gravity - How close does light have to be, to orbit a perfect sphere the size and mass of Earth?

The moon orbits Earth at about $380,\!000 \,\mathrm{km}$ away from it, at around $3,600 \,\mathrm{km}$ an hour.

I was thinking, with light traveling at $300,\!000 \,\mathrm{km/s}$, how close to earth (probably in the $\mathrm{nm}$ range is my guess) would light have to be to Earth to orbit it?

Update: After reading the below answers, here's my reasoning to why I thought it would be in the nanometre range.

I thought that if light was as close as possible to Earth (like, a planck length away or something), Earth's gravity would make it hit Earth immediately, but I forgot that the pull of gravity gets weaker when one is further away from Earth (i.e. at the surface of Earth is still somewhat "far away").

Answer

You can take the Newtonian expression for the orbital speed as a function of orbital radius and see what radius corresponds to an orbital speed of $c$, but this is not physically relevant because you need to take general relativity into account. This does give you an orbital radius for light, though it is an unstable orbit.

If the mass of your planet is $M$ then the radius of the orbit is:

$$ r = \frac{3GM}{c^2} $$

where $G$ is Newton's constant. The mass of the Earth is about $5.97 \times 10^{24}$ kg, so the radius at which light will orbit works out to be about $13$ mm.

Obviously this is far less than the radius of the Earth, so there is no orbit for light round the Earth. To get light to orbit an object with the mass of the Earth you would have to compress it to a radius of less than $13$ mm. You might think compressing the mass of the Earth this much would form a black hole, and you'd be thinking on the right lines. If $r_M$ is the radius of a black hole with a mass $M$ then the radius of the light orbit is $1.5 r_M$.

So you can only get light to orbit if you have an object that is either a black hole or very close to one, but actually it's even harder than that. The orbit at $1.5r_M$ is unstable, that is the slightest deviation from an exactly circular orbit will cause the light to either fly off into space or spiral down into the object/black hole.

If you're interested in finding out more about this, the light orbit round a black hole is called the photon sphere, and Googling or this will find you lots of articles on the subject.