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A study on the oscillatory instability of a hot two-temperature accretion disk including advection
CHINESE ASTRONOMY AND PERGAMON
Chinese Astronomy and Astrophysics 24 (2000)
ASTROPHYSICS
23-29
A study on the oscillatory instability of a hot two-temperature accretion disk including advection t * DING Shi-xue1~2 ‘Department
YANG Lan-tian’
of P h y sits and Institute
of Astrophysics,
WANG Ding-xiong3 Huanzhong
Normal
University.
Wuhan 430079 2Department 3Department
of Physics,
of Physics, Huazhong
Xiangfan
University
College,
of Science
Xiangfan
441053
and Technology,
Wuhan
43007’4
Abstract The radial-azimuthal instability of a hot two-temperature accretion disc with advection is analyzed. After obtaining the dispersion equation, we find that advection and viscous force have an influence on the stability of acoustic modes, but not on the stability of thermal and viscous modes. We also find azimuthal perturbations to affect the stability of all the modes. This model may be useful in explaining the quasi-periodic or periodic luminosity variations in various astronomical objects. Key words:
accretion disk-instabilities-advection-viscous
1.
After the establishment
force
INTRODUCTION
of the theory of accretion disk, the question of the disk’s stability
has become an important topic in the theory. For instability of the accretion disk can well explain the phenomenon of periodic and quasi- periodic light variations in X-ray binaries and active galactic nuclei (AGN). Under the standard model of thin accretion diskf’], viscosity heating reaches equilibrium through radiation cooling. However, if the local radiation cooling rate in the accretion stream is lower than the rate of viscous dissipation, then, according to the energy equation for the accretion disk, the energy that is not radiated away will be stored in the accretion stream in the form of entropy and be transmitted radially inwards. t Supported by National Natural Science Foundation Received 199811-20; revised version 1999-02-02 * A translation of Acta Astrophys. Sin. Vol. 19, No. 4, pp. 369-374,
1999
0,275-1062/00/t - see front matter @ 2000 Elsevier Science B. V. All rights reserved. PII: SO275-1062(00)00022-9
DING
24
Shi-xue
et al. / Chinese
This is known as “advection”.
Astronomy
For example,
and Astrophysics
for an optically
24 (2000)
23-29
thin disk, because
the radiation
cooling rate is very small, a large amount of viscous energy is carried off as entropy by the accreting matter and transported inwards along the radius. Therefore, the effect of advection cannot be neglected. In recent years, many authors have carried out detailed studies of optically thin or thick disks containing advection l2-4l. Advection-dominated accretion disk models have successfully been applied in the explanation of observed low- and highluminosity phenomenal’s]. In 1996, Wu et al.fq studied the local instability of accretion disks containing advection. They pointed out that, for geometrically thin disks, even a very small advection can have a large effect on the accoustic mode&q. In another direction, because the standard accretion disk model of Shakura and Sunyaevl’ was unable to explain the X-ray spectra observed in black hole candidates such as Cyg X1, Shapiro et al. in 1976 set up the optically thin, two-temperature accretion disk model. This model successfully explained the X-ray spectrum between 8 - 500 keV observed in Cyg X-11*1. In recent years this model also been widely used in the study of X-ray binary and AGN modelslg-lll. Recently, Wu studied the instability of two-temperature accretion disks containing advection and obtained some very significant results1121. All the above studies considered only radial perturbations. However, analysis of the instability of geometrically thin disks should, because of shear flow, include both radial and azimuthal perturbations. Mckee carried out a study in this direction on radiationdominated accretion disks and found that the viscous instability that exists when only radial perturbation
is considered
ceases to exist when azimuthal
perturbation
is consideredl131.
