ON EXISTENCE OF HALO ORBITS IN COMPACT OBJECTS SPACETIMES Jiří Kovář Zdeněk Stuchlík & Vladimír Karas Institute of Physics Silesian University in Opava Czech Republic Hradec nad Moravicí, September, 2007 This work was supported by the Czech grant MSM

Introduction ‘Halo orbits’ – non-equatorial circular orbits of constant r and  Stormer problem – charged particle motion in dipole MF, GF and corotating EF (Saturn, planets) Do such orbits exist in strong gravitational fields of compact objects? 1)Neutron stars (pseudo-Newtonian approach) 2)Schwarzschild black hole and test dipole MF (relativistic approach) 3)Kerr-Newman black holes and naked singularities (relativistic approach)

inspiration by classical Dullin’s [2002] studies of Stormer problem particle of mass m and charge q, angular momentum L object of radius R, mass M, rotating with  gravitational field rigidly co-rotating magnetic field induced electric field Neutron star Model

Hamiltonian cylindrical coordinates effective potential stream function switches Neutron star Effective potential

Neutron star Existence of orbits scaling: time distance Hamiltonian effective potential spherical coordinates existence of orbits

Neutron star Character of orbits A: B: C: D:

Neutron star Summary ChargeHalo orbitsGFEF NO YES counter-rotating co-rotating YESNO counter-rotating co-rotating YES

Schwarzschild black hole and dipole MF Model relativistic description particle of charge q and mass m, angular moment L Schwarzschild black hole of mass M with plasma ring of radius R in equatorial plane with electric current I Schwarzschild metric dipole magnetic test field vector potential [Petterson 74]

Schwarzschild BH and dipole MF Effective potential Hamiltonian Hamilton’s equations normalization condition effective potential

Schwarzschild BH and dipole MF Existence of orbits existence of orbits

Kerr-Newman BH and NS Model exact relativistic solution particle of charge q and mass m, angular moment L Kerr-Newman BH (NS) of mass M spin a and charge Q metric magnetic field vector potential

Kerr-Newman BH (NS) Effective potential Hamiltonian Hamilton’s equations normalization condition effective potential

Kerr-Newman BH (NS) Inertial forces formalism equation of motion projection locally non-rotating observer circular motion equations of motion azimuthal velocity

Kerr-Newman BH (NS) Existence of orbits Black holes condition

Kerr-Newman BH (NS) Existence of orbits Extreme black holes condition

Kerr-Newman BH (NS) Existence of orbits Extreme black holes condition

Kerr-Newman BH (NS) Existence of orbits Naked singularities condition

Kerr-Newman BH (NS) Effective potential

Conclusions Existence of orbits Rotating (slowly rotating) neutron stars YES (very approximative model) (pseudo-Newtonian description) Schwarzschild BH with dipole magnetic field YES (relativistic description) (test magnetic field) Kerr-Newman BH YES (NO) (relativistic description) Kerr-Newman NS YES (relativistic description)