Institute of Physics, Silesian University in Opava Gabriel Török GAČR 209/12/P740, CZ.1.07/2.3.00/ Synergy, GAČR G, SGS , Determination of compact object parameters from CO-AUTHORS: Eva Šrámková, Martin Urbanec, Kateřina Goluchová, Andrea Kotrlová, Karel Adámek, Jiří Horák, Pavel Bakala, Marek Abramowicz, Zdeněk Stuchlík, Wlodek Kluzniak, Gabriela Urbancová, Tomáš Pecháček observations of high frequency quasiperiodic oscillations

Outline of our progress report 1.Introduction: neutron star rapid X-ray variability, quasiperiodic oscillations, twin peaks 2.Measuring BH spin from HF QPOs 2.1 BH spin from geodesic models (summary of some older results by Torok et al, 2011, A&A) 2.2 Consideration of a>1 (Kotrlová et al 2014, A&A) 2.3 Nongeodesic effects (Šrámková et al 2014, to be submitted) 3.Measuring NS spin from HF QPOs 3.1 Mass-angular-momentum relations, EoS consideration (summary of Torok et al, ApJ, 2010, 2012, Urbanec et al 2010, A&A) 3.2 Detailed consideration of EoS and spin (Torok et al 2014, in prep.) 3.3 Torus model (Torok et al 2014, in prep.) 3.4 General constraints (Torok et al 2014, A&A) Tento projekt je spolufinancován Evropským sociálním fondem a státním rozpočtem České republiky

density comparable to the Sun mass in units of solar masses temperature ~ roughly as the T Sun more or less optical wavelengths Artists view of LMXBs “as seen from a hypothetical planet” Companion: Compact object: - black hole or neutron star (>10^10gcm^3) >90% of radiation in X-ray LMXB Accretion disc Observations: The X-ray radiation is absorbed by the Earth atmosphere and must be studied using detectors on orbiting satellites representing rather expensive research tool. On the other hand, it provides a unique chance to probe effects in the strong-gravity-field region (GM/r~c^2) and test extremal implications of General relativity (or other theories). T ~ 10^6K Figs: space-art, nasa.gov 1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs

Fig: nasa.gov LMXBs short-term X-ray variability: peaked noise (Quasi-Periodic Oscillations ) Low frequency QPOs (up to 100Hz) hecto-hertz QPOs ( Hz) HF QPOs (~ Hz): Lower and upper QPO mode forming twin peak QPOs frequency power Sco X-1 The HF QPO origin remains questionable, it is often expected that it is associated to orbital motion in the inner part of the accretion disc. Individual peaks can be related to a set of oscillators, as well as to time evolution of the oscillator. 1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs

1.1 Black hole and neutron star HF QPOs Lower frequency [Hz] Upper frequency [Hz] Figure (“Bursa-plot”): after M. Bursa & MAA 2003, updated data 3:2

Figures - Left: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); Right: Torok (2009) 1.1 Black hole and neutron star HF QPOs BH HF QPOs: (perhaps) constant frequencies, exhibit the 3:2 ratio NS HF QPOs: 3:2 clustering, - two correlated modes which exchange the dominance when passing the 3:2 ratio It is unclear whether the HF QPOs in BH and NS sources have the same origin. Amplitude difference Frequency ratio Upper frequency [Hz] Lower frequency [Hz] 3:2

Figures - Left: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); Right: Torok (2009) 1.1 Black hole and neutron star HF QPOs BH HF QPOs: (perhaps) constant frequencies, exhibit the 3:2 ratio NS HF QPOs: 3:2 clustering, - two correlated modes which exchange the dominance when passing the 3:2 ratio It is unclear whether the HF QPOs in BH and NS sources have the same origin. Amplitude difference Frequency ratio Upper frequency [Hz] Lower frequency [Hz] 3:2

There is a large variety of ideas proposed to explain the QPO phenomenon [For instance, Alpar & Shaham (1985); Lamb et al. (1985); Stella et al. (1999); Morsink & Stella (1999); Stella & Vietri (2002); Abramowicz & Kluzniak (2001); Kluzniak & Abramowicz (2001); Abramowicz et al. (2003a,b); Wagoner et al. (2001); Titarchuk & Kent (2002); Titarchuk (2002); Kato (1998, 2001, 2007, 2008, 2009a,b); Meheut & Tagger (2009); Miller at al. (1998a); Psaltis et al. (1999); Lamb & Coleman (2001, 2003); Kluzniak et al. (2004); Abramowicz et al. (2005a,b), Petri (2005a,b,c); Miller (2006); Stuchlík et al. (2007); Kluzniak (2008); Stuchlík et al. (2008); Mukhopadhyay (2009); Aschenbach 2004, Zhang (2005); Zhang et al. (2007a,b); Rezzolla et al. (2003); Rezzolla (2004); Schnittman & Rezzolla (2006); Blaes et al. (2007); Horak (2008); Horak et al. (2009); Cadez et al. (2008); Kostic et al. (2009); Chakrabarti et al. (2009), Bachetti et al. (2010)…] - in some cases the models are applied to both BHs and NSs, in some not - some models accommodate resonances, some do not The ambition /common to several of the authors/ is to relate HF QPOs to orbital motion in strong gravity and infer the compact object properties using the QPO measurements… 1.2. The ambition

