ISCO Calculator

The ISCO radius for a non-rotating black hole is r_ISCO = 6GM/c², equivalent to 3 Schwarzschild radii or 6 gravitational radii. In convenient units, r_ISCO in km equals approximately 8.862 times the mass in solar masses. For a 10 solar mass black hole this gives 88.6 km.

ISCO stands for Innermost Stable Circular Orbit. It is the smallest radius at which a massive particle (such as a gas parcel in an accretion disk) can maintain a stable circular orbit around a black hole. Inside this radius, no stable orbit exists and infalling matter plunges directly to the event horizon.

For a prograde orbit (co-rotating with the black hole spin), the ISCO shrinks as spin increases, reaching about 1.237 gravitational radii at maximum spin (a* = 0.998). For a retrograde orbit (counter-rotating), the ISCO expands, reaching about 9 gravitational radii at maximum spin. The Schwarzschild value of 6 gravitational radii applies at zero spin for both orbits.

A non-rotating 10 solar mass black hole has an ISCO radius of 88.615 km. The orbital period at the ISCO is 4.549 milliseconds. The orbital velocity is about 40.8% of the speed of light. For a spinning black hole the prograde ISCO is smaller, down to 18.269 km at near-maximal spin (a* = 0.998).

Inside the ISCO, any small perturbation causes a circular orbit to become unstable: rather than returning to a circular path, the particle spirals inward and falls to the event horizon on a dynamical timescale. This is analogous to the last stable orbit in Newtonian gravity but arises purely from general relativistic effects. Accretion disk emission is therefore expected to cut off at the ISCO inner edge.

The coordinate orbital velocity at the Schwarzschild ISCO is v = c/sqrt(6) ≈ 0.4082c (40.82% of the speed of light). This is independent of the black hole mass because the ISCO always sits at 6 gravitational radii. For a Kerr prograde ISCO the velocity is higher, approaching c near the event horizon for maximally spinning black holes.

Two main methods are used. X-ray continuum fitting measures the thermal emission spectrum of the inner accretion disk; the inner disk temperature profile depends on the ISCO radius, which depends on spin. Iron K-alpha line reflection spectroscopy measures relativistic broadening of the Fe-K line at 6.4 keV; the line profile carries information about the innermost emitting radius. Both methods yield spin estimates by comparing the measured ISCO to the Kerr formula.

No. The photon sphere at r_ph = 3GM/c² (1.5 r_s) is inside the ISCO at r_ISCO = 6GM/c² (3 r_s) for a Schwarzschild black hole. The photon sphere is the orbit of massless particles (photons); the ISCO is the innermost stable orbit for massive particles. In the Schwarzschild case, r_ISCO = 2 r_ph exactly.

Gas in the disk loses angular momentum through viscous processes and drifts slowly inward. At the ISCO it reaches a turning point: no more quasi-circular orbits are available, so the gas transitions to a nearly radial plunge toward the event horizon. This plunge region emits relatively little radiation compared to the disk above the ISCO, creating the characteristic inner truncation seen in X-ray binary spectra.

Sgr A* (4 million solar masses) has a Schwarzschild ISCO of approximately 35.4 million km (0.237 AU) if non-rotating. If it has moderate spin a* = 0.5, the prograde ISCO shrinks to about 250 million km (0.167 AU). Measurements from near-infrared flares and VLBI imaging suggest Sgr A* has significant spin, though the exact value is uncertain.

Only for general relativistic compact objects. Newtonian gravity has no ISCO concept; all circular orbits are stable in Newtonian mechanics. General relativity introduces the ISCO for any black hole. Neutron stars also have an ISCO in principle, but if their physical radius is larger than the ISCO, the innermost stable orbit of the surrounding disk is truncated at the stellar surface rather than the ISCO.

The ISCO is always at 3 Schwarzschild radii for a non-rotating black hole. The Schwarzschild radius r_s = 2GM/c² is the event horizon, while r_ISCO = 6GM/c² = 3 r_s. Between r_s and r_ISCO lies the photon sphere at 1.5 r_s and the innermost bound circular orbit (IBCO) at 2 r_s. All three are inside the ISCO and inaccessible to stable massive-particle orbits.

A Kerr black hole cannot have spin a* greater than 1 in classical general relativity (the Kerr bound). However, astrophysical spin evolution via disk accretion is limited to a* approximately 0.998 by the Thorne limit (1974): photon capture from the disk carries retrograde angular momentum that prevents the spin from reaching exactly 1. The value 0.998 is therefore the standard near-maximum spin for realistic astrophysical black holes.