In
1992, Cao et a1.[141studied isothermal disks for radial-azimuthal perturbations. In 1995, Wu et a1.1151studied in detail non-isothermal disks for radial-azimuthal perturbations. In 1998 we made a detailed analysis of radial, axial and azimuthal instabilities of the isothermal, thin accretion disk embedded in a three-dimensional magnetic field. This paper is based on the above studies and investigates the radial and azimuthal instabilities for a two-temperature disk containing advection. First we give the basic equations for the accretion disk; next, we use the method of perturbation to derive the dispersion relation results
and carry out a stability
analysis
with numerical
calculation;
lastly
we discuss
the
obtained.
2. BASIC Here, we consider potentia11171,
a non-self
gravitating
EQUATIONS accretion
disk.
We take the pseudo
Newtonian
where M is the mass of the central body, rg the Schwarzschild radius, R isthe distance of the accretion stream from the central body, R = (T~+Z~)'/~ in the cylindrical coordinate system (T, 4, 2). Neglecting the axial component, the hydrodynamic equations after integration in the vertical direction are
DING
Shi-me
et al. / Chinese
Astronomy
and Astrophysics
where v,., v+, s2 are, respectively, the radial, azimuthal Kepler angular velocity defined by fl& = (LW/T~T),=O,
24 (2000)
and angular
25
23-99
velocities,
RK is the
P, T, C represent, respectively, the pressure, temperature and surface density, CV is the heat capacity, and I?3 is a constant depending on ,0. For the two-temperature disk, j3 = l,C, = 3P/2pT,p3 = 513, P = pk(T~ + Te)/mp, TI, T,, mp being the ion and electron temperatures and the ion mass, and F, is the raidal viscous force, defined byll’l F
where u, is the radial viscosity cosities to be of the o-viscosity
Y
= a[.$+~]
the scale height. disk, is takerill
3. PERTURBATION of instability
tions on all the physical
T
aT
(6)
’
coefficient. We assume both the radial and azimuthal visform, V, = vu, v = ac, H, c, being the sonic speed defined
by cf = P/p, and H(= c#/Cl~) being cooling, which, for the twotemperature
For the study
-_~2% a(V,c) aT
AND
we carried
In (5), Q- represents the radiation to be Q- 0: C7/5T11/5.
DISPERSION
out a local instability
EQUATIONS analysis
on assuming
perturba-
C, fl, v,, T to be of the form
quantities,
SQ = Q’exp [i(wt + n+ -
HT)]
,
(7)
wave where A$= 2n/X) is th e radial component of the wave number, n is angular perturbation number. The local approximation implies x < T, and since V, - crcf/~fiK, the vertically integrated equation should satisfy kV, < SzK. This condition can be written as T
x
> H > 2na-
z
H T
.
For a geometrically thin disk, this condition is well satisfied even if we let X/H range from 1 to 80; but for a “slim” disk, where H/T 5 1 we need to have sufficiently small cr. Because of this, the range of A/H has to be rather small; for example, if (Y = 0.001, the range of X/H has to be about from 0.01 to 2. In order to obtain reasonable results, our discussion of second in this paper will be carried out under these limits. Neglecting small quantities order and higher, we have the linear perturbation equations, 6 sv, 6C ~_-ii-=--
c
H
QKT
(9)
26
DING
Shi-me
et al. / Chinese
Astronomy
where 5 = u/n ~,c=I(w-kv,+nR,fLR/R
and Astrophysics
24 (ZOOO) .??-29
K, ii = H/r, c = IcH, g = it2/2i? - 2fi, ii = (K 2 = 2R(2R
rc/Cl~, K being the cyclic frequency number; and q is the ratio of the adevection
+ ~(afi/&));
energy
m(=
]v,.]/c~) is the Mach
to the viscous dissipation
energy,
that
is,
ar - (I-3 cvixr fT For a disk dominated
by radiation
dominated
1.
