Photon Sphere Radius
Radius where gravity bends light into a circular orbit.
Formula
Live Calculator
MembersVariables
Example
It sits at 1.5 times the Schwarzschild radius
How it works
This formula gives the radius of the photon sphere around a non-rotating, uncharged black hole: the distance where light can move on a circular path if it is aimed just right. It comes from general relativity, not Newtonian gravity, because only curved spacetime can make photons follow such an unstable orbit. The radius is proportional to the mass, so more massive black holes have a larger photon sphere.
When to use it
Use this when analyzing light trajectories near a Schwarzschild black hole, especially in discussions of strong gravitational lensing, black hole shadows, or unstable photon orbits. It is also useful for comparing the scale of the photon sphere with the event horizon.
Worked example, step by step
Suppose a black hole has mass M = 10 times the mass of the Sun. Take G = 6.674×10^-11 N·m^2/kg^2, c = 3.00×10^8 m/s, and M_sun = 1.989×10^30 kg. Step 1: Convert the mass: M = 10 × 1.989×10^30 = 1.989×10^31 kg Step 2: Substitute into r_ph = 3GM/c^2 r_ph = 3 × (6.674×10^-11) × (1.989×10^31) / (3.00×10^8)^2 Step 3: Calculate the numerator: 3 × 6.674×10^-11 × 1.989×10^31 ≈ 3.983×10^21 Step 4: Calculate the denominator: (3.00×10^8)^2 = 9.00×10^16 Step 5: Divide: r_ph ≈ 3.983×10^21 / 9.00×10^16 ≈ 4.43×10^4 m Final answer: r_ph ≈ 4.4×10^4 m, or about 44 km
Common mistakes to avoid
- Using this radius for any black hole; it applies to the idealized non-rotating, uncharged case
- Confusing the photon sphere with the event horizon, which is smaller by a factor of 1.5 in Schwarzschild geometry
- Forgetting to keep units consistent, especially when using solar masses without converting to kilograms
Did You Know?
At the photon sphere, in principle you could see the back of your own head as light loops around the black hole.
Share this formula
More in Astronomy & Space
View allEscape Velocity
IntermediateMinimum speed to escape a body's gravity.
Orbital Velocity
IntermediateSpeed of a circular orbit.
Kepler's Third Law
AdvancedRelates orbital period to semi-major axis.
Schwarzschild Radius
AdvancedEvent horizon radius of a black hole.
Hubble's Law
AdvancedRecession velocity of galaxies with distance.
Stefan–Boltzmann Law
AdvancedLuminosity of a star from radius and temperature.