Figs on this page: nasa.gov

Outline

1. Introduction: Quasi-periodic oscillations (QPOs) - Black-hole and neutron star binaries, accretion disks and QPOs

2. Non-linear resonance models 2.1. Orbital resonance model 2.2. Some properties of non-linear resonances 3. Black-hole observations 3.1. The 3:2 phenomenon 3.2. The 1/M scaling

4. Neutron-star observations 4.1. Frequency – frequency fits 4.2. The slope-shift anti-correlation 4.3. Evolution of RMS amplitudes across the resonance point

5. Summary and black hole – neutron star comparison

6. References

Presentation download: www.physics.cz/research in sect. news

1. Basic introduction: Quasiperiodic oscillations in X-ray

Figs on this page: nasa.gov

radio

“X-ray”and visible

1.1. Black hole binaries and accretion disks

Figs on this page: nasa.gov

t

I

Pow

erFrequency

1.2. X-ray observations and QPOs

Light curve:

Power density spectra (PDS):

Figs on this page: nasa.gov

hi-frequencyQPOs

low-frequencyQPOs

frequency

pow

er

1.2. Quasiperiodic oscillations

The rms amplitude = percentual fraction (root mean square fraction) of the peak energy with the respect to the total countrate, the rms amplitude indicates the energy connected to the observed oscillations.

Pow

er

Frequency

1.2. Quasiperiodic oscillations

2. Non-linear resonance models

Figs on this page: nasa.gov

in general relativity, all the three frequencies depend on generic mass as f ~ 1/M

!

2.1. Resonance of epicyclic frequencies (a particular non-linear resonance)

2.1. Resonance of epicyclic frequencies (a particular non-linear resonance)

2.1.1. Resonance of epicyclic frequencies (a particular non-linear resonance)

frequencies are in ratio of small natural numbers [e.g, Landau & Lifshitz, 1976], which must hold also in the case of forced resonances

!

animation by Bursa 2004

• The 1/M scaling in epicyclic model follows from nature of Keplerian motion. On the other hand, frequencies in ratio of small natural numbers come from general properties of non-linear oscillators and have nothing to do with particular choice of resonant frequencies.

• One can check more general properties of non-linear resonances.

2.2. Some properties of non-linear oscillators

2.2. Some properties of non-linear oscillators

!

2.2. Some properties of non-linear oscillators

!

2.2. Some properties of non-linear oscillators

• Of non-linear resonances:

• Eigenfrequencies in ratio of small natural numbers

(Landau&Lifshitz, 1974)

(Abramowicz etal., 2005/6, Török etal. 2006)

• Of particular epicyclic model:

• 1/M scaling

• Preferred 3:2 ratio (Horák 2004)

Well defined predictions

• Lines in plane:

Their anticorrelation:

3. Black-hole hi-frequency QPOs

Figs on this page: nasa.gov, personal archiv

3. Black-hole hi-frequency QPOs: the 3:2 ratio and 1/M scaling

3. Black-hole hi-frequency QPOs: the 3:2 ratio and 1/M scaling

Lachowicz, Czerny & Abramowicz (2006), astro-ph/0607594:

3. Black-hole hi-frequency QPOs

• Of non-linear resonances:

• Eigenfrequencies in ratio of small natural numbers

• Anticorrelated Bursa-lines in frequency-frequency plane

Frequencies seems to be constant,

Bursa lines are not observed.

• Of particular epicyclic model:

• 1/M scaling

• Preferred 3:2 ratio

Predictions vs. observations in the case

Figs on this page: nasa.gov, personal archiv

4. Neutron-star hi-frequency QPOs

kHz frequency – frequency relationfor low mass X-ray binaries

4.1. Frequency – frequency fits

NS

BH

Psaltis et al. 1998

4.1. Frequency – frequency fits

Bursa 2002:

NS are far away from= 1.51,

f-f relations differ forindividual sources

4.1. Frequency – frequency fits

4.2. Frequency – frequency fits: slope-shift anti-correlation

The coefficients of linear fits for twelve neutron-star sources are anticorrelated [Abramowicz et al. 2006]

4.2. Frequency – frequency fits and slope-shift anti-correlation

The coefficients of linear fits for twelve neutron-star sourcespointed to the eigenfrequency ratio equal to 3/2.

Two weakly coupled oscillators with eigenfrequency ratio A^0:

From the data for 12 sources,

4.2. The rms amplitude evolution across the resonance point

For few sources examined so far, the difference of lower and upper rms amplitudes changes its sign when the source pass the exact 3:2 ratio.

The case of 4U 1636:

When the ratio between upper and lower frequency is higher then 3:2, the upper oscillation is stronger.For the ratio of frequencies equal to 3:2, both oscillations have the same amplitudes.When the ratio between upper and lower frequency is lower then 3:2, the lower oscillation is stronger.

4.3. The rms amplitude evolution across the resonance point

When the ratio between upper and lower frequency is higher then 3:2, the upper oscillation is stronger.For the ratio of frequencies equal to 3:2, both oscillations have the same amplitudes.When the ratio between upper and lower frequency is lower then 3:2, the lower oscillation is stronger. (Török & Barret, Horák & Török, 2006 in prep.)

4. Neutron-star hi-frequency QPOs

• Of non-linear resonances:

• Eigenfrequencies in ratio of small natural numbers

• Anticorrelated Bursa-lines in frequency-frequency plane

+ the effect of rms reverse

• Of particular epicyclic model:

• 1/M scaling

• Preferred 3:2 ratio

Predictions vs. observations in the case

+ observation of the effect of the oscillation energy reverse

5. Summary and black hole – neutron star comparison

Both the black hole and neutron star QPOs shows the 3:2 ratio pointing to a resonance. The 1/M scaling indicating the orbital origin of hi-frequency QPOs holds /at least roughly/. The neutron star hi-frequency QPOs forms anticorrelated lines in frequency-frequency plane as predicted from the assumtion of non-linear weakly coupled oscillations.

On the other hand, the black hole (microquasar) hi-frequencies seems to be rather fixed….

5. Summary and black hole – neutron star comparison

The range of frequencies for NS Bursa lines is in hundreds of Hz while the black hole frequencies are fixed. This could be due to scaling of range with the square of dimensionless amplitudes ?

range ~ sq. of , NS range ~ 500 Hz but NS amplitudes are ~ 10 times higherthus BH range should be ~ 100 times lower, i.e. ~ 5Hz

6. References

• Török, Abramowicz, Kluzniak, Stuchlík2006, proc. of Albert Einstein Conf. Paris 2005, astro-ph: 0603847(also contain complete list of references briefly mentioned in this presentation)

• Abramowicz, Barret, Bursa, Horák, Kluzniak, Olive, Rebusco, Török,2006, proc. Of RAGtime 2005, download: ADS or www.physics.cz/research

• Abramowicz, Barret, Bursa, Horák, Kluzniak, Olive, Rebusco, Török2006, submitted to MNRAS

• Lachowicz, Czerny & Abramowicz (2006), astro-ph/0607594

Presentation download: www.physics.cz/research in sect. news