(9)-(12)
disk, q -
we obtain
l)Tv,~]
cooling,
Equating
q is approximately
dispersion + a2C3
(13)
.
zero, while for an advection-
to zero the determinant
a fourth-order ala4
= qL+.~)2
of the linear
set of equations
equation,
+ a3C2
+ u5 = 0 ,
+ a46
(14)
where ar=$, _
a2=(Y[E2(%+27])-(~+3q)g2] a3 = aEg2{0~[-( a4 = 2ifh2cg + dg(
,
$7) + l)( f + ;q) + 21 - i$},+
5 + %)(ae2)277 - icYG& + $(“E2)2
g - 56) + +4
+ 4?pwe3gn
factor
for n = 0,~ = 1, equation
We now carry out numerical thin
NUMERICAL calculations
tw-temperature
disk,
g
- 2fi)
- ($ + $7)k2 + $7 - l)E2} , 1)++ncx3g3e3.
(14) agrees with Wu’s resultsl12l
with
left out.
4.
cally
+ ;l?m(
)
~ezg2(l-q)]+nLieag2~(q-l)(~-2~)+~(q-
It is easily seen that, the heat diffusion
of the
+ $”
3% m +cxg2{-$($
a5 z (LYEz)[~(1-q)hg3-
tures
277”2E4 + $2
we take
8
disk. =
CALCULATION of the dispersion
We take
0.01,(-r
=
fi =
0.01.
i?. =
equation 1,m
=
(14) for different 0.01.
For the geometrically
struc-
for the geometrislim disk,
we take
ti = 0.6, (Y = 0.001. Fig. l(a)
gives the variation
disk with no advection dashed thermal,
lines).
of Re(8)
(q = 0, solid lines)
with wavelength
We take n = 1,~ = 1. The
outward
(0)
for the thin, cooling-dominated
and with a small amount
and inward (I) modes,
curves labelled respectively.
of advection
(q = 0.01,
1, 2, 3, 4 refer to the viscous, We see that the viscous
mode is
DING
Shi-xue
et al. / Chinese
Astronomy
and Aatmphysics
27
24 (.%X70) 29-29
(a) 2
s 4
0.00
-0.05
5o:oo
, LiE
0.00
0.00
50.00
h/H
1.00
Q
1
3’ N /
/
/
m
0.05
(4 / ’ /
3
/
0.00
0.00
d
1,--L
,A_
4’-
-1.00
-
-_ _
-0.05
0.00
l.bO
2.00
0.00
50:oo
m
Q
0.00
$
x/H
F 2
_
3 -
lp
4.00 2.00
1.00 ?JH
Fig. 1
!J; I
0.00
0.00
,
,
,
,
,
,
,
,
I
1
1
,
1.00
I
1
1
,
2.00
h/H
The influence of various parameters on the stability of accretion disks.
(a), (b), (d)--geometrically cooling-dominated
thin, cooling-dominated
disk. (c), (f)-g
disks. (e)-geometrically
eometrically slim, advection-dominated
slim, disk.
28
always stable, the acoustic
DING
Shi-zue
the thermal mode branches
et al. / Chinese
Astronomy
mode, always unstable,
and Astrophysics
24 (2000) 23-29
and even for a small mount
out into two, the 0 mode becoming
unstable,
of advection,
and the I mode,
more stable. Figs. l(b) and l(c) refer to the effect of an azimuthal perturbation on the various modes. We take 7 = 1. The solid an dashed lines correspond to n = 1 and n = 3, respectively. Fig. l(b) is for the thin, cooling-dominated two-temperature disk with q = 0.01. We see that, as the azimuthal perturbation increases, the viscous mode changes from stability to instability, the thermal mode becomes more unstable, the I mode becomes more stable and the 0 mode changes from being unstable
to being stable.
Fig. l(c) is for the slim, advection-
dominated disk with q = 0.99. We see, here, with increasing azimuthal perturbation, the viscous and I modes become more stable, while the 0 mode becomes more unstable. Figs. l(d)-l(f) illustrate the effect of radial viscous force on the various modes (n = 1). The solid and dashed lines correspond to q = 0 and 77 = 1, respectively. Fig. l(d) is for the thin, cooling- dominated two-temperature disk with q = 0.01. Fig. l(e) is for the slim, cooling-dominated disk again with q = 0.01. fig. l(f) is for the slim, advection-dominated disk with q = 0.99. These figures show that a radial viscous force has no effect on the stability character of the viscous and thermal modes, while it makes the acoustic modes more stable.
