Download ON EXISTENCE OF HALO ORBITS IN COMPACT OBJECTS

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