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CELESTIAL NAVIGATION EXERCISES for Class and Home Study
Free downloadable version, without the required pages from the Nautical Almanac or the Sight Reduction Tables
Dominique F. Prinet www.CelestialNavigationBook.com Your comments and recommendations will help improve this book. Please email them to [email protected]. Thank you.
The Exercise Manual for students was designed to facilitate the work of instructors using the free slide presentation available from the website www.CelestialNavigationBook.com. This downloadable version of the Exercise Manual allows students to print the questions offered at regular intervals in the presentation, together with the work-forms guiding the calculations and the answers. Most students taking a course from an instructor following the slide presentation will have the associated book “Celestial Navigation using the Sight Reduction Tables Pub. No. 249”. In order to facilitate the download process, this free version of the
Exercise Manual does not include the data tables required for the calculations since they already are in the Celestial Navigation book. All exercises are for the year 2003. The complete version of the Exercise Manual, with all the data tables, can be purchased in print or electronic format (pdf) from Amazon or retailers of marine books.
Acknowledgments I am grateful to the many students who offered valuable suggestions over the years and in particular to Steven Buchi, Eng., who verified all the answers. I would also like to express my appreciation to Linda Mitsui, Graphic Designer, who has spent countless hours setting up the text, tables and graphics.
iii
CELESTIAL NAVIGATION EXERCISES for Class and Home Study Table of Contents
QUESTIONS Session 1: Sextant Corrections (Exercises 1-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 1 Session 2: Time; Conversions between arcs and Time; GHA and Dec (4-10). . . . . . . . . . . p. 3 Session 3: Interpolations of GHA & Dec; LHA; Lat. by Noon Sight (11-15). . . . . . . . . . . p. 9 Session 4: Sight Reduction Tables (16-19). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 13 Session 5: Review Exercises on the Sun; Plotting (20-28) . . . . . . . . . . . . . . . . . . . . . . . . p. 21 Session 6: Twilight; Moon (29-33). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 29 Session 7: Planets; Stars; Selected Stars; Polaris (34-40) . . . . . . . . . . . . . . . . . . . . . . . . . p. 33 Session 8: Review Exercises, entire course (41-58). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 41 Additional Home Exercises: Home Exercises 1-9 following Session 3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 55 Home Exercises 10 and 11 following Session 4:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 67 The exercises are for the year 2003. The required Almanac and Sight Reduction Tables are given in the Appendix of the book “Celestial Navigation using the Sight Reduction Tables Pub. No. 249”.
ANSWERS Exercise Answers 1-58. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 75 Home Exercise Answers 1-11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 113
continued... www.CelestialNavigationBook.com
v
Table of Contents
Appendix 1: Sight Reduction Work Forms
vi
A1-1 Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 129
A1-2 Noon Sight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 130
A1-3 Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 131
A1-4 Planet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 132
A1-5 Star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 133
A1-6 Selected Stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 134
A1-7 Latitude by Polaris. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 135
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Session 1 of the Slide Presentation Exercises 1-3 Chapters 1 & 2 of the Celestial Navigation book
Sextant Corrections
1
Session 1, Exercises 1-3
1. Sextant corrections, index error Sextant altitude Hs
27° 45.0'
Index error
59° 13.0'
21.0' Off
Hs corrected for index error “On the arc”: – “Off the arc”: +
° .
'
07° 23.7
28.0' On
° .
'
36.4' Off
° .
'
2. Sextant corrections, correction for Dip Sextant altitude Hs corrected for index error
27° 05.7'
Height of eye
DIP (–)
5 9° 03.0'
40 m
4.8 m
– .
'
° .
'
Apparent altitude Ha
6 feet
.
'
° .
'
–
0 7° 23.2'
.
'
° .
'
–
3. Sextant corrections, main correction Body observed
Time of year
October
Apparent altitude Ha Main correction, for the month and Ha: LL + UL – Observed altitude Ho
2
Sun, UL
20° 09.1'
Sun, LL
Sun, UL
June
March
49° 53.9'
.
'
.
° .
'
° .
www.CelestialNavigationBook.com
17° 06.1'
' .
'
'
'
° .
Session 2 of the Slide Presentation Exercises 4-10 Chapters 3 & 4 (beginning) of the Celestial Navigation book
Measures of time; Conversions arcs
time;
Greenwich Hour Angle; Declination for round UTC hours
3
Session 2, Exercises 4-10
4. Sextant corrections, all corrections combined Calculate the “Observed Altitude Ho” of the sun above the horizon, after all sextant corrections. Note the corrections during summer or winter months, and for Upper or Lower Limb.
Body observed
Sun, LL
Sun, UL
Sun, LL
April
July
November
Sextant altitude Hs
37° 59.1'
49° 03.0'
17° 56.6'
Index error
2.7' Off
8.1' On
4.5' Off
2.9 m
3.3 m
9.0 feet
Hs corrected for index error “On the arc”: – “Off the arc”: +
° ' .
° ' .
° ' .
DIP
Apparent altitude Ha
° ' .
° ' .
° ' .
Main correction, for the month and Ha: LL + UL –
Observed altitude Ho
° ' .
Time of year
Height of eye
4
–
' .
' .
–
' .
' .
° ' .
www.CelestialNavigationBook.com
–
' .
' .
° ' .
Session 2, Exercises 4-10
5. Corrections to the chronometer
CHRONOMETER 1 CRONO. CORRECTION UTC TIME
CHRONOMETER 2 CRONO. CORRECTION UTC TIME
CHRONOMETER 3 CRONO. CORRECTION UTC TIME
Chronometer 1:
01:39:24;
02 min 28 s fast
Chronometer 2:
22:09:19;
12 min 39 s fast
Chronometer 3:
18:53:19;
08 min 47 s slow
h
min s
+ –
min s
h
min s
h
min s
+ –
min s
h
min s
h
min s
+ –
min s
h
min s
SLOW FAST
SLOW FAST
SLOW FAST
6. Almanac table “Conversion of Arcs to Times”, to convert arcs (angles) to times A. Determine the time difference corresponding to the angle (arc) between the meridian of Greenwich (longitude: 0°) and that of the boat. This time difference represents the time it takes for the sun to travel from one meridian to the other at 15° of longitude per hour. Boat longitude
123° 08.25' W
068° 28.75' W
041° 37.50' E
Hours and min for the ° of angle
:
min and s for the ' of angle
00 :
:
s
00 :
:
s
00 :
:
s
:
:
s
:
:
s
:
:
s
Total time difference Greenwich to boat
: 00 s
:
: 00 s
www.CelestialNavigationBook.com
:
: 00 s
5
Session 2, Exercises 4-10
7. Almanac pages “Increments and Corrections”, to convert times to arcs (i.e. to angles) Knowing the time it takes for the sun to travel from one meridian to another, we can calculate the angular distance between the meridians, i.e. the difference in longitude between the two meridians. UTC time of sun’s crossing of boat meridian (noon sight) UTC time of sun’s crossing of Greenwich meridian (Almanac) Time for sun’s travel from one meridian to the other
6
19 : 31 : 18
23 : 08: 12
12 : 16 : 00
11: 48: 00
:
:
:
:
01 : 23: 49 12 : 09: 00
(previous UTC day)
:
:
° of longitude for the hours only
°
° and ' of longitude for the minutes and seconds of time
° ' .
° ' .
° ' .
Total = difference in longitude
° ' .
° ' .
° ' .
0 0 . 0 '
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°
0 0 . 0 '
°
0 0 . 0 '
Session 2, Exercises 4-10
8. Prediction of time (UTC) when the sun will cross the boat meridian Knowing, from the Almanac daily pages, the UTC time when the sun crosses the meridian of Greenwich, and given the longitude of the boat, we can determine the UTC time when the sun will cross the meridian of the boat. Time of sun’s crossing of the Greenwich meridian (from the Almanac)
12 : 09 : 28 s
11 : 47 : 46 s
Longitude of the boat
123° 12.50' W
65° 28.75' E
1. Time it will take the sun to travel from Greenwich to the boat, or from the boat to Greenwich, for the ° only of boat longitude 2. Additional time it will take the sun to travel because of the ' (and fractions of ') of the boat longitude
:
+ 0 0 :
: 00 s
:
s
:
s
:
+ 0 0 :
12 : 13 : 11 s
18° 49.25' W
: 0 0 s
:
+ 0 0 :
:
s
:
s
: 00 s
:
s
:
s
3. Total time between Greenwich and the boat longitude
4. Time of sun crossing of the Greenwich meridian (UTC)
5. T otal time from 3. (+ if boat long W; – if boat long E)
+ –
:
:
s
+ –
:
:
s
+ –
:
:
s
UTC time when the sun will cross the boat meridian (4 + or – 5)
:
:
s
:
:
s
:
:
s
:
12 :
09 : 28 s
:
11 :
www.CelestialNavigationBook.com
47 : 46 s
:
12 :
13 : 11 s
7
Session 2, Exercises 4-10
9. Meridian and zone times; determination of longitude 1. What is the UTC time when the sun crosses the meridian of Greenwich, July 1, 2003?__________________________ 2. What is the meridian time when the sun crosses the meridian of Vancouver? _________________________________ 3. On what meridian is the “zone time” based for the province?______________________________________________ 4. When, on July 1, 2003, the sun is at its highest point over English Bay (long: 123º 10.5' W), what is the zone time in BC?
