Innermost stable circular orbit

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. The location of the ISCO, the ISCO-radius ( r i s c o {displaystyle r_{mathrm {isco} }} ), depends on the angular momentum (spin) of the central object. 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. The location of the ISCO, the ISCO-radius ( r i s c o {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, where R S {displaystyle R_{S}} is the Schwarzschild radius of the massive object with mass M {displaystyle M} . Thus, even for a non-spinning object, the ISCO radius is only three times the Schwarzschild radius, R S {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, r i s c o {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 For a massless test particle like a photon, the only possible circular orbit is exactly at the photon sphere, and is unstable. Inside the photon sphere, no circular orbits exist. 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):