5. DISCUSSION We have studied the radial and azimuthal instabilities of two-temperature disks containing advection. The results show that, for radiation cooling dominated, geometrically thin disks, the viscous mode is always stable, and the thermal mode is unstable. The I mode is always stable while the 0 mode is stable for short wavelengths and unstable for long wavelengths. (Fig. l(a)). The same figure shows also that the presence of advection has a large influence on the acoustic mode: even for a small amount of advection, the acoustic mode splits into two branches, the 0 mode becoming more unstable, the I mode, more stable. These conclusions are consistent with the relevant results in Wu’s paper 1121. For the cooling-dominated, slim disk, the thermal mode is always unstable, while the viscous and acoustic modes are always stable (Fig. l(e)). For the advection-dominated slim disk, the viscous and I modes are always stable while the 0 mode is stable in the range of short wavelengths (Fig. l(c)). Meanwhile, we see from Figs. l(b) and l(c) that azimuthal perturbation has a larger effect on the stability of the different modes. For the several radiation cooling dominated, geometrically thin accretion disks, an increase in azimuthal perturbation causes the thermal mode to become more unstable and the I mode to be more stable. It is worth noting that the 0 mode changes from being unstable to being stable, while the viscous mode changes in the opposite direction, from being stable to being unstable. (Fig. l(b)). The stability character of these modes in this case differs somewhat from the results of Fig. l(b) of Wu’s paper, where the 0 mode was shown to be unstable and the viscous mode to be stable. The cause for the difference lies in the fact that, here, we consider not only radial perturbation but also azimuthal perturbation and we neglect the effect of thermal diffusion. For the several advection-dominated geometrically slim disks, an increases in azimuthal perturbation causes the viscous and I modes to be more stable, and the 0 mode to be more unstable. These points
DING
demonstrate
Shi-sue
that
et al. / Chinese
our inclusion
Astronomy
of azimuthal
and Astrophysics
perturbation
24 (2000)
in the study
29
.%?J-89
of two-temperature
accretion disks with advection is meaningful. Besides, Figs l(d)-l(f) show that, for both thin and slim disks, radial viscous force does not change the stability character of the thermal and viscous modes, while it has a larger effect on the acoustic modes, that is, its introduction makes the acoustic modes more stable. The physical reason is that the instability of the acoustic mode propagates along the radial direction and the waves are longitudinal waves. The disk matter is being compressed and stretched in the radial direction, and a radial viscous force dampens the propagation of such a perturbation and dissipates its energy. In sum, when both radial and azimuthal perturbations are considered in regard to the instability of a twotemperature accretion disk containing advection, there exists instability in the acoustic and thermal modes. The former is advantageous for an explanation of the periodic and quasi- periodic light variations in AGN, and the latter may appear in the inner unstable regions of AGN, X-ray binaries and cataclysmic variable stars. For studying light variation
phenomena
in these objects,
further
discussion
on the evolving
structure
of the
disk is very necessary. ACKNOWLEDGEMENT for helpful discussion.
