Photon Orbit Calculator

The Photon Orbit Calculator is an advanced online tool that instantly computes the radius of the photon sphere around a black hole using Einstein’s general theory of relativity. The photon sphere — also known as the photon orbit — is the critical boundary where light can orbit a black hole in an unstable circular path at exactly 1.5 times the Schwarzschild radius.

About the Photon Orbit Calculator

The Photon Orbit Calculator is built strictly according to the principles of general relativity as derived by Karl Schwarzschild in 1916 and further refined in the study of null geodesics around spherically symmetric, non-rotating black holes. The photon sphere exists at r = 3GM/c² = 1.5 rₛ, where photons follow perfectly circular but unstable orbits.

This tool uses the exact, peer-reviewed formula derived from the Schwarzschild metric:

rₚₕ = 3 × (G × M) / c² = 1.5 × rₛ

Importance of the Photon Sphere

The photon sphere is one of the most fascinating predictions of general relativity. Inside this boundary, no light can escape (event horizon at rₛ), while exactly at 1.5 rₛ, light bends so severely that it can complete full circular orbits. This region is responsible for the famous "black hole silhouette" observed by the Event Horizon Telescope in 2019 (M87*) and 2022 (Sgr A*).

Scientific Accuracy & Formula Derivation

The calculation is based on the effective potential for null geodesics in the Schwarzschild spacetime:

V_eff(r) = (1 – 2GM/rc²) × (L²/r²)

The condition for a circular photon orbit occurs when dV_eff/dr = 0 and d²V_eff/dr² = 0, yielding the exact radius:

rₚₕ = 3GM/c²

When and Why You Should Use This Calculator

Use this Photon Orbit Calculator when you are:

• Studying general relativity or astrophysics
• Preparing educational content or lectures
• Researching supermassive black holes (e.g., Sgr A*, M87*, TON 618)
• Simulating gravitational lensing effects
• Writing science blogs or creating YouTube videos

User Guidelines

Real-World Examples

• Sagittarius A* (4.3 million solar masses) → Photon sphere radius ≈ 19.4 million km
• M87* (6.5 billion solar masses) → Photon sphere radius ≈ 29 billion km

Purpose of This Tool

This free Photon Orbit Calculator was created to make advanced general relativity accessible to students, educators, researchers, and space enthusiasts worldwide. No login or software installation required — works instantly in any browser.

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Learn more about the theoretical foundation on Wikipedia: Photon Orbit.

Understanding the Photon Sphere in Depth

The photon sphere represents the innermost stable distance at which a photon can theoretically orbit a black hole forever (though any perturbation causes it to either fall in or escape). This boundary is crucial in strong gravitational lensing, where background light is bent dramatically, creating multiple images and Einstein rings.

In the landmark 2019 EHT image of M87*, the dark shadow is approximately 2.6 times the Schwarzschild radius, with the bright photon ring corresponding exactly to light emitted near the photon sphere at 1.5 rₛ. Understanding this radius is therefore essential for interpreting real black hole images.

Comparison with Other Critical Radii

• Event Horizon: rₛ = 2GM/c²
• Photon Sphere: 1.5 rₛ = 3GM/c²
• Innermost Stable Circular Orbit (ISCO) for matter: 3 rₛ = 6GM/c²

Educational Value

Teachers and professors worldwide use tools like this Photon Orbit Calculator to demonstrate how extreme gravity warps spacetime. Students can instantly see how doubling the black hole mass exactly doubles the photon sphere radius — a direct consequence of the linearity in the Schwarzschild solution.

The calculator supports both solar masses and kilograms, making it suitable for both introductory astrophysics classes and advanced research applications.