NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Bennett's '...Celestial Navigator' --An improved Zn calculation
From: Dave Walden
Date: 2005 Nov 22, 15:04 -0800
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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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.)
Yahoo! FareChase - Search multiple travel sites in one click.
