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PRACTICAL ASTRONOMY
FOR ENGINEERS



PRACTICAL ASTRONOMY
FOR ENGINEERS



BY



FREDERICK HANLEY SEARES

Professor of Astronomy in the University of Missouri
and Director of the Laws Observatory




COLUMBIA, MISSOURI

THE E. W. STEPHENS PUBLISHING COMPANY
1909



COPYRIGHT, 1909

BY
FREDERICK HANLEY SEARES




PREFACE

The following pages represent the result of several years' experience in
presenting to students of engineering the elements of Practical Astronomy.
Although the method and the extent of the discussion have been designed to
meet the specialized requirements of such students, it is intended that the
work shall also serve as an introduction for those who desire a broader knowl-
edge of the subject.

The order of treatment and the methods proposed for the solution of the
various problems have been tested sufficiently to establish their usefulness;
and yet the results are to be regarded as tentative, for they possess neither the
completeness nor the consistency which, it is hoped, will characterize a later
edition. The volume is incomplete in that it includes no discussion of the
principles and methods of the art of numerical calculation a question funda-
mental for an appreciation of the spirit of the treatment. Difficulties inherent
in this defect may be avoided by a careful examination of an article on
numerical calculation which appeared in Popular Astronomy, 1908, pp. 349-367,
and in the Engineering Quarterly of the University of Missouri, v. 2, pp. 171-192.
The final edition will contain this paper, in a revised form, as a preliminary
chapter. The inconsistencies of the work are due largely to the fact that the
earlier pages were in print before the later ones were written, and to the
further fact that the manuscript was prepared with a haste that permitted no
careful interadjustment and balancing of the parts.

The main purpose of the volume is an exposition of the principal methods
of determining latitude, azimuth, and time. Generally speaking, the limit of
precision is that corresponding to the engineer's transit or the sextant. Though
the discussion has thus been somewhat narrowly restricted, an attempt has been
made to place before the student the means of acquiring correct and complete
notions of the fundamental conceptions of the subject. But these can scarcely
be attained without some knowledge of the salient facts of Descriptive
Astronomy. For those who possess this information, the first chapter will
serve as a review; for others, it will afford an orientation sufficient for the
purpose in question. Chapter II blocks out in broad lines the solutions of the
problems of latitude, azimuth, and time. The observational details of these
solutions, with a few exceptions, are presented in Chapter IV, while Chapters
V-VII consider in succession the special adaptations of the fundamental
formulae employed for the reductions. In each instance the method used in
deriving the final equations originates in the principles underlying the subject
of numerical calculation. Chapter III is devoted to a theoretical considera-
tion of the subject of time.

It is not customary to introduce historicat data into texts designed for
the use of professional students; but the author has found so much that is



198988



vi PREFACE

helpful and stimulating in a consideration of the development of astronomical
instruments, methods, and theories that he is disposed to offer an apology for
the brevity of the historical sections rather than to attempt a justification of
their introduction into a work mainly technical in character. To exclude
historical material from scientific instruction is to disregard the most effective
means of giving the student a full appreciation of the significance and bearing
of scientific results. Brief though they are, it is hoped that these sections
may incline the reader toward wider excursions into this most fascinating field.

-The numerical solutions for most of the examples have been printed in
detail in order better to illustrate both the application of the formulae involved
and the operations to be performed by the computer. Care has been taken to
secure accuracy in the text as well as in the examples, but a considerable
number of errors have already been noted. For these the reader is referred
to the list of errata on page 132.

The use of the text should be supplemented by a study of the prominent
constellations. For this purpose the "Constellation Charts" published by the
editor of Popular Astronomy, Northfield, Minnesota, are as serviceable as any,
and far less expensive than the average.

My acknowledgments are due to Mr. E. S. Haynes and Mr. Harlow
Shapley, of the Department of Astronomy of the University of Missouri, for
much valuable assistance in preparing the manuscript, in checking the calcu-
lations, and in reading the proofs.

F. H. SEARES.
LAWS OBSERVATORY,
UNIVERSITY OF MISSOURI,
June, 1909.



CONTENTS



CHAPTER I

INTRODUCTION CELESTIAL SPHERECOORDINATES

PAGE.

