# Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Particles with Electric Charge and a Magnetic Dipole

#### 2.1. Equation of Motion

#### 2.2. Energy Efficiency

## 3. Spin of Kerr BH versus Magnetic Interactions

#### 3.1. In the Same ISCO Radius

#### 3.2. In the Same Accretion Disk Luminosity

## 4. Particle Collisions

#### 4.1. Critical Angular Momentum

#### 4.2. Collisions of Neutral and Electrically Charged Particles

#### 4.3. Collisions of Electrically Neutral and Magnetized Particles

#### 4.4. Charged-Magnetized

#### 4.4.1. Positively Charged Particle–Magnetized Particle with Positive $\beta $

#### 4.4.2. Negatively Charged Particle–Magnetized Particle with Positive $\beta $

#### 4.4.3. Positively Charged Particle–Magnetized Particle with Negative $\beta $

#### 4.5. Particles Collision Having Electric Charge and a Magnetic Dipole

## 5. Particles with Electric Charge and a Magnetic Dipole Moment in Astrophysics

#### 5.1. Neutron Stars and White Dwarfs as Test Particles with a Magnetic Dipole Moment and Electric Charge

#### 5.2. Electron and Protons as Candidates for Particles with Electric Charge and a Magnetic Dipole Moment

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

$\mathit{Gaussian}$ | $\mathit{Geometrized}$ | $\mathit{Conv}.$ | |
---|---|---|---|

$Period$ | 1 s | $2.99\times {10}^{10}$ cm | c |

Mass | 1 g | $7.42\times {10}^{-29}$ cm | $G/{c}^{2}$ |

Electric charge | 1 $statC$ | $2.87\times {10}^{-25}$ cm | $\sqrt{G}/{c}^{2}$ |

Magnetic field | 1 $Gauss$ | $8.16\times {10}^{-15}$ 1/cm | $\sqrt{G}/c$ |

$\mathit{q},\phantom{\rule{4pt}{0ex}}\mathit{e}$ | $\mathit{m},\phantom{\rule{4pt}{0ex}}{\mathit{m}}_{\mathit{e}}$ | $\mathit{\mu},\phantom{\rule{4pt}{0ex}}{\mathit{\mu}}_{\mathit{B}}$ | |
---|---|---|---|