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
5. When it is 13:32 in BC (zone time) on that day, what time is it along the meridian of Greenwich?_________________ 6. Captain Vancouver sets his watch to read 12:00:00 when the sun is at its zenith over English Bay February 9, 1793 (Long. 123º 10.5' W). Exploring the West Coast of Vancouver Island, five days later, he notes that the sun is at its zenith at 12:14:05. He knows that his chronometer loses 1s per day. What name does he give to the Sound which he has just discovered?
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
10. GHA and Dec from the Almanac for the sun’s GP (on the hour only) Read the coordinates of the sun’s GP on the Almanac (GHA and Dec) for the days and times (UTC).
8
Day and time
April 19
June 30
Oct 29
Time (UTC)
14:00:00
05:00:00
17:00:00
GHA
° ' .
° ' .
° ' .
Dec
° ' .
° ' .
° ' .
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Session 3 of the Slide Presentation Exercises 11-15 Chapters 4 (continued), 5, 6, 7 & 8 of the Celestial Navigation book
Interpolation of GHA and Dec for Exact Times of Sights; Local Hour Angle; Latitude by Noon Sight
9
Session 3, Exercises 11-15
11. GHA and Dec from the Almanac for the sun’s GP (interpolation for min & s) Read the coordinates of the sun’s GP on the Almanac (GHA and Dec) for the hours, and interpolate for minutes and seconds of time. Day and time
Feb 02
July 01
Oct 29
Time (UTC)
13 : 15 : 25 s
05 : 06 : 13 s
17 : 28 : 33 s
° .
'
° .
'
Total GHA
° .
Dec for hours only (N or S)
° .
GHA for hours only Increment of GHA for minutes and sec
(d
+ or –
in '/ h)
Increment of Dec for min Total Dec
+
(d =
° .
'
° .
'
'
° .
'
° .
'/ h)
+
(d =
° .
'
° .
'
'
° .
'
'
° .
'
'/ h)
+
(d =
'/ h)
.
'
.
'
.
'
° .
'
° .
'
° .
'
12. Local Hour Angle Given the longitude of the boat, and the GHA of the sun, is the sun east or west of us? In other words, are we taking the sight in the morning or afternoon? How many degrees of longitude is the sun’s GP away from out boat, measured westward, i.e. the Local Hour Angle (LHA). Note: LHA = distance of from the boat longitude to the sun’s GP, counted westward.
If boat longitude is West, LHA = GHA – boat longitude; If boat longitude is East, LHA = GHA + boat longitude.
Add 360º to the GHA in order to allow the subtraction of the west longitude of the boat if required. Remove 360º from the result if the addition of the east longitude of the boat results in a number larger than 360.
10
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Session 3, Exercises 11-15
GHA
158° 52.4'
Boat longitude
123° 49.3' W
19° 23.8'
98° 32.9' W
205° 42.6' 27° 59.5' E
GHA (add 360° if required)
° .
'
° .
'
° .
'
+ longitude E – longitude W
° .
'
° .
'
° .
'
LHA = Total (subtract 360° if required)
° .
'
° .
'
° .
'
AM or PM
13. Latitude by noon sights Determine the boat latitude knowing the maximum sun altitude of the sun over the horizon (Ho max, sun crossing of boat meridian), and the Dec of the sun at the time of sight (calculated from the Almanac). At noon (boat-meridian time), the sun is at its zenith when it crosses the observer’s meridian. A sight on the sun at that instant allows the calculation of the latitude from the sun altitude Ho max: the latitude is the Zenith Distance (ZD = 90° – Ho) plus or minus the Declination of the sun, depending on whether the sun’s declination is North or South of the equator. If the boat is between the equator and the sun’s GP, then the latitude is Dec – ZD. Dec & Lat. “same name”
Dec & Lat. “contrary names”
Lat. > Dec
Lat. < Dec (Tropics)
Lat. = Dec + ZD
Lat. = Dec – ZD
Lat. = ZD – Dec
Approximate boat latitude
50° N
28° N
10° S
Sun’s Declination from Almanac: Dec:
21° 18.3' N
12° 24.0' S
21° 23.0' S
90° =
89° 60.0'
89° 60.0'
89° 60.0'
Measured sun’s altitude Ho
61° 32.4'
49° 45.9'
78° 45.2'
Zenith distance ZD = 90° – Ho
° .
'
° .
'
° .
'
Boat latitude = “Dec + ZD” or “Dec – ZD” or “ZD – Dec”
° .
'
° .
'
° .
'
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11
Session 3, Exercises 11-15
14. Approximate latitude by Polaris Even without the several small corrections from the Polaris Almanac tables, a sight on Polaris can give us our latitude, with an error of, at the most, 1.5°: Lat. = Ho. Altitude of Polaris, after correction of all sextant and sight errors: Ho =
49° 59.5'
'
° .
Latitude of observer
68° 71.2'
89° 00.0'
'
° .
'
° .
15. Approximate longitude by noon sights, knowing the time when the sun crossed the boat meridian Knowing the UTC time of passage of the sun over the Greenwich meridian (from the Almanac), and having established the approximate UTC time of passage of the sun over our local meridian from a series of sights of the sun before, during and after noon (boat-meridian time), we can calculate our approximate longitude: UTC time of sun’s crossing of Greenwich meridian (Almanac)
12 : 16
11: 48
12 : 09
UTC time of sun’s crossing of boat meridian (estimated from a series of sextant sights)
19 : 26
22 : 33
01 : 54
Difference in time (bigger number minus smaller one)
:
:
:
° of longitude for the hours °
12
°
°
° and ' of longitude for the minutes of time
° .
'
° .
'
° .
'
Total = “boat longitude” (E if local noon is earlier; W if local noon is later)
° .
'
° .
'
° .
'
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Session 4 of the Slide Presentation Exercises 16-19 Chapters 9, 10 & 11 of the Celestial Navigation book
Sight Reduction Tables
13
Session 4, Exercises 16-19
16. Azimuth Angle of the sun “Z” (bearing of the sun “Zn”), and calculated altitude “Hc” from the Sight Reduction Tables Pub. No. 249 (without any interpolation). Given an assumed latitude (in round degrees), a Declination of the sun (also in round degrees), and a LHA (also in round degrees) between the assumed longitude of the boat and the sun meridian (GHA), we can read, in the Sight Reduction Tables, the azimuth “Z” of the sun, and calculate its bearing “Zn”. We can also read the calculated altitude “Hc” that the sun would have if we were exactly at our assumed position. Assume that the sun crosses the meridian of Greenwich at exactly 12:00 each day. This allows the calculation of the sun’s GHA without the use of the Almanac tables. Assumed latitude
Date of sight
49° N
22 June
Declination of the sun (approximate, in round degrees)
°
42° N
22 September °
00'
40° S 22 December
°
00'
00'
“Same name” or “Contrary name”? UTC time of the sun sight
4 : 0 0 '
1 7 : 0 0 '
2 0 : 0 0 '
Greenwich Hour Angle of the sun
°
°
°
Assumed longitude of the boat
060° E
Local Hour Angle (angle between the meridians of the boat and of the sun) Hc for the ° of Dec
°
00'
°
180° or 360° if needed (see “N. Lat.” ° & “S. Lat.” cases below) Azimuth angle Z Sun bearing Zn
00'
+/–
° °
015° W
'
°
00'
°
00'
°
00'
°
00'
°
N. Lat.: If LHA greater than 180°............. Zn = Z If LHA less than 180°.................. Zn = 360° – Z S. Lat.: If LHA greater than 180°............. Zn = 180° – Z If LHA less than 180°.................. Zn = 180° + Z
14
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00'
00'
135° W
'
°
00'
°
00'
°
00'
00'
°
00'
00'
°
00'
'
Session 4, Exercises 16-19
17. Bearing of the sun "Z" (sun's azimuth “Zn”), and calculated altitude “Hc” from the Sight Reduction Tables Pub. No. 249 (with interpolation for the ' of Dec). Given an assumed latitude (in round degrees), and a Local Hour Angle (also in round degrees) between the assumed meridian of the boat and the sun meridian, we can read, in the Sight Reduction Tables, the azimuth “Z” of the sun; then calculate the bearing “Zn” of the sun; and calculate the altitude “Hc” that the sun would have if we were exactly at our assumed position. By comparing Hc with Ho (actual altitude of the sun, measured with the sextant), we can tell how far off we are from our assumed position. This is the Marcq Saint Hilaire method. Assumed latitude Declination of the sun for the exact time of the sight “Same name” or “Contrary name”? Local Hour Angle (between the boat and sun meridians) 1. Hc for the ° of Dec 2. Change in ’of Hc for each ° of Dec d = + /– ( '/° ) 3. Interpolation of Hc for ' of Dec (Table 5, last page of HO 249) Total Hc (1 + 3)
42° N
11° 28.3' S
49° N
19° 54.8' S
49° N
21° 09.1' N
43°
325°
292°
° ' .
° ' .
° ' .
+ –
+ –
+ –
(
'/° )
(
'/° )
(
'/° )
+ ' –
+ ' –
+ ' –
° '
° '
° '
°
°
180° or 360° if needed (see “N. Lat.” & “S. Lat.” cases ° below) Azimuth angle Z ° +/– ° Sun bearing (azimuth) Zn
00' 00' 00'
° °
00' 00' 00'
° °
00' 00' 00'
N. Lat.: If LHA greater than 180°............. Zn = Z If LHA less than 180°.................. Zn = 360°– Z S. Lat.: If LHA greater than 180°............. Zn = 180°– Z If LHA less than 180°.................. Zn = 180° + Z
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15
Session 4, Exercises 16-19
18. Choices of “Assumed Lat.” and “Assumed Long.” to simplify calculations using the Sight Reduction Tables Pub. No. 249 Given a DR latitude, we can chose our assumed latitude as the nearest latitude in round degrees. We can chose our assumed longitude in such a way that the Local Hour Angle (angle between the assumed meridian of the boat and the sun meridian) is a round number of degrees: If the boat is WEST of the Greenwich meridian (west longitude), LHA = GHA-Assumed Boat Longitude. We need to choose, for our assumed longitude, the same number of minutes than in the GHA, since the minutes will cancel out by subtraction. In the case of longitude EAST, LHA = GHA + Assumed Boat Longitude. We need to choose, for our assumed longitude, a number of minutes equal to (60' – minutes of GHA): the minutes will cancel out by addition. In either case, the degrees of assumed boat longitude may have to be adjusted by 1 (up or down) so that the assumed position is as close as possible as the DR. DR coordinates: latitude longitude
Assumed latitude
49° 28.3' N 123° 19.7' W ° 00' N
Greenwich Hour Angle (at 011° time of sight) Add 360° to GHA if required to allow subtracting ° the assumed longitude Assumed longitude For W Long, use the ' of GHA For E Long, ° use “60 – ' of GHA” ° Local Hour Angle
37.2' .
'
45° 54.8' N 009° 12.4' E
° 00' N
° 00' S
219° 23.1'
321° 09.1'
° .
' ° . W ° 00'
N. Lat.: If LHA greater than 180°............. Zn = Z If LHA less than 180°.................. Zn = 360°– Z S. Lat.: If LHA greater than 180°............. Zn = 180°– Z If LHA less than 180°.................. Zn = 180° + Z
16
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21° 09.1' S 146° 51.1' W
'
° .
' ° . E ° 00'
'
' . W 00'
Session 4, Exercises 16-19
19. Review exercise: From the Sight Reduction Tables Pub. No 249, we can determine the bearing of the sun “Zn" and calculated altitude “Hc”. By comparing Hc with our observed altitude Ho, we can calculate by how much the assumed arc of circle of position should be moved (Marcq Saint Hilaire method). Sight No: DR latitude DR longitude Sun’s GHA Sun’s Dec (corrected) Observed altitude Ho
# 19
48° 123° 78° 22° 46°
45.3' N 03.9' W 26.5' 35.3' N 05.2'
Find the Local Hour Angle LHA; the coordinates of the assumed position, the computed altitude Hc; the azimuth angle Z; the bearing of the sun Zn; and the intercept (“Towards” or “Away from” the sun’s GP). Use the following work forms, with a blank globe, and the pre-calibrated blank Mercator charts.
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17
Session 4, Exercises 16-19
Sight 19 SIGHT REDUCTION TABLES: INTERCEPT, and BEARING Zn (“Azimuth”) of the SUN Sun’s GHA (Almanac, Daily pages); LHA 1 2
DR LAT DR LONG
N S E W
º º
. .
'
Sun’s GHA
16
15 + 360° if required to allow subtraction in 18
+ 3 6 0° 00 .0 ' º .
17
ASSUMED LONG º from (2) [ ' in 16] if long W W– [60' – ' in 16] if long E E+
º
º
'
18 19
LOCAL HOUR ANGLE 16 + or – 17 Subtract 360° if LHA > 360° Copy LHA to 23
º
'
15
.
'
.
0 0 . 0' 0 0 . 0'
– 3 6 0º
Sight Reduction Tables HO 249 23 24 25
26
27 N. Lat.: { LHA > 180°, …………Zn = Z { LHA < 180°, …………Zn = 360 – Z S. Lat.: { LHA > 180°, …………Zn = 180 – Z { LHA < 180°, …………Zn = 180 + Z
28
LHA (total in 18 above, minus 360° if LHA > 360°) ASSUMED LAT N from DR in 1, rounded off S DEC CORRECTED N S COMPUTED ALT Hc for 23, 24 & 25; same or contrary; for DEC ° only. Note d ( + / – ) in 27, & Z in 31 d (+ / – ' per ° of Dec) CORR’N to Hc for DEC ' table 5 (back of book), for ' in 25 HC CORRECTED for ' of DEC in 25
º
º
º
º
30
INTERCEPT Ho > Hc: T (28 – 29 or 29 – 28) Ho < Hc: A
T/A
31
N. LAT
32
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360º
Z
º
Zn
º
.0
.0
. .
S. LAT
'
.
+ –
º
0 0 . 0'
.0
º
OBS. ALT Ho
0 0 . 0'
29
18
180º
Z
º
Zn
º
'
' ' '
Session 4, Exercises 16-19
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19
Session 4, Exercises 16-19
20
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Session 5 of the Slide Presentation Exercises 20-28 Chapters 1-11 of the Celestial Navigation book
Review of the Sun; Plotting
21
Session 5, Exercises 20-28
20. Interpolate GHA for the exact time of the sight Sight taken on April 21, 2003, at 06:21:33 UTC. 1. Estimate the approximate GHA of the sun. ________________________________________________________ 2. Calculate the exact GHA of the sun.
____________________________
____________________________
____________________________
21. Interpolate Dec for the exact time of the sight Sight taken on August 22, 2003, at 14:06:18 UTC. 1. Estimate the approximate Dec of the sun. _______________________________________ 2. Calculate the exact Dec of the sun.
_______________________________________
_______________________________________
_______________________________________
22. Calculate the LHA
1. B oat at 120º 18.3' W 2. Boat at 147º 58.2' E Sun’s GHA at 105º 13.2' W Sun’s GHA at 255º 08.9' W LHA? LHA?
________________________________
______________________________________
_________________________________
______________________________________
_________________________________
______________________________________
_________________________________
______________________________________
_________________________________
______________________________________
23. Read Hc and Z; calculate Zn (no interpolation) Boat Lat.: 39º N Dec of the sun: 13º N LHA: 301º 1. During what months of the year would the sight have been taken? _____________________________________________________ 2. Was this a morning or afternoon sight? ____________________________________________________________________________________ 3. What was the sun’s exact altitude Hc over the horizon at the time of sight?____________________________________________ 4. What was the sun’s azimuth Z? _____________________________________________________________________________________________ 5. What was the sun’s bearing Zn? _____________________________________________________________________________________________ 22
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Session 5, Exercises 20-28
24. Interpolate Hc for the specific Dec at the time of sight Boat Lat.: 49º N Dec of the sun: 19º 15' S LHA: 034º What is the exact altitude of the sun (Hc) at the time of sight? _________________
25. Plot a Mercator chart for the location of the boat Prepare a Mercator chart centered on 40º N, 141º W, with the parallels one degree apart.
26. Plot a morning sight UTC date: Log: Course: Speed: DR position: Time of sight: Chrono error: Body: Sextant Alt: Index error: Height of eye:
04 July 2003 1,350 NM 250° T approx. 5 kn (use the log to determine the exact distance travelled) 40° 17.0' N; 140° 08.4' W 17 h 06 min 37 s UTC slow, 0 min 42 s Sun LL Hs = 33° 45.3' on the arc, 4.8' 3.1 m
Draw the LOP at 17 h 07 min
27. Plot a noon sight UTC date: 04 July 2003 Log: 1,376 NM Course: 250° T Speed: approx. 5 kn (use the log to determine the exact distance travelled) DR position: 40° 12.0' N; 140° 28.0' W Body: Sun UL Sextant Alt: Hs = 73° 03.2' Index error: off the arc, 3.5' Height of eye: 12.0 ft Draw the noon LOP; mark the noon fix, by advancing the morning LOP.
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23
Session 5, Exercises 20-28
Sight 26
24
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Session 5, Exercises 20-28
Plotting of sights 26, 27 and 28
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25
Session 5, Exercises 20-28
Sight 27
26
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Session 5, Exercises 20-28
28. Plotting an afternoon sight UTC date: Log: Course: Speed: DR position: Time of sight: Chrono error: Body: Sextant Alt: Index error: Height of eye:
05 July 2003 1,409 NM 250° T approx. 5 kn (Use the log to determine the exact distance travelled.) 39° 53.8' N; 141° 03.2' W 03 h 05 min 24 s UTC slow, 0 min 42 s Sun LL Hs = 18° 30.8' off the arc, 2.6' 3.3 m
Draw the LOP at 03 h 06 min; mark the 03 h 06 min fix by advancing the noon LOP (latitude line).
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27
Session 5, Exercises 20-28
Sight 28
28
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Session 6 of the Slide Presentation Exercises 29 - 33 Chapters 12 & 13 of the Celestial Navigation book
Twilight; Moon
29
Session 6, Exercises 29-33
29. Twilight Twilights times for sunset, Nov. 18, 2003, boat at 47° 38.2' N , 125° 18.7' W . a) At boat latitude, along the Greenwich meridian (or along the boat meridian, using the boat-meridian time): Almanac Times Latitude
Sunset
Civil Twilight
Nautical Twilight
At this Lat. #1, on Greenwich meridian
45° N
:
:
:
At this Lat. #2, on Greenwich meridian
50° N
:
:
:
Difference for 5° of Lat.
:
:
:
Difference for 1° of Lat.
:
:
:
Difference for boat ° of Lat. above or below Lats. #1 or 2
:
:
:
:
:
:
Twilight times for an observer at the boat latitude, on the Greenwich meridian (Note: twilight times for an observer on the boat are the same if the boat clock is set at the local (boat) meridian time)
b) Using the times applicable to the time zone: Zone times Sunset
Civil Twilight
Nautical Twilight
Boat longitude
° .
'
° .
'
° .
'
Nearest meridian at the center of the time-zone (every 15º from Greenwich)
° .
'
° .
'
° .
'
Difference in longitude between “boat meridian” and “meridian at the center of the time zone”
° .
'
° .
'
° .
'
Time for sun travel between the two meridians (conversion of arc to time) Twilight times for an observer at the boat latitude on the Greenwich meridian (directly from table in a) Add or remove the time for the sun travel, to obtain the zone-times for the twilights
:
:
:
:
:
:
:
:
:
c) When would you take a sight on stars or planets? __________________
30
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Session 6, Exercises 29-33
30. Moon observed altitude Ho Moon, April 19, 2003; UTC 12:14:35. Assume a measured sextant altitude Hs on LL = 31°18.0', after corrections for index error & Dip, but before the correction specific to the moon. Ho? .
Horizontal Parallax “HP” for April 19, 12:00 Z MOON, LL Measured sextant altitude after corrections
' ° .
'
Apparent altitude Ha for Hs = 31°18.0' (“Altitude Correction Table – Moon”, top table):
+ .
'
Parallax for LL, given HP from above (“Altitude Correction Table – Moon”, lower table):
+ .
'
° .
'
° .
'
° + .
'
+ .
'
° .
'
Hs =
Remove 30' for an UL sight Ho for the moon at 12:14:35
31. Moon GHA Moon, April 19, 2003; UTC 12:14:35. GHA? MOON GHA for 12:00 (Almanac page for April 19) Increment for 14 min 35s (Inc. & Corr. for 14 min & 35s, moon column) Adjustment for actual rotation of the moon, slightly above the minimum of 14°19'/hr “v” = '/hr Total GHA for the moon, at 12:14:35 (sum)
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31
Session 6, Exercises 29-33
32. Moon Declination Moon, April 19, 12:14:35 UTC. Dec? MOON Dec for 12:00 (Almanac page for April 19)
°
.
Increment in Dec for 15 min, using “d” above in “Inc. & Corr.” table for 15 min
+ _
.
Total Declination for the moon around 12:14 UTC
°
.
'
Increment rate for Dec, per hour, around 12:00 “d” = '/h ' '
33. Moon, complete calculations: Ho + GHA + Dec Moon, July 04, 2003; time of sight: 20: 20: 45 UTC; Ha = 31° 48.0' on UL, after all corrections for index error and for Dip, but before all the corrections specific to the moon.
Moon altitude & corrections
Moon GHA
(end of Almanac, Alt Corr. Tables for Moon) 5
Time of sight (after corrections) UTC
10
UL APPARENT ALT. Ha LL
11
12
MAIN CORRECTION to Ha for ° of “App. Alt”, & nearest 10' (top table) Additional CORRECTION for “HP” (from daily Almanac) and for “UL” or “LL”, (same column, bottom table), given the Parallax Correction for day and hour of sight, “HP” = '
13
Subtract 30' for a sight on UL
14
OBSERVED ALT. Ho (10) + (11) + (12) [ – (13) ?]
32
:
(Almanac, Daily pages) :
º
.
+ .
15 '
16
' 17
18
+ .
GHA of the moon For that day, & hours in (5) MAIN INCREMENT in GHA for min & s in (5) (Inc. & Corr. Table, moon) Additional increment “v” for the moon, for min in (5) (Inc. & Corr. Table “v or d”) ( “v” = + '/h)
' – . ' º .
º
' .
+
º
' . ' .
+
GHA TOTAL (18) +(19) + (20)
'
º
' .
º
' .
Moon Declination
(Almanac, Daily pages) 24
DEC of the moon For that day, & hours in (5)
25
INCREMENT in DEC for min & s in 5 (“d” = + / – '/h) (Inc. & Corr. table “v or d”)
26
DEC TOTAL (24) + (25)
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N S
' + – . N S
º
' .
Session 7 of the Slide Presentation Exercise 34 - 40 Chapters 14 to 17 of the Celestial Navigation book
Planets; Stars; Selected Stars; Polaris
33
Session 7, Exercises 34-40
34. Twilight (review) UTC date: DR position:
31 January 2003 47° 40.0' N; 158° 15.0' W
What would be a good time (in UTC) to take a dawn sight of Venus? Nautical Twilight
Civil Twilight
At boat Lat. 48° N, for the observer on Greenwich meridian (UTC)
:
:
:
Add the hrs & min for the degrees of boat longitude
:
:
:
Add the minutes of time for the minutes of boat longitude
:
:
:
Twilight times in UTC for the observer on the boat
:
:
:
Recommended time of sight:______________________________UTC.
34
Sunrise
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Session 7, Exercises 34-40
35. Moon (review) Moon, April 20, 12:06:15 UTC; Ha = 17° 19.9' on LL, after all sextant and Dip corrections. Ho? GHA? Dec?
Moon altitude & corrections
Moon GHA
(end of Almanac, Alt Corr. Tables for Moon) 5 10 11
12
13 14
Time of sight : (after corrections) UTC UL º APPARENT ALT. Ha LL MAIN CORRECTION to Ha for ° of “App. Alt”, & nearest 10' (top table) + Additional CORRECTION for “HP” (from daily Almanac) and for “UL” or “LL”, (same column, bottom table), given the Parallax Correction for day and hour of sight, “HP” = ' + Subtract 30' for a sight on UL – OBSERVED ALT. Ho º (10) + (11) + (12) [ – (13) ?]
(Almanac, Daily pages) 15
: .
'
.
'
16
17
18 . . .
GHA of the moon For that day, & hours in (5) MAIN INCREMENT in GHA for min & s in (5) (Inc. & Corr. Table, moon) Additional increment “v” for the moon, for min in (5) (Inc. & Corr. Table “v or d”) (“v” = + '/h) GHA TOTAL (15) +(16) + (17)
º
.
'
+
º
.
'
+ . º .
'
.
'
'
Moon Declination
' '
(Almanac, Daily pages) 24
' 25 26
DEC of the moon N For that day, & hours in (5) S INCREMENT in DEC for min & s in 5 (“d” = + / – '/h) (Inc. & Corr. table “v or d”) DEC TOTAL N (24) + (25) S
º
+ – . º .
' '
36. Planets (Venus) UTC date: 31 January 2003 Log: 1,250 NM Course: 200° T DR position: 47° 40.0' N; 158° 15.0' W Chrono error: fast, 2 min 20 s Body: Venus Sextant Alt: Hs = 09° 50.2' Index error: off the arc, 10.0' Height of eye: 2.9 m Under a very clear sky rendering the horizon visible in the general direction of the rising sun, the sight on Venus is taken at 16 h 31 min 38 s UTC, slightly before the beginning of the nautical twilight. Using the attached work-form, complete the Sight Reduction calculations, and plot the LOP for the sight on Venus. www.CelestialNavigationBook.com
35
Session 7, Exercises 34-40
Sight 36
36
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Session 7, Exercises 34-40
Plotting of sight 36
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37
Session 7, Exercises 34-40
UTC date: Body: Time of sight: GHA? Dec?
'
+
º
. º . º . º .
'
min
s
15
GHA TOTAL for Aries (13) + (14)
16
SHA of Star* for that day
17
GHA for the Star* (15) + (16)
26
DEC of Star* N S
14
03 July 2003 Vega 13:48:12 UTC
. º .
GHA for Aries for hours of time of sight INCREMENT in GHA for Aries for min & s; Inc. & Corr. table
13
37. Stars, GHA, Dec
º
'
' ' '
38. Stars (Betelgeuse), complete calculations Betelgeuse; 02 Feb 2003, 03:14:22 UTC; DR: 39º15'N, 130º30'W; GHA? LHA? Dec? Zn? 1 2
DR LAT C opy ° to (25) N after rounding S DR LONG E W
. º . º
' '
UTC DATE 5
GHA of Aries (Almanac, Daily pages); GHA of Star (Almanac, Daily pages); LHA of Star 13 14 15 16 17
18 20 21
GHA for Aries for hours in (5) INCREMENT in GHA for Aries for min & s (5); Inc. & Corr. table
+
. º .
