1997MNRAS.285..394I
Mon. Not. R. Astron. Soc. 285, 394-402 (1997)
The oscillatory shape of the stationary twisted disc around a Kerr black
hole
P. B. Ivanov* and A. F. Illarionov*
Astra Space Center of P. N. Lebedev Physical Institute of the Russian Academy of Sciences, Profsoyuznaya st., 84/32, Moscow 117810, Russia
Accepted 1996 September 25. Received 1996 January 17; in original form 1995 June 7
ABSTRACT
We consider the twisted accretion disc around a Kerr black hole and derive the
stationary twist equation in the leading order on small parameters of relative
thickness b, viscosity ex and relativistic corrections rir. We take into account postNewtonian corrections which determine the shape of the stationary twisted
accretion disc in the limit of small viscosity, ex < b4/S • We find the phenomenon of
radial oscillations of the disc inclination angle P(r) due to this correction. The period
6fthe oscillations is about Ar:::::; b -4/\ and decreases with decreasing distance r. We
present a qualitative analysis of the oscillatory behaviour and conclude that the
oscillations are likely to destroy a disc of low viscosity at sufficiently small
r < rf :::::; b- 3P( 00 )8/\. We suggest that the twisted disc might be in a highly turbulent
state with ex :::::; 1 at r < rf •
Key words: accretion, accretion discs - black hole physics - relativity - celestial
mechanics, stellar dynamics.
1 INTRODUCTION
Since the first paper from Bardeen & Petterson (1975) the
configuration of the twisted accretion disc around a Kerr
black hole has been discussed by a number of authors
(Petterson 1977, 1978; Hatchett, Begelman & Sarazin 1981;
Papaloizou & Pringle 1983, hereafter PP; Kumar & Pringle
1985; Kumar 1990; Pringle 1992). PP pointed out that the
equation describing the geometry of a twisted disc (the twist
equation) derived by the previous authors was incorrect
because it did not conserve angular momentum. As was
shown by PP, the perturbations of the matter density and
the velocities should be taken into account for the correct
derivation of the twist equation. The terms corresponding
to such perturbations appear in the twist equation in the
main order due to a resonance in the system, and in conjunction with gravitomagnetic force determine the disc
twist. In particular, the characteristic scale of the disc twist
becomes smaller as the viscosity parameter IX decreases,
contrary to previous claims, and therefore in the case of a
sufficiently low value of IX the post-Newtonian corrections to
the equations of motion should be taken into account
(PP).
*E-mail: [email protected] (PBI);
[email protected] (AFI)
Here we consider the thin (with opening angle tJ < 1)
twisted accretion disc around a Kerr black hole. We derive
and solve the stationary twist equation, and take into
account post-Newtonian relativistic correction. Solution of
the stationary twist equation shows that the post-Newtonian
correction leads to radial oscillations in the disc inclination
angle fJ(r) in the case of low viscosity (IX < tJ4/5: tJ is the
parameter of relative disc thickness, see below). We discuss
the nature of this oscillatory regime in detail. We suppose
that the oscillations may lead to shear instability and effectively increase the viscosity parameter IX.
In Section 2 we introduce the twisting coordinate system
(Petterson 1977, 1978). The use of the twisting coordinate
system formally enables us to consider rather large inclination angles fJ < 1 in contrast to the use of an ordinary cylindrical coordinate system (PP), which is applicable in the
case of very small fJ < tJ only. In Section 3 we write the main
equations of twisted disc accretion. We derive the twist
equation in the main approximation in terms of the small
parameters of disc inclination angle p, viscosity IX, disc opening angle tJ and post-Newtonian corrections rg/r (see Appendix) in Section 4. We analyse the solutions of twist equation
in Section 5. We present a qualitative analysis of oscillatory
behaviour in Section 6. We discuss a possible role of the
shear instability in twisted accretion discs (Coleman &
Kumar 1993) and present the restrictions on the initial
inclination angle in Section 7.
We use below the natural system of units G = c = 1.
