After posting the Kerr ISCO calculator, I got some suggestions for improvements. I may add to this page later if I feel like procrastinating. Grab a red dot and drag. Or, change the value in the $a$ input box below and hit enter. Horizontal axis is $a$, vertical axis is Boyer-Lindquist $r$. I use units where $M=1$. Scroll down for details.

Details

This plot is more or less an interactive version of Fig. 1 of Bardeen, Press, and Teukolsky (1972). All formulas and way more detail may be found in that reference.

The darker shaded region is inside the event horizon. The lighter shaded region denotes the extent of the ergosphere.

Aside from the outer horizon, a quantity with a + (respectively -) sign refers to a prograde (resp. retrograde) orbit. The dimensionless spin parameter is $\chi\equiv a/M$, $-1 \le \chi \le +1$. The following quantites are plotted:

• $r_+$: This is the event horizon. It is given by \[ \frac{r_+}{M} = 1 + \sqrt{1 - \chi^2}. \]

• $r_{ms\pm}$: The marginally stable orbits for massive particles, also known as the ISCO (innermost stable circular orbit). They are given by \begin{align} \frac{r_{ms\pm}}{M} &= 3 + Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}, \\ Z_1 &= 1 + \left(1 - \chi^2\right)^{1/3} \left((1 + \chi)^{1/3} + (1 - \chi)^{1/3}\right),\\ Z_2 &= \sqrt{3 \chi^2 + Z_1^2}. \end{align}

• $r_{ph\pm}$: Photon orbits, given by \[ \frac{r_{ph\pm}}{M} = 2\left[ 1 + \cos \left( \frac{2}{3} \cos^{-1}\mp \chi \right) \right] \]

• $r_{mb\pm}$: Marginally bound orbits, given by \[ \frac{r_{mb\pm}}{M} = 2 \mp \chi + 2 \sqrt{1 \mp \chi} \]

One word of caution. Note that all of the prograde quantities converge to Boyer-Lindquist $r\to M$ as $a\to M$. You can read about how this is a coordinate artifact in Bardeen, Press, and Teukolsky, or in many other places, e.g. Jacobson’s Where is the extremal Kerr ISCO?.