TEXT-BOOK
ON
PRACTICAL ASTRONOMY
BY
GEORGE L. HOSMER
Assist ant Professor of Civil Engineering, Massachusetts Institute of Technology
FIRST EDITION
FIRST THOUSAND
NEW YORK
JOHN WILEY & SONS
LONDON: CHAPMAN & HALL, LIMITED
1910
COPYRIGHT, 1910
BY GEORGE L. HOSMER
Stanhope Jpresa
F. H. CILSON COMPANY BOSTON. U.S.A.
PREFACE
THE purpose of this volume is to furnish a text in Practical Astronomy especially adapted to the needs of civil-engineering students who can devote but little time to the subject, and who are not likely to take up advanced study of Astronomy. The text deals chiefly with the class of observations which can be made with surveying instruments, the methods applicable to astronomical and geodetic instruments being treated but briefly. It has been the author's intention to produce a book which is intermediate between the text-book written for the student of Astronomy or Geodesy and the short chapter on the subject generally given in text-books on Surveying. The subject has therefore been treated from the standpoint of the engineer, who is interested chiefly in obtaining results, and those refinements have been omitted which are beyond the requirements of the work which can be performed with the engineer's transit. This has led to the introduction of some rather crude mathematical processes, but it is hoped that these are presented in such a way as to aid the student in gaining a clearer conception of the prob- lem without conveying wrong notions as to when such short-cut methods can properly be applied. The elementary principles have been treated rather elaborately but with a view to making these principles clear rather than to the introduction of refine- ments. Much space has been devoted to the Measurement of Time because this subject seems to cause the student more difficulty than any other branch of Practical Astronomy. The attempt has been made to arrange the text so that it will be a convenient reference book for the engineer who is doing field work.
For convenience in arranging a shorter course those subjects
iv PREFACE
which are most elementary are printed in large type. The mat- ter printed in smaller type may be included in a longer course and will be found convenient for reference in field practice, par- ticularly that contained in Chapters X to XIII.
The author desires to acknowledge his indebtedness to those who have assisted in the preparation of this book, especially to Professor A. G. Robbins and Mr. J. W. Howard of the Massa- chusetts Institute of Technology and to Mr. F. C. Starr of the George Washington University for valuable suggestions and crit- icisms of the manuscript.
G. L. H.
BOSTON, June, 1910.
TABLE OF CONTENTS
CHAPTER I
THE CELESTIAL SPHERE — REAL AND APPARENT MOTIONS
ART. PAGE
1. Practical Astronomy i
2. The Celestial Sphere i
3. Apparent Motion of the Sphere 3
4. The Motions of the Planets 3
5. Meaning of Terms East and West 6
6. The Earth's Orbital Motion — The Seasons 7
7. The Sun's Apparent Position at Different Seasons 9
8. Precession and Nutation 10
9. Aberration of Light 12
CHAPTER II
DEFINITIONS — POINTS AND CIRCLES OF REFERENCE
10. Definitions 14
Vertical Line — Zenith — Nadir — Horizon — Vertical Circles — Almucantars — Poles — Equator — Hour Circles — Par- allels of Declination — Meridian — Prime Vertical — Eclip- tic — Equinoxes — Solstices — Colures.
CHAPTER III
SYSTEMS OF COORDINATES ON THE SPHERE
11. Spherical Coordinates 18
12. The Horizon System 19
13. The Equator Systems 19
15. Coordinates of the Observer 22
16. Relation between the Two Systems of Coordinates 23
VI TABLE OF CONTENTS
CHAPTER IV
RELATION BETWEEN COORDINATES ART. PAGE
17. Relation between Altitude of Pole and Latitude of Observer. ... 27
1 8. Relation between Latitude of Observer and the Declination and
Altitude of a Star on the Meridian 30
19. The Astronomical Triangle 31
20. Relation between Right Ascension and Hour Angle 36
CHAPTER V MEASUREMENT OF TIME
21. The Earth's Rotation 39
22. Transit or Culmination 39
23. Sidereal Day 39
24. Sidereal Time 40
25. Solar Day 40
26. Solar Time ' 40
27. Equation of Time 41
28. Conversion of Apparent Time into Mean Time and vice versa ... 43
29. Astronomical and Civil Time 44
30. Relation between Longitude and Time 45
31. Relation between Sidereal Tune, Right Ascension and Hour Angle
of any Point at a. Given Instant 48
32. Star on the Meridian 49
33. Relation between Mean Solar and Sidereal Intervals of Time. ... 49
34. Relation between Sidereal and Mean Time at any Instant 52
35. Standard Time 56
36. The Date Line 58
37. The Calendar 59
CHAPTER VI
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC — STAR CATALOGUES — INTERPOLATION
38. The Ephemeris 62
39. Star Catalogues 69
40. Interpolation 69
TABLE OF CONTENTS Vll
/ CHAPTER VII
THE EARTH'S FIGURE — CORRECTIONS TO OBSERVED ALTITUDES
ART. PAGE
41. The Earth's Figure 72
42. Parallax 73
43. Refraction 76
44. Semidiameters 78
45- Dip 79
46. Sequence of Corrections 80
CHAPTER VIII
DESCRIPTION OF INSTRUMENTS — OBSERVING
47. The Engineer's Transit 82
48. Elimination of Errors 83
49. Attachments to the Engineer's Transit — Reflector 86
50. Prismatic Eyepiece 87
51. Sun Glass 87
52. The Portable Astronomical Transit 87
53. The Sextant 88
54. Artificial Horizon 91
55. Chronometer 92
56. Chronograph. 93
57. The Zenith Telescope 94
58. Suggestions about Observing 95
CHAPTER IX
THE CONSTELLATIONS
59. The Constellations 98
60. Method of Naming Stars 98
61. Magnitudes 99
62. Constellations near the Pole 99
63. Constellations near the Equator 100
64. The Planets 102
CHAPTER X
OBSERVATIONS FOR LATITUDE
65. Latitude by a Circumpolar Star at Culmination 103
66. Latitude by Altitude of the Sun at Noon 105
67. Latitude by the Meridian Altitude of a Southern Star 107
Vlll TABLE OF CONTENTS
ART. PAGE
68. Latitude by Altitudes Near the Meridian 108
69. Latitude by Polaris when the Time is Known no
70. Precise Latitude Determinations — Talcott's Method 112
CHAPTER XI
OBSERVATIONS FOR DETERMINING THE TIME
71. Observations for Local Time 114
72. Time by Transit of a Star 114
73. Observations with Astronomical Transit 117
74. Selecting Stars for Transit Observations 117
75. Time by Transit of the Sun 119
76. Time by Altitude of the Sun 120
77. Time by Altitude of a Star 123
78. Tune by Transit of Star over Vertical Circle through Polaris. ... 124
79. Time by Equal Altitudes of a Star 127
80. Time by Two Stars at Equal Altitudes 128
83. Rating a Watch by Transit of a Star over a Range 135
84. Time Service 136
CHAPTER XII
OBSERVATIONS FOR LONGITUDE
85. Methods of Measuring Longitude 139
86. Longitude by Transportation of Timepiece 139
87. Longitude by the Electric Telegraph 140
88. Longitude by Transit of the Moon 141
CHAPTER XIII
OBSERVATIONS FOR AZIMUTH
89. Determination of Azimuth 146
90. Azimuth Mark 146
91. Azimuth by Polaris at Elongation 147
92. Observations near Elongation 149
93. Azimuth by an Altitude of the Sun 151
94. Azimuth by an Altitude of a Star !55
95. Azimuth Observation on a Circumpolar Star at any Hour Angle . 155
96. The Curvature Correction 158
TABLE OF CONTENTS ix
ART. PAGE
97. The Level Correction 158
98. Diurnal Aberration 158
99. Meridian by Polaris at Culmination 161
100. Azimuth by Equal Altitudes of a Star 164
101. Observation for Meridian by Equal Altitudes of the Sun 165
102. Observation of the Sun near Noon 166
103. Combining Observations 167
CHAPTER XIV
NAUTICAL ASTRONOMY
104. Observations at Sea 170
Determination of Latitude at Sea:
105. Latitude by Noon Altitude of the Sun 170
106. Latitude by Ex-Meridian Altitudes 171
Determination of Longitude at Sea:
107. Longitude by the Greenwich Time and the Sun's Altitude. ... 172
108. Longitude by the Lunar Distance 172
109. Azimuth of the Sun at a Given Time 174
no. Azimuth of the Sun by Altitude and Time 175
in. Sumner's Method of Determining a Ship's Position 175
112. Position by Computation 178
TABLES
I. MEAN REFRACTION 184
II. CONVERSION OF SIDEREAL TO SOLAR TIME 185
III. CONVERSION OF SOLAR TO SIDEREAL TIME 186
IV. (A) SUN'S PARALLAX — (B) SUN'S SEMIDIAMETER — (C) DIP OF
HORIZON 187
V. TIMES OF CULMINATION AND ELONGATION OF POLARIS 188
VI. CORRECTION TO OBSERVE ALTITUDE OF POLARIS 189
VII. VALUES OF FACTOR 112.5 X 3600 X sin i" X tan Ze 190
GREEK ALPHABET 190
LIST OF ABBREVIATIONS 191
PRACTICAL ASTRONOMY
CHAPTER I
THE CELESTIAL SPHERE — REAL AND APPARENT
MOTIONS
1. Practical Astronomy.
Practical Astronomy treats of the theory and use of astro- nomical instruments and the methods of computing the results obtained by observation. The part of the subject which is of especial importance to the surveyor is that which deals with the methods of locating points on the earth's surface and of ori- enting the lines of a survey, and includes the determination of (i) latitude, (2) time, (3) longitude, and (4) azimuth. In solving these problems the observer makes measurements of the direc- tions of the sun, moon, stars, and other heavenly bodies; he is not concerned with the distances of these objects, with their actual motions in space, nor with their physical characteristics, but simply regards them as a number of visible objects of known positions from which he can make his measurements.
2. The Celestial Sphere.
Since it is only the directions of these objects that are required in practical astronomy, it is found convenient to regard all heavenly bodies as being situated on the surface of a sphere whose radius is infinite and whose centre is at the .eye of the observer. The apparent position of any object on the sphere is found by imagining a line drawn from the eye to the object, and prolonging it until it pierces the sphere. For example, the apparent position of Si on the sphere (Fig. i) is at Si, which is supposed to be at an infinite distance from C; the position of Sz is Sz, etc. By means of this imaginary sphere all problems
2 PRACTICAL ASTRONOMY
involving the angular distances between points, and angles between planes through the centre of the sphere, may readily be solved by applying the formulae of spherical trigonometry. This device is not only convenient for mathematical purposes, but it is perfectly consistent with what we see, because all celestial objects are so far away that they appear to the eye to be at the same distance, and consequently on the surface of a great sphere.
sr
FIG. i. APPARENT POSITIONS ON THE SPHERE
From the definition it will be apparent that each observer sees a different celestial sphere, but this causes no actual inconve- nience, for distances between points on the earth's surface are so short when compared with astronomical distances that they are practically zero except for the nearer bodies in the solar system. This may be better understood from the statement that if the entire solar system be represented as occupying a field one mile in diameter the nearest star would be about 5000 miles away on the same scale; furthermore the earth's diameter is but a minute fraction of the distance across the solar system, thematic being about 8000 miles to 5,60x3,000,000 miles,* or one 7oo,oooth part of this distance.
* The diameter of Neptune's orbit.
THE CELESTIAL SPHERE 3
Since the radius of the celestial sphere is infinite, all of the lines in a system of parallels will pierce the sphere in the same point, and parallel planes at any finite distance apart will cut the sphere in the same great circle. This must be kept constantly in mind when representing the sphere by means of a sketch, in which minute errors will necessarily appear to be very large. The student should become accustomed to thinking of the appearance of the sphere both from the inside and from an out- side point of view. It is usually easier to understand the spheri- cal problems by studying a small globe, but when celestial objects are actually observed they are necessarily seen from a point inside the sphere.
3. Apparent Motion of the Celestial Sphere.
If a person watches the stars for several hours he will see that they appear to rise in the east and to set in the west, and that their paths are arcs of circles. By facing to the north (in the northern hemisphere) it will be found that the circles are smaller and all appear to be concentric about a certain point in the sky called the pole ; if a star were exactly at this point it would have no apparent motion. In other words, the whole celestial sphere appears to be rotating about an axis. This apparent rotation is found to be due simply to the actual rotation of the earth about its axis (from west to east) in the opposite direction to that in which the stars appear to move.*
4. Motions of the Planets.
If an observer were to view the solar system from a point far outside, looking from the north toward the south, he would see that all of the planets (including the earth) revolve about the sun in elliptical orbits which are nearly circular, the direction of the motion being counter-clockwise or left-handed rotation.
* This apparent rotation may be easily demonstrated by taking a photo- graph of the stars near the pole, exposing the plate for several hours. The result is a series of concentric arcs all subtending the same angle. If the camera is pointed southward and high enough to photograph stars near the equator the star trails appear as straight lines.
PRACTICAL ASTRONOMY
He would also see that the earth rotates on its axis, once per day, in a counter-clockwise direction. The moon revolves around the earth in an orbit which is not so nearly circular, but the motion is in the same (left-handed) direction. The
FIG. 2. DIAGRAM OF THE SOLAR SYSTEM WITHIN THE ORBIT OF SATURN
apparent motions resulting from these actual motions are as follows: The whole celestial sphere, carrying with it all the stars, sun, moon, and planets, appears to rotate about the earth's axis once per day in a clockwise (right-handed) direction. The stars change their positions so slowly that they appear to be fixed in position on the sphere, whereas all objects within the solar system rapidly change their apparent positions among the stars. For this reason the stars are called fixed stars to distinguish them from the planets; the latter, while closely resembling the stars
THE CELESTIAL SPHERE 5
in appearance, are really of an entirely different character. The sun appears to move slowly eastward among the stars at the rate of about i° per day, and to make one revolution around the earth
30"
FIG. 3a. SUN'S APPARENT POSITION AT GREENWICH NOON ON MAY 22, 23,
AND 24, 1910
10 Y IV III
FIG. 3b. MOON'S APPARENT POSITION AT 14^ ON FEB. 15, 16, AND 17, 1910
in just one year. The moon also travels eastward among the stars, but at a much faster rate; it moves an amount equal to its own diameter in about an hour, and completes one revolu-
PRACTICAL ASTRONOMY
tion in a lunar month. Figs. 3a and 3b show the daily motions of the sun and moon respectively, as indicated by their plotted positions when passing through the constellation Taurus. It should be observed that the motion of the moon eastward among the stars is an actual motion, not merely an apparent one like that of the sun. The planets all move eastward among the stars, but since we ourselves are on a moving object the motion we see is a combination of the real motions of the planets around
VIRGO
-10
Spica
-15°
XIII XII
FIG. 4. APPARENT PATH OF JUPITER FROM OCT., 1909 TO OCT., 1910.
the sun and an apparent motion caused by the earth's revolution around the sun; the planets consequently appear at certain times to move westward (i.e., backward), or to retrograde. Fig. 4 shows the loop in the apparent path of the planet Jupiter caused by the earth's motion around the sun. It will be seen that the apparent motion of the planet was direct except from January to June, 1910, when it had a retrograde motion.
5. Meaning of Terms East and West.
In astronomy the terms " east " and " west " cannot be taken to mean the same as they do when dealing with directions in one
THE CELESTIAL SPHERE
plane. In plane surveying " east " and " west " may be con- sidered to mean the directions perpendicular to the meridian line. If a person at Greenwich (England) and another person at the 1 80° meridian should both point due east, they would actu- ally be pointing to opposite points of the sky. In Fig. 5 all four of the arrows are pointing east at the places shown. It will be seen from this figure that the terms " east " and " west " must therefore be taken to mean directions of ro- tation.
6. The Earth's Orbital Motion. — The Seasons.
The earth moves eastward around the sun once a year in an orbit which lies (very nearly) in one plane and whose form is that
FIG. 5. ARROWS ALL POINT EASTWARD
b V FIG. 6. THE EARTH'S ORBITAL MOTION
of an ellipse, the sun being at one of the foci. Since the earth is maintained in its position by the force of gravitation, it moves, as a consequence, at such a speecl in each part of its path that the
8
PRACTICAL ASTRONOMY
,
line joining the earth and sun moves over equal areas in equal times. In Fig. 6 all of the shaded areas are equal and the arcs aa'j bb', cc' represent the distances passed over in the same num- ber of days.*
The axis of rotation of the earth is inclined to the plane of the orbit at an angle of about 66°^, that is, the plane of the earth's equator is inclined at an angle of about 23°^ to the plane of the orbit. This latter angle is known as the obliquity of the ecliptic. (See Chapter II.) The direction of the earth's axis of rotation is nearly constant and it therefore points nearly to the same place in the sky year after year.
The changes in the seasons are a direct result of the inclination of the axis and of the fact that the axis remains nearly parallel
Vernal Equinox (March 21)
Summer Solstice (June 21)
Aphell
Perihelion (Dec. 31)
Winter Solstice (Dec. 21)
Autumnal Equinox (Sept. 22)
FIG. 7. THE SEASONS
to itself. When the earth is in that part of the orbit where the northern end of the axis is pointed away from the sun (Fig. 7) it is winter in the northern hemisphere. The sun appears to be
* The eccentricity of the ellipse shown in Fig. 6 is exaggerated for the sake of clearness ; the earth's orbit is in reality much more nearly circular, the variation in the earth's distance from the sun being only about three per cent.
THE CELESTIAL SPHERE 9
farthest south about Dec. 21, and at this time the days are shortest and the nights are longest. When the earth is in this position, a plane through the axis and perpendicular to the plane of the orbit will pass through the sun. About ten days later the earth passes the end of the major axis of the ellipse and is at its point of nearest approach to the sun, or perihelion. Although the earth is really nearer to the sun in winter than in summer, this has but a small effect upon the seasons; the chief reasons why it is colder in winter are that the day is shorter and the rays of sunlight strike the surface of the ground more obliquely. The sun appears to be farthest north about June 22, at which time summer begins in the northern hemisphere and the days are longest and the nights shortest. When the earth passes the other end of the major axis of the ellipse it is farthest from the sun, or at aphelion. On March 21 the sun is in the plane of the earth's equator and day and night are of equal length at all places on the earth (Fig. 7). On Sept. 22 the sun is again in the plane of the equator and day and night are everywhere equal. These two times are called the equinoxes (vernal and autumnal), and the points in the sky where the sun's centre ap- pears to be at these two dates are called the equinoctial points, or more commonly the equinoxes.
7. The Sun's Apparent Position at Different Seasons.
The apparent positions of the sun on the celestial sphere corresponding to these different positions of the earth .are shown in Fig. 8. As a result of the sun's apparent eastward motion from day to day along a path which is inclined to the equator, the angular distance of the sun from the equator is continually changing. Half of the year it is north of the equator and half of the year it is south. On June 22 the sun is in its most northerly position and is visible more than half the day to a person in the northern hemisphere (/, Fig. 8). On Dec. 21 it is farthest south of the equator and is visible less than half the day (Z>, Fig. 8). In between these two extremes it moves back and forth across the equator, passing it about March 21 and Sept. 22 each year.
10
PRACTICAL ASTRONOMY
The apparent motion of the sun is therefore a helical motion about the axis, that is, the sun, instead of following the path which would be followed by a fixed star, gradually increases or decreases its angular distance from the pole at the same time that it revolves once a day around the earth. The sun's motion eastward on the celestial sphere, due to the earth's orbital motion,
FIG. 8. SUN'S APPARENT POSITION AT DIFFERENT SEASONS
is not noticed until the sun's position is carefully observed with reference to the stars. If a record is kept for a year showing which constellations are visible in the east soon after sunset, it will be found that these change from month to month, and at the end of a year the one first seen will again appear in the east, showing that the sun has apparently made the circuit of the heavens in an eastward direction
8. Precession and Nutation.
While the direction of the earth's rotation axis is so nearly constant that no change is observed during short periods of time, there is hi reality a very slow progressive change in its direction. This change is due to the fact that the earth is not quite spherical in form but is spheroidal, and there is in conse- quence a ring of matter around the equator upon which the sun and the moon exert a force of attraction which tends to pull the plane of the equator into coincidence with the plane of the orbit. But since the earth is rotating with a high velocity and
THE CELESTIAL SPHERE
II
resists this attraction, the actual effect is not to permanently change the inclination of the equator to the orbit, but first to cause the earth's axis to describe a cone about an axis per- pendicular to the orbit, and second to cause the inclination of the axis to go through certain periodic changes (see Fig. 9). The movement of the axis in a conical surface causes the line of intersection of the equator and the plane of the orbit to revolve slowly westward, the pole itself always moving directly toward the vernal equinox. This causes the equinoctial points to move westward in the sky, and hence the sun crosses the equator each spring earlier than it would otherwise; this is known as the
-Plcme-of-Eartlfs-Orbit-
FIG. 9. PRECESSION OF THE EQUINOXES
precession of the equinoxes. In Fig. 9 the pole occupies suc- cessively the positions /, 2 and J, which causes the point V to move to points i, 2 and 3- This motion is but 50". 2 per year, and it therefore requires about 25,800 years for the pole to make one complete revolution. The force causing the precession is not quite constant, and the motion of the equinoctial points is therefore not perfectly uniform but has a small periodic varia- tion. In addition to this periodic change in the rate of the precession there is also a slight periodic change in the obliquity,
12
PRACTICAL ASTRONOMY
called Nutation. The maximum value of the nutation is about 9"; the period is about 19 years. The phenomenon of preces- sion is clearly illustrated by means of the apparatus called the gyroscope. As a result of the precessional movement of the axis all of the stars gradually change their positions with refer- ence to the plane of the equator and the position of the equinox. The stars themselves have but a very slight angular motion, this apparent change in position being due almost entirely to the change in the positions of the circles of reference.