© 1997 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.285..394I
The twisted disc around a Kerr black hole 395
e=D(~+~ow
~-rw~)
or r oqJ oqJ
o~ ,
2 COORDINATE SYSTEM
(4)
r
We use the local cylindrical (twisting) coordinate system
(Petterson 1977) in which the disc equations can be written
in the simplest form. The position of disc element R is
determined by the cylindrical coordinates r, l/I, ~ related to
an inclined cylinder of radius r (see Fig. 1). This cylinder is
taken to be oriented perpendicular to the disc ring with the
same radius. The cylinder orientation is characterized by
two Euler angles: the inclination angle fJ and the precession
angle y. The transformation law from Cartesian coordinate
system x, y, z (where z coincides with the black hole rotational axis andx,y lie in the equatorial plane) to the twisting
coordinate system can be written as
(X)
(rcosl/l)
rSi~l/I =B(fJ)C(y) \~ ,
(1)
where B (fJ) and C(y) are the rotational matrices
0)
(1 0 0)
COS I' sin y
C= ( -siny cosy 0 .
o 0 1
B= 0 cosfJ sinfJ ,
o - sin fJ cos fJ
1 0
II' r oqJ'
e
(5)
=--
(6)
where D = (1 -
~ W) - I
and the value of local twist is
W= 'II; sin qJ - 'II; cos qJ = fJ' sin l/I- fJl" cos l/I,
(7)
and the prime denotes %r.
To perform covariant differentiation in the basis of vectors (4)-(6) it is necessary to evaluate the connection coefficients. Taking into account that we consider the disc twist in
the leading approximation we only need standard cylindrical components r 11"11' = - rrq>q> = 1/r.
(2)
3 BASIC EQUATIONS
In our calculations we use the - y and X components of
unit vector ~ perpendicular to the ring,
The hydrodynamical equations projected on to the basis
(4)-(6) reduce to the continuity equation
'¥1=fJcosy,
(pVi);i=O,
'¥2=fJsiny
(3)
instead of fJ and y, and the angle qJ = y + l/I instead of l/I. This
enables us to consider the case fJ = 0 where the twisting
coordinate system becomes degenerate. We consider all
vectors and tensors to be projected on to the orthonormal
basis that is connected with the twisting coordinate system
(Petterson 1977):
z
\~
,
R
",j--,
,,
,,
(8)
and the equations of motion of viscous fluid
p(vjv;j+Ai)=piPi_pi_tj+pFi,
(9)
where p, P, Vi denote respectively the density, pressure and
velocity components, iP =M/r is the Newtonian potential
and M is the black hole mass, t ij are the components of
viscous stress tensor, and the semicolon denotes differentiation with respect to the basis (4)-(6). Ai denotes postNewtonian corrections to the convective term VjV;j. Fi is the
gravitomagnetic force, which has only one relevant component:
F~
... ,
i=1, 2, 3
4avq>M2
('III sin qJ - '112cos qJ),
r3
(10)
where a is the black hole rotational parameter.
Following PP we divide the matter density p and velocity
components v" vII' into standard parts and perturbations,
,
y
x
(11)
where small perturbations ViI' PI oc (ofJ/or) ~ cos (qJ + qJo).
Substituting equation (11) into the set of basic equations (8)
and (9), and separating the terms with different dependencies on ~ and qJ, we obtain the sets of equations for the
quantities with indices 0 and 1.
The set of equations for the quantities with index 0 is
identical to the standard system for flat disc accretion. We
use the relation
V!o
Figure 1. The r, 1/1, ~ coordinate system shown with respect to the
x, y, z system. Locally, it simply consists of a cylindrical coordinate
system rotated with respect to the x, y, z system over angles fJ and
oiP
-+-=0
r
or
y. y is an angle between the axis OX and the line of intersection of
the disc plane and the OXY plane. P is the projection of the point
R on to the disc ring plane.
to determine the Newtonian vq>o:
vq>o=rQ,
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
(12)
(13)
1997MNRAS.285..394I
396 P. B. Ivanov and A. F. Illarionov
where O=MI!2/r3!2. The radial component of velocity VrO can
be taken from the equation of angular momentum transfer,
3v
vrO- --2r'
(14)
where kinematic viscosity (Shakura & Sunyev 1973)
o:~~
v=-v<p o,
(15)
r
where 0: is the viscosity parameter and ~* is the disc halfthickness. The surface density L = JPo d ~ comes from the
equation of mass conservation,
!VI
(16)
- = -LvrOr,
21t
where !VI is the mass transfer rate. We use the standard
relation
all> =~ all> = _ ~ v!o
o~
r or
The relativistic corrections can be written in the explicit
form (see Appendix)
M eOVrl
A =--+V )
r
r 2 oqJ
<pI'
(25)
A = M (~ OV<p1 _ ~ v ).
