Download Chapter 5 Thin Accretion Disks

Chapter 5 Thin Accretion Disks
Chapter 5 Thin Accretion Disks
1. Disk formation with Roche lobe overflow
In a non-rotating frame, the components of stream velocity at the L1
point along and perpendicular to the instantaneous line of centers
are
1 / 3
v ~ b1 ~ 100m11 / 3 (1  q)1 / 3 Pday
kms 1 ,
and v|| cs ~10 kms-1 for normal stellar
envelope.
1
Chapter 5 Thin Accretion Disks
Implications:
(1) The transferring material has high specific angular momentum, so
that it cannot accrete directly onto the compact star.
(2) The gas stream issuing through the L1 point is supersonic, so that
pressure forces can be neglected and the stream will follow a
ballistic trajectory determined by the Roche potential.
2
Chapter 5 Thin Accretion Disks
The initial trajectory would be an elliptical orbit lying in the binary
plane.
The presence of the secondary causes the orbit to precess slowly.
The stream will therefore intersect itself, resulting in dissipation of
energy.
Meanwhile, the angular momentum is conserved. So the gas will tend
to the orbit of lowest energy for a given
angular moment, i.e. a circular orbit.
3
Chapter 5 Thin Accretion Disks
In most cases the total mass of gas in the disk is so small that we
can neglect the self-gravity of the disk. The circular orbit is then
Keplerian with angular velocity
K(R)=(GM1/R3)1/2.
The radius of this circular orbit is called the circularization radius
Rcirc, which follows from the relation
(GM1Rcirc )1 / 2  b12 .
Then
Rcirc/a = (1+q)(0.5-0.227log q)4.
4
Chapter 5 Thin Accretion Disks
Within the ring of radius of Rcirc, there will be dissipative processes,
e.g. collisions, shocks, viscous dissipation, etc.
These will convert some of the energy of the ordered bulk orbital
motion into internal energy (heat). Eventually some of this energy is
radiated and therefore lost from the gas.
The gas has to sink deeper into the gravitational potential of the
primary, orbiting it more closely. This in turn requires it to lose
angular momentum.
So most of the gas will spiral towards the primary through a series
of approximately circular orbits.
5
Chapter 5 Thin Accretion Disks
The angular momentum is transferred outwards through the disk by
viscous torques.
The outer parts of the ring will gain angular momentum and will spiral
outwards.
The original ring of matter at R = Rcirc will spread to both smaller and
larger radii by this process, to form an accretion disk.
6
Chapter 5 Thin Accretion Disks
To see it in detail, let’s write the conservation equations for the
mass and angular momentum transport in the disk due to radial drift
motion.
The mass of an annulus of (R, R+R) is
(2RR) 2RR,
where H is the disk height and is the surface density.
The angular momentum is then
2RRR2.
7
Chapter 5 Thin Accretion Disks
For the mass of the annulus,

(2RR)  vR ( R, t )2R( R, t )  vR ( R  R, t )2 ( R  R)( R  R, t )
t

 2R ( RvR )
R
In the limit R→0, we get the mass conservation equation
 
R

( RvR )  0
t R
8
Chapter 5 Thin Accretion Disks
The conservation equation of angular momentum is

(2 RRR 2)  vR ( R, t )2 R( R, t ) R 2
t
 vR ( R  R, t )2 ( R  R)( R  R, t )( R  R)2 ( R  R)
 G ( R  R)  G ( R)

G
2
 2R ( RvR R )  R
R
R
where G(R, t) 2RR2′is the viscous torque exerted by the outer
ring (here ´= d/dR, and is the viscosity).
9
Chapter 5 Thin Accretion Disks
In the limit R→0, we get


1 G
2
2
R (R ) 
( RvR R ) 
t
R
2 R .
The equation can be further simplified by use of the mass
conservation equation to be
1
G
1/ 2 
1/ 2
RvR 


3
R
(


R
)
2
2 ( R )' R
R
where we have used
( R)  K ( R)  (GM / R3 )1/ 2 .
10
Chapter 5 Thin Accretion Disks
We finally get the equation governing the time evolution of surface
density in a Keplerian disk,


