Equilibriums of extremely magnetized compact stars with force-free magnetotunnels

Kōji Uryū, Shijun Yoshida, Eric Gourgoulhon, Charalampos Markakis, Kotaro Fujisawa, Antonios Tsokaros, Keisuke Taniguchi, and Mina Zamani

Phys. Rev. D 107, 103016 – Published 9 May 2023

Abstract

We present numerical solutions for stationary and axisymmetric equilibriums of compact stars associated with extremely strong magnetic fields. The interior of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD) conditions, while in the region of negligible mass density the force-free conditions or electromagnetic vacuum are assumed. Solving all components of Einstein’s equations, Maxwell’s equations, ideal MHD equations, and force-free conditions, equilibriums of rotating compact stars associated with mixed poloidal and toroidal magnetic fields are obtained. It is found that in the extreme cases the strong mixed magnetic fields concentrating in a toroidal region near the equatorial surface expel the matter and form a force-free toroidal magnetotunnel. We also introduce a new differential rotation law for computing solutions associated with force-free magnetosphere, and present other extreme models without the magnetotunnel.

Received 16 March 2023
Accepted 19 April 2023

DOI:https://doi.org/10.1103/PhysRevD.107.103016

Physics Subject Headings (PhySH)

Fluids & classical fields in curved spacetime General relativity General relativity equations & solutions

Neutron stars & pulsars

Astrophysical electromagnetic fields

Numerical relativity Numerical simulations in gravitation & astrophysics

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Kōji Uryū ^1,*, Shijun Yoshida^2,†, Eric Gourgoulhon ^3,‡, Charalampos Markakis ^4,5,6,§, Kotaro Fujisawa^7,8,∥, Antonios Tsokaros^9,5,¶, Keisuke Taniguchi ^1,**, and Mina Zamani ^10,††

¹Department of Physics, University of the Ryukyus, Senbaru 1, Nishihara, Okinawa 903-0213, Japan
²Astronomical Institute, Tohoku University, Aramaki-Aoba, Aoba, Sendai 980-8578, Japan
³Laboratoire Univers et Théories, UMR 8102 du CNRS, Observatoire de Paris, Université PSL, Université Paris Diderot, F-92190 Meudon, France
⁴DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
⁵National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
⁶School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
⁷Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan
⁸Department of Liberal Arts, Tokyo University of Technology, 5-23-22 Kamata, Ota, Tokyo 144-8535, Japan
⁹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
¹⁰Department of Physics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

^*uryu@sci.u-ryukyu.ac.jp
^†yoshida@astr.tohoku.ac.jp
^‡eric.gourgoulhon@obspm.fr
^§c.markakis@qmul.ac.uk
^∥fujisawa@resceu.s.u-tokyo.ac.jp
^¶tsokaros@illinois.edu
^**ktngc@sci.u-ryukyu.ac.jp
^††m_zamani@znu.ac.ir

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Issue

Vol. 107, Iss. 10 — 15 May 2023

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Images

Figure 1
Solutions for uniformly rotating extremely magnetized compact stars associated with an electromagnetic vacuum outside and magnetotunnel (EV-MT-UR type). The panels in the first row correspond to the rapidly rotating model EV-MT-UR-1. The left panel: contours of $p / ρ$ (black closed curves), the poloidal magnetic field (orange arrows), color density map for the toroidal magnetic fields (red and blue), and the boundary of the magnetotunnel (green circles) are shown. The contours of $p / ρ$ are drawn at $p / ρ = 0.001$ , 0.002, 0.005, 0.01, 0.02, 0.05, 0.1. The middle panel: the rest mass density $ρ / ρ_{c}$ (red curve) and the angular velocity $Ω / Ω_{c}$ are plotted along the equatorial radius ( $x$ axis). Inset: enlargement of $ρ / ρ_{c}$ near the surface. The right panel: components of the magnetic fields, $B_{pol} = F_{x y}$ (dashed purple curve) and $B_{tor} = - F_{x z}$ (dark green curve) are plotted along the equatorial radius ( $x$ axis). The panels in the second row are the same as the first row but for the slowly rotating model EV-MT-UR-2. In the third row, the first panel from the left, the metric potentials are shown, which are the contours of $ψ$ (green closed curves), the color density map for ${\tilde{β}}_{y}$ (red and blue), the contours of $h_{x z}$ (red and blue curves), and the surface of the star (black closed curve). In the second panel from the left, the components of electromagnetic one form are shown, which are the contours of $A_{ϕ}$ (green curves), the contours of $A_{t}$ [dashed red (positive), purple (zero), blue (negative)], and the surface of the star (black closed curve) for the model EV-MT-UR-1. The third and fourth panels of the third row are the same as the first and the second panels, respectively, but for the model EV-MT-UR-2.
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Figure 2
Same as Fig. 1 but for differentially rotating and extremely magnetized compact stars associated with a magnetosphere and a magnetotunnel, MS-MT-DR-1 (rapidly rotating model) and MS-MT-DR-2 (slowly rotating model).
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Figure 3
Same as Fig. 1 but for differentially rotating and extremely magnetized compact stars associated with an electromagnetic vacuum outside and a magnetotunnel, EV-MT-DR-1 (rapidly rotating model) and EV-MT-DR-2 (slowly rotating model).
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Figure 4
Same as Fig. 1 but for differentially rotating and extremely magnetized compact stars associated with a magnetosphere, MS-DR-1 (supramassive model) and MS-DR-2 (normal mass model), whose toroidal magnetic fields are distributed across the stellar support and magnetosphere.
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