Photon Sphere Radius Calculator

Find the photon sphere radius for any black hole mass, including the orbital period of trapped light, or infer the mass from a known photon orbit radius.

🔵 Photon Sphere Radius Calculator
Black hole mass1.000 M☉
M☉
0.1 M☉50 M☉
Photon sphere radius13.300 km
km
0.1 km500 km
Photon Sphere Radius
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In Metres
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Schwarzschild Radius
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Photon Orbit Period
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Implied Mass
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In Kilograms
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Schwarzschild Radius
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🔵 What is the Photon Sphere?

The photon sphere (also called the light sphere or photon ring radius) is the spherical shell surrounding a non-rotating black hole where gravity is strong enough to force photons into circular orbits. Its radius is rph = 3GM/c2, exactly 1.5 times the Schwarzschild radius (event horizon). The concept was formalized in the study of null geodesics in the Schwarzschild metric and plays a central role in observational black hole physics.

The photon sphere has three major real-world applications. First, it sets the scale of the black hole shadow observed by radio telescopes. The Event Horizon Telescope images of M87* (2019) and Sgr A* (2022) both reveal a bright emission ring surrounding a darker interior: that bright ring is radiation that looped near the photon sphere before escaping, and its apparent angular size is directly tied to rph. Second, the photon sphere determines the gravitational lensing cross-section: light rays with impact parameters smaller than the critical value (b = 3√3 GM/c2) are captured by the black hole. Third, it is used in accretion disk modeling to separate the stable orbit region (outside the ISCO at 6GM/c2) from the plunge region between the photon sphere and ISCO.

A subtle but important point is that photon sphere orbits are unstable. A photon placed exactly at rph with the right tangential velocity travels in a perfect circle, but any infinitesimal perturbation breaks this orbit. Inward-perturbed photons spiral into the singularity; outward-perturbed photons escape to infinity. This instability is why the photon ring appears thin and bright rather than as a broad halo. Photons that pass close to rph wind many times around the black hole before escaping, accumulating exponential amplification with each orbit.

The ratio rph/rs = 3/2 is a universal constant for Schwarzschild black holes of any mass, from microscopic primordial black holes to supermassive ones weighing billions of solar masses. This universality means the photon sphere is always a simple proxy for the event horizon: measuring one gives the other. For rotating Kerr black holes the photon sphere generalizes to distinct prograde and retrograde photon orbits at different radii, and the ISCO and photon orbit both shift inward on the prograde side. The calculator above applies to the non-rotating Schwarzschild case, which is the standard first approximation for most astrophysical calculations.

📐 Formula

rph  =  3GM ÷ c2  =  (3/2) × rs  ≈  4.431 × (M / M☉) km
G = gravitational constant = 6.674 × 10−11 m3 kg−1 s−2
M = mass of the black hole in kg; use M / M☉ with M☉ = 1.989 × 1030 kg for solar masses
c = speed of light = 2.998 × 108 m/s, so c2 = 8.988 × 1016 m2/s2
rs = Schwarzschild radius = 2GM/c2; the photon sphere is always at 1.5 rs
Coordinate orbital period: T = 6√3 π GM / c3 ≈ 1.608 ms × (M / 10 M☉)
Inverse (Find Mass): M = rph × c2 / (3G)
Example: For M = 10 M☉, rph = 3 × 6.674 × 10−11 × 10 × 1.989 × 1030 / 8.988 × 1016 = 44,308 m = 44.308 km

📖 How to Use This Calculator

Steps

1
Select a calculation mode — choose Find Photon Sphere to compute the photon orbit radius from a black hole mass, or Find Mass to infer the mass from a known photon sphere radius.
2
Enter mass or radius — in Find Photon Sphere mode, type a mass in solar masses or drag the slider (range 0.1 to 50 M☉ on the slider; type any value for supermassive objects). In Find Mass mode, enter the photon sphere radius in kilometres.
3
Read the results — the calculator shows the photon sphere radius in human-readable units (km for stellar-mass objects, AU for supermassive ones), the raw value in metres, the Schwarzschild radius for comparison, and the coordinate orbital period of a photon travelling at the light sphere.