3  1/ 2 

( RvR ) 
[R
(R1 / 2 )]
.
t
RR
R R
R
The radial velocity is
3 

1/ 2
vR   1 / 2
(R ) ~
R R
R
11
Chapter 5 Thin Accretion Disks
As an example, the figure shows the spreading of a ring of matter
with a Keplerian orbit at RR0, under the action of viscous torques.
The viscosity is assumed to be constant. The typical timescale of the
ring’s spreading is
tvisc ~ R / vR ~ R 2 /
12
Chapter 5 Thin Accretion Disks
Near the outer edge of the disk Rout ~ (0.70.8) RL1> Rcirc, some other
process must finally remove this angular momentum, and it is likely
that angular momentum is fed back into the binary orbit through
tides exerted by the secondary.
13
Chapter 5 Thin Accretion Disks
2. Viscous torque
How the orbital kinetic energy is converted into heat?
The differential rotation of the gas means that elements on
neighboring streamlines will slide past each other.
Because of thermal and/or turbulent motions, viscous stresses are
generated.
The angular momentum is transported by shear viscosity.
14
Chapter 5 Thin Accretion Disks
We now consider local mechanisms of angular momentum transport.
First consider a uniform gas whose streaming motion is in the
x-direction with velocity u(z).
Consider the x-momentum transport
across an arbitrary plane z = z0 due to
the exchange of fluid elements (or
molecules) between the levels z0,
with the typical turbulent scale
(or mean free path) and speed v.
15
Chapter 5 Thin Accretion Disks
There is no net mass motion across z0, so the average upward and
downward mass fluxes are the same.
During the motion there is no force acting on the turbulent elements,
so the linear momentum (and angular momentum) are conserved. The
upward (downward) moving elements carry with them the
x-momentum from a level ~z0z0.
The net upward x-momentum flux density is
vuz0uz0vu´z0
16
Chapter 5 Thin Accretion Disks
The viscous stress exerted by the gas below on the gas above is
given by
u
Txz  
~   vu '( z0 ) ,
z
where u´= du/dz and  is the dynamical viscosity.
The kinematic viscosity  is defined as , so
v.
17
Chapter 5 Thin Accretion Disks
In the case of molecular transport, and v are the mean free path
and thermal speed of the molecules respectively. In the case of
turbulent motions,  is the characteristic spatial scale of the
turbulence and v is the typical velocity of the eddies.
18
Chapter 5 Thin Accretion Disks
Now consider the similar process in a thin, differentially rotating
disk in polar coordinates (R, ). Here the problem is which should be
conserved, the angular momentum or linear momentum, when gas
elements are constantly exchanged across the surface R = const?
19
Chapter 5 Thin Accretion Disks
If the elements are not interacting with the streaming fluid, and
subject only to external forces (e.g. gravity), the appropriate
assumption is that angular momentum is conserved.
If the effects of surrounding fluid (e.g. pressure gradients) must be
included, the external forces are canceled by bulk rotation and
pressure gradients in a steady state, so stream-wise momentum is
conserved.
The example of the first case is the ring of Saturn, while the later
situation may be applied to accretion disks.
20
Chapter 5 Thin Accretion Disks
Following the same method, the net upward -momentum flux
density is
~ v~[( R   / 2)v ( R   / 2)  ( R   / 2)v ( R   / 2)]   v~R' .
-direction per unit area is
TR   (v / R  v / R)  R' ~  v~R' ,
yielding a kinematic viscosity
 ~ v~
See Hayashi and Matsuda (2001, astro-ph/0102484), Subramanian et al.
(2004, astro-ph/0403362), and Hayashi et al. (2005, Progress of
Theoretical Physics, 113, 1183) for detailed discussions on the angular
momentum transfer.
21
Chapter 5 Thin Accretion Disks
3. The magnitude of viscosity
The momentum transfer in the disk means that there is a force
acting in the -direction on the volume element due to shear
viscosity,
f ~  v
 2v
R
2
~  v
v
R2 .


Comparing this with the inertia terms  (v  )v in
equation leads to the Reynolds number
v2 / R
Rv
inertia
Re 
~

2
viscous  vv / R
v .
22
the
Euler
Chapter 5 Thin Accretion Disks
If Re1, viscous force dominates the flow; if Re1, the viscosity
is dynamically unimportant.
In the case of molecular viscosity due to Coulomb collisions between
charged particles in the ionized gas, given by d and v ~ cs,
1 / 2 5 / 2
Remol ~ 2  1014 (n / 1015 cm-3 )m11 / 2 R10
T4 ,
for a typical accretion disk in an X-ray binary.
Hence molecular viscosity is far too weak to bring about the viscous
dissipation and angular momentum required.
23
Chapter 5 Thin Accretion Disks
A number of hypotheses have been proposed to explain the much
larger effective viscosity in accretion disks. The most important of
these are:
(1) A turbulent viscosity resulting from random small-scale
turbulent fluid motions in the disk, generated by the strong shear in
the differentially rotating disk.
(2) A magnetic viscosity associated with the magnetic Lorentz
force in a disk containing magnetic fields.
(3) Nonlinear (spiral) waves or shocks in the disk.
(4) Outflows from the disk
24
Chapter 5 Thin Accretion Disks
We will discuss the first and second possibilities.
Since little is known about turbulence, the most we can do is to place
plausible limits on turb and vturb.
First, the typical size of the largest eddies cannot exceed the disk
thickness H, so turb  H.
Second, it is unlikely that the turnover velocity vturb is supersonic;
otherwise the turbulent motions would be thermalized by shocks, so
vturb  cs.
25
Chapter 5 Thin Accretion Disks
Hence we can write the viscosity as
= cs H
with 1. This is the famous -prescription of Shakura and Sunyaev
(1973, A&A, 24, 337).
Note that with this semi-empirical approach all our ignorance about
viscosity mechanism has been isolated in, which depends on other
parameters and should not be taken as a constant.
26
Chapter 5 Thin Accretion Disks
The shear stress in a thin, Keplerian disk is
3
3
3
2
TR   R'  cs HK  cs  P ,
2
2
2
so
2 TR

