ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD BLACK HOLES

ORBITAL RESONANCE MODELS OF QPOsORBITAL RESONANCE MODELS OF QPOsIN BRANEWORLDIN BRANEWORLD BLACKBLACK HOLESHOLES

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

RUSGRAV-1313 Russian Gravitational Conference

International Conference on Gravitation, Cosmology and Astrophysics

June 23-28, 2008, PFUR, Moscow, Russia

Outline

1. Braneworld, black holes & the 5th dimension 1.1. Rotating black hole with a tidal charge

2. Quasiperiodic oscillations (QPOs)2.1. Black hole binaries and accretion disks2.2. X-ray observations2.3. QPOs2.4. Non-linear orbital resonance models2.5. Orbital motion in a strong gravity2.6. Properties of the Keplerian and epicyclic frequencies2.7. Digest of orbital resonance models2.8. Resonance conditions2.9. Strong resonant phenomena - "magic" spin

3. Application 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

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 (Phys. Rev. D 71, 104027):

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

where

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

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 (Phys. Rev. D 71, 104027):

– 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 β

1.1. Rotating black hole with a tidal charge

The tidal charge β

– 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

for

for extreme horizon:

The tidal charge β

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

– may take on both positive and negative values !

The effects of the negative tidal charge β

– tends 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

forThis is not allowedin the framework

of general relativity !!

for extreme horizon:


rms – the radius of the marginally stable orbit, implicitly determined by the relation

Stable circular geodesics exist for

Extreme BH:

dimensionless radial coordinate:



BHs

NaS


The effects of the negative tidal charge β:

– tends to increase xh, xph, xmb, xms

– mechanism for spinning up the black hole(a > 1)


BHs

NaS


BHs

NaS


BHs

NaS




2. Quasiperiodic oscillations (QPOs)

Black hole high-frequency QPOs in X-ray

Figs on this page: nasa.gov

radio

“X-ray”and visible

2.1. Black hole binaries and accretion disks


time

Inte

nsity

Pow

erFrequency

2.2. X-ray observations

Light curve:

Power density spectra (PDS):


- a detailed view of the kHz QPOs in Sco X-1

high-frequencyQPOs

low-frequencyQPOs

(McClintock & Remillard 2003)

2.3. Quasiperiodic oscillations





2.4. Non-linear orbital resonance models

– were introduced by Abramowicz & Kluźniak (2000) who considered the resonance between

radial and vertical epicyclic frequency as the possible explanation of NS and BH QPOs

(this kind of resonances were, in different context, independently considered by

Aliev & Galtsov, 1981)

"Standard" orbital resonance models

2.5. Orbital motion in a strong gravity

Parametric resonance

frequencies are in ratio of small natural numbers(e.g, Landau & Lifshitz, 1976), which must hold also in the case of

forced resonances

Epicyclic frequencies depend on generic mass as f ~ 1/M.

2.5. Orbital motion in a strong gravity

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

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

Stable circular geodesics exist for

2.6. Keplerian and epicyclic frequencies

2.6. Properties of the Keplerian and epicyclic frequencies

Local extrema of the Keplerian and epicyclic frequencies:



can have a maximum at

Keplerian frequency:



can have a maximum at

Could it be located above• the outher BH horizon xh

• the marginally stable orbit xms?

Keplerian frequency:

2.6.1. Local extrema of the Keplerian frequency



has a local maximum for all values of spin a

the locations of the local extrema of the epicyclic frequenciesare implicitly given by

2.6.2. Local extrema of the epicyclic frequencies

- only for rapidly rotating BHs

BHs

NaS

2.6.2. Local extrema of the vertical epicyclic frequency

•BHs: one local maximum at for •NaS: two or none local extrema

BHsNaS

2.6.2. Local extrema of the radial epicyclic frequency

BHs

NaS








2.7. Digest of orbital resonance models

2.8. Resonance conditions

– determine implicitly the resonant radius

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

2.9. Strong resonant phenomena - "magic" spin




- spin is given uniquely,

- the resonances could be causally related and could cooperate efficiently

(Landau & Lifshitz 1976)

for special values of BH spin a and brany parameter β strong resonant phenomena

(s, t, u – small natural numbers)

Resonances sharing the same radius


3. Application to microquasars

GRO GRO JJ1655-41655-400

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

From the observed twin peak frequencies and the known limits on the mass M of the central BH, the dimensionless spin a and the tidal charge β 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

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

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

-1 < β < 0.51

3.4. Conclusions

there is not only one specific type of resonance model that could work for both sources simultaneously

-1 < β < 0.51

• Stuchlík, Z. & Kotrlová, A. 2007, in: Proceedings of RAGtime 8/9: Workshops on black holes and neutron stars, Opava, Hradec nad Moravicí, 15–19/19–21 September 2006/2007, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 323-361

• Stuchlík, Z., Kotrlová, A., & Török, G. 2007, in: Proceedings of RAGtime 8/9: Workshops on black holes and neutron stars, Opava, Hradec nad Moravicí, 15–19/19–21 September 2006/2007, S. Hledík and Z. Stuchlík (Opava: Silesian University in Opava), 363-416

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

• Stuchlík, Z. & Kotrlová, A.: Orbital resonances in discs around braneworld Kerr black holes, 2008, subm.

• Kotrlová, A., Stuchlík, Z., & Török, G.: QPOs in strong gravitational field around neutron stars testing braneworld models, 2008, subm.

• 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 Univ. 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

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

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

THANK YOU FOR YOUR ATTENTIONTHANK YOU FOR YOUR ATTENTION

4. References

Documents

ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD BLACK HOLES