1 . The results of astronomical investigations '. 1

2. The apparent phenomena of the heavens 4

3. Relation of the apparent phenomena to their interpretation 5

4. Relation of the problems of practical astronomy to the phenomena of the heavens 7

5. Coordinates and coordinate systems , 8

6. Characteristics of the three systems. Changes in the coordinates 10

7. Summary. Method of treating the corrections in practice ._ 15

8. Refraction 16

9. Parallax 18

CHAPTER II

FORMULAE OF SPHERICAL TRIGONOMETRY TRANSFORMATION OF
COORDINATES GENERAL DISCUSSION OF PROBLEMS

1 0. The fundamental formulae of spherical trigonometry 21

11. Relative positions of the reference circles of the three coordinate systems .... 23

12. Transformation of azimuth and zenith distance into hour angle and declina-
tion 25

13. Transformation of hour angle and declination into azimuth and zenith dis-
tance 29

14. Transformation of hour angle into right ascension, and vice versa 29

1 5. Transformation of azimuth and altitude into right ascension and declination,
and vice versa 31

16. Given the latitude of the place, and the declination and zenith distance of an ,
object, to find its hour angle, azimuth, and parallactic angle 31

17. Application of transformation formulae to the determination of latitude, azimuth,
and time 32

CHAPTER III

TIME AND TIME TRANSFORMATION

1 8. The basis of time measurement 36

1 9. Apparent, or true, solar time 36

20. Mean solar time > 36

21 . Sidereal time ' 37

22. The tropical year 38

23. The calendar. 38

24. Given the local time at any point, to find the corresponding local time at any
other point 39

25. Given the apparent solar time at any place, to find the corresponding mean
solar time, and vice versa 40

26. Relation between the values of a time interval expressed in mean solar and
sidereal units 42

27. Relation between mean solar time and the corresponding sidereal time 44

28. The right ascension of the mean sun and its determination 44

29. Given the mean solar time at any instant to find the corresponding sidereal
time 47

30. Given the sidereal time at any instant to find the corresponding mean solar
time , 48



viii CONTENTS

CHAPTER IV

INSTRUMENTS AND THEIR USE

PAGE.