3 P ,
which means that the viscous stress shouldn’t exceed the gas
pressure in the disk.
27
Chapter 5 Thin Accretion Disks
Another source of shear stress can be a magnetic field.
The material in an accretion disk is usually ionized. This means that
it can support electrical currents. These electrical currents
generate a magnetic field, and the field and current together lead to
a Lorentz force on the gas,
1
1
f L  ( j  B) 
[(  B)  B]    TM
.
c
4
The magnetic stress tensor is given by


1 B2
TM 
(
Î  BB )
4 2
28
Chapter 5 Thin Accretion Disks
The first term is the magnetic pressure, the second term BRB/ is
responsible for magnetic shear stress (magnetic tension).
The idea of MHD turbulence was initially discussed by Velikhov
(1959) and specifically developed by Balbus and Hawley (1991, ApJ,
376, 214; 376, 223).
29
Chapter 5 Thin Accretion Disks
“Weakly magnetized accretion disks are subject to an axisymmetric
shearing instability. ”
“The most important consequence of this instability is that the
mechanism behind a generic means of transport in accretion disks
has been elucidated. The underlying cause of turbulent structure in
accretion disks stems from the tendency of a weak magnetic field to
try to enforce corotation on displaced fluid elements, a behavior
which results in excess centrifugal force at larger radii, and a
deficiency at small radii.”
30
Chapter 5 Thin Accretion Disks
Imagine that a magnetic field line
initially connects two neighboring
annuli in a radial direction, as shown
by the dotted line.
Because these two annuli have differing angular velocities, the field
line will tend to become stretched as the shear proceeds (solid
curve).
The magnetic field will try to oppose the shear, and try to
straighten out, which requires speeding up the outer annulus relative
to the inner annulus, i.e. transferring angular momentum outward.
31
Chapter 5 Thin Accretion Disks
The image shows a cross section
through a magnetized disk in which the
magnetorotational
instability
has
created turbulence. The blue indicates
gas with less than Keplerian angular
momentum; the red is gas with excess
angularmomentum.
(http://www.astro.virginia.edu/~jh8h/).
However, recent MHD simulations of the magnetorotational instability by
Fromang & Papaloizou (2007, A&A, 476, 1113) demonstrate that turbulent
activity decreases as resolution increases.
32
Chapter 5 Thin Accretion Disks
3. Steady thin disks
 Assumptions
(1) Steady disks: the external conditions (e.g. mass transfer rate)
change on timescale much longer than the viscous timescale tvisc
R2/, i.e. /t 0.
(2) Thin disks: the disk height H(R) is much smaller than the radius
R.
(3) Keplerian rotation: the angular velocity of disk material is
Keplerian, i.e. (R, z)K(R) (GM/R3)1/2.
33
Chapter 5 Thin Accretion Disks
Mass conservation
 
R

( RvR )  0
t R
 RvRconstant
 M  2R(vR )
where M is the accretion rate (vR ).
34
Chapter 5 Thin Accretion Disks
Angular momentum conservation


1 G
2
2
R (R ) 
( RvR R ) 
t
R
2 R
G const
 RvR R  

2
2
2
G(R, t)=2RR2
vRconst/(2R3)
The constant is related to the boundary condition.
35
Chapter 5 Thin Accretion Disks
Suppose that the disk extends all the way to the surface R R of
the central star, which is rotating at a rate  <K(R). The disk joins
the star with a boundary layer with width b where the angular
velocity of the disk material decreases from the Keplerian value
K(R) to .
There exists a radius RRb
where |R=Rband
RbK(R) with bR for thin
disks.
36
Chapter 5 Thin Accretion Disks
Then we have
const  2R3(vR )K ( R* )  M (GMR* )1 / 2
M
R* 1 / 2
  [1  ( ) ]