GHA TOTAL for Aries (13) + (14) SHA of Star* for that day GHA for the Star* (15) + (16)
º
º
º
GHA for the Star* (17), or (17) + 360° if required ASSUMED LONG º from (2) [ ' in (19) ] if long W W– [60' – ' in (19)] if long E E+ LOCAL HOUR ANGLE (Star) [(17) or (19)] +/– (20)*
+360º º
N. Lat.: { LHA > 180°, …………Zn = Z { LHA < 180°, …………Zn = 360 – Z S. Lat.: { LHA > 180°, …………Zn = 180 – Z { LHA < 180°, …………Zn = 180 + Z 38
º
UTC TIME
h
Reduction Tables, Pub. No. 249 '
24
'
25
' . ' . ' . 00.0 ' .
26
27
28 29
' º . º 00.0'
LHA (total in (21) above, minus 360° if required)* ASSUMED LAT N from DR in (1), rounded off S DEC of Star* N S COMPUTED ALT Hc for 27, 28 & 29; same or contrary, for Dec ° only. Note d ( + / – ) in 28, & Z in 30 d( + / – ' per ° of Dec) CORR’N to Hc for ' of Dec. Use “Table 5” (back of book) HC CORRECTED for ' of Dec
32
N. LAT
33
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º
Zn
º
0 0 . 0' º 0 0 . 0' ' º .
º
' .0
' + – .0 ' º .0
360º
Z
º
S. LAT
180º
Z
º
Zn
º
Session 7, Exercises 34-40
39. Selected Stars UTC date: DR position:
October 29, 18:15:12 11° 28.0' N; 164° 17.5' E
Hc and Zn of the three best stars for a sight?
1 2
DR LAT C opy ° to (13) after rounding DR LONG
5
UTC times
6
GHA for Aries
7 8
N S E W
. .
º º
' '
:
:
º
INCREMENT in GHA for Aries for min & s (5); Inc. & Corr. table
+
º
GHA TOTAL for Aries (6) + (7)
º
for hours in (5)
+ 360º 9 10 11
12 13
. .
'
.
'
'
00.0
º
.
'
º
.
'
LOCAL HOUR ANGLE Aries (8) or (9) +/– (10) Subtract 360º if required
º
LHA Aries (11), minus 360° if required ASSUMED LAT from DR in (1), rounded off
º
º
GHA for Aries (8), or (8) – 360° if required ASSUMED LONG º from (2) [ ' in (16) ] if long W [60' – ' in (16)] if long E
W– E+
–360º
N S
00.0' 00.0 00.0' 00.0'
STAR NAMES: 21 23
COMPUTED Alt Hc for Stars, from Vol. 1, º given 12 (LHA ) and 13 (Assumed Lat.) COMPUTED Zn for Star, from Vol 1, º given 12 (LHA ) and 13 (Assumed Lat.) Zn
' .0
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º º
' .0
º
' .0
º
39
Session 7, Exercises 34-40
40. Sight with Polaris, complete calculations Celestial body: UTC date: Time of sight: Chrono error:
Polaris June 15, 2003 11:20:42 (UTC) 4 min 36s slow
DR position: 43º 30' N; 128º 20' W Sextant altitude Hs: 43º 32.1' Index error: “on the arc”: 12.5' Height of eye: 2.5m above water
Boat latitude? Exact bearing of Polaris (Zn)? Hint: Boat latitude = Ho (on Polaris) –1º + a0 + a1 +a2
40
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Session 8 of the Slide Presentation Review Exercises 41 – 58 Chapters 14 to 17 of the Celestial Navigation book
Review Exercises: Sun, Twilight, Moon, Planets, Stars, Selected Stars, Polaris
41
Session 8, Review Exercises 41-58
41. Time of sight Chrono 11:02:12 (UTC) Chrono fast, 03 min 30 s UTC corrected time?
CHRONOMETER
h
min s
CRONO. CORRECTION
SLOW + FAST –
min s
UTC TIME
min s
h
42. Sun altitude Ho over the horizon (Almanac, Altitude Correction Table) UTC date: 15 March 2003 Body: Sun LL Sextant altitude: Hs = 38º 00.0' Index error: 12.0' on the arc Height of eye: 2.9 m above water Observed altitude Ho?
SEXTANT ALT Hs
7
INDEX CORRECTION
8
CORRECTED
Off the arc On the arc
Hs
9 HT OF EYE (
6
)
º + – º –
DIP
º
10
APPARENT ALT. Ha
11
MAIN CORRECTION for Ha SUN: Summer or Winter, LL or UL
UL – LL +
12
OBSERVED ALT. Ho
º
. . . . . . .
'
. . . . . . . .
'
' ' ' ' ' '
43. Planet altitude (Almanac, Altitude Correction Table) UTC date: 15 Aug 2003 6 Body: Mars Sextant altitude: Hs = 29º 00.0' 7 Index error: 04.4' off the arc Height of eye: 9.3 feet 8 above water 9 10 11 12 13
42
º
SEXTANT ALT Hs INDEX CORRECTION CORRECTED
Off the arc On the arc
Hs
HT OF EYE (
)
APPARENT ALT. Ha MAIN CORRECTION for Ha In the Almanac, use “Stars & Planets” Additional Correction for MARS or VENUS OBSERVED ALT. Ho
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+ – º
DIP
–
º
– + º
' ' ' ' ' ' '
Session 8, Review Exercises 41-58
44. Moon altitude (end of Almanac, Alt. Corr. Tables for Moon) UTC date: 19 April 2003 Body: Moon UL Time of sight: 04:15:00 UTC Sextant altitude: Hs = 32º 00.0' Index error: 31.0' on the arc Height of eye: 2.9 m above water Observed altitude Ho?
6
SEXTANT ALT Hs
7
INDEX CORRECTION
8
CORRECTED
9
HT OF EYE (
10
APPARENT ALT. Ha
11
12
)
14
OBSERVED ALT. Ho (10) + (11) + (12) [ – (13) ?]
+ . – . º .
'
DIP –
MAIN CORRECTION to Ha for ° of “App. Alt”, & nearest 10' (top table) Additional CORRECTION for “HP” (from daily Almanac), and for sights on “UL” or “LL”, (same column, bottom table) (HP = . ') Subtract 30' for a sight on UL
'
º
Off the arc + On the arc – º
Hs
13
. . . . . .
º
+
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' ' ' ' '
' '
43
Session 8, Review Exercises 41-58
45. Latitude by noon sight UTC date: DR position: Meridian passage of sun: Sextant altitude: Boat latitude at noon?
29 October 2003 46º 10.0' N; 125º 20.0' W 11h 44 min Ho = 30º 29.0' after all corrections
LATITUDE BY NOON SIGHT GREENWICH DATE:___________________; SUN: LL / UL ; LOG:__________; COURSE:___________ 1 2
3 4 5 6
DR LAT DR LONG
N S E W
. ° . °
SUN MERIDIAN PASSAGE in local meridian time (in the Almanac page for the day; approximately 12:00 noon) CONVERT DR ° of LONG to TIME E – (use table “Conversion of Arcs to Time” ) W + CONVERT DR ' of LONG to TIME (use table “Conversion of Arcs to Time”) MER. PASS. in UTC (3 +/– 4 +/– 5) at the boat longitude (pre-calculated, or measured from several sights)
14
DEC of SUN for hours in 6
15
INCREMENT for min in 6 (d = + / – ) (use “Increment and Correction” tables for d)
16
DEC TOTAL Copy to 20
17
ZENITH
18
ALT OBS Ho
19 20 21
44
90° =
89º
ZENITH DISTANCE ZD (90º – Ho), i.e. 17 – 18 DEC from 16 LATITUDE = +/– ZD +/– DEC, +/– 19 +/– 20 N Should match DR Lat. in 1 S
'
h
min
h
min
E – W +
min
s
h
min
s
N S
º .
' '
N S
+ – . º .
'
6 0 .0 '
. ° . N º . S
'
°
º
.
' ' '
'
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Lat. & Dec same name, & Lat. > Dec: Lat. = ZD + Dec; Lat. & Dec same name, & Lat. < Dec: Lat. = Dec – ZD; Lat. & Dec contrary names: Lat. = ZD – Dec
Session 8, Review Exercises 41-58
46. Moon Declination (Almanac, Daily pages) UTC date: 20 April 2003 Body: Moon Time of sight: 04:25:18 UTC Declination? 24 25 26
DEC of the moon N For that day & hours S INCREMENT in DEC for min & s in 5 (“d” = + / – '/h) (Inc. & Corr. table “v or d”) DEC TOTAL N (24) + (25) S
º
.
'
+ –
º
. .
'
º
.
