specific azimuthal
view) and within a relativistic framework (Okazakiet al. angular momentum L˜ of a particle of mass m, given by
1987; Perez et al. 1997; Semerak & Zacek 2000; Kato 2001;
Silbergleit et al.2001;Rodriguez et al.2002;Rezzolla et al. L˜ =uφ =ut(gtφ+gφφΩ/c), (3)
2003; Zanotti et al. 2005; Blaes et al. 2006; O’Neill et al. and thespecific energy E˜, given by
2009; Shi & Li 2000). The oscillations of the accretion
disks may produce the quasi-periodic oscillations (QPO’s) E˜ = ut = ut(gtt+gtφΩ/c), (4)
− −
(Titarchuk & Osherovich 2000; Tagger & Varniere 2006; areconstantsofmotion.Thefour-velocitysatisfiesthecondi-
Fan et al. 2008; Horak et al. 2009) in black hole low-mass tionu uµ = 1.Inastationary axisymmetricgravitational
µ
X-ray binaries. field the mos−t important types of worldlines correspond to
Itisthepurposeofthepresentpapertoproposeafully spatiallycircularorbits,givenbyρ=constant,z=constant
general relativistic model for the description of the oscilla- andΩ=constant,respectively.Inthiscasethefour-velocity
tionsof theparticles in theaccretion disksaround compact is given by
objects in contact with an external heat bath. The oscil-
uµ =ut(1,0,Ω/c,0), (5)
lations of the disk, assumed to interact with an external
medium (the heat bath), are described by a general rela- where
tivistic Langevin equation, which is derived, for the case of 1 Ω −1
a general axisymmetric gravitational field, by considering ut = = E˜ L˜ .(6)
g 2g (Ω/c) g (Ω/c)2 − c
the perturbations of the geodesic equation describing the − tt− tφ − φφ (cid:16) (cid:17)
motionoftestparticlesinstablecircularorbits.Theenergy Fpor circular orbits the four-acceleration aµ =
transportinthepresenceofthestochasticfluctuationsofthe (1/2)g uαuβ ofthetestparticleshasonlyaradialcom-
αβ,µ
−
disk is discussed in detail. As an application of our general ponent,
formalism weconsiderindetailtheverticalstochasticoscil-
1
lationsoftheaccretiondisksinboththestaticSchwarzschild aρ = −2gαβ,ρuαuβ =
(cid:13)c 2002RAS,MNRAS000,1–??
Stochastic oscillations of general relativistic disks 3
1g +g Ω + 1g Ω2 ut 2. (7) represents the stochastic force generated by the interaction
− 2 tt,ρ tφ,ρc 2 φφ,ρ c2 oftheparticlesofthediskwiththeexternalheatbath.Thus
(cid:18) (cid:19)
(cid:0) (cid:1) the motion of the general relativistic massive test particles
The radial component of the force a vanishes for two
ρ inthestochastically perturbeddiskcanbedescribedbythe
particular values Ω of theangular velocity,given by
± following general relativistic Langevin typeequation
Ω± = −gtφ,ρ± gt2φ,ρ−gtt,ρgφφ,ρ. (8) dδVµ +2Γµ uαδVβ+Γµ uαuβδxλ =
c pgφφ,ρ ds αβ αβ,λ
ν(δµ+δV δVµ)(δVα Uα)+ξµ[g;x]. (15)
− α α −
InthefollowingwewilluseEq.(15)toanalyzethephys-
3 STOCHASTIC PERTURBATION OF AN
ical properties of the randomly fluctuatingaccretion disks.
