# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Equation for dip?**

**From:**Lu Abel

**Date:**2006 Oct 02, 19:50 -0700

Frank: You are, as always, perfectly correct. I was more trying to explain why the radar horizon is beyond the visible horizon than to explain refraction due to atmospheric density. I fretted about simplifying things too much. Thanks for filling in the details way better than I ever could have. Lu FrankReedCT@aol.com wrote: > Lu, you wrote: > "The surface of the earth, especially seawater, is a conductive medium > and hence will bend electromagnetic waves." > > This is significant for long wavelength radio waves, but totally irrelevant > to the behavior of visible light. > > "This effect is inversely proportional to the frequency of the > electromagnetic wave. Thus light bends very little and the visual > horizon is an almost negligible amount beyond the geometric horizon.* > > The ground wave effect for visible light is a tad more than "almost > negligible". It's nil, zilch, nada, certainly below one part in a trillion and I > wouldn't be surprised if it's below one part in a trillion trillion (ok, even > that's not zilch, but it's certainly irrelevant). > > The distance to the visible horizon is different from the Euclidean > geometric distance because of the variation in the density of the atmosphere. It gets > thinner as you go up, so rays of light necessarily bend downwards allowing > us to see beyond the distance to the geometric horizon. This variation in > density depends somewhat on the weather. If the temperature in the air rises with > height above the water, and rises steeply enough, the bending due to the > density gradient can easily be much greater than normal and allow us to see many > miles beyond the normal horizon. It's worth nothing that this sort of > temperature gradient also promotes the formation of sea fog. So if you're in an > area noted for sea fog, you can also expect unusual refraction including > anomalous dip and the ability to see far beyond the normal horizon. > > And you wrote in a footnote: > " * Before one of the more erudite people on the list jumps on me, I'm > fully aware that light IS refracted near the horizon and we DO apply > refraction corrections when taking low-angle sights; distance to the > horizon is a much cruder calculation. In fact, I believe the last place > in the 1.17 multiplier has jumped around as people have argued about the > effect of atmospheric refraction." > > Distance to the horizon can be calculated in the same way as dip, range of > visibility, distance by angle to the horizon, etc. The last place in the > multiplier does indeed jump around in the literature because different sources > have made different assumptions about the rate of change in atmospheric density > (which itself depends primarily on the lapse rate in temperature). They're > trying to give a value for a mean state of the weather. > > Some mathy details: > First: the curvature of the surface of the Earth is 1 minute of arc per > nautical mile (this is a direct consequence of the definition of a nautical > mile). What this means is if I place a vector on the ground at the equator > perfectly horizontal at some point P pointing east and then travel 1 n.m. east and > place a vector horizontally on the ground at some point Q, again pointing > east, then the angle between those two vectors is 1 minute of arc. The second > vector is tilted relative to the first vector by 1 minute of arc. Makes sense, > right? > > Now suppose I shoot a light ray (aim a laser) horizontally at sea level from > point P towards point Q. Under average conditions, the curvature of this > light ray will be about 0.15 minutes of arc per nautical mile. Because it is > proportional to horizontal distance traveled, this curvature can be accounted > for by pretending that the light ray does not bend but the Earth is slightly > less curved (bigger Earth radius). More generally, when you work out the > refraction, you'll find that the curvature of the trajectory of a light ray is > c = Q*(alpha0*Re/scaleHt)*(1-h/scaleHt) > where alpha0 is the refraction constant for air, approximately 0.000281 at > standard Temp/Pressure, Re is the actual radius of the Earth, h is height > above sea level, scaleHt is the "scale height" of the atmosphere, and Q is the > usual temperature pressure factor: Q=(P/P0)/(T/T0). The scale height is the > distance over which the atmospheric density "e-folds" or decreases by a factor > of e (base of the natural logs). > > The scale height is usually about 11 kilometers. A very reasonable range of > values to expect for the low level scale height is from 7 to 13 km > (corresponding to low-level lapse rates of +5 and -12 degrees C per km respectively). > If the lapse rate should happen to be as low as -34.1 degrees C per km, the > scale height is infinite and the curvature is zero. That's the case I mentioned > in another message where the lowest layer of the atmosphere has constant > density, not varying with height. In that case, the refraction is zero and it is > just as if there is no air at all. If the lapse rate is positive and rather > high, 129.6 degrees C per km, the scale height is about 1.7 km which implies > a curvature of 1.0 minutes of arc per nautical mile --equal to the curvature > of the Earth's surface. In that case, light rays remain parallel to the > Earth's surface and it is "as if" the curvature of the Earth is zero. > Flat-earthers rejoice! The dip is zero at all heights (assuming that lapse rate is > maintained over the range of heights under consideration). > > Please note: I'm working from my notes from last January and some of the > exact numbers here may be based on slightly different assumptions. The general > conclusions are correct though, I think. > > -FER > 42.0N 87.7W, or 41.4N 72.1W. > www.HistoricalAtlas.com/lunars > > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---