1. Pavel Bakala Eva Šrámková, Gabriel Török and Zdeněk Stuchlík Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic Dipole magnetic field on the Schwarzschild background and related epicyclic frequencies.or On magnetic-field induced non-geodesic corrections to the relativistic precession QPO model frequency relations relativistic precession QPO model frequency relations.

2. Dipole magnetic field on the Schwarzschild background and related epicyclic frequencies  Mass estimate and quality problems of LMXBs kHz QPOs data fits by the relativistic precession QPO model frequency relations  Arbitrary solution: improving of fits by lowering the radial epicyclic frequency  Possible interpretation: The Lorentz force  Frequencies of orbital motion in the dipole magnetic field  Implications for the relativistic precession kHz QPO model  Conclusions

3. Fitting the LMXBs kHz QPO data by relativistic precession frequency relations The relativistic precesion model (in next RP model) introduced by Stella and Vietri, (1998, ApJ) indetifies the upper QPO frequency as orbital (keplerian) frequency and the lower QPO frequency as the periastron precesion frequency. The geodesic frequencies are the functions of the parameters of spacetime geometry (M, j, q) and the appropriate radial coordinate.

4. Fitting the LMXBs kHz QPO data by relativistic precession frequency relations (From : T. Belloni, M. Mendez, J. Homan, 2007, MNRAS) M=2M sun

5. Fitting the LMXBs kHz QPO data by relativistic precession frequency relations Hartle - Thorne metric, particular source 4U 1636-53 Fit parameters: mass, specific angular momentum, quadrupole momentum M=2.65M sun j=0.48 q=0.23

6. The discussed geodesic relation provide fits which are in good qualitative agreement with general trend observed in the neutron star kHz QPO data, but not 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, by the slightly charged hotspots interaction 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. Improving of fits : non-geodesic correction ? Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

7. The relativistic precession model with arbitrary „non-geodesic“ correction The relativistic precession model with arbitrary „non-geodesic“ correction M=1.75 M sun j=0.08 q=0.01 k=0.20 Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

8.  Slowly rotating neutron star, spacetime described by Schwarzschild metric  Dominating static exterior magnetic field generated by intrinsic magnetic dipole moment of the star μ perpendicular to the equatorial plane  Negligible curents and related magnetic field in the disc  Slightly charged orbiting matter Exact calculations of non-geodesics correction induced by the magnetic field of the star.

9.  The equation of equatorial circular orbital motion with the Lorentz force  Components of the four-velocity and the orbital angular frequency Exact calculations of non-geodesics correction induced by the magnetic field of the star.

10.  Aliev and Galtsov (1981, GRG) aproach to perturbate the position of particle around circular orbit  The radial and vertical epicyclic frequencies in the composite of Schwarzschild spacetime geometry and dipole magnetic field Exact calculations of non-geodesics correction induced by the magnetic field of the star.

11.  In the absence of the Lorenz force new formulae merge into well-known formulae for pure Scharzschild case  Localy measured magnetic field for observer on the equator of the star  Model case Exact calculations of non-geodesics correction induced by the magnetic field of the star.

12.  The behavior of the orbital and epicyclic frequencies for tiny charge of orbiting matter  Significant lowering of radial epicyclic frequency  Significant shift of marginaly stable orbit ( ISCO)  Violence of equality of the orbital frequency and the vertical epicyclic frequency Exact calculations of non-geodesics correction induced by the magnetic field of the star.

13.  The behavior of the effective marginaly stable orbit (EISCO)  Constraints for the specific charge of the disc ( R EISCO < 10 M ) Exact calculations of non-geodesics correction induced by the magnetic field of the star.

14.  Lowering of NS mass estimate obtained by the fitting of twin kHz QPO data  Lowering of NS mass estimate obtained from highest observed frequency of the source ( ISCO estimate) Implications for the relativistic precession kHz QPO model

15.  The Lorenz force induced by the presence of the magnetic dipole moment and the small charge of orbiting matter significantly modifies the frequency relation of relativistic precesion QPO model. The same corrections should be valid for other orbital models. Note that in the Schwarschild case the frequency identification of RP model coincides with radial m=1 and vertical m=2 disc oscilations modes.  In the presence of such Lorentz force on the Schwarzschild background the radial epicyclic frequency is lowered down, the position of ISCO is shifted and the equality of orbital and vertical epicyclic frequency is violated.  The presence of such Lorentz force improves NS mass estimate obtained by the fitting LMXBs twin kHz QPO data.  The problems remains : an origin of the such small charge.  In order to fitting a particular source the solution in rotating NS spacetime background (Hartle-Thorne metric) is needed.  Lowering of NS mass estimate obtained from highest observed frequency of the source ( ISCO estimate) Conclusions

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