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If a photon is moving past a black-hole, what is the minimum size of the black-hole that will capture the photon and prevent the photon from going onto orbit around the mass? Do black-holes exist that have a stream of photons orbiting them?

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    $\begingroup$ Your question is a little bit unclear, I suggest to give it a more descriptive title. $\endgroup$ – peterh Jan 21 '18 at 22:17
  • $\begingroup$ I'm a little weak on the physics, but photons can't orbit black holes in the traditional sense of an orbit, they will either bend but escape or spiral in towards the black hole. A true orbit of a photon around a black hole is essentially impossible, unless it's nearly perfectly lined up and that kind of perfect-precision is very rare. $\endgroup$ – userLTK Jan 21 '18 at 23:55
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There isn't really a minimum size. All black holes have an event horizon from which nothing, not even light can escape. The (Schwarzschild) radius of this event horizon is 3km for a black hole of a solar mass and scales linearly with mass. There is a region just outside the event horizon called the photon sphere, within which light can briefly orbit the black hole, but not in a stable fashion. It will either fly off in some other direction or spiral into the black hole.

You can check out this Physics SE question to see a derivation of the impact parameter (the effective radius at which light will be absorbed) and you will see that it is a small numerical factor ($\sqrt{27/4}$ for a non-rotating black hole) times the Schwarzschild radius (the event horizon of a non-rotating black hole), which is indeed proportional to the black hole mass.

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