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Astrophysical relevance of off-equatorial circular orbits near compact objects Jiří Kovář, Zdeněk Stuchlík and Vladimír Karas Institute of Physics Silesian University in Opava Czech Republic "High-energy sources at different time scales" September 29 - October 3, 2008, Kathmandu, Nepal This work was supported by the Czech grant MSM 4781305903 Introduction • • ‘Halo orbits’ – off-equatorial circular orbits of constant r and q (stable orbits) Störmer problem - charged particle motion in dipole MF, GF (and EF) Introduction • Planets (Saturn) [Dullin, Horányi and Howard, 1999, 2002] • Strong gravitational fields of compact objects [Kovář J Stuchlík Z and Karas V, 2008, Class. Quantum Grav. 25] ? 1) Neutron star (pseudo-Newtonian approach) 2) Schwarzschild black hole with test dipole MF (relativistic approach) 3) Kerr-Newman black hole and naked singularity (relativistic approach) 1. Neutron star Model • object of radius R, mass M, rotating with W, corotating dipole MF B0 • particle of mass m and charge q, angular momentum L • pseudo-Newtonian description • gravitational field • rigidly co-rotating magnetic field • induced electric field Pseudo-Newtonian Classical Maxwell 1. Neutron star • Hamiltonian • cylindrical coordinates • effective potential • switches Effective potential Neutron star • scaling: time distance • Hamiltonian • effective potential • spherical coordinates • existence of orbits Existence of orbits 1. Neutron star Character of orbits A: Corotating negative charge B: 1. Neutron star Character of orbits C: Positive charge counterrotating (corotating) D: 1. Neutron star Charge Summary Halo orbits counter-rotating co-rotating counter-rotating co-rotating co-rotating GF EF MF 2. Schwarzschild BH with dipole MF Model • Schwarzschild black hole of mass M with plasma ring of radius R in equatorial plane with electric current I • particle of charge q and mass m, angular momentum L • relativistic description • Schwarzschild metric • dipole magnetic test field vector potential [Petterson 74] 2. Schwarzschild BH and dipole MF • Hamiltonian • Hamilton’s equations • normalization condition • effective potential Effective potential 2. Schwarzschild BH and dipole MF • existence of orbits Existence of orbits 3. Kerr-Newman BH (NS) • Kerr-Newman BH (NS) of mass M, spin a and charge Q • particle of charge q and mass m, angular momentum L • relativistic description • Kerr-Newman metric • magnetic field vector potential Model 3. Kerr-Newman BH (NS) • Hamiltonian • Hamilton’s equations • normalization condition • effective potential [Calvani, de Felice, Fabbri, Turolla, 82] Effective potential 3. Kerr-Newman BH (NS) • equation of motion • projection • locally non-rotating observer • circular motion • equations of motion • azimuthal velocity Inertial forces formalism 3. Kerr-Newman BH (NS) • conditions Existence of orbits 3. Kerr-Newman BH (NS) Effective potential Black hole - inner Naked singularity 3. Kerr-Newman BH (NS) Effective potential Black hole – outer Conclusions Existence of stable halo orbits Rotating (slowly rotating) neutron stars (very approximative model) (pseudo-Newtonian description) Schwarzschild BH with dipole magnetic field (relativistic description) (test magnetic field) Kerr-Newman BH (relativistic description) Kerr-Newman NS (relativistic description) Conclusions 1) Quasi-periodic oscillations 2) Corona Astrophysical relevance Acknowledgement To all the authors of the papers which our study was based on To you Thank you