Innermost Stable Circular Orbit Calculator

The Innermost Stable Circular Orbit Calculator determines the radius of the innermost stable circular orbit (ISCO) around a black hole using Einstein’s General Theory of Relativity. The ISCO marks the closest distance at which a test particle can maintain a stable circular orbit before spiraling into the black hole. This calculator uses the exact Schwarzschild metric solution for non-rotating (Schwarzschild) and rotating (Kerr) black holes.

Calculate ISCO Radius

Black Hole Mass (M)

Black Hole Type

Dimensionless Spin Parameter (a*) | -1 ≤ a* ≤ 1

Output Unit for ISCO Radius

About the Innermost Stable Circular Orbit Calculator

The Innermost Stable Circular Orbit (ISCO) is one of the most important concepts in black hole astrophysics. It represents the smallest radius at which a test particle can orbit a black hole in a stable circular path. Inside the ISCO, any perturbation causes the particle to plunge irreversibly into the event horizon. This boundary is crucial for understanding accretion disks, X-ray binaries, quasi-periodic oscillations (QPOs), and gravitational wave signals from extreme mass-ratio inspirals (EMRIs).

Scientific Foundation & Formulas Used

For a Schwarzschild (non-rotating)) black hole, the ISCO radius is exactly:

r_ISCO = 6 GM / c² = 6 rₛ

where rₛ = 2GM/c² is the Schwarzschild radius. This corresponds to approximately 3 times the event horizon radius.

For a Kerr (rotating) black hole, the ISCO depends on the spin parameter a* = Jc/(GM²), and whether the orbit is prograde or retrograde. The exact analytic expression derived by Bardeen et al. (1972) is:

r_ISCO / (GM/c²) = 3 + Z₂ ∓ √[(3−Z₁)(3+Z₁+2Z₂)]

where

Z₁ = 1 + (1−a*²)^{1/3} [(1+a*)^{1/3} + (1−a*)^{1/3}]
Z₂ = √(3a*² + Z₁²)

The upper sign (∓) corresponds to prograde orbits (a* > 0), the lower to retrograde.

Importance of the ISCO

Defines the inner edge of relativistic accretion disks (e.g., in X-ray binaries and AGN)
Sets the maximum temperature and efficiency of accretion (up to 42% for extremal Kerr vs 6% for Schwarzschild)
Explains high-frequency QPOs observed in black hole candidates
Critical for LISA detections of extreme mass-ratio inspirals
Fundamental test of strong-field general relativity

When and Why You Should Use This Calculator

Use this tool when:

Modeling relativistic jets and accretion flows around supermassive black holes (Sgr A*, M87*)
Interpreting X-ray spectra from stellar-mass black holes (Cygnus X-1, GRS 1915+105)
Planning gravitational-wave observations with LIGO/Virgo/KAGRA or future LISA
Teaching general relativity and black hole astrophysics
Comparing theoretical predictions with Event Horizon Telescope images

User Guidelines

Enter the black hole mass in solar masses or kilograms.
Choose Schwarzschild (a*=0) or Kerr and provide spin |a*| ≤ 0.998 (realistic astrophysical limit).
Select desired output unit.
Click “Calculate” — results appear instantly with physical interpretation.

Real-World Examples

• Sagittarius A* (4.3 × 10⁶ M☉, a* ≈ 0.5–0.9) → ISCO ≈ 4–7 million km)
• M87* (6.5 × 10⁹ M☉) → ISCO ≈ 50–110 billion km (several AU!)
• Cygnus X-1 (≈15 M☉, high spin) → ISCO ≈ 130–200 km

This calculator implements the peer-reviewed analytic formulas from Bardeen, Press & Teukolsky (1972) and subsequent works, ensuring scientific accuracy trusted by researchers worldwide.

For deeper reading, visit the Wikipedia page on Innermost Stable Circular Orbit.

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