9. Aberration of Light.
Another apparent displacement of the stars due to the earth's motion is what is known as aberration. On account of the rapid motion of the earth through space, the direction in which a star is seen by an observer is a result of the combined velocities of the observer and of light from the star. The star always appears to be slightly displaced in the direction in which the observer is actually moving. In Fig. 10, if light moves from C to B in the same length of time that the observer moves from A to B, then C would appear to be in the direction AC. This
FIG. 10
FIG. ii
may be more clearly understood by using the familiar illustra- tion of the falling raindrop. If a raindrop is falling vertically, CB, Fig. n, and while it is falling a person moves from A to B, then, considering only the two motions, it appears to the person' that the raindrop has moved toward him in the direction CA. If a tube is to be held in such a way that the raindrop shall pass through it without touching the sides, it must be held at the
THE CELESTIAL SPHERE 13
inclination of AC. The apparent displacement of a star due to the observer's motion is similar to the change in the apparent direction of the raindrop.
There are two kinds of aberration, annual and diurnal. Annual aberration is that produced by the earth's motion in its orbit and is the same for all observers. Diurnal aberration is due to the earth's daily rotation about its axis, and is different in different latitudes, because the speed of a point on the earth's surface is greatest at the equator and diminishes toward the pole.
If v represents the velocity of the earth in its orbit and V the velocity of light, then when CB is at right angles to AB the displacement is a maximum and
v tan a0 = — >
where a0 is the angular displacement and is called the "constant of aberration." Its value is about 20."$. If CB is not per- pendicular to AB, then
v tan a = — sin B,
where a is the angular displacement and B is the angle ABC.
Problem
Referring to Fig. 2, make a sketch showing the path which Jupiter appears to describe, in the plane of its motion, but considering the earth as a fixed point on the diagram.
CHAPTER II
DEFINITIONS— POINTS AND CIRCLES OF REFERENCE
10. The following astronomical terms are in common use and are necessary in denning the positions of celestial objects on the sphere by means of spherical coordinates.
Vertical Line.
A vertical line at any point on the earth's surface is the direc- tion of gravity at that point, and is shown by the plumb line or indirectly by means of the spirit level (OZ, Fig. 12).
Zenith — Nadir.
If the vertical at any point be prolonged upward it will pierce the sphere at a point called the Zenith (Z, Fig. 12). This point is of great importance because it is the point on the sphere which indicates the position of the observer on the earth's surface. The point where the vertical prolonged downward pierces the sphere is called the Nadir (N', Fig. 12).
Horizon.
The horizon is the great circle on the celestial sphere cut by a plane through the centre of the earth perpendicular to the vertical (NESW, Fig. 12). The horizon is everywhere 90° from the zenith and the nadir. It is evident that a plane through the observer perpendicular to the vertical cuts the sphere in this same great circle. The visible horizon is the circle where the sea and sky seem to meet. Projected onto the sphere it is a small circle below the true horizon and parallel to it. Its dis- tance below the true horizon depends upon the height of the observer's eye above the surface of the water.
Vertical Circles.
Vertical Circles are great circles passing through the zenith and nadir. They all cut the horizon at right angles (HZJ, Fig. 12).
14
POINTS AND CIRCLES OF REFERENCE 1 5
Almucantars.
Parallels of altitude, or almucantars, are small circles parallel to the horizon (DFG, Fig. 12).
Poles.
If the earth's axis of rotation be produced indefinitely it will pierce the sphere in two points called the celestial poles (PP' Fig. 12).
Equator.
The celestial equator is a great circle of the celestial sphere cut by a plane through the centre of the earth perpendicular to
FIG. 12. THE CELESTIAL SPHERE
the axis of rotation (QWRE, Fig. 12). It is everywhere 90° from the poles. A parallel plane through the observer cuts the sphere in the same circle.
1 6 PRACTICAL ASTRONOMY
Hour Circles.
Hour Circles are great circles passing through the north and south celestial poles (PVPf, Fig. 12).
Parallels of Declination.
Small circles parallel to the plane of the equator are called parallels of decimation (BKC, Fig. 12).
Meridian.
The meridian is the great circle passing through the zenith and the poles (SZPL, Fig. 12). It is at once an hour circle and a vertical circle. It is evident that different observers will in general have different meridians. The meridian cuts the horizon in the north and south points (N,S, Fig. 12). The intersection of the plane of the meridian with the horizontal plane through the observer is the meridian line used in plane surveying.
Prime Vertical.
The prime vertical is the vertical circle whose plane is per- pendicular to the plane of the meridian (EZW, Fig. 12). It cuts the horizon in the east and west points (E, W, Fig. 12).
Ecliptic.
The ecliptic is the great circle on the celestial sphere which the sun's centre appears to describe during one year (AMVL, Fig. 12). Its plane is the plane of the earth's orbit; it is inclined to the plane of the equator at an angle of about 23° 27', called the obliquity of the ecliptic.
Equinoxes.
The points of intersection of the ecliptic and the equator are called the equinoctial points or simply the equinoxes. That intersection at which the sun appears to cross the equator when going from the south side to the north side is called the Vernal Equinox, or sometimes the First Point of Aries (V, Fig. 12). The sun reaches this point about March 21. The other inter- section is called the Autumnal Equinox (A, Fig. 12).
Solstices.
The points on the equator midway between the equinoxes are called the winter and summer solstices.
POINTS AND CIRCLES OF REFERENCE \J
Colures.
The great circle through the poles and the equinoxes is called
FIG. 12. THE CELESTIAL SPHERE
the equinoctial colure (PVP', Fig. 12). The great circle through
the poles and the solstices is called the solstitial colure.
i
Questions
1. What imaginary circles on the earth's surface correspond to hour circles? To parallels of declination? To vertical circles?
2. What are the widths of the torrid, temperate and arctic zones and how are they determined?
CHAPTER III
SYSTEMS OF COORDINATES ON THE SPHERE
ii. Spherical Coordinates.
The direction of a point in space may be denned by means of two spherical coordinates, that is, by two angular distances, measured on a sphere along arcs of two great circles which cut each other at right angles. Suppose that it is desired to locate C (Fig. 13) with reference to the plane OAB and the line
B -^^A. PRIMARY
FIG. 13. SPHERICAL COORDINATES
OA, O being the origin of coordinates. Pass a plane OBC through C and perpendicular to OAB: these planes will intersect in the line OB. The two angles which fix the position of C, or the spherical coordinates, are BOC and AOB. These may be regarded as the angles at the centre of the sphere or as the arcs BC and AB. In every system of spherical coordinates the two coordinates are measured, one on a great circle called the primary, and the other on one of a system of great circles at right angles to the primary called secondaries. There are an infinite number of secondaries, each passing through the two poles of the primary. The coordinate measured from the primary is. an arc of a
18
SYSTEMS OF COORDINATES ON THE SPHERE 19
secondary circle; the coordinate measured between the secondary circles is an arc of the primary.
12. Horizon System.
In this system the primary circle is the horizon and the sec- ondaries are vertical circles, or circles passing through the zenith and nadir. The first coordinate of a point is its angular distance above the horizon, measured on a vertical circle; this is called the Altitude. The complement of the altitude is called the Zenith distance. The second coordinate is the angular distance on the horizon between the meridian and the vertical circle through the point; this is called the Azimuth. Azimuth may be reckoned either from the north or the south point and in either direction, like bearings in surveying, but the custom is to reckon it from the south point right-handed from o° to 360° except for stars near the pole, in which case it is more convenient to reckon
Azimuth FIG. 14. THE HORIZON SYSTEM
from the north, and either to the east or to the west. In Fig. 14 the altitude of the star A is BA ; its azimuth is SB.
13. The Equator Systems.
The circles of reference in this system are the equator and great circles through the poles, or hour circles. The first coor- dinate of a point is its angular distance north or south of the
20
PRACTICAL ASTRONOMY
equator, measured on an hour circle; it is called the Declination. Declinations are considered positive when north of the equator, negative when south. The complement of the declination is called the Polar Distance. The second coordinate of the point is the arc of the equator between the vernal equinox and the foot of the hour circle through the point; it is called Right Ascension. Right ascension is measured from the equinox eastward to the hour circle through the point in question ; it may be measured in degrees, minutes, and seconds of arc, or in hours, minutes, and
FIG. 15. THE EQUATOR SYSTEM
seconds of time. In Fig. 15 the decimation of the star S is AS; the right ascension is VA.
Instead of locating a point by means of declination and right ascension it is sometimes more convenient to use declination and Hour Angle. The hour angle of a point is the arc of the
SYSTEMS OF COORDINATES ON THE SPHERE
21
equator between the observer's meridian and the hour circle through the point. It is measured from the meridian westward (clockwise) from oh to 24* or from o° to 360°. In Fig. 16 the declination of the star S is AS (negative); the hour angle is
FIG. 1 6. HOUR ANGLE AND DECLINATION
MA. It is evident that the hour angles of all points on the celestial sphere are always increasing.
These three systems are shown in the following table.
|
Name. |
Primary. |
Secondaries. |
Origin of Coordinates. |
ist coord. |
and coord. |
|
Horizon System |
Horizon Equator |
Vert. Circles Hour Circles |
South point. Vernal Equi- |
Altitude Declin. |
Azimuth Rt. Ascen. |
|
nox. |
|||||
|
Equator Systems • |
M |
u n |
Intersection of Meridian |
" |
Hour Angle |
|
and Equator. |
22
PRACTICAL ASTRONOMY
14. There is another system which is employed in some branches of astronomy but will not be used in this book. The coordinates are called celestial latitude and celestial longitude; the primary circle is the ecliptic. Celestial latitude is measured from the ecliptic just as declination is measured from the equator. Celestial longitude is measured eastward along the ecliptic from the equinox, just as right ascension is measured eastward along the equator. The student should be careful not to confuse celes- tial latitude and longitude with terrestrial latitude and longitude. The latter are the ones used in the problems discussed in this book.
15. Coordinates of the Observer.
The observer's position is located by means of his latitude and longitude. The latitude, which on the earth's surface is the angular distance of the observer north or south of the equator, may be defined astronomically as the declination of the ob- server's zenith. In Fig. 17, the terrestrial latitude is the arc EO,
r'
FIG. 17. THE OBSERVER'S LATITUDE
EQ being the equator and O the observer. The point Z is the observer's zenith, so that the latitude on the sphere is the arc E'Z, which evidently will contain the same number of degrees as EO. The complement of the latitude is called the Co-latitude.
SYSTEMS OF COORDINATES ON THE SPHERE 23
The terrestrial longitude of the observer is the arc of the equator between the primary meridian (usually that of Greenwich) and the meridian of the observer. On the celestial sphere the longi- "1 tude would be the arc of the celestial equator contained between ( two hour circles whose planes are the planes of the two terrestrial^ meridians.
1 6. Relation between the Two Systems of Coordinates.
In studying the relation between different points and circles on the sphere it may be convenient to imagine that the celestial sphere consists of two spherical shells, one within the other.
FIG. 18. THE SPHERE SEEN FROM THE OUTSIDE
The outer one carries upon its surface the ecliptic, equinoxes, poles, equator, hour circles and all of the stars, the sun, the moon and the planets. On the inner sphere are the zenith, horizon, vertical circles, poles, equator, hour circles, and the meridian. The earth's daily rotation causes the inner sphere to revolve,
24 PRACTICAL ASTRONOMY
while the outer sphere is motionless, or, regarding only the apparent motion, the outer sphere revolves once per day on its axis, while the inner sphere appears to be motionless. It is evident that the coordinates of a fixed star in the first equatorial system (Declination and Right Ascension) are practically always the same, whereas the coordinates in the horizon system are continually changing. It will also be seen that in the first equatorial system the coordinates are independent of the ob- server's position, but in the horizon system they are entirely dependent upon his position. In the second equatorial system one coordinate is independent of the observer, while the other (hour angle) is not. In making up catalogues of the positions of the stars it is necessary to use right ascensions and declina- tions in defining these positions. When making observations
E R w
FIG. 19. THE SPHERE SEEN FROM THE EARTH (LOOKING SOUTH)
with instruments it is usually simpler to measure coordinates in the horizon system. Therefore it is necessary to be able to compute the coordinates of one system from those of another. The mathematical relations between the spherical coordinates are discussed in Chapter IV.
SYSTEMS OF COORDINATES ON THE SPHERE 25
Figs. 18, 19, and 20 show three different views of the celestial sphere with which the student should be familiar. Fig.. 18 is the sphere as seen from the outside and is the view best adapted to showing problems in spherical trigonometry. The star 5 has the altitude RS, azimuth S'R, hour angle Mm, right ascension Vm, and declination mS; the meridian is ZMS' . Fig. 19 shows a portion of the sphere as seen by an observer looking southward; the points are indicated by the same letters as in Fig. 18. Fig. 20
•w
FIG. 20. THE SPHERE PROJECTED ONTO THE PLANE OF THE EQUATOR
shows the same points projected on the plane of the equator. In this view of the sphere the angles at the pole (i.e., the angles between hour circles) are shown their true size, and it is therefore a convenient diagram to use when dealing with right ascension and hour angles.
26
PRACTICAL ASTRONOMY
Questions and Problems
1 . What coordinates on the sphere correspond to latitude and longitude on the earth's surface?
2. Make a sketch of the sphere and plot the position of a star having an altitude of 20° and an azimuth of 250°. Locate a star whose hour angle is i6h and whose declination is — 10°. Locate a star whose right ascension is 9** and whose declina- tion is N. 30°.
3. If a star is on the equator and also on the horizon, what is its azimuth? Its altitude? Its hour angle? Its declination?
CHAPTER IV
RELATION BETWEEN COORDINATES
17. Relation between Altitude of Pole and Latitude of Ob- server.
In Fig. 21, SZN represents the observer's meridian; let P be the celestial pole, Z the zenith, E the point of intersection of the meridian and the equator, and N and S the north and south points of the horizon. By the definitions, OZ (vertical) is perpendicular to SN (horizon) and OP (axis) is perpendicular to EO (equator). Therefore the arc PN = arc EZ. By the
E
FIG. 22
definitions, EZ is the declination of the -zenith, or the latitude, and PN is the altitude of the celestial pole. Hence the altitude of the pole is always equal to the latitude of the observer. The same relation maybe seen from Fig. 22, in which P is the north pole of the earth, OH is the plane of the horizon, the observer being at O, EQ is the earth's equator, and OP' is a line parallel to CP and consequently points to the celestial pole. It may readily be shown that ECO, the observer's latitude, equals HOP', the altitude of the celestial pole. A person at the equator would
27
28
PRACTICAL ASTRONOMY
see the north celestial pole in the north point of his horizon and the south celestial pole in the south point of his horizon. If he travelled northward the north pole would appear to rise, its altitude being always equal to his latitude, while the south pole would immediately go below his horizon. When the traveller reached the north pole of the earth the north celestial pole would be vertically over his head.
To a person at the equator all stars would appear to move vertically at the times of rising and setting, and all stars would be above the horizon i2h and below i2h during one revolution
(S.Pole) S
N (N.Pole)
FIG. 23. THE RIGHT SPHERE
of the sphere. All stars in both hemispheres would be above the horizon at some time every day. This is called the " right sphere" (Fig. 23).
If a person were at the earth's pole the celestial equator would coincide with his horizon, and all stars in the northern hemi- sphere would appear to travel around in circles parallel to the horizon; they would be visible for 24* a day, and their altitudes would not change. The stars in the southern hemisphere would never be visible. The word north would cease to have its usual
RELATION BETWEEN COORDINATES
29
meaning, and south might mean any horizontal direction. The longitude of a point on the earth and its azimuth from the Greenwich meridian would then be the same. This is called the "parallel sphere" (Fig. 24).
At all points between these two extreme latitudes the equator cuts the horizon obliquely. A star on the equator will be above
FIG. 24. THE PARALLEL SPHERE
the horizon half the time and below half the time. A star north of the equator will (to a person in the northern hemisphere) be above the horizon more than half of the day; a star south of the equator will be above the horizon less than half of the day. If the north polar distance of a star is less than the observer's north latitude, the whole of the star's diurnal circle is above the hori- zon, and the star will therefore remain above the horizon all of the time. It is called in this case a circumpolar star (Fig. 25). The south circumpolar stars are those whose south polar distances are less than the latitude; they are never visible to an observer in the northern hemisphere. If the observer travels
PRACTICAL ASTRONOMY
north until he is beyond the arctic circle, latitude 66° 33' north, then the sun becomes a circumpolar at the time of the summer solstice. At noon the sun would be at its maximum altitude; at midnight it would be at its minimum altitude but would still be above "the horizon. This is called the " midnight sun."
z
Circumpolars (Never Rise)
FIG. 25. CIRCUMPOLAR STARS
18. Relation between Latitude of Observer and the Declina- tion and Altitude of a Star on the Meridian.
The relation between the latitude, altitude, and declination at the instant when a star is crossing the observer's meridian may be seen from Fig. 26. Let A be a star on the meridian, south of the zenith and north of the equator; then
EZ = L, the latitude,
EA = D, the declination,
SA = h, the altitude,
ZA = z, the zenith distance.
From the figure
or
and
also
ZA = EZ - EA z = L - D
h = 90° - (L - D}; L = 90° - (h - D).
[i]
[2]
RELATION BETWEEN COORDINATES 3 1
If A is south of the equator the declination is considered negative, so the same equation will hold true for this case.
FIG. 26. STAR ON THE MERIDIAN
If the star is north of the zenith, as at B, it wil^be more con- venient to use the polar distance,* = 90° — D.
In this case NP = NB - PB
or L — h — p.
If B is below the pole the equation is
L = h + p.
[3] [4]
19. The Astronomical Triangle.
By joining the pole, zenith, and any star 5 on the sphere by arcs of great circles we obtain a triangle from which the rela- tion existing among the spherical coordinates may be obtained. This triangle is so frequently employed in astronomy and navi- gation that is it called the "astronomical triangle" or the "PZS triangle." In Fig. 27 the arc PZ is the complement of the latitude, or co-latitude; arc ZS is the zenith distance or comple- ment of the altitude; arc PS is the polar distance or complement of the declination; the angle P is the hour angle of the star if .S is west of the meridian, or 360° minus the hour angle if S is east of the meridian; and Z is the azimuth of S, or 360° minus the azimuth, according as S is west or east of the meridian. The angle at S is called the parallactic angle ; it is little used in practical astronomy. If any three parts of this triangle are
PRACTICAL ASTRONOMY
known the other three may be calculated. The fundamental formulae of spherical trigonometry are
cos a = cos b cos c + sin b sin c cos A , sin a cos B = cos b sin c — sin b cos c cos ^4 , sin a sin 5 = sin b sin ^4 .
L5J [6] [7]
If we put A = P, B = S, C = Z, a = 90° - h, b = 90° - L, c = 90° — D, then these become
(sin h = sin L sin D + cos L cos D cos P, [8]
cos /* cos S = sin L cos Z) — cos L sin Z) cos P, [9]
cos A sin 51 = cos L sin P. [10]
If A = P, B = Z, C = S, a = 90° - h, b = 90° - D, c = 90° - "L, then
cos h cos Z = sin D cos Z, — cos D sin L cos P, [n]
cos A sin Z = cos Z) sin P. [12]
If ^ = Z, B = S, C = P, a = 90° - D, b = 90° - L, c = 90° - h, then
sin D = sin L sin h + cos Z, cos h cos Z, [13]
cos D cos 5 = sin L cos ^ — cos Lsin k cos Z, [14]
cos D sin 5 = cos L sin Z. [15]
Other forms may be derived by assigning different values to the parts of the triangle ABC. The formulae given in the following chapters may in nearly all cases be derived from equations [5] to [15].
