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:
1.1. Rotating black hole with a tidal charge
rms – the radius of the marginally stable orbit, implicitly determined by the relation
Stable circular geodesics exist for
Extreme BH:
dimensionless radial coordinate:
1.1. Rotating black hole with a tidal charge
1.1. Rotating black hole with a tidal charge
The effects of the negative tidal charge β:
– tends to increase xh, xph, xmb, xms
– mechanism for spinning up the black hole(a > 1)
2. Quasiperiodic oscillations (QPOs)
Black hole high-frequency QPOs in X-ray
Figs on this page: nasa.gov
time
Inte
nsity
Pow
erFrequency
2.2. X-ray observations
Light curve:
Power density spectra (PDS):
Figs on this page: nasa.gov
- a detailed view of the kHz QPOs in Sco X-1
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
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:
2.6. Properties of the Keplerian and epicyclic frequencies
Local extrema of the Keplerian and epicyclic frequencies:
can have a maximum at
Keplerian frequency:
2.6. Properties of the Keplerian and epicyclic frequencies
Local extrema of the Keplerian and epicyclic frequencies:
can have a maximum at
Could it be located above• the outher BH horizon xh
• the marginally stable orbit xms?
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
2.8. 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
3.1. 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
3.3. 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
3.4. 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