QPO-Based Bayesian Constraints on Charged Particle Dynamics Around Magnetized Schwarzschild Black Holes

TL;DR

This work investigates the dynamics of a magnetized charged particle with a magnetic dipole moment in the external paraboloidal magnetosphere of a Schwarzschild black hole. Using the Hamilton–Jacobi formalism, it derives the effective potential and motion equations, analyzes circular orbits and ISCO shifts under the dipole coupling, and computes Keplerian and epicyclic frequencies that feed QPO modeling within the relativistic precession framework. The study couples disk-radiation properties via the Novikov–Thorne model to the magnetized dynamics, revealing how magnetic field polarity and dipole coupling alter flux, temperature, and luminosity near the ISCO. A Bayesian MCMC analysis with emcee constrains the black hole mass, magnetic-field strength and geometry, dipole coupling, and QPO orbital radius using QPO data from stellar-, intermediate-, and supermassive BHs, highlighting the diagnostic power of QPOs for magnetospheric physics across mass scales. These results demonstrate a coherent framework linking BH magnetospheres, particle dynamics, and timing properties, with implications for interpreting high-frequency QPOs and disk emission in magnetized accretion flows.

Abstract

We study the motion of charged particles with a magnetic dipole moment orbiting a Schwarzschild black hole immersed in an external paraboloidal magnetic field. The interaction between the particle's intrinsic magnetic moment and the black hole magnetosphere is modeled through a dipole coupling, and the equations of motion are derived using the Hamilton-Jacobi formalism. We analyze equatorial circular orbits, the innermost stable circular orbit, and epicyclic oscillations, showing that the magnetic field strength and coupling parameter produce competing effects on orbital stability and fundamental frequencies. These frequencies are applied to model high-frequency quasi-periodic oscillations within the relativistic precession framework. Using observational QPO data from stellar-mass, intermediate-mass, and supermassive black holes, we perform a Bayesian parameter estimation based on Markov Chain Monte Carlo techniques. The analysis constrains the black hole mass, magnetic field strength, field geometry, coupling parameter, and QPO orbital radius, highlighting the role of magnetospheric interactions in shaping both particle dynamics and timing properties of accreting black holes.

Figures (12)

• Figure 1: Effective potential as a function of the external magnetic field strength $B$ and the coupling parameter $\beta$, for fixed values $r=6$ and $L'=6$. The parameters $B$ and $\beta$ exhibit opposite qualitative effects: increasing $B$ lowers the potential, whereas increasing $\beta$ raises it.
• Figure 2: Radial dependence of the specific angular momentum $L'$ for circular orbits of charged magnetized particles for different values of the magnetic field strength $B$ and coupling parameter $\beta$.
• Figure 3: ISCO radius $r_{\rm ISCO}$ as a function of the magnetic field strength $B$ for different values of the coupling parameter $\beta$.
• Figure 4: Specific energy $E'$ as a function of the specific angular momentum $L'$ for different values of the coupling parameter $\beta$ and magnetic field strength $B$.
• Figure 5: Trajectories of charged magnetized particles around a magnetized black hole for different magnetic field strengths and coupling parameters. The black disk represents the black hole, while gray curves denote magnetic field lines.