Nautical Astronomy

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MI R PUBLISHERS

E. II. KPACABIJEB, E. n . XJIR3CTHH

MOPEXO^HAfl ACTPOHOMMfl

H3AATEJIBCTBO «TPAHCnOPT* JIEHHHrPA.Ii;

B. KRASAVTSEV, B. K H L Y U S T IN

NAUTICAL ASTRONOMY TRANSLATED FROM THE RUSSIAN IIV

GEORGE YANKOVSKY

MIR PUBLISHERS

MOSCOW 1970

UDC 522.7+527+529.781+629.123.05 (075.8) = 20

Revised from the 1960 Russian edition

TRANSLITERATION OF THE RUSSIAN ALPHARET Russian

T ransliteration

Russian

A a B 6 B B F r A a E e ‘M )K 3 3 H H K I< JI a

a 1)

P p Cc

M M H H 0 o 11 n

V

T

g d e

v N S ............................................................................................. INTIHHHIOTION .............................................................................................

13 19

PART ONI',. TIIH PRINCIPLES OF SPHERICAL ASTRONOMY AND TIIIO NAUTICAL ASTRONOMICAL ALMANAC (MAE) ('hap ter

Soc. Sec. Sec.

Soc.

( ' ha p t e r

I. The S p he r i c a l Coordi nat es of Celest i al Bo di es .................. I. Tim Celestial S p h e r e ....................................................... 25 2. Coordinates of Celestial Bodies on the Celestial Sphere 3. The Relationship Between the Geographic Latitude of the Observer’s Position and the Spherical Coordinates of Points of the S p h e r e ................................................................... 37 4. Representations of the Celestial Sphere Used in Nautical Astronomy ................................................................................. 2. Conversi on f r o m One S y s t e m

of Sp h e r i c a l

25 30

38

Coordinat es to

Other S y s t e m s ...................................................................

Sec. Soc. Sec.

Chapter

Sec.

Sec. Sec. Sec. Sec.

41 5. Constructing the Celestial S p h e r e .......................... 41 6. Special Grids for Transformation of Coordinates ( Funda­ m e n ta ls ) ......................................................................................... 7. The Astronomical Triangle of a Celestial Body and Its Solution ..................................................................................... 3. A p p a r e n t D i u r n a l M o t i o n of C e l e s t i a l B o d i e s .....................

8. General Characteristics of the Diurnal Motion of Celestial Bodies. Conditions for the Rising and Setting of Bodies. Their Passage Through the Zenith and So Forth . . . 9. Some Problems Associated with the Diurnal Motion of Celestial B o d i e s ......................................................................... 10. Peculiarities in the Apparent Diurnal Motion of Celestial Bodies for an Observer at the Equator and at the Poles 11. Changes in the Coordinates of Bodies Due to Their Appa­ rent Diurnal M o tio n ................................................................. 12. The Apparent D iurnal Motion of CelestialBodies Explained

43 45 53

53 59 63 65 74

C O N TE N TS

0

4. A p p a r e n t A n n u a l M o t i o n of the S u n ..................... 77 Sec. 13. A Characteristic of the Apparent Motion of the Sun. The E c l i p t i c ......................................................................................... Sec. 14. Ecliptic C o o rd in ates.................................................................. , Sec. 15. Geographic (Climatic) Zones. S e a s o n s ................................. Sec. 16. The Diurnal and Annual Motion of the Sun for Observers in Different L a t i t u d e s ............................................ Sec. 17. An Explanation of the Apparent Annual Motion of the Sun Sec. 18. Variations in the Equatorial Coordinates of the Sun . . Sec. 19. Approximate Solution of Problems Associated with the Sun’s M o tio n .............................................................................

Chapter

Chapter

5. Phenomena Ass o c i a t e d wi t h the R e v o l u t i o n and R o t a t i o n of the E a r t h .............................................................................

