The innermost stable circular orbit (often called the ISCO) is the smallest circular orbit in which a test particle can stably orbit a massive object in general relativity.[1] The location of the ISCO, the ISCO-radius ( $${\displaystyle r_{\mathrm {isco} }}$$ ), depends on the angular momentum (spin) of the central object.

The ISCO plays an important role in black hole accretion disks since it marks the inner edge of the disk.

For a non-spinning massive object, where the gravitational field can be expressed with the Schwarzschild metric, the ISCO is located at,

$${\displaystyle r_{\mathrm {isco} }={\frac {6\,GM}{c^{2}}}=3R_{S},}$$

where $${\displaystyle R_{S}}$$is the Schwarzschild radius of the massive object with mass M {\displaystyle M} M. Thus, even for a non-spinning object, the ISCO radius is only three times the Schwarzschild radius, $${\displaystyle R_{S}}$$ , suggesting that only black holes and neutron stars have innermost stable circular orbits outside of their surfaces. As the angular momentum of the central object increases, $${\displaystyle r_{\mathrm {isco} }}$$ decreases.

Circular orbits are still possible between the ISCO and the photon sphere, but they are unstable. The photon sphere has a radius of

$${\displaystyle r={\frac {3\,GM}{c^{2}}}={\frac {3R_{S}}{2}}.}$$

For a massless test particle like a photon, the only possible circular orbit is exactly at the photon sphere, and is unstable.[2] Inside the photon sphere, no circular orbits exist.
Rotating black holes

The case for rotating black holes is somewhat more complicated. The equatorial ISCO in the Kerr metric depends on whether the orbit is prograde (negative sign below) or retrograde (positive sign):

$${\displaystyle r_{\mathrm {isco} }={\frac {GM}{c^{2}}}\left(3+Z_{2}\pm {\sqrt {(3-Z_{1})(3+Z_{1}+2Z_{2})}}\right)}$$

where

$${\displaystyle Z_{1}=1+{\sqrt[{3}]{1-x^{2}}}\left({\sqrt[{3}]{1+x}}+{\sqrt[{3}]{1-x}}\right)}$$
$${\displaystyle Z_{2}={\sqrt {3x^{2}+Z_{1}^{2}}}}$$

with $${\displaystyle x=a/M}$$ as the rotation parameter.[3] As the rotation rate of the black hole increases the retrograde ISCO increases towards $${\displaystyle 9GM/c^{2}}$$ (4.5 times the a=0 horizon radius) while the prograde ISCO decreases towards the horizon radius and appears to merge with it for an extremal black hole (however, this later merger is illusory and an artefact of using Boyer-Lindquist coordinates [4]).

If the particle is also spinning there is a further split in ISCO radius depending on whether the spin is aligned with or against the black hole rotation.[5]
References

Misner, Charles; Thorne, Kip S.; Wheeler, John (1973). Gravitation. W. H. Freeman and Company. ISBN 0-7167-0344-0.
Carroll, Sean M. (December 1997). "Lecture Notes on General Relativity: The Schwarzschild Solution and Black Holes".arXiv:gr-qc/9712019. Bibcode:1997gr.qc....12019C. Retrieved 2017-04-11.
Bardeen, James M.; Press, William H.; Teukolsky, Saul A. (1972). "Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation". The Astrophysical Journal. 178: 347–370. Bibcode:1972ApJ...178..347B. doi:10.1086/151796.
Hirata, Christopher M. (December 2011). "Lecture XXVII: Kerr black holes: II. Precession, circular orbits, and stability" (PDF). Caltech. Retrieved 5 March 2018.

Jefremov, Paul I; Tsupko, Oleg Yu; Bisnovatyi-Kogan, Gennady S (15 June 2015). "Innermost stable circular orbits of spinning test particles in Schwarzschild and Kerr space-times". Physical Review D. 91 (12): 124030.arXiv:1503.07060. Bibcode:2015PhRvD..91l4030J. doi:10.1103/PhysRevD.91.124030. S2CID 119233768.

Leo C. Stein, Kerr calculator V2 [1]

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