- Email: [email protected]
Diskoseismology: Probing black holes and their accretion disks
.nlmmnmmmq
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
Diskoseismology: Probing Black Holes and Their Accretion Disks D. E. Lehr a* aDepartment of Physics, Stanford University, Stanford, CA 94305-4060, USA W e review the relativisticresults for diskoseismic modes of oscillationwhich are trapped within thin accretion disks by non-Newtonian gravitational properties of a black hole. Predicted frequencies are calculated for the potentially most observable modes, "internal gravity" modes and "corrugation" modes. The most definitive property of these two classes of modes is that the resulting eigenfrequencies depend almost entirely upon only the mass and angular m o m e n t u m of the black hole. Such features m a y have been detected by R X T E in the power spectra of the luminosity modulations of the two galactic microquasars, G R S 1915+105 and G R O J165540. In the former system, we consider the possibility that this 67 Hz feature can be attributed to a g - m o d e in an accretion disk about a 10.6 M o (nourotating) to 36.3 M o (maximally rotating) black hole. In the latter system, identificationof the fundamental g - m o d e with the 300 Hz feature implies a black hole angular m o m e n t u m approximately 93% of maximum.
1. I N T R O D U C T I O N Within the region of strong gravity that exists in accretion disks near black holes, general relativity can trap normal modes of oscillation akin to the helioseismology modes that exist within the sun. These modes do not exist in Newtonian gravity; thus, they are a good tool to probe the nature of strong gravitational fields. The resulting spectrum of these so-called "diskoseismic" modes can determine both the mass M and the dimensionless angular momentum parameter a = c J / G M 2 of the central black hole. In addition, the physical structure of the accretion disk can be reflected in the frequencies and widths of these modes. We review the relativistic results for these trapped oscillations and discuss their possible observation by RXTE in the galactic microquasars, GRS 1915+105 and GRO J1655-40. It should be noted that these modes might also be found in an accretion disk surrounding a very weakly-magnetized, compact neutron star, such that the inner part of the disk which supports the modes is not disturbed by the magnetosphere or the stellar surface. The results obtained below apply to disks around slowly rotating neu*This research was supported in part by NASA GSRP Training Grant NGT5-50044. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PlI S0920-5632(98)00238-2
tron stars (a g 0.1) since their external metric is also Kerr through first order in a. (We note that a < 0.2 for most models of rotating neutron
stars.) 2. D I S K O S E I S M I C M O D E S
As first recognized by Kato and Fukue [1], oscillations can be trapped in the inner regions of accretion disks surrounding black holes by the general-relativistic modifications of the radial (~) and vertical (f~±) epicyclic frequencies of free-particle circular-orbit perturbations. These modes of oscillation are perturbations of the disk which are of the form f ( r , z ) e x p [ i ( a t + me)], where r, z , ¢ are cylindrical coordinates. With angular velocity f~(r), the corotation frequency is w(r) = a + mfL Modes are trapped because ~(r) reaches a maximum at small r and vanishes at the inner edge of the disk, thereby creating a resonant cavity. (This behavior in ~(r) is shown in Figure 1 along with other important orbital frequencies.) These modes represent temperature and pressure fluctuations in the disk which consequently modulate the luminosity of the disk at predictable frequencies. Three classes of modes have been identified: (g) Internal (gravity) modes are trapped where
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351 10o
(P) Acoustic (pressure) modes are trapped where w2 > g2. These modes extend a small distance from the inner edge of the disk and might also exist in the large outer region of the disk. These modes have considerable radial displacements.
a=O
10 "1
10-2
~2 ~Zi2
10-3 10 .4 10-5
,~i 5
0
~,
,
,
=
. . . .
=
10 r [GM/c2]
. . . .
15
20
100 a=0.5
10 -1 f~2
10-2
10-3 10 -4
10 .5
,
,
,
0
,
,
,
5
,
,
i
,
10 r [GM/c2]
,
,
,
,
15
,
,
20
100 :~
10 "~
~
10 .2
(c) The simplest corrugation modes are nonradial (m > 1) oscillations with w ~ fl±. When trapped near the inner edge of the disk, they represent nearly incompressible vertical displacements of the disk that resemble a small, tilted inner disk slowly precessing about the spin axis of the black hole. The m = 1 mode has a frequency approximating the Lense-Thirring frequency. 3. F R E Q U E N C Y
CALCULATIONS
a=0.998
10 .3 10 -4 K 2
lO-S
349
0
'
'
' 5
. . . .
