MEASUREMENT OF BRANY BLACK HOLE PARAMETERSMEASUREMENT OF BRANY BLACK HOLE PARAMETERSIN THE FRAMEWORKIN THE FRAMEWORK

OFOF THE ORBITAL RESONANCE MODEL OF THE ORBITAL RESONANCE MODEL OF QPOQPOss

Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, CZECH REPUBLIC

supported byCzech grant

MSM 4781305903

Presentation download:www.physics.cz/researchin section news

Zdeněk Stuchlík and Andrea Kotrlová

Outline

1. Braneworld, black holes & the 5th dimension1.1. Rotating braneworld black holes

2. Quasiperiodic oscillations (QPOs)2.1. Black hole high-frequency QPOs in X-ray2.2. Orbital motion in a strong gravity2.3. Keplerian and epicyclic frequencies2.4. Digest of orbital resonance models2.5. Resonance conditions2.6. Strong resonant phenomena - "magic" spin

3. Applications to microquasars3.1. Microquasars data: 3:2 ratio3.2. Results for GRO J1655-403.3. Results for GRS 1915+1053.4. Conclusions

4. References

1. Braneworld, black holes & the 5th dimension

Braneworld model - Randall & Sundrum (1999):

- our observable universe is a slice, a "3-brane" in 5-dimensional bulk spacetime

1.1. Rotating braneworld black holes

The metric form on the 3-brane

– assuming a Kerr-Schild ansatz for the metric on the brane the solution in the standard Boyer-Lindquist coordinates takes the form

Aliev & Gümrükçüoglu (2005):

– exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld

where

1.1. Rotating braneworld black holes

The metric form on the 3-brane

– assuming a Kerr-Schild ansatz for the metric on the brane the solution in the standard Boyer-Lindquist coordinates takes the form

Aliev & Gümrükçüoglu (2005):

– exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld

where

– looks exactly like the Kerr-Newman solution in general relativity, in which the square of the electric charge Q2 is replaced by a tidal charge parameter .

1.1. Rotating braneworld black holes

The tidal charge

– means an imprint of nonlocal gravitational effects from the bulk space,

– may take on both positive and negative values !

The event horizon:

– the horizon structure depends on the sign of the tidal charge

condition:

for

for extreme horizon and

1.1. Rotating braneworld black holes

The tidal charge

– means an imprint of nonlocal gravitational effects from the bulk space,

– may take on both positive and negative values !

The event horizon:

– the horizon structure depends on the sign of the tidal charge

condition:

for

for extreme horizon and

1.1. Rotating braneworld black holes

The tidal charge

– means an imprint of nonlocal gravitational effects from the bulk space,

– may take on both positive and negative values !

The event horizon:

– the horizon structure depends on the sign of the tidal charge

condition:

forThis is not allowedin the framework

of general relativity !!

for extreme horizon and

1.1. Rotating braneworld black holes

The tidal charge

– means an imprint of nonlocal gravitational effects from the bulk space,

– may take on both positive and negative values !

The effects of negative tidal charge

– tend to increase the horizon radius rh, the radii of the limiting photon orbit (rph), the innermost bound (rmb) and the innermost stable circular orbits (rms) for both direct and retrograde motions of the particles,

– mechanism for spinning up the black hole so that its rotation parameter exceeds its mass. Such a mechanism is impossible in general relativity !

The event horizon:

– the horizon structure depends on the sign of the tidal charge

condition:

forThis is not allowedin the framework

of general relativity !!

for extreme horizon and

2. Quasiperiodic oscillations (QPOs)

Fig. on this page: nasa.gov

Black hole hi-frequency QPOs in X-ray

hi-frequencyQPOs

low-frequencyQPOs

(McClintock & Remillard 2003)

2.1. Quasiperiodic oscillations

2.1. Quasiperiodic oscillations

(McClintock & Remillard 2003)

2.2. Orbital motion in a strong gravity

– the Keplerian orbital frequency– and the related epicyclic frequencies (radial , vertical ):

ν ~ 1/M

Rotating braneworld BH with mass M, dimensionless spin a, and the tidal charge :the formulae for

has a local maximum for all values of spin a- only for rapidly rotating BHs

xms – radius of the marginally stable orbit

Stable circular geodesics exist for

2.3. Keplerian and epicyclic frequencies

- can have a maximum at x = xex !!

Notice, that reality condition must be satisfied

2.3. Keplerian and epicyclic frequencies

Can it be located above• the outher BH horizon xh

• the marginally stable orbit xms?

- can have a maximum at x = xex !!

Notice, that reality condition must be satisfied

2.3. Keplerian and epicyclic frequencies

Can it be located above• the outher BH horizon xh

• the marginally stable orbit xms?

- can have a maximum at x = xex !!