EQUATORIAL ORBIT
In theabsence of any externalperturbation theparticles in
thedisk movealong thegeodesic lines given by 4 STOCHASTIC OSCILLATIONS OF
ACCRETION DISKS
d2xµ dxαdxβ
+Γµ =0, (9)
ds2 αβ ds ds We assume that the heat bath is at rest, so that Uµ =
whereΓµ aretheChristoffelsymbolsassociatedtothemet- g−1/2,0,0,0 . Also we assume that, since the perturba-
αβ tt
ric. If the disk is perturbed, a point particle in the nearby t(cid:16)ionsaresmal(cid:17)l,thevariation ofthegravitational fieldalong
position x′µ must also satisfy a generalized geodesic equa- the z - direction can be neglected. This implies that along
tion the equatorial plane g = 0, but g = 0. Then from
µν,z µν,zz
6
d2x′µ dx′αdx′β 1 the Langevin equation Eq. (32) we obtain the equations of
+Γ′µ = fµ, (10)
ds2 αβ ds ds m motion of the particles in the stochastically perturbed disk
as
where fµ is the external, non-gravitational force, acting on
theparticle.Thecoordinatesx′µaregivenbyx′µ=xµ+δxµ, d2δt +2 Γt +Γt Ω utdδρ =
whereδxµ isasmallquantity.Thus,inthefirstapproxima- ds2 tρ φρ c ds
tion, we obtain for theChristoffel symbols ν δt +(cid:16)δV δVt (δV(cid:17)α Uα)+ξt[g;x], (16)
− α α −
Γ′µ (x+δx)=Γµ (x)+Γµ δxλ, (11)
αβ αβ αβ,λ d2δ(cid:0)ρ (cid:1) Ω dδρ
whereΓµαβ,λ =∂Γµαβ/∂xλ.Thereforetheequationofmotion ds2 +2 Γρtt+Γρtφc ut ds +
for δxµ becomes (cid:16) Ω dδφ(cid:17)
2 Γρ +Γρ ut +
d2δxµ dxαdδxβ dxαdxβ tφ φφ c ds
ds2 +2Γµαβ ds ds +Γµαβ,λ ds ds δxλ =fµ. (12) Γ(cid:16)ρ +2Γρ Ω(cid:17)+Γρ Ω 2 ut 2δρ=
If fµ = 0 we reobtain the equation of perturbation tt,ρ tφ,ρ c φφ,ρ c
of the geodesic line introduced in Shirokov (1973). In the (cid:20)ν(δρ +δV δVρ)δVα+(cid:16)ξρ[g(cid:17);x(cid:21)](cid:0), (cid:1) (17)
following we introduce the four-velocity of the perturbed − α α
motion as δVµ = dδxµ/ds. δVµ satisfies the condition d2δφ Ω dδρ
uµδVµ = 0. We consider that the particles in the disk are ds2 +2 Γφtρ+Γφφρ c ut ds =
in contact with an isotropic and homogeneous heat bath. ν δφ+(cid:16)δV δVφ δVα(cid:17)+ξφ[g;x], (18)
The interaction of the particles with the bath is described − α α
by a friction force, and a random force. With respect to a (cid:0) (cid:1)
comovingobserverframetheheatbathhasanon-vanishing d2δz + Γz +2Γz Ω +Γz Ω 2 ut 2δz=
average four-velocity Uµ. ds2 tt,z tφ,z c φφ,z c
Inthenon-relativisticcasethefriction forceisgivenby ν(δz +(cid:20)δV δVz)δVα+ξz[g;x](cid:16). (cid:17) (cid:21)(cid:0) (cid:1) (19)
fi = mνvi, where ν is the friction coefficient, and vi are − α α
thfrecom−ponentsofthenon-relativisticvelocity.Therelativis- Intheabsenceofthefrictionforceandofthestochastic
ticgeneralization ofthefriction force requirestheintroduc- forcegenerated bytheinteraction betweenthediskandthe
tion of the friction tensor νµ, given by (Dunkel& H¨anggi heatbathwereobtaintheequationsforthefreeoscillations
α
2005a,b, 2009) of accretion disks (Semerak & Zacek 2000).
νµ=νm(δµ+δV δVµ). (13)
α α α
The friction force can beexpressed then as 5 ENERGY TRANSFER IN
STOCHASTICALLY OSCILLATING DISKS
fµ = νµ(δVα Uα)=
fr −ναm(δµ+−δV δVµ)(δVα Uα). (14) Inthefollowing wewillusetheEckartmodelfordissipative
− α α − processes (Weinberg 1972), and we will choose a frame de-
LetusnowintroduceaGaussian stochasticvectorfield finedbythefamilyofobserversmovingwiththenormalized
mξµ[g;x], where g is the determinant of the metric ten- four-velocity vµ = nµ/n parallel to the oscillating matter
sor, defined by the following correlators: ξµ[g;x] =0 and fluid of thedisk, where n2 = g nµnν.
µν
h i −
ξµ[g;x]ξν[g;y] = Dµν[g;x;y], where Dµν[g;x;y] is the We assume that the stochastically oscillating disk con-
h i
noisekerneltensor.Fromaphysicalpointofviewmξµ[g;x] sists of at least one fluid, whose particle number density
(cid:13)c 2002RAS,MNRAS000,1–??
