Paczyński Modulation: Diagnostics of the Neutron Star EOS? Institute of Physics, Silesian University in Opava Gabriel Török, Martin Urbanec, Karel Adámek, Pavel Bakala, Eva Šrámková, Zdeněk Stuchlík CZ.1.07/2.3.00/ Synergy, GAČR 209/12/P740, 202/09/0772, SGS ,

1. Outline 1.Introduction: QPOs 2.NS Compactness C (another introduction) 3.Epicyclic Resonance Model – Falsification using condition for Paczynski Modulation, C < 1 4.General Implications of Paczynski Modulation Mechanism (disc oscillation models): report on a work in progress

density comparable to the Sun mass in units of solar masses temperature ~ roughly as the T Sun more or less optical wavelengths MOTIVATION 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 2. Introduction: QPOs LMXBs

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 feature forming twin peak QPOs frequency power Sco X-1 The HF QPO origin remains questionable, it is most 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 a single oscillator. 2. Introduction: QPOs MOTIVATION

Power Frequency height h width w at ½ h Quality factor Q indicates sharpness of the peak, Q ~ h/w Amplitude r indicates strength of peak variability (its energy) in terms of “rms amplitude” = percentual fraction (root mean square fraction) of the peak energy with the respect to the total countrate (r ~ area under peak) BH QPOs (Galactic microquasars): frequencies up to 500Hz low amplitude and Q : typically up to r~5% and Q~5 NS QPOs: frequencies up to 1500Hz often amplitudes up to r~20% and quality factors up to Q~ Introduction: QPOs

KERR 3. NS Compactness OBLATENESS The influence of NS oblateness on orbital frequenies has been extensively studied in last decade, e.g., Morsink, Stella, 1999, ApJ; Gondek-Rosińska, Stergioulas, Bulik, Kluźniak, Gourgoulhon, A&A (2001); Amsterdamski, Bulik, Gondek-Rosińska, Kluźniak, A&A (2002),… Kluzniak et al., ApJ (1990) Torok et al. (2010),ApJ

KERR OBLATENESS 3. NS Compactness

KERR OBLATENESS C = RNS/Rms 3. NS Compactness

KERR OBLATENESS C = RNS/Rms 1 3. NS Compactness

KERR OBLATENESS C = RNS/Rms NS Compactness

KERR OBLATENESS C = RNS/Rms NS Compactness

KERR OBLATENESS C = RNS/Rms MASS low mass high mass 3. NS Compactness

C = RNS/Rms 3. NS Compactness

a) Observed frequencies are roughly equal to resonant eigenfrequencies. b) Alternatively, there are large corrections to the resonant eigenfrequencies. This for NSs FAILS. Abramowicz et al., Epicyclic Resonance Model for NS QPOs and NS Mass Within the group of non-linear models suggested by Abramowicz and Kluzniak there is one specific (often quted and discussed) model which relates QPOs to the axisymmetric vertical and radial accretion disc oscillations (Abramowicz & Kluzniak 2001). These oscillations have frequencies equal to the vertical and radial frequency of the perturbed geodesic motion. Two distinct simplifications can be than assumed (see Urbanec et al. 2010, for refs): Fig: J. Horák

For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). The solution related to the high mass (i.e. Kerr) approximation thus cannot be trusted. j 3. Epicyclic Resonance Model for NS QPOs and NS Mass

For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). Mass-spin relations inferred assuming Hartle-Thorne metric and various NS oblateness. One can expect that the red/yellow region is allowed by NS equations of state (EOS). q/j2q/j2 j Urbanec et al., (2010), A & A 3. Epicyclic Resonance Model for NS QPOs and NS Mass KERR OBLATENESS

For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). Mass-spin relations calculated assuming several modern EOS (of both “Nuclear” and “Strange” type) and realistic scatter from 600/900 Hz eigenfrequencies. j 3. Epicyclic Resonance Model for NS QPOs and NS Mass Urbanec et al., (2010), A & A

After Abr. et al., (2007), Horák (2005) 4. Paczynski Modulation and NS Compactness Possible relation between the X-ray QPO phenomenon and general relativity Bohdan Paczyński, 1987 ”….suggest that the unsteady flow would make the boundary-layer luminosity variable, possibly giving rise to the X-ray quasi-periodic oscillation (QPO) phenomenon.” REQUIRED CONDITION: C = RNS/Rms < 1

4. Paczynski Modulation and NS Compactness KERR OBLATENESS C = RNS/Rms MASS low mass high mass

(Epicyclic Resonance Model) The condition for modulation is fulfilled only for rapidly rotating strange stars, which most likely falsifies the postulation of the 3:2 resonant mode eigenfrequencies being equal to geodesic radial and vertical epicyclic frequency…. (Typical spin frequencies of discussed sources are about Hz; based on X-ray bursts) 4. Paczynski Modulation and Implied Restrictions Urbanec et al., (2010), A & A

5. Paczynski Modulation – General Implications Almost any disc-oscillation model requires C<1 MASS [MSun] SPIN [Hz] Initial Distribution of NS [C<>1] => Distribution of QPO Sources

5. Paczynski Modulation – General Implications Almost any disc-oscillation model requires C<1 Mass [Msun] SPIN [Hz] MASS [MSun] Initial Distribution of NS (one concrete EoS)

5. Paczynski Modulation – General Implications Almost any disc-oscillation model requires C<1 Mass [Msun] SPIN [Hz] MASS [MSun] Initial Distribution of NS (one concrete EoS)

5. Paczynski Modulation – General Implications Spin [Hz] Mass [Msun] MASS [MSun] SPIN [Hz] Resulting Distribution of QPO sources (the same EoS)

5. Paczynski Modulation – General Implications Spin [Hz] Mass [Msun] MASS [MSun] SPIN [Hz] Resulting Distribution of QPO sources (another example)

5. Paczynski Modulation – General Implications Mass [Msun] Number of Sources SPIN [Hz]

6. Conclusions Mass [Msun] Number of Sources SPIN [Hz]

END Thank you for your attention…