Charged fluid tori with an approximate magnetic self-field around magnetized Schwarzschild black holes: A semi-analytic approach

The gist The total magnetic field comprises a prescribed asymptotically uniform external component and an approximate self-field associated with the azimuthal current in the torus This study investigates toroidal structures formed by a charged, non-self-gravitating perfect fluid orbiting a Schwarzschild black hole immersed in a poloidal magnetic field.

Model Assumptions and Framework

The study models the fluid as a cold, rest-mass-dominated perfect fluid with uniform specific charge and zero conductivity, meaning the current is purely convective The total magnetic field is represented as the superposition of an asymptotically uniform external field and an effective self-field modeled by a regularized dipole approximation to the field of an effective circular current loop. The fluid dynamics are governed by the relativistic Euler equation derived from the conservation of total stress-energy tensor, where the electromagnetic field is decomposed into external and self-field components. Key thermodynamic assumptions include a barotropic polytropic equation of state, specifically setting pressure and internal-energy density small compared to the rest-mass density in the cold-fluid limit.

Integrability and Toroidal Morphology

The existence of solutions depends on satisfying an integrability condition for the Euler equations, which couples the specific angular momentum and specific charge profiles, l(r,θ) and q(r,θ). For a prescribed constant specific charge, this condition simplifies to a first-order quasilinear partial differential equation for the angular-momentum parametrization function u(r,θ). The resulting solutions permit the Euler equations to be integrated consistently to obtain an enthalpy potential w(r,θ). Closed poloidal sections of the equipotential surfaces w(r,θ) = const determine the equilibrium toroidal structure. The center of the torus corresponds to a local maximum of w, located by imposing stationarity conditions ∂rw = 0 and ∂θw = 0.

Analysis of Different Magnetic Field Configurations

The analysis proceeds by examining different magnetic field scenarios to understand how they modify the morphology and domain of existence of charged-fluid tori.

**- External field only: The external field supports closed tori within analytically derived parameter bounds. In the field-free limit, the general integrability condition reduces to the standard integrability structure of an uncharged barotropic perfect fluid. In any of these limiting cases, Eq. (82) becomes r(r −2)u′ = u, with the solution u(r) = u∞ / (1−2r), which corresponds to a nonconstant specific-angular-momentum profile. In the field-free and uncharged-fluid limits, the candidate centers within this restricted class correspond to radial minima of the enthalpy potential. The radially dependent class does not admit equatorially centered closed toroidal equilibrium structures in these limits. In contrast, constant-l profiles form a degenerate, automatically integrable subclass. The latter includes the well-known constant-specificangular-momentum tori [1, 13]. **

**- Asymptotically uniform external field only: When an asymptotically uniform external magnetic field is introduced, the radial integrability equation can instead be solved exactly in terms of the Lambert W function. Its principal branch admits equatorially centered local maxima within the derived radius-dependent parameter bounds and therefore allows the corresponding closed toroidal structures to be constructed. **

**- Approximate current-loop self-field only: For the isolated approximate current-loop self-field, no tori are found within the parameter ranges examined numerically. The polar and radial local-maximum conditions were not satisfied simultaneously within the parameter ranges and loop radii examined numerically. **