$\mathrm{Electron},e$ | $-1$ | 1 | − |

$\mathrm{Proton},p$ | 1 | 1836 | $1.5\xb7{10}^{-3}$ |

$\mathrm{Neutron},n$ | 0 | 1839 | $-{10}^{-3}$ |

## References

- Abbott, B.P. et al. [Virgo and LIGO Scientific Collaborations] Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef][Green Version] - Abbott, B.P. et al. [Virgo and LIGO Scientific Collaborations] Properties of the Binary Black Hole Merger GW150914. Phys. Rev. Lett.
**2016**, 116, 241102. [Google Scholar] [CrossRef][Green Version] - Akiyama, K. et al. [Event Horizon Telescope Collaboration] First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J.
**2019**, 875, L1. [Google Scholar] [CrossRef] - Akiyama, K. et al. [Event Horizon Telescope Collaboration] First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole. Astrophys. J.
**2019**, 875, L6. [Google Scholar] [CrossRef] - Wald, R.M. Black hole in a uniform magnetic field. Phys. Rev. D
**1974**, 10, 1680–1685. [Google Scholar] [CrossRef] - Herrera, L.; Paiva, F.M.; Santos, N.O.; Ferrari, V. Geodesics in the γ Spacetime. Int. J. Mod. Phys. D
**2000**, 9, 649–659. [Google Scholar] [CrossRef] - Herrera, L.; Magli, G.; Malafarina, D. Non-spherical sources of static gravitational fields: Investigating the boundaries of the no-hair theorem. Gen. Relativ. Gravit.
**2005**, 37, 1371–1383. [Google Scholar] [CrossRef][Green Version] - Dey, D.; Joshi, P.S. Gravitational Collapse of Baryonic and Dark matter. arXiv
**2019**, arXiv:1907.12738. [Google Scholar] [CrossRef][Green Version] - Benavides-Gallego, C.A.; Abdujabbarov, A.; Malafarina, D.; Ahmedov, B.; Bambi, C. Charged particle motion and electromagnetic field in γ spacetime. Phys. Rev. D
**2019**, 99, 044012. [Google Scholar] [CrossRef][Green Version] - Boshkayev, K.; Malafarina, D. A model for a dark matter core at the Galactic Centre. Mon. Not. R. Astron. Soc.
**2019**, 484, 3325–3333. [Google Scholar] [CrossRef] - Shaymatov, S.; Ahmedov, B.; Jamil, M. Testing the weak cosmic censorship conjecture for a Reissner-Nordström-de Sitter black hole surrounded by perfect fluid dark matter. Eur. Phys. J. C
**2021**, 81, 588. [Google Scholar] [CrossRef] - Rayimbaev, J.; Shaymatov, S.; Jamil, M. Dynamics and epicyclic motions of particles around the Schwarzschild-de Sitter black hole in perfect fluid dark matter. Eur. Phys. J. C
**2021**, 81, 699. [Google Scholar] [CrossRef] - Bañados, M.; Silk, J.; West, S.M. Kerr Black Holes as Particle Accelerators to Arbitrarily High Energy. Phys. Rev. Lett.
**2009**, 103, 111102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grib, A.A.; Pavlov, Y.V. On particle collisions near rotating black holes. Gravit. Cosmol.
**2011**, 17, 42–46. [Google Scholar] [CrossRef][Green Version] - Jacobson, T.; Sotiriou, T.P. Spinning Black Holes as Particle Accelerators. Phys. Rev. Lett.
**2010**, 104, 021101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Harada, T.; Kimura, M. Collision of an innermost stable circular orbit particle around a Kerr black hole. Phys. Rev. D
**2011**, 83, 024002. [Google Scholar] [CrossRef][Green Version] - Wei, S.W.; Liu, Y.X.; Guo, H.; Fu, C.E. Charged spinning black holes as particle accelerators. Phys. Rev. D
**2010**, 82, 103005. [Google Scholar] [CrossRef][Green Version] - Zaslavskii, O.B. Acceleration of particles as a universal property of rotating black holes. Phys. Rev. D
**2010**, 82, 083004. [Google Scholar] [CrossRef][Green Version] - Zaslavskii, O.B. Acceleration of particles by nonrotating charged black holes? Sov. J. Exp. Theor. Phys. Lett.
**2011**, 92, 571–574. [Google Scholar] [CrossRef][Green Version] - Zaslavskii, O.B. Acceleration of particles by black holes—A general explanation. Class. Quantum Grav.
