As Far as the Eye Can See

We’ve all heard the idiom “As far as the eye can see,” but just how far is this?  If you’re standing on the Earth, how far can you see?

There are many different ways to interpret my original question, so I should have been more specific.  As a friend wrote,

“I frequently see M31 (Andromeda galaxy) which is about 2 million light years. I did NOT see the GRB that was bright enough to be seen -5 billion light years- (visible only for a few seconds).”

I admit, I wasn’t thinking astronomically when I first posted the question.  Since the age of the universe is thought to be 14 billion years, it stands to reason that the absolute farthest the eye could see is 14 billion light years¹.  I was thinking more terrestrially.  If we confine ourselves to the quasi-two-dimensional world that is the surface of the Earth, how far can the eye see?

Consider the Earth to be a sphere of radius R_E = 6400 km.  The eyes of a standing person might be some distance H = 1.7 m off the ground.  By constructing a triangle (see diagram), the Pythagorean theorem predicts the equation

(R_E + H)² = R_E² + X²,

which can be rearranged to solve for the farthest terrestrial distance X one can see

X = [ (R_E + H)² - R_E² ]^1/2

= [ (6400 km + 1.7 m)² – (6400 km)² ]^1/2

= 4.7 km.

This is the distance at which ships start to sink on the horizon.  Note that the higher up you go (i.e. larger H), the farther you can see.  At the top of Mount Everest, you can see 336 km in each direction.

[1] A light year is the distance light travels in one year, so any object further away than this could not be seen since the light would not have had time to reach you.