Relativistic dust accretion of charged particles in Kerr-Newman spacetime

We describe a new analytical model for the accretion of particles from a rotating and charged spherical shell of dilute collisionless plasma onto a rotating and charged black hole. By assuming a continuous injection of particles at the spherical shell and by treating the black hole and a featureless accretion disk located in the equatorial plane as passive sinks of particles, we build a stationary accretion model. This may then serve as a toy model for plasma feeding an accretion disk around a charged and rotating black hole. Therefore, our new model is a direct generalization of the analytical accretion model introduced by E. Tejeda, P. A. Taylor, and J. C. Miller [Mon. Not. R. Astron. Soc. 429, 925 (2013)]. We use our generalized model to analyze the influence of a net charge of the black hole, which will in general be very small, on the accretion of plasma. Within the assumptions of our model we demonstrate that already a vanishingly small charge of the black hole may in general still have a non-negligible effect on the motion of the plasma, as long as the electromagnetic field of the plasma is still negligible. Furthermore, we argue that the inner and outer edges of the forming accretion disk strongly depend on the charge of the accreted plasma. The resulting possible configurations of accretion disks are analyzed in detail.

Figure 1

Sketch of the accretion model. Here $R=\sqrt{r^2+a^2}\sin (\theta )$ and $Z=r\cos (\theta )$.

Figure 2

Radius of the ISCO in the equatorial plane for charged particles in Kerr-Newman spacetime, with very small BH charge ($Q=0$), for $a=0$ (black, dotted), $a=0.1$ (red, solid), and $a=0.5$ (blue, dashed) as a function of $eQ$. Four different solutions arise, traced back to the four combinations of direct or retrograde orbits and $eQ<0$ or $eQ>0$. The radius $r_{\mathrm{ISCO}}$ grows for bigger $|eQ|$ in all cases, but grows significantly faster for the case where the BH and test particle have the same charge.

Figure 3

Schematic plots of possible accretion disk scenarios. Here $r_{{\mathrm{D}}_1}=r_{\mathrm{D}}{|}_{e_{1}Q}$ corresponds to $e_{1}Q<0$ and $r_{{\mathrm{D}}_2}=r_{\mathrm{D}}{|}_{e_{2}Q}$. If the outer edge $r_{\mathrm{D}}$ is smaller than the corresponding ISCO, no accretion disk is formed. For a general description see Case 1 to Case 4 in Sec. 5, where (a) corresponds to Case 1, (b) corresponds to Case 2, (c) and (d) correspond to Case 3, and (e) corresponds to Case 4.

Figure 4

Streamlines, three velocity field in LNRFs (left and right) and density field (right) are plotted for plasma or neutral particles, a negatively charged BH, $r_0=20$, and different initial conditions. The corresponding parameter $eQ$ for protons can be calculated by $e_{p}Q=\frac{{\mu }_e}{{\mu }_p}{e}_{e}Q$. Black, white, and gray streamlines and velocity fields describe electron, proton, and neutral particle motion respectively. The density color bar is given in a logarithmic scale. The initial condition $v_e^x$ is the $x$ component of the three-velocity at $r=r_0$ and $\theta =\pi /2$, given by Eqs. (37)–(39). Two density peaks arise, which can be traced back to the two differently charged particle types of the plasma. Changes in the initial conditions $v_e^{{{\varphi }_0}^{\prime }}$ and $v_e^{r_0^{\prime }}$ and $e_{e}Q$ have a strong effect on all features of the accretion flow. This effect of the initial velocities and $e_{e}Q$ can be studied by comparing the plots from Figs. 4, 5, 6 with each other.

Figure 5

For a detailed description see caption of Fig. 4. A comparison of the plots (a) and (b) with the plot in Fig. 4 shows the influence of the initial velocities $v_e^{r_0^{\prime }}$ and $v_e^{{{\varphi }_0}^{\prime }}$, respectively. In plot (c) the initial value for the $r$ motion is chosen to be very small. This results in a weak particle infall, leading to very small density values and a big effect of the attractive and repulsive electromagnetic forces on the accretion flow.

Figure 6

For a detailed description see caption of Fig. 4. A comparison of plot (a) with plot (c) and a comparison of plot (b) with plot (c) show the influence of the initial velocities $v_e^{{\varphi }_0^{\prime }}$ and $v_e^{r_0^{\prime }}$, respectively. The influence of $e_{e}Q$ on the accretion flow is shown by a comparison with plot (c) and the plot in Fig. 4. In plot (a) only one density peak arises, produced by the accreted electrons. All streamlines of the proton accretion flow reach the BH horizon before hitting the $\theta =\pi /2$ plane and therefore will not create a density peak.

Figure 7

Outer edge $r_{\mathrm{D}}$ of the forming accretion disk as a function of $eQ$ for $r_0=20$ and $a=0.1$. Here $v_e^x$ gives the $x$ component of the three-velocity at $r=r_0$, $\theta =\pi /2$. (a) $v_e^{{{\varphi }_0}^{\prime }}=0.11$, different $v_e^{r_0^{\prime }}$. (b) $v_e^{r_0^{\prime }}=-0.2$, different $v_e^{{{\varphi }_0}^{\prime }}$. The dependence of $r_{\mathrm{D}}$ on $eQ$ increases with growing values of $v_e^{{{\varphi }_0}^{\prime }}$ and $v_e^{r_0^{\prime }}$.

Figure 8

Outer edge $r_{\mathrm{D}}$ of the forming accretion disk as (a) a function of $Q$ for $r_0=20$, $e=0$, $v_e^{{{\varphi }_0}^{\prime }}=0.11$, $v_e^{r_0^{\prime }}=-0.2$, and different values for $a$ and (b) a function of $v_e^{{{\varphi }_0}^{\prime }}$ and $v_e^{r_0^{\prime }}$ for $r_0=20$, $e_{e}Q=0.5$, $a=0.1$. The dust flow is counter-rotating for $a<0$ and corotating for $a>0$. It can be seen in plot (a) that $r_{\mathrm{D}}$ changes only slightly with variation of $a$, even less with variation of $Q$ compared to the changes induced by a variation of the initial velocities $v_e^{{{\varphi }_0}^{\prime }}$ and $v_e^{r_0^{\prime }}$, plotted in (b). This changes become bigger for bigger values of $v_e^{{{\varphi }_0}^{\prime }}$ and $v_e^{r_0^{\prime }}$. The shift of $r_{\mathrm{D}}$ for growing $a$ to bigger values depicts the frame-dragging effect.