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 On magnetic field induced non-geodesics correction to the relativistic orbital and epicyclic frequencies.

2.  Motivation o Mass estimate and quality problems of LMXBs kHz QPOs data fits by the relativistic precession QPO model frequency relations o Arbitrary solution: improving of fits by lowering the radial epicyclic frequency o Possible interpretation: The Lorentz force o Interesting theoretical aspects  Circular orbital motion in a dipole magnetic field on the Schwarzschild background  Corrected orbital and epicyclic frequencies  Complex behaviour of the frequencies, (m)ISCO and stability of the orbits  Origin of the nodal precession  Implications for the relativistic precession QPO model  Conclusions Outline

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 currents and related magnetic field in the disc  Slightly charged orbiting matter Circular orbital motion in a dipole magnetic field on the Schwarzschild background

9.  The equation of equatorial circular orbital motion with the Lorentz force  Two (±) solution for clockwise and counter-clockwise orbital motion  Components of the four-velocity and the orbital angular frequency Circular orbital motion in a dipole magnetic field on the Schwarzschild background

10. Behavior of corrected orbital angular velocity.  Keplerian geodesic limit  The symmetry of ± solutions with respect to simultaneous interchange of Ω orientation and sign of the specific charge. In the next only “+” solution will be analyzed.  Different behavior for attracting and repulsing region of Lorentz force  Repulsive Lorentz force lowers Ω  Ω grows in attractive region  Existence of orbits near the horizon  Opposite orientation of Ω under circular photon orbit in attractive region

11.  Existence of epicyclic behavior implies stability of the circular orbits  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 Epicylic frequencies as a tool for a investigation of a stability of circular orbits

12.  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 Epicylic frequencies as a tool for a investigation of a stability of circular orbits

13. Behavior of the radial and vertical epicyclic frequency  Different regions of stability with respect to radial and vertical perturbations  The radial epicyclic frequency grows with specific charge, while the vertical one displays more complex behaviour.

14. Global stable region  Region of global stability as a intersection of regions of vertical and radial stability.  Significant shift of ISCO orbit, position of magnetic ISCO orbits strongly depends on specific charge.  Critical specific charge q crit lying in the repulsive region  for q> q crit MISCO is given by ω θ =0 curve  for q< q crit MISCO is given by ω r =0 curve  In the attractive region MISCO is shifted away from the neutron star  In the repulsive region the position of MISCO could be shifted toward to horizon  The lowest MISCO(q=q crit ) at 2.73 M with Ω/2π=3124Hz ( M=1.5 M sun, μ=1.06 x 10 -4 m -2 )

15. Different behavior of the corrected frequencies

16. Origin of the nodal precession  Violence of spherical symmetry - equality of the orbital frequency and the vertical epicyclic frequency  Lense – Thirring like nodal precession frequency  Different phase in attractive and repulsive region Repulsive region Attractive region

17.  Desired correction coresponds to the behavior of frequencies for small charge of orbiting matter in attractive region  Significant lowering of radial epicyclic frequency  Significant shift of marginaly stable orbit ( MISCO) away  Weak violence of spherical symmetry Implications for the relativistic precession kHz QPO model

18.  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

19.  The presence of Lorentz force generated by the interaction of dipole magnetic field of the neutron star and the charge of orbiting matter significantly modifies orbital and epicyclic frequencies of circular orbital motion.  Frequencies displays different complex behavior in attractive and repulsive region of Lorentz force.  In the attractive region the MISCO is shifted away from the horizon  Stable circular orbits exist under the circular photon orbit in the repulsive region  New nodal precession origins as the equality of orbital and vertical epicyclic frequency is violated.  The presence of Lorentz force improves NS mass estimate obtained by the fitting LMXBs twin kHz QPO data by relativistic precesion QPO model and the can improve the quality such fits as well. Conclusions

20. References  P. Bakala, E. Šrámková, Z. Stuchlík, G.Török, 2009, Classical and Quantum Gravity, submitted.  P. Bakala, E. Šrámková, Z. Stuchlík, G.Török in COOL DISCS, HOT FLOWS: The Varying Faces of Accreting Compact Objects (Funäsdalen, Sweden). AIP Conference Proceedings, Volume 1054, pp. 123-128 (2008).  P. Bakala, E. Šrámková, Z. Stuchlík, G. Török, in Proceedings of RAGtime 8/9 (Hradec nad Moravici, Czech Republic), Silesian University in Opava. Volume 8/9, pp. 1-10 (2007) Thank you for your atention