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.

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

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

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

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

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

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

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

Sight Reduction Exercise

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

Sun LOP Sight Reduction Sun LL 20 Mar 2013 14 – 15 – 10 00 – 00 5 19 – 15 – 10 2 44 57 6 92 46 4 23 5 43 16 4 4 7 2 0 6 7 - + 43 09 7 103 10 2 19 15 10 3 47 5 106 57 7 92 46 4 14 11 3 15 2 15 2 43 24 9 0 0 00 07 9 1 0 0 3 00 08 2 G g ʘ Navigation Class St Paul All 14 11 3 43 24 9 00 08 2 44 57 6 14 18833 00 13667 44 96000 + 43 44821 160 3 43 26 9 2 0 200

Moon LOP Sight Reduction Moon LL 20 Mar 2013 16 – 05 – 20 00 – 00 5 21 – 05 – 20 6 44 57 5 92 47 1 7 0 32 07 9 4 4 2 0 6 4 - + 32 01 5 29 56 1 21 05 20 1 16 4 31 13 6 92 47 1 298 26 5 58 0 59 4 59 4 33 00 9 0 0 19 18 36 5 0 4 18 36 1 G g ʘ Navigation Class St Paul 12 2 1 1 54 5 1 4 Karen 298 26 5 33 00 9 18 36 1 44 57 5 298 44167 18 60167 44 95833 + 33 01113 096 4 33 00 7 0 2 096

Moon LOP Sight Reduction Moon UL 20 Mar 2013 16 – 30 – 04 00 – 00 5 21 – 30 – 04 7 44 57 5 92 47 1 7 0 36 51 8 4 4 2 0 6 4 - + 36 45 4 29 56 1 21 30 04 7 10 5 37 12 8 92 47 1 304 25 7 55 6 57 5 27 5 37 12 9 0 0 30 0 18 36 5 2 2 18 34 3 G g ʘ 54 5 1 9 12 2 6 2 Navigation Class St Paul Jeff 304 25 7 37 12 9 18 34 3 44 57 5 304 42833 18 57167 44 95333 + 37 17711 101 1 37 10 6 2 3 101

Sun LOP Sight Reduction Sun LL 20 Mar 2013 16 – 45 – 02 00 – 00 5 21 – 45 – 02 11 44 57 6 92 46 4 23 5 26 11 8 13 7 2 0 15 7 - + 25 56 1 133 10 6 21 45 02 11 15 5 144 26 1 92 46 4 51 39 7 14 3 14 3 26 10 4 0 0 00 09 8 1 0 0 8 00 10 6 G g ʘ Navigation Class St Paul John 51 39 7 26 10 4 00 10 6 44 57 6 51 66167 00 17667 44 96000 + 26 17434 119 1 26 10 5 0 1 241

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 44 57 6 92 46 4 21 Mar 2013 23 5 35 38 4 4 7 2 0 6 7 - + 35 31 7 178 40 6 00 25 00 6 15 0 106 07 9 92 46 4 13 21 5 1 4 1 4 35 30 3 0 0 8 11 5 G g ʘ Navigation Class St Paul 281 12 3 ɤ Al 13 21 5 35 30 3 8 11 5 44 57 6 13 35833 8 19167 44 96000 - 35 50327 163 7 35 30 2 0 1 196

Planet Sight Reducation Navigation Class Ben St Paul 20 Mar 2013 17 23 5 19 – 33 – 32 Jupiter 58 42 4 00 – 00 44 57 6 2 0 19 – 33 – 32 92 46 4 13 7 5 + 25 43 2 58 26 1 21 33 1 44 57 6 25 72000 21 55167 44 96000 + 0 0 15 7 00 – 33 – 32 - 15 7 21 Mar 2013 58 26 7 g G ʘ 58 48378 00 110 05 4 0 6 129 5 33 32 8 23 0 58 29 0 + 2 1 + 1 2 00 21 33 0 118 29 6 + 0 1 231 0 0 0 6 2 9 92 46 4 - - + 0 1 0 6 25 43 2 21 33 1 58 26 1

Meridian Transit Terry 21 Mar 2013 13 – 18 – 07 00 – 00 5 18 – 18 – 07 + Terry 44 57 6 92 46 4 92 46 4 88 14 5 18 4 31 9 18 – 18 – 07 5 13 – 18 – 07 + 18 07

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 45 25 5 45 18 8 15 3 45 34 1 + 44 57 6 92 46 4 18 07 ʘ DR LOP 0 30 6 18 1 0 + 0 3 0 30 9 92 46 4 88 14 5 18 4 31 9 18 – 18 – 07 5 13 – 18 – 07 + 45 34 1 44 25 9 0 30 9 44 56 8 44 57 6 1 2

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

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’

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

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

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

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

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

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

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

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°

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’ 1630-1645 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 Kr@p, outside the 3.0 limit & LOP separation only 40° 29 5’ 93°W 55’ 50’ 45’ 40’ 35’

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

Reducing and Plotting Celestial Sights The End