**- Combined external and approximate self-fields: When a combined magnetic field is considered, the resulting local cancellation between the two magnetic components can substantially modify the enthalpy distribution and the morphology of the torus. For a representative sequence examined, the radial extent L = rout − rcusp depends nonmonotonically on the relative self-field amplitude: it initially increases for weak self-fields but decreases again as the approximate self-field becomes comparable to the external field. The resulting framework clarifies how rotation, pressure, and an approximate magnetic self-field enter the equilibrium conditions and provides a useful basis for more self-consistent models. The principal existence results for closed structures within the radially dependent class are summarized in Table 1. Natural extensions therefore include off-equatorial configurations, rotating black-hole backgrounds, finite electrical conductivity, electrostatic self-fields, and a self-consistent coupling between the fluid current and Maxwell’s equations. The paper is organized as follows. The results indicate that within the configurations examined here, the asymptotically uniform external component provides the magnetic background on which the approximate self-field can significantly reshape an otherwise admissible toroidal equilibrium. Finally, a representative numerical solution with ˜I = −4√2r30 ≡ ˜I0, r0 = 18, u∞ = −1.046 is shown in Fig. 2. The resulting local cancellation between the two magnetic components can substantially modify the enthalpy distribution and the morphology of the torus. The final panel shows poloidal cross-sections of the corresponding equipotential surfaces w(R,z) = const. The dashed contour is the limiting closed equipotential surface passing through the cusp, while the thick contour represents a common closed equipotential surface. The purple curves indicate the magnetic field lines determined by the orthonormal components (B rˆ,B θˆ). The magnetic-null configuration corresponds to ˜I = ˜I0, while the dependence on the self-field strength is explored by setting ˜I = k ˜I0. A representative numerical solution with ˜I = −4√2r30 ≡ ˜I0, r0 = rc = 18, u∞ = −1.046 is shown in Fig. 2. The remaining parameter values used to construct this structure are specified in the figure caption. Near r = r0, the self-field is comparable to the external field and can therefore significantly modify the Lorentz-force balance and the resulting toroidal structure. The radial extent L = rout − rccusp depends nonmonotonically on the relative self-field amplitude. At rc = r∗, its lower and upper bounds coincide. On the principal branch, the function ue u/2 is strictly increasing throughout the relevant domain u > −1. Therefore, Eq. (84) maps the lower and upper limits in Eq. (134) to u∞ > 2W0 − (rc −3)√rc / 2(rc−2) 3/2 e - rc-3/2(rc-2). The lower Lambert branch is excluded because it is incompatible with the polar-maximum condition. The upper Lambert branch is excluded because it is incompatible with the polar-maximum condition. At rc = r∗, its lower and upper bounds coincide.

1. Enhanced modeling of magnetic self-fields allows AI systems to predict how an approximate magnetic self-field modifies the morphology and domain of existence of charged-fluid tori by observing how the radial extent varies nonmonotonically with relative self-field strength.

2. Improved understanding of integrability conditions enables AI systems to determine the admissible specific-angular-momentum profiles l(r,theta) from a prescribed profile or vice versa, which is crucial for constructing physically consistent fluid equilibria.

3. Advanced analysis of the combined magnetic field allows AI systems to quantify how the resulting local cancellation between the two magnetic components can substantially modify the enthalpy distribution and the morphology of the torus, providing better predictions for complex astrophysical environments.

4. The framework provides a basis for developing more self-consistent models, suggesting that future AI research could focus on incorporating finite electrical conductivity, electrostatic self-fields, and a self-consistent coupling between the fluid current and Maxwell’s equations.

We study toroidal structures formed by a charged, non-self-gravitating perfect fluid orbiting a Schwarzschild black hole immersed in a poloidal magnetic field. To obtain analytic and semi-analytic solutions, we assume a cold, rest-mass-dominated fluid with uniform specific charge and zero conductivity, so the current is purely convective. This regime is formally opposite to the ideal GRMHD limit. The total magnetic field comprises a prescribed asymptotically uniform external component and an approximate self-field associated with the azimuthal current in the torus. We represent the latter by a regularized dipole approximation to the field of an effective circular current loop. Neglecting the self-field, the method of characteristics yields a general implicit solution of the integrability condition for a function parametrizing the specific-angular-momentum profile. A separately obtained cylindrically dependent class recovers a solution from our earlier work. For the radially dependent class, we integrate the Euler equations, obtain the enthalpy potential, and derive local conditions for equatorially centered tori. We then examine the field-free limit, both magnetic components separately, and their superposition. The external field supports closed tori within analytically derived parameter bounds, whereas no such equilibrium is found for the isolated approximate self-field within the numerically explored domain. In a representative combined-field sequence, the radial extent varies nonmonotonically with relative self-field strength: it increases for weak self-fields and decreases as the self-field becomes comparable to the external field. The results show how an approximate magnetic self-field modifies the morphology and domain of existence of charged-fluid tori.