4 T. Harko and G. Mocanu
flux nµ = nvµ is conserved, nµ = 0, where ; denotes n(ǫ˙+pv˙)+qµ +qµv˙ v Pµν =v ξµν. (28)
;µ ;µ µ− µ;ν µ ;ν
the covariant derivative with respect to the metric, and
Asfor thetransverse momentumqµ it can beobtained
n is the particle number density. Let us now introduce
from the relativistic analogue of Fourier’s law (Weinberg
a Gaussian stochastic tensor field ξ [g;x] defined by
|muν 1972), namely
the correlators ξ [g;x] = 0 and ξ [g;x]ξ [g;y] =
h µν i h αβ |muν i
Nαβµν[g;x;y], respectively, where hi means statistical aver- qµ =−κhµν(T;ν+Tv˙ν), (29)
age (Hu& Verdaguer 2008). The symmetry and positive
whereκisthethermal conductivitycoefficient andT is the
semi-definite property of the noise kernel guarantees that
equilibrium temperature of the system. Eq. (28) is analo-
thestochastic field tensorξ [g;x], orξ for short,is well-
µν µν
gous to the non-relativistic energy balance equation in the
defined. We assume that this general relativistic stochastic
presence of a stochastic force ξ~, given by
tensor describes the fluctuations of the energy-momentum
tensorδTµν ofthematterinthediskduetotheinteraction dǫ
n + ~q Π~ ~v=~v ξ~, (30)
with thethermal bath. HenceδTµν can bewritten as dt ∇· − ·∇ · ·
δTµν =ρvµvν+2q(µvν)+πµν +ξµν, (20) where ǫ is th(cid:0)e inter(cid:1)nal energy, ~q is the heat flux, Π~ is
the total stress tensor, and ~v is the fluid’s three velocity.
where ρ is the energy density, qµ is the transverse momen-
In Eq. (28) only the term containing the four-acceleration
tum,andπµν istheanisotropicstresstensor,asmeasuredby
u˙ does not have a Newtonian counterpart. When ξ~ 0,
anobservermovingwiththeparticleflux.qµandπµν satisfy µ ≡
Eq.(30)reducestothestandardenergybalanceequationin
the relations qµv =0 and πµνv =0, respectively. Within
µ µ non-relativistic fluid mechanics (Landau & Lifshitz 1987).
the framework of this physical interpretation we can define
the internal energy ǫ through the relation n(ǫ+ǫ ) = ρ,
0
where ǫ is an arbitrary constant (Hawking& Ellis 1973).
0
6 EQUATION OF MOTION OF VERTICALLY
The energy-momentum tensor is covariantly conserved, so
that δTµν = 0. By contracting δTµν with the observer’s PERTURBED ACCRETION DISKS
;µ
four velocity v we obtain
µ Inthefollowingwewillconsideronlytheverticaloscillations
(v δTµν) =v δTµν. (21) ofthedisk,which,foranarbitraryaxisymmetricmetric,are
µ ;ν µ;ν described bytheequation
For thefirst term in Eq.(21) we obtain
vµδTµν = ρvν qν +vµξµν. (22) dd2sδ2z + Γztt,z+2Γztφ,zΩc +Γzφφ,z Ωc 2 ut 2δz=
By usi−ng th−e definition of the internal energy and the −ν(δαz +(cid:20)δVαδVz)δVα+ξz[g;z].(cid:16) (cid:17) (cid:21)(cid:0) (cid:1) (31)
conservation of theparticle numberdensity fluxwe obtain
Since δVµ is a small quantity, the equation of motion
(v δTµν) = nǫ˙ qµ +v ξµν +v ξµν, (23) can befurther simplified to
µ ;ν − − ;µ µ;ν µ ;ν
where we have denoted ǫ˙ = ǫ;νvν. The right-hand side in d2δz +νdδz +ω2δz=ξz[g;z], (32)
Eq. (21) can bewritten as ds2 ds ⊥
v δTµν =qµv˙ +v πµν +v ξµν. (24) where
µ;ν µ µ;ν µ;ν
Thus, the local form of the energy balance of the ω2 = Γz +2Γz Ω +Γz Ω 2 ut 2. (33)
⊥ tt,z tφ,z c φφ,z c
stochastically oscillating disk is given, from the fluid ob- (cid:20) (cid:16) (cid:17) (cid:21)(cid:0) (cid:1)
server’s point of view, by The angular frequency with respect to the radial infinity is
obtained as
nǫ˙+qµ +qµv˙ +v πµν =v ξµν. (25)
;µ µ µ;ν µ ;ν
Whenthestochastictermintheenergy-momentumten- Ω2⊥ =ω⊥2/ ut 2 =Γztt,z+2Γztφ,zΩc +Γzφφ,z Ωc 2. (34)
tshoreifsorigmnoδrTed,a=nd(ρb+ypa)ssvuµmvνin+gpthgaµtν,inwethciasncadseefiδnTeµtνhteahkyes- In the(cid:0)foll(cid:1)owing we consider the vertica(cid:16)l os(cid:17)cillations of
µν
thedisksaroundblackholesdescribedbytheSchwarzschild
drostatic pressure p of the fluid as the trace of the stress
tensor πµν, p = (1/3)πµ. This allows us to define the vis- and theKerr geometry, respectively.