'
º
'
47. Star Declination (Almanac, Daily pages) UTC date: 16 June 2003 Body: Dubhe Declination? 24
DEC of Star * for that day
N S
48. Sun’s GHA; assumed longitude; & LHA from the boat to the sun UTC date: 30 June 2003 Body: Sun Time of sight: 17:44:18 UTC DR longitude: 129º 58.3' W GHA? Assumed longitude? LHA? 13
GHA of sun For that day, & hours of sight
º
14
INCREMENT in GHA for min & s of sight; Inc. & Corr. table
+
º
15
16
GHA TOTAL 13 +14 Add 360º if required 15 + 360° if W longitude > GHA, to allow subtraction in 18
17
ASSUMED LONG º from DRº [ ' in 16] if long W [60' – ' in 16] if long E
18
LOCAL HOUR ANGLE 15 or 16 +/– 17
º
+ 3 6 0° º
W– E+
º
º
. . . 00 .0 .
' ' '
'
' . ' 00 .0
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45
Session 8, Review Exercises 41-58
49. Planet GHA; assumed longitude; & LHA from the boat to the planet UTC date: 28 Jan 2003 Body: Venus Time of sight: 17:44:39 UTC DR longitude: 122º 05.0' W Assumed longitude? LHA?
14
15
16
17
GHA of particular planet for that day, and for hours in 5 Record Dec in 23, and d in 24
º
.
INCREMENT in GHA for min º and s in 5 (Inc. and Corr. table) + . Additional adjustment “v” for particular planet, for min in 5 + (Inc. and Corr. table) – . (“v” = + / – '/h) GHA TOTAL º 14 +15 + 16 Copy to 18 .
18
'
19
'
20 ' 21
'
22
GHA from 17
' . + 3 6 0 ° 0 0 . 0' º
Add 360° if W longitude > º GHA, to allow subtraction in 21 ASSUMED LONG º from (2) [ ' in 16] if long W W – º [60' – ' in 16] if long E E+ LOCAL HOUR ANGLE º 19 +/ – 20 Subtract 360º if required – 360° º
.
'
' . 0 0 . 0' 0 0 . 0' 0 0 . 0'
50. Star GHA; assumed longitude; & LHA from the boat to the Star UTC date: 31 Jan 2003 Body: Aldebaran Time of sight: 07:20:44 UTC DR longitude: 068º 30.0' E Assumed longitude? LHA?
13 14
GHA TOTAL for Aries (14) + (15) SHA of Star * 16 for that day GHA for the Star * 17 (16) + (17) Add 360º if required GHA for the Star * 19 (18), or (18)+360° if required ASSUMED LONG º from DRº 20 [ ' in 16] if long W W– [60' – ' in 16] if long E E+ LOCAL HOUR ANGLE (Star) 21 [(18) or (19)] +/– (20) Subtract 360º if required 15
22 46
GHA for Aries for hours of sight INCREMENT in GHA for Aries for min & s (5); Inc. & Corr. table
LHA of the Star *
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º +
º
º
º º +360° º
º
º –360° º
. .
'
. . . 00 .0 .
'
.
'
0 0 . 0' 0 0 . 0' 0 0 . 0'
'
' '
'
Session 8, Review Exercises 41-58
51. Sight Reduction Tables for any celestial object Assume, for the celestial object: LHA: 330º Boat Lat.: 48º 20' N Observed altitude Ho: 16˚ 35.0' Dec: 20˚ 26.3' S after interpolation for minutes of sight time Intercept? 23 24 25
26
27
º
LHA ASSUMED LAT N from DR in 1, rounded off S N DEC CORRECTED S COMPUTED ALT Hc for 22, 23 & 24; “same” or “contrary”, for DEC ° only. Note d ( + / – ) in 27, & Z in 31 d( + / – ) CORR’N to Hc for DEC ' in 24 table 5 (back of book)
28
HC CORRECTED for ' in 24
29
OBS. ALT Ho
30
INTERCEPT Ho > Hc: T Ho < Hc: A
N. Lat.: { LHA > 180°, …………Zn = Z { LHA < 180°, …………Zn = 360 – Z S. Lat.: { LHA > 180°, …………Zn = 180 – Z { LHA < 180°, …………Zn = 180 + Z
º
º
º
0 0 . 0' 0 0 . 0' ' . ' .0
' + – .0 ' º .0 ' º . ' T/A .
360º 31
N. LAT
32
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Z
º
Zn
º
180º S. LAT
Z
º
Zn
º
47
Session 8, Review Exercises 41-58
52. Hc and Zn for Selected Stars UTC date: 04 July 2003 Time of sight: 20:15:12 UTC (after chrono correction) DR position 39º 08.0' N 036º 30.0' W Three best stars? Hc and Zn for each? Star “V”: Star “S”: Star “P”:
5 6 7 8
9 10 11
12 13
Ho = 23º 38.0' Ho = 38º 51.0' Ho = 29º 26.0'
21 22 23
48
DR LAT C opy ° to (13) after rounding
N S
º
'
2
DR LONG
E W
º
. '
UTC times
GHA for Aries for hours in (5) INCREMENT in GHA for Aries for min & s (5); Inc. & Corr. table GHA TOTAL for Aries (6) + (7) Add 360° if W Long. > GHA, to allow subtraction in 11
:
º
.
'
+
.
'
.
'
º
+360º
LOCAL HOUR ANGLE (Aries) (8) or (9) +/– (10) Subtract 360° if LHA > 360°
00.0
.
'
º
.
'
º –360º
LHA Aries (11), or (11– 360°) if required ASSUMED LAT N from DR in (1), rounded off S
.
:
º
GHA for Aries (8), or (8)+360° if required ASSUMED LONG º from (2) [ ' in 16] if long W W– [60' – ' in 16] if long E E+
STAR NAMES: 20
1
º º
0 0 . 0' 0 0 . 0' 0 0 . 0' 0 0 . 0'
º . ' º . ' º . '
OBSERVED ALT. Ho COMPUTED Alt Hc for Stars, from Vol. 1, given Assumed Lat. & LHA of Aries INTERCEPT (20) – (21) or (21) – (20) Ho > Hc: T ; Ho < Hc: A COMPUTED Zn for Star, from Vol 1, given Assumed Lat. & LHA of Aries Zn:
º T/A
' º ' º ' .0 .0 .0 ' .
º
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T/A
º
' .
T/A
º
' .
Session 8, Review Exercises 41-58
53. Twilights (boat-meridian time) Day of sight: July 01, 2003 DR latitude: 49º N. Sunset? Civil & Nautical EVENING twilights? Use the boat-meridian times. Almanac Latitude At this Almanac Lat. #1, on Greenwich meridian At this Almanac Lat. #2, on Greenwich meridian
Sunset
Civil Twilight
Nautical Twilight
º
:
:
:
º
:
:
:
:
:
:
:
:
:
Difference for 5° of Almanac Lat. Difference for 1° of Almanac Lat. Difference for one or two ° of boat Lat. above or below Almanac Lats. #1 & 2 UTC at boat Lat. on Greenwich meridian (= boat-meridian time)
54. Twilights (boat zone-time) Day of sight: Nov. 18, 2003 DR position: 45º 00.0' N 123º 19.3' W Express the sunrise time, and DAWN twilights, in the zone-time of the zone within which the boat is located.
Boat longitude Nearest center-of-time-zone meridian (every 15º from Greenwich) Difference in longitude between boat meridian and center-of-time-zone meridian Time for sun travel between the two meridians (conversion of arc to time)
' .
º º
00.0 ' .
º
:
Nautical Twilight
:
Civil Twilight
: Sunrise
Almanac twilight times (dawn)
:
:
:
Add or remove, to obtain the zone-time for the twilights
:
:
:
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Session 8, Review Exercises 41-58
55. Zone time to UTC, and to boat-meridian time Wrist-watch on the zone time, reading 04:25:37 (no watch error). Boat Long. = 148º 25.0' W UTC time? Boat-meridian time? (no work form: use the blank space below)
56. Latitude by Polaris UTC date: 16 June 2003 Time of sight: 05:25:18 UTC, after chronometer correction DR position: 40º 30.0' N, 125º 10.0' W Sextant altitude: Hs = 40º 15.0' Index error: 10.2 on the arc Height of eye: 4.2 m above water Boat latitude at the time of sight? Exact bearing of Polaris (Zn)?
50
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51
Session 8, Review Exercises 41-58
57. Plotting a chart for the local ocean-area Prepare a plotting sheet to be used for Question 58. Its center should be at Lat. 47˚ N and Long.130˚ W. Label the printed parallels of latitude one degree apart, and do the same for the meridians of longitude.
