Article

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

Journal of Astrophysics and Astronomy (Impact Factor: 0.71). 06/2011; 32(1):189-192. DOI: 10.1007/s12036-011-9026-3

Source: arXiv

**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

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

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**ABSTRACT:**Fractals are a basic tool to phenomenologically describe natural objects having a high degree of temporal or spatial variability. From a physical point of view the fractal properties of natural systems can also be interpreted by using the standard formalism of thermodynamical fluctuations. In the present paper we introduce and analyze the fractal dimension of the Intra - Day- Variability (IDV) light-curves of the BL Lac objects, in the optical, radio and X-ray bands, respectively. A general description of the fluctuation spectrum of these systems based on general thermodynamical principles is also proposed. Based on the general fractal properties of a given physical system, we also introduce the predictability index for the IDV light curves. We have explicitly determined the fractal dimension for the R-band observations of five blazars, as well as for the radio band observations of the compact extragalactic radio source J 1128+5925, and of several X-ray sources. Our results show that the fractal dimension of the optical and X-ray observations indicates an almost pure "Brownian noise" (random walk) spectrum, with a very low predictability index, while in the radio band the predictability index is much higher. We have also studied the spectral properties of the IDV light curves, and we have shown that their spectral index is very closely correlated with the corresponding fractal dimension. - [Show abstract] [Hide abstract]

**ABSTRACT:**We analyze the general relativistic oscillations of thin accretion disks around compact astrophysical objects interacting with the surrounding medium through non-gravitational forces. The interaction with the external medium (a thermal bath) is modeled via a friction force, and a random force, respectively. The general equations describing the stochastically perturbed disks are derived by considering the perturbations of trajectories of the test particles in equatorial orbits, assumed to move along 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 can be formulated in terms of a general relativistic Langevin equation. The stochastic energy transport equation is also obtained. The vertical oscillations of the disks in the Schwarzschild and Kerr geometries are considered in detail, and they are analyzed by numerically integrating the corresponding Langevin equations. The vertical displacements, velocities and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. - [Show abstract] [Hide abstract]

**ABSTRACT:**Observational data in the BVRI bands of the variable BL Lacertae Object S5 0716+714 is discussed from the point of view of its Power Spectral Distribution (PSD). A model of the type $P(f) = \beta f^{-1} [1 + (\frac{f} {\delta}) ^{\alpha -1}]^{-1} + \gamma $ is fitted to the data for four null hypothesis and the Bayesian $p$ parameter for the fits is calculated. Spectral slopes with values ranging from 1.083 to 2.65 are obtained, with medium values for each band of $\bar{\alpha}_B =2.028$, $\bar{\alpha}_V = 1.809$, $\bar{\alpha}_R = 1.932$ and $\bar{\alpha}_I = 1.54$ respectively. These values confirm conclusions of previous studies, namely that the source is turbulent. Two disk models, the standard prescription of the Shakura-Sunyaev disk and magnetized disks exhibiting MagnetoRotational Instability, were discussed. We found that it is unlikely that they explain this set of observational data.