Jiří Kovář, Ondřej Kopáček, Vladimír Karas , Zdeněk Stuchlík

1. OFF-EQUATORIAL MOTION OF CHARGED PARTICLES NEAR COMPACT OBJECTS Institute of Physics Silesian University in Opava Czech Republic Astronomical Institute Czech Academy of Sciences Prague Jiří Kovář, Ondřej Kopáček, Vladimír Karas, Zdeněk Stuchlík

2. Introduction • ‘Halo orbits’– off-equatorial circular orbits of constant r and q • (stable orbits) • problems to deal with • - halo orbitsexistence and their basic features • [Dullin, Horányi and Howard, 1999, 2002] (weak GF) • [Kovář, Stuchlík and Karas, 08, Class. Quantum Gravity] (strong GF) • [Calvani, de Felice, Fabbri, Turolla, 82, Nuevo Cimento] • - relatedoff-equatorial motion • [Karas, Vokrouhlický,92, General Relativity and Gravitation] • [Kopáček, Kovář, Karas and Stuchlík, 09, in preparation] • -astrophysical consequences • [ ? ]

3. Existence of halo orbits

4. Existence of halo orbitsBasic equations • Einstein-Maxwell’s equations

5. Existence of halo orbitsBackgrounds • Geometry • Schwarzschild • Kerr • Kerr-Newmann • Electromagnetic field • test rotating dipole in Schwarzschild • test static dipole in Kerr • test uniform in Kerr • Kerr-Newmann

6. Existence of halo orbitsEffective potential • Hamiltonian • Hamilton’s equations • constants of motion • effective potential Axially symmetric and stationary

7. Existence of halo orbitsNeutron stars A: Corotating negative charge B:

8. Existence of halo orbitsNeutron stars C: Positive charge counterrotating (corotating) D:

9. Existence of halo orbitsNeutron stars

10. Existence of halo orbitsKerr BH with magnetic fields current ring galactic uniform MF

11. Existence of halo orbitsKerr –Newmann BH and NS Black hole - inner Naked singularity

12. Existence of halo orbitsKerr –Newmann BH and NS Black hole – outer [Calvani, de Felice, Fabbri, Turolla, 82]

13. Related trajectoriesConstants of motion • Equation of motion • Separability of equation and searching for constants of motion • numerical integration • Poincaré surfaces of section • - surface of phase-space • - cross sections of the trajectory with another two coordinates • fuzzy structure chaotic motion no additional constant • curve regular motion additional constant

14. Related trajectoriesKerr-Newmann BH and NS

15. Related trajectoriesKerr BH + dipole MF Takahashi, Koyama ApJ,2009

16. Related trajectoriesCharged particles motion Regular motion Chaotic motion Kerr-Newmann NS Kerr BH and dipole MF

17. Summary • We have proved the existence of stable halo (off-equatorial) orbits of charged particles near all of the investigated models of the compact objects. • Except the unique Kerr-Newmann case, the motion of particles along the halo orbits in the studied cases is chaotic, with the degree of chaoticness growing with the growing energy of particles, especially when both the halo lobes are joined in the equatorial plane or allow the inflow of particles into the BH • We expect halo clouds of charged particles to exist near compact objects, regardless the “rough” used models (the single test particle approximation and approximative background description).

18. Acknowledgement RAGtime workshops (Opava)IAU Symposium (Tenerife)