Extended luminous range?

Question: You’re offshore on a perfectly clear night, preparing to make a landfall, and figure out that a lighthouse has a luminous range of, say, 11 miles and a geographic range of 20 miles. Inside 20 miles, the geographic range is such that you should begin to see the light. You could then do what is known as “bobbing the light”setting dividers to the combined geographic range of height of eye and height of light and strike this distance on the chart so that it crosses your track line. But, of course, you can only do this if the luminous range is great enough. If you use a pair of powerful binoculars, can you increase a light’s luminous range? How about using a telescope?

Matt Drukker Williams College, Williamstown, Mass.

Answer: Will using binoculars increase the luminous range of the beacon?

Unequivocally, yes! Boy, that was easy. Unfortunately, for every complex question there are any number of simple, elegant, and incorrect answers. If you wish to know more, we will have to wander through a whole world of ambiguity. The following is just a start.

The mathematics used to calculate luminosity differs with whether the light is a “point source” or an “extended object.” A point source of light is so tiny that it will not appear any larger regardless of how much we attempt to magnify it. Conversely, an extended object is one that can be magnified. The size of lighthouse beacons vary quite a bit. Therefore, we cannot simply insert a “lighthouse” as part of the equation. One would need to know the size of the lens and the luminosity of the lamp, among other things. Some lenses are two feet in diameter; others may be more than 12 feet high. Some emit 100,000 candle power; others more than 20 million candle power.

The Fresnel lenses used in lighthouse beacons were created to keep the beam collimated and bright at the greatest possible distance. This would work great in a vacuum. But the fact is that no matter how “perfect” the atmosphere is to the mariner, it is not perfect to the light rays trying to get through it, and some image brightness will be lost through scatter and absorption.

When viewing extended objects, brightnessat a given magnificationincreases with aperture. A binocular with a 50-millimeter object lens will have the light grasp of 51 eyes, and a 7×50 binocular will have a considerably brighter image than a 7×35. (It is a myth that the light grasp of a binocular is that accumulated by both objective lenses. There is an overall enhancement of acuity. However, this is because the ability to use both eyes increases contrast as much 40 percent.)

It would be great if we could end the story here. Unfortunately, just as the binocular objective allows for increased brightness, so too does its magnification decrease it. A beacon emits only so much light. Each time the image size is increased by a factor of two, the brightness is decreased by a factor of four. Thus, you may want to take “powerful” out of the picture. So, where do we stand at 11 miles?

I have called on Barlow Pepin, a former editor with Sky & Telescope magazine, to do the math. His calculations were based on viewing the beacon as a point source having an unaided visual range of 11 miles with a 7×50 binocular. His assessment is as follows:

“We are told that the lighthouse is invisible to the unaided eye at more than 11 miles distance. Assuming we are looking at the beacon as a point source of lighti.e., not having a visible angular sizeand taking its brightness at a one-mile distance as a standard, the beacon’s brightness at 11 miles (under the inverse-square law) would have dropped to approximately 0.007 of its brightness at one mile. By rough extrapolation using the inverse square, the limit of the light’s visibility would be extended to about 79 miles by viewing it with a 7X50 binocular.”

Thus, the binocular would increase the visibility from 11 miles to the limit of the light’s geographic range of 20 miles. The light wouldn’t be visible beyond the geographic range because the curvature of the earth would block the light from view. But “bobbing” the light is difficult enough with the naked eye if there is any motion to the boat, let alone peering through the small field of view of binoculars. Telescopes are more powerful than binoculars, but the field of view would be so small as to render them useless on a rolling platform the size of a super tanker or an aircraft carrier. One saving grace is that distortion of the light might affect range but not its bearing.

You see, “yes” wasn’t such a bad answer after all.

By Ocean Navigator