1.3 Models and frequency relations considered here Several models imply observable frequencies that can be expressed in therms of combinations of frequencies of the Keplerian, radial and vertical epicyclic oscillations. In the simple case of geodesic approximation and Kerr metric these are

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP2 1.3 Models and frequency relations considered here

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP TD WD ER KR RP1 RP2 MODEL : Characteristic Frequencies Relativistic Precession Stella et al. (1999); Morsink & Stella (1999); Stella & Vietri (2002)] 1.3 Models and frequency relations considered here

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP TD WD ER KR RP1 RP2 MODEL : Characteristic Frequencies Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Tidal Disruption Čadež et al. (2008), Kostič et al. (2009), Germana et al. (2009) 1.3 Models and frequency relations considered here

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP TD ER KR RP1 RP2 MODEL : Characteristic Frequencies Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Warped Disc Resonance a representative of models proposed by Kato (2000, 2001, 2004, 2005, 2008) WD (or torus) 1.3 Models and frequency relations considered here

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP TD ER KR RP1 RP2 MODEL : Characteristic Frequencies Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Epicyclic Resonance, Keplerian Resonance two representatives of models proposed by Abramowicz, Kluzniak et al. (2000, 2001, 2004, 2005,…) WD (or torus) 1.3 Models and frequency relations considered here

RP TD ER KR RP1 RP2 MODEL : Characteristic Frequencies Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Resonances between non-axisymmetric oscillation modes of a toroidal structure two representatives by Bursa (2005), Torok et al (2010) predicting frequencies close to RP model 2. 1 Models relating both of the 3:2 BH QPOs to a single radius WD Here we only focus on the choice of few hot-spot and disc-oscillation models: (or torus)

Here we only focus on the choice of few hot-spot and disc-oscillation models: RP TD ER KR RP1 RP2 MODEL : Characteristic Frequencies WD 1.3 Models and frequency relations considered here

Figure: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); the (advantage of) BH HF QPOs: (perhaps) constant frequencies, exhibit the mysterious 3:2 ratio Upper frequency [Hz] Lower frequency [Hz] The BH 3:2 QPO frequencies are rather stable which imply that they depend mainly on the geometry and not so much on the dirty physics of the accreted plasma. 2.1 BH spin from the geodesic QPO models

RP ERWD, TD Different models associate QPOs to different radii… 2.1 BH spin from the geodesic QPO models

One can easily calculate the frequency.mass functions for each of the models. Spin a 2.1 BH spin from the geodesic QPO models Torok et al., (2011) A&A

And compare the frequency.mass functions to the observation. For instance in the case of GRS (which here well represents all 3:2 microquasars). Spin a 2.1 BH spin from the geodesic QPO models Torok et al., (2011) A&A

And compare the frequency.mass functions to the observation. For instance in the case of GRS (which here well represents all 3:2 microquasars). Spin a 2.1 BH spin from the geodesic QPO models Clearly, different models imply very different spins… Torok et al., (2011) A&A

When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. 2.2 Consideration of a>1 (naked sigularities or superspinars - NaS) Kotrlová et al., (2014) A&A

When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smooth frequency.mass functions of the spin. 2.2 Consideration of a>1 (naked sigularities or superspinars - NaS) BHNaS Kotrlová et al., (2014) A&A

When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smooth frequency.mass functions of the spin. Some of them not (discontinuities and dichotomies appear…) 2.2 Consideration of a>1 (naked sigularities or superspinars - NaS) Kotrlová et al., (2014) A&A

When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smooth frequency.mass functions of the spin. Some of them not (discontinuities and dichotomies appear…) 2.2 Consideration of a>1 (naked sigularities or superspinars - NaS) The case of epicyclic resonance model Kotrlová et al., (2014) A&A

When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smooth frequency.mass functions of the spin. Some of them not (discontinuities and dichotomies appear…) 2.2 Consideration of a>1 (naked sigularities or superspinars - NaS) The case of epicyclic resonance model ZOOM Kotrlová et al., (2014) A&A

2.2.1 Summary of BH spin estimates For black holes, different models imply very different spin values. Except one model, there is always an alternative compatible with existence of a superspinning compact object. Nevertheless, only epicyclic resonance model then implies spin close to unity, while others imply values that are several times higher. Kotrlová et al., (2014) A&A

2.3. Non-geodesic effects consideration within ER model Pressure supported fluid tori Šrámková et al., (2014), in prep.

2.3. Non-geodesic effects consideration within ER model Pressure supported fluid tori – impact of pressure on the resonant frequency Šrámková et al., (2014), in prep. For low spins the results agree with pseudonewtonian case investigated by Blaes et al For high spins, the situation is different. Resonant frequencies are decreasing instead of increasing as the torus thickness rises.