💡 Example Calculations

Example 1 — Solar-Mass Compact Object (1 M☉)

A neutron star or black hole with the mass of the Sun

1
rph = 3 × 6.674 × 10−11 × 1.989 × 1030 / (2.998 × 108)2 = 4,430.8 m.
2
Schwarzschild radius for comparison: rs = (2/3) × 4,430.8 m = 2,953.8 m = 2.954 km. Ratio rph/rs = 1.5 exactly.
3
Coordinate orbital period: T = 6√3 π × 6.674 × 10−11 × 1.989 × 1030 / (2.998 × 108)3 = 1.608 × 10−4 s = 160.838 μs.
rph = 4.431 km (4.431 × 103 m), rs = 2.954 km, period = 160.838 μs
Try this example →

Example 2 — Stellar-Mass Black Hole (10 M☉)

A typical stellar-mass black hole formed from a massive star collapse

1
rph scales linearly with mass: 4.431 km × 10 = 44.308 km.
2
Schwarzschild radius: rs = 2.954 km × 10 = 29.538 km.
3
Photon orbit period: T = 160.838 μs × 10 = 1.608 ms. A photon circles this black hole about 621 times per second.
rph = 44.308 km (4.431 × 104 m), rs = 29.538 km, period = 1.608 ms
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Example 3 — Binary Merger Remnant (30 M☉)

A black hole typical of those detected by LIGO gravitational wave events

1
rph = 3 × 6.674 × 10−11 × 30 × 1.989 × 1030 / 8.988 × 1016 = 132,923 m = 132.923 km.
2
Schwarzschild radius: rs = (2/3) × 132.923 km = 88.615 km.
3
Photon orbit period: T = 160.838 μs × 30 = 4.825 ms. This rapid orbital frequency is in the LIGO audio-frequency band.
rph = 132.923 km (1.329 × 105 m), rs = 88.615 km, period = 4.825 ms
Try this example →

Example 4 — Inverse Mode: Find Mass from Photon Sphere Radius

A photon sphere radius of 132.923 km is observed. What is the black hole mass?

1
Switch to Find Mass mode. Enter rph = 132.923 km.
2
M = rph × c2 / (3G) = 132,923 × 8.988 × 1016 / (3 × 6.674 × 10−11) = 5.967 × 1031 kg.
3
Convert to solar masses: 5.967 × 1031 kg / 1.989 × 1030 kg = 30.000 M☉. Schwarzschild radius = (2/3) × 132.923 = 88.615 km.
Mass = 30.0000 M☉ (5.967 × 1031 kg), rs = 88.615 km
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❓ Frequently Asked Questions