<p r 2 oqJ 4 rl
(26)
Note, that the system (21), (22) is degenerate in the leading
order; therefore, the small viscous terms and the relativistic
corrections should be retained in equations (21) and (22) to
solve them.
Finally, we consider the perturbed ~-component of equation (9). Integrating this component with respect to the
vertical coordinate ~ we can see that the gravitomagnetic
force F, is balanced by the ~-component of Newtonian
gravity force and ~-component of perturbed pressure
gradient OPI/O~ ,
(27)
(17)
r r
take the disc to be isothermal with density profile
po=p*exp[-(e/2~m
(18)
and assume also that the gas pressure dominates over the
radiation pressure. For this case we have approximately
linear dependence of the disc thickness ~ * on large r
(Shakura & Sunyaev 1973),
where the perturbed pressure gradient term is omitted since
this term vanishes after integration over ~. Equation (27)
has the simple form of two forces balancing in the leading
order on small parameters M/r and 0: only. The corresponding equation in general case of a large value of 0: and time
dependence was derived by PP and in the twisting coordinate system (Demiansky & Ivanov 1997).
(19)
where CJ is the disc opening angle.
The quantities with index 1 come from the perturbed
continuity equation (8) and the equations of motion (9).
From the perturbed continuity equation (8) we have
Po oV<p 1
OPI 1 a
--+O-+--(rpOvrl)=O.
r oqJ
oqJ r or
(20)
The perturbed radial component of equation (9) can be
written as
(21)
4 TWIST EQUATION
To obtain the twist equation we proceed as follows. With
the help of equations (7), (12)-(19) and (21)-(26) we find
and solve the equations for the velocity perturbations.
Using equation (20) we express the density perturbations in
terms of perturbed velocities. Substituting the result into
equation (27) we obtain the twist equation.
In detail, substituting (7), (12)-(19) and (23)-(26) into
equations (21) and (22), and subtracting (22) from (21), we
find
~(40:~ OV<PI_ Wv
r
and from the equation for the qJ-component we have
P n{OVrl
o oqJ
2V
_
<pI
+2~A }=-2~ O[,<p.
oqJ
<p
oqJ
o~
The perturbed components of the viscous tensor [rl; and [<<p
in equations (21) and (22) are
-'1 o~
Vrl ,
where '1 = Po v.
<po
)=V
12M
<pI r2 '
(22)
(23)
[rl;=
oqJ
(28)
where in the leading order we have
The r component of the gas pressure gradient 'VrP in (21)
has to be taken into account due to the disc twist:
a
o~
(24)
(29)
Solution of (28) gives
v = _ ~V<j>O_I_{OW +kW}
<pI
40: 1 + e oqJ
,
(30)
where k=3M/o:r determines the relative role of relativistic
corrections.
Now we express the perturbed density PI in terms of
perturbed velocities with the help of continuity equation
(20) and equation (29):
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.285..394I
The twisted disc around a Kerr black hole
397
5
4
~
8
'---'
Q:l.
"'---
3
~
'-'---'
Q:l.
2
2
101
2
3
1 02
4 5
2
3
4 5
10 3
riM
Figure 2. The dependence of the inclination angle PIP ( 00 ) on radius riM for b = 0.1, a = 1 and for the different values of IX. Curves 1, 2 and
3 correspond to 1X=0.01, 0.1 and 1 (for illustration). We have R 21M=20 and R,IM=4.2, 25, 90 for cases 1, 2 and 3 respectively. When IX
becomes comparable or less than b and R2~R, we obtain oscillations.
(31)
Substituting (30) into equation (31), and (31) into equation
(27), and explicitly performing the integration over ~ in (27)
with help of (18), we obtain the twist equation,
o{
22s(OW/OCP)+kW}
1
- - I:bnr
+I:F,=O.
