1. Pavel Bakala,Gabriel Török, Zdeněk Stuchlík and Eva Šrámková Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic Supported by On a multi-resonant origin of high frequency QPOs in the atoll source 4U 1636-53 Ragtime 9

2. Supported by Ragtime 9 On the general--relativity predicted relations matching neutron star kHz QPOs Pavel Bakala,Gabriel Török, Zdeněk Stuchlík and Eva Šrámková Or On a multi-resonant origin of high frequency QPOs in the atoll source 4U 1636-53

3. Orbital motion in a strong gravity 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.

4. Orbital motion in a strong gravity

5. Geodesic motion models: orbital motion in a strong gravity Imply the existence of the periastron and nodal (Lense-Thirring) precession Periastron precession frequency Nodal ( declination ) precession frequency Total precession frequency

6. Testing frequency relations Figs on this page: nasa.gov

7. Data of atoll source 4U-1636-53

8. Testing frequency relations: Stella, Bursa and Total precession

9. 3.2 Fitting the data in the Hartle-Thorne metric

12. Fitting the data in the Hartle-Thorn We consider the external rotating neutron spacetime description given by „ugly“ HARTLE-THORN METRIC. We use slightly modified relations for geodesic frequencies derived by Abramowicz, M.A., Almergren, G.J.E., Kluzniak, W., Thampan, A.V., 2003 astro-ph/0312070.

13. Numbers Figs on this page: nasa.gov

14. Data of atoll source 4U-1636-53

15. Numbers The atoll source 4U 1636-53 (Rough “fit by eye”, in the Schwarzschild case, where the relations coincide.)

16. Numbers – fits and resulting parameters The atoll source 4U 1636-53 Orbital modelMass (M sun ) Angular momentum Quadrupole momentum Stella2.650.4780.228 Bursa2.360.4950.245 Total precession 1.680.0

17. Extension – non geodesics correction Figs on this page: nasa.gov

18. The discussed geodesic relations provide fits which are in good qualitative agreement with general trend observed in the neutron star kHz QPO data. No one of this relations provides really good fits (we checked for the other five atoll sources, that trends are same as for 4U 1636-53) with realistic values of mass and angular momentum with respect to the present knowledge of the neutron star equations of state To check whether some non geodesic influence can resolve the problem above we consider the assumption that the effective frequency of radial oscillations may be lowered, e.g., by the hotspots interaction with the accretion disk or with the neutron star magnetic field. Then, in the possible lowest order approximation, the effective frequency of radial oscillations may be written as where k is a small konstant. Multiresonant orbital model with non geodesics correction

19. Corrected radial epicyclic frequency corresponds to more realistic values of mass Correctéd Total precesssion model gives minimum of χ 2 with non zero angular momentum

20. Total precession model with non geodesics correction

21. Summary Multiresonant orbital model of QPOs provide fits which are in good qualitative agreement with general trend observed in the neutron star kHz QPO data. For realy satisfactory fit we have to consider non geodesics correction. In zero aproximation we introduce corrected radial epicyclic frequency. Stella and Total precession model with non geodesic correction provides better fits and more realistic values of mass and angular momentum. Corrected orbital model Mass (M sun ) Angular momentumQuadrupole momentum Correction parameter k Stella1.790.080.00640.20 Bursa2.250.50.250.13 Total precession 1.690.070.00490.20

22. Bonus : Epicyclic frequencies in dipole magnetic field on a pure Schwarzschild background M=1.5 M sun B=10 7 Gauss R star =4M Q=10 10 Shift of marginally stable orbit

23. Thank you for your atention Figs on this page: nasa.gov