Constraints on the evolution of black hole spin due to magnetohydrodynamic accretion

Masaaki Takahashi and Akira Tomimatsu

Phys. Rev. D 78, 023012 – Published 14 July 2008

Abstract

Stationary and axisymmetric ideal magnetohydrodynamic (MHD) accretion onto a black hole is studied analytically. The accreting plasma ejected from a plasma source with low velocity must be superfast magnetosonic before passing through the event horizon. We work out and apply a trans-fast magnetosonic solution without detailed analysis of the regularity conditions at the magnetosonic point, by introducing the bending angle $β$ of the magnetic field line, which is the ratio of the toroidal and poloidal components of the magnetic field. To accrete onto a black hole, the trans-magnetosonic solution has some restrictions on $β$ , which are related to the field-aligned parameters of the MHD flows. One of the restrictions gives the boundary condition at the event horizon for the inclination of a magnetic field line. We find that this inclination is related to the energy and angular momentum transport to the black hole. Then, we discuss the spin-up/down process of a rotating black hole by cold MHD inflows in a secular evolution time scale. There are two asymptotic states for the spin evolution. One is that the angular velocity of the black hole approaches to that of the magnetic field line, and the other is that the spin-up effect by the positive angular momentum influx and the spin-down effect by the energy influx (as the mass-energy influx) are canceled. We also show that the MHD inflows prevents the evolution to the maximally rotating black hole.

Received 6 December 2007

DOI:https://doi.org/10.1103/PhysRevD.78.023012

Authors & Affiliations

Masaaki Takahashi^*

Department of Physics and Astronomy, Aichi University of Education, Kariya, Aichi 448-8542, Japan

Akira Tomimatsu^†

Department of Physics, Nagoya University, Nagoya 464-8602, Japan

^*takahasi@phyas.aichi-edu.ac.jp
^†atomi@gravity.phys.nagoya-u.ac.jp

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Vol. 78, Iss. 2 — 15 July 2008