The most common cases arising in the practice of surveying are: —
1. Given the declination, latitude, and altitude, to find the azimuth and the hour angle.
2. Given the declination, latitude, and hour angle, to find the azimuth and the altitude.
RELATION BETWEEN COORDINATES
33
In the following formulae
let
and also let
P = the hour angle,
'Z = the azimuth,*
^h = the altitude,
z = the zenith distance,
D = the declination,
p = the polar distance,
L = the latitude,
FIG. 27. . THE ASTRONOMICAL TRIANGLE
For computing P any of the following formulae may be used, sin \ P = y/Vsin H* + (L - D)} sin * [z-(L=M\ , [l6]
V I T r\
, ? \ cos L cos Z> /
* In the formulae which follow Z is reckoned from the north (interior angle) unless otherwise designated.
34
PRACTICAL ASTRONOMY
sin * P = t/fcosjsinfr-j cos L cos Z)
cos * P =
CQS
- />) sin (5 -
cos D cos
vers P =
cos 5 sin (s — h) ,sin (s— L) cos (5 —
cos (L — D) — sin h cos Z cos 'D
[18] [19]
[201
For computing the angle Z (measured from the north point) we have
n
_ ___ i7 _ //sin* (* + L - D) cosH* + L + Z?)\
cos L sn 2
in Z = 4 /sin (5 - A) sin (5 - L)\ * /
sn
cos L cos
\ Z = J(c°sScoS(s-p)\ y V cos L cos h I
cos
tan i Z =
^ ~ sl 5 ~ cos ^ cos (s — p)
„ ^ cos (L + h) + sin Z) ,versZg* = - — r- — - - •
cos L cos h
[22]
[23] [24] [25]
While any of these formulae may be used to determine the angle sought, the choice of formula should depend somewhat upon the precision with which the angle is denned by the function. If the angle is quite small it is more accurately found through its sine than through its cosine; for an angle near 90° the reverse is the case. On account of the rapid variation of the tangent an angle is always more precisely determined by this function than by either the sine or the cosine. The versed sine formulae require the use of both natural and logarithmic functions, but are sometimes convenient.
* In this case Z is reckoned from the south.
RELATION BETWEEN COORDINATES 35
For computing the altitude and azimuth the following for- mulae may be used:
„ cos M tan P ...
tan Z8 = -T— 77 - — . * 26
sin (L — M)
, cos Zg
tan k = tan (L-lff
where M is an auxiliary angle such that tan M = - — ; Zs is
cos P
measured from the south point.
The altitude may also be found from the formula
sin h = cos (L — D) — 2 cos L cos D sin2 \ P [28]
or sin h = cos (L — D) — cos L cos D vers P, [29]
which may be derived from Equa. [8].
If the declination, hour angle, and altitude are given, the azimuth is found from
. ,, . ^cos D sin Z = sin P - — cos h
= sin P cos D sec h. [30]
For computing the azimuth of a star near the pole when the hour angle is known the following formula is frequently used:
sin P
,, tan Z =
cos L tan D — sin L cos P
This equation may be derived by dividing [12] by [n] and then simplifying the result by dividing by cos D.
Given the latitude and declination, find the hour angle and azimuth of a star on the horizon. Putting h = o in Equa. [8] and [13] the results are
cos P = — tan D tan L [32]
~ sin D T -,
and cos Z = -- - • L33
cos L
* For the derivation of this formula see Chauvenet's Spherical and Practical Astronomy, Vol. I, Art. 14.
PRACTICAL ASTRONOMY
A special case of the PZS triangle occurs when a star near the pole (circumpolar) is at its greatest east or west position, known as its greatest elongation. At this time the star's bearing or azimuth is a maximum and its diurnal circle is tangent to the
E HORIZON A N
FIG. 28. STAR AT GREATEST ELONGATON (EAST)
vertical circle through the star (Fig. 28) ; the triangle is conse- quently right-angled at S. The formulae for this case are
D
cos p =
tan L
and \^ (sinZ = sin p sec L. j> [35]
20. Relation between Right Ascension and Hour Angle.
In order to understand the relation between the right ascen- sion and the hour angle of a point, we may think of the equator on the outer sphere as graduated into hours, minutes, and seconds of right ascension, zero being at the equinox and the numbers increasing toward the east. The equator on the inner sphere is graduated for hour angles, the zero being at the observer's meridian and the numbers increasing toward the west. (See Fig. 29.) As the outer sphere turns, the hour marks on the right ascension scale will pass the meridian in the order of the numbers. The number opposite the meridian at any instant shows how far
37
FIG. 29. RIGHT ASCENSION AND HOUR ANGLE
FIG. 30
38 PRACTICAL ASTRONOMY
the sphere has turned since the equinox was on the meridian. If we read the hour angle scale opposite the equinox, we obtain exactly the same number of hours. This number of hours (or angle) may be considered as either the right ascension of the meridian or the hour angle of the equinox. In Fig. 30 the star S has an hour angle equal to AB and a right ascension CB. The sum of these two angles is AC, or the hour angle of the equinox. The same relation will be found to hold true for all positions of 5. The general relation existing between these coordinates is, then,
Hour angle of Equinox = Hour angle of Star + Right Ascen- sion of Star. [36]
Questions and Problems
1. What is the greatest declination a star may have and culminate south of the zenith?
2. What angle does the plane of the equator make with the horizon?
3. In what latitudes can the sun be overhead?
4. What is the altitude of the sun at noon in Boston (42° 21' N.) on December 22?
5. What are the greatest and least angles made by the ecliptic with the horizon at Boston?
6. In what latitudes is Vega (Decl. = 38° 42' N.) a circumpolar star?
7. Make a sketch of the celestial sphere as it appears to an observer in latitude 20° South at the instant the vernal equinox is on the eastern horizon.
8. Derive formula [35].
CHAPTER V MEASUREMENT OF TIME
21. The Earth's Rotation.
The measurement of intervals of time is made to depend upon the period of the earth's rotation on its axis. Although it is probable that this period is not absolutely invariable, yet the variations are too small to be measured, and the rotation is assumed to be uniform. The most natural unit of time for ordinary purposes is the solar day, or the time of one rotation of the earth with respect to the sun's direction. On account of the earth's annual motion around the sun the direction of the reference line is continually changing, and the length of the solar day is not the true time of one rotation of the earth on its ) axis. For this reason it is necessary in astronomical work to make use of another kind of time, based upon the actual period of rotation, called sidereal time (star time).
22. Transit or Culmination.
Every point on the celestial sphere crosses the meridian of an observer twice during one revolution of the sphere. The instant when any point on the celestial sphere is on the meridian of an observer is called the transit, or culmination, of that point over that meridian. When it is on that half of the meridian contain- ing the zenith, it is called the upper transit; when it is on the other half it is called the lower transit. Except in the case of stars near the elevated pole the upper transit is the only one visible to the observer; hence when the transit of a star is men- tioned the upper transit will be understood unless the contrary is stated.
23. Sidereal Day.
The sidereal day is the interval of time between two successive upper transits of the vernal equinox over the same meridian.
39
40 PRACTICAL ASTRONOMY
If the equinox were absolutely fixed in position, the sidereal day as thus denned would be the true period of the earth's rotation; but since the equinox has a slow westward motion caused by the precessional movement of the axis (see Art. 8), the actual interval between two transits of the equinox differs about os.oi from the true time of one rotation. The sidereal day actu- ally used in practice, however, is the one denned above and not the true rotation period. Sidereal days are not used for reckon- ing long periods of time, dates always being in solar days, so this error never becomes appreciable. The sidereal day is divided into 24 hours and each hour is subdivided into minutes and seconds. When the equinox is at upper transit it is OA, or the beginning of the sidereal day (sidereal " noon ").
24. Sidereal Time.
The sidereal time at a given meridian at any instant is the hour angle of the vernal equinox. It is therefore a measure of the angle through which the earth has turned since the equinox was on the meridian, and shows the position of the sphere at the given instant with respect to the observer's meridian.
25. Solar Day.
A solar day is the interval of time between two successive upper transits of the sun's centre over the same meridian. It is divided into 24 hours, each hour being divided into minutes and seconds. When the sun is on the upper side of the meridian (upper transit) it is noon, or oh solar time. When it is on the lower side of the meridian it is midnight.
26. Solar Time.
The solar time at a given meridian at any instant is the hour angle of the sun's centre at that instant. This hour angle is a measure of the angle through which the earth has turned with respect to the sun's direction, and consequently is a measure of the time elapsed since the sun was on the meridian.
Since the earth revolves around the sun in an elliptical orbit in accordance with the law of gravitation, the apparent angular motion of the sun is not uniform, and the days are therefore of
MEASUREMENT OF TIME 41
unequal length at different seasons. In former times, when sun dials were considered sufficiently accurate for measuring time, this lack of uniformity was not important. Under modern conditions, which demand accurate measurement of time by the use of clocks, an invariable unit of time is essential. As a con- sequence, the time adopted for common use is that kept by a fictitious sun, or mean sun, which is conceived to move at a uniform rate along the equator,* its speed being such that it makes one apparent revolution around the earth in the same time as the true sun (i.e., one year). The fictitious sun is so placed that on the whole it precedes the true sun as much as it follows it. The time indicated by the position of the mean sun is called mean solar time, or simply mean time. The time indicated by the position of the real sun is called apparent solar time and is the time shown by a sun dial.
27. Equation of Time.
Since observations made on the sun for the purpose of deter- mining the time can give apparent time only, it is necessary to be able to find at any instant the exact relation between apparent and mean time. The difference between the two, which varies from +i6m to — i6m (nearly), is called the equation of time. This quantity may be found in the Nautical Almanac for each day of the year.
This difference between the two kinds of time is due to several causes, the chief of which are (i) the inequality of the earth's angular motion in the orbit, and (2) the fact that the true sun is on the ecliptic while the mean sun is on the equator. In the winter, when the earth is nearest the sun, the rate of angular motion about the sun must be greater than in summer in order that the radius vector shall describe equal areas in equal inter- vals of time. (See Fig. 6 and Art. 6.) The sun will then appear
* This statement is true in a general way, but the motion is not strictly uniform because the motion of the equinox itself is variable. The angle from the equinox to the " mean sun " at any instant is the sun's " mean longitude " (along the ecliptic) plus periodic terms.
42 PRACTICAL ASTRONOMY
to move eastward in the sky at a faster rate than in summer, and its daily revolution about the earth will be slower. This delays the instant of apparent noon, making the apparent solar days longer than their average, and therefore a sun dial will " lose time." About April i the sun is moving at its average speed and the sun dial ceases to lose time; from this date until about July i the sun dial gains on mean time, making up what it lost between Jan. i and April i. During the other half of the year the process is reversed; the sun dial gains from July i to Oct. i and loses from Oct. i to Jan. i. The maximum difference in time due to this cause is about 8 minutes, either + or — .
The second cause of the equation of time is illustrated by Fig. 31. Assume that point S' (sometimes called the " first
FIG. 31
mean sun") moves uniformly along the ecliptic at the average rate of the true sun; the time as indicated by this point will evidently not be affected by the eccentricity of the orbit. If the mean sun S (also called " the second mean- sun") starts at V, the equinox, at the same instant that S' starts, then the arcs VS and VS' are equal, since both points are moving with the same speed. By drawing hour circles through these two points it will be seen that these hour circles do not coincide except when the points are at the equinoxes or at the solstices. Since the points are not on the same hour circle they will not cross the meridian at the same time, the difference in time being repre-
MEASUREMENT OF TIME
43
sented by the arc aS. The maximum length of aS is about 10 minutes of time, which may be either + or — . The com- bined effect of these two causes, or the equation of time, is shown in the following table.
TABLE A. EQUATION OF TIME FOR 1910.
|
ISt. |
ioth. |
20th. |
30th. |
|
|
January |
+ yn 26s |
+ ym 2?s |
+ II»* 02» |
+ I3TO 22* |
|
February |
+ 13 4-i |
+ 14 24 |
+ IT. S.Q |
|
|
March |
+ 12 ^8 |
+ 10 36 |
+ 7 48 |
+ 4 4^ |
|
April |
+ 4 08 |
+ 1 31 |
- o <S |
— 2 47 |
|
May |
— 2 CS |
— 3 42 |
3 42 |
- 2 48 |
|
June |
— 2 31 |
- o ?7 |
+ i 08 |
+ •* i; |
|
July . |
+ 3 27 |
-'- ^ 01 |
+ 6 06 |
+ 6 16 |
|
August |
+ 6 n |
+ 5 IQ |
+ * 26 |
+ o 46 |
|
September |
+ o oq |
- 2 48 |
— 6 20 |
— o 46 |
|
October |
— 10 o^ |
— 12 4? |
— i? 01 |
— 16 13 |
|
November |
-16 18 |
— 16 02 |
— 14 26 |
— ii 28 |
|
December |
— ii 06 |
— 7 2T. |
— 2 36 |
+ 2 21 |
28. Conversion of Apparent Time into Mean Time and vice versa.
Apparent time may be converted into mean time by adding
or subtracting the equation of time at the instant. Since the
equation of time is given in the Nautical Almanac for Greenwich
noon its value at the desired instant must be found by adding
\ or subtracting the increase or decrease since Greenwich noon.
Example i. Find the mean time of the sun's transit over the meridian of Boston on June 30, 1910. The apparent time at Boston is 12^ oom oos M. at the instant of the transit of the sun's centre, and this corresponds to 4h 44™ iSs apparent • time at Greenwich, since the longitude of Boston is 4^ 44™ i8s west of Greenwich. The equation of time at Greenwich Apparent Noon is 3"* i4s-92 (to be added to apparent time); the hourly change is os.5oo (increasing). The correction to be applied to the equation of time is 4^.74 X os.5oo = 2s-37, making the equation of time at Boston noon 3"* 17*. 29.
L. A. T.* = 12^00™ oos.oo Equa. of. T =
3 17 -29
L. M. T. = 12* 03™ 1 7s. 29
* A list of abbreviations will be found on p. 191.
44 PRACTICAL ASTRONOMY
Example 2. Find the local apparent time at Boston at 2 P.M. (local mean time) Oct. 28, 1910. The Greenwich Mean Time corresponding to 2 P.M. local mean time is 6h 44"* i8s P.M. The equation of time at G. M. N. Oct. 28, 1910, is i6m04s.29 (to be added to mean time); the hourly increase is os.2o8. The correc- tion to the equation of time is 6^.7 4 X o*.2o8 = is.4o. The equation of time at 2 P.M. is therefore i6TO c>5s.69.
L. M. T. = 2h oom oos.oo
Equa. of T. = 16 05 .69
L. A. T. = 2fti6mo5s.69
29. Astronomical and Civil Time.
For ordinary purposes it is found convenient to divide the solar day into two parts of i2h each; from midnight to noon is called A.M. (ante meridiem), and from noon to midnight is called P.M. (post meridiem). The date changes at the instant of midnight. This mode of reckoning time is called Civil Reck- oning. In astronomical work this subdivision of the day is not convenient. For simplicity in calculation the day is divided into 24^, numbered consecutively from oh to 24^. As it is not convenient to have the date change during the night, the astro- nomical date begins at noon or oh. This is called Astronomical Time. In using the Nautical Almanac it should be remembered ^ that it is necessary to change the date and hours to astronomical y time before taking out the desired data. In order to change •* from one kind of time to the other it is only necessary to remem- ber that the astronomical slay begins at noon of the civil day of the same date; that is, in the afternoon the dates and the hours will be the same, but in the forenoon the astronomical date is one day less and the hours are 1 2 greater than in the civil time.
Examples.
Astr. Time May 10, 15* = Civil Time May u, 3* A.M. " Jan. 3, 7* = " « Jan. 3, 7h P.M.
From these examples the following rules may be derived : To change Civil Time to Astronomical Time, If A.M., add i2h and drop i day from date, and drop the A.M. If P.M., drop the P.M.
MEASUREMENT OF TIME 45
To change Astronomical Time to Civil Time*
If less than i2h, mark it P.M.
If greater than i2h, subtract i2h, add i day to date, and mark it A.M.
30. Relation between Longitude and Time.
The hour angle of the sun at any given meridian at a given instant is the local solar time at that meridian, and will be apparent or mean time according as the true sun or the mean sun is considered. The hour angle of the sun at Greenwich at the same instant is the corresponding Greenwich solar time. The difference between the two hour angles is the longitude of the place from Greenwich, expressed either in degrees or in hours according as the hour angles themselves are expressed in degrees or in hours. Similarly the difference in local solar time of any two places at a given instant is their difference in longitude in hours, minutes, and seconds. In Fig. 32, AC is the hour angle of the sun at Greenwich (G), or the Greenwich solar time. EC is the hour angle of the sun at the meridian through P, or the local solar time at P. The difference, AB, is the longitude of P west of Greenwich. It should be observed that the reasoning is exactly the same whether C represents the true sun or the fictitious sun. The same result would also be found if the point C were to represent the vernal equinox. The arc AC would then be the hour angle of the equinox, i.e., the Greenwich Sidereal Time. BC would be the Local Sidereal Time, and AB the difference in longitude. The measurement of longitude is therefore independent of the kind of time used, because in each case the angular distances to A and B are meas- ured from the same point C on the equator, and the difference in these angles does not depend upon the position of this point nor upon the speed with which this point has moved up to the position at C.
* The student may find it helpful to plot the time along a straight line, and to write two sets of numbers, one for Civil Dates and the other for Astronomical Dates.
46
PRACTICAL ASTRONOMY
The difference in the sidereal times at meridian A and meridian B (Fig. 32) is the interval of sidereal time during which a star would go from A to B. Since the star requires 24 sidereal hours to travel from meridian A to meridian A again, the time interval from A to B bears the same relation to 24* that the longitude
Pole
FIG. 32
difference bears to 360°. The difference in the mean solar times at A and B is the number of mean solar hours that the^sun would take to go from A to B, and since the sun takes 24 solar hours to go from A to A again, the time interval from A to B bears the same ratio to 24 solar hours as when sidereal time was used. The difference in longitude is therefore correctly given when either sidereal or solar times are compared.
The method of changing from Greenwich to local time and the reverse is illustrated by the following examples.
Example i. The Greenwich astronomical time is 7*40"* ios.o. Required the local time at a meridian 4h $om 2is.o West.
G. M. T. = 7h 40™ ios.o Long. West = 4 5° 21 -Q
L. M. T. = 2h 49™ 49s.o (P.M.)
MEASUREMENT OF TIME 47
Example 2. The Greenwich mean time is 3^ 20™ i6s.5. Required the local mean time at a place whose longitude is 120° 10' West.
G. M. T. + 24h = 2jh 20™ i6s.5
Long. West = 8h oom 40s.o
L. M. T. = 19* IQW 36s.s
= 7Ai9TO36s.5A.M.
Example 3. The mean time at a place 3^ East longitude is ioh A.M. Required the Greenwich mean time.
L. M. T. = 22h oom oos. o
Long. East = 3^ oom oos.o
G. M. T. = igh oom oos. o
= Jh 00™ 00S. O A.M.
Since a circle may be divided either into 24^ or into 360°, the relation between these two units is constant. From the fact that
24h = 360° we have also ih= 15°,
iw= 15',
is - IS"- The following equivalents are also convenient:
By means of these two sets of equivalents time may be con- verted into degrees, or the contrary, without writing down the intermediate steps. In the following examples the intermediate steps are written down in order to show the process followed.
Example i. Convert 6h 35™ 51* into degrees.
6h = 90°
35W = 32m + 3m = 8° 45' 5is = 48s + 3s = __ .12' 45"
Total = 98° 57' 45"
Example 2. Convert 47° 17' 35" into hours.
47° = 45° -(- 2° = 3A o8m 17' = 15' + 2' = oimo88 35" = 30" + 5" = 02 .33
Total = 3" 09 w i os. 33
48
PRACTICAL ASTRONOMY
It should be observed that the relation 15° = ih is quite independent of the length of time that has elapsed. A star takes one sidereal hour to move over 15° of hour angle; the sun takes one solar hour to move over 15° of hour angle. In the sense in which it is used here, ih means an angle, and not an absolute interval of time.
31. Relation between Sidereal Time, Right Ascension, and Hour Angle of any Point at a Given Instant.
In Fig. 33 the hour angle of the equinox, or local sidereal time at the meridian through P, is the arc A V. The hour angle of
Pole
FIG. 33
the star 5 at the meridian through P is the arc AB. The right ascension of the star 5 is the arc VB. It is evident from the figure that
AV =VB + AB, or (S—.R.+ Pj [37]
where R = the right ascension and P = the hour angle of the point S, and 5 = the sidereal time; or, in words,
Sidereal Time = Right Ascension -\-Hour Angle. [38]
MEASUREMENT OF TIME 49
This relation is a perfectly general one and will be found to hold true for all points on the sphere, provided it is agreed to reckon the sidereal time beyond 24* when necessary. For example, if the hour angle is ioh and the right ascension is 2oh, the resulting sidereal time is 30*. This means that the equinox has made a complete revolution and has gone 6h, or 90°, on the next revolu- tion; the actual reading of the sidereal clock would be 6h. In the reverse case, when it is necessary to subtract 2OA from 6* to obtain the hour angle, the 6h must first be increased by 24* and the right ascension subtracted from the sum to obtain the hour angle, ioh.