77 82 83 84 88 92 95

99

Sec. 20.Annual Parallax ofStars. The Diurnal Parallax . . . . Sec. 21.Stellar A b e rra tio n ....................................................................... Sec. 22. The Essentials of Procession and N u ta tio n ........ 103 Sec. 23. On Variations in the Equatorial Coordinates of Stars

99 101

. . . . 24. Proper Motion of the Moon and Its Explanation . . . . 25. Periods in Lunar M o tio n ......................................... 110 26. Phases and Age of the Moon. Conditions of Seeing . . . 27. Changes in Lunar C o o rd in a te s...........................' . . . . 28. Approximate Solution of Problems Associated with the Motion of the M o o n .................................................................. 29.The Fundamentals of T i d e s .................................... 30.Planetary Motions P r o p e r .....................................................

108 108

Chapter

Sec. Sec. Sec. Sec. Sec. Sec. Sec.

6.

The A p p a r e n t M o t i o n s of the Mo o n and P l a n e t s

7 . The S t e l l a r S k y ........................................................ . . Sec. 31. On the Classification of Stars . . : .................. , . Sec. 32.Star and Constellation Id e n tific a tio n .................................

Chapter

Chapter

Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec.

138 33.Fundamentals of Measuring T i m e ..................................... 34.Sidereal Units. Sidereal T i m e ............................................. 35.The Basic Formula of T i m e ................................................. 36.Apparent Solar D a y ................................................................. 37.Mean Solar Day. Mean Solar T i m e ..................................... 38.Equation of T i m e ..................................................................... 39. Relationship Between Sidereal and Mean Units of Time M easurem ent................................................................................. 40. Converting from Mean Time to Sidereal Time and Vice V e r s a ............................................................ 8.

Measurement

of

106

Ill 115 115 117 118 124 124 129

T i m e .................................................

138 141 142 144 146 149 152 154

CONTENTS

Sec. Sec. Sec. Sec.

41. 42. 45. 44.

7

T im e s on V ariou s M e r i d i a n s ......................................................... Zone Tim e, Legal T im e, S h ip T i m e ............................................. In tern ation al D a te L i n e .................................................................... C a l e n d a r ......................................................................................................

157 160 165 166

('haptcr !). Nautical Astronomical Almanac ( M A E ) ..................... . Sec. 45. A l m a n a c s ..........................................................................................

170 170

Sec. 46. Tim Slniehm/* !' MAC T a b le s for O b ta in in g Hour A n g le s and D e c lin a t io n s of Celestial B o d i e s ............................................ Sor W D eterm ining the Tim e of Transit of B odies, the A rriv al Tim e of a Body at a G iven Hour A ngle and Other P roblem s Sec. 4H. Basic Karts on the Structure and Use of N a u tic a l A lm an acs

of Other C o u n tr ie s.......................................................................... Sec. 40. U sin g

the

MAC of Earlier Y e a r s ....................

.

171 177

183 185

I*AMT TWO. INSTRUMENTS AND TOOLS USED IN NAUTICAL ASTRONOMY Cha p t e r

Sec.

Hi. The (' hronomel er and T i m e k e e p i n g .......................................

193

50. Timekeeping at S e a ............................................................

193

Sec. 51. Designation and Construction of Chronometer and Other Ship T im epieces........................................................................ Sec. 52. Chronometer and Watch Error (C o rrectio n ).................. Sec. 53. The Chronometer Rate and Its V a ria tio n s ...................... Sec. 54. Determination of Chronometer Error by Radio Time Sig­ nals. Programs of Radio Signals. Chronometer Record Book ................................................................ Sec. 55. Care of Chronometers . . . . ,.......................................... Sec. 56. Working with Chronometer and W a t c h ........................... Sec. 57. Care of Ship T im ep ieces....................................................... Ch a p t e r

11. - he Theory and Co ns t r uc t i o n of the M a r i n e S e x t a n t

. . .

Sec. 58. Peculiarities of Marine Angle-MeasuringInstruments . Sec. 59. Theoretical Principles of the Marine S e x t a n t ............... Sec. 60. Zero-Point Correction. Index C o rre c tio n ........................... Sec. 61. Elements of the Marine S e x t a n t ........................................... Sec. 62. Basic Instrumental Errors of the Sextant and Their Redu­ ction. Finding the Index C o rre c tio n ..................................... Cha p t e r

12. M e a s u r i n g A l t i t u d e s of Celest i al Bodi es wi t h a M a r i n e S e x ­ t a n t .................................................................................................

Sec. 63. Methods of Measuring A ltitu d e ........................................... Sec. 64. Measuring Altitudes of Celestial Bodies Above the Visible H o r i z o n ......................................................................................... Sec. 65. Special Cases in Measuring the Altitudes of Celestial Bodies

193 197 200

202 208 209 212 213 213 214 219 222 225 237 237 240 248

CONTENTS

8

Chapter 13. Sextants with Artificial H o r i z o n ........................................ 252 Sec. 66. Essential Theory of Sextants with Artificial Horizon and Integrator or Averager (Type M A C ).................................... 252 Sec. 67. Design Features of the HAG S e x ta n t ......................... 256 Sec. 68. Using Ithe MAC S e x ta n t.................................................. 262 Sec. 69. Fundamentals of the Radio S e x t a n t.............................. 267 Chapter 14. Correcting Sextant-Measured Altitudes ofCelestial Bodies Sec. Sec. Sec. Sec. Sec. Sec. Sec.

272

The Necessity for Correcting MeasuredAltitudes . . . 272 Astronomical R efraction............................................. 272 Dip Short of the Horizon, the Dip ofthe Horizon . . . 276 Essentials of the D ip m eter.......................................... 283 Diurnal P a r a l l a x .............................................................. 288 Semidiameters of Celestial B o d ie s ............................. 290 Correcting the Altitudes of Bodies Measured Above the Visible Horizon ..................................................................... 292 Sec. 77. Correcting Altitudes Measured with Artificial Horizon and Bubble Sextant M A C ................................................. 301 Sec. 78. Errors Involved in Correcting A ltitu d e ............................ 303 Chapter

70. 71. 72. 73. 74. 75. 76.

15. Errors in Observing Altitude at Sea, Methods of Detecting and Reducing T h e m .................................................................

Sec. 79. Errors in Measured Altitudes and Ways of Determining Them ’ ............................................................................................. Sec. 80. Reducing Altitudes of Celestial Bodies to a Single Instant Sec. 81. Reducing Altitudes of Celestial Bodiesto a Single Zenith Sec. 82. Reducing Altitudes of Celestial Bodiesto a Single Decli­ nation Sec. 83. Computing the Mean-Square Error of Altitude from Observed A l t i t u d e s ..................................................................................... Sec. 84. Methods of Checking Measured A ltitu d e s....................... Sec. 85. Ways of Reducing the Effects of Random and Systematic E rrors.................................................... Chapter Sec. Sec. Sec.

305 305 309 311 313 314 316 319

16. The Celestial Globe and Aids That Replace I t ....................

321

86. The Celestial Globe, Designation and Construction . . 87. Solving Problems with the Celestial G lo b e ................... 88. Aids That Replace the Celestial G lo b e ...........................

321 325 329

PART THREE. METHODS OF NAUTICAL ASTRONOMY

Chapter 17. Astronomical Determination of the Compass Correction Sec. 89. The Fundamentals of Astronomical Determination of the Compass Correction....................................................

337 337

CONTENTS

Sec. 90. The Effect of Errors in D.R. Latitude and Longitude on the Azimuth Being Computed of a Celestial Body. Most Favourable Conditions ofO b serv a tio n ................................... Sec. 91. Systematic and Random Errors in the Compass Bearing of a Celestial Body. Conditions for Taking Bearings . . Sec. 92. Determining Compass Correction in the General Case . . Sec. 93. Special Tables for Computing Azimuth (or Bearing) of a Celestial B o d y ............................................................................. Sec. 94. Special Cases in Determining th e . Compass Correction from the Sun and Polaris ............................................... * . See. 95. Compiling Deviation Tables from the Bearings of Cele­ stial B o d ie s ................................................................................. Chapter 18. Essentials of Astronomical Determination of Position at Sea Sec. 96. The Relationship Between the Position of a Ship and the Position of Its Zenith on the S p h ere................................ Sec. 97. General Principles for Determining the Zenith on the Cele­ stial Sphere or the Observer’s Position on the Earth . . Sec. 98. Most Favourable Conditions in Arrangement of Celestial Bodies for Determining Position and for Separate Deter­ mination of Its Coordinates

