1. Navigation
Astronomy – the science that deals with observation and understanding of relative and apparent motion of stars, planets, moon and sun. Navigational astronomy - the part of that science of practical use to navigator.

2. Navigation
Chapter 4
Reducing & Plotting
Celestial Sights by
Law of Cosines

3. Learning Objectives
Calculate altitude (Hc), intercept (a) & azimuth (Zn) using LOC
Plot a single celestial LOP
Plot a fix using two celestial sights

4. Navigation
What bodies require a ‘v’ correction when calculating GHA?

5. Navigational Triangle
Formed by the GP, DR, & Elevated Pole
The sides
Co-latitude (Co-L)
Co-declination (Co-Dec)
Co-Altitude (Co-H)
Internal angles
LHA
Azimuth angle (Z)
Parallactic angle Pn
Co-L
LHA
Co-Dec Z p DR
Co-H GP
dec
Lat
Hour Circle of Body
Observer’s Meridian 5 5

6. Law of Cosines Formula #1 Solve for Calculated Altitude (Hc)
(cos LHA times cos Lat times cos Dec) (sin Lat times sin Dec) sin Hc
plus =
#2 Solve for Azimuth Angle (z)
sin Dec (sin Lat times sin Hc) (cos Lat times cos Hc) cos z
minus divide by = 6 6

7. Labels for z (azimuth angle)
If LHA <180° If LHA >180°
DR Lat: N
N ddd.d° W
N ddd.d° E
DR Lat: S
S ddd.d° W
S ddd.d° E M
West
East m M
West
East m
Elevated pole + shortage direction from you to body

8. Azimuth TRUE direction from DR to GP
z to Zn Conversion
If LHA <180° If LHA >180°
DR Lat: N
Zn = 360° - Z
Zn = Z
DR Lat: S
Zn = 180° + Z
Zn = 180° - Z
Azimuth TRUE direction from DR to GP Pn
LHA DR z GP N W Ps
LHA DR z GP S W Pn Ps
LHA DR z GP S E Pn Zn Pn
LHA DR z GP N E Zn Zn Zn

9. Sight Reduction Exercise

10. LOC Sight Reduction Sight 2 – All: reduce only
Sight 6 – Karen: reduce & plot
Sight 7 – Jeff: reduce & plot
Sight 11 – John: reduce & plot
Sight 13 – Al: reduce & plot
Sight 17 – Ben: reduce & plot
Sight 21 – Terry: reduce & plot

11. Sun LOP Sight Reduction
Sun LL
20 Mar 2013
14 – 15 – 10
00 – 00 5
19 – 15 – 10 2
23 5
4 7
2 0
6 7 - + 19
15 2
15 2
0 0
1 0
0 3 G g ʘ
Navigation Class
St Paul
All +
160 3
2 0
200

12. Moon LOP Sight Reduction
Moon LL
20 Mar 2013
16 – 05 – 20
00 – 00 5
21 – 05 – 20 6
7 0
4 4
2 0
6 4 - + 21
58 0
59 4
59 4
0 0 19
0 4 G g ʘ
Navigation Class
St Paul
12 2
1 1
54 5
1 4
Karen +
096 4
0 2
096

13. Moon LOP Sight Reduction
Moon UL
20 Mar 2013
16 – 30 – 04
00 – 00 5
21 – 30 – 04 7
7 0
4 4
2 0
6 4 - + 21
55 6
57 5
27 5
0 0
30 0
2 2 G g ʘ
54 5
1 9
12 2
6 2
Navigation Class
St Paul
Jeff +
101 1
2 3
101

14. Sun LOP Sight Reduction
Sun LL
20 Mar 2013
16 – 45 – 02
00 – 00 5
21 – 45 – 02 11
23 5
13 7
2 0
15 7 - + 21
14 3
14 3
0 0
1 0
0 8 G g ʘ
Navigation Class
St Paul
John +
119 1
0 1
241

15. Star Sight Reducation Al 13 21 5 13 35833 44 57 6 44 96000 8 11 5
Rigel
20 Mar 2013
19 – 25 – 00
00 – 00 5
00 – 25 – 00 13
21 Mar 2013
23 5
4 7
2 0
6 7 - + 00
1 4
1 4
0 0 G g ʘ
Navigation Class
St Paul ɤ Al -
163 7
0 1
196

