What Is the Photon Sphere, and What Did the Event Horizon Telescope Image Show?

Light normally travels in straight lines. Near a black hole, gravity can bend it so strongly that it curves all the way around, and at one special distance a beam of light can, in principle, circle the black hole forever. That surface is called the photon sphere, and it is one of the three circles on our Three Radii of a Black Hole poster, alongside the event horizon and the innermost stable orbit. It is also the key to understanding the famous orange ring in the Event Horizon Telescope images. This post builds on our explainer Do Black Holes Evaporate?.

What is the photon sphere?

For a non-rotating black hole of mass M, the natural length scale is the gravitational radius

r_g = GM/c²

where G = 6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the Newtonian constant of gravitation (NIST^[1]), M is the mass in kilograms, and c = 299 792 458 m/s is the speed of light (NIST^[2]). The event horizon, the point of no return, sits at the Schwarzschild radius R_S = 2r_g. The photon sphere sits further out:

r_ph = 3GM/c² = 1.5 R_S

At that radius, a photon moving exactly sideways is bent onto a circular orbit. The orbit is unstable. Nudge the photon inward and it spirals into the black hole; nudge it outward and it escapes. The physicist J. L. Synge analysed this escape problem for photons near very compact stars in a 1966 paper (Synge 1966^[3]).

Where does the 3 come from?

In Einstein’s general relativity, light moving around a non-spinning mass feels an effective potential proportional to (1 − 2GM/(rc²))/r². Setting its slope to zero gives a single maximum at r = 3GM/c². A maximum means an unstable balance point, like a ball perched on a hilltop, which is why the photon orbit cannot last.

Why does the photon sphere set the size of the shadow?

Imagine light rays coming toward the black hole from far away, each passing at a different sideways offset b, called the impact parameter. The first paper of the Event Horizon Telescope (EHT) Collaboration on M87 lays out what happens. A non-rotating black hole “has a photon capture radius R_c = √27 r_g”, which is larger than the Schwarzschild radius. Photons with b < R_c “are captured and plunge into the black hole”; photons with b > R_c “escape to infinity”; and photons with b = R_c “are captured on an unstable circular orbit and produce what is commonly referred to as the lensed ‘photon ring’” (EHT Collaboration 2019, Paper I^[4]).

The two numbers fit together. A photon on the circular orbit at r = 3r_g has impact parameter

b_c = r_ph / √(1 − 2GM/(r_ph c²)) = 3r_g / √(1/3) = √27 r_g ≈ 5.2 r_g

So the dark region we see is wider than the photon sphere itself, because gravity magnifies it like a lens. NASA describes the result: a dark zone called the shadow, “roughly twice as big as the black hole’s actual surface” (NASA^[5]). For the shadow’s diameter, 2√27 r_g ≈ 10.4 r_g, compared with an event horizon diameter of 4r_g, the factor is about 2.6.

Gravitational bending of light is the same effect that makes Einstein rings around distant galaxies, which we cover in Einstein Rings and Gravitational Lensing.

What did the EHT image of M87* show?

In 2019 the EHT, a global array of radio telescopes working together as one Earth-sized interferometer at 1.3 mm wavelength, released the first image of a black hole, the giant one at the centre of galaxy M87. The collaboration reported “an asymmetric bright emission ring with a diameter of 42±3 micro-as”, around a central brightness depression, and derived a mass of M = (6.5 ± 0.7) × 10⁹ solar masses (EHT Collaboration 2019^[6]). A microarcsecond (μas) is a millionth of an arcsecond, which is 1/3600 of a degree.

A worked check

The EHT expresses the black hole’s size as an angle on the sky, θ_g = GM/(Dc²), where D is the distance. For M87 they measured θ_g = 3.8 ± 0.4 μas, using a distance of 16.8 ± 0.8 Mpc (EHT Paper I^[4]). The predicted shadow diameter is then

2√27 θ_g ≈ 10.4 × 3.8 μas ≈ 39.5 μas

That is close to the measured 42 ± 3 μas ring. The paper reports a ratio of ring diameter to θ_g of 10.7 to 11.5, slightly more than 10.4, because the bright ring is glowing gas just outside the shadow’s edge, not the edge itself.

