NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Gary LaPook
Date: 2011 Jan 19, 10:23 0800
If you calculate some test azimuths of Polaris by using the Bygrave procedure you will immediately see why it is so insensitive. The first step of the computation is finding an intermediate value ( I labeled it "W", Bygrave called it "y") which is arc tan W = (tan declination / cos hour angle). Since the current declination of Polaris is 89° 18.9', its tan is 83.639. The cos of 1' is 0.999 and the cosine of 89° 59' is 0.0003 so 'W" is limited to the very narrow range of 89.315° to 89.999° ( 89° 18' 54.0" to 89° 59' 59.2" ) no matter what the hour angle is. Since "W" is the largest factor in the calculation of azimuth this narrow range greatly limits the range of the azimuth and makes it very insensitive to changes in hour angle and latitude. Below is a list of the keystrokes for doing this on a calculator from my website at https://sites.google.com/site/fredienoonan/otherflightnavigationinformation/modernbygravesliderule gl  CHECKING YOUR COMPUTATIONS An easy way to check the computation on a Bygrave is to do the same computation on a calculator since this allows you to check the intermediate steps. Just use the standard Bygrave formulas in the three step process following along on the form I have posted. First calculate colatitude and save it in a memory in the calculator. If you are using a value for hour angle that is not a whole number of degrees you might want to make the conversion to decimal degrees and save it in a memory since it will we used twice. If you are using whole degrees then this step is not necessary. Then you calculate "W" using the formula: tan W = tan D / cos H and sum it to the memory where you have saved colatitude which is then X and then make any adjustment necessary to convert X to Y. (If you are just making trials you can avoid this step by your choice of the trial values.) There is no reason to store W itself since it is not used again. You can then convert W to degree and minute format to compare with the Bygrave derived value. Then you compute azimuth angle using the formula: tan Az = (cos W / cos Y ) x tan H. If you want you can also convert Az to degree and minute format to compare with the Bygrave. The last step is to calculate altitude with the formula: tan Hc = cos Az x tan Y. Then convert to degree and minute format to compare with the Bygrave result. (When entering values in the format of degrees minutes seconds, change decimal minutes to seconds, 6 seconds per tenth of a minute, in your head before punching in the assumed latitude, declination and hour angle if necessary.) Using whole degrees for declination, assumed latitude and hour angle, using a TI30 with only 3 memory locations the key strokes are:  (colatitude = 90  Assimed latitude) 90  Assumed Lat = STO 1 (colatitude stored in memory 1)  (tan W = tan D / cos H) Declination tan / H cos = inv tan (computed W) SUM 1 (X now stored in memory 1)(change X to Y if necessary)  (tan Az = (cos W / cos Y ) x tan H) cos (of W from prior step) / RCL 1 (recalls Y from memory 1) cos x H tan = inv tan (computed Azimuth angle)  (tan Hc = cos Az x tan Y) cos (of Az from prior step) x RCL 1 (recalls Y from memory 1) tan = inv tan (computed altitude, Hc) 2nd D.D  DMS (changes Hc in decimal degrees to degrees, minutes and seconds) DONE  On Tue, 1/18/11, P H <pmh099.com> wrote:
