Why is the ISCO at 6M, not at the horizon?

Circular orbits require a minimum of V_eff; as L drops, the minimum and maximum merge and the inflection defines r_ISCO = 6M. Below that, no stable circular timelike orbit exists even though the horizon is at 2M. A frequent misconception is that matter can stably circle just outside the horizon in Schwarzschild spacetime.

What is special about r = 3M for light?

Null geodesics admit an unstable circular photon orbit at r = 3M. Photons on that sphere can skim many times before falling in or escaping, forming the bright ring around the shadow. It is not a solid surface—only an unstable orbit in the effective potential.

What happens if I lower L for a massive particle?

The stable outer circular orbit and unstable inner one approach each other and annihilate at the ISCO. For still smaller L there is no circular-orbit pair—plunging trajectories dominate. That is why thin discs truncate near 6M for a non-spinning black hole in this model.

ISCO & Photon Sphere (V_eff)

Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) test particles in geometric units. Sliding angular momentum L collapses the stable / unstable circular pair into the innermost stable circular orbit r_ISCO = 6M (the inner edge of accretion discs); for photons the unstable photon sphere r = 3M defines the inner ring of black-hole shadow images.

About this model

In Schwarzschild geometry the radial motion of test particles follows an effective potential V_eff(r) that depends on angular momentum L (geometric units). For massive (timelike) particles, sliding L merges the stable and unstable circular-orbit extrema until they meet at the innermost stable circular orbit r_ISCO = 6M—the inner edge of thin accretion discs. For photons (null geodesics) an unstable circular orbit sits at the photon sphere r = 3M, which sets the bright photon ring in black-hole shadow images. Assumptions: non-spinning hole, test particles, and a 1D effective-potential picture without full ray-traced images. Vary L to watch circular-orbit pairs coalesce at the ISCO and compare the photon potential barrier at 3M.

Who it's for: Advanced general relativity, high-energy astrophysics, and black-hole phenomenology courses.

Key terms

effective potential
ISCO
photon sphere
Schwarzschild metric
circular orbit
black-hole shadow

How it works

Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) geodesics, in geometric units G = c = M = 1. Slide angular momentum L to watch stable / unstable circular orbits collide at the innermost stable circular orbit r_ISCO = 6M; for photons the unstable photon sphere sits at r = 3M, the inner ring of black-hole shadows.

Frequently asked questions

Why is the ISCO at 6M, not at the horizon?: Circular orbits require a minimum of V_eff; as L drops, the minimum and maximum merge and the inflection defines r_ISCO = 6M. Below that, no stable circular timelike orbit exists even though the horizon is at 2M. A frequent misconception is that matter can stably circle just outside the horizon in Schwarzschild spacetime.
What is special about r = 3M for light?: Null geodesics admit an unstable circular photon orbit at r = 3M. Photons on that sphere can skim many times before falling in or escaping, forming the bright ring around the shadow. It is not a solid surface—only an unstable orbit in the effective potential.
What happens if I lower L for a massive particle?: The stable outer circular orbit and unstable inner one approach each other and annihilate at the ISCO. For still smaller L there is no circular-orbit pair—plunging trajectories dominate. That is why thin discs truncate near 6M for a non-spinning black hole in this model.

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About this model

How it works

Frequently asked questions

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About this model

How it works

Frequently asked questions