We thank
Dr WU Xue-bing
of Beijing Astronomical
References 1
Shakura
2
Abrmowin
N. I., Sunyaev
R. A., A&A,
1973, 24, 317
3
Narayan
4
Chen X., MNRAS,
5
Narayan
R., Mcclintock
6
Narayan
R., Yi I., Machadevan
7
Wu X. B., Li Q. B., ApJ, 1996, 469, 776
M. A., Chen X., Kato S. et al., ApJ, 1995, 438, L37 R., Yi I., ApJ, 1994, 428, L13
8
Shapiro
9
Kusunose
1995, 275, 641 J. E., Yi I., ApJ, 1996,457,
S. L., Lightman M., Takahara
R., Nature,
A. P., Eardley F., PASJ,
10
Write T. R., Light-
11
Luo C., Liang E. P., MNRAS,
12
Wu X. B., MNRAS,
13
M&e
14
Cao X. W., A&A,
821
1995, 374,623
D. N., ApJ, 1976, 204, 187 (SLE)
1988, 40, 709
A. P., ApJ, 1989, 40, 1024 1994, 266, 386
1997, 292,113
M. R., A&A,
1990, 235, 521 1992, 262, 350
15
Wu X. B., Li Q. B., Zhao Y. H. et al., ApJ, 1995, 442, 736
16
DING
17
Paczyanski
18
Papaloizzou
Shi-xue,
YANG
B., Wiita
Lan-tian, P. J., A&A,
J. C. B., Stanley
LU Ye et al., AApS
1998, 18(4),
1980, 88, 23
G. Q. G., MNRAS,
1986, 220, 593
352
Observatory
Chinese Astronomy and Astrophysics 24 (2000)
ASTROPHYSICS
23-29
A study on the oscillatory instability of a hot two-temperature accretion disk including advection t * DING Shi-xue1~2 ‘Department
YANG Lan-tian’
of P h y sits and Institute
of Astrophysics,
WANG Ding-xiong3 Huanzhong
Normal
University.
Wuhan 430079 2Department 3Department
of Physics,
of Physics, Huazhong
Xiangfan
University
College,
of Science
Xiangfan
441053
and Technology,
Wuhan
43007’4
Abstract The radial-azimuthal instability of a hot two-temperature accretion disc with advection is analyzed. After obtaining the dispersion equation, we find that advection and viscous force have an influence on the stability of acoustic modes, but not on the stability of thermal and viscous modes. We also find azimuthal perturbations to affect the stability of all the modes. This model may be useful in explaining the quasi-periodic or periodic luminosity variations in various astronomical objects. Key words:
accretion disk-instabilities-advection-viscous
1.
After the establishment
force
INTRODUCTION
of the theory of accretion disk, the question of the disk’s stability
has become an important topic in the theory. For instability of the accretion disk can well explain the phenomenon of periodic and quasi- periodic light variations in X-ray binaries and active galactic nuclei (AGN). Under the standard model of thin accretion diskf’], viscosity heating reaches equilibrium through radiation cooling. However, if the local radiation cooling rate in the accretion stream is lower than the rate of viscous dissipation, then, according to the energy equation for the accretion disk, the energy that is not radiated away will be stored in the accretion stream in the form of entropy and be transmitted radially inwards. t Supported by National Natural Science Foundation Received 199811-20; revised version 1999-02-02 * A translation of Acta Astrophys. Sin. Vol. 19, No. 4, pp. 369-374,
1999
0,275-1062/00/t - see front matter @ 2000 Elsevier Science B. V. All rights reserved. PII: SO275-1062(00)00022-9
DING
24
Shi-xue
et al. / Chinese
This is known as “advection”.
Astronomy
For example,
and Astrophysics
for an optically
24 (2000)
23-29
thin disk, because
the radiation
cooling rate is very small, a large amount of viscous energy is carried off as entropy by the accreting matter and transported inwards along the radius. Therefore, the effect of advection cannot be neglected. In recent years, many authors have carried out detailed studies of optically thin or thick disks containing advection l2-4l. Advection-dominated accretion disk models have successfully been applied in the explanation of observed low- and highluminosity phenomenal’s]. In 1996, Wu et al.fq studied the local instability of accretion disks containing advection. They pointed out that, for geometrically thin disks, even a very small advection can have a large effect on the accoustic mode&q. In another direction, because the standard accretion disk model of Shakura and Sunyaevl’ was unable to explain the X-ray spectra observed in black hole candidates such as Cyg X1, Shapiro et al. in 1976 set up the optically thin, two-temperature accretion disk model. This model successfully explained the X-ray spectrum between 8 - 500 keV observed in Cyg X-11*1. In recent years this model also been widely used in the study of X-ray binary and AGN modelslg-lll. Recently, Wu studied the instability of two-temperature accretion disks containing advection and obtained some very significant results1121. All the above studies considered only radial perturbations. However, analysis of the instability of geometrically thin disks should, because of shear flow, include both radial and azimuthal perturbations. Mckee carried out a study in this direction on radiationdominated accretion disks and found that the viscous instability that exists when only radial perturbation
is considered
ceases to exist when azimuthal
perturbation
is consideredl131.
