For a Schwarzschild black hole, the photon sphere lies at how many Schwarzschild radii?

For a Schwarzschild black hole, the photon sphere lies at 1.5 Schwarzschild radii.

A Schwarzschild black hole is nonrotating and uncharged. Its photon sphere is the spherical region where light can travel on circular paths around the black hole. The radius of that orbit is three times the Schwarzschild radius, so it lies at 1.5 Schwarzschild radii when distance is measured from the center using the horizon radius as the unit.

The distinction between these numbers causes frequent confusion. The Schwarzschild radius is the event-horizon radius, equal to 2GM/c². The photon sphere is outside the event horizon at 3GM/c², or three-quarters of the way farther out than the horizon radius itself. Light on the exact circular orbit is unstable: a tiny disturbance sends it either outward or into the black hole.

The photon sphere is also related to the black-hole shadow, but it is not identical to the shadow or the event horizon. Light rays passing near this region can loop around the hole and be strongly bent. For a rotating Kerr black hole, the photon-orbit structure becomes more complicated and depends on direction and spin.