Modelling the IDV Emissions of the BL Lac Objects with a Langevin Type Stochastic Differential Equation

Abstract

In this paper, we introduce a simplified model for explaining the observations of optical intra-day variability (IDV) of the
BL Lac Objects. We assume that the source of the IDV are the stochastic oscillations of an accretion disk around a supermassive
black hole. The stochastic fluctuations on the vertical direction of the accretion disk are described by using a Langevin
type equation with a damping term and a random, white noise type force. Furthermore, preliminary numerical simulation results
are presented, which are based on the numerical analysis of the Langevin stochastic differential equation.

KeywordsLangevin type stochastic differential equation–BL Lac objects

Full text

Page 1

arXiv:1103.0362v1 [astro-ph.HE] 2 Mar 2011
J. Astrophys. Astr. (0000) 00, 000–000
Modeling the IDV emissions of the BL Lac Objects with a
Langevin type stochastic differential equation
C. S. Leung1∗, J. Y. Wei1†, T. Harko2‡, Z. Kovacs2§
1National Astronomical Observatories, Chinese Academy of Sciences
20A Datun Road, Chaoyang District, Beijing, China
2Department of Physics and Center for Theoretical and Computational Physics,
The University of Hong Kong, Pok Fu Lam Road, Hong Kong, PR China
2011 March 3
Abstract. In this paper, we introduce a simplified model for explain-
ing the observations of the optical intraday variability (IDV) of the BL
Lac Objects. We assume that the source of the IDV are the stochas-
tic oscillations of an accretion disk around a supermassive black hole.
The Stochastic Fluctuations on the vertical direction of the accretion
disk are described by using a Langevin type equation with a damping
term and a random, white noise type force. Furthermore, the prelim-
inary numerical simulation results are presented, which are based on
the numerical analysis of the Langevin stochastic differential equation.
Key words:
Objects
Langevin type stochastic differential equation, BL Lac
1.Introduction
Intraday variability is usually defined asthe variation of thedifferential photometric
magnitude greater than 3 times of sigma value. However, if the variation is greater
than 5 times of sigma value is the best and reliable (Wagner and Witzel 1995 ;
Fan, J. H. 2005 ; Gupta et al. 2008). There are many observations for the intraday
variation in difference bands, and to explain the observational data many theoretical
models have also been proposed for explaining these interesting phenomena. A
cellular automaton model was introduced initially by Kawaguchi et al. (1998), and
further developed in Tang (1999).
It is the purpose of the present paper to propose an alternative model for the
∗E-mail:astrosinghk@yahoo.com.hk
†Email:wjy@bao.ac.cn
‡E-mail:harko@hkucc.hku.hk
§E-mail:zkovacs@hku.hk
1

Page 2

2
explanation of the observed intra-day variability in BL Lac objects and for other
similar transient events. The basic physical idea of our model is that the source of
the intraday variability can be related to some stochastic oscillations of the disk,
triggered by the interaction of the disk with the central supermassive black hole,
as well as with a background cosmic environment, which perturbs the disk. To
explain the observed light curve behavior, we develop a model for the stochastic
oscillations of the disk, by taking into account the gravitational interaction with the
central object, the viscous type damping forces generated in the disk, and a stochas-
tic component which describes the interaction with the cosmic environment. The
mathematical model is formulated in terms of a stochastic, Langevin type differ-
ential equation, describing the stochastic oscillation of the accretion disk. This
stochastic oscillation model can reproduce the aperiodic light curves associated
with transient astronomical phenomena. Usually the IDV phenomena is explained
by using the jet model. There are definitely interactions between the jet and the
accretion disk. The stochastic fluctuations of the disk can influence the jet through
some energy transfer, and they can represent a source for the jet perturbations.
2. Vertical thin disk undamped oscillations
We consider the oscillations of the disk as a whole body under the influence of
gravity. The disk is considered thin in the sense discussed by Shakura and Sun-
yaev (1973). We assume that at the center of the disk we have a compact object
of mass M. We approximate the distribution of the surface density in the geo-
metrically thin disk by the formulae Σ = Σ0= constant,Rin≤ r ≤ Radj, and
?
Shakura and Sunyaev 1976; Titarchuk and Osherovich 2000), where Rinis the in-
nermost radius of the disk, Radjis an adjustment radius in the disk, and Routis the
outer radius of the disk. The index γ of the surface density can be either 3/5 or
3/4.
The vertical oscillations of the disk as a whole can be described by the equation
of motion Mdd2z/dt2+ FG(z) = 0, or, equivalently,
Σ = Σ0
r
Radj
?−γ,Radj≤ r ≤ Rout, respectively, (Shakura and Sunyaev 1973;
d2z
dt2+ ω2
0z = 0,
(1)
where
ω2
0= 4π2ν2
0=(2 − γ)GM
RinR2
adj
?
Rout
Radj
?γ−2?
1 −
γ
γ + 1
Rin
Radj
?
.
(2)
3. Stochastic fluctuations of accretion disks
In the following, we consider a model in which we assume that the accretion disk
around a massive central object is immersed in a fluctuating fluid flow. For sim-
plicity describe the accretion disk as a macroscopic sphere of radius a. In order