32. Star on the Meridian.
At the instant when the star is on the meridian its hour angle is OA and the equation becomes
Sidereal Time = Right Ascension; [39]
that is, the right ascension of a star equals the local sidereal time at which that star crosses the meridian. (See Art. 20, p. 36.)
33. Relation between Mean Solar and Sidereal Intervals of Time.
It has already been stated that on account of the earth's orbital motion the sun has an apparent eastward motion among the stars of nearly i° per day. This eastward movement of the sun makes the intervals between the sun's transits greater by nearly 4"* than the intervals between the transits of the equinox, that is, the solar day is nearly 4™ longer than the sidereal day. In Fig. 34 let C and C' be the positions of the earth on two consecutive days. When the observer is at O it is local noon. After the earth makes one complete rotation, the observer will be at O', and the sidereal time will be exactly the same as it was the day before when he was at O. But the sun's direction is now C'O", so the earth must turn through the angle O'C'O" in order to bring the sun again on the observer's meridian. Since this angle is about i° it takes about 4™ longer to complete the solar day than it does to complete the sidereal day. Since
5° PRACTICAL ASTRONOMY
each kind of day is subdivided into hours, minutes, and seconds, all of these units in solar time will be proportionally longer than the corresponding units of sidereal time. If two clocks, one regulated to mean solar time and the other to sidereal time, were started at the same instant, both reading oh, the sidereal clock would immediately begin to gain on the solar clock, the gain
FIG. 34
being exactly proportional to the time interval, that is, about ios per hour, or more nearly 3"* 56* per day.
In order to find the exact relation between the two kinds of time it should be observed that the number of sidereal days in the year is exactly one greater than the number of solar days, because the sun comes back to the equinox at the end of one year. The length of the tropical* year is found to be 365.2422
* The tropical year is the interval of time between two successive passages of the sun over the vernal equinox. The sidereal year is the interval between two passages of the sun across the hour circle through a fixed star on the equator. On account of the movement of the equinox caused by precession, the tropical year is about 2OTO shorter than the sidereal year.
MEASUREMENT OF TIME 51
mean solar days. The relation between the two kinds of day is therefore
366.2422 sidereal days = 365.2422 solar days, [40]
or i sidereal day = 0.99726957 solar day, [41]
and i solar day = 1.00273791 sidereal days. [42]
Equations [41] and [42] may be written
24h sidereal time = (24* — 3™ 55S.9O9) mean solar time, 24* mean solar time = (24* + 3 m 56^.555) sidereal time.
These equations may be put in more convenient form for com- putation by expressing the difference in time as a correction to be applied to any interval of time to change it from one kind of unit to the other. If Im is a mean solar interval and 7S the corresponding number of sidereal units, then
Is = Im + .00273791 X Im [43]
and Im = Is - .00273043 X /s. [44]
Tables II and III are constructed by multiplying different values of Im and I8 by these constants. More extended tables may be found in the Nautical Almanac. The use of Tables II and III is illustrated by the following examples.
Examples.
Reduce 9^ 23'" 5is.oof sidereal time to the equivalent number of solar units. From Table II, opposite 9^ is the correction — im 28S.466; opposite 23™ in the 4th column is —3". 768; and opposite 5is in the last column is os.i39. The sum of these three partial corrections is — im 32S.373, which is the amount to be subtracted from gh 23"* 5is.o to reduce it to the equivalent solar interval, gh 22™ 1 8s. 627.
Reduce jh iom solar time to sidereal time. The correction for 7 , Table III, is + im o8s.995, and for iom is is.643. The sum, im 1 0^.638, added to 7* iom gives 7* uw ios.638 of sidereal time.
This reduction may be made approximately by the following rule: the correction equals io8 per hour diminished by i8 for
PRACTICAL ASTRONOMY
every 6* in the interval. The correction for 6h would be 6 X io8 — is = 59s. This rule is based on a change of 3"* 56* per day. For changing solar into sidereal the error is os.o23 per hour; for sidereal into solar the error is os.oo4 per hour.
It should be kept in mind that the .conversion of time discussed in this article concerns the change from one kind of unit to another, like changing from yards to metres, and is not the same as changing from the local sidereal time to the local solar time at a particular instant.
34. Relation between Sidereal Time and Mean Solar Time at any Instant.
If in Fig. 33, Art. 31, the point B is taken to represent the mean sun, then equation [37] becomes
S = Rs + Ps, [45]
where Rs and Ps are the right ascension and the hour angle of the mean sun at the instant considered. Ps is the local mean time by the definition given in Art. 26. If the equation is written
S - Ps = Ra, [46]
then, since the value of the right ascension Rs does not depen upon the time at any particular meridian, but only upon the absolute instant of time considered, it is evident that the differ- ence between sidereal time and mean time at any instant is the same for all places on the earth. The actual values of S and Pa will of course be different at different meridians, but the difference between the two is a constant for all places for the given instant. In order that Equa. [45] shall hold true it is essential that Rs and Ps shall refer to the same position of the sun, that is, to the same absolute instant of time. The right ascension of the sun obtained from the Nautical Almanac is its value at the instant of the Greenwich Mean Noon preceding, that is, at the beginning of the astronomical day at Greenwich.*
* The dates are always in mean solar days, not in sidereal days.
MEASUREMENT OF TIME
53
To reduce this right ascension to its value at the desired instant it is necessary to multiply the hourly increase in the right ascen- sion of the mean sun by the number of solar hours elapsed since the instant of Greenwich Mean Noon. The hourly increase in the right ascension of the mean sun is constant and is evidently equal to the correction in Table III, for the difference between sidereal and solar time is caused by the sun's motion, and the amount of the difference for any number of hours is exactly equal to the increase in the right ascension. If it is desired to find the increase for any number of solar hours, Table III should be used; for sidereal hours use Table II. Equation [45] may be written
5 = Rs + Ps + C, [47]
where Rs refers to the instant of the preceding local mean noon, and C is the correction (Table III) to reduce Ps to a sidereal interval, or to reduce Rs to its value at the time Ps.
In Fig. 35 suppose that the sun S and a star 5" passed the meridian M at the same instant, and at the mean time Ps it is
M
FIG. 35
desired to compute the sidereal time. Since the sun is moving at a slower rate than the star, it will describe the arc MS ( = Pa)
54 PRACTICAL ASTRONOMY
while the star moves from M to S'. The arc SS', or C, repre- sents the gain of sidereal on mean time in the mean time interval MS or Ps. But S' is the position of the sun at noon, so that VSf is the sun's right ascension at the preceding mean noon, or Rs. The right ascension desired is VS, so Rs must evidently be increased by the arc SS', or C.
If it is desired to find the mean solar time corresponding to a given instant of local sidereal time,, the equation is
Sidereal internal from noon = S — Rs, [48]
or Mean time = Ps = S — Rs — C', [49]
where C' is the correction from Table II to reduce 5" — Rs to a solar interval, and represents the increase in the sun's right ascension in 5 — Rs sidereal hours. Examples.
To find the Greenwich Sidereal Time corresponding to Greenwich Mean Time -9* 22m i8s.6o on Jan. 7, 1907. The right ascension of the mean sun at Greenwich Mean Noon is found from the Nautical Almanac to be 19^ 03"* 36s-38. The cor- rection to reduce 9* 22™ i88.6o to sidereal time (Table III) is +im 328-37. Then, applying Equa. [47],
Rs = i9A03"»368.38
Ps = 9 22 18 .60
C = _ i 32 -37
S = 2Sh 27m 27S.3S
Sidereal Time = 4* 27"* 27^.35
To find the Greenwich Mean Solar Time when the Greenwich Sidereal Time is 4h 27™ 27*.3S on Jan. 7, 1907.
S = 2&h 27m 27S.35
Rs = 19 03 36 .38
5 - R9 = 9 23 50 .97
C' (Table H) = -i 32.37
Mean Time = 9* 22TO i8*.6o
If the change from sidereal to solar time (or vice versa) is to be made at any meridian other than Greenwich, the right ascension of the sun for local noon must be found by multiplying the increase per solar hour by the number of solar hours since Green- wich noon, that is, by the number of hours in the longitude, and
MEASUREMENT OF TIME 55
adding this to the value of Rs from the Almanac if the place is west of Greenwich, subtracting if east.* The correction may be taken from Table III. If the sidereal time in the above example is assumed to be the time at a meridian 5A (75°) west of Greenwich, the computation would be modified as follows:
R,f = igh 03™ 368.38
Correction for 5^ longitude = 49 .28
Rs = 19 04 25 .66
S = 28 27 27 .35
5 — Rs = 9 23 01 .69 C - i 32 .24
Local Mean Time = gh 2im 29^45
It is evident that at the instant of mean noon Ps = o and Rs = S. At mean noon, therefore, the sidereal time equals the right ascension of the mean sun. This quantity will be found in the Almanac under both headings, " Sidereal Time of Mean Noon" and " Right Ascension of the Mean Sun." (See
P- 65.)
The reduction of mean solar time to sidereal time, or the re- verse, may be made also by first changing the given local time to the corresponding instant of Greenwich time, then making the transformation as before, and finally changing back to the meridian of the place. Take, for example, the case given on page 54.
Local Sidereal Time = 28^ 27m27s.35
Longitude = 5 oo oo
Greenwich Sidereal Time = 33 27 27 .35
Rs at Gr. M. Noon = 19 03 36 .38
Sidereal Interval from Noon = 14 23 50 .97
C' = —2 21 .52
Greenwich Mean Time = 14 21 29 .45
Longitude = 5 oo oo Local Mean Time = gh 2iOT29s.45
The result agrees with that obtained by the former method. This method is quite as simple as the preceding, especially when
* It should be remembered that the sun's R. A. is always increasing.
56 PRACTICAL ASTRONOMY
Standard Time is to be computed, for the final correction will always be a whole number of hours. Care should be taken always to use the right ascension of the sun at the noon preced- ing the given time. Suppose that the instant of ioh A.M. May 5 is to be converted into sidereal time, the longitude of the place being 4h 4.4™ i8s west. Civil time ioh A.M. May 5 = Astr. time 22h May 4. If the first method is followed, the right ascension of the sun employed should be that of noon May 4. If the reduction is made by first changing to Greenwich time, then 22* + 4h 44m i8s = 26* 44™ i8s May 4 = 2h 44™ i8s May 5. The right ascension for the latter case would be that for noon of May 5.
35. Standard Time.
From the definition of mean solar time it will be seen that at any given instant the solar times at two places will differ from each other by an amount depending upon the difference in the longitudes. All places will have different local times except where they happen to be on the same meridian. Previous to the year 1883 it was customary in this country for each large city or town to use the mean time at its own meridian, and for all other places in the vicinity to adopt the same time. Before railroad travel became extensive this change of time from one point to another caused no great difficulty, but with the in- creased amount of railroad and telegraph business these frequent and irregular changes in time became so inconvenient that in 1883 a uniform system of time was adopted in the United States. The country is divided into time belts each theoretically 15° in width; these are known as the Eastern, Central, Mountain and Pacific time belts, and places in these belts use the mean local time of the 75°, 90°, 105° and 120° meridians respectively. The time at the 60° meridian is called Atlantic time and is used in the Eastern Provinces of Canada. The actual positions of the dividing lines between these belts depend upon the positions of the principal cities and the railroads (see Fig. 36), but the change of time from one belt to another is always exactly one hour. The
MEASUREMENT OF TIME
57
58 PRACTICAL ASTRONOMY
minutes and seconds of all clocks are the same as the minutes and seconds of the Greenwich clock. When it is noon at Green- wich it is 8 A.M. Atlantic time, 7 A.M. Eastern time, 6 A.M. Central time, 5 A.M. Mountain time, and 4 A.M. Pacific time.
The change from local to standard time, or the contrary, con- sists in expressing the difference in longitude between the local meridian and the standard meridian in units of time, and adding or subtracting this correction, remembering that the farther west a place is, the earlier it is in the day at any given instant of time.
Examples.
Find the standard time at a place 71° west of Greenwich when the local time is 4h 20™ oos P.M. In longitude 71° the standard time would be that of the 75° meridian. The difference in longitude is 4° = i6m . Since the standard meridian is west of the 71° meridian, the time is i6m earlier than the local time. The standard time is therefore 4* 04"* oo8 P.M.
Find the local time at a place 91° west of Greenwich when the Central time is 9* OOTO oos A.M. The difference in longitude is i° = 4m. Since the place is west of the standard meridian, the time is earlier. The local time is therefore Sh 56™ oos A.M.
Standard time is used not only in the United States but in a majority of the countries of the world; in nearly all cases these systems of standard time are based on the meridian of Green- wich as the prime meridian. Germany, for example, uses the local mean time at the meridian ih east of Greenwich; Japan uses that of the meridian 9* east of Greenwich ; Turkey, 2h east cf Greenwich, etc.
36. The Date Line.
If a person were to start at Greenwich at the instant of noon and travel westward rapidly enough to keep the sun always on his meridian he would get back to Greenwich 24* later, but his own (local) time would not have changed but would have remained noon all the time. In travelling westward at a slower rate the same thing occurs, only in a longer interval of time.
MEASUREMENT OF TIME 59
The traveller has to set his watch back every day in order to keep it regulated to the meridian at which his noon occurs. As a consequence, his watch has recorded one day less than it has actually run, and his calendar is one day behind that of a person who remains at Greenwich. If the traveller goes east he has to set his watch ahead every day, and after circumnavigating the globe his calendar is one day ahead of what it should be. In order that the calendar may be everywhere uniform, it is agreed to change the date at the meridian 180° from Greenwich. When- ever a ship crosses the 180° meridian going westward, a day is omitted from the calendar, and when going eastward a day is repeated. In practice the change is made at midnight near the 180° meridian, not at the instant of crossing. The date line actually used does not follow the 180° meridian in all places, but is deflected so as not to separate the Aleutian islands, and in the South Pacific ocean it passes east of several groups of islands so as not to change the date formerly used in these islands.
37. The Calendar.
Previous to the time of Julius Caesar the calendar was based upon the lunar month, and, as this resulted in a continual change in the date at which the seasons occurred, the calendar was fre- quently changed in an arbitrary manner in order to keep the seasons in their places, the result being extreme confusion in the dates. In the year 45 B.C. Julius Caesar reformed the calendar and introduced one based upon a year of 365! days, since called the Julian calendar. The \ day was taken care of by making the year contain 365 days, except every 4th year, called leap year, which contained 366; the extra day was added to February in such years as were divisible by 4. The year was begun on Jan. i ; previously it had begun in March. Since the year con- tains actually 365^ 5^ 48 m 46*, this difference of nm 14* caused a gradual change in the dates at which the seasons occurred. After many centuries the difference had accumulated to about 10 days, so in 1582 Pope Gregory XIII ordered that the calendar should be corrected by dropping ten days and that future dates
6o
PRACTICAL ASTRONOMY
should be computed by omitting the 366th day in those leap years which occurred in century years not divisible by 400; that is, such years as 1700, 1800 and 1900 should not be counted as leap years. This is the calendar used at the present time.
Questions and Problems
1. (a) Prove by direct computation of sidereal time from Fig. 37 that
R + P = 24h + S,
in which R and P are the right ascension and hour angle of the star S, and S is the sidereal time, or hour angle of V.
(b) Prove the same relation when V is at the point V. (See Art. 31, p. 48.)
2. Prove that the difference in longitude of two points is independent of the kind of time used, by selecting two points at which the solar time differs by say 3A, and then converting the solar time at each place into sidereal time.
FIG. 37
FIG. 38
3. Make a design for a horizontal sun dial for a place whose latitude is 42° 21' N. The gnomon ad (Fig. 38), or line which casts the shadow on the horizontal plane, must be parallel to the earth's rotation axis; the angle which the gnomon makes with the horizontal plane therefore equals the latitude. The shadow lines for the hours (X, XI, XII, I, II, etc.) are found by passing planes through the gnomon and finding where they cut the horizontal plane of the dial. The vertical plane adb coincides with the meridian and therefore is the noon (XII*) line. The other planes make, with the vertical plane, angles equal to some multiple of 15°. In finding the trace dc of one of these planes on the dial it should be observed that the foot of the gnomon, d , is a point common to all such traces. In order to find another point c on any trace, or shadow line, pass a plane abc through some point a on the gnomon and perpendicular to it. This plane (the plane of the equator) will cut an east and west line cc on the dial. If a line be drawn in this plane making an
MEASUREMENT OF TIME 6 1
angle of n X 15° with the meridian plane, it will cut ce at a point c which is on the shadow line. Joining c with the foot of the gnomon gives the required line.
In making a design for a sun dial it must be remembered that the west edge of the gnomon casts the shadow in the forenoon and the east edge in the afternoon; there will be of course two noon lines, and the two halves of the diagram will be symmetrical and separated from each other by the thickness of the gnomon. The dial may be placed in position by levelling the horizontal surface and then com- puting the watch time of apparent noon and turning the dial so that the shadow is on the XIIA line at the calculated time.
Prove that the horizontal angle bdc is given by the relation
tan bdc = tan P sin L, in which P is the sun's hour angle and L is the latitude.
4. Why are the sun's and moon's right ascension always increasing?
5. The local apparent time at a point A is ioh 30"* A.M. If the equation of time is + 3"* 25^.8, what is the local mean time? What is the astronomical mean time at the given instant? Assuming the longitude of A to be 95° West, what is the Greenwich Mean Time? What is the Central Standard Time? What is the local mean time at the same instant at a point B in longitude 110° W.? If the right ascension of the mean sun at G. M. N. is i8A4iTOois.6, what is the local sidereal time? What is the Greenwich Sidereal Time?
CHAPTER VI
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC — STAR CATALOGUES — INTERPOLATION
38. The Ephemeris.
In the problems previously discussed it has been assumed that the coordinates of celestial objects and various other data men- tioned are known to the computer. These data consist of results calculated from observations made with large instruments at the principal observatories; these results are published by the government several years in advance in the American Ephemeris and Nautical Almanac.* The Almanac contains the declinations and the right ascensions of the sun, moon, planets and stars, as well as the angular semidiameters, horizontal parallaxes, the equation of time, and other data required in astronomical cal- culations. Since all of these quantities vary with the time, their values are usually given for equidistant intervals of Green- wich time or of Washington time.
The Almanac is divided into three parts. Part I is computed for the meridian of Greenwich, and is arranged especially for the convenience of navigators. Part II is computed for the meridian of Washington, and is arranged chiefly for the con- venience of astronomers. Part III contains the data for pre- dicting phenomena, such as eclipses, occultations, etc. At the end of the book are certain tables computed especially for the use of the navigator and the surveyor.
The first page of Part I is headed "At Greenwich Apparent
* Similar publications by other governments are: the Nautical Almanac (Great Britain), the Berliner Astronomisches Jahrbuch (Germany), the Con- naissance des Temps (France), and the Almanaque Nautico (Spain).
The word " ephemeris " means a table of coordinates of a celestial body given for equidistant intervals of time.