3
R
The viscous dissipation rate is independent of viscosity.
G' 9 GM 3GMM
R* 1 / 2
D( R ) 
  3 
[1  ( ) ]
3
4R 8
R
8R
R
37
Chapter 5 Thin Accretion Disks
The luminosity produced by the disk between R1 and R2 is
L( R1 , R2 )  2
R2
R1
3GMM 1
2 R* 1 / 2
1
2 R* 1 / 2
D( R)2RdR 
{ [1  ( ) ]  [1  ( ) ]}
2
R1
3 R1
R2
3 R2
Let R1R and R2 we obtain the luminosity of the whole disk
GMM 1
Ldisk 
 Lacc
2 R*
2
38
Chapter 5 Thin Accretion Disks
 The vertical (z-direction) structure of the disk
Assume that there is no motion in the z-direction, the hydrostatic
equilibrium equation is
1 P 
GM
 [ 2
]
2 1/ 2 .
 z z ( R  z )
For a thin disk (zR) this becomes
1 P
GMz
 3
 z
R ,
39
Chapter 5 Thin Accretion Disks
cs2 GMH
or H  R3
for a typical scale-height H of the disk.
cs ( R)
H
R

vK ( R)
csvK
(the local Keplerian velocity is highly supersonic for a thin disk).
40
Chapter 5 Thin Accretion Disks
The radial velocity is highly subsonic
M
3
R* 1 / 2 1 
H
 vR 

[1  ( ) ] ~ ~ cs  cs
.
2R 2 R
R
R
R
Now consider the radial component of the Euler equation
2
vR v 1 P GM
vR
 
 2 0
R R  R R
2
2
2
v
v
c
v
   s  K 0
 R
R R R
2
R
vRcs vK
 vvK
41
Chapter 5 Thin Accretion Disks
 The emitted spectrum
Assume that energy transport is radiative, and the disk is optically
thick, i.e.
  h R (  ,Tc )   R  1
(where R is the Rosseland mean opacity), the radiation field is
locally very close to the blackbody, the flux of radiant energy
through the surface z  constant is given by
16T 3 T 4 4
F ( z)  
~
T ( z)
3 R  z
3
42
Chapter 5 Thin Accretion Disks
The energy balance equation is
F
 Q
z
or
H
F ( H )  F (0)   Q  ( z )dz  D( R)
0
4
4
T

T
( H ) , it becomes approximately
If c
4 4
Tc  D( R)
3
43
Chapter 5 Thin Accretion Disks
Or the effective temperature of the disk is
T 4 ( R)  D( R)
3GMM
R* 1/ 2 1/ 4
T ( R)  {
[1  ( ) ]}
3
8R
R
For R R*,
T  T* ( R / R* )3 / 4
where
3GMM 1 / 4
T*  (
)
3
8R*
1/ 4
 4.1  10 4 M 16
M 11 / 4 R93 / 4 K
1/ 4
 1.3  10 7 M 17
M 11 / 4 R63 / 4 K
44
Chapter 5 Thin Accretion Disks
If we neglect the effect of the atmosphere of the disk, the
spectrum emitted by each element of area of the disk is the
blackbody with temperature T(R)
2h 3
1
I  B [T ( R)]  2 h / kT ( R )
c e
1
The flux observed at a distance of D is
F  
Rout
R*
3
4

h
cos
i

I (2 RdR cos i / D 2 ) 
c2 D2

Rout
R*
RdR
eh / kT ( R )  1
where i is the angle between the line of sight and the normal to the
disk plane.
45
Chapter 5 Thin Accretion Disks
The spectrum is shown in the figure.
If hkT(Rout), B =2kT2/c2 F2
If hkT*, B =2kT3c-2ehkT  F2ehkT
If kT(Rout)hkT*,
F   1 / 3 

0
x5 / 3dx
1/ 3


ex  1
46
Chapter 5 Thin Accretion Disks
 Structure of the standard -disk
The equations of the steady disk
  H
H  cs R3 / 2 /(GM )1 / 2
   R (  , Tc )
R
M
[1  ( * )1/ 2 ]
3
R
P  cs2
 