In
1992, Cao et a1.[141studied isothermal disks for radial-azimuthal perturbations. In 1995, Wu et a1.1151studied in detail non-isothermal disks for radial-azimuthal perturbations. In 1998 we made a detailed analysis of radial, axial and azimuthal instabilities of the isothermal, thin accretion disk embedded in a three-dimensional magnetic field. This paper is based on the above studies and investigates the radial and azimuthal instabilities for a two-temperature disk containing advection. First we give the basic equations for the accretion disk; next, we use the method of perturbation to derive the dispersion relation results
and carry out a stability
analysis
with numerical
calculation;
lastly
we discuss
the
obtained.
2. BASIC Here, we consider potentia11171,
a non-self
gravitating
EQUATIONS accretion
disk.
We take the pseudo
Newtonian
where M is the mass of the central body, rg the Schwarzschild radius, R isthe distance of the accretion stream from the central body, R = (T~+Z~)'/~ in the cylindrical coordinate system (T, 4, 2). Neglecting the axial component, the hydrodynamic equations after integration in the vertical direction are
DING
Shi-me
et al. / Chinese
Astronomy
and Astrophysics
where v,., v+, s2 are, respectively, the radial, azimuthal Kepler angular velocity defined by fl& = (LW/T~T),=O,
24 (2000)
and angular
25
23-99
velocities,
RK is the
P, T, C represent, respectively, the pressure, temperature and surface density, CV is the heat capacity, and I?3 is a constant depending on ,0. For the two-temperature disk, j3 = l,C, = 3P/2pT,p3 = 513, P = pk(T~ + Te)/mp, TI, T,, mp being the ion and electron temperatures and the ion mass, and F, is the raidal viscous force, defined byll’l F
where u, is the radial viscosity cosities to be of the o-viscosity
Y
= a[.$+~]
the scale height. disk, is takerill
3. PERTURBATION of instability
tions on all the physical
T
aT
(6)
’
coefficient. We assume both the radial and azimuthal visform, V, = vu, v = ac, H, c, being the sonic speed defined
by cf = P/p, and H(= c#/Cl~) being cooling, which, for the twotemperature
For the study
-_~2% a(V,c) aT
AND
we carried
In (5), Q- represents the radiation to be Q- 0: C7/5T11/5.
DISPERSION
out a local instability
EQUATIONS analysis
on assuming
perturba-
C, fl, v,, T to be of the form
quantities,
SQ = Q’exp [i(wt + n+ -
HT)]
,
(7)
wave where A$= 2n/X) is th e radial component of the wave number, n is angular perturbation number. The local approximation implies x < T, and since V, - crcf/~fiK, the vertically integrated equation should satisfy kV, < SzK. This condition can be written as T
x
> H > 2na-
z
H T
.
For a geometrically thin disk, this condition is well satisfied even if we let X/H range from 1 to 80; but for a “slim” disk, where H/T 5 1 we need to have sufficiently small cr. Because of this, the range of A/H has to be rather small; for example, if (Y = 0.001, the range of X/H has to be about from 0.01 to 2. In order to obtain reasonable results, our discussion of second in this paper will be carried out under these limits. Neglecting small quantities order and higher, we have the linear perturbation equations, 6 sv, 6C ~_-ii-=--
c
H
QKT
(9)
26
DING
Shi-me
et al. / Chinese
Astronomy
where 5 = u/n ~,c=I(w-kv,+nR,fLR/R
and Astrophysics
24 (ZOOO) .??-29
K, ii = H/r, c = IcH, g = it2/2i? - 2fi, ii = (K 2 = 2R(2R
rc/Cl~, K being the cyclic frequency number; and q is the ratio of the adevection
+ ~(afi/&));
energy
m(=
]v,.]/c~) is the Mach
to the viscous dissipation
energy,
that
is,
ar - (I-3 cvixr fT For a disk dominated
by radiation
dominated
1.