62
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 63
Noon," and contains the following data for that instant for each day in the month: right ascension and declination of the sun and their hourly changes, sun's semidiameter, time of semi- diameter passing the meridian, and the equation of time with its hourly change. The second page is headed "At Greenwich Mean Noon," and contains the right ascension and declination of the sun with their hourly changes, the equation of time with its hourly change, and the right ascension of the mean sun or sidereal time at mean noon. These pages are the ones to be', used by the navigator or the surveyor when making observa-( tions on the sun. Whether Table I or Table II shall be used in any given case depends upon whether apparent time or mean time is the more convenient. If the declination, right ascension, or equation of time is required for the instant of the sun's transit over any meridian, then the local apparent time is noon and the Greenwich Apparent Time is equal to the west longitude of the place. The desired quantity is found by taking out its value for the instant of Greenwich Apparent Noon and increasing or decreasing it by the hourly change. multiplied by the number of hours in the longitude. If the quantity is to be found for some instant of local mean time or Standard Time, then the Greenwich Mean Time may be readily found, and it is therefore more con- venient to compute the required value from that given for Green- wich Mean Noon. If Local Time is used, the Greenwich Time is found by adding the longitude; if Standard Time is used, the Greenwich Time is found by adding 5^, 6h, etc., according to the time belt indicated. The tabular quantity is then corrected for the time elapsed since Greenwich Mean Noon. Tables I and II for the month of January, 1912, are shown on pages 64 and 65. The third page of the Almanac contains data not usually required by the surveyor. On the fourth page are the semidiameter, horizontal parallax, and time of transit of the moon, with their hourly changes. On account of the rapidity with which the semidiameter and parallax vary, they are given for both Green- wich noon and midnight. The next eight pages contain the
64
PRACTICAL ASTRONOMY
JANUARY, 1912 AT GREENWICH APPARENT NOON
|
. |
THE SUN'S |
-j |
|||||||
|
g |
W.S |
Equa- |
|||||||
|
Day of the, week. |
"o |
Apparent right ascen- |
- 1 Diff. for i |
Apparent declination. |
Diff. for i hour. |
Semi- diam- |
;al time ol meter pas meridian |
tion of time, to be add- ed to ap- |
Diff. for i hour. |
|
• |
sion. |
hour. |
eter. |
"'•I |
parent time. |
||||
|
Q |
K |
||||||||
|
h m s |
$ |
0 ' II |
" |
' " |
i |
m s |
s |
||
|
Mon. |
i |
18 42 21.37 |
11-053 |
S. 23 5 43.7 |
+ 11.19 |
16 17.89 |
71.09 |
3 13-30 |
i -193 |
|
Tues. |
2 |
1 8 46 46.49 |
i i . 039 |
23 i i-5 |
12-34 |
16 17.90 |
71.04 |
3 41-79 |
1.179 |
|
Wed. |
3 |
18 51 11.26 |
11.025 |
22 55 51.6 |
13-49 |
16 17.90 |
71.00 |
4 9-03 |
1.165 |
|
Thur. |
4 |
18 55 35-68 |
11.009 |
22 50 14-4 |
+ 14-63 |
16 17.91 |
70.95 |
4 37-71 |
1.149 |
|
Frid. |
5 |
18 59 59-71 |
10.992 |
22 44 IO.O |
15-75 |
16 17.91 |
70.90 |
5 5-ii |
I-I33 |
|
Sat |
6 |
19 4 23-33 |
10-975 |
22 37 38.5 |
16.87 |
16 17.90 |
70.84 |
5 32.li |
1. 115 |
|
Sun. |
7 |
19 8 46.53 |
10-957 |
22 30 40-3 |
+ 17-98 |
16 17.87 |
70.78 |
5 58.67 |
1-097 |
|
Mon. |
8 |
19 13 9-28 |
10-939 |
22 23 15-3 |
19-09 |
16 17.84 |
70.71 |
6 24.78 |
1.077 |
|
Tues. |
9 |
19 17 31-55 |
10.919 |
22 IS 23.9 |
20.19 |
16 17.80 |
70-64 |
6 50.41 |
1-057 |
|
Wed. |
10 |
19 21 53-31 |
10.896 |
22 7 6.3 |
+ 21.28 |
16 17.76 |
70.57 |
7 15-54 |
1.036 |
|
Thur. |
ii |
19 26 14-54 |
10.873 |
21 58 22.7 |
22.36 |
16 17.72 |
70.50 |
7 40.15 |
1.014 |
|
Frid. |
12 |
19 30 35-23 |
10.850 |
21 49 13.3 |
23-43 |
16 17.67 |
70.42 |
8 4.21 |
0.990 |
|
Sat |
13 |
19 34 55-33 |
10.826 |
21 39 38.4 |
+ 24-49 |
16 17.61 |
70.34 |
8 27.70 |
0.966 |
|
Sun. |
14 |
19 39 14-83 |
10.800 |
21 29 38.3 |
25-53 |
16 17-55 |
70.26 |
8 50.59 |
0.941 |
|
Mon. |
15 |
19 43 33-71 |
10-773 |
21 19 I3.I |
26.57 |
16 17.48 |
70.17 |
9 12.86 |
0.914 |
|
Tues. |
16 |
19 47 51-96 |
10.746 |
21 8 23.3 |
+ 27-59 |
16 17.41 |
70.08 |
9 34-48 |
0.887 |
|
Wed. |
*7 |
19 52 9-54 |
10.717 |
20 57 9.2 |
28.59 |
16 17-34 |
69.98 |
9 55-42 |
0.859 |
|
Thur. |
18 |
19 56 26.42 |
10.688 |
20 45 31.0 |
29-58 |
16 17.26 |
69.88 |
10 15.68 |
0.830 |
|
Frid. |
19 |
20 o 42.58 |
10.658 |
20 33 29.1 |
+ 30-56 |
16 17.18 |
69.78 |
10 35-24 |
0.800 |
|
Sat. |
20 |
20 4 58.02 |
10.628 |
20 21 3-9 |
31-53 |
16 17.10 |
69.68 |
10 54.08 |
0.769 |
|
Sun. |
21 |
20 9 12.71 |
10-597 |
20 8 15.7 |
32.48 |
16 17.01 |
69-57 |
ii 12.17 |
0-738 |
|
Mon. |
22 |
20 13 26.64 |
10.565 |
19 55 4-8 |
+33-41 |
16 16.92 |
69.47 |
ii 29.49 |
0.706 |
|
Tu s. |
23 |
20 17 39-8o |
10-533 |
19 41 31.7 |
34-33 |
16 16.82 |
69.36 |
ii 46.04 |
0.673 |
|
Wed. |
24 |
20 21 52.16 |
10.500 |
19 27 36.7 |
35-24 |
16 16.72 |
69.26 |
12 1. 80 |
0.640 |
|
Thur. |
25 |
2O 26 3.72 |
10.465 |
19 13 20.0 |
+ 36.13 |
16 16.62 |
69.15 |
12 16.76 |
0.607 |
|
Frid. |
26 |
2O 30 14.46 |
10.430 |
18 58 42.3 |
37-oi |
16 16.52 |
69.04 |
12 30.91 |
0-573 |
|
Sat |
27 |
20 34 24.38 |
10.396 |
18 43 43-9 |
37-86 |
16 16.41 |
68.93 |
12 44.24 |
0-538 |
|
Sun. |
28 |
20 38 33-48 |
10.361 |
18 28 25.0 |
+ 38-70 |
16 16.29 |
68.82 |
12 56.74 |
0.504 |
|
Mon. |
29 |
20 42 41-74 |
10.326 |
18 12 46.2 |
39-52 |
16 16. 16 |
68.71 |
13 l8.4I |
0.469 |
|
Tues. |
30 |
2O 46 49.16 |
10.291 |
17 56 48.0 |
40.32 |
16 16.03 |
68.60 |
13 19-25 |
0-434 |
|
Wed. |
31 |
20 50 55.74 |
10.256 |
17 40 30.7 |
41.11 |
16 15.90 |
68.48 |
13 29.25 |
0.400 |
|
Thur. |
32 |
20 S5 1-49 |
10.222 |
S. 17 23 54.5 |
+41.89 |
16 15.76 |
68.37 |
13 38.41 |
0.365 |
Note. — The mean time of semidiameter passing may be found by subtracting o*.ig from the sidereal time. The sign + prefixed to the hourly change of declination indicates that south decli- nations are decreasing.
JANUARY, 1912 AT GREEN'WICH MEAN NOON
|
Day of the week. |
Day of the month. |
THE SUN'S |
Equation of time, to be sub- tracted from mean time. |
Diff. for i hour. |
Sidereal time, or right ascension of mean sun. |
|||
|
Apparent right ascen- sion. |
Diff. for i hour. |
Apparent declination. |
Diff. for i hour |
|||||
|
Mon. Tues. Wed. |
i 2 3 |
h m s 18 42 20.78 18 46 45-81 18 51 10.50 |
^ i i . 049 11-035 II.O2I |
S. 23 5 44.3 23 I 2.2 22 55 52-5 |
+ 11. 18 12.33 13-47 |
m s 3 13-24 3 41-72 4 9-85 |
s i-i93 1.179 1.165 |
h m s 18 39 7-54 18 43 4. 10 18 47 0.66 |
|
Thur. Frid. Sat. |
4 6 |
18 55 34.84 18 59 58.79 19 4 22.33 |
II. 006 10.990 10.972 |
22 50 15.5 22 44 11.3 22 37 40.1 |
+ 14.61 15-74 16.86 |
4 37-62 5 S-oi 5 32-00 |
1.149 I-I33 1-115 |
18 50 57.22 18 54 53-78 18 58 50.33 |
|
Sun. Mon. Tues. |
7 8 9 |
19 8 45-45 19 13 8.12 19 17 30.30 |
10.953 10.934 10.914 |
22 30 42.0 22 23 17.3 22 15 26.2 |
+ 17-97 19.08 20.17 |
5 58.56 6 24.66 6 50.29 |
1.097 1.077 1-057 |
19 2 46.89 19 6 43.45 19 10 40.01 |
|
Wed. Thur. Frid. |
10 ii 12 |
19 21 51.99 19 26 13.15 19 3° 33-77 |
10.893 10.870 10.847 |
22 7 8.8 21 58 25.5 21 49 16.4 |
+ 21.26 22.34 23-41 |
7 15-42 7 40.02 8 4.08 |
1.036 1.014 0.990 |
19 14 36.57 19 18 33-13 19 22 29.68 |
|
Sat Sun. Mon. |
13 14 IS |
19 34 53- 81 19 39 13-25 19 43 32.07 |
10.822 10.797 10.770 |
21 39 41.8 21 29 42.0 21 19 17.2 |
+ 24-47 25-51 26.55 |
8 27-57 8 50-45 9 12-71 |
0.966 0.940 0.914 |
19 26 26.24 19 30 22.8o 19 34 19-36 |
|
Tues. Wed. Thur. |
16 i? 18 |
19 47 50-25 19 52 7-76 19 56 24.58 |
10-743 10.715 10.686 |
21 8 27.7 20 57 13-9 20 45 36.1 |
+ 27-57 28.58 29-57 |
9 34-33 9 55-28 10 15.54 |
0.887 0.859 0-830 |
19 38 15-92 19 42 12.48 19 46 9.03 |
|
Frid. Sat. Sun. |
19 20 21 |
20 o 40.70 20 4 56.09 20 9 10.73 |
10.656 10.626 10.595 |
20 33 34.6 2O 21 9.7 20 8 21.8 |
+ 30-55 31-52 32-47 |
10 35-10 10 53-94 n 12.03 |
0.800 0.769 0.738 |
19 50 5-59 19 54 2.15 19 57 58-71 |
|
Mon. Tues. Wed. |
22 23 24 |
20 13 24.62 20 I? 37.74 20 21 50.06 |
10-563 10.530 10.497 |
19 55 n-2 19 41 38.4 19 27 43.7 |
+ 33-40 34-32 35-23 |
II 29.36 ii 45-91 12 1.68 |
0.706 0-673 0.640 |
20 i 55.27 20 5 51.82 20 9 48.38 |
|
Thur. Frid. Sat. |
25 26 27 |
20 26 1.58 2O 3O 12.29 20 34 22.18 |
10.463 10.429 10-395 |
19 13 27.4 18 58 50.0 18 43 51.9 |
+ 36.12 37-00 37-85 |
12 16.65 12 30.80 12 44-13 |
0.607 0-573 0-538 |
20 13 44-94 20 17 41.50 20 21 38.05 |
|
Sun. Mon. Tues. Wed. |
28 2Q 30 31 |
20 38 31.25 20 42 39.48 20 46 46.87 2° So 53-44 |
10.360 10.326 10.291 10.256 |
18 28 33.4 18 12 55.0 i? 56 57-o 17 40 39.9 |
+ 38-69 39-Si 40.31 41.10 |
12 56.64 13 8.31 13 19-15 13 29.16 |
0.504 0.469 0-434 0.400 |
20 25 34 -61 20 29 31.17 20 33 27.72 20 3? 24.28 |
|
Thur. |
32 |
20 54 59-17 |
IO.22I |
S. 17 24 4.0 |
+ 41-88 |
13 38.33 |
0-365 |
20 41 20.84 |
|
Note. — The semidiameter for mean noon may be assumed the same as that for ap- parent noon. The sign 4- prefixed to the hourly change of declination indicates that >outh declinations are decreasing. |
Diff. for i Hour, +98-8s6s. (Table III.) |
66
PRACTICAL ASTRONOMY
MEAN PLACES OF STARS, 1912.
WASHINGTON, JANUARY I<*.OO6.
|
Name of star. |
Magni- tude. |
Right Ascension. |
Annual Variation. |
Declination |
Annual Variation. |
|
v/ 33 Piscium |
4-7 |
h m s o o 40.808 |
i + 3.o7l<« |
o ' // — 6 II en. 4? |
rt -f- 20 176 |
|
. a Andromedae (Alphe- ratz) |
2 2 |
o 3 co 162 |
2 OO5? |
4-28 3.6 16 t;o |
19 880 |
|
v ft Cassiopeiae |
2 4 |
O A 28 CO4. |
•2 l834 |
4- ?8 3O "CT O7 |
19 862 |
|
v € Phoenicis |
•2.O |
O 4 ^6.83.2 |
^.O?2O |
— 46 i3 ?8.o6 |
19 848 |
|
2 2 Andromedae |
e i |
O C 44. C7I |
3 I088 |
4-4C 34 C7 27 |
|
|
y Pegasi |
2.Q |
o 8 42 161 |
+ "3 0861 |
4- IA 4.1 3O 7? |
* O *J 4~ 20 02 1 |
|
<r Andromedae |
4 1 |
Oil 4? 6l 1 |
•2 I2OO |
4- -26 17 CO C2 |
10 063 |
|
t Ceti |
1 8 |
O 14. ?6 674 |
3O?7O |
— o i 8 42 04 |
IO O7A |
|
f Tucanae |
45 |
O 1C 2O 74? |
1 lAO1? |
— 6c 23 2O 80 |
|
|
44 Piscium |
6 o |
O 2O ? ? 4.64 |
•2 O742 |
+1 27 8 CO |
|
|
ft Hydri |
2.Q |
o 21 8 620 |
+ 5 2O4.3 |
— 77 A A CO 48 |
4- 2O 27O |
|
at Phoenicis |
2 4. |
O 21 ?6 2 ?4 |
2 O73I |
— 42 47 I OC |
TQ CCI |
|
1 2 Ceti |
6 0 |
o 25 32 885 |
-2 0621 |
— 4. 26 36 2C |
|
|
I? Ceti t |
52 |
•2 0877 |
|||
|
f Cassiopeiae |
•2 7 |
O 3.2 3 748 |
3 3.272 |
4- C3 24 J.C 87 |
10 843 |
|
TT Andromedae. . . |
4 4. |
4- 3 TO7O |
|||
|
e Andromedae |
4. e |
O ?? 1\4.I'?I |
3..I63.8 |
4-28 50 2 68 |
IO C73 |
|
5 Andromedae |
2. C |
O ^4 'I?. I 2.8 |
7 2OI4. |
4- 3O 22 4.C 08 |
TO 721 |
|
a Cassiop. (Schedir}.-\ ft Phoenicis. |
var. A 6 |
o 35 30-339 |
3-3852 |
+ 56 3 17-55 |
19-774 |
|
ft Ceti |
2 . 2 |
O 3Q IO. ^8l |
+ 3 0126 |
— 18 28 9 82 |
4- 10 7oc |
|
o Cassiopeiae .... |
4 7 |
TO 738 |
|||
|
21 Cassiopeise . |
c 6 |
TO 7l8 |
|||
|
£ Andromedae. |
A 2 |
||||
|
17 Cassiopeise "\ |
* 6 |
O J."? d.6. 12^ |
3, 6116 |
-J- C7 2O CO C4 |
IO 2OC |
|
8 Piscium |
4 6 |
4- TO 63T |
|||
|
X Hydri . |
So |
||||
|
20 Ceti |
4 O |
o J.8 "?o cr^A |
3 064.1 |
— T 77 18 A7 |
IO COC |
|
y Cassiopeiae |
2 2 |
O Cl 27 2J.7 |
3. tlQ^S |
4- OO T/L 2C C6 |
IO C3Q |
|
fji Andromedae. |
o rT cT Rrc |
TO c6c |
|||
|
a Sculptoris |
-1 A |
O CJ. 21 Ol8 |
+ 2 8908 |
4- IO.4.72 |
|
|
43 H. Cephei |
4 C |
O ^6 71 2J.4 |
7 C72C |
4- 8 c 4.7 81? |
IO.43C |
|
e P'scium |
|||||
|
ft Phcenicis .... f |
19 288 |
||||
|
H Cassiopeiae. . . |
r 7 |
17 7C2 |
|||
|
17 Ceti |
3. 6 |
T J. O *77C |
|||
|
ft Andr~ xlae |
2 4. |
||||
|
T Pisciui. |
4. 7 |
i 6 48 60*5 |
|||
|
f Piscium f |
c 6 |
T O 7 O?8 |
|||
|
K Tucanae f |
50 |
||||
|
f Piscium |
5^ |
T T3 T C C2C |
|||
|
v Piscium |
4. 7 |
18 o8c |
|||
|
B Ceti |
5 8 |
T TO 77 ACC |
— 8 78 17 87 |
18 633 |
|
|
5 Cassiopeiae |
2 8 |
3 8083 |
|||
|
y Phcenicis |
3 A |
2 6o8l |
— j.7 46 8 c8 |
||
|
38 Cassiopeiae |
6 o |
I 2J. 7O 74O |
+ 69 48 43 86 |
4-18 622 |
|
|
17 Piscium |
37 |
I 26 J.6 7O7 |
T8 623 |
||
|
a Ursae Min.(Po/am)f |
2.1 |
i 27 51.07* |
+ 27.8225 |
+ 88 50 10.77 |
+ 18.594 |
13 Ceti. dup. 5w.s. 6*».2, o*.3. a Cassiop.. var. irreg 2"*.2,2"».8 17 Cassiop.. comp.7"*.6, 5" s. pr.
/3 Phoenicis, dup. 4m.i,4m.i,i*. £ Piscium, star 6m.s, 24" n.f.
K Tucanae, comp. 7m, 6" n.
a TJrsse Min., star gm, 18" s. pr.
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 67
APPARENT PLACES OF STARS, 1912.
