This has been part of the science of navigation since the earliest days, when the first vessels sailed off and dropped below the horizon to those waving goodby on shore. In fact, in crystal clear air (no pollutants) and total absence of clouds, it also does not matter if the tip has a light on it or not. That light will dissapear from sight when it passes beyond your geographic range to the light.
Note this says your geographic range, not the geographic range of the object, because this geographic range (Rg) depends not only on the height of the object above the water (h), but also the height of the observer's eye above the water (HE) as well. Those standing on shore will lose sight of the ship sailing away before those watching from a nearby hill top.
The question at hand is: what is the formula for Rg, and how does it depend on h and HE, and maybe other things as well.
If we expand the conditions above to say not only spherical earth (of average radius R), but also no atmosphere, then we can compute this distance with basic trig to get:
Rg = sqrt (2R) x [ sqrt (h) + sqrt (H) ].
The tricky part is the units. We (American navigators) want h to be in feet and Rg in nmi, so we must tweak the conversion factors.
We know the earth radius because we know the circumference (c) is 360º and each degree is 60 nmi, and c = 2 pi R, so R = 360 x 60/2pi = 3438 nmi. We also know that there are 6074 ft per nmi (the exact answer is 1852 meters). Then we use feet everywhere and convert the final answer to nmi to get:
Thus Rg = (1/6074) x sqrt (2 x 3438 x 6074) x [ sqrt (h) + sqrt (H) ], or
Rg = 1.06 x [ sqrt (h) + sqrt (H) ],
for a spherical earth with no atmosphere.
But there is indeed an atmosphere and that atmosphere tends to bend the light rays up and over the horizon a small amount, so the actual theoretical range in a typical atmosphere is about 10% larger than this estimate, and now we have reached the trivial aspects promised by the title.
When we first started actively teaching navigation in the early 80s and even extending at least into the 90s, official navigation sources did not agree on how much this refraction correction should be to account for the typical conditions. The 1.06 was replaced with something between 1.14 and 1.17. The sources available in, for example, the USCG license exam room included: Bowditch (1.17), the USCG Light List (1.14), and the NOAA Coast Pilot (1.15).
We did not pay too much attention to that, other than noting they did not agree, because in our teaching we approximate this range even more, and simply drop the leading factor completely. The working value of Rg we use in our teaching (and easily remember) is about [ sqrt (h) + sqrt (H) ], with no factor at all.
That advice remains what we recommend today. We can always add that it might be in some cases about 10% bigger, but do not count on it. In fact, in those days we spent quite a bit of time sailing away from bright red empty fuel containers to see how long we could indeed see them with binoculars, and did other measurements from landmarks. We concluded that any disturbance of the water or deterioration of the visibility dropped this theoretical range at least 10%. In waves of any size, it drops even more. Thus the easy to memorize value is good enough, especially because this is important to know by heart... it it tells you how far you can see someone how might have fallen over board, or how close you have to be to see a landmass. The more this is "less like a formula" the better.
I write this note now because by accident I have just realized and checked that now all the official sources have agreed that this factor is 1.17. That does not make it any more dependable than it was as a range of values, but it does mean we have to go back over our own materials where we might have referred to the "official value" we were replacing with our 1.0. That official value we are replacing should now everywhere read 1.17. Thus the "official value" is:
Rg = 1.06 x [ sqrt (h) + sqrt (H) ].
To be fair, in our approximation we are addressing one specific aspect of this concept. In other applications the value is more important, which has led to quite a bit of study. It is, for example, more important in cel nav than in coastal navigation, and sets the limits on how accurate we might predict the time of sunrise and sunset, to name a couple.
One of the leading experts on this subject is Andrew Young at San Diego State University. This link to his work cited above has many links related to this topic. The uncertainty in the value comes from the fact that refraction, especially near the surface, depends on many things—not just air temperature, but also on sea state and wind, and probably the water temperature as well. That first meter above the water surface that these light rays must traverse is a very complex bit of atmosphere. He is also an expert on the closely related topics of mirages and the green flash.
In sort, geographic range remains about a ± 10% number. It helps to keep that in mind when we use a piloting technique like bobbing a light, which can determine a remarkably accurate moment. It is just when we convert that moment to a distance off that the light that we should recall this limitation.
So now we are off to replace any 1.14 or 1.15 we find with 1.17, while still recommending 1.00, with maybe an optimistic or special circumstances addition of 10%.
Ironically, though all official sources now agree on 1.17, the USCG exam room is still using tests and reference books based on the early values that maintain the inconsistency.
|2015 Coast Pilot
|2015 Light List
|circa 1990 Coast Pilot
|circa 1990 Light List
To its credit, in a sense, Bowditch has had 1.17 since at least 1980.
For completeness with regard to the introduction, geo range of a light is affected by fog or pollution as it can create a glow that indicates the light before you can actually see it, and a low ceiling of clouds can give rise to a loom over the light that might extend its visibility tremendously. The loom of a bright city, well below the horizon, might be seen 60 miles or more offshore.