µ
cous stress tensor Pµν as (Weinberg1972)
Pµν = πµν+phµν, (26) 6.1 Disk oscillation frequencies in the
− Schwarzschild geometry
where hµν is the orthogonal projection operator to the ob-
server’s four-velocity defined as hµν =gµν +vµvν. The vis- The general form of a static axisymmetric metric can be
given as (Binniet al. 2005)
cous stress tensor can be obtained in a general form as
Pµν =λ σµν +2v˙(µvν) , (27) ds2=−e2Uc2dt2+e−2U e2γ dρ2+dz2 +ρ2dφ2 . (35)
where λ(cid:2)is the viscosit(cid:3)y coefficient and σµν = v(µ;ν) − Forthe Schwarzschild sol(cid:2)ution(cid:0) (cid:1) (cid:3)
(1/3)v;ααhµν. By defining the invariant specific volume v as 1 ρ2+(M +z)2+ ρ2+(M z)2 2M
the inverse of the particle number density n it follows that U = ln − − , (36)
hµνvµ;ν = vµµ = nv˙. Then for the equation of the local en- 2 pρ2+(M +z)2+pρ2+(M −z)2+2M
ergy balance we obtain and p p
(cid:13)c 2002RAS,MNRAS000,1–??
Stochastic oscillations of general relativistic disks 5
1 (M +z)2+ρ2+ (M−z)2+ρ2 2−4M2 ̟3 = 4Rσδ−3δ+6 2(8M3−3MRσ2 −Rσ3)δ+2 +
γ = 2lnhp 4 (M +z)2+pρ2 (M z)2+iρ2 ,(37) 4Rσδ−3δ+6 (cid:2)3(6M2+3MRσ+Rσ2)δ+(cid:3)ρ2+3Mρ4 ,
− (48)
(cid:2) (cid:3)
respectively, whepre M is the mapss of the central compact
object in geometrical units (Binniet al. 2005). The usual and
Schwarzschild line element in (ct,r,θ,φ) coordinates is re- ̟ = 4R δ3δ10
coveredbymakingthetransformationsρ=√r2 2Mrsinθ 4 − σ − + ×
− (8M3 3MR2 R3)δ +3M(2M +R )ρ2 ,
and z =(r M)cosθ. − σ− σ + σ
−
ForanaccretiondiskintheSchwarzschildgeometrythe (49)
(cid:2) (cid:3)
radial frequency at infinity and the proper frequency are
respectively,withδ = ρ2+σ2 M andR = ρ2+σ2,
given by ± ± σ
respectively. In the limit a=0, R = M2+ρ2 and δ =
e4U−2γ p σ p ±
Ω2⊥(ρ,z,M)= 1 ρU,ρU,zz, (38) ERqσ.±(4M0),gaivnidngafttheerossocmilelaatilognebfrraeqicumenacnypipouftlahteioSnc,hwwearrzesccohviledr
−
and disk.