**2011**, 28, 105010. [Google Scholar] [CrossRef] - Kimura, M.; Nakao, K.I.; Tagoshi, H. Acceleration of colliding shells around a black hole: Validity of the test particle approximation in the Banados-Silk-West process. Phys. Rev. D
**2011**, 83, 044013. [Google Scholar] [CrossRef][Green Version] - Bañados, M.; Hassanain, B.; Silk, J.; West, S.M. Emergent flux from particle collisions near a Kerr black hole. Phys. Rev. D
**2011**, 83, 023004. [Google Scholar] [CrossRef][Green Version] - Igata, T.; Harada, T.; Kimura, M. Effect of a weak electromagnetic field on particle acceleration by a rotating black hole. Phys. Rev. D
**2012**, 85, 104028. [Google Scholar] [CrossRef][Green Version] - Frolov, V.P. Weakly magnetized black holes as particle accelerators. Phys. Rev. D
**2012**, 85, 024020. [Google Scholar] [CrossRef][Green Version] - Patil, M.; Joshi, P. Kerr naked singularities as particle accelerators. Class. Quantum Grav.
**2011**, 28, 235012. [Google Scholar] [CrossRef] - Atamurotov, F.; Ahmedov, B.; Shaymatov, S. Formation of black holes through BSW effect and black hole-black hole collisions. Astrophys. Space Sci.
**2013**, 347, 277–281. [Google Scholar] [CrossRef] - Liu, C.; Chen, S.; Ding, C.; Jing, J. Particle acceleration on the background of the Kerr-Taub-NUT spacetime. Phys. Lett. B
**2011**, 701, 285–290. [Google Scholar] [CrossRef][Green Version] - Shaymatov, S.R.; Ahmedov, B.J.; Abdujabbarov, A.A. Particle acceleration near a rotating black hole in a Randall-Sundrum brane with a cosmological constant. Phys. Rev. D
**2013**, 88, 024016. [Google Scholar] [CrossRef] - Shaymatov, S.; Ahmedov, B.; Stuchlík, Z.; Abdujabbarov, A. Effect of an external magnetic field on particle acceleration by a rotating black hole surrounded with quintessential energy. Int. J. Mod. Phys. D
**2018**, 27, 1850088. [Google Scholar] [CrossRef] - Abdujabbarov, A.; Rayimbaev, J.; Turimov, B.; Atamurotov, F. Dynamics of magnetized particles around 4-D Einstein Gauss–Bonnet black hole. Phys. Dark Universe
**2020**, 30, 100715. [Google Scholar] [CrossRef] - Abdujabbarov, A.A.; Ahmedov, B.J.; Shaymatov, S.R.; Rakhmatov, A.S. Penrose process in Kerr-Taub-NUT spacetime. Astrophys. Space Sci.
**2011**, 334, 237–241. [Google Scholar] [CrossRef][Green Version] - Okabayashi, K.; Maeda, K.I. Maximal efficiency of the collisional Penrose process with a spinning particle. II. Collision with a particle on the innermost stable circular orbit. Prog. Theor. Exp. Phys.
**2020**, 2020, 013E01. [Google Scholar] [CrossRef][Green Version] - McKinney, J.C.; Narayan, R. Disc-jet coupling in black hole accretion systems—II. Force-free electrodynamical models. Mon. Not. R. Astron. Soc.
**2007**, 375, 531–547. [Google Scholar] [CrossRef] - Anderson, J.L.; Cohen, J.M. Gravitational Collapse of Magnetic Neutron Stars. Astrophys. Space Sci.
**1970**, 9, 146–152. [Google Scholar] [CrossRef] - de Felice, F.; Sorge, F. Magnetized orbits around a Schwarzschild black hole. Class. Quantum Grav.
**2003**, 20, 469–481. [Google Scholar] [CrossRef] - de Felice, F.; Sorge, F.; Zilio, S. Magnetized orbits around a Kerr black hole. Class. Quantum Grav.
**2004**, 21, 961–973. [Google Scholar] [CrossRef] - Frolov, V.P.; Shoom, A.A. Motion of charged particles near a weakly magnetized Schwarzschild black hole. Phys. Rev. D
**2010**, 82, 084034. [Google Scholar] [CrossRef][Green Version] - Aliev, A.N.; Özdemir, N. Motion of charged particles around a rotating black hole in a magnetic field. Mon. Not. R. Astron. Soc.
**2002**, 336, 241–248. [Google Scholar] [CrossRef][Green Version] - Abdujabbarov, A.; Ahmedov, B. Test particle motion around a black hole in a braneworld. Phys. Rev. D
**2010**, 81, 044022. [Google Scholar] [CrossRef][Green Version] - Shaymatov, S.; Atamurotov, F.; Ahmedov, B. Isofrequency pairing of circular orbits in Schwarzschild spacetime in the presence of magnetic field. Astrophys. Space Sci.
