Zdeněk StuchlíkGabriel Török, Petr Slaný, Andrea Kotrlová, Jiří Kovář

Multi-resonant models of quasi-periodic oscillations in black hole and neutron star systems

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

1. Introduction2. Orbital resonance models- 2.1 Orbital resonance models- 2.2 Parametric and forced resonance - 2.3 Gravitational excitation of QPOs in neutron star systems

3. Orbital multi-resonance models- 3.1 More instances of one resonance occuring at more specific radius- 3.2 More resonances sharing one specific radius (strong resonant

phenomena)- 3.3 Two resonances excited at two resonance radius („Ugly“ case)

4. Extended resonance model with humpy - included oscillations5. Resonance model with halo orbits6. Orbital resonance model for braneworld compact objects- 6.1 Braneworld uniform density compact stars- 6.2 Rotating braneworld black holes

7. Conclusions

OutlinePresentation download: www.physics.cz/research (news)

• Introduction:

Quasi-periodic oscillations (QPOs) in X-ray from the NS an BH systems and their resonant models

Figs on this page: nasa.gov

1.1 Quasiperiodic oscillations

Motivation

• Many Galactic black hole and neutron star sources in low X-ray mass binaries show QPOs (quasi periodic oscillations) in their observerd X-ray fluxes, i.e., peaks in the Fourier variability power density spectra (PDS). QPOs attracted a lot of attention mostly because of their high frequencies. Frequencies of some QPOs are in the kHz range which corresponds to orbital frequencies just a few gravitational radii away from the central black hole or neutron star.

PDS (power density spectra) of the binary Sco X-1 (neutron star):

hi-frequency(kHz) QPOs

low-frequencyQPOs

pow

er

1.1 Quasiperiodic oscillations

Motivation

• Many Galactic black hole and neutron star sources in low X-ray mass binaries show QPOs (quasi periodic oscillations) in their observerd X-ray fluxes, i.e., peaks in the Fourier variability power density spectra (PDS). QPOs attracted a lot of attention mostly because of their high frequencies. Frequencies of some QPOs are in the kHz range which corresponds to orbital frequencies just a few gravitational radii away from the central black hole or neutron star.

PDS (power density spectra) of the microquasar GRO 1655-40 from two different observations

hi-frequency(kHz) QPOs

1.2 kHz quasiperiodic oscillations: BH and NS

2. Orbital and orbital resonance models of QPOs

Figs on this page: nasa.gov

General belief dominating in the astrophysical community links the kHz QPOs to the orbital motion near the inner edge of an accretion disc. Class of models relates kHz QPOs to orbital resonances….

2.1 Orbital resonance model

Imply the existence of the periastron and nodal (Lense-Thirring) precession

Stella, L. \& Vietri, M. 1999, Phys. Rev. Lett., 82, 17related the kHz QPOs to the Keplerian and periastron precession of the blobs close to the inner edge of an accretion disc.

- Relativistic precession model

• Epicyclic frequencies- Test particle (in the gravitational field) on a circular orbit:

Geodesic motion models: orbital motion in a strong gravity

Relativistic precession model(Stella, L. \& Vietri, M. 1999, Phys. Rev. Lett., 82, 17)relates the kHz QPOs to the frequencies of geodesic motion(Keplerian and periastron precession of the blobs close to the inner edge of an accretion disc).

Resonance modelKluzniak, W., Abramowicz, M. A., 2000, Phys. Rev. Lett. (submitted); Klu\'zniak, W., \& Abramowicz, M. A., 2001, Acta Physica Polonica B 32, 3605 [http://th-www.if.uj.edu.pl/acta/vol32/t11.htm]

Orbital resonance model relates the kHz QPOs to disc oscillation modes corresponding to the frequencies of geodesic motion.

2.1 Orbital resonance model

2.1 Orbital resonance model

• Analogy – springy pendulum

0 0 22 δωδδrωrδ r

0)cos(12 thωδ r

Mathieu:

- parametric resonance:Energy overflow between modes

2.2 Parametric and forced resonance

•Parametric resonance

Because of behaviour of epicyclic frequencies (ωr < ωθ),n = 3 is the lowest allowed value.Parametric resonance gives the 3:2 ration naturally.

•Forcer resonance

Forced resonance gives generally any rational ratio,combinational frequencies are allowed.Here we are interested only on such combinations which can give the observed 3:2 ratio.

