Expert: Courtney Seligman - 5/19/2009

Question
I'd like to know how far away I can see objects in the daytime sky, such as balloons. Assuming an endless sky with daytime lighting conditions and the object having high contrast. For example, if a round balloon 100' in diameter is floating in the sky, how many miles away would it have to get before it was too small to see with the naked eye?

Is there an equation to calculate this?  

Answer
There are two possible answers to this question, depending upon what you mean by 'seeing' it. One, which would pertain to stars, meteors and other bright objects which are seen not by their size but by their light, and another, which would pertain to objects 'big enough' to see as having some actual size. Since you specify the size of the balloon, I presume you are referring to the latter.

The smallest object which can be seen by the human eye is about 1/50th of a degree in size. At that apparent size, it would appear to be a point. To see it as having some real diameter, it would have to be a little larger -- perhaps 1/30th of a degree, to be sure that it is the size of the object you are looking at, rather than just a point of light.

In this small range of sizes, the distance at which the object is discernible is about 57 times the reciprocal of the size in degrees, which means in this case, about 1500 times the size of the object to be certain of its not being a point, to perhaps twice that distance, if you don't need to be quite so certain. For a 100-foot diameter balloon, that would be about 30 miles to be reasonably certain of seeing it as other than a point, with the certainty of detection falling to zero between 50 and 60 miles away.