2.3. Non-geodesic effects consideration within ER model For low spins the results agree with pseudonewtonian case investigated by Blaes et al For high spins, the situation is different. Resonant frequencies are decreasing instead of increasing as the torus thickness rises. Pressure supported fluid tori – impact of pressure on the resonant frequency Šrámková et al., (2014), in prep.

3.1 NS mass-angular momentum relations from HF QPO data

We have considered the relativistic precession (RP) twin peak QPO model to estimate the mass of central NS in Circinus X-1 from the HF QPO data. We have shown that such an estimate results in a specific mass–angular-momentum (M–j) relation rather than a single preferred combination of M and j. Later we confronted our previous results with another binary, the atoll source 4U 1636–53 that displays the twin peak QPOs at very high frequencies, and extend the consideration to various twin peak QPO models. In analogy to the RP model, we find that these imply their own specific M–j relations. Torok et al., (2010) ApJ Torok et al., (2012) ApJ

RP MODEL (4U ): Color-coded map of chi^2 [M,j,10^6 points] well agrees with rough estimate given by simple one-parameter fit. M= M s [1+0.75(j+j^2)], M s = 1.78M_sun Best chi^2 Torok et al., (2012) ApJ. 3.1 NS mass-angular momentum relations from HF QPO data

Several other models and sources Torok et al., (2012) ApJ.

3.1 NS mass-angular momentum relations from HF QPO data Model Model atoll source 4U Z-source Circinus X-1 Mass Mass R NS Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K 1.8M Sun [1+0.7(j+j 2 )] < r ms 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K 2.2M Sun [1+0.7(j+j 2 )] < r ms X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  1.8M Sun [1+(j+j 2 )] < r ms 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r 2.5M Sun [1+0.7(j+j 2 )] < r ms 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms 3M Sun < r ms

One can calculate M-j relations from EoS and spin frequency and compare these to the results based on QPOs. 3.2 Detailed consideration of EoS and NS spin Torok et al., (2014) in prep.

One can calculate M-j relations from EoS and spin frequency and compare these to the results based on QPOs. 3.2 Detailed consideration of EoS and NS spin Torok et al., (2014) in prep.

3.2 Detailed consideration of EoS and NS spin Another possibility is to INFER the spin from the QPO model. When EoS are considered directly for QPO modelling, the M-J degeneracy is broken and QPO models provide chi-square minima. RP MODEL Torok et al., (2014) in prep.

3.2 Detailed consideration of EoS and NS spin Another possibility is to INFER the spin from the QPO model. When EoS are considered directly for QPO modelling, the M-J degeneracy is broken and QPO models provide chi-square minima. RP MODEL X-ray burst VERY GOOD AGREEMENT ! Torok et al., (2014) in prep.

3.3 Torus Model The models which we have considered so far always require tuning of parameters in order to provide good fits of the high- frequency sources data. Here we attempt to fit the data with “modified RP model” which does not contain any new free parameters.

3.3 Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequency of the m=1 radial epicyclic oscillations of the torus.

3.3 Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequency of the m=1 radial epicyclic oscillations of the torus. RP MODEL BEST FITS TORUS MODEL Torok et al., (2014) in prep.

3.3 Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequency of the m=1 radial epicyclic oscillations of the torus. RP MODEL BEST FITS TORUS MODEL CUSP TORUS Torok et al., (2014) in prep.

3.4 General constraints SPIN [Hz] Number of ISCO-NS [relative units] The ISCO-NS distribution has the peaks at the values of the spin which can be very different from the peak in the distribution of all NS. High M -> peak at the original value of spin Low M -> peak at the high value of spin Inter. M -> two peaks – POSSIBLE DISTRIBUTION OF QPO SOURCES ! Torok et al., (2014) A&A Letters

3.5 Summary of NS spin estimates When only Kerr or Hartle-Thorne spacetime is assumed, the HF QPO data and their individual models imply mass-spin relations instead of the preferred combinations of these quantities. The degeneracy is broken when the EoS are considered. The relativistic precession model implies NS spin frequency in a perfect agreement with the observation (4U ). The modified RP model (torus model) provides good fits of the data without any tuning of parameters. Strong constrainst are possible even from very general assumptions.

END Thank you for your attention…