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There are a lot of questions about photons orbitting a black hole, but I'm interested if a photon can orbit a galaxy. Our Milky Way for example has a radius of 52000 light years (according to Google) so a photon would need to travel 52000 years to complete a circle - perhaps enough time to bend its path even for weak gravity?

Likewise, is it possible for a photon to orbit around a galaxy cluster?

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No.

The milky way has a lot of mass — about 1.5 trillion times the mass of the sun.

For light to orbit it would have to follow a null geodesic, ie the photon sphere at 1.5 times the Schwarzschild radius. As light always travels at the speed of light, it can't go into any other orbit by going slower.

The Schwarzschild radius of the milky way would be about 0.5 light years, so the photon sphere would be 0.75 light years. This means that light could only orbit a body with a mass of 1.5-trillion-suns if that mass was compressed to a sphere with a radius less than 0.75 light years. Light could not orbit around the galaxy, because the galaxy has a radius of more than 0.75 lightyears.

A similar argument can be made for any stellar cluster.

And as Rob notes, even after you've collapsed the milky way into a black hole, it still wouldn't have stable photon orbits, for reasons given in the linked question.

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  • $\begingroup$ Are there no non-circular orbits for light? It can't change speed, but it can go farther and less far and change direction, which you'd think would lead to some kind of orbit. $\endgroup$
    – user253751
    Aug 25, 2022 at 18:25
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    $\begingroup$ No. It's an unstable equilibrium. If light is further out than the circular orbit, it's bent less, which means it goes further out, which means it gets bent less, and so on and so forth out to infinity. (Ditto in the other direction.) $\endgroup$
    – TLW
    Aug 25, 2022 at 19:04
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    $\begingroup$ If you squint a bit you can kind of consider gravitational lensing to be an odd sort of analogy to hyperbolic orbits... $\endgroup$
    – TLW
    Aug 25, 2022 at 19:04
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    $\begingroup$ @user253751: If you're familiar with the concept of the "effective potential" for central-force motion, the answer becomes clearer. As in Newtonian physics, you can construct effective potentials both for massless and for massive particles moving around a Schwarzschild black hole. The effective potential for massive particles can have a local minimum, and oscillations around that minimum correspond to non-circular orbits. The effective potential for massless particles only has a local maximum, meaning that the only bounded orbits are circular and unstable. $\endgroup$ Aug 26, 2022 at 15:00

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