What is the photon sphere radius formula for a black hole?+
The photon sphere radius is r_ph = 3GM/c², where G = 6.674 × 10−11 m³ kg−1 s−2, M is the mass in kg, and c = 2.998 × 108 m/s. In convenient units, r_ph in kilometres equals approximately 4.431 times the mass in solar masses. It is always exactly 1.5 times the Schwarzschild radius r_s = 2GM/c².
How does the photon sphere relate to the Schwarzschild radius?+
The photon sphere sits at r_ph = 3GM/c², while the Schwarzschild radius (event horizon) is at r_s = 2GM/c². Their ratio is exactly 3/2: the photon sphere is always 50% further from the singularity than the event horizon. This 3/2 factor is a universal constant for non-rotating black holes, independent of mass.
What happens to light at the photon sphere?+
At the photon sphere, photons travel in circular orbits. However, these orbits are unstable: a photon at exactly r_ph follows a perfect circle, but any tiny perturbation breaks the orbit. Inward perturbations send the photon into the black hole; outward perturbations send it escaping to infinity. In practice, near-critical photons wind many times around the black hole, forming the bright photon ring visible in Event Horizon Telescope images.
Is the photon sphere the same as the event horizon?+
No. The event horizon (Schwarzschild radius r_s = 2GM/c²) is the point of no return where escape velocity equals the speed of light. The photon sphere (r_ph = 3GM/c²) is outside the event horizon at 1.5 times its radius. Light at the photon sphere can still escape to infinity, but only in an unstable circular orbit that breaks under any perturbation.
What is the photon orbit period for a 10 solar mass black hole?+
A 10 solar mass black hole has a photon sphere radius of 44.308 km. The coordinate orbital period is T = 6√3π GM/c³ = 1.608 milliseconds. For comparison, a 1 solar mass compact object has a period of about 161 microseconds. The period scales linearly with mass: doubling the mass doubles the period.
Can we observe the photon sphere with telescopes?+
Not directly, but its effect is visible. The Event Horizon Telescope images of M87* and Sgr A* show a bright photon ring surrounding the dark shadow. This ring is radiation that looped near the photon sphere multiple times before escaping. Its apparent angular diameter is approximately 10 times the Schwarzschild radius, tightly coupled to the photon sphere geometry.
How does the photon sphere differ for rotating black holes?+
For non-rotating Schwarzschild black holes, the photon sphere is a single sphere at r = 3GM/c². For rotating Kerr black holes, the photon orbit splits into a prograde orbit (co-rotating, closer to the horizon) and a retrograde orbit (counter-rotating, farther out). For a maximally spinning black hole, the prograde photon orbit shrinks to r = GM/c² while the retrograde orbit expands to r = 4GM/c².
What is the photon sphere radius of Sagittarius A*?+
Sgr A*, the Milky Way's central black hole, has a mass of approximately 4 million solar masses. Its photon sphere radius is r_ph = 4.431 km × 4 × 106 = 17.7 million km (0.118 AU). Its photon orbital period is T = 0.161 ms × 4 × 106 = 643 seconds (about 10.7 minutes).
Is the photon sphere inside or outside the innermost stable circular orbit (ISCO)?+
The photon sphere at r_ph = 3GM/c² is inside the ISCO. For a non-rotating Schwarzschild black hole, the ISCO is at r_ISCO = 6GM/c² = 3r_s, which is twice as far from the singularity as the photon sphere. Any massive particle orbiting between the photon sphere and the ISCO will spiral into the black hole rather than maintaining a stable orbit.
Does every black hole have a photon sphere?+
Every non-rotating (Schwarzschild) black hole has a photon sphere at exactly 3GM/c². Rotating Kerr black holes have analogous photon orbits, though they are not spherically symmetric and depend on spin. Charged Reissner-Nordstrom black holes also have a photon sphere whose radius depends on charge. The photon sphere is a universal feature of compact objects described by general relativity.
What is the apparent shadow size compared to the photon sphere?+
The apparent shadow radius as seen by a distant observer is r_shadow = 3√3 GM/c² = √3 × r_ph, which is approximately 2.6 times the Schwarzschild radius. This shadow is larger than the event horizon because the photon capture cross-section includes trajectories that loop around the photon sphere. The photon ring marks the bright edge of this shadow.
Why is the photon sphere important for gravitational lensing?+
The photon sphere determines the critical impact parameter b_c = 3√3 GM/c². Light rays from a distant source with impact parameter smaller than b_c are captured by the black hole. Rays with impact parameters just above b_c loop near the photon sphere before escaping, producing a bright photon ring on the far side of the lens. This strong-field lensing effect is distinct from the weak-field lensing of distant galaxy clusters and carries direct information about the spacetime geometry near the horizon.
Tags: Photon Sphere Light Ring Black Hole Photon Orbit Calculator Formula Free Online General Relativity

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What is the photon sphere radius formula for a black hole?

The photon sphere radius is r_ph = 3GM/c², where G = 6.674e-11 m³ kg⁻¹ s⁻², M is the mass in kg, and c = 2.998e8 m/s. In convenient units, r_ph in km equals approximately 4.431 times the mass in solar masses. It is always exactly 1.5 times the Schwarzschild radius r_s = 2GM/c².

How does the photon sphere relate to the Schwarzschild radius?

The photon sphere is at r_ph = 3GM/c², while the Schwarzschild radius (event horizon) is at r_s = 2GM/c². Their ratio is exactly 3/2, so the photon sphere always sits 50% further from the singularity than the event horizon. This 3/2 factor is a universal constant for non-rotating black holes.

What happens to light at the photon sphere?

At the photon sphere, photons travel in circular orbits. However, these orbits are unstable: a photon at exactly the photon sphere radius follows a perfect circle, but any tiny perturbation causes it to spiral either into the black hole or out to infinity. In practice, photons make many partial loops near the photon sphere before escaping, forming the bright photon ring seen in black hole images.

Is the photon sphere the same as the event horizon?