**2014**, 350, 413–419. [Google Scholar] [CrossRef] - Shaymatov, S.; Dadhich, N.; Ahmedov, B.; Jamil, M. Five-dimensional charged rotating minimally gauged supergravity black hole cannot be over-spun and/or over-charged in non-linear accretion. Eur. Phys. J. C
**2020**, 80, 481. [Google Scholar] [CrossRef] - Shaymatov, S.; Malafarina, D.; Ahmedov, B. Effect of perfect fluid dark matter on particle motion around a static black hole immersed in an external magnetic field. Phys. Dark Universe
**2021**, 34, 100891. [Google Scholar] [CrossRef] - Shaymatov, S.; Vrba, J.; Malafarina, D.; Ahmedov, B.; Stuchlík, Z. Charged particle and epicyclic motions around 4 D Einstein–Gauss–Bonnet black hole immersed in an external magnetic field. Phys. Dark Universe
**2020**, 30, 100648. [Google Scholar] [CrossRef] - Shaymatov, S.; Dadhich, N. On overspinning of black holes in higher dimensions. Phys. Dark Universe
**2021**, 31, 100758. [Google Scholar] [CrossRef] - Shaymatov, S.; Atamurotov, F. Geodesic Circular Orbits Sharing the Same Orbital Frequencies in the Black String Spacetime. Galaxies
**2021**, 9, 40. [Google Scholar] [CrossRef] - Piotrovich, M.Y.; Silant’ev, N.A.; Gnedin, Y.N.; Natsvlishvili, T.M. Magnetic Fields of Black Holes and the Variability Plane. arXiv
**2010**, arXiv:1002.4948. [Google Scholar] - Eatough, R.P.; Falcke, H.; Karuppusamy, R.; Lee, K.J.; Champion, D.J.; Keane, E.F.; Desvignes, G.; Schnitzeler, D.H.F.M.; Spitler, L.G.; Kramer, M.; et al. A strong magnetic field around the supermassive black hole at the centre of the Galaxy. Nature
**2013**, 501, 391–394. [Google Scholar] [CrossRef][Green Version] - Shannon, R.M.; Johnston, S. Radio properties of the magnetar near Sagittarius A* from observations with the Australia Telescope Compact Array. Mon. Not. R. Astron. Soc.
**2013**, 435, L29–L32. [Google Scholar] [CrossRef] - Dallilar, Y.; Eikenberry, S.S.; Garner, A.; Stelter, R.D.; Gottlieb, A.; Gandhi, P.; Casella, P.; Dhillon, V.S.; Marsh, T.R.; Littlefair, S.P.; et al. A precise measurement of the magnetic field in the corona of the black hole binary V404 Cygni. Science
**2017**, 358, 1299–1302. [Google Scholar] [CrossRef][Green Version] - Baczko, A.K.; Schulz, R.; Kadler, M.; Ros, E.; Perucho, M.; Krichbaum, T.P.; Böck, M.; Bremer, M.; Grossberger, C.; Lindqvist, M.; et al. A highly magnetized twin-jet base pinpoints a supermassive black hole. Astron. Astrophys.
**2016**, 593, A47. [Google Scholar] [CrossRef][Green Version] - Prasanna, A.R. General-relativistic analysis of charged-particle motion in electromagnetic fields surrounding black holes. Nuovo C. Riv. Ser.
**1980**, 3, 1–53. [Google Scholar] [CrossRef] - Kovář, J.; Stuchlík, Z.; Karas, V. Off-equatorial orbits in strong gravitational fields near compact objects. Class. Quantum Grav.
**2008**, 25, 095011. [Google Scholar] [CrossRef][Green Version] - Kovář, J.; Kopáček, O.; Karas, V.; Stuchlík, Z. Off-equatorial orbits in strong gravitational fields near compact objects II: Halo motion around magnetic compact stars and magnetized black holes. Class. Quantum Grav.
**2010**, 27, 135006. [Google Scholar] [CrossRef] - Shaymatov, S.; Patil, M.; Ahmedov, B.; Joshi, P.S. Destroying a near-extremal Kerr black hole with a charged particle: Can a test magnetic field serve as a cosmic censor? Phys. Rev. D
**2015**, 91, 064025. [Google Scholar] [CrossRef][Green Version] - Dadhich, N.; Tursunov, A.; Ahmedov, B.; Stuchlík, Z. The distinguishing signature of magnetic Penrose process. Mon. Not. R. Astron. Soc.
