Black Hole ISCO Calculator

Calculate the innermost stable circular orbit (ISCO) around a Schwarzschild or Kerr black hole from its mass and dimensionless spin a*, in gravitational radii, Schwarzschild radii, and kilometers — with a live horizon/ISCO/disk diagram and the accretion efficiency it implies.

Black hole ISCO calculator

The innermost stable circular orbit (ISCO) is the smallest radius where orbiting gas can stay on a stable circular path before plunging into the black hole. For a non-spinning (Schwarzschild) hole it's exactly 6 GM/c²; spin things up and it depends strongly on direction — prograde spin drags the ISCO in toward the horizon, all the way down to just 1 GM/c² at maximal spin, while retrograde spin pushes it out to 9 GM/c².

Mass (M)

Dimensionless spin a* (−1 retrograde … +1 prograde)

−1 (max retrograde)0 (Schwarzschild)+1 (max prograde)

r_ISCO = 6 r_g = 3 r_s

Event horizon at 2 r_g · accretion efficiency η ≈ 5.7%

Drawn to a fixed scale in gravitational radii, so the circles genuinely shrink or grow as spin changes. The disk is truncated for illustration — real accretion disks extend much farther out than shown.

ISCO radius as a function of spin a* (positive = prograde). A single continuous curve — a* < 0 uses the retrograde branch, a* > 0 the prograde branch.

ISCO radius88.6 km

Event horizon radius29.533 km (1 r_s)

If this spin were prograde6 r_g

If this spin were retrograde6 r_g

Gas doesn’t fall straight into a black hole — it spirals in through an accretion disk, orbiting stably right up until it crosses one specific radius: the innermost stable circular orbit, or ISCO. Inside it, no circular orbit is stable at all, and matter plunges the rest of the way in almost immediately. Where that radius sits — and how close it can get to the event horizon — depends entirely on how fast the black hole is spinning, which is why the ISCO turns out to be one of the most observationally useful numbers in black hole astrophysics.

Schwarzschild: the non-spinning case

For a black hole with no spin at all, the ISCO sits at a clean multiple of the gravitational radius r_g = GM/c²:

r_{\rm ISCO} = 6\ \frac{GM}{c^2} = 3\ r_s

where r_s = 2GM/c² is the Schwarzschild radius (the event horizon itself, for a = 0). This number — 6 gravitational radii, 3 Schwarzschild radii — is worth memorizing; it’s the baseline every spinning case is measured against.

Kerr: spin changes everything

Real astrophysical black holes spin, and spin drags spacetime around with it (frame dragging). For a black hole with dimensionless spin a* = a/M ∈ [−1, 1], the ISCO radius follows the Bardeen–Press–Teukolsky (1972) formula:

\frac{r_{\rm ISCO}}{r_g} = 3 + Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}

Z_1 = 1 + (1-a_*^2)^{1/3}\left[(1+a_*)^{1/3} + (1-a_*)^{1/3}\right], \quad Z_2 = \sqrt{3a_*^2 + Z_1^2}

(minus for prograde orbits, plus for retrograde). Plug in a* = 0 and both signs collapse to exactly 6, recovering Schwarzschild. Push a* toward +1 (maximal prograde, disk co-rotating with the hole) and the ISCO shrinks all the way to 1 r_g — frame dragging lets orbits survive stably almost down to the horizon itself. Push a* toward −1 (maximal retrograde) and it’s pushed out to 9 r_g instead, since a counter-rotating orbit fights the dragging of spacetime the whole way in. This calculator uses the sign of a* directly to choose which branch applies, so sliding smoothly from −1 to +1 traces one continuous curve: 9 → 6 → 1 r_g.

Why this changes accretion efficiency

Gas doesn’t just orbit at the ISCO — it releases gravitational binding energy as radiation on its way there, and then falls through the rest of the way without radiating much more. The fraction of rest-mass energy radiated away, the accretion efficiency η = 1 − E_ISCO (E_ISCO being the specific orbital energy at the ISCO), depends only on spin:

a = 0 (Schwarzschild):* η ≈ 5.7%
a ≈ 0.998 (the Thorne limit* — roughly the maximum spin a black hole can reach by steadily accreting from a thin disk, since the accreted gas itself carries angular momentum that limits further spin-up): η ≈ 32%
a = 1 (extremal, mathematical limit):* η ≈ 42.3%

For comparison, nuclear fusion converts only about 0.7% of rest mass to energy. A rapidly spinning black hole’s accretion disk is a dramatically more efficient engine than any nuclear process — which is exactly why measuring a black hole’s spin (often via how far in its X-ray-emitting disk extends, i.e. via the ISCO) is such a direct observational probe of its physics.

Reading the diagram

The circular schematic draws the event horizon, the ISCO, and a (illustratively truncated) accretion disk to a single fixed scale in gravitational radii — so as you move the spin slider, the circles genuinely shrink and grow relative to each other rather than just displaying different numbers. The rotation glyph at the center scales with spin magnitude and flips direction for retrograde vs. prograde. The second chart shows the full textbook curve of ISCO radius against spin, with your current setting marked on it.