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Stochastic oscillations of general relativistic disks

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

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Stochastic oscillations of general relativistic disks

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  1. 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)

  2. Object of study thin accretion disks around compact astrophysical objects which are in contact with the surrounding medium through non-gravitational forces http://www.astro.cornell.edu/academics/courses/astro101/herter/lectures/lec28.htm e.g. AGN

  3. 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 4h GM, A. Marcu – accepted in Astronomische Nachrichten Theory cannot explain the fast variability BL Lac 0716+714 Or the Power Spectral Distribution

  4. 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

  5. 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 • Numerical solution to the eom for displacement, velocity, luminosity (LC) • implement the BBK integrator (Brunger et al. 1984) • Determine the PSD of the LC • use .R software (Vaughan 2010)

  6. 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)

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

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

  9. 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

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

  11. Approximations

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

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

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

  15. Simulations – the equations collection of standard Wiener processes

  16. Simulations – BBK integratorBrunger et al. (1984) Z, Normal Gaussian variable

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

  18. Simulations – dimensionless variables

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

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

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

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

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

  24. Conclusions Tested the effect of a heath bath on vertical oscillations of accretion disks • 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 • 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

  25. 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.

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

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