Orbital Period Near Black Hole Calculator

Calculate the Orbital Period of an Object Near a Black Hole

Black Hole Mass

Orbital Radius (in Schwarzschild radii) Stable circular orbits exist only for r ≥ 6 Rs (Innermost Stable Circular Orbit for Schwarzschild black hole)

Enter values and click Calculate

About the Orbital Period Near Black Hole Calculator

The Orbital Period Near Black Hole Calculator is a precise, scientifically accurate online tool that calculates the orbital period of a test particle (or star) in a stable circular orbit around a non-rotating (Schwarzschild) black hole using full general relativistic formulas derived from Einstein’s field equations.

Unlike Newtonian gravity, where orbital period follows Kepler’s third law (T² ∝ r³), near a black hole spacetime is strongly curved and the orbital period is dramatically affected by gravitational time dilation and frame-dragging effects. This calculator uses the exact general-relativistic expression for the Keplerian orbital period in Schwarzschild metric.

Scientific Foundation – The Formula

For a Schwarzschild black hole of mass M, the Schwarzschild radius is:
Rs = 2GM/c²

The orbital period T for a circular geodesic orbit at radius r (expressed in units of Rs) is:
T = 2π √(r³ Rs / 8) = π Rs √(r³ / 2) (proper time for the orbiting object)
In terms of solar masses and seconds:
T ≈ 15.5 × M/M☉ × √(r/Rs)³ seconds

At the Innermost Stable Circular Orbit (ISCO) r = 6 Rs, the orbital period reaches its minimum value of approximately 93 × (M/M☉) seconds. For a 10 solar mass black hole, this is only ~15 minutes; for a supermassive black hole of 4 million solar masses (like Sagittarius A*), the ISCO period is ~33 minutes.

Why This Calculator Matters

Understanding orbital periods near black holes is crucial in several astrophysical contexts:

X-ray binaries – Stellar-mass black holes accreting from companion stars show quasi-periodic oscillations (QPOs) linked to orbital motion near the ISCO.
Extreme Mass Ratio Inspirals (EMRIs) – Future space-based gravitational-wave detectors like LISA will detect small objects spiraling into supermassive black holes; accurate orbital period evolution is essential for waveform modeling.
Event Horizon Telescope observations – The “shadow” and photon ring of M87* and Sgr A* are influenced by light bending from orbits at ~5–10 Rs.
Testing General Relativity in the strong-field regime.

When & Why You Should Use This Tool

Use this Orbital Period Near Black Hole Calculator when you are:

Researching or teaching general relativity and black hole astrophysics
Writing science articles, blog posts, or educational content about black holes
Modeling accretion disks, hot spots, or quasi-periodic oscillations
Preparing for astronomy or physics exams or public outreach events
Curious about how time itself slows down near the event horizon

User Guidelines for Accurate Results

Enter the black hole mass in kilograms, solar masses, million, or billion solar masses.
Enter the orbital radius in units of the Schwarzschild radius (r/Rs ≥ 6 for stable circular orbits).
The result is the orbital period as measured by a distant observer (coordinate time). For the proper time experienced by the orbiting object, the period would be shorter due to gravitational time dilation.
For spinning (Kerr) black holes, the ISCO can be as close as 1 Rs and periods are shorter – this calculator assumes non-spinning case for maximum clarity and simplicity.

Fun fact: If you place Earth in a circular orbit at exactly 6 Rs around a 10 solar mass black hole, one orbit would take only about 0.026 seconds from the perspective of a distant observer – faster than any known pulsar!