16. Planet Sight Reducation
Navigation Class
Ben
St Paul
20 Mar 2013 17
23 5
19 – 33 – 32
Jupiter
00 – 00
2 0
19 – 33 – 32
13 7 5 + +
0 0
15 7
00 – 33 – 32 -
15 7
21 Mar 2013 g G ʘ 00
0 6
129 5 +
2 1 +
1 2 00 +
0 1
231
0 0
0 6
2 9 - - +
0 1
0 6

17. Meridian Transit Terry 21 Mar 2013 13 – 18 – 07 00 – 00 5 18 – 18 – 07+
Terry 18
18 – 18 – 07 5
13 – 18 – 07 +

18. Meridian Transit Terry ʘ 21 Mar 2013 13 – 18 – 07 00 – 00 5
18 – 18 – 07 + 21
Terry
23 5
2.0
4.7
6.7
0.0
15 3 + ʘ DR
LOP 18
1 0 +
0 3 18
18 – 18 – 07 5
13 – 18 – 07 +
1 2

19. Plotting LOPs Must plot sight to determine its LOP.
Before GPS, plotted Estimated Position (EP)
EPs – now associated only with DR positions
Sight accuracy (SErr)– associated with GPS or KP positions. 19

20. Sight plotted from DR position
Navigation Class
Plotting 2
St Paul
20’
Sight plotted from DR position
10’
1415 Sun
45°N ʘ
1415
50’
40’
30’ EP
L 45° 59.5’ N Lo 92° 45.4’ W 20
20’
10’
93°W
50’
40’
30’
20’

21. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
St Paul
Plotting 2
Sight plotted from GPS position
1415 Sun
1415 GPS
a 2.0nm A 200°
SErr 2.0 21

22. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
St Paul
Plotting 6
Sight plotted from GPS position
1605 Moon
a 0.2nm T 096°
1605 GPS
SErr 0.2 22

23. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
St Paul 7
Sight plotted from GPS position
1630 GPS
a 2.3nm T 101°
1630
Moon
SErr 2.3 23

24. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
St Paul 11
Sight plotted from GPS position
1645
Sun
1645 GPS
a 0.1nm A 241°
SErr 0.1 24

25. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
St Paul
1925 GPS
1925 Rigel
SErr 1.1 13
Plotting
Sight plotted from GPS position
a 0.1nm T 196° 25

26. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
Plotting 17
St Paul
Sight plotted from GPS position
1934
Jupiter
1934 GPS
a 2.9nm A 231°
SErr 2.9 26

27. Sight plotted from GPS position
10’
20’
50’
40’
30’
93°W ʘ
Navigation Class
Plotting 21
St Paul
Sight plotted from GPS position
a 1.2nm South
1318 GPS
1318
Sun
SErr 1.2 27

28. Celestial Fixes Sights on two or more different bodies
if time between sights < 20 min – Fix
if time interval longer – RFix
Reliability of a fix affected by:
Number of observations
Azimuths of observations
Optimum cuts:
2-body fix = 90°
3-body fix = 60°
Minimum cuts for fix:
2-body fix > 45°
3-body fix > 30°

29. Kr@p, outside the 3.0 limit & LOP separation only 40°
Navigation Class
Plotting
7 & 11
St Paul
DOUBLE SCALE
#7 – 1630 Moon 2.3nm T; Zn 101°
#11 – 1645 Sun 0.1nm A; Zn 241°
10’ 5’
Moon
1645
Sun
45°N
55’
50’
45’
Plot 1645 Position
Plot 1645 LOP
Re-plot 1630 LOP from 1645 Position
Intersection of two LOPs is RFix
1645 position to 1645 RFix = SErr ʘ
1645 GPS ʘ
1630 GPS ʘ
1645 RFix
1630
Moon
SErr 3.5
outside the 3.0 limit & LOP separation only 40° 29 5’
93°W
55’
50’
45’
40’
35’

30. Next week bring your plotting tools & handout
Navigation
Next week bring your plotting tools & handout
CHAPTER 5
NASR Reduction

31. Reducing and Plotting Celestial Sights
The End