Stochastic oscillations of general relativistic disks Gabriela Mocanu Babes-Bolyai University, Romania Stochastic oscillations of general relativistic disks, Tibor Harko, GM, accepted in MNRAS, (this morning)

Object of study thin accretion disks around compact astrophysical objects which are in contact with the surrounding medium through non-gravitational forces s/astro101/herter/lectures/lec28.htm e.g. AGN

4h Theory cannot explain the fast variability Or the Power Spectral Distribution GM, A. Marcu – accepted in Astronomische Nachrichten Purpose estimate the effect of this interaction on the luminosity of a GR disk temporal behaviour (Light Curve - LC) Power Spectral Distribution (PSD) of the LC BL Lac

Power spectral distribution (PSD) correlation function of the (stochastic) process; not accessible, but interesting power spectral distribution; accessible Importance of time lag in the analyzed observational time-series

Means Analytical derivation of the GR equation of motion (eom) of a vertically displaced plasma element. Displacement occurs as a consequence of a stochastic perturbation Brownian motion framework; Langevin-type equation Determine the PSD of the LC use.R software (Vaughan 2010) Numerical solution to the eom for displacement, velocity, luminosity (LC) implement the BBK integrator (Brunger et al. 1984)

Schematic picture illustrating the idea of disk oscillation. The disk as a whole body oscillates under the influence of the gravity of the central source. (Newtonian approx)

We assume the disk as a whole is perturbed - restoring force The equation of motion for the vertical oscillations Surface mass-density in the disk; model dependent

Chaterjee et al. (2002) Massive point like object Why is this approach valid? Slowly varying influence of the stellar aggregate Rapidly fluctuating stochastic force <- discrete encounters with individual stars independent Potential of aggregate distribution Dynamical friction (Newtonian approx) Brownian motion framework; Langevin-type equation

Analytical proof that for a Plummer stellar distribution the motion of the massive particle is a Brownian motion (Chaterjee et al. 2002) Numerical simulations compared to N-Body simulations (Chaterjee et al. 2002) A correct theory of relativistic Brownian motion may be constructed a covariant stochastic differential equation to describe Brownian motion a phase space distribution function for the diffusion process Dunkel & Hanggi (2005a, 2005b, 2009) This approach is conceptually correct

What we did Rotating axisymmetric compact GR object Choose a family of observers moving with velocity n – particle number density

Approximations

Unperturbed equatorial orbit What is this ? Perturbed orbit e.o.m. for displacement

4 - velocity of the perturbation 4 - velocity of the heat bath Gaussian stochastic vector field, rapidly varying Friction, slowly varying noise kernel tensor

Equation of motion Vertical oscillations of the disk Proper frequency for vertical oscillations; metric dependent Dynamical frictionStochastic interaction Velocity of the perturbed disk is small Assumptions

Simulations – the equations collection of standard Wiener processes

Simulations – BBK integrator Brunger et al. (1984) Z, Normal Gaussian variable

Luminosity of a stochastically perturbed disk Total energy per unit mass of a stochastically perturbed oscillating disk

Simulations – dimensionless variables

Observational data x(t).R software Observed x(t) Vaughan (2010) input output

BBK integrator ζ=100 ζ=250 ζ=500 Vertical displacement Perturbation velocity Schwarzschild BH

.R software, bayes.R scriptBBK integrator ζ=100 ζ=250 ζ=500 Schwarzschild BH Luminosity PSD of luminosity

Kerr BH ζ=100 ζ=250 ζ=500 Vertical displacement Perturbation velocity a=0.9

ζ=100 ζ=250 ζ=500 Kerr BH Luminosity PSD of luminosity a=0.9

Numerical solution to the e.o.m. for displacement, velocity, luminosity (LC) implemented the BBK integrator Determined the PSD of the LC used.R software Analytical derivation of the equation of motion (eom) of a vertically displaced plasma element. Displacement occurs as a consequence of a stochastic perturbation Brownian motion framework; Langevin-type equation Conclusions Tested the effect of a heath bath on vertical oscillations of accretion disks

We obtained a PSD with spectral slope very close to -2 consistency check of the proposed algorithm In this framework: the amplitude of the luminosity and the PSD slope do not depend sensibly on rotation The amplitude of oscillations is larger for smaller friction closer to the horizon.

Future work? Radial oscillations – tricky problem of angular momentum transfer What does an ordered/disordered Magnetic field do to the PSD?