kTc 4 4
P

Tc
mp 3c
   cs H
4Tc4 3GMM
R* 1 / 2

[1  ( ) ]
3
3
8R
R
47
Chapter 5 Thin Accretion Disks
The disk may be composed of a number of distinct regions
(a) PrPg, T ff (T  0.4 cm2g-1)
3 T M
H (cm) 
f  1.6  104 M16 f
8c
 (gcm -3 )  23 1M 162 M11/ 2 R83 / 2 f 2
2
vR (cms -1 )  44M16
M11 / 2 R85 / 2 f
T (K)  4.2  10 
6
1 / 4
M
1/ 8
1
R83 / 8
where f(R*/R)1/2.
48
Chapter 5 Thin Accretion Disks
(b) Pg Pr, Tff
1/ 5
H (cm)  8.0  105 1 / 10M16
M17 / 20R821/ 20 f 1 / 5
 (gcm -3 )  1.9  104 7 / 10M162 / 5 M111/ 20R833/ 20 f 2 / 5
vR (cms -1 )  1.0 105 4 / 5 M 162 / 5 M11/ 5 R82 / 5 f 3 / 5
T (K)  5.9  105 1 / 5 M162 / 5 M1
3 / 10
R89 / 10 f 2 / 5
The boundary between regions (a) and (b) lies on the radius
16/ 21
Rab (cm)  2.5 106  2 / 21M16
M17 / 21 f 16/ 21
49
Chapter 5 Thin Accretion Disks
(c) Pg Pr
ff T (R =ff = 6.61022T-7/2 cm2g-1)
3 / 20
9 / 8 3 / 20
H (cm)  1.27  108 1 / 10M16
M13 / 8 R10
f
11/ 20
 (gcm -3 )  4.6  108 7 / 10M16
M15 / 8 R1015 / 8 f 11/ 20
3 / 10
vR (cms -1 )  2.7  104 4 / 5 M16
M11 / 4 R101 / 4 f 7 / 10
T (K)  1.4  104 1 / 5 M163 / 10M1 R103 / 4 f 3 / 10
1/ 4
The boundary between regions (b) and (c) lies at
Rbc (cm)  2.9 108 M162 / 3M11/ 3 f 2 / 3
50
Chapter 5 Thin Accretion Disks
51
Chapter 5 Thin Accretion Disks
4. Steady disks: confrontation with observation
Accretion disks:
Inner regions: closely related to the compact star.
Outer regions: T  106 K, radiating predominantly in UV, optical and
IR.
To study the outer regions of the disks observationally, we require
that the light in one or more of these parts of the spectrum is
dominated by the disk contribution.
52
Chapter 5 Thin Accretion Disks
 Galactic X-ray sources
(1) HMXBs
37
38
The donor stars are O or B giants or supergiants, Lopt
ergs-1, much higher than the UV and optical luminosity of the disks.
(2) LMXBs and CVs
The donor stars are late type, low-mass, faint stars, the accreting
compact stars are neutron stars (black holes) and white dwarfs
respectively.
The processes of mass transfer are similar in these two types of
systems, but Lopt,LMXB 100 Lopt,CV
53
Chapter 5 Thin Accretion Disks
This means that re-absorption of X-rays in LMXB disks is very
important.
So CVs are the best candidates for testing the theory of steady
thin disks.
54
Chapter 5 Thin Accretion Disks
Evidence of circular motion of accreting material from eclipse of a
double-peaked emission line from the
optically thin gas in a CV
(i) and (iv) Outside eclipse the line appears
double-peaked because of the nearly
circular motion in the disk around the white
dwarf.
(ii) The advancing side of the disk is eclipsed
first, leading to the disappearance of the
blueward component of line.
55
Chapter 5 Thin Accretion Disks
(iii) As the eclipse proceeds this side of the disk re-emerges and the
receeding side is eclipsed, leading to the loss of the redward
component and the re-appearance of the blueward component.
56
Chapter 5 Thin Accretion Disks
Doppler tomography of binary
accretion disks
(from Steeghs et al. 2004, AN,
325, 185)
57
Chapter 5 Thin Accretion Disks
Comparing the predicted spectra from optically thick disks with
observations
Method: eclipsing mapping the surface brightness distribution in an
accretion disk.
Because
of
the
temperature
distribution in the disk, the light at
short
wavelengths
is
strongly
concentrated towards the central disk
regions, while for long wavelengths the
brightness distribution is almost
uniform outside the central regions.
58
Chapter 5 Thin Accretion Disks
Hence if we observe a CV with a sufficiently high orbital inclination
that the companion star eclipses the central regions of the disk,
there should be a deep and sharp eclipse at short wavelengths and a
shallower broader one at long wavelengths.
59
Chapter 5 Thin Accretion Disks
The figure shows the effective temperature distribution given by
maximum-entropy deconvolution, compared with the theoretical
temperature distribution for various values of mass transfer rates.
60
Chapter 5 Thin Accretion Disks
5. Irradiation of accretion disks
The disks in LMXBs are probably heated by X-ray irradiation by the
central accretion source.
If the central source with X-ray luminosity Lx can be regarded as a