**2018**, 478, L89–L94. [Google Scholar] [CrossRef][Green Version] - Pavlović, P.; Saveliev, A.; Sossich, M. Influence of the vacuum polarization effect on the motion of charged particles in the magnetic field around a Schwarzschild black hole. Phys. Rev. D
**2019**, 100, 084033. [Google Scholar] [CrossRef][Green Version] - Shaymatov, S. Magnetized Reissner–Nordström black hole restores cosmic censorship conjecture. Int. J. Mod. Phys. Conf. Ser.
**2019**, 49, 1960020. [Google Scholar] [CrossRef] - Shaymatov, S.; Jamil, M.; Jusufi, K.; Bamba, K. Constraints on the magnetized Ernst black hole spacetime through quasiperiodic oscillations. Eur. Phys. J. C
**2022**, 82, 636. [Google Scholar] [CrossRef] - Zhang, J.; Xie, Y. Probing a black-bounce-Reissner-Nordström spacetime with precessing and periodic motion. Eur. Phys. J. C
**2022**, 82, 854. [Google Scholar] [CrossRef] - Zhou, T.Y.; Xie, Y. Precessing and periodic motions around a black-bounce/traversable wormhole. Eur. Phys. J. C
**2020**, 80, 1070. [Google Scholar] [CrossRef] - Zhang, J.; Xie, Y. Probing a self-complete and Generalized-Uncertainty-Principle black hole with precessing and periodic motion. Astrophys. Space Sci.
**2022**, 367, 17. [Google Scholar] [CrossRef] - Deng, X.M. Geodesics and periodic orbits around quantum-corrected black holes. Phys. Dark Universe
**2020**, 30, 100629. [Google Scholar] [CrossRef] - Lin, H.Y.; Deng, X.M. Rational orbits around 4 D Einstein–Lovelock black holes. Phys. Dark Universe
**2021**, 31, 100745. [Google Scholar] [CrossRef] - Rayimbaev, J.R. Magnetized particle motion around non-Schwarzschild black hole immersed in an external uniform magnetic field. Astrophys. Space Sci.
**2016**, 361, 288. [Google Scholar] [CrossRef] - Rayimbaev, J.; Turimov, B.; Palvanov, S. Plasma magnetosphere of slowly rotating magnetized neutron star in branewold. Int. J. Mod. Phys. Conf. Ser.
**2019**, 49, 1960019. [Google Scholar] [CrossRef] - Rayimbaev, J.; Figueroa, M.; Stuchlík, Z.; Juraev, B. Test particle orbits around regular black holes in general relativity combined with nonlinear electrodynamics. Phys. Rev. D
**2020**, 101, 104045. [Google Scholar] [CrossRef] - Rayimbaev, J.; Bardiev, D.; Abdulxamidov, F.; Abdujabbarov, A.; Ahmedov, B. Magnetized and Magnetically Charged Particles Motion around Regular Bardeen Black Hole in 4D Einstein Gauss–Bonnet Gravity. Universe
**2022**, 8, 549. [Google Scholar] [CrossRef] - Rayimbaev, J.; Abdujabbarov, A.; Jamil, M.; Ahmedov, B.; Han, W.B. Dynamics of test particles around renormalization group improved Schwarzschild black holes. Phys. Rev. D
**2020**, 102, 084016. [Google Scholar] [CrossRef] - Rayimbaev, J.; Abdujabbarov, A.; Jamil, M.; Han, W.B. Dynamics of magnetized particles around Einstein-Æther black hole with uniform magnetic field. Nuclear Phys. B
**2021**, 966, 115364. [Google Scholar] [CrossRef] - Juraeva, N.; Rayimbaev, J.; Abdujabbarov, A.; Ahmedov, B.; Palvanov, S. Distinguishing magnetically and electrically charged Reissner–Nordström black holes by magnetized particle motion. Eur. Phys. J. C
**2021**, 81, 124078. [Google Scholar] [CrossRef] - Shaymatov, S.; Narzilloev, B.; Abdujabbarov, A.; Bambi, C. Charged particle motion around a magnetized Reissner-Nordström black hole. Phys. Rev. D
**2021**, 103, 124066. [Google Scholar] [CrossRef] - Abdulxamidov, F.; Benavides-Gallego, C.A.; Han, W.B.; Rayimbaev, J.; Abdujabbarov, A. Spinning test particle motion around a rotating wormhole. Phys. Rev. D
**2022**, 106, 024012. [Google Scholar] [CrossRef] - Bokhari, A.H.; Rayimbaev, J.; Ahmedov, B. Test particles dynamics around deformed Reissner-Nordström black hole. Phys. Rev. D
**2020**, 102, 124078. [Google Scholar] [CrossRef] - Abdujabbarov, A.; Rayimbaev, J.; Atamurotov, F.; Ahmedov, B. Magnetized Particle Motion in γ-Spacetime in a Magnetic Field. Galaxies
**2020**, 8, 76. [Google Scholar] [CrossRef] - Narzilloev, B.; Rayimbaev, J.; Abdujabbarov, A.; Bambi, C. Charged