58. Plotting LOPs and Advanced LOPs on the chart (no calculations required) a) On the plotting sheet prepared for Exercise 57, plot the morning sun bearings Zn, and the corresponding Lines of Position, after correcting for the intercept. b) Advance the morning LOP to noon, and establish the noon fix. c) Advance the noon LOP to the evening, and establish the evening fix. Morning Sight (Plotting only): Assumed Position AP1: 48ºN; 129º 10.0W Course: 230º T Speed: 10 kn Log: 2,350 NM Time of sight on the sun: 05:30 (boat-meridian time) True bearing of the sun: Zn = 100º Intercept for AP1: A, 12' Noon Sight (Plotting only): Course: Speed: Log: Time of sight on the sun: Calculated Lat.:
230º T 10 kn 2,415 NM (distance travelled during the morning: 65 NM) 12:03 (boat-meridian time) 47º 35.0' N
Evening Sight (Plotting only): Assumed Position AP3: 47ºN; 131º 32.0' W Course: 230º T Speed: 10 kn Log: 2,480 NM (distance travelled during the afternoon: 65 NM) Time of sight on the sun: 18:30 (boat-meridian time) True bearing of the sun: Zn = 260º Intercept for A3: T, 08'
52
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53
Home Session 1
Home Exercises 1-9 following Session 3 on the Interpolation of GHA and Dec for Exact Times of Sights; Local Hour Angle; and Latitude by Noon Sight.
55
Home Exercises 1-9 following Session 3
1. James Cook sun sights On June 22, 1770, Captain Cook furled all sails and took a sun sight, using his new Harrison chronometer H4. In his Nautical Almanac, he noted that, on that day, the sun was crossing the Greenwich meridian at 12:08. After correcting for the chronometer error, he established the UTC time of his first sight to be 10:08. He measured the altitude of the sun above the horizon and, after correcting his sextant for index error, Dip, and the main correction (1/2 diameter, parallax, & refraction), he found Ho = 38º 40'. From a noon sight, he determined his latitude to be approximately 15º 00.0' N. Later-on that day, he took another sight on the sun, at 16:08 UTC. He found Ho = 54º 40'.
a) What are the coordinates of the points on earth over which the sun was when he took the sights (GP of the sun) in the morning and in the afternoon?
Morning sun Lat. (Dec): ________________
Morning sun Long. (GHA): _____________
Afternoon sun Lat. (Dec): _______________
Afternoon sun Long. (GHA): ____________
b) What are the radii of his two circles of position? AM sight 90º
89º 60'
º
Ho
º
ZD
PM sight
89º
'
º
'
º
60'
' '
NM for the degrees of angle NM for the minutes of angle Total radius
c) Where is he?
56
Use the attached world map, with its scale in NM valid near the tropics.
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Home Exercises 1-9 following Session 3
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57
Home Exercises 1-9 following Session 3
2. Cook’s sextant corrections On June 22, Cook’s morning sight was: Body: Sextant Alt: Index error: Height of eye:
Hs
Sun LL Hs = 40° 50.2' off the arc, 10.2' 9.0 m above water
Observed altitude Ho after all corrections?
Index error. “On the arc”: – “off the arc”: + Hs corrected for index error Ht of the eye: Apparent altitude Ha Main correction, given the month and Ha: LL + UL – Observed altitude Ho
.
'
. º . DIP – . º .
'
. º .
'
º
' ' '
'
3. Estimate the sun’s GP Over which point on earth would you guess that the sun might be on any October 11 at 09:20 UTC, assuming the sun crosses the meridian of Greenwich at 12:00 on that day? Latitude? _______________________ Longitude?______________________ Use a world map to find the name of the nearest island, known for its spices and its sultan with 100 wives.
4. Interpolation of GHA What was the GHA of the sun on April 21, 2003 at 14:25:48 UTC? GHA for hours only Increment of GHA for minutes and sec Total GHA
58
º º + º
. . .
' ' '
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Home Exercises 1-9 following Session 3
5. Interpolation of Dec What was the Dec of the sun on Oct. 28, 2003 at 16:44:52 UTC Dec for hours only (N or S) (d + or – in ’/ h ) Increment of Dec for min Total Dec
º
.
(d =
'/ h)
'
. º .
'
6. Estimation of position After three weeks at sea in your trusty life raft somewhere on the North Atlantic at a latitude between 30º and 35º N, you take a sight on the sun at a time when you think it is highest above the horizon (top of the curve). Your wrist watch, on UTC, reads July 19, 2003 at 16:32. You remember reading that, on that day, the sun crossed the meridian of Greenwich at 12:12. You measure Ho = 78º. Any hope?
Sun crossing of boat meridian
:
Sun crossing of Greenwich meridian
:
Duration of sun travel
:
Longitude
º
90º Ho
º
ZD = 90º – Ho
º
Dec (estimated)
º
Lat. = Dec + ZD
º
7. Local Hour Angle (LHA) GHA Boat longitude AM or PM (boat time) GHA (add 360° if required) + longitude E – longitude W LHA = Total (subtract 360° if required)
138° 49.3' 123° 49.3' W
. º . º
º
.
29° 33.8' 98° 27.9' W
. ' º . ' º
' º
.
215° 47.6' 37° 49.5' E
. ' º . ' º
' º
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.
' '
'
59
Home Exercises 1-9 following Session 3
8. Review Exercise Celestial body: UTC date: DR position: Time of sight: Chrono error: Measured sextant altitude Hs: Index error: Height of eye:
Sun, LL, 04 July 2003 40º 45.0' N; 019º 01.0' W 15:19:04 (UTC) 1 min 20s (slow) 45º 37.1' 4' on the arc 2.9 m above water
Corrected UTC time? Hs corrected for index error? Ha, after correction for Dip? Observed Ho, after main correction? Exact Dec of the sun at the time of the sight? Exact GHA of the sun at the time of the sight? Local Hour Angle? Use the form attached.
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Home Exercises 1-9 following Session 3
Calculation of the Local Hour Angle GREENWICH DATE:___________________; LOG:__________; COURSE:___________ 1 2
N º S . E º W .
DR LAT DR LONG
' '
Time of sight (UTC ) 3
CHRONOMETER
4
CRONO. CORRECTION
5
UTC TIME
h
min
s
SLOW + FAST –
min
s
min
s
h
Sun altitude (Almanac, Altitude Correction Table) 6
SEXTANT ALT Hs
º
7
Off the arc INDEX CORRECTION On the arc
+ – º
8
CORRECTED
Hs )
DIP
9
HT OF EYE (
–
10
APPARENT ALT. Ha
º
11
MAIN CORRECTION for Ha SUN: Summer or Winter, LL or UL
12
OBSERVED ALT. Ho
UL – LL + º
. . . . . . .
'
Sun’s Declination (Almanac, Altitude Correction Table)
'
of sun N 20 DECLINATION For that day, and hours in S 21 CORR. to DEC for minutes in 5; d (+ / –
'
CORRECTED 22 DEC 20 +/– 21
'
N S
. + ) – . º .
º
' ' '
' ' '
Sun’s GHA (Almanac, Daily pages); LHA 13
14 15
GHA of sun for that day, and for hours in 5 Record "Dec" in 20, and increment "d" in 21 INCREMENT in GHA for min & s of sight; Inc. & Corr. table GHA TOTAL 13 +14
º º + º + 3 6 0°
16 17 18
19
Add 360° if W longitude > GHA to allow subtraction in 18 DR LONG from 2 E+ W– LOCAL HOUR ANGLE 15 or 16 +/– 17 Minus 360 º if required
º
LHA
º
º º – 3 6 0°
' . ' . ' . 0 0 . 0' ' . ' . ' . 0 0 . 0' ' .
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Home Exercises 1-9 following Session 3
9. Estimated longitude, and calculated latitude, by noon sight A graph is often used to determine the maximum sun altitude over the horizon (Ho max, crossing of the boat meridian), and the (approximate) time of meridian crossing. Use the attached graph to estimate Ho max on Feb. 01, 2003, as well as the time when the sun crossed the boat meridian. The sun Declination will need to be calculated for the exact time of sight.
9A. Estimation of longitude Longitude = [time for the sun to travel from the Greenwich meridian to the boat meridian] x 15º/h.
Multiply the hours of travel of the sun by 15º/h. Multiply the minutes of travel of the sun by (15º/60)ºh. Or use the Almanac (Appendix 2).
1. E stimate the UTC time when the sun crossed the boat meridian, i.e. when the sun reached Ho max (use the attached graph). For increased precision, draw a horizontal line (constant altitude) and take the middle of the segment between the two points of the intersection with the curve. Time when the sun crossed the boat meridian: ____ h____min
UTC.
2. Note the time when the sun crossed the Greenwich meridian (Almanac, 01 Feb 2003) ____h____min
UTC
3. Calculate the total time of travel of the sun between the meridians of Greenwich and of the boat. 4. From this time of travel at 15˚/h, determine the longitude. Use the Almanac (“Increments and Corrections”) to determine the arc corresponding to the minutes and seconds of travel. Boat-meridian time (UTC) Greenwich meridian time (UTC) Sun travel time (difference)
Time for sun travel from Greenwich to the boat ° of longitude for the hours only ° and ' of longitude for the minutes and seconds of time Total = longitude 62
h min – h min h min
h º º º
min 00.0 ' ' . ' .