point, the flux crossing the disk surface is
Lx
F
(1   ) cos
2
4R
where
is the albedo (~0.9, de
Jong et al. 1996, A&A, 314, 484),
and
is the angle between the
local disk normal and the direction
of the incident radiation.
61
Chapter 5 Thin Accretion Disks
   / 2   
where
tan dH/dR, and tanH/R.
Since dH/dR and H/R1 for thin disks, we have
cos  sin(   )  tan  tan  dH / dR  H / R
The effective temperature Tirr resulting from irradiation is
Lx (1   ) H d ln H
T 
( )[
 1]
2
4 R  R d ln R
4
irr
62
Chapter 5 Thin Accretion Disks
The effective temperature of the disk is a combination the
irradiation temperature and the viscous temperature
Teff4  Tirr4  Tvis4
In the outer part of the disk where Tirr >> Tvis, the structure of the
disk changes as (Fukue, 1992, PASJ, 44, 669)
1/ 7
H/R  1.2  102 (1   )1 / 7 M16
M13 / 7 R*61 / 7 R102 / 7
 (gcm -3 )  7.4  108 (1   )3 / 7  1M164 / 7 M111/ 14R*36/ 7 R1033/ 14
1 / 14
vR (cms -1 )  8.7  103 (1   )2 / 7 M162 / 7 M15 / 14R*62 / 7 R10
T (K)  1.2  104 (1   )1 / 7 M162 / 7 M1 R*62 / 7 R103 / 7
1/ 7
Note that the disk height H changes from HR9/8 to HR9/7.
63
Chapter 5 Thin Accretion Disks
If the accretor is a black hole, the irradiating source is the inner
region of the accretion disk, and there is an extra factor ~H/R to
the irradiating flux, see Sanbuichi et al. (1993, PASJ, 45, 443) for
details.
64
Chapter 5 Thin Accretion Disks
 Evidence for X-ray irradiation in LMXB disks
van Paradijs & McClintock (1994, A&A, 290, 133) show that there is
a strong relation between the absolute magnitudes in optical of
LMXB disks and the X-ray luminosities. This can be explained as
follows.
In the temperature range encountered in
LMXB disks the visual surface brightness Sv
of a blackbody emitter approximately varies
as Sv T 2. So we have
2/3
Lv  L1x/ 2a  L1x/ 2 Porb
.
65
Chapter 5 Thin Accretion Disks
6. Time dependence and stability
 Reasons for studying time-dependent disks
(1) To check that the steady disk models are stable against smaller
perturbations;
(2) To get information about disk viscosity from time-dependent
disk behavior.
66
Chapter 5 Thin Accretion Disks
 Typical timescales
(1) Dynamical timescale, the timescale on which inhomogeneities on
the disk surface rotate, or hydrostatic equilibrium in the
vertical direction is established.
t ~ R / v ~ K1
(2) Viscous timescale, the timescale on which matter diffuses
through the disk under the effect of viscous torques.
tvisc ~ R 2 / ~ R / vR
(3) Thermal timescale, the timescale for re-adjustment to thermal
equilibrium.
tth ~ cs2 / D( R) ~ ( H / R) 2 tvisc
67
Chapter 5 Thin Accretion Disks
We have the following relation
t ~ tth ~  ( H / R)2 tvisc
Or numerically
t ~  tth ~ (100 s)M11/ 2 R103/ 2 ,
tvisc ~ (3 105 s) 4/ 5 M163/10 M11/ 4 R105/ 4 .
68
Chapter 5 Thin Accretion Disks
 Thermal instability
The thermal equilibrium at a given radius R in the disk is defined by
the equation Q
Q , where Q and Q are the heating and
cooling rates per unit surface respectively, or
9
T  2K
.
8
4
eff
Since R,,the thermal equilibrium equation can be
represented as a Teff() relation. This relation forms an S–curve on
the (, Teff) plane.
69
Chapter 5 Thin Accretion Disks
Each point on the (, Teff) S–curve represents an accretion disk’s
thermal equilibrium at a given radius R.
The middle branch of the S–curve corresponds to thermally unstable
equilibrium.
A stable disk equilibrium can be
represented only by a point on the
lower cold or the upper hot branch
of the S–curve.
70
Chapter 5 Thin Accretion Disks
This means that the surface density in the cold state must be lower
than the maximal value on the cold branch:
1.14
max  13.4c0.83 M10.38 R10
gcm2 ,
whereas the surface density in the hot state must be larger than
the minimum value on this branch:
1.11
min  8.3 h0.77 M10.37 R10
gcm2 .
For thermal instability, since tvisctth, and t < tth, we can assume
thatconstant during the growth time and the vertical structure
of the disk can respond rapidly towards hydrostatic equilibrium.
71
Chapter 5 Thin Accretion Disks
The disk is thermally unstable on the middle branch because
radiative cooling varies slower with temperature than viscous
heating
d ln Teff4
d ln Tc
d ln Fvisc