Is the bright ring the photon sphere?

Not exactly, and this is a common confusion. The orange ring is mostly hot, glowing plasma lensed around the black hole. Inside it, general relativity predicts a much thinner feature, the photon ring, made of light that looped the hole one or more times. NASA’s 2019 visualization explains that this “photon ring” is “composed of multiple rings, which grow progressively fainter and thinner, from light that has circled the black hole two, three, or even more times before escaping” (NASA^[7]).

Michael Johnson and colleagues showed that these subrings are “an infinite sequence of self-similar subrings”, and that they should leave “strong and universal signatures on long interferometric baselines”, which could allow precise measurements of mass and spin in future (Johnson et al. 2020^[8]). The same paper describes the 2019 image as “dominated by a bright, unresolved ring”, so picking out the subrings is a job for future observations with longer baselines.

What about our own galaxy’s black hole?

In 2022 the EHT published an image of Sagittarius A*, at the centre of the Milky Way. It showed “a bright, thick ring with a diameter of 51.8 ± 2.3 μas”, consistent with a black hole of about 4 × 10⁶ solar masses (EHT Collaboration 2022^[9]). Sgr A* is more than a thousand times less massive than M87*. Because the angular size scales as θ_g = GM/(Dc²), similar ring sizes on the sky mean Sgr A* must be correspondingly closer. For more black-hole physics to wear, see our Relativity & Cosmos tees, including Spaghetta, the spaghettification tee.

FAQ

Could you stand on the photon sphere and see the back of your head?

In principle, light sent exactly sideways at r = 3GM/c² would circle around. In practice the orbit is unstable, and the tidal forces and radiation near a real black hole make it a thought experiment.

Do spinning black holes have a photon sphere?

They have a more complicated photon region. EHT Paper I notes that for spinning (Kerr) black holes, the capture radius changes with the ray’s orientation, so the shadow is not necessarily a perfect circle.

Is the photon sphere related to Hawking radiation?

They are separate ideas. Hawking radiation is a quantum effect at the horizon; the photon sphere is pure classical general relativity. See our post The Black Hole Information Paradox for the quantum side.

References

1. NIST CODATA 2022 recommended values: Newtonian constant of gravitation. https://physics.nist.gov/cgi-bin/cuu/Value?bg
2. NIST CODATA 2022 recommended values: speed of light in vacuum (exact). https://physics.nist.gov/cgi-bin/cuu/Value?c
3. J. L. Synge (1966), “The Escape of Photons from Gravitationally Intense Stars,” MNRAS 131, 463–466 (DOI 10.1093/mnras/131.3.463; metadata verified via Crossref). https://doi.org/10.1093/mnras/131.3.463
4. Event Horizon Telescope Collaboration (2019), Paper I full text (arXiv:1906.11238v1), §2 on the photon capture radius. https://arxiv.org/pdf/1906.11238
5. NASA Science, “Black Hole Anatomy”. https://science.nasa.gov/universe/black-holes/anatomy/
6. Event Horizon Telescope Collaboration (K. Akiyama et al.) (2019), “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” ApJL 875, L1 (arXiv:1906.11238; DOI 10.3847/2041-8213/ab0ec7). https://arxiv.org/abs/1906.11238
7. F. Reddy, “NASA Visualization Shows a Black Hole’s Warped World,” NASA (25 September 2019). https://www.nasa.gov/universe/nasa-visualization-shows-a-black-holes-warped-world/
8. M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong, S. Hadar, D. Kapec, R. Narayan, A. Chael, C. F. Gammie, P. Galison, D. C. M. Palumbo, S. S. Doeleman, L. Blackburn, M. Wielgus, D. W. Pesce, J. R. Farah & J. M. Moran (2020), “Universal interferometric signatures of a black hole’s photon ring,” Science Advances 6, eaaz1310 (arXiv:1907.04329). https://arxiv.org/abs/1907.04329
9. Event Horizon Telescope Collaboration (K. Akiyama et al.) (2022), “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” ApJL 930, L12 (arXiv:2311.08680; DOI 10.3847/2041-8213/ac6674). https://arxiv.org/abs/2311.08680

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.