(9)-(12)
disk, q -
we obtain
l)Tv,~]
cooling,
Equating
q is approximately
dispersion + a2C3
(13)
.
zero, while for an advection-
to zero the determinant
a fourth-order ala4
= qL+.~)2
of the linear
set of equations
equation,
+ a3C2
+ u5 = 0 ,
+ a46
(14)
where ar=$, _
a2=(Y[E2(%+27])-(~+3q)g2] a3 = aEg2{0~[-( a4 = 2ifh2cg + dg(
,
$7) + l)( f + ;q) + 21 - i$},+
5 + %)(ae2)277 - icYG& + $(“E2)2
g - 56) + +4
+ 4?pwe3gn
factor
for n = 0,~ = 1, equation
We now carry out numerical thin
NUMERICAL calculations
tw-temperature
disk,
g
- 2fi)
- ($ + $7)k2 + $7 - l)E2} , 1)++ncx3g3e3.
(14) agrees with Wu’s resultsl12l
with
left out.
4.
cally
+ ;l?m(
)
~ezg2(l-q)]+nLieag2~(q-l)(~-2~)+~(q-
It is easily seen that, the heat diffusion
of the
+ $”
3% m +cxg2{-$($
a5 z (LYEz)[~(1-q)hg3-
tures
277”2E4 + $2
we take
8
disk. =
CALCULATION of the dispersion
We take
0.01,(-r
=
fi =
0.01.
i?. =
equation 1,m
=
(14) for different 0.01.
For the geometrically
struc-
for the geometrislim disk,
we take
ti = 0.6, (Y = 0.001. Fig. l(a)
gives the variation
disk with no advection dashed thermal,
lines).
of Re(8)
(q = 0, solid lines)
with wavelength
We take n = 1,~ = 1. The
outward
(0)
for the thin, cooling-dominated
and with a small amount
and inward (I) modes,
curves labelled respectively.
of advection
(q = 0.01,
1, 2, 3, 4 refer to the viscous, We see that the viscous
mode is
DING
Shi-xue
et al. / Chinese
Astronomy
and Aatmphysics
27
24 (.%X70) 29-29
(a) 2
s 4
0.00
-0.05
5o:oo
, LiE
0.00
0.00
50.00
h/H
1.00
Q
1
3’ N /
/
/
m
0.05
(4 / ’ /
3
/
0.00
0.00
d
1,--L
,A_
4’-
-1.00
-
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m
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x/H
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lp
4.00 2.00
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Fig. 1
!J; I
0.00
0.00
,
,
,
,
,
,
,
,
I
1
1
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I
1
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The influence of various parameters on the stability of accretion disks.
(a), (b), (d)--geometrically cooling-dominated
thin, cooling-dominated
disk. (c), (f)-g
disks. (e)-geometrically
eometrically slim, advection-dominated
slim, disk.
28
always stable, the acoustic
DING
Shi-zue
the thermal mode branches
et al. / Chinese
Astronomy
mode, always unstable,
and Astrophysics
24 (2000) 23-29
and even for a small mount
out into two, the 0 mode becoming
unstable,
of advection,
and the I mode,
more stable. Figs. l(b) and l(c) refer to the effect of an azimuthal perturbation on the various modes. We take 7 = 1. The solid an dashed lines correspond to n = 1 and n = 3, respectively. Fig. l(b) is for the thin, cooling-dominated two-temperature disk with q = 0.01. We see that, as the azimuthal perturbation increases, the viscous mode changes from stability to instability, the thermal mode becomes more unstable, the I mode becomes more stable and the 0 mode changes from being unstable
to being stable.