2IXr2 or
1 +e
ox
1 +e
ox
15 2
(34)
where Jv are the Bessel functions, r is the gamma function,
Yt = (2/3) exp (i1t/4)C tI7;r3/2 and C = 32IXa/b 2.1 This solution in
the limit ofx-+ 00 (r-+O) is
w
~K(r) exp {(r/R t )-3/4} exp {i [y( 00) - ~: +
(;J -3/4]} ,
(35)
where the coefficient K (r)
scale
~ constant.
The characteristic
(33)
where we introduce new variables w='I't+i'l'2=.8expiy
instead of W and x = (r/M) -t/2. Note, that the analogous
equation obtained in the previous works (PP; Kumar &
Pringle 1985) does not take into account the relativistic
correction term [(1 +ik)/(1 +k2)].
5 ANALYTIC SOLUTIONS
The shape of twist equation (33) is determined by the relation between two small parameters IX and 15 (see equation 40
below). In general, the twist equation can be solved numerically (see Figs 2 and 3), but for the limiting cases of
IX » 15 and IX « 15 the twist equation has analytic solutions.
First we consider the case of the large values of IX. In this
case we can neglect the relativistic correction term k in
equation (33), and the analytical solution of (33) with
IWe consider the case a > 0 hereafter.
w = (Yt/2)t/3w (00) r(2/3) {J -1/3 (Yt) - e- iIti3J I/3 (YI)}'
(32)
Transforming the functions in the brackets in equation (32)
with the help of the equation (14)-(16), we find the twist
equation in the final form,
~ {1 + ik Ow} + i32IXa wx = 0,
boundary conditions w(O)=O and w(r-+oo)-+w(oo) is
expressed in terms of Bessel functions:
(36)
describes two effects: the monotonic exponential decrease
of the inclination angle .8 with decreasing r and the precession y (r) of the disc rings in the direction of black hole
rotation (Bardeen & Petterson 1975).
When we consider the case of small IX we neglect the unity
in the equation (33) in comparison with k. The approximate
solution of equation (33) drastically changes in this case:
instead of the well-known effect of a smooth decrease of
inclination angle .8 we have the radial oscillations .8(r). The
oscillating solution has the form
w =yf2[2 -3/Sr(2/5)w (oo)J -3/S(Y2) +AJ3/S (Y2)],
(37)
whereY2=(6/5) c~I7;rS/2, C*=32a/3b 2 and A is some constant determined by the inner boundary condition. In the
ordinary case when the factors affecting the disc inclination
are determined by the large scales we can set A = O. When
r-+O we have
© 1997 RAS, MNRAS 285,394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
->
......
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~
9
0
N
.j:>.
v.>
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f
0.2
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~
0.0
0.5
c/
-0.4
-1.0
-0.8
-0.6
-0.4
-0.2
D
-¥2
0.0
A,B
-1.0
-0.4
0.4
-¥2
1.0
J
(max I 'I'll:» max 1'1'21). The number of peaks (5) on curve 1 in Fig. 2 corresponds to the number of max 1'1'11 and to the growth of y = arctan ('I'21'I'1) on 51t. (b) The same as (a) but IX =0.1.
The amplitude of oscillations is suppressed with respect to the previous case due to the growth of the viscosity. (c) The case of large viscosity IX= 1. The oscillations are damped .
,;
(c)
-¥2
2
j
-0.2
Figure 3. The result of the integration of the twist equation presented in a parametric form ['I'I (r) = Re w, '1'2 (r) = 1m w]. The points A, B, C, and D correspond to the radii riM = 2, 10,
100 and 1000 respectively. We take £5=0.1 below. (a) The case of 1X=0.01, corresponding to the oscillating curve 1 in Fig. 2. At small r the 'spiral' is strongly elongated
-2
-6
-3
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0
;:=
~.
is''
~
~
~
;:=
I=:l..
l::l
~
;:=
0
0
l::l
~
~
~
D./J
'-<
@
~
00
\0
(,;.)
