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    Latitude by Polaris
    From: Ron Jones
    Date: 2016 Mar 26, 15:58 -0700

    The 2002 Edition of Bowditch omitted an explanition of how the Polaris correction tables in in the Nauticak Almanac were constructed.

    The following was taken from the 1977 Edition of Bowditch article 2027, pages 553 & 554 and may be of interest to any one interested in how the table were constructed:

    Another special case of finding latitude is available in most of the northern hemisphere, it utilizes the fact that Polaris is less than from the north celestial pole.   Since Polaris  is never far from the north celestial pole, its observed altitude (Ho) , with suitable correction , is the observer's latitude.   When Polaris is on the upper branch of observer's meridian ( LHA = 0° ) the observer's latitude is equal to the observed altitude minus the polar distance ( p ).   When Polaris is on the lower branch of the observer's meridian ( LHA = 180° ) the observer's latitude is equal to the observed altitude plus the polar distance   ( p ).   When the LAH of Polaris is not  or 180°  the Polaris correction is approximately defined by the polar distance ( p ) times the cosine of LHA. Thus the correction is a function of the LHA of Polaris, and hence also of the LHA of Aries, insofar as the difference between these quantities (the SHA of Polaris) can be considered a constant.   Although this method provides sufficient accuracy, a higher degree of accuracy can be obtained by the use of the Polaris correction tables contained in the Nautical Almanac.

    The Nautical Almanac tables are based on the following formula:

         Latitude - Ho = p•cos(h) + (p/2)•sin(p)•sin2(h)•tan(latitude)

                        where    p = polar distance of Polaris = 90° -Declination of Polaris 

                                        h = Local Hour Angle of Polaris = LHA of Aries + SHA of Polaris

    The value of a0, which is a function of LHA Aries only, is the value of both terms of the above formula calculated for the mean values of SHA and Dec. of Polaris, for a mean latitude of 50° , and adjusted by the addition of a constant ( 58.8' ).  The value of a1 which is a function of LHA of Aries and latitude, is the excess of the value of the second term over its mean value for latitude 50°, increased by a constant         ( 0.6' ) to make it always positive.  The value of a2 , which is a function of LAH Aries and date, is the correction to the first term for the variation of Polaris from its adopted mean position increased by a constant ( 0.6' ) to make it positive.  The sum of the added constants is , so that: 

            Latitude =  corrected sextant altitude -1° +a0 +a1 +a2

    The Nautical Almanac table at the top of each Polaris correction page (274→276) is entered with LHA Aries, and the first correction ( a0 ) is taken out by single interpolation.  The second and third corrections  ( a1 and a2 ) are taken from the double entry tables without interpolation, using the LAH Aries column with the latitude for the second correction ( a1 ) and with the month for the third correction ( a2 ).

     

       
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