Dynamics Of Charged Particles Moving Around Magnetized Kerr Black Hole

The study of dynamics around black hole can describes the properties of the space-times around the black hole and can help researchers understand the theory of accretion disk,which is important significance.The effective potential on the equatorial plane does not require solving the equations of motion,yet it reflects the laws of motion of the system.Kerr black holes are surrounded by uniform magnetic fields,and induced charges are generated by the parallel orientation of the spin of the black hole and the magnetic field.In this context,we focus on the one-dimensional effective potential of the test particle on the equatorial plane.The effective potential allows researchers to discuss the effects of small changes of parameters on the stable circular orbit,the unstable circular orbit and the innermost stable circular orbit of the test particle.We find that the energy of the test particle at the unstable circular orbit increases with an increase in the black hole spin and the angular momentum of the particle,but decreases with an increase of one of the induced charge parameter and the magnetic field parameter.The radius of the stable circular orbit on the equatorial plane also increases,whereas those of the innermost stable circular orbits decreases.Separating the motion of the test particle into two different classes according to the spin of the black hole and the sign of the angular momentum of the charged particle,the larger the spin the more obvious the distinction is,and the radius of the retrograde orbits tends to be larger than that of the direct orbits.The two-dimensional effective potential is also of great importance.Looking at the whole figure,we find that the lowest point of the potential energy is often near the equatorial plane,which is consistent with the results obtained from the one-dimensional effective potential.The photon effective potential allows researchers to obtain the unstable circular orbits of photons,from which we can classify the light around the black hole and help to reconstruct the bright ring of the black hole.Numerical integration schemes for investigating the motion of test particles on the non-equatorial plane are indispensable.Explicit algorithm exhibit good behavior in the hamiltonian error for a long time integration.This article extend the applicability of the explicit integrator and construct the explicit algorithm in the case of Kerr black holes with induced charges and in the case of Kerr Newman black holes.The non-spin black hole constructing explicit method can be decomposed directly,however,this algorithm fail to work in the spin black hole.In order to crack this hard nut,researchers can introduce the time transformation function to modify the equation of Hamiltonian and then calculate the analytical solution of it.Compared with the Runge-Kutta integrators,the algorithm exhibit a good long-term behavior in the energy error.And this method is better than the implicit midpoint method in computational cost.The orbital dynamics of charged particles on the non-equatorial plane are traced by the constructed time-transformed expanded phase space explicit algorithm.The motion of charged test particles around Kerr black holes is roughly divided into two categories.One case is the external environment where no magnetic field exists.The other category is the Kerr black holes surrounded by an external electromagnetic field.The magnetic field can be considered as tidal environments,destroying the integrability of these spacetime and the dynamics of the test particle can be chaotic in some circumstances.We focus on the second category.Using the Chaos Detection Index Poincare Section、Lyapunov and the Fast Lyapunov Indicators,it is shows that the dynamics sensitivity depends on small variations in the several parameters,namely,the induced charge parameter,the magnetic field parameter,the energy of the particle,and the angular momentum.Chaos is more likely to appear when the induced charge and magnetic parameters as well as the energy of the test particle increase.However,as the angular momentum of the particles increases,the chaotic properties are weakened.When the dragging effects of the spacetime increase,the chaotic properties are not always weakened under some circumstances.