Circular motion and fundamental frequencies of charged particles in dipole magnetic field around magnetized Bocharova--Bronnikov--Melnikov--Bekenstein black holes

Abstract

We examine the field produced by a stationary current loop encircling a static Bocharova--Bronnikov--Melnikov--Bekenstein (BBMB) black hole and the response of charged test particles. From the dipole projection of Maxwell's equations, we obtain closed-form interior and exterior radial functions, and we test them against a multipole expansion of the localized source whose radial functions are also elementary. Every horizon-regular mode vanishes at the degenerate horizon $r_{+}=M$, so the magnetic flux through the horizon is zero. For a loop at $r_0=10M$, the dipole errors in the equatorial potential and polar field remain below $10\%$ for $r\lesssim 0.42r_0$ and $r\gtrsim 3.02r_0$, but reach about $40\%$ and $80\%$ close to the loop. Using the resulting potential, we study the effective potential, circular orbits, the innermost stable circular orbit (ISCO), a binding-energy proxy, and epicyclic frequencies. Because the dipole field reverses its equatorial direction across the loop, a positive magnetic coupling $\mathcal{B}$ acts oppositely on the two sides. Above ${\mathcal{B}}_c\simeq 0.0119$, the prograde radially stable orbits split into two bands separated by an unstable gap. At $\mathcal{B}=0.2$, the ISCO moves from $4M$ to $2.97M$, the exterior band begins at $45.46M$, and the ISCO binding-energy proxy grows from $8.14\%$ to $23.39\%$, compared with $5.72\%$ for neutral particles around a Schwarzschild black hole. For negative coupling, exterior orbits can be radially stable but vertically unstable. Higher multipoles preserve the large-coupling trends but shift the near-loop values appreciably, so the near-loop results apply to the projected source rather than to a thin ring.