Photon Sphere Radius Calculator

Calculator

Calculation mode From mass From Schwarzschild radius

Choose the input you already know.

Mass

Enter a positive value.

Mass unit kg Solar masses (M☉) Earth masses (M⊕) Jupiter masses (M♃)

Common astrophysical mass choices.

Schwarzschild radius

Use a positive radius value.

Radius unit m km cm mm

Pick the unit matching your source.

Decimal places 2 3 4 5 6 7 8 9 10

Controls displayed precision.

Calculate Reset

Formula used

For a Schwarzschild black hole (non-rotating, uncharged), the photon sphere radius is:

• Gravitational radius: r_g = GM / c²
• Schwarzschild radius: r_s = 2GM / c² = 2r_g
• Photon sphere radius: r_ph = 3GM / c² = 3r_g = 1.5r_s

Here, G is the gravitational constant, c is the speed of light, and M is mass.

How to use this calculator

1. Select a calculation mode: from mass or from Schwarzschild radius.
2. Enter your value and choose its unit.
3. Pick decimal places for output formatting.
4. Press Calculate to display results above the form.
5. Use Download CSV or Download PDF after calculation.

Example data table

Values are rounded and assume the Schwarzschild case.

Article

1) What the photon sphere represents

The photon sphere is the radius where light can orbit a compact object in unstable circular paths. In the Schwarzschild case, these orbits exist at a single scale set by mass, so the radius is a clean benchmark for gravitational lensing strength, shadow size, and how quickly photons are deflected.

2) Core scaling with mass

This calculator uses r_ph = 3GM/c², so the photon sphere grows linearly with mass. Using standard constants, one solar mass gives r_ph ≈ 4.430 km, while the Schwarzschild radius is r_s ≈ 2.953 km. Their ratio is always 1.5 in this non-rotating model.

3) Relationship to the event horizon

The event horizon sits at r_s, but the photon sphere lies outside it at 1.5r_s. Photons passing near r_ph can loop around the object, producing multiple images and high magnification. Because the orbit is unstable, tiny perturbations send light either outward or into the horizon.

4) Practical numbers for stellar black holes

For a 10 M_☉ black hole, the calculator returns r_s ≈ 29.533 km and r_ph ≈ 44.300 km. These lengths are comparable to the size of a small city, which is why timing signals and gravitational waves are essential for probing such compact systems rather than direct imaging.

5) Supermassive examples and unit conversion

For Sagittarius A* (about 4.3 million solar masses), r_ph is roughly 19,049,039 km, which is ≈ 0.127 AU. For an M87*-scale mass near 6.5 billion solar masses, r_ph is about 28,795,058,975 km (≈ 192.48 AU). The calculator reports meters, kilometers, and AU for quick comparison.

6) Using Schwarzschild radius as an input

If you already know r_s from a simulation or a dataset, choose the Schwarzschild mode. The tool applies r_ph = 1.5r_s and also back-calculates the equivalent mass using M = r_sc²/(2G). This is useful for validating tables and unit conversions.

7) Limits of the model

Real astrophysical black holes can rotate. Rotation shifts the photon orbit radii and makes them depend on direction (prograde vs retrograde), so a single radius no longer tells the whole story. Charge is typically negligible in nature, but it also modifies the photon sphere in idealized solutions.

8) How to interpret outputs for reporting

Use r_ph to communicate “light trapping” scale, and include r_s to anchor the horizon size. The fixed ratio r_ph/r_s = 1.5 is a quick consistency check. Export CSV for datasets and print to PDF when preparing lab notes, articles, or classroom material.

FAQs

1) Is the photon sphere the same as the event horizon?

No. The event horizon is at r_s. The photon sphere is outside it at 1.5r_s, where light can orbit in unstable paths.

2) Why does the ratio r_ph/r_s always equal 1.5 here?

In the Schwarzschild model, r_s = 2GM/c² and r_ph = 3GM/c². Dividing gives a constant 3/2, independent of mass.

3) Can light orbit forever at the photon sphere?

In theory, a perfectly tuned photon could orbit, but the orbit is unstable. Any tiny disturbance makes the photon escape outward or fall inward.

4) Which input mode should I use?

Use “From mass” when you know the object’s mass. Use “From Schwarzschild radius” when your source provides r_s directly and you want r_ph and mass consistency checks.

5) Does rotation change the photon sphere radius?

Yes. Rotating black holes have different photon orbit radii depending on direction. This calculator targets the non-rotating case, which is a common baseline for comparisons.

6) Why do you show AU for very large masses?

For supermassive objects, radii can reach billions of kilometers. AU provides a familiar astronomical scale and makes comparisons to planetary distances easier to read.

7) Are the constants adjustable?

This version uses standard physical constants internally. If you need alternate constants for a specific convention, you can edit the constant values near the top of the file.