90° (second quadrant), then sin A and cosec A are positive, and the other functions are negative. If should be noted that the azimuth always enters into the astro­ nomical triangle of a body in semicircular units; therefore, if the n/ 1in ii Ih is given in quadrantal or circular units, it must first be con\erfed to semicircular reckoning, and only then can the formula Ih» Investigated. h, The hour angle (t) in the astronomical triangle of a body may he west nr east, but not greater than 180°, which means in the first ••I Mirund trigonometric quadrants; therefore, if a west hour angle imi'Mler limn 180° is indicated, it must first be converted to east, less Hmu IHO . If the hour angle, west or east, is less than 90°, then all H" Irlgimnmolric functions are positive; if the hour angle t > 90°, •hen Min f a fid cosec t are positive, and all the other functions are 11* il a 11 ve,

•I I'nrallactic angle (q) may be either less or greater than 909. i AI ler investigating the signs of the formulas, note the follo^ hie 11 ns n result of investigation the signs of the first and second lei him ul the formula are different, then the sign of the left-hand •hie id Ilie form'ula (the function being computed) will be the same •i • Ihe M|gn nf the greater term; if the signs of the terms are the same,.

48 C O N V E R S I O N F R O M ONE S Y S T E M O F S P H E R . C O O R D I N A T E S TO O T H E R S

then the sign of the desired function will coincide with it. If the triangle is right-angled or quadrantal, the formulas for the desired quantities will be single-termed and their signs will be found imme­ diately after investigation. 8. When computing any part of the astronomical triangle by a formula that has heen investigated as to sign, one should remember: (a) when computing altitude, that the altitude is never greater than 90° and that a negative altitude indicates reduced height; (b) when computing azimuth, that the azimuth computed from the astronomical triangle of a celestial body is always in semicircular units, which numerically may be less or more than 90°. Compute the numerical value of the azimuth according to the investigation, and then give it a name; here, the first letter of the name of the azimuth will always be the same as that of the latitude, the second will either be the same as the practical hour angle that enters into the triangle (west or east), or, if the hour angle is not given, it will depend on whether the body is in the western or eastern half of the sphere. To check the correctness of computations, we can use: (a) intermediate checking of the values of the chosen logarithms; (b) approximate checking of the results obtained by constructing the celestial sphere (or on the basis of the relations between the sides and angles of the triangle, or in other ways); (c) exact checking of the final results by special check formulas which relate the obtained results to the given data. Checking will he illustrated in examples. Let us consider the principal cases of solving the astronomical triangle that occur in nautical astronomy. I. Given: cp, 5 and t ; to find h and A. Depict the astronomical triangle in the general form (Fig. 12) and label the knowns and unknowns. Applying the formulas of the cosine of a side and of the four adjacent parts,* we get cos (90° - h) = cos (903- c p ) . cos (90° — 6) + + sin (90° — cp) •sin (90° — 6) •cos t and cot A •sin t = cot (90° — 8 ) •sin (90° — cp) — cos (90° — cp) •cos t Simplifying and isolating the unknowns, we finally get sin h = sin cp •sin 6 + cos cp •cos 6 •cos t cot A = tan 8 •cos cp •cosec I — sin cp •cot t * See Appendix III.

( 2 . 1; ( 2 . 2)

7. A S T R O N O M IC A L T R I A N G L E

OF C E L E ST IA L BODY

49

To check, apply the formula of sines: sin A ______sin t sin (90°— 6) — sin (90 — h) or

sin A •cos h = sin t •cos 6 h'rom now on, when computing the azimuth, we will also make of this formula in the form

umo

sin A = sin t •cos 8 -seek

t

(2.3)

Example 2 . Given:

90°; therefore, the arc A ' chosen from the tables by the liiUMrltlun of c o ti4 ' w ill be 180° — A . After computing the azimuth A = H7"2il'.4t we give it the name N-E, since the latitude is north and the hour xiigln In east.

