1. Gabriel Török* Relating high-frequency QPOs and neutron-star EOS *Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic The presentation refers to work in progress and draws mainly from the collaboration with M. Abramowicz*, P.Bakala*, M. Bursa, D. Barret, J. Horák, J. Miller, M. Urbanec*, and Z. Stuchlík*

2. 1. Low-mass X-ray binaries (LMXBs), accretion discs, variability density comparable to the Sun mass in units of solar masses temperature ~ roughly as the T Sun moreless 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 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

3. 2. Short-term and rapid variability, kHz QPOs Fig: nasa.gov LMXBs short-term X-ray variability: peaked noise (Quasi-Periodic Oscillations) Low frequency QPOs (up to 100Hz) hecto-hertz QPOs (100-200Hz) kHz QPOs (~200-1500Hz): Lower and upper QPO mode forming twin peak QPOs frequency power Sco X-1 kHz QPO origin remains questionable, it is often expected that they are associated to the orbital motion in the inner part of the disc. Individual peaks can be related to a set of oscillators as well as to a time evolution of an oscillator.

4.  Several models have been proposed. Most of them relate QPOs to the orbital motion in inner parts of accretions disc. For instance,  Relativistic precession model, Stella, Vietri, 1999, relates the kHz QPOs to the frequencies of geodesic motion.  Some models relate the kHz QPOs to resonance between disc oscillation modes corresponding to the frequencies of geodesic motion (Kluzniak, Abramowicz, 2001) In next we focuse mostly on frequency identification given by relativistic precession model, (But we note that, in Schwarzschild spacetime, this identification correspond to m= -1 radial and m= -2 vertical disc oscillation modes as well.) 3. Orbital models of kHz QPOs

5. *For simplicity we consider Kerr spacetime on next few slides (while finaly we apply a more realistic approach needed for rotating neutron stars). One can solve above equations in order to obtain frequency relations U ( L ) which can be compared to those observed. * 4. Frequency relations given by the relativistic precession model

6. M=1.4M_sun, j=0 M=2M_sun, j=0 M=1.4M_sun, j=0.3 From the definition equations, there is always a unique curve for each different combination of (M, j). On the other hand, it is the practical question whether one can obtain curves which are rather similar for rather different combinations of (M, j).

7. M s =1M_sun M s =1.5M_sun M s =2M_sun M s =2.5M_sun M s =3M_sun M = M s (1+0.9*j) j = 0.3 For a given mass M s of the non-rotating neutron star there is a set of similar curves given by the relation M ~ M s (1+0.9*j). M = M s, j = 0 4. Frequency relations given by the relativistic precession model

8. 5. Fitting the data It was previously noticed that the RP model fits the data qualitatively well but always with non-negligible residuals (which arise especially on the top part of the correlation). It is often quoted that the model implies a high angular momentum (j>0.25) for which the residuals are somewhat lower (but still significant). Here we suggests that a fit for the non-rotating neutron star with only free parameter M s implies a rough mass-angular-momentum relation M ~ Ms(1+0.9*j). related to a “family of best fits” giving comparable chi^2. We investigate this suggestion for the source 4U 1636-53.

9. 5. Fitting the data The best fit of 4U 1636-53 data for j = 0 is reached for M s = 1.77 M_sun, which implies M= M s (1+0.9*j), M s = 1.77M_sun Expected inaccuracy of this rough relation (still analytic formulae)

10. 5. Fitting the data The best fit of 4U 1636-53 data for j = 0 is reached for M s = 1.77 M_sun, which implies M= M s (1+0.9*j), M s = 1.77M_sun Expected inaccuracy of this rough relation (still analytic formulae) Color-coded map of xi^2 for Hartle-Thorne metric [M,j, q; 10^9 points; 40hours on IBM Blade server] well agrees with rough estimate given by simple one-parameter fit. chi^2 ~ 300 chi^2 ~ 400

11. 6. EOS From X-ray burst lighcurves, the spin frequency was estimated (Sthromayer et. al, 1998-2005) to be either 290 or 580Hz. We calculated relevant NS configurations for several EOS. EOS 580Hz EOS 290Hz

12. 7. Two more QPO models We calculated also predictions of two other QPO model (frequency identifictions):  Epicyclic model (one of models by Abramowicz & Kluzniak): L = r, U =   Warped disc oscillations model (S. Kato, 2008): L = 2(  - r ), U = 2 K - r EOS 580Hz EOS 290Hz

13. 7. Conclusions  Relativistic precession model: M ~ 1.77 M_sun (1+0.9*j), considering EOS M ~1.9-2 M_sun, j ~ 0.15 (280Hz) or M ~2-2.3 M_sun, j ~ 0.25 Neutron star radii below ISCO  Warped disc oscillation model: M ~ 2.3 M_sun (1+0.75*j) considering EOS M ~2.4 M_sun, j ~ 0.15 (280Hz), no solution for 580Hz Neutron star radii below ISCO  Epicyclic model: M ~ 0.95 M_sun (1+ j) considering EOS M ~1.1 M_sun, j ~ 0.2 (280Hz), M ~1.2-1.3 M_sun, j ~ 0.4 (280Hz), Neutron star radii highly above ISCO but slightly below the relevant resonant orbit R NS ~ 8 – 10M vs. R 3:2 ~ 10-11M  The above results are preliminary and still require further investigation but the main outlined quantitative differences between predictions of individual models are robust.