Possible strong resonant phenomena exciting QPOs
in some black hole systems
Andrea Kotrlová, Zdeněk Stuchlík & Gabriel Török
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-74601 Opava, Czech Republic
"magic" spin a = 0.983
Black Holes in Kraków,
17th May 2007
Supported by
MSM 4781305903
Outline
1. Motivation:
Quasiperiodic oscillations (QPOs) in X-ray from the NS an BH systems
- Black hole and neutron star binaries, accretion disks and QPOs
2. Non-linear orbital resonance models
2.1. Orbital motion in a strong gravity
2.2. Standard orbital resonance models
3. "Exotic" multiple resonances at the common orbit
3.1. Triple frequencies and black hole spin a
a) at different radii *)
b) at the common radius (strong resonant phenomena) **)
3.2. Necessary conditions
3.3. Classification
- "magic" value of the black hole spin a = 0.983
4. A little "gamble"
- The Galaxy centre source Sgr A* as a proper candidate system
5. Conclusions
6. References
*) Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole
spin, in preparation
**) Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena
1. Motivation
Quasi-periodic oscillations (QPOs) in X-ray from the NS an BH systems
Fig. on this page: nasa.gov
1.1. Black hole binaries and accretion disks
radio
“X-ray”
and visible
Figs on this page: nasa.gov
1.2. X-ray observations
I
Light curve:
t
Power
Power density spectra (PDS):
Frequency
Figs on this page: nasa.gov
power
1.3. Quasiperiodic oscillations
hi-frequency
QPOs
low-frequency
QPOs
frequency
2. Non-linear orbital resonance models
2.1. Orbital motion in a strong gravity
2.1. Orbital motion in a strong gravity
a=0
non-rotating BH
a~1
rotating BH
2.2. "Standard" 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)
2.2. "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.
2.2. "Standard" orbital resonance models
3. "Exotic" multiple resonances at the common orbit
Fig. on this page: nasa.gov
3.1. Triple frequencies and black hole spin a
From
a)
we can determine spin for various versions of the resonance model:
Two resonances at different radii
for special values of spin  common top, bottom, or mixed frequency
 two frequency pairs reduce into a triple frequency ratio set
"top identity"
"bottom identity"
"middle identity"
- possibility of highly precise determination of spin – given by the types of the two resonances and the ratios
quite independently of the BH mass M (but not uniquely, as the same frequency set could
correspond to more than one concrete spin a).
b)
Resonances sharing the same radius
for special values of a  strong resonant phenomena – allow direct resonances at a given radius
(s, t, u – small natural numbers)
- for each triple frequency ratio set spin is given uniquely,
- the resonances could be causally related and could cooperate efficiently (Landau & Lifshitz 1976).
3.2. Necessary conditions
Strong resonant phenomena
- only for special values of spin a
• we consider BH when a ≤ 1  restriction on allowed values of s, t, u
• we have to search for the integer ratios s:t:u at x ≥ xms
and
at the same radius
 condition:
 an explicit solution determining the relevant radius for any triple frequency ratio set s:t:u
and the related BH spin:
The solutions have been found for s ≤ 5 since the strength of the resonance and the resonant frequency width
decrease rapidly with the order of the resonance (Landau & Lifshitz 1976)
s:t:u = 3:2:1,
4:2:1, 4:3:1, 4:3:2,
5:2:1, 5:3:1, 5:3:2, 5:4:1, 5:4:2, 5:4:3.
3.2. Necessary conditions
A

 
   
s:t:u = 3:2:1,
B
4:2:1, 4:3:1, 4:3:2,
5:2:1, 5:3:1, 5:3:2, 5:4:1, 5:4:2, 5:4:3.
C
D
E


direct
resonances
realized
only with
combinational
frequencies
3.2. Classification
"Magic" spin a = 0.983
A)
• arises for the so called "magic" spin am = 0.983
• the Keplerian and epicyclic frequencies are in the lowest possible ratio at the common radius
• any of the simple combinational frequencies coincides with one of the frequencies
and are in the fixed small integer ratios
• the only case when the combinational frequencies (not exceeding
orbital frequencies
) are in the same ratios as the
• we obtain the strongest possible resonances when the beat frequencies enter the resonances satisfying
the conditions
3.2. Classification
"Magic" spin a = 0.983
A)
• arises for the so called "magic" spin am = 0.983
• the Keplerian and epicyclic frequencies are in the lowest possible ratio at the common radius
• any of the simple combinational frequencies coincides with one of the frequencies
and are in the fixed small integer ratios
• the only case when the combinational frequencies (not exceeding
orbital frequencies
) are in the same ratios as the
• we obtain the strongest possible resonances when the beat frequencies enter the resonances satisfying
the conditions
3.2. Classification
B)
• the combinational frequencies give
additional frequency ratios
• we can obtain the other three frequency ratio sets
• the four observable frequency ratio set is possible
C)
• we can generate triple frequency sets
involving the combinational frequencies
• two sets of four frequency ratios are possible
• we could obtain one set of five frequency ratio
3.2. Classification
D)
• this case leads to the triple frequency ratio sets
• and one four frequency ratio set
E)
• we can obtain the triple frequency ratio sets
• the related four frequency ratio sets
• and one five frequency ratio set
4. A little "gamble"
Possible application to the Sgr A* QPOs
Figs on this page: nasa.gov
4. A little "gamble"
The Galaxy centre source Sgr A* as a proper candidate system
The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005):
a) Considering the standard epicyclic resonance model:

- it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis
of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):
b) Assuming the "magic" spin
(Sgr A* should be fast rotating), with the frequency ratio
at the sharing radius
and identifying



- it meets the allowed BH mass interval at its high mass end.
c) Using other versions of the multi-resonance model  best fit is for
at two different radii
having common bottom frequency
, with resonances

The model should be further tested, more precise frequency measurements are very important.

4. A little "gamble"
Ghez et al. (2005)
Errors of frequency measurements
The mass of the BH is related to the magnitude
of the observed frequency set, not to its ratio.
more precise measurement of the QPOs frequencies
 more precise determination of the BH mass,
method can work only accidentally, for the properly
taken values of spin
 precision of frequency measurement is crucial
for determination of the BH mass.
errors of frequency measurements
 errors in the spin determination (depends on the concrete resonances occurring at a given radius)
5. Conclusions
Conditions for strong resonant phenomena could be realized only for high values of spin (a ≥ 0.75)
 idea probably could not be extended to the NS (where we expect a < 0.5).
Allowing simple combinational frequencies (not exceeding
)  observable QPOs with:
the lowest triple frequency ratio set
for the "magic" spin a = 0.983,
but also for a = 0.866, 0.882, 0.962 (if the uppermost frequencies are not observed for some reasons),
four frequency ratio set
five frequency ratio set
for a = 0.866, 0.882 and 0.962,
for a = 0.882, 0.962.
It is not necessary that all the resonances are realized simultaneously and that the full five (four) frequency
set is observed at the same time.
Generally, there exist few values of the spin a and the corresponding shared resonance radius allowed for a
given frequency ratio set  detailed analysis of the resonance phenomena has to be considered and further
confronted with the spin estimates coming from
spectral analysis of the BH system,
line profiles,
orbital periastron precession of some stars moving in the region of Sgr A*,
very promising: studies of the energy dependencies of high-frequency QPOs determining the
QPO spectra at the QPO radii (Życki, P. T., Niedzwiecki, A., & Sobolewska, M. A. 2007, Monthly
Notices Roy. Astronom. Soc., in press). (Życki et al. 2007)
6. References
Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena
Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole
spin, in preparation