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.

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

Motivation o Mass estimate and quality problems of LMXBs kHz QPOs data fits by the relativistic precession QPO model frequency relationso Arbitrary solution: improving of fits by lowering the radial epicyclic frequencyo Possible interpretation: The Lorentz forceo 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

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.

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

Slowly rotating neutron star, spacetime described by Schwarzschild metricSlowly rotating neutron star, spacetime described by Schwarzschild 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

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

Slightly charged orbiting matterSlightly charged orbiting matter

Circular orbital motion in a dipole magnetic field on the Schwarzschild background

The equation of equatorial circular orbital motion with the Lorentz forceThe equation of equatorial circular orbital motion with the Lorentz force

Two (±) solution for clockwise and counter-clockwise orbital motionTwo (±) solution for clockwise and counter-clockwise orbital motion Components of the four-velocity and Components of the four-velocity and the orbital angular frequencythe orbital angular frequency

Circular orbital motion in a dipole magnetic field on the Schwarzschild background

Behavior of corrected orbital angular velocity. Keplerian geodesic limit Keplerian geodesic limit

The symmetry of ± solutions with respect to simultaneous interchange of The symmetry of ± solutions with respect to simultaneous interchange of ΩΩ orientation and sign of the specific charge. In the next only “+” solution orientation and sign of the specific charge. In the next only “+” solution will be analyzed.will be analyzed.

Different behavior for attracting and repulsing region of Lorentz forceDifferent behavior for attracting and repulsing region of Lorentz force

Repulsive Lorentz force Repulsive Lorentz force lowers lowers ΩΩ

Ω Ω grows in attractive grows in attractive regionregion

Existence of orbits near Existence of orbits near the horizonthe horizon

Opposite orientation of Opposite orientation of ΩΩ under circular photon orbit in under circular photon orbit in attractive regionattractive region

Existence of epicyclic behavior implies stability of the circular orbitsExistence of epicyclic behavior implies stability of the circular orbits

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

Epicylic frequencies as a tool for a investigation of a stability of circular orbits

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

Epicylic frequencies as a tool for a investigation of a stability of circular orbits

Behavior of the radial and vertical epicyclic frequency Different regions of stability with respect to radial and vertical Different regions of stability with respect to radial and vertical perturbationsperturbations

The The radial epicyclic frequency grows with specific charge, while the radial epicyclic frequency grows with specific charge, while the vertical one displays more complex behaviour.vertical one displays more complex behaviour.

Global stable region Region of global stability as a intersection of regions of vertical and radial Region of global stability as a intersection of regions of vertical and radial stability. stability.

Significant shift of ISCO orbit, position of magnetic ISCO orbits strongly Significant shift of ISCO orbit, position of magnetic ISCO orbits strongly depends on specific charge.depends on specific charge.

Critical specific charge qCritical specific charge qcritcrit lying lying

in the repulsive regionin the repulsive region for q> qfor q> qcrit crit MISCO is given by MISCO is given by

ωωθθ=0 curve =0 curve for q< qfor q< qcrit crit MISCO is given by MISCO is given by

ωωrr=0 curve=0 curve

In the attractive region In the attractive region MISCO is shifted away from MISCO is shifted away from the neutron starthe neutron star

In the repulsive region the In the repulsive region the position of MISCO could be shifted position of MISCO could be shifted toward to horizontoward to horizon

The lowest MISCO(q=qThe lowest MISCO(q=qcritcrit) at 2.73 ) at 2.73

M with M with ΩΩ/2/2ππ=3124Hz =3124Hz ( M=1.5 M( M=1.5 Msun , sun , μμ=1.06 x 10=1.06 x 10-4 -4 mm-2-2))

Different behavior of the corrected frequencies

Origin of the nodal precession Violence of Violence of spherical symmetry - spherical symmetry - equality of the orbital frequency and the equality of the orbital frequency and the vertical epicyclic frequencyvertical epicyclic frequency

Lense – Thirring like nodal precession frequencyLense – Thirring like nodal precession frequency

Different phase in attractive and repulsive regionDifferent phase in attractive and repulsive region

Repulsive region

Attractive region

Desired correction coresponds to the behavior of frequencies for small Desired correction coresponds to the behavior of frequencies for small charge of orbiting matter in attractive regioncharge of orbiting matter in attractive region

Significant lowering of radial epicyclic frequencySignificant lowering of radial epicyclic frequency

Significant shift of marginaly stable orbit ( MISCO) awaySignificant shift of marginaly stable orbit ( MISCO) away

Weak violence of spherical symmetryWeak violence of spherical symmetry

Implications for the relativistic precession kHz QPO model

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 presence of Lorentz force generated by the interaction of dipole The presence of Lorentz force generated by the interaction of dipole magnetic field of the neutron star and the charge of orbiting matter magnetic field of the neutron star and the charge of orbiting matter significantly modifies orbital and epicyclic frequencies of circular orbital significantly modifies orbital and epicyclic frequencies of circular orbital motion.motion.

Frequencies displays different complex behavior in attractive and repulsive Frequencies displays different complex behavior in attractive and repulsive region of Lorentz forceregion of Lorentz force..

In the attractive region the MISCO is shifted away from the horizonIn the attractive region the MISCO is shifted away from the horizon

Stable circular orbits exist under the circular photon orbit in the repulsive Stable circular orbits exist under the circular photon orbit in the repulsive regionregion

New nodal precession origins as the equality of orbital and vertical New nodal precession origins as the equality of orbital and vertical epicyclic frequency is violated.epicyclic frequency is violated.

The presence of Lorentz force improves NS mass estimate obtained by the 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 fitting LMXBs twin kHz QPO data by relativistic precesion QPO model and the can improve the quality such fits as well.can improve the quality such fits as well.

Conclusions

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)

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