e2U−2γ
ω2 (ρ,z,M)= U , (39)
⊥ 1 2ρU ,zz
,ρ
− 7 VERTICAL STOCHASTIC OSCILLATIONS
respectively.EstimatingΩ2⊥ intheequatorialplaneatz=0 OF PERTURBED ACCRETION DISKS
3
gives Ω2 = M/ M + M2+ρ2 , while for ω2 we
⊥|z=0 ⊥ By assuming that the velocity of the perturbed disk oscil-
obtain (cid:16) p (cid:17) lations is small, we can approximate ds as ds ≈ cdt, where
M t is the time coordinate. We also assume that the external
ω2 (M,ρ)= . (40)
⊥ perturbative force is a function of time only, and neglect
ρ2 M2+ρ2 2M2 M+ M2+ρ2
− its possible spatial coordinate dependence. In this case the
p (cid:16) p (cid:17) equation of motion of the disk in contact with a heat bath
follows from Eq. (32) and is given by
6.2 Disk oscillation frequencies in the Kerr
geometry d2δz dδz
+cν +c2ω2 (ρ,M,a)δz=c2ξz(t), (50)
dt2 dt ⊥
The line element of a stationary axisymmetric spacetime is
given by theLewis-Papapetrou metric (Binni et al. 2005) where the oscillation frequency of the disk is given by
Eq.(40) in theSchwarzschild case, and by Eq.(45) in Kerr
ds2=e2(γ−ψ) dρ2+dz2 +ρ2e−2ψdφ2 e2ψ(cdt ωdφ)2,(41) case, respectively. The random force ξz(t) has a zero mean
− −
and a variance ξz(t)ξz(t′) = D/c2 δ(t t′), where
where γ and ψ(cid:0) are funct(cid:1)ions of z and ρ only. The metric h i −
δ(t t′)istheDiracdeltafunctionandD>0.Byintroduc-
functionsgeneratingtheKerrblackholesolutioninaxisym- ingt−hevelocityδVz =dδz/δtofthe(cid:0)vertic(cid:1)aldiskoscillation,
metric form are given by
Eq.(50) can be written in the form
ψ= 1ln(R++R−)2−4M2+ a2/σ2 (R+−R−)2, (42) dδz=δVzdt, (51)
2 (R +R +2M)2+(a2/σ2)(R R )2
+ − (cid:0) (cid:1) +− − dδVz = cνδVzdt c2ω2 (ρ,M)δzdt+D1/2dW(t), (52)
1 (R++R−)2 4M2+ a2/σ2 (R+ R−)2 − − ⊥
γ = ln − − , (43)
2 4R R where W(t)is a collection of standard Wiener processes.
+(cid:0)− (cid:1) A Wiener process W(t)>0, is a one-parameter family
aM (R++R−+2M) (R+ R−)2 4σ2 of Gaussian random variables, with expectations zero and
ω= − − , (44)
− σ2 (R +R )2 4M2+(a2/σ2)(R R )2 covariances E(W(s)W(t)) = min s,t . Because the W(t)
+ − − (cid:2) +− −(cid:3) { }
areallGaussian,thisinformationsufficestodeterminejoint
where M and a are the mass and the specific angular mo-
probabilities. Alternatively, W(t) may be viewed as a ’ran-
mentum of the compact object, R = ρ2+(z σ)2and
± ± dom’continuous function with W(0)=0. A Wiener process
σ = √M2 a2. The usual form of the Kerr line ele-
− p maybegeneratedatconsecutivegridpointstnbyW(0)=0,
ment in Boyer-Lindquist coordinates (t,r,θ,φ) is recov-
W(t )=W(t )+(t t )Z ,and Z isasetofin-
n n−1 n n−1 n n
ered by performing the coordinates transformation ρ = − { }
dependentstandardGaussian randomvariables(withmean
√r2 2Mr+a2sinθ and z=(r M)cosθ, respectively.
− − 0 and variance 1) (Skeel& Izaguirre 2002).
ThefrequencyofthediskoscillationsintheKerrgeom-
In order to numerically integrate Eqs. (51) and (52)
etry is given by
we consider the Bru¨nger - Brooks - Karplus (BBK)
ω2 (ρ,M,a)= M(̟1a+̟3), (45) scheme (Bru¨nger et al. 1984). If the time step is taken
⊥ ̟2a+̟4 asc2∆ωt2,(wρ,eMd,ean)oδtze ∆δztn2.≈Thδez(rne∆cutr)r,enanced rweelatdioefinneforfnth=e
where we havedenoted − ⊥ n
BBK integrator is
̟ = 4M3/2R δ3δ13/2
1 σ − + × 2 1 g/2 1
8M(2M +Rσ)δ++2(5M +3Rσ)ρ2 , (46) δzn+1 = 1+g/2δzn− 1+−g/2δzn−1+ 1+g/2fn+
̟2 =−8M(cid:2)3/2Rσδ−3δ+19/2(4Mδ++2Rσδ++ρ2)(cid:3), (47) 1+1g/2 D∆t3 1/2Zn, (53)
(cid:0) (cid:1)
(cid:13)c 2002RAS,MNRAS000,1–??
