A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Computing azimuth with the Bygrave in special cases.
From: Gary LaPook
Date: 2010 Feb 09, 04:52 -0800
From: Gary LaPook
Date: 2010 Feb 09, 04:52 -0800
I have found a way to calculate azimuth when the declination is off the bottom of the cotangent scale of the Bygrave. I have been using a method of approximating the azimuth in this case which produces usable azimuths, agreeing within less than one degree with the correct azimuth, but I was disappointed that I did not have an exact solution. (See excerpts of my prior posts below and the links to prior posts). My new exact method is to go immediately to the second procedure outlined in the special rules for this situation by exchanging declination and latitude which brings the values within the range of the cotangent scale. You then compute the altitude and the azimuth at the geographic position (G.P.) The altitude is the same calculated at the G.P. and at the observer's position (O.P.) but the azimuth calculated at the G.P. is not the same. But we can use this azimuth to derive the azimuth at the O.P. by using the law of sines. sin a / sin A = sin b / sin B = sin c / sin C so: sin co-lat / sin azimuth at G.P. = sin co-declination / sin azimuth at observer's position which can be re-written as: cos lat / sin Az at G.P = cos dec / sin Az at O.P. An easy way to solve for azimuth at the O.P. using a normal slide rule or a calculator is to divide sin Az at G.P by cos lat and then multiply by cos dec to produce sin Az at O.P. sin Az at O.P =( sin Az at G.P. / cos Lat ) cos dec But cos dec is approximately equal to 1 since dec is between 0 and 50' making cos dec = 1.0 ~.99989 so we can drop that term simplifying it further to Sin Az at O.P = sin Az at G.P. / cos Lat So now I am happy that I have found an exact method but I don't think it is worth the extra effort in practice since the approximate method produces the same azimuth, usually within a fraction of a degree. gl ----------------------------------------------------------------------------------------------------------------------------------------------- This rule comes from the original Bygrave instructions. Even though the original Bygrave cotangent scale was marked every minute of arc it would be difficult to take out intermediate values and it would be expected that a user would take either the one above or the one below any intermediate reading. Bygrave recognized that the azimuth becomes critical as to the determination of altitude when the azimuth is near 90�, and any small error in its determination will have a large effect on the computed altitude. As you noted, the altitudes are computed to a higher level of precision than is needed for plotting the LOP so for this use such small errors can be ignored but can't be ignored for the determination of altitude when the azimuth is near 90�. So, when azimuth is greater than 85� you start with the normal procedure but when the azimuth comes up greater than 85� you use that azimuth for plotting the resulting LOP but you stop the process at that point and do not go on to calculating altitude. You stop and go back to the beginning on another form and this time you interchange the latitude and the declination. You use the same process and compute an azimuth and then go on to compute altitude. This second azimuth is what would exist at the other corner of the triangle and so is not correct at the observer's position and is disregarded when plotting the LOP. In most cases the azimuth computed this second time will not exceed 85� so any error in it will not critically affect the altitude. It is possible in some cases that you will still get an azimuth the second time that exceeds 85� but all you can do in this case is recognize that there might be a greater error in the altitude than normal. I hadn't thought of it before, but is obvious when you do think about it, that the altitude calculated at either the observer's position or at the geographic position of the body must be the same. Since the same calculation will yield the great circle distance between these points, which is the length of this leg, and must be the same length when calculated from either end. But the azimuth will be different which is why you use the first calculated azimuth and ignore the second. If the computed azimuth is greater than 85� the computed altitude will lose accuracy even though the Az is accurate. For azimuths in this range even rounding the azimuth up or down one half minute can change the Hc by ten minutes. So you use the azimuth but you compute altitude by interchanging declination and latitude and then doing the normal computation. You discard the azimuth derived during the computation of altitude and use the original azimuth. When declination is less than 55' on my version (less than 20' on the original) you can't compute "W" because you start the process with declination on the cotangent scale. In this case, Bygrave says to use the same process as when the azimuth exceeds 85�, you simply interchange declination and latitude and compute altitude. But Bygrave didn't tell us how to calculate azimuth in this case. In my testing I have found a method that produces quite accurate azimuths. You simply skip the computation of "W" and simply set "W" equal to declination. The worst case I have found is that the azimuth is within 0.9� of the true azimuth but most are much closer. If the declination is less than one degree and the latitude is also less than one degree, follow this procedure and also assume a latitude equal to one degree. After you have computed the Az you then follow the same procedure discussed above for azimuths exceeding 85� by interchanging the latitude and declination and then computing Hc. http://www.fer3.com/arc/m2.aspx?i=107947 http://www.fer3.com/arc/m2.aspx?i=107414 http://fer3.com/arc/img/106329.bygrave%20manual.pdf