1~0 . . . . r
[ G M / c
1~5 . . . .
20
2]
Figure 1. The radial dependence of the square of the fundamental free-particle frequencies that govern the modes of the disk: Keplerian (fl), and radial (~) and vertical (fl±) epicyclic. Three values of the black hole angular momentum parameter a = cJ/GM 2 are chosen.
w2 < a2, in the region where ~; achieves its maximum value and where the disk is hottest. The lowest modes can have significant vertical displacements and relatively large radial extents Ar ~ GM/c 2. We believe that these modes will produce the greatest luminosity modulations in the disk and are the most robust; thus, they could be the most observable class of modes. The nature of these modes is fundamentally different from stellar g - m o d e s and is not completely understood.
We have analyzed adiabatic perturbations to thin accretion disks in the Kerr metric [2]. Following the general relativistic perfect fluid perturbation formalism of Ipser and Lindblom [3], we expressed the adiabatic oscillations of all physical quantities in terms of a single scalar potential 5V(r, z) =_ 5p/p governed by a secondorder partial differential equation. The stationary (O/cOt = 0) and axisymmetric (cO/cO¢ = 0) unperturbed accretion disk was specified by the relativistic a-model [4,5]. The radial component of the velocity of the fluid was neglected, as was the gravitational field of the disk to good approximation. Since the effective radial wavelengths are significantly smaller than r, we adopted a WKB approach which allows approximate separation of the governing equations into their radial and vertical dependences. The radial component of the fluid perturbations satisfies the WKB relation
dr------5- +
•
]
- 1 (~2 _ w2)W = 0, (1)
where a(r) is inversely proportional to the speed of sound at the midplane and q(r) is the slowlyvarying separation function. The eigenfunction W(r) is proportional to a radial derivative of the potential ~V and to the radial component of the fluid displacement.
350
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
From the radial WKB equation (1) above we can see where each class of mode is trapped ( W - l d 2 W / d r 2 < 0). The p-modes, defined by @(f~±/w) 2 < 1, are trapped where w2 > ~2; g-modes, defined by @(f~±/w) 2 > 1, are trapped where w2 < ~2. The c-modes are defined by • (f~±/w) 2 -~ 1 and may in principal be trapped in either region. The separated equation governing the vertical component involves a more complicated secondorder linear operator which contains the vertical buoyancy (Brunt-V~s~il~i) frequency Nz. The frequencies of all classes of modes are proportional to 1/M, but their dependences on the angular momentum of the black hole are quite different [6]. In principal this would allow one to measure the angular momentum of the black hole if more than one type of mode were to be detected in the same source. Alternatively, if one could infer the mass of the black hole through the motion of a companion which feeds the disk, one could determine the angular momentum of the black hole by a single mode observation.
minosity to the Eddington (limiting) luminosity. Higher axial g-modes with m > 0 have a somewhat different dependence on a than the radial modes, shown in Figure 2(b).
3.5
To analytically approximate the eigenfunctions and eigenfrequencies of the lowest g-modes, which are most relevant observationally, we obtained WKB solutions to the separated radial and vertical equations of fluid perturbations. From the symmetry of the governing equations, it is sufficient to consider eigenfrequencies a < 0 and axial mode integers m > 0. The resulting frequencies, f = -a/27r, of the lowest radial (m = 0) g-modes are given by f = 714 (1 - en#)(Mo/M ) F(a) nz,
;"
(2)
Here F(a) is the known, monotonically increasing function of the black hole angular momentum parameter a shown in Figure 2(a). The properties of the disk enter only through the small correction term en#, which involves the disk thickness 2h(r) and the radial (n) and vertical (j) mode numbers, with $ ~ 1. Typically h/r ~ 0.1 L/LEd~ for a radiation-pressure dominated optically thick disk region, where L/LEad is the ratio of the lu-
'
i
'
I
,
'
'
l
'
'
'
I
,
,
,
L
'
I
,
'
'
I
'
'
'
I
,
,
,
3 2.5 0
2 1.5 1 0.5
,
0
0.2
,
0.4
0.6
0.8
a
,
,
,
i
,
i , , ,
16
,.;=
3.1. G - m o d e s
'
12 m=3
m=2
m=l
4
0
0.2
0.4
0.6
0.8
a
Figure 2. (a) The dependence of the maximum radial (m = 0) eigenfrequency on the black hole angular momentum parameter a = c J / G M 2, relative to its value at a = 0. (b) The ratio of the maximum eigenfrequency of some higher m modes to that of the radial mode.