• i'/n

50 C O N V E R S I O N F R O M ONE S Y S T E M O F S P H E R . C O O R D I N A T E S TO O T H E R S

(p—52°19' .7N 5 = 14°43'.8S *= 66°30'.4E

9.89846 9.40528

sin sin

I Arg

9.30374 0.06847

9.78614 9.98549 9.60058

COS tan CSC

9.78614 9.41980 0.03758

II 9.37221 9.16393 p si nk 8.53614

i Arg

9.24352 0.29310

cos COS COS

fc = l° 5 8 '.2

sin

9.89846

cot

9.63816

II 9.53662 a 0.17875 cot A' 9.71537

180° —A = 62°33' .6 i4 = N 117°26' .4E

When computing the altitude we see that the logarithm II ;> I and, hence, II term > I term in absolute value, and since sin h = — I + II, the computed altitude will be positive. Check: (a) Intermediate checking is possible for the second and third lines of the scheme; indeed, log sin 6— log cos 6 = log tan 6 and log cos t + log cosec t = log cot t Performing the indicated operations we get 9.41979 ^ 9.41980 and 9.63816 = 9.63816 (b) Applying the checking formula (2.3), we get sin A cos h

9.94817 9.99974

sin t cos 8

9.96242 9.98549

9.94791

=

9.94791

which shows that the computations are correct.

Formula sin2 y (haversine z or hav z).

For computing h, formula (2.1) does not always ensure sufficient accuracy, especially when working with four-place tables. For this reason, for altitudes greater than 30°, use is sometimes made of ano­ ther formula in which in place of sin x we apply the more precise fun­ ction sin2 y *. In formula (2.1), replace h by 90° — z and apply the trigonometric formula cos x = 1 — 2 sin2 y ; we get cos z = sin cp•sin 8 + cos cp•cos 8 •cos t * See Appendix IV.

7. A S T R O N O M IC A L T R I A N G L E

OF C E L E S T IA L BODY

51

nr

1 — 2 sin2 y = sin cp •sin 6 + cos cp •cos 8 ^1 — 2sin2y Tuking into account that we obtain

sin c p -sin 5 + coscp -cos 6 —-cos (cp — 6),

1 — 2 sin2y = 1 — 2 sin2

— 2 cos cp•cos 6 •sin2 —

iimlt finally, sin2 y = sin2-2y-^- -f cos cp-cos 6 -sin2 y

(2.4)

When performing computations with this formula, no investiga­ tion is required since both terms are always positive; when working with logarithms, always use tables for sums (a). The values of log sin2 y , in Table 5a and 6, MT-63, are given in a special side roliimn. It should be borne in mind that for cp and 8 of contrary onmes, the formula will contain: cp — (—6) = cp + 6, which is a *nin; for cp and 6 of same name, we have the difference cp — 6, the mnnller value being subtracted from the greater.I. Example 3. G i v e n : cp= 52°12'. 5N, hHermine h.

/ -11°52'.7 ip = 52°12'.5

ft

12°22'.6

ft

39°49'.9

sin2 cos cos

8.02965 9.78731 9.98979

I

7.80675 1.25784

Arg

/i = /i9°2' . 1

6 =

12°22' .6N, * = 11°52'.7W .

