Photon Sphere and ISCO

Photon Sphere and ISCO

The Schwarzschild geometry supports several special orbital radii beyond the event horizon. Two of the most important are the photon sphere and the innermost stable circular orbit (ISCO).

Photon Sphere

At $r_{\text{ph}} = \tfrac{3}{2} r_s = 3GM/c^2$, photons can orbit the black hole in unstable circular orbits. This radius is called the photon sphere. Although these orbits are unstable (a tiny perturbation sends the photon spiralling in or escaping), the photon sphere determines the apparent size of a black hole's shadow.

$r_{\text{ph}} = \frac{3GM}{c^2} = \frac{3}{2} r_s$

Black Hole Shadow

An observer far away sees a dark disk (the shadow) whose apparent radius is larger than the photon sphere due to gravitational lensing:

$r_{\text{shadow}} = 3\sqrt{3}\,\frac{GM}{c^2}$

This formula was used to predict the apparent size of M87* before the Event Horizon Telescope imaged it in 2019.

ISCO — Innermost Stable Circular Orbit

For massive particles, circular orbits are possible for $r > r_s$, but stable orbits only exist for:

$r_{\text{ISCO}} = \frac{6GM}{c^2} = 3 r_s$

Inside the ISCO, any perturbation causes matter to spiral inward. This radius sets the inner edge of accretion disks around black holes.

Your Task

Implement these functions with all constants defined inside each function:

• photon_sphere_radius(M) — returns $r_{\text{ph}} = 3GM/c^2$
• isco_radius(M) — returns $r_{\text{ISCO}} = 6GM/c^2$
• shadow_radius(M) — returns $r_{\text{shadow}} = 3\sqrt{3}\,GM/c^2$