3.2. C - m o d e s
To analytically obtain approximate eigenfrequencies for the c-modes, we solved the separated WKB perturbation equations in the regime • (f~±/w) 2 -~ 1. The c-modes we have stud-
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
ied are nonradial (m > 1), nearly incompressible oscillations trapped in the very innermost region of the disk with eigenfrequencies lal mid(re) - 12±(rc), where rc is the upper radial bound of the mode. Typically, rc is not much greater than the innermost radius of the disk. For m = 1 and la/r½1 << 1 the c - m o d e eigenfrequency is approximately the Lense-Thirring frequency evaluated at the radius rc: 2a
lal ~ r~"
(3)
The physical structure of this mode resembles a tilted inner disk which slowly precesses about the black-hole spin axis. Since the mode is nearly incompressible, there is little temperature or pressure fluctuation in the disk. The mode is observable, however, since the projected area of the inner disk changes with time, thereby modulating either direct emission or scattered coronal photons. 4. O B S E R V A T I O N S 4.1. G R S 1 9 1 5 + 1 0 5 Observations with RXTE [7] have recently detected a narrow (Q - f / A f ,,, 20), weak ( m s variability ~ 0.3%- 1.6%) feature at f = 67 Hz in the black hole candidate GRS 1915+105. What distinguishes this peak from others that have recently been detected at high frequencies is the fact that it did not change its frequency as the source luminosity varied. During the time that thc feature was detectable, the luminosity of the source varied by a factor of two, but the frequency changed by less than 3%. This is clearly predicted by equation (2). The amplitude of the peak was greatest at the highest x-ray energies, which indicates that the feature is most likely being produced in the inner, hottest regions of the disk. These results are what we would expect if the feature is produced by a g - m o d e oscillation. If we identify this observation with the fundamental g-mode, equation (2) predicts a black hole mass of 10.6 M® if it is nonrotating to 36.3 M® if it is maximally rotating. Other aspects of this identification are explored by Nowak et al. [8].
351
4.2. G R O J1655-40 Evidence of a high-frequency QPO near 300 Hz was detected by RXTE in the second galactic microquasar, GRO J1655-40 [9]. Should we identify this feature with a diskoseismic oscillation, we will have to discuss its relation to the strength of the power-law component of the disk spectrum. (The QPO only appears in a combined power spectrum from the seven "hardest" observations. Within the framework of diskoseismology, this might be achieved if the Compton y parameter increases with decreasing radius within a high-temperature atmosphere or "corona".) Nevertheless, identification of this 300 Hz feature with the fundamental g - m o d e oscillating in an accretion disk surrounding the 7.0 M® black hole (as determined from spectra of the companion star [10]) implies that its angular momentum is 93% of maximum. REFERENCES
1. S. Kato and J. Fukue, PASJ, 32, 377 (1980). 2. C.A. Perez, A.S. Silbergleit, R.V. Wagoner and D.E. Lehr, ApJ, 476, 589 (1997). 3. J.R. Ipser and L. Lindblom, ApJ, 389, 392 (1992). 4. I.D. Novikov and K.S. Thorne, in Black Holes, eds. C. DeWitt and B.S. DeWitt (New York: Gordon & Breach), 343 (1973). 5. D.N. Page and K.S. Thorne, ApJ, 191, 499 (1974). 6. C.A. Perez, Ph.D. thesis, Stanford University (1993). 7. E.H. Morgan, R.A. Remillard and J. Greiner, ApJ, 482, 993 (1997). 8. M.A. Nowak, R.V. Wagoner, M.C. Begelman and D.E. Lehr, ApJ, 477, Lgl (1997). 9. R.A. Remillard, E.H. Morgan, J.E. McClintock, C.D. Bailyn, J.A. Orosz and J. Greiner, Proceedings of the 18th Texas Symposium on Relativistic Astrophysics (1996 December), eds. A. Olinto, J. Frieman and D. Schramm (World Scientific Press), in press (1997); astro-ph/9705064. 10. J.A. Orosz and C.D. Bailyn, ApJ, 477, 876 (1997).