FOR THE UPPER TRANSIT AT WASHINGTON.
|
33 Piscium. |
a Andromedae |
/tf Cassiopeia |
6 Phoenicis |
|||||
|
Mag. 4.7 |
Mag. 2.2 |
Mag. 2.4 |
Mag. 3.9 |
|||||
|
Mean so- lar date. |
Right |
Declina- |
Right |
Declina- |
Right |
Declina- |
Right |
Declina- |
|
Ascension |
tion S. |
Ascension |
tion N. |
Ascension |
tion N. |
Ascension |
tions |
|
|
h m |
0 / |
h m |
o / |
/; m |
o ' |
h m |
0 / |
|
|
O O |
- 6 ii |
0 3 |
+ 28 36 |
o 4 |
+ 58 39 |
0 4 |
-46 13 |
|
|
s |
" |
i |
" |
f |
" |
s |
" |
|
|
Jan. 0.2 |
49-31 I0 |
65-8 6 |
49-23 x |
22-3 |
26.88 |
65-8 |
56.67 |
77-3 |
|
10.2 |
49-21 |
66.4 |
49-io ; |
21-5 j |
26-57 3' |
65-1 tl |
56.48 9, |
76.9 4 |
|
20. 2 |
49-12 g |
66.8 |
48.97 3 |
2O.4 |
26.28 29 |
63-9 I6 |
56.31 |
76.0 9 |
|
30.1 |
49-04 '. |
67.1 3 |
48.86 |
19.0 |
26.01 |
62'3 o |
56.16 ; |
74-7 I3 |
|
Feb. 9 . i |
48.98 |
67.2 |
48.76 " |
17-5 5 |
25-77 24 0 ' ' 19 |
to-*:: |
56.04 9 |
73-0 H |
|
19.1 |
48.94 |
67-1 2 |
48.69 |
16.0 |
25.58 z |
57-9 a. |
55-95 |
70.9 |
|
29.1 |
48.92 |
66.9 |
48.65 4 |
14.4 |
25.45 I |
55-4 2J |
55-90 , |
68.6 23 |
|
Mar. 10. o |
48.93 |
66.4 |
48.65 ° |
12.9 |
25-39 |
55-89 |
65-9 27 |
|
|
20.0 |
48.98 J |
65-7 I |
48.69 J |
ii. 5 4 |
25-41 |
5°- 1 2 |
55-92 |
63.0 2' |
|
30.0 |
49.06 |
48.77 I3 |
10.4 |
25-5° c |
47-6 H |
56.01 x9 |
60.0 3° |
|
|
Apr. 9 . o |
49-i8 H |
63.6 |
48.90 |
9 9-5 |
25.68 \ |
45-3 2 |
56.14 |' |
56.8 3 |
|
18.9 |
49-34 20 |
62.2 4 |
49.08 Ic |
9-0 f |
25-93 . |
43-3 .. |
56.33 : |
53-6 32 |
|
28.9 |
49-54 , |
60.6 ; |
49-30 11 |
8.8 |
26.25 * |
41.8 J |
56.57 n |
50-4 32 |
|
May 8.9 |
49.78 4 |
58.8 |
49-56 |
9-° 2f. |
26.63 |
40-7 " |
56.85 2° |
47-4 3° |
|
18.8 |
50.05 27 J •> 29 |
i |
49.85 29 3 32 |
9.6 IO |
27-07 4g |
40. i |
44-5 29 |
|
|
28.8 |
50.34 ,_ |
54.7 |
10.6 |
27-55 , |
40. i |
57-54 3 |
41.8 27 |
|
|
June 7.8 |
50.65 3 |
52.6 |
50.51 34 |
11.9 13 |
28.05 » |
40-5 . |
57-93 3 |
39-4 24 |
|
17.8 |
50.97 : |
50.5 |
50.86 3S |
13.6 1? |
28.57 |
41.5 ! |
o 4 ' |
37-3 2I |
|
27.7 |
51.30 33 |
48.4 |
51.21 3* |
15-6 2' |
29.08 s |
43-° 2 |
58 -76 !! |
35-6 I7 |
|
July 7.7 |
51.62 3 |
46.4 ' |
51.56 3 |
J7'8 ! |
29.58 s° |
45-0 |
59.18 4 |
34-4 I2 |
|
31 |
9 |
33 |
24 |
^ 48 |
. 8 |
|||
|
17.7 |
51,93 2g |
44-5 6 |
51.89 |
20.2 |
30.06 |
47-3 27 |
59.58 |
33-6 |
|
27.7 |
42.9 |
52.19 3! |
22.7 2 |
30.50 4 |
59-96 3 |
33-3 3 |
||
|
Aug. 6 . 6 |
5^47 \\ |
41-4 |
52.47 |
25.2 |
30.89 39 |
53-0 3° |
60.31 3 |
33-5 |
|
16.6 |
52.70 ; |
40.2 |
52.71 ,4 |
27- 8 2< |
-j 2, 34 |
56'2 L |
60. 61 , |
34-1 6 |
|
26.6 |
s2-^ ;§ |
39-3 I |
52.91 i6 |
30-3 |
31-55 22 |
59-6 34 34 |
60.87 2 ' 20 |
35-2 " I ? |
|
Sept. 5.5 |
53.05 |
38.7 |
53.07 |
32-8 |
3L72 |
63-0 |
6l.07 |
36.7 |
|
53.16 'J |
38.3 |
53-19 " |
35-' |
31-87 |
66.5 3S |
61.21 J |
38.5 l8 |
|
|
25-5 |
53.24 |
38-2 |
53.27 |
31.96 |
69.9 34 |
61.30 ? |
40.5 20 |
|
|
Oct. 5.5 |
53.28 ' |
38-3 |
53-31 |
39-2 2° |
31-99 , |
73-2 33 |
61.33 I |
42-7 2 |
|
15-4 |
53.28 ° |
38.6 3 |
53.32 I |
31-96 3 |
76.3 29 |
7 |
45-o 23 2 2 |
|
|
25-4 |
53.26 |
39- 1 |
53.29 |
42.3 ' |
31-88 |
79-2 26 |
61.24 XI |
47.2 |
|
Nov. 4 . 4 |
53.21 I |
39-7 - |
53.24 : |
43-5 " |
1 4 ^ i 74 |
49-3 2I |
||
|
14.4 |
53.15 8 |
40.4 ' |
53.17 |
44-4 |
31-55 2 |
84.0 |
5o . QO> "i! |
51.2 J9 |
|
24.3 |
53.07 |
41- 1 |
53-07 |
44-9 |
2 2 31-33 26 |
85.7 ;.7 |
60. . 7 |
52.8 l6 |
|
Dec. 4.3 |
52.98 *o |
41.8 7 |
52.96 J |
45-2 I |
87-0 I3 |
60. >J 29 |
54-1 is |
|
|
14-3 |
52.88 |
42-5 6 |
52.84 |
45-i |
30.79 „ |
87.8 |
60.43 |
54-9 . |
|
24.2 |
52.78 j; |
52.71 |
44-8 3 |
30-49 ,0 |
88.0 |
60.23 IQ |
55-2 3 |
|
|
34-2 |
52.68 " |
«'.7 6 |
52.58 3 |
44.1 |
3o.i9 3° |
87.6 4 |
60.04 |
55-i |
|
Sec 8, Tan 5 |
i . 006 — o 109 |
i. 139 +o. 545 |
1.923 +1.643 |
I . 446 — I. 044 |
||||
|
Mean Place |
49s- 898 59". 45 |
SO8. 162 16". 59 |
288.504 51". 97 |
568.832 58". 96 |
||||
|
DVa,Dcoa |
0.00 +0.01 |
O . OO — 0 . 04 |
0 . 00 — 0 . 1 1 |
o.oo +0.07 |
||||
|
D\f*8, Da)5 |
+ 0.4 o.o |
+ 0.4 o.o |
+ 0.4 o.o |
+ 0.4 o.o |
68 PRACTICAL ASTRONOMY
moon's right ascension and declination for every hour of Green- wich Mean Time, together with the changes per minute of time. Following the ephemeris of the sun and moon for the twelve months are the ephemerides of the planets. The values given for Greenwich Mean Noon are for oh of the astronomical date or 12 M of the civil date. The word " apparent " used in these tables indicates that the correction for aberration has been applied to the coordinates, giving the position of the object as actually seen by the observer except for the effect of parallax and refraction. The " differences for i hour " are in all cases the rates of change at the instant, that is, the differential coeffi- cients, not the actual differences between the consecutive tabular values.
Part II contains the following three lists of stars, the first headed " Mean places of Stars, 19— "; the second, "Apparent Places of Circumpolar Stars, 19— "; and the third, "Apparent Places of Stars, 19— "; all of these are computed for the instant of transit over the meridian of Washington (5* o8TO 15*. 78 West of Greenwich). A list of south circumpolar stars is also given. The tables given on pages 66 and 67 of this volume are extracts from the first and third of these tables in the Almanac. The first table contains the coordinates of about 800 stars referred to the "mean equinox " at the beginning of the year, that is, to the position that the equinox would occupy at the beginning of the solar * year if it were not affected by small periodic terms of the precession. The second table gives the coordinates of about 1 5 north circumpolar stars. Precession causes the coordi- nates of circumpolar stars to vary more rapidly than those of equatorial stars; the coordinates are therefore given for every day in the year. The hours and minutes of right ascension and the degrees and minutes of declination are at the head of the column ; the column contains only the seconds. In the third table
* The year here referred to, called also the Besselian fictitious year, is one used in computing star places; it begins when the sun's mean longitude is 280°, that is, when the R. A. of mean sun is i8h 40"*, which occurs about Jan. i.
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 69
are about 800 stars, the coordinates for which are given for every ten days. The only other table in Part II of particular interest to the surveyor is that headed " Moon Culminations." This table contains the data needed when determining longitude by observ- ing transits of the moon. (See Art. 88, p. 141.)
In the latter part of the Almanac will be found the following useful tables: I, Times of Culmination and Elongation of Polaris; II, Conversion of Sidereal Time into Mean Time; III, Conver- sion of Mean Time into Sidereal Time; IV, Latitude by an Altitude of Polaris; V and VI, Azimuths of Polaris; VII, Intervals for 5 Cassiopeia and f Ursa Majoris. (See Art. 99, p. 161.)
39. Star Catalogues.
When it is necessary to make observations on stars not given in the list in the Ephemeris, their positions must be taken from one or more of the star catalogues. These give the mean place of the star at some definite epoch, such as the beginning of the year 1890, or 1900, together with the necessary data for reducing to the mean place of any other year. This data is usually obtained by combining the observations made at different obser- vatories and at different times, so that changes in the star's coordinates are accurately determined. After the position in the catalogue has been brought up to the mean place for the desired year, the apparent place of the star for the exact date of the observation is computed by means of the formulae and tables given in Part II of the Ephemeris. For ordinary observations made by the surveyor the list of stars given in the Ephemeris is always sufficient, but in special kinds of work, such as finding latitude by Talcott's Method, many other stars must be used.
40. Interpolation.
When taking data from the Ephemeris it is general necessary, in order to obtain the value for a particular instant, to interpolate between values of the function for stated times. In some cases this may be simple interpolation, in which the function is assumed to vary uniformly between the two values given and the desired value found by direct proportion. When the difference for one
70 PRACTICAL ASTRONOMY
hour is given, this rate of change at the given instant may be assumed to hold good between the given value and the following one. Since this is not usually quite true, it will be more accurate to interpolate from the nearest given value in the table. The change for one hour is to be multiplied by the number of hours between the given time and the tabular time. This correction is either added or subtracted, according to whether the function is increasing or decreasing and whether the preceding or follow- ing tabular value is used.
Example.
At Greenwich Mean Noon. Feb. Sun's declination Diff. ih
1 Si7024'o4".o + 4i".88
2 17 07 09 .8 42 .64
It is desired to find the declination at the instant 22* G. M. T. Feb. i. Since this is much nearer to the moon of Feb. 2 than of Feb. i, it will be more accurate to multiply 42". 64 by 2h and add this to 17° 07' 09". 8 (since the declination is decreasing). The result is S 17° 08' 35".!. By working forward from the value on Feb. i the result is S 17° 08' 3 2" . 6. By using a more exact formula the result is found to be S 17° 08' 35".o.
If the successive values of the " diff. for ih " or " diff. for im" have large differences, and if a precise value of the function is desired, it will be necessary to interpolate between the given values of the differential coefficients to obtain the rate of change at the middle of the interval over which we are interpolating, and to use this interpolated rate of change in computing the correction.
Example.
Time R. A. of the Moon . Diff. for IOT
o* 4ft46mna.49 2s. 54 2 1
i 4 48 44.06 2.5436
If it is desired to find the right ascension for oh 40™, the " diff. for im" to be used is that for the instant oh 20™, the middle
THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 71
of the interval from oh to oh 40™. This value lies one third of the way from 2s. 5421 to 2S.5436, or 2S.5426. The correction to the R. A. at oh is 2S.5426 X 4OTO = 101.70* = im 4is.7o, the required R. A. being 4* 47"* 53s. 19.
For general interpolation formulae the student is referred to Chauvenet's Spherical and Practical Astronomy, Vol. I, to Doo- little's Practical Astronomy or to Hayford's Geodetic Astronomy.
Questions and Problems
1. Compute the sun's apparent declination when the M. L. T. is 8h 30™ A.M., Jan. 16, 1912, at a place 85° west of Greenwich (see p. 65).
2. Compute the right ascension of the mean sun at local mean noon Jan. 10, 1912, at a place 96° 10' west of Greenwich.
3. Compute the equation of time for local apparent noon Jan. 30, 1912, at a place 20° east of Greenwich.
4. Explain the relation between the sun's angular semidiameter and the time of the semidiameter passing the meridian.
5. What is the relation between the " right ascension of the mean sun " and the " apparent right ascension of the sun " on Jan. i, 1912?
CHAPTER VII
THE EARTH'S FIGURE — CORRECTIONS TO OBSERVED
ALTITUDES
41. The Earth's Figure.
The earth's form is approximately that of an oblate spheroid whose shortest axis is the axis of rotation. The actual figure deviates slightly from that of a perfect spheroid, but for most astronomical purposes these deviations may be disregarded. Each meridian may therefore be considered as an ellipse, and the equator and all parallels of latitude as circles. The semi- major axis of the meridian ellipse is about 3962.80 miles, and the semi-minor axis is 3949.56 miles in length. The length of i° of latitude at the equator is 68.704 miles; at the pole it is 69.407 miles.
In locating points on the earth's surface by means of coordi- nates there are three kinds of latitude to be considered. The latitude as found by astronomical observation is dependent upon the direction of gravity as indicated by the spirit levels of the instrument, and is affected by any abnormal deviations of the plumb line* at this point; the latitude as found directly by ob- servations is called the astronomical latitude. The geodetic latitude is the latitude that would be found by observation if the plumb line were normal to the surface of the spheroid taken to represent the earth's figure, that is, if all of the irregularities of the surface were smoothed out. Evidently the geodetic latitude cannot be directly observed but must be found by com- putation. The geocentric latitude is the angle between the plane of the equator and a line drawn from the centre of the earth to the point on the surface. In Fig. 39 the line AD is normal to
* These deviations are small, averaging about 3" or 4", but in some cases deviations of nearly 30" are found.
72
THE EARTH'S FIGURE
73
the earth's surface at A, and the angle ABE is the geodetic latitude of A. If the plumb line coincides with AD, this is also the astronomical latitude. The angle ACE is the geocen- tric latitude. The difference between the two, or angle BAC,
is called the angle of the vertical, or the reduction of latitude.
The geocentric latitude is always less than the observed lati- tude by an angle which varies from about o° n' 30" in latitude 45° to zero at the equator and the poles. Whenever observations are made at any point on the earth's surface it is necessary to reduce the observed values to their values at the earth's centre before they can be combined with other data referred to the centre. In making this reduction the geocentric latitude must be used if the exact position of the observer with reference to the centre is to be computed. For most of the observations treated in the following chapters it will not be necessary to consider the spheroidal shape of the earth; it will be sufficiently exact to regard it as a sphere.
42. Parallax.
The coordinates of a celestial object as given in the Ephemeris are referred to the centre of the earth, while the coordinates
74
PRACTICAL ASTRONOMY
obtained by observation are necessarily measured from a point on the surface, and must be reduced to the centre. The case most frequently occurring in practice is that in which the altitude of an object is observed and the geocentric altitude is desired. For all objects except the moon the distance of the body is so great that it is sufficiently accurate to regard the earth as a
FIG. 40
sphere. In Fig. 40, the angle ZOS is the observed zenith dis- tance, or the complement of the observed altitude, and ZCS is the true zenith distance. This apparent displacement of the object on the celestial sphere is called parallax. The effect of parallax is simply to decrease the altitude without altering the azimuth of the body, provided the spheroidal form of the earth be disregarded. The difference in direction between the lines OS and CS, or the angle OSC, is the parallax correction. In the triangle OSC, angle COS may be considered as known, since the altitude or complement of ZOS is observed. The distance OC is the semidiameter of the earth (3956.1 miles), and CS is the
THE EARTH'S FIGURE' 75
distance from the earth's centre to the centre of the body ob- served. Solving this triangle,
or
sinS = smZOSX~ [50]
It is evident that the parallax correction will be zero at the zenith and a maximum at the horizon. For the maximum, when ZOS = 90°,
. e OC r ,
sin 5 = — > [51]
which is the same for all places on the earth's surface if the earth is regarded as a sphere. If Ph represents the maximum or horizontal parallax, then equation [50] may be written
sin S = sin Ph sin z
= sin Ph cos //, [52]
where h is the apparent altitude of the object. But S and Ph are usually very small angles, and the error is negligible if the sines are replaced by their arcs.* Equation [52] then becomes
S" = Ph" cos h, [53]
where S" and Ph" are both in seconds of arc.
For the moon the mean value of the horizontal parallax is about o° 57' 02" f; for the sun it is 8". 8; for the fixed stars it is
* The sine may be expressed as a series as follows:
x3 x5 sin x = x — Y + T~ ~ ' ' ' [54]
Replacing sin x by x amounts to neglecting all terms after the first. Whether the error will be appreciable in any given case may be determined by computing the value of the first of the neglected terms. If x = i° the neglected terms are less than .005 of i% of x. The error in an angle of i° would be less than o".2. The moon is the only object whose parallax is nearly as large as i°, so that for all other objects this approximation is usually allowable. Similarly for cos x = i, the terms neglected are those of the series
' I • • [55]
L 11
t The moon's mean distance is 238,800 miles; the sun's mean distance is 92,900,000 miles.
76
PRACTICAL ASTRONOMY
too small to be detected. The horizontal parallaxes of objects in the solar system are given in the Nautical Almanac.* For the parallax of the sun for different altitudes see Table IV (A).
43. Refraction.
Refraction is the term applied to the bending of a ray of light by the atmosphere as it passes from a celestial object to the observer's eye. On account of the increasing density of the layers of air the rays of light coming from any object are bent downward into a curve, and consequently when the rays enter the eye they have a greater inclination to the horizon than they did before entering the atmosphere. For this reason all objects appear higher above the horizon than they actually are. In
FIG. 41
Fig. 41, S is the true position of a star and 5" its apparent position. The light from 5 is bent into a curve aO, and the star is seen in the direction of the tangent ObS'. The angle which must be subtracted from the altitude of Sf to obtain the altitude of 61 is called the refraction correction. This angle is really the angle SOS', but on account of the great distance of celestial objects
* On account of the spheroidal form of the earth the equatorial diameter is the greatest and the parallax at the equator is a maximum; the parallaxes are therefore given in the Ephemeris under the heading " Equatorial Horizontal Parallax."
THE EARTH'S FIGURE 77
and the small angle of refraction the correction may be con- sidered as the angle SbS'. From the figure it is evident that
ZcS = ZOS' + S'bS,
or z' = z + r, [56]
where z' = the true and z = the apparent zenith distance and r = the refraction correction. The approximate law of astro- nomical refraction may be deduced by assuming that the bend- ing all occurs at point b. The general law of refraction, when a ray enters a refracting medium, is expressed by the equation
sinz' = n sin z, [57]
where n is the index of refraction of the given medium; for air its value is roughly about 1.0003.
Substituting from Equa. [56],
sin (z + r) = n sin z, [58]
Expanding, sin z cos r + cos z sin r = n sin z. [59]
Since r is a small angle (never greater than 40') it is allowable to put cos r = i and sin r = r; then
sin z + r cos z = n sin z,
and r cos z = (n — i) sin z,
or r = (n — i) tan z. [60]
Replacing n by 1.0003 and dividing by sin i" to reduce r from circular measure to seconds of arc,
„ f.oooO
r" = -± ^- tan 2
(.000,005)
= 60" tan z
= 60" cot A, [61]
where /t is the apparent altitude.*
The value of n varies considerably with the temperature and the pressure of the air, so that equation [61] must be considered as giving only a rough approximation to the true refraction.
* " Apparent" is used here simply to distinguish between the direction of the star as actually seen and the direction unaffected by refraction. In speaking of parallax, the word " apparent " has a different meaning, and in case of aberration, still another meaning.
7§ PRACTICAL ASTRONOMY
For high altitudes this formula is nearly correct, but for altitudes under 10° it is not sufficiently exact. If both sides of the equa- tion are divided by 60 so that r is reduced to minutes, we have the extremely simple relation that the refraction in minutes equals the natural cotangent of the altitude. For altitudes measured with an engineer's transit this formula is close enough for alti- tudes greater than about 10°. For more accurate values of the refraction Table I may be used. From the table it will be seen that the refraction correction is zero at the zenith, about i' at an altitude of 45°, and about 6° 34' at the horizon.*
The following formula, due to Professor George C. Comstock, gives very accurate values of the refraction for altitudes greater than 20°, and is sufficiently accurate for all field observations made with surveyors' instruments.
983 b , r , -,
r = I ° cot h, [62]
460 + t
in which b is the barometer reading in inches, and / is the tem- perature in Fahrenheit degrees. Example.
Altitude 30°, barometer 29. im', thermometer 81° F. log. 983 = 2.9926 log. 29.1 = 1.4639
460° colog. 541 = 7.2668
81 cot h = 0.2386
541° 1.9619
r = 9i".6 = i'3i".6
44. Semidiameters.
The discs of the sun and moon are circular, and their angular semidiameters are given for each day in the Ephemeris. Since measurements can only be taken to the edge, or limb, the altitude of the centre of the object is obtained by making a correction
* The sun's diameter is about 3 2', slightly less than the refraction on the horizon; when the sun has actually gone below the horizon at sunset the entire disc is still visible on account of the 34' increase in its apparent altitude due to atmospheric refraction.
THE EARTH'S FIGURE
79
equal to the semidiameter. The apparent angular semidiameters given in the Ephemeris may be affected in two ways, one by the change in the observer's distance because he is on the earth's surface, the other by the difference in the amount of refraction correction on the upper and lower edges of the disc.
The semidiameter given in the Ephemeris is that as seen from the centre of the earth. When the object is in the zenith the observer is nearly 4000 miles nearer than when it is in the hori- zon. The moon is about 240,000 miles distant from the earth, so that the semidiameter is increased by about ^V part, or about 16".
The vertical diameter of an object appears to be less than its horizontal diameter because the refraction lifts the lower edge more than it does the upper edge. The disc then presents the appearance of an ellipse. When the sun is rising or setting, the contraction is most noticeable. This contraction of the semi- diameter does not affect the correction to an observed altitude, but must be taken into account when the distance is measured between the moon's limb and a star or a planet. (See Art. 108.)
For the angular semidiameter of the sun on the first day of each month see Table IV (B).
