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In February I offered a version of the Bygrave celestial navigation
sight reduction formulas. Though valid for all combinations of latitude,
declination, LHA, and altitude (including negative altitude), I was not
happy with some deviations from standard practice. For instance, my
azimuth angle was defined differently from the normal sense of Z.

Here are a new set of rules which conform to the traditional conventions
of celestial navigation. For convenience on the slide rule, ALL NEGATIVE
SIGNS ARE IGNORED. For instance, tan -10° = -.176, but you calculate as
if it were +.176. Similarly, arc tangents are always in the range 0 - 90°.

It's not necessary to normalize LHA to a certain range. For example, 10°
and 350° work equally well.

This Bygrave variant is intended for the conventional straight slide
rule. It requires that you know how to read and set the full range of
possible angles on the trig scales. See my "360 degree slide rule trig"
series:

http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37154
http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37223
http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-feb-2017-g38237


FORMULAS AND RULES

W = arctan(tan dec / cos LHA)

If LHA is between 90 and 270, replace W with its supplement: W = 180 - W

If latitude and declination have same name
     X = 90 - lat + W
If contrary name
     X = 90 - lat - W
In either case lat is always positive.

If X is not 0 to 180, add or subtract 180 to make it so. If the
adjustment is necessary, ALTITUDE IS NEGATIVE.

Z = arc tan(tan LHA * cos W / cos X)

If EXACTLY ONE of these is true
     altitude negative
     X less than 90
replace Z with its supplement: Z = 180 - Z

Azimuth angle Z is reckoned in the conventional sense: zero at the north
(south) pole if lat is north (south), and increasing east (west) if the
body is east (west) of the meridian.

Hc = arc tan(cos Z * tan X)
Apply negative sign if applicable.

There are two places where an arc tangent is (in some cases) replaced
with its supplement. But if you ignore signs, tangents of supplementary
angles are equal. E.g., tan 10° = tan 170°. Therefore you can read the
supplement on the T scale without any calculation.


ACCURACY TEST

I worked these 10 examples on a 10 inch slide rule. The problems were
generated at random by a program which distributes stars uniformly on
the celestial sphere, and observer positions uniformly on the Earth
(both hemispheres). It excludes altitudes greater than 80 or within 5°
of zero, and latitudes greater than 70.

  51.7575 lat
  -5.1860 dec
  27.9717 LHA
   5.86   W (not so easy to read on some rules)
  32.382  X
147.86   Z (error = .113°)
  28.25   Hc (error = -.009°)

  -1.1816 lat
  47.0243 dec
213.7008 LHA
127.78   W
141.038  X (alt neg)
152.25   Z (error = -.032)
-35.60   Hc (error = -.002)

  22.1846 lat
  43.1575 dec
333.0564 LHA
  46.44   W
114.255  X
  40.50   Z (error = .033)
  59.37   Hc (error = .005)

-47.9567 lat
-58.8701 dec
296.4248 LHA
  74.96   W
117.003  X
  49.00   Z (error = -.012)
  52.16   Hc (error = -.038)

-11.1965 lat
  30.3719 dec
304.7456 LHA
  45.80   W
  33.004  X
129.82   Z (error = -.058)
  22.58   Hc (error = -.035)

  40.7939 lat
  -3.3744 dec
259.7277 LHA
161.70   W
  67.506  X (alt neg)
  85.82   Z (error = -.005)
-10.00   Hc (error = .025)

-50.1163 lat
-51.0710 dec
  97.9351 LHA
  96.37   W
136.254  X
  47.78   Z (error = .039)
  32.74   Hc (error = -.033)

  65.9394 lat
-17.9946 dec
   9.2966 LHA
  18.20   W
   5.861  X
171.11   Z (error = -.024)
   5.79   Hc (error = .021)
That one required cosines of small angles.

-28.7610 lat
-35.6695 dec
233.0997 LHA
129.86   W
  11.099  X (alt neg)
  41.02   Z (error = -.099)
  -8.42   Hc (error = .035)

  -9.1664 lat
-54.4142 dec
  10.8386 LHA
  54.86   W
135.694  X
   8.76   Z (error = -.001°)
  43.94   Hc (error = -.016°)

Root mean squared altitude error = 2.5 arc minutes.

   

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