Off-equatorial orbits in strong gravitational fields near compact objects -- II: halo motion around magnetic compact stars and magnetized black holes

Off-equatorial circular orbits with constant latitudes (halo orbits) of electrically charged particles exist near compact objects. In the previous paper, we discussed this kind of motion and demonstrated the existence of minima of the two-dimensional effective potential which correspond to the stable halo orbits. Here, we relax previous assumptions of the pseudo-Newtonian approach for the gravitational field of the central body and study properties of the halo orbits in detail. Within the general relativistic approach, we carry out our calculations in two cases. Firstly, we examine the case of a rotating magnetic compact star. Assuming that the magnetic field axis and the rotation axis are aligned with each other, we study the orientation of motion along the stable halo orbits. In the poloidal plane, we also discuss shapes of the related effective potential halo lobes where the general off-equatorial motion can be bound. Then we focus on the halo orbits near a Kerr black hole immersed in an asymptotically uniform magnetic field of external origin. We demonstrate that, in both the cases considered, the lobes exhibit two different regimes, namely, one where completely disjoint lobes occur symmetrically above and below the equatorial plane, and another where the lobes are joined across the plane. A possible application of the model concerns the structure of putative circumpulsar discs consisting of dust particles. We suggest that the particles can acquire a small (but non-zero) net electric charge, and this drives them to form the halo lobes.

💡 Research Summary

The paper investigates the existence and stability of off‑equatorial circular orbits—so‑called “halo orbits”—of electrically charged test particles in the strong‑field regime of general relativity. The authors extend their previous pseudo‑Newtonian analysis by treating the full relativistic problem for two astrophysically relevant configurations: (i) a rotating magnetic compact star (magnetic dipole aligned with the rotation axis) described by a Schwarzschild‑like metric with a test dipole field, and (ii) a Kerr black hole immersed in an asymptotically uniform magnetic field (Wald’s solution).

The theoretical framework starts from the super‑Hamiltonian
(H=\frac12 g^{ij}(\pi_i-\tilde q A_i)(\pi_j-\tilde q A_j)),
with conserved specific energy (E) and angular momentum (L) due to stationarity and axial symmetry. From this Hamiltonian the two‑dimensional effective potential (V_{\rm eff}(r,\theta)) is derived. Circular halo orbits correspond to stationary points of (V_{\rm eff}) (∂_r V = ∂θ V = 0). Stability requires a positive‑definite Hessian (∂^2{rr}V>0 and det H>0).

To complement the Hamiltonian approach, the authors employ a “force formalism” obtained by projecting the Lorentz equation onto the locally non‑rotating frame (LNRF). This yields a balance equation between gravitational‑centrifugal‑Coriolis forces (G, Z, C) and electromagnetic forces (E, M). Solving the radial and polar components simultaneously leads to a cubic equation for the orbital velocity (v_h). Real solutions with (|v_h|<1) give physically admissible halo motions; the corresponding specific charge (q_h) and angular momentum (L_h) follow from the balance equations.

Magnetic star case.
The spacetime outside the star is approximated by the Schwarzschild‑like line element
(ds^2=-(1-2/r)dt^2+(1-2/r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)).
A rotating dipole field co‑rotating with angular velocity (\Omega) is introduced via the vector potential
(A_t=\frac{3}{8}\Omega M R\sin^2\theta,\quad A_\phi=-\frac{3}{8}M R\sin^2\theta),
where (R) is a function of the stellar radius. The effective potential becomes
(V_{\rm eff}= -\frac{3}{8}qM\Omega R\sin^2\theta+\sqrt{1-2/r},\Bigl