# NavList:

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

**Re: Advancing position circles: Huxtable vs. Zevering**

**From:**Lars Bergman

**Date:**2006 Sep 25, 13:50 +0200

George Huxtable wrote in NavList 1314 (with a correction in 1323): "But what solutions do NavList members arrive at, I wonder, if they pick up that same challenge, using any method they think appropriate?" The problem was defined as: "1. An observer, at position P1, measures the altitude of a star S1, at(Dec1 = 0, GHA1 = 0), to be 30 degrees. 2. Then he travels due North by 60 nautical miles (= 1 degree), to P2. 3. From there, he observes another star S2 (then at Dec2 = N 1 degree, GHA2 = W 45 degrees) to be at an altitude of 45 degrees. Where on Earth is he then?" --- Assume position P1 to be located at Lat, Long. Then P2 will be at Lat+1d, Long. We use the altitude formula sinAlt=sinLat*sinDec+cosLat*cosDec*cosLHA, where LHA=GHA+Long, GHA is counted westwards and longitude eastwards. Now, at P1 sin30d=sinLat*sin0d+cosLat*cos0d*cos(0d+Long) which can be simplified to 1/2=cosLat*cosLong At P2 we have sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d*cos(45d+Long) Now let's make a guess: Long=-45d, i.e. 45 degrees westerly longitude. With this guess we find that sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d=(cosLat-cos(Lat+2d)+cos(Lat+2 d)+cosLat))/2=cosLat The last equation can be simplified to 1/sqrt(2)=cosLat, and then Lat=+/-45d. This result satisfies the equation at P1 as well, and thus we are quite sure it is correct, but in order to be fully convinced we can verify the solution(s) by calculating the altitudes: sinAlt1=sin45d*sin0d+cos45d*cos0d*cos(0d-45d)=(cos45d)^2=1/2 => Alt1=30d sinAlt1=sin(-45d)*sin0d+cos(-45d)*cos0d*cos(0d-45d)=(cos45d)^2=1/2 => Alt1=30d sinAlt2=sin46d*sin1d+cos46d*cos1d*cos(45d-45d)=cos45d=1/sqrt(2) => Alt2=45d sinAlt2=sin(-44d)*sin1d+cos(-44d)*cos1d*cos(45d-45d)=cos45d=1/sqrt(2) => Alt2=45d The altitudes are correct, thus the guess was right and the two solutions are: P1 45N, 45W P2 46N, 45W or P1 45S, 45W P2 44S, 45W --- Lars 59N 18E --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---