Motion of charged particles near a weakly magnetized Schwarzschild black hole

We study motion of a charged particle in the vicinity of a weakly magnetized Schwarzschild black hole and focus on its bounded trajectories lying in the black hole equatorial plane. If the Lorentz force, acting on the particle, is directed outward from the black hole, there exist two qualitatively different types of trajectories; one is a curly motion and another one is a trajectory without curls. We calculated the critical value of the magnetic field for the transition between these two types. If the magnetic field is greater than the critical one, for fixed values of the particle energy and angular momentum, the bounded trajectory has curls. The curls appear as a result of the gravitational drift. The greater the value of the magnetic field, the larger is the number of curls. We constructed an approximate analytical solution for a bounded trajectory and found the gravitational drift velocity of its guiding center.

Figure 1

Motion of a charged particle in a uniform magnetic field in a flat space-time. Points $A$ and $B$ define the maximal and the minimal distances of the particle from the coordinate origin $O$. (a) $\mathcal{L}<0$. (b) $\mathcal{L}>0$. Points $C$ and $D$ are the turning points where $\dot{\varphi }=0$.

Figure 2

Motion of a charged particle in the mutually orthogonal uniform magnetic and weak gravitational fields. The dashed line illustrates the motion of the guiding center.

Figure 3

Effective potential for $\ell >0$ and $b=1/2$. Curve 1 corresponds to $\ell \approx 1.18$. There are no circular orbits. Curve 2 corresponds to $\ell \approx 2.07$. There is the innermost stable circular orbit defined by $\rho ={\rho }_{+}\approx 1.59$. Curve 3 corresponds to $\ell \approx 3.22$. There are unstable and stable circular orbits corresponding to the minimum and the maximum of the effective potential. For the bounded trajectory corresponding to the energy ${\mathcal{E}}_{+}$, one has $\rho \in [{\rho }_1,{\rho }_2]$.

Figure 4

Dependence of the innermost stable circular orbits $\rho ={\rho }_{\pm }$ on $b$. For each value of $b$, one has ${\rho }_{+}<{\rho }_{-}$. For $b\to +\infty$, one has ${\rho }_{+}\to 1$ and ${\rho }_{-}\to (5+\sqrt{13})/4\approx 2.15$.

Figure 5

Angular momenta ${\ell }_{\pm }$ for the innermost stable circular orbits as functions of $b$. (a) $\ell >0$, the Lorentz force is repulsive. (b) $\ell <0$, the Lorentz force is attractive; the curve $\ell ={\ell }_{-}(b)$ is monotonically decreasing. For $b=0$ we have ${\ell }_{\pm }=\pm \sqrt{3}$, which corresponds to a Schwarzschild black hole. For $b\to +\infty$ we have ${\ell }_{\pm }\to \pm \infty$.

Figure 6

Types of a bounded trajectory corresponding to $\ell >0$. Arrows illustrate direction of motion of a charged particle. Circular arcs represent the stable circular orbit defined by $\rho ={\rho }_{\underset{+}{min}}$. (a) ${\mathcal{E}}_{+}>{\mathcal{E}}_{*}$. Points $A$ correspond to $\rho ={\rho }_1$, and point $B$ corresponds to $\rho ={\rho }_2>{\rho }_{*}$. Points $C$ and $D$ are turning points, where $d\varphi /d\sigma =0$. (b) ${\mathcal{E}}_{+}={\mathcal{E}}_{*}$. Points $A$ correspond to $\rho ={\rho }_1$, and point $B$ is a turning point, which corresponds to $\rho ={\rho }_2={\rho }_{*}$, where $d\varphi /d\sigma =0$. (c) ${\mathcal{E}}_{+}\in [{\mathcal{E}}_{\underset{+}{min}},{\mathcal{E}}_{*})$. Points $A$ correspond to $\rho ={\rho }_1$, and point $B$ corresponds to $\rho ={\rho }_2<{\rho }_{*}$.

Figure 7

${\mathcal{E}}_{\max }^2$, ${\mathcal{E}}_{\min }^2$, and ${\mathcal{E}}_{*}^2$ as functions of $\ell$ for a fixed value of $b$. Point $A$, where the curves ${\mathcal{E}}_{\max }^2(\ell )$ and ${\mathcal{E}}_{\min }^2(\ell )$ join, corresponds to the innermost stable circular orbit. In the domain I between these curves bounded trajectories have no curls. In the domain II between the curves ${\mathcal{E}}_{*}^2$ and ${\mathcal{E}}_{\max }^2$ the trajectories are of the curly type. They degenerate into unstable circular orbit at point $B$. Bounded motion occurs in the regions I and II. This particular plot is constructed for $b=1/2$. The qualitative behavior for different values of $b$ is the same.