Using a refractor telescope at 200x, how close will an object appear at four miles away? Is there a table out there enabling you to calculate apparent seeing by filling in known variables?
This is an interesting question!
Let's first ask; What does "200x" mean exactly?
If we set a photocopy machine to 1.5x or 2x we get a paper copy that we can set next to the original and compare them directly. With a ruler we can confirm that something that's 1 inch on the original is 1.5 inches or 2.0 inches on the copy.
But how big or how much closer something looks is a more subjective because our vision system is not flat. Our vision works with angles rather than distances.
For small angles like several degrees or so, we can use the small angle approximation fairly well, so let's go with that.
When a telescope is focused far away, near infinity, perhaps 100 times farther away than the length of the telescope itself, we can treat 200x as the angular amplification of 200.
Something in the distance (e.g. 4 miles away) that's 0.01 degrees wide will appear 200 x 0.01 = 2 degrees wide through the telescope.
If we move that object 200 times closer, it will also appear almost exactly 2 degrees wide.
So as long as we are considering things far away and small, we can say that something that appears 200x larger through the telescope would appear to be the same size as if it were 200x closer.
4 miles times 5280 = 21,120 feet. Divide that by 200 and we get 105.6 feet.
At 21,120 feet I would appear about 0.015 degrees tall. Viewed through a 200x telescope I would appear about 3 degrees tall. If I moved to 105.6 feet away, I would also appear about 3 degrees tall.
This works as long as we use the small angle approximation and our telescope is focused on things far away, say 100 times the length of the telescope or farther. The same 200x telescope would not necessarily provide exactly 200x angular amplification if they eyepiece were pulled out so far that one could focus on something across the room.