Brown ring around outside of my limbal ring, what is this?

Racial melanosis: Pigmented conjunctiva (the white of the eye) is most often called racial melanosis in dark-skinned individuals and is normal, but not so much in fair complexions, in which conjunctival melanoma is possible. In the cornea a brown ring could be a Kayser-Fleischer ring, a sign of copper deposition in Wilson's disease (a rare disorder). A serum ceruloplasmin level is diagnostic.