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**Physics, Astronomy & Cosmology / Why the discrepancy between orbital velocity and photon sphere radius?**

« **on:**12/09/2010 12:44:30 »

The formula for an approximation of orbital velocity around a mass M (where the mass of the orbiting 'body' is negligible) is:

ref: http://en.wikipedia.org/wiki/Orbital_velocity

...and the formula for the Photon Sphere radius is:

ref: http://en.wikipedia.org/wiki/Photon_sphere

...and finally, the formula for the Schwarzchild Radius (of a non-rotating BH) is:

ref: http://en.wikipedia.org/wiki/Schwarzschild_radius

Where (using mks units):

Using these values, the Schwarzchild Radius for an Earth sized BH is: 0.0088718 (m) i.e. a little under 9 mm

...and using this as the radius to find the orbital velocity we get 211985280 (m/s), which is < 'c'

However, the radius of the Photon sphere, where the orbital velocity must be 'c', turns out to be: 0.01330768 (m), which is well outside the Schwarzchild radius.

So can anyone explain why a negligible mass object orbiting a BH just outside its event horizon will be travelling more slowly than the light in the Photon Sphere, which is orbiting it quite a bit further away?

ref: http://en.wikipedia.org/wiki/Orbital_velocity

...and the formula for the Photon Sphere radius is:

ref: http://en.wikipedia.org/wiki/Photon_sphere

...and finally, the formula for the Schwarzchild Radius (of a non-rotating BH) is:

ref: http://en.wikipedia.org/wiki/Schwarzschild_radius

Where (using mks units):

*G*= the Gravitational constant = 6.674 x 10^{-11}*M/m*= the mass of the orbited body (we'll use the mass of the Earth here, just for fun) = 5.9736 × 10^{24}*c*= the speed of light = 299792458Using these values, the Schwarzchild Radius for an Earth sized BH is: 0.0088718 (m) i.e. a little under 9 mm

...and using this as the radius to find the orbital velocity we get 211985280 (m/s), which is < 'c'

However, the radius of the Photon sphere, where the orbital velocity must be 'c', turns out to be: 0.01330768 (m), which is well outside the Schwarzchild radius.

So can anyone explain why a negligible mass object orbiting a BH just outside its event horizon will be travelling more slowly than the light in the Photon Sphere, which is orbiting it quite a bit further away?