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19:00
•
:10
:20
•
:30
:40
•
:50
20:00
•
:10
•
:20
:30
• • •
:40
Home Exercise 9 Noon sight, 01 Feb 2015
:50
•
21:00
:10
•
:20
:40
:50
•
Time of sights (UTC)
:30
•
22:00
24º 30.0’
25º 00.0’
25º 30.0’
26º 00.0’
26º 30.0’
27º 00.0’
Ho Altitude (degrees)
Home Exercises 1-9 following Session 3
63
Home Exercises 1-9 following Session 3
9B. Calculation of latitude 1. Estimate the maximum height of the sun above the horizon (Ho max) from the graph
Ho max: ______º______. 0'
2. Calculate the Declination of the sun at the time of the sight from the Almanac (01 Feb 2003).
13
14 15
DECLINATION of sun For Feb. 01, for the UTC time of the sight (hours only) N S d (+ / – ) ; CORR. to DEC for the minutes of the time of sight DEC CORRECTED N S
º + – º
3. Calculate the boat latitude from this noon sight. Approximate boat latitude (DR)
64
50° N
Sun’s Declination from Almanac: Dec
º
90° =
Measured highest sun altitude Ho (at noon, boat-meridian time)
º
.
'
Zenith Distance ZD = 90° – Ho
º
.
'
Boat latitude = “Dec + ZD” or “Dec – ZD” or “ZD – Dec”
º
.
'
' . 8 9 ° 6 0 . 0 '
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' ' '
Home Exercises 1-9 following Session 3
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66
Home Session 2
Home Exercises 10 and 11 following Session 4 on the Sight Reduction Tables
67
Home Exercises 10 and 11 following Session 4
10 & 11. S ight Reduction Tables: Assumed Lat. & Long.; calculation of Zn from Z; calculation of Hc by interpolation; calculation of the intercept; plotting. From the Sight Reduction Tables Pub. No 249, we can determine the bearing of the sun (sun’s azimuth “Zn”) and calculated altitude “Hc”. By comparing Hc with our observed altitude Ho, we can calculate by how much the assumed arc of circle of position should be moved. Sight No: DR latitude DR longitude Sun’s GHA Sun’s Dec (corrected) Observed altitude Ho
Home Exercise 10 39° 18.3' N 35° 33.5' E 355° 27.8' 13° 26.3' S 30° 17.6'
Home Exercise 11
38° 44.8' S 178° 39.4' W 133° 41.3' 08° 12.3' N 27° 10.8'
Find the Local Hour Angle (LHA); the coordinates of the assumed position, the computed altitude Hc; the azimuth angle Z; the bearing (azimuth) of the sun’s Zn; and the intercept (“Towards” or “Away from” the sun’s GP). Use the work forms attached, with a blank globe, and the pre-calibrated blank Mercator charts.
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Home Exercises 10 and 11 following Session 4
Home Exercise 10A SIGHT REDUCTION TABLES: INTERCEPT, and BEARING Zn (“Azimuth”) of the SUN 1 2
DR LAT N º S DR LONG E º W
. .
' '
Sun’s GHA (Almanac, Daily pages); LHA Sun’s GHA
16
16 + 360° if required to allow subtraction in 18
17
ASSUMED LONG º from (2) [ ' in 16] if long W W– [60' – ' in 16] if long E E+
18
19
' . + 3 6 0° 00.0 ' º . º
15
LOCAL HOUR ANGLE 16 + / – 17 Subtract 360° if LHA > 360°
' . º 00.0' – 3 6 0° 0 0.0' º
Copy LHA (18 or 19) to 23
0 0.0'
Sight Reduction Tables HO 249 23 24 25
26
27 28 N. Lat.: { LHA > 180°, …………Zn = Z { LHA < 180°, …………Zn = 360 – Z S. Lat.: { LHA > 180°, …………Zn = 180 – Z { LHA < 180°, …………Zn = 180 + Z
LHA (total in 18 above, minus 360° if LHA > 360°) ASSUMED LAT N from DR in 1, rounded off S N DEC CORRECTED S COMPUTED ALT Hc for 23, 24 & 25; same or contrary, for DEC ° only. Note d ( + / – ) in 27, & Z in 31 d (+ / – ') CORR’N to Hc for DEC ' table 5 (back of book), for ' in 25 HC CORRECTED for ' of DEC in 25
29
OBS. ALT Ho
30
INTERCEPT
Ho > Hc: T Ho < Hc: A
º
00.0' º 00.0' ' º .
º
' + – .0 ' º .0 ' º . T/A .
360º 31
N. LAT
32 www.CelestialNavigationBook.com
Z
º
Zn
º
' .0
180º S. LAT
Z
º
Zn
º 69
Home Exercises 10 and 11 following Session 4
Plotting of sight 10
70
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Home Exercises 10 and 11 following Session 4
Home Exercise 11A SIGHT REDUCTION TABLES: INTERCEPT, and BEARING Zn (“Azimuth”) of the SUN 1 2
DR LAT N º S DR LONG E º W
. .
' '
Sun’s GHA (Almanac, Daily pages); LHA Sun’s GHA
16
16 + 360° if required to allow subtraction in 18
17
ASSUMED LONG º from (2) [ ' in 16] if long W W– [60' – ' in 16] if long E E+
18
19
' . + 3 6 0° 00.0 ' º . º
15
LOCAL HOUR ANGLE 16 + / – 17 Subtract 360° if LHA > 360°
' . º 00.0' – 3 6 0° 0 0.0' º
Copy LHA (18 or 19) to 23
00.0'
Sight Reduction Tables HO 249 23 24 25
26
27 28
N. Lat.: { LHA > 180°, …………Zn = Z { LHA < 180°, …………Zn = 360 – Z S. Lat.: { LHA > 180°, …………Zn = 180 – Z { LHA < 180°, …………Zn = 180 + Z
LHA (total in 18 above, minus 360° if LHA > 360°) ASSUMED LAT N from DR in 1, rounded off S DEC CORRECTED N S COMPUTED ALT Hc for 22, 23 & 24; same or contrary, for DEC ° only. Note d ( + / – ) in 27, & Z in 31 d( +/– ') CORR’N to Hc for DEC ' table 5 (back of book), for ' in 25 HC CORRECTED for ' of DEC in 25
29
OBS. ALT Ho
30
INTERCEPT
Ho > Hc: T Ho < Hc: A
º
00.0' º 00.0' ' º .
º
' + – .0 ' º .0 ' º . T/A .
360º 31
N. LAT
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Z
º
Zn
º
' .0
180º S. LAT
Z
º
Zn
º
71
Home Exercises 10 and 11 following Session 4
Plotting of sight 11
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ANSWERS
ANSWERS to the CLASS AND HOME EXERCISES
Class Exercise Answers 1-58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 Home Exercise Answers 1-11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
75
Exercise Answers 1-58
1.
2.
3.
76
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Exercise Answers 1-58
4.
5.
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77
Exercise Answers 1-58
6.
7.
78
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8.
9.
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Exercise Answers 1-58
10.
11.
80
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Exercise Answers 1-58
12.
13.
14.
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Exercise Answers 1-58
15.
16.
82
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Exercise Answers 1-58
17.
18.
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Exercise Answers 1-58
19A.
84
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19B. Plotting of Sight #19
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85
Exercise Answers 1-58
20.
21.
22.
86
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23.
24.
25.
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Exercise Answers 1-58
26. 26A. Calculations
88
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26B. Plotting
Exercise 26
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89
Exercise Answers 1-58
27. 27A. Calculations
90
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27B. Plotting
Exercises 26, 27
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91
Exercise Answers 1-58
28. 28A. Calculations
92
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Exercise Answers 1-58
28B. Plotting
Exercises 26, 27, 28
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93
Exercise Answers 1-58
29.
94
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30.
31.
32.
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Exercise Answers 1-58
33.
96
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34.
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97
Exercise Answers 1-58
35.
98
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36. 36A. Calculations
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Exercise Answers 1-58
Sight 36B. Plotting
100
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37.
38.
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101
Exercise Answers 1-58
39.
102
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Exercise Answers 1-58
40.
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103
Exercise Answers 1-58
41.
42.
43.
104
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44.
45.
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105
Exercise Answers 1-58
46.
47.
48.
49.
106
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50.
51.
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Exercise Answers 1-58
52.
108
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53.
54.
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Exercise Answers 1-58
55.
110
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56.
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111
57. & 58
Exercise Answers 1-58
112
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Home Exercise Answers 1 - 11
113
Home Exercise Answers 1-11
1.
114
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115
Home Exercise Answers 1-11
2.
3.
4.
116
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5.
6.
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Home Exercise Answers 1-11
7.
118
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Home Exercise Answers 1-11
8.
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9. 9A. Estimation of Longitude.
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9A. Plotting
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9B. Calculation of Latitude
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10. 10A. Calculations
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Plotting of sight 10
10B. Plotting
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11. 11A. Calculations
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Plotting of sight 11
11B. Plotting
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Appendix 1 Sight Reduction Work Forms
Appendix 1: Sight Reduction Work Forms
A1-1
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Appendix 1: Sight Reduction Work Forms
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A1-2
Appendix 1: Sight Reduction Work Forms
A1-3
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A1-4
Appendix 1: Sight Reduction Work Forms
A1-5
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A1-6
Appendix 1: Sight Reduction Work Forms
A1-7
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