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
Diskoseismology: Probing Black Holes and Their Accretion Disks D. E. Lehr a* aDepartment of Physics, Stanford University, Stanford, CA 94305-4060, USA W e review the relativisticresults for diskoseismic modes of oscillationwhich are trapped within thin accretion disks by non-Newtonian gravitational properties of a black hole. Predicted frequencies are calculated for the potentially most observable modes, "internal gravity" modes and "corrugation" modes. The most definitive property of these two classes of modes is that the resulting eigenfrequencies depend almost entirely upon only the mass and angular m o m e n t u m of the black hole. Such features m a y have been detected by R X T E in the power spectra of the luminosity modulations of the two galactic microquasars, G R S 1915+105 and G R O J165540. In the former system, we consider the possibility that this 67 Hz feature can be attributed to a g - m o d e in an accretion disk about a 10.6 M o (nourotating) to 36.3 M o (maximally rotating) black hole. In the latter system, identificationof the fundamental g - m o d e with the 300 Hz feature implies a black hole angular m o m e n t u m approximately 93% of maximum.
1. I N T R O D U C T I O N Within the region of strong gravity that exists in accretion disks near black holes, general relativity can trap normal modes of oscillation akin to the helioseismology modes that exist within the sun. These modes do not exist in Newtonian gravity; thus, they are a good tool to probe the nature of strong gravitational fields. The resulting spectrum of these so-called "diskoseismic" modes can determine both the mass M and the dimensionless angular momentum parameter a = c J / G M 2 of the central black hole. In addition, the physical structure of the accretion disk can be reflected in the frequencies and widths of these modes. We review the relativistic results for these trapped oscillations and discuss their possible observation by RXTE in the galactic microquasars, GRS 1915+105 and GRO J1655-40. It should be noted that these modes might also be found in an accretion disk surrounding a very weakly-magnetized, compact neutron star, such that the inner part of the disk which supports the modes is not disturbed by the magnetosphere or the stellar surface. The results obtained below apply to disks around slowly rotating neu*This research was supported in part by NASA GSRP Training Grant NGT5-50044. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PlI S0920-5632(98)00238-2
tron stars (a g 0.1) since their external metric is also Kerr through first order in a. (We note that a < 0.2 for most models of rotating neutron
stars.) 2. D I S K O S E I S M I C M O D E S
As first recognized by Kato and Fukue [1], oscillations can be trapped in the inner regions of accretion disks surrounding black holes by the general-relativistic modifications of the radial (~) and vertical (f~±) epicyclic frequencies of free-particle circular-orbit perturbations. These modes of oscillation are perturbations of the disk which are of the form f ( r , z ) e x p [ i ( a t + me)], where r, z , ¢ are cylindrical coordinates. With angular velocity f~(r), the corotation frequency is w(r) = a + mfL Modes are trapped because ~(r) reaches a maximum at small r and vanishes at the inner edge of the disk, thereby creating a resonant cavity. (This behavior in ~(r) is shown in Figure 1 along with other important orbital frequencies.) These modes represent temperature and pressure fluctuations in the disk which consequently modulate the luminosity of the disk at predictable frequencies. Three classes of modes have been identified: (g) Internal (gravity) modes are trapped where
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351 10o
(P) Acoustic (pressure) modes are trapped where w2 > g2. These modes extend a small distance from the inner edge of the disk and might also exist in the large outer region of the disk. These modes have considerable radial displacements.
a=O
10 "1
10-2
~2 ~Zi2
10-3 10 .4 10-5
,~i 5
0
~,
,
,
=
. . . .
=
10 r [GM/c2]
. . . .
15
20
100 a=0.5
10 -1 f~2
10-2
10-3 10 -4
10 .5
,
,
,
0
,
,
,
5
,
,
i
,
10 r [GM/c2]
,
,
,
,
15
,
,
20
100 :~
10 "~
~
10 .2
(c) The simplest corrugation modes are nonradial (m > 1) oscillations with w ~ fl±. When trapped near the inner edge of the disk, they represent nearly incompressible vertical displacements of the disk that resemble a small, tilted inner disk slowly precessing about the spin axis of the black hole. The m = 1 mode has a frequency approximating the Lense-Thirring frequency. 3. F R E Q U E N C Y
CALCULATIONS
a=0.998
10 .3 10 -4 K 2
lO-S
349
0
'
'
' 5
. . . .