1997MNRAS.285..394I
1997MNRAS.285..394I
The twisted disc around a Kerr black hole
W=
2)112(r)-1/8 35
(~
Rz
2- / r(2/S) W (00) cos
[(r)-5/4
1t]
R
+20'
399
-vP
z
z
(38)
where
(39)
o
is the characteristic scale of relativistic oscillations.
From the condition R z > R I we find that the twist disc
oscillations take place when
(40)
and solution (37) approximately describes the disc twist on
any scales r <R z.
6 OSCILLATORY REGIME
Now we consider the origin of perturbations in the disc and
clarify the physical meaning of the oscillations in the inclination angle. First we need to consider the trajectories of gas
particles r(t) in the stationary disc (see Fig. 3). We rewrite
the equations of motion (21) and (22) in the dynamical form
using the standard leading order relation, a/at = Q (a/a qJ ).
Neglecting the viscous corrections in (21) and (22), we
obtain
(41)
(42)
where t1 =r - ro is the elliptical deviation of the trajectory
from a circle (A = vrl ) and dot denotes the time derivative.
All background quantities are taken at ro (we omit this index
below), we use the explicit form for relativistic corrections
(2S) and (26) and equation (23) for the pressure gradient.
Integrating (42) over t and substituting the result into (41),
we obtain
(43)
If one neglects the relativistic correction term 6M/r
and the pressure gradient in (43), the trajectories are the
stationary ellipses with small arbitrary eccentricities
t1 =e sin (Qt + qJo). In the absence of the pressure gradient
term the relativistic correction leads to Einstein precession
of such ellipses and to the absence of stationary orbits.
In the isothermal case the pressure gradient VP is perpendicular to the lines of equal unperturbed density Po in
the leading order. In the twisted discs these lines have some
inclination to the particle orbit plane, therefore we should
take into account the non-zero projection of the pressure
gradient on to the orbit plane. In Fig. 4 we show the parallelogram-like cross-section of the disc element at radius rand
angle cp=1t/2 by the ZOY plane. The disc surface (solid
lines) is inclined with respect to the r-axis due to non-zero
---
-
Figure 4. The cross-section of the disc element at radius rand
angle </>=rc/2 by the plane ZOY is shown in the case of p' <0. The
Z-axis coincides with the direction of the black hole rotation. A and
P are the corresponding positions of the apocentre and the pericentre of the particle trajectories, TA and Tp. Vectors - VP and
- V)' are the pressure gradient (taken with minus) and its r projection acting to the matter at the point P.
'1';. The case shown, '1'1 = P> 0, 'I';=P' < 0, corresponds to
the position of the pericentre of the trajectories at point P
(at e > 0) and the apocentre at point A (e < 0), see equation
(44) below. The curves Tp and TA are the trajectories of gas
particles at the top and bottom surfaces of the disc. The
non-zero projection of the pressure gradient on to the r-axis
produces the force - VrP acting on the gas.
The pressure gradient term (in the absence of the relativistic correction that occurs in 43) leads to the resonance
and to the absence of any stationary solution of (43). Taking
into account both effects, we find that the relativistic correction suppresses the resonance and allows trajectories to
be aligned ellipses t1 = e sin Qt with small eccentricity e,
where
r'l';
e=-e.
6M
(44)
Thus the twisted disc consists of the ellipses with small
eccentricities e, which are the odd functions of height e and
depend on r due to the disc twist r'l';. The dependence of
eccentricities on r leads to the density perturbation PI.
When the eccentricities increase with decreasing r the particle trajectories concentrate near the apocentres of the
ellipses producing the density excess PI > 0 there, and
deviate near the pericentre of the ellipses producing the
lack of density PI < 0 (see Fig. Sa). In the opposite case of
decreasing eccentricity with decreasing r we have PI > 0
near pericentre and PI < 0 near apocentre (see Fig. Sb).
Now we consider the force balance equation (27). Both
terms in (27) depend on qJ identically. We consider force
balance for the angle cp = 1t/2, where we may have either the
apocentre (at e'l'; > 0) or the pericentre (at
e'l'; < 0). At this point F, > 0 (when '1'1> 0, a > 0), and we
need to have the negative e projection of Newtonian force,
F grav = SdePI (a «P/o 0 = - Fl;' Since the Newtonian acceleration o«P/oe is always directed to the central plane, e =0, we
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.285..394I
400 P. B. Ivanov and A. F. Illarionov
Finally, we estimate the ,-dependence of Fgmv and of F,.
