Expanded solutions of force-free electrodynamics on general Kerr black holes

In this work, expanded solutions of force-free magnetospheres on general Kerr black holes are derived through a radial distance expansion method. From the regular conditions both at the horizon and at spatial infinity, two previously known asymptotical solutions (one of them is actually an exact solution) are identified as the only solutions that satisfy the same conditions at the two boundaries. Taking them as initial conditions at the boundaries, expanded solutions up to the first few orders are derived by solving the stream equation order by order. It is shown that our extension of the exact solution can (partially) cure the problems of the solution: it leads to magnetic domination and a mostly timelike current for restricted parameters.

Figure 1

Illustrations of the invariant $F^2$ at different radial distances $r$ and different poloidal angles $\theta =q\pi /2$. The parameter $c=1$ and the spin parameter $a=0.8M$. The line groups are (1) $\alpha =1$ and $k=1.3$, (2) $\alpha =0.1$ and $k=1.49$, and (3) $\alpha =1$ and $k=2$.

Figure 2

Illustrations of $J^2$ at different distances $r$ and different angles $\theta =q\pi /2$. The parameters are chosen to be $c=1$ and $a=0.8M$. Left panel: (1) $\alpha =1$ and $k=1.3$, and (2) $\alpha =1$ and $k=3$. Right panel: $\alpha =0.1$ and $k=1.49$.