A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: dip, dip short, distance off with buildings, etc.
From: Frank Reed CT
Date: 2006 Jan 11, 00:29 EST
From: Frank Reed CT
Date: 2006 Jan 11, 00:29 EST
Bill, you asked: "Rather than pay sleuth working backwards from your estimate of horizon distance of 4-5 miles (statute for Lake Michigan charts, or nautical miles?), do you recall an approximation of your height of eye for the beach shots? " About fifteen feet. As for the skyscraper heights, I'm not satisfied with them yet so I'm gonna do a few more trials in order to put some error bounds on the results. In another message you wrote: "It would appear Bowditch significantly underestimates lift due to refraction by today's standards." Well, I wouldn't say it that way. The "problem", such as it is, is that the tables in Bowditch are calculated for a specific value of the terrestrial refraction --around 0.16 or so. Now of course, that's better than nothing. But the point I want to make is that we can use different values for the refraction because we can calculate the actual value of the terrestrial refraction that exists under verious atmospheric conditions. Specifically the refraction rate is given by Q*[(1.79km)/s]*(1-h/s) where h is the mean height of the ray, s is the "scale height", Q is the usual temperature and pressure factor given by Q=(P/1010mb)/(T/283deg celsius). [Note that the factor of (1-h/s) can be dropped in most cases of interest for navigation.] The "scale height" applies just to the lower part of the atmosphere, say, the lowest kilometer, where the refraction is taking place and it can be calculated based on the temperature gradient or "lapse rate". The scale height is typically around 10km, but it can be infinite if the lapse rate close to the ground is -34.1 degress per km (which is rare but possible). If the scale height is infinite, it means that the atmosphere has constant density as a function of height in the layer near the ground, and of course an atmosphere of constant density does not refract light rays. The lapse rate can also be positive and the scale height can be 5km or 2km or lower. Notice that the scale height governs most of the possible variaton in the terrestrial refraction, though it also depends on variations in temperature and pressure to a lesser extent. "Remember good old 4134 from the first term? 3438 / (1 - 0.1864). Coincidence? (Radius = 3438 nm)" Yep. Good sleuthing on those constants! Now it's time to derive the equation from scratch. There are several approaches to this. Let's start with the refraction-free version and then apply the substitution on the radius of the Earth. Draw yourself an arc of a circle representing the curved surface of the Earth. Mark a point A at height h above the surface for the observer's position, mark point B at height H for the object's position at some distance from A (H and h should be roughly comparable heights and both should be much less than the radius of the Earth, and mark C for the center of the Earth at the center of the arc. Now draw a triangle connecting all three points. The lengths of the two long sides are R+h and R+H. The length of the other side doesn't matter. We want the length of the arc along the Earth's surface connecting the base of h to the base of H. This length is proportional to the angle in the triangle at the Earth's center, C. Let's call that angle phi. With a sextant we are measuring the altitude of point B from point A so that means that we know the angle in the triangle at point A, let's call that gamma. Finally the angle at the third corner, B, is related to the other two --it's just 180 degrees-(gamma+phi). OK so far? Now apply the law of sines to the big triangle. And expand using the rules for sums of angles. You should get this equation: (R+h)/(R+H) = sin(phi)*cot(gamma)+cos(phi). Using the relationship between the measured altitude and gamma, you can replace cot(gamma) with -tan(alt). And really, we're done at that point. The resulting equation lets us solve for phi for any given values of R,h,H and measured alt. Of course since phi is buried inside two trig functions, that used to be considered a bit of a problem from a calculational standpoint. We can eliminate that problem by assuming that phi is a "small angle". We can then replace sin(phi) by phi and cos(phi) by 1-(1/2)*phi^2. When you do that, you get a quadratic equation in phi as follows: (1/2)*phi^2 + tan(alt)*phi - (H-h)/R = 0. We can solve for phi using the quadratic formula (you know the one... x=(-b+/-sqrt(b^2-4ac))/2a...). And finally, the distance is R*phi. Throw in a factor of 3438 to convert to minutes of arc. Note that none of this involved refraction. Finally, finally, substitute for the radius of the Earth using R=R/(1-beta) to account for refraction. Voila. Table XV. -FER 42.0N 87.7W, or 41.4N 72.1W. www.HistoricalAtlas.com/lunars