d ln Tc ,
so that when the central temperature Tc in an annulus of the disk
initially in thermal equilibrium is increased by a small perturbation,
Tc will rise further because the cooling rate is inadequate.
72
Chapter 5 Thin Accretion Disks
For example, consider the regions of the disk where gas pressure
dominates pressure.
In general we can write the opacity as
 R  Tcn
so that
 ~  R H  2Tcn / H .
Since
H  cs  Tc1 / 2 ,
we get
Teff4  Tc4 /   Tc9 / 2  n2 .
73
Chapter 5 Thin Accretion Disks
So the left hand side of the inequality
d ln Teff4
d ln Tc
d ln Fvisc

d ln Tc
is 9/2n.
From the-prescription we havecsH Tc and Fvisc  Tc.
So the inequality becomes
9/2n<1, or n7/2.
74
Chapter 5 Thin Accretion Disks
Further analysis gives
Teff  
13 2 n
4( 7  2 n )
.
This relation shows that Teff/<0, i.e. the disk is unstable in
regions when 7/2 < n < 13/2.
75
Chapter 5 Thin Accretion Disks
In fact the values of n in the unstable range will always occur in
hydrogen ionization zone, i.e. wherever Teff is close to the local
hydrogen ionization temperature TH~6500 K.
Hydrogen is predominantly neutral
when T<TH, and R increases rapidly
with temperature, i.e., n > 13/2.
For T > TH, hydrogen is essentially
fully ionized, and n takes the
Krammers’ value –3.5.
So the opacity changes abruptly when
T~ TH.
76
Chapter 5 Thin Accretion Disks
(Figure from Menou, K. 2001, ApJ, 559, 1032)
77
Chapter 5 Thin Accretion Disks
 Limit cycle behavior
During an outburst a point representing a local accretion disk’s state
moves in the (, Teff) plane as shown in the figure. A point out of the
S–curve is out of thermal equilibrium.
In the region to the right of the S–
curve heating dominates cooling, so
that the temperature increases
and the system-point moves up
towards the hot branch.
78
Chapter 5 Thin Accretion Disks
On the left to the S–curve is the case opposite and the point moves
down towards the cool branch.
These upward and downward motions take place in thermal time
since they correspond to the heating and cooling of a disk’s ring.
During decay from outburst and during the quiescent phase of the
outburst cycle, the system-point moves along, respectively the
upper and lower branches in viscous time.
79
Chapter 5 Thin Accretion Disks
 Dwarf novae
Dwarf novae are erupting cataclysmic variables (CVs).
In these binary systems outbursts take place in the accretion disk,
which is formed around the central white dwarf by matter
transferred from low-mass, Roche-lobe filling companion star.
Dwarf novae include three tupes: U Gem, SU UMa, and Z Cam, named
after their prototypes.
80
Chapter 5 Thin Accretion Disks
All three types of dwarf novae show normal outbursts and only SU
UMa stars also show superoutbursts.
Normal outbursts have amplitudes of 2-5 magnitudes and last 2-20
days. The recurrence times are typically from ~10 days to years.
(figure
from
http://observe.arc.nasa.gov/nasa/space/stellardeath/stellardeath_
4b.html)
81
Chapter 5 Thin Accretion Disks
Superoutbursts have amplitudes brighter by ~0.7 magnitude, lasting
~5 times longer, and their recurrence time is longer than that of
normal outbursts.
(http://vsnet.kusastro.kyoto-u.ac.jp/vsnet/DNe/wxcet.html)
82
Chapter 5 Thin Accretion Disks
The disk instability model for dwarf novae uses the limit cycle
behavior expected if the disk contains regions of partial ionization,
i.e., the mass transfer rate is lower than the critical mass transfer
rate given by Teff(Rout) = TH
M cr  3  109 ( P / 3 hr)2 M  yr-1
In quiescence (R, t) lies between min and
to increases outwards.
83
max at each R and tends
Chapter 5 Thin Accretion Disks
An outburst is triggered once rises above max at some radius.
The disk annulus at that point makes the transition to the hot state;
the mass and heat diffuse rapidly into the adjacent annuli,
stimulating them to make the same transition.
This leads to the propagation of heating fronts both inwards and
outwards from the initial instability.
The inward moving front propagates at a velocity cs.
84
Chapter 5 Thin Accretion Disks
In fact if has a single constant value no large outburst results,
because the resulting S–curve is rather narrow, i.e., max/min  2.
Consequently the heating front does not propagate very far through
the disk before the cooling wave begins to sweep inwards, shutting
off the outbursts before it develops fully.
It is usually adopted that hon the upper (hot) branch of the
S–curve and c on the lower (cold) branch.
85
Chapter 5 Thin Accretion Disks
 Soft X-ray transients (X-ray novae): effect of irradiation
Accretion disks in low-mass X-ray binaries also subject to the
thermal instability.
However, the irradiation of the disk has enhanced the effective
temperature of the disk,
so that the required mass transfer rate is considerably lower than
the rate given by the dwarf nova condition Teff (Rout) < TH.
86
Chapter 5 Thin Accretion Disks
For an LMXB disk irradiated by a point source the criterion is
where C is defined in
Stability limits and parameters of Low Mass X-ray Binaries.
Filled circles represent steady (i.e. non-transient) LMXBs
containing neutron stars. The two asterisks correspond to
two neutron-star LMXBTs. Diamonds represent black-hole
LMXBTs with known recurrence times and down-pointing
triangles those where only the lower limits for the recurrence
time are known. The up-pointing triangle corresponds to
GRO J1655-40 with the recurrence time between the 1994
and 1996 outbursts. From astro-ph/0102072.
87
Chapter 5 Thin Accretion Disks
7. Tilted/warped accretion disks in XRBs
 Observational clues of tilted
accretion disks in XRBs
(1) The super-orbital
variabilities observed in a
number of X-ray binaries
have long been interpreted
as due to precession of a
tilted accretion disk.
(2) Jet precession
SS433 (164 days)
GRO J1655-40 (3 days)
CAL 83 (~69 days)
88
Chapter 5 Thin Accretion Disks
 Driving mechanisms for disk tilt/warp
(1) irradiation-driven wind (Schandl & Meyer 1994)
(2) radiation pressure (Pringle 1996)
(3) stellar magnetic field (Lai 1999, 2003)
89
Chapter 5 Thin Accretion Disks
 Self-induced warping of accretion disks
When an accretion disk is illuminated by a radiation source at its
center, a twist or warp in the disk will be induced, because the
surface of a warped disk is illuminated by a central radiation source
in a non-uniform manner.
Provided that the disk is optically thick, radiation received at a
particular point on the disk surface is reemitted from that same
point in the direction of the normal to the surface at that point, the
back-reaction of the emitted radiation gives rise to an uneven
distribution of forces on the disk surface.
90
Chapter 5 Thin Accretion Disks
The effect of these forces on a given annulus of the disk is to induce
a torque on that annulus about the disk center.
The effect of such a torque is to change the angular momentum of
the annulus and so to change the twist of the disk at the radius.
91
Chapter 5 Thin Accretion Disks
According to Pringle (1996, MNRAS, 281, 357), radiation driven
warping occurs at radii
R
2  2
 8 ( )
RS