6 T. Harko and G. Mocanu
wherewehavedenotedg=cν∆t.Thestartingprocedureof
theBBK integrator is given by
g 1 D∆t3 1/2
δz =δz + 1 V ∆t+ f + Z , (54)
1 0 − 2 0 2 0 4 0
(cid:16) (cid:17) (cid:18) (cid:19)
where δz and V are the initial displacement and velocity,
0 0
respectively. The velocity δVz can be obtained as
δz δz
δVz = n− n−1. (55)
n ∆t
By defining
1 dδz 2 1
E = + c2ω2 (ρ,M,a)δz2, (56)
2 dt 2 ⊥
(cid:16) (cid:17)
as the total energy per unit mass of the oscillating disk,
theluminosityperunitmassLofthedisk,representingthe
energylostbythediskduetoviscousdissipationandtothe
presence of the random force, is given by
dE dδz 2 dδz
L= =cν c2 ξz(t). (57)
− dt dt − dt
(cid:16) (cid:17)
7.1 The Schwarzschild stochastic disk
In the case of an accretion disk in the static Schwarzschild
geometry of a black hole, the proper frequency of the disk
oscillationsω2 (M,ρ)isgivenbyEq.(40).Byexpressingthe
⊥
radial distance ρ in terms of the gravitational radius M so
that ρ=nM, n=constant>6, we obtain
Figure 1. Variation of the dimensionless coordinate δZ of the
ω2 = 1 . (58) stochasticallyoscillatingSchwarzschilddiskasafunctionofθfor
⊥ M2 (n2 2)√1+n2 2 acentralobjectwithmassM =M10M⊙andζ=100(redcurve),
− − ζ=2.5×100(bluecurve),andζ=5×100(greencurve).Inthe
By int(cid:2)roducing aset of dim(cid:3)ensionless variables (θ,δZ), upperFiguren=7,whileinthelowerFiguren→∞.
defined as t=(M/c)θ and δz =MδZ, respectively, and by
denoting the product νM as ζ, the equation of motion of
aBrownian-typeequationofmotion,ofavolumeelementof
thestochastic oscillating disk becomes
thematterinthedisk.ForsuchasituationweexpectaPSD
d2δZ +ζdδZ + 1 δZ =ηz(θ), (59) slope of −2, and this result in fact represents a consistency
dθ2 dθ (n2 2)√1+n2 2 check of thestochastic simulations.
− −
where ηz(θ)=Mξz(t).
The dimensionless luminosity Lθ can be written as 7.2 The Kerr stochastic disk
L =ζ dδZ 2 dδZηz(θ), (60) By denoting ρ = nM, a = kM and δ = M√n2+1 k2,
θ dθ − dθ with n > 0 and k [0,1], the oscillation frequency o−f the
(cid:16) (cid:17) ∈
where L =ML/c3. Kerrdisk, given byEq. (45),can be represented as
θ
The variations of the dimensionless coordinate δZ, of ω2(n,k)
thestochasticvelocityofthedisk,andtheluminosityofthe ω⊥2 = M21ω2(n,k), (61)
stochastically perturbed disk are represented, for a central 2
object with mass M =1010M , for n=7 and n , and where
⊙
→∞
for several values of ζ, in Figs. 1 - 3, respectively. ω2(n,k) = 3n4+n2√1+δ
Inordertodescribethestochasticcharacteristicsofthe 1 ×
2k(5+3δ)+3√1+δ 6+3δ+δ2
physical parameters of the oscillating disk from a quantita- −
tive point of view we compute the Power Spectral Distri- (cid:2)2(1+δ)3/2 4k(2+(cid:0) δ)+ (cid:1)(cid:3)
bution(PSD)oftheluminosity (Vaughan2010)(fordetails × −
see the Appendix). The numerical values of the slopes of √1+δ 8+h3δ2+δ3 , (62)
the PSD give an insight into the nature of the mechanism −
leadingtotheobservedvariability.Thecomputationsofthe and (cid:0) (cid:1)i
PSD have been done by using the .R software and the as-
wsuhmicphtiwonasHap:pPlie(df)to=tβhfe−tαim,ewsheerrieesαobantadinβedarbeycuosnisntgantthse, ω22(n,k) = (1+δ) n2−δ(1+δ)(3+δ) 2×(δ−
BBK integrator. The results for the Schwarzschild case are (cid:2) (cid:3)
showninFig.4.Inthecaseconsideredinthepresentpaper, k+(1+δ)3/2 2 4(1+δ)+n2(3+δ)
+
theluminosity is determined bythe evolution, according to [n2 δ(1+δ)(3+δ)]2
(cid:2) − (cid:3) (cid:2) (cid:3)
(cid:13)c 2002RAS,MNRAS000,1–??
Stochastic oscillations of general relativistic disks 7
Figure 2. Variation of the dimensionless velocity Vd = dδZ/δθ Figure 3. Dimensionless luminosity Lθ of the stochastically
of the stochastically oscillating Schwarzschild disk as a function oscillating Schwarzschild disk for a central object with mass
ofθforacentralobjectwithmassM =M10M⊙ andforζ=100 M = M10M⊙ and ζ = 100 (red curve), ζ = 2.5×100 (blue
(red curve), ζ =2.5×100 (blue curve), and ζ = 5×100 (green curve),andζ=5×100(greencurve),respectively. Intheupper
curve),respectively.IntheupperFiguren=7,whileinthelower Figuren=7,whileinthelowerFiguren→∞.
