Pavel Bakala Eva Šrámková, Gabriel Török and Zdeněk Stuchlík

1. Dipole magnetic field on the Schwarzschild background and related epicyclic frequencies. or On magnetic-field induced non-geodesic corrections to the relativisticprecession QPO model frequency relations. 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

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 M=2Msun (From : T. Belloni, M. Mendez, J. Homan, 2007, MNRAS)

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=2.65Msun j=0.48 q=0.23

6. Fitting the LMXBs kHz QPO data by relativistic precession frequency relations Improvingoffits : non-geodesiccorrection ? 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.

7. Fitting the LMXBs kHz QPO data by relativistic precession frequency relations Therelativisticprecessionmodel witharbitrary „non-geodesic“ correction M=1.75 Msun j=0.08 q=0.01 k=0.20

8. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • Slowlyrotating neutron star, spacetimedescribed by Schwarzschildmetric • Dominating static exteriormagneticfieldgenerated by intrinsicmagneticdipole moment ofthe star μperpendicular to theequatorial plane • Negligiblecurentsandrelatedmagneticfield in thedisc • Slightlychargedorbitingmatter

9. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • Theequationofequatorialcircular orbital motionwiththeLorentzforce • Componentsofthefour-velocityandthe orbital angularfrequency

10. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • AlievandGaltsov (1981, GRG) aproach to perturbatethepositionofparticlearoundcircular orbit • Theradialandverticalepicyclicfrequenciesin thecompositeofSchwarzschildspacetime geometry anddipolemagneticfield

11. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • In the absence ofthe Lorenz forcenewformulaemergeintowell-knownformulaeforpureScharzschild case • Localymeasuredmagneticfieldforobserver on theequatorofthe star • Model case

12. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • Thebehaviorofthe orbital andepicyclicfrequenciesfortinychargeoforbitingmatter • Significantloweringofradialepicyclicfrequency • Significantshiftofmarginalystable orbit ( ISCO) • Violenceofequalityofthe orbital frequencyandtheverticalepicyclicfrequency

13. Exact calculations of non-geodesics correction induced by the magnetic field of the star. • Thebehavioroftheeffectivemarginalystable orbit (EISCO) • Constraintsforthespecificchargeofthedisc ( REISCO < 10 M )

14. Implications for the relativistic precession kHz QPO model • Loweringof NS massestimateobtained by thefittingoftwin kHz QPO data • Loweringof NS massestimateobtainedfromhighestobservedfrequencyofthesource ( ISCO estimate)

15. Conclusions • The Lorenz forceinduced by the presence ofthemagneticdipole moment andthesmallchargeoforbitingmattersignificantlymodifiesthefrequencyrelationofrelativisticprecesion QPO model. Thesamecorrectionsshouldbevalidforother orbital models. Notethat in theSchwarschild case thefrequencyidentificationof RP model coincideswithradial m=1 andvertical m=2 discoscilationsmodes. • In the presence of such Lorentzforce on theSchwarzschild background theradialepicyclicfrequencyislowereddown, thepositionof ISCO isshiftedandtheequalityof orbital andverticalepicyclicfrequencyisviolated. • The presence of such Lorentzforceimproves NS massestimateobtained by thefittingLMXBstwin kHz QPO data. • Theproblemsremains : anoriginofthe such smallcharge. • In order to fitting a particularsourcethesolution in rotating NS spacetime background (Hartle-Thornemetric) isneeded. • Loweringof NS massestimateobtainedfromhighestobservedfrequencyofthesource ( ISCO estimate)

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