How can I see a cables from 55 miles away?

[Annie Bynnol]

I live on a hill at 700 feet and I can see the aircraft warning lights of Belmont TV transmitter at 1 100 feet which is 55 miles away with my naked eye.
With a small pair of binoculars I can see the mast which is tubular steel.
In the morning I can sometimes see the several sets of cables that hold it up.
The Sun rises in that direction but it is not in silhouette.
How can I see these cables from such a distance?

 

[PhanthomJay]

What do you have for binocs 10 X 50? That would make the tower appear 5 miles away and the larger the second number, the greater the brightness. I'm guessing that the cables might be rather large in diameter (say over 1 inch or 25 mm) and that they are aluminum coated steel...the aluminum is very shiny and reflective...the sun might be hitting it at an optimum angle against a darkened sky...that is nonetheless quite a distance to see them...I'm no expert here on binoculars, so just guessing based on the reflective properties of aluminum. If the cables were rusting steel, forget about seeing them.

 

[Andy Resnick]

<snip>How can I see these cables from such a distance?

In imaging, there is an important difference between 'detecting' and 'resolving' an object. Detection of sub-resolution objects is limited only by the signal-to-noise ratio, for example bright stars against the night sky, while resolving an object is limited by diffraction- can you tell if there is a single star as opposed to two closely spaced stars?

 

[marcusl]

If you see it in the morning only, it may be that you are seeing the sun reflecting off of the cables. You are not resolving the cables themselves in this case, merely seeing brightness against a dimmer background. In the same way, I bet that you are not able to see the lamps when they are off but you can see red light when they are on. This is another way of expressing what Andy Resnick said above.

The apparent size of the light you see is related to the size of the lens you use to look, which we can estimate. To use the example of 10x50, the 50 refers to a 50 mm lens (this is a set of large binoculars). The angular resolution is\beta=1.22*\lambda/D in radians, where λ is the wavelength of light and D is the lens diameter. Taking λ=500 nm and D=0.05 m gives β=1.2e-5 radians. Your tower is R=88 km away, and the apparent width of the cable is Rβ=1.1 m with 10x50 binoculars. That cable looks approximately 3.5 feet thick, even though it's probably a fraction of an inch in reality. A smaller lens would make it look even thicker.

In other words, the light you are seeing shows up because it's bright, and your small lens makes it look like the smallest object the lens can resolve at that distance--which turns out to be quite large!