where RS=2GM/c2 is the Schwarzschild radius of the central star,
 2/ where 2 is the vertical kinematic viscosity coefficient, and
  L / Mc 2 .
92
Chapter 5 Thin Accretion Disks
It is very likely that disks in LMXBs are unstable to warping, but it is
very unlikely to occur in white dwarf binaries.
For example, for a 1 M⊙ accreting white dwarf with radius R~5108
cm, and 1, warping occurs only for radii R51013 cm, whereas for a
1 M⊙ neutron star with R~106 cm, warping occurs for radii R3108
cm.
93
Chapter 5 Thin Accretion Disks
The figures show the numerically simulated results of disk warping
by Wijers and Pringle (1998, MNRAS, 308, 207). Left: the shape of
a disk undergoing warping. Right: the behavior of the inclination of
the outer disk with different dimensionless strength of radiation
field.
94
Chapter 5 Thin Accretion Disks
8. Tides and resonances
At the outer edge the disk experiences the tidal torque exerted by
the companion star. The accretion disk is cut off at the tidal radius,
Rtide  0.9R1, where R1 is the primary’s Roche lobe radius.
Under certain circumstances (q0.25-0.33), tidal force causes the
orbit of disk material to be eccentric, and precess on a period
slighter longer than Porb.
95
Chapter 5 Thin Accretion Disks
Resonance occurs in a disk when the frequency of radial motion of a
particle in the disk is commensurate with the angular frequency of
the secondary star as seen by the
particle.
This condition ensures that the particle
will always receive a ‘kick’ from the
secondary at exactly the same phase of
its radial motion, so allowing the
cumulative effect of repeated kicks to
build up and affect the motion
significantly.
96
Chapter 5 Thin Accretion Disks
If the mean angular frequency in a given orbit is  (measured in a
non-rotating frame), and the orbit precesses at an apsidal
precession frequency , the epicyclic frequency for the particle to
return to the same radial distance is . The particle sees the
orb. Thus the
resonance occurs when
k()j(orb)
where k and j are positive integers.
97
Chapter 5 Thin Accretion Disks
Assume that the orbit of the disk material is close to Keplerian, the
radii Rjk of the resonant orbits near the j:k commensurability is
j  k 2/3
R jk  (
) (1  q) 1 / 3 a
j
This can be compared with the tidal
radius Rtide shown in the figure.
Resonant orbits can only exist for
sufficiently small mass ratio q; the j 
3, k  2 resonances are responsible for
superhumps in SU UMa systems only
for q0.3.
98
Chapter 5 Thin Accretion Disks
References
1. Frank, J., King, A., and Raine, D. 2002, Accretion power in
astrophysics
2. Achterberg, A. 1996, Accretion in astrophysics
3. Lasota, J. P. 2001, astro-ph/0102072
99