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    Bennett's '...Celestial Navigator' --An improved Zn calculation
    From: Dave Walden
    Date: 2005 Nov 22, 15:04 -0800
    With some trepidation, I raise again the question of using '...Celestial Navigator' to obtain Azimuth.  Using the infamous Huxtable example:
    Dec=55-30
    LHA=54-30
    Alt=61-30
    George H. didn't give the corresponding Lat, but it can be found to be:
    Lat=60-18

    First going through the altitude calculation using the Bennett work form, on page 168, to generate altitude from given Dec, LHA, and Lat.

    line 13 Local Hour Angle          54-30 -> 8841
    line 14 DR Latitude             N 60-18 -> 3974
    line 15 Declination             N 55-30 -> 3217
                                              ______
    line 16 (theta=28-04)                 SUM 16032 -> RES  11760
    line 17 Latitude ~ Declination
            (ABS(Lat-Dec))             4-48  ------------>    351
                                                           ______
    line 18 Computed Altitude         61-30.5 <------  ALT  12111

    Now for the new method to calculate Zn.  In a sentence, use Bennett's table 'backwards' substituting Alt for Dec, and Dec for Alt.  The final Z will be the LHA value.  Continuing with the infamous example from above:

    remember, substitute Dec for Alt
    line 18 Computed Altitude         55-30  --------> ALT  17587
    now, Lat~Dec becomes Lat~Alt                           ______
    line 17 Latitude ~ Declination     1-12  -------->         22
    now, calculate what RES must be for sum to equal ALT, (17587-22)
    line 16  (theta=34-29)                SUM 13763 <- RES  17565
                                              ______
    now, we have the sum of three, we know two, so we can solve for the third.
    remember, substitute Alt for Dec
    line 15 Declination              61-30.5 -> 4187
    note, line 14 is the same as above
    line 14 DR Lat                 N 60-18   -> 3974
    for SUM to be correct, line 13 must be 13763-4187-3974
    note use top of column LHA value as Z
    line 13 Local Hour Angle           75-07 <- 5602
    now, we apply our one rule, if LHA(the real LHA)<180, Zn=360-Z, else Zn=Z
    So, Zn=360-(75-07)=284-53  Exactly the ATAN2 formula result!

    Note, there is! a typo, which I don't recall seeing mentioned before, in Bennett's response to Huxtable: "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 last number should be 284.8, as above.

    Pretty slick, eh?  
    (Some adjustments of signs for special cases are left to the reader as an exercise.)


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