3 1 . Instruments used by the engineer 50

TIMEPIECES

32. Historical 50

33. Error and rate 51

34. -Comparison of timepieces 52

35. The care of timepieces 58

THE ARTIFICIAL HORIZON

36. Description and use 59

THE VERNIER

37. 'Description and theory 59

38. Uncertainty of the result 60

THE ENGINEER'S TRANSIT

39. Historical 61

40. Influence of imperfections of construction and adjustment 62

4 1 . Summary of the preceding section 71

42. The level 71

43. Precepts for the use of the striding level 72

44. Determination of the value of one division of a level 73

45. The measurement of vertical angles 77

46. The measurement of horizontal angles 80

47. The method of repetitions 81

THE SEXTANT

48. Historical and descriptive 85

49. The principle of the sextant 86

50. Conditions fulfilled by the instrument 87

5 1 . Adjustments of the sextant 88

52. Determination of the index correction 89

53. Determination of eccentricity corrections 90

54. Precepts for the use of the sextant , 91

55. The measurement of altitudes 91

CHAPTER V

THE DETERMINATION OF LATITUDE

56. Methods 95

1. MERIDIAN ZENITH DISTANCE

57. Theory 96

58. Procedure 96

2. DIFFERENCE OF MERIDIAN ZENITH DISTANCES TALCOTT'S METHOD

59. Theory 97

60. Procedure 98

3. ClRCUMMERIDIAN ALTITUDES

61. Theory 99

62. Procedure . . 101



CONTENTS ix

4. ZENITH DISTANCE AT ANY HOUB ANGLE
63. Theory



PAGE.
. . 102
. 103



64. Procedure

5. ALTITUDE OF POLABIS

104

65. Theory

66. Procedure



67. Influence of an error in time .............................................. J

CHAPTER VI

THE DETERMINATION OF AZIMUTH

68. Methods ...................................................

1. AZIMUTH OF THE SUN

1 AQ

69. Theory .........................................................

70. Procedure .................................................................. *

2. AZIMUTH OF A CIRCUMPOLAB STAB AT ANY HOUB ANGLE

71. Theory ....................................................................

72. Procedure ..................................................................

3. AZIMUTH FROM AN OBSERVED ZENITH DISTANCE

113



73. Theory

74. Procedure



75. Azimuth of a mark 1

76. Influence of an error in the time H*

CHAPTER VII

THE DETERMINATION OF TIME

77. Methods 116

1. THE ZENITH DISTANCE METHOD

78. Theory J

79. Procedure 118

2. THE METHOD OF EQUAL ALTITUDES

80. Theory 1

8 1 . Procedure ^

3. THE MERIDIAN METHOD

82. Theory 1

83 . Procedure 123

4. THE POLABIS VERTICAL CIRCLE METHOD
SIMULTANEOUS DETERMINATION OF TIME AND AZIMUTH

84. Theory 126

85. Procedure



129



ERRATA 132



INDEX



133



PRACTICAL ASTRONOMY
FOR ENGINEERS

CHAPTER I

INTRODUCTION CELESTIAL SPHERE COORDINATES.

1. The results of astronomical investigations. The investigations of
the astronomer have shown that the universe consists of the sun, its attendant
planets, satellites, and planetoids; of comets, meteors, the stars, and the
nebulae. The sun, planets, satellites, and planetoids form the solar system,
and with these we must perhaps include comets and meteors. The stars
and nebulae, considered collectively, constitute the stellar system.

The sun is the central and dominating body of the solar system. It is
an intensely heated luminous mass, largely if not wholly gaseous in consti-
tution. The planets and planetoids, which are relatively cool, revolve about
the sun. The satellites revolve about the planets. The paths traced out in
the motion of revolution are ellipses, nearly circular in form, which vary
slowly in size, form, and position. One focus of each elliptical orbit coin-
cides with the center of the body about which the revolution takes place.
Thus, in the case of the planets and planetoids, one of the foci of each orbit
coincides with the sun, while for the satellites, the coincidence is with the
planet to which they belong. In all cases the form of the path is such as
would be produced by attractive forces exerted mutually by all members of
the solar system and varying in accordance with the Newtonian law of
gravitation. In addition to the motion of revolution, the sun, planets, and
some of the satellites at least, rotate on their axes with respect to the stars.

The planets are eight in number. In order from the sun they are :
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. Their
distances from the sun range from thirty-six million to nearly three thousand
million miles. Their diameters vary from about three thousand to nearly
ninety thousand miles. Nevertheless, comparatively speaking, they are small,
for their collective mass is but little more than one one-thousandth that of
the sun.

The planetoids, also known as small planets or asteroids, number six
hundred or more, and relatively to the planets, are extremely small bodies
so small that they are all telescopic objects and many of them can be seen
only with large and powerful instruments. Most of them are of compara-
tively recent discovery, and a considerable addition to the number already
known is made each year as the result of new discoveries. With but few
exceptions their paths lie between the orbits of Mars and Jupiter.

The only satellite requiring our attention is the moon. This revolves
about the earth with a period of about one month, and rotates on its axis
once during each revolution. Although one of the smaller bodies of the
solar system it is, on account of its nearness, one of the most striking.

l



PRACTICAL ASTRONOMY



The solar and stellar systems are by no means coordinate parts of the
universe. On the contrary, the former, vast as it is, is but an insignificant
portion of the latter, for the sun is but a star, not very different on the
average from the other stars whose total number is to be counted by hun-
dreds of millions; and the space containing the entire solar system, includ-
ing sun, planets, satellites, and planetoids, is incredibly small as compared
with that occupied by the stellar system. To obtain a more definite notion
of the relative size of the two .systems consider the following- illustration :
Let the various bodies be represented by small spheres whose diameters
and mutual distances exhibit the relative dimensions and distribution through
space of the sun, planets, and stars. We shall thus have a rough model of
the universe, and to make its dimensions more readily comprehensible let
the scale be fixed by assuming that the sphere representing the sun is two
feet in diameter. The corresponding diameters of the remaining spheres and
their distances from the central body are shown by the following table.