1~0 . . . . r
[ G M / c
1~5 . . . .
20
2]
Figure 1. The radial dependence of the square of the fundamental free-particle frequencies that govern the modes of the disk: Keplerian (fl), and radial (~) and vertical (fl±) epicyclic. Three values of the black hole angular momentum parameter a = cJ/GM 2 are chosen.
w2 < a2, in the region where ~; achieves its maximum value and where the disk is hottest. The lowest modes can have significant vertical displacements and relatively large radial extents Ar ~ GM/c 2. We believe that these modes will produce the greatest luminosity modulations in the disk and are the most robust; thus, they could be the most observable class of modes. The nature of these modes is fundamentally different from stellar g - m o d e s and is not completely understood.
We have analyzed adiabatic perturbations to thin accretion disks in the Kerr metric [2]. Following the general relativistic perfect fluid perturbation formalism of Ipser and Lindblom [3], we expressed the adiabatic oscillations of all physical quantities in terms of a single scalar potential 5V(r, z) =_ 5p/p governed by a secondorder partial differential equation. The stationary (O/cOt = 0) and axisymmetric (cO/cO¢ = 0) unperturbed accretion disk was specified by the relativistic a-model [4,5]. The radial component of the velocity of the fluid was neglected, as was the gravitational field of the disk to good approximation. Since the effective radial wavelengths are significantly smaller than r, we adopted a WKB approach which allows approximate separation of the governing equations into their radial and vertical dependences. The radial component of the fluid perturbations satisfies the WKB relation
dr------5- +
•
]
- 1 (~2 _ w2)W = 0, (1)
where a(r) is inversely proportional to the speed of sound at the midplane and q(r) is the slowlyvarying separation function. The eigenfunction W(r) is proportional to a radial derivative of the potential ~V and to the radial component of the fluid displacement.
350
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
From the radial WKB equation (1) above we can see where each class of mode is trapped ( W - l d 2 W / d r 2 < 0). The p-modes, defined by @(f~±/w) 2 < 1, are trapped where w2 > ~2; g-modes, defined by @(f~±/w) 2 > 1, are trapped where w2 < ~2. The c-modes are defined by • (f~±/w) 2 -~ 1 and may in principal be trapped in either region. The separated equation governing the vertical component involves a more complicated secondorder linear operator which contains the vertical buoyancy (Brunt-V~s~il~i) frequency Nz. The frequencies of all classes of modes are proportional to 1/M, but their dependences on the angular momentum of the black hole are quite different [6]. In principal this would allow one to measure the angular momentum of the black hole if more than one type of mode were to be detected in the same source. Alternatively, if one could infer the mass of the black hole through the motion of a companion which feeds the disk, one could determine the angular momentum of the black hole by a single mode observation.
minosity to the Eddington (limiting) luminosity. Higher axial g-modes with m > 0 have a somewhat different dependence on a than the radial modes, shown in Figure 2(b).
3.5
To analytically approximate the eigenfunctions and eigenfrequencies of the lowest g-modes, which are most relevant observationally, we obtained WKB solutions to the separated radial and vertical equations of fluid perturbations. From the symmetry of the governing equations, it is sufficient to consider eigenfrequencies a < 0 and axial mode integers m > 0. The resulting frequencies, f = -a/27r, of the lowest radial (m = 0) g-modes are given by f = 714 (1 - en#)(Mo/M ) F(a) nz,
;"
(2)
Here F(a) is the known, monotonically increasing function of the black hole angular momentum parameter a shown in Figure 2(a). The properties of the disk enter only through the small correction term en#, which involves the disk thickness 2h(r) and the radial (n) and vertical (j) mode numbers, with $ ~ 1. Typically h/r ~ 0.1 L/LEd~ for a radiation-pressure dominated optically thick disk region, where L/LEad is the ratio of the lu-
'
i
'
I
,
'
'
l
'
'
'
I
,
,
,
L
'
I
,
'
'
I
'
'
'
I
,
,
,
3 2.5 0
2 1.5 1 0.5
,
0
0.2
,
0.4
0.6
0.8
a
,
,
,
i
,
i , , ,
16
,.;=
3.1. G - m o d e s
'
12 m=3
m=2
m=l
4
0
0.2
0.4
0.6
0.8
a
Figure 2. (a) The dependence of the maximum radial (m = 0) eigenfrequency on the black hole angular momentum parameter a = c J / G M 2, relative to its value at a = 0. (b) The ratio of the maximum eigenfrequency of some higher m modes to that of the radial mode.
