Black Holes & Schwarzschild Geodesics: Photon Sphere & ISCO

Q: What is the Schwarzschild radius?

The Schwarzschild radius r_s = 2GM/c² is the critical radius below which light cannot escape a given mass. For the Sun it is about 3 km; for Earth about 9 mm; for the supermassive black hole M87* about 120 AU. Karl Schwarzschild found this exact solution to Einstein's field equations in 1916, just weeks after general relativity was published.

Q: What is the difference between the photon sphere and the ISCO?

The photon sphere sits at 1.5 Schwarzschild radii, where light itself can orbit in an unstable circle — this is what creates the glowing ring seen in Event Horizon Telescope images. The innermost stable circular orbit (ISCO) for massive matter sits farther out, at 3 Schwarzschild radii for a non-rotating black hole; accretion disks end there because matter spirals inward rapidly once inside it.

Q: What did the Event Horizon Telescope actually image?

In April 2019 the EHT released the first image of a black hole's shadow: the supermassive black hole M87*, 6.5 billion solar masses, 55 million light-years away. Eight radio telescopes on four continents combined via very long baseline interferometry to resolve a bright ring about 42 microarcseconds across — matching the predicted 5.2 Schwarzschild-radii shadow diameter — around a darker central shadow.

The Schwarzschild radius

Karl Schwarzschild found the first exact solution to Einstein's field equations in 1916, just weeks after general relativity was published. For any mass M there is a critical radius below which light cannot escape:

r_s = 2GM / c²

Sun (1 M☉)  → ~3 km        Earth → ~9 mm
M87* (6.5×10⁹ M☉) → ~120 AU

Stars roughly 8–20 solar masses that run out of fusion fuel collapse in under a second; if the remnant core exceeds about 3 M☉ (the Tolman-Oppenheimer-Volkoff limit) it forms a black hole rather than a neutron star. Supermassive black holes, millions to billions of M☉, are found at the centres of almost all large galaxies and grow by accretion and mergers over cosmic time.

live demo · a precessing geodesic near a compact mass● LIVE

Photon sphere, ISCO, and orbital precession

General relativity predicts that even light curves in a strong gravitational field. At exactly 1.5 Schwarzschild radii — the photon sphere — a photon can circle the black hole indefinitely, but the orbit is unstable: a tiny perturbation sends it spiralling in or flying out. Massive matter has its own limit farther out, the innermost stable circular orbit (ISCO) at 3 Schwarzschild radii for a non-rotating black hole — accretion disks end there, since anything crossing inward spirals in rapidly.

Photon sphere:  r_ph  = 1.5 · r_s
ISCO:           r_ISCO = 3.0 · r_s
Observed shadow diameter ≈ 5.2 · r_s

Orbits between these two radii precess — their ellipses rotate slowly with each pass, a relativistic effect too small to notice around the Sun but dramatic near a compact mass. Time dilation is equally extreme: a distant observer sees a clock hovering near the event horizon tick infinitely slowly, while the infalling observer crosses the horizon in finite proper time and simply cannot report what they find inside.

Hawking radiation and the EHT image

Stephen Hawking showed in 1974 that black holes are not entirely black: quantum field theory in curved spacetime predicts thermal emission at a temperature T_H = ħc³/(8πGMk_B), around 6×10⁻⁸ K for a solar-mass hole — so faint that evaporation would take roughly 10⁶⁷ years, far longer than the age of the universe. In April 2019 the Event Horizon Telescope — eight radio telescopes on four continents combined via very long baseline interferometry — released the first image of a black hole's shadow: M87*, 6.5 billion solar masses, 55 million light-years away, showing a bright ring about 42 microarcseconds across (matching the predicted 5.2 r_s shadow) around a darker central shadow.

Frequently asked questions

What is the Schwarzschild radius?

r_s = 2GM/c² is the critical radius below which light cannot escape a given mass — about 3 km for the Sun, 9 mm for Earth, and ~120 AU for the supermassive black hole M87*.

What is the difference between the photon sphere and the ISCO?

The photon sphere at 1.5 r_s is where light itself can orbit unstably, creating the EHT's glowing ring. The ISCO at 3 r_s is the innermost stable orbit for massive matter — accretion disks end there.

What did the Event Horizon Telescope actually image?

The 2019 image of M87*'s shadow — a bright ring about 42 microarcseconds across, matching the predicted 5.2 Schwarzschild-radii shadow diameter, produced by eight radio telescopes combined via VLBI.

Try it live

Everything above runs in your browser — open Schwarzschild Geodesics and watch orbits precess, approach the photon sphere, cross the ISCO, and spiral inward with real RK4 integration.

▶ Open Schwarzschild Geodesics simulation