Figuren→∞.
4k k+(1+δ)3/2
1 , (63)
n2 δ(1+δ)(3+δ) −
−(cid:2) (cid:3) )
respectively.
Then the vertical dimensionless equation of motion of
thestochastically oscillating Kerrdisk is given by
d2δZ δZ ω2(n,k)
+ζ + 1 δZ =ηz(θ). (64)
dθ2 dθ ω2(n,k)
2
In Figs. 5-7 we have represented the variations of the
dimensionless coordinate δZ, of the velocity V = dδZ/dθ,
d
and of the luminosity for a stochastically oscillating Kerr
disk for a rotating massive central object with mass M =
1010M and a = 0.9M, for n = 7, and for different values
⊙
of ζ.
In Fig. 8 we show the PSD of the luminosity of the
stochastically oscillatingdiskaroundaKerrblackholewith
mass M =1010M and a=0.9M, respectively.
⊙
Figure4.PSDoftheluminosityofthestochasticallyoscillating
Schwarzschild disk for a central object of mass M = M10M⊙.
8 DISCUSSIONS AND FINAL REMARKS In the upper Figure n = 7 and α = 2.076. In the lower Figure
n→∞andα=2.070.Theredlinesrepresentthefitsaccording
In the present paper we have introduced a mathematical toH.
model for the description of the perturbations of a thin ac-
cretion disk in contact with an exterior stochastic medium
(athermalbath).Asaresult oftheinteractionbetweenthe
mediumandthedisk,themotionoftheparticlesinthedisk
has essentially a stochastic nature. To obtain theequations
(cid:13)c 2002RAS,MNRAS000,1–??
8 T. Harko and G. Mocanu
Figure 8. PSD of the luminosityfor a stochastically oscillating
diskaroundaKerrblackholewithmassM =M10M⊙ anda=
0.9M,forn=7.TheslopeofthePSDis2.046.
Figure 5. Variation of the dimensionless coordinate δZ of the ofmotionoftheperturbeddiskwehavegeneralized theap-
stochasticallyoscillatingdiskasafunctionofθforarotatingKerr
proachintroducedinShirokov(1973)andSemerak & Zacek
black hole with mass M = M10M⊙ and a = 0.9M, for n = 7, (2000), by introducing in the equations of the perturbed
and for different values of ζ: ζ =100 (red curve), ζ =2.5×100
geodesic lines a dissipative viscous term, and a stochastic
(bluecurve), andζ=5×100(greencurve),respectively.
force term,respectively.Theequationsofthemotion of the
fluctuating disk have been obtained in both the equatorial
and the vertical plane, as well as the general stochastic en-
ergytransferequation.Wehavestudiedindetailthevertical
oscillations ofthedisk,andwehaveshownthatinthiscase
themotion isdescribedbyastandard Langevintypediffer-
ential equation for an harmonic oscillator, with a geometry
dependentoscillation frequency.By numerically integrating
the equation of motion we have obtained the vertical dis-
placements, velocities and the luminosity of the disk in the
cases of both Schwarzschild and Kerrgeometries.
Fluctuations of accretion disks around super-massive
black holes may have important astrophysical applications.
The emission of Galactic Black Hole Binaries (BHBs) and
Active Galactic Nuclei (AGN) displays a significant aperi-
odicvariabilityonabroadrangeoftime-scales.ThePower
Figure6.VariationofthedimensionlessvelocityVd=dδZ/dθof Spectral Density of such variability is generally modeled
thestochasticallyoscillatingdiskasafunctionofθforarotating with a power law, P(f) f−α, where power-law index α
KerrblackholewithmassM =M10M⊙anda=0.9M,forn=7, keeps a constant value in∝a certain range of f, but changes
and for different values of ζ: ζ =100 (red curve), ζ =2.5×100
among different ranges. At high frequencies, the PSDs of
(bluecurve), andζ=5×100(greencurve),respectively.
bothBHBsandAGNspresentasteepslopewithα 2.On
∼
thecontrary,belowabreakfrequency,typicallyatafewHz
for BHBs, they flatten to a slope with α 1, representing
∼
the so-called flicker noise (King et al. 2004). However, the
observed power spectra often deviate from the form f−1.