Page 3

3
to describe the stochastic processes in oscillating disks, we replace the equation of
motion of the disk, given by Eq. (1), with a stochastic Langevin type differential
equation, which can be written as
d2z
dt2+ ξdz
dt+ ω2
0z =F(t)
Md
,
(3)
where the term ξdz/dt, ξ =constant, takes into account viscous dissipation. The
fluctuating force F(t) is independent of z, and it varies extremely rapidly as com-
pared to z. Since F(t) is very irregular, we assume that its mean value is zero,
?F(t)? = 0.
By defining
E =Md
2dt
as the total energy of the oscillating disk, the luminosity of the disk, representing
the energy lost by the disk due to viscous dissipation and to the presence of the
random force, is given by
?dz
?2
+1
2Mdω2
0z2− zF(t),
(4)
dE
dt
= −L = 6πηH
?dz
dt
?2
+ zdF
dt.
(5)
As an example of application of this model to a concrete astrophysical sys-
tem let’s consider the case of a thin disk around a compact general relativistic ob-
ject of mass M = 106M⊙. We assume that the adjustment radius of the disk is
Radj= 75Rin, while the outer radius of the disk is located at Rout= 250Rin. By
assuming γ = 3/5 we obtain first from Eq. (2) the frequency of oscillations of the
unperturbed disk as ω2
of the disk in the absence of any dissipative or random processes is T = 18.739
hours. For the mass of the disk we take Md= 1.58 × 1034g = 7.9M⊙. For the
dumping coefficient ξ we obtain ξ = 6πHη/Md= 2.65×10−6s−1, where for the
viscosity coefficient we have assumed the value η = 1013erg s/cm3. Therefore the
equation of motion of the stochastically oscillating disk is given by
0= 1.89 × 10−8s−2. The corresponding oscillation period
d2z
dt2+ 2.65 × 10−6dz
dt+ 1.89 × 10−8z = 6.329 × 10−35F(t).
(6)
The results of the numerical integration of Eq. (3) are represented in Figs. 1 and
2.
4.Conclusions
We have developed a theoretical model describing the stochastic oscillations of the
disk. A comparison of the model predictions with the observational data of IDV
from different sources will be done in a forthcoming paper. For the case of the
ADAF disks, we can also apply the Langevin equation in the vertical direction,

Page 4

4
-50
-40
-30
-20
-10
0
10
20
30
40
0 0.5 1 1.5
t [104 s]
2 2.5 3 3.5
F(t)/MD [cm/s2]
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.5 1 1.5
t [104 s]
2 2.5 3 3.5
z(t) [106 cm]
Figure1. ThestochasticforceF(t)/Mdcm/s2(leftfigure)andz (rightfigure)asa function
of time.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3 3.5
vz(t) [103 cm/s]
t [104 s]
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5
t [104 s]
2 2.5 3 3.5
L(t) [1038 erg/s]
Figure 2. The stochastic velocity (left figure) and the luminosity of the disk (right figure)
as a function of time.
since this type of disks also has some vertical oscillation modes. Although the
geometry of the ADAF disk is geometrically thick and optically thin, it can also
provide a source for the disturbances of the jet.The effect of the jets in the frame-
work of the present model and the effects of the relativistic corrections to the model
will be also analyzed.
References
Fan J. H. 2005, Chin. J. Astron. Astrophys., S5, 213.
Gupta A. C., Fan J. H., Bai J. M. and Wagner S. J. 2008, Astrophys. J., 135, 1384.
Kawaguchi T., Mineshige S., Umemura M., and Turner, E. L. 1998, Astrophys.J, 504, 671.
Shakura N. I. and Sunyaev R. A. 1973, Astron. Astrophys, 24, 33.
Shakura N. I. and Sunyaev R. A. 1976, Mon. Not. R. Astron. Soc., 175, 613.
Tang W. S. 1999,Study of power spectrumfluctuationin accretiondisc by cellular automa-
ton, HKU-e-thesis
Titarchuk, L. and Osherovich, V. 2000, Astrophys. J., 542, L111
Wagner S. J. and Witzel A. 1995, ARA&A, 33, 167.