particle motion around non-singular black holes in conformal gravity in the presence of external magnetic field. arXiv
**2020**, arXiv:2005.04752. [Google Scholar] [CrossRef] - Narzilloev, B.; Rayimbaev, J.; Shaymatov, S.; Abdujabbarov, A.; Ahmedov, B.; Bambi, C. Can the dynamics of test particles around charged stringy black holes mimic the spin of Kerr black holes? Phys. Rev. D
**2020**, 102, 044013. [Google Scholar] [CrossRef] - Shaymatov, S.; Ahmedov, B. Overcharging process around a magnetized black hole: Can the backreaction effect of magnetic field restore cosmic censorship conjecture? Gen. Relativ. Gravit.
**2023**, 55, 36. [Google Scholar] [CrossRef] - Vrba, J.; Abdujabbarov, A.; Kološ, M.; Ahmedov, B.; Stuchlík, Z.; Rayimbaev, J. Charged and magnetized particles motion in the field of generic singular black holes governed by general relativity coupled to nonlinear electrodynamics. Phys. Rev. D
**2020**, 101, 124039. [Google Scholar] [CrossRef] - Novikov, I.D.; Thorne, K.S. Astrophysics of black holes. In Black Holes (Les Astres Occlus); Gordon & Breach: New York, NY, USA, 1973; pp. 343–450. [Google Scholar]
- Bian, W.H.; Zhao, Y.H. Accretion Rates and the Accretion Efficiency in AGNs. Publ. Astron. Soc. Jpn.
**2003**, 55, 599–603. [Google Scholar] [CrossRef] - Penrose, R. Gravitational Collapse: The Role of General Relativity. Nuovo C. Riv. Ser.
**1969**, 1, 252. [Google Scholar] - Wagh, S.M.; Dhurandhar, S.V.; Dadhich, N. Revival of the Penrose process for astrophysical applications. Astrophys. J.
**1985**, 290, 12–14. [Google Scholar] [CrossRef] - Tursunov, A.; Kološ, M.; Abdujabbarov, A.; Ahmedov, B.; Stuchlík, Z. Acceleration of particles in spacetimes of black string. Phys. Rev. D
**2013**, 88, 124001. [Google Scholar] [CrossRef][Green Version] - Stuchlík, Z.; Hledík, S.; Truparová, K. Evolution of Kerr superspinars due to accretion counterrotating thin discs. Class. Quantum Grav.
**2011**, 28, 155017. [Google Scholar] [CrossRef] - Abdujabbarov, A.A.; Tursunov, A.A.; Ahmedov, B.J.; Kuvatov, A. Acceleration of particles by black hole with gravitomagnetic charge immersed in magnetic field. Astrophys. Space Sci.
**2013**, 343, 173–179. [Google Scholar] [CrossRef][Green Version] - Atamurotov, F.; Shaymatov, S.; Sheoran, P.; Siwach, S. Charged black hole in 4D Einstein–Gauss–Bonnet gravity: Particle motion, plasma effect on weak gravitational lensing and centre-of-mass energy. J. Cosmol. Astropart. Phys.
**2021**, 2021, 045. [Google Scholar] [CrossRef] - Lacroix, T. Dynamical constraints on a dark matter spike at the Galactic centre from stellar orbits. Astron. Astrophys.
**2018**, 619, A46. [Google Scholar] [CrossRef] - Nucita, A.A.; De Paolis, F.; Ingrosso, G.; Qadir, A.; Zakharov, A.F. Sgr A*: A Laboratory to Measure the Central Black Hole and Stellar Cluster Parameters. Publ. Astron. Soc. Pac.
**2007**, 119, 349–359. [Google Scholar] [CrossRef] - Ghez, A.M.; Salim, S.; Hornstein, S.D.; Tanner, A.; Lu, J.R.; Morris, M.; Becklin, E.E.; Duchêne, G. Stellar Orbits around the Galactic Center Black Hole. Astrophys. J.
**2005**, 620, 744–757. [Google Scholar] [CrossRef][Green Version] - Ghez, A.M.; Morris, M.; Becklin, E.E.; Tanner, A.; Kremenek, T. The accelerations of stars orbiting the Milky Way’s central black hole. Nature
**2000**, 407, 349–351. [Google Scholar] [CrossRef][Green Version] - Mori, K.; Gotthelf, E.V.; Zhang, S.; An, H.; Baganoff, F.K.; Barrière, N.M.; Beloborodov, A.M.; Boggs, S.E.; Christensen, F.E.; Craig, W.W.; et al. NuSTAR Discovery of a 3.76 s Transient Magnetar Near Sagittarius A*. Astron. J. Lett.
**2013**, 770, L23. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The radial dependence of the effective potential for magnetized and charged test particles.

