Magnetic reconnection as a mechanism for energy extraction from rotating black holes

Spinning black holes store rotational energy that can be extracted. When a black hole is immersed in an externally supplied magnetic field, reconnection of magnetic field lines within the ergosphere can generate negative energy (relative to infinity) particles that fall into the black hole event horizon while the other accelerated particles escape stealing energy from the black hole. We show analytically that energy extraction via magnetic reconnection is possible when the black hole spin is high (dimensionless spin $a\sim 1$) and the plasma is strongly magnetized (plasma magnetization ${\sigma }_0>1/3$). The parameter space region where energy extraction is allowed depends on the plasma magnetization and the orientation of the reconnecting magnetic field lines. For ${\sigma }_0\gg 1$, the asymptotic negative energy at infinity per enthalpy of the decelerated plasma that is swallowed by a maximally rotating black hole is found to be ${\epsilon }_{-}^{\infty }\simeq -\sqrt{{\sigma }_0/3}$. The accelerated plasma that escapes to infinity and takes away black hole energy asymptotes the energy at infinity per enthalpy ${\epsilon }_{+}^{\infty }\simeq \sqrt{3{\sigma }_0}$. We show that the maximum power extracted from the black hole by the escaping plasma is $P_{\mathrm{extr}}^{\max }\sim 0.1M^2\sqrt{{\sigma }_0}{w}_0$ (here, $M$ is the black hole mass and $w_0$ is the plasma enthalpy density) for the collisionless plasma regime and one order of magnitude lower for the collisional regime. Energy extraction causes a significant spindown of the black hole when $a\sim 1$. The maximum efficiency of the plasma energization process via magnetic reconnection in the ergosphere is found to be ${\eta }_{\max }\simeq 3/2$. Since fast magnetic reconnection in the ergosphere should occur intermittently in the scenario proposed here, the associated emission within a few gravitational radii from the black hole is expected to display a bursty nature.

Figure 1

Schematic illustration of the mechanism of energy extraction from a rotating black hole by magnetic reconnection in the black hole ergosphere. A configuration with antiparallel magnetic field lines adjacent to the equatorial plane is favored by the frame-dragging effect of the rapidly spinning black hole (panels (a) and (b) portray meridional and equatorial views, respectively), and the resulting equatorial current sheet is prone to fast plasmoid-mediated magnetic reconnection (see circular structures in the zoomed-in region [26]). Magnetic reconnection in the plasma that rotates in the equatorial plane extracts black hole energy if the decelerated plasma that is swallowed by the black hole has negative energy as viewed from infinity, while the accelerated plasma with a component in the same direction of the black hole rotation escapes to infinity. The outer boundary (static limit) of the ergosphere is indicated by the short-dashed lines in both panels. In panel (b), long-dashed and solid lines indicate magnetic field lines below and above of the equatorial plane, respectively. Finally, the dashed lines in the zoomed region indicate the two magnetic reconnection separatrices intersecting at the dominant magnetic reconnection $X$-point.

Figure 2

Energy at infinity per enthalpy ${\epsilon }_{+}^{\infty }$ (gray line) and ${\epsilon }_{-}^{\infty }$ (orange line) for maximal energy extraction conditions ($a,r/M\to 1$ and $\xi \to 0$). Energy extraction requires ${\sigma }_0>1/3$. For ${\sigma }_0\gg 1$, ${\epsilon }_{+}^{\infty }\simeq \sqrt{3{\sigma }_0}$ (dash-dotted black line) and ${\epsilon }_{-}^{\infty }\simeq -\sqrt{{\sigma }_0/3}$ (dashed black line).

Figure 3

Regions of the phase-space $\{a,r/M\}$ where the energies at infinity per enthalpy from Eq. (34) are such that $\mathrm{\Delta }{\epsilon }_{+}^{\infty }>0$ (gray area) and ${\epsilon }_{-}^{\infty }<0$ (orange to red areas), for a reconnecting magnetic field having orientation angle $\xi =\pi /12$ and different values of the magnetization parameter ${\sigma }_0\in \{1,3,10,30,100\}$. The area with ${\epsilon }_{-}^{\infty }<0$ increases monotonically as ${\sigma }_0$ increases. The solid black line indicates the limit of the outer event horizon, Eq. (9), the dashed black line represents the limiting corotating circular photon orbit, Eq. (12), while the dash-dotted black line corresponds to the innermost stable circular orbit, Eq. (13). The limit $r/M=2$ corresponds to the outer boundary of the ergosphere at $\theta =\pi /2$.

Figure 4

Regions of the phase-space $\{a,r/M\}$ where the energies at infinity per enthalpy from Eq. (34) are such that $\mathrm{\Delta }{\epsilon }_{+}^{\infty }>0$ (gray area) and ${\epsilon }_{-}^{\infty }<0$ (green areas), for plasma magnetization ${\sigma }_0=100$ and different values of the orientation angle $\xi \in \{\pi /20,\pi /12,\pi /6,\pi /4\}$. Other lines are the same as in Fig. 3. The area with ${\epsilon }_{-}^{\infty }<0$ increases monotonically as $\xi$ decreases.

Figure 5

$P_{\mathrm{extr}}/w_0=-{\epsilon }_{-}^{\infty }{A}_{\mathrm{in}}{U}_{\mathrm{in}}$ as a function of the dominant $X$-point location $r/M$ for a rapidly spinning black hole with $a=0.99$ and reconnection inflow four-velocity $U_{\mathrm{in}}=0.1$ (i.e., collisionless reconnection regime). ${\epsilon }_{-}^{\infty }$ is evaluated using Eq. (34), while $A_{\mathrm{in}}=(r_E^2-r_{\text{ph}}^2)$. We have also set $M=1$. Different colors (from indigo to red) refer to different plasma magnetizations (from ${\sigma }_0=10$ to ${\sigma }_0={10}^5$) and $\xi =\pi /12$ (top panel) or different orientation angles (from $\xi =\pi /6$ to $\xi =0$) and ${\sigma }_0={10}^4$ (bottom panel). The vertical dashed line indicates the limiting circular orbit $r_{\mathrm{ph}}(a=0.99)$.

Figure 6

Efficiency $\eta$ of the reconnection process as a function of the dominant $X$-point location $r/M$ for a reconnection layer with upstream plasma magnetization ${\sigma }_0=100$ and reconnecting magnetic field having orientation angle $\xi =\pi /20$. Different colors (from indigo to red) refer to different black hole spin values (from $a=0.9$ to $a=1$).

Figure 7

Power ratio $P_{\mathrm{extr}}/P_{\mathrm{BZ}}$ as a function of the plasma magnetization ${\sigma }_0$ for a black hole with dimensionless spin $a=0.99$ and a reconnecting magnetic field having orientation angle $\xi =\pi /12$. Different colors (from indigo to red) refer to different dominant $X$-point locations $r/M\in \{1.7,1.6,1.5,1.4,1.3\}$. We considered $U_{\mathrm{in}}=0.1$ (i.e., collisionless reconnection regime), $A_{\mathrm{in}}=(r_E^2-r_{\text{ph}}^2)$, and $\kappa =0.05$.