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    Azimuth Tables and Refraction
    From: George Bennett
    Date: 2003 May 4, 15:41 +1000

    Discussion of the Azimuth Tables in The Complete On-Board Celestial Navigator.

    With reference to the examples used by George Huxtable:

    (1)   If the LHA is 54°, the azimuth is found from the opposite side of the table, see explanation p19, Step 2.

    (2)   The data in the examples is incomplete. To resolve the azimuth quadrant ambiguity, the procedure via the Prime Vertical Altitude should be followed. The tangent formula, heeding the signs of the numerator and the denominator, does not have  this disadvantage. The Weir diagrams are also free from this defect.


                                      Tan Az =                 -Sin LHA                             .

                                                          Cos Lat*Tan Dec –Sin Lat*CosLHA


                              Lat + N, -S    :      Dec +N, -S     :         LHA  0° - 360°)


    In the two examples chosen to highlight the shortcomings of the Azimuth Table all three variables are in error by 0.5° (± 1¢)and the circumstances are in the vicinity of the Prime Vertical. In these extreme, but possible, situations the azimuth derived from its sine is somewhat uncertain as will be seen from an inspection of the Table. If, however, the Tables are interpolated (X=460) the azimuth is found to be 255° or 285° (not 075° or 105°) which compares favourably with the results from direct calculation of 255.3° and 254.8°.

    The user of the book is not informed that this situation can arise. In the examples given in the book it is implied that all values are rounded off to the nearest degree. I have used the tables on innumerable occasions, checking the results by calculator, without this problem occurring. Nevertheless, I accept that a note to this effect should be included. I thank George Huxtable for drawing my attention to this situation.


    The formula that is used in the Nautical Almanac is from my 1982 paper in the British Journal of Navigation (Vol 35, No2) and quoted previously, is the dominant first term of an accurate representation of refraction, which accords with Garfinkel’s algorithm with a maximum error of 0.015¢ in the altitude range of 0° to 90°. An error of less than a second of arc should satisfy most needs.


                  RM = R¢M – 0.06sin (14.7 R¢M  + 13)



                 R¢M  = cot ( h + 7.31/(h +4.4))

    A number of algorithms were quoted in that paper, none of which attained that accuracy. An additional formula was given for calculating the change in refraction for non-standard temperatures and pressures. The maximum error over the range of            –20°C to 40°C and 970mb to 1050mb was 0.2¢ (cf Bowditch 3.5¢)

    George Bennett.   

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