Forexample,thePSDofCygX-1iswelldescribedwiththe
form f−1 in the soft state, but in the hard state it exhibits
theformf−1.3 (Gilfanov 2010).Inparticular,itwasshown
that the power-law index is around α = 0.8 1.3 both in
−
thesoft stateofBHBsandinnarrow-lineSeyfert1galaxies
(Janiuk & Czerny 2007).
Inthecase ofthepresentmodel ofthediskoscillations
undertheeffect of somestochastic forces ξ(t),theresulting
Power SpectralDensity isof theform P(f) f−2. Thisre-
∝
sult can beunderstood qualitatively bynotingthatP(f)
∝
exp(ift) tξ(t′)dt′ dt if−1 texp(ift)ξ(t)dt, which
oFsicgilulartein7g.dVisakriaastioanfuonfctthioenluomf θinfoosritayrLoθtaotifngthKeesrtrocbhlaacstkichaolllye Rgives P hfR−2, since ξi(t) i∝s a stocRhastic variable. This re-
∝
with mass M = M10M⊙ and a = 0.9M, for n = 7, and for sult is generally also valid for harmonically oscillating, un-
different values of ζ: ζ = 100 (red curve), ζ = 2.5×100 (blue dampeddisks,andconsequentlyitisindependentofthegen-
curve),andζ=5×100(greencurve),respectively. eral relativistic corrections to theequation of motion, since
these only change the oscillation frequency of the compact
object - disk system. However, as pointed out previously,
observationsofBHBsandAGNshaveshownthatthespec-
(cid:13)c 2002RAS,MNRAS000,1–??
Stochastic oscillations of general relativistic disks 9
tral index is not α = 2, but it is either α 1, or, more ACKNOWLEDGMENTS
≈
generally, α (0.8,2). Such a dispersion of the power-law
∈ We would like to thank to the anonymous referee for com-
indexrevealsthattheremustexistsomeothermechanisms,
ments and suggestions that helped us to significantly im-
different from purely stochastic oscillations, which may be
prove the manuscript. The work by T. H. was supported
responsible for the observed value of α, like, for example,
by a grant from the Research Grants Council of the Hong
hydrodynamic fluctuations, magnetohydrodynamic turbu-
KongSpecialAdministrativeRegion,China.G.M.acknowl-
lence,magneticflares,densityfluctuationsinthecorona,or
edgesthefinancialsupportoftheSectoralOperationalPro-
variations of the accretion rate, caused by small amplitude
gramme for Human Resources Development 2007-2013, co-
variations in theviscosity (Kinget al. 2004).
financed by the European Social Fund, under the project
numberPOSDRU/107/1.5/S/76841 withthetitle”Modern
In amodel introduced in Takeuchi& Mineshige (1997)
Doctoral Studies: Internationalization and Interdisciplinar-
aperiodic X-ray fluctuations are thought to originate from
ity”. G. M. would like to thank Vienna University of Tech-
instabilitiesoftheaccretiondiskaroundablackhole.Tode-
nology, Institute of Theoretical Physics, for its hospitality
scribethesetypeoffluctuationsacellularautomatonmodel
duringthetime when most of this work was written.
was proposed (Mineshige et al. 1994). In this model, a gas
particle is randomly injected into an accretion disk sur-
rounding a black hole. When the mass density of the disk
exceedssome critical valueat acertain point,an instability
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APPENDIX
In statistical analysis, if X is some fluctuating quantity,
withmean µandvarianceσ2,thenacorrelation function
for quantityX is definedas
(X µ)(X µ)
R(τ)= h s− s+τ − i.
σ2
whereX isthevalueofX measuredattimesand de-
s
hi
notesaveragingoverallvaluess.Basedonthecorrelation
functionthePowerSpectralDistribution(PSD)isdefined
as
+∞
P(f)= R(τ)e−ı2πfτ.
Z−∞
It is straightforward to see the importance of the PSD
in terms of the ”memory” of a given process. For exam-
ple, if X is the measured luminosity of a perturbed disk,
the slope of the PSD of a time series of X provides an
insightintothedegreeofcorrelationtheunderlyingphys-
icalprocesseshavewiththemselves.Thesystemneedsad-
ditional energy to fluctuate, and this mechanism is best
explainedfortheBrownian motion,in whichcasetheen-
ergy is thermal.Brownian motion produces aPSD of the
form P(f) f−2.
∼
However, assessment of the PSD slope from one single
observational time-series is non-trivial. In such cases it is
veryhelpfultousededicated software, such asthestatis-
tical software .R (Vaughan 2010). Basically, the input to
this software consists of one time series. Any assumption
aboutthebehaviourofthesourcethatproducedthistime
(cid:13)c 2002RAS,MNRAS000,1–??