NavList:

Antoine, you wrote:
"But I do not know of any practicable method of reliably checking that this 5.66' refraction value is reasonably realistic given what we know of the way light rays are refracted between 2 points at finite distances within the Earth Atmosphere."

The light that I can shine probably won't make you happy. When you're looking at nearly horizontal rays of light, their refraction is usually termed "terrestrial refraction". There are no reliable "formulae" for terrestrial refraction that can provide better than minute of arc accuracy except when detailed atmospheric structure data are available. Knowing the lapse rate in the lower troposphere helps a great deal, but even when there are recent soundings (by balloon, for example), even these are approximate and subject to rapid, frequent changes.

Celestial navigation by altitudes measured off natural horizons (usually meaning sea horizons) is directly limited by the uncertainty and unpredictability of terrestrial refraction. The sea horizon rises and falls by as much as a minute of arc in seemingly normal weather and even more in unusual weather conditions. And the sea horizon is typically only 3 to 5 miles away on small craft.

Mountains rising from beyond that natural horizon up to dozens of miles away can be shifted by several minutes of arc with no other visual cue that anything unusual is happening. In fact, one of the best ways to detect phenomena in the atmospheric lapse rate is to watch for shifts in distant mountains relative to foreground features.

Wouldn't it be nice if we could find some observable pattern of temperatures and pressures or other phenomena that we could use to adjust the surface-skimming terrestrial refraction... Astronomers and physics-oriented fans of celestial navigation have been seeking this prize for at least two centuries. There has been almost no progress that I can find.

Last weekend I ran a small celestial navigation workshop at Mystic Seaport Museum (Mystic, Connecticut). More classes later this Spring (see ReedNavigation.com if anyone is interesed). We travelled to Stonington Point a few miles from the museum on Saturday near local noon. In the distance the hills of Montauk Point on Long Island were clearly visible. I could see features with known elevations of about 100 feet at a range of about 16 nautical miles. My own height of eye was about 16 feet. Visibility range is sum of distance to horizon for observer and target, which, like dip, is proportional to square root of height, and under average conditions is approximately
    R ≅ 1.16×√ht(feet) n.m.,
so for this case it should have implied a maximum visibility range of 1.16×(10+4) or 16 nautical miles. That's a perfect match, and that's the problem. The hills at Montauk were clearly visible with ups and down and features, all several minutes of arc above the natural horizon. That topography is not always visible, and, in fact, Montauk is only "sometimes" visible from Stonington Point. It's a result of unusual refraction lifting the distant hills by a couple of minutes of arc.

So that's no fun, right? You wanted some math to chew on. I wouldn't want to leave you starving, so I should point you at Andrew Young's page on "circulating rays" which is the key phenomenon to grapple with:
   https://aty.sdsu.edu/explain/atmos_refr/bending.html.

There are links there including one to the paper by Auer & Standish that outlines a general method involving a variable change that facilitates numerical integration of refraction in arbitrary atmosphere models (it's not too hard to code, and I've used it many times myself --but see PS). So you build your own refraction model and dial in whatever lapse rate and low-level troposphere structure appeals to you. Getting back to that Moon photo in Piedmont, you could consider the range of possible atmospheric conditions that would allow a photo like that to be captured...

Frank Reed

PS: A. Young in his bibliography says that they do not use the key invariant (in their year 2000 paper) that makes the integration clever. I'm sure I've seen the invariant version elsewhere. Might be worth digging around...