0)cos(12 thωδ r

snnv

vrr ...3,2,1 ,2

)cos()h(rin mslinear ter-non2 tr

numbers integral small are qp, where,rq

p

tm

fA

2

0

220

Scatter

2

ε

m

fA

2/1

22

0)(

2

1

2

1

hseA ts

22

02

2

0 42

14

2

1

hh

Forced resonance (linear regime)

Forced resonance with dissipation

Friction parameter λ:

Parametric resonance

Scatter condition:

order:

2.2 Parametric and forced resonance

Resonance frequency scatter

4

Threshold 2

1 :1

00

Khhn

20 A

220

2

0

2/

AmfA

Amplitude dependence of eigenfrequency:

κ............given by non-linear effect

12130

32 !12 nhn nnn

nK

n hh:n /10 1

2.2 Parametric and forced resonance

higher order:

Forced resonance:„Pumping energy“ - linear regime(Landau & Lifshitz, 1976)

2.2 Parametric and forced resonance

)(12 tFm

xx exc

)cos()( tftF eexc

Fext (t) … restoring force

fe. … (amplitude)

γ … frequency of the restoring force

β … phase

)sin( 2

)cos( 0

tt

m

ftax e

)cos()cos(220

tm

ftax p

Forced resonance condition:

2.2 Parametric and forced resonance

Cumulative growing of the oscilation amplitude A:

2

2/1

3

6 ;

2 ;

6 ;

r

Mmf

r

M

t

f

f

r

A

grr

g

e

r

310r

A

2.3 Gravitational excitation of QPOs in neutron star systems

Investigated hypothesis: The gravitational perturbations caused either by

- the surface features- mountains- magnetically supported accretion columns

- quadrupole deformations

- the binary companion

may be relevant as an excitation mechanism which may also “feed“ the resonance.

Talk - Stuchlík, Konar, Miller, Hledík (2007 submited)

Gravitational force by isolated mountain and binary companion:

2.3 Gravitational excitation of QPOs in neutron star systems

Gravitational forces by symmetric accretion column:

2.3 Gravitational excitation of QPOs in neutron star systems

Example of gravitational force by binary companion – Fourier analysis

The analysis presented for the binary companion can be applied also to black hole systems.

2/32

2cossin21cos)(

d

Rt

d

R

d

GMtF BBB

BBV

2.3 Gravitational excitation of QPOs in neutron star systems

Quadrupole moments and related perturbative mountain mass.

Estimates

Temperature variations (due to accretion) – outer crust:Bildsten (1998)

inner crust: (Ushomirsky etal, 2000)

Relativistic effects (Haskell etal, 2007)

Hybrid star cores (Owen 2005)

Strange (quark) stars (Owen 2005)

NSqmad MM 810

NSqmad MM 710

NSguad MxM 6105

QSguad MxM 4102

2.3 Gravitational excitation of QPOs in neutron star systems

Crystaline color superconducting cores

(Hashell, Anderson, Jones & Samuelsson, 2007)

LIGO measurements put the limits

In agreements with estimates for enhacement of QPOs.

Internal magnetic field included induced deformations and magnetic accretion columns:

External magnetic field too strong to let survive quasi-Keplerian disc in regions of interest for exciting kHz QPOs

QSqmad MM 310

NSguad MM 610

2.3 Gravitational excitation of QPOs in neutron star systems

Figs on this page: nasa.gov

3. Multi-resonance models

Phenomenologically, there are two possibilities in the resonance models:one eigenfrequency pair hypothesis or,more eigenfrequency pairs hypothesis…

• Orbital resonance models involving Keplerian and epicyclic oscillations:

– more instances of one resonance occuring (excited) at (or close to) morespecific radii

– more resonances sharing one specific radius strong resonant phenomena – black holes with a specific spin

– more resonances occuring (excited) at (or close to) more specific radii(the “ugly” case)

• Extended resonance model with hump-induced oscillations

• Resonance model with halo orbits

3. Orbital multi-resonance models

3.1 More instances of one resonance occuring at more specific radius

Investigated hypothesis: the NS twin peak QPOs originate in a resonance between two modes having time-dependent eigenfrequencies determined by the frequencies of geodesic motion.

Talks - Bakala,Torok, Stuchlík, Urbanec - Urbanec, Stuchlík, Torok, Bakala, Čermák

Relativistic precession model:

Total precession model:

For group of sources some of possible frequency relations considered in the Hartle-Thorne metric implies the neutron star massM ~ 1.5-2M_sun and j ~ 0-0.2.

4U 1636-53

} ~0.3M

3.1 More instances of one resonance occuring at more specific radius

rKlKu ,

rlKu ,

For group of sources some of possible models (frequency relations) considered in the Hartle-Thorne metric implies the neutron star mass M ~ 1.5-2M_sun and j ~ 0-0.2. Concrete models give concrete restrictions to the neutron star structure.

Genetic Algorithm

3.1 More instances of one resonance occuring at more specific radius

3.2 More resonances sharing one specific radius

Investigated hypothesis: For a special values of black hole spin more resonances can occur at the same radius. Strong resonance phenomena may arise when the Keplerian and epicyclic frequencies are in the lowest possible ratio.

