NavList:

"The dip short is to a horizon .55 nautical miles away from a height of eye of 40 feet. "

There's something rather neat about this distance, quite by coincidence, of course. At that distance, the observer height of eye in feet is almost exactly equal to the dip short in minutes of arc. In terms of sensitivity, a one foot error in estimating height of eye yields a one minute error in dip short. It's difficult to estimate height of eye that accurately in typical practical situations. Since I was doing some dip short tests yesterday, I'm going to take this as an opportunity to ramble on about the topic for a bit.

Ignoring the "drop off" from the curvature of the Earth at greater distances, and assuming we would never use a dip short angle greater than about 3°, the dip short angle for distances less than a mile or two is nothing more than the ratio of observer height of eye to distance:
  d.s. = h/D.
If D is is in nautical miles and h is in feet, then to keep the ratio in the same units, we have
  d.s. = h/(6076·D),
and if we want the result in minutes of arc, we multiply by that magic number 3438 (minutes of arc in a "unit angle" or one radian):
  d.s. = (3438/6076)·h/D = 0.566·h/D,
and if D happens to be 0.566 nautical miles, then d.s.=h as promised (h in feet, d.s. in minutes of arc).

Note that this is the dip short for distances up to a mile or two. Beyond that the unrefracted drop-off angle is exactly equal to the distance divided by two. That's how much the curved surface of the Earth is falling away from a flat plane tangent to the surface at the observer's location. This relationship applies at any distance, including two miles or ten miles away from the observer and also a thousand miles or halfway around the world, too. That's surprising at first, but when you draw it out it makes good sense.

Imagine you want to build a whimsical signpost in your garden with signs showing the direction to nearby points and a few distant locations, too. One sign says "barn" and points east, another says "town" and points north. The next sign says "Manhattan" and points WSW. The last two signs say "Bermuda" and "Hawaii". The first points SSE and the Hawaii sign points a bit north of west. But Bermuda and Hawaii are below the horizon so the signs should point somewhat downward. How much? Well, from my location, Bermuda is about 630 nautical miles away and Hawaii (the big island) is about 4350 nautical miles away. That's 10.5° distance to Bermuda and 72.5° to Hawaii. So if I face west (actually about 10° north of west for the straight line to Hawaii from here), I should look 36.25° below horizontal to see Hawaii. That's where my Hawaii sign should point. And my Bermuda sign points SSE 5.25° below horizontal. In fact, even my sign pointing at Manhattan should be angled 2.0° downward.

To correct the "easy" dip short (above) for drop-off angle, I have to add half the distance in nautical miles. But when we can actually see these points (unlike Hawaii and Bermuda above) refraction reduces that angle slightly, making the Earth "less round" from the point of view of optical measurements in this category (surface-skimming rays of light). So instead of adding half the distance, I should add (1-beta)·D/2 where beta is the roll rate of light rays --the minutes of arc that they refract downward per nautical mile. ·This is weather-dependent (and in an unpredictable way) bue a reasonable number for beta is in the range 0.12 to 0.25 (larger values are unusual but not rare!). A reasonable average value then for the refracted drop-off angle is 0.42·D. Finally the complete dip short correction is:
  d.s. = 0.566·h/D + 0.42·D.
Note that you'll find many resources with ridiculous numbers of digits in the coefficients here. Those are misleading!

If we take the derivative of the dip short with respect to distance and then set the result to zero, we find that the dip short reaches a minimum angle below the true horizon and then starts to increase again. The distance at which this occurs is identical to the distance to the visible sea horizon, which makes good sense. Of course, we can't see those points beyond the horizon, so the minimum is the end of the line for the dip short equation:
  d.s. = 0.566·h/D + 0.42·D , if D<1.16·sqrt(h).

Finally, if we want to experiment with refraction variability (experiment in the math, that is), we can keep beta in the game:
  d.s. = 0.566·h/D + (1-beta)·D/2 , if D<sqrt(1.13/(1-beta)) · sqrt(h).
Note that if beta=1, the Earth looks flat as far as optical observations are concerned. The drop-off term in the dip short goes to zero, and the distance to the horizon becomes infinite. This can happen when there is a temperature inversion. In fact, beta can be greater than one, too. It's also just barely possible for beta to be zero (the atmosphere is on the verge of hydrostatic instability at this point) and in that case light rays follow straight lines and the drop-off correction is the pure geometric drop-off, D/2.

Frank Reed