OBJECT


DIAMETER


DISTANCE


Sun


2 feet






Mercury


0.08 inch


83 feet


Venus


0.21 inch


155 feet


Earth


0.22 inch


215 feet


Mars


o. 12 inch


327 feet


Jupiter


2.42 inch


1116 feet


Saturn


2. 02 inch


2048 feet


Uranus


0.97 inch


4118 feet


Neptune


0.91 inch


6450 feet


Nearest Star


Unknown


nooo miles



Tt will be seen that the distance of the outermost planet from the sun
is represented in the model by about a mile and a quarter. On the same
scale, the distance of the nearest star, the only one included in the table,
is approximately equal to one-half the circumference of the earth. When
it is remembered that this object is but one of perhaps two hundred million
stars, the vast majority of which are probably at least one hundred times
more distant, and further that each of these stars is a sun as our owin sun,
the very subordinate position of the solar system becomes strikingly ap-
parent.

The fact that the sun is similar in size and chemical composition to
millions of other stars at once raises the question as to whether they too
are not provided with attendant systems of planets and satellites. A de-
finite answer is wanting, although analogy suggests that such may well be
the case. Bodies no larger than the planets and shining only by reflected
light would be quite invisible, even in the most powerful telescopes, when
situated at distances comparable with those separating us from the stars.



, INTRODUCTION 3

We do know, however, that in many instances two or more stars situated
relatively near each other revolve about their common center of gravity thus
forming binary or multiple systems. The discovery and study of these
systems constitutes one of the most interesting- and important lines of modern
astronomical investigation.

The distances separating the various members of the solar system are
such that the motions of the planets and planetoids with respect to the sun,
and of the satellites relative to their primaries, produce rapid changes in
their positions as seen from the earth. The stars are also in motion and
the velocities involved are very large, amounting occasionally to a hundred
miles or more per second of time, but to the observer on the earth, their
relative positions remain sensibly unchanged. The distances of these ob-
jects are so great that it is only when the utmost refinement of observation
is employed and the measures are continued for months and years, that any
shift in position can be detected even for those which move most rapidly.
With minor exceptions, the configuration of the constellations is the same
as it was two thousand years ago when the observations upon which are
based the earliest known record of star positions were made.

To the casual observer there is not a great deal of difference in the ap-
pearance of the stars and the planets. The greater size and luminosity of
the former is offset by their greater distance. In ancient times the funda-
mental difference between them was not known, and they were distinguished
only by the fact that the planets change their positions, while relatively to
each other the stars are apparently fixed. In fact the word planet means
literally, a moving or wandering star, while what appeared to the early ob-
servers as the distinguishing characteristic of the stars is shown by the fre-
quent use of the expression fixed stars.

The nebulae are to be counted by the hundreds of thousand's. They con-
sist of widely extended masses of luminous gas, apparently of simple chemical
composition. They are irregularly distributed throughout the heavens, and
present the greatest imaginable diversity of form, structure, and brightness.
Minute disc like objects, rings, double branched spirals, and voluminous
masses of extraordinarily complex structure, some of which resemble closely
the delicate high-lying clouds of our own atmosphere, are to be found among
them. The brightest are barely visible to the unaided eye, while the faintest
tax the powers of the largest modern telescopes. Their distances are of the
same order of magnitude as those of the stars, and, indeed, there appears
to be an intimate relation connecting these two classes of objects, for there
is evidence indicating that the stars have been formed from 1 the nebulae
through some evolutionary process the details of which are as yet not fully
understood.

The preceding paragraphs give the barest outline of the interpretation
which astronomers have been led to place upon the phenomena of the
heavens. The development of this conception of the structure of the universe
forms the major part of the history of astronomy during the last four cen-



4 PRACTICAL ASTRONOMY

turies. Many have contributed toward the elaboration of its details, but
its more significant features are due to Copernicus, Kepler, and Newton.

Although the scheme outlined above is the only theory thus far formu-
lated which satisfactorily accounts^for the celestial phenomena in their more
intricate relations, there is another conception of the universe, one far earlier
in its historical origin, which also accounts for the more striking phenomena.
This theory bears the name of the Alexandrian astronomer Ptolemy, and,
as its central idea is immediately suggested b}' the most casual examination
of the motions of the celestial bodies, we shall now turn to a consideration
of these motions and the simple, elementary devices which can be used for
their description.

