Circular motion of charged test particles in a weakly magnetized Taub--NUT background: From the equatorial slice to self-consistent conical orbits

Abstract We study circular motion of charged test particles around a Taub--NUT black hole with Manko--Ruiz parameter $C$, immersed in a weak external magnetic field introduced via Wald's prescription. Because the Taub--NUT metric is not reflection-symmetric about the equator once $l\neq 0$, generic charged orbits lie on cones $x=\cos\theta\neq 0$ rather than on the equatorial plane. We first analyse \emph{constrained} circular orbits obtained by imposing $x=\dot x=0$, exhibit in closed form the residual angular constraint that such orbits violate, and derive the circularity and marginal-stability conditions on the imposed slice. We then remove the constraint: the radial and angular conditions are solved simultaneously for self-consistent circular orbits on cones $x=x_0$, and the true ISCO is obtained from the stability of these orbits. For $B=0$ the cone sits at $x_0=2lE/\tilde L$ with $\tilde L=L-2lCE$, and the constrained ISCO radius and energy coincide \emph{exactly} with the self-consistent ones; for $B\neq0$ the difference grows as $B^{2}$ and stays below $10^{-3}M$ over the parameter range studied. The Manko--Ruiz parameter is pure gauge at $B=0$, where it only relabels the angular momentum, and enters physical quantities at $B\neq0$ only through the electrostatic part $A_t\propto lCB$ of the Wald potential. Increasing $B$ monotonically decreases $r_{\mathrm{ISCO}}$; the sign of $eB\mathcal{L}_{\rm eff}$ splits the branches, with anti-Larmor orbits reaching closer to the horizon; and $C$ contributes only at order $eBl$.