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Figure 1
Trans-fast magnetosonic inflow and outflow solutions (thick bold curves) started from the separation surface with zero velocity (labeled “inj”). (a) The square of the Alfvén Mach number $M^{2}$ vs radius $r / m$ and (b) the radial 4-velocity of the flow $u^{r}$ vs radius $r / m$ . The solution is plotted on the equatorial plane ( $θ = π / 2$ ) in Schwarzschild geometry ( $a = 0$ ), where the magnetic field configuration is assumed with $ξ (r, θ) = 1.0$ . The flow parameters are given as $\hat{E} = 1.15$ , $m Ω_{F} = 0.7 Ω_{\max }$ , $\hat{L} Ω_{F} = 0.6$ , where $Ω_{\max }$ is the maximum value of $Ω_{F}$ , and is given as the double roots of $α = 0$ . (If $Ω_{F} > Ω_{\max }$ , no light surfaces exist along a magnetic field line considered.) The broken curve shows the $M^{2} = M_{AW}^{2}$ curve (left) and the $| u^{r} | = u_{AW}^{r}$ curve (right), and the dotted curve shows the $M^{2} = M_{FM}^{2}$ curve (left) and the $| u^{r} | = u_{FM}^{r}$ curve (right). The crossing points of these curves with the flow solution labeled by “A” and “F” are the Alfvén and fast magnetosonic points, respectively. The thin curves started from the light surface and approached to the $A = 0$ line are unphysical solutions of the quadratic equation.Reuse & Permissions
Figure 2
The square of the Alfvén Mach number $M^{2}$ vs radius $r / m$ for trans-fast magnetosonic inflow solutions (thick bold curve) on the equatorial plane ( $θ = π / 2$ ) in Kerr geometry ( $a = 0.8 m$ ). The flow parameters are given as $\hat{E} = 1.0$ , $m Ω_{F} = 0.5 Ω_{\max }$ , $\hat{L} Ω_{F} = - 0.5$ (left) and $\hat{L} Ω_{F} = 0.09$ (right). The magnetic field configuration is assumed by Eq. (a2) discussed in Sec. . The right-hand figure shows an unphysical solution, where the Alfvén Mach number $M^{2}$ diverges at the location of $A = 0$ . The black spots (right) indicate $M^{2} = \infty$ . The dotted curve shows the $M^{2} = M_{AW}^{2}$ curve, and the solid curve shows the $M^{2} = M_{FW}^{2}$ curve. The crossing points of these curves with the flow solution labeled by “A” and “F” are the Alfvén and fast magnetosonic points, respectively. The thin curves started from the light surface and approached to the $A = 0$ line are unphysical as inflow solutions.Reuse & Permissions
Figure 3
(Top) The possible range of $\tilde{L} Ω_{F}$ for (a) type I and (b) type II cases. The vertical axis is $\tilde{L} Ω_{F}$ , while the horizontal axis is the Alfvén radius. Curves $\tilde{L} Ω_{F} = Y (r_{A})$ (marked by $[Y_{A}]$ ) and $k (r_{A}) = 0$ (marked by $[k_{A}]$ ) are plotted. Two crossing points between the $\tilde{L} Ω_{F} = Y (r_{A})$ curve and $\tilde{L} Ω_{F} = constant$ line in the type I case (left) show the inner and outer Alfvén radii. On the other hand, one Alfvén radius exists in the type II case (right). The hatched region bounded by $k_{A} = 0$ shows the forbidden region for the parameter. For type I case the range of $(\tilde{L} Ω_{F})_{\min } < \tilde{L} Ω_{F} < 1$ is acceptable as a trans-Alfvénic solution, while for type II case the ranges of $\tilde{L} Ω_{F} < (\tilde{L} Ω_{F})_{\max }$ and $1 < \tilde{L} Ω_{F}$ are acceptable. (Bottom) Curves $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})^{\pm}$ (marked by $[+]$ and $[-]$ ) and $\tilde{L} Ω_{F} = Y$ (marked by $[Y]$ ) for (c) type I and (d) type II cases. Some accretion solutions with certain $\tilde{L} Ω_{F}$ values are shown by horizontal arrows, where the crossing point with the $\tilde{L} Ω_{F} = Y$ curve (i.e., $Y = Y_{A}$ ) indicates the Alfvén point (marked by the filled square). When the solution crosses the $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})^{\pm}$ curve, where $A = 0$ , the Mach number of the solution diverges there; that is, no physical accretion solution exits (the horizontal arrow marked by the cross). The shaded regions show the $A < 0$ regions. Note that the $k > 0$ region is enclosed in the $A < 0$ region. In these schematic diagrams, we set up the field configuration $ξ^{2}$ by Eq. (a1) for type I and Eq. (a2) for type II near the equator as typical models.Reuse & Permissions
Figure 4
The horizon’s boundary condition on $\tilde{L} Ω_{F}$ for (a) type I of $a = 0.3 m$ and (b) type II of $a = 0.9 m$ . The bold thick curves denote the $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})_{H}^{\pm} (χ)$ curves (marked $[+]$ and $[-]$ , respectively) with ${\hat{E}}^{2} = 1.0$ , $Ω_{F} = 0.5 Ω_{\max } = 0.096 23 / m$ , and $θ = π / 2$ . The shaded region corresponds to the $A_{H} < 0$ region, which is forbidden as a black hole accretion solution. The hatched regions in (a) show the $\tilde{L} Ω_{F} > 1$ and $\tilde{L} Ω_{F} < 0$ regions (unphysical). The $\tilde{L} Ω_{F} = Y_{H}$ line is denoted by the dotted line and labeled by $[δ S_{H}]$ . The horizontal broken line marked by $[δ ω_{H}]$ shows the $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})_{ω}$ line. Although, in (b), it seems $(\tilde{L} Ω_{F})_{H}^{+} < (\tilde{L} Ω_{F})_{ω} < (\tilde{L} Ω_{F})_{H}^{-}$ in this plot, we can find the case $(\tilde{L} Ω_{F})_{ω} < (\tilde{L} Ω_{F})_{H}^{+} < Y_{H}$ for larger ( $- χ$ ) values. For the range $(\tilde{L} Ω_{F})_{ω} < \tilde{L} Ω_{F} < (\tilde{L} Ω_{F})_{H}^{+}$ , the spin-up is caused for type II.Reuse & Permissions
Figure 5
Summary of the $\tilde{L} Ω_{F}$ ranges for the spin-up/down of a black hole in (a) type I case and (b) type II case. The value of $\tilde{L} Ω_{F}$ for acceptable MHD accretion is restricted by at least the boundary condition at the event horizon ( $A_{H} > 0$ ) and the condition of the Alfvén point for existing in the $A > 0$ region. The shaded forbidden region is the same as that of Fig. 4. The labels $[A *]$ , $[δ ω_{H}]$ , and $[δ S_{H}]$ show the $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})_{A *}$ line, the $\tilde{L} Ω_{F} = (\tilde{L} Ω_{F})_{ω}$ line, and the $\tilde{L} Ω_{F} = Y_{H}$ line, respectively. The hatched region is forbidden by the condition at the Alfvén point. For type I case, the range of $(\tilde{L} Ω_{F})_{\min } < \tilde{L} Ω_{F} < (\tilde{L} Ω_{F})_{A *}$ is acceptable, and the MHD inflows with $θ_{A} > (θ_{A})_{cr}$ always cause the hole’s spin-up.Reuse & Permissions

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