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

http://www.astro.cornell.edu/academics/courses/astro101/herter/lectures/lec28.htm

e.g. AGN

4h

Theory cannot explain the fast variability

Or the Power Spectral Distribution

)2,0(,)( ffP

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 0716+714

Power spectral distribution (PSD)

2

)(

sss XX

R

correlation function of the (stochastic) process;

not accessible, but interesting

deRfS fi2)()(

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)

outRh

out

in

R

RG dr

zr

rrGMzzF

2

322 )(

)(2)(

0)(2

2

zFdt

zdM Gd

,0202

2

zdt

zd

adj

in

adj

out

adjino R

R

R

R

RR

GM

11

)2(2

22

)sin()( 0tAtz

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 processDunkel & Hanggi (2005a, 2005b, 2009)

This approach is conceptually correct

What we did

22222 2 dzdgdgcdtdgcdtgds ttt

zct ,,, Rotating axisymmetric compact GR object

Choose a family of observers moving with velocity

nnu / n – particle number density

0; n

1uu

.const.constz .const

Approximations

0,,0,1c

uu t

2//2

1

cgcggu

ttt

t

fmds

dx

ds

dx

ds

xd 1'''

'2

2

Unperturbed equatorial orbit

What is this ?

02

2

ds

dx

ds

dx

ds

xd

Perturbed orbit

xxx '

e.o.m. for displacement

fx

ds

dx

ds

dx

ds

xd

ds

dx

ds

xd ,2

2

2

f

UVVVmf fr

ds

xdV

U

4 - velocity of the perturbation

4 - velocity of the heat bath

xg;

0; xg

yxgDygxg ;;;;

Gaussian stochastic vector field,

rapidly varying

Friction, slowly varying

noise kernel tensor

0,0,0,2/1 ttgU

zgzds

zd

ds

zd z ;22

2

Equation of motion

22

,,,2 2 tz

zzzt

zztt u

cc

Vertical oscillations of the disk

Proper frequency for vertical oscillations;

metric dependent

Dynamical friction Stochastic interaction

Velocity of the perturbed disk is small cdtds )(t

Assumptions

Simulations – the equations

tczaMcdt

zdc

dt

zd z 2222

2

,,

''2

ttc

Dtt zz

dtVzd z

)(,, 2/122 tdWDzdtaMcdtVcVd zz

collection of standard Wiener processes

Simulations – BBK integrator Brunger et al. (1984)

nnnnn ZtDg

fg

zg

gz

gz

2/1311 2/1

1

2/1

1

2/1

2/1

2/1

2

tnzzn

222 ,, tzMacf nn

tcg

0

2/13

0001 42

1

21 Z

tDftV

gzz

t

zzV nnzn

1

Z, Normal Gaussian variable

2222

,,2

1

2

1zaMc

dt

zdE

tdt

zdc

dt

zdc

dt

dEL z 2

2

Luminosity of a stochastically perturbed disk

Total energy per unit mass of a stochastically perturbed oscillating disk

Simulations – dimensionless variables

6, nnM

M

ct

M

zZ

M

dZdVd /

tM zz

3/ cMLL

Observational data x(t)

.R softwareαfβ=P(f)H :

βα,y

Observed x(t)

Vaughan (2010)

pB

input

output

H)dy,x|p(yy=H,x|yE

gr6

BBK integrator

ζ=100ζ=250ζ=500

Vertical displacementPerturbation velocity

Schwarzschild BH

M1010

gr6

M1010

.R software, bayes.R scriptBBK integrator

ζ=100ζ=250ζ=500

Schwarzschild BHLuminosity

PSD of luminosity

gr6

M1010

076.2

07.2

Kerr BH

0 2000 4000 6000 8000 10 000 12 000

2

1

0

1

2

3

4

Z

0 2000 4000 6000 8000 10 000 12 000

0 .5

0 .0

0 .5

1 .0

V d

ζ=100ζ=250ζ=500

gr6

Vertical displacement

Perturbation velocity

M1010

a=0.9

ζ=100ζ=250ζ=500

gr6

Kerr BH

0 2000 4000 6000 8000 10 000 12 000

0

20

40

60

80

100

120

L

Luminosity

PSD of luminosity

M1010

046.2

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

ConclusionsTested 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?