3.2. C - m o d e s
To analytically obtain approximate eigenfrequencies for the c-modes, we solved the separated WKB perturbation equations in the regime • (f~±/w) 2 -~ 1. The c-modes we have stud-
D.E. Lehr/Nuclear Physics B (Proc. Suppl.) 69/1-3 (1998) 348-351
ied are nonradial (m > 1), nearly incompressible oscillations trapped in the very innermost region of the disk with eigenfrequencies lal mid(re) - 12±(rc), where rc is the upper radial bound of the mode. Typically, rc is not much greater than the innermost radius of the disk. For m = 1 and la/r½1 << 1 the c - m o d e eigenfrequency is approximately the Lense-Thirring frequency evaluated at the radius rc: 2a
lal ~ r~"
(3)
The physical structure of this mode resembles a tilted inner disk which slowly precesses about the black-hole spin axis. Since the mode is nearly incompressible, there is little temperature or pressure fluctuation in the disk. The mode is observable, however, since the projected area of the inner disk changes with time, thereby modulating either direct emission or scattered coronal photons. 4. O B S E R V A T I O N S 4.1. G R S 1 9 1 5 + 1 0 5 Observations with RXTE [7] have recently detected a narrow (Q - f / A f ,,, 20), weak ( m s variability ~ 0.3%- 1.6%) feature at f = 67 Hz in the black hole candidate GRS 1915+105. What distinguishes this peak from others that have recently been detected at high frequencies is the fact that it did not change its frequency as the source luminosity varied. During the time that thc feature was detectable, the luminosity of the source varied by a factor of two, but the frequency changed by less than 3%. This is clearly predicted by equation (2). The amplitude of the peak was greatest at the highest x-ray energies, which indicates that the feature is most likely being produced in the inner, hottest regions of the disk. These results are what we would expect if the feature is produced by a g - m o d e oscillation. If we identify this observation with the fundamental g-mode, equation (2) predicts a black hole mass of 10.6 M® if it is nonrotating to 36.3 M® if it is maximally rotating. Other aspects of this identification are explored by Nowak et al. [8].
351
4.2. G R O J1655-40 Evidence of a high-frequency QPO near 300 Hz was detected by RXTE in the second galactic microquasar, GRO J1655-40 [9]. Should we identify this feature with a diskoseismic oscillation, we will have to discuss its relation to the strength of the power-law component of the disk spectrum. (The QPO only appears in a combined power spectrum from the seven "hardest" observations. Within the framework of diskoseismology, this might be achieved if the Compton y parameter increases with decreasing radius within a high-temperature atmosphere or "corona".) Nevertheless, identification of this 300 Hz feature with the fundamental g - m o d e oscillating in an accretion disk surrounding the 7.0 M® black hole (as determined from spectra of the companion star [10]) implies that its angular momentum is 93% of maximum. REFERENCES
1. S. Kato and J. Fukue, PASJ, 32, 377 (1980). 2. C.A. Perez, A.S. Silbergleit, R.V. Wagoner and D.E. Lehr, ApJ, 476, 589 (1997). 3. J.R. Ipser and L. Lindblom, ApJ, 389, 392 (1992). 4. I.D. Novikov and K.S. Thorne, in Black Holes, eds. C. DeWitt and B.S. DeWitt (New York: Gordon & Breach), 343 (1973). 5. D.N. Page and K.S. Thorne, ApJ, 191, 499 (1974). 6. C.A. Perez, Ph.D. thesis, Stanford University (1993). 7. E.H. Morgan, R.A. Remillard and J. Greiner, ApJ, 482, 993 (1997). 8. M.A. Nowak, R.V. Wagoner, M.C. Begelman and D.E. Lehr, ApJ, 477, Lgl (1997). 9. R.A. Remillard, E.H. Morgan, J.E. McClintock, C.D. Bailyn, J.A. Orosz and J. Greiner, Proceedings of the 18th Texas Symposium on Relativistic Astrophysics (1996 December), eds. A. Olinto, J. Frieman and D. Schramm (World Scientific Press), in press (1997); astro-ph/9705064. 10. J.A. Orosz and C.D. Bailyn, ApJ, 477, 876 (1997).