**Figure 2.**Radial profiles of the specific energy (

**left panel**) and angular momentum (

**right panel**) for magnetized and charged particles at the circular orbits.

**Figure 3.**The dependence of the ISCO radius on the parameters ${\omega}_{B}$ (

**left panel**) and $\beta $ (

**right panel**).

**Figure 4.**Possible values of parameters ${\omega}_{B}$ and $\beta $ for a ISCO radius between 5 and 10 M.

**Figure 5.**Dependence of the energy efficiency on the magnetic-coupling parameter $\beta $ for different values of the parameter $\omega $.

**Figure 6.**Degeneracy between parameter $\beta $ and a, i.e., for a given value of $\beta $ there exists a Kerr geometry with a given value of spin parameter $a/M$ that has the same ISCO radius for various combinations of parameter ${\omega}_{B}$. The point to be noted here is that $a/M$ represents a black hole rotation (spin) parameter.

**Figure 7.**Degeneracy between the parameter $\beta $ and the spin parameter of Kerr BH, providing the same energy efficiency for different values of parameter $\omega $.

**Figure 8.**The dependence of critical angular momentum on parameter $\beta $ with the different values of ${\omega}_{B}$.

**Figure 9.**Figure shows the radial dependence of ${\mathcal{E}}_{\mathrm{cm}}$ of the collisions of two positively and negatively charged particles with electrically neutral particles on the different values of the charge.

**Figure 10.**The same as Figure 9 but for collisions of neutral-magnetized particles with positive magnetic dipole particles.

**Figure 11.**The same as Figure 9 but for collisions of positively charged particles with no magnetic dipole-positive magnetized particles with no charge.

**Figure 12.**The same as Figure 9 but for the collisions of negatively charged particles with non-dipole-positive magnetized particles with no charge.

**Figure 13.**The same as Figure 9 but for collisions of positively charged particles with non-dipole-negative magnetized particles with no charge.

**Figure 14.**The same as Figure 9 but for the collisions of two positively charged particles with a magnetic dipole for different values of ${\omega}_{B}$ (

**left panel**) and $\beta $ (

**right panel**).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rayimbaev, J.; Shaymatov, S.; Abdulxamidov, F.; Ahmedov, S.; Begmatova, D.
Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields. *Universe* **2023**, *9*, 135.
https://doi.org/10.3390/universe9030135

**AMA Style**

Rayimbaev J, Shaymatov S, Abdulxamidov F, Ahmedov S, Begmatova D.
Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields. *Universe*. 2023; 9(3):135.
https://doi.org/10.3390/universe9030135

**Chicago/Turabian Style**

Rayimbaev, Javlon, Sanjar Shaymatov, Farrux Abdulxamidov, Saidmuhammad Ahmedov, and Dilfuza Begmatova.
2023. "Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields" *Universe* 9, no. 3: 135.
https://doi.org/10.3390/universe9030135