Celestial Navigation – Spring Fix


Last week in Toronto the weather turned from Siberian winter to balmy spring. So I decided to head down to Harbourfront with my sextant and take some measurements. I have covered all the basics of CelNav in previous posts (Ref.1/9). Remember one bad solar X flare and we are back to slide rules & sextants :)! I used an excellent location on the tip of Trillium Park that has a maximum view of the lake surface from East to West. I sat and took measurements for several hours as well as soaking up the Sun and vitamin D. I was hoping to see the Moon as well, but it was only 16% illuminated and was not visible in the light haze.

Sun Measurements

Fig.1 Sun Measurements 15:05UTC – 17:50UTC

The Sun measurements were taken from 11:05DST to 13:50DST as shown in Figure 1. The table shows the time in UTC (DST+4hrs), Hs sextant altitude, Ho altitude, Hc altitude, Intercept and Azimuth. NavAlgos (Ref.10) was used to quickly determine Ho, Hc, Zn and Intercept. Two measurements at 15:50 & 17:30 were chosen in the top & bottom section of the table to be used to calculate a fix. Although they only differ by 41deg, they should be representative of the method. A Meridian Passage was also determined. Several different methods were used: 2 Altitude Direct Calculation (ScicosLab), AP3270, NavAlgos and NAO. Ha, Ho and Hc are determined as follows:

Hs = Sextant Altitude
Ha = Apparent Altitude = Hs +/- Instrument Error +/- I (Sextant Index Error) – D (Dip Error)
Ho = Observed Altitude = Ha – R + PA (Parallax in Altitude) +/- S (Semi Diameter for Sun/Moon)
Hc = Theoretical Altitude = Determine from GP/AP use Trig or NAO/AP3270
Intercept = Ho – Hc, Positive move toward GP, Negative move away from GP

I = Sextant Index Error = 1min “on” = -1min
Dip = -3min (Heye=3m/Table in Nautical Almanac)
R = 0.76min at 15:50UTC & 0.64min at 17:30UTC (Table in Nautical Almanac)

Fix – 2 Altitude Solution ScicosLab

Fig.2 ScicosLab 2 Altitude Direct Solution
Fig.3 Nautical Almanac Sun Apr16-2023

Figure 2 shows a direct spherical geometry solution for the fix based on the two Sun altitudes Ho1 & Ho2. A is the SunGP at 15:50UTC and B is the SunGP at 17:30UTC. D is the Observer location. Zd1=(90deg-Ho1) & Zd2=(90-Ho2) are two great circle radial arms from GP1 and GP2 intersecting at D. The SunGPs are determined by interpolation of the Nautical Almanac Figure 3.

Fix – AP3270 Solution

Fig.4 AP3270 Sight Reduction Sun 15:50UTC & 17:30UTC
Fig.5 AP3270 Marcq St.Hilaire Plot LOP1 & LOP2

Figure 4 shows the AP3270 sight reduction for 15:50UTC & 17:30UTC. Figure 5 shows the Marcq St.Hilaire plot of two lines of position formed from Intercept1/Zn1 & Intercept2/Zn2 intersecting at the fix.

Fix – NavAlgos Solution

Fig.6 NavAlgos 2Altitude (Red) & Marcq St.Hilaire (Blue) Solution Map

Figure 6 shows the NavAlgo solutions for the two sights. There are two solutions shown on the plot. The 2 altitude geometric solution (red) and Marcq St.Hilaire solution (blue).

Fix – NAO Solution

Fig.7 NAO Sight Reductions 15:50UTC & 17:30UTC

Figure 7 shows the NAO sight reductions. The Hc values differ quite a bit from AP3270/NavAlgos in this case.

Fix – Meridian Passage

Fig.8 Meridian Passage Measurements
Fig.9 Meridian Passage Calculations

Figure 8 shows an expanded view of the Sun measurements close to the Meridian passage. We can get a rough estimate of our position by comparing the curve maximum to the Nautical Almanac. The curve maximum occurs at 17:25UTC at Hs=56.38deg. Figure 4 shows the calculations. The estimated Latitude is 1.6′ off and the estimated Longitude is 1deg 53.2min off. The Sun moves at 15deg/hr=15min/min=1min/4secs so as it hovers at the maximum there is a lot of room for error in longitude with this method.

Fix Comparisons

Known Position43deg 37′ 39″N79deg 24′ 47″W
2 Altitude ScicosLab43deg 38.5’N
Error = 0.85′
79deg 23.3’W
Error = -1.5′
(Marcq St.Hilaire)
43deg 39.5′
Error = 1.85′
79deg 23′
Error = -1.8′
(2 altitude)
43deg 38.5’N
Error = 0.85′
79deg 23.2’W
Error = -1.6′
(Marcq St.Hilaire)
43deg 39.4’N
Error = 1.75′
79deg 22.1’W
Error = -2.7′
Meridian Passage43deg 38.5’N
Error = 0.85′
81deg 18.1’W
Error = 1deg 53.2
Fig.10 Summary of Different Methods

Figure 10 shows a summary of the results. The two altitude geometric solution along with AP3270 and NavAlgos give close results. I didn’t plot the Marcq St. Hilaire drawing for NAO because the Hc sight reductions were too far off. Of the two paper solutions, AP3270 is far more accurate. The longitude in the Meridian Passage was way off because I didn’t take more readings around the maximum to give a better time result.

Fig.11 YouTube Video Celestial Navigation – Spring Fix


#1. – “Celestial Navigation Basics Land Sea & Air – Course_d”

#2. – “Celestial Navigation Basics Land Sea & Air – Course_c”

#3. – “Celestial Navigation Basics Land Sea & Air – Course_b”

#4. – “Celestial Navigation Basics Land Sea & Air – Course_a”

#5. – “Celestial Navigation Basics – Fix

#6. – “Celestial Navigation Basics – Hc Calculated Altitude”

#7. – “Celestial Navigation Basics – Ho Observed Altitude”

#8. – “Celestial Navigation Basics – Ha Apparent Altitude”

#9. – “Celestial Navigation Basics – Hs Sextant Altitude”

#10. – “Navigational Algorithms”, Andrés Ruiz González

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By Jeremy Clark

Jeremy Clark is a Senior Telecommunications Engineer and Advanced Amateur Radio Operator VE3PKC. He is the author of E-Books on Telecommunications, Navigation & Electronics.