Pavel Bakala Martin, Urbanec, 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

Non-Geodesic Orbital and Epicyclic Frequencies in Vicinity of Slowly Rotating Magnetized Neutron Stars.

Mass estimate and quality problems of LMXBs kHz QPOs data fits by the frequency relations based on orbital QPO models

Improving of fits by lowering the radial epicyclic frequency

Frequencies of orbital motion of slightly charged particles in the dipole magnetic field

Existence of epicyclic frequencies as stability conditions

Schwarzschild case

Lense –Thiring case

Conclusions

Non-Geodesic Orbital and Epicyclic Frequencies in Vicinity of Slowly Rotating Magnetized Neutron Stars.

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. Wide class of QPO similar models could be constructed.

The geodesic frequencies are the functions of the parameters of spacetime geometry (M, j, q) and the appropriate radial coordinate.

Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

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

M=2Msun

Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

Hartle - Thorne metric, particular source 4U 1636-53Fit parameters: mass, specific angular momentum, quadrupole momentum

M=2.65Msun

j=0.48q=0.23

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

)0.1(~ krr wherewhere k k is a small konstant is a small konstant..

Improving of fits : non-geodesic correction ?Improving of fits : non-geodesic correction ?

Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

The relativistic precession The relativistic precession model with model with arbitrary „non-geodesic“ correctionarbitrary „non-geodesic“ correction

M=1.75 Msun

j=0.08q=0.01k=0.20

Fitting the LMXBs kHz QPO data by relativistic precession frequency relations

Frequencies of orbital motion of slightly charged test particles in the dipole magnetic field

Slowly rotating neutron star, spacetime described by Lense-Thiring metricSlowly rotating neutron star, spacetime described by Lense-Thiring metric

Dominating static exterior magnetic field generated by Dominating static exterior magnetic field generated by intrinsic magnetic intrinsic magnetic dipole moment of the star dipole moment of the star μμ perpendicular to the equatorial planeperpendicular to the equatorial plane

The same angular component as in the pure Schwarzschild case The same angular component as in the pure Schwarzschild case

Electric/time component induced only by rotation (frame-draging). Electric/time component induced only by rotation (frame-draging).

Constants aConstants at0,1t0,1 are function of are function of starstar´s´s parameters parameters J,J,ΩΩ

Negligible curents and related magnetic field in the disc Negligible curents and related magnetic field in the disc

Slightly charged orbiting matterSlightly charged orbiting matter

<< for ions of hydrogen<< for ions of hydrogen

Relation between spin and angular frequency Relation between spin and angular frequency

Intrinsic magnetic dipole momentIntrinsic magnetic dipole moment

Frequencies of orbital motion of slightly charged test particles in the dipole magnetic field

Relation between spin and angular frequency Relation between spin and angular frequency

Intrinsic magnetic dipole momentIntrinsic magnetic dipole moment

Frequencies of orbital motion of slightly charged test particles in the dipole magnetic field

Aliev and Galtsov (1981, GRG) aproach to perturbate the position of Aliev and Galtsov (1981, GRG) aproach to perturbate the position of particle around circular orbit particle around circular orbit

The The radial and vertical epicyclic frequencies radial and vertical epicyclic frequencies in the composite of in the composite of Schwarzschild spacetime geometry and dipole magnetic fieldSchwarzschild spacetime geometry and dipole magnetic field

Exact calculations of non-geodesics correction induced by the magnetic field of the star.

In the absence of the Lorenz force new formulae merge into well-known In the absence of the Lorenz force new formulae merge into well-known formulae for pure Scharzschild caseformulae for pure Scharzschild case

Localy measured magnetic field for observer on the equator of the starLocaly measured magnetic field for observer on the equator of the star

Model case Model case

Exact calculations of non-geodesics correction induced by the magnetic field of the star.

The behavior of the orbital and epicyclic frequencies for tiny charge of The behavior of the orbital and epicyclic frequencies for tiny charge of orbiting matter orbiting matter

Significant lowering of radial epicyclic frequencySignificant lowering of radial epicyclic frequency

Significant shift of marginaly stable orbit ( ISCO)Significant shift of marginaly stable orbit ( ISCO)

Violence of equality of the orbital frequency and the vertical epicyclic Violence of equality of the orbital frequency and the vertical epicyclic frequencyfrequency

Exact calculations of non-geodesics correction induced by the magnetic field of the star.

The behavior of the effective marginaly stable orbit (EISCO)The behavior of the effective marginaly stable orbit (EISCO)

Constraints for the specific charge of the disc ( RConstraints for the specific charge of the disc ( REISCO EISCO < 10 M ) < 10 M )

Exact calculations of non-geodesics correction induced by the magnetic field of the star.

Lowering of NS mass estimate obtained by the fitting of twin kHz QPO data Lowering of NS mass estimate obtained by the fitting of twin kHz QPO data

Lowering of NS mass estimate obtained from highest observed frequency Lowering of NS mass estimate obtained from highest observed frequency of the source ( ISCO estimate)of the source ( ISCO estimate)

Implications for the relativistic precession kHz QPO model

The Lorenz force induced by the presence of the magnetic dipole moment The Lorenz force induced by the presence of the magnetic dipole moment and the small charge of orbiting matter significantly modifies the frequency and the small charge of orbiting matter significantly modifies the frequency relation of relativistic precesion QPO model. The same corrections should be relation of relativistic precesion QPO model. The same corrections should be valid for other orbital models. Note that in the Schwarschild case the valid for other orbital models. Note that in the Schwarschild case the frequency identification of RP model coincides with radial m=1 and vertical frequency identification of RP model coincides with radial m=1 and vertical m=2 disc oscilations modes.m=2 disc oscilations modes.

In the presence of such Lorentz force on the Schwarzschild background In the presence of such Lorentz force on the Schwarzschild background the radial epicyclic frequency is lowered down, the position of ISCO is the radial epicyclic frequency is lowered down, the position of ISCO is shifted and the equality of orbital and vertical epicyclic frequency is violated.shifted and the equality of orbital and vertical epicyclic frequency is violated.

The presence of such Lorentz force improves NS mass estimate obtained The presence of such Lorentz force improves NS mass estimate obtained by the fitting LMXBs twin kHz QPO data.by the fitting LMXBs twin kHz QPO data.

The problems remains : an origin of the such small charge.The problems remains : an origin of the such small charge.

In order to fitting a particular source the solution in rotating NS spacetime In order to fitting a particular source the solution in rotating NS spacetime background (Hartle-Thorne metric) is needed. background (Hartle-Thorne metric) is needed.

Lowering of NS mass estimate obtained from highest observed frequency Lowering of NS mass estimate obtained from highest observed frequency of the source ( ISCO estimate)of the source ( ISCO estimate)

Conclusions

Thank you for your atention

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