Extreme BHs:

Notice, that reality condition must be satisfied

2.3. Keplerian and epicyclic frequencies

2.3. Keplerian and epicyclic frequencies

2.3. Keplerian and epicyclic frequencies

2.3. Keplerian and epicyclic frequencies

2.4. Digest of orbital resonance models

2.4. Digest of orbital resonance models

2.5. Resonance conditions

– determine implicitly the resonant radius

– must be related to the radius of the innermost stable circular geodesic

2.5. Resonance conditions

2.5. Resonance conditions

2.5. Resonance conditions

2.5. Resonance conditions

2.5. Resonance conditions

2.5. Resonance conditions

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

2.6. Strong resonant phenomena - "magic" spin

3. Applications to microquasars

GRO GRO JJ1655-41655-400

3. Applications to microquasars

GRS 1915+105GRS 1915+105

3.1. Microquasars data: 3:2 ratio

Törö

k, A

bra

mow

icz,

Klu

znia

k,

Stu

chlík

20

05

3.1. Microquasars data: 3:2 ratio

Törö

k, A

bra

mow

icz,

Klu

znia

k,

Stu

chlík

20

05

3.1. Microquasars data: 3:2 ratio

Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio

Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio

Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio

Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio

Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge can be related assuming a concrete version of the resonance model.

The most recent angular momentumestimates from fits of spectral continua:

GRO J1655-40: a ~ (0.65 - 0.75)GRS 1915+105: a > 0.98

a ~ 0.7

- Shafee et al. 2006

- McClintock et al. 2006

- Middleton et al. 2006

3.2. Results for GRO J1655-40

3.2. Results for GRO J1655-40

Shafee et al. 2006

McC

linto

ck &

Rem

illard

20

04

Possible combinations of mass and spin predicted by individual resonance models for the high-frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J1655-40.

3.2. Results for GRO J1655-40

Possible combinations of mass and spin predicted by individual resonance models for the high-frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J1655-40.

Shafee et al. 2006

McC

linto

ck &

Rem

illard

20

04

3.2. Results for GRO J1655-40

The only model which matches the observational constraintsis the vertical-precession resonance (Bursa 2005)

Possible combinations of mass and spin predicted by individual resonance models for the high-frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J1655-40.

Shafee et al. 2006

McC

linto

ck &

Rem

illard

20

04

3.2. Results for GRO J1655-40

3.3. Results for GRS 1915+105

3.3. Results for GRS 1915+105

McC

linto

ck &

Rem

illard

20

04

3.3. Results for GRS 1915+105

estimate 1

1 - Middleton et al. 2006

McC

linto

ck &

Rem

illard

20

04

3.3. Results for GRS 1915+105

estimate 1

estimate 2

2 - McClintock et al. 2006

1 - Middleton et al. 2006

McC

linto

ck &

Rem

illard

20

04

3.3. Results for GRS 1915+105

3.4. Conclusions

3.4. Conclusions

β = 0

3.4. Conclusions

-1 < β < 0.51 (βmax for a = 0.7)

3.4. Conclusions

-1 < β < 0.51

3.4. Conclusions

-1 < β < 0.51

3.4. Conclusions

-1 < β < 0.51

3.4. Conclusions

-1 < β < 0.51

3.4. Conclusions

-1 < β < 0.51

3.4. Conclusions

there is no specific type of resonance model that could work for both sources simultaneously

-1 < β < 0.51

THANK YOUTHANK YOUFOR YOUR ATTENTIONFOR YOUR ATTENTION

4. References

• Abramowicz, M. A. & Kluzniak, W. 2004, in X-ray Timing 2003: Rossi and Beyond., ed. P. Karet, F. K. Lamb, & J. H. Swank, Vol. 714 (Melville: NY: American Institute of Physics), 21-28

• Abramowicz, M. A., Kluzniak, W., McClintock, J. E., & Remillard, R. A. 2004, Astrophys. J. Lett., 609, L63

• Abramowicz, M. A., Kluzniak, W., Stuchlík, Z., & Török, G. 2004, in Proceedings of RAGtime 4/5: Workshops on black holes and neutron stars, Opava, 14-16/13-15 October 2002/2003, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 1-23

• Aliev, A. N., & Gümrükçüoglu, A. E. 2005, Phys. Rev. D 71, 104027

• Aliev, A. N., & Galtsov, D. V. 1981, General Relativity and Gravitation, 13, 899

• Bursa, M. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September 2004/2005, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 39-45

• McClintock, J. E. & Remillard, R. A. 2004, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge University Press)

• McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, Astrophys. J., 652, 518

• Middleton, M., Done, C., Gierlinski, M., & Davis, S. W. 2006, Monthly Notices Roy. Astronom. Soc., 373, 1004

• Randall, L., & Sundrum, R. 1999, Phys. Rev. Lett. 83, 4690

• Shafee, R., McClintock, J. E., Narayan, R., et al. 2006, Astrophys. J., 636, L113

• Stuchlík, Z. & Török, G. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September 2004/2005, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 253-263

• Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena, submitted

• Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole spin, in prep.

• Török, G., Abramowicz, M. A., Kluzniak,W. & Stuchlík, Z. 2005, Astronomy and Astrophysics, 436, 1

• Török, G. 2005, Astronom. Nachr., 326, 856