Talk – Kotrlová, Stuchlík, Torok

for special values of black hole spin strong resonant phenomena

(s, t, u – small natural numbers)

- spin is given uniquely,

- the resonances could be causally related and could cooperate efficiently (Landau & Lifshitz 1976)

3.2 More resonances sharing one specific radius

Triple frequencies and black hole spin a

• 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 ) are in the same ratios as the orbital frequencies

• we obtain the strongest possible resonances when the beat frequencies enter the resonances satisfying the conditions

3.2 More resonances sharing one specific radius

"Magic" spin a = 0.983

3.2 More resonances sharing one specific radius

Investigated hypothesis: For a special values of black hole spin more resonances can occur at the same radius. Strong resonance phenomena may arise when the Keplerian and epicyclic frequencies are in the lowest possible ratio.

3.2 "Exotic" multiple resonances at the common orbit

Triple frequencies and black hole spin a

a) Two resonances at different radii

- 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)

for special values of spin common top, bottom, or mixed frequency

two frequency pairs reduce into a triple frequency ratio set

3.3 Two resonances excited at two resonance radius („Ugly“ case)

Triple frequencies and black hole spin a "top identity"

"bottom identity"

3.3 Two resonances excited at two resonance radius („Ugly“ case)

"middle identity"

two special cases

Triple frequencies and black hole spin a

3.3 Two resonances excited at two resonance radius („Ugly“ case)

4. Extended resonance model with humpy - included oscillations

Investigated hypothesis:excitation of the epicyclic oscillations by the processes related to the LNRF orbital velocity hump arising in the case of near-extreme Kerr black holes.

Talk – Slaný, Stuchlík, Torok

Investigated hypothesis: excitation of the epicyclic oscillations by the processes related to the LNRF orbital velocity hump arising in the case of near-extreme Kerr black holes:

Rcritttt

Rh

rrrrRcrit

ggg

ddgRR

~2

~

~

2

~d max,|~

4. Extended resonance model with humpy - included oscillations

In the relevant regions, for increasing rotational parameter the ratios of the epicyclic frequencies to the humpy frequency tend to the ratios of small integers. The regions are close to the 3:1 and 4:1 epicyclic resonant orbits.

4. Extended resonance model with humpy - included oscillations

Neutron star with corotating aligned dipole magnetic field and induced electric fieldPseudo-Newtonian approach - Paczynski-Wiita gravitational potential - classical magnetic dipole field vector potential

2D effective potential:

Neutron star and particle scaled parameters:Gravitational and electric field switches:

a) Existence of halo orbits

Talk - Kovář, Stuchlík, Karas (paper in preparation)

5. Resonance model with halo orbits

=>

Radial and vertical epicyclic frequencies

Neutron starM=1.5MSUN

R=10 km Beq=108 G7x103 rad/s

Dust particlem=4x10-15 kgq=10-14 C

Paper – Stuchlík, Kovář (in preparation)

R

V

=-10R =-5R =0 =5R =10R

=-10R

=-5R

=0

=10R

=5R

5. Resonance model with halo orbits

6.1 Braneworld uniform density compact stars - influence of the tidal charge parameter on QPOs

Parameters - mass M and a "tidal" charge b instead of an usual "electric" charge Q2:

• results for the atoll source 4U 1636-53

Parameters - mass M, dimensionless spin a, tidal charge b:

6.2 Rotating braneworld black holes - influence of the tidal charge parameter on QPOs

Strong resonant phenomena - "magic" spin

6.2 Rotating braneworld black holes - influence of the tidal charge parameter on QPOs• results for microquasar GRO J1655-40

LINE PROFILES OF re=5M ROTATING CIRCLE AROUND BLACK HOLE-DIRECT IMAGE(talk:Schee,Stuchlík)

6.2 Rotating braneworld black holes

LINE PROFILES OF re=5M ROTATING CIRCLE AROUND BLACK HOLE-INDIRECT IMAGE(talk:Schee,Stuchlík)

6.2 Rotating braneworld black holes

7. Conclusions

Present: Measured eigenfrequencies of QPOs+ properly chosen orbital resonance

black hole (neutron star) parameters

• Magnetic restoring force: complicated intraction of the magnetic field and the disc material

• Gravitational restoring force: could act to some extend independently of detailed disc or blob structure

• The same magnitude of quadrupole moment deformations (and related gravitationallyperturbing mass) used for explaining limiting pulsar perionds (300Hz-600Hz) together with limits coming from LIGO measurements seems to be effective for giving a viable mechanism for exciting QPOs observed in NS (QS) systems.

7. Conclusions

Future:

Mathematical theory of resonances+ physical theory of accretion disks

connection of resonance parameters & physical properties of disks

Observed details of QPOs (scatter of resonant frequencies, details of Török´s energy switch effect, etc.) imply restrictions on physical properties of accretion disk (or internal structure of neutron stars)