No. The event horizon (Schwarzschild radius r_s = 2GM/c²) is the point of no return where escape velocity equals the speed of light. The photon sphere (r_ph = 3GM/c²) is outside the event horizon, at 1.5 times its radius. Light at the photon sphere can still escape to infinity, but only just barely in an unstable orbit.

What is the photon orbit period for a 10 solar mass black hole?

A 10 solar mass black hole has a photon sphere radius of 44.308 km. The coordinate orbital period is T = 6√3π GM/c³ = 1.608 milliseconds. For comparison, a 1 solar mass object has a period of about 161 microseconds. The period scales linearly with mass.

Can we observe the photon sphere directly?

Not directly, but the photon sphere's gravitational influence is visible. The Event Horizon Telescope images of M87* and Sgr A* show a bright ring of emission surrounding a dark shadow. This photon ring is radiation that orbited the black hole multiple times near the photon sphere before escaping. The ring diameter is approximately 2.6 times the Schwarzschild diameter, closely related to the photon sphere geometry.

How does the photon sphere differ for rotating black holes?

For non-rotating Schwarzschild black holes, the photon sphere is a single sphere at r = 3GM/c². For rotating Kerr black holes, the photon orbit splits into a prograde orbit (co-rotating with the spin, closer to the horizon) and a retrograde orbit (counter-rotating, farther out). The prograde ISCO for a maximally spinning black hole approaches r = GM/c², while the retrograde photon orbit can extend to r = 4GM/c².

What is the photon sphere radius of Sagittarius A*?

Sgr A*, the Milky Way's central black hole, has a mass of approximately 4 million solar masses. Its photon sphere radius is r_ph = 4.431 km × 4e6 = 17.7 million km (0.118 AU). The Event Horizon Telescope resolved emission at scales of a few Schwarzschild radii, which corresponds closely to the photon sphere scale.

Is the photon sphere inside or outside the innermost stable circular orbit (ISCO)?

The photon sphere at r_ph = 3GM/c² is inside the ISCO. For a non-rotating Schwarzschild black hole, the ISCO is at r_ISCO = 6GM/c² = 3 r_s. So the ISCO sits twice as far from the singularity as the photon sphere. Any massive particle orbiting between the photon sphere and the ISCO will spiral into the black hole.

Does every black hole have a photon sphere?

Every Schwarzschild (non-rotating) black hole has a photon sphere at exactly 3GM/c². Rotating Kerr black holes have analogous photon orbits, though they are not spherically symmetric. Charged (Reissner-Nordstrom) black holes also have a photon sphere, but its radius depends on the charge. The photon sphere is a universal feature of compact objects described by general relativity.

What is the shadow size of a black hole compared to its photon sphere?

The apparent shadow radius seen by a distant observer is r_shadow = 3√3 GM/c² = √3 × r_ph, which is approximately 2.6 times the Schwarzschild radius. This shadow is larger than the event horizon because the photon capture cross-section includes trajectories that loop around the photon sphere. The photon ring marks the bright edge of this shadow.

Can neutrinos or gravitational waves orbit at the photon sphere?

Massless particles (zero rest mass) travel on null geodesics and can orbit at the photon sphere, just like photons. Gravitational waves are also massless and follow the same null geodesics in the geometrical optics limit. Neutrinos have extremely small but non-zero mass, so their orbits are timelike rather than null, meaning they orbit slightly outside the photon sphere for the same angular momentum.

📌 Quick Tips

💡The photon sphere is always at 1.5 times the Schwarzschild radius, regardless of the black hole mass. For a 10 solar mass black hole, r_ph is 44.3 km while r_s is 29.5 km.
💡Light at the photon sphere is in an unstable orbit. Any small inward perturbation sends the photon into the black hole; any outward perturbation sends it escaping to infinity.
💡The black hole shadow seen by the Event Horizon Telescope is bounded by the photon capture cross-section, which is closely linked to the photon sphere geometry, not the event horizon itself.
💡For rotating (Kerr) black holes, the photon sphere splits into a prograde and retrograde orbit. This calculator applies to the non-rotating Schwarzschild case only.
💡The photon orbit period scales linearly with mass: a 1 solar mass compact object has a period of about 161 microseconds, while a billion solar mass black hole has a period of roughly 4.6 hours.