2. The apparent phenomena of the heavens. The observer who goes
forth under the star-lit sky finds himself enclosed by a hemispherical vault
of blue which meets in the distant horizon the seemingly flat earth upon
which he stands. The surface of the vault is strewn with points of light
of different brightness, whose number depends upon the transparency of the
atmosphere and the brightness of the moon, but is never more than two or
three thousand. A fewi hours observation shows that the positions of the
points are slowly shifting in a peculiar and definite manner. Those in the
east are rising from the horizon while those in the west are setting. Those
in the northern heavens describe arcs of circles in a counter-clockwise di-
rection about a common central point some distance above the horizon.
Their distances from each other remain unchanged. The system moves as
a whole.

The phenomenon can be described by assuming that each individual
point is fixed to a spherical surface which rotates uniformly from' east to
west about an axis passing through the eye of the observer and the central
point mentioned above. The surface to which the light-points seem at-
tached is called the Celestial Sphere. Its radius is indefinitely great. Its
period of rotation is one day, and the resulting motion of the celestial bodies
is called the Diurnal Motion or Diurnal Rotation.

The daylight appearance of the heavens i's not unlike that of the night
except that the sun, moon, and occasionally Venus, are the only bodies to
be seen in the celestial vault. They too seem to be carried along with the
celestial sphere in its rotation, rising in the east, descending toward the
west, and disappearing beneath the horizon only to rise again in the east ;
but if careful observations be made it wall be seen that these bodies can
not be thought of as attached to the surface of the sphere, a fact most easily
verified in the case of the moon. Observations upon successive nights show
that the position of this object changes with respect to the stars. A con-
tinuation of the observations will show that it apparently moves eastward
over the surface of the sphere along a great circle at such a rate that an
entire circuit is completed in about one month. A similar phenomenon in
the case of the sun manifests itself by the fact that the time at which any
o-iven star rises does not remain the same, but occurs some four minutes



INTRODUCTION 5

earlier for each successive night. A star rising two hours after sunset on a
given night will rise approximately l h 56 m after sunset on the following
night. The average intervals for succeeding nights will be l h 52 m , l h 48 m , l h
44 m , etc., respectively. That the stars rise earlier on successive nights shows
that the motion of the sun over the sphere is toward the east. Its path is a
great circle called the Ecliptic. Its motion in one day is approximately one
degree, which corresponds to the daily change of four minutes in the time
of rising of the stars. This amount varies somewhat, being greatest in
January and least in July, but its average is such that a circuit of the
sphere is completed in one year. This motion is called the Annual Motion
of the Sun.

With careful attention it will be found that a few of the star-like points
of light, half a dozen more or less, are exceptions to the general rule which
rigidly fixes these objects to the surface of the celestial sphere. These are
the planets, the wandering stars of the ancients. Their motions with respect
to the stars are complex. They have a general progressive motion toward
the east, but their paths are looped so that there are frequent changes in di-
rection and temporary reversals of motion. Two of them, Mercury and
Venus, never depart far from the sun, oscillating from one side to the other in
paths which deviate but little from the ecliptic. The paths of the others also
lie near the ecliptic, but the planets themselves are not confined to the
neighborhood of the sun.

The sun, moon, and the planets therefore appear to move over the surface
of the celestial sphere with respect to the stars, in paths which lie in or near
the ecliptic. The direction of motion is opposite, in general, to that of the
diurnal rotation. The various motions proceed quite independently. While
the sun, moon, and planets move over the surface of the sphere, the sphere
itself rotates on its axis with a uniform angular velocity.

These elementary facts are the basis upon which the theory of Ptolemy
was developed. It assumes the earth, fixed in position, to be the central
body of the universe. It supposes the sun, moon, and planets to revolve
about the earth in paths which are either circular or the result of a com-
bination of uniform circular motions;; and regards the stars as attached to
the surface of a sphere, which, concentric with the earth and enclosing the
remaining members of the system, rotates from east to west, completing a
revolution in one day.

3. Relation of the apparent phenomena to their interpretation. The re-
lation of the apparent phenomena to the conception of Ptolemy is obvious,
and their connection with the scheme outlined in Section 1 is not difficult to


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