Electron-positron cascade in magnetospheres of spinning black holes

We quantitatively study the stationary, axisymmetric, force-free magnetospheres of spinning (Kerr) black holes (BHs) and the conditions needed for relativistic jets to be powered by the Blandford-Znajek mechanism. These jets could be from active galactic nuclei, blazars, quasars, microquasars, radioactive galaxies, and other systems that host Kerr BHs. The structure of the magnetosphere determines how the BH energy is extracted, e.g., via the Blandford-Znajek mechanism, which converts the BH rotational energy into Poynting flux. The key assumption is the force-free condition, which requires the presence of plasma with the density being above the Goldreich-Julian density. Unlike neutron stars, which in principle can supply electrons from the surface, BHs cannot supply plasma at all. The plasma must be generated in situ via an electron-positron cascade, presumably in the gap region. Here we study varying conditions that provide a sufficient amount of plasma for the Blandford-Znajek mechanism to work effectively.

Figure 1

The BH radius is set to one. The blue/gray regions and the red/yellow regions signify the plasma densities. The red, solid line is the surface where ${\rho }_{GJ}$ goes to zero. The green, dash-dotted line is the ergosphere. The light gray, long-dashed lines represent the inner and outer light cylinders. And finally, the dark gray, short-dashed lines display the geometry of the magnetic field lines.

Figure 2

The electric field in the gap and the Lorentz factor of the charges in the gap as a function of position inside the gap. The $x$ axis has been normalized by the BH radius. This result is for a maximally spinning BH of mass $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$. The top curve shown in black represents the Lorentz factor, and the orange curve represents the electric field.

Figure 3

The charge density as a function of position inside the gap. The blue curve represents the charge density that is moving away from the BH, and similarly the orange curve represents the inward moving charge density. The $x$ axis has been normalized by the BH radius. This result is for a maximally spinning BH of mass $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$.

Figure 4

The outgoing photon energy flux as a function of position inside the gap. This result is for a maximally spinning BH of mass $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$. The $x$ axis has been normalized by the BH radius.

Figure 5

Lorentz factor versus polar angle. This solution is for a BH mass of $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$.

Figure 6

The outgoing energy flux from the upscattered photons as a function of polar angle. This solution is for a BH mass of $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$.

Figure 7

The width of half of the gap normalized at the axis of rotation versus the polar angle. An exponential fit of all three masses is $1+5.2\times {10}^{-4}{e}^{7.4\theta }$. This demonstrates that the efficiency of the cascade process as a function of polar angle, while the gap is thin, is invariant relative to the mass of the BH.

Figure 8

The outgoing photon energy density normalized at the axis of rotation versus the polar angle. An exponential fit of all three magnetic fields is $1-0.013e^{4.5\theta }$.

Figure 9

Lorentz factor of the gap versus polar angle. The ten curves represent the change in width as polar angles increase going away from the axis of rotation for ten different spins and their corresponding fits represented with dashed lines. From the top down the spins are 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, and 0.1. And similarly, the fits from the top down are $1900-6.3e^{5.0\theta }$, $940-9.5e^{4.0\theta }$, $1100-11e^{4.0\theta }$, $1200-13e^{4.0\theta }$, $1300-12e^{4.1\theta }$, $1400-15e^{4.0\theta }$, $1500-14e^{4.1\theta }$, $1600-11e^{4.4\theta }$, $1700-14e^{4.2\theta }$, and $1700-10e^{4.5\theta }$.

Figure 10

The distance between the BH horizon and the inner edge of the gap as a function of spin for a BH of mass $10^7{M}_{\odot }$ with a magnetic field strength of $10^4\mathrm{G}$ and an ambient energy density of $10^6\mathrm{ergs}/{\mathrm{cm}}^3$.

Figure 11

The $\star$ is a place holder that represents the maximum Lorentz factor, the maximum electric field, the gap width, and the photon energy flux. Each physical quantity is normalized to its minimum value and then plotted with respect to the spin of the BH on a log-log scale.

Figure 12

The $\star$ is a place holder that represents the maximum Lorentz factor, the maximum electric field, the gap half-width, and the photon energy flux. Each physical quantity is normalized to its minimum value and then plotted with respect to the background energy density on a log-log scale.

Figure 13

Gap width increased by an order of magnitude and the radius is set to one, for a maximally spinning BH. The plasma density is displayed in red and green which correspond to positive and negative charge densities, respectively. It can be seen that as the BH’s spin decreases the gap width increase and the plasma density around the gap decreases.

Figure 14

A comparison of the spectral transition from the inner boundary of the gap through the center of the gap (bold dashed line) and to the outer boundary of the gap (bold solid line). Starting at 12% of the gap width after the inner (closest to the BH) boundary (bottommost curve), 14 spectral lines are shown. The top spectral transition plot is for $U_b={10}^5\mathrm{ergs}/{\mathrm{cm}}^3$. The bottom spectral transition plot is for $U_b={10}^6\mathrm{ergs}/{\mathrm{cm}}^3$.

Figure 15

The curve for Sgr ${\mathrm{A}}^{*}$ is at the top and followed by M87. Next is MCG-6-30-15 and NGC 3783. They are followed by 1H0707-495. Next Mrk 79, Mrk 335, and SWIFT $\mathrm{J}2127.4+5654$ are clustered together. They are followed by NGC 7469 and Fairall 9. The values for mass, spin, and energy density in Eq. (36) are listed in Table 2.

Figure 16

The BH radius of a maximally spinning BH has been set to one, and the gap widths have been left to scale. M87 has a luminosity of $2.7\times {10}^{42}\mathrm{ergs}/\mathrm{s}$, mass of $10^{9.5}{M}_{\odot }$, a spin of 0.65, and a magnetic field of 15 G. Sgr ${\mathrm{A}}^{*}$ has a luminosity of $10^{37}\mathrm{ergs}/\mathrm{s}$, mass of $10^{6.6}{M}_{\odot }$, a spin of 0.65, and a magnetic field of 30 G.