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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?