a

9.06459 0.02335

s in s - i

9.08794

sin 2

z = 4C)°57'.9

,

II. (liven 66°33'S, polar night. The earth arrives in position III on 23.09. The sun is projected on the autumnal equinox (in the region of the constellation Virgo); all phenomena will be similar to position 1, but in the nor­ thern hemisphere autumn will be setting in. Finally, the earth will arrive in position IV on 22.12. The sun will appear at the point of winter solstice V (in the region of the constellation Sagittarius). The straight line connecting the centres of the earth and sun will cross the Tropic of Capricorn (cp = 23°27'S) where the sun will be in the zenith; the southern hemisphere will bo more illuminated and summer will set in; winter will begin in the northern hemisphere. As may be seen from Fig. 33, the sun’s rays intersect the earth’s axis (which retains a constant direction in space) at angles from 90° to zh66°33' or with the equator from 0° to ±23°27'; this explains the variation of the sun’s declination on the sphere from 0° to ±23°27'N and S. Let us now see how the earth’s velocity in orbit varies, and conse­ quently how the longitude of the sun changes on the sphere. Accord­ ing to Kepler’s second law, the earth’s speed is greatest when closest to the sun (point p of the orbit). This point is the perihelion (Greek: peri—about, hellos—sun)* of the orbit which the earth enters on 30, the period 30 is dropped. 4 ) M — epact — 3 = 8, where 3 is an empirical coefficient. Thus, in 1958 M = 8 (more precisely, 7). SEC. 29. THE FUNDAMENTALS OF TIDES

Observations of the ocean level show that in the general case it attains two peaks a day (high water or high tide) and two mini­ ma (low water or low tide). These phenomena are of a wave nature and lag daily about 50 minutes, thus indicating their relationship with the movements of the moon. Celestial mechanics gives a dyna­ mic explanation of the tides. The flood tide is due to the attraction of particles of water in the oceans by the moon and sun, and since the tidal force is inversely proportional to the cube of the distance, the attraction of the moon is 2.2-fold that of the sun. The tidal wave forms on the terrestrial meridian nearest the moon, and also on the meridian opposite it (Fig. 46), and follows the moon. Howe­ ver, the tidal maximum does not occur at the time of upper or lower transit of the moon, but later by an amount known as the lunitidal interval. This tidal lag is due to the friction of the particles of water, the shape of the sea bottom and coastline, and to other local causes. The size of the tidal wave depends on the position of the moon (L) relative to the sun (S) and the earth (T). The largest tides occur during new and full moon (Fig. 46) and are called spring tides, the

APPARENT

118

M O T IO N S

OF

MO ON

A ND

PLANETS

smallest occur at quadrature and are known as neap tides. Tides are characterized by the time and height of high and low water. To determine the times of flood and ebb, use is made not of the vari­ able lunitidal interval, but of a constant interval called the High S -0

I2

Water Full and Change, which is the arithmetic mean of the luniti­ dal intervals in the mean equinoctial syzygy. Tidal data are given in special tables of tides and other manuals. The tides are very important in navigation and are therefore stu­ died in oceanography and in pilotage (tidal aids and how to work with them). SEC. 30. PLANETARY MOTIONS PROPER

In ancient times it was already known that not all stars are fixed. Some of them—which the ancient astronomers called planets— Data on PJanets

/

E ccentri­ city

Mercury Venus

9 ?

4,840 12,400

4".7-12".9 9.9-65.2

0.2056 0.0068

Earth

5

12,756

—

0.0167

Mars Asteroids } Jupiter Super­ Saturn ior Uranus Neptune Pluto

5 03 U u hJ

cd bJO 03 > 8

cd 64 >5

CM CO VP lO

tH tH

^-1

Fi£. 52

31.

05 C M LO O o - 1- 1 vf< vt* Q 05 00 /C

CO -* o +

CO Ot> 05 LO LO 00 £; z

CO 05 00 co o o 7 '1 1 »o 05 05.

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OF

oCO C CO O C CM O o o o o i 1 + + rH CO O L C CM O co

t— LO 05 c— vf L O 05 00 CM CO CO CO

CO CO CM CM 00 LO rCM 05 o ro CO 05 oo o o o o o O O o 1 1 1 i 1 1 1 i 1

CO O L o'

oo o

00

1

1

1

05 CO o LO CO o t— LO CO v-t< CO vf LO CO _< o 05 CO CO CO 05 C M O CO CM ”"rH

t— C O LO L O CO co

05 05 CM C O t'- L LO O o CM C O r- CO oo o

t< CO CO CO r- CO vC O CO CM o CM CO CM >o F CM O CO L C O O iO LO C 00 V ^■h ^th

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