From equations (17), (30) and (31) we have Fgravcx
CI~r'l';'and from equation (10) F,CXC z,-3.5'1'1' where C I,
C 2 :::::: constant. Therefore we write the approximate expression for the balance equation in the form
(a)
p
+A'A
(46)
where the ratio C 2/C I should be positive for the validity of
the inequalities (21). Obviously we have the oscillations of
the inclination angle with frequency CX,-9/4 sharply increasing with decreasing r (as described by equation 38).
7 RESTRICTIONS
(b)
A'
Solutions (34) and (37) are applicable only for sufficiently
small values of initial inclination angle f3 ( (0). The shear
velocities (29) and (30) are proportional to f3', increase with
increasing of f3( (0) and lead to a shear instability (Kumar
1990; Coleman & Kumar 1993) when
p'p+
A
(47)
Figure 5. Two possible configurations of aligned ellipses. In our
case we have a black hole at the focus. (a) The eccentricity
increases with decreasing impact parameter. This leads to a concentration of trajectories ( + ) at apocentres A', A and a deviation
( - ) at pericentres P', P. (b) The eccentricity decreases. This leads
to the concentration of trajectories ( + ) at pericentres P', P and
deviation ( - ) at apocentres A', A.
where VI = .Jv~ I + V;I and Vs = ( ~ *1,) v"'o is the sound velocity.
In the case of large ex> 15 4/5 (see equation 40), we find that
solution (34) is stable whenz
f3( (0) < 2ex.
(48)
In the opposite case of small viscosity the shear velocities
increase with decreasing' due to oscillations (see equations
29, 30 and 37, 38), and always lead to instability at small
radii:
z
(49)
The relativistic oscillations begin when, <Rz, and we find
from the condition < R z that the oscillating disc is stable
when the initial angle is small:
'f
f3( (0) < 215 4/5 •
o
Figure 6. The same as Fig. 3. C is the position of the centre of
gravity on a disc element shifted to the positive direction ~ > 0 from
the geometrical centre due to the density perturbations PI' Fg,av is
the ~ component of Newtonian force, and F, is the gravitomagnetic
force.
should have PI> 0 when ~ > 0 and PI < 0 when ~ < 0 (see
Fig. 6). Let us consider the case '1'1 > 0, '1'; < 0 in detail. For
this case we have the pericentre for ~ > 0 and the apocentre
for ~ < 0 at q> = n12. As follows from our previous discussion
the eccentricity cx '1'; has to decrease with decreasing' in
order to provide the necessary distribution of the perturbed
density and then we have '1';' < O. The cases '1'1> 0, '1'; > 0;
'1'1 < 0, '1'; < 0; '1'1 < 0 and '1';> 0 can be considered by the
same way. In general we have
'1';' < 0 when '1'1 > 0 and
'I'~>
0 when '1'1 < O.
(45)
(50)
The other restrictions come from stability analysis of the
twisted discs in the case of ex < b. Papaloizou & Lin (1995)
showed that twist waves propagate over a Newtonian disc.
In our case of a relativistic twisted accretion disc this effect
also takes place (Demianski & Ivanov 1997). The effect of
the persistent sources of these waves might prevent the disc
relaxation to our stationary oscillatory solution (37) if the
amplitude of the waves is greater than the amplitude of the
oscillations.
8 DISCUSSION
We have derived the stationary twist equation for the
twisted accretion disc around a Kerr black hole in the twisting coordinate system. We take into account the post2Note that this result differs from a 2 proportionality (Kumar 1988;
Coleman & Kumar 1993).
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1997MNRAS.285..394I
The twisted disc around a Kerr black hole
Newtonian relativistic corrections to the equations of
motion.
We have found the radial relativistic oscillations of the
inclination angle when IX < (j4/5 for the stationary twisted
accretion disc with low viscosity.
In general, let us consider a low viscous twisted accretion
disc in a slightly modified Newtonian potential and some
force F~ acting perpendicular to the disc rings. The modification of the potential leads to the precession of the free
particle orbital axis which is characterized by precession
vector WI while the presence of F~ leads to the precession
of the orbit plane which is characterized by precession vector W 2• We suppose that if the precessions corotate
[(WI· ( 2) > 0] we have oscillations of the inclination angle.