45- Dip-
If altitudes are taken from the sea horizon, as when observing on board ship with the sextant, the measured altitude must be diminished by the angular dip of the sea horizon below the true horizon. In Fig. 42 suppose the observer to be at 0; the true
horizon is OB and the sea horizon
FIG. 42
OH. Let OP = h, the height in
feet above the surface; PC = R, the radius of the earth; and
D, the angle of dip.
— B
80 PRACTICAL ASTRONOMY
Then cos Z> = -^-j-^ • [63]
D2
Putting cos D = i — — , neglecting other terms in the series,
D2 h h , , v
= — (nearly).
R + h R
Replacing R by its value in feet, 20,884,000, and dividing by sin i' to reduce D to minutes,
V
V
- X sin i'
2
[64]
This shows the amount of dip unaffected by refraction. The effect of refraction is to apparently lift the horizon, and the dip affecting the observed altitude is therefore less than that given by the formula. If the coefficient 1.064 is taken as unity, the formula is nearer the truth and is simpler, although still some- what too large. Table IV (C) , based on a more exact formula, will be seen to give smaller values. For ordinary sextant observations made at sea, where the greatest precision is not required it is sufficient to take the dip in minutes equal to the square root of the height of the eye in feet, that is,
D' = Vh ft. [65]
46. Sequence of Corrections.
Strictly speaking, the corrections to the latitude should be made in the following order :
(i) Instrumental corrections; (2) dip (if at sea); (3) refraction; (4) semidiameter; (5) parallax. In practice, however, it is not always necessary to follow this order exactly. At sea the cor- rections are often taken together as a single " correction to the altitude." Care should be taken to use the refraction correction
THE EARTH'S FIGURE 8 1
for the limb observed, not for the centre, for if the altitude is small the two will differ appreciably.
Problems
1. Compute the sun's mean horizontal parallax. The sun's mean distance is 92,900,000 miles; for the earth's radius see Art. 41. Compute the sun's parallax at an altitude of 60°.
2. Compute the moon's mean horizontal parallax. The moon's mean distance is 238,800 miles; for the earth's radius see Art. 41. Compute the moon's parallax at an altitude of 45°,
CHAPTER VIII DESCRIPTION OF INSTRUMENTS
47. The Engineer's Transit.
The engineer's transit is an instrument for measuring hori- zontal and vertical angles. For the purpose of discussing the theory of the instrument it may be regarded as a telescopic line of sight having motion about two axes at right angles to each other, one vertical, the other horizontal. The line of sight is determined by the optical centre of the object glass and the intersection of two cross hairs* placed in its principal focus. The vertical axis of the instrument coincides with the axes of two spindles, one inside the other, each of which is attached to a horizontal circular plate. The lower plate carries a graduated circle for measuring horizontal angles; the upper plate has two verniers, on opposite sides, for reading angles on the circle. On the top of the upper plate are two uprights, or standards, supporting the horizontal axis to which the telescope is attached and about which it rotates. At one end of the horizontal axis is a vertical arc, or a circle, and on the standard is a vernier, in contact with the circle, for reading the angles. The plates and the horizontal axis are provided with clamps and slow-motion screws to control the motion. On the upper plate are two spirit levels for levelling the instrument, or, in other words, for making the vertical axis coincide with the direction of gravity.
The whole instrument may be made to turn in a horizontal plane by a motion about the vertical axis, and the telescope may be made to move in a vertical plane by a motion about the horizontal axis. By means of a combination of these two
* Also called wires or threads; they are either made of spider threads or are lines ruled upon glass.
82
DESCRIPTION OF INSTRUMENTS 83
motions, vertical and horizontal, the line of sight may be made to point in any desired direction. The motion of the line of sight in a horizontal plane is measured by the angle passed over by the index of the vernier along the graduated horizontal circle. The angular motion in a vertical plane is measured by the angle on the vertical arc indicated by the vernier attached to the standard. The direction of the horizon is denned by means of a long spirit level attached to the telescope. When the bubble is central the line of sight should lie in the plane of the horizon. To be in perfect adjustment, (i) the axis of each spirit level * should be in a plane at right angles to the vertical axis; (2) the horizontal axis should be at right angles to the vertical axis; (3) the line of sight should be at right angles to the horizontal axis; (4) the axis of the telescope level should be parallel to the line of sight, and (5) the vernier of the vertical arc should read zero when the bubble is in the centre of the level tube attached to the telescope. When the plate levels are brought to the centres of their tubes, and the lower plate is so turned that the vernier reads o° when the telescope points south, then the vernier readings of the horizontal plate and the vertical arc for- any position of the telescope are coordinates of the horizon system (Art. 12). If the horizontal circles are clamped in any position and the telescope is moved through a complete revolution, the line of sight describes a vertical circle on the celestial sphere. If the telescope is clamped at any altitude and the instrument turned about the vertical axis, the line of sight describes a cone and traces out on the sphere a circle of equal altitudes, or an almucantar.
48. Elimination of Errors.
It is usually more difficult to measure an altitude accurately with the transit than to measure a horizontal angle. While the precision of horizontal angles may be increased by means of repetitions, in measuring altitudes the precision cannot be
* The axis of a level may be defined as a line tangent to the curve of the glass tube at the point on the scale taken as the zero point, or at the centre of the tube.
84 PRACTICAL ASTRONOMY
increased by repeating the angles, owing to the construction of the instrument. The vertical arc usually has but one vernier, so that the eccentricity cannot be eliminated, and this vernier often does not read as closely as the horizontal vernier. One of the errors, which is likely to be large, but which may be elimi- nated readily, is that known as the index error. The measured altitude of an object may differ from the true reading for two reasons: first, the zero of the vernier may not coincide with the zero of the circle when the telescope bubble is in the centre of its tube; second, the line of sight may not be horizontal when the bubble is in the centre of the tube. The first part of this error can be corrected by simply noting the vernier reading when the bubble is central, and applying this as a correction to the measured altitude. To eliminate the second part of the error the altitude may be measured twice, once from the point on the horizon directly beneath the object observed, and again from the opposite point of the horizon. In other words, the instru- ment may be reversed (180°) about its vertical axis and the vertical circle read in 'each position while the horizontal cross hair of the telescope is sighting the object. The mean of the two readings is free from the error in the sight line. Evidently this method is practicable only with an instrument having a complete vertical circle. If the reversal is made in this manner the error due to non-adjustment of the vernier is eliminated at the same time, so that it is unnecessary to make a special deter- mination of it as described above. If the circle is graduated in one direction, it will be necessary to subtract the second reading from 180° and then take the mean between this result and the first altitude. In the preceding description it is assumed that the plate levels remain central during the reversal of the instrument, indicating that the vertical axis is truly vertical. If this is not the case, the instrument should be relevelled before the second altitude is measured, the difference in the two altitude readings in this case including all three errors. If it is not de- sirable to relevel, the error of inclination of the vertical axis may
DESCRIPTION OF INSTRUMENTS 85
still be eliminated by reading the vernier of the vertical circle in each of the two positions when the telescope bubble is central, and applying these corrections separately. With an instru- ment provided with a vertical arc only it is essential that the axis of the telescope bubble be made parallel to the line of sight, and that the vertical axis be made truly vertical. To make the axis vertical without adjusting the levels themselves, bring both bubbles to the centres of their tubes, turn the instrument 180° in azimuth, and then bring each bubble half way back to the centre by means of the levelling screws. When the axis is truly vertical, each bubble should remain in the same part of its tube in all azimuths. The axis may always be made vertical by means of the long bubble on the telescope; this is done by set- ting it over one pair of levelling screws and centring it by means of the tangent screw on the standard; the telescope is then revolved about the vertical axis, and if the bubble moves from the centre of its tube it is brought half way back by means of the tangent screw, and then centred by means of the levelling screws. This process should be repeated to test the accuracy of the levelling; the telescope is then turned at right angles to the first position and the whole process repeated. This method should always be used when the greatest precision is desired, because the telescope bubble is much more sensitive than the plate bubbles.
If the line of sight is not at right angles to the horizontal axis, or if the horizontal axis is not perpendicular to the vertical axis, the errors due to these two causes may be eliminated by com- bining two sets of measurements, one in each position of the instrument. If a horizontal angle is measured with the vertical circle on the observer's right, and the same angle again observed with the circle on his left, the mean of these two angles is free from both these errors, because the two positions of the horizontal axis are placed symmetrically about a true horizontal line,* and
* Strictly speaking, they are placed symmetrically about a perpendicular to the vertical axis.
86 PRACTICAL ASTRONOMY
the two directions of the sight line are situated symmetrically about a true perpendicular to the rotation axis of the telescope. If the horizontal axis is not perpendicular to the vertical axis the line of sight describes a plane which is inclined to the true vertical plane. In this case the sight line will not pass through the zenith, and both horizontal and vertical angles will be in error. In instruments intended for precise work a striding level is provided, which may be set on the pivots of the horizontal axis. This enables the observer to level the axis or to measure its inclina- tion without reference to the plate bubbles. The striding level should be used in both the direct and the reversed position and the mean of the two results used in order to eliminate the errors of adjustment of the striding level itself. If the line of sight is not perpendicular to the horizontal axis it will describe a cone whose axis is the horizontal axis of the instrument. The line of sight will in general not pass through the zenith, even though the. horizontal axis be in perfect adjustment. The instrument must either be used in two positions, or else the cross hairs must be adjusted. Except in large transits it is not usually practicable to determine the amount of the error and allow for it.
49. Attachments to the Engineer's Transit. — Reflector.
When making star observations with the transit it is necessary to make some arrangement for illuminating the field of view. Some transits are provided with a special shade tube into which is fitted a mirror set at an angle of 45° and with the central portion removed. By means of a lantern held at one side of the telescope light is reflected down the tube. The cross hairs appear as dark lines against the bright field. The stars can be seen through the opening in the centre of the mirror. If no special shade tube is provided, it is a simple matter to make a substitute, either from a piece of bright tin or by fastening a piece of tracing cloth or oiled paper over the objective. A hole about | inch in diameter should be cut out, so that the light from the star may enter the lens. If cloth or paper is used, the lan- tern must be held so that the light is diffused in such a way as
DESCRIPTION OF INSTRUMENTS 87
tQ render the cross hairs visible. The light should be held so as not to shine into the observer's eyes.
50. Prismatic Eyepiece.
When altitudes greater than about 55° to 60° are to be meas- ured, it is necessary to attach to the eyepiece a totally reflecting prism which reflects the rays at right angles to the sight line. By means of this attachment altitudes as great as 75° can be measured. In making observations on the sun it must be remembered that the prism inverts the image, so that with a transit having an erecting eyepiece with the prism attached the apparent lower limb is the true upper limb; the positions of the right and left limbs are not affected by the prism.
51. Sun Glass.
In making observations on the sun it is necessary to cover the eyepiece with a piece of dark glass to protect the eye from the sunlight while observing. The sun glass should not be placed in front of the objective. If no shade is provided with the instrument, sun observations may be made by holding a piece of paper behind the eyepiece so that the sun's image is thrown upon it. By drawing out the eyepiece tube and varying the distance at which the paper is held, the images of the sun and the cross hairs may be sharply focussed. By means of this device an observation may be quite accurately made after a little practice.
52. The Portable Astronomical Transit.
The astronomical transit differs from the surveyor's transit chiefly in size and in the manner of support. The diameter of the object glass may be anywhere from 2 to 4 inches, and the focal length from 24 to 48 inches. The instrument is set upon a stone or brick pier. The cross hairs usually consist of several vertical hairs (say n or more) instead of a single one as in the surveyor's transit. The motion in altitude is controlled by means of a clamp and a tangent screw. The azimuth motion is usually very small, simply enough to allow adjustments to be made, as the transit is not used for measuring horizontal angles. The axis is levelled or its inclination measured by means of a sensitive striding level.
On account of the high precision of the work done with the astronomical transit the various errors have to be determined with great accuracy, and corresponding corrections applied to the observed results. The transit is chiefly used in the plane
88
PRACTICAL ASTRONOMY
of the meridian for determining the times of transit of stars. The principal errors determined and allowed for are (i) azimuth, or deviation from the true meridian; (2) inclination of the horizontal axis; (3) collimation, or deviation of the sight line from the true perpendicular to the rotation axis. The corrections to reduce an observed time to the true time of transit across the meridian are given by formulae [66] to [68]. These corrections would apply equally well to observations with the engineer's transit, and serve to show the relative magnitudes of the errors for different positions of the objects observed.
Azimuth correction = a cos h sec D, [66]
Level correction = b sin h sec D, [67]
Collimation correction = c sec D, [68]
where a, b and c are the errors in azimuth, inclination and collimation respectively (expressed in seconds of time), and h is the altitude and D the declination of the star observed. From these formulae Table B has been computed. It is assumed that the instrument is i', or 4", out of the meridian (a = 4s); that the axis is inclined i', or 4s, to the horizon (b = 4*); and that the sight line denned by the middle (or the mean) wire is i', or 4s, to the right or left of its true position (c=48). The numbers in the table show the effect of these errors at different altitudes and declinations.
TABLE B. ERROR IN OBSERVED TIME OF TRANSIT (IN SECONDS OF TIME) WHERE a, b OR c = i'.
|
Declinations. |
||||||||||||
|
2 |
h |
0° |
10° |
20° |
30° |
40° |
50° |
60° |
70° |
80° |
h |
<2 |
|
K w |
0° |
o*.o |
os.o |
OS.O |
0s. 0 |
o*.o |
os.o |
os.o |
0s. 0 |
os.o |
90° |
§ |
|
W |
||||||||||||
|
c |
10 |
0.7 |
0.7 |
0.8 |
0.8 |
0.9 |
I . I |
1.4 |
2 .O |
4.0 |
80 |
|
|
1 |
20 |
1.4 |
1.4 |
1.4 |
1.6 |
1.8 |
2. I |
2-7 |
4.0 |
7-9 |
70 |
| |
|
"o |
3° |
2.0 |
2.O |
2. I |
2-3 |
2.6 |
3-i |
4.0 |
5-8 |
, "-5 |
60 |
3 |
|
g |
40 |
2.6 |
2.6 |
2.7 |
3-o |
3-4 |
4-0 |
S-2 |
7-5 |
14.8 |
5° |
j> |
|
^ |
5° |
3.1 |
3.1 |
3-3 |
3-6 |
4-0 |
4-8 |
6.1 |
9.0 |
17.6 |
40 |
V |
|
•a |
60 |
3-5 |
3-5 |
3-7 |
4-0 |
4-5 |
5-4 |
6.9 |
10. I |
19.9 |
3° |
3 |
|
| |
7° |
3-8 |
3-8 |
4.0 |
4-4 |
4-9 |
5-8 |
7-5 |
II. 0 |
21.6 |
20 |
S3 |
|
80 |
3-9 |
4.0 |
4.2 |
4.6 |
5-2 |
6.1 |
7-9 |
"•5 |
22.7 |
10 |
||
|
90 |
4-o |
4-1 |
4.2 |
4.6 |
S-2 |
6.2 |
8.0 |
11.7 |
23.0 |
o |
Note. — Use the bottom line for the collimation error.
53. The Sextant.
The sextant is an instrument for measuring the angular dis- tance between two objects, the angle always lying in the plane
DESCRIPTION OF INSTRUMENTS
89
through the two objects and the eye of the observer. It is particularly useful at sea because it does not require a steady support like the transit. It consists of a frame carrying a graduated arc, AB, Fig. 43, about 60° long, and two mirrors / and H, the first one movable, the second one fixed. At the center of the arc, 7, is a pivot on which swings an arm IV, 6 to 8 inches long. This arm carries a vernier V for reading the
angles on the arc AB. Upon this arm is placed the index glass /. At H is the horizon glass. Both of these mirrors are set so that their planes are perpendicular to the plane of the arc AB, and so that when the vernier reads o° the mirrors are parallel. The half of the mirror H which is farthest from the frame is unsilvered, so that objects may be viewed directly through the glass. In the silvered portion other objects may be seen by reflection from the mirror I to the mirror H and thence to point O. At a point near 0 (on the line HO) is a telescope of low power for viewing the objects. Between the two mirrors
90 PRACTICAL ASTRONOMY
and also to the left of H are colored shade glasses to be used when making observations on the sun. The principle of the instru- ment is as follows : — A ray of light coming from an object at C is reflected by the mirror / to H, where it is again reflected to O. The observer sees the image of C in apparent coincidence with the object at D. The arc is so graduated that the reading of the vernier gives directly the angle between OC and OD. Drawing the perpendiculars FE and HE to the planes of the two mirrors, it is seen that the angle between the mirrors is a — )8. Prolonging CI and DH to meet at O, it is seen that the angle between the two objects is 2 a — 2 /3. The angle between the mirrors is therefore half the angle between the objects that appear to coincide. In order that the true angle may be read directly from the arc each half degree is numbered as though it were a degree. It will be seen that the position of the vertex O is variable, but since all objects observed are at great distances the errors caused by changes in the position of 0 are always negligible in astronomical observations.
The sextant is in adjustment when, (i) both mirrors are per- pendicular to the plane of the arc; (2) the line of sight of the telescope is parallel to the plane of the arc; and (3) the vernier reads o° when the mirrors are parallel to each other. If the vernier does not read o° when the doubly reflected image of a point coincides with the object as seen directly, the index cor- rection may be determined and applied as follows. Set the vernier to read about 30' and place the shades in position for sun observations. When the sun is sighted through the tele- scope two images will be seen with their edges nearly in contact. This contact should be made as nearly perfect as possible and the vernier reading recorded. This should be repeated several times to increase the accuracy. Then set the vernier about 30' on the opposite side of the zero point and repeat the whole operation, the reflected image of the sun now being on the opposite side of the direct image. If the shade glasses are of different colors the contacts can be more precisely made. Half
DESCRIPTION OF INSTRUMENTS 91
the difference of the two (average) readings is the index correc- tion. If the reading off the arc was the greater, the correction is to be added to all readings of the vernier; if the greater reading was on the arc, the correction must be subtracted,
In measuring an altitude of the sun above the sea horizon the observer directs the telescope to the point on the horizon ver- tically under the sun and then moves the index arm until the reflected .image of the sun comes into view. The sea horizon can be seen through the plain glass and the sun is seen in the mirror. The sun's lower limb is then set in contact with the horizon line. In order to be certain that the angle is measured to the point vertically beneath the sun, the instrument is tipped slowly right and left, causing the sun's image to describe an arc. This arc should be just tangent to the horizon. If at any point the sun's limb goes below the horizon the altitude measured is too great. The vernier reading corrected for index error and dip is the apparent altitude of the lower limb above the true horizon.
54. Artificial Horizon.
When altitudes are to be measured on land the visible horizon cannot be used, and the artificial horizon must be used instead. The surface of any heavy liquid, like mercury, molasses, or heavy oil, may be used for this purpose. When the liquid is placed in a basin and allowed to come to rest, the surface is perfectly level, and in this surface the reflected image of the sun may be seen, the image appearing as far below the horizon as the sun is above it. Another convenient form of horizon con- sists of a piece of black glass, with plane surfaces, mounted on a frame supported by levelling screws. This horizon is brought into position by placing a spirit level on the glass surface and levelling alternately in two positions at right angles to each other. This form of horizon is not as accurate as the mercury surface but is often more convenient. The principle of the artificial horizon may be seen from Fig. 44. Since the image seen in the horizon is as far below the true horizon as the sun is
92 PRACTICAL ASTRONOMY
above it, the angle between the two is 2 h. In measuring this angle the observer points his telescope toward the artificial horizon and then brings the reflected sun down into the field of view by means of the index arm. By placing the apparent lower limb of the reflected sun in contact with the apparent upper limb of the image seen in the mercury surface, the angle measured is twice the altitude of the sun's lower limb. The two points in contact are really images of the same point. If the telescope inverts the image, this statement applies to the upper limb. The index correction must be applied before the angle is
Sextant
FlG. 44
divided by 2 to obtain the altitude. In using the mercury hori- zon care must be taken to protect it from the wind, otherwise small waves on the mercury surface will blur and distort the image. The horizon is usually provided with a roof -shaped cover having glass windows, but unless the glass has parallel faces this introduces an error into the result. A good substitute for the glass cover is one made of fine mosquito netting. This will break the force of the wind if it is not blowing hard, and does not introduce errors into the measurement.