Fig. l(c) is for the slim, advection-
dominated disk with q = 0.99. We see, here, with increasing azimuthal perturbation, the viscous and I modes become more stable, while the 0 mode becomes more unstable. Figs. l(d)-l(f) illustrate the effect of radial viscous force on the various modes (n = 1). The solid and dashed lines correspond to q = 0 and 77 = 1, respectively. Fig. l(d) is for the thin, cooling- dominated two-temperature disk with q = 0.01. Fig. l(e) is for the slim, cooling-dominated disk again with q = 0.01. fig. l(f) is for the slim, advection-dominated disk with q = 0.99. These figures show that a radial viscous force has no effect on the stability character of the viscous and thermal modes, while it makes the acoustic modes more stable.
5. DISCUSSION We have studied the radial and azimuthal instabilities of two-temperature disks containing advection. The results show that, for radiation cooling dominated, geometrically thin disks, the viscous mode is always stable, and the thermal mode is unstable. The I mode is always stable while the 0 mode is stable for short wavelengths and unstable for long wavelengths. (Fig. l(a)). The same figure shows also that the presence of advection has a large influence on the acoustic mode: even for a small amount of advection, the acoustic mode splits into two branches, the 0 mode becoming more unstable, the I mode, more stable. These conclusions are consistent with the relevant results in Wu’s paper 1121. For the cooling-dominated, slim disk, the thermal mode is always unstable, while the viscous and acoustic modes are always stable (Fig. l(e)). For the advection-dominated slim disk, the viscous and I modes are always stable while the 0 mode is stable in the range of short wavelengths (Fig. l(c)). Meanwhile, we see from Figs. l(b) and l(c) that azimuthal perturbation has a larger effect on the stability of the different modes. For the several radiation cooling dominated, geometrically thin accretion disks, an increase in azimuthal perturbation causes the thermal mode to become more unstable and the I mode to be more stable. It is worth noting that the 0 mode changes from being unstable to being stable, while the viscous mode changes in the opposite direction, from being stable to being unstable. (Fig. l(b)). The stability character of these modes in this case differs somewhat from the results of Fig. l(b) of Wu’s paper, where the 0 mode was shown to be unstable and the viscous mode to be stable. The cause for the difference lies in the fact that, here, we consider not only radial perturbation but also azimuthal perturbation and we neglect the effect of thermal diffusion. For the several advection-dominated geometrically slim disks, an increases in azimuthal perturbation causes the viscous and I modes to be more stable, and the 0 mode to be more unstable. These points
DING
demonstrate
Shi-sue
that
et al. / Chinese
our inclusion
Astronomy
of azimuthal
and Astrophysics
perturbation
24 (2000)
in the study
29
.%?J-89
of two-temperature
accretion disks with advection is meaningful. Besides, Figs l(d)-l(f) show that, for both thin and slim disks, radial viscous force does not change the stability character of the thermal and viscous modes, while it has a larger effect on the acoustic modes, that is, its introduction makes the acoustic modes more stable. The physical reason is that the instability of the acoustic mode propagates along the radial direction and the waves are longitudinal waves. The disk matter is being compressed and stretched in the radial direction, and a radial viscous force dampens the propagation of such a perturbation and dissipates its energy. In sum, when both radial and azimuthal perturbations are considered in regard to the instability of a twotemperature accretion disk containing advection, there exists instability in the acoustic and thermal modes. The former is advantageous for an explanation of the periodic and quasi- periodic light variations in AGN, and the latter may appear in the inner unstable regions of AGN, X-ray binaries and cataclysmic variable stars. For studying light variation
phenomena
in these objects,
further
discussion
on the evolving
structure
of the
disk is very necessary. ACKNOWLEDGEMENT for helpful discussion.
We thank
Dr WU Xue-bing
of Beijing Astronomical
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