In the opposite case (for example the oppositely rotating
black hole a < 0) we have the smooth non-oscillating
dependence of p on r.
The shear velocities lead to the shear instability at any
distances r when p( 00 ) > IX for the large values of IX, and
when r < rf for the low values of IX. This instability may
generate the turbulence in the disc and effectively increases
the IX parameter. When VI ~ V, at the height of disc atmosphere
the increment of instability growth on scale
is
comparable with the dynamical frequency Q (Coleman &
Kumar 1993). The calculations (Canuto, Goldman &
Hubickyj 1984) show that in this case the turbulent motion
develops and the resulting value of the IX parameter is about
1. Therefore the twisted accretion disc might be in a highly
turbulent state with IX '" 1 at radii smaller than rt . In this case
the oscillatory shape of the disc is damped.
e*,
e*
401
APPENDIX
For the derivation of the post-Newtonian corrections (25)
and (26) we should calculate rand cp components of
4-acceleration F= 'YuU (U is 4-velocity) in the corresponding approximation. We write the post-Newtonian metric in
the form
2M)2(
2M) dr 2-de 222
ds 2(
= 1---;dt - 1 +--;-r dcp ,
(AI)
where we neglect the twisting of the coordinate system and
the black hole rotation which are unimportant for our
purpose.
Using (AI) we calculate the cp and r components of F:
OU'"
1 0 2
F"'=U"'--+U'--r U"',
ocp
r2 or
From the normalization condition U·U. = 1
we have
(A2)
(IX
= 0, 1, 2, 3)
M r\U",)2
UO=l+-+---.
r
(A4)
2
Substituting (A4) into (A3) we obtain
(AS)
ACKNOWLEDGMENTS
We are grateful to D. A. Kompaneets for helpful discussions. We also thank M. Abramowicz, A. M. Beloborodov,
M. Demianski and G. S. Bisnovatyi-Kogan for fruitful conversations and D. I. Novikov for the help in computations.
PBI thanks the Department of Astronomy and Astrophysics
of Chalmers Technological Institute for hospitality. The
present paper is supported in part by the Russian Foundation of Fundamental Research (Grants 93-02-17059,
9302-02-2929, 95-02-06063), and in part by Grant MEZOOO
from International Science Foundation.
As was done in the main text we divide U, F into a main
part with index 0 (which describes the circular motion in the
field of post-Newtonian source) and a perturbed part with
index 1. From the equation F~ = 0 we determine the motion
of free particle on the circular orbit:
U;f=n(l+
3~).
(A6)
For the perturbed part we have, in the linear approximation,
(A7)
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Demiansky M., Ivanov P. B., 1997, A&A, submitted
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F;=U;f~
U~ -2U;fU j r(1- 3M).
ocp
r
(A8)
Projecting (A7) and (A8) on to an orthonormal basis, we
obtain
Wt =(l-
~)dt,
W'=(l+
~)dr, w~=de,
w"'=rdcp,
(A9)
and substituting (A6) we obtain
{o
F'=Q
-U'-2U"'+A'
1
ocp I
I
A
A
A
}
,
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
(AI0)
1997MNRAS.285..394I
402 P. B. Ivanov and A. F. Illarionov
_
{u~ 0 _
}
F"'=n
-+-U"'+A'"
1
2 OqJ 1
,
(All)
where
M
Ar=-
(3--+U'"
ou~ _)
r 2 OqJ
(AlO) and (All) correspond to the equations (21) and (22)
respectively.
In our calculations the post-Newtonian corrections
appear only in the combination (see 21, 22 and 28)
OA"')
_
of'''1 =n (Ar-2B=F'-2(A12)
1
OqJ
OqJ
,
(A14)
l'
(A13)
therefore for our purpose it is sufficient to replace the
components of the 4-velocity in (A12) and (A13) by their
classical limit,
(AIS)
and
Fit> U;
correspond to the basis (A9). The equations
to obtain (25) and (26).
© 1997 RAS, MNRAS 285, 394-402
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System