55. Chronometer.
The chronometer is simply an accurately constructed watch with a special form of escapement. Chronometers may be
DESCRIPTION OF INSTRUMENTS 93
regulated for either sidereal or mean time. The beat is usually a half second. Those designed to register the time on chrono- graphs are arranged to break an electric circuit at the end of every second or every two seconds. The 6oth second is dis- tinguished either by the omission of the break at the previous second, or by an extra break, according to the construction of the chronometer. Chronometers are usually hung in gimbals to keep them level at all times; this is invariably done when they are taken to sea. It is important that the temperature of the chronometer should be kept as nearly uniform as possible, be- cause fluctuation in temperature is the greatest source of error. Two chronometers of the same kind cannot be directly com- pared with great accuracy, os.i or os.2 being about as close as the difference can be estimated. But a sidereal and a solar chro- nometer can easily be compared within a few hundredths of a second. On account of the gain of the sidereal on the solar chronometer, the beats of the two will coincide once in about every 3 m 05*. If the two are compared at the instant when the beats are apparently coincident, then it is only necessary to note the seconds and half seconds, as there are no fractions to be estimated. By making several comparisons and reducing them to some common instant of time it is readily seen that the comparison is correct within a few hundredths of a second. The accuracy of the comparison depends upon the fact that the ear can detect a much smaller interval between the two beats than can possibly be estimated when comparing two chronome- ters whose beats do not coincide.
56. Chronograph.
The chronograph is an instrument for recording the time kept by a chronometer and also any observations the times of which it is desired to determine. A piece of paper is wrapped about a cylinder, which is revolved by a mechanism at a uniform rate. A pen in contact with the paper is held on an arm, connected with the arma- ture of an electro-magnet, in such a way that the pen draws a continuous line which has notches in it corresponding to the breaks in the circuit made by the chro- nometer. By means of this instrument the time is represented accurately on the sheet as a linear distance. If it is desired to record the instant when any event
94
PRACTICAL ASTRONOMY
^
occurs, such as the passage of a star over a cross hair, the observer presses a tele- graph key which breaks the same circuit, and a mark is made on the chronograph sheet. The instant of the observation may be scaled from the record sheet with great precision.
57. The Zenith Telescope.
The zenith telescope is an instrument designed for making observations for latitude by a special method devised by Capt. Andrew Talcott, and which bears his name. The instrument consists of a telescope having a vertical and a horizontal axis like the transit; the telescope is attached to one end of the horizontal axis in- stead of at the centre. The essential features of the instrument are (i) a microm- eter, placed in the focus of the eyepiece, for
^^ \ r^*+^ measuring small differences in zenith distance,
and (2) a sensitive spirit level, attached to a small vertical circle on the telescope tube, for measuring small deflections of the vertical axis. The telescope is used in the plane of the meridian. There are two stops whose positions can be regulated so that the telescope may be quickly shifted, by a rotation about the ver- tical axis, from the north meridian to the south meridian. The observation consists in measuring with the micrometer the difference in zenith distance of two stars, one north of the zenith and one south, which culminate - within a few minutes of each other, and in taking readings of the spirit level at the same time the micrometer settings are made. A FIG. 45. THE ZENITH TELESCOPE diagram of the instrument in the two posi- tions is given in Fig. 45. The inclination of
the telescope to the vertical is not changed between the two observations, so it is essential that the zenith distances of the two stars should be so nearly equal that both will come within the range of the micrometer screw, usually 30' or less. The principle involved in this method may be seen from Fig. 46. From the observed zenith distance of the star Ss the latitude is
and from the star Taking the mean,
L = Dn — zn. L = HA, + /?„) + i (*. - zn).
[69]
The latitude is therefore the mean of the declinations corrected by half the djffer- ence of the zenith distances. The declination may be computed from the star catalogues, and the difference in zenith distance may be very accurately measured with the micrometer screw. It is evidently essential that the telescope should
DESCRIPTION OF INSTRUMENTS
95
have the same inclination to the vertical in each case. If the inclination changes, however, the amount of this change is accurately determined from the level readings already mentioned (see Art. 70).
FIG. 46
58. Suggestions about Observing.
The instrument used for making such observations as are described in this book will usually be either the engineer's transit or the sextant. In using the transit care must be taken to give the tripod a firm support. It is well to set the transit in position some time before the observations are to be begun; this allows the instrument to assume the temperature of the air and the tripod legs to come to a firm bearing on the ground. The observer should handle the instrument with great care, par- ticularly during night observations, when the instrument is likely to be accidentally disturbed. In reading angles at night it is important to hold the light in such a position that the graduations on the circle are plainly visible and may be viewed along the lines of graduation, not obliquely. By changing the position of the lantern and the position of the eye it will be found that the reading varies by larger amounts than would be expected when reading in the daylight. Care should be taken not to touch the graduated silver circles, as they soon become tarnished. The lantern should be held so as to heat the instru- ment as little as possible, and so as not to shine into the observer's eyes. Time may be saved and mistakes avoided if the program of observations is laid out beforehand, so that the observer knows just what is to be done and the proper order of the different
96
PRACTICAL ASTRONOMY
steps. The observations should be arranged so as to eliminate instrumental errors, usually by means of reversals; but if this is not practicable, then the instrument must be put in good adjustment. The index correction should be determined and applied, unless it can be eliminated by the method of observing. In observations for time it will often be necessary to use an ordinary watch. If there are two observers, one can read the time while the other makes the observations. If a chronometer is used, one observer may easily do the work of both, and at the
c sec. h
FIG. 47
same time increase the accuracy. In making observations by this method (called the " eye and ear method ") the observer looks at the chronometer, notes the reading at some instant, say at the beginning of some minute, and, listening to the half-second beats, carries along the count mentally and without looking at the chronometer. In this way he can note the second and estimate the fraction without taking his attention from the star and cross hair. After making his observation he may check his count by again looking at the chronometer to see if the two agree. After a little practice this method can be used easily and accurately. In using a watch it is possible for one observer to make the observations and also note the time, but it cannot be done with any such precision as with the chronometer, be- cause on account of the rapidity of the ticks (5 per second), the observer cannot count the seconds mentally. The observer
DESCRIPTION OF INSTRUMENTS
97
must in this case look quickly at his watch and make an allow- ance, if it appears necessary, for the time lost in looking up and
taking the reading.
Problems
i. Show that if the sight line makes an angle c with the perpendicular to the horizontal axis (Fig. 47) the horizontal angle between two points is in error by the angle
c sec h' — c sec h",
where h' and h" are the altitudes of the two points.
•i tan h
FIG. 48
2. Show that if the horizontal axis is inclined to the horizon by the angle * (Fig. 48) the effect upon the azimuth of the sight line is i tan h, and that an angle is in error by
i (tan h' - tan h"),
where &' and h" are the altitudes of the points.
CHAPTER IX THE CONSTELLATIONS
59. The Constellations.
A study of the constellations is not really a part of the subject of Practical Astronomy, and in much of the routine work of observing it would be of comparatively little value, since the stars used can be identified by means of their coordinates and a knowledge of their positions in the constellations is not essential. If an observer has placed his transit in the meridian and knows approximately his latitude and the local time, he can identify stars crossing the meridian by means of the times and the alti- tudes at which they culminate. But in making occasional observations with small instruments, and where much of the astronomical data is not known to the observer at the time, some knowledge of the stars is necessary. When a surveyor is be- ginning a series of observations in a new place and has no accu- rate knowledge of his position nor the position of the celestial sphere at the moment, he must be able to identify certain stars in order to make approximate determinations of the quantities sought.
60. Method of Naming Stars.
The whole sky is divided in an arbitrary manner into irregular areas, all of the stars in any one area being called a constellation and given a special name. The individual stars in any constel- lation are usually distinguished by a name, a Greek letter,* or a number. The letters are usually assigned in the order of brightness of the stars, a being the brightest, /3 the next, and so on. A star is named by stating first its letter and then the name of the constellation in the (Latin) genitive form. For instance,
* The Greek alphabet is given on p. 190.
THE CONSTELLATIONS 99
in the constellation Ursa Minor the star a is called a Ursa Minoris; the star Vega in the constellation Lyra is called a Lyra. When two stars are very close together and have been given the same letter, they are often distinguished by the numbers i, 2, etc., written above the letter, as, for example, a2 Capricorni, meaning that the star passes the meridian after a1 Capricorni. * 61. Magnitudes.
The brightness of stars is shown on a numerical scale by their magnitudes. A star having a magnitude i is brighter than one having a magnitude 2. On the scale of magnitudes in use a few of the brightest stars have fractional or negative magnitudes. Stars of the fifth magnitude are visible to the naked eye only under favorable conditions. Below the fifth magnitude a tele- scope is usually necessary to render the star visible.
62. Constellations Near the Pole.
The stars of the greatest importance to the surveyor are those near the pole. In the northern hemisphere the pole is marked by a second-magnitude star, called the polestar, Polaris, or a Ursa Minoris, which is about i° 10' distant from the pole at the present time (1910). This distance is now decreasing at the rate of about one-third of a minute per year, so that for several centuries this star will be close to the celestial north pole. On the same side of the pole as Polaris, but much farther from it, is a constellation called Cassiopeia, the five brightest stars of which form a rather unsymmetrical letter W (Fig. 49). The lower left-hand star of this constellation, the one at the bottom of the first stroke of the W, is called 5, and is of importance to the surveyor because it is very nearly on the hour circle passing through Polaris and the pole; in other words its right ascension is nearly the same as that of Polaris. On the opposite side of the pole from Cassiopeia is Ursa Major, or the great dipper, a rather conspicuous constellation. The star f , which is at the bend in the dipper handle, is also nearly on the same hour circle as Polaris and 5 Cassiopeia. If a line be drawn on the sphere
100 PRACTICAL ASTRONOMY
between 8 Cassiopeia and f Ursa Majoris, it will pass nearly through Polaris and the pole, and will show at once the position of Polaris in its diurnal circle. The two stars in the bowl of the great dipper on the side farthest from the handle are in a line which, if prolonged, would pass near to Polaris. These stars are therefore called the pointers and may be used to find the polestar. There is no other star near Polaris which is likely to be confused with it. Another star which should be remembered is /3 Cassiopeia, the one at the upper right-hand corner of the W. Its right ascension is very nearly OA and therefore the hour circle through it passes nearly through the equinox. It is possible then, by simply glancing at 0 Cassiopeia and the polestar, to estimate approximately the local sidereal time. When /3 Cassiopeia is vertically above the polestar it is nearly OA sidereal time; when the star is below the polestar it is 1 2h sidereal time ; half way between these positions, left and right, it is 6h and iSh, respectively. In intermediate positions the hour angle of the star ( = sidereal time) may be roughly estimated.
63. Constellations Near the Equator.
The principal constellations within 45° of the equator are shown in Figs. 50 to 52. Hour circles are drawn for each hour of R. A. and parallels for each 10° of declination. The approxi- mate declination and right ascension of a star may be obtained by scaling the coordinates from the chart. The position of the ecliptic, or sun's path in the sky, is shown as a curved line. The moon and the planets are always found near this circle because the planes of their orbits have only a small inclination to the earth's orbit. A belt extending about 8° each side of the ecliptic is called the Zodiac, and all the members of the solar system will always be found within this belt. The constellations along this belt, and which have given the names to the twelve " signs of the Zodiac," are Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces. These constellations were named many centuries ago, and the
FlG. 49. CONSTELLATIOJ
MAPI
BOUT THE NORTH POLE
FIG. 50. PRINCIPAL FIXED STARS BETWEE:
MAP II
:CLINATIONS 45° NORTH AND 45° SOUTH
»*.
2* *'
JULY
JUNE
•1C
3d
CANES VENATI
-— t
CORONA
-fa
HERCUES
20
BOREALIS
/ BOOTES
P-
.-
8 V
COMA BEF
XVII
SERPENS
*a e-t-
XVI
I XV
XIV
OPHIUCHUS
V. *
VIRGO
so
Li.
L.y
SCORPIO
FIG. 51. PRINCIPAL FIXED STARS BETWEI
DECLINATIONS 45° NORTH AND 45° SOUTH
NOVEMBER
OCTOBER
SEF
40-
LACERTA
30
ANDROMEDA
10
PEGASUS
8 PISCES
XXIII
XXII
XXI
10
-ft
/ AQUARIUS
CETUS
CARRICORNU
30
PISCIS AUSTRALIS
FIG. 52. PRINCIPAL FIXED STARS BETWEI
MAP IV
AUGUST
JULY
)ECLINATIONS 45° NORTH AND 45° SOUTH
FlG. 53. CONSTELLATIO
ABOUT THE SOUTH POLE
THE CONSTELLATIONS IOI
names have been retained, both for the constellations themselves and also for the positions in the ecliptic which they occupied at that time. But on account of the continuous westward motion of the equinox, the " signs " no longer correspond to the con- stellations of the same name. For example, the sign of Aries extends from the equinoctial point to a point on the ecliptic 30° eastward, but the constellation actually occupying this space at present is Pisces. In Figs. 50 to 52 the constellations are shown as seen by an observer on the earth, not as they would appear on a celestial globe. On account of the form of pro- jection used in these maps there is some distortion, but if the observer faces south and holds the page up at an altitude equal to his colatitude, the map represents the constellations very nearly as they will appear to him. The portion of the map to be used in any month is that marked with the name of the month at the top; for example, the stars under the word " February " are those passing the meridian in the middle of February at about 9 P.M. For other hours in the evening the stars on the meridian will be those at a corresponding distance right or left, according as the time is earlier or later than 9 P.M. The approxi- mate right ascension of a point on the meridian may be found at any time as follows: First compute the R. A. of the sun by allowing 2h per month, or more nearly 4 per day for every day since March 23, remembering that the R. A. of the sun is always increasing. Add this R. A. to the local mean time and the result is the sidereal time or right ascension of a star on the meridian.
Example. On October 10 the R. A. of the sun is 6 X 2h + 17 X 4TO = 13* o8m. At gh P.M. (local mean time) the sidereal time is 13* o8w + gh oom = 22h oSm. A star having a R. A. of 22* o8m wgould therefore be close to the meridian at 9 P.M.
Fig. 53 shows the stars about the south celestial pole. There is no bright star near the south pole, so that the convenient methods of determining the meridian by observations on the polestar are not practicable in the southern hemisphere.
102 PRACTICAL ASTRONOMY
64. The Planets.
In using the star maps, the student should be on the lookout for planets. These cannot be placed on the maps because their positions are rapidly changing. If a bright star is seen near the ecliptic, and its position does not correspond to that of a star on the map, it is a planet. The planet Venus is very bright and is never very far from the sun; it will therefore be seen a little before sunrise or a little after sunset. Mars, Jupiter, and Saturn are outside the earth's orbit and therefore revolve around the earth. Jupiter is the brightest, and when looked at through a small telescope shows a disc like that of the full moon, and four satellites can usually be seen all lying nearly in a straight line. Saturn is not as large as Jupiter, but in a telescope of moderate power its rings can be distinguished, or at least the planet looks elongated. Mars is reddish in color and shows a disc.
CHAPTER X OBSERVATIONS FOR LATITUDE
IN this chapter and the three immediately following are given the more common methods of determining latitude, time, longi- tude, and azimuth with small instruments. Those which are simple and direct are printed in large type, and may be used for a short course in the subject. Following these are given, in smaller type, several methods which, although less simple, are very useful to the engineer; these methods require a knowledge of other data which the engineer must obtain by observation, and are therefore better adapted to a more extended course of study.
65. Latitude by a Circumpolar Star at Culmination.
This method may be used with any circumpolar star, but Polaris is the best one to use, when it is practicable to do so, because it is of the second magnitude, while all of the other close circumpolars are quite faint. The observation consists in measuring the altitude of the star when it is a maximum or a minimum, or, in other words, when it is on the observer's me- ridian. This altitude may be obtained by trial, and it is not necessary to know the exact instant when the star is on the meridian. The approximate time when the star is at culmina- tion may be obtained from Table V or by formulae [39] and [49]. It is not necessary to know the time with accuracy, but it will save unnecessary waiting if the time is known approximately. In the absence of any definite knowledge of the time of culmina- tion, the position of the pole star with respect to the meridian may be estimated by noting the positions of the constellations. When S Cassiopeia is directly above or below Polaris the latter is at upper or lower culmination. The observation should be begun some time before one of these positions is reached. The hori-
103
104 PRACTICAL ASTRONOMY
zontal cross hair of the transit should be set on the star* and the motion of the star followed by means of the tangent screw of the horizontal axis. When the desired maximum or minimum, is reached the vertical arc is read. The index correction should then be determined. If the instrument has a complete vertical circle and the time of culmination is known approximately, it will be well to eliminate instrumental errors by taking a second altitude with the instrument reversed, provided that neither observation is made more than 4 m or 5 m from the time of culmi- nation. If the star is a faint one, and therefore difficult to find, it may be necessary to compute its approximate altitude (using the best known value for the latitude) and set off this altitude on the vertical arc. The star may be found by moving the telescope slowly right and left until the star comes into the field of view. Polaris can usually be found in this manner some time before dark, when it cannot be seen with the unaided eye. It is especially important to focus the telescope carefully before attempting to find the star, for the slightest error of focus may render the star invisible. The focus may be adjusted by look- ing at a distant terrestrial object or, better still, by sighting at the moon or at a planet if one is visible. If observations are to be made frequently with a surveyor's transit, it is well to have a reference mark scratched on the telescope tube, so that the objective may be set at once at the proper focus.
The latitude is computed from Equa. [3] or [4]. The true altitude h is derived from the reading of the vertical circle by applying the index correction with proper sign and then subtract- ing the refraction correction (Table I). The polar distance is found by taking from the Ephemeris (Table of Circumpolar Stars) the apparent declination of the star and subtracting this from 90°.
* The image of a star is practically a point of light; if the telescope were perfect it would be actually a point, but, owing to the imperfections in the corrections for aberration, the image, even though perfectly distinct, has an appreciable width. The image of the star should be bisected with the horizontal cross hair.
OBSERVATIONS FOR LATITUDE 105
Example i.
Observed altitude of Polaris at upper culmination = 43° 37'; index correction = +30"; declination = +88° 44' 35".
Vertical circle = 43° 37' oo"
Index correction = +30
Observed altitude =43 37 30
Refraction correction = i 02
True altitude = 43 36 28
Polar distance = i 15 25
Latitude = 42° 21' 03"
Since the vertical circle reads only to i' the resulting value for the latitude must be considered as reliable only to the nearest i'.
Example 2.
Observed altitude of 51 Cephei at lower culmination = 39° 33' 3°"j index correction = o"; declination = -f- 87° n' 25".
Observed altitude = 39° 33' 30" Refraction correction = i 10
True altitude =39 32 20
Polar distance = 2 48 35
Latitude = 42° 20' 55"
\A 66. Latitude by Altitude of Sun at Noon.
The altitude of the sun at noon (meridian passage) may be determined by placing the line of sight of the transit in the plane of the meridian and observing the altitude of the upper or lower limb of the sun when it is on the vertical cross hair. The watch time at which the sun will pass the meridian may be computed by converting i2h local apparent time into Standard or locaJ mean time (whichever is used) as shown in Arts. 28 and 35. Usually the direction of the meridian is not known, so the maxi- mum altitude of the sun is observed and assumed to be the same as the meridian altitude. On account of the sun's changing declination the maximum altitude is not quite the same as the meridian altitude; the difference is quite small, however, usually a fraction of a second, and may be entirely neglected for obser- vations made with the engineer's transit or the sextant. The maximum altitude of the upper or lower limb is found by trial,
106 PRACTICAL ASTRONOMY
the horizontal cross hair being kept tangent to the limb as long as it continues to rise. When the observed limb begins to drop below the cross hair the altitude is read from the vertical arc and the index correction is determined. The true altitude of the centre of the sun is then found by applying the corrections for index error, refraction, semidiameter, and parallax. In order to compute the latitude it is necessary to know the sun's declina- tion at the instant the altitude was taken. If the longitude of the place is known approximately (say within half a degree) the declination may be taken from the Nautical Almanac for the instant of Greenwich Apparent Noon and increased or decreased by the hourly change multiplied by the number of hours in the longitude. If the place is west of Greenwich the correction is to be added algebraically; if the place is east, it is to be subtracted. If the longitude is not known, but the Greenwich mean time is known, as would be the case if the timepiece kept either Green- wich time or Standard time, the declination may be computed by noting the watch time of the observation as nearly as possible and correcting the declination as follows: take out the declina- tion at Greenwich Mean Noon, and increase it by the hourly change multiplied by the number of hours since Greenwich Mean Noon. The latitude is then found from Equa. [2].
Example i.
Observed maximum altitude of sun's lower limb, Jan. 8, 1906, = 25° 06'; index correction = +i'; the longitude is 4>»44»*i8s (= 71° 04' 5) west; the declina- tion of the sun at Greenwich Apparent Noon = S 22° 19' 33"; hourly change = + i9".59; the semidiameter = 16' 17".
Observed altitude = 25° 06'. o Decl. at G. A. N. = - 22° 19' 33" Index correction + i .o 19". 59 X 4A- 74 = +i 33
25 07 . o Decl. at L. A. N. = — 22° 18' oo"
Refraction correction = —2.0
25 05 . o Semidiameter = +