In this post we will look at how to generate a fix from observations of two Celestial Bodies. In previous posts we looked at Hc (Ref.9), Ho (Ref.8), Ha (Ref.7) and Hs (Ref.6). Now we are in a position to use Ho & Hc from two measurements to get our fix. The Marcq St. Hilaire method will be used to solve for our position and AP3270 (Ref.4) tables for sight reduction. The complete details are contained in Ref.1. In order to get the widest sighting angles, our site is moved slightly west from HF1 to HF1a. This gives us an Eastern sighting angle of 131deg (Fig.1) and Western sighting angle of 214deg (Fig.2). The two Celestial Bodies used are the Sun and Moon.
The two measurements taken on Aug13th_2020 were:
EP: Lat = 44deg N Long =79deg W
Hs_moon_13:24UTC=64deg 4min
Hs_sun_15:30UTC=51deg 57min
Heye = 3m
Sextant Index Error = 3min “off limb”
Temp = 25degC Sunny Clear Sky & Horizon Slightly Hazy
Moon Occulted on East Side
The Fix was determined in several ways. The first method was graphical using AP3270 for sight reduction and Marcq St. Hilaire. The second method was the 2 altitude method with a direct trigonometric solution by the intersection of the 2 circles of position. The third method was a check using Admiralty NavPac. The results are very close and shows the power of the Marcq St. Hilaire method.
Posn(GPS): Lat =43deg 37.6min N Long =79deg 24.8min W
Posn(MSH): Lat = 43deg 36.1min N Long = 79deg 24.7min W
Posn(ScicosLab): Lat = 43deg 35.8min N Long = 79deg 24.8min W
Posn(NaPac): Lat = 43deg 35.8min N Long = 79deg 24.9min W
Let’s consider the reduction of the Moon reading first.
EP = 44degN 79degW
GHA_13:24UTC (Nautical Almanac):
86deg 53.3min at 13:00UTC + 5deg 43.6min Increment for 24min + 4.5min v correction = 92deg 41.4min
DEC_13:24UTC (Nautical Almanac):
20deg 48.7minN + 2.7min d correction = 20deg 51.4min
LHA = 92deg 41.4min – 78deg 41.4min = 14deg
AP = 44deg N 78deg 41.4min W (adjust Long so that LHA is integral)
Extract Hc,d,Z from Volume3/page35: Lat=44deg Dec=20deg 51.4min LHA=14deg
Hc=63deg 19min, d=55min, Z=149deg
Table 5 dec min(51.4min) = 47min
Hc=64deg 6min Z=149deg
Dip=3min, Index Error=+3min
Ha=Ho-Dip+IE=64deg 4min -3min+3min=64deg 4min
R=0.5min SD=15.1min PA=0.404deg=24.2min
Ho=Ha-R+PA+SD=64deg 42.8min
Intercept=64deg 42.8min – 64deg 6min = +36.8min
Zn=211deg
Now the reduction of the Sun reading.
EP = 44degN 79degW
GHA_15:30UTC:
43deg 48.8min at 15:00UTC + 7deg 30′ Increment 30min=51deg 18.8min
DEC_15:30UTC:
14deg 23.8minN – 0.4min Correction=14deg 23.4minN
LHA = 51deg 18.8min – 79deg 18.8min = 332deg
Extract Hc,d,Z from Volume3/page35: Lat=44deg Dec=14deg 23.4min LHA=332deg
Hc=51deg 40min, d=50min, Z=132.5deg
Table 5 dec min(23.4min) = 19.5min
Hc=51deg 59.5min, Z=132.5deg
Dip=3min, Index Error=+3min
Ha=Ho-Dip+IE=51deg 57min -3min+3min=51deg 57min
R=0.7min SD=15.8min PA=0min
Ho=Ha-R+PA+SD=51deg 57min-0.7min+15.8min=52deg 12.1min
Intercept=52deg 12.1min – 51deg 59.5min = +12.6min
Zn=132.5deg
Fig.4 shows the completed AP3270 sighting form, Fig.5 shows the ScicosLab solution for the two altitude problem and Fig.6 shows the complete solution spreadsheet.
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References
#1. “Celestial Navigation Basics & Equipment”, Clark Telecommunications, 2019, ISBN 9780988049086
https://www.clarktelecommunications.com/celnav.htm
#2. – “Nautical Almanac 2020 Commercial Edition”, Paradise Cay Publications, ISBN 9781951116033
https://www.paracay.com/nautical-almanac/
#3. – “Rapid Sight Reduction Tables for Navigation”, NP303/AP3270
http://astro.ukho.gov.uk/nao/publicat/ap3270.html
#4. – “Rapid Sight Reduction Tables for Navigation”, Pub.No.249
https://www.thenauticalalmanac.com/Pub.%20No.%20249.html
#5. – “Navigational Algorithms”, Andres Ruiz González
https://sites.google.com/site/navigationalalgorithms/
#6. – “Celestial Navigation Basics – Hs Sextant Altitude”
#7. – “Celestial Navigation Basics – Ha Apparent Altitude”
#8. – “Celestial Navigation Basics – Ho Observed Altitude”
#9. – “Celestial Navigation Basics – Hc Calculated Altitude”
