Electrically charged matter in rigid rotation around magnetized black hole

Jiří Kovář, Petr Slaný, Claudio Cremaschini, Zdeněk Stuchlík, Vladimír Karas, and Audrey Trova

Phys. Rev. D 90, 044029 – Published 13 August 2014

Abstract

We study charged-fluid toroidal structures surrounding a nonrotating charged black hole immersed in a large-scale, asymptotically uniform magnetic field. In continuation of our former study on electrically charged matter in approximation of zero conductivity, we demonstrate the existence of orbiting structures in permanent rigid rotation in the equatorial plane and charged clouds hovering near the symmetry axis. We constrain the range of parameters that allow stable configurations and derive the geometrical shape of equipressure surfaces. Our simplified analytical study suggests that these regions of stability may be relevant for trapping electrically charged particles and dust grains in some areas of the black hole magnetosphere and thus important in some astrophysical situations.

Received 3 June 2014

DOI:https://doi.org/10.1103/PhysRevD.90.044029

Authors & Affiliations

Jiří Kovář^*, Petr Slaný, Claudio Cremaschini, and Zdeněk Stuchlík

Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo náměstí 13, CZ-74601 Opava, Czech Republic

Vladimír Karas and Audrey Trova

Astronomical Institute, Academy of Sciences, Boční II, CZ-141 31 Prague, Czech Republic

^*Jiri.Kovar@fpf.slu.cz

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Issue

Vol. 90, Iss. 4 — 15 August 2014

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Images

Figure 1
Illustration of black hole with mass $M$ and charge $Q$ immersed in asymptotically homogeneous magnetic field $\vec{B}$ encircled by a toruslike equatorial configuration and a polar cloud.
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Figure 2
Illustration of binary system of a magnetized star with mass $M^{'}$ and a black hole with mass $M$ and charge $Q$ orbiting around a common center of mass in distances $d^{'}$ and $d$ . The dipole-type magnetic field $\vec{B}$ of a magnetized star appears to be almost uniform in close vicinity of a distant black-hole companion, which is encircled by a toruslike equatorial configuration and a polar cloud.
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Figure 3
Exemplary equatorial torus (left) and polar cloud (right) presented by means of poloidal sections of the function $h$ , corresponding to the solutions (37) and (40) of Eqs. (13). The thick curves denote zero contours of the $h$ function, corresponding to the contours of zero equipressure and equidensity surfaces, bounding the tiny torus centered in the equatorial plane (left) and the polar cloud centered on the axis of symmetry. The equatorial torus solution, centered at $r_{c} = 8$ is characterized by the parameters $B ≐ 8.78 \times 10^{- 11}$ , $Q = - 10^{- 11}$ , $k_{0} = 10^{- 11}$ $ω = 1 / 100$ and $h_{0} = 0.1134551345$ leading to a relatively small torus with the edges in the equatorial plane at $r_{in} ≐ 7.985$ and $r_{out} ≐ 8.015$ . The polar cloud solution, centered at $r_{c} = 5$ , then corresponds to the parameters $B = 10^{- 11}$ , $Q ≐ 4.1 7^{- 11}$ , $k_{0} = 10^{11}$ , $ω = - 1 / 1000$ and $h_{0} = 0.10313150$ , leading also to a relatively small cloud with the edges in the axis of symmetry at $r_{in} ≐ 4.976$ and $r_{out} = 5.024$ . The matter characterized in both the cases by $Γ = 5 / 3$ . The related specific charge and density profiles are illustrated in Fig. 4.
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Figure 4
Exemplary equatorial torus (top) and polar cloud (low) presented by means of specific charge $q_{s}$ and density $ρ$ profiles determined by relations (45) and (43), corresponding to the equatorial solution (37) and polar solution (40) for the $h$ function. The corresponding $h$ -function contours outside the torus and cloud, and parameters of the solution are given in Fig. 3; the $h$ -function contours inside the structures are presented in the right column of this figure. The density reach its maximum $ρ_{\max} \approx 10^{- 19}$ at the center of the equatorial torus $r_{c} = 8$ for the polytropic coefficient $κ = 10^{5}$ ; the specific charge descents outwards from the black hole, being of the order of $10^{9}$ . In the case of polar cloud, we have $ρ_{\max} \approx 10^{- 19}$ in the center $r_{c} = 5$ , for the chosen coefficient $κ = 10^{6}$ , and the specific charge being of the order $10^{10}$ throughout the polar cloud.
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Figure 5
Modifications of the exemplary equatorial torus presented in Fig. 4 through the $κ$ parameter (top) and $h_{0}$ parameter (low). The torus is presented through specific charge $q_{s}$ and density $ρ$ profiles determined by relations (45) and (43), corresponding to the equatorial solution (37) for the $h$ function (with its contours presented in the right column of the figure), with the parameters $r_{c} = 8$ , $B ≐ 8.78 \times 10^{- 14}$ , $Q = 10^{- 14}$ , $ω = 1 / 100$ , $k_{0} = 10^{- 14}$ and $Γ = 5 / 3$ . The profiles illustrated in the top part of the figure correspond to $h_{0} = 0.1134551345$ and $κ = 10^{9}$ , which yields the maximal density $ρ_{\max} \approx 10^{- 25}$ in the center of the torus with the edges at $r_{in} ≐ 7.985$ and $r_{out} ≐ 8.015$ , and the specific charge of the order $10^{12}$ . The profiles illustrated in the low part of the figure correspond to $h_{0} = 0.11345511319$ and $κ = 10^{5}$ , giving rise to the maximal density $ρ_{\max} \approx 10^{- 23}$ in the center of the torus with the edges at $r_{in} ≐ 7.999$ and $r_{out} = 8.001$ , and to the specific charge of the order $10^{12}$ .
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